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J. Chem. Phys. 154, 144305 (2021); https://doi.org/10.1063/5.0046577 154, 144305 © 2021 Author(s).Time-resolved photoelectron imaging of complex resonances in molecular nitrogen Cite as: J. Chem. Phys. 154, 144305 (2021); https://doi.org/10.1063/5.0046577 Submitted: 04 February 2021 . Accepted: 19 March 2021 . Published Online: 09 April 2021 Mizuho Fushitani , Stephen T. Pratt , Daehyun You , Shu Saito , Yu Luo , Kiyoshi Ueda , Hikaru Fujise , Akiyoshi Hishikawa , Heide Ibrahim , François Légaré , Per Johnsson , Jasper Peschel , Emma R. Simpson , Anna Olofsson , Johan Mauritsson , Paolo Antonio Carpeggiani , Praveen Kumar Maroju , Matteo Moioli , Dominik Ertel , Ronak Shah , Giuseppe Sansone , Tamás Csizmadia , Mathieu Dumergue , N. G. Harshitha , Sergei Kühn , Carlo Callegari , Oksana Plekan , Michele Di Fraia , Miltcho B. Danailov , Alexander Demidovich , Luca Giannessi , Lorenzo Raimondi , Marco Zangrando , Giovanni De Ninno , Primož Rebernik Ribič , and Kevin C. Prince ARTICLES YOU MAY BE INTERESTED IN Understanding the structure of complex multidimensional wave functions. A case study of excited vibrational states of ammonia The Journal of Chemical Physics 154, 144306 (2021); https://doi.org/10.1063/5.0043946 Velocity map imaging of ions and electrons using electrostatic lenses: Application in photoelectron and photofragment ion imaging of molecular oxygen Review of Scientific Instruments 68, 3477 (1997); https://doi.org/10.1063/1.1148310 Gas-phase studies of chemically synthesized Au and Ag clusters The Journal of Chemical Physics 154, 140901 (2021); https://doi.org/10.1063/5.0041812The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Time-resolved photoelectron imaging of complex resonances in molecular nitrogen Cite as: J. Chem. Phys. 154, 144305 (2021); doi: 10.1063/5.0046577 Submitted: 4 February 2021 •Accepted: 19 March 2021 • Published Online: 9 April 2021 Mizuho Fushitani,1 Stephen T. Pratt,2,a) Daehyun You,3 Shu Saito,3Yu Luo,3Kiyoshi Ueda,3,b) Hikaru Fujise,1Akiyoshi Hishikawa,1,4 Heide Ibrahim,5 François Légaré,5Per Johnsson,6 Jasper Peschel,6Emma R. Simpson,6Anna Olofsson,6Johan Mauritsson,6 Paolo Antonio Carpeggiani,7 Praveen Kumar Maroju,8Matteo Moioli,8Dominik Ertel,8Ronak Shah,8Giuseppe Sansone,8 Tamás Csizmadia,9Mathieu Dumergue,9N. G. Harshitha,9 Sergei Kühn,9Carlo Callegari,10 Oksana Plekan,10 Michele Di Fraia,10 Miltcho B. Danailov,10 Alexander Demidovich,10 Luca Giannessi,10,11 Lorenzo Raimondi,10Marco Zangrando,10,12 Giovanni De Ninno,10,13 Primož Rebernik Ribi ˇc,10,13 and Kevin C. Prince10,14,c) AFFILIATIONS 1Department of Chemistry, Nagoya University, Furo-cho, Chikusa, Nagoya, Aichi 464-8602, Japan 2Chemical Sciences and Engineering Division, Argonne National Laboratory, Lemont, Illinois 60439, USA 3Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai, Miyagi 980-8577, Japan 4Research Center for Materials Science, Nagoya University, Furo-cho, Chikusa, Nagoya, Aichi 464-8602, Japan 5Institut National de la Recherche Scientifique, Centre Énergie Matériaux Télécommunications, Varennes, Québec J3X 1S2, Canada 6Lund University, Department of Physics, Lund, Sweden 7Technische Universität Wien, Institut für Photonik, Gußhausstraße 27-29, 1040 Wien, Austria 8Albert-Ludwigs-Universität, Stefan-Meier-Strasse 19, 79104 Freiburg, Germany 9ELI-ALPS, ELI-HU Non-Profit, Ltd., Wolfgang Sandner utca 3, Szeged H-6728, Hungary 10Elettra-Sincrotrone Trieste, SS 14, km 163.5, in Area Science Park, 34149 Basovizza, Trieste, Italy 11INFN, Laboratori Nazionali di Frascati, Via Enrico Fermi 54, 00044 Frascati (Roma), Italy 12IOM-CNR, SS 14, km 163.5, in Area Science Park, 34149 Basovizza, Trieste, Italy 13University of Nova Gorica, Vipavska 13, SI-5000 Nova Gorica, Slovenia 14Centre for Translational Atomaterials, and Department of Chemistry and Biotechnology, Swinburne University of Technology, Melbourne, Australia a)E-mail: stpratt@anl.gov b)E-mail: kiyoshi.ueda@tohoku.ac.jp c)Author to whom correspondence should be addressed: prince@elettra.eu ABSTRACT We have used the FERMI free-electron laser to perform time-resolved photoelectron imaging experiments on a complex group of resonances near 15.38 eV in the absorption spectrum of molecular nitrogen, N 2, under jet-cooled conditions. The new data complement and extend the earlier work of Fushitani et al. [Opt. Express 27, 19702–19711 (2019)], who recorded time-resolved photoelectron spectra for this same group of resonances. Time-dependent oscillations are observed in both the photoelectron yields and the photoelectron angular distributions, providing insight into the interactions among the resonant intermediate states. In addition, for most states, we observe an exponential decay of the photoelectron yield that depends on the ionic final state. This observation can be rationalized by the different lifetimes for the interme- diate states contributing to a particular ionization channel. Although there are nine resonances within the group, we show that by detecting individual photoelectron final states and their angular dependence, we can identify and differentiate quantum pathways within this complex system. Published under license by AIP Publishing. https://doi.org/10.1063/5.0046577 .,s J. Chem. Phys. 154, 144305 (2021); doi: 10.1063/5.0046577 154, 144305-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp I. INTRODUCTION Molecular Rydberg states provide a rich environment for the investigation of intramolecular dynamics.1,2In particular, for inter- mediate values of the principal quantum number, n ∼5–15, the electronic, vibrational, and rotational level spacings become com- parable, leading to complex interactions that can provide insight into the flow of energy and angular momentum within the isolated molecule.1,2Such flows are an inherent part of chemical reactions, and their study can provide insight into the nature of chemical reactivity. Spectroscopic studies have been particularly effective at reveal- ing the coupling among molecular degrees of freedom, as well as the mechanisms that drive dynamical processes such as internal con- version, intersystem crossing, and dissociation.3,4Over the last few decades, time-domain studies using ultrafast lasers have provided a complementary perspective of these mechanisms.5–11More recently, ultrafast sources in the vacuum ultraviolet (VUV) have been devel- oped based on high-harmonic generation, nonlinear frequency mix- ing, and free-electron lasers (FELs).12These sources allow direct excitation of the excited states of interest, and a number of new techniques have been developed to characterize the excited state structure and dynamics.6–11The present paper reports new ultrafast experiments using a combination of tunable, single-mode VUV light from the FERMI free-electron laser, a near-infrared (NIR) 795-nm (1.560 eV) probe laser, and photoelectron velocity map imaging to determine time-resolved photoelectron energy and angular distri- butions for a complex region of the spectrum of molecular nitrogen, N2. This work builds on and extends a previous study of N 2by Fushi- taniet al. ,13who recorded time-dependent photoelectron spectra in the same region. The spectroscopy and dynamics of N 2have long been of inter- est due to their importance in understanding and modeling the Earth’s upper atmosphere, the atmosphere of other planets and their moons, and interstellar chemistry.14,15The first ionization energy of N 2is at 15.580 726 ±0.000 002 eV,16and above ∼12.6 eV, the absorption spectrum rapidly becomes quite complex as a result of strong Rydberg–valence interactions, as well as inter- actions between the Rydberg series converging to the N 2+X2Σg+ ground state and the low-lying A2Πuexcited state.16–25In par- ticular, long vibrational progressions are observed for the b1Πu and b′1Σu+valence states, as well as shorter progressions for the (X2Σg+)3pπc1Πu, (X2Σg+)3pσc′1Σu+, and (A2Πu)3sσo1Πustates. The interactions among these states were characterized in a clas- sic set of papers by Lefebvre-Brion,17Dressler,18,19and Carroll and Collins.20At higher energy, transitions to higher members of the (X2Σg+)npπ1Πu, (X2Σg+)npσ1Σu+, and (A2Πu)nsσ1Πuseries are observed, along with members of the (X2Σg+)nf, (A2Πu)ndσ, (A2Πu)ndπ, and (A2Πu)ndδseries.21–25Using a combination of very high-resolution absorption spectroscopy at room temperature and in supersonic beams and calculations based on multichannel quantum defect theory, Huber, Jungen, and their co-workers were able to provide a comprehensive analysis of the rotationally resolved spectra up to ∼15.44 eV.21,22,24,25 There have been a number of recent time-resolved photoelec- tron studies of the Rydberg states of N 2.13,26–28Zipp et al.26used ultrafast multiphoton excitation of the (X2Σg+)4f complex and a time-delayed ionization probe to study ℓ-uncoupling of the Rydbergelectron from the rotating molecular frame. Here, the time depen- dence of the photoelectron angular distribution provided a signature of the uncoupled electron motion. Somewhat closer to the present experiments, Marceau et al.27used an ultrafast VUV source and time-resolved photoelectron spectroscopy to probe the region near 14 eV that includes the interacting b′1Σu+, v′= 13 valence band and the c 4′1Σu+, v′= 4 Rydberg band. The coherent excitation of the two bands leads to oscillations in the intensity of the selected photo- electron peaks, reflecting the energy splitting between the interacting states. In this example, the intermediate state was ionized by a two- photon process, and a resonant or near-resonant intermediate state was found to enhance the sensitivity to oscillations in selected ion- ization channels. In a study closely related to the present work, Trabs et al.29performed an ultrafast time-resolved photoelectron imaging study of the interacting Rydberg and valence states of NO. As dis- cussed below, the similarities and differences between the present study and that earlier work help provide a context for the present findings. Recently, Fushitani et al.13used high-harmonic generation to produce VUV light to excite N 2at∼15.380 eV with a bandwidth of ∼0.070 eV and a time-delayed 800-nm (1.550 eV) probe to record FIG. 1 . An energy level diagram of the states in the region of the VUV pump photon energy. The right-hand side shows the oscillator strengths for each level extracted from Refs. 24 and 25. The Rydberg states are labeled Λ+(v)nℓλ, where Λ+(=X+, A+) denotes the electronic state of the ion core, v denotes the vibrational quantum number, n and ℓdenote the principal and orbital angular momentum quantum numbers for the Rydberg electron, and λdenotes the projection of ℓon the internuclear axis. The full width half maximum of the present VUV pulse was 0.023 eV, while that in the earlier experiments of Fushitani et al. was 0.070 eV. At lower energy, the next state corresponds to the X+(0)8pσstate at 15.333 eV, while at higher energy, the next state corresponds to the X+(2)5pσstate at 15.403 eV (see Ref. 25). J. Chem. Phys. 154, 144305 (2021); doi: 10.1063/5.0046577 154, 144305-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp time-dependent photoelectron energy distributions with a resolu- tion of∼94 fs. The pump transition coherently excited both Rydberg and valence bands, and the photoelectron spectra showed time- dependent oscillations that provided information on the coupling between the Rydberg states, which are shown schematically in Fig. 1. In this figure and in what follows, we use the labeling scheme adopted by Fushitani et al.13Specifically, the Rydberg states are labeledΛ+(v)nℓλ, whereΛ+denotes the electronic state of the ion core, X+or A+; v denotes the vibrational quantum number; n and ℓ denote the principal and orbital angular momentum quantum num- bers for the Rydberg electron; and λdenotes the projection of ℓon the internuclear axis. In the present study, we extend the work of Fushitani et al.13 by re-examining the same spectral region using photoelectron imag- ing to detect both the time-dependent photoelectron energy and angular distributions. The new experiments expand on the earlier work in two significant ways. First, the measurement of photoelec- tron angular distributions provides complementary information on the interaction among the pumped levels that is not revealed in the time-dependent ion yields or electron yields alone. Second, super- sonic cooling of the sample in a molecular beam reduces the dephas- ing due to rotational effects in the room temperature sample of the earlier experiments. This cooling results in better defined coherent oscillations that can be followed over longer time delays. In what follows, the details of the experiment are discussed in Sec. II. Next, the relevant portion of the absorption spectrum of N 2 is described in more detail, and the basic principles and expectations for the time-dependent photoelectron energy and angular distribu- tions are presented in Sec. III. The experimental results and their implications are presented in Sec. IV. This paper concludes with a discussion of potential extensions of the work, as well as of poten- tial theoretical studies that could provide additional insight into the present findings. II. EXPERIMENT The experiments were performed using FEL-1 at the FERMI free-electron laser facility.30The FERMI FEL is a seeded laser, and it was operated at a wavelength of 80.56 nm (15.390 eV) for the experiments described here. The FEL, beam-delivery system, and optical arrangement for the pump–probe experiments have been described previously.31–34The present experiments were performed using the velocity-map imaging photoelectron spectrometer in the low-density matter end station,35which has also been described pre- viously.31,32Only the particular details of the present experiment are included here. The FEL was operated at 50 Hz, and the VUV pulse dura- tion was∼140 fs full-width at half-maximum (FWHM), as mea- sured by using sidebands in the NIR+VUV ionization of Xe. The pulse energy at the exit of the undulator was estimated to be ∼25μJ and 5–6μJ in the interaction region, assuming the calculated trans- mission of the photon transport optics.31The pulse energy mon- itors of FERMI are based on ionization of nitrogen and so do not measure the pulse energy below the ionization threshold. The PRESTO spectrometer33was used to measure the shot-by-shot spec- tra in this experiment, which were then integrated to give a value of intensity. The spectrometer was calibrated against gas ionizationdetectors at shorter wavelength where these detectors function. This calibration was then corrected for the spectrometer efficiency as a function of wavelength. The bandwidth of the light was 0.023 eV (FWHM). The VUV light was focused with the active optics of the beamline to an approximately rectangular spot of dimensions 94×125μm2by observing the spot on a fluorescent screen at the focus. The 795-nm (1.560 eV) probe pulse duration was 55 fs (FWHM), and the pulse energy was set to 21 μJ on the target for all of the present measurements. No significant multiphoton ion- ization was observed for pulse energies up to 500 μJ, and in the data, only one weak feature was observed and assigned to absorp- tion of two IR photons. The probe beam was focused to a 40 μm diameter spot (laser field intensity: 1 ×1013W/cm2) overlapping the VUV beam. These pulse energies and focusing conditions resulted in a strong pump–probe signal while minimizing multiphoton effects from either beam alone. The VUV and NIR beams were both lin- early polarized along the same axis, which was parallel to the plane of the imaging detector. This geometry facilitates the reconstruction of the photoelectron images. The sample was introduced into the interaction region as a skimmed molecular beam produced by expanding 6 bars of pure N 2 into the source chamber via a room-temperature Even–Lavie valve with an orifice of nominal diameter 100 μm. Although we have no direct measurement of the rotational temperature of the sample, it can be estimated by using the data and analysis of Aoiz et al.36on similar expansions of N 2. With the present stagnation pressure and nozzle diameter, the sample translational and rotational tempera- tures are both expected to be below 10 K. The14N nucleus has spin 1 so that14N2exists in two forms, with a population ratio of 2:1, and they do not interconvert readily under the present conditions.37The even:odd rotational levels of the X1Σg+ground state have 2:1 statisti- cal weights.37At 10 K, the rotational distribution peaks at J′′= 0, and only 5% of the population is in levels with J′′= 2. This observation is important because it sets the timescale for rotational coherences that could be observed in the ionization process. In particular, excitation from J′′= 2 could produce rotational coherences in the ionization process by interfering ionization pathways with intermediate states with J′= 1 and 3. Using an approximate rotational constant,38 B′= 2 cm−1, the splitting between these two intermediate states is ∼20 cm−1, which would lead to coherences with oscillation periods of 1.7 ps. This timescale is just outside the range of the present exper- iments, but a fraction of a period may be observable as a slow modu- lation of the overall signal. Rotational coherences from J′′= 1 would have even longer periods. Thus, the present discussion will focus on vibronic (or rovibronic) interactions coupling different electronic and vibrational levels. The time-dependent photoelectron images were recorded by scanning the time delay between the VUV and NIR pulses from neg- ative delays to 1.4 ps in 25 fs steps. Additional images were recorded every 3 laser shots in which the gas valve was not synchronized with the FEL pulse to allow the subtraction of background from the data. Four independent measurements of the time-dependent signal were summed to yield the final dataset. The data were reconstructed by using the BASEX approach of Dribinski et al.39The photoelectron energy scale was calibrated by using Xe as a calibration gas along with the known energies of the N 2photoelectron peaks obtained from the ionization energy16and vibrational constants of N 2+.38The electric field in the interaction region of the imaging spectrometer J. Chem. Phys. 154, 144305 (2021); doi: 10.1063/5.0046577 154, 144305-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp corresponds to ∼10.5 V/cm, which effectively reduces the ionization energy by∼19.8 cm−1.40 For two-photon ionization in the weak-field electric-dipole approximation with linearly polarized light, the photoelectron angu- lar distribution from a randomly oriented, non-chiral sample must take the form8,9 I(Δt,Ek,θ)=σ(Δt,Ek) 4π[1 +β2(Δt,Ek)P2(cosθ) +β4(Δt,Ek)P4(cosθ)], (1) whereΔtis the time delay, Ekis the photoelectron energy, θis the angle between the polarization axis and the electron velocity vector, σis the cross section, βiare the angular distribution parameters, and Pi(cosθ) are the Legendre polynomials. In general,β2andβ4values of zero correspond to isotropic dis- tributions. Distributions for positive values of β2andβ4are more strongly peaked along the polarization axis of the ionizing laser, while distributions for negative values are peaked more strongly per- pendicular to this axis. The present analysis yields these angular dis- tribution parameters as well as a quantity (which is referred to below as the electron or ionization yield) proportional to the cross section. These time-dependent quantities are the principal observables of the present experiments and analysis. There are two sources of time-dependent behavior in the total ionization yield. First, there is a time dependence due to the inter- mediate state lifetime that results in the decay of the ionization yield with an increase in delay of the probe pulse. In principle, this time dependence is not expected to appear in the βiparameters, which describe the shape of the angular distribution and are inde- pendent of the ionization yield; however, some exceptions to this are discussed below. The second time dependence is due to interfer- ence among alternative ionization pathways, leading to oscillations as a function of the pump–probe time delay. This dependence can appear in both the ionization yields and the angular distribution parameters. The exponential decay of the electron yields for different pho- toelectron peaks can be fitted to determine decay constants and lifetimes. The oscillatory time-dependent waveforms can be Fourier transformed from the time domain to the frequency (or energy) domain to yield the splittings between intermediate states in the ionization process. To do this, the total electron yield was deter- mined and, after suitable normalization, subtracted from each curve of photoelectron intensity. This time-dependent signal was extended by zero-padding (that is, adding a number of zeros) to extend the range of the trace, to make the number of time points a power of two, and to increase the density of interpolated points in the Fourier transform. As discussed below, each time trace was mul- tiplied by a window function that strongly suppresses the con- tribution from negative time delays and from times when the VUV and NIR pulses overlap while retaining all of the signals for long time delays. For the present data, the combination of the number of time steps and the sampling rate limits the expected resolution of the Fourier transform to ∼24 cm−1. In what fol- lows, we discuss both the exponential decay and the oscillatory behavior.III. BACKGROUND A. Experimental considerations As in the work of Fushitani et al. ,13the present experiments are focused on a region of the single-photon excitation spectrum of N 2between∼15.36 and 15.40 eV. This is a complicated portion of the spectrum that includes transitions to nine different vibronic bands. Fortunately, Jungen et al.24performed a detailed analysis of these bands and the interactions among them by using a combina- tion of rotationally resolved, jet-cooled absorption spectroscopy and theoretical calculations based on multichannel quantum defect the- ory. Figure 1 shows a schematic diagram of the vibronic bands in this region, along with the corresponding band oscillator strengths extracted by Huber et al.25The bands in this region are clustered into two groups. The A+(1)3dδ1Πu, X+(0)8f, and X+(1)6pπbands form the low-energy group centered at ∼15.373 eV; the transition to the b′1Σu+, v′= 34 valence band has also been predicted to lie near this energy but has not been observed.21,24The X+(4)4pπ, X+(2)5pπ, X+(0)9pπ, and X+(0)9pσbands form the second group at ∼15.386 eV. The X+(1)6pσband appears weakly midway between these two groups. As discussed by Jungen et al. ,24the vibronic states in this region are significantly mixed among each other through both homoge- neous and heterogeneous interactions. Jungen et al.24indicated that most of the observed oscillator strength in this region is carried by the A+(1)3dδ1Πulevel and that the other levels gain their intensity through interactions with this level. The interactions among all the states play an important role in the evolution of the time-dependent coherences and photoelectron spectra and asymmetry parameters described below. In the previous work of Fushitani et al. ,13the VUV light was centered at 15.38 eV with a bandwidth of 0.070 eV FWHM. This energy lies between the two groups of bands and the bandwidth allows the coherent excitation of the entire group of vibronic bands. In the present experiment, the photon energy was set by scanning the VUV wavelength and maximizing the total two-color signal for a delay of the NIR of 1.35 ps, yielding a VUV wavelength of 80.56 nm, or 15.39 eV with an estimated error of ±0.01 eV. This error is determined by the precision of the seed wavelength, which was 241.68 ±0.1 nm. This energy is just on the high energy side of the higher energy group of bands. Alternatively, we can calibrate the photon energy based on published wavelengths and absorption cross sections. We calculated the predicted cen- tral wavelength by convoluting the absorption spectrum from Jun- genet al.24with the VUV envelope to model the expected wave- length dependence of the total ionization signal in the absence of interference effects in the two-photon ionization signal. This approach yields an energy of 15.374 eV, but it neglects potential interference effects in the two-photon ionization signal, as well as intermediate-state lifetime effects due to the delayed probe pulse. The latter effects might influence the relative contributions of dif- ferent intermediate states to the ionization signal and thus shift the measured intensity maximum. Averaging the results of the two approaches, we conclude that the center of the FEL pulse envelope is 15.382 ±0.010 eV. While the 0.023 eV bandwidth of the FEL pulse is significantly smaller than that of the table-top source of Fushitani et al. ,13it still allows the coherent excitation of all nine bands in this region. J. Chem. Phys. 154, 144305 (2021); doi: 10.1063/5.0046577 154, 144305-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp A small change in the center energy would simply result in a differ- ent set of initial amplitudes for each of the vibronic levels. That is, each level would be weighted by its oscillator strength and its posi- tion on the intensity profile of the VUV pulse envelope. This uncer- tainty makes it difficult to perform detailed theoretical modeling of the observed time-dependent data. B. Theoretical considerations In principle, the coherent excitation of nine different vibronic levels (not to mention the associated rotational levels) could result in a very complex time dependence for the observables. In the present experiments, however, the particular detection channel and observ- able act as filters that limit which states can contribute to the inter- ference and oscillatory behavior. The nature of this filtering is differ- ent for the electron yields and the angular distribution parameters. While these ideas are known,27,41we briefly describe them in the context of the present experiments. A model system with two intermediate states provides a frame- work for the present observations (Fushitani et al. 2019).13We con- sider four scenarios, which are illustrated schematically in Fig. 2. First, we consider coherent excitation of two intermediate states, |1 ⟩ and |2⟩, that do not interact with each other [Fig. 2(a)]. The resultant wavepacket will be expressed as Ψ(t)=a1∣1⟩e−iE1t/̵h+a2|2⟩e−iE2t/̵h, (2)where the coefficients ai(i= 1, 2) are determined by the transition moments from the ground state and the intensity distribution of the excitation pulse. The energies of the intermediate states | i⟩are given by Ei. If the two intermediate states are ionized to different (electronic or vibrational) ionic states such that the photoelectrons do not overlap in energy, the photoelectron intensity and angular distributions will be time independent. Next, in Fig. 2(b), we consider the case in which the two intermediate levels |1 ⟩and |2⟩interact to form the eigenstates given by ψ+=c1∣1⟩+c2∣2⟩ (3) and ψ−=c2∣1⟩−c1∣2⟩. (4) The coherent excitation of these two levels results in a wavepacket described by Ψ(t)=a+ψ+e−iE+t/̵h+a−ψ−e−iE−t/̵h =a+(c1∣1⟩+c2∣2⟩)e−iE+t/̵h+a−(c2∣1⟩−c1∣2⟩)e−iE−t/̵h =(a+c1e−iE+t/̵h+a−c2e−iE−t/̵h)∣1⟩ +(a+c2e−iE+t/̵h−a−c1e−iE−t/̵h)∣2⟩. (5) In this case, even if the ionization of levels |1 ⟩and |2⟩leads to different final states, there will be two paths to each continuum, i.e., FIG. 2 . Schematic illustration of the four cases considered for yields and pho- toelectron angular distributions (PADs). Green vertical arrows: FEL excitation; red or blue vertical arrows: NIR ioniza- tion. Horizontal lines represent the ener- gies of states; GS is the ground state; |1 ⟩ and |2 ⟩represent non-interacting excited states;ψ±represent interacting excited states; and the blue and red ellipses rep- resent the continuum states. The Gaus- sian curves represent the bandwidth of the FEL radiation. (a) Non-interacting intermediate states. (b) Interacting inter- mediate states. (c) Non-interacting inter- mediate states, ionization to the same continuum. (d) Interacting states, ioniza- tion to the same continuum. J. Chem. Phys. 154, 144305 (2021); doi: 10.1063/5.0046577 154, 144305-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp via |+⟩and |−⟩. Since the two mixed states have different energies, their relative phases will vary with time, thus modifying the super- position coefficients in each channel and producing time-dependent oscillations in the photoelectron yields. However, because each final state is only populated by ionization of either |1 ⟩or |2⟩, the pho- toionization dynamics are always the same, and the angular distri- butions are still time independent, even though the yields in the two channels oscillate. An extension of this model to a system with three interacting states is described in the supplementary material. That model shows how additional states result in additional oscillation frequencies. In the third example, Fig. 2(c), we again consider non- interacting levels |1 ⟩and |2⟩as in Eq. (2) and Fig. 2(a) but with ionization from both levels into the same continuum (and the photo- electron peaks overlap in energy). Because there are then two path- ways to the same final state and because |1 ⟩and |2⟩have different energies, the relative phase between the two paths is time dependent. This situation produces time-dependent ion yields, as well as time- dependent photoelectron angular distributions resulting from the time-dependent amplitudes of the interfering partial waves. This sit- uation occurs in many experiments involving rotational coherences or hyperfine coherences in the intermediate state.42–44 Finally, we consider the case in which the intermediate states are mixed and ionization into the same ionization continuum is possible [Fig. 2(d)]. In this case, as in the previous one, both the elec- tron yield and the angular distributions can show oscillations whose frequency is determined by the splitting between the two levels. In principle, each pair of intermediate levels excited by the pump photon will produce an oscillation frequency in the elec- tron yield and angular distributions if the right conditions are met. With nine intermediate levels (ignoring the rotational structure), this would result in 36 different frequencies. Here, however, two general propensities for the photoionization of Rydberg states result in a considerable simplification. First, the ion core of the Rydberg state generally acts as a spectator in the photoionization process, and it is preserved upon ionization. Second, the potential curves of Rydberg states are generally very similar to the corresponding ionic states, and the Franck–Condon factors for photoionization are thus often nearly diagonal.45If we assume that both assumptions are strictly valid, the number of possible interfering paths to each final state is greatly reduced. Furthermore, if most of the Rydberg state interactions are weak, a small number of oscillation frequencies are expected to dominate. Finally, we note that for closely spaced lev- els, the relevant oscillation frequencies will be outside the resolution (∼24 cm−1) of the experiment, which is determined by the length of the time-delay scans. While these propensities for the ionization step are expected to be generally applicable, there are several ways that they can break down. First, with respect to the preservation of the ion core upon photoionization, it is, in principle, possible for the ion core, rather than the Rydberg electron, to absorb the NIR photon. The corresponding photon energy is close to that of the N 2+A2Πu ←X2Σg+transition, particularly when one accounts for vibra- tional excitation.15,37Such processes are similar to “isolated core excitation” techniques used to extract information on the spectra of ions from Rydberg studies.40,46–48More generally, configuration interaction can result in ionizing transitions that look like two- electron, core-switching transitions. Second, because the potentialcurves of the Rydberg states are not perfectly parallel with those of their corresponding ion cores (the Rydberg electron does affect the motion of the nuclei), the Franck–Condon factors for ion- ization will not be perfectly diagonal. Thus, ionizing transitions withΔv≠0 will result in potential weaker interferences with both Δv = 0 andΔv≠0 transitions from other intermediate levels. Such Δv≠0 transitions also allow the possible excitation of autoioniz- ing resonances converging to the X+(v>5) and A+(v>1) levels at the two-photon energy. In principle, such resonances can decay into any of the open ionization continua, providing an additional pathway for interference that would likely lead to a very differ- ent photoelectron angular distribution from the direct ionization processes. Unfortunately, the autoionizing states of N 2with the required gerade symmetry have not been extensively explored.28,49 The accurate theoretical treatment for each of these possibilities requires considerable care, and it is beyond the scope of the present work. One additional mechanism can produce time-dependent angu- lar distributions even without true interference effects. If there is one ionization process with an oscillating electron yield and a time independent angular distribution and a second ionization process that is time independent in both the yield and angular distribu- tion, the angular distribution for two processes together will show a time dependence resulting from the varying contribution of the process with the time-dependent electron yield. For example, if there is a time-independent isotropic ionization signal ( β2=β4 = 0) and a second ionization process with a time-dependent ion yield and a constant β2= 1, the observed β2is the incoherent sum of the two processes and will follow the ion yield and oscil- late between 1 and a value closer to 0. Such a mechanism has been proposed by Trabs et al.29to explain time-dependent angular distri- butions in two-photon ionization via Rydberg–valence mixed states in NO. IV. RESULTS Figure 3(a) shows an example of both the raw image and the reconstructed (inverted) image for two-color, VUV+NIR photoion- ization with a time delay of 950 fs. The reconstructed image allows the determination of both the angular distribution parameters, β2 andβ4, and integration over all angles allows the determination of the photoelectron yield as a function of the electron momentum or kinetic energy. Figure 3(b) shows a slice through the photoelec- tron yield data at a delay of 950 fs. This yield is similar to that observed previously by Fushitani et al. ,13but some differences also exist. In particular, although the resolution of the present electron yield spectrum is not as high as that in the earlier study, it is clear that there are some differences in the relative intensities of different final state bands. For example, in Fig. 3(b), we conclude that the strong, lowest energy feature corresponds to the A+(1) photoelectron band blended with the slightly higher energy X+(5) band. This feature is considerably stronger than the corresponding feature in the early work.13The differences between the two spectra are most likely a result of the slightly different photon energies and bandwidths of the VUV pump beams, which will affect not only the intermediate state preparation but also the importance of any resonant feature (i.e., autoionizing levels) at the two-photon (VUV+NIR) energy in the continuum. J. Chem. Phys. 154, 144305 (2021); doi: 10.1063/5.0046577 154, 144305-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . (a) The raw (left) and reconstructed (right) photoelectron images following two-color, VUV+NIR ionization with a time delay of 950 fs. The εindicates the polarization direction of the VUV and NIR pulses, and the different final states are indicated in the reconstructed image. (b) The energy-calibrated photoelectron spectrum extracted from the reconstructed images by integrating over all angles. Two other differences with the photoelectron spectrum of Fushitani et al.13deserve mention. First, a weak feature at 1.6 eV in the present data appears at t = 0 and decays with an increase in time delay. This feature is attributed to the above-threshold ioniza- tion associated with the A+(1)/X+(5) band, that is, ionization of the intermediate states to populate the A+(1)/X+(5) levels by two (rather than one) 795 nm photons. Second, we have not determined the source of the shoulder on the low-kinetic-energy side of the X+(2) peak. By recording the image as a function of the delay between the VUV and NIR pulses, the time dependence of the photoelec- tron spectrum and angular distribution parameters can be deter- mined. Figure 4 shows these photoelectron yields and βvalues as a function of the photoelectron energy and the time delay, as well as the corresponding Fourier transforms. Clear peaks are observed in the photoelectron yields along the energy axis, corresponding to the production of specific vibronic levels of the N 2+X2Σg+and A2Πustates. Along the time-delay axis, each of these peaks shows time-dependent behavior produced by the coherences prepared by the VUV pulse, as well as by the decay of the intermediate state population. Along the energy axes, the β2andβ4plots show a structure similar to the electron yields, corresponding to the pro- duction of different vibronic levels of the ion. Along the delay axes, theβ2andβ4plots show similar oscillations to the electron yields, resulting from the interference among different ionization pathways. In the Fourier transform analysis below, we consider only the N2dynamics when the VUV and NIR pulses are not temporally overlapped. In this manner, the information is extracted from the N2wavepackets without any influence of the strong NIR pulses.When the pulses are temporally overlapped, the physical process is described by dressed states of the target, which are ionized non- sequentially. Such a treatment is beyond the scope of the present work. We note that there are some weak features in the photoelec- tron images at negative time delays to about −100 fs, that is, when the IR pulse arrives before the FEL pulse. These features are assigned to non-sequential two-photon processes of the partially overlapping pulses. The full-width at half-maximum of the cross correlation of the two pulses is estimated to be ∼150 fs from the sidebands in the Xe photoelectron spectrum. Using the measured width of the IR pulse, 55 fs, and assuming that the cross correlation width is the sum in quadrature of the two pulse widths, we derived the estimated FEL pulse width of 140 fs. Time traces for the electron yield and βparameters for each final state were obtained by integrating over the range of electron kinetic energies corresponding to that state at each time step. These traces are shown in Fig. 5. Some, but not all, of the electron yield curves show a smooth time-dependent decay reflecting the effective lifetime of the intermediate state(s) contributing to each photoelec- tron peak. Similar behavior was also noted by Fushitani et al.13Some of theβicurves also show a similar smooth decay that is discussed in more detail below. In addition, most of the electron yield and βi curves show an oscillatory behavior superimposed on the smooth decay. Perhaps the most surprising aspect of Fig. 5 is the relative simplicity of the oscillations observed in both the photoelectron yields and the angular distributions, especially given the potentially large number of ionization processes that could contribute to the signal. This observation clearly indicates that only a small num- ber of ionization pathways dominate for each final state. In most cases, well-defined oscillations are observed in both the electron J. Chem. Phys. 154, 144305 (2021); doi: 10.1063/5.0046577 154, 144305-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . (a) The photoelectron yield, (b) the β2value, and (c) the β4value for two-color, VUV+NIR ionization as a function of the photoelectron energy and the time delay between the VUV and NIR pulses. The trace to the right of (a) shows the total electron yield as a function of delay. The assignments of the final ionic levels are given at the top of each frame. [(d)–(f)] The Fourier transform as a function of electron kinetic energy for (d) the photoelectron yield, (e) the β2value, and (f) the β4value. yields andβparameters. The oscillations observed in the electron yields are similar to those observed by Fushitani et al. ,13,50but the present oscillations are better defined and more persistent than in the earlier work, an observation that most likely results from the lower rotational temperature here. In what follows, we first discuss the overall decay (or lack thereof) of the electron yield, and then, we discuss the oscillatory behavior of the electron yields and the βi parameters. V. DISCUSSION A. Decay of the electron yield In their earlier study,50Fushitani and Hishikawa fitted the decay of the X+(2) photoelectron yield to a bi-exponential form with fast and slow time constants of 290(40) fs and 9(7) ps, respec- tively. In the present experiments, the limited range of time delays only allows the fitting of a single (fast) time constant. In particu- lar, analysis of the present total electron yield for delays between500 and 1250 fs gives a time constant of 450(50) fs. Fitting the present decay curves of individual photoelectron peaks is chal- lenging. Even ignoring the oscillatory behavior, for the most part, the decays are not smooth exponentials, and the oscillations and phases of the oscillations add to the difficulty of extracting mean- ingful fits. Nevertheless, Table I shows the resulting time constants for both the full (0–1350 fs) traces and the time-delayed (200– 1350 fs) portions when the VUV and NIR pulses do not overlap. Values with uncertainties greater than 100% are not included. In most instances, the time constants extracted from the full traces are shorter than those extracted from the delayed portions of the trace. In principle, the different decay times for different photoelec- tron peaks are another aspect of the filtering associated with each detection channel. If, for example, a given final state is only pro- duced by ionization via long-lived intermediate states, the decay constant of the photoelectron signal is expected to be much longer than if the final state is only reached via short-lived intermediate J. Chem. Phys. 154, 144305 (2021); doi: 10.1063/5.0046577 154, 144305-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . Time-delay scans obtained by integrating the signal for the electron yield across individual photoelectron peaks (top, blue) and averaging the corresponding β2and β4values weighted by the electron yield across each peak for the following final states: (a) X+(0), (b) X+(1), (c) X+(2), (d) X+(3). (e) X+(4), and (f) A+(1)/X+(5). In frame (f), the β2andβ4values from the blended A+(1) and X+(5) peaks are separated into two components corresponding to the low and high energy sides of the photoelectron peak. The oscillation of the β2values for these two components have the same frequency and opposite phases. The gray curve in the electron yield plots corresponds to the window function used in the Fourier transforms shown in Fig. 4. states. In the energy region of interest, the largest contribution to the decay rate comes from the predissociation of the A+(1)3dδ1Πu state.24As can be seen in the work of Jungen et al. ,24the correspond- ing widths in the high-resolution absorption spectrum are quite TABLE I . Decay time constants for selected photoelectron peaks. Values in paren- theses for the two traces are uncertainties. Final state Full trace (0–1350 fs) Delayed tracea(200–1350 fs) X+(0) No decay No decay X+(1) 732(200) ... X+(2) 247(200) ... X+(3) 449(200) 834(86) X+(4) 871(200) ... A+(1)/X+(5) 610(200) 266(39) aValues are not reported for time constants with uncertainties greater than 100%.large for the A+(1)3dδ1Πustate; the states that mix most strongly with the A+(1)3dδ1Πustate also have increased widths, while the states that mix only weakly with it have quite narrow widths. Recent high-resolution absorption spectra of jet-cooled N 2recorded using the VUV Fourier-transform spectrometer51at Synchrotron SOLEIL yield widths of ∼2.2 cm−1for transitions to individual rotational lev- els of the A+(1)3dδ1Πustate, but the lines are not fully resolved in those spectra. Those spectra also suggest that there may be a broader, very weak underlying continuum from the state responsible for the predissociation. In contrast, some of the lines in the high-resolution absorption spectra, such as those associated with transitions to the X+(1)6pπand X+(0)9pπstates,51display resolution-limited widths of∼0.3 cm−1. These two widths correspond to lifetimes of 2.4 ps and 17.6 ps, respectively. The X+(0) photoelectron intensity in Fig. 5(a) actually grows slightly with increasing delay before returning to approximately its initial value at the end of the time window. The lack of decay in this channel is not surprising because the transitions associated with J. Chem. Phys. 154, 144305 (2021); doi: 10.1063/5.0046577 154, 144305-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the three intermediate states with X+(0) ion cores [i.e., the X+(0)8f, X+(0)9pπ, and X+(0)9pσstates] that are expected to contribute most strongly to the X+(0) photoelectron signal all have resolution-limited linewidths and thus much longer lifetimes than would be observable on the timescale of the present experiment. All of the remaining photoelectron peaks have decay time con- stants of less than 1 ps. The shortest time constants are found for the A+(1)/X+(5) and X+(2) peaks, corresponding to ∼250–270 fs. This observation is consistent with the results of Fushitani and Hishikawa,50who found a decay time for the X+(2) photoelectron signal of 290(40) fs. Note that the time constant determined from the full A+(1)/X+(5) trace is larger than that determined from the delayed trace as a result of a broad maximum in the trace near Δt = 0 where the VUV and NIR pulses overlap, which is not fit- ted well by the exponential form. This behavior may also result from an interference oscillation with a long period outside the win- dow of the experiment. The fast decay of the A+(1) and X+(2) intensities is not unexpected due to the fast predissociation of the A+(1)3dδintermediate state and its relatively strong mixing with the X+(2)5pπintermediate state.24The somewhat longer lifetimes for the other photoelectron peaks most likely reflect the weaker mixing of the relevant intermediate states with the predissociated A+(1)3dδ state. The trends in the decay times for the different photoelec- tron peaks are consistent with expectations, and the decay time determined for the X+(2) signal is consistent with the earlier mea- surements of Fushitani and Hishikawa. However, the decay times determined from the photoelectron signal seem surprisingly short compared to the expectations based on the linewidths in the high- resolution absorption measurements. For example, the decay time of 266 fs for the A+(1) signal implies a width of ∼19.8 cm−1, which is significantly larger than the 2.2 cm−1widths observed in the absorption measurements. This discrepancy may reflect contribu- tions from weak absorption directly to the state responsible for the predissociation of the A+(1) state or contributions from higher rota- tional levels that predissociate faster as a result of a heterogeneous perturbation.4 In addition to the photoelectron yield curves, some of the βi curves in Fig. 5 also show an overall decay (or growth) in the βival- ues with time. Two possible explanations for this behavior can be proposed that are also based on the decay of the intermediate-state populations. First, if there are ionization pathways via two non- interacting intermediate states leading to the same continua [see Figs. 2(a) and 2(c)] and these two states have different lifetimes, the relative contribution of the two processes to the measured βiwill vary, ultimately reaching the value of the state with the longer life- time. Alternatively, if there is an isotropic background contributing to the electron signal, it will produce a βi= 0 contribution to the overall signal. If the true signal has a significant positive or nega- tiveβivalue, the overall βivalue will decay from close to that value (depending on the relative magnitude of the background) at t = 0 to βi= 0 at long times when the intermediate state has decayed. B. Oscillatory behavior of the electron yields and angular distributions As discussed in Sec. III, of the nine vibronic levels in the bandwidth of the pump laser, three levels—X+(0)8f, X+(0)9pπ, andX+(0)9pσ—have X+(0) cores. The propensity rules for ionization then suggest that ionization via all three levels will result in X+(0), and interference among the corresponding ionization pathways may produce oscillations in both the electron yields and the βiparame- ters. This example is thus an illustration of the fourth scheme dis- cussed in Fig. 2(d). The term energies for the X+(0)8f, X+(0)9pπ, and X+(0)9pσlevels are 123 991, 124 081.45, and 124 108.28 cm−1, respectively,24leading to energy differences of 26.8, 90.5, and 117.3 cm−1, respectively. These differences will be modified somewhat when the rotational structure is considered. The X+(0)9pπand X+(0)9pσlevels interact with each other via the ℓ-uncoupling oper- ator.4In general, p–f interactions are expected to be weak, but all three levels may interact with the A+(1)3dδ1Πulevel and thus can interact with each other in second order. In principle, the inter- action with the A+(1)3dδ1Πulevel could in itself produce oscilla- tions in the electron yields in the X+(0) photoelectron band, but it would not be expected to affect the βivalues because ionization from that level would primarily produce the A+(1) state of the ion [see Fig. 2(b)]. The band oscillator strengths for the three X+(0) bands have been determined by Huber et al.25The X+(0)8f and X+(0)9pπbands have a nearly equal oscillator strength of ∼0.0013, while the X+(0)9pσ value is∼0.000 58. Two additional factors will affect the relative strengths of the three transitions from the ground state. First, the X+(0)8f level gains essentially all of its intensity through interaction with A+(0)3dδ1Πu, and this interaction is strongest for the lowest rotational levels.24Because the experimental rotational temperature is expected to be quite low, the X+(0)8f band may effectively be much stronger than expected relative to the X+(0)9pπand X+(0)9pσ bands. Second, the center wavelength of the VUV pulse envelope will also influence the relative importance of the different intermediate levels. Figure 5(a) shows the time-dependent traces for the X+(0) pho- toelectron band. The electron yield curve is relatively smooth, and it appears to show a single oscillation between 0.20 and 1.35 ps, starting low and reaching a maximum at ∼0.675 ps. This oscilla- tion has a frequency of ∼20 cm−1, which is too low to be effec- tively captured with the present resolution in the Fourier trans- form. Nevertheless, the shape is reproduced reasonably well by a sine function with a frequency corresponding to the difference in term energies for X+(0)9pπand X+(0)9pσlevels (27.8 cm−1),24and with a phase shift of ∼−2.54 rad. The source of this phase shift has not been determined. Note that because the X+(0)9pπand X+(0)9pσlevels interact and because ionization of both states leads to the X+(0) level of the ion [i.e., the case illustrated in Fig. 2(d)], the oscillations could be caused by the interactions between levels or by the interference between ionization pathways. The relatively small magnitude of the oscillation suggests that the overall effect is small. In contrast to the X+(0) electron yield, the X+(0)β2andβ4val- ues show significant oscillations at a higher frequency. Aside from the region in which the VUV and NIR pulses overlap, the β2andβ4 curves in Fig. 5(a) show similar oscillations that are approximately in phase, beginning at smaller values (more isotropic angular distri- butions), growing, and then oscillating with β2ranging from ∼0.4 to 0.7 andβ4ranging from ∼0.0 to 0.25. For the X+(0) final state, the Fourier transform of the β2curve shows a broad feature centered at 105 cm−1, while the Fourier transform of the β4curve shows a J. Chem. Phys. 154, 144305 (2021); doi: 10.1063/5.0046577 154, 144305-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp strong feature at 109 cm−1. The similarities of the two curves suggest that the difference in the frequency simply reflects the large uncer- tainties in the measurements. The observed splittings are reasonably close to the splittings between the X+(0)8f to X+(0)9pπand X+(0)8f to X+(0)9pσterm energies (90.5 and 117.3 cm−1, respectively), sug- gesting that ionization via all three states contributes to the X+(0) photoelectron signal. Figure 5(b) shows the time-dependent electron signal in the X+(1) peak, along with those for the corresponding β2andβ4param- eters. The two potential intermediate levels with X+(1) ion cores correspond to the X+(1)6pπand X+(1)6pσlevels, with term ener- gies differing by 93.2 cm−1.24These levels are weakly mixed by the ℓ-uncoupling operator.4Although the oscillations are not clearly vis- ible Fig. 5(b), the Fourier transform of the electron yield shows a clear peak at 128 cm−1. Figure 5(c) shows the time traces for the X+(2) photoelec- tron signal and the corresponding βivalues. Here, both the yield and theβ2value show strongly modulated oscillations that reach a peak about 75 fs after t = 0, while the β4value is relatively con- stant. The X+(2)5pπlevel is the only intermediate within the band- width of the VUV pulse with an X+(2) core, but this level is mixed with the A+(1)3dδ1Πulevel. The Fourier transforms for the X+(2) yield and the β2andβ4time traces show peaks at 85–90 cm−1. Theβ2curve shows a second peak at 41 cm−1. For reference,24the A+(1)3dδ1Πulevel lies 88.8 cm−1below the X+(2)5pπlevel, in good agreement with the higher frequency feature. A peak at 84 cm−1was also observed in the corresponding Fourier transform of the photo- electron signal of the previous study.13A simple three-level model taking the X+(4) 4pπlevel into account in addition to the A+(1) 3dδand X+(2) 5pπlevels quantitatively explains the appearance of the peak. The model also suggests that the relative phase of the wavefunctions can be inferred from the oscillation phase of the time- trace, as discussed for the two-level model13(see the supplementary material). Figure 5(d) shows the time-dependent traces for the X+(3) photoelectron peak. Here, there is no evidence for any significant oscillations in the electron yield or in the β2andβ4parameters, and the electron yield for X+(3) shows only the exponential decay described above. The X+(3) photoelectron peak is quite small, which is not unexpected because there are no levels with the X+(3) core within the bandwidth of the pump laser. The lack of oscillations sug- gests that a single weak process, for example, photoionization with Δv =±1, leads to the observed signal. Figure 5(e) shows the time-dependent traces for the X+(4) pho- toelectron peak. The electron yield shows a clear oscillation that is out of phase with that of the A+(1) peak.13As in the X+(2) data, here the electron yield and β2values show strong in-phase oscillations, with the first peak occurring at ∼50 fs after t = 0. The corresponding β4curve is again quite flat. The X+(4)4pπlevel is the only intermedi- ate with an X+(4) core within the VUV bandwidth, but, as discussed by Jungen et al. ,24the X+(4)4pπlevel interacts with the A+(1)3dδ1Πu level as well, which has a term energy that is 108.2 cm−1lower. This value is in good agreement with the Fourier transforms of the time traces, which yield a strong peak at 111–113 cm−1. This interaction would not explain the oscillations in β2unless photoionization of the A+(1)3dδ1Πulevel also populated the X+(4) state of the ion. One way in which this would be possible involves the autoionization mechanism discussed in more detail below.The lowest energy band in the photoelectron energy spectrum of Fig. 3(b) is assigned to the unresolved X+(5) and A+(1) bands in the order of decreasing photoelectron energy. The two bands appear to have comparable intensities, but we note that the A+(1) band was much stronger than the X+(5) band in the earlier study by Fushi- taniet al.13As discussed in Sec. IV, this difference may result from small differences in the pump wavelength and two-photon energy in the two experiments. Although the A+(1)3dδ1Πulevel lies within the pump bandwidth, there are no levels with X+(5) ion cores in this neighborhood. This observation suggests that the stronger X+(5) sig- nal in the present experiment may result from the autoionization of resonances lying at the two-photon energy because the impor- tance of such processes could depend strongly on the pump wave- length. Vibrational autoionization of Rydberg series converging to X+(≥6) is expected to preferentially populate the N 2+X+(5) level as a result of the vibrational propensity rule.52Electronic autoion- ization of Rydberg states converging to A+(≥2) to populate X+(<6) levels is expected to be faster than vibrational autoionization of the same states to populate A+(<2) levels. The former process is expected to populate a range of X+(v) levels, while the latter is expected to preferentially populate A+(1). Figure 5(f) shows the time traces for these two components of the low-energy peak. We have integrated over the low- and high- energy regions of the band, and we avoided the region near the center where both components are contributing to the signal. The electron yields show similar time dependences, peaking at t = 0 and displaying regular oscillations. The two β2curves show oscillations with similar frequencies, but the comparison in Fig. 5(f) shows that the A+(1) (lower energy) component is 180○out of phase with the X+(5) (higher energy) component. The Fourier transforms of the β2 andβ4curves for the high-energy component show peaks at 127 and 111 cm−1, respectively, while those of the βcurves for the low-energy component show no obvious peaks. The A+(1) yield peaks at t = 0 and oscillates ∼180○out of phase with the X+(4) yield, as observed in the previous study. The Fourier transforms of both sets of elec- tron yields show a peak at 118–120 cm−1for A+(1) and 113 cm−1 for X+(4). These values for the X+(4) and A+(1) photoelectron yields were both 110 cm−1in the earlier work of Fushitani et al.13 This difference might reflect the somewhat different distribution of intermediate states prepared by the pump transition. VI. CONCLUSIONS In the present work, time-resolved photoelectron energy and angular distributions were recorded following two-color, two- photon ionization of molecular nitrogen via a complex group of nine Rydberg and valence states centered around 15.38 eV exci- tation energy. The most notable aspect of the present work is the remarkable simplification of the observed behavior that results from detecting the ionization process via a particular detection channel. For example, the ionization propensities for different intermediate states lead to the result that by monitoring the photoelectron signal for a particular final state, one can selectively enhance the detec- tion of a specific ionization pathway or set of pathways. Similarly, by focusing on oscillations in the photoelectron angular distribu- tions, one can selectively enhance only those pathways that produce the same continuum state. While such filtering has been discussed J. Chem. Phys. 154, 144305 (2021); doi: 10.1063/5.0046577 154, 144305-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp previously,27,41the potential complexity of the present ionization process makes the observed simplicity of the results particularly striking. For the present experiments, we have been able to build on the detailed analysis of the high-resolution absorption spectrum of N2,21,22,24,25which has allowed us to correlate the time-dependent behavior with the frequency domain data. The full power of the present approach will be revealed through the study of systems that are not nearly as well-characterized. Furthermore, while the present data clearly point to the important interactions and ionization path- ways, realization of the full value of the approach will require some improvements in the experimental data. The first requirement is the need to extend the time-delay scans to much longer times. In par- ticular, extending the delays to 15–20 ps would allow the detailed examination of the rotational structure and corresponding inter- actions, as well as of the nature of the electronic coherences.27,29,53 Such data would allow a more detailed comparison with both high- resolution absorption measurements and the multichannel quan- tum defect analysis of Jungen et al.24Characterization of both the VUV and NIR wavelength dependence of the photoelectron data would also provide a more complete picture of the energetics and dynamics, particularly with respect to the role of autoionizing reso- nances and their influence on the time-dependent dynamics. Such states may ultimately prove valuable in controlling interferences and ionization dynamics. While there is considerable work to be done, the present study provides an important step in the applica- tion of time-resolved photoelectron imaging using FEL sources to the spectroscopy and dynamics of small molecules. SUPPLEMENTARY MATERIAL See the supplementary material for a description of a three- level model for the time-resolved photoelectron intensity and for the photoelectron angular distributions. Figure S1 provides a graph- ical description of the Fourier transform procedure, and Figure S2 provides a summary of the results of the Fourier transforms. AUTHORS’ CONTRIBUTIONS All authors contributed equally to the data acquisition and treatment. This paper was drafted by S.T.P., refined in consulta- tion with K.U. and K.C.P., and then circulated to all authors who contributed comments and constructive criticism. ACKNOWLEDGMENTS We would like to thank the staff of the FERMI facility for their technical support and for ensuring the smooth operation of the experiments. S.T.P. was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Divi- sion of Chemical Sciences, Geosciences, and Biosciences under Con- tract No. DE-AC02-06CH11357. D.Y. acknowledges the Grant-in- Aid of the Tohoku University Institute for Promoting Graduate Degree Programs Division for Interdisciplinary Advanced Research and Education for support. K.U. acknowledges the Research Pro- gram of “Dynamic Alliance for Open Innovations Bridging Human, Environment and Materials” and the IMRAM project for support. H.I. and F.L. acknowledge funding from the NSERC and FRQNT. P.J. and J.P. acknowledge the support of the Swedish ResearchCouncil and the Swedish Foundation for Strategic Research. J.M., A.O., and E.R.S. would like to acknowledge support from the Swedish Research Council (VR) and the Crafoord Foundation. J.M. and A.O. would like to acknowledge support from the Wallenberg Center for Quantum Technology (WACQT) funded by the Knut and Alice Wallenberg Foundation (Grant No. KAW 2017.0449). E.R.S. would like to acknowledge support from the Royal Phys- iographic Society of Lund. G.S., P.K.M., and M.M. acknowledge funding from the European Union’s Horizon 2020 research and innovation program under Marie Sklodowska-Curie Grant Agree- ment No. 641789 MEDEA. G.S., D.E., and R.S. acknowledge fund- ing from the Deutsche Forschungsgemeinschaft (DFG) [IRTG CoCo (2079) and QUTIF SA 3470/2]. The ELI-ALPS project (No. GINOP- 2.3.6-15-2015-00001) was supported by the European Union and co-financed by the European Regional Development Fund. M.F. and A.H. acknowledge support from the Cooperative Research Pro- gram of “Network Joint Research Center for Materials and Devices” (Grant No. 20191083). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1Molecular Rydberg Dynamics, edited by M. S. Child (Imperial College Press, London, 1997). 2M. S. 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5.0035425.pdf

J. Chem. Phys. 154, 064106 (2021); https://doi.org/10.1063/5.0035425 154, 064106 © 2021 Author(s).A periodic equation-of-motion coupled- cluster implementation applied to F- centers in alkaline earth oxides Cite as: J. Chem. Phys. 154, 064106 (2021); https://doi.org/10.1063/5.0035425 Submitted: 29 October 2020 . Accepted: 11 January 2021 . Published Online: 11 February 2021 Alejandro Gallo , Felix Hummel , Andreas Irmler , and Andreas Grüneis ARTICLES YOU MAY BE INTERESTED IN Local embedding of coupled cluster theory into the random phase approximation using plane waves The Journal of Chemical Physics 154, 011101 (2021); https://doi.org/10.1063/5.0036363 r2SCAN-D4: Dispersion corrected meta-generalized gradient approximation for general chemical applications The Journal of Chemical Physics 154, 061101 (2021); https://doi.org/10.1063/5.0041008 r2SCAN-3c: A “Swiss army knife” composite electronic-structure method The Journal of Chemical Physics 154, 064103 (2021); https://doi.org/10.1063/5.0040021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp A periodic equation-of-motion coupled-cluster implementation applied to F-centers in alkaline earth oxides Cite as: J. Chem. Phys. 154, 064106 (2021); doi: 10.1063/5.0035425 Submitted: 29 October 2020 •Accepted: 11 January 2021 • Published Online: 11 February 2021 Alejandro Gallo,a) Felix Hummel, Andreas Irmler, and Andreas Grüneis AFFILIATIONS Institute for Theoretical Physics, TU Wien, Wiedner Hauptstraße 8–10/136, Vienna 1040, Austria a)Author to whom correspondence should be addressed: alejandro.gallo@tuwien.ac.at ABSTRACT We present an implementation of the equation of motion coupled-cluster singles and doubles (EOM-CCSD) theory using periodic bound- ary conditions and a plane wave basis set. Our implementation of EOM-CCSD theory is applied to study F-centers in alkaline earth oxides employing a periodic supercell approach. The convergence of the calculated electronic excitation energies for neutral color centers in MgO, CaO, and SrO crystals with respect to the orbital basis set and system size is explored. We discuss extrapolation techniques that approxi- mate excitation energies in the complete basis set limit and reduce finite size errors. Our findings demonstrate that EOM-CCSD theory can predict optical absorption energies of F-centers in good agreement with experiment. Furthermore, we discuss calculated emission energies corresponding to the decay from triplet to singlet states responsible for the photoluminescence properties. Our findings are compared to experimental and theoretical results available in the literature. Published under license by AIP Publishing. https://doi.org/10.1063/5.0035425 .,s I. INTRODUCTION Density Functional Theory (DFT)1,2using approximate exchange and correlation energy density functionals is arguably the most successful ab initio approach to compute materials prop- erties. Its application goes beyond ground state properties by providing a reference or a starting point for methods that treat excited-state phenomena explicitly. In this context, theories such as Time-Dependent Density Functional Theory (TD-DFT)3,4and theGW approximation5are widely used to tackle excited states in molecules and solids.6,7Nonetheless, they often suffer from a strong dependence on the DFT reference calculation. In the case of TD-DFT, results depend strongly on the choice of the approx- imate exchange and correlation density functional. Similarly, the so-called non-self-consistent G0W0quasiparticle energies depend strongly on the Kohn–Sham orbital energies, whereas fully self- consistent GW calculations are not as often performed and do not necessarily improve upon the accuracy compared to G0W0.8 To compute charge neutrality preserving optical absorption ener- gies from the electron addition and removal energies obtainedin the GW framework, it is necessary to account for the exciton binding energy. Excitonic effects are often approximated using the Bethe–Salpeter equation (BSE).9We note that despite the high level of accuracy and efficiency of GW-BSE calculations,10many choices and approximations have to be made in practice that are difficult to justify in a pure ab initio framework. Therefore, it seems worth- while to explore alternative methods that are less dependent on DFT approaches. Coupled-Cluster (CC)11–13formulations are widely used in the field of molecular quantum chemistry for both the ground state and excited states via the Equation of Motion Coupled-Cluster (EOM- CC) formalism.14Ground state CC theories such as Coupled Clus- ter Singles Doubles (CCSD) and perturbative triples [CCSD(T)]15,16 have become one of the most successful methods in molecules in terms of their systematically improvable accuracy and computa- tional efficiency. Likewise, EOM-CC methods are routinely applied to molecular systems with great success.17–21However, we stress that the computational cost of CC theories is significantly larger than that of Green’s function based methods mentioned above. Nonetheless, several studies have focused on making use of these J. Chem. Phys. 154, 064106 (2021); doi: 10.1063/5.0035425 154, 064106-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp wavefunction methods also in solids to study ground and excited state properties.22–24While the Equation of Motion (EOM) type methods are well understood and benchmarked in finite systems, this is less so for periodic systems, where ongoing efforts are made toward applications in solids. Previous applications of the EOM type methods have focused on electronic band structures using the Ionization Potential EOM-CC (IP-EOM-CC) and Elec- tron Attachment EOM-CC (EA-EOM-CC) extensions23–26as well as its Electron Excitation EOM-CC (EE-EOM-CC) extensions,27,28 all of which are based on Gaussian basis sets. For local phenom- ena such as defect excitation energies, several studies have been performed employing cluster models of the periodic structures.29,30 One of the main challenges in these calculations is to achieve a good control over the finite basis set and system size errors, which is often achieved using extrapolation techniques. In this manuscript, we study excited state properties of point defects in solids computed on the level of EE-EOM-CC. Understanding impurities in solids is important for both theoretical and practical reasons. Lattice defects affect bulk properties of the host crystal, and both the understand- ing of ground and excited-state properties is essential for these sys- tems.31,32Here, we focus on color centers in the alkaline earth oxide crystals MgO, CaO, and SrO in the rock salt structure. Removing an oxygen atom from these systems results in the so-called F-centers that can be filled by 2 ( F0), 1 ( F+), or 0 ( F2+) electrons. The corre- sponding one-electron states are stabilized by the Madelung poten- tial of the crystal and their electron density is, in general, localized in the cavity formed by the oxygen vacancy. These defects are typ- ically produced by neutron irradiation33or additive colorization.34 Much effort has been made to elucidate the exact mechanism of the luminescence of F-centers in MgO, CaO, and SrO.35,36The ground and excited state properties of these vacancies are of importance for a wide range of technological applications, including color center lasers. Furthermore, vacancies of oxides are of general importance for understanding their surface chemistry and related properties. In this work, we will concentrate on the diamagnetic F0-center. The trapped electrons can be viewed as a pseudoatom embedded in a solid, where the optical absorption and emission between ground and low-lying excited states is characterized by the electron trans- fer between 1 sand 2 sor 2pone-electron states. Initial theoreti- cal studies of these defects were already performed in the 1960s and 1970s using effective Hamiltonians.37–39Modern ab initio stud- ies of the F0center in MgO have employed cluster approaches in combination with quantum chemical wavefunction based meth- ods,30fully periodic supercell approaches in combination with the GW-BSE approach,40,41or quantum Monte Carlo calculations.42In this work, we seek to employ a periodic supercell approach and a novel implementation of Equation of Motion CCSD (EE-EOM- CCSD) theory using a plane wave basis set. In addition to the F- center in MgO, we will also study F-centers in CaO and SrO. We note that EE-EOM-CCSD theory is exact for ground and excited states of two electron systems and is therefore expected to yield very accurate results for the F0center in alkaline earth oxides. We will discuss different techniques to correct for the finite basis set and supercell size errors and demonstrate that EE-EOM-CCSD theory can be used to compute accurate absorption and emission energies compared to experiment without the need for adjustable parameters and the ambiguity caused by the choice of the starting point.The following is a summary of the structure of this work: In Sec. II, we give a brief overview of the employed theoretical and computational methods used to compute excitation energies includ- ing extrapolation techniques that are needed to approximate to the complete basis set and infinite system size limit. Section III presents the obtained results of the defect calculations and draws a compari- son between this work and the available experimental and theoretical results from the literature. II. THEORY AND METHODS We start this section by giving a brief description of the employed CC methods followed by a discussion of the computa- tional details. A. CCSD theory The CC approximation is based on an exponential ansatz for the electronic wavefunction13,43acting on a single Slater determinant ∣0⟩, ∣ΨCC⟩=eˆT∣0⟩, (1) where the cluster operator consists of second-quantized neutral excitation operators, ˆT=∑ μtμˆτμ, tμ∈C, (2) with μlabeling the excitation configurations. For instance, when considering only singles and doubles excitations (CCSD), the unre- stricted CCSD cluster operator is given by ˆT=∑ a,ita iˆa† aˆai+1 4∑ a,b,i,jtab ijˆa† aˆa† bˆajˆai, (3) where the set of indices { a,b,c,. . .} denote virtual or unoccu- pied spin orbitals and { i,j,k,. . .} denote occupied spin orbitals. Orbitals are occupied or unoccupied with respect to the reference Slater determinant ∣0⟩, which may come from a Hartree–Fock (HF) or a DFT calculation. Here, we will restrict the discussion to the case of CCSD. Applying the CC ansatz to the stationary many-body electronic Schrödinger equation results in ¯H∣0⟩=e−ˆTˆHeˆT∣0⟩=ECC∣0⟩, (4) where ECCis the coupled cluster energy, and we have implicitly defined the similarity transformed Hamiltonian ¯H. The state ∣ΨCC⟩ is parametrized by the coefficients tμ, which can be obtained by pro- jection. In the case of CCSD, one projects the Schrödinger equation onto the singles and doubles sections of the Hilbert space, ECC=⟨0∣¯H∣0⟩, (5) 0=⟨0∣ˆa† iˆaa¯H∣0⟩, (6) 0=⟨0∣ˆa† iˆa† jˆabˆaa¯H∣0⟩. (7) J. Chem. Phys. 154, 064106 (2021); doi: 10.1063/5.0035425 154, 064106-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Equations (5)–(7) are a set of coupled non-linear equations in terms of the amplitudes ta iandtab ijthat are solved by iterative methods. B. EE-EOM-CCSD theory A common way to obtain excited states based on the CC theory is through diagonalization of the similarity transformed Hamilto- nian ¯Hin a suitable subspace of the Hilbert space.14We are going to present the neutral variant of this approach, also called electronically excited equation of motion, for which the number of electrons is conserved. In consequence, restricting from now on again the anal- ysis to singles and doubles excitations, the ansatz for an excited state ˆR∣ΨCC⟩is ˆQˆHˆR∣ΨCC⟩=ˆQˆHˆReˆT∣0⟩=ERˆQˆR∣ΨCC⟩, (8) where ˆR=r0+∑ a,ira iˆa† aˆai+1 4∑ a,b,i,jrab ijˆa† aˆa† bˆajˆai, rμ∈C, (9) is a linear excitation operator and ERis its excitation energy and ˆQ is the projector onto the singles and doubles excitations manifold of the Hilbert space, that is, ˆQ=∑ a,i∣a i⟩⟨a i∣+1 4∑ a,b,i,j∣ab ij⟩⟨ab ij∣. (10) Equation (8) is equivalent to a commutator equation only involv- ing ¯Hand the excitation energy difference ΔERbetween ERand the correlated ground state ECC, [ˆQ¯H,ˆR]∣0⟩=(ˆQ¯HˆR)c∣0⟩=ΔERˆQˆR∣0⟩. (11) It is worthwhile noting that the commutator on the left-hand side means that only connected diagrams need to be considered in the CI expansion, which is denoted by the parentheses () c. Equation (11) motivates the name equation of motion due to its resemblance to the time-dependent Heisenberg picture differential equation for the time evolution of an operator. C. Computational methods and details Here, all EE-EOM-CCSD calculations of defective supercells employ a HF reference. The HF calculations are performed using the Vienna ab initio simulation package (VASP)44and a plane wave basis set in the framework of the projector augmented wave (PAW)45 method. The energy cutoff for the plane wave basis set is 900 eV. The defect geometries have been relaxed on the level of DFT-PBE, start- ing from a defective geometry with the corresponding equilibrium lattice constant (MgO—4.257 Å, CaO—4.831 Å, and SrO—5.195 Å), keeping the lattice vectors and volume fixed. In this work, we study defective 2 ×2×2, 3×3×3, and 4 ×4×4 fcc supercells containing 15, 53, and 127 atoms, respectively. The oxygen vacancy results in an outward relaxation of the alkaline earth atoms away from the cav- ity created by the oxygen vacancy. This outward relaxation strongly FIG. 1 . Geometry of the neutral F-center in MgO. Red and orange spheres corre- spond to oxygen and magnesium atoms, respectively. The yellow isosurface was computed from the localized electronic states in the bandgap of MgO that origi- nates from the two trapped electrons. δmeasures the displacement along the A1g vibrational mode with the Mg atoms out of their equilibrium position in the bulk structure and was deliberately chosen larger for this figure to emphasize the effect of lattice relaxation. overlaps with the vibrational mode A1gand is illustrated in Fig. 1. While the DFT-PBE calculations have been carefully checked for convergence with respect to the k-point mesh used to sample the first Brillouin zone, all HF and post-HF calculations employ the Γ-point approximation. We have implemented Unrestricted CCSD (UCCSD) and EE-EOM-CCSD in the coupled cluster for solids (cc4s) code that was previously employed for the study of various ground state prop- erties of periodic systems.22,46The employed Coulomb integrals and related quantities were calculated in a completely analogous manner. Our UCCSD implementation is based on the interme- diate amplitude approach of Stanton et al.47On the other hand, our EE-EOM-CCSD implementation uses intermediates for the similarity transformed Hamiltonian e−ˆTˆHeˆTfrom the works of Stanton et al.14and Shavitt et al.48We use the Cyclops Tensor Framework (CTF)49for the implemented computer code, which enables an automated parallelization of the underlying tensor contractions. The similarity transformed Hamiltonian is a non-Hermitian operator, and therefore, left and right eigenvectors differ in general. The diagonalization of the similarity transformed Hamiltonian is done using a generalized Davidson solver.50,51This solver enables the calculation of EE-EOM-CCSD energies by approximating the left eigenvector space by the right eigenvector space. For the ini- tial guess of the eigenvectors, we use the one-body HF excita- tion energies and corresponding Slater determinants. The UCCSD and EE-EOM-CCSD calculations have been performed using only a small number of active HF orbitals around the Fermi energy of the employed supercells. Most occupied orbitals at low ener- gies are frozen and the same applies to all unoccupied orbitals above a certain cutoff energy. The following sections summarize the J. Chem. Phys. 154, 064106 (2021); doi: 10.1063/5.0035425 154, 064106-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp benchmarks of the implemented EE-EOM-CCSD code and inves- tigate the convergence behavior of the computed excitation ener- gies with respect to the number of active orbitals as well as system size. 1. Benchmark results In the following, we discuss benchmark results of our EE- EOM-CCSD implementation and outline our approach to identify the spin multiplicity attributed to the excited states. To verify the implemented expressions, we have compared the computed EE- EOM-CCSD excitation energies to results computed using a well- established quantum chemical code NWCHEM.52As most quan- tum chemical codes (including NWCHEM) employ atom-centered Gaussian basis sets, it was also necessary to implement an inter- face that reads the orbital coefficients from NWCHEM and employs the LIBINT253library to compute corresponding integrals. As test systems, we have selected the neon atom and the water molecule in the aug-cc-pvdz basis. All computed UCCSD energies using HF and DFT reference determinants achieve excellent agreement (eight significant digits) between both codes. For EE-EOM-CCSD calcula- tions, the singlet states computed by NWCHEM were also obtained using our EE-EOM-CCSD implementation with similar conver- gence behavior and in excellent agreement (eight significant dig- its). We can identify the triplet and the singlet state in our output by using spin-flip EE-EOM-CCSD54and comparing the degener- acy of the states computed with and without spin-flip excitations. In future work, we will implement the direct computation of the spin expectation value. 2. Orbital basis convergence of excited states All presented findings in this section have been obtained for the F-center in MgO. However, the corresponding findings for CaO and SrO are qualitatively identical unless stated explicitly. We first seek to investigate the character of the employed HF orbitals and the convergence of the computed excitation energies with respect to the canonical orbital basis set size. The HF orbitals have been computed for a defective 2 ×2×2 MgO supercell con- taining 15 atoms. Figure 2 depicts the energy levels around the Fermi energy and isosurfaces of charge densities computed for the defect states. The occupied state with the highest one-electron energy cor- responds to the occupied defect state and its orbital energy is located in the gap of the bulk crystal. Its charge density is well localized in the cavity created by the oxygen vacancy. In the thermodynamic limit (big supercells or dense k-meshes), the direct and fundamen- tal gap of pristine MgO is 15.5 eV on the level of HF theory,55 which is significantly larger than the experimental gap of about 7.8 eV. The neglect of correlation effects in HF theory overestimates bandgaps for a wide range of simple semiconductors and insulators. The orbital ordering between defect and bulk states depicted in Fig. 2 is qualitatively identical to the one observed for CaO and SrO. How- ever, we stress that in contrast to MgO, CaO and SrO exhibit an indirect bandgap with a conduction band minimum at the Brillouin zone boundary. We note that the supercells investigated in this work contain up to 127 atoms, corresponding to more than 1000 valence electrons. The computational cost of EE-EOM-CCSD theory scales as O(N6), where Nis some measure of the system size. In particular, the cost FIG. 2 . Occupied and virtual HF energy levels. The red levels correspond to the defect states, and the corresponding isosurfaces of the charge densities are depicted. for some of the most important tensor algebraic operations scales asO(N4 vN2 o)andO(N2 vN4 o), where NoandNvrefer to the num- ber of occupied and virtual orbitals, respectively. Additionally, the memory footprint of our implementation scales as O(N4). Due to the steep scaling of the computational cost, an explicit treatment of all electrons on the level of EE-EOM-CCSD becomes intractable and renders it necessary to freeze a large fraction of the occupied and vir- tual HF states. In the following, we will investigate the convergence of the computed excitation energies with respect to the number of active virtual and occupied states. We first investigate the convergence of EE-EOM-CCSD excita- tion energies with respect to the virtual orbital basis set. Among the 61 occupied spatial HF orbitals, we keep only the four orbitals active with the highest energy. Furthermore, we only investigate excited many-electron states with ra iexcitation amplitudes that correspond to a significant charge transfer from the occupied s-like defect state to the virtual p-like defect states, as illustrated in Fig. 2. Figure 3 depicts the convergence of the EE-EOM-CCSD excitation energies that we assign to local excitations of the F-center. In passing, we note that EE-EOM-CCSD theory predicts a number of excited states that describe electronic excitations with charge transfer from the defect to bulk states, which will not be explored in this work. The elec- tronic ground state of the neutral F-centers studied in this work is a singlet state. The lower and higher excitation energies shown in Fig. 3 correspond to singlet–triplet and singlet–singlet transi- tion energies, respectively. We observe for both excitation energies a 1/(Nv+No) convergence to the complete basis set limit. This behav- ior is not unexpected and agrees with the convergence of ground state energies. Furthermore, we note that a similar convergence was observed for EE-EOM-CCSD exciton energies of bulk materials28 and for excitation energies of the cytosine molecule.56We note that it might seem advantageous to replace HF virtual orbitals with a different type of orbitals; for example, natural orbitals to acceler- ate the convergence. However, we have found that these orbitals will mostly accelerate the convergence of the ground state energy, while J. Chem. Phys. 154, 064106 (2021); doi: 10.1063/5.0035425 154, 064106-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Basis set extrapolation of the lowest EE-EOM-CCSD excitation energies corresponding to excitations of the F-center defect in MgO. All computed ener- gies have been fitted against 1/( Nv+No), where NvandNoare the number of virtual and occupied orbitals used, respectively. The lower and upper panels correspond to a singlet–triplet and a singlet–singlet transition, respectively. This extrapolation has been obtained for a supercell composed of eight Mg and seven O atoms. introducing large basis set incompleteness errors in the convergence of excitation energies. In this work, we will employ a 1/( Nv+No) extrapolation to approximate excitation energies in the complete basis set limit of all systems. Figure 4 shows the employed basis set extrapolation for identi- cal transitions in a larger 4 ×4×4 supercell. We note that the slope of the excitation energy extrapolation is significantly steeper compared to the 2 ×2×2 supercell shown in Fig. 3. This can be attributed FIG. 4 . Basis set extrapolation of the lowest EE-EOM-CCSD excitation energies corresponding to excitations of the F-center defect in MgO using a 4 ×4×4 supercell. The fit has been performed ignoring the first four data points. States, energies, and fit are to be interpreted as in Fig. 3.to the smaller number of virtual orbitals relative to the complete basis set size for the given plane wave cutoff energy. Therefore, we ignore the first four points in the extrapolation for all systems in the 4×4×4 supercell. In the case of CaO and SrO, the basis set con- vergence of the excitation energies is qualitatively identical, and we employ the same orbital basis set sizes in all extrapolations. We now investigate the convergence of the EE-EOM-CCSD excitation energies with respect to the number of active occu- pied orbitals, keeping a virtual orbital basis set consisting of 10 unoccupied orbitals and employing a 2 ×2×2 supercell only. Figure 5 depicts the convergence of the lowest defect excitation energy (singlet–triplet transition) with respect to the size of the active occupied orbital space. The horizontal axis at the bottom shows the number of active occupied orbitals. The horizontal axis at the top of Fig. 5 shows the corresponding lowest HF orbital energy. Our findings demonstrate that the excitation energy increases with respect to the number of active occupied orbitals and is well con- verged to within a few meV using more than about 25 occupied orbitals. However, a comparison between the converged result and a minimal active occupied orbital space, consisting of the occupied defect orbital only, reveals that such a truncation introduces exci- tation energy errors of about 120 meV. We note that one-electron states with relative energies below −50 eV exhibit Mg 2 pand 2 s character and are therefore expected to be negligible for the com- puted excitation energies. From the above findings, we conclude that the excitations studied in the present work exhibit a significantly larger error from the virtual orbital basis truncation than from the occupied orbital basis truncation. Due to the computational cost of EE-EOM-CCSD calculations, we will therefore extrapolate the exci- tation energy to the complete basis set limit while using only four occupied orbitals. FIG. 5 . Convergence of the EE-EOM-CCSD excitation energy for the singlet–triplet transition at the F-center of MgO with respect to the number of inactive/frozen occupied orbitals in the EE-EOM-CCSD calculation. For the employed supercell, the HF calculations have been performed using 61 occupied and 10 virtual orbitals. The top horizontal axis shows the lowest HF energy of the included active occupied orbital relative to the occupied defect state. All orbitals with a lower energy have not been included in the respective EE-EOM-CCSD calculation. J. Chem. Phys. 154, 064106 (2021); doi: 10.1063/5.0035425 154, 064106-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 3. System size convergence of excitation energies Having discussed the basis set convergence of the computed EE-EOM-CCSD excitation energies, we now turn to the discussion of their convergence with respect to the supercell size. Excitation energies are intensive quantities. However, their convergence with respect to the system size can sometimes be extraordinarily slow. We have computed the F-center singlet–triplet and singlet–singlet transition energies for three different supercell sizes containing 15, 53, and 127 atoms. Table I lists the computed excitation ener- gies for all systems using different supercell sizes. The excitation energies have been obtained using four active occupied orbitals only and extrapolated to the complete basis set limit, as discussed in Secs. II C 1 and II C 2. We note that the excitation energies converge monotonously for MgO with the increase in the supercell size but show some non- monotonic behavior for the other two systems studied. This can be explained by the fact that CaO and SrO exhibit a conduction band minimum at the Brillouin zone boundary. The electronic states at the conduction band minimum are therefore only accounted for when using supercells that are constructed from even-numbered multi- ples of the fcc unit cell. Neglecting these important states around the Fermi energy leads to a significant overestimation of the excitation energies for the excited singlet states, as can be seen by comparing the results obtained for the 3 ×3×3 supercell to the findings for the 2×2×2 and 4 ×4×4 supercells. Here, we seek to remove the remaining finite size errors of the excitation energies by performing an extrapolation to the infinite system size limit assuming a 1/ Nconvergence, where Nis the total number of electrons in each supercell. This approach is in agreement with the procedures that are applied to ground state energy calcu- lations.22,57For the sake of consistency, we employ only 2 ×2×2 and 4 ×4×4 supercells for the extrapolation for all three studied systems. TABLE I . Convergence of the F-center excitation energies in MgO, CaO, and SrO for increasing supercell sizes. TDL corresponds to the extrapolated thermodynamic limit estimate of the respective excitation energies, assuming a 1/ Nconvergence and employing the energies of the 2 ×2×2 and 4 ×4×4 supercells. Here, Nstands for a measure of the system size. In this case, the number of electrons is used. All energies are in units of eV. System Supercell3T1T MgO 2 ×2×2 7.009 8.522 3×3×3 4.866 6.571 4×4×4 4.038 5.646 TDL 3.660 5.281 CaO 2 ×2×2 3.224 3.338 3×3×3 2.951 4.025 4×4×4 2.081 3.157 TDL 1.936 3.134 SrO 2 ×2×2 2.324 2.413 3×3×3 2.404 3.155 4×4×4 1.332 2.351 TDL 1.206 2.343Our findings show that the excitation energies decrease signif- icantly with the increase in the supercell size in the case of MgO. Changing the supercell size from 2 ×2×2 to a 4 ×4×4 cell results in a lowering of the excitation energies by almost 3 eV. This relatively slow convergence is expected to originate from the strongly delocal- ized excited defect states of the neutral F-center in MgO. We note in passing that the excitation energies of the F-centers in CaO and SrO exhibit a significantly faster convergence with respect to the sys- tem size. We attribute this behavior to a more localized character of the excited F-center in CaO and SrO compared to MgO that might be explained by the significantly smaller size of the cavity formed by the oxygen vacancy in MgO compared to CaO or SrO. III. RESULTS In this section, we describe the photochemical process of absorption and emission at the F-center of alkaline earth oxides. We first discuss the energies of the electronically excited defect states as a function of the atomic displacements along the A1gvibrational mode in MgO to introduce the emission model. Next, we present our results for the absorption and emission of the F-center in MgO, where problems in the interpretation of the experimentally observed luminescence band are discussed additionally. We end this section with a discussion of the results for CaO and SrO. A. Absorption and emission process in F-centers Our analysis of the emission process is based on a Franck– Condon58,59description of the defect. This is a common approach to treat emission processes in solids and molecules.39,60,61Figure 6 FIG. 6 . Configuration curve along the phonon A1gmode for the excited states of theF-center in MgO (as shown in Fig. 1). The1ACCcurve represents the singlet UCCSD ground state and the upper curves depict the EE-EOM-CCSD excited states. Dashed lines represent the EE-EOM-CCSD states that do not play a role for our discussion but are included for completeness. The energies presented are energy differences between the excited state energies and the UCCSD energy. The calculation was done for a 4 ×4×4 cell containing 127 atoms, four active electrons, and 64 virtual orbitals. J. Chem. Phys. 154, 064106 (2021); doi: 10.1063/5.0035425 154, 064106-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp shows the configuration coordinate diagram along an approxi- mate A1gvibrational mode for the most important EE-EOM-CCSD excited states and the UCCSD ground state singlet1ACC. We approx- imate the atomic displacement along the A1gmode by increasing the outward displacement of the alkaline earth atoms, as depicted in Fig. 1, and keeping all other atomic positions of the employed 4×4×4 supercell fixed. The configuration curve has been com- puted only for MgO but serves as a qualitatively identical model for CaO and SrO. Within this picture, the absorption is given by the optically allowed transition of1ACC→1Tat the ground state geometry in Fig. 6. Taking into account the Franck–Condon approx- imation, once the F-center is in the excited singlet state, a relaxation of the atoms along the A1gvibrational mode sets off, which could induce a crossing in the configuration curve with the excited triplet state3T. Luminescence is then achieved through the transition 3T→1ACC. From the above discussion and the fact that the mini- mum of the3Tstate is close to the minimum of the ground state, we conclude that the absorption and emission energies can there- fore be well approximated using the energy differences computed in the equilibrium structure of the electronic ground state for the F-center. B. MgO TheF-center in MgO was first discovered by Wertz et al.66in its positively charged variant ( F+-center) by electron spin resonance measurements, showing a strong localization of the electrons in oxy- gen vacancies. A host of experimental results followed and with it a better understanding of the absorption and luminescence mecha- nisms.32,67,68Experimental and theoretical studies have shown that the Mg atoms relax in an outward direction from the vacancy.69,70 By using a semi-empirical model, Kemp and Neeley37predicted an optical absorption energy of 4.73 eV in good agreement with exper- imental findings of 4.95 eV.67,71The luminescence band of the F+ center was measured at around 3.15 eV,72while for the F0center, a luminescence of 2.4 eV was predicted from temperature depen- dent measurements of the absorption spectrum in conjunction with a simplified Huang–Rhys model approach.71 Using EE-EOM-CCSD in combination with the outlined extrapolation techniques yields an absorption and emission energy of 5.2 eV and 3.66 eV, respectively. Previous many-body ab initio calculations using GW-BSE,40quantum Monte Carlo42methods, and CASPT230agree with our results for both absorption and emission to within about 0.4 eV, as summarized in Table II. The calculated absorption energies are in good agreement with exper- imental measurements of 5.0 eV. We note, however, that the GW results (excluding the BSE exciton binding energy) obtained for different levels of self-consistency and DFT references exhibit a significant variance ranging from 4.48 eV to 5.4 eV. Conse- quently, GW-BSE absorption energies are strongly dependent on the DFT reference. Furthermore, we stress that a direct comparison of the computed emission energies between the quantum chemi- cal approaches (EE-EOM-CCSD and CASPT2) and QMC or GW is complicated by the fact that the latter approaches do not con- sider the emission process of de-excitation from the excited triplet states. Instead, the emission energies computed using QMC and GW-BSE correspond to the decay from the excited singlet state in its relaxed geometry along the A1gmode. Nonetheless, from theTABLE II . Obtained results from this work for the absorption and emission energies of the F-centers in MgO, CaO, and SrO. The EE-EOM-CCSD results are extrapolated to the complete basis set and infinite supercell size limit in order to allow for a direct comparison between theory and experiment. The GW gaps do not correspond to optical excitation energies but are included for comparison. All energies are in eV units. System Method Absorption Emission MgO EE-EOM-CCSD 5.28 3.66 Expt.625.0 2.4 QMC425.0(1) 3.8(1) CASPT2305.44 4.09 G0W0@LDA0-BSE404.95 3.4 G0W0@LDA0405.4 G0W0@PBE414.48 GW 0@PBE414.71 GW@PBE415.20 CaO EE-EOM-CCSD 3.13 1.93 Expt.63,643.02 1.93 Expt.623.1 2.05–2.01 TD-DFT@B3LYP653.52 2.1 G0W0@PBE413.20 GW 0@PBE413.53 GW@PBE413.87 SrO EE-EOM-CCSD 2.34 1.2 Expt.622.4 results shown in Fig. 6, we conclude that these different emission energies are expected to agree to within the errors made by other approximations. The measured experimental emission at 2.4 eV34and its inter- pretation are the topic of an ongoing debate. Initially, this peak has been attributed to the F-center and common interpretations have ranged from a singlet–singlet transition to a3T1u→1A1gtransi- tion.38,62However, it was first suggested by Edel et al.34,73,74that this band results from a recombination process similar to recom- bination processes in semiconductors. Edel and co-workers argued that the three-electron vacancy F−recombines with the F+-center. Rinke et al.40have suggested that the 2.4 eV emission is produced when electrons in the defect orbitals recombine with the valence holes that can be produced by intense UV light irradiation. The cre- ation of these holes is possibly also related to the concentration of H−impurities that are commonly present in MgO samples, espe- cially when these have been thermochemically reduced.33,75–78The presence of H−impurities in MgO could account for the long-lived luminescence through a hopping mechanism of the electrons from H−to H−impurities until they encounter an F-center. However, it is not immediately clear from the ab initio calculations thus far if these states are orbital and spin triplets or otherwise, as has been proposed in experimental evidence and symmetry arguments.62It has been noticed, however, that the strength of the 2.4 eV band is temperature dependent as well as F-center and H−concentra- tion dependent.78Typically, neutron irradiation produces mainly J. Chem. Phys. 154, 064106 (2021); doi: 10.1063/5.0035425 154, 064106-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp F+-centers, while electron irradiation or additive colorization induces mainly F-centers.34Rinke et al. argue that given the fact that the positions of the absorption band for the Fand F+cen- ters are almost identical, it is to be expected that this is also the case for the emission. Even though similar luminescence peaks for these centers have been predicted in Ref. 40, no substructure in the emission band can be observed experimentally (unlike in the absorption band). Here, we propose a different interpretation of this observation. We suggest that the F-center does not, in fact, lumi- nesce. Indeed, modern theoretical computations seem to agree on the fact that the 2.4 eV band does not belong to the F-center lumi- nescence process. We stress that all theoretical results for the emis- sion energy summarized in Table II range from 3.4 eV to 4.09 eV. Moreover, there is a strong photoconversion from Finto F+- centers,79suggesting that before the F-center has a chance to lumi- nesce, a conversion into F+happens followed by an absorption of the F+-center since the absorption band for it is similar to the F-band. Our calculations show that the excitation energy for the singlet state in the F-center of MgO converges very slowly with respect to the system size, indicating that the optically excited state is signif- icantly more delocalized than the ground state. This could make a photoconversion into F+significantly more likely and therefore corroborates our interpretation. C. CaO and SrO Historically, one of the best studied F-centers in the alkaline earth oxides is the one in CaO.38The identification of the F-center’s charged state is made easier by the fact that, unlike for MgO, the absorption band is different for the FandF+centers. Furthermore, we note that the lattice constant of CaO is significantly larger than that for MgO, which leads to a reduced confinement of the trapped charges and shifts the absorption band to lower energies. Early the- oretical and experimental investigations have interpreted the 2.0 eV emission band to be a transition from a spin and orbital triplet3T1u into the ground state singlet1A1g.38,63,80However, a1T1u→1A1g transition is also possible at a slightly higher energy. In general, the CaO luminescence mechanism has been found to be a combination of a singlet–singlet and a triplet–singlet transition, which are acti- vated at different temperatures.63,64Since the excited triplet state lies slightly below in energy from the excited singlet state, there is a pop- ulation conversion at temperatures of around 600 K. Specifically, at low temperatures up to 300 K, one measures a transition at around 1.98 eV, whereas as the temperature increases, the excited singlet gets populated and a much more rapid luminescence gets gradually triggered at around 2 eV.63,64 Using EE-EOM-CCSD in combination with the outlined extrapolation techniques yields an absorption and emission energy of 3.13 eV and 1.93 eV for the F-center in CaO, respectively. To the best of our knowledge, only one TD-DFT result can be found in the literature for this system, predicting an absorption energy and emission energy of 3.52 eV and 2.1 eV, respectively. Table II also summarizes two different experimental estimates, showing that the EE-EOM-CCSD and TD-DFT@B3LYP calculations agree with experiment to within 0.1 eV and 0.5 eV, respectively. We note again that GW results for the absorption energy obtained for different levels of self-consistency show a significant variance ranging from 3.2 eV to 3.87 eV and cannot be compared directly to the experimentdue to the neglect of the exciton binding energy. We note that our quantum chemical results have been obtained using periodic bound- ary conditions, whereas previous calculations have been carried out using a cluster model approach.30,65 Finally, we turn to the discussion of the F-center in SrO. This system exhibits an even larger lattice constant and the absorp- tion and emission energies are shifted to even lower energies compared to MgO and CaO. However, the F-center in SrO is qualitatively very similar to the CaO case, and agreement of EE- EOM-CCSD in both cases with experimental values is excellent. To the best of our knowledge, there exist only experimental esti- mates of the absorption energy with about 2.4 eV, whereas no measurements for the emission band are known to the authors. We report the results for the absorption1ACC→3Tand emis- sion3T→1ACCin the infinite supercell size limit in Table II. We hope that this prediction will be verified experimentally in the future. IV. CONCLUSIONS In this work, we have presented a novel implementation of the UCCSD and EE-EOM-CCSD methods for periodic systems using a plane wave basis set and applied them to the F-center in the alka- line earth oxides MgO, CaO, and SrO. The implementation was tested on molecular systems, and we have verified it by comparing against well-established quantum chemistry codes for a number of molecular and atomic systems. Convergence of the calculated exci- tation energies with respect to the basis set and size of the simulation cell is crucial for reliable predictions in periodic systems. We have presented a framework to obtain basis-set and finite-size corrected excitation energies by freezing the number of occupied orbitals in a controlled fashion and extrapolating to the complete basis set and infinite system size limit. We have calculated EE-EOM-CCSD absorption and emission energies of the F-center in MgO, CaO, and SrO, accounting for finite basis set and system size errors using extrapolation techniques. The obtained results are in good agreement with previous calculations (where available)30,40–42,65and with experimental data.62In addition, a prediction for the emission band of the F-center in SrO has been made. Furthermore, we provide additional evidence for the assign- ment of the 2.4 eV band in MgO crystals to recombination processes, and we propose a new interpretation of the previous results by sug- gesting that the F-center in MgO does not luminesce. However, further work is needed to clarify the nature of these transitions. The achieved level of accuracy for the calculated EE-EOM- CCSD absorption and emission energies shows that this method has the potential to significantly expand the scope of currently avail- able ab initio techniques for the study of defects. However, further improvements for the corrections to the finite basis and system size errors are urgently needed to allow for a more extensive and detailed study of defects in solids on the level of EE-EOM-CCSD theory. ACKNOWLEDGMENTS The authors thankfully acknowledge the support and funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant J. Chem. Phys. 154, 064106 (2021); doi: 10.1063/5.0035425 154, 064106-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Agreement No. 715594). The computational results presented have been achieved, in part, using the Vienna Scientific Cluster (VSC). 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5.0038284.pdf

J. Chem. Phys. 154, 054309 (2021); https://doi.org/10.1063/5.0038284 154, 054309 © 2021 Author(s).Optoelectronic properties of diketopyrrolopyrrole homopolymers compared to donor–acceptor copolymers Cite as: J. Chem. Phys. 154, 054309 (2021); https://doi.org/10.1063/5.0038284 Submitted: 20 November 2020 . Accepted: 05 January 2021 . Published Online: 04 February 2021 Ulrike Salzner COLLECTIONS Paper published as part of the special topic on Special Collection in Honor of Women in Chemical Physics and Physical Chemistry ARTICLES YOU MAY BE INTERESTED IN Vibronic coupling in the first six electronic states of pentafluorobenzene radical cation: Radiative emission and nonradiative decay The Journal of Chemical Physics 154, 054313 (2021); https://doi.org/10.1063/5.0039923 Excitonic and charge transfer interactions in tetracene stacked and T-shaped dimers The Journal of Chemical Physics 154, 044306 (2021); https://doi.org/10.1063/5.0033272 Applying Marcus theory to describe the carrier transports in organic semiconductors: Limitations and beyond The Journal of Chemical Physics 153, 080902 (2020); https://doi.org/10.1063/5.0018312The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Optoelectronic properties of diketopyrrolopyrrole homopolymers compared to donor–acceptor copolymers Cite as: J. Chem. Phys. 154, 054309 (2021); doi: 10.1063/5.0038284 Submitted: 20 November 2020 •Accepted: 5 January 2021 • Published Online: 4 February 2021 Ulrike Salznera) AFFILIATIONS Department of Chemistry, Bilkent University, 06800 Bilkent, Ankara, Turkey Note: This paper is part of the JCP Special Collection in Honor of Women in Chemical Physics and Physical Chemistry. a)Author to whom correspondence should be addressed: salzner@fen.bilkent.edu.tr ABSTRACT Diketopyrrolopyrrole ( DPP ) is a component of a large number of materials used for optoelectronic applications. As it is exclusively used in combination with aromatic donors, the properties of its homopolymers are unknown. Because donor–acceptor character has been shown for other systems to reduce bandwidths, DPP homopolymers should have even larger conduction bands and better n-type conductivity than the thiophene-flanked systems, which have exceptional n-type conductivity and ambipolar character. Therefore, a theoretical study was carried out to elucidate the properties of the unknown DPP homopolymer. Calculations were done with density functional theory and with the complete active space self-consistent field method plus n-electron valence state perturbation theory for the dynamic correlation. Poly-DPP is predicted to have radical character and an extremely wide low-lying conduction band. If it were possible to produce this material, it should have unprecedented n-type conductivity and might be a synthetic metal. A comparison with various unknown donor–acceptor systems containing vinyl groups and thienyl rings with a higher concentration of DPP than the known copolymers reveals how donor–acceptor substitution reduces bandwidths and decreases electron affinities. Published under license by AIP Publishing. https://doi.org/10.1063/5.0038284 .,s INTRODUCTION Diketopyrrolopyrrole ( DPP ) based conducting polymers are among the most successful systems for organic optoelectronic devices.1In particular, their high electron affinities (EAs) and wide conduction bands lead to unusually high electron conductivities2 and allow for the realization of ambipolar conductors.3Recently, DPP systems were shown to undergo singlet fission4,5and to be good candidates for thermoelectrics.6 DPP flanked by two phenyl rings was first reported in the literature as a very stable brightly colored pigment.7Often, the abbreviation DPP is used for the entire system including the aro- matic rings. The phenyl rings can be replaced with other aromatic units, for instance, thiophene (T).8,9Polymerization of T-DPP-T affords ambipolar conducting polymers.10,11T-DPP-T -systems have donor–acceptor character, with DPP being the acceptor and thio- phene being the donor. The aromatic units can be varied, andadditional acceptors may be combined with DPP in order to fine-tune the properties of the polymers. However, the properties of poly-DPP homopolymers seem to be unknown as a literature search forDPP homooligomers or DPP polymers gave no results. Since the known methods for the synthesis of DPP systems yield only species flanked with aromatic donors, a completely different synthetic approach might be necessary. Before starting such an endeavor, it is desirable to establish whether homopolymers of DPP are likely to have any advantage over the readily available donor–acceptor systems. The donor–acceptor concept was originally conceived with the aim of decreasing bandgaps and simultaneously increasing bandwidths.12Theoretical studies have shown, however, that prop- erties of donor–acceptor systems are between those of the corre- sponding homopolymers.13–16While bandgaps can indeed be tuned over a wide range, bandwidths decrease with an increasing energy difference between a donor and an acceptor. This is accompanied J. Chem. Phys. 154, 054309 (2021); doi: 10.1063/5.0038284 154, 054309-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp by localization, usually of the conduction band states and a decrease in the intensity of the first band in the absorption spectrum.16 The localization of the conduction band leads to a decrease in n-type conductivity even if the electron affinity of the system is high.14These findings suggest that DPP homopolymers should have higher electron affinities and wider conduction bands than T-DPP-T donor–acceptor copolymers. Thus, pure DPP polymers could have interesting opto-electronic properties in terms of n-type conductivity, ambipolar transport, and stability of n-doped poly- mers under ambient conditions. To elucidate these possibilities, a detailed theoretical investigation of pure DPP oligomers and of a variety of donor–acceptor systems with a higher DPP content than commonly used was performed. The results regarding these so far unknown systems are indeed promising, and it is hoped that some of these systems will eventually be synthesized and exploited experimentally. METHODS Structures of DPP noligomers with n = 1–24, of methyl substi- tuted DPP (DPPM n) with n = 1–12, of vinyl and thiophene spaced DDPM (DPPM-V n) and ( DPPM-T n) with n = 2–12, and of bithio- phene spaced DPPM (T-DPPM-T n) with n = 1–7 (see Scheme 1 for structures of trimers) were optimized with density functional theory (DFT) employing the B3P8617,18functional with 30%19,20of Hartree–Fock exchange and the 6-31G∗21basis set. Results with this functional are reliable and compare well with those obtained withrange-separated hybrid functionals.20Wavefunction stability was checked for all species. Appropriate closed- and open-shell (broken symmetry) calculations were used for band structure predictions, for singlet–triplet splittings ( ΔES–T), and for excited state calcula- tions with time-dependent DFT (TDDFT). The DFT ground state singlet–triplet energy differences were corrected with Yamaguchi’s spin projection formula,22 ΔES−T=2Jab=EBS−ET ⟨S2⟩T−⟨S2⟩BS, (1) where E BSand⟨S2⟩BSare the energy and expectation value of the spin operator of the broken symmetry calculations. For DPP oligomers, highly spin-contaminated open-shell wavefunctions resulted. Although the expectation values of the spin operator, ⟨S2⟩, are not clearly defined for DFT calculations, the deteriorating values indicate that the open-shell DFT wavefunc- tions of longer DPP oligomers are problematic. These systems were, therefore, investigated additionally with complete active space self- consistent field (CASSCF) calculations using the DFT structures. Occupations of natural orbitals23and fractional occupation num- bers from finite temperature DFT with a smear temperature of 11 000 K for B3P86 with 30% HF exchange24were analyzed to deter- mine which orbitals should be included in the active space. How- ever, the populations with both approaches decrease very gradually without any obvious gap. Including all occupied natural orbitals with occupancies of less than 1.98 e and all unoccupied molecular SCHEME 1 . Trimers of the repeat units investigated. J. Chem. Phys. 154, 054309 (2021); doi: 10.1063/5.0038284 154, 054309-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp orbitals with more than 0.02 e as suggested in the ORCA man- ual25would lead to huge active spaces including σ-orbitals. There- fore, two series of test calculations were carried out. The first one used active spaces of four electron and four orbitals [CASSCF(4,4)] for all systems. The longest oligomer that could be treated at this level is 12- DPP . The second series was done with active spaces that include all electrons and orbitals arising from the highest occupied molecular orbitals (HOMOs) and the lowest unoccupied molecu- lar orbitals (LUMOs) of the repeat units. This leads to increasing active spaces [CASSCF(2,2) for monomers, CASSCF(4,4) for dimers, and so on] and includes all orbitals that eventually form the valence and conduction bands of the corresponding polymers. (The terms valence and conduction bands will also be used in the following for oligomers.) The longest oligomer that could be handled in this way is 6-DPP with an active space containing 12 electrons and 12 orbitals. 8-DPP would require 16 electrons and 16 orbitals, which is beyond the available computational resources. Finally, more extended active spaces were tested for short oligomers. CASSCF(12,12) calculations for3-DPP and4-DPP showed that the orbital occupancies of the additional low-lying orbitals were significantly higher and those of the additional high lying orbitals were significantly lower than those of the orbitals, forming valence and conduction bands. The excita- tion energies differed by less than 0.15 eV, showing that larger active spaces are not needed. Dynamic correlation was included via n-electron valence state perturbation theory (NEVPT2).26–28Dynamic correlation is abso- lutely essential as the excitation energies and the order of the excited states undergo significant changes when dynamic correlation is added. State specific calculations were done for singlet and triplet ground states, and ΔES–Twere calculated as energy differences. Sin- glet and triplet excitation energies were calculated with state averag- ing over three triplet states and the number of singlet states required to include the first strongly allowed singlet state [e.g., nine singlets for6-DPP with CASSCF(12,12)]. The number of singlet states that has to be included is so high because CASSCF without dynamic cor- relations incorrectly places several dark states below the singlet state. This is especially pronounced with the larger active spaces. The diradical character (y) of the oligomers was determined from the state averaged CASCF(4,4) NEVPT2 calculations accord- ing to the formula of Kamada et al. ,29 y=1−√ 1−(E1S−E1T E2S)2 , (2) where E1S is the energy of the allowed singlet state, E1T is the energy of the first triplet state, and E2S is the energy of the first doubly excited singlet state. To handle the large systems, the resolution of identity (RI) approximation30for the Coulomb term and the chain of sphere (COSX) approximation31of the exchange term were used. The basis sets were the split valence polarized (SVP) basis sets of the Karl- sruhe group32and the auxiliary basis set by Weigend,33which is required for the RI method. The DFT calculations were carried out using Gaussian 1634and ORCA.35,36That both program packages lead to the same results was verified. The finite temperature DFT and CASSCF/NEVPT2 calculations were done with ORCA.RESULTS DPP oligomers are planar due to hydrogen bridges between the close lying N–H and C = =O groups. Conjugation along the back- bone is extremely strong, leading to interring double bonds and exchange of single and double bond positions in the DPP units com- pared to the monomer (Scheme 2). The HOMO–LUMO energy gap decreases rapidly. The exchange of single and double bond positions leaves unpaired electrons at the terminal DPP units. As a result, the closed-shell wavefunction becomes unstable for the trimer and the tetramer has already an ⟨S2⟩value of 1.0. Because DFT cal- culations might be unreliable for such systems, CASSCF/NEVPT2 calculations were done as well. In Table I, ΔES–T from B3P86-30%/6-31G∗and CASSCF/ NEVPT2/SVP ground state calculations and CASSCF/NEVPT2/SVP state averaged excited state calculations are compared. The values with active spaces (4, 4) and (2n, 2n), with n being the number of repeat units, are reported next to each other. The ⟨S2⟩values of the DFT calculations deteriorate rapidly for singlet and triplet states starting with 4-DPP . For 8-DPP , one can no longer speak of singlet or triplet states. The DFT ΔES–Twere, therefore, cor- rected with Yamaguchi’s spin projection formula [Eq. (1)].22The spin-projected ΔES–Tdecrease from 1.73 eV to 0.41 eV from the monomer to dodecamer and then increase slightly to 0.45 eV for 24-DPP . Without spin projection, ΔES–Tdecreases smoothly to 0.10 eV for the dodecamer and then stays constant up to 24- DPP . The CASSCF/NEVPT2 ground state ΔES–Talso decrease at first, reach a minimum for 4–5 repeat units, and then increase with both active spaces. This unphysical37chain length dependence occurs at the same chain length where the oligomers develop open-shell character in the DFT calculations. Simultaneously, the ground state CASSCF wavefunction develops significant doubly excited charac- ter as the HOMO–LUMO gaps approach 0. Starting with 10- DPP , it was not possible, despite many attempts, to converge to the ground state wavefunction in which the electronic configuration with two doubly occupied lowest energy active orbitals and two unoccupied highest energy active orbitals (2200) has the largest contribution. Instead, (2020) or (2110) “ground states” are obtained for 10- DPP and 12- DPP . This observation can be rationalized by the vanishing HOMO–LUMO gaps. The CASSCF/NEVPT2 triplet excitation energies decrease monotonically from 2.06 eV to 0 eV from the monomer to decamer and dodecamer. Note, however, that wavefunctions of the latter two are mixtures of many electron configurations. The CASSCF/NEVPT2 calculations reveal, therefore, that ΔES–Tof DPP systems with ten repeat units and more is 0 and that ground and doubly excited states are close in energy. The diradical character y increases from very small values for the closed-shell (according to DFT) oligomers to 0.549 for 8- DPP . For the longer oligomers, E1T approaches zero, and E2S is smaller than E1S. Therefore, the bracket in Eq. (2) is larger than 1, the root argument is negative, and the root becomes imaginary. The value of y is, however, close to 1, as expected for diradicals. AllDPP oligomers give rise to a strong singlet excited state at low energy. Below lie two triplet states and one dipole for- bidden singlet state. A very weakly allowed feature is predicted by DFT at slightly lower energy and with CASSCF/NEVPT2 at slightly higher energy than the strong state. With DFT, the weakly J. Chem. Phys. 154, 054309 (2021); doi: 10.1063/5.0038284 154, 054309-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp SCHEME 2 . Structures of isolated DPP and of the inner rings of 24-DPP , CH 2-capped 24-DPP ,12-DPPM ,12-DPPM-V , 12-DPPM-T, and 7- T-DPPM-T . allowed state is a HOMO-1-LUMO plus HOMO-LUMO+1 tran- sition. With CASSCF/NEVPT2, it is also a HOMO-1-LUMO plus HOMO-LUMO+1 transition, but there is an additional double exci- tation character. The strong state is dominated by HOMO–LUMO transitions with both methods, and the absorption energies decrease with chain length. The CASSCF/NEVPT2 values with both activespaces differ by less than 0.1 eV and are lower than the DFT val- ues by 0.03 eV–0.4 eV. Considering the catastrophic spin con- tamination, DFT excitation energies are remarkably close to the CASSCF/NEVPT2 values. Thus, broken symmetry TDDFT exci- tation energies are at least qualitatively acceptable. The further comparisons are done with DFT calculations only. TABLE I .⟨S2⟩values of the B3P86-30% (DFT) “singlet” and “triplet” wavefunctions, energy difference between spin-projected DFT and CASSCF/NEVPT2 (CAS) ground state ΔES–T, CASSCF/NEVPT2 triplet excitation energies (3exc), B3P86-30% and CASSCF/NEVPT2 singlet excitation energies (1exc), diradical character y at CAS(4,4), and DFT ΔSCF IEs and EAs of DPP oligomers in eV. ΔES–T3exc1exc y IE EA ⟨S2⟩S0/T0 DFT CAS(4,4)/(2n,2n) DFT CAS(4,4)/(2n,2n) CAS(4,4) DFT DPP 0/2.0 1.73 2.13/2.16 2.30/2.06 3.65 3.65/3.38 8.54 0.95 2-DPP 0/2.0 1.01 1.59 1.34 2.55 2.49 0.086 8.06 2.65 3-DPP 0.4/2.1 0.79 0.41/0.91 1.13/1.00 2.11 2.04/2.02 0.085 7.76 3.43 4-DPP 1.0/2.2 0.61 0.31/0.11 0.57/0.63 1.83 1.55/1.50 0.213 7.53 3.92 5-DPP 1.3/2.2 0.52 0.15/0.18 0.45/0.35 1.67 1.30/1.24 0.282 7.42 4.18 6-DPP 1.5/2.3 0.47 0.25/0.43 0.31/0.24 1.56 1.09/1.18 0.393 7.35 4.36 8-DPP 1.9/2.6 0.42 0.79/ . . . 0.10/ . . . 1.41 0.82/ . . . 0.549 7.27 4.59 10-DPP 2.2/2.7 0.41 0.00/ . . . 0.02/ . . . 1.32 / . . . 7.24 4.73 12-DPP 2.6/3.1 0.41 0.00/ . . . 0.00/ . . . 1.26 / . . . 7.24 4.81 16-DPP 3.3/4.2 0.43 / . . . /. . . 1.19 / . . . 7.23 4.92 20-DPP 4.0/4.4 0.44 / . . . /. . . /. . . 7.24 4.98 24-DPP 4.7/5.1 0.45 / . . . /. . . /. . . 7.25 5.01 J. Chem. Phys. 154, 054309 (2021); doi: 10.1063/5.0038284 154, 054309-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The DFT ionization energies (IEs) reach constant values at 10- DPP and are, therefore, converged. The final value of 7.25 eV is rela- tively high compared to other conjugated polymers such as polythio- phene, for instance, which has an IE of 6.0 eV at the same level of the- ory. The electron affinities (EAs) converge more slowly but increase between 20-DPP and24-DPP by only 0.03 eV. The predicted gas phase EA of the polymer of over >5 eV is huge, and the transport gap, IE-EA,38is very small, 2.24 eV. The large EA makes DPP oligomers excellent candidates for n-type transport under ambient conditions because anions would be stable against oxidation by oxygen and moisture. However, there would most likely be unintentional doping because the EA exceeds 4.5 eV.39 In Fig. 1, DFT orbital energies of DPP and thiophene oligomers with up to 24 conjugated double bonds along the backbone are compared. In the middle of the graph, orbital energies of the cooligomer 4-T-DPPM-T , which also has 24 conjugated dou- ble bonds, are the inserted. DPP oligomers have much higher EAs and IEs than thiophene oligomers. The valence band of4-T-DPPM-T is close to that of 12-thiophene, and the conduction band lies midway between that of 12-thiophene and 12-DPP and is much narrower. Thus, the high EA and good n-type conductiv- ity of poly- T-DPPM-T are inherited from DPP and not a result of donor–acceptor character. The bandgap of 4-T-DPPM-T , 2.0 eV, is almost the same as that of 12-DPP , 1.9 eV, but much smaller than that of 12-thiophene, 3.0 eV. Occupied and unoccupied energy lev- els are shifted upward compared to 12-DPP so that the copolymer is easier to oxidize but less stable in the reduced form. Therefore, if it were possible to make poly- DPP , they were expected to have a much higher electron mobility and better n-type conductivity than poly-T-DPPM-T . The unpaired electrons that appear at the terminal DPP units can be paired by end-capping the oligomers with double bonded groups, the simplest being CH 2. To test the effect of end-capping on the ⟨S2⟩value, closed-shell and broken symmetry DFT calculations were carried out for CH 2-capped DPP oligomers. Trimer and tetramer are closed-shell species, but the pentamer FIG. 1 . Orbital energies of thiophene oligomers (left), 4-T-DPPM-T (middle, green markers), and DPP oligomers (right). J. Chem. Phys. 154, 054309 (2021); doi: 10.1063/5.0038284 154, 054309-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II . Comparison of negative HOMO and LUMO energies (IP Koop and EA Koop), bandwidths of valence (BW val) and conduction bands (BW con), bandgap (E g), and E S–Tin eV, and ⟨S2⟩of different oligomers with 24 conjugated or close to 24 double bonds (23 for DPPM-V and 26 for DPPM-T). IEKoop EA Koop BW val BW con Eg ES–T ⟨S2⟩ 12-DPP 6.97 5.07 1.34 3.36 1.90 0.41 2.59 CH 2-12-DPP-CH 2 6.95 5.08 1.48 3.21 1.87 0.80 1.56 12-DPPM 6.47 4.50 1.22 2.46 1.97 0.90 1.34 8-DPPM-V 6.21 4.24 1.08 2.44 1.97 0.36 2.38 V-8-DPPM-V 6.17 4.23 1.08 2.28 1.94 0.36 2.46 6-DPPM-T 5.77 3.97 1.08 1.69 1.80 0.81 0.74 4-T-DPPM-T 5.73 3.70 0.67 0.95 2.03 1.02 0 and hexamer start breaking symmetry. The dodecamer has an ⟨S2⟩ value of 1.6, less than the uncapped dodecamer with ⟨S2⟩= 2.6. Hence, end-capping does not make DPP oligomers closed-shell sys- tems but reduces the spin contamination of the wavefunctions con- siderably. It is interesting to note that the orbital energies differ very little from those of the uncapped isomers. Therefore, end-capping would be a route to increase the stability of pure DPP systems. Although the data for the DPP oligomers are very promising, there are two issues that might prevent experimental realization. First, the open-shell diradicalic character might render the systems unstable despite their large EA and IE. Second, the planar hydrogen- bridged structure indicates that the longer oligomers and polymers are probably insoluble. To address these problems, the effect of alkyl substitution to improve solubility is tested with the introduction of methyl groups on nitrogen, leading to DPPM oligomers. Methyl groups are used to simulate the steric effect of longer alkyl chains as their effects on the electronic structure are very similar. The sta- bility issues and non-planarity resulting from methyl substitution are addressed with vinyl and thienyl spacers, which lead to DPPM- VandDPPM-T oligomers. Properties for oligomers with close to 24 double bonds along the backbone are compared in Table II. The absorption spectra of the same oligomers are presented in Fig. 2DPPM oligomers are closed-shell systems up to the tetramer. Pentamer and hexamer are almost pure singlets with ⟨S2⟩values of 0.08 and 0.19. Spin contamination gets severe for longer oligomers, but the ⟨S2⟩value reaches only 1.34 for 12-DPPM . Because DFT and CASSCF/NEVPT2 results do not differ dramatically for DPP oligomers with much larger ⟨S2⟩values, the calculation was all done with DFT. DPPM oligomers are non-planar with dihedral angles between theDPPM units of 44○. The non-planarity reduces conjugation along the backbone and prevents the inversion of single and dou- ble bonds that was observed for DPP oligomers compared to iso- lated DPP (Scheme 2). Figure 3 compares the energy levels of DPP andDPPM oligomers. Methyl substitution pushes valence and con- duction bands up, reducing IE (0.5 eV) and EA (0.6 eV). The reduced conjugation in DPPM decreases the conduction bandwidth by 0.9 eV. The bandgap and excitation energy to the strongly allowed state of 12- DPPM increase by less than 0.1 eV compared to those of 12-DPP (Fig. 2). To restore planarity, it is sufficient to introduce a vinyl spacer between the DPPM units to reduce the steric repulsion between the methyl and carbonyl groups. This creates donor–acceptor oligomers DPPM-V . Upon return to planarity, the single and double bond lengths along the backbone invert again (Scheme 2), and the spin FIG. 2 . TDDFT absorption spectra of closed-shells 12- DPP , 12-DPP , 12-DPPM , 8-DPPM -V, 6- DPPM-T , and 4- T-DPPM-T . J. Chem. Phys. 154, 054309 (2021); doi: 10.1063/5.0038284 154, 054309-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Orbital energies of DPPM andDPP oligomers. contamination increases due to the unpaired electrons in the termi- nal units. Figure 4 reveals that vinyl-spacers push the valence and conduction bands up by 0.2 eV–0.3 eV and decrease the valence and conduction bandwidths slightly but have no effect on the bandgap. Thus, although the DPPM-V oligomers are planar, the bandwidth is not increased because the system has donor–acceptor charac- ter. The absorption maximum shifts by 0.07 eV to lower energy, and the oscillator strength increases (Fig. 2). Adding vinyl groups at the ends of the oligomers in the same fashion as with thio- phene rings (Scheme 1) does not alter the energy levels as the entry for V-8-DPPM-V below that for 8-DPPM-V in Table II shows. DPPM-T oligomers are closed-shell up to the trimer. Then, the ⟨S2⟩value increases to 2.0 for the dodecamer. Thienyl spacers are intrinsically aromatic and resist single and double bond inversion. As a result, there is almost no bond length alternation in DPPM-T oligomers. Reducing bond length alternation is one of the strategies to reduce bandgaps in organic polymers.40It might also be benefi- cial for charge carrier transport. The IEs and EAs of 12- DPPM-T are reduced by over 1 eV compared to DPP oligomers (Table II). The EA of the polymer would be slightly above 4 eV, which is the ideal value for n-doping and electron transport.39The valence band- width is reduced by only ∼0.3 eV compared to that of 12- DPP, but the conduction bandwidth is halved. Such a bandwidth decrease is generally observed for donor–acceptor systems because interaction between orbitals is inversely proportional to the energy difference between them. DPPM-T systems have the smallest bandgaps among the oligomers studied, but the absorption maximum of 6-DPPM-T lies 0.04 eV above that of 12-DPPM . FIG. 4 . Orbital energies of DPPM andDPPM-V oligomers. FIG. 5 . Orbital energies of DPPM-T andT-DPPM-T oligomers. Figure 5 compares the energy levels of DPPM-T andT-DPPM- Toligomers. All T-DPPM-T oligomers that were investigated (up to the heptamer) are closed-shell species. The bandgap is larger than those of all other systems. The second thienyl spacer reduces the valence bandwidths of 4-T-DPPM-T compared to that of 6- DPPM- Tfrom 1.2 eV to 0.8 eV and the conduction bandwidth from 1.7 eV to 1.1 eV. The conduction bandwidth estimated for the poly-T- DPPM-T is, therefore, less than 1/3 that of poly-DPP and about half that of poly-DPPM-T . The EA would be ∼0.3 eV lower than that of poly-DPPM-T . The absorption spectrum is blue-shifted compared to those of the other systems. DISCUSSION For the majority of copolymers used in organic electronics, the properties of the corresponding homopolymers are known as well. DPP is the exception as the original synthesis produced the DPP unit with attached phenyl groups. Different aryl groups can replace the phenyl groups, but to the best of the author’s knowledge, DPP homooligomers or homopolymers have never been reported. The present theoretical investigation reveals that DPP homooligomers and all derivatives included in this study with less than two aromatic spacers are open-shell di- or poly-radicals. The reason for this rad- ical character is that the conjugation along the backbone of DPP oligomers is extremely strong. As a consequence, the single and dou- ble bond positions switch upon the increase in the chain length, leaving an unpaired electron at each terminal unit. This electron can be bound by end-capping the CH 2-groups, but the HOMO– LUMO gaps are still so small that symmetry breaking occurs by HOMO–LUMO mixing. As the bandgap is 0 for the decamer and longer oligomers according to CASSCF/NEVPT2 calculations, poly- DPP could be a synthetic metal. This is in contrast to other small bandgap systems, where symmetry breaking opens the bandgap. The radical nature of these systems can lead, of course, to stability issues. However, diradicals are increasingly used in organic electron- ics, and exceptional conductivities have been reported.41–43Mod- erate diradical character has also been demonstrated to facilitate singlet fission4,44–48and is essential for increasing nonlinear optical properties.49–53 Based on the earlier work on donor–acceptor systems,13–16 the conduction bandwidth of poly-DPP was expected to be huge because poly-T-DPPM-T has a wider conduction band than J. Chem. Phys. 154, 054309 (2021); doi: 10.1063/5.0038284 154, 054309-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 6 . Comparison of the valence and conduction bands of 12- DPP , CH 2-capped 12-DPP , 12-DPPM , 8-DPPM -V, 6- DPPM -T, and 4- T-DPPM-T . polythiophene. The comparison of the orbital energies of oligomers with 24 or very close to 24 double bonds along the backbone in Fig. 6 nicely demonstrates the effect of donor–acceptor substitu- tion on the bandwidth. It might look like using fewer energy lev- els for the larger repeat units may bias the results, but it is clearly visible from Fig. 5 that not much increase in bandwidths occurs for the longer T-DPPM-T -oligomers and, therefore, also for the polymer. Because the wavefunctions of the open-shell systems are highly spin-contaminated, multi-reference calculations were carried out in addition to DFT. Use of the CASSCF method to compare a series of molecules with different electronic structures and sizes on equal footing is not straightforward. The DPP monomer has already 12 π- electrons and requires an active space of 12 electrons and 10 orbitals if all π-electrons should be included in the active space. The cor- responding active space of the dimer is, therefore, already beyond the capacity of most current computers. CASSCF with the density matrix renormalization group54approach can handle larger active spaces but requires huge amounts of disk space. The comparison of the two series, CASSCF(4,4) and CASSCF(2n,2n), shows that the results agree within 0.2 eV–0.3 eV including all properties. There- fore, it is probably safe to assume that the non-dynamic correlation effects do not reach far beyond the frontier orbitals and that the cal- culations with both active spaces are sufficient. It is puzzling that ground state ΔES–Tvalues with both active spaces do not decrease monotonically as expected but produce a minimum at 4–5 DPP units and then start increasing and fluctuating. This trend is not observed when the triplet excitation energies are calculated with the same active spaces. Another disadvantage of the CASSCF/NEVPT2 method is that with state averaging, the excitation energies depend on the number of states calculated. Therefore, there is an arbitrari- ness in the results that makes accurate theoretical prediction of properties very difficult. It is encouraging, however, that there are no dramatic differences between the CASSCF/NEVPT2 and bro- ken symmetry DFT results so that the use of broken symmetry DFT can be justified despite the very bad expectation values of the spin operator. CONCLUSIONS Homopolymers of DPP are strongly conjugated and have extremely wide low-lying conduction bands and small or zerobandgaps. Therefore, they would be stable in the n-doped form and should be very good electron conductors. Because of their small HOMO–LUMO gaps, symmetry breaking occurs, and diradical character increases with chain lengths. This might further increase conductivity but will also lead to stability issues. Since IE and EA are very high, it might be possible to stabilize the radicals with branched alkyl groups on nitrogen to prevent radical recombination. Introduction of vinyl or thienyl groups creates donor–acceptor systems with less radical character but with smaller conduction bandwidths and lower EAs. Thus, optoelectronic properties of DPP systems can be adjusted with donor spacers from small bandgap semiconductors to almost zero bandgap systems with infrared absorption maxima. Donor–acceptor substitution induces localization of electrons on the acceptor groups. 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5.0038799.pdf

J. Appl. Phys. 129, 075104 (2021); https://doi.org/10.1063/5.0038799 129, 075104 © 2021 Author(s).Type-III Dirac fermions in HfxZr1−xTe2 topological semimetal candidate Cite as: J. Appl. Phys. 129, 075104 (2021); https://doi.org/10.1063/5.0038799 Submitted: 26 November 2020 . Accepted: 04 February 2021 . Published Online: 19 February 2021 Sotirios Fragkos , Polychronis Tsipas , Evangelia Xenogiannopoulou , Yerassimos Panayiotatos , and Athanasios Dimoulas COLLECTIONS Paper published as part of the special topic on Topological Materials and Devices ARTICLES YOU MAY BE INTERESTED IN Ultrafast investigation and control of Dirac and Weyl semimetals Journal of Applied Physics 129, 070901 (2021); https://doi.org/10.1063/5.0035878 Topological semimetals from the perspective of first-principles calculations Journal of Applied Physics 128, 191101 (2020); https://doi.org/10.1063/5.0025396 Dirac equation perspective on higher-order topological insulators Journal of Applied Physics 128, 221102 (2020); https://doi.org/10.1063/5.0035850Type-III Dirac fermions in Hf xZr1−xTe2topological semimetal candidate Cite as: J. Appl. Phys. 129, 075104 (2021); doi: 10.1063/5.0038799 View Online Export Citation CrossMar k Submitted: 26 November 2020 · Accepted: 4 February 2021 · Published Online: 19 February 2021 Sotirios Fragkos,1,2,a) Polychronis Tsipas,1 Evangelia Xenogiannopoulou,1 Yerassimos Panayiotatos,2 and Athanasios Dimoulas1,3,a) AFFILIATIONS 1Institute of Nanoscience and Nanotechnology, National Center for Scientific Research ‘Demokritos ’, 15310 Athens, Greece 2Department of Mechanical Engineering, University of West Attica, 12241 Athens, Greece 3Department of Physics, National and Kapodistrian University of Athens, 15772 Athens, Greece Note: This paper is part of the Special Topic on Topological Materials and Devices. a)Authors to whom correspondence should be addressed: s.fragkos@inn.demokritos.gr ;s.fragkos@uniwa.gr ; and a.dimoulas@inn.demokritos.gr ABSTRACT Topological semimetals host interesting new types of low-energy quasiparticles such as type-I and type-II Dirac and Weyl fermions. Type-III topological semimetals can emerge exactly at the border between type-I and II, characterized by a line-like Fermi surface and a flatenergy dispersion near the topological band crossing. Here, we theoretically predict that 1T-HfTe 2and 1T-ZrTe 2transition metal dichalco- genides are type-I and type-II DSMs, respectively. By alloying the two materials, a new Hf xZr1−xTe2alloy with type-III Dirac cone emerges at x = 0.2, in combination with 1% in-plane compressive strain. By imaging the electronic energy bands with in situ angle-resolved photo- emission spectroscopy of this random alloy with the desired composition, grown by molecular beam epitaxy on InAs(111) substrates, weprovide experimental evidence that the t οp of type-III Dirac cone lies at —or very close to —the Fermi level. Published under license by AIP Publishing. https://doi.org/10.1063/5.0038799 I. INTRODUCTION Topological 3D Dirac semimetals (DSMs) and Weyl semime- tals (WSM), often called “3D graphenes, ”exhibit Dirac-like cones with linear dispersion in all three dimensions in k-space. 1DSMs are classified either as type-I DSM1with untilted or slightly tilted Dirac cone and a point-like Fermi surface or as type-II DSM.1In the latter case, the Dirac cone is overtilted producing a finite Fermi surface consisting of electron and hole pockets which cross at theDirac point (DP) ( Fig. 1 ). Yet, a type-III DSM 2–4merges as a theo- retical possibility exactly at the borders between type-I and type-II,characterized by a unique line-like Fermi surface and a flat energy dispersion along one direction in the Brillouin zone (BZ) ( Fig. 1 ). The Fermi surface topology depends sensitively on the posi- tion of the Fermi level relative to the energy degeneracy points(Dirac points) of the DSM/WSM. Near the Dirac point, the systemcan undergo an electronic (Lifshitz) phase transition 5,6under the influence of external stimuli (e.g., temperature, pressure, etc.). At the critical conditions for the transition, the Fermi surface topologycan drastically change7–10with important consequences in elec- tronic and thermoelectric transport. Most commonly observed is a temperature driven Lifshitz transition from a hole-like to electron- like Fermi surface,7e.g., in ZrTe 57,8and HfTe 59accompanied with resistivity and thermopower anomalies at the critical temperature.An anomalous Nernst effect has also been observed 10in TaP and TaAs WSMs, which has been correlated to a Lifshitz transition. Aquite different Lifshitz transition is expected in a type-III DSM. The system can easily make the transition from a type-I to type-II DSM associated with a big change in the Fermi surface from apoint-like to a needle-like configuration, likely accompanied byresistivity and thermopower changes as in the case of ZrTe 5.8The latter changes could be accessible to (thermo)electric transport experiments; therefore, there is a strong motivation to search fortype-III DSMs (and WSMs). This search is also motivated by thespectacular predictions, 2,3that type-III DSMs (i.e., black phospho- rous,11Zn2In2S5)12could be the solid state (or fermionic) analog of the black hole event horizon potentially generating “HawkingJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 075104 (2021); doi: 10.1063/5.0038799 129, 075104-1 Published under license by AIP Publishing.radiation ”at relatively high “Hawking temperature, ”2showing promise for new exotic physics and applications. Although atype-III Dirac crossing was only a theoretical possibility up to now, experimental evidence has been recently obtained in strained epi- taxial SnTe, 13between its two uppermost valence bands, 1.83 eV below the Fermi level using synchrotron ARPES and also in artifi-cial photonic orbital graphene lattices. 14 Prototypical topological Dirac semimetals are 3D crystal struc- tures and are typically grown as bulk crystals,15,16usually suffering from heteroepitaxial defects which yield discontinuous films withpoor crystalline quality. Discovering and engineering topologicalsemimetals from the family of two-dimensional transition metaldichalcogenides (TMDs) could open the way for exploitation of their topological properties by fabricating thin epitaxial films and devices on suitable substrates via van der Waals epitaxy. Previousreport from our team 17,18provide the first experimental evidence by angle-resolved photoemission spectroscopy (ARPES) that few- layer 1 Τ-HfTe 217and 1T-ZrTe 218epitaxially grown by molecular beam epitaxy (MBE) are 3D DSMs. More specifically, we observedthat linearly dispersing bands along ΓK and ΓM directions in the plane of the film cross at the Fermi level indicating the existence ofDirac fermions even down to the ultimate 2D limit of 1 ML with the DP located at —or very close to —Fermi level, which is notably different than theory which predicts the DP well above it. The question about the possibility of topological properties in HfTe 2and ZrTe 2remains open. Synchrotron ARPES studies on single crystals of K and Cr doped HfTe 219,20and ZrTe 2,21respec- tively, show no clear signs of Dirac-like features compared to the epitaxially grown films,17,18while others, by combining synchro- tron ARPES22on 1T-ZrTe 2single crystals with DFT calculations suggest that ZrTe 2is a DSM. From the aforementioned discussion, it is concluded that doping by intercalation changes drastically both crystal and electronic structure of these materials, so an asso- ciated change of the topological properties of HfTe 2and ZrTe 2 cannot be excluded. Despite the controversy, evidence is accumulat-ing in favor of non-trivial topology in these materials. Magnetotransport measurements indicate that both HfTe 223and ZrTe 224,25are topological materials. A recent theoretical study26using a newly developed formalism known as topological quantum chemistry27predicts that both HfTe 2and ZrTe 2are topological semimetals. Note, however, that others28,29identified HfTe 2as DSM but ZrTe 2as topological crystalline insulator. The range of applications of topological materials including ZrTe 2and HfTe 2is widely open. Several of these materials have large spin Berry curva- ture as a combined result of their electronic band structure and large spin –orbit coupling. Consequently, they have large spin hall conductivity30,31which makes them suitable candidates for charge to spin conversion in combined topological/ferromagnet spin orbittorque (SOT) devices. This could lead to all electrically driven SOT spintronic devices for storage and processing of information. Replacing heavy metals like Pt or Ta typically used in SOT devicesby two-dimensional TMDs, it is expected that the energy efficiencyfor charge to spin conversion will be improved. 31This has already been demonstrated for the WTe 2/permalloy system.32Other ditel- lurides like HfTe 2and ZrTe 2may follow a similar trend manifesting their suitability for spintronics applications. In a similar perspec-tive, a recent work 24reports anomalous hall effect of ZrTe 2in prox- imity with a magnetic material, which could be considered as asign that a quantum anomalous Hall effect (QAHE) is possible in this system as already predicted for topological/ferromagnetic mate- rials combinations. This opens the possibility to utilize the dissipa-tionless chiral edge states of a QAHE system to create energyefficient spintronic devices without the need of a magnetic field. In a different perspective, 33it has been predicted that monolayer 1T-HfTe 2nanosheets are highly selective toward the sensing of environmental hazardous NO gas which also plays an active role inseveral physiological processes, and thus, could have importantapplications in the area of environment and medicine. In this paper, our first-principles calculations reveal that HfTe 2and ZrTe 2are type-I and type-II DSMs, respectively, which creates the prospect that by alloying the two materials, a newHf xZr1−xTe2type-III DSM material will emerge. After a systematic investigation of Hf xZr1−xTe2energy bands as a function of compo- sition and strain, a type-III Dirac cone with a line-like Fermi surface is predicted to form at 20% concentration of Hf in combi-nation with 1% in-plane compressive strain. Furthermore, byimaging the electronic energy bands with in situ ARPES of this layered compound at the desired composition x = 0.2, grown by MBE on the InAs(111) substrate, we provide experimental evidence that the t οp of type-III Dirac cone lies at —or very close to —the Fermi level. II. FIRST-PRINCIPLES CALCULATIONS HfTe 2and ZrTe 2belong to the 2D layered 1T octahedral family with space group P/C223m1(No. 164). In order to reveal the topological nature of the Dirac cones in these materials, we presentab initio calculations of electronic structure and perform symmetry analysis. The first-principles calculations were performed using the Vienna Ab initio Simulation Package 34,35(VASP). The generalized- gradient approximation with Perdew –Burke –Ernzerhof36(PBE) parameterization was used as exchange-correlation functional. Ourstudy of the electronic band structure is based on the experimental lattice constants of Refs. 17and18, measured by synchrotron x-ray diffraction, instead of the equilibrium ones from DFT calculations. FIG. 1. Schematic illustration of the different types of DSMs. Type-III is at the border between type-I and type-II, characterized by a line-like Fermi surface and flat energy dispersion at the Fermi level.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 075104 (2021); doi: 10.1063/5.0038799 129, 075104-2 Published under license by AIP Publishing.The kinetic energy cutoff was set at 500 eV, and the reciprocal space was sampled using the Monkhorst –Pack scheme37employing a 11 × 11 × 11 k-point mesh. Spin –orbit coupling (SOC) was included for the band structure calculations. We used Hf dorbitals, Zrdorbitals, and Te porbitals to construct Wannier functions using the Wannier90 code,38,39and the band structures and Fermi surfaces were obtained by the WannierTools software.40 A. Electronic band structure of HfTe 2and ZrTe 2 VASP2Trace code26was used to examine the topology and cal- culate the irreducible representations at the high-symmetry points of VASP wavefunctions. The decomposition of these irreducible representations for both HfTe 2and ZrTe 2, moving along the Γ-A direction, were deduced from the BANDREP program27,41,42of the Bilbao Crystallographic Server. From these calculations[Figs. 2(a) and 2(b)], it is inferred that gapless crossing occurs because the two bands belong to different irreducible representations (/C22Δ 6and/C22Δ4/C22Δ5) which prohibits hybridization and gap opening. As aresult, a pair of fourfold degenerate Dirac nodes is generated which is protected by C3rotational symmetry along the caxis43at (0,0, ±0.036 c*) and (0,0, ±0.228 c*) for HfTe 2and ZrTe 2, respectively. The crossings are created via band inversion43at positions which are symmetrically placed with respect to Γalong the Γ–A(kz) direction of the BZ, thus defining a pair of DPs. In Figs. 2(c) and2(d),t h e electronic band structures around the DPs are visualized in the kz–kx plane. Further calculations of the 2D and 3D Fermi surfaces are pre- sented in Fig. 3 . By setting the chemical potential at the position of the Dirac points, a point-like Fermi surface [the inset of Figs. 3(a) and3(c)] is revealed in the case of HfTe 2characteristic to a type-I Dirac semimetal. On the other hand, needle-like electron and holepockets are formed in the case of ZrTe 2[Figs. 3(b) and 3(d)], which touch at the DP, indicating that ZrTe 2is type-II DSM. It should be mentioned that, unlike Weyl semimetals, Dirac semimetals do not possess Berry phase or Berry curvature since the three-dimensional Dirac nodes are not chiral since they are thesum of two Weyl nodes of opposite chirality at the same energy FIG. 2. Band structure of (a) HfT e 2and (b) ZrT e 2along the Μ-Γ-Αdirection of the BZ, where valence and conduction bands which belong to different irreducible repre- sentations cross each other. (c) and (d) show the energy dispersion in the kz–kxplane near the crossings for HfT e 2and ZrT e 2, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 075104 (2021); doi: 10.1063/5.0038799 129, 075104-3 Published under license by AIP Publishing.and momentum. Thus, the topological charge is zero everywhere in momentum space within the Brillouin zone. In addition, unlike Weyl semimetals, Dirac semimetals are protected only by underly-ing crystallographic symmetry, otherwise they become gapped. Inour case, this role is played by the C 3rotational symmetry.B. The Hf 0.2Zr0.8Te2type-III Dirac semimetal state By alloying the two materials (HfTe 2and ZrTe 2), a new HfxZr1−xTe2material with type-III Dirac cone with a line-like Fermi surface could emerge at a certain composition. The elec- tronic band structure of the Hf xZr1−xTe2alloy was calculated by a FIG. 3. The 2D [(a) and (b)] and 3D [(c) and (d)] Fermi surface of HfT e 2and ZrT e 2when the chemical potential is set to nearly coincide with the DP energy. The point-like features in the inset of (c) indicate a type-I DSM for HfT e 2. The needle-like e−and h+pockets in ZrT e 2[the inset of (d)] define the DP at the touching point indicating type-II DSM.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 075104 (2021); doi: 10.1063/5.0038799 129, 075104-4 Published under license by AIP Publishing.FIG. 4. (a) Systematic investigation of Hf xZr1−xTe2energy bands as a function of composition and (b) of compressive strain, where the orange, blue, and red insets show the evolution of the Dirac cone (type-II, type-III, and type-I, respectively). (c) Electronic band structure of Hf 0.2Zr0.8Te2with 1% in-plane compressive strain along the Μ-Γ-Αdirection. The inset indicates the type-III band crossing between valence and conduction bands featuring a line-like Fermi surface. (d) The 3D band s tructure near the crossing point. (e) and (f) The 2D and 3D Fermi surfaces of Hf 0.2Zr0.8Te2, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 075104 (2021); doi: 10.1063/5.0038799 129, 075104-5 Published under license by AIP Publishing.linear interpolation of tight-binding model matrix elements44–46of HfTe 2and ZrTe 2generated by the Wannier90 code,38,39that is, in order to construct the tight-binding model of Hf xZr1−xTe2random alloy for each matrix element tij,HfxZr1/C0xTe2we used a linear interpo- lation of the form tij,HfxZr1/C0xTe2¼x/C1tij,HfTe 2þ(1/C0x)/C1tij,ZrTe 2, where xis the Hf concentration and since the two materials have very similar crystal structures, the experimental lattice constants of Ref. 18were used. The proposed approach could be sensitive to a possible asymmetry between the Zr and Hf radii. Note, however,that the van der Waals radii 47of Hf and Zr are 2.53 and 2.52 Å, respectively, so their difference is negligible. In addition, if we con- sider the Wigner –Seitz radii provided by the PBE pseudopoten- tials,36then Hf and Zr have the same value of 1.614 Å. This is also true for the empirical atomic radii with a value of 1.55 Å.48 Therefore, the radii of Hf and Zr are very similar and we expect nobig asymmetry in size that will adversely affect our calculations. Tight-binding elements contain all important information 44–46for an interpolation to cover all changes of the electronic structurebetween the two end point materials. Furthermore, Wannier func-tions are critical for the study and classification of topologicalmaterials, 26,27,49–52thus making this approach suitable and reliable for calculating their alloys. Previously, the same method was used to investigate the electronic band structure of BiTlSe 1−xSxtopologi- cal insulator44and of the transition metal dichalcogenide Mo xW1 −xTe2topological WSM46and was in excellent agreement with their ARPES measurements. A systematic investigation of Hf xZr1−xTe2energy bands as a function of the composition was made [ Fig. 4(a) ]. Hf doping is not efficient enough to completely flatten the valence band around thecrossing point. The optimum value of x = 0.2 produces a slope dE/ dk =−0.001 eV/Å −1, a value quite close to the ideal of a type-III crossing. Therefore, we applied an in-plane compressive strain[Fig. 4(b) ] resulting in an out-of-plane expansion of the film and the van der Waals gap, which in turn weakens the interlayer interactionand flattens the bands along the Γ-Αdirection. More precisely, by applying 1% in-plane compressive strain, the /C22Δ 6valence band is flat (dispersionless) near the DP while the conduction /C22Δ4/C22Δ5band dis- perses downward crossing the flatband [ Figs. 4(d) and5(c)]. Plotting the 2D [ Fig. 4(e) ]a n d3 D[ Fig. 4(f) ] Fermi surfaces, line-like electron and hole pockets are visible and cross at (0,0, ±0.332 c*), which satis- fies the condition for the formation of type-III Dirac cones. Therefore, the in-plane strain could be considered as a switchingmechanism for shifting from type-I to type-II DSM as indicated inFig. 4(b) . It should be reminded that such a change is highly desir- able since it constitutes an electronic Lifshitz phase transition with important consequences in electronic and thermoelectric transport(see Sec. I). The crystal structures of 1T HfTe 2, ZrTe 2, and Hf 0.2Zr0.8Te2 are illustrated in Fig. 5 , where the Hf atomic positions in Hf0.2Zr0.8Te2are randomly selected. III. GROWTH AND ELECTRONIC BAND IMAGING OF Hf0.2Zr0.8Te2ALLOY In this section, we provide experimental evidence that few- layer Hf xZr1−xTe2compound can be grown with the desired com- position x = 0.2 using MBE on InAs(111) substrates. The InAs(111)/Si(111) substrates were chemically cleaned in a 5 N HF solu- tion in isopropyl alcohol for 5 min to etch the surface oxide and subsequently rinsed in isopropyl alcohol for 30 s in order to avoidreoxidation of the substrate. An annealing step at 400 °C in ultra-high vacuum (UHV) follows to get a clean and flat InAs(111)surface observed by reflection high-energy electron-diffraction (RHEED). Where appropriate, mild Ar +sputtering was used (E ≈ 1.5 keV, p ≈2×1 0−5mbar, t ≈30 s) prior the annealing step to obtain a clean surface as evidenced by a 2 × 2 reconstruction inRHEED pattern attributed to In surface vacancies. 53It should be noted that starting with a clean 2 × 2 InAs(111) reconstructed surface is an important requirement in order to obtain a good registry of the epitaxial film with the substrate with an in-plane ori-entation. The same procedure was previously applied in ZrTe 2,18 MoTe 2,54and TiTe 255two-dimensional transition metal dichalcoge- nides. The films are grown under Te-rich conditions in an UHV MBE (DCA) vertical chamber. The base pressure of the system is ∼5×1 0−10Torr. Te (99.999%) is evaporated from Knudsen cell. Hf 99.9% (metal basis excluding Zr, Zr nominal 2%) and Zr 99.8%(metal basis excluding Hf, Hf nominal 4%) metals areco-evaporated from two different electron guns. ARPES measure- ments were carried out at room temperature with a 100 mm hemi- spherical electron analyzer equipped with a 2D CCD detector(SPECS) without breaking the vacuum. The He I (21.22 eV) reso-nant line is used to excite photoelectrons. The energy resolution of the system is better than 40 meV with a polar angle step of 1°. After the cleaning process, a 2 × 2 reconstruction of the InAs (111) surface is observed by RHEED [ Fig. 6(a) ], as expected for a clean (oxygen-free) In-terminated InAs(111). The streaky patternsof 17 monolayers Hf 0.2Zr0.8Te2films along the [1 /C2210] and [11 /C222] InAs azimuths indicate smooth, well ordered surfaces with no rota- tional domains, aligned in-plane with the InAs(111) substrate,which is characteristic of vdW epitaxial growth. The surface mor-phology of the 17 monolayers Hf 0.2Zr0.8Te2film on InAs(111) is examined by in situ room-temperature ultra-high vacuum scanning tunneling microscopy (UHV-STM). The scanning conditions are V = 200 mV and I = 400 pA. Figure 6(b) shows a 500 × 500 nm2 area scan of the sample. It can be inferred that Hf 0.2Zr0.8Te2is grown in the form of two-dimensional islands with an averagesurface roughness of ∼3.5 Å, which is consistent with the InAs atomic step previously observed in epitaxial ZrTe 218and MoTe 2.54 The band structure of 17 layers Hf 0.2Zr0.8Te2is imaged along theΓM direction of the BZ ( Fig. 7 ). The valence band exhibits a Dirac-like cone dispersion with the cone tip touching the Fermi FIG. 5. Schematic illustration of HfT e 2, ZrT e 2, and Hf 0.2Zr0.8Te2crystal struc- tures. The Hf atoms are randomly placed inside the Hf 0.2Zr0.8Te2crystal.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 075104 (2021); doi: 10.1063/5.0038799 129, 075104-6 Published under license by AIP Publishing.level which is similar to what has been observed in epitaxial HfTe 217and ZrTe 2.18On the other hand, these observations are notably different than the theoretical predictions that the DP is located around 0.63 eV above the E F. This is also in contrast to what is observed in the band structure of HfTe 2and ZrTe 2bulk single crystals imaged by synchrotron ARPES.19–22This differencemay be attributed to a substrate induced effect in the epitaxial few- layer film. A possible doping of the epitaxial films originating from the InAs(111) substrate could shift the Fermi level upward, allow- ing the observation of the linear band dispersions up to the tip ofthe Dirac cone. The cone-like band overlaps with a parabolic onelocated at the M point, indicating semimetalic character. The k x−ky constant energy contours are also measured, where parts of the Fermi surface at different binding energies below the Fermi level are imaged. The conical shaped band around the zone center ( Γ point) is visible and indicated with white dashed lines which inter-sect at the Dirac point at the Fermi level. Using E = ℏv Fk, the Fermi velocity vFcan be estimated to be ∼0.6 × 106m/s, a value similar to that obtained in the case of HfTe 2and ZrTe 2epitaxial films.17,18 IV. CONCLUSIONS In this work, ab initio calculations indicate that HfTe 2is a type-I DSM with a point-like Fermi surface, while ZrTe 2is a type-II DSM with needle-like electron and hole pockets touching atthe DP. A new alloy Hf xZr1−xTe2is predicted with a type-III Dirac cone emerging at x = 0.2 with a line-like Fermi surface. An in-planecompressive strain is also applied in order to obtain the type-III phase. This strain could be used to switch between the two types of DSM. Epitaxial Hf 0.2Zr0.8Te2films are successfully grown on InAs (111) substrates by MBE. ARPES measurements indicate thatHf 0.2Zr0.8Te2films have semimetalic character with the valence bands exhibiting a linear dispersion in k-space, thus indicating topological Dirac semimetal behavior. The top of Dirac cone touches the Fermi level, which is notably different than theorywhich predicts the DP at an energy well above the Fermi level.Although, further studies are needed with synchrotron ARPES in order to visualize the k zdispersion of the energy bands by varying the photon energy, these observations provide evidence that the FIG. 6. (a) RHEED patterns of the InAs(111) substrate and 17 monolayers Hf 0.2Zr0.8Te2films along the InAs[1 /C2210] and [11 /C222] azimuths. (b) 500 × 500 nm2STM image of 17 monolayers Hf 0.2Zr0.8Te2on InAs(111). FIG. 7. ARPES spectra and kx−kyenergy contour plots at different binding energies of 17 layers Hf 0.2Zr0.8Te2along the ΓΜ direction of the BZ using the He I resonance line at 22.21 eV . The white dashed lines are guides to the eye that indicate the cone-like dispersion of the valence bands which intersect at theDirac point.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 075104 (2021); doi: 10.1063/5.0038799 129, 075104-7 Published under license by AIP Publishing.type-III topological DP, predicted by theoretical calculations, natu- rally lies at —or very close to —the Fermi level. ACKNOWLEDGMENTS This work was supported by the Horizon 2020 projects SKYTOP “Skyrmion —Topological Insulator and Weyl Semimetal Technology ”(FETPROACT-2018-01, No. 824123) and MSCA ITN SMART_X 860553, the FLAG-ERA project MELoDICA, theHellenic Foundation for Research and Innovation and the General Secretariat for Research and Technology, under Grant No. 435 (2D-TOP). We also acknowledge the computational time grantedfrom the Greek Research and Technology Network in the NationalHigh-Performance Computing facility ARIS under project“2D-TOP ”and we thank Professor Phivos Mavropoulos for discussions. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding authors upon reasonable request. REFERENCES 1N. P. Armitage, E. J. Mele, and A. Vishwanath, Rev. Mod. Phys. 90, 015001 (2018), and references therein. 2G. E. Volovik, JETP Lett. 104, 645 (2016). 3G. E. Volovik and K. Zhang, J. Low Temp. Phys. 189, 276 (2017). 4T. Mizoguchi and Y. Hatsugai, J. Phys. Soc. Jpn. 89, 103704 (2020). 5I. M. Lifshitz, Sov. Phys. JETP 11, 1130 (1960). 6G. E. Volovik, Phys. Usp. 61, 89 (2018). 7Y. Zhang, C. Wang, L. Yu, G. Liu, A. Liang, J. Huang, S. Nie, X. Sun, Y. Zhang, B. Shen, J. Liu, H. Weng, L. Zhao, G. Chen, X. Jia, C. Hu, Y. Ding, W. Zhao, Q. Gao, C. Li, S. He, L. Zhao, F. Zhang, S. Zhang, F. Yang, Z. Wang, Q. Peng, X. 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5.0045091.pdf

Appl. Phys. Lett. 118, 072405 (2021); https://doi.org/10.1063/5.0045091 118, 072405 © 2021 Author(s).Electrical and optical characterizations of spin-orbit torque Cite as: Appl. Phys. Lett. 118, 072405 (2021); https://doi.org/10.1063/5.0045091 Submitted: 23 January 2021 . Accepted: 01 February 2021 . Published Online: 16 February 2021 Hanshen Huang , Hao Wu , Tian Yu , Quanjun Pan , Bingqian Dai , Armin Razavi , Kin Wong , Baoshan Cui , Su Kong Chong , Di Wu , and Kang L. Wang COLLECTIONS Paper published as part of the special topic on Spin-Orbit Torque (SOT): Materials, Physics, and Devices ARTICLES YOU MAY BE INTERESTED IN Spin–orbit torque and Dzyaloshinskii–Moriya interaction in perpendicularly magnetized heterostructures with iridium Applied Physics Letters 118, 062409 (2021); https://doi.org/10.1063/5.0035769 Spin-orbit torques: Materials, physics, and devices Applied Physics Letters 118, 120502 (2021); https://doi.org/10.1063/5.0039147 Field-free and sub-ns magnetization switching of magnetic tunnel junctions by combining spin-transfer torque and spin–orbit torque Applied Physics Letters 118, 092406 (2021); https://doi.org/10.1063/5.0039061Electrical and optical characterizations of spin-orbit torque Cite as: Appl. Phys. Lett. 118, 072405 (2021); doi: 10.1063/5.0045091 Submitted: 23 January 2021 .Accepted: 1 February 2021 . Published Online: 16 February 2021 Hanshen Huang, Hao Wu,a) Tian Yu,Quanjun Pan, Bingqian Dai,Armin Razavi, KinWong, Baoshan Cui,Su Kong Chong, DiWu, and Kang L. Wang AFFILIATIONS Department of Electrical and Computer Engineering, Department of Physics and Astronomy, and Department of Material Science and Engineering, University of California, Los Angeles, California 90095, USA Note: This paper is part of the Special Topic on Spin-Orbit Torque (SOT): Materials, Physics and Devices. a)Author to whom correspondence should be addressed: wuhaophysics@ucla.edu ABSTRACT To further reduce the energy consumption in spin–orbit torque devices, it is crucial to precisely quantify the spin–orbit torque (SOT) in different materials and structures. In this work, heavy metal/ferromagnet and heavy metal/ferrimagnet heterostructures are employed as themodel systems to compare the electrical and optical methods for the SOT characterization, which are based on the anomalous Hall effect andthe magneto-optical Kerr effect, respectively. It is found that both methods yield the consistent SOT strength for the current-driven magneti- zation switching measurements and the harmonic measurements. Our results suggest that the optical method is a feasible and reliable tool to investigate SOT, which is a powerful way to develop insulator-based magnetic systems in the future. Published under license by AIP Publishing. https://doi.org/10.1063/5.0045091 Spin–orbit torque (SOT) is a powerful way to manipulate mag- netization electrically, which exhibits great potential innext-generation magnetic memory and logic applications. SOT- induced magnetization switching has been realized in abundant materials and structures, including heavy metal/ferromagnet hetero-structures, 1–3heavy metal/magnetic insulator heterostructures,4–7 topological insulator/ferromagnet (ferrimagnet) heterostructures,8–12 and magnetic topological insulators.13–15Also, much effort has been made to realize field-free SOT switching,16–21which paves the way for all-electrical SOT applications. Conventionally, SOT is characterized electrically based on the anomalous Hall effect (AHE) or the tunneling magnetoresistance(TMR) effect. 22–24However, the electrical method is limited by its complex signal origins and choice of materials. On the one hand, electrical signals originating from sources other than SOT, such as the current-induced thermal effects and the unidirectional magne-toresistance, 25,26may obscure the characterization. On the other hand, for magnetic insulators (MIs), the electrical detectionstrongly relies on the interfacial proximity effect between the MIs and the conductors, resulting in an extremely small signal and instability in the SOT characterization. 27–29Third, the optical method is a powerful way to detect the ultrafast SOT dynamicsdown to sub-picosecond. 30In this work, both electrical transport and optical methods are employed to characterize the SOT in ferromagnetic and ferrimagneticheterostructures. To be more specific, films with stack of Ta(5)/ CoFeB(1.1)/MgO(2)/Ta(2) and Ta(5)/GdFeCo(4)/MgO(2)/Ta(2) (unit in nm) are used as our model structures. Figures 1(a) and1(b) show the lattice structure for CoFeB and GdFeCo, respectively. It is knownthat CoFeB is a typical ferromagnet, in which the spin of Co and Featoms points in the same direction, while for the ferrimagnetic GdFeCo, the spin sublattices of Gd and FeCo are antiferromagnetically coupled due to the atomic-level interfacial magnetic anisotropy, wherethe exchange magnetic resonance mode gives rise to the ultrafastferrimagnetic spin dynamics beyond 100 GHz. 31–33The films are grown by the magnetron sputtering with the base pressure lower than 5 /C210/C08Torr. The ferrimagnetic GdFeCo is deposited by co-sputtering Gd and CoFe targets, where its composition is controlledby tuning the sputtering power. To further enhance the perpendicularmagnetic anisotropy (PMA) of Ta/CoFeB/MgO/Ta, post-annealing is carried out at 250 /C14C. These films are then patterned into 20 lm /C2130lm Hall-bar devices by standard photolithography combined with the dry etching method. The electrical transport measurements are based on the AHE, where a Keithley 2612a source meter is used for generating the writingpulse current with a width of 1 ms, and the magnetization of the Appl. Phys. Lett. 118, 072405 (2021); doi: 10.1063/5.0045091 118, 072405-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apldevices is subsequently read by applying a small reading current of 0.1 mA to minimize the thermal effects. The optical measurements areconducted based on the magneto-optic Kerr effect (MOKE) in the polar configuration, 34as shown in Fig. 1(c) , where the rotated polari- zation angle of the reflected light is proportional to the out-of-planecomponent of magnetization ( M z). The optical measurement setup is shown in Fig. 1(d) . A laser light source of 410 nm is deployed for optical measurements. To further improve the signal-to-noise ratio, a photoelastic modulator (PEM) is adopted and its modulation rate iskept at 100 kHz. The incident light is linearly polarized and focused onto the devices using a microscope (20 /C2/0.42 NA, spot size /C2410lm). The reflected light is separated into spolarized light and p polarized light by a Wollaston prism and then collected by a balancedphotodiode. The Kerr rotation angle and, thus, magnetization of the devices are demodulated and detected using a lock-in amplifier. All measurements are conducted at room temperature. Magnetic materials with PMA are preferred for high-density and low-energy consumption memory applications. 35We first evaluate the PMA for Ta/CoFeB/MgO/Ta and Ta/GdFeCo/MgO/Ta by measuring their in-plane saturation field (hard axis),36from which the anisotropy fields ( Hk) are estimated to be 1470 Oe and 2731 Oe, respectively. The PMA is also identified by measuring anomalous Hall resistance ( Rxy) as a function of magnetic field along the out-of-plane direction ( Hz), as shown in Figs. 2(a) and2(b), where the sharp switching in the Rxy- Hzloops clearly indicates the strong PMA. Similar results are obtained in the polar MOKE measurements, where the magnetization informa- tion is represented by the Kerr rotation angle ( hK) instead of Rxy.A s seen in Figs. 2(c) and2(d),t h ehK-Hzloops resemble the results in the electrical transport method. Unlike the electrical transportmeasurement that averages the magnetization across the whole device, the optical method detects local magnetization beneath the laser spot, and we attribute the slight coercivity difference between the electrical transport and optical measurements for GdFeCo to the magnetic inho-mogeneity across the device. By applying a fixed in-plane magnetic field ( H x¼55 Oe) to break the symmetry between up and down magnetizations, the current- induced SOT could switch the perpendicular magnetization determin- istically.18This deterministic SOT switching is characterized by both electrical transport and optical methods. As indicated by the changes inRxyinFigs. 3(a) and3(d), the magnetization is switched by sweeping the writing current. The critical switching current densities for Ta/CoFeB/MgO/Ta and Ta/GdFeCo/MgO/Ta estimated from Rxy-Je loops are 8.2 /C2106Ac m/C02and 1.1 /C2107Ac m/C02, respectively. It is also shown in Figs. 3(a) and3(d)that the polarity of the SOT-induced magnetization switching reverses correspondingly as we reverse the in-plane bias field, which confirms the SOT characteristic. In the opti- cal measurements, the readout of the SOT switching is accomplishedby recording the h K-Jeloops. As shown in Figs. 3(b) and3(e),t h ehKvs Jecurve is similar to the RxyvsJecurve by the electrical method, suggesting that hKis equivalent to Rxyin characterizing the SOT switching. Nevertheless, the optical method further provides us the ability to visualize the magnetic domain changes during the SOT switching. This is demonstrated in Figs. 3(c) and3(f)by implementing the same principle as the above polar MOKE measurements but using a charge- coupled device (CCD) to obtain a spatial resolution of magnetic domains. The contrast of the MOKE image represents the magnetic domain state, i.e., the bright and dark contrast corresponds to the up FIG. 1. Lattice structure of CoFeB (a) and GdFeCo (b); the blue balls stand for CoFe atoms, and red balls stand for Gd atoms. (c) Polar MOKE configuration. (d) Sch ematic of the polar MOKE setup for the optical measurement.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 072405 (2021); doi: 10.1063/5.0045091 118, 072405-2 Published under license by AIP Publishingand down magnetic domains, respectively. The MOKE images not only characterize the initial and final states of the magnetic domains in the devices but also visualize the whole switching process. For exam-ple, in CoFeB heterostructures [ Fig. 3(c) ], the magnetic domain nucle- ates at the right end of the Hall bar device first. As we increased the writing current density gradually, the domain wall propagates from the right side to the left side to realize the full magnetization switching. The SOT strength is further investigated by the electrical trans- port method and the optical MOKE method, respectively. In the elec-trical transport measurements, conventional first and second harmonic signals are measured by sweeping the magnetic field in the film plane. A sinusoidal modulation current at a frequency of 273 Hzis mixed with the writing current ( J e¼2.46/C2106Ac m/C02and Je¼1.67/C2106Ac m/C02for the CoFeB and GdCoFe stacks, respec- tively). During the electrical transport measurements, the laser contin- uously illuminates the samples to ensure the same thermal conditionas that in the optical measurements. Figures 4(a)–4(d) show the har- monic signals measured by the electrical transport method for Ta/CoFeB/MgO/Ta and Ta/GdCoFe/MgO/Ta, respectively. When the applied field H xis larger than the anisotropy field Hk,t h em a g n e t i z a - tion is aligned along the in-plane direction and the out-of-plane oscil-lation is mainly due to the sinusoidal current modulation-induced damping-like torque. Thus, the tilting angle can be expressed as 37–39DhM¼/C0HDL jHxj/C0Hksinxt; (1) where xis the current frequency, HDLis the damping-like effective field, and Hkis the anisotropy field. As the planar Hall effect is negligi- ble in these two heterostructures, the Hall resistance can be written as Rxy¼RAHEcoshþDhM ðÞ sinxt; (2) where RAHE is the anomalous Hall resistance and his the angle between magnetization and the z-axis direction h¼p 2/C0/C1.E x p a n d i n g Eq.(2)and taking the thermal contribution RSSEinto account, we can obtain the second harmonic Hall resistance as follows: R2x xy¼/C0RAHE 2HDL jHxj/C0HkþRSSEHx jHxjþRoffset ; (3) where Roffsetis the offset background. By fitting the experimental data with Eq. (3), it is found that the damping-like effective fields ( HDL)a r e 3.09 Oe and 2.54 Oe for the CoFeB and GdFeCo heterostructures, respectively, which implies that the SOT coefficient ( vSOT¼HDL=Je) and the effective spin Hall angle ( jhSHj¼ð 2jejMstF=/C22hÞvSOT)a r e 1.26/C210/C06Oe A/C01cm2and 0.04 for the Ta/CoFeB stack, and 1.52/C210/C06Oe A/C01cm2and 0.03 for the Ta/GdFeCo stack, respectively. FIG. 2. Hysteresis loops of (a) Ta/CoFeB/MgO/Ta by the transport measurement. (b) Ta/GdFeCo/MgO/Ta by the transport measurement. (c) Ta/CoFeB/MgO/Ta by t he optical measurement. (d) Ta/GdFeCo/MgO/Ta by the optical measurement.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 072405 (2021); doi: 10.1063/5.0045091 118, 072405-3 Published under license by AIP PublishingA similar setup as shown in Fig. 1(d) is used for determining these parameters optically. The first harmonic measurement is the same as that conducted in electrical transport measurements except now hKis recoded and the lock-in frequency is chosen as the PEM modulationfrequency. The optical counterpart of the second harmonic measure- ment is also known as the differential Kerr method ( DhK). Following t h ed e r i v a t i o ni nt h e supplementary material of reference,4the Kerr rotation hKcan be expanded using the Taylor expansion as FIG. 3. SOT-induced magnetization switching measured by transport and optical methods for Ta/CoFeB/MgO/Ta (a)–(c) and Ta/GdFeCo/MgO/Ta (d)–(f). The mag netic domain states at different current densities indicate the domain nucleation and domain wall propagation switching mechanism. FIG. 4. First harmonic (a) and (c) and second harmonic Hall resistance (b) and (d) by the transport method for Ta/CoFeB/MgO/Ta (a) and (b) and Ta/GdFeCo/MgO/T a (c) and (d). Kerr rotation (e) and (g) and differential Kerr rotation (f) and (h) by the optical method for Ta/CoFeB/MgO/Ta (e) and (g) and Ta/GdFeCo/MgO/Ta (f ) and (h).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 072405 (2021); doi: 10.1063/5.0045091 118, 072405-4 Published under license by AIP PublishinghK¼hK0þDhKsinxt; (4) where hK0is the Kerr rotation angle without applying current and xis the current frequency. By setting the zero net torque condition, we havem/C2H¼0, where mandHare the magnetization and magnetic field, respectively. Since the second order magneto-optic effect is muchsmaller than that of the first order, we only consider the first-orderterm here. Thus, we are able to obtain the expression similar to R 2x xyin the electrical transport measurement for DhK, DhK¼fHDL jHxj/C0Hk; (5) where fis the magneto-optic coefficient. Based on the results of the optical method, as shown in Figs. 4(e)–4(h) , the damping-like effective fields are 3.60 Oe and 3.32 Oe for Ta/CoFeB/MgO/Ta and Ta/GdFeCo/MgO/Ta, respectively, which are similar to those found bythe electrical transport method and, thus, yield a similar SOT coeffi-cient v SOT(1.47/C210/C06Oe A/C01cm2for the Ta/CoFeB stack and 1.99/C210/C06Oe A/C01cm2for the Ta/GdFeCo stack) and the effective spin Hall angle jhSHj(0.05 for the CoFeB stack and 0.04 for the GdFeCo stack). To explore more advantages of the optical method, we further investigate the spatial distribution of SOT by the differential Kerr m e t h o d .W ed i v i d et h eH a l lb a rd e v i c ei n t ofi f t e e nr e p r e s e n t a t i v ea r e a sas shown in Figs. 5(a) and5(b)and used the differential Kerr method to extract the effective spin Hall angle in each area. Consequently, weobtain the spatial mapping of the effective spin Hall angle, which shows how the effective spin Hall angle fluctuates across the device. From the results, we can see that there is not much fluctuation in the CoFeB het-erostructure. However, in the GdFeCo heterostructure, the effectivespin Hall angles in the central regions (the second row) of the sampleare 10% smaller than the edge areas (the first and the third row). One possible reason is that the CoFeB stack is less sensitive to temperature, which is not the case for the GdFeCo stack. When the sinusoidal currentis injected into the current channel for the differential Kerr measure-ment, there will be a temperature distribution across the cross section ofthe device, which means that the temperature is higher in the middle of the device. According to reference, 40for the FeCo-rich GdFeCo stack,saturation magnetization Msis much larger with the increasing temper- ature. Therefore, the obtained SOT effective field and the calculated effective spin Hall angle are smaller. Besides, the effective spin Hall angle in the arm area of the Hall bar device (the first and the fifth col-umn) is a bit larger than those in the cross area (the second and thefourth column), which can be attributed to the higher current density inthe arm area due to the shunting effect. 41In addition, we perform all optical methods on the unpatterned CoFeB stack and GdFeCo stackfilms as well ( supplementary material ). We can clearly observe the switching processes by MOKE imaging and optically extract the effec-tive spin Hall angle by the differential Kerr method. This shows the fea-sibility of using all the optical method on unpatterned films, which cansimplify the measurement by reducing the need for device fabrication. In summary, we measure a series of magnetic and SOT prop- erties, including the hysteresis loops, current-driven SOT switch-ing, and first and second harmonic SOT signals, in Ta/CoFeB/ MgO/Ta and Ta/GdFeCo/MgO/Ta heterostructures by both the electrical transport and the optical MOKE methods. Current-driven deterministic SOT switching is demonstrated by both theelectrical transport Hall measurements and optical MOKE mea-surements. The magnetic domain switching process during theSOT switching is also visualized by the MOKE imaging technique,which suggests that the SOT switching is dominated by the mag-netic domain nucleation and domain wall propagation process.The SOT strength is further quantified by the harmonic measure-ments based on both the electrical and the optical signals. It is demonstrated that the optical method gives consistent SOT param- eters as those obtained in the conventional electrical transportmeasurements, which would be a powerful tool for investigatingthe SOT in insulating magnets in the future. See the supplementary material for MOKE imaging of the switching process, Kerr rotation, and differential Kerr rotation mea-sured on unpatterned films. This work was supported by the NSF under Award Nos. 1935362, 1909416, 1810163, and 1611570, the Nanosystems FIG. 5. Spatial mapping of the effective spin Hall angle on (a) Ta/CoFeB/MgO/Ta and (b) Ta/GdFeCo/MgO/Ta heterostructure Hall bar devices (the lower left co rner is not measured).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 072405 (2021); doi: 10.1063/5.0045091 118, 072405-5 Published under license by AIP PublishingEngineering Research Center for Translational Applications of Nanoscale Multiferroic Systems (TANMS), the U.S. Army ResearchOffice MURI program under Grant Nos. W911NF-16-1-0472 andWN911NF-20-2-0166, and the Spins and Heat in Nanoscale Electronic Systems (SHINES) Center funded by the U.S. Department of Energy (DOE), under Award No. DE-SC0012670. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1L. Liu, C. F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336(6081), 555–558 (2012). 2C. F. Pai, L. Liu, Y. Li, H. W. Tseng, D. C. 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5.0044693.pdf

J. Chem. Phys. 154, 104106 (2021); https://doi.org/10.1063/5.0044693 154, 104106 © 2021 Author(s).Parameterization of a linear vibronic coupling model with multiconfigurational electronic structure methods to study the quantum dynamics of photoexcited pyrene Cite as: J. Chem. Phys. 154, 104106 (2021); https://doi.org/10.1063/5.0044693 Submitted: 18 January 2021 . Accepted: 16 February 2021 . Published Online: 08 March 2021 Flavia Aleotti , Daniel Aranda , Martha Yaghoubi Jouybari , Marco Garavelli , Artur Nenov , and Fabrizio Santoro COLLECTIONS Paper published as part of the special topic on Quantum Dynamics with ab Initio Potentials ARTICLES YOU MAY BE INTERESTED IN Classical molecular dynamics The Journal of Chemical Physics 154, 100401 (2021); https://doi.org/10.1063/5.0045455 Relaxation dynamics through a conical intersection: Quantum and quantum–classical studies The Journal of Chemical Physics 154, 034104 (2021); https://doi.org/10.1063/5.0036726 Comparing (stochastic-selection) ab initio multiple spawning with trajectory surface hopping for the photodynamics of cyclopropanone, fulvene, and dithiane The Journal of Chemical Physics 154, 104110 (2021); https://doi.org/10.1063/5.0045572The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Parameterization of a linear vibronic coupling model with multiconfigurational electronic structure methods to study the quantum dynamics of photoexcited pyrene Cite as: J. Chem. Phys. 154, 104106 (2021); doi: 10.1063/5.0044693 Submitted: 18 January 2021 •Accepted: 16 February 2021 • Published Online: 8 March 2021 Flavia Aleotti,1 Daniel Aranda,2 Martha Yaghoubi Jouybari,2 Marco Garavelli,1,a) Artur Nenov,1,b) and Fabrizio Santoro2,c) AFFILIATIONS 1Dipartimento di Chimica Industriale “Toso Montanari,” Università di Bologna, Viale del Risorgimento 4, 40136 Bologna, Italy 2Istituto di Chimica dei Composti Organometallici (ICCOM-CNR), Area della Ricerca del CNR, Via Moruzzi 1, I-56124 Pisa, Italy Note: This paper is part of the JCP Special Topic on Quantum Dynamics with Ab Initio Potentials. a)Electronic mail: marco.garavelli@unibo.it b)Electronic mail: artur.nenov@unibo.it c)Author to whom correspondence should be addressed: fabrizio.santoro@pi.iccom.cnr.it ABSTRACT With this work, we present a protocol for the parameterization of a Linear Vibronic Coupling (LVC) Hamiltonian for quantum dynamics using highly accurate multiconfigurational electronic structure methods such as RASPT2/RASSCF, combined with a maximum-overlap dia- batization technique. Our approach is fully portable and can be applied to many medium-size rigid molecules whose excited state dynamics requires a quantum description. We present our model and discuss the details of the electronic structure calculations needed for the param- eterization, analyzing critical situations that could arise in the case of strongly interacting excited states. The protocol was applied to the simulation of the excited state dynamics of the pyrene molecule, starting from either the first or the second bright state (S 2or S 5). The LVC model was benchmarked against state-of-the-art quantum mechanical calculations with optimizations and energy scans and turned out to be very accurate. The dynamics simulations, performed including all active normal coordinates with the multilayer multiconfigurational time- dependent Hartree method, show good agreement with the available experimental data, endorsing prediction of the excited state mechanism, especially for S 5, whose ultrafast deactivation mechanism was not yet clearly understood. Published under license by AIP Publishing. https://doi.org/10.1063/5.0044693 .,s I. INTRODUCTION Quantum dynamical (QD) simulations in polyatomic molecules are often run with reduced dimensionality models, gen- erated predetermining the most important coordinates on the grounds of chemical intuition. This approach is advantageous since it strongly reduces the computational effort necessary to generate high-dimensionality potential energy surfaces (PESs) and to run QD in many dimensions with traditional methods.1On the other side, recent methodological advances have made possible also thepropagation of wavepackets (WP) in many dimensions, open- ing the route to a non-phenomenological description of deco- herence and energy redistribution. The methods of reference in this field are probably the multiconfigurational time-dependent Hartree (MCTDH)2–4and its multilayer (ML) extension (ML- MCTDH).5–8They are extremely effective, even for nonadiabatic problems, especially if the coupled PESs have some simple func- tional form, like a low-order Taylor expansion in normal coordi- nates. Hamiltonians that use these simplified PESs are often referred to as model vibronic coupling Hamiltonians.9,10They use a diabatic J. Chem. Phys. 154, 104106 (2021); doi: 10.1063/5.0044693 154, 104106-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp representation and quadratic expansions for the diagonal and off- diagonal PESs. If no other approximation is invoked, the above definition describes what is known as the quadratic vibronic cou- pling (QVC) Hamiltonian. However, it is usually further assumed that all diagonal PESs share the same normal modes and frequen- cies (usually taken all equal to the ones of the initial state before photo-excitation) and that off-diagonal terms are linear functions of the coordinates. These assumptions lead to the so-called linear vibronic coupling (LVC) model. LVC is the simplest Hamiltonian that can describe Conical Intersections (CoIs) and their multidi- mensional extensions (intersection seams), and in fact, it can be seen as a generalization to many states and modes of the two-state two-mode model adopted long-time ago to investigate the CoI prob- lem.11Model vibronic Hamiltonians have been quite successful to introduce the effect of interstate couplings in electronic spectra and to clarify the main features of a nonadiabatic dynamics around a CoI.9,10Despite the “model” attribute, they can be adopted also for accurate descriptions of realistic problems, especially if the inves- tigated molecules are rigid and/or the timescale of interest is very short (∼100 fs). As a matter of fact, in the last decade, they have been employed in the study of fast intersystem crossings in metal–organic complexes,12–15inππ∗/nπ∗decays in nucleobases,16–20and also to couple QD simulations with an explicit description of the environ- ment.18,21It is further worth noting that also popular models for excitonic problems essentially belong to the same family of Hamil- tonians,22–26but they (also) include off-diagonal constant terms so that CoIs cannot occur and the adiabatic PES are characterized by avoided crossings. It became increasingly evident that even in the ultrafast regime, the QD can be drastically dependent on the parameters of the vibronic Hamiltonians, especially if the investigated system is char- acterized by several coupled quasi-degenerate states. In fact, the rate and yield of the predicted non-radiative processes can be totally different employing different Density Functional Theory (DFT) functionals19,20or even different descriptions of the environment.18 These findings highlight the necessity to work out effective proto- cols to parameterize model Hamiltonians with electronic structure methods as accurate as possible. We recently proposed a method based on a maximum- overlap diabatization to parametrize LVC Hamiltonians with time- dependent DFT (TD-DFT) calculations,19which is very effective also for several excited states (10–20) and molecules with many degrees of freedom (100).27From the point of view of electronic calculations, it only requires the ability to run single-point calcu- lations and compute the overlap between electronic wavefunctions (WFs) at geometries displaced along the normal modes. There- fore, in principle, it is suitable for many electronic structure meth- ods, and indeed, it is inspired by a procedure formerly proposed for configuration interaction (CI) WFs.28Moreover, its computa- tional cost is similar to that required to obtain the numerical gra- dients of all the involved states, and since the necessary calcula- tions are embarrassingly parallel, even accurate methods can be adopted. Multiconfigurational methods based on complete active space self-consistent field (CASSCF) and subsequent perturbative cor- rections (CASPT2) and their generalized extensions RASSCF and RASPT2 are, at the state of the art, among the most reliable electronic structure methods for computational photophysics andphotochemistry. One of their major qualities is the capability to treat with similar accuracy states with different nature, including charge- transfer and double-excited states that challenge TD-DFT, provided the active space is properly selected. However, the dependence of the results on the active space composition, on the number of electronic states, and on the form of the zeroth order Hamiltonian makes LVC parameterization based on the CASSCF/CASPT2 protocol a rather intricate task. In particular, the formulation of the Fock operator in the construction of the zeroth order Hamiltonian has spawned several flavors of the perturbative correction, multi-state (MS),29 extended multi-state (XMS),30and, more recently, extended dynam- ically weighted31CASPT2, as well as the single-state single-reference and multi-state multi-reference variations of the MS-CASPT2.32 In this contribution, we present, at the best of our knowl- edge, the first LVC Hamiltonian parameterized with (X)MS- RASPT2/RASSCF calculations for a medium-size molecule, such as pyrene. Pyrene is an interesting molecule that exhibits absorption bands of different bright states with a clear vibronic structure in the deep UV. Its photoinduced dynamics is characterized by the ultrafast internal conversion (IC) to the lowest dark excited state. While the IC process from the first bright excited state (320 nm) has been studied in detail both experimentally33–36and theoreti- cally,37,38the IC process from the second excited state has been addressed only recently with transient absorption and bidimensional and photoelectron spectroscopy.36,39,40Thanks to the unprecedented time-resolution (down to 6 fs), transient spectroscopy has allowed to resolve quantum beatings due to the motion of the vibrational WP in the excited state. Still, the picture of the IC mechanism from the second bright state is incomplete. Picchiotti et al.39and Noble et al.40recognized the involvement of intermediate dark states, but their role in the IC is not well understood yet. We will study the decay dynamics of pyrene photoexcited to either its first or second bright states, adopting LVC Hamiltoni- ans that fully account for the couplings of the lowest seven excited states and include all the active nuclear coordinates (49). We will evaluate the reliability of LVC PES by recomputing energies at rele- vant points of the dynamics, such as minima and energy-accessible CoIs. Moreover, we will investigate in depth the dependence of the QD results on different parameterizations of the Hamiltonian obtained with different active spaces and different implementa- tions of the perturbative corrections. A parameterization of an LVC Hamiltonian is, actually, a much more stringent test of the sta- bility of the computational protocol than the computation of the vertical excitations and/or of the numerical gradients, and we will analyze our results to enunciate few recommendations for future studies. II. METHODOLOGY: THE LINEAR VIBRONIC COUPLING MODEL We consider a ndimensional diabatic basis, | d⟩= (|d1⟩, |d2⟩, . . ., |dn⟩), and the following expression of the Hamiltonian: H=∑ i(K+Vdia ii(q))∣di⟩⟨di∣+∑ i,j>iVdia ij(q)(∣di⟩⟨dj∣+∣dj⟩⟨di∣), (1) where qis the column vector of the ground state (GS) dimension- less normal coordinates. According to the Linear Vibronic Coupling J. Chem. Phys. 154, 104106 (2021); doi: 10.1063/5.0044693 154, 104106-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp (LVC) model, the kinetic (K) and potential (V) terms have the following form: K=1 2pTΩp, (2) Vdia ii(q)=E0 i+λT iiq+1 2qTΩq, (3) Vdia ij(q)=λT ijq, (4) where Ωis the diagonal matrix of the GS normal-modes frequen- cies, pis the vector of the conjugated momenta, and Tindicates the standard transpose operation for matrices. Therefore, the diag- onal terms of the potential energy Vdia ii(q)are described in the harmonic approximation, and they share the same frequencies as the GS. The linear terms in the Hamiltonian represent the dia- batic energy gradients λiiand the inter-state diabatic couplings λij(i≠j). The LVC Hamiltonian is parameterized by defining diabatic states | di⟩to be coincident with the adiabatic reference states | ai⟩at a reference geometry. We choose the GS minimum as reference. At displaced geometries, diabatic states are defined so to remain as sim- ilar as possible to the reference states | a(0)⟩. This idea was already proposed by Cimiraglia et al.28for configuration-interaction WFs and then extended to TD-DFT by Neugebauer et al.41and by some of us.19More precisely, we follow the derivation presented in Ref. 19, and for each displaced geometry 0+Δα(since now on Δα), we compute the adiabatic states | a(Δα)⟩and the matrix S(Δα) of their overlaps with | a(0)⟩, Sij(Δα)=⟨ai(0)∣aj(Δα)⟩. (5) The transformation matrix Dthat defines the diabatic states at Δα, ∣d⟩=∣a(Δα)⟩D(Δα), (6) is then obtained as D=ST(SST)−1 2, (7) where for brevity the dependence on Δαis not explicitly reported. In Eq. (7), a Löwdin orthogonalization is used to account for the fact that the set of the computed adiabatic states at the displaced geometries is finite and therefore not complete. At each displaced geometry the computed adiabatic energies form a diagonal matrix Vad(Δα)=diag(Ead 1(Δα),Ead 2(Δα),. . ., Ead n(Δα))and the diabatic potential terms are simply Vdia(Δα)=DT(Δα)Vad(Δα)D(Δα). (8) Therefore, the gradients λiiand couplings parameters λijcan be obtained from numerical differentiation with respect to each qα, λij(α)=∂Vdia ij(q) ∂qα≃Vdia ij(Δα)−Vdia ij(−Δα) 2Δα. (9) In the following, the normal coordinates qand frequencies Ω were obtained at the second order perturbation theory level (MP2), whereas the energies Ead i(Δα)of the adiabatic states at each dis- placed geometry and their overlap Swith the wave functions at the reference geometry were obtained at the RASSCF/RASPT2 level.The vibronic wavefuction is defined in terms of the diabatic basis as | Ψ(q,t)⟩=∑i|di⟩|Ψi(q,t)⟩, and the time evolution is computed by solving the time-dependent Schrödinger equation, i̵h∂∣Ψi(q,t)⟩ ∂t=H∣Ψi(q,t)⟩. (10) In the following, we will investigate the time evolution of the population of the diabatic states. For state iat time t, it is simply Pi(t) =⟨Ψi(q,t)|Ψi(q,t)⟩. III. COMPUTATIONAL DETAILS A. Electronic structure calculations Pyrene is a highly symmetric molecule (D 2hsymmetry) with 26 atoms and 72 normal modes (see Tables S1–S3 in the supplementary material). For the parameterization of the LVC Hamiltonian, we have identified our diabatic states with the low- est seven excited adiabatic states at the S 0equilibrium geometry, belonging to four different irreducible representations: one state in Ag, two in B 3u, one in B 2u, and three in B 1g. Then, we have displaced the atoms along each normal coordinate (obtained at MP2/ANO- L-VDZP level) both in the positive and in negative direction and calculated two main quantities: excitation energies and WF overlaps ⟨Sref i∣Sdispl j⟩between all the eigenstates at the displaced and refer- ence geometry (details on the WF overlap calculations at different geometries are given in the supplementary material). These data are then utilized to parameterize the LVC Hamiltonian according to Eqs. (7)–(9). We note that while energy gradients are present only along symmetry conserving (A g) modes, interstate couplings also exist along modes belonging to B 1g, B2u, and B 3uirreducible representations, which decrease the symmetry of the system, as indi- cated in Table I. 23 modes do not couple the electronic states of interest and are, therefore, excluded from the model. Our previ- ous experience in the parameterization of the LVC Hamiltonian from TD-DFT indicates that a shift Δ= 0.1 in dimensionless coor- dinates guarantees accurate and robust results.19,42Since diabatic states are built so to preserve at all geometries their electronic char- acter, in the following, they will be named with the D 2hsymmetry labels of the adiabatic states they coincide with at the S 0mini- mum. Adiabatic states, on the contrary, will be denoted with the usual nomenclature S xwith x= 1, 2, . . ., 7 in order of increasing energy. It is worthy to remark that different diabatization techniques are actually possible.43A strategy based on a one-shot computa- tion of energy, gradients, and nonadiabatic coupling vectors with multireference Configuration Interaction Singles (CIS) and Config- uration Interaction Singles and Doubles methods has been recently presented and implemented in Surface Hopping Including Arbi- trary Couplings (SHARC) code.44“Energy-based” methods, which rely only on energies and not on WFs, are also very attractive, and their simplicity makes them well suited to be applied also in com- bination with accurate and time-consuming electronic-structure methods such as CASSCF,45Extended Multi-Configuration Quasi- Degenerate Perturbation Theory (XMCQDPT2),46and Equation- of-Motion Coupled-Cluster Singles and Doubles (EOM-CCSD).47 Their implementation is very straightforward when each mode can only couple two states,46while in the more general case, they require a fitting of the parameters, e.g., minimizing the root mean square J. Chem. Phys. 154, 104106 (2021); doi: 10.1063/5.0044693 154, 104106-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Coupling of the reference states along symmetry-breaking modes. Forbidden interactions in D 2hsymmetry are possible between states falling in the same irreducible representation of the lower point groups. Irreducible representation Point group at Classification of D 2hstates of modes displaced geometries into new irreducible representations B3u C2v A1: 1A g, 1B 3u, 2A g, 2B 3u B1: 1B 2u, 1B 1g, 2B 1g, 3B 1g B2u C2v A1: 1A g, 1B 2u, 2A g B2: 1B 3u, 1B 1g, 2B 3u, 2B 1g, 3B 1g B1g C2h Ag: 1A g, 2A g, 1B 1g, 2B 1g, 3B 1g Bu: 1B 3u, 1B 2u, 2B 3u deviation of the original ab initio and the model adiabatic PES at a representative number of points. The method we apply here is com- putationally demanding but is fully general. Moreover, being based on the overlaps of the WFs, it allows a direct and detailed control of the electronic character of the diabatic PESs. Electronic structure calculations with D 2hand with reduced symmetry were performed at the RASPT2/RASSCF/ANO-L-VDZP level of theory. The calculations encompass the lowest eight roots of pyrene, which, due to the use of symmetry, fall in different irre- ducible representations. Three active spaces were used: a minimal one consisting of the frontier eight πand eight π∗orbitals (full- π), with up to quadruple excitations [denoted as RAS(4, 8 ∥0, 0∥4, 8)] and two extended active spaces encompassing four and eight extra-valence virtual orbitals of π∗character with a higher angu- lar quantum number, denoted RAS(4, 8 ∥0, 0∥4, 12) and RAS(4, 8∥0, 0∥4, 16), respectively. The RASSCF scheme in which allmolecular orbitals are put in RAS1 and RAS3 (leaving RAS2 empty) has been benchmarked previously, demonstrating the need of a high RAS1/RAS3 excitation level.48The “empty RAS2” active space construction recipe has already shown to give accurate results for pyrene.39We note that the extra-valence orbitals, despite bearing some resemblance to Rydberg orbitals, are not suitable for describ- ing Rydberg states (not present among the states below 5 eV). Their only role is to capture more dynamic correlation at the RASSCF level, which has been shown to significantly improve the agreement with experimental data.49–51Figure 1 shows the active orbitals. In all calculations, on top of the RASSCF results, we have applied different types of perturbative corrections: either single state (SS), multi state (MS), or extended multi state (XMS) RASPT2, always using an imaginary shift of 0.2 a.u. and setting the IPEA shift to zero. For a more compact notation, each calculation will FIG. 1 . Active orbitals for pyrene in D 2hsymmetry, for each irreducible representation (top label; representations A g, B1g, B2u, and B 3uhave no active orbitals). Bottom row (dark gray): πorbitals (RAS1), middle row (light gray) π∗orbitals (RAS3), and top row (white): virtual orbitals with higher angular momentum (RAS3). The orbitals marked with∗were excluded from the MS(8:12) and XMS(8:12) calculations. J. Chem. Phys. 154, 104106 (2021); doi: 10.1063/5.0044693 154, 104106-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp be labeled SS( n:m), MS( n:m), or XMS( n:m) depending on the type of perturbative correction, where nandmrefer to the number of orbitals in RAS1 and RAS3, respectively. For calculations with D 2h symmetry (at the reference geometry and along A gmodes), we rely on SS(8:16) energies which are virtually identical to MS results when the states are energetically separated and more accurate than XMS energies that rely on an average Fock operator. The only exception are the three close lying states belonging to the B 1girreducible rep- resentation for which MS(8:16) and XMS(8:16) energies were also evaluated. The SS(8:16) energies at the reference geometry were used as a uniform reference. For calculations with lower symmetry, we rely on (extended) multistate energies and WFs with reduced active space [i.e., (X)MS(8:12)] due to the interaction of near-degenerate states (forbidden at D 2hsymmetry) and the increase of computa- tional effort. To allow for consistency, the change of energy along symmetry-reducing modes, evaluated at the (X)MS(8:12) level, was added to the reference SS(8:16) energies. The only exception are A1states at geometries with C 2vsymmetry obtained by displacing along B 3umodes, which were computed at the (X)MS(8:16) level as smaller active spaces were found to give nonphysically large inter- state couplings. Overlaps were computed with the perturbatively modified WFs, obtained either at the (X)MS(8:12) or (X)MS(8:16) level. Further details on the calculations of the overlaps are given in Sec. III of the supplementary material. All the QM compu- tations were performed with OpenMolcas,52,53applying Cholesky decomposition.B. QD calculations ML-MCTDH wavepacket propagation2–8was performed with the Quantics package.54,55The method is also implemented in the original MCTDH code distributed upon request by Meyer and co- workers at Heidelberg University. The seven lowest energy excited states and the 49 (out of 72) normal coordinates with the appro- priate symmetry to have non-vanishing couplings were included for all the LVC parametrized diabatic PESs. The dimension of the primitive basis set, the number of single particle functions, and the structure of the ML-MCTDH trees are shown in Sec. IV of the supplementary material for each type of calculation, together with some convergence tests (Fig. S9). We used a variable mean field (VMF) scheme with a fifth-order Runge–Kutta integrator of 10−7 accuracy threshold. The wavepackets were propagated for a total time of 2 ps. IV. RESULTS AND DISCUSSION A. Energy calculations The lowest seven excited states of pyrene belong to four irre- ducible representations (Table II). Among these states, we iden- tify two optically bright states—1B 2uwith dominant configuration H(OMO)→L(UMO) and 2B 3uwith dominant configurations H − 1→L + H→L + 1—as well as several dark states. Importantly, the lowest excited state is optically dark and, thus, responsible for TABLE II . Vertical excitation energies and transition dipole moment module (TDM) at the reference geometry for the first seven excited states of pyrene, obtained with the full- πactive space (8:8) and with the extended active spaces (8:12) and (8:16). States are labeled according to the irreducible representations of the D 2hpoint group. In the third column are reported the most relevant configuration state functions (CFSs) describing each state (see Fig. 1 for the representation of the involved orbitals). The last column reports the experimental adiabatic transition energies in the gas phase56,57for bright states or of two-photon absorption experiments in apolar solvent58for dark states. The (8:16) active space results are all reported relative to the SS(8:16) ground state value. TDM Energy (eV) Experimental State Label CSFs (Debye) SS(8:8) SS(8:12) SS(8:16) MS(8:16) XMS(8:16) ΔE0-0(eV) S0 1Ag GS . . . 0.00 0.00 0.00 . . . . . . . . . S1 1B3uH→L + 1 0.00 3.23 3.22 3.23 . . . . . . 3.3657 H-1→L S2 1B2u H→L 1.83 3.55 3.69 3.75 . . . . . . 3.8456 S3 1B1gH→L + 2 0.00 4.11 4.13 4.16 4.00 4.10 4.1258 S4 2Ag(H→L)20.00 4.30 4.35 4.32 . . . . . . 4.2958 S5 2B3uH→L + 1 1.73 4.18 4.35 4.43 . . . . . . 4.6656 H-1→L S6 2B1g H-2→L 0.00 4.28 4.46 4.56 4.64 4.48 4.5458 H→L + 2 S7 3B1g H-3→L 0.00 4.73 4.77 4.82 4.89 4.85 4.9458 H→L + 3 J. Chem. Phys. 154, 104106 (2021); doi: 10.1063/5.0044693 154, 104106-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE III . Comparison between SS-RASPT2/RASSCF(4, 8 ∥0, 0∥4, 8)/ANO-L-VDZP optimized minima (OPT) and LVC model minima for the adiabatic excited states of pyrene: reorganization energy λfor each structure and RMSD between the two Cartesian structures for each state. The reorganization energies were obtained as the difference in energy between the reference geometry and the corresponding minimum at the SS(8:16) level (OPT) or by projecting the SS(8:16) gradient onto the normal modes (LVC, see the supplementary material). S1 S2 S3 S4 S5 S6 OPT39LVC OPT39LVC OPT LVC OPT LVC OPT39LVC OPT LVC λ(eV) 0.08 0.09 0.10 0.10 0.19 0.16 0.26 0.18 0.05 0.06 0.22 0.24 RMSD 0.005 0.005 0.004 0.012 0.005 0.010 the characteristic fluorescence of pyrene of hundreds of nanosec- onds.59,60We note the presence of a doubly excited state of A gsym- metry in the vicinity of the second bright state evidencing the need of multiconfigurational methods. The vertical excitation energies at the reference geometry, obtained at different levels of theory, are reported in Table II. The full-π(8:8) active space shows both quantitative and qualitative dif- ferences with respect to the stronger correlated (8:12) and (8:16) active spaces. Indeed, while the energies of states such as 2A g, 1B 1g, and 1B 3uare already converged with respect to the active space size, the remaining states (in particular, both bright states 1B 2uand 2B 3u) exhibit strong dependence on the active space size, being red-shifted by 0.2 eV–0.3 eV at the SS(8:8) level with respect to SS(8:16). As a consequence of the unbalanced description, the energy order of the states changes as a function of the active space (Table II) with profound consequences for the QD simulations. The trend in the(8:8)-(8:12)-(8:16) sequence evidences that energies are not fully converged even with the largest active space, but they show an asymptotic behavior. Accordingly, comparison with the experimen- tal gas-phase data56–58shows that the computed transition energies of the bright states are underestimated. The SS(8:16) set provides closest agreement, thus implicitly supporting the predicted state order. Concerning the type of perturbative correction, the SS- variation of the RASPT2 method is the best approximation with D 2h symmetry where states of the same irreducible representation are far apart in energy and do not mix. Only in the case of the B 1girre- ducible representation, (X)MS-RASPT2 energies were considered due to the proximity of the electronic states. Indeed, the three meth- ods predict energies that deviate by up to 0.16 eV. XMS-RASPT2, whose use is advocated for near-degenerate and strongly interact- ing electronic states,61is found to deviate only marginally from the TABLE IV . Vertical excitations at the reference geometry: deviation from the reference D 2h-SS(8:16) values (reported in the first row) at different levels of theory. Positive and negative deviations larger than 0.10 in the absolute value are highlighted in bold and italic, respectively. For each symmetry, states of the same irreducible representation fall into the same RASPT2/RASSCF calculation. C 2v(1) and C 2v(2) refer to the reduced symmetry along modes B 3uand B 2u, respectively. Deviation from reference energy (eV) Absolute mean Standard Symmetry Level of theory S 1 S2 S3 S4 S5 S6 S7 deviation (eV) deviation (eV) D2h SS(8:16) 3.23 3.75 4.16 4.32 4.43 4.56 4.82 . . . . . . Irreducible rep. A 1 B1 B1 A1 A1 B1 B1 C2v(1) MS(8:12) 0.04 −0.05 −0.19 0.06 −0.06 0.08 0.09 0.082 0.12 MS(8:16) −0.01 −0.02 −0.17 0.02 −0.03 0.10 0.08 0.061 0.10 XMS(8:12) 0.05 −0.11 −0.04 0.05 −0.05 −0.04 0.11 0.064 0.09 XMS(8:16) 0.00 −0.08 −0.03 0.03 0.00 −0.02 0.09 0.036 0.06 Irreducible rep. B 2 A1 B2 A1 B2 B2 B2 C2v(2) MS(8:12) 0.02 0.02 −0.23 0.04 −0.05 0.10 0.10 0.080 0.13 MS(8:16) −0.01 0.02 −0.20 0.01 −0.02 0.10 0.08 0.063 0.11 XMS(8:12) 0.01 0.00 −0.08 0.12 0.06 −0.06 0.09 0.060 0.09 XMS(8:16) −0.02 0.01 −0.06 0.09 −0.03 −0.06 0.06 0.047 0.07 Irreducible rep. B u Bu Ag Ag Bu Ag Ag C2h MS(8:12) 0.06 −0.06 −0.22 0.03 −0.02 0.11 0.10 0.086 0.14 MS(8:16) 0.02 0.02 −0.19 −0.03 −0.01 0.11 0.07 0.064 0.11 XMS(8:12) 0.07 −0.10 −0.06 0.02 0.00 −0.04 0.09 0.055 0.08 XMS(8:16) 0.03 −0.03 −0.05 0.06 0.00 −0.06 0.04 0.039 0.06 J. Chem. Phys. 154, 104106 (2021); doi: 10.1063/5.0044693 154, 104106-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp SS-RASPT2 results. Eventually, considering the computational cost and the small error, SS(8:16) was used to calculate the energies along symmetry-conserving normal modes. At the S 0equilibrium geometry, all the excited states show a gradient only along the totally symmetric A gmodes. With the numerical gradients at hand, within the displaced harmonic oscil- lator approximation, we can predict the structures of the minima of the adiabatic states and the reorganization energies λ(details in the supplementary material). Interestingly, we obtain small reor- ganization energies (up to ∼0.3 eV, Table III), which reflects the rigidity of the pyrene molecule and justifies the harmonic approx- imation underlying the LVC model. The predicted structures and reorganization energies are in a very good agreement with the results from explicit optimizations at the SS-RASPT2/RASSCF(4, 8 ∥0, 0∥4,8)/ANO-L-VDZP level39[i.e., SS(8:8), Table III].62Taking into con- sideration the reorganization energies resolves the apparent dis- agreement between experiment and theory regarding the energetic order of 2B 3uand 2B 1g(Table II). Two-photon absorption exper- iments put the 2B 1g(4.54 eV) below the second bright state 2B 3u (4.66 eV) at the respective excited minimum. When the reorgani- zation energies—predicted as ∼0.05 eV for 2B 3uand 0.23 eV for 2B1g(Table III)—are considered, the state order is inverted in the Franck–Condon (FC) point. B. Wavefunction overlap calculations Vibronic coupling between the considered diabatic states is observed both along totally symmetric A gmodes and along the FIG. 2 . Vertical excitation energies at the reference geometry calculated with the reduced symmetries of the B 3umodes (top right), B 2umodes (bottom left), and B 1gmodes (bottom right). In the top left panel are reported the reference D 2h-SS(8:16) energies. Full circles = S 0and bright states; empty circles = dark states. Vertical dotted lines connect states of the same irreducible representation for each point group and level of theory. The horizontal full lines set the reference D 2h-SS(8:16) energies. J. Chem. Phys. 154, 104106 (2021); doi: 10.1063/5.0044693 154, 104106-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp symmetry-decreasing modes belonging to the B 3u, B2u, and B 1girre- ducible representations (Table I). As noted earlier, in D 2hsymmetry, electronic states of the same irreducible representation are energet- ically well separated, which results in a weak interaction (coupling). On the other hand, displacement along symmetry-lowering modes allows also for interactions that were forbidden in D 2hsymmetry: this is particularly evident in the case of the first bright state S 2, which is the only B 2ustate in D 2hsymmetry and otherwise would never be depopulated. Symmetry-lowering results in variable group- ing of the states in irreducible representations of lower point groups. This requires a different state averaging along each of the three symmetry-decreasing sets of normal modes, which affects both the RASSCF and RASPT2 results, in particular, in the case of XMS- RASPT2, which relies on an average Fock operator. Moreover, the presence of close lying states requires the use of (X)MS-RASPT2 corrections. Because of this, the level of theory of the WF overlap calculations must be accurately selected for each irreducible repre- sentation of each point group so as to balance between computa- tional cost and accuracy of the description. To assess the reliability of the reduced symmetry calculations in reproducing the electronic structure with the same precision as the D 2hcalculations, the elec- tronic structure at the reference geometry was computed with each of the lower symmetries. Table IV and Fig. 2 show the deviation of the adiabatic energies at the (X)MS(8:12) and (X)MS(8:16) lev- els from the reference D 2h-SS(8:16) values when the symmetry is reduced. The agreement with the reference values is generally good, with XMS-RASPT2 being more accurate than MS-RASPT2, which tends to overestimate the energy splitting and WF mixing in the case of strongly interacting states. Comparing the two active spaces, it is evident how the energies are sensitive to the degree of electronic cor- relation, with the (8:16) results being more faithful to the reference energies than the (8:12) ones, both for MS- and XMS-RASPT2. Thus, it is obvious that the best choice would be to calculate all the WF overlaps (necessary for the LVC parameterization) with the larger active space, but this is computationally very demanding. To bal- ance between computational cost and accuracy of the description, we have computed the wavefunction overlaps at the (X)MS(8:12) level,except for critical situations (i.e., strongly interacting states), where we have used (X)MS(8:16), and that will now be discussed. For each group of symmetry-reducing modes, we can identify a pair of close lying states, which require particular attention, to make sure that the new state averaging scheme retains the relative state order and energy gaps as at the reference D 2hgeometry: S 4/S5along B3umodes [ ΔED2h SS(8:16)= 0.11 eV], S 5/S6along B 2umodes [ ΔED2h SS(8:16) = 0.13 eV], and S 3/S4along B 1gmodes [ ΔED2h SS(8:16)= 0.16 eV]. Table V shows the average, maximum, and minimum WF overlap (absolute value) for each critical couple of states. For S 6–S5(along B 2umodes) and S 4–S3(along B 1gmodes), the (8:12) energy splitting is always overestimated with respect to the reference one, and the WF over- laps are consequently small, with XMS-RASPT2 being more accu- rate than MS-RASPT2. Even though from the theoretical point of view the overestimation of the energy gap is conceptually as wrong as its underestimation, from the practical point of view, a larger energy gap (which results in a smaller diabatic coupling in the final Hamiltonian) is not as dramatic as a too small energy gap since artificially large diabatic couplings can make the QD calculations much more problematic. On the contrary, the case of S 4and S 5 states along B 3umodes (i.e., A 1representation, see Fig. 2) is more critical: (X)MS(8:12) reduces the energy gap until near-degeneracy of the two states, producing an unphysically high WF overlap (and diabatic coupling, see Fig. S3 in the supplementary material for the correlation between accuracy of the ΔE and wavefunction mixing). Table V shows that at the MS(8:12) level, they are perfectly degener- ate, resulting in an average WF overlap of about 0.40. On the other hand, increasing the active space, the energy gap increases, getting closer to the reference D 2h-SS(8:16) value, and the S 5–S4mixing is significantly reduced [0.012 at MS(8:16) and 0.006 at XMS(8:16) level, see Table V]. In conclusion, the (X)MS(8:12) WF overlaps represent a fair compromise between computational time and accuracy, except for the states of A 1representation along B 3umodes (C 2vsymmetry), for which the bigger active space is needed to avoid artificially high S5/S4overlaps. For comparison, we have produced three sets of data TABLE V . Energy gap and WF overlaps along symmetry reducing modes (average absolute value, minimum and maximum absolute values) between states S 5–S4(top), S 6–S5(middle), and S 4–S3(bottom) calculated with different symmetry and level of theory. Deviation from⟨Sref i∣Sdispl j⟩ Modes Symmetry Level of theory ΔE (eV) reference ΔE (eV) Average Min Max S5–S4 B3u C2v(1) MS(8:12) 0.00 −0.11 0.395 0.137 0.613 MS(8:16) 0.09 −0.02 0.012 0.001 0.044 XMS(8:12) 0.01 −0.10 0.080 0.001 0.262 XMS(8:16) 0.09 −0.02 0.006 8 ×10−50.020 S6–S5 B2u C2v(2) MS(8:12) 0.27 0.14 0.025 3 ×10−40.070 XMS(8:12) 0.13 0.00 0.029 0.001 0.112 S4–S3 B1g C2h MS(8:12) 0.41 0.25 0.030 0.005 0.090 XMS(8:12) 0.25 0.09 0.010 0.001 0.033 J. Chem. Phys. 154, 104106 (2021); doi: 10.1063/5.0044693 154, 104106-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp for the LVC parameterization: one in which all the overlaps were computed at the XMS(8:12) level and two sets in which the B 3u-A1 states were computed with the bigger active space [i.e., MS(8:16) or XMS(8:16)]. C. Accuracy of the LVC PES The three different parameterizations of the LVC Hamiltonian will be named from now on LVC MS(16), LVC XMS (12), and LVC XMS (16) depending on the highest level of theory employed for the com- putation of the WF overlaps [MS(8:16), XMS(8:12) or XMS(8:16), respectively]. Figure 3 compares scans of the LVC MS(16)diabatic PESs along A gcollective coordinates leading from the S 0minimum to the minima of the different LVC diabatic PESs (solid lines) with the energies of the corresponding adiabatic states recomputed at the D2h-SS(8:16) level (scattered points). The comparison shows that LVC PESs are remarkably accurate, especially for the lower energy states. Some inaccuracies arise for 3B 1gand 2B 3ualong the coor- dinate connecting the S 0and the 1B 1gminima (Fig. 3, middle left FIG. 3 . Scans of the LVC MS(16)diabatic potential energy surfaces (dashed lines) along collective A gcoordinates connecting the 1A gequilibrium geometry with the minima of the LVC diabatic states and corresponding adiabatic energies computed at the SS(8:16) level (hollow circles). Note that although the SS(8:16) states are adiabatic, they are distinguished by symmetry, which explains the observed cross- ings and justifies that for each symmetry, LVC adiabatic energies are very similar to LVC diabatic ones.panel). This is connected with the degeneracy, at distorted geome- tries, with a higher lying “intruder” state at the RASSCF level that is influencing the CASPT2 correction. We emphasize that upon (X)MS-CASPT2 correction, the “intruder” states blue-shift above 5 eV, which evidences that their involvement at the RASSCF level is merely an artifact of the unbalanced description of the electronic states when dynamic correlation is not considered. To have a closer look at the performance of the LVC model in the minima, we consider the LVC MS(16)parameterization and recomputed the SS(8:16) energies at all the diabatic minima located with the LVC model. Data in Table S8 of the supplementary mate- rial show that LVC and RASPT2 energies are extremely similar. The largest differences for a state in its own minimum are seen for 2A g and 2B 1gand are 0.04 eV. At each minimum, also the energies of the other states are quite similar with the partial exceptions of states 2B3uand 3B 1g, which, far from their own minimum, can show an interaction with higher lying states at the RASSCF level not included in the model, as mentioned previously. With the LVC model it is also possible to analytically determine the lowest energy crossing of pairs of diabatic states in D 2hsymme- try. Note that since in D 2hoff-diagonal couplings among states of the same symmetry are possible, diabatic and adiabatic LVC states do not coincide, and therefore, these crossings do not correspond, rigorously speaking, to CoIs between adiabatic states. However, we already showed that mixings between states of the same symmetry are minimal when the D 2hpoint group is applied. Table VI reports the LVC and SS(8:16) energies of all states at crossings with ener- gies lower than 4.5 eV (i.e., accessible from 2B 3u, whose vertical excitation energy is 4.43 eV). For crossing up to 4.5 eV, the agree- ment is remarkably good. RASPT2 confirms that these structures correspond to points of quasi-degeneracy, and in most of the cases, also the LVC absolute energy is correct up to few hundredths of eV. In particular, LVC correctly predicts that the 1B 1g/2B 3ucross- ing actually corresponds to a quasi-triple CoI involving also the 2A g state and reproduces the absolute energies up to 0.02 eV. A fur- ther quasi-triple CoI involving the 1B 3u, 1B 2u, and 1B 1gstates (pro- posed previously based on orbital analysis and CoI search39) is also confirmed. In this case, however, LVC overestimates the energy by ∼0.10 eV–0.15 eV. Considering diabatic crossings at higher energy (check Table S9 in the supplementary material), LVC predictions are still rather reliable, but, as expected, differences with respect to RASPT2 energies increase. Interestingly, LVC correctly predicts that at 1B 3u/2A gcrossing, four states are found in <0.17 eV (i.e., also 1B 2u and 1B 1g), suggesting that a quasi-fourfold CoI might exist in the proximity of that structure. D. Dynamics of electronic populations Figure 4 shows the time evolution of the electronic populations up to 2 ps after the initial photo-excitation to either the first (1B 2u) or the second (2B 3u) bright states according to the LVC MS(16)and LVC XMS (16)parameterizations (results with LVC XMS (12)are given in Figure S14 of the supplementary material). The insets report a close- up of the same data in the first 100 fs. LVC MS(16)and LVC XMS (16) Hamiltonians deliver similar predictions: 1B 2udecays essentially on the lowest state 1B 3u, while after an initial excitation to 2B 3u, we observe a fast ( <20 fs) rise of a transient population of some inter- mediate states, followed by a only slightly slower population of the J. Chem. Phys. 154, 104106 (2021); doi: 10.1063/5.0044693 154, 104106-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE VI . Diabatic [LVC MS(16)] and adiabatic [RASPT2, SS(8:8)] energies (eV) of pyrene at a number of crossing points between LVC diabatic states. Bold characters highlight states that are quasi-degenerate (data for higher energy crossings are in Table S9 of the supplementary material). States S1 S2 S3 S4 S5 S6 S7 CoI Methods 1B 3u 1B2u 1B1g 2Ag 2B3u 2B1g 3B1g 1B3u/1B 2u LVC 4.20 4.20 4.42 4.53 5.37 4.76 5.64 RASPT2 4.16 4.16 4.37 4.37 5.72 5.07 5.07 1B3u/1B 1g LVC 4.43 4.46 4.43 4.81 5.57 4.84 5.73 RASPT2 4.27 4.29 4.33 4.60 5.71 4.81 5.41 1B2u/1B 1g LVC 4.04 4.25 4.25 4.68 5.18 4.68 5.37 RASPT2 3.88 4.12 4.18 4.56 5.57 4.63 5.14 1B1g/2B 3u LVC 3.20 3.89 4.45 4.50 4.45 4.80 4.91 RASPT2 3.21 3.89 4.47 4.49 4.46 4.88 4.96 2Ag/2B 3u LVC 3.17 3.80 4.27 4.40 4.40 4.64 4.83 RASPT2 3.17 3.80 4.27 4.39 4.40 4.65 4.82 2B3u/2B 1g LVC 3.19 3.67 4.06 4.20 4.40 4.40 4.79 RASPT2 3.19 3.68 4.06 4.20 4.41 4.42 4.75 first bright state 1B 2u, which reaches its maximum population ( ∼0.5) in 100 fs and then slowly decays toward 1B 3u. The intermediate population of 1B 2uis consistent with the two-step interpretation of Borrego-Varillas et al. who reported transient signatures of 1B 2u when pumping the second bright state.36Moreover, the delayeddecay to the lowest excited state (on a 0.5 ps time scale) observed after excitation to 2B 3uagrees with experimental time constants reported in the literature.36,39,40 A closer analysis highlights some differences. For an excita- tion to 1B 2u, the decay to 1B 3uis faster according to LVC MS(16) FIG. 4 . Dynamics of the populations of the diabatic electronic states obtained by initially exciting the wavepacket on 1B2u(left) or 2B 3u(right) states for the LVC MS(16)[panels (a) and (b)] and LVCXMS (16)[panels (c) and (d)] param- eterizations. The insets highlight the dynamics in the first 100 fs. J. Chem. Phys. 154, 104106 (2021); doi: 10.1063/5.0044693 154, 104106-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE VII . Norm of the diabatic coupling vectors for MS(8:16) and XMS(8:16) parameterizations. Bold numbers highlight differences between the two parameterizations that have a remarkable impact on the population dynamics. MS(8:16) XMS(8:16) State 1B 3u 1B2u 1B1g 2Ag 2B3u 2B1g 3B1g 1B3u 1B2u 1B1g 2Ag 2B3u 2B1g 3B1g 1B3u 0.159 0.159 1B2u 0.043 0.184 0.030 0.184 1B1g 0.199 0.196 0.257 0.111 0.096 0.257 2Ag 0.108 0.116 0.049 0.257 0.105 0.043 0.058 0.256 2B3u 0.108 0.124 0.027 0.054 0.126 0.072 0.176 0.096 0.028 0.126 2B1g 0.087 0.096 0.037 0.235 0.152 0.266 0.056 0.126 0.059 0.109 0.146 0.267 3B1g 0.175 0.238 0.042 0.089 0.077 0.073 0.143 0.105 0.028 0.089 0.093 0.046 0.037 0.142 than according to LVC XMS (16). Thereby, the LVC MS(16)dynamics agrees better with experiments, uniformly assigning a sub-100 fs time constant to the S 2→S1IC. Analysis of the couplings (Table VII) suggests that this finding can partially arise from the larger cou- pling predicted by LVC MS(16)(norm: 0.043 eV) than by LVC XMS (16) (norm: 0.030 eV) and mainly due to the contribution of mode 60: 0.025 eV in LVC MS(16)and 0.010 eV in LVC XMS (16). However, further motivations will be highlighted below. For an excitation to 2B 3u, the initial decay ( ∼10 fs) is toward 2B1gand 2A gaccording to LVC MS(16)and toward 2B 1g, 1B 1g, and directly 1B 2uaccording to LVC XMS (16). These differences can be attributed to corresponding differences in the pattern of the cou- plings reported in Table VII. Indeed, the couplings of 2B 3uwith 1B1gand 1B 2uare remarkably larger according to LVC XMS (16). On the contrary, the coupling of 2B 3uwith 2A gis larger according to LVC MS(16). The latter also predicts a much larger coupling of the higher-energy state 2B 1gwith 2A gexplaining why, despite its energy, 2B 1ggains some transient population, which, according to LVC MS(16), reaches slightly larger values and decays at a slightly slower rate than in the case of LVC XMS (16). Analysis of Fig. 4 suggests that after photoexcitation to 1B 2u the dynamics is quite simple, being essentially characterized by a progressive (approximatively mono-exponential) flow of population from 1B 2uto the lowest-energy state 1B 3u. This is not surprising con- sidering that at the FC position, the third state, 1B 1gis∼0.5 eV higher in energy than 1B 2u. However, Table VII shows that 1B 1gis strongly coupled to both 1B 2uand 1B 3ustates. More specifically, the norm of its coupling to these two states is, respectively, more than three [LVC XMS (16)] and more than four [LVC MS(16)] times larger than the direct 1B 1g/1B 2ucoupling. A small transient population on 1B 1gis actually seen in Fig. 4 for the Hamiltonian with the larger couplings [LVC MS(16)]. In Fig. 5, we investigate in greater detail the impact on the 1B 2u→1B3utransfer of the existence of 1B 1gand the higher energy states. In order to do that, we compare the dynamics includ- ing all the seven coupled states (seven-states model) with a number of reduced models in which some states were removed: the two- state model “1B 2u+ 1B 3u,” the three-state model “1B 2u+ 1B 3u+ 1B1g,” and the six-state model obtained including all states except 1B1g. Differences are striking: according to the two-state model, the population transfer is much slower, smaller in amplitude, and shows large oscillations. Including also 1B 1g, the population transfer becomes much faster (even more than in the seven-states model) andirreversible, without any significant quantum beating. However, higher energy states also play a role. This is shown considering the six-state model in which 1B 1gis removed. In the six-state-model, the predicted population flow from 1B 2uto 1B 3uis, in fact, simi- lar to what is obtained with the complete seven-state model. Actu- ally, in the long-time limit, 1B 3ureaches even a higher population, although the transfer is slower in the first 500 fs (this is better shown by a zoomed-in view of the figure reported in Fig. S10 in the supplementary material). In summary, the existence of 1B 1ghas a dramatic impact on the 1B 2u→1B3utransfer, much larger than what one could hypothesize looking at the small transient popula- tion it acquires. Its main role, in fact, is to provide an alternative and very effective coupling channel between the two lowest states. On the short-time scale, the effect of 1B 1gis partially contrasted by the higher-energy states, which slow down the rise of the pop- ulation of 1B 3u. On the long-time scale, however, according to the seven-state model, 1B 1gmaintains a weak population ( ∼3%). If such a state is not included in the calculation, this small population flows FIG. 5 . Dynamics of the populations of the diabatic electronic states after an initial excitation on 1B 2u. Comparison of the results obtained with the complete seven- states model and with a number of reduced-dimensionality models in which some electronic states are removed from the LVC MS(16)Hamiltonian. J. Chem. Phys. 154, 104106 (2021); doi: 10.1063/5.0044693 154, 104106-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 6 . Diabatic LVC potential ener- gies at the average position of the wavepacket obtained for an initial pho- toexcitation to 1B 2u(left) or 2B 3u(right) with the LVC MS(16)Hamiltonian. A com- parison with the adiabatic energies, very similar, is shown in Fig. S11. to 1B 3u, making the yield of this state even larger (six-state model). 1B1gand higher-energy states play a qualitatively similar role also according to the XMS(8:16) parameterization, but couplings with 1B1gare smaller. In conclusion, the faster 1B 2u→1B3udecay pre- dicted by LVC MS(16)with respect to LVC XMS (16)is not only due to the larger direct coupling (as discussed above) but also, for a signif- icant part, due to the larger couplings of both states with 1B 1g(see Table VII). Figure 6 plots the diabatic LVC PES at the average position of the WP as a function of time according to the LVC MS(16)Hamilto- nian [results for LVC XMS (16)are very similar and are given in Fig. S12 of the supplementary material]. It shows that at all times, S 1and S 2 are well separated in energy and rather distant from two pairs of close-lying states, namely, S 3–S4, and S 5–S6. Interestingly, these data indicate that the average position of the WP does not encounter con- ical intersections. This finding, together with the smooth changes of the electronic populations, suggests that the picture that better describes the dynamics is not a ballistic movement of the WP toward a CoI. On the contrary, we observe a gradual transfer due to the fact that vibrational states of the upper electronic states are embedded in (and coupled to) a denser manifold of vibrational states of the lower-energy electronic states. Actually, the possible occurrence of fast population transfers in QD even in cases where CoIs are inac- cessible has been recently discussed in the literature.63While this mechanism could be anticipated for an initial excitation to 1B 2u, since the initial potential energy of the WP is 3.75 eV (Table II) and the lowest 1B 1g/1B 2ucrossing is at ∼4.2 eV (Table VI), it is note- worthy that the same picture applies also for an initial excitation to 2B3ualthough several crossings between diabatic states are reachable at this energy, including the (quasi) triple-crossings 1B 3u/1B 2u/1B 1g and 1B 1g/2A g/2B 3u. Finally, Fig. 7 reports the expectation values of all the total- symmetric modes as a function of time for the LVC MS(16)Hamil- tonian. The results for LVC XMS (16)are shown in Fig. S13 and are very similar. Both starting from 1B 2uand 2B 3u, the dynamics is dominated by the oscillations of four modes: two CC stretchings with frequencies 1456 cm−1(mode 52) and 1669 cm−1(mode 62) and two lower frequency modes corresponding to a breathing mode with frequency 593 cm−1(mode 17) and to an in-plane elongation along the long molecular axis with frequency of 406 cm−1(mode 8). These modes agree with Raman signatures of 1B 2uand 2B 3u,64,65 and their involvement is consistent with the analysis of excited statevibrational coherences resolved recently in transient absorption spectra with ultrahigh time-resolution (6 fs).39It is noteworthy that despite the involvement of multiple electronic states coupled dif- ferently to the A gvibrational modes, the dynamics of the average FIG. 7 . Time evolution of average position of the A gmodes for excitation on 1B 2u (top) or 2B 3u(bottom) excited states. Only the modes with largest displacement are labeled (LVC MS(16)Hamiltonian). J. Chem. Phys. 154, 104106 (2021); doi: 10.1063/5.0044693 154, 104106-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp position along individual modes shows only minor deviations in the first 500 fs aside from mode 62, which shows a characteristic shift and damping. We conclude this section mentioning that LVC XMS (12)predicts a very different dynamics (Fig. S14 in the supplementary material), characterized by the fact that both starting from 1B 2uand 2B 3u, the states 2B 3uand 2A gbehave similarly, with very similar populations at all times. Such a peculiar behavior can be explained with the very large coupling between these two states predicted at this level of theory (Table S10 in the supplementary material). V. CONCLUSIONS In this contribution, we have combined highly accurate, mul- ticonfigurational electronic structure methods such as RASPT2/ RASSCF with a maximum-overlap diabatization technique to parameterize a LVC Hamiltonian for QD. As a case study, we have applied our protocol to the fast QD of pyrene photoexcited to either the first or the second bright state. The rigidity of this molecule justifies the LVC approximation to describe the potential energy sur- faces. Yet, its electronic structure and the large number of modes make necessary the inclusion of many electronic states and the devel- opment of an effective diabatization protocol to build the vibronic Hamiltonian. From the point of view of the electronic structure the- ory, several characteristics of pyrene require the adoption of mul- ticonfigurational methods, such as the presence of a state with a high contribution from a double excitation (2A g) and the difficulty of many TD-DFT functionals in reproducing the relative order of the lowest-energy states.66–68The involvement in the dynamics of the 2A gstate makes also problematic the usage of methods such as ADC(2) or CC2 since although they have shown remarkable accu- racy for single excitations in organic molecules, they are not accu- rate for double-excited states.69,70The parameterization based on RASPT2/RASSCF makes also our LVC Hamiltonian suitable for the simulation, in the near future, of transient absorption spectra. To this end, in fact, the computation of transition dipoles with the pos- sible final states reached by the absorption of the probe is required, which have an increased probability to show a significant double- excited character.71To the best of our knowledge, this is the first reported example of LVC parameterization based on energies and WFs overlaps computed with RASPT2/RASSCF electronic structure calculations. Our results evidence that it is not a “black box” proce- dure. While, in principle, the RASTP2/RASSCF protocol is able to describe states with different nature on an equal footing, large active spaces, beyond the full- πset of orbitals, are needed to achieve this. Therefore, benchmarking is essential for assuring the convergence of the excited state energies with respect to the active space size.72The undertaking is nowadays possible even for relatively big systems, thanks to flexible approaches to the construction of the active space such as the generalized active space (GAS)SCF/GASPT2 approach73 or the generalized multi-configuration quasi-degenerate perturba- tion theory (GMCQDPT),74as well as modern day CI solvers such as the density matrix renormalization group (DMRG)75and the full configuration interaction quantum Monte Carlo (FCIQMC),76 to name a few, which allow to handle active spaces with many tens of orbitals. Another critical point to address is the various fla- vors of the perturbative correction each one with its strengths andweaknesses. Our result indicates that SS-RASPT2 should be used for the energy calculations whenever the electronic states are far apart in energy. On the other hand, MS- and XMS-RASPT2 ener- gies and WFs more reliable in the case of close-lying, interacting states. In particular, perturbatively modified WFs should be used in the maximum-overlap diabatization procedure. Finally, when sym- metry can be applied to reduce the computational cost, attention is advised in regard to biases introduced by the RASSCF/RASPT2 protocol in the calculation of coupling parameters along symmetry- reducing normal modes. Exemplarily, the unbalanced description of two close lying states (i.e., 2A g, already well described with the full- πactive space, and the 2B 3uthat shows a strong dependence on the active space size) could result in non-physically large vibronic cou- plings as demonstrated by LVC parameterization at the XMS(8:12) level. In spite of these complications, benchmarking of diabatic PESs obtained with our model [parameterized at adequate levels such as MS(8:16) and XMS(8:16)] against RASPT2 calculations proves that the LVC Hamiltonian can be highly accurate, being also able to predict the structure and energy of both excited state minima and crossings between the states included in the model. LVC XMS (16)and LVC MS(16)dynamics are qualitatively similar. Still, both for an ini- tial excitation to 1B 2uand to 2B 3u, LVC MS(16)predicts that the decay from 1B 2uto 1B 3uis remarkably faster. These differences point out that, at the state of the art, even quite sophisticated electronic struc- ture methods cannot guarantee the computation of precise decay times. On the one side, this result witnesses the necessity to use accurate methods even for the parameterization of simple vibronic Hamiltonians such as LVC. On the other side, it documents the necessity of further efforts in the development of electronic structure methods for excited states of medium size molecules. The QD simulations indicate that after an initial photoexcita- tion to 1B 2u(S2), the population progressively flows to 1B 3u(S1). In particular, the population growth with a sub-100 fs time constant predicted by the LVC MS(16)Hamiltonian agrees very well with exper- imental observations.34,36Quite interestingly, this transfer is strongly affected by the existence of higher-energy states, especially 1B 1g, even if it lies ∼0.5 eV above the bright state in the Franck–Condon region. This finding highlights that in order to obtain robust QD results, it is necessary to adopt LVC models including a sufficiently large number of diabatic states. Direct excitation of the second bright state 2B 3u(S5) leads to its ultrafast (sub-100 fs) depopulation in favor of a number of intermediate states, especially 1B 2u, followed by a much slower progressively decay to 1B 3u(S1), supporting the mech- anism proposed based on recent experimental findings.36Rather surprisingly, in both QD simulations, population transfers occur smoothly and in an ultrafast manner even if the average position of the WP never get really close to crossing points of the diabatic (and adiabatic) states. In particular, the 1B 2u(S2)→1B3u(S1) trans- fer was found to occur on a sub-100 fs time-scale even if the CoI lies ∼0.4 eV above the FC point. This observation can be rationalized by coupling between vibrational levels, rather than ballistic motion toward a CoI. In the light of this finding, the question arises whether semi-classical trajectory-based approaches, which treat nuclei clas- sically, are capable of capturing the ultrafast nature of the internal conversion. Finally, it is noteworthy that the protocol for the parametriza- tion of LVC Hamiltonians from RASPT2/RASSCF is fully general J. Chem. Phys. 154, 104106 (2021); doi: 10.1063/5.0044693 154, 104106-13 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp and ready to be applied to other interesting problems, like the ultrafast internal conversion in photoexcited nucleobases.19Further- more, the protocol is straight-forwardly extendable to incorporate spin–orbit couplings to describe inter-system crossing.77 SUPPLEMENTARY MATERIAL See the supplementary material for pyrene normal modes and frequencies, adiabatic excited state minima with the LVC displaced harmonic oscillator model, adiabatic overlap matrices, ML-MCTDH trees and convergence tests, population dynamics of models with reduced number of electronic states, diabatic and adiabatic ener- gies for the diabatic states minima and conical intersections esti- mated by LVC, diabatic and adiabatic potential energy surfaces at the average position of the wavepacket for the LVC MS(16)and LVC XMS (16)Hamiltonians, average position of the A gmodes during wavepacket propagations for the LVC XMS (16)Hamiltonian, compari- son of the population dynamics with the LVC MS(16), LVC XMS (12), and LVC XMS (16)Hamiltonians, and norm of the diabatic coupling vectors for LVC XMS (12)parameterization. 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5.0045712.pdf

J. Chem. Phys. 154, 124303 (2021); https://doi.org/10.1063/5.0045712 154, 124303 © 2021 Author(s).Quantum reactive scattering calculations for the cold and ultracold Li + LiNa → Li2 + Na reaction Cite as: J. Chem. Phys. 154, 124303 (2021); https://doi.org/10.1063/5.0045712 Submitted: 28 January 2021 . Accepted: 02 March 2021 . Published Online: 22 March 2021 Brian K. Kendrick COLLECTIONS Paper published as part of the special topic on Quantum Dynamics with ab Initio Potentials ARTICLES YOU MAY BE INTERESTED IN Signature of shape resonances on the differential cross sections of the S(1D)+H 2 reaction The Journal of Chemical Physics 154, 124304 (2021); https://doi.org/10.1063/5.0042967 How important are the residual nonadiabatic couplings for an accurate simulation of nonadiabatic quantum dynamics in a quasidiabatic representation? The Journal of Chemical Physics 154, 124119 (2021); https://doi.org/10.1063/5.0046067 Direct time delay computation applied to the O + O 2 exchange reaction at low energy: Lifetime spectrum of species The Journal of Chemical Physics 154, 104303 (2021); https://doi.org/10.1063/5.0040717The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Quantum reactive scattering calculations for the cold and ultracold Li + LiNa →Li2+ Na reaction Cite as: J. Chem. Phys. 154, 124303 (2021); doi: 10.1063/5.0045712 Submitted: 28 January 2021 •Accepted: 2 March 2021 • Published Online: 22 March 2021 Brian K. Kendricka) AFFILIATIONS Theoretical Division (T-1, MS B221), Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA Note: This paper is part of the JCP Special Topic on Quantum Dynamics with Ab Initio Potentials. a)Author to whom correspondence should be addressed: bkendric@lanl.gov ABSTRACT A first-principles based quantum dynamics study of the Li + LiNa( v= 0, j= 0)→Li2(v′,j′) + Na reaction is reported for collision energies spanning the ultracold (1 nK) to cold (1 K) regimes. A full-dimensional ab initio potential energy surface for the ground electronic state of Li 2Na is utilized that includes an accurate treatment of the long-range interactions. The Li + LiNa reaction is barrierless and exoergic and exhibits a deep attractive potential well that supports complex formation. Thus, significant reactivity occurs even for collision tempera- tures approaching absolute zero. The reactive scattering calculations are based on a numerically exact time-independent quantum dynamics methodology in hyperspherical coordinates. Total and rotationally resolved rate coefficients are reported at 56 collision energies and include all contributing partial waves. Several shape resonances are observed in many of the rotationally resolved rate coefficients and a small res- onance feature is also reported in the total rate coefficient near 50 mK. Of particular interest, the angular distributions or differential cross sections are reported as a function of both the collision energy and scattering angle. Unique quantum fingerprints (bumps, channels, and ripples) are observed in the angular distributions for each product rotational state due to quantum interference and shape resonance contri- butions. The Li + LiNa reaction is under active experimental investigation so that these intriguing features could be verified experimentally when sufficient product state resolution becomes feasible for collision energies below 1 K. Published under license by AIP Publishing. https://doi.org/10.1063/5.0045712 .,s I. INTRODUCTION The experimental techniques for cooling and trapping molecules at cold ( T<1 K) and ultracold ( T<1 mK) tem- peratures continue to advance at a remarkable pace.1–16The first experimental demonstration of a controlled ultracold chemical reac- tion was reported for the benchmark KRb + KRb system several years ago.3Recent experimental techniques now routinely prepare cold molecules in a specific initial rovibrational state and can now also probe and detect specific intermediate complexes and prod- uct states.14,15Product angular distributions of cold non-reactive (inelastic) collisions have also been recently measured for the HD + H 2and HD + He systems.5,12,13Reactive systems involving het- eronuclear alkali metal dimers such as KRb, NaK, NaRb, and LiNa are under active experimental investigation, and we antici- pate that the measurement of fully state-resolved initial and product state rate coefficients and angular cross sections will soon become feasible.3,6,8,9,14,15Thus, accurate first-principles based theoreticalstate-to-state quantum mechanical calculations of these systems are needed. For these systems, the reaction path is barrierless and exoer- gic so that significant reactivity can occur even for collision energies approaching absolute zero. At these extremely low temperatures, quantum mechanical (e.g., interference) effects dominate and the tiny relative kinetic energy allows for the possibility to steer and control these reactions via the application of external fields,17,18 the preparation of a particular initial quantum state,3,4,8,9,16or the choice of the relative orientation of the colliding partners (i.e., stereodynamics).13,19–22 Accurate ultracold quantum dynamics calculations are very challenging for a number of reasons: (1) despite the small relative collision energy, the density of states in the interaction region can be quite large due to the relatively heavy masses of the nuclei and the deep attractive potential wells, (2) the tiny collision energies and long-range van der Waals interactions require a much larger computational grid, (3) an accurate and stable numerical method capable of treating a large dynamic range is required, and (4) an J. Chem. Phys. 154, 124303 (2021); doi: 10.1063/5.0045712 154, 124303-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp accurate full-dimensional ab initio potential energy surface (PES) is required that includes a smooth and accurate treatment of the long-range region as well as accurate asymptotic diatomic poten- tials. As the collision energy approaches the ultracold regime, by definition, only a single partial wave contributes to the scattering process. This property reduces the size of the calculations signifi- cantly. However, at the higher cold collision energies approaching 1 K, several partial waves contribute and these must also be included in the theoretical calculations. For these reasons, the majority of first- principles based quantum mechanical calculations have been limited to a single partial wave, which is sufficient for the ultracold regime: these include O + OH,23,24H + H 2(also its isotopic variants),25–32Li + LiYb,33K + KRb,34,35and Li + LiNa.36For the H + H 2and O + OH reaction systems, the calculations have been extended into the cold energy regime by including several partial waves. However, until now, the treatment of the heavier alkali metal (non-hydrogen) systems has been limited to the ultracold regime and a single partial wave. In this work, we report accurate first-principles based quan- tum reactive scattering calculations for the Li + LiNa( v= 0, j= 0) →Li2(v′,j′) + Na reaction. The calculations are based on a time- independent coupled-channel (CC) formulation using hyperspher- ical coordinates. A recently developed state-of-the-art ab initio PES for Li 2Na is used, which includes an accurate treatment of the long- range interactions and diatomic potentials.36This PES was recently used in a quantum mechanical calculation for the ultracold Li + LiNa reaction but was limited to one partial wave (i.e., l= 0).36In the present study, the collision energies span the entire range from Ec= 1 nK to 1 K and include all the required partial waves (i.e., up to l= 10 atT= 1 K). Both the total and rotationally resolved rate coefficients are reported as a function of the collision energy. Particular focus of the present work is on the angular distributions or differential cross sections (DCSs), which are also reported as a function of the collision energy and scattering angle. Intriguing quantum interference effects are seen in the DCSs as a function of both energy and scattering angle that give rise to prominent features: channels, sharp dips, and curved ridges. Furthermore, some of these features appear at small collision energies T≈1 mK where only two or three partial waves contribute. These features or “quantum fingerprints” are unique to each Li 2(v′,j′) product state. As recently shown in a series of theoret- ical studies,23–29the unique properties of ultracold and cold chemical reactions can lead to a dramatic enhancement or suppression of the reactivity due to quantum interference effects. These quantum inter- ference effects could be exploited by experimentalists to control the reaction outcome. We show here for the first time how these quan- tum interference effects manifest in the angular distributions of a reactive alkali system under active experimental investigation. Section II discusses the computational methodology, numeri- cal parameters, and the ab initio PES used to perform the calcula- tions. The quantum reactive scattering results are then presented in Sec. III, which include the total and rotationally resolved rate coef- ficients and angular distributions. Section IV concludes the paper with summarizing discussion and future outlook. II. METHODOLOGY The quantum reactive scattering calculations are performed using a time-independent coupled-channel approach inhyperspherical coordinates and are reviewed in Sec. II A. The methodology makes no dynamical approximations and is numeri- cally exact for a given PES. A first-principles based ab initio Born– Oppenheimer PES for the ground electronic state of Li 2Na is used, which includes an accurate treatment of the long-range interac- tions and diatomic potentials. This PES is discussed in Sec. II B, and the numerical convergence and various parameters used in the calculations are discussed in Sec. II C. A. Quantum reactive scattering in hyperspherical coordinates In the theoretical description of molecules, the conventional Born–Oppenheimer approximation is typically used, where the Schrödinger equation for the nuclear motion is expressed in terms of a ground state electronic PES, which is a function of the internu- clear distances. In the present work, we consider triatomic molecules for which there are three nuclei and three internuclear distances. The three-dimensional electronic PES is assumed to be calculated separately for each fixed internuclear distance of interest and is dis- cussed in detail in Sec. II B. The Born–Oppenheimer equation for the nuclear motion (relative to the center of mass of the triatomic system) is given by [−̵h2 2μ∇2+V(x)]ψ(x)=Eψ(x), (1) where the gradient operator ∇spans the six nuclear coordinates denoted by x=(x,ˆx)and x and ˆxdenote the three internuclear dis- tances and three Euler angles, respectively. The three Euler angles specify the orientation of the molecular body frame relative to the space-fixed frame and are typically denoted by ˆx=α,β,γ. The three- body reduced mass for the nuclei is μ, and Especifies the total energy of the molecule. The corresponding Born–Oppenheimer total molecular wave function is given by Ψtot=ψ(x)ϕ(r; x)ψN, (2) whereψ(x) is the nuclear motion wave function of Eq. (1), ϕ(r) is the electronic ground state wave function, and ψNis the total nuclear spin wave function for the three nuclei. The electronic wave function and PES ( V) are assumed to be previously computed and satisfy the electronic Schrödinger equation, h(r; x)ϕ(r; x)=V(x)ϕ(r; x), (3) where hdenotes the electronic Hamiltonian operator, which is a function of all the electronic coordinates (collectively denoted by r) and the three internuclear distances x. In the present work, we seek numerically exact quantum mechanical solutions of Eq. (1) for a given total energy Eand ground electronic state PES. We choose to work in hyperspherical coordi- nates that have several advantages: (1) a simultaneous and demo- cratic treatment of all three arrangement channels, (2) a straightfor- ward treatment of the identical nuclei exchange symmetry, and (3) a single radial coordinate that facilitates a numerical separation of the problem. In addition, two variants of hyperspherical coordinates are utilized: (a) the adiabatically adjusting principal axis hyperspherical (APH) coordinates of Pack and Parker37and (b) the Delves channel- centered hyperspherical coordinates.38The APH coordinates are J. Chem. Phys. 154, 124303 (2021); doi: 10.1063/5.0045712 154, 124303-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp optimal in the interaction region where all three arrangement chan- nels overlap and are mixed, whereas the Delves hyperspherical coor- dinates are optimal outside the interaction region where the three arrangement channels are well separated and uncoupled. Both sets of coordinates share the same radial coordinate, which makes the transformation from the APH to Delves hyperspherical coordinates straightforward to implement.37 The APH coordinates are denoted by ρ,˜θ, andϕ, whereρ is the radial coordinate and corresponds to a symmetric stretch motion, ˜θis the polar coordinate (related to the θof Pack and Parker via˜θ=π−2θ) and corresponds to a bending motion, and ϕis the azimuthal coordinate and corresponds to an internal pseudo- rotational motion. These three APH coordinates can be expressed in terms of the three internuclear distances.37Collectively, these three plus the three Euler angles specify the six internal degrees of free- dom of a triatomic molecule. For more detailed descriptions of these coordinates, see Refs. 37 and 39. Expressing Eq. (1) in terms of the APH coordinates in the interaction region gives [−̵h2 2μρ5∂ ∂ρρ5∂ ∂ρ+ˆΛ2 2μρ2+V(ρ,˜θ,ϕ)]ψ=Eψ, (4) where the operator ˆΛis the grand angular momentum operator. The explicit expression for this operator is given elsewhere and will not be duplicated here.40It includes the J2 x,J2 y,J2 z, and the Coriolis cou- pling term Jz∂ ∂ϕ. The PES strongly couples the three hyperspherical coordinates. Nevertheless, we can perform a numerical separation of Eq. (4) by first solving the five-dimensional (5D) angular problem on a discretized grid of fixed ρvalues. That is, we split Eq. (4) into two parts. First, we solve the 5D angular equation given by ⎡⎢⎢⎢⎣ˆΛ2 2μρ2 ξ+15̵h2 8μρ2 ξ+V(ρξ,˜θ,ϕ)⎤⎥⎥⎥⎦ΦJMpq t(w;ρξ)=εJpq t(ρξ)ΦJMpq t(w;ρξ), (5) whereρξdenotes the fixed (discretized) value of ρand the ΦJMpq t are the 5D hyperspherical (angular) functions. The solutions are labeled by the integer t,pdenotes inversion parity, qdenotes the particle exchange symmetry that is relevant for triatomic molecules with identical nuclei, and w≡(˜θ,ϕ,α,β,γ). In practice, we expand the angular solutions in terms of a primitive basis of Jacobi polynomials in˜θand complex exponential functions in ϕ.40A hybrid approached is used based on a Discrete Variable Representation (DVR) in ˜θand a Finite Basis Representation (FBR) in ϕ. The hybrid approach has several advantages, as discussed in detail in Ref. 40: (1) it allows for the truncation of the basis set via the Sequential Diagonalization Truncation (SDT) method, (2) it accurately treats the Eckart sin- gularities associated with body-frame coordinates, and (3) it gives a real-symmetric Hamiltonian matrix even for non-zero total angu- lar momentum J. Once the Hamiltonian matrix is constructed in the hybrid basis, it is diagonalized in parallel using the parallel version of ARPACK.41The lowest Nchsolutions are computed and used as a basis for solving the one-dimensional (1D) radial equation in ρ. That is, we expand the wave function ψin Eq. (4) in terms of the angular solutions of Eq. (5) at each fixed value of ρ=ρξ, ψJMpq i(x)=4√ 2Nch ∑ tρ−5/2ζJpq it(ρ)ΦJMpq t(w;ρξ), (6)where iand tdenote the coupled-channel indices ( iand t= 1, 2, 3, ...,Nch). The radial coefficients ζJpq itare computed numerically from the propagation of the resulting set of coupled-channel (CC) radial equations, [∂2 ∂ρ2+2μ ̵h2E]ζJpq it(ρ)=2μ ̵hNch ∑ t′⟨ΦJMpq t∣Hc∣ΦJMpq t′⟩ζJpq it′(ρ), (7) where the potential coupling matrix is defined as ⟨ΦJMpq t∣Hc∣ΦJMpq t′⟩=⟨ΦJMpq t∣ρ2 ξ ρ2εJpq t(ρξ)+˜V(ρ,˜θ,ϕ) −ρ2 ξ ρ2˜V(ρξ,˜θ,ϕ)∣ΦJMpq t′⟩. (8) The coupled-channel equations (7) and (8) are solved using John- son’s log-derivative method42,43and the optimal number of coupled channels ( Nch) is determined from convergence studies. This value and other parameters used in the calculations [such as the size of the primitive basis sets used to solve the angular equation (5)] will be discussed in Sec. II C. The Delves hyperspherical equations are similar to the APH equations (4)–(8) and are explicitly given in Ref. 38 and will not be duplicated here. By construction, in the Delves region, the three arrangement channels are well separated and decoupled so that the solutions for each channel can be computed separately. Further- more, the angular or rotational part of the Delves hyperspherical functions is analytic and can be expressed in terms of the well- known spherical harmonics. Thus, only a 1D numerical solution of the vibrational degree of freedom is required. An efficient Numerov method44is used to compute the vibrational wave functions for each value of j(the diatomic rotational quantum number) and l(the orbital angular momentum quantum number). The rovibrational solutions are then used as the basis for solving the Delves version of the coupled channel equations (7) and (8). To summarize, the quantum reactive scattering calculations are performed in a series of steps: (1) the APH surface (angular) func- tions are computed on a discretized grid from ρ=ρitoρ=ρmatch , (2) the Delves rovibrational functions are computed for each chan- nel on a discretized grid from ρ=ρmatch toρ=ρf, (3) the over- lap matrix between the APH and Delves functions is computed at ρ=ρmatch , (4) the APH log-derivative matrix is propagated from ρ= ρitoρ=ρmatch , (5) the overlap matrix from step (3) is used to trans- form the APH log-derivative matrix into the Delves basis, (6) the Delves log-derivative matrix is then propagated from ρ=ρmatch toρ= ρf, (7) the reactance Kmatrix is computed at ρ=ρfusing the Delves log-derivative matrix, Delves basis, and asymptotic Jacobi basis, and, finally, (8) the full state-to-state Smatrix is computed from the K matrix. The integral cross sections, rate coefficients, and DCSs can then be computed from the Smatrix using standard expressions.37 The APH and Delves solutions in steps (1) and (2) are indepen- dent of the collision energy and only need to be computed once for each value of total angular momentum J, inversion parity p=±, and exchange symmetry q= even/odd. The log-derivative propagation (steps 4–6) must be repeated for each collision energy and used in steps 7 and 8 to compute the scattering matrix as a function of col- lision energy. Many of the calculations (i.e., for each J,p, and q) are J. Chem. Phys. 154, 124303 (2021); doi: 10.1063/5.0045712 154, 124303-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp independent of each other and can be performed simultaneously if sufficient computational resources are available. B.Ab initio electronic potential energy surface for Li 2Na A state-of-the-art ab initio based electronic PES for the Li 2Na molecule was recently reported in Ref. 36. These first-principles based theoretical calculations include the treatment of both the ground and first excited electronic doublet states (2A′and2B′) of the Li 2Na molecule. We give only a brief summary of the ab initio calculations here (see Ref. 36 for a more detailed description). The ab initio calculations were performed using the MOLPRO45quan- tum chemistry package at over 40 000 internuclear geometries. Core potentials were used for the filled electron shells and include polar- ization and core–valence interactions. The valence electrons were treated using a multi-configurational self-consistent field (MCSCF) method to compute an initial set of configuration state functions (CSFs). The CSFs were then used with a large active space to per- form multi-reference configuration interaction (MRCI) calculations for the two lowest adiabatic PESs of Li 2Na. The ab initio points were fit using a reproducing kernel Hilbert space technique. An accurate treatment of the long-range interactions was also included through the introduction of a three-body dispersion term and a smooth switching function that connects the short range three-body poten- tial with the long-range three-body dispersion term. The dynami- cal effects of using a non-additive three-body long-range potential are discussed in Ref. 33. In addition, the MOLPRO computed ab initio potential curves for Li 2and LiNa were replaced with spectro- scopically accurate diatomic potentials. The non-adiabatic derivative coupling matrix was also computed and used to obtain a two-state diabatic potential matrix. In the present work, we restrict our quan- tum dynamics treatment to the single adiabatic ground electronic 2A′state of Li 2Na. Figure 1 plots the adiabatic ground electronic2A′state PES of Li2Na at several fixed values of the hyperradius ρ. All energy con- tours are relative to the bottom of the asymptotic potential of the Li2+ Na product and the dashed contour line at 2.23 ×103K denotes the total energy of the reactant LiNa( v= 0,j= 0). The PES is symmet- ric across the horizontal axis ( ϕ= 0) due to the symmetry associated with the two identical Li nuclei. Thus, wave functions of even and odd exchange symmetry correspond to even or odd symmetry with respect toϕ→−ϕ. Asymptotically, these wave functions correlate with the even or odd rotational levels of Li 2. Starting with large ρ= 25.0 a 0[panel (d)], we see that the three arrangement channels are isolated and well separated. As ρdecreases to ρ= 16.0 a 0[panel (c)], the LiNa and Li 2arrangement channel regions have spread out, and we see that the barrier between them is smallest near the equa- tor, which corresponds to collinear geometries. Thus, the minimum energy reaction path lies along a collinear approach of Li with LiNa. For smaller ρ= 12.0 a 0[panel (b)], the arrangement channels have merged and the region surrounded by the dashed contour encom- passes all three arrangement channels. In the interaction region ρ= 8.5 a 0[panel (a)], the three arrangement channels are now entirely mixed. The region lying within the dashed contour spans a large area surrounding the north pole of the hypersphere and includes two symmetric deep potential wells. The red dot in panel (a) denotes the location of the C 2vconical intersection (CI), whichoccurs between the ground2A′and first excited2B′electronic states. In the present work, we ignore the excited electronic state and its associated non-adiabatic and geometric phase effects (see Ref. 36 for a non-adiabatic quantum dynamics calculation that includes these effects). In summary, the Li + LiNa →Li2+ Na reaction proceeds along a barrierless collinear reaction path. In the interaction region, the reactants “fall” into a deep attractive potential well and form an intermediate complex. This complex subsequently decays into the various open Li 2(v′,j′) product rovibrational channels. The quan- tum mechanical calculation of the associated product state-resolved rate coefficients and angular distributions is presented in Sec. III. C. Numerical parameters The accuracy of the APH surface function solutions computed by diagonalizing the Hamiltonian operator in Eq. (5) depends upon the size of the primitive basis sets in ˜θandϕspecified by the inte- gers lmaxand mmax, respectively.40The accuracy also depends on the energy cutoff ( Ecut) used in the SDT procedure. A series of convergence studies are performed at several fixed values of ρto determine a set of optimal values for these parameters. Typically, larger basis sizes are required as ρincreases due to the localization of the channels (see Fig. 1). Several full scattering calculations are then performed to also verify the convergence of the cross sections with respect to lmax,mmax, and Ecut. The convergence studies are performed for zero total angular momentum J= 0, and the same parameters are then used for all J>0. The optimal values for lmax used in the calculations are 99, 119, 127, and 127, and for mmax, they are 220, 240, 260, and 280 for the four ranges of ρ: 6.0≤ρ≤11.024 a0, 11.024<ρ≤19.544 a 0, 19.544<ρ≤23.374 a 0, and 23.374 <ρ ≤33.034 a 0, respectively. The basis size in ˜θis given by n˜θ=lmax+ 1. Forϕ, the basis size is given by nϕ= 2mmax+ 1 and nϕ= 2mmax + 2 for even and odd J, respectively. In the APH region, the hyper- radius was discretized at 144 logarithmically spaced points between ρi= 6.0 andρf= 33.034 a 0inclusive. The total dimension of the APH angular Hamiltonian matrix that is diagonalized at each ρbefore applying the basis reduction via SDT is given by n=n˜θnϕnΩ, where nΩdenotes the number of Ω blocks for a given value of J. The value of Ω denotes the projection of Jon the body-frame zaxis and is even (odd) for even (odd) inversion parity p.40We note that in this work, the initial reactant rotational state is j= 0 so that J=j+l=l(where lis the orbital angular momentum of Li about LiNa) and J+pis always even. Thus, only the Jpvalues given by Jp= 0+, 1−, 2+, 3−, ...contribute to the cross sections. The values of Ω for both even and odd Jspan the range Ω = + J,J−2,J−4,J−6,...,−Jand nΩ=J+ 1. Thus, the total dimension of the angular matrix increases linearly with J. For J= 0, the total matrix dimensions in the four regions ofρgiven above before applying the SDT are 44 100, 57 720, 66 688, and 71 808, respectively. After SDT, the maximum matrix size in each region is significantly reduced to 13 008, 9412, 8401, and 4814, respectively. All these matrix dimensions scale as nΩ(i.e., lin- early with J). Thus, for the maximum value of Jp= 10+considered in this work, the largest matrix dimension was 13 008 ×11 = 143 088. Clearly, the SDT is critical to making the calculations feasible. The value of the SDT energy cutoff ( Ecut) is also optimized at each ρand is typically about 1.5 times larger than the highest lying eigenvalue of interest. For reference, the values of Ecutat the center of each of the four regions of ρlisted above are 1.09, 0.658, 0.752, and 0.746 J. Chem. Phys. 154, 124303 (2021); doi: 10.1063/5.0045712 154, 124303-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . Contour plots of the ab initio adiabatic PESs for the ground2A′electronic state of Li 2Na for several values of hyperradius: ρ= 8.5, 12.0, 16.0, and 25.0 a 0in panels (a)–(d), respectively. A stereographic projection of the hypersphere is plotted with the north pole ( θ= 0) centered at the origin. The value of the hyperspherical azimuthal angle (ϕ) is indicated by the radial blue dashed lines. The contours lie between 15 ×103and−5×103K inclusive and the dashed contour at 2.23 ×103K indicates the energy of the Li + LiNa ultracold collision. The red dot in panel (a) marks the location of the C 2vconical intersection between the ground and excited (not plotted) electronic states. All energies are relative to the minimum energy of the asymptotic adiabatic ground electronic state of Li 2+ Na. eV, respectively (relative to the bottom of the asymptotic Li 2+ Na potential). Fortunately, the large matrices are very sparse, and only, the diagonal Ω blocks and the off-diagonal coupling between their nearest neighbors (i.e., Ω ±2) are dense. In addition, only a rela- tively small percentage of the lowest-lying eigensolutions are needed for the coupled-channel equations. Thus, these matrices can be diag- onalized using an efficient sparse matrix diagonalization techniquesuch as the Implicitly Restarted Lanzcos Method (IRLM), as imple- mented in the ARPACK library.41The IRLM diagonalization scales asn2 SDT×Nch, where nSDTis the dimension of the reduced SDT matrix and Nchis the number of required eigensolutions (or coupled chan- nels, see below). The matrix diagonalization at all 144 ρvalues can be done independently. Furthermore, the calculations for all values of Jpand each exchange symmetry q=even /odd can also be performed J. Chem. Phys. 154, 124303 (2021); doi: 10.1063/5.0045712 154, 124303-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Number of coupled channels for each value of total angular momentum J. JpAPH region Delves region 0+800 500 1−1600 1000 2+2400 1500 3−3200 2000 4+4000 2500 5−4800 3000 6+5600 3500 7−6400 4000 8+7000 4500 9−7600 5000 10+8000 5250 separately. Thus, all these matrix diagonalizations can be performed simultaneously if sufficient computational resources are available. Once the eigensolutions have been computed and stored on disk, the overlap and potential coupling matrices are then computed at eachρfor use in the log-derivative coupled-channel propagation [see Eqs. (7) and (8) and Refs. 30, 40, and 46 for additional details]. The primary convergence parameter in the log-derivative prop- agation is the number of coupled-channels Nch(i.e., the number of angular eigensolutions described in the preceding paragraph). The J= 0 scattering calculations are performed for several values ofNchto determine the optimal value. For J>0, the optimal value forNchscales approximately linearly with respect to nΩ, that is, Nch≈N0 ch×nΩ(where N0 chdenotes the J= 0 value). The dense linear algebra required by the propagation involves matrix–matrix multi- plication and matrix inversion at each step. Thus, the computational cost of the log-derivative propagation scales as N3 ch. However, the calculations at each collision energy and for each value of Jpandexchange symmetry q=even /odd can be done separately. In addi- tion, for large values of Nch>2500, the linear algebra can be done more efficiently using the parallel ScaLAPACK library.47If sufficient resources are available, the parallel ScaLAPACK library enables the treatment of much larger values of Nchand therefore larger J. At ρmatch = 33.034 a 0, the APH log-derivative matrix is transformed into the Delves basis using the overlap matrix computed between the APH angular functions and the Delves channel-centered rovi- brational functions. The log-derivative propagation is then contin- ued into the asymptotic region until ρ=ρf. The final value of ρ is also determined from convergence studies, and in this work, it was chosen to be ρf= 144.574 a 0. Several full scattering calculations are performed for J= 0 and various values of ρf(and alsoρmatch ) to verify convergence. Thus, the coupled-channel propagation in the Delves region spans 338 uniformly spaced values of ρbetween ρ= 33.034 and ρ= 144.574 a 0inclusive. Table I lists the number of channels used in the log-derivative propagation in the APH and Delves regions as a function of J. From Table I, we see that the num- ber of Delves channels is less than the number of APH channels. In the APH interaction region, the deep potential wells give rise to a larger number of open channels relative to the asymptotic Delves region. Another convergence parameter is the number of substeps used in each ρsector. For a uniform grid, the sector at ρχis defined as the regionρχ±Δρ/2. Each sector is subdivided into smaller propaga- tion steps Δρ/nstep, where nstepis an integer determined from conver- gence studies. For the APH and Delves regions, the optimal values of nstepare 100 and 200, respectively. The propagation across all of the APH and Delves sectors was performed at 56 logarithmically spaced collision energies spanning the range from 1 nK to 1 K (see the tri- angles in Fig. 2). The total scattering energy is given by E=Ec+E0, where Ecdenotes the relative collision energy of LiNa + Li and E0 denotes the rovibrational ground state energy of LiNa( v= 0, j= 0) relative to the bottom of the asymptotic Li 2+ Na potential well (in this work, E0= 0.192 002 eV or 2.228 ×103K). The largest collision energy considered here is Ec= 1 K, which gives E= 0.192 088 3 eV or 2.229×103K. FIG. 2 . Total rate coefficients (thick black solid curves) are plotted as a function of collision energy for the Li + LiNa( v= 0, j= 0)→Li2+ Na reaction summed over all the product rovibrational states of Li 2for even [panel (a)] and odd [panel (b)] exchange symmetry. The contributions from each individual partial wave from l= 0 to l= 10 are also plotted (the even and odd partial waves are plotted in red and blue, respectively). The partial wave sums are the thin black curves and demonstrate convergence as the sum is extended to include all l= 0–10. The 56 collision energies are indicated by the open black triangles. The inset in each panel shows the total rate coefficient plotted on a linear xy-scale near 50 mK. J. Chem. Phys. 154, 124303 (2021); doi: 10.1063/5.0045712 154, 124303-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II . Delves and Jacobi rovibrational quantum numbers for Li 2and LiNa. aLi2 Delves v,jmax Jacobi v,jmax Open v,jmaxbLiNa Delves v,jmax Jacobi v,jmax 0, 62(63) 0, 68(69) 0, 40(41) 0, 62(63) 0, 72(72) 1, 58(59) 1, 64(65) 1, 34(35) 1, 57(57) 1, 67(68) 2, 54(55) 2, 60(61) 2, 28(27) 2, 51(52) 2, 63(63) 3, 50(49) 3, 56(57) 3, 18(17) 3, 45(45) 3, 57(57) 4, 44(45) 4, 52(53) 4, 37(38) 4, 52(52) 5, 40(39) 5, 48(47) 5, 28(28) 5, 45(45) 6, 34(33) 6, 42(43) 6, 13(14) 6, 38(38) 7, 26(25) 7, 36(37) 8, 14(15) 8, 30(31) aThe jmaxvalue indicates the maximum value of j= 0, 2, 4, . . .,jmaxfor even symmetry and the value in parenthesis indicates the maximum value for odd symmetry j= 1, 3, 5, . . .,jmax. bThe jmaxvalue indicates the maximum value of all j= 0, 1, 2, . . .,jmax. The Delves hyperspherical functions used in the Delves prop- agation step discussed above are computed using an efficient 1D Numerov method for the vibrational wave functions at each value of jand l. As mentioned in Sec. I, the rotational part of the wave func- tions is analytic. The number of rovibrational quantum numbers is determined from energy considerations. All the locally open chan- nels and a significant number of closed channels must be included to ensure good unitarity. The Delves basis for J= 0 includes 500 rovibrational functions, which span nine vibrational levels of Li 2(v = 0–8) and eight for LiNa( v= 0–7) up to a total energy of ∼0.394 213 eV. Table II lists all the rotational quantum numbers used for each vibrational level. The rovibrational states computed for the asymp- totic Jacobi basis are also listed. The number of rotational states for each vibrational level of Li 2is given by jmax/2 + 1 and ( jmax+ 1)/2 for even and odd symmetry, respectively. For LiNa, all values of j are included for each symmetry, so the number of states in each case is given by jmax+ 1. From Table II, we see that by summing up these expressions for each Delves vibrational level, there are 200 even states for Li 2and 300 for LiNa giving a total of 500. Similarly, for the odd states, we find 196 for Li 2and 304 for LiNa. The Jacobi basis functions are computed in the same way as the Delves functions but satisfy the asymptotic Jacobi diatomic Schrödinger equation for each arrangement channel.37The Jacobi rovibrational quantum numbers are also listed in Table II. The open Jacobi channels label the initial and final states of the scattering matrix SJ p q f←i(E), where i= (τi,viji, li) and f= (τf,vfjf,lf) label the initial and final quantum numbers: τ is the arrangement channel, vand jare the diatomic vibrational and rotational quantum numbers, and lis the orbital angular momen- tum. In this work, τiandτflabel the reactant LiNa and product Li2arrangement channels, respectively. In Sec. III, the product Li 2 quantum numbers will also be denoted by primes v′and j′. Asymp- totically, for J=l= 0, there are 65 and 63 open Jacobi channels for Ec≤1 K and even and odd symmetry, respectively. These open chan- nels include the even and odd states of the initial reactant channel LiNa( v= 0, j= 0). The rovibrational quantum numbers for the open Li2channels are listed in the third column of Table II. We note that the number of Delves channels also increases approximately linearly with J(see Table I), but the number of Delves channels is much smaller than the number of APH channels (the large numberof APH channels is due to the deep potential well in the interaction region). III. RESULTS The full-dimensional numerically exact ab initio based quan- tum reactive scattering results for the Li + LiNa( v= 0,j= 0)→Li2(v′, j′) + Na reaction are presented below. The calculations include all the partial waves needed to obtain fully converged state-to-state cross sections at 56 collision energies between 1 nK and 1 K. The rate coefficients are computed by multiplying the cross sections σfiby the relative collision velocity v:Kfi=vσfi. The exchange symmetry asso- ciated with the identical6Li nuclei is rigorously treated by using the appropriately symmetrized quantum mechanical wave functions. All the results include the appropriate nuclear spin statistical factors: 2/3 and 1/3 for even and odd exchange symmetry, respectively. We note that asymptotically, the even (odd) exchange symmetry correlates with the even (odd) rotational levels of Li 2. The total rate coefficients are presented in Sec. III A, and convergence with respect to the par- tial wave sum is demonstrated. A representative set of rotationally resolved rate coefficients are presented in Sec. III B. Notable shape resonances and other features are identified and the primary partial wave(s) that are responsible for the features are indicated. The angu- lar distributions or DCSs are presented in Sec. III C as a function of both the collision energy and scattering angle. Significant oscil- lations and other striking features (i.e., sharp spikes, deep channels, and ridges) due to quantum interference and shape resonances are observed. A. Total rate coefficients The total rate coefficients for the Li + LiNa( v= 0, j= 0)→Li2 + Na reaction summed over all product v′and j′states of Li 2are plotted in Figs. 2(a) and 2(b) for even and odd exchange symmetry, respectively. The individual contributions from each partial wave (l= 0–10) are also plotted in color: red (blue) for the even (odd) partial waves. The 56 collision energies are indicated by the black triangles superimposed on the l= 0 (red) curve. The total rate coef- ficients (summed over all 11 partial waves) are the thick black solid J. Chem. Phys. 154, 124303 (2021); doi: 10.1063/5.0045712 154, 124303-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp curves in panels (a) and (b). The rate coefficients for each partial sum are also indicated by the thin solid black curves. The plots in Fig. 2 show that summing over the 11 partial waves (i.e., l= 0–10) are suf- ficient for obtaining fully converged results up to 1 K. It is clear that the Wigner regime where only the s-wave (i.e., l= 0) contributes occurs for collision temperatures below 10−4K. At these ultracold temperatures, the rate coefficients approach a constant (non-zero) value. We note that the rate coefficients of odd symmetry are approx- imately a factor of two smaller than the even ones due to the factor of two difference in the nuclear spin statistical factors. The total rate coefficients increase more or less monotonically with the collision energy. A small but noticeable undulation (bump) is seen near 50 mK due to an l= 3 shape resonance (see Sec. III B). The inset in each panel of Fig. 2 plots a zoomed in view of the total rate coefficient near the resonance (note the linear scale in both the x and y axes). B. Rotationally resolved rate coefficients A representative set of rotationally resolved rate coefficients are plotted in Figs. 3 and 4 for even and odd exchange symmetry, respectively. Each of the four panels (a)–(d) in these two figures cor- responds to the product Li 2vibrational states v′= 0, 1, 2, and 3, respectively. The product Li 2rotational quantum number j′is indi- cated for each curve. As expected, the rate coefficients approach aconstant value in the Wigner regime where only s-wave scattering occurs (i.e., for collision energies below about 0.1 mK). As shown in several previous ultracold studies, the ultracold rate coefficients can be dramatically enhanced or suppressed due to quantum interfer- ence effects.23,25,36Thus, the smallest and largest ultracold rate coef- ficients in Figs. 3 and 4 are the result of significant destructive and constructive interference, respectively. This quantum interference occurs between two primary reaction pathways: (1) a direct path- way and (2) a looping pathway that encircles the conical intersection [see the red dot in Fig. 1(a)]. Due to the unique properties of ultra- cold collisions, the relative phase shift of the scattering amplitudes associated with these two pathways approaches an integral multiple ofπ.23,25Thus, the quantum interference can approach its maxi- mum constructive or destructive values if the magnitudes of these two scattering amplitudes are comparable.23,25At higher collision energies, the rate coefficients typically increase in magnitude and notable features such as bumps, steps, and shoulders are observed due to underlying shape resonances associated with the various par- tial waves. Several of the most notable features are labeled by the partial wave(s) responsible. For example, in panel (a) of Fig. 3, a prominent l= 3 shape resonance occurs for j′= 14 (black long-short dashed curve) near 50 mK. A series of two shape resonances due to l= 2 and 5 appear for j′= 2 (black dashed curve) near 20 mK and 0.2 K, respectively. A prominent l= 2 shape resonance also occurs FIG. 3 . Rotationally resolved rate coefficients are plotted as a function of the collision energy for the Li + LiNa( v= 0, j= 0)→Li2(v′,j′) + Na reaction and even exchange symmetry. Panels (a)–(d) plot the selected rate coefficients for v′= 0, 1, 2, and 3, respectively. The j′value is indicated next to each curve. Some of the curves are plotted in red to help distinguish them. Notable shape resonances are visible in many of the curves and the partial wave(s) responsible for the feature are indicated. J. Chem. Phys. 154, 124303 (2021); doi: 10.1063/5.0045712 154, 124303-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . (a)–(d) Same as in Fig. 3 except for the odd exchange symmetry. forj′= 24 (red solid curve) in the 20–40 mK range. Shape resonance features can also be seen in panels (b)–(d) of Fig. 3. In particular, a high energy resonance feature is predicted for j′= 0 in both pan- els (c) and (d) near 0.6 K due to l= 6, 7, and 8. Highlighting a few notable features for odd symmetry, we see in panel (a) of Fig. 4 that a notable shape resonance occurs for j′= 13 (black dashed-dotted curve) due to the l= 2 partial wave near 20 mK. In panel (b) of Fig. 4, a series of two l= 1 and l= 3 shape resonances are seen for j′= 3 (black dashed curve) near 5 and 50 mK, respectively. An l= 3 shape resonance is also seen for j′= 15 (black dashed-dotted curve) and a l= 2 resonance occurs for j′= 21 (red solid curve). As shown in Fig. 3, a high energy resonance feature is predicted near 0.6 K for v′ = 3 and j′= 1 [black solid curve in panel (d)] due to l= 7 and 8. C. Differential cross sections The appearance of intriguing quantum phenomena (i.e., inter- ference and shape resonances) reported above in the rotationally resolved rates coefficients become even more noticeable in the angu- lar distributions or DCSs. The integration of the DCSs over the scat- tering angle in computing the integral cross sections and rate coef- ficients averages out a significant portion of these quantum effects. Figure 5 plots the DCSs for v′= 0 and even symmetry for the same product rotational states presented in panel (a) of Fig. 3. That is, panels (a)–(f) in Fig. 5 correspond to j′= 0, 2, 14, 24, 34, and 36, respectively. The DCSs are plotted as a function of both collisionenergy and scattering angle. The scattering angle is the relative angle between the reactant Li–LiNa and product Na–Li 2kvectors. Thus, forward and backward scattering correspond to θ= 0○and 180○, respectively. A quick scan over the six panels immediately reveals fascinating features in the DCSs that are unique to each product state: high-frequency choppy oscillations are seen in panel (a), sharp downward spikes in the forward scattering region are seen in pan- els (b) and (c), also notable bumps are seen in the backward regions in panels (b) and (c) due to the l= 2 and l= 3 shape resonances, a deep channel (due to the l= 1 partial wave) is seen in panel (d) that starts in the forward scattering region at low energies and curves into larger angles at higher energies, a significant dip is seen in the back- ward scattering region in panels (e) and (c) due to l= 1 and (a) due to l= 3, and a very large downward spike is seen in the forward scatter- ing region in panel (f) due to the l= 1 partial wave. These features are due to constructive and destructive quantum interference between the two reaction pathways mentioned above, shape resonances, and quantum interference between the partial waves themselves. The DCSs for v′= 1 and even symmetry are plotted in Fig. 6, and the product rotational states correspond to those in panel (b) of Fig. 3. That is, panels (a)–(f) in Fig. 6 correspond to j′= 0, 2, 10, 14, 16, and 34, respectively. Again, we see unique structure in the DCSs for each product state: in panel (a), a notable bump in the backward region associated with the l= 2 shape resonance, a depression near θ= 130○and 20 mK is also seen in panel (a) due to primarily l= 1 and 4, a broad symmetric ridge (bump) is seen in panel (b) in both J. Chem. Phys. 154, 124303 (2021); doi: 10.1063/5.0045712 154, 124303-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . Differential cross sections for the Li + LiNa( v= 0, j= 0)→Li2(v′= 0, j′) + Na reaction are plotted as a function of the collision energy and scattering angle for the even exchange symmetry. Panels (a)–(f) correspond to j′= 0, 2, 14, 24, 34, and 36, respectively. Significant features due to quantum interference effects are observed. J. Chem. Phys. 154, 124303 (2021); doi: 10.1063/5.0045712 154, 124303-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 6 . Same as in Fig. 5 except for v′= 1, and panels (a)–(f) correspond to j′= 0, 2, 10, 14, 16, and 34, respectively. J. Chem. Phys. 154, 124303 (2021); doi: 10.1063/5.0045712 154, 124303-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 7 . Same as in Fig. 5 except that panels (a)–(c) correspond to v′= 2 and j′= 0, 8, and 16, respectively, and panels (d)–(f) correspond to v′= 3 and j′= 0, 8, and 18, respectively. J. Chem. Phys. 154, 124303 (2021); doi: 10.1063/5.0045712 154, 124303-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the forward and backward regions associated with an l= 1 shape resonance, a notable dip due to l= 1 and a ridge due to l= 2 and 3 are seen in the backward region in panel (c), several sharp downward spikes are seen in the forward region in panels (d) and (e) (the lowest energy spikes are due to l= 1), and a broad channel due to l= 1 and a ridge due to the l= 2 shape resonance are seen in the forward region in panel (f). The DCSs for even symmetry with v′= 2 and j′= 0, 8, and 16 are plotted in the panels (a)–(c) of Fig. 7. These three product states cor- respond to those plotted in panel (c) of Fig. 3. Two bumps are seen in the backward region of panel (a). The broad bump at a low energy (near 5 mK) is due to the l= 1 shape resonance and the narrow bump at a high energy ( ≈0.2 K) is due to the overlapping l= 6 and 7 shape resonances. The low energy channel feature in the forward region in panel (b) near 0.2 mK is due to l= 1 and the sharp downward spike at 50 mK is due to l= 3. The broad dip at a low energy in the forward region in panel (c) is due to l= 1 and the sharp dip in the backward region is due to l= 2. The bump in the backward region of panel (c) is due to l= 3. The DCSs for even symmetry with v′= 3 and j′= 0, 8 and 18 are plotted in panels (d)–(f) of Fig. 7. These three product states correspond to those plotted in panel (d) of Fig. 3. The first shallow dip at a low energy in the forward region in panel (d) is from l= 1 and the sequence of ripples extending to higher energies corresponds to several higher partial waves. The first dip and narrow downward spike in the forward region of panel (e) correspond to l= 1 and 3, respectively. In panel (f), the first and second downward spikes correspond to l= 2 and 3, respectively. IV. CONCLUSIONS In summary, a recently developed first-principles based ab initio PES for the adiabatic ground electronic state of Li 2Na was used to perform numerically exact quantum reactive scatter- ing calculations for the Li + LiNa( v= 0, j= 0)→Li2(v′,j′) + Na reaction. This PES includes an accurate treatment of the long-range three-body interactions and spectroscopically accurate asymptotic diatomic potential curves.36A time-independent coupled-channel methodology based on hyperspherical coordinates was used to per- form the full-dimensional scattering calculations at 56 collision energies spanning nine orders of magnitude from ultracold (1 nK) to cold (1 K) temperatures. The scattering calculations include all the partial waves necessary to obtain fully converged cross sections (with respect to the partial wave sum) over the full energy range for the first time. The calculations also include a rigorous quantum mechanical treatment of the identical particle exchange symmetry due to the two identical6Li nuclei. The total and rotationally resolved rate coefficients are com- puted as a function of collision energy for both even and odd exchange symmetries. The total rate coefficients approach a constant (non-zero) value in the ultracold limit and increase monotonically with the increase in collision energy. A small bump is observed in the total rate coefficients for collision energies near 50 mK due to an l= 3 shape resonance. The rotationally resolved rate coefficients exhibit more structure and several bumps and other features are observed as a function of energy due to the contributions from vari- ous partial wave shape resonances. The unique properties associated with ultracold collisions leads to significant quantum interference effects that ultimately determine the asymptotic value of the rota-tionally resolved rate coefficients as the collision energy approaches absolute zero. Particular focus of the present work is on the inves- tigation of the angular distributions or DCSs as a function of both the collision energy and scattering angle. Significant quantum inter- ference and resonance structure are observed in the DCSs, which are unique to each product Li 2rovibrational state. These unique “quantum fingerprints” encode the underlying quantum dynamics and molecular interactions (e.g., potential wells, multiple reaction pathways, rotational anisotropy, and long-range interactions) and are experimentally measurable. Although we expect that improve- ments to the PES will be required to obtain quantitative agree- ment with future state-resolved experiments, the current theoreti- cal study clearly demonstrates that fascinating quantum phenomena and structure will be present in product state-resolved experimen- tal cross sections. In future theoretical studies, we plan to investigate the effects of the excited electronic state and its associated geometric phase and non-adiabatic coupling on the angular distributions. This will require the use of a two-state diabatic approach.30,36 The quantum effects reported here are unique to the ultra- cold/cold energy regime and provide an unprecedented opportunity for probing and understanding fundamental molecular interactions, reaction mechanisms, and quantum phenomena. One can envision exploiting these quantum effects to produce extremely sensitive sen- sors and measuring devices. They also provide a potential avenue for the realization of quantum control technologies via the selection of a particular initial and/or final state, the preparation of a specific ori- entation between the colliding partners (i.e., stereodynamics), or the application of external fields. We hope that this work will help stim- ulate future theoretical and experimental studies on the fascinating energy regime below 1 K. SUPPLEMENTARY MATERIAL See the supplementary material for plots of the differential cross sections of the odd exchange symmetry. ACKNOWLEDGMENTS This work was performed under the auspices of the U.S. Department of Energy under Project No. 20170221ER of the Labo- ratory Directed Research and Development Program at Los Alamos National Laboratory. This research used resources provided by the Los Alamos National Laboratory Institutional Computing Program. Los Alamos National Laboratory is operated by Triad National Secu- rity, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy (Contract No. 89233218CNA000001). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1R. V. Krems, Phys. Chem. Chem. Phys. 10, 4079 (2008). 2L. D. Carr, D. DeMille, R. V. Krems, and J. Ye, New J. Phys. 11, 055049 (2009). 3S. Ospelkaus, K.-K. Ni, D. Wang, M. H. G. de Miranda, B. Neyenhuis, G. Quéméner, P. S. Julienne, J. L. Bohn, D. S. Jin, and J. Ye, Science 327, 853 (2010). 4B. K. Stuhl, M. T. Hummon, and J. Ye, Annu. Rev. Phys. Chem. 65, 501 (2014). J. Chem. Phys. 154, 124303 (2021); doi: 10.1063/5.0045712 154, 124303-13 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 5W. E. Perreault, N. Mukherjee, and R. N. Zare, Science 358, 356–359 (2017). 6T. M. Rvachov, H. Son, A. T. 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5.0042608.pdf

J. Appl. Phys. 129, 155305 (2021); https://doi.org/10.1063/5.0042608 129, 155305 © 2021 Author(s).Low temperature growth of stress-free single phase α-W films using HiPIMS with synchronized pulsed substrate bias Cite as: J. Appl. Phys. 129, 155305 (2021); https://doi.org/10.1063/5.0042608 Submitted: 01 January 2021 . Accepted: 29 March 2021 . Published Online: 20 April 2021 Tetsuhide Shimizu , Kazuki Takahashi , Robert Boyd , Rommel Paulo Viloan , Julien Keraudy , Daniel Lundin , Ming Yang , and Ulf Helmersson ARTICLES YOU MAY BE INTERESTED IN Optimization of HiPIMS discharges: The selection of pulse power, pulse length, gas pressure, and magnetic field strength Journal of Vacuum Science & Technology A 38, 033008 (2020); https:// doi.org/10.1116/6.0000079 Enhanced interlayer coupling and efficient photodetection response of in-situ grown MoS 2– WS2 van der Waals heterostructures Journal of Applied Physics 129, 155304 (2021); https://doi.org/10.1063/5.0040922 Crystal facet engineering for production of TiO 2/WO 3 photoelectrodes with superior energy storage capacity Journal of Applied Physics 129, 155303 (2021); https://doi.org/10.1063/5.0042707Low temperature growth of stress-free single phase α-W films using HiPIMS with synchronized pulsed substrate bias Cite as: J. Appl. Phys. 129, 155305 (2021); doi: 10.1063/5.0042608 View Online Export Citation CrossMar k Submitted: 1 January 2021 · Accepted: 29 March 2021 · Published Online: 20 April 2021 Tetsuhide Shimizu,1,2,a) Kazuki Takahashi,2Robert Boyd,1 Rommel Paulo Viloan,1 Julien Keraudy,1 Daniel Lundin,1 Ming Yang,2and Ulf Helmersson1 AFFILIATIONS 1Plasma and Coatings Physics Division, IFM Materials Physics, Linköping University, Linköping SE 581-83, Sweden 2Department of Mechanical Systems Engineering, Graduate School of Systems Design, Tokyo Metropolitan University, 6-6, Asahigaoka, Hino-shi, 191-0065 Tokyo, Japan a)Author to whom correspondence should be addressed: tetsuhide.shimizu@liu.se ABSTRACT Efficient metal-ion-irradiation during film growth with the concurrent reduction of gas-ion-irradiation is realized for high power impulse magnetron sputtering by the use of a synchronized, but delayed, pulsed substrate bias. In this way, the growth of stress-free,single phase α-W thin films is demonstrated without additional substrate heating or post-annealing. By synchronizing the pulsed sub- strate bias to the metal-ion rich portion of the discharge, tungsten films with a ⟨110 ⟩oriented crystal texture are obtained as compared to the ⟨111 ⟩orientation obtained using a continuous substrate bias. At the same time, a reduction of Ar incorporation in the films are observed, resulting in the decrease of compressive film stress from σ=1 . 8 0 –1.43 GPa when switching from continuous to synchronized bias. This trend is further enhanced by the increase of the synchronized bias voltage, whereby a much lower compressive stressσ= 0.71 GPa is obtained at U s= 200 V. In addition, switching the inert gas from Ar to Kr has led to fully relaxed, low tensile stress (0.03 GPa) tungsten films with no measurable concentration of trapped gas atoms. Room-temperature electrical resistivity is correlated with the microstructural properties, showing lower resistivities for higher Usand having the lowest resistivity (14.2 μΩcm) for the Kr sputtered tungsten films. These results illustrate the clear benefit of utilizing selective metal-ion-irradiation during film growth as aneffective pathway to minimize the compressive stress induced by high-energetic gas ions/neutrals during low temperature growth ofhigh melting temperature materials. © 2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/ ).https://doi.org/10.1063/5.0042608 I. INTRODUCTION Tungsten (W) thin films are receiving a great amount of attention for many different industrial applications due to theirrefractory properties such as high physical stability and chemicalinertness. 1In particular, the equilibrium A2 bcc tungsten, denoted α-W, has attractive properties like the highest melting tem- perature ( Tm= 3422 °C) and the lowest thermal expansion coefficient (4.5 × 10−6K−1) among metals, together with a low electrical resistiv- ity (5.28 μΩcm). From a mechanical point of view, it exhibits consid- erable hardness and toughness at high temperatures.1These unique features has led to its use in the metallization of integrated circuits,2–4as contact plugs and as diffusion barriers.5Tungsten has also been considered for use in x-ray reflection masks6and as electrode material for high-temperature surface acoustic wave devices.7Nanostructured tungsten films are also interesting as a protective coating.8–10Its high density (19.3 g/cm3), low sputtering yield, and low coefficient of electron emission also make tungsten a candidate for the primary plasma-facing materials in fusion reactors.11,12Recently, a metastable form of tungsten ( β-W with the A15 cubic structure13)w a sf o u n dt o be a good candidate for spin –orbit torque applications due to its large spin Hall angle and high resistivity (150 –350μΩcm).14 However, β-W also shows a high tensile stress that occurs due to theJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 155305 (2021); doi: 10.1063/5.0042608 129, 155305-1 ©A u t h o r ( s )2 0 2 1phase transformation to α-W that occurs at moderate temperatures13 making it unsuitable for many applications.15 To avoid the formation of β-W and to obtain primary α-W, a sufficiently high surface mobility needs to be maintained duringfilm growth. 10At low growth temperatures, a route to increase surface mobility is to apply momentum transfer from particles impinging the growing film.16Monte Carlo simulations have shown that for a relatively low working gas pressure of ∼0.2 Pa, the average kinetic energy of sputtered tungsten is as high as 20 eV.17 This energy is sufficient to enhance the surface mobility of tungstento overcome the surface diffusion barrier leading to crystalline growth of the α-W phase. 16,18However, bombardment by high- energy sputtering gas atoms/ions can cause film stress due to theincorporation of excess atoms in the film. 19Most of the previously reported primary α-W films, with thicknesses above 50 nm, show compressive in-plane stress in the range of 3 –7G P a .3,4,16,18,20This will limit the thickness of tungsten films to ∼150 nm due to film failure.2These deleterious effects can be minimized by optimizing the energy of impinging particles, mainly ions, to below that of thebulk lattice displacement threshold. In early studies by Kao et al., 21 compressive stress was significantly reduced from 4.8 to 1 GPa by increasing the flux of Ar ions with energy of 100 –150 eV. However, Ar entrapment in interstitial sites could not be completely sup-pressed and thus the zero-stress condition was not achieved. To overcome the issue with compressive stress induced by rare gas incorporation in the films, an alternative way is to efficiently irradiate with metal ions of the growth material. 22This can be achieved by using ionized-PVD, such as high power impulse mag-netron sputtering (HiPIMS). 23Key features of HiPIMS are (1) ioni- zation of a large fraction of the sputtered metal flux and (2) time separation between metal- and gas-ion fluxes incident at the sub- strate.24In the recent reports by Velicu et al.25and Engwall et al.,26 a clear advantage of utilizing HiPIMS has been demonstrated in the growth of dense tungsten coatings with film thickness between 400and 1200 nm, respectively. The films, however, contained compres- sive stress of ∼2 GPa. The second feature of HiPIMS (time separa- tion of ion arrivals to the substrate) enables the further preciseselective tuning of metal-ion energy and momentum transferduring the film growth without introducing film stresses throughminimizing the incorporation of gas ions by synchronizing the pulsed substrate bias with the metal-rich portion of HiPIMS pulses. This was first demonstrated by Greczynski et al . 27showing film densification, microstructure enhancement, surface smoothening,and decreased film stress with no measurable Ar incorporation in Ti 1−xAlxN films. Significant enhancement of surface mobility by efficient metal-ion-irradiation was demonstrated also by the deposi-tion of the high melting temperature material Hf ( T m= 2233 °C) on unheated substrates, showing epitaxial growth of HfN thin films onMgO(001) at very low T h(<0.10).28 To clarify the impact of metal-ion-irradiation during film growth, and at the same time, the reduction of gas atoms andgas-ion-irradiation, the present work investigates the low tempera-ture growth of tungsten films by the use of a synchronized pulsedsubstrate bias in HiPIMS discharges. The synchronization of the negative substrate bias pulse with the metal-rich portion of the HiPIMS discharges were selected based on the time evolution ofthe ion fluxes in the pulsed sputtering process, as investigated byoptical emission spectroscopy (OES) and ion mass spectrometry. The effect of the nature of the incident ions, gas vs metal ions, and their energies have been studied by comparing tungsten filmgrowth in argon (Ar) discharges under different substrate biasconfigurations; continuously applied bias and synchronizedpulsed bias with negative bias voltage ( U s)v a l u e sr a n g i n g between 50 and 200 V. The effect of reducing the number back- reflected Ar gas atom was also explored by using krypton (Kr) asan alternative working gas. II. EXPERIMENTAL All experiments were performed in a high vacuum stainless- steel chamber with a base pressure of ∼10 −4Pa. A planar circular unbalanced magnetron with a tungsten (99.999% in purity) diskwith a diameter of 75 mm and a thickness of 5 mm was used asthe sputtering target. Ar or Kr gas with a purity of 99.997% was introduced into the chamber through a mass-flow controller at a constant flow rate of 100 sccm and was maintained at a constantworking pressure of 1.0 Pa by adjusting the pumping speed viathe main gate valve. Unipolar HiPIMS pulses, 100 μs in length, were supplied by a HiPSTER 1 pulsing unit fed by a 1 kW HiPSTER 1-DCPSU DC power supply (Ionautics AB, Sweden). The pulse frequency wasadjusted in the range of 100 –150 Hz to maintain an average power in the range of 250 –280 W. The target current and voltage characteris- tics were recorded and monitored with a Tektronix TBS 2104 digital oscilloscope connected directly to the HiPSTER pulsing unit. To analyze the time evolution of HiPIMS discharges in a Ar atmosphere, especially the development of the plasma compositionclose to the target, time-resolved optical emission spectroscopy(OES) was performed using an optical monochromator system MS3501i (SOL Instruments Ltd.) with a grating of line density of 1200 l/mm (which has an optical resolution of 0.06 nm) operatingwith an intensified-CCD photodetector, iStar ICCD DH320T(Andor Technology Ltd.). The emission spectra were collected through a collimator quartz lens with an aperture of 9.5 mm and a quartz optical fiber capable of transmitting in the ultraviolet spec-tral region, mounted in the deposition chamber with the line ofsight parallel with and 25 mm from the target surface. To obtainthe temporal evolution of neutral and ionized species in the plasma composition at a given set of discharge conditions, representative emission lines were carefully selected based on their intensity, exci-tation energy, and transition probability. Table I summarizes a list of selected Ar I, Ar II, W I, and W II lines with details of the exci-tation and transition of these lines. 29 To obtain the time evolution of the selected line intensities during HiPIMS discharges, the ICCD camera was triggered syn-chronously with the HiPIMS pulsing unit with a gate width of 3 μs. The delay time with respect to the onset of the voltage pulse to thecathode varied up to 120 μsi n4 μs intervals. The signal-to-noise ratio was optimized by accumulating the spectra acquired during 100 consecutive pulses for each time point. In situ time-dependent mass and energy analyses of the ion flux incident at the substrate plane were carried out using a Hiden Analytical EQP 300 spectrometer. The mass spectrometer orifice, located at a distance of 100 mm from target center, was electricallyJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 155305 (2021); doi: 10.1063/5.0042608 129, 155305-2 ©A u t h o r ( s )2 0 2 1grounded during the experiments. Ion energy distribution func- tions (IEDFs) were scanned from 1 to 50 eV for W+and W2+, and from 1 to 30 eV for Ar+and Ar2+ions, in 1 eV steps, while sputter- ing under the conditions stated above. All measurements were carried out with the same global spectrometer settings obtained bycalibrating the mass spectrometer to Ar +ions to allow comparisons between the different ion species. However, it should be noted thatno attempts have been made to adjust the measurements for spec- trometer performance at different ion energies, such as focal length of the electrostatic lenses and acceptance angles of the samplingorifice. 30In order to acquire time-resolved data, the detector gate of the mass spectrometer was synchronized with the target-pulseonset, triggering by a transistor –transistor logic pulse sequence gen- erated at the output of the HiPSTER synchronization unit. The detector gate width was set to 10 μs and the delay time with respect to the onset of the voltage pulse to the cathode was varied from50μs up to 400 μsi n1 0 μs intervals. The total acquisition time per data point was 10 s, implying that data were collected during 100 consecutive pulses for the used pulse frequency of 100 Hz. All time- resolved data presented were corrected for the ion time-of-flight(TOF) within the mass spectrometer, following the procedure ofBohlmark et al. and listed in Table II for the lowest and highest incident kinetic energies. 24 Tungsten films were grown onto Si (001) substrates with a 100 nm-thick thermally grown oxide layer. Prior to the depositions,the substrates were cleaned using successive rinses in ultrasonicbaths of acetone followed by isopropanol and finally blown dry with N 2. They were then mounted on a substrate stage located at a distance of 100 mm from the target surface, the same as for themass spectrometer orifice. To investigate the effect of gas- vs metal-ion acceleration, the depositions were conducted using different substrate bias configurations: floating bias, continuously applied (DC) bias, and synchronized pulsed bias, as summarized in Table III .F o rthe synchronized bias configuration, a negative pulsed bias U s= 50 V with a duration of 100 μs, i.e., the same pulse duration as the target voltage pulse, is applied with a time delay Δtof 60 μsf r o m the onset of the applied target voltage pulse. The selected Δtensures that the metal ion flux at the substrate is maximized, as determinedby the plasma characterization demonstrated in Sec. III A .A sf o ra bias scheme for gas-dominated ion irradiation, deposition using con- tinuously applied DC bias at the same level of U s=5 0V w a s p e r - formed for comparison. In addition, the impact of various metal ionenergies during film growth was also studied by increasing thesynchronized bias voltage to U s= 100 and 200 V using the same timing. All depositions above were performed at the same discharge conditions, resulting in a peak current density JD,peak≈0.75 A/cm2in the Ar atmosphere. The effect of reducing backscattering of inert working gas atoms at the cathode target was also explored by using Kr instead of Ar. Discharge conditions for the Kr atmosphere was selected to obtain the same average power of 280 W but a lower peakcurrent density J D,peak≈0.45 A/cm2due to the voltage limitation of the power s upply. A deposition time of 30 min was used for all depositions, and the resulting film thicknesses are listed in Table III . Higher deposi- tion rate (as seen by a higher averaged film thickness) for the Krprocess is attributed to the higher sputtering yield of tungsten byKr +ions ( SWKr+≈1.35 at 950 eV accelerated across the cathode sheath) compared to Ar+ions ( SWAr+≈0.74 at 800 eV). Crystal structure analysis of the deposited samples were carried out by high-resolution x-ray diffraction (XRD) using a PANalyticalEmpyrean diffractometer, equipped with a PIXcel-3D detector, usingCu-Kα 1radiation ( λ= 0.154 059 7 nm). θ–2θscans between 35° and 120° were performed with line-focus and an x-ray mirror with a two- bounce Ge monochromator as an incident-side and a parallel platecollimator as a diffracted-side optics. The observed strong α-{110} film texture (2 θ= 40.265°) motivated detailed scans in a narrower range of 39° –41° with a 0.002° step size. Both θ–2θand grazing inci- dent (GI) scans, at an incident angle of 1°, were performed. Diffraction peaks from the GI scan were utilized to estimate the grainsizes as determined by applying Scherrer ’se q u a t i o n . The same XRD instrument but with different optics, point focus, a four-bounce Ge (220) monochromator as an incident-side and a three-bounce Ge (220) monochromator as a diffracted-side, was used to study the macroscopic residual stresses in the depositedfilms from the curvature of the single-crystal substrate. The curva-ture was assessed by the change in orientation of a diffracting crys- tallographic plane at two different location of the sample surface. 31 The Stoney formula for anisotropic single-crystal Si (001) was used to extract residual coating stress from the measured substrate curva-ture. Here, uniform plane stress in the film was assumed. 32 XRD pole figure measurements were systematically carried out onα-{110} and α-{200} diffraction peaks in order to determine the main texture component. Point focus and x-ray lens as incidentoptics and the same diffracted-side optics as for the θ–2θscans were used. The scans were performed at ψ-angles (angle between the normal to the sample surface and the normal to the diffracting planes) ranging from 0° to 85° with a 5° step and w-angles (rota- tional angle around the normal to the sample surface) between 0°and 360° with a 5° step. The integration time was 1 s per point.TABLE I. List of the monitored OES lines with corresponding wavelength λ, the transition strength Aki, and the energy of lower Eiand upper Ektransition levels. OES lines λ(nm) Aki(×108)( s−1) Ei(eV) Ek(eV) Ar I 763.51 0.25 11.54 13.17 Ar II 427.75 0.80 18.45 21.35W I 255.13 1.78 0.00 4.86W II 276.42 0.48 0.00 4.48 TABLE II. Ion species, mass/charge ( m/z) ratios, incident ion-kinetic-energy ranges, corrected TOF values, and energy-averaged TOF values. Ion species m/zKinetic energy range (eV) TOF ( μs) Average TOF ( μs) 40Ar+40 1 –100 75 –79 77.0 40Ar2+20 1 –100 53 –56 54.5 184W+184 1 –100 162 –170 166.0 184W2+92 1 –100 114 –120 117.0Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 155305 (2021); doi: 10.1063/5.0042608 129, 155305-3 ©A u t h o r ( s )2 0 2 1The microstructures of selected W samples were characterized by electron microscopy. The scanning electron microscopy (SEM)analysis was done using a Zeiss LEO 1550 field emission gun instru- ment. Prior to analysis, cross sections of the films were prepared by first immersing into liquid nitrogen followed by fracturing. The top-ography of the thin films was analyzed both by plan-view SEMimages and by atomic force microscopy (AFM) analyses using aVeeco Dimension 3100 instrument with a Nanoscope IIIa control- ler, operating in tapping mode under ambient conditions. Root mean square (RMS) roughness was determined from AFM imageswith an area of 1 × 1 μm 2with the image analysis tools of the WSxM ver. 4.0 software. Selected thin films were chosen for more detailed microstruc- tural investigations by (scanning) transmission electron microscopy (TEM). Prior to analysis, suitable plan-view and cross-sectionalspecimens were prepared by using a focused ion beam (FIB) instru-ment (Zeiss 1540). Cross-sectional specimens were prepared by thetraditional lift out approach. 33For plan-view, a sample was prepared by first cleaving to give a wedge shape specimen then using the FIB to thin a 5 × 7 μm region below the surface of the film. All analyses were performed using a FEI Tecnai G2 TF 20 UT instrument oper-ated at 200 keV. STEM images were collected with the annular detector spanning the range 80 –260 mrad. All image analysis was performed using the Digital Micrograph software V3 (Gatan, CA),including fast Fourier transmission (FFT) and inverse FFT (FFT −1) processing, Both FFT and FFT−1are well established methods applied to the post-processing of HRTEM images.34FFT provides an effective diffraction pattern from a HRTEM image, from which selected diffraction spots are used to form a processed image, pro-viding spatial information on specific domains and defects. 35 Room-temperature electrical resistivities ρwere determined by multiplying the thickness and the sheet resistance of the W film measured by using a four-point probe SRS-4 (Astellatech, Inc.). Thicknesses were determined from step profiles between a coatedand a non-coated area by using a Keyence NANOSCALE VN-8000hybrid microscope. The obtained value was also confirmed fromcross-sectional images from the SEM analyses, showing a good agreement between the both measurements. To correlate the amount of incorporation of Ar as an impurity acting as electronscattering centers, 36relative difference of Ar contents in the films between different bias configurations were additionally analyzed by energy dispersive x-ray spectroscopy (EDS). The EDS measure- ments were taken in the same system as microstructure analysisoperated at an accelerating voltage of 10 kV. The samples were tilted 40° from the vertical so that they are normal to the EDSdetector. The recorded values are an average of ten measurements taken from randomly selected regions magnified by 1000 (approxi- mate a size of 300 × 225 μm) with errors determined by the stand- ard deviation of the mean. III. RESULTS A. Plasma characterization Prior to the film growth experiments, HiPIMS plasma dis- charges in the Ar atmosphere were characterized in order to designthe substrate bias strategy. Figure 1 shows typical target current I c and voltage Ucwaveforms acquired. The evolution of the discharge current is characterized by an initial peak followed by a slowdecrease or a plateau. The onset of the current increase is affectedby the applied U cwith steeper current rise for higher voltages, indi- cating efficient ionization of the Ar gas.37After the current peak, a current plateau is observed for Uc≥600 V. The level of the current plateau increases with Ucup to 30 A for Uc= 900 V. This transition atUc= 600 V is likely due to the discharge mode transition from a working gas sputtering regime toward a working gas-sustained self- sputtering or self-sustained self-sputtering mode.37–39 The ion and neutral compositions of the plasma as evaluated with time-resolved OES measurements are presented in Fig. 2 . The line intensities are shown for neutrals (Ar I and W I) and for singlyionized ions (Ar II and W II) in a process operated at U c= 800 V with a peak current of ∼35 A ( ∼0.76 A/cm2in peak current density). Ar I increases rapidly at the beginning of the pulse andpeaks at t∼4μs(I cpeaks at t∼15μs). When Ar I starts to decay, Ar II, W I, and W II increase and reache a maximum at t∼35μs. After this, the emission lines converge to plateau levels until the pulse current is terminated. According to Hala et al. , the detected emission from neutral Ar at the early stage of the HiPIMS pulse isdue to electron impact excitation involving fast secondary electronsaccelerated in the developing cathode sheath. 40,41The increase of Ar II emission lines is attributed to the ionization of working gas atoms in collision with these energetic electrons. Simultaneously, the increase of W I and W II intensities indicates the injection ofmetal into the discharge after having been sputtered by workinggas ions. The sudden decay of the Ar I is a sign of gas rarefaction of the working gas, 42which can be particularly pronounced when sputtering heavy elements like tungsten.43TABLE III. Conditions for HiPIMS deposition of W at different substrate bias voltage configurations. Conditions (i) (ii) (iii) (iv) (v) (vi) Working gas Ar Ar Ar Ar Ar Kr JD,peak (A/cm2) 0.75 0.75 0.75 0.75 0.75 0.45 Usmode ⋯ Cont. Sync. (aΔt:6 0μs) Sync. ( Δt:6 0μs) Sync. ( Δt:6 0μs) Sync. ( Δt:6 0μs) Us(V) Floatingb50 50 100 200 50 Film thickness (nm)c528 512 662 574 462 841 aΔt: Time delay of synchronized substrate bias from the onset of the target voltage. bEstimated to 3 V. cAveraged film thickness after 30 min deposition.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 155305 (2021); doi: 10.1063/5.0042608 129, 155305-4 ©A u t h o r ( s )2 0 2 1To characterize the gas and metal ion fluxes to the substrate plane, in situ time-resolved mass spectroscopy measurements were performed at a distance d= 100 mm away from the target surface. Time-dependent ion flux intensities are presented in Fig. 3 for Ar+, W+,a n dW2+obtained using the same discharge condition as used for the OES measurements. A small intensity signal from the Ar2+ flux is also detected (data not shown) with a measured ion intensity<3% of the total measured ion intensity. Each data point representsthe number of counts integrated during 10 μsi n t e r v a l s ,a n d t=0 corresponds to the onset of the HiPIMS pulse. The ion intensityduring the early part of the HiPIMS pulses is dominated by Ar +fol- l o w e db yar a p i di n c r e a s ei nt h eW+intensity and a decrease of Ar+ starting at t∼25μs. The W+intensity reaches a maximum at t∼70μs after which it gradually decays and the Ar+intensity again increases. The time for the W+peak agrees with the W II OES peak taking an estimated time-of-flight (TOF) of 23 μs into account (10 eV W+traveling the 75 mm from the position where the OES spectrum was obtained). After the pulse has been switched off att=1 0 0 μs, the Ar +flux intensity significantly increases with a delay after pulse off due to a TOF estimated to be 20 μsf o r5e VA r+trav- eling the ∼100 mm distance from the target region and reaches maximum intensity at t∼150μs. The increasing intensity of Ar+ during the plasma afterglow phase can be explained by refilling of Ar neutrals followed by electron impact ionization by a relativelyhigh density remanent plasma. Poolcharuansin and Bradley investi- gated Ti/Ar HiPIMS discharges and recorded afterglow electrons with an effective temperature ( T eff)o f∼0.2 eV, which exhibited a slow decay rate of several milliseconds, having a density of2×1 0 9cm−3even at the end of the pulse off-time.41 To get more insight into the ion flux, detailed time-evolutions of the ion energy distribution functions (IEDFs) were obtained and illustrated in Fig. 4 . The first W+ions are detected at t=2 4–34μs FIG. 2. A typical time evolution of the target current Icand of OES line intensi- ties from W I (255.13 nm), W II (276.42 nm), Ar I (763.51 nm), and Ar II(427.75 nm) during a 100 μs long, U c= 800 V HiPIMS pulse. FIG. 3. Time evolution of the energy-integrated ion flux intensity of W+,W2+, and Ar+recorded at the substrate position d= 100 mm during a HiPIMS dis- charge of W in 1 Pa Ar, with the initial cathode voltage Uc= 800 V , a pulse dura- tion of 100 μs, and a pulse repetition frequency of 100 Hz. FIG. 1. The applied discharge voltage Ic(upper panel) and the resulting discharge current Uc(lower panel) waveforms during HiPIMS discharges sputtering of a W target in an Ar atmosphere of 1 Pa. Pulse repetition frequency used is 100 Hz.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 155305 (2021); doi: 10.1063/5.0042608 129, 155305-5 ©A u t h o r ( s )2 0 2 1[as seen in Fig. 4(a) ], and a broad energy distribution with a peak at around 15 eV is seen at t=3 4–44μs. After that, the IEDFs gradually decrease in energy with time and only consist of thermalized ionsaccelerated to about 3 eV by the plasma potential after the termina-tion of the HiPIMS pulse. These temporal evolutions of IEDFs aremainly due to the difference in TOF of ions with different energies 44 and are consistent with previous mass spectrometry investigations ofHiPIMS discharges. 45The origin of the high- and low-energy ions in typical HiPIMS discharges are described elsewhere.46,47 The time evolution of the Ar+IEDF, as shown in Fig. 4(b) , has a different trend as compared to W+. The first Ar+ions are detected already at t=1 3μs and at t=2 3–33μs their intensity begins to increase, and the energy peak position shifts up toaround 2 eV. In the time interval t∼33–53μs, the IEDF decreases in intensity after which it increases again with a broadening of theenergy distribution up to 13 eV until t∼110μs, which corresponds to the time directly after the pulse termination. Shortly after the end of the pulse, only the low-energy peak remains as it graduallynarrows down to an energy peak centered around 2 eV. The broadening of the energy distribution coincides with the time when the high-energy flux of W +develops. This can be attributed to the heating of the Ar gas by energy exchange through collisionswith energetic sputter-ejected W neutrals, or by Ar reflected atthe target, followed by immediate ionization. 24,48,49Although the time these reflected Ar neutrals spend in the dense plasma region would be very short, reflection of Ar onto the target would occurwith a high probability due to the large mass difference. 50This latter effect was also found to be significant in HiPIMS dischargesusing a Ta target, as report by Rudolph et al. 51a n ds h o u l dl i k e l y be taken into account also here.52 Overall, the time evolution of the different species investigated by OES and mass spectrometry reveals that depletion of neutral Arthrough gas rarefaction occurs at the same time as increasing fluxesof energetic sputter-ejected W neutrals are observed. As a result, metal ions (W ++W2+) become dominant between t∼30–120μs, having more than 50% of the recorded total ion intensity. In partic-ular, at t∼50–60μs when the Ar gas ion intensity reaches a local minimum, the metal ion intensities (W ++W2+) represent ∼90% of the total recorded ion intensity with the major contribution, ∼80% due to W+. Based on these results, the delay time of the synchron- ized pulsed bias was selected to be Δt=6 0μs (taking into account the TOF for the metal ions to reach the substrate region), as sum-marized in Table III and described in Sec. II. B. Crystal phase and texture analysis Typical XRD out-of-plane θ–2θscans ( ψ= 0°) from films grown using the different bias configurations are shown in Fig. 5 .F o r purpose of clarity, the diffractograms are offset vertically and theintensities are displayed on a logarithmic scale to enhance the low intensity part of the diffractogram. Regardless of the bias configura- tions used, the films exhibit the α-W (bcc) phase with a predominant α-{110} crystal plane texture, which is commonly reported in the lit- erature for the growth of α-W films by sputtering. 9,14,18,21,26,53 Although the presence of the β-W phase cannot be excluded directly from these data due to the overlap between α-{110} peak (2θ= 40.265°) and β-{210} diffraction peak (2 θ= 39.866°), any signifi- cant amount of the β-W phase could be ruled out from the pole figures and the electron diffraction patterns from TEM images shown later in this section. Besides the dominating α-W peaks, a weak {222} peak is observed for the films deposited using a continuous bias atU s= 50 V and a {211} peak for the film deposited using synchronized bias at Us=2 0 0Va sw e l la sf o rt h ef i l md e p o s i t e du s i n gK rg a s . High-resolution θ–2θout-of-plane scans of the α-{110} peak are shown in Fig. 6 . The Bragg angle for the unstrained reflection is shown as a dotted line in the figure. The XRD scans reveal changesin the α-{110} peak intensity, I 110,t h ef u l lw i d t ha th a l fm a x i m u m (FWHM), and peak position depending on the choice of bias config-u r a t i o na n dw o r k i n gg a s .A ni n c r e a s eo ft h e I 110is demonstrated from 1.3 × 103cps at floating conditions to 2.3 × 103cps by applying continuous Us= 50 V, which is further increased to 4.9 × 103cps by synchronizing a pulsed bias with the metal-rich portion of theHiPIMS pulse. The results clearly indicate enhanced crystalline quality for tungsten films grown under synchronized U sconfigura- tion. Moreover, increasing Usin synchronized bias configuration FIG. 4. Time evolution of the IEDFs of (a) W+and (b) Ar+from 100 μs long, Uc= 800 V pulses, at a distance of 100 mm from the target surface. The IEDFs are acquired in 10 μs time intervals from t= 0 (pulse ignition) to t= 320 μs.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 155305 (2021); doi: 10.1063/5.0042608 129, 155305-6 ©A u t h o r ( s )2 0 2 1resulted in I110=7 . 2×1 03cps for Us=100 V and I110=1 . 7×1 04cps forUs= 200 V. The latter intensity is an order of magnitude higher than that obtained with continuous Us. The tungsten film grown in the Kr discharge exhibits a rather low intensity of I110=2 . 0×1 03cps. To get a better understanding of the variations in both the FWHM and shift of the α-{110} peak position, the FWHM value Г2θand the crystal lattice parameters are plotted in Fig. 7 .T h e unstrained lattice parameter for α-W is indicated with a red dotted line as a reference. By changing the bias mode from con-tinuous to synchronized U smode, Г2θdecreases from 0.75° to 0.68° even under the same Us= 50 V and decreases further to Γ2θ= 0.57° by increasing synchronized Usto 200 V. Moreover, by changing the working gas to Kr, Γ2θdecreases to 0.45°. A similar decreasing trend of the lattice parameter from 0.3202 nm at float-ingU smode to 0.3193 nm by applying synchronized Usmode was observed. A further decrease to ∼0.3176 nm was obtained by increasing the synchronized Uslevel and by using Kr as working gas. These results clearly illustrate the reduction of the compressivestress levels achieved upon changing the U smode. To confirm the stress state in the films, residual stresses were measured utilizing thesubstrate curvature of the thermally oxidized Si (001) substrate and calculating the stress using the Stoney formula. 32Obtained results inFig. 7 clearly demonstrate the relaxation in compressive stress from σ=−1.79 ± 0.11 to σ=−1.43 ± 0.15 GPa upon changing the Usmode from continuous to synchronized mode. Increasing the synchronized Uslevel contributes to further reduction to σ=−0.71 ± 0.04 GPa at Us= 200 V. Moreover, extremely low tensile stress with σ= 0.03 ± 0.08 GPa was obtained using the synchronized bias mode and Kr. To study the film texture in more detail, XRD pole figures of the {110} and {200} planes in α-W were obtained. Typical α-{110} pole figures for the W films deposited at continuous and synchron-ized U s= 50 V using Ar and Kr are displayed in Fig. 8 . The results clearly show the dramatic changes in film texture as bias mode and FIG. 6. Series of high-resolution θ–2θout-of-plane XRD scans of the α-(110) peaks from the W films grown on thermally oxidized Si (001) substrates with dif- ferent substrate bias configurations. The red dotted vertical line represents theexpected peak position for the unstrained α-(110) crystal plane. FIG. 7. FWHM Г2θ, lattice parameters, and residual stress of the W films grown on thermally oxidized Si (001) substrate with different Usconfigurations calculated from the diffraction peak of α-(110). The red dotted line represents the lattice parameter for an unstrained α-W crystal. FIG. 5. Series of θ–2θx-ray diffractograms scans performed with Cu –Kαradia- tion, from the W films grown on thermally oxidized Si (001) substrate with differ-entV sconfigurations. Red solid and gray dotted vertical lines represent bulk diffraction peak position of the α- and β-W phases, respectively [International Center for Diffraction Data-Powder Diffraction File (ICDD) No. 00-004-0806 forα-W and No. 00-047-1319 for β-W].Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 155305 (2021); doi: 10.1063/5.0042608 129, 155305-7 ©A u t h o r ( s )2 0 2 1working gas are changed. To display these variations in A compre- hensive way using radial intensities, sectional profiles of the pole figure intensities are given in Fig. 9 . Negative and positive degrees indicate integrated intensity values obtained from different 180°sectors. The indices given in the top of the figure indicate theout-of-plane direction of the crystal in case a peak appears at that position. The intensity profile for the continuous U s= 50 V config- uration reveals peaks at ψ∼± 35° and ±55°, for the α-{110} and {200} diffracting planes, respectively. These angles match the angleof 35.26° between ⟨110⟩and ⟨111⟩and the angle of 54.74° between ⟨200⟩and ⟨111⟩in a cubic structure. This illustrates a ⟨111⟩fiber orientation of the film in the growth direction. It has been reported in growth of tungsten films, which is believed to be due to ionchanneling. 9,54For a bcc lattice, the channeling direction is known as more favorable in ⟨111⟩.55The other peak at the center of the α-{110} pole figure and the shoulder at around ψ∼±45° in the α-{200} (angle between ⟨200⟩and ⟨110⟩in cubic crystals) indicates the presence of a secondary fiber texture component of α-⟨110⟩, which is similar to what has been reported by Girault et al.9 By changing to the synchronized bias mode, the ⟨110⟩-fiber texture becomes gradually more dominant as the bias voltage isincreased. This can be concluded from the increasing peak intensityat the center and at ±60° ( ψ∼±60° corresponds to the angle between the equivalent directions of ⟨110⟩)i nt h e α-{110} in Fig. 9(a) ,a n da t ψ∼±4 5 ° i n t h e α-{200} plot with increasing synchronized U slevel up to 200 V [ Fig. 9(b) ]. Instead, the intensity correlation with FIG. 8. Typical α-W {110} pole figures of the W films grown with (a) continuous bias Us= 50 V , (b) synchronized bias Us= 50 V using Ar, and (c) synchronized bias Us= 50 V using Kr.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 155305 (2021); doi: 10.1063/5.0042608 129, 155305-8 ©A u t h o r ( s )2 0 2 1⟨111⟩-orientation for the higher synchronized bias voltages of Us= 100 and 200 V becomes less pronounced. For the Kr sputtered tungsten film, a completely different trend is shown. Both the {110} and {200} pole figures show that thefilm does not have such a strong fiber texture as the films grownusing Ar. A broadening of the intensity from the center to around ψ∼± 35°in the α-{110} pole figure are due to grain growth shifting toward the ⟨211⟩direction. The studied pole figures also allow us to investigate the possi- ble misinterpretation of the film texture as being a presence of theβ-W phase in the films. If we instead of α-{110} assume β-{210} in the pole figure, which has an overlapping with α-{110} at 2 θ∼40°, radial intensity peaks should be seen at ψ∼±37° or ±67°, corre- sponding to the angle between equivalent directions of ⟨210⟩. These peaks could not be observed, and therefore we conclude thatthe films contain no detectable amount of β-W. C. Microstructure observation SEM micrographs of fracture cross sections of selected thin films in Fig. 10 show a columnar growth for all of the films depos- ited in this study. The film grown at U s= 50 V has the narrowest columns for the Ar deposited sample having a 50 ± 15 nm in average width near the film substrate interface increasing to 100 ± 27 nmnear the film surface, indicating competitive columnar growth. Filmsgrown at U s= 100 and 200 V clearly have wider columns showing 137 ± 27 and 198 ± 52 nm in average throughout the film thickness, respectively. The films grown using Kr also show a columnar grain structure throughout the film thickness, although it is difficult toestimate the width from the SEM images. However, in the latterpart of this section, the columnar width was estimated to bearound 50 –100 nm from plan-view SEM and the cross-sectional STEM images in Figs. 11 and12. To further investigate this change in grain growth between Ar and Kr sputtered films, microstructural characterizations of the surfa-ces were carried out on the films grown in synchronized bias mode at U s= 50 V using both Ar and Kr as working gas. Figure 11 shows plan-view SEM images for both samples and their correspondingAFM images and associated sectional profiles. Both SEM and AFMshow streaked or rippled features on the Ar-grown films with a sizeof less than 20 nm [ Figs. 11(a) and11(b) ]. These nanoridge patterns resemble structures reported by Singh et al . who explained their formation as a result of anisotropic diffusion of sputtered W parti-cles over the α-W (100) surface. 55The sizes of the grains are in the range of 100 –200 nm, which correlate well with the columnar width observed by SEM ( Fig. 10 ). The Kr sputtered film consists of elongated grains with a faceted surface [ Figs. 11(c) and 11(d) ] with 10 –50 nm width and 100 –200 nm length. These facets contrib- ute to the rougher surface observed for Kr sputtered films as com-pared to the case of Ar. The RMS value, as determined by AFM, isapproximately a factor of three higher for the Kr sputtered film. In-depth observations of the microstructure of the films were carried out using (S)TEM analysis of plan-view samples preparedto visualize the microstructure of the upper part of the films (top50–70 nm region). Figures 12(a) ,12(b) ,12(d) and12(e) show plan- view high angle annular dark-field (HAADF)-STEM micrographs and corresponding selected-area electron diffraction (SAED) FIG. 9. Comparison of the sectional profiles of (a) α-W {110} and (b) {200} pole figures of the W films grown with different bias configurations. The profile was obtained in the section between w= 0 and 180°. Black dotted vertical lines rep- resent the expected angle between the subjected crystal plane and denoted ori-entation. The indices given on the top are the crystal orientations perpendicular to the substrate in case ⟨110⟩or⟨200⟩intensities, respectively, are observed at these positions.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 155305 (2021); doi: 10.1063/5.0042608 129, 155305-9 ©A u t h o r ( s )2 0 2 1patterns from the films grown under synchronized Us= 50 V using Ar and Kr as the working gas. The SAED pattern in Fig. 12(a) from the film deposited by using Ar contains the intense α-110 diffrac- tion ring, together with several diffraction rings from lattice planes (211, 220, 222, and 321), which can be fitted to the expected tung- sten structure, in agreement with the θ–2θx-ray diffractograms scans shown in Fig. 5 . The α-110 ring in the SAED patterns exhibit spots uniformly distributed around the rings, indicating that, while the 110 grains are well aligned along the growth direction, they are randomly distributed azimuthally. The principle contrast mechanismin HAADF-STEM is mass thickness. However, the presence of defects gives a bright contrast.56Here, constant composition and thickness can be assumed and the principle contrast mechanism is from thepresence of defects. HAADF-STEM micrographs in Fig. 12(b) exhibit two different types of grains, low-contrast grains with low defect con-centration and high-contrast grains with a high defect concentration. It is expected that the low-contrast grains correspond to the rippled grains in the SEM observations seen in Fig. 11 ,i fi ti sa s s u m e dt h a t this nano-ridged surface is attributed to the α-W {100} crystal plane. 55 The high-contrast grains in between the rippled grains causes the for- mation of sub-grains roughly 10 –20 nm wide, as can be seen from the cross-sectional HAADF-STEM image in Fig. 12(c) . SAED patterns for the films deposited by Kr in Fig. 12(d) shows no clearly pronounced direction as seen in the intense (110) ring inAr sputtered films. Instead, the presence of a wider variety of diffrac-tion rings with localized diffraction spots along the rings indicates randomly oriented grain growth, which is in agreement with the results in the pole figure measurement in Fig. 10 . In addition, the grain structure observed in Fig. 12(e) resembles the observation from Figs. 11(c) and 11(d) , showing elongated grains. Cross-sectional STEM in Fig. 12(f) reveals no evidence for the formation of sub- grains seen for the Ar deposited film in Fig. 12(c) . Additionally, EDS analysis were performed on the cross-sectional samples shown inFigs. 12(c) and12(f) for both Ar and Kr sputtered films. Although compositional quantification and its spatial mapping are difficult, qualitatively the presence of O was clearly confirmed in both cases. Detailed analysis of high-resolution TEM micrographs, shown inFig. 13 , was carried out to investigate the localized concentration of defects. Starting from the FFT of the HRTEM images in Figs. 13(a) and13(f) ,m a s k sw e r ea p p l i e do ns e l e c t e ds p o t sf o rA r[ Fig. 13(b) ] and Kr sputtered films [ Fig. 13(g) ] to highlight a specific set of atomic planes from each grain. Then, to identify the location of spe-cific defects, an inverse FFT algorithm was performed to reconstructfiltered real space images, which were superimposed on the HRTEMimages as shown in Figs. 13(c) and13(h) . These results highlight the positions of specific defects as shown in insets 1 and 2, where differ- ent regions display defects such as lattice distortions and dislocationsi nt h em a s k e da t o m i cp l a n e s[ Figs. 13(d) ,13(e) ,13(i),a n d 13( j) ]. The HRTEM analysis show that the grain boundaries are fully densei nt h ec a s eo fA rs p u t t e r e df i l m s ,w h i l eu n d e r d e n s es t r u c t u r e sa r e shown in the case of Kr. The densification of the grain boundaries leads to the formation of defects in their vicinity, which appear topropagate into the grain, as can be seen in Fig. 13(d) for the Ar sput- tered film. 57The presence of these defects will also result in electron scattering and increase the electrical resistivity of the film, as clearly indicated in Sec. III D , while in the case of Kr sputtered films in Figs. 13(i) and13( j) , defects are observed but with much lower concentration. Overall, a clear evidence of residual defectsnear the grain boundaries are demonstrated for Ar sputtered films, while the grains for the Kr sputtered film appears to have a microstructure with much lower defect density. D. Electrical resistivity The electrical resistivities of the tungsten films deposited under the different bias configurations are presented in Fig. 14 . The resistivity decreases from 44.2 to 34.7 μΩcm by changing the FIG. 10. Fracture cross-sectional SEM micrographs from the Ar sputtered W films grown with synchronized Usof (a) 50, (b) 100, (c) 200 V , and (d) by using Kr gas with sync. Us=5 0V .Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 155305 (2021); doi: 10.1063/5.0042608 129, 155305-10 ©A u t h o r ( s )2 0 2 1bias mode from continuous to synchronized, and it further decreases to 18.3 μΩcm by increasing the synchronized bias voltage from Us=5 0t o2 0 0V .T h ef i l mg r o w nu s i n gK re x h i b i t st h el o w e s tr e s i s t i v - ity, 14.2 μΩcm. This is obtained without intentional heating of the substrate or post-annealing. Since all films in this investigation have the same crystal phase, α-W, the difference in resistivity between the films must arise from electrons scattering on external surfaces (topsurface and film/substrate interface) or on film defects such as grainboundaries and lattice imperfections as can be seen in Figs. 12 and13. To analyze the contributions to the difference in resistivity between bias configurations, we focused on the role of Ar impuritiesin the films, as they are known to increase the resistivity by acting aselectron scattering centers. 36By using a resistivity sensitivity to the incorporation of Ar impurities, SAr=9 . 1 μΩcm/at. %, reported by Meyer et al .,36concentrations of Ar impurities can be inverselycalculated using a model proposed by Ligot et al. from the total resis- tivity obtained above.6Details of the calculations are described in the Appendix . To compare the relative trend between the different bias configurations, calculated values are normalized by that from the floating bias configurations, as plotted with open diamond symbols inFig. 14 . Experimental measurements of the Ar content were also performed (EDS) and denoted with filled diamond symbols in thefigure. The correlation between the Ar impurity values calculatedfrom the resistivity measurements and measured Ar values are clear except for the highest synchronized bias voltage of 200 V. IV. DISCUSSION The results presented in Sec. IIIdemonstrate the importance of the choice of substrate bias configuration and working gas (Ar FIG. 11. (a) Plan-view SEM image with (b) corresponding AFM image with its cross-sectional profile for a W film deposited using Ar gas. The same is shown in (c) an d (d) for W films deposited using Kr as working gas. Both films were deposited in synchronized bias mode with Us=5 0V .Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 155305 (2021); doi: 10.1063/5.0042608 129, 155305-11 ©A u t h o r ( s )2 0 2 1or Kr) during W film growth. The selection of ion species for ion bombardment and their energies has a clear effect on defectcontent and physical properties of the grown films. The time-resolved plasma characterization reveals that the sput- tering process initially generates a gas-dominated ion flux that shifts to metal dominated later in the pulse. After the discharge voltage is switched off, the flux again becomes gas-dominated. The shift froma gas-dominated to a metal dominated flux is understood to be dueto a preferential ionization of sputtered particles as compared to Ar during the HiPIMS pulse due to decreasing electron temperature with increasing discharge current (mainly reduces the ionizationefficiency of Ar 38) and due to working gas rarefaction,58which is particularly pronounced when sputtering heavy elements like tung-sten. 59In the present work, this alteration of ion fluxes was used to selectively accelerate the metal ions by changing the bias configura-tions from continuous to synchronized pulsed bias mode. Based on the mass spectroscopy measurements shown in Fig. 3 , 60% of the total ion intensity was W ions at the time period 60 –160μsf r o mt h e onset of the HiPIMS pulse. By applying the substrate bias duringthis time period, the contribution of W ions dominates during film growth. On the other hand, Ar ions dominate the accelerated flux in the continuous bias mode, as 68% of the total ion intensity consisted FIG. 12. SAED (a) and HAADF-STEM (b) images taken from the plan-view specimen of the W film deposited using Ar gas, and its corresponding cross-sectional image (c). (d) –(f) are the respective images from the W film deposited with using Kr gas. Indexing of the SAED patterns is as indicated.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 155305 (2021); doi: 10.1063/5.0042608 129, 155305-12 ©A u t h o r ( s )2 0 2 1of Ar ions, as confirmed by a time averaged acquisition at same dis- charge conditions. The above observations allow the correlation between the time evolution of plasma characteristics and film growth by firstaddressing the evolutions of crystal textures. Factors related with texture evolution for polycrystalline W films have been discussed in the literature, addressing the importance of the minimization of total free energy60along with other factors, such as an ion channel- ing effect,9,54,61surface stress/strain, and surface energy modifica- tion due to mixing effects, which could be enhance at nanometric scales.9The former, surface energy minimization effect, is generally driven by thermodynamics and is well known to preferentiallypromote the growth of a ⟨110⟩texture. 9,14,18,21,26,53 For films grown using continuous bias, the ⟨111⟩texture com- ponents are found as shown in Fig. 9 , contrary to the thermodynami- cally favored ⟨110⟩texture. By considering the fact that Ar ions dominate the incident flux in the continuous bias scheme, this couldoriginate from the ion channeling effect, 9,54which explains the higher survival rate for the open crystallographic direction (the ⟨111⟩in a bcc lattice) that channels the incident ions into the lattice with less lattice distortion and a lower sputtering yield as a result.62 By changing the bias configuration to synchronized mode, where the preferential acceleration of W ions becomes dominant,transformation to a ⟨110⟩fiber texture is clearly observed, as shown inFig. 9 . In this situation, efficient momentum transfer by self-ion-irradiation can be expected due to the perfect mass match between the collision partners. Through recoil collisional andforward sputtering processes during self-ion-irradiation, displaciveknock-on motion at surface atomic migration events will be initiated combining with atomic relaxation, or “recrystallization, ”processes. 63 Such processes can be expected to lead to the generation of islands FIG. 13. Plan-view high-resolution TEM images of (a) Ar deposited W films with (b) its respective FFT image and (c) reconstructed FFT−1image superimposed on the HRTEM image magnified at selected area in (a). The reconstructedimages were produced using the highlighted regions with a red circle in the FFT in (b). (d) and (e) are magnifications of reconstructed FFT−1indicated as insets 1 and 2 in (c). (f) –( j) are the respective images from the W film deposited with Kr gas. Both films were deposited in synchronized bias mode with Us=5 0V . FIG. 14. Electrical resistivity (left) and normalized concentrations of Ar impuri- ties in the W films (right) grown with different bias configurations. The bulk resis- tivity is also shown for comparison (red dotted horizontal line). Calculated Ar impurities [open diamond symbols denoted as Ar impurities (Cal.)] are estimatedby a model proposed by Ligot et al .6Ar atomic concentrations determined by EDS analysis are also shown for comparison [filled diamond symbols denoted as Ar impurities (Exp.)] Both of the plotted values of Ar impurities are normal- ized with the Ar concentrations of the film grown under floating biasconfigurations.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 155305 (2021); doi: 10.1063/5.0042608 129, 155305-13 ©A u t h o r ( s )2 0 2 1with low-energy {110} crystallographic planes to minimize the surface and interface energy during island growth.64Recrystallization during the coalescence of small clusters is also known to lead to theformation of highly textured grains more easily. 65 Moreover, by increasing the synchronized bias voltage to Us= 100 and 200 V, the ⟨110 ⟩texture becomes stronger and columnar grains become larger, resulting in lattice relaxation with significant decrease of compressive stress as seen in Figs. 6 and7. In addition to the above kinetically induced displacement, this could also be attributed to temperature-induced structuralchanges, for instance, due to ion-induced thermal spikes during film growth. 61This thermally induced process may result in sec- ondary recrystallization, also called abnormal grain growth, inwhich the degree of texture is further enhanced with a muchlarger in-plane grain size. 66 The above observation of the film growth events from the texture evolution can be correlated with stress formation and relaxation mechanisms at the different bias configurations. First,the highest compressive stress ( σ=−1.79 ± 0.11 GPa) was found for the W films grown using continuous bias, as shown in Fig. 7 . This value is very close to the value found by Engwall et al. for HiPIMS α-W films ( σ∼−2.5 GPa at a thickness of 500 nm) deposited at comparable experimental conditions (working pres-sure of 0.93 Pa, U c:−646 V, Us:−45 V).26High compressive stress was also reported for most of the previously reported α-W films in the range of 3 –7 GPa.3,4,16,18,20,21 The reason for such a high compressive stress is due to the high flux fraction of energetic species reaching the substrate duringfilm growth. This flux is mainly composed of gas atoms reflectedback from the target and plasma ions accelerated across the sub- strate sheath (mainly W +and Ar+ions). The impact of these inci- dent energetic particles produces recoil implantation into the filmof surface atoms and entrapment of working gas atoms, and conse-quently building up compressive stress into the film. Through aseries of knock-on mechanisms, the subsurface would also be affected inducing the creation of point defects at grain boundaries and/or interstitial sites above a certain energy threshold. 67 The contribution of mean energy deposited per incoming particle Edep(eV) to the stress evolution from a high-energy vapor flux is expressed by Colin et al. , as a sum of the average energy of sputtered atoms, that of backscattered gas atoms and the acceler- ated energy of ions at substrate sheath, /C22Ei.68A gradual increase of compressive stress up to σ∼−2.5 GPa was demonstrated with increasing Edepup to∼80 eV/atom in the case of Ta films, due to the increase in the number of defects created with increasing deposited energy.68In this regard, the contribution of the energy induced by the backscattered gas atoms is large in the case of thepresent study, as we used a heavy-mass target ( m W=1 8 3 . 8 4a m u ) as compared with the mass of gas atoms Ar ( mAr=3 9 . 9 5a m u ) .69 According to the model reported by Matsui et al.,52an estimate of the maximum backscattering energy Eb,Arof Ar+ions in 180° reflections from a W target atoms yields a maximum Ar backscat-tering energy E b,Ar= 330 eV, assuming a kinetic energy of the inci- dent Ar+ions of 800 eV. Due to the elastic collisions during transport, this energy can be somewhat lower than the above esti- mation.70However, it is still high as compared to the contribution of incident ions entering the substrate sheath, as the averagekinetic energy is in the range of /C22E0 Arþ/C253e V f o r A r+and /C22E0 Wþ/C25 13 eV for W+ions, as estimated from our mass spectrometry mea- surements. This high-energy vapor flux will contribute to anincrease in the mean energy per deposited particle, leading to theproduction of gas entrapment at the subsurface level. By changing the bias configuration to synchronized bias, the compressive stress decreased from σ=−1.79 ± 0.11 to −1.43 ± 0.15 GPa as shown in Fig. 7 . The stress reduction can be explained by the reduction of trapped Ar in the films, as seen fromthe∼20% reduction in the amount of Ar in the film when chang- ing the bias configuration ( Fig. 14 ). By the use of synchronized bias, incident ions at the substrate during the remainder of the pulse, mainly composed by Ar ions as shown in Fig. 3 ,h a v ea n energy /C22E i/C2510 eV (assuming the substrate at floating potential). This is below the bulk lattice-atom displacement threshold, lower-ing the rate of Ar gas entrapment. However, there is still some gas entrapment due to the energetic backscattered Ar neutrals, which is unaffected by changing the bias configurations. This is clearly seenin the sub-domain structure as shown in Figs. 12(b) and12(c) , and in the lattice distortions shown in Fig. 13(d) and13(e) . Of greater importance when discussing the stress evolution in this study is a significant drop of compressive stress to σ=−0.71 ± 0.04 GPa by the increase of synchronized bias voltage up toU s= 200 V ( Fig. 7 ). Based on the above discussion in the evolution of crystal textures, thermally induced processes seem to be contribut- ing to the film growth at this bias configuration, as the relaxation of crystal lattice and coarsening of the grains were observed as shown inFigs. 7 and 10. Shallowly implanted and trapped gas atoms are, in general, known to be unstable and can be annihilated by diffusiontoward the nearest underdense region if sufficient energy is provided, e.g., by ion irradiation. 71In the present case, the shallowly implanted hyperthermal W ions selected by the synchronized bias efficientlytransfer enough energy to trigger diffusion of entrapped Ar atomstoward underdense regions, i.e., the free surface of the growingfilm. 67This leads to the desorption of Ar inclusions from the growing film, as evidenced by the significant reduction of Ar con- centration ( ∼50% reduction compared when using continuous bias) as shown in Fig. 14 . Evidence of the thermally activated dif- fusion by ion-induced processes is also seen from the largercolumnar grain widths for films deposited using synchronized bias U s= 100 and 200 V as shown in Fig. 10 . If the trapped region of the Ar gas atoms is in the vicinity of the grain boundaries, it isknown to diffuse not only toward the free surface but also towardgrain boundaries and to enhance grain boundary motion causing grain coarsening during the film growth. 72Consequently, even at the concurrent production of gas entrapment during film growth, ther-mally induced events attributed to the efficient moment transfer bythe selective acceleration of the metal ions rich incident flux contrib-ute to the desorption of rare gas atoms, and thus, the total stress of the films decreases with increasing synchronized bias voltage. An alternative route to decrease film stress is to simply sup- press the origin of Ar gas entrapment, i.e., backscattered Ar neu-trals, by using heavier sputtering gas as presented in the case ofKr sputtered films. Effects of the selection of sputtering inert gas in W film growth has been thoroughly studied in ion beam sput- tering by Hoffman et al . 50By the use of Kr gas instead of Ar under the primary ion energy of 600 eV, trapped gas compositionJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 155305 (2021); doi: 10.1063/5.0042608 129, 155305-14 ©A u t h o r ( s )2 0 2 1was significantly reduced from 1.97 to 0.06 at. %, resulting in a reduc- tion of residual compressive stress from −2.7 to 1.3 GPa. In the present case, by assuming the incident Kr+energy of 950 eV, the cor- responding maximum backscattering energy for Kr+isEb,Kr= 133 eV. This is 1/3 of the value estimated in Ar process, and it will clearly con-tribute to reduce the production of rare gas entrapment, leading to a reduced fraction of point defects in the films as we confirmed by the HRTEM and inverse FFT analysis in Figs. 13(f) –13( j) .T h i s is, in fact, also shown by the almost unstrained crystal lattice withextremely low tensile stress σ= 0.03 GPa ( Fig. 7 ) resulting in the lowest resistivity of 14.2 μΩcm for Kr sputtered films ( Fig. 14 ). This was achieved without increasing the substrate potential to U s= 200 V as in the case of Ar sputtered films. One important aspect concerning the Kr sputtered films is the slightly narrower columnar growth with lesser degree of pre-ferred orientation as can be seen in Figs. 8 –12. One reason for limited grain coarsening during growth of Kr sputtered films can be explained by the low production rate of defects. Atwater et al. investigated grain growth mechanisms enhanced by ion bombard-ment during growth of Ge, Si, and Au films. 72In this report, they highlighted the importance of thermal migration of bombardment- generated defects across the boundary by showing the proportional relation between grain boundary motion and the defect concentra-tion at the boundary. Another possible explanation for limiting grain growth could be due to impurities, 73in particular oxygen, which still is present at the base pressure used in the present study ( ∼8.0 × 10−4Pa). The influence of incorporation of atmospheric contaminants,such as oxygen, on the changes in film structure and grain orien-tation are summarized as a function of a ratio of oxygen flux to that of deposited atoms by using a structure zone model. 74In the present case, the incident O to W flux ratio JO/JWtoward the sub- strate can be roughly estimated from the above base pressure toJ O/JW∼2.1 × 1019/2.9 × 1019= 0.73. Here, the O flux was derived from the following classical equation:75 J¼P/ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2πmkTp ¼2:7/C21022P, (1) where Pis the pressure, mis the oxygen molecular mass (mO2= 31.99 amu), kis the Boltzmann constant, and Tis the gas temperature, assuming to be room temperature. The W flux was estimated from the density of the W film (19.05 g/cm3, as mea- sured by XRR (X-ray reflectometry) for 50 nm Kr HiPIMS sput-tered films), and the coating volume of the Kr sputtered sample(15 mm × 15 mm × 842 nm). The above estimated value is in good agreement with the range where 3D equiaxial grains with random orientations were obtained for pure Al films ( J O/JAl∼10−1–1).74 In this growth regime, residual oxygen in the atmosphere can be incorporated and it segregates into the grain boundaries, eventuallylimiting grain coarsening during coalescence and film growth. 74 With slightly higher oxygen concentration levels, initial coalescenceof discrete small crystals of random orientation can be also affected,resulting in grains with random orientation. By balancing with theeffects from ion-induced-irradiation events as discussed above, the resulting texture can still remain with columnar growth extending through the films, but with a lesser degree of preferred orientation.Complementary TEM-EDS analysis from the Kr sputtered sample also showed the evidence of O presence in the film. The incorporation of the oxygen is likely affecting the resis- tivity value of Kr sputtered films, ρ= 14.2 μΩcm, as shown in Fig. 14 . Despite of its high crystal quality with extremely low residual stress, the obtained resistivity is higher than the bulk value (5.28 μΩcm) as well as epitaxially grown films annealed at 850 °C (6.3 μΩcm for 300 nm-thick films) 76and DC magnetron sputtered films at UHV conditions (12 μΩcm for 150 nm-thick films).2This can be caused by the incorporation of oxygen into the grain boundary, which generates new electron diffusion centers.6By reducing the inclusion of oxygen into the films, by, e.g., operating under UHV conditions, highly ⟨110⟩textured W can be expected leading to a further reduced electrical resistivity. V. CONCLUSION An effective pathway to minimize the ion-induced compres- sive stress during low temperature film growth of high- Tmmaterials is investigated in this work. It is shown that stress-free, unstrainedsingle phase α-W films can be obtained, without utilizing post- annealing, by HiPIMS through the means of a synchronized pulsed substrate bias that selectively enhances the energy of the metal portion of the ion bombardment. Plasma diagnostics using time-resolved optical emission spec- troscopy and mass ion spectroscopy clearly demonstrated a timeevolution of the plasma ion-composition reaching the substrate. The ion-composition is initially and finally gas dominated, but there is a window ranging from t∼30–120μs after the HiPIMS pulse where metal ions are dominating, having more than 50% ofthe total ion intensity. This is due to a combination effect of effi-cient ionization of sputtered particles during HiPIMS pulse and a strong working gas rarefaction. By applying a synchronized pulsed substrate bias at the metal dominated portion, films that normally(using a continuous bias) achieve a ⟨111⟩transform to highly ⟨110⟩ textured films. We also find that by increasing the synchronized negative pulsed substrate voltage from U s= 50 to 200 V, there is a general increase of columnar size, a higher degree of preferred ori-entation, and a lattice relaxation resulting in a lower film compres-sive stress. Concurrent reduction of Ar incorporation by theincrease of U ssuggested the contribution of the efficient momen- tum transfer by the selective W ion irradiation to kinetic collision cascades and the consequent thermally induced events, which helpanneal out defects and desorb trapped noble gas atoms. The quality of the films can be further enhanced by the sup- pression of the high-energetic backscattered Ar gas atoms by shift- ing to Kr. In this case, a fully relaxed crystal lattice with almost no stress, only σ= 0.03, was demonstrated. This is all achieved without the need for substrate heating or post-annealing, that are normallyneeded to achieve comparable film properties. Finally, it is concluded that the large contribution of W ions dominated fluxes and their selective acceleration will provide effi- cient momentum transfer, which enhances surface adatom mobili-ties and consequent annihilation effects even at low depositiontemperatures. This will potentially open up more efficient pathways for reducing ion-induced point defects and compressive stress for pure metallic films even for the ultra-thin (few tens of nm-thick)Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 155305 (2021); doi: 10.1063/5.0042608 129, 155305-15 ©A u t h o r ( s )2 0 2 1range for, e.g., interconnect materials for semiconductor metalliza- tion like Cu, Co, and Ru.77 ACKNOWLEDGMENTS This work was supported by the Swedish Research Council (No. VR 2018-04139), the Swedish Government Strategic ResearchArea in Materials Science on Functional Materials at LinköpingUniversity (Faculty Grant SFO-Mat-LiU No. 2009-00971), and the Japan Society for the Promotion of Science (JSPS), for a Fund for the Promotion of Joint International Research (No.17KK0136). Theauthors would also like to thank Professor Hiroyuki Kousaka atGifu University for support in the usage of ICCD camera for theOES measurements, Ana Beatriz Chaar at Linköping University for assistance with the OES measurements, Yoshikazu Teranishi at Tokyo Metropolitan Industrial Technology Research Institute, andHidetoshi Komiya at Tokyo Metropolitan University for assistancewith the SEM observations. Petter Larsson at Ionautics AB is also greatly appreciated for the technical support. APPENDIX: INVERSE CALCULATION OF AR IMPURITIES FROM TOTAL ELECTRIC RESISTIVITY We are interested to see if we can draw any conclusion on which of the remaining factors that dominate the observed resistivityvalues. We have the XRD-measurements that, through the peak broadening, give us an estimate of the defect content of the films and film impurities are expected to be dominated by atoms from theresidual gas in the vacuum system (mainly water) and from the sput-tering gas (Ar or Kr). It is generally observed that reactive impurities, such as O and OH, in the grain boundaries, 16while noble gas atoms can be included in the W lattice through energetic implantation.36 Mayadas and Shatzkes proposed a model assuming polycrystallinethick films consisting of a columnar grain with an average diameterd. 78In their model, the scattering probability of electron waves at the grain boundaries is taken into account by the reflection coefficient R. The intrinsic resistivity ρintc a nt h e nb ed e s c r i b e db y ρint¼ρ01/C03 2αþ3α2/C03α2/C03α3ln 1þ1 α/C18/C19 /C20/C21/C01 ,( A 1 ) where α¼λe d/C1R 1/C0R,( A 2 ) Here, ρ0is the bulk resistivity and λeis the electron mean free path (19.1 nm for bcc W at 293 K76). Hence, ρintof the W films can be described using the two variables ρint¼ρ0F(R,d). To esti- mate the intrinsic resistivity ρintfor the present W films under dif- ferent bias configurations, dis set to be the Scherer size estimated from GI-XRD scans. To establish an upper level of the effect from the grain boundaries, we assume Rto be 0.65, which is the highest value reported for W films that we can find.6 If we make the bold assumption ( Ris constant for these films) that the deviation between the experimental and calculated resistivities are due to lattice defects, we can follow Ligot et al. that the increase of resistivity due to the incorporation ofimpurities, ρimp, can be approximated to be the sum of effects from different impurities,6 ρimp¼Sn/C2at:%(n), (A3) where nis an atomic element and Snis the sensitivity of resistiv- ity for that element in the W. The dominating lattice impurity in the present W films is believed to be the noble gas. Ligot et al . gave the value of SAr=9 . 1 μΩcm/at. %. By assuming total resis- tivity obtained in the resistivity measurement is a sum of ρintand ρimp, atomic concentrations of Ar in the W films can be inversely estimated using Eqs. (A1) –(A3) . 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9.0000096.pdf

AIP Advances 11, 025113 (2021); https://doi.org/10.1063/9.0000096 11, 025113 © 2021 Author(s).Jahn–Teller distortions in (Co1–xCux)Cr2O4 (x = 0.5, 0.25) nanoparticles: Structural, magnetic and electronic properties Cite as: AIP Advances 11, 025113 (2021); https://doi.org/10.1063/9.0000096 Submitted: 15 October 2020 . Accepted: 22 January 2021 . Published Online: 08 February 2021 P. Mohanty , C. J. Sheppard , B. P. Doyle , E. Carleschi , and A. R. E. Prinsloo COLLECTIONS Paper published as part of the special topic on 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials and 65th Annual Conference on Magnetism and Magnetic Materials ARTICLES YOU MAY BE INTERESTED IN Jahn–Teller distorted Cu 1−xNixCr2O4 (x = 0, 0.5, 1) nanoparticles Surface Science Spectra 27, 024015 (2020); https://doi.org/10.1116/6.0000444 Multiferroic nanoparticles of Ni doped CoCr 2O4: An XPS study Surface Science Spectra 27, 014003 (2020); https://doi.org/10.1116/1.5127349 Magnetoelectric and multiferroic properties of spinels Journal of Applied Physics 129, 060901 (2021); https://doi.org/10.1063/5.0035825AIP Advances ARTICLE scitation.org/journal/adv Jahn–Teller distortions in (Co 1–xCux)Cr2O4(x=0.5, 0.25) nanoparticles: Structural, magnetic and electronic properties Cite as: AIP Advances 11, 025113 (2021); doi: 10.1063/9.0000096 Presented: 6 November 2020 •Submitted: 15 October 2020 • Accepted: 22 January 2021 •Published Online: 8 February 2021 P. Mohanty,1,a)C. J. Sheppard,1 B. P. Doyle,2 E. Carleschi,2 and A. R. E. Prinsloo1 AFFILIATIONS 1Cr Research Group, Department of Physics, University of Johannesburg, P.O. Box 524, Auckland Park, Johannesburg 2006, South Africa 2ARPES Group, Department of Physics, University of Johannesburg, P.O. Box 524, Auckland Park, Johannesburg 2006, South Africa Note: This paper was presented at the 65th Annual Conference on Magnetism and Magnetic Materials. a)Author to whom correspondence should be addressed: pankajm@uj.ac.za ABSTRACT The spinel ferrimagnetic compound CoCr 2O4demonstrates a spin spiral ( TS) ordering at 25 K, as well as an anomaly at 15 K termed as lock-in transition ( TL). From crystallographic perspective CoCr 2O4retains the cubic phase down to 11 K. On the other hand, the normal spinel CuCr 2O4crystallizes into the cubic phase above 850 K, below which Jahn-Teller (J-T) activity of the Cu reduces the crystal symmetry by transforming it to a tetragonal phase. Contraction of CuO 4tetrahedra towards the formation of a square planar structure accounts for the tetragonal to orthorhombic structural transition at ∼130 K associated with the ferrimagnetic Curie temperature ( TC). Considering the differences in crystal structure and magnetism of these two compounds, the current work investigates the modification in crystal structure and magnetic behaviour by mixing Co site with Cu in CoCr 2O4. To achieve this, (Co 1–xCux)Cr 2O4(x=0.5, 0.25 and 0.75) nanoparticles were prepared by chemical routes. X-ray diffraction (XRD) revealed the retention of cubic structure for the samples calcined at a temperature of 600○C for x=0.25 and 0.5. On the other hand, J-T distortion becomes prominent for x=0.75. Hence, only the compositions with x=0.25 and 0.5 were studied in detail as unusual cubic phase retention is observed in these compounds. The temperature dependent magnetization studies revealed that the TCvalues of both the samples, 103 K for (Co 0.5Cu 0.5)Cr 2O4and 99 K for (Co 0.75Cu 0.25)Cr 2O4, compare well with the value reported for CoCr 2O4. However, the feature related to TSis quite prominent for x=0.25, whereas it is suppressed for x=0.5. The electronic properties of the cations associated with these compounds, probed using X-ray photoelectron spectroscopy (XPS), indicate that Cu and Co mostly has a 2 +oxidation state whereas that of Cr is 3 +. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/9.0000096 I. INTRODUCTION Multiferroic CoCr 2O4forms a normal cubic spinel structure of the form AB 2O4(A=tetrahedral site occupied by Co2+ions, B =octahedral site occupied by Cr3+ions).1The compound undergoes a magnetic phase transition to ferrimagnetic ordering from param- agnetic phase at Curie temperature, TC≈93 K.1The most fascinat- ing features of this compound relates to the magnetic transitions that occur in the temperature range below the TC. These transitions observed at the temperatures associated with the spiral ordering and lock-in transitions, that occur at TS≈25 K and TL≈15 K.1Experi-mental findings evidenced that the origin of the multiferroicity stems from the conical spin modulation below TS.1In order to examine the low temperature magnetic structure, temperature dependent neu- tron diffraction measurements of the polycrystalline samples were carried out.2Lawes et al.2confirmed the stability of the cubic phase up to 11 K and indicated the signature of complex magnetic order- ing in this compound.2The observed low temperature magnetic transitions were also substantiated through heat capacity ( Cp) and dielectric measurements.2 On the other hand CuCr 2O4, having a normal spinel structure with Cu2+atAsite, demonstrates orbital ordering.3The structure AIP Advances 11, 025113 (2021); doi: 10.1063/9.0000096 11, 025113-1 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 1. (a) XRD patterns measured for nanosized (Co 1–xCux)Cr2O4withx=0.5 and 0.25. TEM images of nanostruc- tured (b) (Co 0.5Cu0.5)Cr2O4and (c) (Co 0.75Cu0.25)Cr2O4where correspond- ing insets show SAED. undergoes a cubic to compressed tetragonal symmetry on cooling at 850 K (=TJT) because of the cooperative Jahn-Teller (J-T) order- ing, and a tetragonal to orthorhombic structural change below 130 K at the onset of its transition from paramagnetic to ferrimagnetic regime, at the Curie temperature ( TC).3,4The Cu2+and Cr3+ions have 3 d9and 3 d3spin configurations, respectively. The cooperative J-T effect destabilizes the crystal structure leading to a splitting of the triply degenerate t2gorbital into a non-degenerate orbital at elevated energy and doubly degenerate orbitals at lower energy.3 The orbital ordering in CuCr 2O4is expected because of the exis- tence of unpaired 3 delectrons related to the Cu2+ion occupying thexyorbital.3The orthorhombic crystal distortion of the J-T effect induced tetragonal CuCr 2O4phase persists concomitantly with fer- rimagnetic phase transition at 130 K, as evidenced by temperature dependent high-resolution synchrotron XRD results.4Cpmeasure- ments of CuCr 2O4show a transition at 130 K and an additional anomaly at 155 K, indicating a second magnetic phase.4Tomiyasu et al.5explored this new magnetic structure by conducting neutron diffraction studies within 155 K and 125 K and described it as almost collinear magnetic arrangement in pyrochlore lattice formed by Cr.However, under 125 K, a ferrimagnetic component of non-collinear nature evolves due to the Cu-Cr interaction.5 Looking at the uniqueness of both CoCr 2O4and CuCr 2O4 motivated the current work to investigate the crystal structure, mag- netism and electronic behaviour with Co site substituted by J-T active Cu ions. Besides the effect of cationic site substitution, the possible role of particle size on crystal structure is also investigated. II. EXPERIMENTAL METHODS Chemical sol-gel method was used to synthesize powders of (Co 1−xCux)Cr 2O4nanoparticles, with x=0.25, 0.5 and 0.75, using 0.5 M stock solutions of commercially available respective metal nitrates. After gelation the gel residue was dried over a hotplate. Separate portions of the dried powder were calcined in a box fur- nace at 600○C and 800○C for 1 hour, respectively. X-ray diffraction (XRD) with Cu– Kαradiation was employed for structural charac- terization and phase identification and microstructure was analyzed using transmission electron microscopy (TEM). Magnetic field and temperature dependent magnetization studies were carried with a 14 T Cryogenic Measurement System having VSM insert. Room FIG. 2. XRD patterns depicting cubic to tetragonal splitting of (220) peak as function of xand calcination temperature: (a) 600○C and (b) 800○C. AIP Advances 11, 025113 (2021); doi: 10.1063/9.0000096 11, 025113-2 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv temperature electronic properties were probed using X-ray pho- toelectron spectroscopy (XPS) experiments employing a SPECS PHOIBOS 150 electron energy analyser, and monochromatised photon source produced by Al– Kαradiation with energy 1486.71 eV. III. RESULTS AND DISCUSSION Fig. 1(a) depicts the X-ray diffraction (XRD) plots of (Co 1−xCux)Cr 2O4(x=0.5 and 0.25) powders calcined at 600○C that show reflections related to CoCr 2O4(JCPDS: 221084) with Fd3m symmetry. The broadening of XRD peaks as observed in the present case is ascribed to the nano size effect.6The Scherrer equation,6 was used to calculate the crystallite size for (Co 0.5Cu 0.5)Cr 2O4and (Co 0.75Cu 0.25)Cr 2O4nanoparticles that turns out to be ∼11 nm and ∼18 nm, respectively. Careful analyses of the obtained microscopic pictures of (Co 0.5Cu 0.5)Cr 2O4and (Co 0.75Cu 0.25)Cr 2O4powder sam- ples confirm the non-uniform size distribution (Figs. 1(b) and (c)). Formation of ring like shapes composed of spots as observed in selected area electron diffraction (SAED) (insets of Figs. 1(b) and (c)) confirms crystalline nature of the nanoparticles. Estimation of the average particle sizes, obtained from the TEM results in the present work, yields 32(6) nm for (Co 0.5Cu 0.5)Cr 2O4and 20(8) nm for (Co 0.75Cu 0.25)Cr 2O4nanoparticles. Comparison of the particlesize with the crystallite size indicates the polycrystalline nature of (Co 0.5Cu 0.5)Cr 2O4particles. Here it is important to note that the CuCr 2O4prefers to crys- tallize in the tetrahedral structure below 850 K due to the cooper- ative J-T effect.3,4Increasing the Cu content ( x=0.75) resulted in the onset of the tetragonal phase; as indicated from the splitting of (220) peak for the sample calcined at 600○C (Fig. 2(a)).4To probe the role of calcination temperature on structure, the three powder samples were further calcined at 800○C. The selected XRD pattern around (220) plane is shown in Fig. 2(b). It is noted from this that the (Co 0.75Cu 0.25)Cr 2O4sample does not show any splitting of the peak related (220) reflection which is indicative of J-T distortion despite calcination.4It is observed in Fig. 2 that calcination of the (Co 0.5Cu 0.5)Cr 2O4sample at 800○C results in the splitting of the (220) peak. Consequently, it is evident that both Co concentration and calcination temperature determines the crystal structure of this kind of chromite samples. As the XRD results provide evidence of the unusual cubic phase retention of the x=0.5 and 0.25 samples calcined at 600 ○C further magnetic and electronic properties will focus only on these samples. Temperature (T)dependent magnetization (M) experiments were done for both zero ( MZFC) and field ( MFC) cooled conditions. For both measurements the applied probing FIG. 3. Temperature ( T) dependent magnetization ( M) for nanosized: (a) (Co 0.5Cu0.5)Cr2O4and (b) (Co 0.75Cu0.25)Cr2O4powders calcined at 600○C. Applied field ( μ0H) dependent magnetization ( M) for nanosized: (c) (Co 0.5Cu0.5)Cr2O4and (d) (Co 0.75Cu0.25)Cr2O4calcined at 600○C. AIP Advances 11, 025113 (2021); doi: 10.1063/9.0000096 11, 025113-3 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv field was 0.1 T. Figs. 3(a) and (b) show the temperature depen- dent magnetization data obtained for the samples (Co 0.5Cu 0.5)Cr 2O4 and (Co 0.75Cu 0.25)Cr 2O4calcined at 600○C, where TCwas cal- culated following the reported methods7–9and found to be 103 K and 99 K, respectively. MZFC for both samples becomes neg- ative at a particular temperature identified as the compensation point, Tcomp, below TC.10,11(Co 0.5Cu 0.5)Cr 2O4compound shows Tcomp≈84 K and (Co 0.75Cu 0.25)Cr 2O4sample indicates Tcomp≈79 K (Figs. 3(a) and (b)). The feature related to TSbecomes prominent for (Co 0.75Cu 0.25)Cr 2O4(Fig. 3(b)). The difference between MFCand MZFCis significant for x=0.5, indicating more frustration in the system when compared to x=0.25.8Hence, it appears that the frus- tration possibly drives the cubic to tetragonal phase transition for samples calcined at 800○C.4,5 Additionally, M(μ0H) measurements were done at distinct temperatures and are depicted in Figs. 3(c) and (d). Significant enhancement of coercivity occurs with increasing xfrom 0.25 to 0.5, however, the magnetization does not saturate in line with previous findings for undoped and Ni substituted CoCr 2O4nanoparticles.11,12 The ferrimagnetic hysteresis observed at 10 K for (Co 0.5Cu 0.5)Cr 2O4 shows an abrupt kink marked by arrows in Fig. 3(c) around zero field that can be compared to the observation of a wasp-waist like feature as detected in Ni substituted CoCr 2O4nanoparticles.11With increasing temperature the coercivity decreases and at 103 K (close to the TC) the M(μ0H) plot indicates a superparamagnetic phase with non-linear behaviour of magnetization and zero-coercivity.11,12 The asymmetric feature seen in the hysteresis loops at 10 K can arise due to several reasons such as the bi- or multimodal distri- bution of magnetic granules with dissimilar coercivities,13the con- currence of hard and soft kind of magnetic entities,14or due to the differences in the super-exchange interaction that can result as a consequence of spinel structure.15In a similar spinel compound CoFe 2O4nanoparticles this behaviour was attributed to the pres- ence of particles belonging to two separate groups with dissimi- lar anisotropies,16or the presence of canted spins that can favour the spin re-orientation transitions that can occur at low temper- ature.17Both the samples demonstrate decrease in coercivity with advancement in temperature and the hysteresis loops approach asuperparamagnetic type curve when measured at temperature less than TC.11,12However, in the current scenario the occurrence of an anomaly in the hysteresis at 10 K is attributed to the Cu substitution that resembles the characteristic feature of CuCr 2O4(Fig. 3(d)).5 Substituting Cu at the Co site leads to an enhancement in the probability that the cationic oxidation state can change and that can alter the magnetic exchange interactions.11Fig. 4 show the 2p XPS core levels of Co and Cu. The spin orbit splitting (SOS) ≥15.2 eV for Co 2p (Fig. 4(a)) and the presence of a strong satellite confirm the dominance of the Co2+oxidation state.18,19However, the presence of weak satellites at a binding energy (BE) of approximately 10 eV higher than the main spin orbit peaks, suggests the presence of frac- tional Co3+ions. In the case of the Cu 2p core level, the presence of a broad O KVV Auger line with a centroid at ∼975 eV BE signifi- cantly affects the shape of the background in the Cu 2p 1/2BE region (not shown here). Therefore the focus is on the Cu 2p 3/2BE region (Fig 4(b)). The main peak is composed of a more intense feature at ∼933.6 eV with a clear shoulder on the lower BE side corresponding to 2+and 1 +/0+oxidation states, respectively. The strong char- acteristic satellite feature centred at ∼940.5 eV confirms the Cu2+ majority oxidation state in both compounds.18The majority of Cr ions are found to be in the 3 +oxidation state in both compounds (not shown in this work). Smart and Greenwald,20showed that different exchange cou- pling mechanisms in the distorted orthorhombic structure play a key role in the magnetostructural coupling. As observed, antiferromag- netism can only be achieved by two conditions.20Firstly, the crystal structure should have two magnetic sublattices, Ahaving nearest neighbours only on sublattice B, and vice versa.20Secondly, the exchange integral, J, between neighbouring magnetic atoms should be negative leading to an antiparallel alignment.20For a particular temperature perfect antiparallel ordering is highly sensitive towards thermal agitation.20The value of Jis a function of interatomic sep- aration, r. In addition, in the process of building the crystal, the exchange interaction will arrange the atoms to increase ∣J∣, conse- quently the magnetic atoms either pull closer together or push apart on the basis of the sign of d J/dr.19Since the number of nearest neigh- bours increases with decreasing temperature, more deformation is FIG. 4. Core level XPS results obtained for the (Co 1–xCux)Cr2O4(x=0.5, 0.25) samples calcined at 600○C: (a) Co 2p and (b) Cu 2p. AIP Advances 11, 025113 (2021); doi: 10.1063/9.0000096 11, 025113-4 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv expected at low temperature.20Hence, the population of Co atoms atAsites manipulates the exchange interaction so that it retains the cubic phase up to x=0.75. IV. CONCLUSION Sol-gel method was used to synthesize nanosized (Co 1−xCux)Cr 2O4with x=0.5, 0.25 and 0.75. XRD results confirm crystallinity and cubic phase retention up to x=0.5 for the samples calcined at 600○C. Further increasing xor the calcination temperature to 800○C leads to a tetragonal distortion in crystal structure (for x≥0.5) due to J-T effect. Ferrimagnetic TCof (Co 0.5Cu 0.5)Cr 2O4samples was observed to be higher in comparison to the (Co 0.75Cu 0.25)Cr 2O4powders. Ferrimagnetism persist below T<TCfor both the samples having a non-saturated magnetization. M(μ0H) experiments performed at 10 K for both the samples demonstrated a kink-like feature which is attributed to CuCr 2O4. In addition, a superparamagnetic feature was observed at the onset of TC. XPS data revealed that Co and Cu core levels contain majority 2 +oxidation states of the cations. The cubic phase, as well as the TC, is found to be strongly controlled by the Asite cations. ACKNOWLEDGMENTS Financial grant from the SANRF (Grant No. 93551, 120856, 126978, 93205, 119314, 90698, 126911), and the funding from URC and FRC of University of Johannesburg (UJ) is acknowledged. The utilization of the NEP Cryogenic Physical Properties Measurement System at UJ, procured from the funding of SANRF (Grant No: 88080) and UJ, South Africa, is acknowledged. Authors thank the Spectrum Analytical Facility within FoS at UJ for characterizations. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request.REFERENCES 1Y. Yamasaki, S. Miyasaka, Y. Kaneko, J.-P. He, T. Arima, and Y. Tokura, Phys. Rev. Lett. 96, 207204 (2006). 2G. Lawes, B. Melot, K. Page, C. Ederer, M. A. Hayward, Th. Proffen, and R. Seshadri, Phys. Rev. B 74, 024413 (2006). 3E. Jo, B. Kang, C. Kim, S. Kwon, and S. Lee, Phys. Rev. B 88, 094417 (2013). 4M. R. Suchomel, D. P. Shoemaker, L. Ribaud, M. C. Kemei, and R. Seshadri, Phys. Rev. B 86, 054406 (2012). 5K. Tomiyasu, S. Lee, H. Ishibashi, Y. Takahashi, T. Kawamata, Y. Koike, T. Nojima, S. Torii, and T. Kamiyama, arXiv:1803.06447. 6B. D. Cullity, Elements of X-Ray Diffraction (Addison-Wesley, Reading, MA, 1978). 7P. Mohanty, C. J. Sheppard, A. R. E. Prinsloo, W. D. Roos, L. Olivi, and G. Aquilanti, J. Magn. Magn. Mat. 451, 20 (2018). 8P. Mohanty, A. M. Venter, C. J. Sheppard, and A. R. E. Prinsloo, J. Magn. Magn. Mat. 498, 166217 (2020). 9P. Mohanty, S. Chowdhury, R. J. Choudhary, A. Gome, V. R. Reddy, G. R. Umapathy, S. Ojha, E. Carleschi, B. P. Doyle, A. R. E. Prinsloo, and C. J. Sheppard, Nanotechnology 31, 285708 (2020). 10L. Néel, Ann. Phys. 3, 137 (1948). 11P. Mohanty, A. R. E. Prinsloo, B. P. Doyle, E. Carleschi, and C. J. Sheppard, AIP Advances 8, 056424 (2018). 12L. Kumar, P. Mohanty, T. Shripathi, and C. Rath, Nanosci Nanotechnol Lett 1(3), 199–203 (2009). 13A. P. Roberts, Y. Cui, and K. L. Verosub, J. Geophys. Res. 100, 17909 https://doi.org/10.1029/95jb00672 (1995). 14P. K. Pandey, R. J. Choudhary, and D. M. Phase, Appl. Phys. Lett. 103, 132413 (2013). 15A. U. Rahman, M. A. R. K. Maaz, S. Karim, K. Hayat, and M. M. Hasan, J. Nanopart. Res. 16, 1 (2014). 16M. Chithra, C. N. Anumol, B. Sahu, and S. C. Sahoo, J. Magn. Magn. Mater. 401, 1 (2016). 17S. T. Xu, Y. Q. Ma, G. H. Zheng, and Z. X. Dai, Nanoscale 7, 6520 (2015). 18J. F. Moulder, in Handbook of X-Ray Photoelectron Spectroscopy , A Refer- ence Book of Standard Spectra for Identification and Interpretation of XPS Data, edited by J. Chastain and R. C. King (Physical Electronics Division, Perkin-Elmer Corporation, Eden Prairie, MN, 1992), p. 261. 19P. Mohanty, G. B. Geetha, E. Carleschi, C. J. Sheppard, and A. R. E. Prinsloo, Surface Science Spectra 27, 014003 (2020). 20J. Smart and S. Greenwald, Nature 166, 523 (1950). AIP Advances 11, 025113 (2021); doi: 10.1063/9.0000096 11, 025113-5 © Author(s) 2021

5.0038694.pdf

J. Chem. Phys. 154, 074109 (2021); https://doi.org/10.1063/5.0038694 154, 074109 © 2021 Author(s).Too big, too small, or just right? A benchmark assessment of density functional theory for predicting the spatial extent of the electron density of small chemical systems Cite as: J. Chem. Phys. 154, 074109 (2021); https://doi.org/10.1063/5.0038694 Submitted: 25 November 2020 . Accepted: 20 January 2021 . Published Online: 18 February 2021 Diptarka Hait , Yu Hsuan Liang , and Martin Head-Gordon ARTICLES YOU MAY BE INTERESTED IN The Devil’s Triangle of Kohn–Sham density functional theory and excited states The Journal of Chemical Physics 154, 074106 (2021); https://doi.org/10.1063/5.0035446 r2SCAN-3c: A “Swiss army knife” composite electronic-structure method The Journal of Chemical Physics 154, 064103 (2021); https://doi.org/10.1063/5.0040021 r2SCAN-D4: Dispersion corrected meta-generalized gradient approximation for general chemical applications The Journal of Chemical Physics 154, 061101 (2021); https://doi.org/10.1063/5.0041008The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Too big, too small, or just right? A benchmark assessment of density functional theory for predicting the spatial extent of the electron density of small chemical systems Cite as: J. Chem. Phys. 154, 074109 (2021); doi: 10.1063/5.0038694 Submitted: 25 November 2020 •Accepted: 20 January 2021 • Published Online: 18 February 2021 Diptarka Hait,1,2,a) Yu Hsuan Liang,1 and Martin Head-Gordon1,2,a) AFFILIATIONS 1Kenneth S. Pitzer Center for Theoretical Chemistry, Department of Chemistry, University of California, Berkeley, California 94720, USA 2Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA a)Authors to whom correspondence should be addressed: diptarka@berkeley.edu and mhg@cchem.berkeley.edu ABSTRACT Multipole moments are the first-order responses of the energy to spatial derivatives of the electric field strength. The quality of density functional theory prediction of molecular multipole moments thus characterizes errors in modeling the electron density itself, as well as the performance in describing molecules interacting with external electric fields. However, only the lowest non-zero moment is translationally invariant, making the higher-order moments origin-dependent. Therefore, instead of using the 3 ×3 quadrupole moment matrix, we utilize the translationally invariant 3 ×3 matrix of second cumulants (or spatial variances) of the electron density as the quantity of interest (denoted byK). The principal components of Kare the square of the spatial extent of the electron density along each axis. A benchmark dataset of the principal components of Kfor 100 small molecules at the coupled cluster singles and doubles with perturbative triples at the complete basis set limit is developed, resulting in 213 independent Kcomponents. The performance of 47 popular and recent density functionals is assessed against this Var213 dataset. Several functionals, especially double hybrids, and also SCAN and SCAN0 predict reliable second cumulants, although some modern, empirically parameterized functionals yield more disappointing performance. The H, Li, and Be atoms, in particular, are challenging for nearly all methods, indicating that future functional development could benefit from the inclusion of their density information in training or testing protocols. Published under license by AIP Publishing. https://doi.org/10.1063/5.0038694 .,s I. INTRODUCTION Quantum chemistry methods have seen increasingly widespread use over the last few decades, and now permit sufficiently accu- rate computation of energies ranging in scale from total atomization energies1to extremely weak noncovalent interactions.2In partic- ular, density functional theory3–5(DFT) is ubiquitously employed to study medium to large systems as it provides an acceptable bal- ance between accuracy and computational cost. Although exact in theory,6,7the practical usage of DFT almost always entailsthe use of computationally tractable density functional approx- imations (DFAs) within the Kohn–Sham (KS) formalism.8It is often remarked that DFAs are not really systematically improv- able the way the exact coupled cluster (CC) hierarchy is, but the best-performing DFAs on a given rung of Jacob’s ladder9 statistically improve upon lower rungs,5,10–12indicating that greater complexity could (but is not guaranteed to) lead to better accu- racy. Caution is still warranted while applying KS-DFT to systems with substantial levels of delocalization error13–16or multireference character,17–19but DFAs are quite effective overall at predicting J. Chem. Phys. 154, 074109 (2021); doi: 10.1063/5.0038694 154, 074109-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp ground state relative energies associated with main-group chem- istry,5,10and even organometallic chemistry.20,21Indeed, modern DFAs often surpass the accuracy of single-reference wave func- tion approaches such as second-order Møller–Plesset perturba- tion theory (MP2) or CC singles and doubles (CCSD) for such problems.11,22,23 This is perhaps unsurprising, as DFA development has his- torically emphasized improved prediction of such chemically relevant energies, out of the belief that this leads to better approx- imations to the exact functional. However, the exact functional ought to map the exact density to the exact energy, and it is entirely possible for a DFA to obtain a reasonable energy from a relatively inaccurate density via hidden cancellation of errors. It has, indeed, been suggested that modern DFAs predict worse densities than older, less parameterized models,24leading to con- siderable discussion22,25–34in the scientific community about den- sity predictions from DFT. It is certainly plausible that greater complexity in the functional form creates additional avenues for incorrect behavior (as can be seen in wave function theory as well35,36), even without accounting for overfitting to empirical data. Separate from the debate about “true paths” for DFA devel- opment,24we note that densities are important for a very practical reason. The electron density controls the response of the elec- tronic energy to external electric fields, such as those that might be encountered in condensed phases or in spectroscopic simula- tions. Density errors are thus related to energy errors under a different one-particle potential (that DFT should still be formally capable of addressing6). The application of DFT beyond ground state gas-phase chemical physics is thus reliant upon DFT yield- ing reasonable densities, or at least a characterization of when and how various DFAs fail. This has led to investigations on the performance of DFT for predicting dipole moments22,37–41and polarizabilities,37,40,42–45as they represent the linear and quadratic responses, respectively, of the energy to a spatially uniform electric field. The natural next step would be to assess the linear response of the energy to a (spatially) constant electric field gradient, which is equivalent to the molecular quadrupole moment Q. Indeed, there have been a number of assessments of the quality of various den- sity functionals for evaluating molecular quadrupole moments in the past,46–49albeit before the development of many modern func- tionals. At the same time, efforts have been made to converge high- quality coupled cluster theory calculations of quadrupole moments with respect to the basis set,50–52setting the stage for benchmark assessments. Qis, however, not translationally invariant for polar systems, making it a somewhat unsuitable metric. We thus decided to investigate the performance of DFT in predicting the second spa- tial cumulant of the electron density ( K), instead, which is a trans- lationally invariant quantity connected to Qand the dipole moment μ.Kthus relates the density to the response of the energy to elec- tric fields that linearly vary with spatial coordinates. Mathemati- cally,Kis also the variance of the electron density. To the best of our knowledge, there have been no previous studies that explicitly focus on the prediction of Kby DFT or MP/CC methods. How- ever, a related quantity had previously been studied with Hartree– Fock (HF)53and found to be useful in predicting steric effects of substituents.II. SECOND SPATIAL CUMULANT OF ELECTRON DENSITY A. Definition Let us consider a system with electron density ρ(r) and total number of electrons N=∫ρ(r)dr. The spatial probability density p(r) for a single electron is p(r)=ρ(r) N. (1) The alternative definition of p(r) in terms of the wave function ∣Ψ⟩is p(r)=⟨r∣(Tr N−1[∣Ψ⟩⟨Ψ∣])∣r⟩=Γ1(r,r) N, (2) which corresponds to the scaled diagonal elements of the one par- ticle density operator, Γ1(r,r′) that results from tracing out N−1 degrees of freedom from the full density operator ΓN=∣Ψ⟩⟨Ψ∣. For a single Slater determinant, this is equivalent to averaging the square of each occupied spin orbital. The first and second spatial moments of the electron density are, consequently, ⟨r⟩=∫rp(r)dr, (3) ⟨rrT⟩=∫(rrT)p(r)dr. (4) The second spatial cumulant of the electron density is thus K=⟨(δr)(δr)T⟩=⟨rrT⟩−⟨r⟩⟨r⟩T, (5) which is equivalent to the spatial (co)variance of the probability distribution p(r). The individual components are thus of the form, Kxx=⟨x2⟩−⟨x⟩2=∫x2p(r)dr−(∫xp(r)dr)2 , (6) Kxy=⟨xy⟩−⟨x⟩⟨y⟩=∫xyp(r)dr−(∫xp(r)dr)(∫yp(r)dr), (7) etc. in terms of the individual Cartesian directions x,y,z.Kis thus translationally invariant, and represents the “width” (or “spread”) of the electron density. We also note that the eigenvalues of Kare rotationally invariant. In addition, while we have only focused on the second-order cumulant in this work, higher-order cumulants of p(r) can similarly be readily defined and utilized for problems where they might be relevant. J. Chem. Phys. 154, 074109 (2021); doi: 10.1063/5.0038694 154, 074109-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp B. Connection to multipole moments If the system has nuclear charges { ZA} at positions { RA}, the dipole moment μand (Cartesian/non-traceless) quadrupole moment Qare μ=∑ AZARA−∫rρ(r)dr=∑ AZARA−N⟨r⟩, (8) Q=∑ AZA(RA)(RA)T−∫(rrT)ρ(r)dr =∑ AZA(RA)(RA)T−N⟨rrT⟩. (9) Therefore, the second cumulant can be expressed as K=1 N(∑ AZA(RA)(RA)T−Q) −1 N2(∑ AZARA−μ)(∑ AZARA−μ)T . (10) C. Physical interpretation Let us begin with the case of atoms, and suppose that the single electron probability distribution for an (spherical) atom at (0, 0, 0) isp1(r). Spherical symmetry indicates that ∫xp1(r)dr=∫yp1(r)dr=∫zp1(r)dr=0, (11) ∫xyp 1(r)dr=∫yzp 1(r)dr=∫xzp 1(r)dr=0, (12) ∫x2p1(r)dr=∫y2p1(r)dr=∫z2p1(r)dr=η2. (13) This implies that Kxx=Kyy=Kzz=η2. Therefore,√ 3ηis an effec- tive atomic radius associated with the spatial extent of the electron density, much like a covalent or van der Waals radius. In particular, for the H atom, ηis the Bohr radius (1 a.u. = 0.529 Å). For molecules without symmetry, Kis not generally a diagonal 3×3 matrix, and, likewise, Kxx≠Kyy≠Kzz.Kcan be diag- onalized to yield principal axes and 3 principal components, very analogous to the molecular inertia tensor associated with the nuclei, I=∑ AmA(RA)(RA)T. The eigenvalues of Kthereby give the square of the spatial extent of the electron density along each principal axis. Systems where both a C n(n>1) axis of rotation and σvplane(s) of symmetry are present (i.e., C nvor higher symmetry) have the prin- cipal axes defined by molecular symmetry, and we shall later choose members of our dataset based on this convenient simplification. D. Behavior vs system size The behavior of Kvs system size could be useful in character- izing the utility of this quantity. A simple case to consider is a 1D lattice of Mnoninteracting He atoms, placed at (0, 0, 0), (0, 0, a),(0, 0, 2 a). . .(0, 0, ( M−1)a). Let each atom have spatial extent ηas in Eq. (13). Subsequently, p(r) for the supersystem is given by p(r)=1 MM−1 ∑ m=0p1(r−maˆz), (14) which is the average of the individual single electron probability distributions. Transverse to the lattice vector aˆz, we thus have ⟨x2⟩=1 MM−1 ∑ m=0∫x2p1(r−maˆz)dr =1 MM−1 ∑ m=0∫x2 mp1(rm)drm=η2, (15) ⟨x⟩=1 MM−1 ∑ m=0x∫p1(r−maˆz)dr =1 MM−1 ∑ m=0∫xmp1(rm)drm=0, (16) where rm=r−maˆz. Therefore, Kxx=η2=Kyyand is invariant vs system size. However, parallel to the lattice vector aˆz, we obtain ⟨z2⟩=1 MM−1 ∑ m=0∫z2p1(r−maˆz)dr =1 MM−1 ∑ m=0∫(zm+ma)2p1(rm)drm =1 MM−1 ∑ m=0(∫z2 mp1(rm)drm+m2a2∫p1(rm)drm + 2am∫zmp1(rm)drm) =1 MM−1 ∑ m=0(η2+m2a2)=η2+(M−1)(2M−1) 6a2, (17) ⟨z⟩=1 MM−1 ∑ m=0∫zp1(r−maˆz)dr =1 MM−1 ∑ m=0∫(zm+ma)p1(rm)drm =1 MM−1 ∑ m=0(∫zmp1(rm)drm+ma∫p1(rm)drm) =1 MM−1 ∑ m=0ma=M−1 2a, (18) ∴Kzz=⟨z2⟩−⟨z⟩2=η2+(M2−1) 12a2. (19) Kzzthus grows as O(M2) vs the system size M. However, this increase is entirely due to geometric/structural factors [i.e., only J. Chem. Phys. 154, 074109 (2021); doi: 10.1063/5.0038694 154, 074109-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp dependent on the lattice spacing aand not on the electronic contri- bution from p1(r)]. Thus, differences between Kzzcomputed by var- ious methods will be independent of M, and will solely be a function of the subsystem densities { p1(r)}. This is not true in the interacting subsystems limit, but the analysis nonetheless reveals a significant contribution to Kfrom molecular geometry alone. A similar analysis for 2D (square) and 3D (cubic) lattices shows thatKiigrows as O(M) and O(M2 3), respectively, vs the number of identical noninteracting subsystems M(where ˆiis parallel to the lat- tice vectors), due to geometric factors, while the electronic contribu- tion intrinsically arising from p1(r) remains constant. Consequently, differences in Kshould be size-intensive in the non-interacting limit. However, relative error in Kwould shrink with increasing M(as the reference value in the denominator would increase). We consequently only consider absolute errors of the form K−Kref vs a reference value Kref. This stands in contrast to properties like dipole moments and static polarizabilities, which are properly size- extensive and thus suitable for relative/percentage error based met- rics. It is also possible to look at standard deviations (i.e.,√Kii) instead of variances, but the geometric factors would prevent the errors from being size-intensive in that case. III. DATASET We have investigated Kfor 100 small main-group systems (listed in Table I) for which it was possible to get highly accurate benchmark values with CC singles and doubles with perturbative TABLE I . The 100 species in the dataset, sorted by whether they are not spin- polarized (NSP) or spin-polarized (SP). NSP SP AlF Cl 2 Mg AlH 2 NH Ar ClCN Mg 2 BH 2 NH 2 BF ClF N 2 BO NO 2 BH 2Cl FCN NH 3 BS NP BH 2F H 2 NH 3O Be Na BH 3 H2O NaCl BeH Na 2 BHF 2 HBO NaH C 2H NaLi BeH 2 HBS Ne CF 2 O2 C2H2 HCCCl OCl 23CH 2 O3 C2H4 HCCF PH 3 CH 3 OF 2 CH 2BH HCHO PH 3O CN P CH 3BO HCN SCl 2 F2 P2 CH 3Cl HCl SF 2 H PCl CH 3F HF SH 2 H2CN PF CH 3Li HNC SO 2 HCHS PH CH 4 He SiH 3Cl HCP PH 2 CO LiBH 4 SiH 3F Li PO 2 CO 2 LiCl SiH 4 N S 2 CS LiF SiO NCl SO-trip CSO LiH NF SiH 3 NF 2triples [CCSD(T)54] at the complete basis set (CBS) limit. The sys- tems were chosen from the combined datasets considered in Refs. 22 and 43 such that Kwas diagonal for a fixed coordinate system for all (spatial symmetry preserving) electronic structure methods. Asymmetric systems with nondiagonal Kwere not included for sim- plicity. Linear molecules with a single πelectron/hole (such as OH) were also not considered, as real valued orbitals would be incapable of predicting a cylindrically symmetric orbital of πsymmetry,55,56 and would spuriously lead to Kxx≠Kyy. Complex valued orbitals would therefore be necessary to describe such species with proper symmetry within a DFT framework.55 The complete dataset consists of 59 not spin-polarized (NSP) and 41 spin-polarized (SP) systems. The NSP vs SP classifica- tion was done on the basis of whether the stable HF solution has ⟨S2⟩= 0 or not. NSP species are thus unambiguously closed-shell, and are therefore more likely to be “easier” for single-reference quantum chemistry methods like KS-DFT or MP2. Indeed, the size of the (T) correction to Kwas quite small for nearly all species (NSP or SP, as discussed later in Sec IV A), indicating that the multireference character of the chosen systems (if any) did not strongly influence Kpredictions and that CCSD(T) is likely to be an adequately accurate benchmark. Furthermore, none of these systems have pathological, delocalization driven qualitative fail- ures,22,41,57–59making them reasonable choices for understanding the behavior of DFAs in the regime where they are expected to work well. The error ϵi,min an individual component Km iifor a given molecule m(vs a reference value Km ref, ii) is ϵi,m=Km ii−Km ref, ii. (20) The cumulative errors over all molecules and directions are thus 1. Root mean square error (RMSE):⌟roo⟪⟪op ⌟roo⟪mo⟨⌟roo⟪mo⟨⌟roo⟪⟨o⟪1 3NN ∑ m=1(ϵ2 x,m+ϵ2 y,m+ϵ2 z,m). 2. Mean error (ME):1 3NN ∑ m=1(ϵx,m+ϵy,m+ϵz,m). 3. Maximum absolute error (MAX): max (∣ϵi,m∣)∀i∈{x,y,z}and ∀m∈Z+,m≤N. All quantities are reported in atomic units (a.u.) unless specified otherwise. A. Computational details All calculations were performed with the Q-Chem 5 package,60 using fixed geometries obtained from Refs. 22 and 43 (also provided in the supplementary material). We examined the performance of 47 DFAs spanning all five rungs of Jacob’s ladder (as can be seen from Table II), ensuring reasonable representation at each level. Individ- ual DFAs were selected based on (perceived) popularity, recency, and performance over other benchmark datasets.5,10,11,22,43 Kwas obtained from Qcomputed for this work and μwas obtained from Ref. 22. Qfor self-consistent field (SCF) approaches like HF and non-double hybrid DFAs were found analytically via J. Chem. Phys. 154, 074109 (2021); doi: 10.1063/5.0038694 154, 074109-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II . Errors in K(in a.u.) predicted by various DFAs for the species in the dataset. Positive ME values indicate less compact (i.e., more diffuse) densities, relative to the benchmark. RMSE RMSE Method Class Full NSP SP ME MAX Method Class Full NSP SP ME MAX CCSD WFT 0.003 0.002 0.003 −0.001 0.022 SCAN076Rung 4 0.008 0.004 0.011 0.000 0.047 MP2 WFT 0.022 0.004 0.035 0.001 0.312 B97-277Rung 4 0.009 0.004 0.013 0.001 0.047 RMP2 WFT 0.010 0.004 0.015 0.002 0.074 TPSSh78Rung 4 0.009 0.004 0.013 0.002 0.053 HF WFT 0.027 0.015 0.039 0.007 0.301 PBE079Rung 4 0.009 0.005 0.014 0.003 0.052 RHF WFT 0.020 0.015 0.026 0.006 0.091 B9780Rung 4 0.010 0.006 0.013 0.005 0.049 HSE-HJS81Rung 4 0.010 0.005 0.014 0.003 0.051 SPW9282,83Rung 1 0.022 0.017 0.028 0.011 0.129 SOGGA11-X84Rung 4 0.010 0.005 0.015 0.004 0.057 Slater82Rung 1 0.047 0.038 0.057 0.034 0.275 LRC- ωPBEh85Rung 4 0.011 0.005 0.016 0.002 0.061 PW6B9586Rung 4 0.011 0.006 0.016 0.003 0.084 BPBE87,88Rung 2 0.013 0.008 0.017 0.004 0.062 ωM05-D89Rung 4 0.011 0.007 0.015 0.005 0.064 mPW9190Rung 2 0.014 0.010 0.019 0.006 0.069 M0691Rung 4 0.011 0.007 0.016 0.001 0.058 B97-D392Rung 2 0.016 0.010 0.022 0.007 0.103 ωB97X–V93Rung 4 0.012 0.008 0.016 0.006 0.066 PBE88Rung 2 0.016 0.012 0.021 0.008 0.079 ωB97X-D94Rung 4 0.012 0.006 0.017 0.003 0.064 N1295Rung 2 0.020 0.013 0.028 0.006 0.120 M06-2X91Rung 4 0.013 0.006 0.019 0.003 0.104 BLYP87,96Rung 2 0.021 0.016 0.026 0.012 0.112 HFLYP96Rung 4 0.014 0.013 0.015 −0.004 0.051 SOGGA1197Rung 2 0.048 0.035 0.062 0.024 0.500 M1198Rung 4 0.014 0.009 0.019 0.005 0.083 B3LYP99Rung 4 0.014 0.009 0.019 0.007 0.068 SCAN100Rung 3 0.009 0.005 0.012 0.002 0.052 CAM-B3LYP101Rung 4 0.014 0.010 0.019 0.007 0.070 MS2102Rung 3 0.010 0.004 0.014 0.000 0.072 M08-HX103Rung 4 0.015 0.009 0.020 0.009 0.079 TPSS104Rung 3 0.010 0.006 0.015 0.003 0.065 ωB97M-V105Rung 4 0.022 0.013 0.030 0.013 0.127 mBEEF106Rung 3 0.011 0.005 0.017 −0.001 0.074 MN15107Rung 4 0.035 0.013 0.052 0.013 0.231 M06-L108Rung 3 0.014 0.010 0.018 −0.006 0.068 revM06-L109Rung 3 0.015 0.013 0.018 0.002 0.091 DSD-PBEPBE110Rung 5 0.005 0.003 0.007 0.002 0.032 B97M-V111Rung 3 0.026 0.013 0.037 0.011 0.157 XYGJ-OS112Rung 5 0.006 0.003 0.008 0.002 0.028 M11-L113Rung 3 0.038 0.023 0.052 0.005 0.332 PTPSS114Rung 5 0.006 0.003 0.009 0.002 0.032 MN15-L115Rung 3 0.040 0.024 0.055 0.022 0.278 XYG3116Rung 5 0.006 0.003 0.009 0.002 0.032 ωB97M(2)11Rung 5 0.008 0.005 0.010 0.003 0.037 B2GPPLYP117Rung 5 0.008 0.005 0.011 0.004 0.042 ωB97X-2(TQZ)118Rung 5 0.008 0.006 0.010 0.005 0.044 B2PLYP119Rung 5 0.010 0.007 0.013 0.005 0.047 integrating over ρ(r).Qfor other methods (MP2, CC, or dou- ble hybrid DFT) was found via a central, two point finite differ- ence approach using a constant electric field gradient of 2 ×10−4 a.u. (similar to Ref. 22). Finite difference errors thus introduced appear to be quite small, as the RMS deviation between analytic and finite-difference CCSD/aug-cc-pCVTZ61–64Kiiis 2.2×10−4a.u. Comparison between wave function theory and various DFAs are only meaningful at the CBS limit due to different rates of basis set convergence. HF/aug-cc-pCV5Z Kiiwere assumed to be at the CBS limit as it only has an RMS deviation of 2.5 ×10−4a.u. vs aug-cc-pCVQZ results (which should be sufficiently small, in light of the empirically observed exponential convergence of HF ener- gies vs cardinal number of the basis set65,66). Similarly, DFT/aug- pc-467–72values were assumed to be at the CBS limit for functionals from Rungs 1–4 of Jacob’s ladder. The virtual orbital dependent correlation contribution Kcorr(in MP2/CC/double hybrids) wasextrapolated to the CBS limit via the two point extrapolation for- mula73Kcorr, iin=Kcorr, iiCBS+A n3from aug-cc-pCVTZ ( n= 3) and aug- cc-pCVQZ ( n= 4), which is adequate for dipoles22,73and appears to also be adequate for quadrupole moments.50–52 Local exchange-correlation integrals for all DFT calculations were computed over a radial grid with 99 points and an angular Lebedev grid with 590 points for all atoms. Non-local correlation was evaluated on an SG-1 grid.74Unrestricted (U) orbitals were employed for all CC (except Be, where UHF breaks spatial sym- metry) and non-double hybrid DFT calculations. MP2 is known to yield non N-representable densities when spin-symmetry breaks,75 leading to poor dipole22and polarizability43predictions. Conse- quently, MP2 with both restricted (R) and unrestricted orbitals were carried out, and the restricted variant was found to yield a signifi- cantly smaller RMSE. Double hybrid calculations were subsequently J. Chem. Phys. 154, 074109 (2021); doi: 10.1063/5.0038694 154, 074109-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp carried out with only R orbitals. Stability analysis was performed in the absence of fields to ensure that the orbitals correspond to a min- imum, and the resulting orbitals were employed as initial guesses for finite field calculations (if any). The frozen-core approximation was not employed in any calculation. IV. RESULTS AND DISCUSSIONS A. Full dataset The errors in DFA predictions for the full dataset are reported in Table II, along with errors for the wave function methods CCSD, MP2, and HF. Considering the wave function theories (WFTs) first, we see that CCSD has the lowest RMSE of 0.003 of all the meth- ods considered, and gives fairly similar performance across the NSP and SP subsets. MP2 gives good performance for the NSP dataset, but N-representability failures75lead to significantly worse results over the SP subset. The use of R orbitals ameliorates this consid- erably, with RMP2 having a quite improved RMSE of 0.010 over the full dataset. RMP2 is nonetheless still somewhat challenged by open- shell systems (especially, NF and NCl), where the artificial enforce- ment spin-symmetry appears to be suboptimal. We further note that MP3 is also expected to have the same N-representability failures as MP2 and is therefore unlikely to lead to significant improvements over MP2. This in fact has been observed for μ, although alternative orbital choices could lead to better MP performance.23 HF performs quite poorly due to the lack of correlation, hav- ing an RMSE of 0.027 (which can be considered as a ceiling for reasonable DFA performance). Based on the mean error (ME), HF has a tendency to make the variance too large (i.e., HF densities are too diffuse). The SP species are much more challenging than NSP, and RHF actually reduces RMSE substantially to 0.020. However, this is almost solely on the account of two challenging alkali metal containing species (Na 2and NaLi) and the RMSE for HF and RHF are quite similar upon their exclusion. Nonetheless, this serves as a warning that spin-symmetry breaking might compromise property predictions, despite being the natural approach for improving the energy. In particular, spin-symmetry breaking in diatomics such as Na2or NaLi results in densities similar to two independent atoms vs a bonded molecule, leading to larger widths than the restricted solution. Coming to the DFAs, we observe that Rung 1 local spin-density approximation (LSDA) functionals fare considerably worse than CCSD or RMP2. Bare Slater (LSDA) exchange has a rather large RMSE of 0.047, while inclusion of correlation reduces RMSE sub- stantially to 0.022 for SPW92. The SP subset is significantly more challenging in both cases, with an RMSE that is almost double the corresponding NSP value. The minimally parameterized nature of the LSDA functionals means that the SPW92 RMSE of 0.022 can also be considered as a reasonable reference for judging DFAs against. In fact, the HF, RHF, and SPW92 RMSEs collectively sug- gest that DFAs with RMSE larger than 0.02 are not really accurate for predicting K. Moving up Jacob’s ladder to Rung 2 generalized gradi- ent approximations (GGAs), we observe that all but one of the functionals investigated improve upon SPW92. SOGGA11 is the exception, being essentially as bad as bare Slater exchange. On theother hand, BPBE and mPW91 have substantially lower RMSEs, indicating considerable improvement over Rung 1. However, the SP RMSE continues to be nearly twice the NSP one for all GGAs. Rung 3 meta-GGAs (mGGAs) further improve upon predic- tions, with the best mGGA (SCAN) predicting a quite low RMSE of 0.009, while MS2 and TPSS are also fairly reasonable. It is worth noting that all three were developed with a heavy emphasis on non-empirical constraints and were specifically fit to the H atom (one of the most challenging species in the dataset, as discussed later). In contrast, modern mGGAs developed principally by fitting to empirical benchmark data appear to fare worse, with MN15-L, M11-L, and B97M-V being particularly disappointing as they per- form worse than SPW92. Nonetheless, it is worth noting that the modern, empirically fitted mBEEF functional yields a respectable performance. Hybrid functionals on Rung 4 of Jacob’s ladder do not sig- nificantly improve upon mGGAs. The best-performer is SCAN0, which only marginally improves upon the parent SCAN functional. Similarly, TPSSh only slightly improves upon TPSS. This general behavior stands in contrast to the case of dipole moments, where hybrid functionals strongly improve upon lower rungs.22Some other decent performers are B97-2 (which was partially fitted to den- sities), PBE0, B97, HSE-HJS, and SOGGA11-X, which also hap- pen to be hybrid GGAs. PW6B95 is the best-performing hybrid mGGA that was not already based on an existing Rung 3 func- tional with good performance. Interestingly, HFLYP improves sig- nificantly upon HF, mostly by the virtue of dramatically reducing SP errors (in large part because it does not break spin-symmetry for NaLi or Na 2). SP errors are roughly 2–3 times the NSP errors for nearly all other hybrid functionals. It is also worth noting that modern ωB97M-V and MN15 functionals fare particularly poorly, yielding performance worse than SPW92. On the other hand, several empirically fitted functionals like M06 and ωB97X-V yield quite respectable performance (albeit significantly worse than SCAN0). Rung 5 double hybrid functionals reduce error significantly vs hybrids, with DSD-PBEPBE approaching CCSD levels of RMSE for NSP species (although SP species are much more challenging). Even B2PLYP (the poorest performer, and coincidentally, the oldest) per- forms similarly to a good hybrid functional such as PBE0. Most Rung 5 functionals also improve upon RMP2 (especially for the SP subset), indicating perceptible benefit from the local exchange-correlation contributions. It is also worth noting that the recently developed ωB97M(2) functional yields mediocre performance, despite being one of the best-performers for energy predictions11(although it represents an enormous improvement over the parent ωB97M-V functional). The positive ME values in Table II also indicates that most DFAs predict slightly less compact densities (especially LSDA). This is perhaps not too surprising in light of delocalization error present in the studied functionals.15However, the connection between den- sity compactness and delocalization error is not always straightfor- ward, as shown later. Indeed, it can be seen that the local M06-L and mBEEF functionals systematically predict too small K, which is contrary to expectations based on delocalization error alone. HFLYP, however, has a negative ME, consistent with overlocal- ization of density that is expected from 100% HF exchange. On the other hand, HF systematically overestimates K, which seems J. Chem. Phys. 154, 074109 (2021); doi: 10.1063/5.0038694 154, 074109-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp puzzling at first glance as HF should overlocalize if anything.15,120 However, the lack of correlation in HF leads to artificial symme- try breaking where densities become more “atom-like” than “bond- like,” leading to spurious overestimation of Kfor NaLi and Na 2. In fact, the lack of correlation hinders electrons from occupying the same region of space and could lead to wider “spread” in the electron density. Comparison of the results obtained earlier for dipole moments22indicates some similarities and differences. Ascending Jacob’s ladder leads to improved predictions in both cases, and sev- eral modern, empirically designed functionals are found to perform relatively poorly. In fact, some of the functionals that were found to be among the top performers for their rung in the dipole study (SCAN, DSD-PBEPBE, PBE0, SOGGA11-X, etc.) continue to per- form well. The major differences are that hybrid functionals do not significantly improve upon mGGAs and some functionals that are good for Kpredictions do not fare as well for dipoles. A clear exam- ple of this is SCAN0 faring worse at predicting μthanωB97M-V, despite the former being the best hybrid for predicting K(while the latter is among the worst). Nonetheless, the ability of several func- tionals to predict both Kandμwith low error is encouraging, as that indicates reasonable performance for problems involving external electric fields. It is also worthwhile to identify what species in the dataset are most challenging for DFAs. Ordering the molecules in the dataset by the first quartile of RMS error (over Cartesian axes) across dif- ferent DFAs reveals that there is a break in the distribution after the first 10 species. These difficult cases are Be, H, Li, NaLi, Na 2, BeH, LiH, H 2, BeH 2, and BF in descending order of difficulty. Sev- eral of these cases are also known to be challenging for dipole moment22and static polarizability43predictions. NaLi, in particular, was already known to be a very challenging case for many DFAs, and it is unsurprising that the analogous Na 2is also challenging. Both of these alkali metal dimers feature long bonds ( ∼3 Å) and are SP at equilibrium geometry, suggesting some multireference character(analogous to the isovalent case of stretched H 2). In addition, Na 2 and NaLi are outliers with respect to the RMS deviation of CCSD Kvs the CCSD(T) benchmark, having ∼0.01 deviation (while the next largest deviation is about half of that, for CH 3Li). Near-exact calculations with the adaptive sampling configuration interaction method121,122(with the small cc-pVDZ basis64), however, indicate that the species are not particularly multireference (a single deter- minant has ∼90% weight in the total wave function) and CCSD(T) is sufficiently accurate for K(as shown in the supplementary material). Furthermore, the three atoms represent the most chal- lenging cases overall, with Li having a first quartile RMSE ∼35% larger than NaLi. B. The case of challenging atoms It is thus worthwhile to examine these atoms in greater detail for better insight into functional behavior. H has only one electron and thus all DFA errors are definitionally self-interaction driven. It is thus interesting to observe that the error in Kdoes not appear to correlate well with delocalization error in many cases, as can be seen from Fig. 1. Perhaps the clearest example of this is a compar- ison between the local M06-L functional and the range separated hybridωB97M-V, with the former surprisingly predicting a smaller H atom than exact quantum mechanics, while the latter overesti- mates the size to essentially the same extent as SPW92 (contrary to the behavior seen for delocalization error15). Similarly, M06-2X predicts a larger H atom than M06, despite having twice the HF exchange (54% vs 27%). It is thus apparent that the local exchange- correlation components of several modern density functionals lead to significant errors for the size of the H atom, which are counter- intuitive from the perspective of delocalization error (or fraction of HF exchange present). On the other hand, if we only consider functionals that use the same local exchange-correlation components hybridized with vary- ing amounts of HF exchange (such as PBE/PBE0), we find that error FIG. 1 .Kiipredicted for the H atom by various DFAs (ordered by their position in Table II). The dashed line is the analytic value. J. Chem. Phys. 154, 074109 (2021); doi: 10.1063/5.0038694 154, 074109-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 .Kiipredicted for the Li atom by various DFAs. The dashed line is the benchmark [CCSD(T)/CBS] value and the HF value is 2.070. decreases with the increase in the HF exchange contribution. This is on account of the functional becoming strictly more HF like. It is also worth noting that the double hybrid functionals appear to have much lower error than hybrids, potentially due to relatively smaller contributions from local exchange-correlation. While the cancella- tion of errors in larger systems is likely to make this less of an issue (as can be seen from other species in the dataset), future density functional development would likely benefit from including the size and polarizability43of the H atom as soft constraints. Li superficially resembles H if the core electrons are neglected. However, Fig. 2 shows that the error Kpredictions are quite dif- ferent. Most functionals underestimate the size of the diffuse alkali metal atom, predicting a more compact density than the benchmark. Delocalization errors are normally expected to lead to less compactdensities due to self-interaction. This can indeed be seen from the overly repulsive mean-field electron–electron repulsion potentials predicted by many DFAs.123,124The Li atom is also not multirefer- ence, and thus the reason behind the size underestimation by many functionals is not entirely clear. HFLYP not being an outlier with respect to overlocalization indicates that delocalization is unlikely to be a major factor (and bare HF in fact overestimates the size). Nonetheless, most of the modern functionals that fare poorly for H (B97M-V, MN15-L, MN15, and ωB97M-V) continue to fare poorly for Li and overestimate the size, revealing considerable scope for improvement. The size of the Li atom is thus another reasonable choice as a soft constraint for future functional fitting. In contrast to the preceding two cases, Fig. 3 shows that most methods overestimate the size of the Be atom by a similar amount, FIG. 3 .Kiipredicted for the Be atom by various DFAs. The dashed line is the benchmark [CCSD(T)/CBS] value and the HF value is 1.444. J. Chem. Phys. 154, 074109 (2021); doi: 10.1063/5.0038694 154, 074109-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp with some poor performing outliers. In fact, the best mGGA and hybrid have ∼0.001 variation in Kii(while the best double hybrid is better by ∼0.005). This relatively uniform performance is likely a consequence of the challenges faced by DFAs accounting for the multireference character of the Be atom (it has 0.37 effectively unpaired electrons125). It is nonetheless worth noting that the worst performers are recently parameterized modern functionals. It is not entirely clear whether the Be atom should feature in datasets used for functional training because of the partial multireference charac- ter of this problem, but it should definitely be employed in test sets to gauge fit quality. V. STRETCHED H 2 The behavior of DFAs in predicting Kin non-equilibrium configurations is potentially interesting. We only investigate the case of H 2to avoid qualitative failures associated with delocal- ization error for polar bond22,41,57,58dissociations. We, further- more, restrict ourselves to a few well-behaved functional at Rung 4 and below to avoid incomplete spin-polarization based prob- lems.45Double hybrid functionals are not considered due to N-representability problems in UMP275and divergence of RMP2 with bond stretching (indicating that they are quite unsuitable for bond dissociation problems). Figure 4(a) shows that Kzzgrows quadratically with the internuclear distance r, and Fig. 4(b) there- fore only plots the error in Kzzfor clarity. All of the single-reference methods encounter a derivative discontinuity at the Coulson– Fischer (CF) point126(which is the onset of spin-polarization) due to the orbital stability matrix being singular at that point. The behav- ior right beyond this point is the most interesting as the two elec- trons are fairly strongly correlated in this regime. The DFAs other than SPW92 yield remarkably similar behavior in this region, sys- tematically overestimating Kzzto some extend, and thus predicting more “atom” like densities as opposed to a still partially bondedmolecule. HF exhibits the same qualitative behavior, but has much larger errors due to the complete absence of correlation [indeed, the systematic overestimation past the CF point can be clearly observed in Fig. 4(b) as well]. Similar overestimation in Kis seen for the Be atom, which also has partial multireference character. Consis- tent systematic overestimation of Kby several well-behaved func- tionals with varying levels of delocalization error could, in fact, potentially be a signature of strong correlation as the electrons are forced to be less compact in the absence of explicit (i.e., post mean- field) correlation that allows them to occupy the same region of space. VI. DISCUSSION AND CONCLUSION The objective of this work was to characterize the electron den- sity of small molecules using a scalar metric that goes beyond the first moment of the electron density (i.e., dipole moments). Since the molecular quadrupole moment is origin-dependent for molecules with non-zero charge or dipole, we elected instead to characterize the 3×3 matrix of second cumulants, or spatial variances, of the electron density, K. The eigenvalues of Kare thus a measure of the square of the characteristic extent of the molecular density along each principal axis. We produced a benchmark dataset, Var213, that contains 213 benchmark values of K, from 100 small molecules, evaluated with CCSD(T)/CBS correlation based on extrapolation toward the complete basis set limit from aug-cc-pCVTZ and aug-cc-pCVQZ calculations, combined with HF/aug-cc-pCV5Z. These reference values were then used to assess Kpredictions from 47 density func- tional approximations (DFAs). Broadly, the molecules studied here are relatively straightforward in terms of their electronic structure. According to Ref. 5, this dataset should be considered “easy” for DFAs, rather than “difficult” due to the absence of strong correlation or delocalization effects. FIG. 4 . Behavior of Kzzfor stretched H 2. J. Chem. Phys. 154, 074109 (2021); doi: 10.1063/5.0038694 154, 074109-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The results show that it is possible to obtain quite reason- ableKvalues in many cases, even with modern density functionals. However, non-empirical functionals, especially at the mGGA level via SCAN, and at the hybrid level via SCAN0, seem to fare better than more empirically parameterized models. This constitutes useful independent validation of the quality of electron densities from these functionals. Double hybrid Rung 5 functionals yield the best overall performance: indeed all double hybrids tested match or exceed the performance of SCAN, and only the early B2PLYP double hybrid fails to outperform SCAN0. Interestingly, the performance of the modern MN15-L, MN15, B97M-V, and ωB97M-V functionals over the studied dataset is dis- appointing, suggesting that the vastness of the meta-GGA space111 has offered these functionals the opportunity to predict reason- able energies from fairly poor densities. There is an interesting les- son to be drawn from this outcome, particularly for the minimally parameterized meta-GGA-based functionals such as B97M-V and ωB97M-V whose 12–16 parameter form was inferred from the use of large number of chemically relevant energy differences. Specifi- cally, such forms are evidently not fully constrained by the data used to generate them. Independent data such as Kmay, indeed, be very useful in attempting to develop improved data-driven functional designs. Overall, our Kbenchmark tests show that the performance of the best-performing functional strictly improves on ascending Jacob’s ladder, indicating that extra complexity has the potential to improve behavior. The use of this dataset and other density based information (especially the H, Li and Be atoms) for future func- tional development could thus provide sufficient soft constraints to yield improved Kvalues, and, by implication, electron densities as well. The results obtained here may thereby contribute to offering a route to better computationally tractable approximations to the exact functional. AUTHORS’ CONTRIBUTIONS D.H. and Y.H.L. contributed equally to this work. SUPPLEMENTARY MATERIAL See the supplementary material for geometries of species studied (zip) and computed values and analysis (xlxs). ACKNOWLEDGMENTS This research was initially supported by the Director, Office of Science, Office of Basic Energy Sciences, the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. Completion of the project was based on the work performed by the Liquid Sunlight Alliance, a DOE Energy Innovation Hub, supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award No. DE-SC0021266. Y.H.L. was funded via the UC Berkeley College of Chemistry summer research program.DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material. REFERENCES 1A. Karton, S. Daon, and J. M. L. Martin, Chem. Phys. Lett. 510, 165 (2011). 2J. Rezac and P. Hobza, J. Chem. Theory Comput. 9, 2151 (2013). 3A. D. Becke, J. Chem. Phys. 140, 18A301 (2014). 4R. O. Jones, Rev. Mod. Phys. 87, 897 (2015). 5N. Mardirossian and M. Head-Gordon, Mol. Phys. 115, 2315 (2017). 6P. Hohenberg and W. Kohn, Phys. 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9.0000161.pdf

AIP Advances 11, 025010 (2021); https://doi.org/10.1063/9.0000161 11, 025010 © 2021 Author(s).DFT-based calculations of the magnetic hyperfine interactions at Cd sites in RCd (R = rare earth) compounds with the FP-LAPW ELK code Cite as: AIP Advances 11, 025010 (2021); https://doi.org/10.1063/9.0000161 Submitted: 28 October 2020 . Accepted: 15 January 2021 . Published Online: 05 February 2021 L. S. Maciel , A. Burimova , L. F. D. Pereira , W. L. Ferreira , T. S. N. Sales , V. C. Gonçalves , G. A. Cabrera- Pasca , R. N. Saxena , and A. W. Carbonari COLLECTIONS Paper published as part of the special topic on 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials and 65th Annual Conference on Magnetism and Magnetic Materials ARTICLES YOU MAY BE INTERESTED IN WIEN2k: An APW+lo program for calculating the properties of solids The Journal of Chemical Physics 152, 074101 (2020); https://doi.org/10.1063/1.5143061 Coercivity enhancement of hot-deformed NdFeB magnet by doping R 80Al20 (R = La, Ce, Dy, Tb) alloy powders AIP Advances 11, 025001 (2021); https://doi.org/10.1063/9.0000028 Local inspection of magnetic properties in GdMnIn by measuring hyperfine interactions AIP Advances 11, 015322 (2021); https://doi.org/10.1063/9.0000037AIP Advances ARTICLE scitation.org/journal/adv DFT-based calculations of the magnetic hyperfine interactions at Cd sites in RCd (R = rare earth) compounds with the FP-LAPW ELK code Cite as: AIP Advances 11, 025010 (2021); doi: 10.1063/9.0000161 Presented: 3 November 2020 •Submitted: 28 October 2020 • Accepted: 15 January 2021 •Published Online: 5 February 2021 L. S. Maciel,1 A. Burimova,1 L. F. D. Pereira,1W. L. Ferreira,1T. S. N. Sales,1V. C. Gonçalves,1 G. A. Cabrera-Pasca,2 R. N. Saxena,1 and A. W. Carbonari1,a) AFFILIATIONS 1Instituto de Pesquisas Energéticas e Nucleares, IPEN-CNEN/SP, 05508-000 São Paulo, SP, Brazil 2Faculdade de Ciências Exatas e Tecnologia, Universidade Federal do Pará, Abaetetuba, PA 68440-000, Brazil Note: This paper was presented at the 65th Annual Conference on Magnetism and Magnetic Materials. a)Author to whom correspondence should be addressed: carbonar@ipen.br ABSTRACT In the work here reported, we have calculated magnetic hyperfine interactions in rare-earth (R) intermetallic compounds by using the free open-source all-electron ELK code. The RCd (R = Ce, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb) series was chosen as a test system because an almost complete set of experimental data on the hyperfine parameters at Cd sites was acquired through the time differential per- turbed angular correlation (TDPAC) spectroscopy as previously reported. Moreover, results on magnetic hyperfine field ( Bhf) from WIEN2k code were also reported allowing a qualitative comparison analysis. We emphasize that the utilized version of ELK accounted for the contact field only. Yet, as it is the only contribution expected for Cd site in RCd compounds, the calculated Bhfvalues are in reasonable agreement with the experimental results. The Spin-orbit coupling when taken into account led to a decrease in deviation from experimental data. Addition, the Hubbard-like term was revealed crucial in order to make Bhfpredictions for CeCd, suggesting that this behavior may be associated with a weaker 4 felectron localization in Ce. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/9.0000161 .,s I. INTRODUCTION Magnetic materials present singular properties which have their origin at the atomic level where the electronic contributions play a crucial role throughout. In particular, hyperfine interactions (nuclei and magnetic and/or electric potentials from electron cloud interactions) investi- gations provide valuable informations on the parameters that char- acterizes magnetic systems, for example, magnetic exchange interac- tions,1,2the order and temperature transition of magnetic phases,1,3 the spin-wave excitations,4,5and magnetic relaxation processes6 among others. Although measurements knowledge of those hyperfine fields could give a widely comprehension7about a system, first princi- ples calculations based on the density functional theory (DFT)8are desired to analyse more deeply their outcomes.In the DFT, the operators at ground state are functional of the electronic density ( ρ(r)) which is obtained from solving self- consistently a set of equations called Kohn-Sham scheme8,9 Several computer codes based on DFT have been developed to calculate the solid electronic band structure. In this work, we present the magnetic hyperfine field ( Bhf) at Cd nuclei in RCd com- pounds (where R is a rare-earth element) evaluated by the free open- source all-electron ELK code10and compare the results11with both experimental values and those calculated by the WIEN2k code.12 Elk project was released a few years ago with a user-friendly philosophy and growing significantly since then, despite that, the number of magnetic studies using it is scarce, especially those regarding hyperfine interactions. On the other hand, WIEN2k is one of the most popular codes available and a widespread useful tool to improve magnetic knowledge. Although the two codes use the same AIP Advances 11, 025010 (2021); doi: 10.1063/9.0000161 11, 025010-1 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv methodology for calculations of Bhf,13this work might shed light in the comparison between their performance and reproducibility on ab initio calculations. Rare-earth (R) elements differ in the number of well-shielded 4felectrons which are responsible for the magnetism in compounds consisting of rare earth element as well as non-magnetic atoms. The main feature of rare-earth elements is that the orbital magnetic con- tribution is strong in relation to spin and intensively varies with the number of 4 felectrons. Such RCd compounds (where Cd is a non- magnetic atom) crystallize in the high-symmetry cubic CsCl - type of structure (space group Pm-3m), which favors the experimental magnetic hyperfine interaction investigation due to the absence of electric quadrupole interactions11as well as reduces the time for first-principles calculations. Furthermore, those compounds are great candidates for the Bhf successful theoretical outcomes since have single and simple fer- romagnetic structure. In addition, since cadmium is diamagnetic, the only contribution to Bhfat Cd nucleus comes from the core polarization of the transferred field from the rare-earth host and, consequently, the major Bhfcomponent is yielded from selectrons spin - called Fermi Contact contribution. It is known that there is a shortcoming in the electronic band structure codes for Bhfcalcula- tions.14Nevertheless, they are state-of-the-art of ab initio calculation in solids and the Fermi Contact contribution has evaluated better than the orbital contribution to Bhf.15The total hyperfine magnetic field is given by Bhf=Bc+Borb+Bsp, (1) where Bcis the Fermi contact term, Borbis the field associated with the on-site orbital moment and Bspis the dipolar field from the on- site spin density. II. FIRST-PRINCIPLES CALCULATIONS In order to investigate the hyperfine interactions in the RCd series, the theoretical spin band structure calculations based on the DFT formalism have been performed with the full potential lin- earized augmented plane wave (FP-LAPW)16plus local orbitals, as implemented in the ELK code. In the context of the many-body sys- tems under Bloch periodic boundary conditions, the aim of DFT calculations is to predict the ground-state energy whereby function- als of the electronic density. Hence, the problem is solved by find- ing the spin electron density that minimizes the total energy which, according to Hohenberg and Kohn9is given by E[ρ]=Ts[ρ]+Eei[ρ]+EH[ρ]+Eii[ρ]+Exc[ρ], (2) where, ρis the the electronic density (for spin polarized calcula- tions [ ρ]→[ρ↑,ρ↓]),Ts[ρ] is the single particle kinetic energy, Eei[ρ] correspond to the Coulomb interaction between electrons and nuclei, the Eii[ρ] term express the interaction between nuclei, EH[ρ] is the Hartree component of the electron-electron interaction and Exc[ρ] represents the exchange and correlation effects whose were treated using the generalized gradient approximation (GGA) using the Perdew-Burke-Ernzerhof (PBE96) parametrization.17The computational details are now summarized. The radius of the non overlapping muffin-tin spheres centered in the atomic positions were 2.8 and 2.6 a.u. for the rare-earth and Cd specimens, respectively. The optimized FP-LAPW parameters are: (i) Kmax= 8/RCd MT, where Kmaxis the biggest modulus of the FIG. 1 . Variation of DFT total energy for GdCd versus volume (circles) and a cubic fit to the data (solid line). cutoff wave vector of the plane wave basis in the interstitial region andRCd MTis cadmium muffin-tin radius; (ii) The mesh of k-points in the irreducible Brillouin zone of the RCd series was set to 15 ×15 ×15. Once the majority RCd elements has ferromagnetic order at low temperatures,18,19ferromagnetic structure and, as a conse- quence, 2-atoms single cell (one R and one Cd) have been took into account. The crystallographic informations for modeling the RCd unit cells was obtained in the experimental work of Buschow.18 We have considered the fully optimized RCd equilibrium structures, whose optimization data were fitted using the Birch-Murnaghan equation of state (EOS) given by20 E(η)=E0+9B0V0 16(η2−1)2(6 +B′ 0(η2−1)−4η2), (3) where the V0(volume at zero pressure), B0(bulk modulus at zero pressure), B′ 0(pressure derivative) and E0(total energy at zero pressure) are fitting parameters and η=(V/V0)1/3. For a case study considering the GdCd system, according to inset image of Fig. 1, the lattice parameter equilibrium predicted by FP-LAPW is about 1.37% greater than the experimental value previ- ously reported by Buschow.18This difference for all RCd series is in the range of 1-2% with respect to experimental lattice param- eter. Regarding the computed values of V0,B0and B′ 0, we can observe that they are within the expected range of those predicted by DFT calculations.21,22It is worthwhile mention that the influ- ence of the spin-orbit coupling in our optimization process it was irrelevant. III. RESULTS AND DISCUSSION Results of the magnetic hyperfine field for the whole RCd series calculated by ELK code ( BELK hf) taking into account the spin-orbit coupling are displayed in Table I. These magnetic hyperfine fields are compared with those calculated by WIEN2k ( BW2k hf) and the experi- mental data ( BEXP hf), both extracted from Cavalcante et al.11As can be AIP Advances 11, 025010 (2021); doi: 10.1063/9.0000161 11, 025010-2 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv TABLE I . Theoretical values of the total magnetic hyperfine field in Cd position deter- mined by first-principles calculations using ELK code ( BELK hf) and compared with values calculated by WIEN2k ( BW2k hf) and experimental values ( BEXP hf(0)) taken from Ref. 11. The sign of the experimentally determined BEXP hfis unknown. Theoretical results Compound BELK hf(T) BW2k hf(T) BEXP hf(0) (T) CeCd -6.94 -5.75 5.10 PrCd -14.51 -13.92 - NdCd -21.27 -15.89 17.4 PmCd -24.28 - - SmCd -22.93 -21.98 21.1 EuCd -28.51 - - GdCd -38.15 -35.11 30.8 TbCd -26.40 -25.67 25.6 DyCd -20.03 -17.91 20.0 HoCd I -12.64 -11.74 10.3 HoCd II -12.64 -11.74 11.1 ErCd -7.06 -6.22 6.9 TmCd -2.76 - - YbCd -0.83 - - seen, we obtained a good hyperfine field prediction for all the RCd series. It is important to mention that for CeCd, considering only spin-orbit coupling in the calculations, the difference was about 63% higher when compared with BEXP hf. This discrepancy is attributed to the high degree of hybridization between the 4 fand 5 delectrons as shown in the partial density of states (PDOS) (Fig. 2). To take into account this effect, the on-site correlation through the Hubbard parameter Uwas used in our calculation for CeCd and the result ofBELK hf=-6.94 T is now compatible to the experimental value of BEXP hf=-5.10 T. For this calculation, the values of U= 0.6 eV and J = O, as reported by Mestnik-Filho et al.23This is verifies in the work of Cavalcante et al. ,24where the Hubbard model describe correctly the effect of strong correlations of the 4 felectrons in the rare-earth elements, leading to a considerable improvement of the agreement between theoretical and experimental hyperfine fields. The PDOS for CeCd, DyCd, and ErCd (see Fig. 2) reveals that the 4 fband for Dy and Er are almost isolated from the 5 dband whereas in CeCd there is a hybridization between them (-0.5 Ha to 0.1 Ha) indicating a stronger correlation, reinforcing the necessity of taking into account the U parameter in the calculations. These predictions suggest that the polarization mechanism may be differ- ent in CeCd as compared with DyCd and ErCd. Compounds with rare-earth atoms from Gd to Yb may have the same mechanism where the 4 fspins couple indirectly through the polarization of conduction s-electrons ( f-scoupling). On the other hand, in CeCd and possibly in the compounds with Pr, Nd, Pm, and Sm the cou- pling occurs between fand ⁀{d} electrons ( f-dcoupling). The mag- netic measurements18performed on polycrystalline RCd materials, reveals different coupling mechanisms in light (Ce, Pr, Nd, Pm, Sm) and heavy (Gd, Tb, Dy, Ho, Er) rare-earth compounds. Buschow considered that the so-called RKKY interaction is responsible for FIG. 2 . Partial density of states (PDOS) of CeCd, DyCd, and ErCd relative to Fermi energy (E = 0). the coupling between the heavy rare-earth spins in which the con- duction s-electrons are polarized by the 4 felectrons acquiring an oscillatory character.18 The Fig. 3 shows the values of the Bhfcalculated in this work via ELK code compared with those reported by Cavalcante et al.11 There is agreement between experimental and our theoretical results with both following a linear dependence on the spin projection ( g −1)Jon the total angular momentum Jof the rare-earth element. The calculated ELK values are slightly greater than the TDPAC experimental results and WIEN2k values for almost the whole series. This small difference is probably due to the exchange and correlation approximation which is the GGA for ELK calculations whereas the WIEN2k calculations used the local density approximation (LDA)11 as well as some difference in the local base set of plane waves. More- over, we believe that the use of Uparameter could improve the results. For CeCd the calculations of GGA+U have improved the value of Bhffrom -13.68 T for spin-orbit coupling to -6.94 T. It is noteworthy, nonetheless, that a faster and easier to use code could provide almost the same results for the Bhfthan a complex and AIP Advances 11, 025010 (2021); doi: 10.1063/9.0000161 11, 025010-3 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 3 . Spin dependence of experimental (Ref. 11) and theoretical values, calcu- lated by ELK code (this work) and by WIEN2k code (Ref. 11). The experimental values are the saturation values of Bhfat 0 K. Straight lines are the linear fit to the values. time-consuming calculation. In addition, we predict the Bhffor PmCd, EuCd, TmCd, and YbCd completing the whole rare-earth series of RCd compounds. IV. SUMMARY We perform DFT calculations for the RCd series to investi- gate the behavior of the magnetic hyperfine interactions at Cd sites and the reliability of the FP-LAPW code ELK. The first-principles calculations were carried out with spin polarization and spin orbit coupling. Calculations for CeCd were also performed by taking the Hubbard parameter into account. Results of density of states shows that the spin transfer mechanism for CeCd and probably for the rest of light rare-earth (Pr, Nd, Pm, Sm) compounds may occur through polarization of delectrons whereas for heavy rare-earth (from Gd to Yb) it is realized by the polarization of conduction s-electrons. The values of the Bhfat Cd nuclei for the whole series of RCd were calculated by the ELK code. Even those compounds not experimen- tally measured such as PrCd, PmCd, EuCd, TmCd, and YbCd follow the linear dependence with the spin projection ( g−1)J. The values ofBhfare a little higher than the experimental ones, which can be improved by performing calculations adding the U parameter, in order to taking into account the local correlation from RE 4f elec- trons as demonstrated in the case of CeCd. The high correlation was observed experimentally in the case of RAg where 4f electrons shown anomalous behavior that was attributed by hybridization between 4f-5d electrons.24Our work revealed that the ELK version used is a good tool in order to determine the hyperfine field in mag- netic systems with localized magnetic moments and weak electronic hybridization. ACKNOWLEDGMENTS Partial financial support for this work was provided by Fun- dação de Amparo a Pesquisa do Estado de São Paulo (FAPESP)under grant 2014/140001-1. AWC and RNS acknowledge the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for financial support in a form of research fellowship (grant 304627/2017-8). AB greatly acknowledges the financial support of FAPESP (grant 2019/15620-0). TSNS acknowledge financial support from CNEN. LS acknowledge CNPq for financial support in a form of scholarship (grant 134233/2017-4) and WLF acknowledge finan- cial support from Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES). LFDP expresses his thanks for support by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) in a form of the Programa de Capacitação Institucional (PCI) fellowship (grant 444323/2018-0). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1B. Bosch-Santos, A. W. Carbonari, G. A. Cabrera-Pasca, M. S. Costa, and R. N. Saxena, J. Appl. Phys. 113, 17E124 (2013). 2B. Bosch-Santos, A. W. Carbonari, G. A. Cabrera-Pasca, and R. N. Saxena, J. Appl. Phys. 115, 17E124 (2014). 3A. L. Lapolli, R. N. Saxena, J. Mestnik-Filho, D. M. T. Leite, and A. W. Carbonari, J. Appl. Phys. 101, 09D510 (2007). 4G. A. Cabrera-Pasca, A. W. Carbonari, R. N. Saxena, B. Bosch-Santos, J. A. H. Coaquira, and J. A. Filho, J. Alloys and Compounds 515, 44–48 (2012). 5G. A. Cabrera-Pasca, A. W. Carbonari, B. Bosch-Santos, J. Mestnik-Filho, and R. N. Saxena, J. Phys.: Condens. Matter 24, 416002 (2012). 6G. A. Cabrera-Pasca, B. Bosch-Santos, A. Burimova, E. L. Correa, and A. W. Carbonari, AIP Advances 10, 015223 (2020). 7A. W. Carbonari, J. Mestnik-Filho, and R. N. Saxena, Defect and Diffusion Forum 311, 39–61 (2011). 8P. Hohenberg and W. Kohn, Phys. Rev. 136, B864–B871 (1964). 9W. Kohn and L. J. Sham, Phys. Rev. 140, A1133–A1138 (1965). 10See http://elk.sourceforge.net/ for, The Elk Code. 11F. H. M. Cavalcante, O. F. L. S. L. Neto, H. Saitovitch, J. T. P. D. Cavalcante, A. W. Carbonari, R. N. Saxena, B. Bosch-Santos, L. F. D. Pereira, J. Mestnik-Filho, and M. Forker, Phys. Rev. B 94, 064417 (2016). 12P. Blaha, K. Schwarz, F. Tran, R. Laskowski, G. K. H. Madsen, and L. D. Marks, The Journal of Chemical Physics 152, 074101 (2020). 13S. Blügel, H. Akai, and P. H. Dederichs, Phys. Rev. B 35, 3271–3283 (1987). 14K. Lejaeghere, G. Bihlmayer, T. Björkman, P. Blaha, S. Blügel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. Dal Corso et al. , Science 351, aad3000 (2016). 15P. Novák, J. Kuneš, W. Pickett, W. Ku, and F. Wagner, Physical Review B 67, 140403 (2003). 16E. Sjöstedt, L. Nordström, and D. J. Singh, Solid State Communications 114, 15–20 (2000). 17J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865–3868 (1996). 18K. H. J. Buschow, J. Chem. Phys 61, 4666–4670 (1974). 19F. H. M. Cavalcante, L. F. D. Pereira, A. W. Carbonari, J. Mestnik-Filho, and R. N. Saxena, J. Alloys and Compounds 660, 148–158 (2016). 20F. Birch, Physical review 71, 809 (1947). 21L. Petit, Z. Szotek, J. Jackson, M. Lüders, D. Paudyal, Y. Mudryk, V. Pecharsky, K. Gschneidner, Jr., and J. B. Staunton, Journal of Magnetism and Magnetic Materials 448, 9–12 (2018). 22G. Pagare, H. Devi, S. S. Chouhan, and S. P. Sanyal, Computational materials science 92, 178–184 (2014). 23J. Mestnik-Filho, L. F. D. Pereira, M. V. Lalic, and A. W. Carbonari, Physica B 389, 73–76 (2007). 24F. H. M. Cavalcante, L. F. D. Pereira, A. W. Carbonari, J. Mestnik-Filho, and R. N. Saxena, Journal of Alloys and Compounds 660, 148–158 (2016). AIP Advances 11, 025010 (2021); doi: 10.1063/9.0000161 11, 025010-4 © Author(s) 2021

5.0039069.pdf

Appl. Phys. Lett. 118, 052406 (2021); https://doi.org/10.1063/5.0039069 118, 052406 © 2021 Author(s).Enhancement of spin–orbit torque in WTe2/perpendicular magnetic anisotropy heterostructures Cite as: Appl. Phys. Lett. 118, 052406 (2021); https://doi.org/10.1063/5.0039069 Submitted: 30 November 2020 . Accepted: 21 January 2021 . Published Online: 02 February 2021 Wenxing Lv , Hongwei Xue , Jialin Cai , Qian Chen , Baoshun Zhang , Zongzhi Zhang , and Zhongming Zeng COLLECTIONS Paper published as part of the special topic on Spin-Orbit Torque (SOT): Materials, Physics, and Devices ARTICLES YOU MAY BE INTERESTED IN Large damping-like spin–orbit torque and perpendicular magnetization switching in sputtered WTe x films Applied Physics Letters 118, 042401 (2021); https://doi.org/10.1063/5.0035681 Spin-orbit torques: Materials, physics, and devices Applied Physics Letters 118, 120502 (2021); https://doi.org/10.1063/5.0039147 Voltage-controlled spin–orbit torque switching in W/CoFeB/MgO Applied Physics Letters 118, 052409 (2021); https://doi.org/10.1063/5.0037876Enhancement of spin–orbit torque in WTe 2/perpendicular magnetic anisotropy heterostructures Cite as: Appl. Phys. Lett. 118, 052406 (2021); doi: 10.1063/5.0039069 Submitted: 30 November 2020 .Accepted: 21 January 2021 . Published Online: 2 February 2021 Wenxing Lv,1,2Hongwei Xue,3Jialin Cai,2 Qian Chen,2Baoshun Zhang,2 Zongzhi Zhang,3,a) and Zhongming Zeng2,a) AFFILIATIONS 1Physics Laboratory, Industrial Training Center, Shenzhen Polytechnic, Shenzhen, Guangdong 518055, People’s Republic of China 2Key Laboratory of Multifunctional Nanomaterials and Smart Systems, Suzhou Institute of Nano-Tech and Nano-Bionics, CAS, Suzhou, Jiangsu 215123, People’s Republic of China 3Department of Optical Science and Engineering, Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), andShanghai Ultra-Precision Optical Manufacturing Engineering Center, Fudan University, Shanghai 200433, People’s Republic of China Note: This paper is part of the Special Topic on Spin-Orbit Torque (SOT): Materials, Physics and Devices. a)Authors to whom correspondence should be addressed: zzzhang@fudan.edu.cn and zmzeng2012@sinano.ac.cn ABSTRACT Spin–orbit torque (SOT), exerted to a ferromagnet from an adjacent non-magnetic layer, has been widely considered as a promising strategy to realize spintronic devices with high energy efficiency, endurance, and speed. Much effort has been devoted to the search for materials and structures that can generate strong SOTs. Recent investigations showed that two-dimensional (2D) transition metal dichalcogenides providethe potential to produce strong enough SOTs to manipulate the magnetic devices due to rich spin-dependent properties. Here, we presentthe study of SOT in WTe 2/ferromagnet with perpendicular magnetic anisotropy devices, and an enhancement of SOT efficiency with the thickness of WTe 2is observed, which may be ascribed to the spin absorption at the WTe 2/Ta interface and the spin Hall effect. This work demonstrates the possibility of manipulating magnetization by 2D materials and an avenue for engineering spintronic devices based on 2Dmaterials. Published under license by AIP Publishing. https://doi.org/10.1063/5.0039069 Spin-orbit torque (SOT) has attracted intensive attention since it offers a promising mechanism for realizing devices with energy effi- cient operation, higher endurance, and reliability. 1–7In a SOT device, the electrons flowing in a non-magnetic spin-source layer with strongspin–orbit coupling are spin polarized, and this polarized spin currenttransfers its angular momentum to the neighboring ferromagneticlayer. Thus, SOT efficiency, defined as the ratio of the spin currentabsorbed by the ferromagnetic layer to the charge current in the non- magnetic layer, becomes a key factor when building energy efficient SOT devices, which consequently spurs much interest in seeking newmaterials or artificial structures to achieve stronger SOTs. 8–14 Recently, research studies have proved that two-dimensional (2D) transition metal dichalcogenides (TMDs) provide a rich platformfor investigating spin-dependent physics due to their unique properties including well-defined atomically thin layered structure characteristics, reduced crystal structure symmetry, and strong spin–orbitcoupling. 15,16More excitingly, it is found that when combined with ferromagnetic films forming TMD/ferromagnet heterostructures,many exotic magnetic phenomena, such as strong Rashba effects in the (WSe 2)MoS 2/CoFeB bilayer17and WS 2/Permalloy,18giant enhancement of magnetic anisotropy in MoS 2/Pt/[CoNi] 2 multilayers,19high spin Hall efficiency in WSe 2/Ta/Co 60Fe20B20/MgO multilayers,20and high spin Hall conductivity in PtTe 2/Py bilayer,21 will be introduced, which motivates the 2D TMD-based heterostruc-ture research to explore the potential applications of TMDs in spin-tronic devices. Among them, WTe 2, a member of TMD family, has been widely acknowledged as a spin-source material in WTe 2/ferro- magnet heterostructures due to its strong spin–orbit coupling, brokeninversion symmetry, and type-II Weyl semimetal nature. A number ofintriguing properties have been proved in WTe 2/ferromagnet hetero- structures. For instance, N /C19eel-type skyrmion is observed in WTe 2/ Fe3GeTe 2heterostructures induced by the large interfacial Appl. Phys. Lett. 118, 052406 (2021); doi: 10.1063/5.0039069 118, 052406-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplDzyaloshinskii-Moriya interaction at the interface ascribed to the strong spin–orbit interaction in WTe 2.22Field-free current-induced magnetization switching owing to Dzyaloshinskii-Moriya interaction,23 significantly enhanced spin conductivity due to spin-momentum lock-ing effects, 24and especially, an out-of-plane spin–orbit torque (SOT) induced by inversion symmetry breaking of WTe 225,26are demon- strated in WTe 2/Permalloy bilayers. All these indicate that WTe 2is a promising candidate for realizing low-power consumption spintronic devices. In this work, we fabricated WTe 2/ferromagnet heterostructures with perpendicular magnetic anisotropy (PMA) since the SOT-drivendevices with PMA can take advantage of high-density integration,high-speed processing, and low-power consumption. 3Amorphous fer- romagnetic rare earth-transition metal film TbCo was employed as theferromagnet because it can exhibit strong PMA when the antiferro-magnetically coupled magnetic moments of rare earth and transitionmetals are nearly balanced, which is beneficial for the design of nano-scaled memory devices with high thermal stability. The PMA of theferromagnet in the heterostructure was characterized by hysteresisloops, and an enhancement of spin–orbit torque efficiency with thethickness of WTe 2was observed using the harmonic measurement, which may be mainly due to the enhanced spin absorption at theWTe 2/Ta interface and the spin Hall effect. Our results are helpful for constructing advanced spintronic devices that can be further applica-ble in logic, memory, and neuromorphic computing. 27,28 The WTe 2/PMA heterostructure device was fabricated as follows: First, a thin WTe 2flake was mechanically exfoliated from a high- quality WTe 2crystal (HQ graphene) onto a thermally oxidized silicon substrate with 280 nm SiO 2. The exfoliation procedure was performed inside a nitrogen-filled glovebox with H 2Oa n dO 2concentrations of <1 ppm. Then, the ferromagnet composed of Ta (1 nm)/TbCo (4 nm)/AlO x(3 nm) thin films was deposited on the whole substrate b ym a g n e t r o ns p u t t e r i n gi nw h i c ht h eA l O xcapping layer was obtained by natural oxidation of a metallic Al layer. After the deposi-tion, the WTe 2/[Ta (1 nm)/TbCo (4 nm)/AlO x(3 nm)] heterostructure was patterned to 3 lm/C29.5lm Hall bar by electron beam lithography(EBL) and Ar ion milling. Finally, Ti(10 nm)/Au(60 nm) contacts were defined using EBL and deposited using electron beam evaporation(EBE). The AFM measurements (Icon Dimension 3100, USA) were car- ried out to characterize the surface morphology and thickness of the WTe 2thin flake in these samples. The Raman spectra of the WTe 2 flake were measured with a laser radiation of 532 nm and a power of 10lW. Magnetic hysteresis loops of [Ta/TbCo/AlO x] thin films were obtained using a vibration sample magnetometer (VSM, Lakeshore,USA). The anomalous Hall effect (AHE) resistance R AHwas obtained by sweeping the external magnetic field along the z-axis. Harmonic measurements were carried out to investigate the spin–orbit effect,where first and second harmonic signals were collected by two lock-in amplitudes. Two types of control samples are investigated without (type A) and with (type B) a WTe 2layer, as shown in Fig. 1(a) .T h ea n g u l a r - dependent Raman spectra of WTe 2collected with a crystal orientation are shown in Fig. 1(b) .F r o m Fig. 1(b) , it is seen that the polar diagram of the relative intensity of the Raman mode of A 1at (/C24211 cm/C01)i s different, which is in good agreement with the previous studies23,25,29 and indicates that the longitude axis of the investigated devices isbetween the a- and b-axes of WTe 2.Figure 1(c) shows the optical microscopy image of one typical fabricated device. Before the Hall bar device fabrication process, we performed the VSM measurements to characterize the static properties of the Ta/ TbCo/AlO xmultifilm. Figure 2(a) shows the out-of-plane and in-plane hysteresis loops of the Ta/TbCo/AlO xmultifilm at room tem- perature, respectively. The more-squared shape and larger coercivity Hc(/C243600 Oe) in the out-of-plane loop [the red line in Fig. 2(a) ] signify a very good PMA of the Ta/TbCo/AlO xmultifilm. We further study the magnetism in Ta/TbCo/AlO xfilms by probing the Hall resis- tance, Rxy, under an external magnetic field, l0H, applied perpendicu- lar to the film plane. For the magnetic material, Rxyis composed of two parts as follows: Rxy¼RNHþRAH; (1) FIG. 1. (a) The stacks investigated in this work. (b) The representative angle-dependent intensities of the A g(A1) mode at /C24211 cm/C01. (c) Optical images of the WTe 2film and corresponding WTe 2/Ta/TbCo/AlO xdevice.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 052406 (2021); doi: 10.1063/5.0039069 118, 052406-2 Published under license by AIP Publishingwhere RNHrepresents the normal Hall resistance, e.g., the resistance contribution from the Lorenz force ( RNH¼R0l0H), and RAHis the anomalous Hall resistance contributed from the magnetization of themeasured device ( R AH¼RSM).Figure 2(b) shows RAHfor the Ta/ TbCo/AlO xdevice. A clear hysteresis in RAHwas observed. Such hys- teresis reflects the hysteresis in M; the remnant Matl0H¼0m e a n s spontaneous magnetization and therefore the long-range magnetic order, e.g., PMA in the Ta/TbCo/AlO xdevice. By comparing Fig. 2(a) with Fig. 2(b) ,t h ef a c tt h a t Hcextracted from the anomalous Hall effect [see Fig. 2(b) ]i sm u c hs m a l l e rt h a n Hcextracted from the hys- teresis loop [see the red line in Fig. 2(a) ] is noticed. Similar reduction of the switching magnetic field is also observed in Co-Fe-B/MgO nanodots as the size of nanodots decreases, which can be wellexplained by the influence of Laplace pressure applied on the domain wall at the edges of the nanodots where damages are introduced by the patterning process. 30In addition, the oxidation of Tb can lead to reduced H cas well.31–33Thus, we believed that the decreased H chere is closely related to the damages such as edge roughness, intermixingof the interfaces, or oxidation of the layers introduced by the fabrica- tion process, especially, Ar ion milling. Figure 3 displays the anomalous Hall effect of the WTe 2/Ta/ TbCo/AlO xheterostructure. Obviously, the anomalous Hall effectwith hysteresis is indicative of good PMA. However, when in contrast to the pure Ta/TbCo/AlO xdevice, PMA was found to be greatly reduced after the introduction of WTe 2according to the smaller Hc (/C2454 Oe), which is contrary to the enhanced PMA performed in MoS 2/Pt/[Co/Ni] 2heterostructures19and similar to the weakened PMA observed in WSe 2/Ta/CoFeB heterostructures.20The PMA qual- ity can be influenced by a lot of extrinsic factors such as deposition conditions, interfacial roughness, oxidation of the ferromagnet, and intrinsic factors such as hybridization at the heavy metal/transition fer- romagnetic metal interface.19In this work, the deposition parameters are identical for all samples. The substrate roughness was improved for the WTe 2/Ta/TbCo/AlO xdevice owing to the atomically flat surface of WTe 2, which in principle supports the induction of PMA. Meanwhile, the hybridization at the interface of heavy metal (Ta)/transition ferromagnetic metal may be affected by the W or Te atoms during the Ar ion milling process and further the PMA. To shed light on the impacts of the WTe 2underlayer on the SOTs, harmonic Hall measurements were performed,34and the mea- surement configuration is schematically depicted in Fig. 4(a) .F o rt h e harmonic measurements, an ac current with a fixed frequency of 135.5 Hz was applied along the xdirection to exert SOTs to the mag- netization, causing the magnetization to oscillate around the equilib- rium position and further generating harmonic signals. SOTs have two components, namely, damping-like torque and field-like torque,which can be equivalently described as effective fields, that is, damping-like ( H DL) and field-like ( HFL) effective fields. During the measurements, the external l0Hw a ss w e p ta l o n ge i t h e rt h e xory direction to extract HDLorHFLsince HDLorHFLis parallel or perpen- dicular to the applied ac current direction, respectively. HDL(HFL)c a n be expressed by harmonic Hall measurements as follows:15,34 HDL FLðÞ¼/C02@V2x @HxðyÞ/C30@2Vx @H2 xðyÞ; FIG. 2. (a) Out-of-plane and in-plane magnetic hysteresis loops of the Ta/TbCo/ AlO xmultifilm. (b) Anomalous Hall effect of the Ta/TbCo/AlO xdevice at room temperature. FIG. 3. Anomalous Hall effect of the WTe 2/Ta/TbCo/AlO xdevice at room temperature. FIG. 4. (a) Measurement schematic for the second harmonic measurement. (b) First and second harmonic anomalous Hall voltages as a function of an externalmagnetic field for the WTe 2/Ta/TbCo/AlO xdevice. (c) Damping-like HDLand (d) field-like HFLeffective fields at different current densities.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 052406 (2021); doi: 10.1063/5.0039069 118, 052406-3 Published under license by AIP Publishingwhere Vxand V2xare the first and second harmonic Hall voltages, respectively. Figure 4(b) shows the first and second harmonic voltages of the WTe 2/Ta/TbCo/AlO xdevice as a function of l0H.It is clear that the Vxsignal exhibits a quadratic relationship, while the V2xsig- nal shows a linear dependence. By fitting the experimental Vxand V2xvsHxðyÞcurves under different Jacvalues, the corresponding HDL andHFLdata, as shown in Figs. 4(c) and4(d), can be obtained, respec- tively. Both effective fields display good linearity with Ja, revealing weak contribution from the thermal effect.35The SOT fields per cur- rent density are obtained for vDL¼HDL/Jac¼5.1/C210/C010Oe/(A cm/C02) and vFL¼HFL/Jac¼2.6/C210/C010Oe/(A cm/C02)b yl i n e a r l yfi t t i n g HDL(HFL)v sJaccurves. Generally, the SOT can be estimated by the spin torque efficiency defined as n¼/C02jejMstF /C22hHDL Jac,w h e r e eis the electron charge, /C22hi st h e reduced Plank constant, t Fis the thickness of the ferromagnetic layer, and Msis the saturation magnetization. The Msvalue of the 4 nm TbCo film is estimated to be 35 emu/cc obtained from the hysteresis loop. Note that, if using this value, the calculated nis usually small in comparison with the actual value because of the increased Msof the patterned device, which may result from the preferential oxidation of rare-earth metal Tb during the fabrication process31–33and the natural oxidation process of the Al layer. As shown in Fig. 5 ,t h ed a m p i n g - l i k e nincreases with WTe 2thickness. Similar enhancement of spin Hall efficiency has been reported in the WSe 2/Ta/CoFeB/MgO/Ta stack, which accounts for by the suppression of the spin back diffusion since a large fraction of these unwanted spins are absorbed at the WSe 2/Ta interface; in other words, the spin flow injected into the CoFeB layer was increased.20Qiuet al. have demonstrated an enhancement of spin torque in the heavy metal/ferromagnet/Ru multilayer resulted fromthe spin absorption by the top Ru layer. 36As the thickness of the Ta l a y e r( 1 n m )i no u rW T e 2/Ta/TbCo/AlO xheterostructure is smaller than the spin diffusion length in Ta, it is reasonable to assume a spin backflow at the Ta/TbCo interface. Thus, the increase in nmay be due to the enhanced spin absorption at the WTe 2/Ta interface if taking the unique spin properties of WTe 2such as unconventional spin-to- charge conversion37and nanosecond spin relaxation29into account. Another possible mechanism for this enhancement might result fromthe spin Hall effect of WTe 2, similar to the behaviors observed indifferent Pt/ferromagnet structures, in which the ratio Js/JNM(Jsrepre- sents the spin current and JNMrepresents the charge current) increases with Pt thickness in the case of Pt thickness smaller than its spin diffu- sion length.38–40 In conclusion, we have demonstrated a reduction of PMA and an enhancement of SOT efficiency in the Ta/TbCo/AlO xheterostructure by inserting the WTe 2underlayer. The SOT efficiency dependence of WTe 2thickness indicates that the enhancement of SOT efficiency may be a consequence of the enhanced spin absorption at the WTe 2/Ta interface and the spin Hall effect. These findings may suggest a strat- egy for integrating 2D materials with strong coupling into spintronic devices. This work was supported by the National Natural Science Foundation of China (Nos. 51732010, 11974379, and 11874120). 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5.0037568.pdf

J. Chem. Phys. 154, 054312 (2021); https://doi.org/10.1063/5.0037568 154, 054312 © 2021 Author(s).Argon tagging of doubly transition metal doped aluminum clusters: The importance of electronic shielding Cite as: J. Chem. Phys. 154, 054312 (2021); https://doi.org/10.1063/5.0037568 Submitted: 14 November 2020 . Accepted: 11 January 2021 . Published Online: 04 February 2021 Jan Vanbuel , Piero Ferrari , Meiye Jia , André Fielicke , and Ewald Janssens ARTICLES YOU MAY BE INTERESTED IN Stability of cationic silver doped gold clusters and the subshell-closed electronic configuration of AgAu 14+ The Journal of Chemical Physics 153, 244304 (2020); https://doi.org/10.1063/5.0033487 Gas-phase studies of chemically synthesized Au and Ag clusters The Journal of Chemical Physics 154, 140901 (2021); https://doi.org/10.1063/5.0041812 Theoretical insights into the thermal reduction of N 2 to NH 3 over a single metal atom incorporated nitrogen-doped graphene The Journal of Chemical Physics 154, 054703 (2021); https://doi.org/10.1063/5.0039338The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Argon tagging of doubly transition metal doped aluminum clusters: The importance of electronic shielding Cite as: J. Chem. Phys. 154, 054312 (2021); doi: 10.1063/5.0037568 Submitted: 14 November 2020 •Accepted: 11 January 2021 • Published Online: 4 February 2021 Jan Vanbuel,1 Piero Ferrari,1 Meiye Jia,1 André Fielicke,2 and Ewald Janssens1,a) AFFILIATIONS 1Quantum Solid-State Physics, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium 2Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4 −6, 14195 Berlin, Germany and Institut für Optik und Atomare Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany a)Author to whom correspondence should be addressed: ewald.janssens@kuleuven.be ABSTRACT The interaction of argon with doubly transition metal doped aluminum clusters, Al nTM 2+(n= 1–18, TM = V, Nb, Co, Rh), is studied experimentally in the gas phase via mass spectrometry. Density functional theory calculations on selected sizes are used to understand the argon affinity of the clusters, which differ depending on the transition metal dopant. The analysis is focused on two pairs of consecutive sizes: Al6,7V2+and Al 4,5Rh2+, the largest of each pair showing a low affinity toward Ar. Another remarkable observation is a pronounced drop in reactivity at n= 14, independent of the dopant element. Analysis of the cluster orbitals shows that this feature is not a consequence of cage formation but is electronic in nature. The mass spectra demonstrate a high similarity between the size-dependent reactivity of the clusters with Ar and H 2. Orbital interactions provide an intuitive link between the two and further establish the importance of precursor states in the reactions of the clusters with hydrogen. Published under license by AIP Publishing. https://doi.org/10.1063/5.0037568 .,s INTRODUCTION The interaction of noble gases with clusters provides a rich, albeit indirect source of information on a variety of cluster properties. Foremost, the interaction is in most cases governed by weak charge-induced-dipole forces (adsorption energies E ads ≈0.1 eV–0.2 eV), which makes noble gases ideal messenger atoms for gas phase action spectroscopic techniques, such as infrared multiple photon dissociation and vibrational predissociation spec- troscopy.1,2By virtue of the weakness of the interaction, the noble gas can be considered a spectator atom, with a negligible influ- ence on the geometry and vibrational modes of the bare clus- ters.3Exceptions do exist, for which the presence of the argon tag(s) should be explicitly taken into account to correctly inter- pret the infrared spectra and infer the cluster geometry.4,5For some small molecular complexes, which primarily involve transition metal or gold atoms, noble gases can form strong covalent bonds (Eads≈1 eV).6,7Besides their role as messengers in action spectroscopy, noble gases are also used in mass spectrometric studies as a structural probe. As transition metal atoms typically bind argon stronger than the atoms of main group elements do, argon tagging experiments elegantly demonstrated encaging of transition metal atoms in doped silicon and aluminum clusters.8,9Differences in propensity toward argon attachment, depending on cluster dimensionality, have also been exploited to titrate planar isomers out of a molecular beam of gold clusters.10 Early experimental studies showed that (transition) metal clus- ter reactivity patterns toward closed-shell molecules, such as H 2, N2, and CH 4, exhibit very similar features.11–13Moreover, for clus- ters of transition metals (Co nFen, Ni n, and Nb n), there is a strong correspondence between these reactivity patterns and the propen- sity of the clusters to form complexes with argon.14,15For hydrogen, this correspondence can be explained by a two-step adsorption pro- cess. The first step involves a physisorbed hydrogen molecule in a precursor complex, which could explain the similarity with argon J. Chem. Phys. 154, 054312 (2021); doi: 10.1063/5.0037568 154, 054312-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp adsorption; the second step concerns the hydrogen dissociation. A stronger hydrogen physisorption in the first steps provides more time for the hydrogen molecule to probe the potential energy sur- face for a dissociative channel and thereby a higher overall reaction rate.13 Precursors are important in a variety of reactions, such as the activation of nitrogen on Fe(111)16and the dissociation of oxygen on Pt(111).17They provide an explanation for the often observed but rather counterintuitive increase in reaction rate with decreas- ing temperature, similar to what is observed for ion–molecule reac- tions: the lifetime of the (weakly bound) precursor decreases with increasing temperature or, in other words, the adsorbent is more likely to desorb before the (dissociation) reaction occurs. Although precursors are a key ingredient in the field of reaction dynamics, such states are typically short-lived and therefore difficult to detect experimentally.18 Several studies have alluded to the importance of a precursor state for the dissociative adsorption of hydrogen on transition metal clusters of, for example, iron and niobium, in order to explain the temperature-dependence of the experimentally determined reaction rates.19,20Similarly, the dehydrogenation of hydrocarbons by nio- bium clusters was hypothesized to be preceded by a strongly bound precursor complex.21The similar size-dependent interaction of TM n (TM = Co, Fe, Ni) with hydrogen and argon was explained by the higher polarizability of clusters with a smaller HOMO–LUMO gap.13Not only do clusters with a small HOMO–LUMO gap trans- fer more easily electrons to dissociate the hydrogen molecule, they also bind argon stronger due to higher polarizability. For small alu- minum clusters, Al n(n= 2–6), in contrast, calculations by Upton found no significant correlation between the HOMO–LUMO gaps and the reactivity toward hydrogen.22Orbital symmetry arguments, however, are more in line with the observed size-dependency of the reaction rates; the activation barrier seems to be dominated by Pauli repulsion between the σorbital of H 2and the cluster orbitals.23Sim- ilarly, no clear correlation between the ionization energy and the electron affinity with the reaction rate coefficients of Al nCo clusters (n= 10, 12, 15) was found.24 In this paper, the interaction of argon with doubly transition metal doped aluminum clusters, Al nTM 2+(n= 1–18; TM = V, Nb, Co, Rh), is studied in the gas phase via mass spectrometry. Density functional theory calculations for selected species are used to rationalize the size-dependent fraction of argon-tagged clusters and their similarity to the reactivity patterns of H 2. Analysis of the electronic structure of these selected clusters shows that orbital interactions provide an intuitive link between the two and further establish the importance of a precursor state in the reaction with hydrogen. MATERIALS AND METHODS Experimental Doubly transition metal doped aluminum clusters Al nTM m+ (n= 1–18; TM = V, Nb, Co, Rh) are produced by laser abla- tion in a dual-target dual-laser vaporization source, described in detail in Ref. 25. To form the Ar complexes, 1% of Ar is mixed in the He carrier gas. By cooling the cluster source to a temperature of 110 K, up to two Ar atoms could be attachedto the Al nTM m+clusters. To form the hydrogenated complexes, in contrast, hydrogen gas was injected into the nozzle of the cluster source through a separate valve, at a backing pressure of 1 bar. After formation, the cluster beam is collimated by a 1 mm skimmer before it enters to a reflectron time-of-flight mass spectrometer. Computational Density function theory (DFT) calculations are performed with the Gaussian 09 software package26using the Perdew-Burke- Ernzerhof (PBE) functional.27This functional was shown to perform well to calculate the properties of pure and doped aluminum clus- ters by comparison with CCSD simulations and experimental data.28 For the Al nV2+and Al nRh2+(n= 1–12), the structures were taken from Refs. 29–31. In these earlier studies, extensive searches for low- energy isomers were conducted. Al nTM 2+(n= 12–14; TM = V, Nb, Co, Rh) clusters and their Ar-tagged counterparts are studied for the first time, and global optimizations were carried out using the CALYPSO methodology.32The low-lying isomers were opti- mized for a range of possible spin multiplicities with the extensive Def2-TZVP basis set.33For the argon complexes Al 4,5Rh2+⋅Ar and Al6,7V2+⋅Ar, Ar adsorption on different sites was considered. To calculate Ar adsorption energies at the PBE/Def2-TZVP level, the D3 version of Grimme’s dispersion corrections with Becke–Johnson damping (D3BJ) was used.34Wiberg bond indexes were computed in order to characterize the TM–TM and TM 2–Al n+interactions.35 MASS SPECTROMETRIC RESULTS The mass spectra of the argon-tagged Al nTM 2+(TM = V, Rh, Nb, Co; n= 1–18) clusters have been recorded. The spec- tra are complex due to the presence of up to three dopant atoms and up to two argon tagging atoms. In order to facili- tate the analysis of the size-dependent abundance of argon com- plexes, the fractional distribution F(Ar p) of formed complexes, defined as [F(Arp)]=I(AlnTM 2Arp+) ∑2 i=0I(AlnTM 2Ari+), (1) with I(AlnTM 2Arp) being the abundance of the cluster species in the mass spectrum, is extracted from the mass spectra and plotted in Fig. 1. The mass spectra are given in the supplementary material. Singly doped Al nTM+clusters were also formed. However, the fol- lowing analysis will focus on the double doped species. The main reasons for this selection are the following: (1) in Al nTM+clus- ters with TM = Co and Rh, argon complexes are hardly visible in mass spectra and (2) correlations between Ar and H 2adsorption are less pronounced for the singly doped clusters. The fractional dis- tribution of the Al nTM+clusters is presented in the supplementary material. A great deal of similarity is noticeable between the argon affin- ity of V 2and Nb 2doped cationic aluminum clusters on the one hand, and the Co 2and Rh 2doped cationic aluminum clusters on the other hand. For V 2and Nb 2, all clusters with n<14 have compa- rable amounts of one and two Ar atoms adsorbed (with the excep- tion of Al 7V2+), while for Co 2and Rh 2doped Al n+clusters only J. Chem. Phys. 154, 054312 (2021); doi: 10.1063/5.0037568 154, 054312-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . Fractional distribution of argon-tagged complexes with two (a) vanadium, (b) niobium, (c) cobalt, and (d) rhodium dopants. For sizes larger than n= 13, the total fraction of argon complexes decreases significantly, independent of the dopant element. the smaller ( n= 1–7 for Co 2,n= 1–4 for Rh 2) clusters have two Ar-tags that account for a fraction of at least 5%. The argon affin- ity for both Co 2and Rh 2doped Al n+clusters increases again for n= 11–13. In our previous work on the interaction of doubly vana- dium doped aluminum clusters with hydrogen,30it was found that both vanadium atoms occupy a surface position for all cal- culated sizes ( n= 1–12). The low coordination of both transi- tion metal dopants could explain why a considerable fraction of the Al nV2+clusters have two argon tags. If this line of reason- ing is correct, also for Al nNb 2+, both dopants should be located at the surface, while for Al nCo2+and Al nRh2+, there should be a structural transformation in which both transition metal dopants gradually become inaccessible to the argon. For singly rhodium doped aluminum clusters, the calculated structures in Ref. 36 indeed show that for n>3 and with the exception of n= 7, the rhodium occupies a highly coordinated position and is somehow geomet- rically shielded.9For the doubly doped clusters, however, for all calculated structures, at least one of the rhodium atoms takes a surface position.31Thus, geometric shielding of the dopants does not seem to be the correct interpretation for the size-dependent Ar adsorption in Fig. 1. As will be discussed later, electronic instead ofgeometric shielding effects are more important to explain the observed trends. A striking feature can be observed for all Al nTM 2+clusters in Fig. 1, and for sizes n>13, there is, independent of the dopant element in question, a drastic decrease in the fraction of doubly doped aluminum clusters that form argon complexes. Although one could be tempted to attribute this decrease in Ar affinity to cage formation, i.e., the geometric shielding of the transition metals due to a structural transformation in which the dopant atom moves from a surface to an internal position (cf. Refs. 8 and 9), it seems counterintuitive that both dopants already occupy an internal posi- tion at smaller sizes as for a single dopant. For singly doped vana- dium clusters, the critical size for encapsulation is n= 16.9,37For example, experimental evidence for transition metal doped silicon clusters, Si nTM 1,2+(TM = Ti, V, Cr, Co), suggests that the onset of cage formation for two dopant atoms occurs at sizes that are 6–7 atoms larger than the critical sizes for encapsulation of a single dopant atom.8 ANALYSIS AND DISCUSSION Ar-tagging of Al nRh2+and Al nV2+(n= 1–14) clusters In Fig. 2, the fractional distribution of Ar-tagged Al nRh2+and AlnV2+(n= 1–14) clusters is plotted above the inverse of the calcu- lated distance between the two rhodium and two vanadium atoms in the clusters, 1 /d(TM−TM). The TM–TM distances are taken from the lowest-energy structures, which are plotted in the supplementary material. As can be seen in this figure, there is a strong correlation between the two quantities for the Rh 2doped clusters [panels (a) and (b)]. The bonding between rare gases and transition metals is gov- erned by a balance between strong but short-range Pauli repulsion and the weaker, long-range electrostatic forces. Intuitively, a larger Rh–Rh interatomic distance implies decreased electronic charge density between the two Rh atoms (bonding state) and increased charge density at the periphery of Rh–Rh (antibonding state). As the structures of the clusters indicate that both Rh atoms prefer highly coordinated positions, only the peripheral region is accessible to the argon. Increased electron density at the periphery facilitates a larger overlap with the argon orbitals and results in stronger Pauli repul- sion, pushing the argon atom further away from the cluster and reducing the electrostatic ion-induced dipole binding energy. In contrast to Al nRh2+, the correlation between the V–V inter- atomic distance and the propensity for Ar attachment on Al nV2+ is not obvious [panels (c) and (d)]. In particular, the low argon affinity of Al 7V2+cannot be related to an enhanced V–V dis- tance. The lack of such correlation does not come as a big sur- prise, as the preceding analysis hinges on the fact that only the periphery of the TM–TM bond is accessible to the argon, whereas for the V 2doped aluminum clusters, both vanadium atoms are at the cluster surface. This intuitive argument is made more com- prehensible by means of two pairs of consecutive sizes: Al 4,5Rh2+ and Al 6,7V2+. Case study: Al 4Rh2+and Al 5Rh2+ The sudden decrease in the fraction of Ar-tagged Al nRh2+clus- ters and 1/ d(Rh−Rh)in going from size n= 4 to n= 5 motivates J. Chem. Phys. 154, 054312 (2021); doi: 10.1063/5.0037568 154, 054312-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Comparison between the fractional distribution of Ar-tagged (a) AlnRh2+⋅Arpand (c) Al nV2+⋅Arpclusters ( n= 1–14, p= 1, 2), and the inverse dis- tance between the two (b) rhodium and (d) vanadium atoms, 1/ d(TM−TM), in those clusters. a detailed electronic structure analysis. In Fig. 3(a), the structures of Al 4Rh2+and Al 5Rh2+are shown together with their Ar-tagged counterparts. As bond distances are governed by and sensitive to the distribution of electronic charge density, inspection of the molecu- lar orbitals should be able to shed more light on the 20% increase in d(Rh−Rh), the 10% increase in the distance between Ar and the Rh atom to which it attaches d(Ar−Rh), and the concomitant decrease inEB(Ar) of about 40% from n= 4 to n= 5. Figure 3(b) shows selected frontier molecular orbitals of the Ar-tagged Al 4Rh2+and Al 5Rh2+that have electron density on both argon and the metal cluster. For n= 4, which has a doublet spin state, the first (second) row shows the majority spin α(minority spin β) HOMO and HOMO −1. For n= 5, which is a singlet, the HOMO, HOMO −1, and HOMO −2 orbitals are shown. All these orbitals have a node between the argon atom and the cluster, indicating that they are repulsive in nature and prevent the argon from further approach- ing the cluster. A second observation is that there is a clear differ- ence between the two sizes with respect to the orientation of the argon valence p-orbitals relative to the Ar–Rh internuclear axis. For Al4Rh2+⋅(Ar), three of the four p-orbitals shown in Fig. 3(b) are oriented perpendicular to the internuclear axis, i.e., they are of π∗ symmetry. Only the majority spin HOMO has σ∗symmetry with the Ar p-orbital oriented parallel to the internuclear axis. For n= 5, the FIG. 3 . (a) The structures of Al 4Rh2+and Al 5Rh2+(left) together with their Ar- tagged counterparts (right). Aluminum atoms in gray, rhodium atoms in dark green, and argon in cyan. (b) Selected molecular orbitals of Al 4Rh2+⋅Ar and Al 5Rh2+⋅Ar. For Al 4Rh2+⋅Ar, which is a doublet, spin up ( α) and spin down ( β) orbitals are plotted. Isosurfaces are plotted for a density of 0.015 e/Å3. contributing Ar p-orbitals of HOMO −1 and HOMO −2 are oriented parallel to the Ar–Rh internuclear axis. As the orbital overlap in the σ∗configuration is larger than in the π∗orientation, the Pauli repul- sion is larger; as a consequence, the Ar–Rh interatomic distance is larger for n= 5. J. Chem. Phys. 154, 054312 (2021); doi: 10.1063/5.0037568 154, 054312-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp A third observation, which ties the story of the correlation in Fig. 2 together, is that the HOMO −1 and HOMO −2 orbitals of Al 5Rh2+⋅(Ar), shown in Fig. 3(b), have a node between the two rhodium atoms. The HOMO −2 orbital mainly consists of the antisymmetric combination of rhodium d z2orbitals that hybridize with aluminum s- and p-orbitals. Such a bonding draws electronic density away from the bond center toward the periphery of the inter- nuclear Rh–Rh axis. As a result, the Rh–Rh bond strength decreases and the Rh–Rh interatomic distance increases. The increased elec- tron density at the periphery gives rise to additional Pauli repulsion, pushing the argon further away. The Ar binding energy therefore decreases, in agreement with the experimental fractional distribu- tion in Fig. 2(a). A similar kind of electronic shielding was observed for singly transition metal doped silicon clusters, Si nTM+(n= 5–10, TM = Cr, Mn, Cu, and Zn), interacting with Ar,38with the difference that for those silicon clusters, the shielding was mainly due to the s-electrons of the transition metal dopants, while here the shielding is caused by Rh d z2orbitals. Case study: Al 6V2+and Al 7V2+ To investigate the underlying reason for the abrupt decrease in argon affinity for the V 2doped clusters at n= 7, the molecular orbitals of Al 6V2+and Al 7V2+are analyzed. The geometries of both clusters and the shapes of their HOMOs are shown in Fig. 4. For FIG. 4 . Geometries (left) and highest occupied molecular orbitals (HOMOs, right) of Al 6V2+⋅Ar and Al 7V2+⋅Ar.n= 6, which is open shell, both αand βHOMOs are shown. Despite small differences in V–V and Ar–V bond lengths between n= 6 and n= 7, the mechanism described for Al 4,5Rh2+is likely not the reason for the large difference in argon affinity. The orbitals on the right- hand side in Fig. 4, however, clearly show that the vanadium atoms of Al 7V2+are shielded by its HOMO orbital. For Al 6V2+, the vana- dium atoms are more accessible, and the Ar–V repulsion is due to the overlapping orbitals of π∗symmetry. Analysis of Löwdin charges suggests significant electron transfer to the vanadium atoms for n= 7 compared to n= 6. While in n= 7, the vanadium dopants have a partial charge of −0.78e, and in n= 6, the dopants only have −0.16e (see the supplementary material). Such a charge transfer can significantly alter the electrostatic potential of the cluster, which, in turn, affects the induced dipole moment of the argon atom and hence its binding energy. The elec- trostatic potential of Al 4Rh2+, Al 5Rh2+, Al 6V2+, and Al 7V2+is plot- ted in Fig. 5 on top of an iso-surface of electronic charge den- sity ( ρe=0.001 e/a3 0). The electrostatic potentials of Al 4Rh2+and Al5Rh2+are quite similar, with a slightly lower potential at the less coordinated (more to the right in Fig. 5) rhodium atom for n= 5 compared to n= 4. For the vanadium doped clusters, on the other hand, there is a clear difference between n= 6 and n= 7, with a more positive potential near the vanadium atoms for Al 6V2+. Argon binds stronger to cations than to anions for atoms as well as clusters.39,40 The lower argon affinity for anionic clusters has been attributed to electron “spillout”41,42and thus increased Pauli repulsion. The elec- trostatic potential is thus consistent with the stronger argon bind- ing to Al 6V2+than to Al 7V2+. Note that although the Al 4Rh2+and Al5Rh2+electrostatic potentials are higher around the higher coor- dinated rhodium atom (more to the left in Fig. 5), the DFT calcu- lations clearly indicate that the argon preferably binds to the least coordinated rhodium atom. This again evidences the importance of Pauli repulsion: the charge density in-between the atoms derives FIG. 5 . Electrostatic potentials of Al 4Rh2+, Al5Rh2+(left), Al 6V2+, and Al 7V2+(right) on top of an isosurface of the electronic charge density ( ρe=0.001 e/a3 0). J. Chem. Phys. 154, 054312 (2021); doi: 10.1063/5.0037568 154, 054312-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp from the aluminum s and p electrons, which have a stronger over- lap with the argon valence orbitals, and hence results in a larger repulsion. The combined results of the case studies show that the rela- tive contributions of Pauli repulsion and electrostatic attraction in the argon—-cluster interaction is subtle. Both effects are not inde- pendent from one another. Therefore, the preceding analysis can be summarized as follows: increased electron charge density at the location of the transition metal dopants lowers the electrostatic potential and at the same time increases Pauli repulsion, result- ing in a lowering of the binding energy of the argon atom and decreasing the abundance of the argon-tagged species in the mass spectra. Analysis of Al nTM2+(n= 12, 13, 14; TM = V, Nb, Co, Rh) The calculated structures of the lowest-energy isomers of Al12–14 TM 2+(TM = V, Nb, Co, Rh) are shown in Fig. 6. While for the rhodium and cobalt doped clusters, one of the dopants assumes an internal position, both transition metals remain at the surface for the vanadium and niobium doped clusters, with the exception of Al 14V2+. As anticipated, encapsulation is not the expla- nation for the observed decrease in argon affinity for the larger sizes: none of the Al 14TM 2+aluminum frameworks encapsulates both dopants. There must be another reason for the observed decrease in the abundance of the argon-tagged clusters for n>13 in Fig. 1. To investigate the hypothesis of electronic shielding, i.e., that unfavorable overlap of electronic orbitals increases Pauli repulsion and decreases the binding energy, each of the structures in Fig. 6 was reoptimized after adding an argon atom. In all Al 12–14 TM 2+ clusters, argon was found to bind stronger to the least coordinated transition metal atom. The calculated binding energies EB(Ar) are plotted in the top row in Fig. 7. The calculations predict for all TM dopants a larger binding energy (more negative) for n= 12 and 13 as compared to n= 14, consistent with a higher abundance of FIG. 6 . Lowest-energy isomers of Ar-tagged Al 14TM2+(TM = V, Nb, Co, Rh) clusters. For all sizes, at least one of the dopant atoms occupies a surface position. FIG. 7 . Calculated properties of Al 12–14TM2+(TM = V, Nb, Co, Rh) clusters. First row: argon binding energy, EB(Ar). Second row: distance between the two transi- tion metals, d(TM–TM). Third row: Wiberg bond index of the TM–TM interaction, W(TM–TM). the argon complexes. The second row in Fig. 7 presents the inter- atomic d(TM–TM) distances. A distinct correlation between the bond distance and the argon binding energy can be seen. Corre- lation, however, does not imply causation; both quantities could be brought about by another underlying, intrinsic property of the cluster. Moreover, the fact that the decrease in EB(Ar) and increase ind(TM–TM) occur independently for each of the four transi- tion metals strongly suggests that the underlying cause is related to the aluminum host. Wiberg bond indexes ( W) were computed for the clusters. The third row in Fig. 7 presents the TM–TM bond indexes. Although not perfect, a correlation between d(TM–TM) and W(TM–TM) is observed, with shorter TM–TM interatomic distances corresponding to a higher bond order. These observations ask for a closer examination of the elec- tronic structure of the clusters. For n= 13, the orbital overlap between argon and the cluster is of π∗symmetry, whereas for n= 14, all occupied frontier orbitals with electron density on both Ar and the TM dopants are of σ∗symmetry. Plots of the rele- vant molecular orbitals for n= 13 and n= 14 are presented in Fig. 8. J. Chem. Phys. 154, 054312 (2021); doi: 10.1063/5.0037568 154, 054312-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 8 . Occupied frontier orbitals with electron density on both Ar and the TM of AlnTM2+(n= 13, 14; TM = V, Nb, Co, Rh). On the side of each molecular orbital plot, the cluster geometry is presented. While the frontier orbitals explain why the Ar binding energy is lower for n= 14 as for n= 13, they do neither explain why the decreased Ar affinity occurs exactly at this size nor why the transi- tion size is independent of the kind of TM dopant. One pertinent question remains, namely, what is the role of the aluminum frame- work in the observed reactivity pattern? The striking feature that for all dopants, the abundance of argon-tagged clusters decreases atn= 14, which is a magic size of cationic aluminum clusters,43 is an indication for an electronic shell structure effect. However, four factors render electron counting and shell structure identifica- tion in transition metal doped aluminum clusters non-trivial. The first is that the number of valence electrons of transition metals that delocalize depends on size and structure.44The second factor is that not only transition metals but also aluminum can exhibit mixed valence.45A third factor is that the nuclear charge of the alu- minum atoms is not well screened, and so the ionic background plays a larger role.46Finally, the fourth factor is that the ionic background potential is disturbed by the presence of heteroatom dopants.47The orbitals of n= 13 and 14 in Fig. 8 indicate that the HOMO orbitals of the rhodium and cobalt doped aluminum clus- ters resemble 2P states of the spherical jellium model. As the 2P shells are more sensitive to the potential background in the inte- rior of the cluster, the filling of the 2P shells might explain why the two rhodium atoms approach each other after n= 10; one rhodium atom is “pulled” inside, thereby lowering the energy of the 2P orbitals.47The HOMO of Al 13V2+resembles a 2P jellium orbital as well. The HOMO of n= 14, on the other hand, is resem- bling a 1G jellium orbital, indicating a shell closure between n= 13 andn= 14.Similar reactivity pattern of Al nRh2+and Al nV2+ toward H 2and Ar: Precursor state Figure 9 shows the fractional distribution of the argon- tagged [panels (a) and (c)] and hydrogenated [panels (b) and (d)] doubly rhodium and vanadium doped aluminum clusters. Although there is not a one-to-one correspondence, the reactiv- ity pattern of Al nRh2+toward hydrogen and the trends in their propensity to form argon complexes are very similar: Al 1–4Rh2+ and Al 11–13 Rh2+clusters readily form argon and hydrogen com- plexes, whereas other sizes do not. Exceptions are Al 7Rh2+and Al14Rh2+, which do not attach any argon but are reactive toward hydrogen. For the V 2doped aluminum clusters, a less pronounced but nevertheless similar size-to-size variation can be discerned. Up to n= 13, all clusters can be tagged with one and even two argon atoms, except for Al 7V2+, which is also unreactive toward hydrogen. Al 4V2+ and Al 5V2+, in contrast, do adsorb argon but barely any hydrogen. For sizes larger than n= 13, there is a sudden decrease in the frac- tional distribution for Ar adsorption, which, as argued before, is due to electronic shielding. Similarly, there is a gradual decrease in the reactivity of the Al nV2+clusters toward hydrogen for n>13. It was already noted in earlier work that for n= 14 and n= 15, an equal amount of one and two hydrogen molecules were adsorbed onto the cluster, whereas most other Al nV2+clusters only adsorb a single H 2.30 In previous works, the importance of a possible precursor state in the hydrogenation reaction mechanism of doped alu- minum clusters was discussed.29,30,36The main argument that leads to this interpretation is the correlation of calculated molecular hydrogen adsorption energies, H–H distances in the molecularly adsorbed complexes, and the abundance of the hydrogenated clus- ters. Although for the smallest singly rhodium doped clusters (n<4) and larger doubly doped rhodium clusters ( n= 12, 13), the infrared multiple photon dissociation spectra indicated that the hydrogen adsorbs molecularly on the dopant atom in a side-on configuration,29,36for other abundant hydrogenated complexes, e.g., Al7RhH 2+, Al 11Rh2H2+, and the vanadium doped aluminum clus- ters, the infrared absorption bands were assigned to the vibrational modes of single hydrogen atoms at different sites. DFT calcula- tions also predict that dissociative adsorption for these clusters is energetically preferred. Therefore, the formation and stability of the molecularly adsorbed complex seem rate-limiting for the dissocia- tive adsorption of hydrogen, or in other words, the reaction likely proceeds via a precursor state. Based on this argument and the analy- sis of the interaction of argon with the doubly transition metal doped clusters, the similarity between the propensity to adsorb hydrogen and argon can be due to two related reasons. First, the HOMOs that result in less shielding of the transition metal atoms, i.e., those that exhibit locally π-symmetry are also less repulsive for the σorbital of H2. Additionally, these orbitals overlap with the anti-bonding σ∗- orbital of hydrogen and therefore activate H 2and lead to the forma- tion of a Kubas-complex and/or dissociative adsorption of hydrogen. The correlation, however, is not one-to-one; there are clear differ- ences between the hydrogen reactivity and argon affinity patterns in Fig. 9 (e.g., for clusters, Al 1–3Rh+, Al 7Rh2+, Al 4,5V2+, and the larger vanadium doped clusters). These differences, however, are not sur- prising; even though the H 2adsorption on the clusters is strongly J. Chem. Phys. 154, 054312 (2021); doi: 10.1063/5.0037568 154, 054312-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 9 . Fractional distribution of (a) Ar and (b) hydrogen complexes of Al nRh2+ (n= 1–18) clusters. Fractional distribu- tion of (c) Ar and (d) hydrogen com- plexes of Al nV2+(n= 1–18) clusters. influenced by a precursor state in which hydrogen is adsorbed molecularly, it is also dependent on the subsequent dissociation of H2. This additional step is evidently absent when the clusters interact with Ar. CONCLUSIONS In summary, the interaction of argon with doubly transition metal doped aluminum clusters, Al nTM 2+(n= 1–18, TM = V, Nb,Co, Rh), was studied experimentally in the gas phase via mass spec- trometry. Selected sizes were analyzed computationally using den- sity functional theory calculations. Although the argon affinity of the clusters is size- and dopant-dependent, there is a similarity between V and Nb on the one hand, and Co and Rh on the other. A detailed analysis of the rhodium and vanadium doped aluminum cluster electronic structure shows that the abundance of argon–cluster com- plexes in the mass spectra is sensitive to electronic shielding of the transition metal dopants. For Rh, the distance between the two tran- sition metals is indicative of the charge density location along the J. Chem. Phys. 154, 054312 (2021); doi: 10.1063/5.0037568 154, 054312-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp internuclear axis (center vs periphery) and hence the presence or absence of shielding. For the doubly vanadium doped aluminum clusters, both transition metals bind at the surface and the TM–TM distance is less sensitive to charge transfer. The argon affinity for all AlnTM 2+(TM = V, Nb, Co, Rh) clusters decreases abruptly at n= 14, which seems to be related to electronic shell closings and not to encapsulation of the TM atoms. Finally, the similarity between the fractional distribution of argon–cluster complexes and hydrogenated clusters is proposed to be related to the presence of a precursor state on the PES of the cluster–hydrogen reaction: HOMOs that result in less shield- ing of the transition metal atoms from argon are also less repul- sive toward the σorbital of H 2. Second, due to their overlap with the anti-bonding σ∗orbital of hydrogen, these orbitals lead to the stabilization of a Kubas-complex and/or dissociative adsorption of hydrogen. SUPPLEMENTARY MATERIAL See the supplementary material for (i) representative mass spectra, (ii) the calculated lowest-energy structures of Al nV2+and AlnRh2+(n= 2–14) clusters, (iii) mass spectrometric fractions of Ar and H 2on cationic single V and Rh doped clusters, and (iv) a partial charge analysis of selected clusters. ACKNOWLEDGMENTS This work was supported by the KU Leuven Research Council (Project No. C14/18/073) and the Research Foundation—Flanders (FWO, Project No. G0A05.19N). J.V. and P.F. acknowledge the FWO for a Ph.D. Fellowship and Postdoctoral grant, respectively. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1K. R. Asmis, M. Brümmer, C. Kaposta, G. Santambrogio, G. von Helden, G. Meijer, K. Rademann, and L. Wöste, Phys. Chem. Chem. Phys. 4, 1101 (2002). 2J. Oomens, B. G. Sartakov, G. Meijer, and G. von Helden, Int. J. Mass Spectrom. 254, 1 (2006). 3P. Gruene, D. M. 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5.0037876.pdf

Appl. Phys. Lett. 118, 052409 (2021); https://doi.org/10.1063/5.0037876 118, 052409 © 2021 Author(s).Voltage-controlled spin–orbit torque switching in W/CoFeB/MgO Cite as: Appl. Phys. Lett. 118, 052409 (2021); https://doi.org/10.1063/5.0037876 Submitted: 16 November 2020 . Accepted: 24 January 2021 . Published Online: 03 February 2021 Jinsong Xu , and C. L. Chien ARTICLES YOU MAY BE INTERESTED IN Enhancement of spin–orbit torque in WTe 2/perpendicular magnetic anisotropy heterostructures Applied Physics Letters 118, 052406 (2021); https://doi.org/10.1063/5.0039069 Spin-orbit torques: Materials, physics, and devices Applied Physics Letters 118, 120502 (2021); https://doi.org/10.1063/5.0039147 Electrical and optical characterizations of spin-orbit torque Applied Physics Letters 118, 072405 (2021); https://doi.org/10.1063/5.0045091Voltage-controlled spin–orbit torque switching in W/CoFeB/MgO Cite as: Appl. Phys. Lett. 118, 052409 (2021); doi: 10.1063/5.0037876 Submitted: 16 November 2020 .Accepted: 24 January 2021 . Published Online: 3 February 2021 Jinsong Xua) and C. L. Chiena) AFFILIATIONS Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA a)Authors to whom correspondence should be addressed: jxu94@jhu.edu and clchien@jhu.edu ABSTRACT Voltage control of magnetism and spintronics have been highly desirable but rarely realized. In this work, we show voltage-controlled spin–orbit torque (SOT) switching in W/CoFeB/MgO films with perpendicular magnetic anisotropy (PMA) with voltage administeredthrough SrTiO 3with a high dielectric constant. We show that a DC voltage can significantly lower PMA by 45%, reduce switching current by 23%, and increase the damping-like torque as revealed by the first- and second-harmonic measurements. These are characteristics that are prerequisites for voltage-controlled and voltage-select SOT switching spintronic devices. Published under license by AIP Publishing. https://doi.org/10.1063/5.0037876 Electrical control of magnetism and spin phenomena without using a magnetic field has been a long-standing goal in spintronics. The discovery of spin-transfer-torque (STT) earlier accommodates electrical switching of ferromagnetic (FM) entities with spin-polarized currents.1,2More recently, the advent of spin–orbit torque (SOT) via the Rashba effect and spin Hall effect (SHE) using pure spin current has been demonstrated for switching magnetization in ferromagnet (FM)/heavy metal (HM) bilayers with geometries that are not possible in conventional STT devices.3–8A pure spin current has the distinct attributes of delivering spin angular momentum with a minimal num- ber of charge carriers in metals and no charge carriers in insulator. With different heavy metals (HMs), SOTs of various magnitudes and sign have achieved in switching FM entities with in-plane anisotropy (IPA) and also perpendicular magnetic anisotropy (PMA), if addi- tional measure is taken to break the up/down symmetry. It is of great interest to explore the voltage reduction of SOT switching current and especially to achieve voltage-controlled switch- ing so that only the specific device among many can be selected by voltage to execute switching on demand.9–11It has been shown that v o l t a g ec a nm o d i f yt h em a g n e t i ca n i s o t r o p yo fs o m eF Ml a y e r sv i a inducing surface charges12–14or modifying the orbit occupancy at the oxide/metal interface.15–19As a result of the modification of interface anisotropy or exchange bias by the electric field, electric-field-assisted magnetization switching has been reported in magnetic tunnel junc- tions,20–25as well as modulation of spin Hall nano-oscillator.26–28 There have been some attempts in voltage-control SOT switching,29–33 but with very small effects and obscured underlying physics.In this work, we report the efficient voltage control of SOT switching in the well-known W/CoFeB/MgO thin films with PMA, where the gate voltage is applied via SrTiO 3with a high dielectric con- stant. The coercivity field ( Hc) and critical current for switching ( Ic) are reduced by 45% and 23%, respectively, by a gate voltage of a few volts. The gate voltage not only affects anisotropy as alluded above but also the efficiency of SOT switching via the pure spin current as revealed by the first- and second-harmonic measurements. Only a gate voltage of a suitable sign results in reduced magnetic anisotropy field (Hk) but increased damping-like SOT effective field ( HDL), leading to different reductions in HcandIc. We pattern Hall bar devices for this study as schematically shown inFig. 1(a) , where the W(2)/CoFeB(1)/MgO(1.5)/TaO x(1) multilayered films (where the numbers in parentheses are thicknesses in nm) have been made by sputtering at room temperature on Si wafer with a 300 nm SiO 2layer. To achieve PMA, the as-grown films were annealed in an out-of-plane field of 1 T at 280/C14Ci nv a c u u mf o r3 0 m i n .W e used photolithography and ion milling to fabricate the Hall bar struc- ture with 8- lm wide channels. A SrTiO 3layer of 200 nm in thickness was sputtered onto the device as the dielectric layer at 77 K to preserve PMA, before the final Au top layer. More information about the device fabrication can be found in the supplementary material ,N o t e1 .T h e gate voltage is applied between the top Au electrode and the CoFeB layer to generate an electric field. The high dielectric constant of SrTiO 3enables large electric field without leakage current.10,34The suit- able range of applied gate voltage varies from /C02V t o 2 0V ; b e y o n d this range, either the devices break down or the PMA is compromised. Appl. Phys. Lett. 118, 052409 (2021); doi: 10.1063/5.0037876 118, 052409-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplA single CoFeB layer has in-plane anisotropy (IPA) because of the large demagnetization factor 4 pMs,w h e r e Msis the magnetization. However, when sandwiched between a HM and a suitable oxide, W/CoFeB/MgO acquires a surface anisotropy of /C04Ks/Mst,w h e r e Ks is the surface anisotropy constant. With a positive Ks, at a sufficiently small thickness t, the surface anisotropy alters the total anisotropy from IPA to PMA as in our sample of W(2)/CoFeB(1)/MgO(1.5).35 One may exploit the delicate PMA with a gate voltage and achieve voltage-control switching. We use anomalous Hall effect (AHE) to demonstrate PMA in the W/CoFeB/MgO multilayers. As shown in Fig. 1(b) ,t h e r ei sas h a r p square AHE hysteresis loop with an out-of-plane magnetic field l0Hz, confirming PMA in the W/CoFeB/MgO heterostructures. Under agate voltage, the width of the AHE hysteresis loop varies as shown by the red (at /C02V ) a n d b l u e ( a t 2 0V ) c u r v e s i n Fig. 1(b) , revealing the systematic modification of the coercivity field by the gate voltage, asshown in Fig. 1(c) . The coercivity l 0Hcmonotonically decreases from 5.8 mT to 3.2 mT when varying Vgatefrom/C02Vt o2 0V ,a i d e db yt h e large dielectric constant of SrTiO 3.I ng e n e r a l ,t h ec o e r c i v i t yfi e l d depends on both the magnetic anisotropy and microstructure of the materials.36,37Here, the low gate voltage mainly changes the magnetic anisotropy. Previous studies have shown that the electric field canmodify the orbital hybridization between the Fe 3dorbitals and the O 2porbitals to alter magnetic surface anisotropy. 15–19The unidirec- tional change of Hcwith both negative and positive gate voltages is consistent with this scheme. As we will discuss later, the gate voltage affects Hkonly by about 10% but resulting in a much larger change of coercivity Hcby 45%. This modification of the coercivity by a gate voltage enables voltage-controlled magnetic field switching. For example, when a 4:2m T l0Hpulsefield pulse is applied opposite to the magnetization, the magnetization of CoFeB is readily switchable at Vgate¼20 V but remains unchanged at /C02 V (see the supplementary material ,N o t e2 ) because the 4 :2m T l0Hpulseis larger than l0Hcat 20 V but smaller thanl0Hcat/C02V .In addition to voltage-controlled magnetic field switching, we explore the prospect of gate voltage on SOT switching. As it is wellknown that because the spin index of the pure spin current is parallelto the interface, the SOT is incapable of switching a PMA layer with- out some means to break the up/down symmetry, such as a magnetic field is also applied along the current direction. The SOT switchingbehavior of W/CoFeB/MgO at V gate¼/C02 V and 20 V with an external field of 20 mT along the current direction is shown in Fig. 2(a) .W e note that there are asymmetry and intermediate states of the SOT switching hysteresis loops for one current sweeping direction. These are likely due to the different processes of domain wall depinning andexpansion in the Hall cross between the two current sweepingdirections. 38The width of the SOT switching loop at Vgate¼20 V is narrower than that at Vgate¼/C02 V, indicating the gate tunability of the critical current Icrequired to switch magnetization ( Icis defined as the current required to switch Rxyto 0). The gate voltage dependence ofIcis shown in Fig. 2(b) ,w h e r e Icdecreases from 0.73 mA (3.0/C21010Am/C02)t o0 . 5 6 m A( 2 :3/C21010Am/C02)b y/C2423% when varying Vgatefrom/C02 V to 20 V. The monotonic decrease in Ic,e v e n when Vgatechanges sign, illustrates that it is an electric field effect influencing the charge transfer at the CoFeB/MgO interface. The mod-ification of the switching current by a gate voltage clearly demonstratesvoltage-controlled SOT switching. For example, when a current pulse I pulseof magnitude 0 :75 mA (3 :1/C21010Am/C02)o ft h ea p p r o p r i - ate polarity is applied to W/CoFeB/MgO, the magnetization ofCoFeB is readily switchable at V gate¼20 V but not at /C02 V (see the supplementary material ,N o t e3 ) . There are two terms of SOT in a HM/FM/oxide heterostructure, aM/C2rþbM/C2ðr/C2MÞ, known as the field-like torque and the damping-like torque, respectively. The effect of the two terms can be viewed as the field-like effective field HFL/C24rand the damping-like effective field HDL/C24r/C2M. In general, the SOT switching efficiency depends on the intrinsic magnetic properties of the FM layer andI c/MsHk=HDL:3,39To determine the gate voltage dependence of the two SOT terms, we perform first- and second-harmonic measure- ments. Referring to Fig. 1(a) ,a nA Ce x c i t a t i o nc u r r e n t Iacwith a mag- nitude of 0.3 mA (1 :25/C21010Am/C02) is applied along the xdirection; the transverse in-phase first-harmonic ( V1x) and out-of-phase second-harmonic ( V2x) voltages along the ydirection are simulta- neously measured by two lock-in amplifiers.40Figure 3(a) shows the manner V1xevolves as an external magnetic field Hxis applied along thexdirection (the actual magnetic field is applied with a small out- of-plane tilting angle to ensure coherent domain wall rotation). FIG. 1. (a) Schematic of the Hall bar device with a layer order shown on the right. (b) Anomalous Hall effect measured at Vgate¼/C0 2 V (red) and 20 V (blue) with an out-of-plane magnetic field. (c) Gate voltage dependence of the coercivity field. FIG. 2. (a) SOT-induced magnetization switching at Vgate¼/C0 2 V (red) and 20 V (blue) with an assisted field of l0Hx¼20 mT. (b) Gate voltage dependence of the critical current Icrequired to switch magnetization.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 052409 (2021); doi: 10.1063/5.0037876 118, 052409-2 Published under license by AIP PublishingThe value of V1xdecreases faster as a function of HxatVgate¼20 V (blue circles) than that at Vgate¼/C02 V (red circles), indicating a weaker magnetic anisotropy at a larger gate voltage. To quantify Hk, we follow the Stoner–Wohlfarth model, V1x¼IacRxy/RAHEMz Ms¼RAHEHkþHzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi HkþHz ðÞ2þH2 xq ;(1) where RAHE is the anomalous Hall coefficient, with the extracted values of HkandRAHEshown in Figs. 3(b) and3(d).A c r o s st h eg a t e voltage range of /C02 V to 20 V, Hkchanges by about 10%, which con- tributes to the reduction of Icbecause of the aforementioned linear dependence. This modification of Hkby gate voltage is usually attrib- uted to the change of the dstate electron density of CoFeB,16–19,41 which appears to be responsible for the small change of RAHE as a result of the Fermi level change. We have also observed similarbehavior when the magnetic field is applied along the ydirection (shown in the supplementary material ,N o t e4 ) .B a s e do nt h eg a t e - dependent R AHEvsHxcurves, the voltage control magnetic anisotropy coefficient is estimated to be about 30 fJ V/C01m/C01(see the supplementary material , Note 5), which is similar to previous reports.23,42 The SOT effective fields HDLandHFLcan be obtained by3,40,43 HDL FLðÞ¼/C02BxyðÞ62nByxðÞ 1/C04n2; (2) where the 6sign corresponds to the 6Mzstate, BxyðÞ/C17@V2x @HxyðÞ=@2V1x @H2 xðyÞ, andn¼RPHE RAHEis the ratio of the planar Hall effect (PHE) coefficient RPHEto the anomalous Hall coefficient RAHE.T oe x t r a c t RPHE,w eapply a constant in-plane magnetic field l0Hext¼600 mT, which is larger than the perpendicular anisotropy field l0Hk/C24200 mT, and sweep the angle ubetween the current direction and the magnetic field d i r e c t i o n[ s h o w ni n Fig. 3(c) ]. In this configuration, as the magnetization is nearly in-plane, Rxymainly comes from PHE, which is proportional to sin2u. There is also a small contribution from AHE caused by the unin- tentional small out-of-plane titling angle h0, which is proportional to sinu.T h e r e f o r e ,w efi tt h ed a t au s i n g RPHEsin2uþRAHEsinh0sinu [solid lines in Fig. 3(c) ]. We find that RPHEis about 1 X, which is much smaller than the previously measured RAHE(/C2410X), therefore n¼RPHE RAHE is about 0.1 in the applied gate voltage range. The second-harmonic measurement results are shown in Fig. 4(a) . When the magnetic field is applied along the current direction (xdirection), the slopes of V2xas a function of field are the same for both magnetization states 6Mz[Fig. 4(a) , upper panel], while they have an opposite sign when the field is applied perpendicularto the current direction ( ydirection) [ Fig. 4(a) , lower panel]. The solid curves are the results of linear fit to the data. At a larger gatevoltage (blue curves), V 2xhas a larger slope with respect to the magnetic field, which provides a hint that there might be a largerSOT. By combining the results of first- and second-harmonic mea-surements and planar Hall measurements, we can calculate theSOT quantitatively, and the results are shown in Fig. 4(b) . Because of the above-mentioned inverse proportionality between H DLand Ic, we focus on the damping-like SOT effective field HDL(there is some variation of field-like torque at different gate voltages, but noclear trend is observed; see the supplementary material , Note 6). As seen in Fig. 4(b) ,H DLincreases with Vgateby about 10% in the applied gate voltage range, which causes the reduction of Ic. This modification of HDLby gate voltage may result from the alteration of the interfacial Rashba effect,30spin mixing conductance, and/or the change of effective spin Hall angle, which depends on theinterfacial electronic structure and scattering rates influenced bythe electric field. 29,32,44,45Together with the 10% decrease in the magnetic anisotropy Hk, they can account for the /C2423% reduction ofIc. We have observed similar behaviors in other devices (see the supplementary material , Note 7). Therefore, a gate voltage can tune both the magnetic anisotropy and SOT, resulting in the mod-ification of the critical current for switching.FIG. 3. (a) First-harmonic measurements at Vgate¼/C0 2 V (red) and 20 V (blue). Solid (open) circles represent magnetization along the þ(/C0)zdirection. The solid curves are best-fit to the data using Eq. (1)with an AC excitation current of 0.3 mA. (b) Gate voltage dependence of magnetic anisotropy l0Hk, extracted from the first- harmonic data using Eq. (1). (c) RxyatVgate¼/C0 2 V (red) and 20 V (blue) as a function of the in-plane angle /at a constant in-plane magnetic field l0Hext¼600 mT with the solid curves as the best-fits. (d) Gate voltage dependence of anomalous Hall coefficient RAHEand planar Hall coefficient RPHE, extracted from (a) and (c), respectively. FIG. 4. (a) Second-harmonic measurement with the magnetic field applied along thex(upper panel) and ydirections (lower panel) at Vgate¼/C0 2 V (red) and 20 V (blue). Solid (open) circles represent magnetization pointing along the þ(/C0)z direction. The solid lines are the linear fits to the data. (b) Gate voltage dependence of the damping-like SOT effective field l0HDL, extracted from the first- and second- harmonic measurements with the AC excitation current at 0.3 mA (1 :25/C21010A m/C02). The solid line is the linear fit to the data.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 052409 (2021); doi: 10.1063/5.0037876 118, 052409-3 Published under license by AIP PublishingIn conclusion, we demonstrate the voltage control of field switching and SOT switching in perpendicularly magnetized W/CoFeB/MgO mul- tilayers, facilitated by the usage of SrTiO 3with a large dielectric constant. The coercivity and critical current are reduced by 45% and 23%, respec- t i v e l y ,w i t hag a t ev o l t a g eu n d e r2 0 V .T h eh a r m o n i cm e a s u r e m e n t s show that the reduction of critical current originates from both the alter-ation of magnetic anisotropy and damping-like SOT effective field. The tunability of coercivity and critical current by a gate voltage demonstrate the prospects of voltage-controlled switching for future energy efficient memory devices. 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5.0036302.pdf

Appl. Phys. Lett. 118, 062903 (2021); https://doi.org/10.1063/5.0036302 118, 062903 © 2021 Author(s).Electric-polarization-driven magnetic phase transition in a ferroelectric–ferromagnetic heterostructure Cite as: Appl. Phys. Lett. 118, 062903 (2021); https://doi.org/10.1063/5.0036302 Submitted: 03 November 2020 . Accepted: 27 January 2021 . Published Online: 10 February 2021 Dier Feng , Ziye Zhu , Xiaofang Chen , and Jingshan Qi ARTICLES YOU MAY BE INTERESTED IN Tunable magnetism in ferroelectric α-In2Se3 by hole-doping Applied Physics Letters 118, 072902 (2021); https://doi.org/10.1063/5.0039842 Electric field control of molecular magnetic state by two-dimensional ferroelectric heterostructure engineering Applied Physics Letters 117, 082902 (2020); https://doi.org/10.1063/5.0012039 Exchange interactions in topological/antiferromagnetic heterostructures Applied Physics Letters 118, 062410 (2021); https://doi.org/10.1063/5.0039741Electric-polarization-driven magnetic phase transition in a ferroelectric–ferromagnetic heterostructure Cite as: Appl. Phys. Lett. 118, 062903 (2021); doi: 10.1063/5.0036302 Submitted: 3 November 2020 .Accepted: 27 January 2021 . Published Online: 10 February 2021 Dier Feng,1Ziye Zhu,2Xiaofang Chen,1,a)and Jingshan Qi1,a) AFFILIATIONS 1School of Physics and Electronic Engineering, Jiangsu Normal University, Xuzhou 221116, People’s Republic of China 2School of Engineering, Westlake University, Hangzhou 310027, People’s Republic of China a)Authors to whom correspondence should be addressed: chenxf@jsnu.edu.cn andqijingshan@jsnu.edu.cn ABSTRACT Magnetoelectric coupling is of great interest recently to both understand the fundamental physics and device applications. Materials with strong magnetoelectric coupling, high Curie temperature, and large electric polarization are still rare. We suggest a heterostructure that combines the known memory effect of the switchable ferroelectric In 2Se3[Adv. Funct. Mater. 2019, 29, 1808606] with a van der Waals bonded two-dimensional (2D) metal-organic framework (MOF) film. The magnetic ground state of this MOF can be changed from an anti-ferromagnetic state to a ferromagnetic through hole-doping. We use first-principles calculations to show that in such a heterostructure,adequate doping differences to cause this phase transition are expected from the changes in the interfacial charge transfer between the MOF and In 2Se3when the polarization direction of the In 2Se3is reversed. This and similar 2D heterostructures may, therefore, provide both a fas- cinating material platform for understanding the fundamental physics of magnetoelectric coupling and a strategy for designing spin-current-based nonvolatile memory structures. Published under license by AIP Publishing. https://doi.org/10.1063/5.0036302 Controlling the magnetism of a system by an electric field has been challenging but of great importance for developing magneto-electric devices. One of the platforms for controlling the magnetism byan electric field is multiferroic materials. They play a curial role inhigh-density nonvolatile memories and low-cost electronic devices. 1–7 Generally speaking, the magnetoelectric coupling mechanisms includetwo types. Type-I multiferroics are renowned for their robust magne-tism and polarization, but the magnetoelectric coupling is usually quiteweak because of the different origin of magnetism and electric polari-zation. 8–13For example, in BiFeO 3, the electric polarization originates from the lone pair electrons of Bi ions, while the magnetism comes from the partial filling of the d-orbitals of Fe ions. By contrast, in type- II multiferroics, the polarization is directly induced by a magneticorder, thus leading to a strong magnetoelectric coupling, although theelectric polarization is usually too weak. 14–16For example, in TbMnO 3, the noncollinear helical spin order of Mn ions breaks the space-inversion symmetry and thus couples with ferroelectric order,explained by the KNB theory 17and DM interaction theory.18,19 Therefore, multiferroic materials with the large polarization and strongmagnetoelectric coupling are still rare to date. It is highly demandingto find more multiferroic systems or develop different magnetoelectric coupling mechanisms. Recently, many 2D multiferroics are predicted theoretically by introducing electric polarization into 2D magnetic materials or intro-ducing magnetism into 2D ferroelectric materials, such as the halogen-decorated phosphorene bilayer, 20CH 2OCH 3-functionalized germa- nene,21transition-metal-doped group-IV monochalcogenides,22 C6N8Ho r g a n i cn e t w o r k ,23graphitic bilayer,24SrMnO 3thin films,25 CuMP 2X6(M¼C r ,V ;X ¼S, Se) monolayer,26CrBr 3monolayer,27 VOCl 2,28hyper-ferroelectric metals,29Hf2VC 2F2,16ReWCl 6 monolayer,30and so on. In addition to single-phase 2D materials, van der Waals (vdW) heterostructures30,31provide us a fascinating struc- tural platform for exploring different mechanisms of magnetoelectric coupling by combining 2D ferroelectric with magnetic layered materi-als. Very recently, the magnetoelectric coupling effects are realized inseveral vdW heterostructures, such as the Cr 2Ge2Te6/In2Se3,32FeI2/ In2Se3,33and bilayer CrI 3/Sc2CO 2.34 Here, based on the first-principles calculations, we propose a differ- ent magnetoelectric coupling mechanism by heterostructure engineering.We validate this concept by studying a 2D vdW heterostructure Appl. Phys. Lett. 118, 062903 (2021); doi: 10.1063/5.0036302 118, 062903-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplconsisting of 2D magnetic metal-organic framework (MOF) and ferro- electric In 2Se3. Our idea is to change the relative electrostatic potential difference between In 2Se3and MOF and band alignments of heterostruc- tures by reversing the polarization direction of In 2Se3.Thus, this leads to t h ec h a r g et r a n s f e rb e t w e e nI n 2Se3and MOF, as schematically shown in Fig. 1 . When the charge transfer is very small, the magnetic MOF main- tains its intrinsic magnetic ground state. But when the charge transfer is large enough to change the exchange interaction, a magnetic phase tran- sition will occur in the magnetic MOF. Furthermore, by the first-principles calculations, we confirm that the different magnetic states,antiferromagnetic (AFM) or ferromagnetic (FM) state, can be indeedrealized in MOF V 2C18H12by reversing the polarization direction of In2Se3in heterostructure V 2C18H12/In2Se3. This can be well understood from the interfacial charge transfer between V 2C18H12and In 2Se3,r e s u l t - ing in significant hole doping in V 2C18H12. Our predicted system can be called as the magnetoelectric heterostructure. Compared with single- phase multiferroic systems, it is a more flexible approach to realize mag- netoelectric coupling by the interfacial effect and, thus, provides morepossibilities for the realization of multiferroic devices. The magnetoelectric heterostructure consists of 2D ferroelectric In 2Se3and magnetic MOF V 2C18H12,a ss h o w ni n Fig. 2 .I nc o n t r a s tt o inorganic 2D magnetic materials, MOFs are much easier to achievelong-range magnetic order due to the diverse functionality by tuningthe metal centers and functional groups. 35M2C18H12(M is the transi- tion metal atom) is a kind of MOF material with fascinating magnetic and topological properties.36–38Considering the lattice matching and relative electrostatic potential difference between In 2Se3and MOF, we chose V 2C18H12to construct heterostructure with In 2Se3. Monolayer In2Se3has been proved to be a 2D room-temperature ferroelectric material with the spontaneous out-of-plane polarization.39,40The movement of the middle Se atom (circled in Fig. 2 )b r e a k st h ei n v e r - sion symmetry and, thus, leads to a spontaneous out-of-plane polari-zation with 0.094 eA ˚/unit cell. 41Two opposite polarization states are denoted as the ferroelectric state with up polarization (FE up) and the ferroelectric state with down polarization (FE dn). The energy barrier is 0.066 eV/unit cell if the ferroelectric polarization is reversed.41 Therefore, In 2Se3is an excellent ferroelectric substrate with large intrinsic polarization and suitable transition energy barrier. Thein-plane lattice constants of free monolayer In 2Se3and V 2C18H12are 4.106 A ˚and 11.90 A ˚, respectively. Thus, we use a 3 /C23 supercell of In2Se3to match with V 2C18H12with a lattice mismatch of 3.4%. Calculations were performed by using density functional theory (DFT) calculations,41,42as implemented in Vienna ab initio Simulation Package (VASP).43The core–valence interaction is treated by the pro- jector augmented-wave method,44and the exchange–correlation func- tional is treated using the Perdew–Burke–Ernzerhof (PBE) parameterization of generalized gradient approximation (GGA).45The kinetic cutoff energy is set to 400 eV in all calculations, and all thestructures are fully relaxed until the energy and force are less than10 /C05eV and 10/C02eV/A˚, respectively. The Brillouin zone is sampled FIG. 1. Schematic diagram of the physical mechanism of the magnetic phase transition driven by ferroelectric polarization in a two-dimensional magnetoele ctric heterostructure V2C18H12/In2Se3. FIG. 2. (a) and (b) are the side views of the V 2C18H12/In2Se3heterostructure with up polarization (FE up) and downward polarization (FE dn) configurations, respec- tively. The black thick arrows denote the direction of the out-of-plane electric polari-zation in In 2Se3. (c) and (d) are the top views and spin densities of FE upand FE dn states, respectively. Blue and red contours represent two different spin states.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 062903 (2021); doi: 10.1063/5.0036302 118, 062903-2 Published under license by AIP Publishingusing a 9 /C29/C21 Monkhorst–Pack grid.46The GGA þUm e t h o dw i t h an effective Hubbard U eff¼U-J¼2e V ( U ¼3e V a n d J ¼1e V ) f o r the consideration of the strong Coulomb interaction is performed on the transition metal V atom.47We also check out our conclusion by varying U efffrom 1 to 4 eV (U ¼2, 3, 4, 5 eV and J ¼1e V ) . I n a d d i - tion, we take the semiempirical Grimme method to describe vdW interactions.48 First, we study the stacking configurations between V 2C18H12 and In 2Se3. After optimizing the positions of all atoms, we find three possible local minimum, energies, corresponding to different possible ways of stacking the MOF on the In 2Se3. The system is energetically most favorable when the V atoms sit at the top site of Se, as shown in Fig. 2 . The preferred stacking does not depend on the polarization of the In 2Se3. The other two configurations with higher energy are shown in Fig. S1 and Table S1 of the supplementary material . In the follow- ing, we will focus on the most stable configuration. In the FE upphase as shown in Fig. 2(a) , the nearest distance between V and Se atoms is 2.73 A ˚, while in the FE dnphase, this distance is 2.64 A ˚, indicating the stronger interfacial interaction, as shown in Fig. 2(b) . Calculated out- of-plane electric dipole moments are 0.735 and /C02.003 eA ˚/supercell for the FE upand FE dnphases, respectively. In order to compare the heterostructures with intrinsic In 2Se3(0.094 eA ˚/unit cell), we average the dipole moments to each unit cell of In 2Se3, and the values are 0.082 and /C00.222 eA ˚/unit cell for the FE upand FE dnphases, respec- tively. It is indicated that the charge transfer should be negligible for the FE upphase due to the almost equal dipole moments. For the FE dn phase, the large difference in dipole moments indicates that the chargetransfer has an important effect on the electric polarization of the whole heterostructure system. However, two ferroelectric phases withopposite polarization directions can be switched by reversing the external vertical electric field. The total energy difference between the two different ferroelectric states is 384 meV/supercell (43 meV/unit cell for In 2Se3). This energy difference indicates the relative stability between two different ferroelectric states. The asymmetric interfacialcoupling in ferroelectric heterostructures may distort the ferroelectric double-well potential and suppress one of the two potential wells of In 2Se3,s u c ha si nt h eF e 3GeTe 2/In2Se3heterostructure.49However, the ferroelectric bistability can be recovered when the In 2Se3layer is thicker than three unit cells. Therefore, multilayer In 2Se3may be used for maintaining 2D ferroelectric bistability in ferromagnetic/ferroelec- tric vdW heterostructures. The most important finding is that when the In 2Se3is in the FE up polarization configuration, the ground state of monolayer V 2C18H12is the AFM order, while for the FE dnpolarization configuration, the ground state becomes the FM order. Therefore, the magnetic order is coupled with electric polarization, resulting in a significant magneto-electric coupling. The spin densities of the FE upand FE dnphases are shown in Figs. 2(c) and2(d), respectively. It is clearly seen that the magnetic moments are mainly localized at V atoms of V 2C18H12.V atom is located at the center of a triangular lattice of three C atoms, as shown in Fig. 3(a) . According to the ligand field theory, in a D 3hsym- metric ligand field, the dorbital splits into three groups: doubly degen- erate dx2/C0y2and dxy,dz2and doubly degenerate dxzand dyz,a ss h o w n inFig. 3(b) . Since two delectrons of V occupy the two lowest dxzand dyzorbitals, each V atom possesses a magnetic moment of 2.0 lB. When the V 2C18H12is placed on the surface of the In 2Se3, we find that the hybridization and mixing of atomic orbitals do not change thefilling modes of electrons in a triangular ligand field and the magnetic moments of V atoms. Generally, the experimental values are slightly lower than this theoretical value of V3þdue to the canted magnetic moments or the presence of a small antiferromagnetic component, for example, in AgVP 2Se6.50In addition, we find that the V atom magnet- izes the surrounding three C atoms, which have opposite small mag- netic moments. The change in the magnetic order is closely related to the elec- tronic structure. Band structures and PDOS of the V 2C18H12/In2Se3 are shown in Fig. 3 . For the FE upconfiguration, the V 2C18H12/In2Se3 is a type-II semiconductor with a bandgap of 0.5 eV, as shown in Fig. 3(c). The highest valence band (HVB) is contributed from the V and C atoms of the V 2C18H12[seeFig. 3(e) ] and is almost flat, indicating the strong localization of magnetic moment, while the lowest conduc- tion band (LCB) comes from the In 2Se3. For the FE dnconfiguration, the bandgap is closed due to the polarization reversal and relative movement of bands. The LCB from the In 2Se3touches the HVB from the V 2C18H12, forming a type-III band alignment, as shown in Fig. 3(d) . Such a band crossing indicates the existence of the large charge transfer between the V 2C18H12and In 2Se3. Therefore, the dis- tinction of electronic and magnetic properties between the FE upand FEdnstates is attributed to different polarization directions and interfa- cial interactions. In addition, a significant structural change has taken p l a c ef o rt h eV 2C18H12when the heterostructures are formed. For the free-standing V 2C18H12, two V atoms are buckled and have a vertical distance of about 0.94 A ˚in the out-of-plane direction (see Fig. S2 of thesupplementary material ). However, in the V 2C18H12/In2Se3,t w oV atoms move to the In 2Se3during structural optimization due to strong interfacial interaction and finally keep at the same height, as shown inFigs. 2(a) and2(b). In order to reveal the effect of the charge transfer on the magne- tism, we further evaluate electron distribution by the Bader chargeanalysis. For the FE upphase, there are only 0.03 electrons transferred from the In 2Se3to V 2C18H12, resulting in slight electron doping in the V2C18H12. In contrast, there are 0.22 electrons transferred from the V2C18H12to In 2Se3for the FE dnphase, resulting in significant hole doping in the V 2C18H12. Thus, hole doping could be an impor- tant reason for the magnetic phase transition from an AFM to FM state for the V 2C18H12/In2Se3.I na d d i t i o n ,w efi n dt h a ta b o u t9 0 % of transferred holes locate at C and H atoms (see Table S2 of the supplementary material ), indicating the importance of a hexagonal carbon ring for the magnetic coupling between V atoms. The distance between two V atoms is too large (about 6.92 A ˚) to directly couple with each other, and, therefore, V atoms are indirectly coupled by the intermediate carbon rings. In the V 2C18H12, because V atoms are directly bonded to C atoms, the magnetic coupling between V and Catoms is directly AFM coupling. Therefore, the magnetic coupling between two C atoms of the hexagonal carbon ring determines the magnetic coupling between two V atoms connected with them, as schematically shown in Fig. 4(a) .I nt h eF E upphase, due to extremely small charge transfer, the V 2C18H12keeps in the AFM ground state. In contrast, in the FE dnphase, the large hole doping induced by the charge transfer drives the magnetic interaction between C atoms to change from an AFM to FM state, and, thus, the magnetic coupling between V atoms is also transformed from the AFM to FM state. Therefore, the magnetic coupling between V atoms can be tuned by hole doping.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 062903 (2021); doi: 10.1063/5.0036302 118, 062903-3 Published under license by AIP PublishingTo further clarify this mechanism, we calculate the magnetic cou- pling of the free-standing V 2C18H12under uniform hole doping, as shown in Fig. 4(b) .W ec a ns e et h a tw i t h o u td o p i n g ,t h eA F Ms t a t ei s more stable than the FM state. However, with the increase in hole con- centration, the energy difference between AFM and FM states is get-ting smaller and smaller, indicating that the AFM state becomes moreand more unstable. Different from the V 2C18H12/In2Se3,t h ef r e e - standing V 2C18H12is still in an AFM state under the doping concen- tration of 0.22 hole. The possible reason is that the doping in the free-standing V 2C18H12is uniform and simulated by removing elec- trons from the system and using a homogeneous background chargeto maintain charge neutrality, while the doping in the V 2C18H12/ In2Se3is realized by the charge transfer between the V 2C18H12andIn2Se3, depending on the local interlayer interaction. By the Bader charge analysis, we find that about 75% of holes are located at the car-bon ring (see Table S3 of the supplementary material ). However, when the hole doping concentration exceeds 0.4 hole/cell, a magnetic phase transition in the free-standing V 2C18H12from AFM to FM state occurs, which is qualitatively consistent with the heterostructure.Therefore, the hole doping has an important effect on magnetic cou-pling between two C atoms of the hexagonal carbon ring, resulting in magnetic phase transition from an AFM to FM state. This mechanism is similar to magnetic coupling between two edges of zigzag graphenenanoribbons under carrier doping. Previous studies have shown thatthe two edges of zigzag graphene nanoribbon are AFM coupling andthe ground state can be transformed from AFM order to FM order by FIG. 3. (a) Top view of V 2C18H12with two triangular lattices of C atoms shown by dotted blue lines. (b) Splitting of the 3 dlevel of the V atom in a triangular ligand field. Band structures of V 2C18H12/In2Se3for FE up(c) and FE dn(d) polarization configurations. The PDOS of V 2C18H12/In2Se3for FE up(e) and FE dn(f) polarization configurations.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 062903 (2021); doi: 10.1063/5.0036302 118, 062903-4 Published under license by AIP Publishingcarrier doping along with the decrease in bandgap and the increase in the one-electron energy.37,51In zigzag graphene nanoribbons, edge carbon atoms carry small magnetic moments, about 0.2 lB/atom and from the edge to the inside, the magnetic moments are getting smaller and smaller.52Similarly, in MOF-V 2C18H12, carbon atoms have also a small magnetic moment of about 0.06 lB/atom. The magnetic moments on carbon atoms have the same order of magnitude for gra-phene nanoribbon and V 2C18H12, and the magnetic coupling in two systems can be controlled similarly by hole doping. Therefore, we con- clude that the polarization direction of In 2Se3dramatically changes the band alignment and charger transfer in the V 2C18H12/In2Se3and, thus, drives the magnetic phase transition of the V 2C18H12from the AFM to FM state. Based our prediction, a nonvolatile spin field effect transistor can be suggested, shown in Fig. S3 of the supplementary material . In this device, the electric field is applied by the gate voltage to control the out-of-plane ferroelectric polarization in In 2Se3.T h e n , the AFM–FM transition in MOF can be tuned by reversing the direc-tion of an electric field. For the FM state, the high spin-polarized cur- rent can be detected by bias voltage, while the AFM state corresponds to low spin-polarized current transport. Therefore, a nonvolatile spin-polarized transport current can be controlled by applying an electricfield. In summary, based on first-principles calculations, we propose a strategy to realize strong magnetoelectric coupling by stacking 2Dmagnetic V 2C18H12with ferroelectric In 2Se3.T h em a g n e t i cg r o u n d state of the V 2C18H12changes from an AFM to FM state as the polari- zation direction of In 2Se3is reversed. The underlying physical mecha- nism is the interfacial charge transfer between the V 2C18H12and In2Se3, leading to a significant hole doping in the V 2C18H12. The hole doping further changes the exchange interaction between the localmagnetic moments and, thus, leads to a magnetic phase transition.The proposed 2D heterostructure and magnetoelectric coupling mech-anism not only enrich the 2D multiferroic family but also offer astrategy for exploring magnetoelectric coupling in low-dimensional nanostructures by heterostructure engineering. See the supplementary material for details on the structures of three stacking configurations, the structure of the free-standing V 2C18H12and band structures of the FM state without U value and AFM state with U eff¼2 eV, the schematic diagram of the proposed prototype device realizing electric field control of AFM and FM states,spin densities of zigzag graphene nanoribbon and the variation ofenergy difference between the AFM state and FM state with the con-centration of hole doping, total energy of three stacking configurationsfor two ferroelectric states, Bader charge analysis of heterostructureV 2C18H12/In2Se3, and the free-standing V 2C18H12w i t h0a n d0 . 2h o l e doping for AFM state. This work is supported by the National Natural Science Foundation of China (Project Nos. 11974148, 11674132, and11804127). DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material . REFERENCES 1J. F. Scott, Nat. 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5.0046664.pdf

J. Chem. Phys. 154, 120901 (2021); https://doi.org/10.1063/5.0046664 154, 120901 © 2021 Author(s).Two-dimensional terahertz spectroscopy of condensed-phase molecular systems Cite as: J. Chem. Phys. 154, 120901 (2021); https://doi.org/10.1063/5.0046664 Submitted: 05 February 2021 . Accepted: 09 March 2021 . Published Online: 29 March 2021 Klaus Reimann , Michael Woerner , and Thomas Elsaesser COLLECTIONS Paper published as part of the special topic on Coherent Multidimensional Spectroscopy This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN Reflections on electron transfer theory The Journal of Chemical Physics 153, 210401 (2020); https://doi.org/10.1063/5.0035434 Transient 2D IR spectroscopy from micro- to milliseconds The Journal of Chemical Physics 154, 104201 (2021); https://doi.org/10.1063/5.0045294 Classical molecular dynamics The Journal of Chemical Physics 154, 100401 (2021); https://doi.org/10.1063/5.0045455The Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp Two-dimensional terahertz spectroscopy of condensed-phase molecular systems Cite as: J. Chem. Phys. 154, 120901 (2021); doi: 10.1063/5.0046664 Submitted: 5 February 2021 •Accepted: 9 March 2021 • Published Online: 29 March 2021 Klaus Reimann,a) Michael Woerner,b) and Thomas Elsaesserc) AFFILIATIONS Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie, 12489 Berlin, Germany Note: This paper is part of the JCP Special Topic on Coherent Multidimensional Spectroscopy. a)Electronic mail: reimann@mbi-berlin.de b)Electronic mail: woerner@mbi-berlin.de c)Author to whom correspondence should be addressed: elsasser@mbi-berlin.de ABSTRACT Nonlinear terahertz (THz) spectroscopy relies on the interaction of matter with few-cycle THz pulses of electric field amplitudes up to mega- volts/centimeter (MV/cm). In condensed-phase molecular systems, both resonant interactions with elementary excitations at low frequencies such as intra- and intermolecular vibrations and nonresonant field-driven processes are relevant. Two-dimensional THz (2D-THz) spec- troscopy is a key method for following nonequilibrium processes and dynamics of excitations to decipher the underlying interactions and molecular couplings. This article addresses the state of the art in 2D-THz spectroscopy by discussing the main concepts and illustrating them with recent results. The latter include the response of vibrational excitations in molecular crystals up to the nonperturbative regime of light–matter interaction and field-driven ionization processes and electron transport in liquid water. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0046664 .,s I. INTRODUCTION Molecular systems in the condensed phase display a variety of low-energy excitations in the frequency range from some 10 GHz to 30 THz. Collective molecular motions, inter- and intramolecu- lar vibrations, and coupled nuclear-electronic degrees of freedom such as soft modes give rise to dipole-allowed absorption and/or Raman bands, most of which exhibit complex and partly overlapping line shapes. While linear dielectric, far-infrared, and Raman spectro- scopies have been developed over decades and reached a high level of sophistication,1,2nonlinear THz methods have opened a new field of research, in which the nonlinear low-frequency response, the intrin- sic dynamics of excitations, their couplings, and the interaction with a thermal bath are addressed most directly.3–7This rapidly devel- oping field has benefitted from major progress in generating THz few-cycle pulses with electric field amplitudes reaching the MV/cm range8–11and from the introduction of new nonlinear spectroscopies in which THz pulses induce a nonlinear response of a solid or a molecular ensemble.The concept of two-dimensional correlation spectroscopy goes back to nuclear magnetic resonance12and has been transferred to the optical domain to study vibrational excitations in the mid- infrared range and/or electronic and vibronic excitations in the visible and ultraviolet. Both pump–probe and three-pulse photon- echo methods have been applied to follow the nonlinear response of a molecular system in third order in the optical field, mostly under conditions of resonant excitation. Detailed accounts of 2D third-order spectroscopy have been given in Refs. 13–16. Two- dimensional THz (2D-THz) spectroscopy, introduced some ten years ago,17–19goes well beyond the third-order limit. It allows for mapping nonlinearities up to the regime of nonperturbative light–matter interaction, where the coupling to the external THz electric field is comparable to or even stronger than interactions within the molecular system.7Both resonant and nonresonant interactions and field-driven processes of charge transport have been studied by 2D-THz spectroscopy, which is inherently phase- resolved and gives access to absolute rather than relative optical phases. J. Chem. Phys. 154, 120901 (2021); doi: 10.1063/5.0046664 154, 120901-1 © Author(s) 2021The Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp Early work in 2D-THz spectroscopy has focused on method development and applications to electronic excitations in solids. Here, the large transition dipoles, e.g., of intersubband transitions in semiconductor quantum wells,17,18,20facilitated the generation of a quasi-resonant nonlinear response with moderate peak elec- tric fields in the range of 1–50 kV/cm. More recent applications of 2D-THz spectroscopy in spectroscopic and transport studies of solids, metamaterials, and semiconductor devices have been reported in Refs. 21–34. The rapid development of generation schemes for THz pulses with MV/cm amplitudes has strongly widened the application range of 2D-THz methods, now including vibrational and/or phononic excitations and, in particular, a broad range of nonresonant nonlinear interactions. This article gives an account of the current state of this field with a focus on molecular excitations and processes in the condensed phase. It combines a pre- sentation of the method and experimental aspects with a discussion of recent prototypical applications. The content of this article is organized as follows: Section II summarizes methods for THz pulse generation in laser-driven sources (Sec. II A) and field-resolved detection (Sec. II B), including a short description of linear THz spectroscopy. Section III intro- duces the basic concepts of 2D-THz spectroscopy and the experi- mental implementation (Sec. III A), complemented by a discussion of data handling and analysis (Sec. III B). A comparison to related 2D methods such as nonlinear 2D Raman-THz and multicolor 2D spectroscopy is included as well (Sec. III C). A prototypical study of soft-mode excitations in molecular crystals of aspirin is presented in Sec. IV, followed by a discussion of ionization and electron trans- port in water driven by strong THz fields (Sec. V). Conclusions and an outlook on future developments are given in Sec. VI. II. FEW-CYCLE THz PULSES: GENERATION AND CHARACTERIZATION A. Laser-driven generation of THz pulses Nonlinear THz spectroscopy relies on ultrashort THz pulses with an electric field amplitude sufficient to induce a nonlinear response in the sample under study. Such pulses have been gener- ated with electrically biased photoconductive switches and antenna structures or by nonlinear optical frequency conversion of optical pulses from the visible and near-infrared to the THz range. Irradi- ation of a photoconductive switch made from GaAs, InP, or other semiconductors by a femto- to picosecond optical pulse induces short-lived transient currents, which radiate electric fields at THz frequencies. The maximum amplitude of the THz field is set by the DC bias field of the switch and reaches values on the order of 10 kV/cm for large-area emitters. While photoconductive switches driven by femtosecond laser oscillators are widely applied in com- mercial devices for linear THz spectroscopy, their applicability in nonlinear experiments is limited because of the comparably small peak electric fields. Plasmas generated with intense femtosecond laser pulses in dilute gases emit strong few-cycle THz pulses.8,37,38Two-color gen- eration of a plasma in air or nitrogen, e.g., by the fundamental and second-harmonic frequency of an amplified femtosecond laser pulse, generates a unidirectional transient plasma current of elec- trons, which radiates a very broad frequency spectrum includingpronounced THz components (up to 70 THz).39Peak electric fields of up to 400 kV/cm have been generated at a kilohertz repetition rate with 25-fs pulses from an amplified Ti:sapphire laser system.8 This type of source has been applied in a number of nonlinear THz experiments, although plasma instabilities result in fluctuations of the electric field amplitudes and the spatial beam parameters. Optical difference frequency mixing or rectification in nonlinear crystals and/or at surfaces with a high second-order nonlinearity χ(2)is the predominant method for generating few-cycle THz pulses with high electric field amplitudes. Here, the difference frequency of two synchronized incoming pulses or between different components in the broad spectrum of a single femtosecond pulse is generated, preferentially in a phase-matched conversion process, in which the phase velocity of the THz pulse needs to match the group velocity of the pump pulses. Beyond the THz frequency range discussed here, this method has been widely applied to generate pulses in the mid-infrared up to frequencies on the order of 100 THz. Inorganic semiconductors with a zincblende crystal structure such as GaAs, GaP, InP, CdTe, ZnTe, and the ferroelectric LiNbO 3 are standard materials to cover a frequency range up to some 6 THz with electric field amplitudes7up to several hundred kV/cm. The introduction of non-collinear rectification in LiNbO 3using pump pulses with a tilted wavefront10,40allows for reaching peak electric fields of several MV/cm at frequencies between 0.2 and 1.5 THz. The time-dependent electric field of a THz pulse gener- ated with this method is shown in Fig. 1(a), and the correspond- ing intensity spectrum is shown in Fig. 1(b). Drawbacks of this method are the spatial and temporal break-up of the optical pump due to angular dispersion, which limits the conversion efficiency, and the limited quality of the generated THz beam. Very recent work has addressed such issues by both numerical simulations of different interaction geometries and experiments.41It should be noted that few-cycle pulses with electric field amplitudes up to 100 MV/cm have been generated9in the frequency range from 10 to 70 THz. Organic nonlinear crystals consisting of molecular chro- mophores have received substantial interest for THz generation because of their high second-order nonlinearities.42,43Figures 1(c) and 1(d) display the time-resolved electric field and the intensity spectrum of pulses generated by driving a 400- μm thick DSTMS crystal (DSTMS: 4-N,N-dimethylamino-4′-N′-methylstilbazolium 2,4,6-trimethylbenzenesulfonate) with 25-fs pulses from an ampli- fied Ti:sapphire laser.44The THz spectrum covers the range from 0.1 to 30 THz. The narrow features in the spectrum originate from radiation on vibrational resonances and correspond to the long-lived oscillations of the electric field in the time-domain. The frequency conversion process is only partially phase-matched. However, due to the very high optical nonlinearity, an appreciable conversion effi- ciency is possible even without phase-matching. The comparably low optical damage thresholds of organic nonlinear crystals at higher pulse repetition rates eventually limit the attainable electric field amplitudes. B. Field-resolved detection and measurement Phase-resolved detection of THz pulses, i.e., the measurement of the electric field as a function of time, is a key ingredient of J. Chem. Phys. 154, 120901 (2021); doi: 10.1063/5.0046664 154, 120901-2 © Author(s) 2021The Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp FIG. 1 . THz pulses generated by optical rectification of femtosecond optical pulses in [(a) and (b)] LiNbO 3and [(c) and (d)] the organic crystal DSTMS. Panels (a) and (c) show the electric field as a function of time, and panels (b) and (d) show the intensity spectra of the pulses. Note the different frequency scales in (b) and (d). (e) THz absorption spectrum of a 15 μm thick water sample with a high peak absorbance of 2.8 (blue line, taken from Ref. 35). The absorbance A=−log(T) (T: intensity transmission of the sample) derived from a time-domain measurement with the broadband pulses shown in panels (c) and (d) is plotted as a function of frequency. The dashed line gives a reference spectrum from the literature.36 2D-THz spectroscopy. The standard method is free-space electro- optic (EO) sampling, which exploits the change in refractive index induced by the THz field in an electro-optic crystal.45This change is mapped by the polarization change in a femtosecond probe pulse, the duration of which needs to be much smaller than the THz period. Measuring the polarization change for different delay times between the THz transient and the probe pulse allows for reconstructing the time-dependent THz field and, in calibrated setups, the absolute electric field amplitudes. Widely used electro-optic materials include ZnTe, GaP, GaSe, LiTaO 3, and LiNbO 3. A detection bandwidth on the order of 100 THz has been demonstrated with 10–50 μm thick ZnTe and GaSe crystals.46,47The smallest electric fields measured by EO sampling are on the order of 1 V/cm, as shown in studies of vacuum fluctuations of the electric-field.48,49Further details and lit- erature can be found in Ref. 7. A setup for EO sampling is presented in Sec. III A (Fig. 4).Time-domain THz methods have found broad application in linear mid- and far-infrared spectroscopy. A field-resolved detection scheme allows for deriving both the real and imaginary parts of the linear dielectric function from the THz field transmitted through a sample. Moreover, the fact that electric fields are detected rather than intensities as in conventional far-infrared spectroscopy makes the study of optically thick samples possible. The electric field amplitude decreases only proportional to the square root of the intensity, e.g., a reduction of intensity to 1/100 corresponds to a reduction of 1/10 in electric field amplitude only. This aspect is relevant for aqueous solutions, e.g., of biomolecules where the background absorption of water represents a major contribution to the overall absorption spectrum. To illustrate the potential of field-resolved spectroscopy, Fig. 1(e) shows the absorption spectrum of a 15 μm thick film of neat water (blue line), derived from a time-domain THz measure- ment35with the broadband pulses shown in Figs. 1(c) and 1(d). The spectrum, which is in good agreement with literature data (dashed line, Ref. 36), displays the prominent L2 absorption band around 20 THz (600 cm−1) due to librations of water molecules, and the weaker stretching band of OH ⋯O hydrogen bonds between 5 and 6 THz. The peak absorbance in this spectrum has a value of∼2.8, corresponding to a very small intensity transmission of 1.5×10−3, which is typically beyond the detection range of con- ventional Fourier spectrometers. III. TWO-DIMENSIONAL THz SPECTROSCOPY A. Concept and experimental implementation Two-dimensional THz spectroscopy is based on the interac- tion of a pulse sequence consisting of two or three THz pulses with a sample and a measurement of the total transmitted THz electric field in amplitude and phase. If the response of the sample is per- fectly linear, the total transmitted field Elin ABCof, e.g., three pulses (A, B, and C) is equal to the sum of the electric fields of the individual pulses, Elin ABC=Elin A+Elin B+Elin C. (1) Accordingly, 2D-THz spectroscopy only yields additional informa- tion in the case of a nonlinear response of the sample. In 2D-THz spectroscopy, one exploits the nonlinearities resulting from interac- tion with all pulses applied. For two pulses [Fig. 2(a)], the nonlinear response determines the electric field ENL(τ,t) given by17 ENL(τ,t)=EAB(τ,t)−EA(τ,t)−EB(t). (2) In this equation, τis the time delay between the two pulses A and B,tis the real time, i.e., the time axis along which the electric field is measured by electro-optic sampling, EA(τ,t) is the transmitted electric field when only pulse A is incident on the sample, EB(t) is the transmitted field when only pulse B is incident on the sample, andEAB(τ,t) is the transmitted field with both pulses present. The single-pulse nonlinearities are contained in EA(τ,t) and EB(t) so that they are removed by Eq. (2). J. Chem. Phys. 154, 120901 (2021); doi: 10.1063/5.0046664 154, 120901-3 © Author(s) 2021The Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp FIG. 2 . Calculated electric field transients of (a) a two-pulse and (b) a three-pulse sequence. The transients labeled A, B, and C are the incident pulses before inter- action with the sample. EABandEABCare the transients after transmission through the sample when two [in (a)] or three [in (b)] pulses have interacted with the sample. ENLis the nonlinear signal according to Eqs. (2) and (3). The nonlinear response was calculated for an ensemble of two-level systems. For three pulses [Fig. 2(b)], one has to subtract both one- and two-pulse nonlinearities to recover ENL(τ,T,t) according to50 ENL(τ,T,t)=EABC(τ,T,t) −EAB(τ,T,t)−EBC(T,t)−EAC(τ,T,t) +EA(τ,T,t)+EB(T,t)+EC(t). (3) The delay between pulses B and C is T, often called waiting time. The meaning of EABC(τ,T,t) is analogous to the two-pulse case. For two pulses, the nonlinear susceptibilities contributing to ENLareχ(2),χ(3), and higher; for three pulses, they are χ(3),χ(4), and higher. In general, ann-pulse scheme of this type will show nonlinearities of order n and higher. Nonlinearities of even and odd orders can be distinguished by looking at the spectrum of the nonlinear field. For input fields of a frequencyω, even-order nonlinearities will result in nonlinear fields with frequencies 0, 2 ω, 4ω,. . ., whereas odd-order nonlinearities generate fields with ω, 3ω, 5ω,. . .. Thus, a nonlinear signal with the frequencyωin two-pulse experiments cannot result from a second- order nonlinearity but originates from a third- or higher odd-order nonlinearity. Most third-order nonlinearities can be measured with only two pulses. The different contributions to the total nonlinear signal can be analyzed and separated with density matrix theory describing the nonlinear light–matter interaction.51The signal components corre- spond to different pathways in Liouville space. Such pathways have been visualized by diagrammatic techniques such as double-sided Feynman diagrams,51multi-level schemes,52or sequences of arrows in 2D frequency space.19 Reproducible femtosecond delays between the THz pulses and synchronization with the read-out pulse for electro-optic samplingare essential for 2D-THz spectroscopy. Typically, all pulses are derived from a single mode-locked laser oscillator. There are vari- ous schemes to generate the two or three THz pulses for 2D-THz spectroscopy as schematically illustrated in Fig. 3 for two pulses. In scheme I, a single THz pulse is generated, split in two replicas, and recombined after variable time delays. In scheme II, the beam from the pump laser is split and then sent to two separate THz genera- tors G. In this way, it is possible to have different frequencies of the two beams for two-color 2D spectroscopy. Scheme III requires only a single THz generator G but has the disadvantage that the gener- ation process leads to nonlinear electric fields according to defini- tions (2) and (3), in particular during the time overlap of the pump pulses. Key components of schemes I and II are the THz beam split- ters S and D in I and D in II. An ideal beam splitter should gen- erate a reflected pulse with Er(t) =a E(t−δr) and a transmitted pulse Et(t) =b E(t−δt) from the incident THz pulse E(t) with a2 +b2= 1. Here,δtandδrare time delays upon reflection and trans- mission. Components coming close to this ideal are thin pellicle beam splitters, plate-type beam splitters, and wire grid polarizers. The thin plastic substrate of a pellicle introduces weak dispersion- induced changes in the transmitted transients but may display con- siderable absorption at certain THz frequencies. Disadvantages of pellicles are their fragility and their high sensitivity to acoustic noise and air currents. Thicker plate-type beam splitters need to possess low THz dispersion and absorption. Suitable materials are undoped diamond, silicon, and germanium. Unwanted reflections from the back surface of beam splitter plates can be suppressed by antire- flection coatings.53–56Another solution is to use the beam splitter under Brewster’s angle for ppolarization and add a metallic coating on one side for the required reflectivity.57Wire grid polarizers con- sist of parallel thin metallic wires. For wavelengths large compared to the distance between neighboring wires, they transmit electric fields perpendicular to the direction of the wires and reflect electric fields parallel to the wires. After a wire grid beam splitter D, the two beams will be orthogonally polarized. To obtain the same polariza- tion, one can introduce an additional polarizer between D and the sample. FIG. 3 . Upper part: schemes for generating a sequence of two THz pulses from a visible/near-infrared laser. G consists of a nonlinear crystal for THz generation including a filter to remove the residual laser light, and S and D are beam splitters. The laser beam is shown in red, and the two THz beams are shown in blue and green. Bottom part: overlapping divergent THz beams A and B as an alternative to using a beam splitter D in schemes I and II. J. Chem. Phys. 154, 120901 (2021); doi: 10.1063/5.0046664 154, 120901-4 © Author(s) 2021The Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp In both schemes I and II, the beam splitters D can be replaced by having the two beams parallel and close to each other (lower part of Fig. 3). Because of the generally high divergence of THz beams, they will overlap after some propagation length. For instance, an initial beam waist of 5 mm (typical aperture of non- linear crystals, e.g., GaSe) at a frequency of 1 THz (wavelength λ= 300μm) results in a minimum divergence of 38 mrad. After a propagation length of 30 cm, the two beams overlap nearlyperfectly. This method can easily be extended to three beams arranged, e.g., in a triangular pattern (Fig. 4). In such geometries, essentially no losses occur, i.e., in scheme I for two (three) beams, each beam has, upon hitting the sample, one half (one third) the energy of the initially generated THz pulse outside the time over- lap of the two pulses. In contrast, the pulse energies on the sample are only one quarter (one ninth) of the initial energy for ideal beam splitters D. FIG. 4 . Setup for three-pulse measurements. BS are beam splitters, Q are quartz glass windows for the vacuum chamber, NL are nonlinear crystals for THz generation (GaSe), ZnTe is the electro-optic crystal, λ/4 is a quarter-wave plate, WP is a Wollaston polarizer, and PD1 and PD2 are silicon photodiodes. The detected signal is the difference of the outputs of PD1 and PD2. Details are discussed in the text. J. Chem. Phys. 154, 120901 (2021); doi: 10.1063/5.0046664 154, 120901-5 © Author(s) 2021The Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp A setup for three pulses using scheme II is shown in Fig. 4.50 The pump pulses are generated with a Ti:sapphire oscillator- amplifier system. This particular setup includes two multipass amplifiers, both seeded from the same oscillator to ensure synchro- nization. In this scheme, the shape of the two amplified pulses can be set independently with acousto-optic pulse shapers, permitting both phase and amplitude changes. The settings are chosen for opti- mum generation of the required THz pulses,58a feature particularly important for two-color measurements. The setup of Fig. 4 uses pulses from the oscillator as read-out pulses in electro-optic sampling. They are shorter than the amplified pulses (12 fs vs 20–25 fs) and, thus, allow for measuring THz pulses at higher frequency (40 THz vs 25 THz).7,59Moreover, the smaller pulse-to-pulse fluctuations improve the signal-to-noise ratio. While the temporal jitter between oscillator and amplifier output is typi- cally on the order of a few femtoseconds, temporal drifts of the read- out relative to the THz pulse represent a challenge in this scheme. Due to the long optical path length through the amplifiers (on the order of 10 m), even a very small temperature difference between the path of the oscillator pulse and the path of the amplified pulse can lead to significant drifts. For example, a temperature difference of 0.1 K between parts of the optical table will lead to a change in the distance of 10 μm, equivalent to a time difference of 33 fs. It is, how- ever, possible to correct for such drifts after the measurement (see Sec. III B). The read-out and the THz pulse overlap in a thin (thickness ≈10μm) electro-optic ZnTe crystal of (110) orientation. The thin crystal is mounted onto a 0.5 mm-thick (100) ZnTe crystal, which shows no electro-optic effect.60In this way, the thin electro-optic crystal is mechanically more stable, and even more importantly, the time range over which reflections from the back surface of the crystal are absent is extended. The THz electric field changes the dielectric tensor of the (110) ZnTe crystal, which, in turn, leads to a change in the polarization state of the initially linearly polarized read-out pulse. This change is converted by the quarter-wave plate and the polarizer into a difference between the signals of two balanced pho- todiodes. The difference signal is proportional to the THz electric field at this instant of time. It is digitized for every laser shot and stored in a PC, which also detects the settings of all choppers (see below) and moves the various delays according to a predetermined sequence. The THz generation stages, the sample, and the EO sampling setup are placed in a closed chamber, which is evacuated with a scroll and a turbomolecular pump for removing gases such as CO 2and water vapor. In this way, absorption and temporal distortions of the THz pulses are suppressed. At a pressure of 10−3mbar, the gas con- centrations are reduced from their values at atmospheric pressure by six orders of magnitude. At pressures of the order of 10−6mbar, it is possible to perform measurements at low temperatures by mounting the sample on the cold finger of a helium-flow cryostat. Mechanical choppers synchronized to the 1-kHz repetition rate of the Ti:sapphire amplifiers are introduced in all beams to distin- guish the THz pulses in the pulse sequence. The chopper in beam A works with half the repetition rate (500 Hz), that in beam B works with one quarter (250 Hz), and that in beam C works with one eighth (125 Hz). In this way, chopper A transmits one pulse and blocks one, B transmits two and blocks two, and C transmits four and blocks four (lower part of Fig. 4). In this way, all possible combinations ofthe three pulses A, B, and C are measured. Varying the delays T, τ, and t, one obtains the single-pulse transients EA(T,τ,t),EB(T,t), andEC(t); the two-pulse transients EAB(T,τ,t),EBC(T,t), and EAC(τ, t); the three-pulse transient EABC(T,τ,t); and the nonlinear signal [Eq. (3)]. B. Data analysis Two-dimensional THz spectra are derived from the electric field ENL(τ,T,t) at a fixed waiting time Tby a 2D Fourier trans- form along τandt, giving the signal ENL(ντ,νt) as a function of the excitation frequency ντand the detection frequency νt. For illustrat- ing this concept and discussing key aspects of data analysis, we use a 2D-THz dataset recorded in the three-pulse setup of Fig. 4 with the narrow-gap semiconductor InSb.50,61 InSb is a direct-gap III–V semiconductor with a bandgap of Eg= 0.17 eV, corresponding to a frequency of ν=Eg/h= 41 THz.62 There is a TO two-phonon resonance at 10 THz, as observed in the higher-order Raman spectrum. The THz pulses with a center fre- quency of 21 THz are neither resonant to the bandgap nor to the two-phonon resonance. The THz peak field strength E≈50 kV/cm and the extraordinarily large interband transition dipole dcv=e⋅(4 nm) at the Γpoint ( e: elementary charge)62,63result in a Rabi frequency Ω Rabi=E d cv/̵h= 30 THz, i.e., comparable to the THz carrier frequency. As a result, the 2D-THz experiments are in the strongly nonperturbative regime of light–matter interaction, which is characterized by the simultaneous occurrence of nonlinear polarizations of different orders χ(n). In particular, the nonpertur- bative regime allows for multiple interactions of a single THz pulse with the InSb crystal. In Refs. 50 and 61, we reported the ultrafast dynamics of two-phonon coherences and signals related to multiple two-photon interband excitations of electron–hole pairs. In the fol- lowing, we concentrate on the dynamics of two-phonon coherences in InSb. In Fig. 5(a), the sum of electric field transients EA(τ,T,t) +EB(T,t) +EC(t) transmitted through the InSb sample is shown in a contour plot as a function of the coherence time τand real time tfor the waiting time T= 35 fs. The corresponding nonlin- ear signal ENL(τ,T,t) derived from Eq. (3) is plotted in panel (b). In accordance with causality, a nonvanishing ENL(τ,T,t) sets in only with or after the last pulse in the timing sequence. For better ori- entation, the center of pulse A is indicated by the orange dashed line, which intersects the horizontal τ= 0 line (black) at t=−T. In Fig. 5(c), the contour plot of the amplitude | ENL(ντ,νt)| is displayed forT= 35 fs. The circles of different sizes and colors indicate the positions of relevant signals in the 2D frequency space. As a function of detection frequencyνt, strong nonlinear signals occur in the spectral range of the driving pulses 15 THz <νt<25 THz (large circles) and at the two-phonon resonance νt= 10 THz (small circles). The former sig- nals have been discussed in detail in Ref. 61. In the following, we focus on the small circles at the two-phonon resonance νt= 10 THz. As a function of excitation frequency ντ, we observe significant non- linear signals for ντ= 0 (blue circle), ντ=−νt(black circle), and ντ= +νt(red circle). The signal at ( ντ,νt) = (0, 10) THz is a nonlinear signal of at least 11th order in the THz field, as has been discussed in detail in Ref. 50. For τ>0, i.e., for the pulse sequence [A( τ,T,t), B(T,t), C( t)], pulse A induces a complete two-photon absorption J. Chem. Phys. 154, 120901 (2021); doi: 10.1063/5.0046664 154, 120901-6 © Author(s) 2021The Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp FIG. 5 . Two-dimensional spectroscopy on InSb using three THz pulses as shown in Fig. 2(b). (a) Contour plot of the sum of electric field transients EA(τ,T,t) +EB(T,t) +EC(t) transmitted through the InSb sample as a function of the coherence time τand real time tfor the waiting time T= 35 fs. (b) Nonlinear signal ENL(τ,t) for T= 35 fs according to Eq. (3). The orange dashed line indicates the center of pulse A. The linear amplitude scales of the 2D scans range over (a) ±76.5 kV/cm and (b) ±30 kV/cm. (c) Contour plot of the amplitude | ENL(ντ,νt)|, which is the 2D Fourier transform of ENL(τ,t). The colored circles indicate the position of relevant signals in the 2D frequency space. (d) Selecting the peak at the two-TO-phonon resonance (small blue circle at ντ= 0,νt= 10 THz) by a 2D-Gaussian filter allows for back-transforming the E2-Phonon (τ,t) signal to the time domain. Red transients in (e) and (f) show ENL(τ= 500 fs, t) and E2-Phonon (τ= 500 fs, t), respectively. Gray circles: signals after artificially adding white noise to the measured ENL(τ,t) shown in panel (b). (g) ENL(τset,tset) as a function of the set-point values τsetandtsetgiven by the computer controlling the experiment. Part of the results shown here have been presented in Refs. 50 and 61. event, i.e., it creates an electron–hole pair by four interactions with its electric field. After the coherence time τ, pulse B creates a two-phonon coherence via an impulsive third-order Raman exci- tation, which evolves during the waiting time T. Pulse C generates a second electron–hole pair, again requiring four interactions and thereby transferring the two-phonon coherence to the final elec- tronic state. After the third pulse, the two-phonon coherence evolves along the real time tand emits radiation via its optical transition dipole. Selection of the peak at ( ντ,νt) = (0, 10) THz by a 2D-Gaussian filter and subsequent Fourier back-transform into the time-domain gives the signal field E2-phonon (τ,t), which is plotted in Fig. 5(d) as a function of τandt. Panel (f) shows a cut of this signal field for τ= 500 fs. The Fourier filtering by the 2D Gaussian results in an extraordinarily low noise level below <0.1 kV/cm. For compari- son, a cut of the total signal field ENL(τ,t) forτ= 500 fs and T = 35 fs is shown in Fig. 5(e). Here, a noise level of ≈2 kV/cm is estimated from the signal at t<0, which should vanish for causal- ity reasons. This analysis shows that 2D-Fourier filtering can con- siderably increase the signal-to-noise ratio of narrow signal peaks in the 2D-THz spectrum. To further illustrate the power of 2D- Fourier filtering, we added artificially white noise to the measured ENL(τ,t) shown in panel (b), which results in the gray dots in pan- els (e) and (f). Here, the total nonlinear signal ENL(τ= 500 fs, t) is barely detectable. In contrast, the 2D-filtered nonlinear two-phonon coherence E2-phonon (τ= 500 fs, t) has a noise level of ≈0.3 kV/cm [gray dots for t<0 in panel (f)] allowing for a clear experimental characterization of this nonlinear contribution.We now address the correction of long-term delay drifts between the three THz pulses A, B, and C derived from the out- put of the Ti:sapphire amplifiers in Fig. 4 and the read-out pulse for EO sampling, which is derived from the mode-locked oscillator. In Fig. 5(g), the nonlinear signal field ENL(τset,tset) is plotted as a func- tion of the uncorrected times τsetandtset, as given by the computer controlling the delay stages. The time drifts in this raw signal lead to slight distortions of the shape and the period of the signal wave- fronts. Such distortions can easily be corrected as the timing of the individual three pulses is measured simultaneously during a com- plete 2D scan, applying the pulse sequences shown in the lower part of Fig. 4. Using a 2D linear interpolation, one can reconstruct the nonlinear signal on a regular equidistant ( τ,t) 2D grid, resulting in the 2D signal ENL(τ,t) shown in Fig. 5(b). The separation of nonlinear signals with respect to their order in the THz field and the assignment to different pathways in Liou- ville space (cf. Sec. III A) require distinct non-overlapping signal peaks in the frequency-domain 2D spectrum. In the nonperturba- tive regime of both resonant and nonresonant light–matter cou- pling, this condition may not be fulfilled. Resonantly excited Rabi flops, e.g., of intersubband transitions,17include signal components up to a very high nonlinear order and display overlapping 2D sig- nal peaks at high driving fields. Electrons coherently driven in the conduction band of a semiconductor by a strong nonresonant THz field display a nonperturbative response and emit a multitude of harmonics of the fundamental driving frequency that are difficult to separate.64,65In general, an interplay of resonant and nonreso- nant contributions to the overall nonlinear response results in highly J. Chem. Phys. 154, 120901 (2021); doi: 10.1063/5.0046664 154, 120901-7 © Author(s) 2021The Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp complex 2D-THz spectra. Here, the variation of the phase and polar- ization of the THz pulses and, in anisotropic samples, measure- ments with different sample orientations can help decipher the 2D spectra. C. Related 2D methods There are a number of related 2D methods that involve at least one THz pulse in a sequence of ultrashort pulses and/or rely on the detection of coherent THz emission from an excited sample. In the present context, 2D Raman-THz,66,672D THz-Raman,68–70 and THz-infrared-visible spectroscopy71are relevant, all so far being applied in the third-order limit of light–matter interaction. Two-dimensional Raman-THz spectroscopy of liquids gives insight into the dynamics and couplings of intermolecular degrees of freedom. The method involves a femtosecond pulse in the visible or near-infrared for the Raman interaction, which is second order in the electric field, and a THz pulse, which induces a coherent polar- ization by a single interaction with the transition dipole of a low- frequency excitation. The THz emission of the sample represents the third-order nonlinear signal, which is measured as a function of real time t2in amplitude and phase by EO sampling. The second rele- vant time axis is the temporal separation t1of the Raman and the THz pulse. Two interaction sequences have mainly been applied in exper- iments on liquid water and aqueous salt solutions.66,67,72–75For pos- itive t1, the Raman pulse interacts with the sample first and gen- erates a Raman coherence, which is translated to a dipole-induced coherence by an interaction with the delayed THz pulse. The latter coherence gives rise to THz dipole emission, the nonlinear signal. For negative t1, the THz pulse comes first and generates a dipole coherence, which is switched to a Raman coherence by two interac- tions with the delayed Raman pulse. The THz signal field is emitted at a time t2after the THz pulse. Two-dimensional Raman-THz spectroscopy of water and aqueous salt solutions has revealed echo-like signals in the range of water–water hydrogen-bond modes. A detailed account on such results has been given in Ref. 67. The ultrafast structural fluctua- tions of aqueous systems set a time window, in which a particular local structure exists and rephasing of the vibrational excitations of different environments is possible. As a result, 2D Raman-THz spec- troscopy represents a direct probe of the complex intermolecular dynamics in liquids. A correct theoretical description of the nonlin- ear Raman-THz response requires polarizable water models, which can be benchmarked by comparison with 2D spectra.76–79 Two-dimensional THz-Raman spectroscopy implies two inter- actions with THz pulses and one interaction with the near-infrared Raman pulse.69,80In a three-level system, interaction with the first THz pulse generates a coherence on the 1–2 transition, which the second THz pulse at a delay t1transfers to the 2–3 transition. This coherent excitation is read out after a time interval t2by a near- infrared Raman pulse on the 3–1 transition. The coherences during the periods t1andt2allow for generating 2D spectra as a function oft1andt2or—after a 2D Fourier transform—as a function of the corresponding frequencies ν1andν2. Anharmonic couplings of low- frequency modes in liquid CHBr 3, CCl 4, and CBr 2Cl2have been studied by 2D THz-Raman spectroscopy.68,69,80In part, these results have been re-analyzed in Ref. 81.Third-order THz-infrared-visible spectroscopy71aims at elu- cidating couplings between low- and high-frequency vibrational modes and involves a THz pulse, a femtosecond mid-infrared pulse, and a sub-picosecond near-infrared or visible pulse. The THz and the mid-infrared pulse are both in resonance with vibrational tran- sitions so that each generates a vibrational coherence. The third interaction with the visible pulse transfers the coherent polariza- tion to the visible spectral range and, thus, generates a coherent emission at the frequencies ( νVIS+νIR±νTHz). There is a delay tbetween the THz pulse and the mid-infrared and visible pulses, the latter with a fixed temporal delay to each other. The emit- ted electric field is heterodyned with a local oscillator for sensitive phase-resolved detection. From this signal, 2D spectra are calcu- lated as a function of ν1, which is in the THz range and derived from a Fourier transform along t, and as a function of the mid- infrared frequency ν2, which is obtained by subtracting νVISfrom the frequency of the nonlinear signal. Details of data process- ing and analysis have been given in Ref. 82. THz-infrared-visible spectroscopy has been applied to phonons in solids and to liquid water. We note that there are a number of 2D experiments in which a coherent THz emission has been recorded after interaction of the sample with pulses at higher frequencies. A recent example is a study of TO phonon coherences in bulk GaAs.83 IV. SOFT-MODE EXCITATIONS IN POLAR MOLECULAR CRYSTALS Soft modes are lattice vibrations with a particularly strong cou- pling to the electronic system of a polar and/or ionic crystal. Exci- tation of a soft mode is connected with a pronounced relocation of electronic charge, thus directly affecting the electric polarization of the material. Vice versa, the coupling to electronic degrees of freedom enhances the vibrational oscillator strength significantly. Prototypical systems displaying soft modes are displacive ferro- electrics, e.g., with a perovskite lattice structure, and other ionic crystals. The interplay of soft-mode excitations and changes in electronic charge density has been revealed in femtosecond x-ray diffraction experiments following the transient charge densities in space and time.85,86Much less is known on the intrinsic nonlin- ear response of soft modes,87in particular in molecular materials. In the following, we discuss results from 2D-THz spectroscopy, which give detailed insight into soft mode nonlinearities in aspirin crystals.84 The crystal structure of aspirin consists of layers, in which pairs of aspirin molecules are arranged as cyclic hydrogen-bonded dimers [Fig. 6(a)]. The layers are connected via hydrogen bonds between the acetyl groups of neighboring molecules. Theoretical calculations of the electronic and vibrational structure of aspirin have shown that a number of vibrational modes in the THz range display a significant coupling to the electronic system.88Among them is the rotation of the methyl groups at a frequency of 1.1 THz, much lower than that of a free CH 3rotator. Figure 6(b) shows the linear THz absorption of a polycrystalline aspirin sample at a temperature of 80 K as measured with weak THz pulses. The methyl rotation gives rise to the weak absorption shoulder around 1 THz. A 2D-THz experiment was per- formed with two pulses A and B, the spectra of which are shown in J. Chem. Phys. 154, 120901 (2021); doi: 10.1063/5.0046664 154, 120901-8 © Author(s) 2021The Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp FIG. 6 . (a) Unit cell and crystal structure of aspirin. (b) Linear absorption of a polycrystalline aspirin pellet measured at a temperature of 80 K together with the amplitude spectra of THz pulses A (green) and B (orange). The weak feature around 1.1 THz represents the soft-mode absorption. (c) Contour plot of the nonlinear 2D spectrum |ENL(ντ,νt)|. The colored ovals indicate the positions ( ντ,νt) of relevant signals in the 2D frequency space. The red oval encircles the A pump–Bprobe signal at (ντ= 0,νt). (d) Contour plot of EApu-Bpr (τ,t) (amplitude scale: ±150 V/cm) from a back transform of the 2D Fourier-filtered spectrum. The green dashed line indicates the center of pump pulse A. (e) Spectrally resolved A pump–Bprobe signal (symbols) at a delay time of τ= 1.6 ps [indicated by the red horizontal line in panel (f)]. The dashed line gives the calculated field-induced response dominated by the soft mode, and the solid line gives the minor contributions of the modes at 1.8 and 2.3 THz. (f) Contour plot of the spectrally resolved A pump–Bprobe signal as a function of the probe frequency νtand pump–probe delay τ. The results shown here have been presented in Ref. 84. Fig. 6(b). The maximum electric field amplitudes of pulses A and B are 25 and 50 kV/cm, respectively. The 2D-THz spectrum shown in Fig. 6(c) exhibits different pump–probe and photon-echo signals, among which the A pump – Bprobe signal atντ= 0 is the strongest one (oval boundary). Applying a 2D Fourier filter (see Sec. III B) and transforming the signal back to the time domain gives the contour plot of Fig. 6(d), which shows an oscillation along twith a frequency below 2 THz, the maximum of the pulse spectra in Fig. 6(b). Moreover, there is a phase shift com- pared to the field of pulse B (not shown). To assess this transient in more detail, it was Fourier-transformed along the real time tto generate the spectrally resolved pump–probe signal as a function of νtand the delay time τbetween the two pulses [Figs. 6(e) and 6(f)]. The symbols in panel (e) give the pump–probe signal at a delay time τ= 1.6 ps, which shows a bleaching (increased transmission) for fre- quenciesνt<1.4 THz and an enhanced absorption (decreased trans- mission) at higher frequencies. This behavior persists up to delay times of several picoseconds [Fig. 6(f)]. Pulse A in the 2D experiment induces both a population change in the v = 0 and v = 1 states of the vibrations within the pump spec- trum and a nonlinear polarization. The population changes result in a third-order [ χ(3)] nonlinear response with a transmission increaseon the v = 0–1 transition and a transmission decrease on the anhar- monically redshifted v= 1–2 transition. The spectral envelope of the measured pump–probe signal in Figs. 6(e) and 6(f) with a blueshifted transmission decrease demonstrates that such population-induced signals play a minor role here. Instead, the signal is dominated by a pump-induced blueshift of the v= 0–1 transition of the soft mode and represents a response beyond the χ(3)regime. The observed behavior reflects the transient local-field response of the aspirin crystal. Due to the strong enhancement of the soft- mode transition dipole by coupling to the electronic system, there are strong dipole–dipole couplings between different aspirin sites, a concomitant electric polarization, and a strong Lorentz electric field. In thermal equilibrium, the electrostatic energy of the crystal is minimized, resulting in a pronounced redshift of the soft-mode frequency compared to its value without a Lorentz field. The tran- sition frequency of 1.1 THz observed in the linear THz absorption spectrum is a result of this redshift. Excitation by pulse A in the 2D-THz experiment reduces the contribution of the excited molecules to the overall polarization. This nonlinear saturation of electric polarization reduces the local electric field and shifts the soft mode back to a higher frequency, i.e., a blueshift arises. This nonperturbative mechanism governs the line shapes of the J. Chem. Phys. 154, 120901 (2021); doi: 10.1063/5.0046664 154, 120901-9 © Author(s) 2021The Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp pump–probe signals in Figs. 6(e) and 6(f) and strongly dominates over the population-induced signals. This interpretation is sup- ported by an analysis of the photon-echo signals in the 2D-THz spectrum [Fig. 6(c)], which has been presented in Ref. 84. The photon-echo signals display pronounced components at negative delay times τ, which originate from non-instantaneous contribu- tions of Lorentz fields with the time structure of a free-induction decay. The 2D-THz results reveal a nonlinear optical response of the aspirin soft mode in the nonperturbative regime, even for mod- erate THz driving fields of some 50 kV/cm. This fact reflects the sensitivity of collective electric properties, here the local field, to a comparably weak external stimulus. A similar behavior is expected for a large range of vibrational and/or phononic excitations with a pronounced coupling to the electronic system. The interplay of vibrational excitations and electronic properties is highly relevant for optically induced phase transitions of crystal structure and for transiently steering macroscopic electric properties of functional materials. V. FIELD-INDUCED IONIZATION AND ELECTRON TRANSPORT IN LIQUID WATER Intermolecular and/or collective molecular motions together with librations, i.e., hindered rotations, give rise to the complex absorption spectrum of liquid water in the frequency range from a few GHz to some 30 THz [Fig. 1(e)]. The broad unstructured envelopes and the substantial spectral overlap of different absorption bands make an analysis of linear spectra in terms of the underly- ing degrees of freedom highly demanding. This fact has generated a quest for nonlinear time-resolved spectroscopies, by which the dif- ferent excitations can be discerned in a better way and couplings between modes can be identified. A first proposal for nonlinear 2D-THz spectroscopy of liquid water was published in the 1990s.89Recent work in 2D Raman- THz and THz-infrared-visible spectroscopy has started to address the properties of intermolecular vibrations (see Sec. III C). There has been related work studying the nonlinear Kerr effect in a sim- ilar frequency range.90–95Until very recently, however, there have been no genuine 2D-THz results on liquid water, partly due to the insufficient experimental sensitivity and comparably small transi- tion dipole moment of intermolecular and collective modes. In the following, we discuss very recent results of a 2D-THz study of liquid water, which applies THz pulses with peak electric field amplitudes in a range from 100 kV/cm to 2 MV/cm to study the nonperturbative response of the liquid.96 The experimental geometry is sketched in Fig. 7(a). A phase- locked pair of pulses A and B separated by a delay τinteracts with a 50-μm thick water jet. As the water jet cannot be operated in vac- uum, it was placed in an atmosphere of dry nitrogen, together with the 2D-THz setup. The transmitted pulses are recorded by EO sam- pling (EOS). A transmitted pulse pair detected this way is shown in Fig. 8(a). The pulses cover a spectral range from 0.2 to 2 THz. The nonlinear signal E NL(τ,t) is given by Eq. (2). In Fig. 8(b), the field E AB(τ,t) is plotted as a function of τand t, while panel (d) shows the nonlinear signal field E NL(τ,t). In the 2D frequency- domain [Fig. 8(c)], the strong pump–probe signal generated with the FIG. 7 . (a) Schematic of the 2D-THz experiment with phase-locked THz pulses A and B separated by delay τ, the water jet (blue) in transmission geometry, and the path toward the electro-optic sampling (EOS) detector. (b) Fluctuating electric fields in liquid water as calculated from a molecular dynamics simulation for elec- trons at the position of the second-highest occupied orbital 3 a1(HOMO–1). (c) Schematic diagram of field-induced electron tunneling. (d) Schematic of electron propagation and localization. stronger pump pulse A and the weaker probe pulse B (A–B signal) is complemented by the much weaker B–A pump–probe signal. This observation points to a highly nonlinear dependence of ENLon the pump field, which has been studied in a series of measurements with pump pulses A of different peak field strengths EA. Data for different values of EAand a pump–probe delay of τ= 7 ps are summarized in Fig. 8(e). Here, the orange lines show the transmitted probe pulse B measured without excitation and the blue lines show the nonlinear signal field. The results of Fig. 8(e) demonstrate a threshold behavior of the nonlinear pump–probe signal. For field amplitudes of pulse A below the threshold Eth= 225 kV/cm, the nonlinear signal (blue lines) van- ishes for all delay times τ. Above the threshold, E NL(τ= 7 ps, t) shows a dispersive shape. It reaches considerable amplitudes up to 100 kV/cm. As seen from the total transmitted probe field (dashed lines), the nonlinear response leads to an increase in amplitude and a phase shift along the taxis, i.e., there is a nonlinear change in both the real and imaginary part of the refractive index of the excited water jet. A quantitative analysis discussed in detail in Ref. 96 gives a decrease in the real and imaginary parts by some 10%, with a flat spectral envelope distinctly different from the absorption spectrum shown in Fig. 1(e). The strong changes in the real and imaginary parts of the refractive index together with their spectra demonstrate that exci- tations of intermolecular vibrations giving rise to comparably weak absorption bands [molar extinction coefficient ε(1 THz) = 1.67 M−1 cm−1] play a minor role for the observed nonlinear response. The same holds for the nonlinear Kerr effect in the THz range, which is connected with changes in the refractive index several orders of J. Chem. Phys. 154, 120901 (2021); doi: 10.1063/5.0046664 154, 120901-10 © Author(s) 2021The Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp FIG. 8 . Time-resolved nonlinear THz response of liquid water. (a) THz pulses EA(τ= 6 ps, t) and EB(t) transmitted through a 50 μm-thick liquid water jet. (b) Contour plot of the 2D-THz scan EAB(τ,t) transmitted through the water sample as a function of real time tand delay time τbetween pulses A and B. (c) 2D-Fourier transform ENL(ντ,νt) ofENL(τ,t) as a function of detection frequency νtand excitation frequency ντ. The 2D spectrum shows a predominant A pump–Bprobe signal (blue ellipse) and a much weaker Bpump–Aprobe signal (gray arrow). (d) Contour plot of the A pump–Bprobe signal gained by Gaussian 2D-Fourier filtering and back transform to the 2D time domain. (e) Cuts of the time-resolved A pump–Bprobe signal (blue lines) at a fixed delay time τ= 7 ps for different peak amplitudes E Aof pulse A between 170 and 500 kV/cm. The probe field E B(t) is shown as an orange line. The sum EB+ENL(dashed lines) shows that electron generation leads to both a phase shift to earlier times tand an amplitude increase in probe pulse B. magnitude smaller than that found here.96–98Instead, the nonlin- ear response is governed by the generation of solvated electrons in the presence of the THz field, strongly modifying both the real and imaginary parts of the refractive index in a wider frequency range. Water molecules in the liquid phase have an electric dipole moment d≈2.9 D, which gives rise to a very strong electric field in the hydrogen bond network of H 2O. Due to thermally activated molecular motions at ambient temperature, such fields display fluctuations in a time range from ∼50 fs to several picosec- onds. An electric field trajectory from molecular dynamics simula- tions is shown in Fig. 7(b), where the electric field projected on the 3a1electron orbital of a water molecule is plotted as a function of time. Individual fluctuation peaks reach amplitudes on the order of 100–200 MV/cm. At such high external fields, the binding potential of elec- trons in the water molecule is strongly distorted and tunneling ionization becomes possible, as schematically shown in Fig. 7(c). As both the amplitude and direction of the fluctuating electric field change on an ultrafast time scale, recombination processes of a released electron with its H 2O+parent ion at the ionization site strongly dominate over successful ionization events, resultingin a very low concentration of released electrons on the order of some 10−7M. This scenario changes dramatically if one superim- poses a directed THz field on the fluctuating electric field. Now, the THz field can drive the released electron away from the ion- ization site and, thus, induce a persistent charge separation. After charge separation and transport, electrons equilibrate in their own solvation shell on a time scale of a few picoseconds [Fig. 7(d)].99 The presence of such solvated electrons changes the dielectric func- tion or refractive index in the THz range and, thus, causes the observed strong nonlinear response. This response is in the regime of nonperturbative light–matter interaction under nonresonant conditions. Independent evidence for the generation of solvated electrons comes from the observation of the characteristic absorption band of solvated electrons around 700 nm after interaction of the water sample with the strong THz field.96The strength of this absorption with a known molar extinction coefficient100allows for estimating the electron concentration c e. One derives c e= 5×10−6M for a THz peak field of 500 kV/cm and c e= 2×10−5M for 1.9 MV/cm. The results in Fig. 8(e) demonstrate a threshold electric field Eth= 225 kV/cm of pulse A for the generation of solvated elec- trons. This threshold originates from the requirement of a persistent J. Chem. Phys. 154, 120901 (2021); doi: 10.1063/5.0046664 154, 120901-11 © Author(s) 2021The Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp spatial separation of the electron from its parent ion by the trans- port process along the THz field. The electron needs to take up a ponderomotive (or kinetic) energy exceeding the ionization energy of a water molecule of ∼11 eV. This energy is provided by the local THz field in the liquid, which is, due to the polarity of the water molecules, roughly two times higher than the external field. Thus, an external field of some 250 kV/cm corresponds to a local field of 500 kV/cm at which the ponderomotive energy reaches a value of 11 eV, the water ionization energy. Below this threshold, recombi- nation of the electron with its parent ion prevails, while for even higher fields, the electron either travels larger distances in the liq- uid or generates secondary electrons by impact ionization of water molecules. The present 2D-THz study of water in the nonperturbative regime reveals a novel aspect of the strong fluctuating electric fields in polar liquids, namely, their potential for inducing tunneling ion- ization processes. In combination with strong THz fields, mobile electrons can be generated and their transport and localization pro- cesses can be steered by tailoring the THz fields in amplitude and time. Such an approach may allow for studying a new regime of charge transport in liquids. VI. CONCLUSIONS Two-dimensional THz spectroscopy has undergone a rapid development and represents an important addition to the portfo- lio of multidimensional spectroscopies. It allows for studying reso- nant excitations and field-driven processes up to arbitrary nonlin- ear orders, i.e., it clearly exceeds the third-order limit. In partic- ular, the response of a molecular ensemble under conditions of a nonperturbative light–matter interaction becomes accessible. There are a number of directions for future developments. First, a more systematic application of 2D-THz spectroscopy and related 2D methods to intermolecular modes in liquids and low-frequency phonons in crystalline and disordered solids appears promising. Sec- ond, the investigation of charge transport processes in molecular systems driven by nonresonant THz fields holds potential for elu- cidating frictional forces and the interplay between delocalized and localized states of electrons, protons, and ions. Third, field-induced changes in vibrational transition frequencies, i.e., a vibrational Stark effect induced by strong THz fields, will allow for mapping electric fields in liquids and biomolecules at their intrinsic THz fluctua- tion frequencies and, thus, to go beyond steady-state electrostatics. Work along those lines will benefit from the ongoing rapid devel- opment of THz sources and detection systems and the implemen- tation of more complex pulse sequences in multidimensional THz spectroscopy. ACKNOWLEDGMENTS We thank our colleagues who were involved in the research reviewed here, in particular Klaus Biermann, Benjamin P. Fingerhut, Giulia Folpini, Christos Flytzanis, Ahmed Ghalgaoui, and Carmine Somma. We acknowledge funding from the Deutsche Forschungs- gemeinschaft (Grant No. WO 558/14-1) and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 833365).DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1Broadband Dielectric Spectroscopy , edited by F. Kremer and A. Schönhals (Springer, Berlin, 2003). 2Infrared and Raman Spectroscopy: Methods and Applications , edited by B. Schrader (Verlag Chemie, Weinheim, 1995). 3C. W. Luo, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Phase-resolved nonlinear response of a two-dimensional electron gas under femtosecond intersubband excitation,” Phys. Rev. Lett. 92, 047402 (2004). 4P. Gaal, W. Kuehn, K. Reimann, M. Woerner, T. Elsaesser, and R. 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5.0035366.pdf

APL Mater. 9, 030904 (2021); https://doi.org/10.1063/5.0035366 9, 030904 © 2021 Author(s).Ferroelectric/multiferroic self-assembled vertically aligned nanocomposites: Current and future status Cite as: APL Mater. 9, 030904 (2021); https://doi.org/10.1063/5.0035366 Submitted: 28 October 2020 . Accepted: 28 January 2021 . Published Online: 10 March 2021 Oon Jew Lee , Shikhar Misra , Haiyan Wang , and J. L. MacManus-Driscoll COLLECTIONS Paper published as part of the special topic on 100 Years of Ferroelectricity - a Celebration ARTICLES YOU MAY BE INTERESTED IN A new era in ferroelectrics APL Materials 8, 120902 (2020); https://doi.org/10.1063/5.0034914 Strain-gradient effects in nanoscale-engineered magnetoelectric materials APL Materials 9, 020903 (2021); https://doi.org/10.1063/5.0037421 Ferroelectric field effect transistors: Progress and perspective APL Materials 9, 021102 (2021); https://doi.org/10.1063/5.0035515APL Materials PERSPECTIVE scitation.org/journal/apm Ferroelectric/multiferroic self-assembled vertically aligned nanocomposites: Current and future status Cite as: APL Mater. 9, 030904 (2021); doi: 10.1063/5.0035366 Submitted: 28 October 2020 •Accepted: 28 January 2021 • Published Online: 10 March 2021 Oon Jew Lee,1,a) Shikhar Misra,2 Haiyan Wang,2,3 and J. L. MacManus-Driscoll4 AFFILIATIONS 1Faculty of Science and Marine Environment, Universiti Malaysia Terengganu, 21030 Kuala Nerus, Malaysia 2School of Materials Engineering, Purdue University, West Lafayette, Indiana 47907, USA 3School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA 4Department of Materials Science and Metallurgy, University of Cambridge, 27 Charles Babbage Rd., Cambridge CB3 OFS, United Kingdom Note: This paper is part of the Special Topic on 100 Years of Ferroelectricity––A Celebration. a)Author to whom correspondence should be addressed: oonjew@umt.edu.my ABSTRACT Even a century after the discovery of ferroelectricity, the quest for the novel multifunctionalities in ferroelectric and multiferroics continues unbounded. Vertically aligned nanocomposites (VANs) offer a new avenue toward improved (multi)functionality, both for fundamental understanding and for real-world applications. In these systems, vertical strain effects, interfaces, and defects serve as key driving forces to tune properties in very positive ways. In this Perspective, the twists and turns in the development of ferroelectric/multiferroics oxide–oxide and unconventional metal–oxide VANs are highlighted. In addition, the future trends and challenges to improve classic ferroelectric/multiferroic VANs are presented, with emphasis on the enhanced functionalities offered by existing VANs, as well as those in emerging systems. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0035366 I. INTRODUCTION Complex metal oxides offer a spectrum of tunable function- alities, including ferroelectricity, ferromagnetism, magnetoelectric (ME), superconductivity, ionic conductivity, dielectricity, and plas- monic resonance.1–7Incorporating complex metal oxides into ver- tically aligned nanocomposite (VAN) thin films provides another handle to tune properties in very positive ways. Improved func- tionalities of these complex metal oxides are realized and coupling between different functions in the composite films induces novel multi-functionalities.8 Of all the aforementioned functionalities, the phenomenon of ferroelectricity is of high topical interest. Ferroelectric materials have constantly attracted scientific interest owing to their switch- able spontaneous electric polarization by an electric field appli- cation.9Apart from spontaneous polarization switching, they also exhibit piezoelectric, pyroelectric, dielectric, and electro-optic prop- erties,10and this is only the start. When ferroelectric research isfurther extended to include magnetic coupling, hence producing multiferroicity, this further degree of freedom is highly promis- ing for yet more technological applications. Both ferroelectrics and multiferroics have been widely investigated for countless appli- cations such as supercapacitors, efficient energy harvesting and storage, ultralow power logic-memory devices, high power elec- tronic transducers, solid state electrocaloric cooling/heating devices, microwave electronics, and non-volatile memories.5,11–13As part of the drive toward device miniaturization, the unique VAN architecture has improved the existing functionalities in a trans- formative way, beyond the state-of-the-art ferroelectrics such as Pb1−xZrxTiO 3.14 This article begins with a short overview of VANs (their pos- sible structures and proven salient properties) and is followed by the early works of ferroelectric/multiferroic VANs. Then, recent advances of oxide–oxide and unconventional metal–oxide ferroelec- tric/multiferroic VANs are highlighted. Finally, their future perspec- tive and challenges are outlined. APL Mater. 9, 030904 (2021); doi: 10.1063/5.0035366 9, 030904-1 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm II. THE ANATOMY OF VANS VANs are self-assembled nanostructures in which two immis- cible phases grow epitaxially on the substrate. Fabrication of VANs is made easy via physical thin film deposition methods. Both tem- plated growth and spontaneous ordered growth can occur.6,8The former involves the utilization of templating methods such as anodic aluminum oxide (AAO) templates,15block copolymers,16 lithography,17and focus ion beams.18,19Spontaneous growth offers synergistic advantages to control deposition precisely, minimize defects, and overcome the complication of templated growth that implies extra step of template removal.20Hence, the spontaneous growth is rather facile and low-cost and hence favorable for device integration. With proper selection, the host matrix nucleates and grows epi- taxially on the crystallographically matched substrate. The host and epiphyte phases are vertically coupled through an epitaxial relation- ship or through domain matching epitaxy (DME).7Figure 1 shows that three dimensional (3D) VANs are comprised of diverse pat- terns with tunable shape, size, and distribution. Their diversity is reflected in various structures such as circular nanocolumn matrix, facet nanocolumn matrix, nanomazes, and nanocheckerboard. In the most typical configuration, one phase (the one closely struc- turally matched with the substrate) forms the matrix phase, and the other forms pseudo-1D nanopillars or nanodomains (such as circular, rectangular, pyramidal, hexagonal, octagonal, and triangu- lar shapes1,6,20) embedded in the matrix [Figs. 1(a) and 1(c)]. This structure is referred to as the 1–3 configuration and is the most com- monly used one. The growth origin of 1–3 configurations is driven by the minimization of the total free energy that depends on ther- modynamic epitaxial stability, growth kinetics, and phase ratio.1,8,21 According to phase field simulations,1competing interfacial inter- actions and elastic moduli that arise between matrix and embedded phases, as well as the interaction of these phases with the substrate,are vital. It is found that circular pillars with small diameters mini- mize the surface energy. On the other hand, the elastic strain energy is dominant in the case of faceted domain nanopillars.1The phase has a higher interfacial energy (lower wettability) on the substrate such that it nucleates and grows into nanopillars. The other phase, with a lower interfacial energy, undergoes layer-by-layer growth and becomes the planar matrix.21Beyond a critical thickness of ∼20 nm (above which the role of the substrate is strongly diminished), the stiffer phase in the film dominates the strain of softer phase.22 Depending on the epitaxial stability consideration, the role of each phase as the matrix and pillar can be reversed when grown on differently oriented substrates. For instance, the CoFe 2O4(CFO) phase in BFO–CoFe 2O4(BFO–CFO) VANs evolves from faceted pyramidal nanodomains and subsequently formed into nanomaze structures on (001), (111), and (110) SrTiO 3(STO) substrates.20In addition, growth kinetics and phase ratios are the key factors in tuning the feature size of nanodomains, phase segregation, and sto- ichiometry of VANs.1,23As a result, smaller nanodomains merge into belt-shaped or T-shaped nanostructures, as observed in the nanomaze morphology [Fig. 1(d)].21 On the contrary, nanocheckerboard VANs [Fig. 1(b)] originate from highly ordered (pseudo-) spinodal decomposition growth.6,8 Again, the well-matched crystallography and sufficient growth kinetics between the decomposing phases and substrate are prereq- uisites for the formation of nanocheckerboards.20 Depending on the overall film thickness, VANs display higher interface-to-volume ratio by up to two orders of magnitude22as compared to the conventional multilayer counterparts of the 2–2 configuration [Fig. 1(e)]. Unlike the lateral thin films, the strain exerted by a substrate has inherent limitation owing to their crit- ical film thickness of a few tens of nm. The 1D nanopillars in the VANs, on the other hand, extend perpendicularly throughout the entire thickness of the film, even up to the micrometer thickness.22,24 Furthermore, for thinner films ( <30 nm), the in-plane (IP) and FIG. 1. 3D schematic illustration of vertical aligned nanocomposites (1–3 type) that comprise two immiscible constituents grown simultaneously on a substrate with (a) circular nanocolumn matrix, (b) nanocheckerboard (c) facet nanocolumn matrix, (d) nanomaze, and (e) conventional multilayer thin films (2–2 type). APL Mater. 9, 030904 (2021); doi: 10.1063/5.0035366 9, 030904-2 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm out-of-plane (OP) strain in the VAN can be tuned independently via the separate lateral substrate and vertical pillar strain control mechanisms.4The high interface-to-volume ratio in addition to uni- form strain control imparted by the fine dimensions of the struc- ture22enables unprecedented possibilities to improve the physical properties of the VANs via strain-, interface-, and defect-control. Vertical strain engineering, stemming from lattice mismatch and elastic coupling, is critical to promote competing interactions among spin, charge, orbital, and lattice degrees of freedom, polarization domain, and magnetic moment.1,10Subsequently, these competing interactions are intimately coupled to ferroelectricity and multi- ferroic properties. Furthermore, the vertical heterointerfaces and defects (notably oxygen vacancies) in VANs also adversely affected the polarization, magnetoelectric coupling strength, and dielectric relaxation.1,25Hence, a delicate balance between the VAN morphol- ogy and the fundamental couplings is vital to expand the realm of potential multifunctionalities in ferroelectrics and multiferroics, as discussed in Sec. III. III. OVERVIEW OF FERROELECTRIC/MULTIFERROICS AND PROPERTY ENHANCEMENTS OFFERED BY VANS Extensive research has been focused on oxide–oxide based VAN thin films, with functionalities including ferromagnetic, ferro- electric, multiferroic, dielectric, semiconducting, superconducting, and nonlinear optical effects.11,24,26–35An overview of the property enhancements offered by VAN structures for tuning ferroelectrics whether used alone or coupled with ferromagnets in artificial mag- netoelectrics is shown in Fig. 2. The property enhancements offered by ferroelectric VANs are shown, as well as the properties achieved in artificial multiferroic VANs, which incorporate ferroelectrics. In summary, by exploiting the enhanced property advantages offered by VAN architectures, ferroelectric VANs with enhanced Curietemperatures ( Tcs),24,36lower leakage,37,38fine dimensions,6,8and higher polarization12,39offer the prospect of higher density non- volatile random-access memory. The increased Tcs, higher tem- perature coefficient capacitance (TCC), and increased piezoelectric coefficients12offer the prospect of higher operation temperature Pb-free piezoelectrics for energy harvesting, actuation, and energy storage. In addition, the increased dielectric tunability without increasing the loss at the same time [or high commutation qual- ity factor (CQF)]13,36offers the prospect of improved tunable microwave filters. Both enhanced optical anisotropy responses in terms of dielectric permittivity and polarization offer the prospect of ultrasensitive and broadband optical sensors, signal processors, and lasers.13,40,41 Multiferroics underpin the emergence of more than one pri- mary ferroic order, including ferromagnetism, ferroelectric, fer- roelastic, and ferrotoroidic in a phase but also not limited to antiferromagnetism and magnetoelectricity.5The coexistence of fer- romagnetic and ferroelectric orders is pivotal to induce electric polarization by an applied magnetic field or vice versa to medi- ate the magnetization through the electric field. However, these orders often exclude each other owing to their intrinsic electronic structure. Ferroelectricity relies on transition metals with empty d orbitals, whereas the magnetism involves partially filled dorbitals. Single-phase multiferroic materials such as BiFeO 3, TbMnO 3, and Cr2O3are scarce in nature and do not offer large magnetoelectric effects at room temperature.1,5On the other hand, artificial mul- tiferroics with moderate performance at room temperature can be made by combining ferroelectric and ferromagnetic (magnetostric- tive) phases in a VAN architecture. Artificial multiferroic VANs offer exemplary characteristics with large magnetoelectric coeffi- cients42and spin-driven room temperature ferroelectricity,43which are of great interests for non-volatile memory and energy harvesters with ultralow power ( ∼100 mV).5The electric field induced ferro- magnetic resonance characteristics44are also beneficial for tunable FIG. 2. Schematic representation of the main interactions (gray) that can take place in ferroelectric-based VAN systems leading to enhanced ferroelectric properties (red), improved dielectric properties (yellow), novel artificial multiferroics and effects when combined with magnetic systems (blue), and improved properties general to all systems (purple). APL Mater. 9, 030904 (2021); doi: 10.1063/5.0035366 9, 030904-3 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm radiofrequency (RF)/microwave magnetic devices and resonators that rely on robust spontaneous polarization switching and tunable ferromagnetic resonance. IV. EARLY WORKS ON CLASSIC FERROELECTRIC AND MULTIFERROIC VANS There has been a considerable volume of work on ferroelec- tric and multiferroic VAN over the past 15 years. The earliest work on ferroelectric VAN focused on the classic ferroelectric, BaTiO 3–Sm 2O3(BTO–SmO). The stiffer SmO nanopillars exerted a>2% vertical tensile strain on the BTO phase, forcing the tetrag- onal structure of BTO to be maintained at a high temperature (up to 800○C).24This also opened a path for the BTO–SmO VANs to work as piezoactuators, as an increased longitudinal piezoelec- tric coefficient ( d33) of 45 pm V−1–50 pm V−1was achieved up to 250○C.12Additionally, owing to the unusual strain states induced in the BTO, a high transverse piezoelectric coefficient ( d31) was achieved,12outperforming the state-of-the-art Pb 1–xZrxTiO 3films at>200 pm V−1(as tabulated in Table I). The leakage current of the BTO–SmO VANs was reduced by a factor of 15 because of the formation of highly crystalline BTO in the composite.24Liet al.suggested that both vertical strain and interfaces have active roles to trap oxygen vacancies (defects), which suppress the carrier mobility and trigger a trap-controlled space-charge-limited current (SCLC) mechanism in the BTO–SmO films.25 Similar enhancements in Tcfrom 130○C to 616○C and d33 of∼80 pmV−1were observed in more recent studies on (111)- oriented BTO–SmO VANs that were grown on 0.5 wt. % Nb- doped (111) SrTiO 3substrates.45The origin of enhanced ferro- electric and piezoelectric properties lies in the presence of unusual strain states also known as the auxetic-like effect. Such an auxetic- like effect is achieved because there is IP and OP expansion of the BTO matrix once the stiff SmO nanopillars shrink upon cooling.8Ba0.6Sr0.4TiO 3–Sm 2O3(BSTO–SmO) VANs also agree with the earlier BTO–SmO work that vertical strain induces strong enhancements of Tcand dielectric tunability, which scale inversely with the dielectric loss and leakage current. BaTiO 3–CeO 2 (BTO–CeO) VANs also show the similar behavior whereby both Tcand remnant polarization ( Pr) are enhanced. Khatkhatay et al. reported a slightly enhanced Tcof 175○C and saturation polariza- tion ( Ps) of 29 μC cm−2in BTO–CeO 2, demonstrating a precise vertical strain tuning (tensile strain of 0.6%) by the CeO 2nanopillar phase.46 TABLE I. Classic oxide–oxide ferroelectric and multiferroic VANs that have been developed. VAN System Enhanced functionalities References Classic ferroelectric oxide–oxide Circular nanocolumn- matrixBaTiO 3–Sm 2O3/Nb–STO (100)●Increase Tcup to 800○C 12,24●Increase d33of 45–50 pmV−1up to 250○C ●d31exceeds PZT films at 200 pm V−1 ●Reduce leakage current to 2.2 ×10−7A cm−2 Ba0.6Sr0.4TiO 3–Sm 2O3/SRO/STO (100) ●High tunability (75%) scales inversely with dielectric loss (0.01)13 BaTiO 3–CeO 2/SRO/STO (100) ●Increase Tcup to 175○C and 150○C46BaTiO 3–CeO 2/SrRuO 3/sto???n/Si ●Increase in remnant polarization up to 29 μC cm−2 BTO-NiO/Nb-STO (100) ●Enhancement of ∼240% in the dielectric constant 95 Classic multiferroic oxide–oxide Circular/Rectangular nanocolumn- MatrixBaTiO 3–CoFe 2O4/SRO/STO●Achieved Psof 23 μC cm−2 47,57 ●Obtained d33of 50 pm V−1 ●Direct coupling between BTO and CFO NanocheckerboardBiFeO 3–Sm 2O3/Nb–STO (100)●Reduced leakage current density of three orders of magnitude lower 37,38 ●Reduced the dielectric loss to 0.016 BiFeO 3–Eu 2O3/SRO/STO (100)●Reduced leakage current <1.25 mAcm-249BiFeO 3–BaZrO 3/SRO/STO (100) Rectangular nanocolumn matrixBiFeO 3–CoFe 2O4/SRO/STO (100)●Sharp 180○PFM domain switching in the BFO matrix with multiple wring–rewriting poling35,42●ME coefficient of 20 mV cm−1Oe−1measured by a magnetic cantilever method APL Mater. 9, 030904 (2021); doi: 10.1063/5.0035366 9, 030904-4 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm Looking now into the early works of the classic multiferroic VANs, Zheng et al. pioneered the fabrication of BaTiO 3–CoFe 2O4 (BTO–CFO) VANs and demonstrated the cross-control of ferro- electrics and magnetism from BTO perovskite and CFO spinel, respectively.47The elastic interaction of BTO and CFO at the vertical epitaxial interfaces facilitated a strong coupling between the elec- trical polarization and magnetic order, leading to magnetoelectric properties. Motivated by the promising magnetoelectric coupling of this classic multiferroic BTO–CFO system, a vast number of clas- sic multiferroic VANs such as PbTiO 3–CoFe 2O4, BiFeO 3–CoFe 2O4, BiFeO 3–Sm 2O3, BiFeO 3–Eu 2O3, and BiFeO 3–BaZrO 3have been widely studied ever since.37,38,48–50The remarkable functionalities of some of the multiferroic VAN systems are tabulated in Table I. V. HIGHLIGHTS OF RECENT WORK ON OXIDE–OXIDE FERROELECTRIC VANS A. Strain-, interface-, and defect-driven ferroelectric related functionalities Dynamic polarization in the classic ferroelectric VAN has not been widely studied. However, dynamic polarization (polarization retention) has been studied in BSTO–SmO VAN films.39The ulti- mate aim of high polarization retention (either thermal or tempo- ral) is to achieve infinite switching without degradation. However,polarization retention is limited by depolarization, inhomogeneous field distributions, defects and thermal depolarization, or back- switching of the poled state.51,52Figure 3(a) shows enhanced thermal polarization retention, R(T), of BSTO–SmO VAN films as the SmO fraction, x, is increased. Such a phenomenon aligns with the man- ifestation of a high density of vertical interfaces (TbIn.−2) between the BSTO nanopillars and SmO matrix.39These vertical interfaces not only act as surfaces for ferroelectric domains to nucleate but also act as domain pinning centers. This is distinctive because the domain motion can be confined and, simultaneously, the polariza- tions are retained for multiple switching cycles at elevated tempera- tures. Apart from dynamic polarization, a high Prof 12.1 μC cm−2 (3×greater than that of the standard BSTO films) and d33of 88.1 pm V−1–81.4 pm V−1up to 120○C were achieved in the BSTO–SmO VANs. In terms of recent Tcenhancements and strain control effects, this has been effectively demonstrated in SrTiO 3–Sm 2O3 (STO–SmO) with Tcraised to>300○C, the highest ever reported forTcin STO [Fig. 3(b)].36In addition, in this system, a trade-off between dielectric tunability and loss is mitigated. A high dielectric tunability (49%) and low tangent loss ( ≤0.01) were achieved. This is desirable for impedance in high-power phase shifters and accelera- tors. The highest CQF reported ( >2800), this factor integrating tun- ability and loss as a figure-of-merit, was reported in these STO–SmO VANs. FIG. 3. BSTO–SmO, STO–SmO and STO–MgO VANs and their ferroelectric and dielectric properties. (a) Temperature dependence of polarization retentions of the BSTO–SmO VANs with different SmO fractions, x. Reproduced with permission from Lee et al. , Adv. Mater. Interfaces 4(15), 1700336 (2017). Copyright 2017 WILEY-VCH.39 (b) Temperature dependence of relative permittivity of the STO–SmO VAN at different frequencies, demonstrating an increment of Tcup to 305○C. Reproduced with per- mission from Sangle et al. , Nanoscale 10(7), 3460 (2018). Copyright 2018 The Royal Society of Chemistry.36(c) PFM image, (d) polarization switching mapping, and (e) piezo-response at locations A, B, and C of the STO–MgO VAN with scale bars of 200 nm. The existence of ferroelectricity in the STO–MgO VAN is consistent with (f) out-of-plane strain, εzz, distribution and polarization Pzobtained from phase field modeling. Reproduced with permission from Enriquez et al. , Nanoscale 12(35), 18193 (2020). Copyright 2020 The Royal Society of Chemistry.53 APL Mater. 9, 030904 (2021); doi: 10.1063/5.0035366 9, 030904-5 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm Furthermore, to the STO–SmO VANs mentioned above, more recent work on SrTiO 3–MgO (STO–MgO) VANs53has shown room temperature ferroelectricity. Phase field modeling was performed to predict strain induced ferroelectricity in STO–MgO theoreti- cally, followed by local in situ characterizations via piezoresponse force microscopy (PFM). Figures 3(c)–3(f) show the signature fer- roelectricity of STO–MgO, also cross-checked by piezoresponse force microscopy (PFM), which shows the spatial resolution and functional ferroelectric response. Inhomogeneous polarization of the STO–MgO films stemming from a strain induced ferroelectric STO fraction and non-ferroelectric MgO nanopillars was resolved via polarization switching mapping, local hysteresis piezo-response loops at A, B, and C sites, and phase field modeling. Optical second harmonic generation (SHG) also provided the perfect complement for probing ferroelectricity in STO–MgO VANs due to its sensitivity toward C4v symmetry. B. Ferroelectric–ferromagnetic couplings in multiferroic VANs Recently, the popular multiferroic VAN system, BTO–CFO, has been revisited by Chen et al.54Figures 4(a) and 4(b) show the PFM amplitude mapping of non-ferroelectric CFO nanopillars (blue dots) embedded in a polarized BTO matrix (red area) and the discrepancy between the butterfly loops under zero and 2 kOe in-plane applied magnetic fields, indicating the presence of direct ME coupling or magnetic field control of ferroelectricity. The result- ing direct ME coefficient of 390 mVcm−1Oe−1at 8 kOe is higher than others (10 mVcm−1Oe−1–100 mVcm−1Oe−1). The work cor- roborates and expands on what Zavaliche et al. reported earlier, i.e., electric field induced magnetization reversal via magnetic force microscopy (MFM) in the BFO–CFO VANs.55We now turn to a new room temperature multiferroic VAN comprised of ordered ferrimagnetic spinel α-LiFe 5O8(LFO) nanopillars in a matrix of ferroelectric perovskite BiFeO 3(BFO).56 As shown in Figs. 4(c)–4(e), the coexistence of ferroelectric polar- ization and magnetic order can be observed and compared via PFM phase poling, local PFM, and MFM images acquired at the same area. Both characterizations were carried out at room temperature. As we focus on the LFO nanopillars, their piezophase response is suppressed and, conversely, the magnetic response is enhanced. Apart from PFM and MFM, the magnetic switching induced by electric poling can be identified by x-ray magnetic circular dichroism (XMCD) as evidenced in the BFO–CFO and BTO–CFO systems.57,58 A key challenge to multiferroic VANs is the relatively high elec- trical conductivity in these systems arising from some conduction of the ferromagnetic (or ferrimagnetic) nanopillars, as well as the leaky ferroelectric matrix. Leakage can also arise from the vertical inter- faces in the VAN films. Reduced leakage current density has been reported in both multiferroic VANs (BFO–CFO and BFO–EuO) and non-multiferroic VANs (BFO–SmO and BFO–BZO, as tab- ulated in Table I). The leakage current density of BFO–SmO VANs was reduced by three orders of magnitude in comparison with as-deposited pure BFO films.37,38Oxygen vacancies at the vertical interface, serving as donors, were believed to contribute mainly to the leakage reduction. Furthermore, tuning of vertical interface conduction has been described in the BFO–CFO VANs. Hsieh et al. reported a pathway to control local conduction due to oxygen vacancies at the BFO/CFO interface by a combina- tion of conductive atomic force microscopy (c-AFM), PFM, and Kelvin force microscopy (KFM).50The local conduction was con- trollable by applying negative and positive tip biases on the as-grown BFO–CFO VANs (down polarization) once the BFO matrix was FIG. 4. Ferroelectricity–ferromagnetism coupling in the BTO–CFO and BFO–LFO VANs grown STO (100) sub- strates. (a) PFM amplitude mapping of the BTO–CFO VAN at a 2 kOe magnetic field and (b) amplitude–voltage butter- fly loop of BTO–CFO VAN before and after the application of an external magnetic field of 2 kOe. Reproduced with per- mission from Chen et al. , Adv. Sci. 6(19), 1901000 (2019). Copyright 2019 WILEY-VCH.54(c) PFM phase image taken after poling the BFO–LFO film using ±6 V. (d) Local PFM and (e) MFM acquired at the same area indicate the coexis- tence of ferroelectric and ferromagnetic whereby the piezo phase response of LFO nanopillars is suppressed and their magnetic response is enhanced, respectively. Reproduced with permission from Sharma et al. , Adv. Funct. Mater. 30(3), 1906849 (2020). Copyright 2020 WILEY-VCH.56 APL Mater. 9, 030904 (2021); doi: 10.1063/5.0035366 9, 030904-6 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm pooled. Figure 5(a) illustrates that the positively charged oxygen vacancies (yellow dots) are attracted by the negative tip bias and accumulated on the top surface, thus suppressing the vertical con- duction. After switching the polarization to upward [Fig. 5(b)], the conductive vertical interface diminishes as evidenced by the KFM image with a negative surface potential. In parallel to the control of vertical conduction in BFO–CFO VANs, intriguing topological domain switching dynamics of BFO nanodomains has attracted broad attention.59,60Figure 5(c) shows the reconstructed polarization vector mapping of a BFO nan- odomain from the PFM piezoresponse signal components using a MATLAB program. The BFO nanodomain exhibits a polarization configuration, which consists of stripe domains and convergent and divergent domain structures. Tian et al. found that the convergent and divergent domains were likely driven by the inhomogeneous strains exerted by vertical interface and substrate.60As a result, these inhomogeneous strains generate flexoelectric rotations and influence the polarization direction. Strong leakage reduction in multiferroic VANs was also reported in the system Na 0.5Bi0.5TiO 3–CoFe 2O4(NBT–CFO). The leakage was reduced by using the less leaky NBT ferroelectric and by creating a current-blocking p–n junction between the NBT and one of the electrodes. This system has enabled a strong converse (elec- tric field control of magnetism) ME effect of 1.25 ×10−9sm−1to be achieved at room temperature.61Research on new multiferroic VANs has been expanded to combine ferroelectric BaTiO 3with multiferroic YMnO 3(YMnO 3 is antiferromagnetic and concurrently exhibits geometric-driven ferroelectric) and ferrimagnetic yttrium iron garnet (YIG) Y3Fe5O12.62,63The combinations prove the possibility to diversify the ME coupling for realizing RF/microwave devices with low noise and energy consumption. In addition, the newly developed BaTiO 3–La 0.7Sr0.3MnO 3(BTO–LSMO) VANs exhibit tunable anisotropic optical bandgaps, unveiling their potential toward an entirely new direction in non-linear optics.64 C. Spin-driven room temperature ferroelectric in SmMnO 3–(Bi,Sm) 2O3 Type-II multiferroic materials have exquisitely coupled mag- netic and ferroelectric orders via breaking of the inversion sym- metry. However, no spontaneous room-temperature ferroic prop- erties have been observed in the type-II multiferroics, orthorhombic RMnO 3, so far.65,66 Choi and co-workers incorporated stiff (Bi,Sm) 2O3nanopil- lars in a SmMnO 3matrix, forming nanomaze VANs with clear phase separation in the plan-view scanning transmission elec- tron microscopy (STEM) high-angle annular dark-field (HAADF) images [Figs. 6(a) and 6(b)].43These (SmMnO 3)0.5((Bi,Sm) 2O3)0.5 (SMO–BSO) VANs demonstrated spin-driven ferroelectricity at FIG. 5. Conduction modulation and domain switching in the BFO–CFO VANs on (100) SrTiO 3and (110) LaAlO 3 substrates, respectively. (a) Schematic diagram showing that the local conduction due to oxygen vacancies at the BFO/CFO interface is controllable by applying negative and positive tip bias. The oxygen vacancies (yellow dots) are attracted by the negative tip bias and accumulated on the top surface of BFO. P represents the direction of ferroelec- tric polarization. Reproduced with permission from Hsieh et al. , ACS Nano 7(10), 8627 (2013). Copyright 2013 Amer- ican Chemical Society.50As evidenced by (b) PFM, CAFM, and KFM images: the out-of-plane phase orientation of BFO matrix is initially downward and possesses a conductive interface. Once the BFO is poled downward, the conduc- tive vertical interphase as shown in the CAFM diminishes, whereas the KFM image depicts negative surface poten- tial. (c) The domain switching of BFO is reconstructed by phase angle mapping, indicating (i) stripe domains, (ii) center-convergent domains, and (iii) center-divergent domains. Reproduced with permission from Tian et al. , Adv. Electron. Mater. 5(7), 1900012 (2019). Copyright 2019 WILEY-VCH.60 APL Mater. 9, 030904 (2021); doi: 10.1063/5.0035366 9, 030904-7 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm FIG. 6. SMO–BFO nanomaze VAN and room temperature and its intrinsic ferroelectricity. [(a) and (b)] Plan-view STEM-HAADF image of SMO–BFO VAN nanomaze structure over a wide area. A regular pattern of laterally connected BSO pillars in an SMO matrix is observed. (c) Amplitude (circle and blue) and phase (square and red) of the PFM signal as a function of bias voltage. The inset AFM image shows the surface morphology of the film. (d) PFM phase contrast after multiple ±5 V writing–rewritings at room temperature. The phase contrast remains after 24 h. (e) PUND measurements with 9 V (E =500 kV cm−1) for capturing both switching (∗) and non-switching (ˆ) polarizations. The net switching polarization (2 PR) is∼3.9μC cm−2. (f) Room temperature optical SHG signals, which indicate p-wave (red) and s-wave (blue) of the second harmonic electric field with an incidence angle of 45○. Reproduced with permission from Choi et al. , Nat. Commun. 11(1), 2207 (2020). Copyright 2020 Springer Nature.43 room temperature with a large d33of 6.7 pmV−1, 24 h remained PFM polings at ±5 V, and net switching remnant polarization (2Pr) of 3.9 μC cm−2[Figs. 6(c)–6(e)]. A consistent result was observed from the SHG polar plots, indicating a breaking of cen- trosymmetry of SMO [Fig. 6(f)]. In addition, a clear ferromagnetic behavior was observed together with an enhanced ferromagnetic transition temperature ( Tc) of 90 K and a saturation moment of 1.02 μBMn−1(at 10 K). In contrast, bulk SmMnO 3exhibits a ground state of paraelectricity with ferroelectric Tc<40 K and A-typeantiferromagnetism with a Néel temperature ( TN) of 60 K. The rea- son behind the Tcenhancement was the creation of large in-plane compression and out-of-plane tensile strains ( −3.6% and+4.9%, respectively) exerted on the SMO, leading to the change in the Mn–O bond angle and length. As proven by density functional the- ory (DFT) calculations, the reduction of the Mn–O bond length and the increase in the Mn–O–Mn bond angle are in good agreement with the proposed mechanism of Mn–Mn exchange interactions. These enhanced interactions explain the room APL Mater. 9, 030904 (2021); doi: 10.1063/5.0035366 9, 030904-8 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm temperature ferroelectric polarization and enhanced ferromag- netism in the SMO–BSO VANs. VI. FUTURE OUTLOOK Self-assembled oxide-based VAN thin films offer an interest- ing way toward achieving novel physical properties and functionality coupling ranging from ferromagnetic, ferroelectric, superconduct- ing, and nonlinear optical effects as discussed above. These prop- erties arise via coupling between charge, orbital, spin, and lattice degrees of freedom. However, due to the limited range of avail- ability of structures in terms of crystallinity and morphology, a greater design flexibility and a structural complexity along with ver- satile growth techniques are needed for developing next generation integrated photonic and electronic devices. This can be achieved by several approaches such as designing VANs with unconven- tional phases, substituting one of the oxide phases with a metallic phase (oxide-metal nanocomposite), and fabricating self-assembled supercell-based nanocomposites. Each of these approaches is dis- cussed in more detail in Secs. VI A–C. A. Unconventional VANs VANs also provide a platform to couple unconventional phases with vertical strain present along their interfaces to enhancetheir properties. Figure 7 shows a VAN structure composed of a new tetragonal tungsten bronze (TTB) phase, which is one of the most promising classes of ferroelectric materials.67The VAN structure shows the presence of a new TTB phase of LiNb 6Ba5Ti4O30 (LNBTO), with a secondary phase, LiTiO 2(LTO), that is present as vertical nanopillars embedded in the LNBTO matrix. Figure 7(a) presents the cross section STEM image showing the presence of a typical VAN structure. The main phase is determined to be LNBTO, while the darker contrast at the phase boundaries corresponds to LTO. This is further confirmed by the plan view STEM image in Fig. 7(b), showing a unique nanonecklace-like structure. Such an interesting structure combines the new TTB phase with the VAN design, resulting in enhanced multifunctional properties including ferroelectricity, anisotropic dielectric function, and strong optical nonlinearity. Figure 7(c) shows the non-linear optical properties as measured using the SHG as a function of the polarization angle of the inci- dent beam. The enhanced SHG signal can be attributed to (1) highly non-centrosymmetric crystal structure of the LNBTO phase and (2) the interface between LNBTO and LTO. Furthermore, Fig. 7(d) shows the ferroelectric hysteresis loop with low leakage, having Ps of 26 μC cm−2and Prof 8.1 μC cm−2for a coercive field ( Ec) of 74.6 kV cm−1, which are comparable or higher than those of the constituent ferroelectric materials such as BTO or LNO. Such FIG. 7. LiNb 6Ba5Ti4O30(LNBTO)–LiTiO 2(LTO) system. (a) Cross section and (b) plan-view STEM image of the LNBTO–LTO thin film nanocomposite showing the typical VAN structure with the darker phase corresponding to LTO, being segregated at the phase boundaries. (c) SHG intensity vs incident polarization angle with output polarization fixed at 0○(P-out) and (d) polarization hysteresis loop. Reproduced with permission from Huang et al. , ACS Appl. Mater. Interfaces 12(20), 23076 (2020). Copyright 2020 American Chemical Society.67 APL Mater. 9, 030904 (2021); doi: 10.1063/5.0035366 9, 030904-9 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm unconventional VANs provide a new design pathway toward further multifunctionality and its compatible use with Si-based photonic devices. B. Ferroelectric and multiferroic oxide–metal VANs Recently, research has been focused on oxide–metal VAN hybrid nanostructures due to their practical device applications, which include aforementioned applications but in addition, opti- cal metamaterials.40,68–73One of the first metallic-oxide VANs has been demonstrated by Vidal et al. in the Co–CeO 2system with anisotropic magnetization having its easy axis along the OP direc- tion.74Little work has been done by other groups on metallic-oxide VAN thereafter. However, in recent years, there have been numer- ous works. Figure 8(a) shows recent work of a representative two- phase Fe-BTO VAN with Fe nanopillars embedded within the BTO matrix.75Both the cross section and plan-view STEM images along with the energy dispersive spectroscopy (EDS) elemental mapping [Figs. 8(a1)–8(a3)] confirm the highly ordered growth of Fe nanopil- lars with a diameter of ∼5 nm. Such an ordered structure can be attributed to the effective strain compensation along the vertical and lateral between the Fe and BTO matrix. The ferroelectric nature of the composite is shown by the PFM phase and amplitude switching curves in Fig. 8(a4). The presence of the Fe phase also contributes to the anisotropic ferromagnetic properties, thus classifying it as an artificial multiferroic material. Greater design flexibility and structural complexity is required to develop next generation integrated electronic devices. This canbe achieved through the three-phase nanocomposite design and growth. Figure 8(b) shows such a self-assembled ordered three- phase Au–BTO–ZnO nanocomposite, fabricated by depositing an Au–BTO VAN template layer followed by the BTO–ZnO VAN.40 Both the cross section and plan-view STEM images along with the EDS elemental mapping shows a mix of Au particles, capping the ZnO nanowires, as well as Au pillars, embedded in the BTO matrix along with the highly ordered growth of both ZnO and Au phases. This unique “nanoman”-like structure is enabled by the combina- tion of template-assisted Vapor Liquid Solid (VLS) and two-phase epitaxy growth mechanisms. Besides the interesting nonlinear and anisotropic optical properties, the three-phase nanocomposite is fer- roelectric, which is confirmed by the PFM phase switching map, with a d33piezoelectric coefficient of 19.4 pm V−1greater than the two-phase BTO–ZnO VAN ( ∼10 pm V−1). Such multi-phase struc- tures open new possibilities in design, growth, and engineering of other systems toward increased control over light–electron–matter interaction at the nanoscale. C. Beyond VANs to self-assembled layered supercell structures While VANs have been vastly explored for ferroelectric and multiferroic properties, there are also a new family of materi- als with a self-assembled layered supercell (LSC) structure that have room temperature multiferroic properties. These are of the form of self-assembled superlattices, as shown in Fig. 9. Such lay- ered materials have been shown to be formed from Aurivillius FIG. 8. (a) Fe–BaTiO 3(BTO) system. (a1) Cross section STEM image and (a2) the corresponding EDS elemental map showing the Fe pillars embedded in the BTO matrix, (a3) plan-view STEM image showing the distribution of Fe pillars, and (a4) phase and amplitude switching curves measured using PFM. Reproduced with permission from Zhang et al. , Mater. Today Nano 11, 100083 (2020). Copyright 2020 Elsevier Ltd.75(b) Au–BaTiO 3–ZnO ordered three phase nanocomposites. (b1) Cross section STEM image and (a2) the corresponding EDS elemental map showing the ordered Au pillars and Au nanoparticles capping the ZnO nanowires in a unique “nanoman” structure, (b3) plan-view STEM image showing the spatial ordered pillar distribution, and (b4) PFM phase switching map. Reproduced with permission from Misra et al. , Adv. Mater. 31(7), 1806529 (2018). Copyright 2018 WILEY-VCH.40 APL Mater. 9, 030904 (2021); doi: 10.1063/5.0035366 9, 030904-10 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm FIG. 9. (a) Bi 2(AlMn)O 6(BAMO) layered supercell structure (LSC). (a1) Cross-sectional STEM image of the BAMO thin film deposited on STO (001) substrate. Inset shows the SAED pattern, (a2) HR-STEM image showing the LSC structure with a three-atom-thick Bi-based slab and one single Al/Mn based layer, (a3) piezoelectric coefficient d33vs tip bias hysteresis loop at room temperature, and (a4) IP and OP magnetization hysteresis (M −H) loops of the BAMO LSC at 300 K. Reproduced with permission from Li et al. , Nano Lett. 17(11), 6575 (2017). Copyright 2017 American Chemical Society.32(b) BiWO 3–CoFeO 3(BWO–CFO) system. (b1) Cross section TEM image of the BWO–CFO system showing the CFO nanocones embedded in the layered BWO matrix, (b2) atomic-scale high-resolution STEM image of the BWO–CFO interface, (b3) amplitude and phase switching behavior of BWO–CFO, and (b4) IP and OP magnetization hysteresis loops of BWO–CFO and pure CFO at 300 K. Reproduced with permission from Wang et al. , Mater. Res. Lett. 7(10), 418 (2019). Copyright 2019 Taylor and Francis.78(c) Bi 3MoM TO9(T=Fe, Mn, Co, and Ni) LSC system. (c1) Cross- sectional TEM image and (c2) HR-TEM image of the Bi 3MoFeO 9thin film, (c3) phase (blue) and amplitude (red) switching curves, and (c4) in-plane magnetic hysteresis loops of BMoM TO films measured at 300 K. Reproduced with permission from Gao et al. , Nanoscale 12, 5914 (2020). Copyright 2020 The Royal Society of Chemistry.77 and Ruddlesden–Popper phases, and these compositions have var- ious applications in piezoelectricity, superconductivity, water split- ting, and thermoelectricity. For ferroelectric and multiferroics, Bi-based layered oxide structures have been widely explored, includ- ing Bi 2FeMnO 6, Bi 2AlMnO 6, and Bi 2NiMnO 6. They show room temperature multiferroicity arising from the coupling of ferroelec- tricity from the lone-pair electrons of Bi cations and ferromag- netism, which arises from the coupled Mn–M (M =Fe, Al, and Ni) cations.27,31–33,76Figure 9 shows three different such LSC structures, namely, Bi 2AlMnO 6(BAMO), Bi 2WO 6–CoFe 2O4(BWO–CFO), and Bi 3MoMO 9(M=Fe, Mn, Co, Ni; BMMO), along with their multiferroic properties.32,77,78 Figures 9(a1)–9(c1) shows the STEM image of the three differ- ent Bi-based LSC structures: BAMO, BWO–CFO, and BMFO hav- ing lattice planes parallel to the substrate.32,77,78All the films show a highly epitaxial growth along the (00 l) out-of-plane direction. Figures 9(a2)–9(c2) shows the high-resolution STEM (HR-STEM) images of the LSC structures. Interestingly, the BAMO structureshows the supercell structure with three-atom-thick Bi-based slab and one single Al/Mn based layer. BWO–CFO VANs presents a mix of LSC and VAN structure with the CFO phase being embedded as a “nanocone”-like structure in the layered BWO matrix. The BMFO VANs shows a LSC with domain boundaries separating the Fe-rich and Mo-rich domain, and the LSC structures show the presence of the Aurivillius phase consisting of alternating Bi–O layers and M–O layers. Interestingly, all the LSC structures show a coupled ferroelectric [Figs. 9(a3)–9(c3)] and ferromagnetic response [Figs. 9(a4)–9(c4)], thereby characterizing them as multiferroics. All the films show a well-defined ferroelectric hysteresis loop measured locally using the PFM, while the ferromagnetic character is confirmed by the M–H curves. The inherent anisotropy in LSC is confirmed by the different IP and OP ferromagnetic response, which is attributed to the presence of magnetic elements. More interestingly, most of the LSC structures show a strong IP magnetic response pos- sibly due to the easy IP magnetocrystalline axis for the layered APL Mater. 9, 030904 (2021); doi: 10.1063/5.0035366 9, 030904-11 © Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm structure. Therefore, by combining an Aurivillius phase with an appropriate perovskite transition metal oxide, an alternative way to explore new materials systems using LSC structures is pro- vided. This gives further flexibility in VAN design and property tuning. VII. OUTLOOK Regarding the future research directions, selected focus direc- tions in the field of ferroelectric VANs are as follows: (1) Modulation of defects such as oxygen vacancies by carefully controlling the depo- sition parameters. Oxygen vacancy migration plays a very important role in the tuning the conductance and changing the resistive char- acteristics of the films, which are useful in practical applications such as memristor and neuromorphic computing devices.79–81Therefore, it is critical to carefully control the oxygen vacancy concentration by introducing dopants and changing the deposition conditions. (2) Having a non-centrosymmetric structure, ferroelectric materials can also be used in non-linear optical devices including applications such as low-loss waveguides, wavelength conversion, and electro-optic modulators.82,83The use of VAN design provides further design flex- ibility in terms of property and microstructure tuning by varying the lateral and vertical strain state in the film. However, such photonic device applications require uniform deposition over a larger area than that can be achieved with pulsed laser deposition (PLD). There- fore, alternate deposition techniques such as sputtering or chemical vapor deposition (CVD) should be also be explored for VAN growth for scaling up device fabrication. There is some progress in this direction already.84,85(3) Recently, various data science techniques and machine learning (ML) models have been developed in the field of text and data mining, molecular modeling and microscopy image analysis for the accelerated materials discovery, data-driven materials design, and gaining insights into the material synthesis parameter.86–91However, most of the ML tools developed are for specific material systems and mostly for single phase materials with challenges in the consideration of complex materials interfaces and heterostructures.92–94Therefore, similar techniques can be extended toward the exploration of future VAN systems including the two- phase and three-phase nanocomposites having interesting electrical, magnetic, and optical applications. AUTHORS’ CONTRIBUTIONS This manuscript was written through contributions of all authors. O.J.L. and S.M. contributed equally. All authors have given approval to the final version of the manuscript. ACKNOWLEDGMENTS O.J.L. acknowledges the support from the Ministry of Edu- cation, Malaysia, under the Fundamental Research Grant Scheme (Grant No. FRGS/1/2018/STG07/UMT/02/7). S.M. and H.W. acknowledge the support from the U.S. National Science Founda- tion for the VAN processing, characterization (Grant Nos. DMR- 1565822 and DMR-2016453), and device related efforts (Grant No. ECCS-1902644). J.L.M.-D. acknowledges the Royal Academy of Engineering (Grant No. CIET1819-24), the Leverhulme Trust (Grant No. RPG-2015-017), EPSRC (Grant Nos. EP/N004272/1, EP/T012218/1, EP/P007767/1, and EP/M000524/1), and the IsaacNewton Trust in Cambridge [minute 16.24(p) and Grant No. RG96474]. DATA AVAILABLITY Data sharing is not applicable to this article as no new data were created or analyzed in this study. 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9.0000119.pdf

AIP Advances 11, 025305 (2021); https://doi.org/10.1063/9.0000119 11, 025305 © 2021 Author(s).Study of structural, transport and magneto- crystalline anisotropy in La1−xSrxMnO3 (0.30 ≤ x ≤ 0.40) perovskite manganites Cite as: AIP Advances 11, 025305 (2021); https://doi.org/10.1063/9.0000119 Submitted: 15 October 2020 . Accepted: 15 January 2021 . Published Online: 03 February 2021 Ganesha Channagoudra , Shalabh Gupta , and Vijaylakshmi Dayal COLLECTIONS Paper published as part of the special topic on 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials and 65th Annual Conference on Magnetism and Magnetic Materials ARTICLES YOU MAY BE INTERESTED IN Structural, magneto transport and magnetic properties of Ruddlesden–Popper La2-2x Sr1+2x Mn2O7 (0.42≤x≤0.52) layered manganites AIP Advances 11, 025331 (2021); https://doi.org/10.1063/9.0000109 Electronic structure and thermoelectric properties of half-Heusler alloys NiTZ AIP Advances 11, 025304 (2021); https://doi.org/10.1063/5.0031512 Enhancement of low-frequency spin-orbit-torque ferromagnetic resonance signals by frequency tuning observed in Pt/Py, Pt/Co, and Pt/Fe bilayers AIP Advances 11, 025206 (2021); https://doi.org/10.1063/9.0000066AIP Advances ARTICLE scitation.org/journal/adv Study of structural, transport and magneto-crystalline anisotropy in La 1−xSrxMnO 3(0.30 ≤x≤0.40) perovskite manganites Cite as: AIP Advances 11, 025305 (2021); doi: 10.1063/9.0000119 Presented: 2 November 2020 •Submitted: 15 October 2020 • Accepted: 15 January 2021 •Published Online: 3 February 2021 Ganesha Channagoudra,1Shalabh Gupta,2and Vijaylakshmi Dayal1,a) AFFILIATIONS 1Department of Physics, Maharaja Institute of Technology Mysore (Aff. VTU-Belagavi), Karnataka 571 477, India 2Division of Material Science and Engineering, Ames Laboratory, U.S. Department of Energy, Ames, Iowa 50010, USA Note: This paper was presented at the 65th Annual Conference on Magnetism and Magnetic Materials. a)Author to whom correspondence should be addressed: drvldayal@gmail.com ABSTRACT In this paper we present structural, transport, magnetic and magneto-crystalline anisotropy of La 1−xSrxMnO 3(х=0.30, 0.33, 0.36, and 0.40) synthesized using the solid-state reaction method. The X-ray diffraction pattern of the samples is well indexed to the rhombohedral structure with R 3c space group. Lattice parameters and unit cell volume are found to decrease monotonically upon increasing Sr2+concentration at La3+site. X-ray photoelectron spectroscopy confirms the presence of Mn4+and Mn3+, the ratio of which (Mn4+/Mn3+) increases from 0.44 to 0.64 with increasing Sr2+concentration. From the ρ(T) and M(T) data analysis, we find that T MIand M Sare larger for x =0.33 composition, suggesting a weakening of the double-exchange interaction due to competing super-exchange interactions in x =0.30, 0.36 and 0.40 samples. We calculated MCA constant (K 1) by fitting the saturation magnetisation data at a higher magnetic field using the law of approach model, M=MS[1-a/H-b/H2]-χH with K 12=(105/8)bMs2and is in the order of 105erg/cm3. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/9.0000119 I. INTRODUCTION Perovskite manganite La 1−xAxMnO 3(A=Ca, Ba, and Sr) has been thoroughly investigated due to room temperature fer- romagnetic behaviour in few compositions, both from the fun- damental and the applied perspective.1,2Mixed valent Mn3+and Mn4+ions are developed at the Mn-site when trivalent rare-earth (La3+) ions in LaMnO 3are replaced by the divalent alkaline-earth elements (Ca2+, Ba2+, Sr2+, etc.). Depending on the ratio of the Mn3+and Mn4+ions, the double exchange (DE) interaction3,4and the competition between the DE and super-exchange (SE) interac- tions can be corelated to the significant enhancement/decrement in the electrical conductivity and magnetization.5,6Among the var- ious perovskite manganites, La 1−xSrxMnO 3(LSMO) in the com- position range 0.30 ≤x≤0.40, exhibits interesting properties such as;colossal magnetoresistance, high spin-polarized current ( ∼90%) and high Curie temperature ( ∼370K) due to coupling between the charge, spin and orbital degrees of freedom, making it a potential candidate for fundamental research and industrial applications.7,8In fact, the characteristic behaviour and transitions in these composi- tions also depend on the degree of the lattice distortion in the crystal with varying concentration of Sr2+.4Another important aspect from the point of view of the application of ferromagnets is magneto- crystalline anisotropy (MCA), which is an intrinsic property of a material independent of grain size and shape and is basically depen- dent on the crystallographic orientation.9Anisotropy constants in polycrystalline materials can be obtained from magnetisation mea- surements, since magnetisation close to saturation follows the law of approach to saturation.10The aim of this work is to study elec- trical, magnetic and magneto-crystalline anisotropic properties in AIP Advances 11, 025305 (2021); doi: 10.1063/9.0000119 11, 025305-1 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv La1−xSrxMnO 3with different concentration of Sr2+(x=0.30, 0.33, 0.36, and 0.40). II. EXPERIMENTAL The polycrystalline La 1−xSrxMnO 3with different composi- tions were synthesized using the conventional solid-state reaction method. The raw precursors; La 2O3, SrCo 3, and MnO 2supplied by Sigma Aldrich (99.9% purity) were used as starting materials. The stoichiometric amount of precursors have been thoroughly mixed and milled for 8 hrs using agate mortar and pestle. The powder had been calcined at 800○C for 24 hrs. The calcined powder was then milled for 8 hrs and recalcined for 24 hrs at 1050○C. The pellets were prepared by applying a 6-tonnes uniaxial pressure using a commer- cially made KBr press. Finally, the pellets were sintered at 1350○C for 36 hrs. The structural phase formation of the synthesized sam- ples was confirmed by the X-ray diffraction measurement technique using the Bruker D8 advance X-ray diffractometer equipped with Cu-K αsource of wavelength λ=1.5406 Åand the lattice parameters were determined by the Rietveld refinement of the XRD data using Fullprof ™software.11The X-ray photoelectron spectroscopy (XPS) was employed to probe the mixed valence state of the samples. The XPS was recorded using Al-K αradiation source with an energy value of 1486.6 eV. The temperature-dependent resistivity measurement was carried out using four probe technique. Magnetic measurement was performed using SQUID-VSM. III. RESULTS AND DISCUSSIONS Room temperature XRD along with Rietveld refinement pat- tern of LSMO with different compositions are depicted in Fig. 1(a-d) and found to be in good agreement with the rhombohedral struc- ture (JCPDS card No. 89-4466).12For each sample, the refinement of the XRD pattern taking rhombohedral structure (R 3c space group) showed good agreement between experimental and Rietveld pat- terns. The tolerance factor, lattice parameters obtained from the refinement and the calculated unit cell volume are specified in Table I. Table I shows that the lattice parameters and the unit cell volume decrease monotonically with an increase in the concentra- tion of Sr2+in the samples. Here, it can be understood that by substituting Sr2+(ri=1.44Å) at the La3+(ri=1.36Å) site, the equivalent amount of larger Mn3+(ri=0.645Å) ions gets converted into smaller Mn4+(ri=0.530Å) ions. Besides, the radii difference between La3+and Sr2+ions (0.08 Å) is smaller compared to radii dif- ference between Mn3+and Mn4+(0.11Å), thus the change in B-site radii is dominant and the change in A-site radii does not account for a decrease in unit cell volume with an increasing Sr2+at La3+site.13 Overall, the Sr2+doping minimises the distortion of the MnO 6octa- hedra due to increase in tolerance factor ( τ) resulting in a decrease in the lattice parameters. Similar findings are reported in PSMO and PCMO.14,15 X-ray photoelectron spectroscopy (XPS) was performed for the Mn2p element of x =0.30 and 0.40 samples to determine the mixed valence ratio of manganese ions (Mn4+/Mn3+). The fitted XPS plot using XPSPEAK 4.1 software with Shirley background are shown in Fig. 2(a) and (b), respectively. The spin–orbit interaction of Mn2p doublet spectra corresponds to Mn2p 3/2and Mn2p 1/2at about FIG. 1. XRD pattern along with Rietveld refinement of (a) x =0.30, (b) x =0.33, (c) x=0.36 and (d) x =0.40 samples. 642.20 eV and 653.60 eV, respectively for x =0.30, while at 641.89 eV and 653.24 eV, respectively for x =0.40. Consequently, the separa- tion between the doublets is approximately 11.3(4) eV, which is in agreement with the earlier report.16It can also be observed from the Fig. 2 that each Mn2p peaks when fitted, de-convolutes into only two peaks corresponding to the Mn4+and Mn3+ions. Consider- ing Mn2p 3/2peak, the higher binding energy peak located at around 644.47 (644.315) eV for x =0.30 (x=0.40) corresponds to the Mn4+, while lower one at around 642.19 (642.81) eV for x =0.30 (x=0.40) corresponds to the Mn3+. The percentage concentration of each ions in the samples was determined by calculating the area under the peak and was found to be 69.41 (60.83) % for Mn3+in x=0.30 (x=0.40) and 30.58 (39.17) % for Mn4+in x=0.30 (x=0.40). A higher con- centration of Mn3+ions than Mn4+and also an increase in Mn4+ ions with an increase of Sr2+, the valence ratio Mn4+/Mn3+increases from 0.44 to 0.64 with respect to Sr2+concentrations x =0.30 to AIP Advances 11, 025305 (2021); doi: 10.1063/9.0000119 11, 025305-2 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv TABLE I. Lattice parameter, unit cell volume, crystalline size (S), resistivity at T MIand T MI. Saturated magnetization (M S) and magneto-crystalline anisotropy constant (K 1) at temperature 100K and 300K. La1−xSrxMnO 3 x=0.30 Ref. 18 x =0.33 x =0.36 x =0.40 a (Å) 5.503(9) 5.497(5) 5.491(4) 5.482(5) b (Å) 5.503(9) 5.497(5) 5.491(4) 5.482(5) c (Å) 13.358(1) 13.357(1) 13.353(6) 13.349(3) V (Å3) 404.655(7) 403.685(0) 402.684(1) 401.250(6) τ 0.9791 0.9817 0.9842 0.9876 ρ(mΩ-cm) at T MI 43.156 56.065 76.980 117.42 TMI(K) 368 378 367 366 MS(emu/gm) at 100K 85.87 86.35 85.18 84.35 MS(emu/gm) at 300K 59.67 61.60 59.13 54.59 K1(105erg/cm3) at 100K 1.08 0.83 0.76 1.02 K1(105erg/cm3) at 300K 0.52 0.30 0.35 0.32 0.40. This in turn may influence the magnetic and transport proper- ties of the LSMO samples. Further, Mn2p XPS spectra for both the compositions do not have a peak near 639 eV, showing the absence of Mn2+ions, which in essence suggests that the samples are not FIG. 2. De-convoluted Mn2p XPS spectrum of (a) x =0.30, (b) x =0.40. Circle indicates experimental data and solid lines represent fitted data.oxygen deficient. Additionally, an analysis of the XPS spectrum of La3d and Sr3d elements has been carried out (not shown). The XPS peak of La3d 5/2at 834.5 eV and Sr3d at 133.6 eV confirms the pres- ence of La3+and Sr2+respectively in the samples in accordance with Han et al.17 The temperature dependent resistivity of La 1−xSrxMnO 3 (x=0.33, 0.36, 0.40) is measured at a temperature range of 300 K ≤T<400 K using four probe technique are shown in Fig. 3(a)-(c) and for x=0.30 refer Channagoudra et al.18(published). T MIand resistivity at the corresponding T MIvalue are tabulated in Table I. The table indicates that, the x =0.33 sample shows the maximum TMIin the series. In addition, T MIshifts to lower temperature side, regardless of whether the Sr2+concentration increases or decreases. Similar findings are reported by Urushibara et al.19in the phase diagram study of La 1−xSrxMnO 3. It is well known that ferromag- netic metallic behaviour in manganites is due to a DE interaction between Mn3+and Mn4+and its weakening may be understood due to competing interaction between DE interaction (Mn3+-O2−- Mn4+) and the super-exchange interactions (Mn3+-O2--Mn3+or Mn4+-O2--Mn4+), which in turn reduced the T MIfor x=0.33, 0.36, and 0.40 samples. The temperature dependent FC and ZFC magnetization (M-T) plot at magnetic field 500 Oe and the magnetic (M-H) hysteresis loop measured at 100 and 300 K are shown in Fig. 4(a) and (b), respectively. It can be observed that magnetization increases with decreasing temperature and gradually saturates at lower tempera- tures, indicating the transition from paramagnetic (PM) to ferro- magnetic (FM) state with decreasing temperature. The published literature19observed that the PM-FM transition temperature (T C) of the LSMO samples for Sr2+compositions 0.30 ≤x≤0.40 was approximately 370 K. Due to the limitation of measurements, the magnetic measurements in this study range from 5 K to 350 K. The saturation magnetization (M s) at 5 K varies with Sr2+and is shown in inset of Fig. 4(a). In accordance with the temperature-dependent resistivity analysis, x =0.33 shows higher magnetization which can attributed to the DE interaction (Mn3+-O2−-Mn4+) due to the pres- ence of adequate amounts of Mn4+. In comparison, the relatively lower magnetization value in the samples on either side of the series can be understood due to the weakening of the DE inter- action (Mn3+-O2−-Mn4+) and the growing competing interaction AIP Advances 11, 025305 (2021); doi: 10.1063/9.0000119 11, 025305-3 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 3. Temperature dependent resistivity of (a) x =0.33, (b) x =0.36, (c) x =0.40 and (d) Variation of T MIand resistivity at corresponding T MIwith respect Sr2+ concentration. between the DE interaction (Mn3+-O2−-Mn4+) and the SE interac- tion (Mn3+-O2--Mn3+or Mn4+-O2--Mn4+).20 The magnetic hysteresis loop (M-H) of each sample measured at 100 K and 300 K up to 1T magnetic field are shown in Fig. 4(b). The low coercivity and low remnant magnetization at both temper- atures signify the soft ferromagnetic nature of the samples. Accurate saturation magnetization (Ms) and magneto-crystalline anisotropy (MCA) constant (K 1) can be determined by applying the law of approach to saturation for the M-H loop to higher fields (H ≫HC) given by;21 M=MS[1−a H−b H2]−χH, (1) FIG. 4. Magnetic properties of LSMO, (a). M-T curve of LSMO measured at mag- netic field 500 Oe. Variation of M Sat 5K with respect Sr2+concentration shown in inset of (a). (b) Field dependent magnetization (M-H loop) at 100 K and 300 K. Fittings of law of approach to saturation at both temperatures shown in the inset (b). here,a His corresponds to the inclusions/micro-stress,b H2corre- sponds to the contribution of MCA) and xHis the forced mag- netization term. The constant b for a cubic crystal is given by;21 b=8 105×K2 1 M2s, (2) where, K 1is the MCA constant of uniaxial crystal. The values of M S and K 1at both 100 K and 300 K obtained using the best fit to the Eq. (1) are mentioned in the Table I. For each sample, the obtained significant MCA constant K 1value is of the order 105erg/cm3, a value comparable with the spinel ferrites such as Fe 3O4, CoFe 2O4, etc.22Also, for each sample, the value of K 1at 100 K is higher than at 300 K, suggesting that the samples at 100 K are more magnetically ordered than at 300 K. The magnetic materials with strong MCA AIP Advances 11, 025305 (2021); doi: 10.1063/9.0000119 11, 025305-4 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv of the order 105-106erg/cm3, offers superior qualities for magne- toresistive random access memories (MRAM) and multifunctional spintronic devices and have been successfully commercialized in the last decade.22 IV. CONCLUSION Perovskite manganites La 1−xSrxMnO 3(x=0.30, 0.33, 0.36, and 0.40) have been synthesized by standard solid-state route and their structural, electrical and magnetic properties have been stud- ied. The XRD pattern refined using Rietveld technique reveals that each sample has rhombohedral symmetry with the R 3c space group. The improvement of Mn4+/Mn3+valent ratio with an increase in the concentration of Sr2+, the double-exchange interaction pre- dominates in the x =0.33 sample, which contributes to the maxi- mum value of T cand M s. The value of K 1at 100 K is higher than at 300 K due to high magnetic ordering at low temperature. The obtained result suggests that the LSMO samples in the composition range 0.30 ≤x≤0.33 are a good candidate for room temperature multifunctional devices. ACKNOWLEDGMENTS This work is supported by SERB-DST, New Delhi (EMR/2016/005424) and partially by UGC DAE CSR, Indore (CSR-IC/CRS-89/2014-2019) granted to V.D. G.C. gratefully acknowledges SERB-DST for SRF. We gratefully acknowledge Dr. Mukul Gupta and Layanta Behera for XRD, Dr. Uday Desh- pande for XPS, Dr. Alok Banerjee and Kranti Kumar for magnetic measurements and Dr. Rajeev Rawat for resistivity measurement at UGC-DAE CSR, Indore Centre, India. The measurement was partially carried out by Division of Materials Sciences and Engi- neering of Basic Energy Sciences Program of the Office of Science of the U.S. Department of Energy and we gratefully acknowledge Professor (Dr.) V. K. Pecharsky. Authors are indebted to Maharaja Research Foundation, MITM Campus for facilities to support our research. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1J. B. Goodenough, “Theory of the role of covalence in the perovskite-type manganites [La,M(II)]MnO 3,” Phys. Rev. 100, 564–573 (1955). 2J. M. D. Coey, M. Viret, and S. von Molnár, “Mixed-valence manganites,” Adv. Phys. 58, 571–697 (2009).3C. Zener, “Interaction between the d-shells in the transition metals. II. Ferro- magnetic compounds of manganese with perovskite structure,” Phys. Rev. 82, 403 (1950). 4A. V. Deshmukh, S. I. Patil, S. M. Bhagat, P. R. Sagdeo, R. J. Choudhary, and D. M. Phase, “Effect of iron doping on electrical, electronic and magnetic proper- ties of La 0.7Sr0.3MnO 3,” J. Phys. D.: Appl. Phys. 42, 185410 (2009). 5G. Channagoudra, A. K. Saw, and V. Dayal, “Low temperature spin polar- ized tunnelling magneto-resistance in La 1-xCaxMnO 3(x=0.375 and 0.625) nanoparticles,” Emergent Mater. 3, 45–49 (2020). 6V. Dayal and S. Keshri, “Structural and magnetic properties of La0.67Ca0.33Mn (1-x)FexO3(x=0-0.07),” Solid State Commun. 142, 63–66 (2007). 7P. Schlottmann, “Spin, charge, orbital and lattice degrees of freedom in quasi- cubic manganites: A brief review,” Phys. B Condens. Matter. 404, 2699–2704 (2009). 8C. N. R. Rao, “Charge, spin, and orbital ordering in the perovskite manganates, Ln1-xAxMnO 3(Ln=Rare Earth, a =Ca or Sr),” J. Phys. Chem. B 104, 5877–5889 (2000). 9L. Sagnotti, Magnetic Anisotropy (Springer Science and Business Media, 2011). 10R. Grössinger, “A critical examination of the law of approach to saturation. I. Fit procedure,” Phys. Status Solidi 66, 665–674 (1981). 11C. Frontera and J. Rodríguez-Carvajal, “FULLPROF as a new tool for flipping ratio analysis,” Phys. B. 335, 219–222 (2003). 12D. Varshney and M. A. Dar, “Structural and magneto-transport properties of (1-x)La 0.67Sr0.33MnO 3(LSMO) +(x)BaTiO 3(BTO) composites,” J. Alloys Compd. 619, 122–130 (2015). 13K. McBride, J. Cook, S. Gray, S. Felton, L. Stella, and D. Poulidi, “Evaluation of La1-xSrxMnO 3(0≤x<0.4) synthesised via a modified sol-gel method as mediators for magnetic fluid hyperthermia,” CrystEngComm 18, 407–416 (2016). 14V. Prashanth Kumar, P. Srinivas Reddy, S. Sripada, and C. Vishnuvardhan Reddy, “Preparation and characterization of Pr 1-xSrxMO 3(PSMO) (0 ≤x≤0.6) nanomaterials,” Mater. Today Proc. 3, 3709–3712 (2016). 15A. T. Kozakov, A. G. Kochur, V. G. Trotsenko, A. V. Nikolskii, M. El Marssi, B. P. Gorshunov, and V. I. Torgashev, “Valence state of cations in manganites Pr1-xCaxMnO 3(0.3≤x≤0.5) from x-ray diffraction and x-ray photoelectron spectroscopy,” J. Alloys Compd. 740, 132–142 (2018). 16O. Pana, M. L. Soran, C. Leostean, S. MacAvei, E. Gautron, C. M. Teodor- escu, N. Gheorghe, and O. Chauvet, “Interface charge transfer in polypyrrole coated perovskite manganite magnetic nanoparticles,” J. Appl. Phys. 111, 044309 (2012). 17S. W. Han, J. D. Lee, K. H. Kim, H. Song, W. J. Kim, S. J. Kwon, H. G. Lee, C. Hwang, J. I. Jeong, and J. S. Kang, “Electronic structures of the CMR perovskites R1-xAxMnO 3(R=La, Pr; A=Ca, Sr, Ce) using photoelectron spectroscopy,” J. Korean Phys. Soc. 40, 501–510 (2002). 18G. Channagoudra, A. K. Saw, and V. Dayal, “Significant enhancement of magnetization in 0.67Pb(Mg 1/3Nb 2/3)O3-0.33PbTiO 3/La 0.70Sr0.30MnO 3 heterostructure,” Thin Solid Films 709, 138132 (2020). 19A. Urushibara, Y. Moritomo, T. Arima, A. Asamitsu, G. Kido, and Y. Tokura, “Insulator-metal transition and giant magnetoresistance in La 1-xSrxMnO 3,” Phys. Rev. B 51, 14103–14109 (1995). 20Y. Tokura and Y. Tomioka, “Colossal magnetoresistive manganites,” J. Magn. Magn. Mater. 200, 1–23 (1999). 21M. Suliga and K. Chwastek, “Effect of stress in drawn wires on magnetization curves in the saturation region,” Acta Phys. Pol. A. 135, 243–245 (2019). 22C. D. Graham and B. D. Cullity, Introduction to Magnetic Materials , 2nd ed. (John Wiley & Sons, Inc., 2008). AIP Advances 11, 025305 (2021); doi: 10.1063/9.0000119 11, 025305-5 © Author(s) 2021

5.0037615.pdf

AIP Conference Proceedings 2324 , 030027 (2021); https://doi.org/10.1063/5.0037615 2324 , 030027 © 2021 Author(s).Whirling of a Jeffcott rotor on a superelastic SMA shaft Cite as: AIP Conference Proceedings 2324 , 030027 (2021); https://doi.org/10.1063/5.0037615 Published Online: 25 February 2021 M. Ashiqur Rahman , S. M. Mohiuddin Siddique , and M. Mahbubur Rahman ARTICLES YOU MAY BE INTERESTED IN Buckling of pushrods made of superelastic shape memory alloy AIP Conference Proceedings 2324 , 030008 (2021); https://doi.org/10.1063/5.0037613 Self-consistent clustering analysis for modeling of theromelastic heterogeneous materials AIP Conference Proceedings 2324 , 030029 (2021); https://doi.org/10.1063/5.0038297 Experimental study on the fabrication and mechanical properties of medium-density particleboards from coconut coir AIP Conference Proceedings 2324 , 030028 (2021); https://doi.org/10.1063/5.00375641 Whirling of a Jeffcott Rotor on a Superelastic SMA Shaft M Ashiqur Rahmana), S M Mohiuddin Siddique and M Mahbubur Rahman Department of Mechanical Engineering, Bangladesh University of Engineering and Technology, Dhaka - 1000, Bangladesh a) Corresponding Author: ashiq@me.buet.ac.bd Abstract: Response of shafts made of superelastic shape memory alloy is analyzed emphasizing following points during whirling: mass eccentricity ratio, whirl speed ratio, spin ratio, damping ratio, asynchronous m ode, and, the effect of material non -linearity. A mathematical model is developed and material non -linearity issue is incorporated in terms of non -linear stress -strain curves. Next computer codes are developed in Matlab and shaft responses during steady -state whirl are predicted. It is found that the Jeffcott rotor system behaves as a single degree of freedom system. Naturally, i ncreased damping greatly reduces whirling amplitude at resonance. Some new results are found. For example, for asynchronous whirl, resonance does not occur at a spin ratio of unity. As material non -linearity is taken into account, some portion of the shaft is found to experience stresses of different magnitude in tension and compression. Moreover, the maximum deflection of a shaft is found to be much larger in comparison to the case of linearly elastic shaft material. Finally, by experiment inelastic behavior of an SS shaft is demonstrated due to whirling. INTRODUCTION Whirling is defined as the rotation of the plane made by the bent shaft and the line of centers of the bearing [Thomson et al. (2011)] . The phenomenon of whirling of shaft is often explained in terms of a “Jeffcott rotor model” that consists of a simply supported flexible shaft with a rigid thin disc mounted at the mid-span (Figure 1). Generally, two modes of whirling (synchronous and asynchronous) can be observed in various rotating machines (Figure 2). In the synchronous mode, shaft’s own spin speed ( ω) and the whirl / orbital speed (𝛳̇) are equal. However, they are unequ al in case of asynchronous whirl mode . As pointed out in literature [1, 3 -6, 11 -12 & 15 -19], whirling results from various causes such as: a) Mass eccentricity (rotating unbalance), b) Lack of initial straightness of the shaft, c) Non -homogenous material, d) Unbalanced magnetic pull in case of electrical machinery e) Lubricant viscosity, f) Shaft material’s initial stiffness and number of supports, g) Unbalanced centrifugal forces, h) Hysteresis characteristics of shaft materials etc. In this study whirling is studied considering eccentric mass centre of the rotor on the shaft. A large number of studies are reported in the literature dealing with the whirling of rotating shaft, out of which only a few are discussed. Swanson et al. (2005) studied whirling for different modes. Nelson (2007) and Y oon et al. (2013) studied whirling in terms of lateral and torsional vibrations of shafts. Kolenda and Marynarki (2012) analyzed whirling of asymmetric shaft under constant lateral force. Shyong et al. (2014) developed analytical solution for whirling spe eds and mode shapes of a distributed -mass shaft with arbitrary rigid discs. Rajiv (2017) dealt with the gyroscopic effect on whirl. Previous studies of Jeffcott rotor mostly dealt with synchronous whirl considering steady -state vibration and involving lin early elastic shaft materials that follows Hooke’s law. Therefore, present research aims to focus following unexplored but important points concerning whirling of superelastic SMA shafts: asynchronous mode of whirl and the effect of material non -linearity (that is, the shaft material has a non - linear stress -strain relation and Hook’s law cannot be applied) on the shaft’s response. GOVERNING EQUATIONS & SOLUTION STRATEGY A typical Jeffcott rotor model is shown in Figure 1. A single disc of mass m symmetrica lly located on a shaft supported by two bearings where for the present study, t he mass center of the disc is at a distance e Proceedings of the 13th International Conference on Mechanical Engineering (ICME2019) AIP Conf. Proc. 2324, 030027-1–030027-10; https://doi.org/10.1063/5.0037615 Published by AIP Publishing. 978-0-7354-4068-5/$30.00030027-12 (eccentricity) from the geometric center S of the disc. S is deflected by an amount r from the intersecting point of plane of the d isc and centre line of bearings. (a) (b) FIGUR E 1. Shaft geometry defining whirling of a rotating shaft (a) & (b) . FIGURE 2. Definition of whirl parameters. Whirl speed ratio, 𝜆=𝛳̇ 𝜔, λ = 1 for synchronous whirl, 𝜆≠1 for asynchronous whirl. FIGURE 3. Circular cross -section of the shaft. It is assumed that the shaft (i.e., the line e = SG) to be rotating at a constant speed ω, and in the general case, the line r be whirling at speed 𝛳̇ that is not equal to ω. Aside from the restoring force ( kr) of the shaft, 030027-23 it is assumed that a viscous damping force (𝑐𝑟̇) to be acting at S. The exciting forces are induced by the motion itself. The non -linear governing equations of motion resolved in the radial and tangential directions then become −𝑘𝑟−𝑐𝑟̇=𝑚[𝑟̈−𝑟𝛳̇2−𝑒𝜔2𝑐𝑜𝑠(𝜔𝑡−𝛳)] (1) −𝑐𝑟𝛳̇=𝑚[𝑟𝛳̈−2𝑟̇𝛳̇−𝑒𝜔2𝑠𝑖𝑛(𝜔𝑡−𝛳)̇] (2) The symbols used denote, ω = shaft spin, 𝛳̇= shaft whirl speed, e = mass centre eccentricity, kr = restoring force due to bending stiffness, 𝑐𝑟̇= radial damping force, 𝑐𝑟𝛳̇= tangential damping force. Derivations of non-dimensional radial displacement and dynamic force are briefly shown below. Interested readers may refer to Siddique [2018] to see all the steps of derivation. Mathematical Modeling for Steady -state Synchronous and Asynchronous Whirl For steady -state motion, 𝛳̈=𝑟̈=𝑟̇=0 Let 𝛳̇=𝜆𝜔, where λ is a real number. Therefore, 𝜆=1 stands for synchronous whirl. While, 𝜆≠1 represents asynchronous whirl. Now, the problem reduces to that of a SDOFS in r. Thus we have, 𝜆= Whirl ratio =𝛳̇ 𝜔 and, 𝛳̇=𝜆𝜔 (3) Integrating, 𝛳=𝜆𝜔𝑡 −𝜙 (4) or, 𝜔𝑡−𝛳=𝜔𝑡(1−𝜆)+𝜙 (5) Where φ refers to constant of integration. The phase angle between eccentricity e and whirling amplitude r is 𝛿=𝜔𝑡−𝛳=𝜔𝑡(1−𝜆)+𝜙 (6) Incorporating Equations (3 -5) in the governing equations (1 -2) and simplifying, the non -dimensional displacement is 𝑟 𝑒=𝜆2𝛽2 √(1−𝜆2𝛽2)2+(2𝜁𝛽𝜆)2 (7) Equation for the phase angle ( δ) 𝑡𝑎𝑛 𝛿=2𝜁𝛽𝜆 1−𝜆2𝛽2 (8) The dynamic load is given by, 𝑃𝑑=√(𝑘𝑟)2+(𝑐𝜆𝜔𝑟 )2 (9) Rearranging in non -dimensional form, 𝑃𝑑 𝑘𝑟=√1+(2𝜁𝛽𝜆)2 (10) Another non -dimensional form is, 𝑃𝑑 𝑚𝑔=√(𝑘𝑟)2+(𝑐𝜆𝜔𝑟 )2 𝑚𝑔 (11) Where, ωn = √𝑘 𝑚 is the natural frequency, spin ratio is 𝛽=𝜔 𝜔𝑛, damping ratio is 𝜁=𝑐 𝑐𝑐 , critical damping coefficient, 𝑐𝑐=2𝑚𝜔𝑛 030027-34 Handling of Material Non -linearity For superelastic SMA h aving highly non-linear stress -strain relations Rahman et al. (2007, 20 08, and 2009) derived the moment – curvature ( M – 𝛥) and reduced modulus – curvature ( Er – 𝛥) relations for a rectangular beam section. It should be noted that shape of cross -section plays a n important role in determining response of beams and columns ma de of SMA as pointed out by Rahman et al. (2007, 2008 and 2009). Therefore, material non -linearity issue of a SMA shaft (obviously having a circular cross -section) is quite different from that of SMA beams of rectangular cross -section. Circular cross -sections need more involvement because width ( b) changes with height (Figure 3). No closed form solutions exist and numerical integration technique becomes essential for the evaluation of M – 𝛥 and Er – 𝛥 relations [Siddique (2018)]. The whole mathematical pr ocedure is elaborately shown in [Siddique (2018)]. Following the same procedure as in [Siddique (2018)] and using the actual true stress -strain diagram (Figure 4) Er – 𝛥 (Figure 5 ) and M – 𝛥 (Figure 6) relations are obtained for superelastic SMA shaft. Next 2 computer codes (one to handle the governing e quation and another to handle material non -linearity) are developed in Matlab and shaft responses during steady -state whirl are predicted. FIGURE 4. Actual stress -strain curve of superelastic SMA (dia = 2 mm), showing non-linear and asymmetric behavior in tension and compression [Rahman, M. A. (2001)]. RESULTS AND DISCUSSION Modulus of elasticity ( E) of SMA in the parent phase is 65 GPa [Rahman (2001)]. From Figure 5, it is observed that Er = E = 65 GPa up to Δ = 0.012. Therefore, superelastic SMA shaft will exhibit linearly elastic behavior up to Δ = 0.012. For Δ > 0.012 , Er starts decreasing as material non -linearity effect starts because the stress applied on the shaft material exceeds proportional limit of the stress -strain cur ve (Figure 4). The lowest value of Er is 16.9 GPa for a total strain of 0.226. In such a case the shaft will exhibit the least bending stiffness as Er <<E. If M – 𝛥 curve (Figure 6) is analyzed, it is seen that up to 𝛥 = 0.012 bending moment curve is a straight line but after that it is increasing with a decreasing slope. Bending moment is 0.3041 Nm corresponding to 𝛥 = 0.012. So, M threshold = 0.3041 Nm, is the threshold value after which shaft exhibits material non -linearity. More specifically, if the value of 𝛥 is more than 0.012, shaft undergoes strain which is no more proportional to stress. -2500-2000-1500-1000-500050010001500 -0.2 -0.1 0 0.1True Strain True Stress(MPa) 030027-45 In Figure 6, the maximum bending moment is M = 1.5 Nm. According to linearly elastic model (classical Euler beam theory) for the same bending moment correspon ding elastic strain ( 𝛥elastic) will be only 0.059. But by considering material non -linearity effect, for M = 1.5 Nm, actual total strain ( 𝛥actual) is 0.226 as seen from Figure 6. This shows actual strain is much larger than assumed by linear elastic mode l. In turn, this point proves the necessity of taking into account the material non -linearity effect. It should be mentioned here that upon unloading, SMA can recover large strain by virtue of super/pseudo elasticity through a hysteresis. Therefore, large deformation for SMA can be termed as non -linearly elastic. However, conventional engineering material like stainless steel (SS) invariably shows large inelastic/plastic strain upon unloading due to material non -linearity effect [Rahman (2001)]. Next, resul ts concerning whirling of SMA shafts for different conditions (steady, synchronous and asynchronous etc.) are discussed below. Values of damping ratio ( ζ) are chosen as 0.05, 0.30 and 0.5. It is assumed damping will be quite low ( ζ = 0.05) when the shaft w hirls in air. However, if the shaft whirls in liquid (water or, oil) other values ( ζ = 0.1, 0.3 and 0.5) might be more realistic. FIGURE 5: Reduced modulus vs. Δ curve of superelastic SMA shaft. FIGURE 6: Bending moment vs. Δ curve of superelastic SMA shaft. Steady -state Synchronous and Asynchronous Whirl Considering Linearly Elastic Materials This section deals with linearly elastic (SMA remains in parent austenite phase) response of shaft material. Solutions are obtained for followin g specifications of the shaft: Shaft length ( L) = 200 mm, shaft dia (d) = 2 mm, disc mass, ( m) = 300 g, modulus of elasticity ( E) = 65 GPa, linearly elastic bending stiffness of the shaft ( k = 48EI/L3) = 306.31 N/m, eccentricity of disc mass ( e) = 4.5 mm, natural frequency ( ωn) = 31.95 rad/s. 030027-56 FIGURE 7. Non-dimensional whirling amplitude (𝑟/𝑒) vs. spin ratio ( β) for different damping ratio ( ζ ) for synchronous whirl condition. FIGURE 8. Non-dimensional dynamic force (𝑃𝑑/𝑚𝑔)vs. spin ratio β for different damping ratio ( ζ ) for synchronous whirl condition. Synchronous Whirl From Figure 7 non-dimensional whirling amplitude ( r/e) vs. spin ratio ( β) relation can be seen. It is observed that increased damping greatly reduces r/e. The maximum r/e is 10 for ζ = 0.05 and it is only 1.09 in case of ζ = 0.5. In case of low damping ratio ( ζ = 0.05 & 0.1), high whirling amplitude is observed at spin ratio of unity, because of resonance. For large value of spin ratio, r/e remains constan t which is approximately equal to unity for any damping ratio. Non-dimensional dynamic force ( Pd/mg) vs. spin ratio ( β) considering different damping ratio can be viewed in Figure 8. Here, it is observed that, for low damping ratio ( ζ = 0.05 & 0.1) at β=1, forces are very high. Another interesting finding is that all curves intersect for a specific value of spin ratio ( β = 1.414). However, for β > 1.414 forces notably decrease. But in case of high damping ratio ( ζ = 0.3 and 0.5) damping forces contribute more to the dynamic force ( Pd/mg) acting on the shaft so that dynamic force is increasing monotonously for β > 1.414. 024681012 0 1 2 3 4 5r/e βξ = 0.05 ξ = 0.1 ξ = 0.3 ξ = 0.5Maximum 𝒓/𝒆= 10 030027-67 Asynchronous Whirl FIGURE 9. Non-dimensional whirling amplitude ( r/e) vs. spin ratio β for different asynchronous w hirl conditions, ζ = 0.05. FIGURE 10. Non-dimensional dynamic force (𝑃𝑑/𝑚𝑔) vs. spin ratio (𝛽) for different asynchronous whirl conditions, ζ = 0.05. Non-dimensional whirling amplitudes ( r/e) as a function of spin ratio ( β) for different whirl speed ratio (λ) are shown in Figure 9 (for low damping). It is observed that at resonance the maximum r/e is 10, independent of whirl speed ratio. Theoretically , the maximum r/e should be 10 for the specified values [Siddique (2018)]. Inter estingly, resonance occurs at different spin ratio for different values λ. Higher the whirl speed ratio, lower the value of spin ratio that corresponds to the maximum r/e. For example, the maximum r/e occurs at spin ratio of unity for whirl speed ratio of unity. Whereas, it occurs at spin ratio of 0.5 for λ = 2. Results of steady -state synchronous whirl are placed in the same Figures for comparison only. It is also observed that as spin ratio increases beyond resonance, r/e approaches unity for any value of whirl speed ratio. This point is theoretically verified in Siddique (2018). Non-dimensional dynamic force ( Pd/mg) is a function of damping ratio, whirl speed ratio and spin ratio. From Figure 1 0 it is observed that resonance occurs at different spin ratio but the maximum force is same and is independent of whirl speed ratio. However, increased damping reduces Pd/mg at a great extent at resonance. Steady -state Synchronous and Asynchronous Whirl Co nsidering Material Non - linearity With reference to Figure 10, the maximum value of non -dimensional dynamic force acting at the shaft center due to whirl is 4.71 and corresponding total force ( P) acting at the mid - span is 16.80 N. Therefore, corresponding maximum bending moment ( M = PL/4) at the shaft center is 0.84 Nm, much more than the threshold value ( M threshold = 0.3041 Nm). From the bending moment v ersus shaft length diagram (Figure 1 1) of superelastic SMA shaft under consideration, length of the sec tion from 0.0362m (Point A) to 0.1638m (point B) is experiencing non - linearly elastic deformation because corresponding bending moment is higher than M threshold . Corresponding value of Δ for threshold bending moment is 0.012. Corresponding strain ε2 (comp ressive) is 0.005725 and strain ε1 (tensile) is 0.006178. While, corresponding compressive stress from Figure 4 (σ – ε curve of SMA shaft) is 362.81 MPa and tensile stress is 389.42 MPa. The maximum compressive stress and tensile stress at shaft mid -span ( point C) is 917.47 MPa and 606.54 MPa, respectively. 030027-78 Range of spin ratio causing deformation beyond proportional limit can be determined from Figure 1 1. For example, M threshold persists for β = 0.79 to 1.35. Again, if high damping ratio ( ζ =0.5) is considered, shaft spin ratio greater than 1.73 will make M greater than M threshold . Figure 1 2 shows M/EI versus shaft length. Since reduced modulus ( Er) is known, therefore, the term M/ErI can also be evaluated for any point on the shaft axis. Therefore, M/ErI vs. shaft length is incorporated in Figure 1 2 to analyze the material non -linearity effect. Next, using Figure 1 2, whirling shaft’s elastic curves with and without material linearity effect can be known by the application of t wo theorems of area moment method (Figure 1 3). Figure 1 3 shows the predicted shapes of the shaft; the maximum deflection at shaft mid - span considering the material as linearly elastic is 54.9 mm but it is 127.8 mm when material non -linearity is considered. Effect of Reduced Modulus ( Er) on Jeffcott Rotor’s Natu ral Frequency ( ωn) can be assumed from the above discussion. Points on the shaft will have different bending moments when it is whirling. When bending moment at any point along the shaft length exceeds M threshold , corresponding Young’s modulus will be lo wer than that for initial linearly elastic model. This will affect the natural frequency as explained below, 𝑘=𝑃 𝛿=48𝐸𝐼 𝐿3 and, 𝜔𝑛=√𝑘 𝑚=√48𝐸𝐼 𝑚𝐿3 . For λ = 1, resonance occurs when ω = ωn. As E reduces to Er, ωn will also reduce to √48𝐸𝑟𝐼 𝑚𝐿3. Obviously, resonance will occur at a lower speed than that expected by linearly elastic model. FIGURE 11. Bending moment ( M) vs. shaft length ( L) w.r.t. Figure 10. FIGURE 12. Bending moment ( M/EI ) vs. shaft length. 00.20.40.60.81 00.04 0.08 0.12 0.16 0.2Moment, M(Nm) Shaft Length, L(m)A BC 01020304050 0 0.05 0.1 0.15 0.2M/EI (m-1) Shaft Length (m)Linearly elastic Non-linearly elastic 030027-89 FIGURE 13. Shaft deflection vs. shaft length considering both linear and non - linear models w . r. t. Figure 1 2. Code Validation An unsteady -state whirl problem that has exact solution [Thomson et al. 2011] was simulated by Siddique [2018] using the same code developed for this research. Simulation results exactly match with the exact solution in Thomson et al. (2011). Experiment A set-up made of MS plate with a 0.5 hp AC motor (single phase), is constructed mainly to demonstrate whirling of shafts and material non -linearity effect. If, the threshold bending moment is exceeded and if conventional engineering material (for example, s tainless steel) is used in place of superelastic SMA as shaft material, inelastic deformation of shaft material (material non -linearity effect) is observed (Figure 1 4). Interested reader s may refer to [Siddique (2018) ] for more information about experiment . (a) (b) FIGURE 14: Whirling shaft’s shapes (a) During experiment and (b) After experiment [Material : SS, shaft dia = 8mm, disc mass ( m) = 250 g, ω = 2298 rpm, β = 2.29]. CONCLUSIONS Steady -state response of a superelastic SMA shaft has been analyzed comprehensively considering effect of material non -linearity and all practically possible modes of whirl of a Jeffcott r otor due to eccentricity of mass. Soundness of the developed mathematical scheme and computer code has been demonstrated. In terms of linearly elastic material, it is found that increased damping greatly reduces whirling amplitude ( r/e). For synchronous wh irl maximum r/e is at spin ratio of unity, because of resonance. However, for large value of spin ratio, value of r/e approaches to unity. Interestingly, all Pd/mg vs. β curves intersect each other for a specific value of spin ratio β = 1.414. Shaft respon se is notably different on two sides of the intersecting point. 00.040.080.120.160.2 0 0.04 0.08 0.12 0.16 0.2Shaft deflection, (m) Shaft Length (m)Linearly elastic With reduced modulus 030027-910 As far as asynchronous whirl is concerned, interestingly, resonance takes place at different β for different values of whirl speed ratio. Higher the whirl speed ratio, lower the value of β that corresponds to the maximum r/e. As material non -linearity is taken into account, it is found that when whirl amplitude is large enough and actual strain is much larger than assumed by linear elastic model. Some portion of the shaft is subjected to stress es (different magnitude in tension and compression) beyond proportional limit. Another important point: resonance is likely at a lower speed than that expected by classical linear elastic model. The following recommendations can be made for future works b ased on the experience gained while achieving the set objectives of this research : (1) Jeffcott Rotor shafts’ cross -sections are not always perfectly circular due to manufacturing imperfections. Effect of imperfect cross -sections on whirling can be considered in future studies. (2) Simply supported end -conditions of the shafts have been considered here. Different boundary conditions can be considered for further analysis. ( 3) Geometric non -linearity issue can be considered in particular for slender s hafts . REFERENCES 1. Kole nda J. Marynarki A. W., (2012), ROK LIII NR 3 (190). 2. Nelson M., (2007), International Journal of COMADEM, 10(3), pp. 2 -10. 3. Rahman, M. A., (2001), “Behavior of Superelastic SMA Columns Under Compression and Torsion.” Ph.D. Thesis, Tohoku University, Sendai, Japan, 2001. 4. Rahman, M. A., Kows er, M. A. (2007), Smart Materials & Structures (IOP Publishing Ltd., UK) 16, pp. 531 -540. 5. Rahman, M. A., Akanda , S. R., Hossain, M . A. (2008), Journal of Intelligent Material Systems and Structures (SAGE Publications, UK) , Feb 2008: vol. 19pp.243 -252. 6. Rahman M. A. and Kowser M.A., (2009), Meccanica, no. DOI 10.1007/s11012 -009-9270 -7. 7. Rajiv T., (2017), “Rotor Systems: Analysis and Identification”, Taylor & Francis. 8. Rao, S.S. (2012), Mechanical Vibrations, Miami: Pearson Education, Inc. 9. Siddique, S. M. M., 2018. “ Analysis of Synchronous and Asynchronous Whirl of Jeffcott Rotor Considering Material Non -linearity”, M. Sc. Engg. Thesis, Department of Mechanical Engineering, Bangladesh University of Engineering & Technology (BUET), Dhaka. 10. Shyoung J. W., Tang F. L. , Shaw H. J. (2014), J. Appl. Mech 81(3), 0345031 –03450310, doi: 10.1115/1.4024670. 11. Swanson E., Powell C. D., Weissman S. “A Practical Review of Rotating Machinery Critical Speeds and Modes.” Sound & Vibration 2005. 12. Thomson W.T., Dahleh M.D., Pad manabhan, C ., (2011), “Theory of Vibration with Applications”, Pearson Education, Inc. 13. Yoon, S.Y., Lin, Z., Allaire, P.E. (2013), (XXI), 275 P., Springer.com/978 -1-4471 -4239 -3. 030027-10

5.0035768.pdf

Appl. Phys. Lett. 118, 062403 (2021); https://doi.org/10.1063/5.0035768 118, 062403 © 2021 Author(s).Highly efficient charge-to-spin conversion from in situ Bi2Se3/Fe heterostructures Cite as: Appl. Phys. Lett. 118, 062403 (2021); https://doi.org/10.1063/5.0035768 Submitted: 31 October 2020 . Accepted: 18 January 2021 . Published Online: 08 February 2021 Dapeng Zhu , Yi Wang , Shuyuan Shi , Kie-Leong Teo , Yihong Wu , and Hyunsoo Yang COLLECTIONS Paper published as part of the special topic on Spin-Orbit Torque (SOT): Materials, Physics, and Devices ARTICLES YOU MAY BE INTERESTED IN Magnetization switching induced by spin–orbit torque from Co 2MnGa magnetic Weyl semimetal thin films Applied Physics Letters 118, 062402 (2021); https://doi.org/10.1063/5.0037178 Large spin Hall angle enhanced by nitrogen incorporation in Pt films Applied Physics Letters 118, 062406 (2021); https://doi.org/10.1063/5.0035815 Enhancement of spin–orbit torque in WTe 2/perpendicular magnetic anisotropy heterostructures Applied Physics Letters 118, 052406 (2021); https://doi.org/10.1063/5.0039069Highly efficient charge-to-spin conversion from in situ Bi2Se3/Fe heterostructures Cite as: Appl. Phys. Lett. 118, 062403 (2021); doi: 10.1063/5.0035768 Submitted: 31 October 2020 .Accepted: 18 January 2021 . Published Online: 8 February 2021 Dapeng Zhu, YiWang, Shuyuan Shi,Kie-Leong Teo, Yihong Wu, and Hyunsoo Yanga) AFFILIATIONS Department of Electrical and Computer Engineering, National University of Singapore, 117576 Singapore Note: This paper is part of the Special Topic on Spin-Orbit Torque (SOT): Materials, Physics and Devices. a)Author to whom correspondence should be addressed: eleyang@nus.edu.sg ABSTRACT Topological insulators (TIs) show bright prospects in exerting spin–orbit torques (SOTs) and inducing magnetization switching in the adjacent ferromagnetic (FM) layer. However, a variation of the SOT efficiency values may be attributed to the ex situ deposition of the FM layer or the complex capping/decapping processes of the protection layer. We have employed an in situ fabrication of Bi 2Se3/Fe heterostructures and investigated the SOT efficiency by spin torque ferromagnetic resonance. An enhanced SOT efficiency and large effectivespin mixing conductance have been obtained especially below 100 K as compared with ex situ methods. The enhancement of the SOT efficiency is attributed to a much thinner interfacial layer (0.96 nm) in the in situ case and thus the enhanced interface spin transparency. Our results reveal the crucial role of interface engineering in exploring highly efficient TI-based spintronic devices. Published under license by AIP Publishing. https://doi.org/10.1063/5.0035768 Current induced spin–orbit torques (SOTs) provide efficient elec- trical manipulation and switching of magnetization, 1–4which are prom- ising for developing next-generation memory and logic devices.5,6The SOT efficiency is a figure of merit to describe the interconversion between a charge and a spin current, and a high SOT efficiency is of key importance for realizing energy efficient magnetic devices. Recently, dueto the spin-momentum-locked topological surface states (TSSs), 7–9three dimensional topological insulators (3D-TIs) have attracted much atten- tion for achieving highly efficient charge-spin interconversion and thus the current induced magnetization switching in the adjacent ferromag- netic (FM) layer.10–21So far, the SOT efficiency in the TI/FM hetero- structures has been characterized by using different techniques, such as spin torque ferromagnetic resonance (ST-FMR),10–12second harmonic magnetometry,13,14and spin pumping.15–18However, a large variation of the SOT efficiency values ranging from /C240.01 to 3.5 has been reported even in the same TI material Bi 2Se3.10,11,16,17,19 Recently, a few works have demonstrated the important role of the interface in the resultant SOT efficiency in heavy metal (HM)/FMbilayers, where the interface might affect the SOTs through the spin mixing conductance of the interface 22–24and the spin relaxation at the interface.25,26However, the role of interface between the topological insulators, such as Bi 2Se3, and the FM layer in TI/FM heterostructures has not been studied. Moreover, most of the SOT studies in the Bi 2Se3/ FM heterostructures so far have used the ex situ deposition of the FMlayer with breaking vacuum and TI film transfer or involving the complex capping/decapping processes for the Se protection layer.10,11,17,19,20Therefore, different Bi 2Se3/FM interface qualities can be obtained using these ex situ fabrication methods, which might be one of the possible reasons for the scattered SOT efficiency valuesreported in Bi 2Se3. In this work, we have employed an in situ film growth of Bi 2Se3/ Fe heterostructures in a molecular beam epitaxy (MBE) system andinvestigated the SOT efficiency of the heterostructures by the ST-FMR technique. We find that the effective SOT efficiency ( h jj)i s/C240.7 at 300 K, and it increases with decreasing temperature and reaches /C244.9 at 20 K. An interfacial SOT efficiency ( kTSS) is also evaluated to extract the TSS originated SOT efficiency, which is determined to be /C241.53 nm/C01at 20 K. Both hjjandkTSSare significantly enhanced as compared with ex situ methods,10,11,16,17,19which is attributed to a bet- ter quality interface and thus the enhanced interface spin transparencyin the in situ grown Bi 2Se3/Fe heterostructures. Bi2Se3thin films and Bi 2Se3/Fe heterostructures were grown on Al2O3(0001) substrates by an MBE system ( supplementary material , S1).Figure 1(a) shows the atomic force microscopy (AFM) image of a 10 quintuple-layer (QL) Bi 2Se3film. Typical triangular terraces with a step height of /C241 nm can be clearly observed, and the root mean square (RMS) surface roughness is /C240.7 nm, indicating a good film quality. A 10 QL Bi 2Se3film was patterned into standard Hall bar Appl. Phys. Lett. 118, 062403 (2021); doi: 10.1063/5.0035768 118, 062403-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apldevices by photolithography and ion milling for the electrical transport measurements. Figure 1(b) shows the temperature dependent resistiv- ity (q)o ft h eB i 2Se3film measured by the four-probe method. We find thatqdecreases as the temperature decreases and remains almost con- stant below 30 K. The sheet carrier concentration ( n2D) at different temperatures is shown in the inset of Fig. 1(b) , which has a weak tem- perature dependence of /C245.3/C21013cm/C02. The temperature depen- dence of qandn2Dshows the typical characteristics of Bi 2Se3.27,28 ST-FMR devices with substrates/Bi 2Se3(10 QL)/Fe (16 nm)/ MgO (2 nm)/SiO 2(4 nm) were fabricated using the methods reported previously.11,29Figure 2(a) shows the schematic of the device structure and the radio frequency (rf) circuit of the ST-FMR measurements. The channel width Wand the gap Lbetween the ground ( G)a n ds i g n a l (S) electrodes are varied to make the device impedance to be /C2450X. Figure 2(b) s h o w st h em e a s u r e dS T - F M Rs i g n a l so fat y p i c a lB i 2Se3/Fe device at different frequencies and room temperature. The ST-FMRspectrum can be decomposed to a symmetric component and an anti- symmetric component, 2,10,11Vmix¼VSFSðHexÞþVAFAðHexÞ,w h e r e Vmixis the detected ST-FMR signal, FSðHexÞ¼D2=½ðHex/C0H0Þ2þ D2/C138is a symmetric Lorentzian function centered at resonant field H0 with a linewidth D,FAðHexÞ¼DðHex/C0H0Þ=½ðHex/C0H0Þ2þD2/C138is an antisymmetric Lorentzian function, and VSandVAare the weight factors for the symmetric and antisymmetric components of thespectrum, respectively. Figure 2(c) shows a representative fitting of the measured ST-FMR signal. The fitted VSvalue is correlated with the in-plane SOT efficiency of the Bi 2Se3/Fe heterostructures. Similar measurements have been also performed for a control device of a Fe single layer. As shown in the inset of Fig. 2(c) ,t h e symmetric component of the ST-FMR signal from the Fe control sample is much smaller than that of the Bi 2Se3/Fe device, support- ing a dominant role of Bi 2Se3in the SOTs observed in the Bi 2Se3/Fe heterostructures. T h ee f f e c t i v eS O Te f fi c i e n c y hjjin the Bi 2Se3/Fe heterostructures can be calculated using the method established in Refs. 10and11and is shown in Fig. 3(a) at different temperatures. Two devices ( in situ devices 1 and 2) have been measured, and they exhibit a similar behav- ior.hjjis/C240 . 7a t3 0 0 Ka n dr e m a i n sa l m o s tc o n s t a n ta st h et e m p e r a - ture decreases down to /C24100 K. However, hjjincreases with further decreasing temperature, and hjjreaches to /C244.9 at 20 K. The observed SOT efficiency in the Bi 2Se3/Fe heterostructures is one to two orders of magnitude higher than the reported values in HMs and Rashba interfaces.2,30,31As a reference, the ST-FMR data from the Bi 2Se3 (10 QL)/Fe (16 nm) control device fabricated using an ex situ method by sputtering are also shown in Fig. 3(a) , with overall smaller hjjvalues compared to those of in situ devices especially below 100 K. The steep increase in hjjwith decreasing temperature below 100 K is different from the reported temperature dependence of the charge-to-spin conversion efficiency in the HMs, which is usually independent or weakly dependent on the temperature.29,32However, a similar temperature dependent behavior has been reported previously by spin-related signal detection in the TI-based devices using other techniques. For example, by spin injection/detection measurements, Li et al. found that the charge-current-induced spin polarization signals increase with decreasing temperature below 150 K in Bi 2Se3.33By the polarized optical spectroscopic technique, Mclver et al. also reported that the TSS-related photocurrent signals increase with decreasing temperature below 60 K in Bi 2Se3.34In addition to the TSS, the bulk states (BSs) and/or two dimensional electron gas (2DEG) Rashba states inevitably coexist in the Bi 2Se3system, forming parasitic charge current channels.35,36These extra current channels shunt the charge current passing through the TSS and thus decrease the TSS contribu- tion and the SOT efficiency in Bi 2Se3. It has been reported that the contributions of BS and/or 2DEG Rashba states decrease with decreas- ing temperature in Bi 2Se3.37Therefore, the suppression of the current shunting from the BS and/or 2DEG Rashba states is a possible reason for the abrupt increase in hjjat low temperatures. In addition, the probability of inelastic scatterings at the interface also decreases at low temperatures, conserving the spin polarizations at the surface conduc- tion states,38which can also contribute to the increase in the SOT effi- ciency at low temperatures. In the above SOT efficiency analyses, hjjis evaluated by using a uniform charge current density JC(A/cm2)i nt h ee n t i r eB i 2Se3 layer. On the other hand, in order to extract the SOT efficiency originated only from the TSS channel, an interfacial SOT efficiency kTSS¼JS/JC-TSS is estimated, where JC-TSS (A/cm) is the two dimen- sional surface charge current density flowing in the TSS.20Since the SOT efficiency shows a much enhanced value at low tempera- tures, our following analyses and discussions are mainly focused on the result at the representative temperature of 20 K. First, we estimate the carrier concentration in each of the TSS ( nTSS), 2DEG FIG. 1. Properties of the 10 QL Bi 2Se3film. (a) Typical AFM image. (b) Temperature dependences of the resistivity ( q) and sheet carrier concentration (inset).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 062403 (2021); doi: 10.1063/5.0035768 118, 062403-2 Published under license by AIP Publishing(n2DEG ), and BS ( n2D-Bulk ) channels, as well as the current shunting of the BS and 2DEG Rashba states ( supplementary material , S5).20 As shown in Fig. 3(b) ,nTSS,n2DEG ,a n d n2D-Bulk are obtained to be /C241.72/C21013,8 . 5/C21012, and 2.8 /C21012cm/C02respectively, which agree well with the reported values.39,40Then, we extract the inter- facial SOT efficiency kTSS,20and the resultant kTSSis/C241.53 nm/C01 at 20 K, which is much higher than the reported charge-to-spin conversion efficiency of /C240.01–0.8 nm/C01in the Bi 2Se3/FM hetero- structures fabricated using ex situ methods.16,20It should be noted that the actual kTSSm a yb ee v e nh i g h e rt h a n1 . 5 3 n m/C01 because the Rashba splitting states can induce opposite spin polar- izations and partially cancel the spin accumulations induced bythe TSS. 10,41 As mentioned above, the SOT efficiency can be significantly affected by the spin transmission across the TI/FM interface, which can be quantified with the effective spin mixing conductance g"# eff.23,24 g"# effcan be extracted from the broadening of the ST-FMR signal line- width using equations g"# eff¼4pMSt glBðaBi2Se3=Fe/C0aFeÞand a¼c 2pfD, where Msandtare the saturation magnetization and thickness of Fe, respectively, gis the g-factor, lBis the Bohr magneton, and fis the fre- quency of rf currents.23,24g"# effis determined to be /C242.63/C21020m/C02 (7.65 /C21019m/C02) at 20 K (300 K) in the Bi 2Se3/Fe heterostructures.Table I shows that the g"# effof our in situ fabricated Bi 2Se3/Fe is much higher than that of the ex situ fabricated Bi 2Se3/Fe and other Bi 2Se3/ FM samples using ex situ methods. Previous electrical transport studies have revealed that air expo- sure can result in n-type doping to Bi 2Se3, and the Shubnikov–de Haas (SdH) oscillation measurements have shown that the TSS wasdegraded by air exposure. 42,43It has been also found that the Se resid- uals or decomposition of Bi 2Se3can be easily induced in the decapping process of the Se protection layer due to the inappropriate decappingtemperature or nonuniform heating. 44On the contrary, the in situ film deposition can avoid the air exposure of the TSS and eliminatethe subtle Se decapping processes and thus can reduce the possible film degradations and surface contaminations. To better understand the interface of the in situ andex situ samples, we further performed high-resolution transmission electron microscopy (HRTEM) andenergy dispersive x-ray spectroscopy (EDX) measurements on thecross-sectional thin film samples of the in situ andex situ fabricated Bi 2Se3/Fe. As can be observed from the HRTEM results shown in Figs. 4(a) and4(b), both the in situ andex situ films have an interfacial layer with atomic intermixing/diffusion, but the thickness of the inter-facial layer for the in situ film is much thinner than the ex situ one, and the interfacial layer of the ex situ film shows a crystalline structure distinctively different from that of Bi 2Se3. In addition, more Se spreads FIG. 2. (a) ST-FMR bilayer and measurement circuit with the illustration of the SOTs in the Bi 2Se3/Fe heterostructure. (b) ST-FMR signals of a Bi 2Se3(10 QL)/Fe (16 nm) device measured at frequencies of 6–12 GHz at 300 K. The power of the applied rf current from a signal generator (SG) is 15 dBm. (c) The fitting of a typical S T-FMR signal of the Bi 2Se3/Fe device with a symmetric Lorentzian ( VSFS) component and an antisymmetric Lorentzian ( VAFA) component. The inset of (c) shows the symmetric Lorentzian (VSFS) component of the ST-FMR signal from a control sample with a Fe single layer.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 062403 (2021); doi: 10.1063/5.0035768 118, 062403-3 Published under license by AIP Publishingin the interfacial region for the ex situ sample than for the in situ one from the EDX results shown in Fig. 4(c) . Therefore, our in situ sample fabrication process can induce a better quality interface, which conse- quently provides an enhanced effective spin mixing conductance and SOT efficiency. The effective spin mixing conductance g"# effobtained above describes the net spin currents injected from the Bi 2Se3layer to the adjacent Fe layer, which includes all the aspects in the spin transmis- sion process, such as the spin memory loss (SML), interface spin trans- mission, and spin back flow. According to a theoretical model,25the effective spin mixing conductance g"# effcan be expressed as g"# eff¼h 1 g"#þe2 h2lsf rR/C3 lsf rdsinhdþR/C3coshdtanhd lsf/C0/C1i/C01 ,w h e r e g"#is the interfacial spin mixing conductance determined by the matching of the electronic bands in the two materials on either sides of the interface, lsfanddare the spin diffusion length and thickness of Bi 2Se3,r e s p e c t i v e l y , R/C3is the effective interface specific resistance, dis the SML parameter, eis the electronic charge, and his the Planck constant. If there is no SML, i.e., d¼0, by using the measured conductivity and the reported spin diffu- sion length of Bi 2Se3,16,18we obtain a negative interface spin mixing conductance g"#(supplementary material ,S 6 ) .T h i sn e g a t i v e g"#value seems not physical.16–18This result indicates that just the electronic band mismatching and spin back flow cannot fully explain the spin transmission process across the interface in the Bi 2Se3/FM heterostruc- tures. The additional effect of SML, which describes the spin relaxation at the interface, needs to be further considered. We find that a signifi- cant SML with d/C240.8/C02.7 (lower bound) exists in the in situ fabri- cated Bi 2Se3/Fe heterostructures by using the reported R/C3values of 0.1–3.1 f Xm2(supplementary material , S6), which is much higher FIG. 3. (a) Temperature dependent effective SOT efficiency hjj.In situ devices 1 and 2 represent two ST-FMR devices of the in situ fabricated Bi 2Se3(10 QL)/Fe (16 nm) heterostructures. The ex situ device represents the control device of the ex situ fabricated Bi 2Se3(10 QL)/Fe (16 nm) heterostructure by sputtering. (b) Analysis results of the sheet carrier concentration in each of the TSS ( nTSS), 2DEG (n2DEG ), and BS ( n2D-Bulk ) channels, as well as the interfacial SOT efficiency kTSS originated from the TSS channel. TABLE I. Estimated values of the effective spin mixing conductance ( g"# eff) and SOT efficiency ( hjj) for the in situ and ex situ fabricated Bi 2Se3/Fe heterostructures in this work, as well as those reported from other Bi 2Se3/FM heterostructures fabricated using conventional ex situ methods. Sample structure g"# eff(m–2) hjj References Bi2Se3/Fe (300 K) 7.65 /C210190.7 This work ( in situ ) Bi2Se3/Fe (20 K) 2.63 /C210204.9 This work ( in situ ) Bi2Se3/Fe (300 K) 1.07 /C210190.2 This work ( ex situ ) Bi2Se3/Fe (20 K) 2.63 /C210190.8 This work ( ex situ ) Bi2Se3/Py (RT) 1.514 /C210190.0093 16 Bi2Se3/CoFeB (RT)1.2–26 /C210190.021–0.43 17 Bi2Se3/YIG (RT) 3–7 /C210180.0007–0.022a18 Bi2Se3/Py (RT) 6.09 /C21019b2 10 aThe hjjvalues were converted from the reported inverse Edelstein effect length kIEE (1.1–35 pm) and spin diffusion length lsf(1.6 nm) from Ref. 18using the equation hjj¼kIEE/lsf. bThe g"# effvalue was estimated from the reported Mstvalue (14.2 mA) and the extracted linewidths DBS=Py(3.3 mT) and DPy(1.5 mT) for the 8 GHz ST-FMR data of the Bi2Se3(8 nm)/Py(16 nm) and Py (16 nm) samples, respectively, from Ref. 10, using the equation g"# eff¼2Msct fglBDBS=Py/C0DPy/C0/C1. FIG. 4. (a) and (b) Cross-sectional HRTEM images of the in situ and ex situ fabri- cated Bi 2Se3/Fe heterostructures, respectively. (c) Comparison of EDX line-scan results for the in situ and ex situ fabricated Bi 2Se3/Fe heterostructures with the Bi2Se3/Fe interfacial region denoted by light blue color.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 062403 (2021); doi: 10.1063/5.0035768 118, 062403-4 Published under license by AIP Publishingthan the reported dvalue of 0 /C00.96 in the normal metal systems.45In this sense, the high quality of the interface between the TI and FMlayers is crucial to decrease the SML and thus realize the higher spin transparency and larger SOTs. We can roughly estimate the interfacial thickness ( t i)t ob e /C240.96 nm and 1.68 nm for the in situ andex situ fabricated Bi 2Se3/Fe samples, respectively, from the HRTEM results. Assuming that the two samples have the same interfacial spin diffusion length ( lsf-i),dwould be about 1.75 times higher for the ex situ Bi2Se3/Fe sample than that of the in situ case using d¼ti=lsf/C0i.45Therefore, more efficient SOTs in TI/FM heterostructures can be anticipated through appropriate interface engineering such as matching the electronic bands and reducing the contamination and interdiffusion at the interface, which will benefit for driving the efficient domain wall motion and mag-netization switching in the TI-based SOT applications. In summary, we have investigated the SOT efficiency in the in situ fabricated Bi 2Se3/Fe heterostructures by the ST-FMR technique. A significantly enhanced SOT efficiency has been observed as com- pared with those fabricated using ex situ methods. We find that the effective spin mixing conductance at the interface is significantly enhanced in the in situ Bi2Se3/Fe bilayers. Our observations reveal that the interface can play an important role in the SOTs of TI/FM hetero-structures and suggest that the interface engineering may serve as a tuning knob for exploring highly efficient TI-based spintronic devices. See the supplementary material for sample preparation and mea- surement methods, weak antilocalization effects in the Bi 2Se3film, power dependent ST-FMR signals, comparison of the field-like and damping-like SOT, evaluation of the interfacial SOT efficiency, and evaluation of the spin memory loss. This research was supported by the SpOT-LITE programme (A/C3STAR Grant No. 18A6b0057) through RIE2020 funds and Singapore Ministry of Education AcRF Tier 1 (R-263-000-D60-114). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V. Costache, S. Auffret, S. Bandiera, B. Rodmacq, A. Schuhl, and P. 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J. Appl. Phys. 129, 064305 (2021); https://doi.org/10.1063/5.0039777 129, 064305 © 2021 Author(s).Energetics and magnetism of topological graphene nanoribbons Cite as: J. Appl. Phys. 129, 064305 (2021); https://doi.org/10.1063/5.0039777 Submitted: 07 December 2020 . Accepted: 27 January 2021 . Published Online: 11 February 2021 Mari Ohfuchi , and Shintaro Sato ARTICLES YOU MAY BE INTERESTED IN Two types of corner states in two-dimensional photonic topological insulators Journal of Applied Physics 129, 063104 (2021); https://doi.org/10.1063/5.0039586 High thermoelectrical figure of merit in chiral topological superconductor junctions Journal of Applied Physics 129, 064306 (2021); https://doi.org/10.1063/5.0039910 Ultrafast investigation and control of Dirac and Weyl semimetals Journal of Applied Physics 129, 070901 (2021); https://doi.org/10.1063/5.0035878Energetics and magnetism of topological graphene nanoribbons Cite as: J. Appl. Phys. 129, 064305 (2021); doi: 10.1063/5.0039777 View Online Export Citation CrossMar k Submitted: 7 December 2020 · Accepted: 27 January 2021 · Published Online: 11 February 2021 Mari Ohfuchia) and Shintaro Sato AFFILIATIONS Fujitsu Laboratories Ltd. and Fujitsu Limited, 10-1 Morinosato-Wakamiya, Atsugi, Kanagawa, 243-0197, Japan a)Author to whom correspondence should be addressed: mari.ohfuti@fujitsu.com ABSTRACT The topological properties of graphene nanoribbons (GNRs) have received a significant amount of attention in emerging fields such as spintronics and quantum computing. This study is focused on the energetics and magnetism of symmetry-protected junction state arrays,which are realized in the alternating periodic structures of two topologically different armchair GNRs. We found that the antiferromagneticstates require at least eight unit cells for each segment of the periodic armchair GNRs, where the armchair GNRs whose numbers of carbonatoms in a row are seven and nine are connected with a junction structure. We also found the junction structure that provides more stable antiferromagnetic states. Furthermore, we propose an end (armchair GNRs/vacuum interface) structure to avoid disturbing the global topological properties of the junction state array. This means that if the topological end states (non-trivial phases of the Su, Schrieffer, andHeeger model or Majorana fermions) exist, they are properly formed at the endmost junctions without the requirement for extra effort suchas long end extension. We believe that this study can add new guidelines and challenges for realizing graphene-based quantum computing. Published under license by AIP Publishing. https://doi.org/10.1063/5.0039777 I. INTRODUCTION The experimental demonstration in the bottom-up synthesis of graphene nanoribbons (GNRs) has introduced new possibilitiesand challenges for their electronics applications. 1Atomically precise structures based on precursor monomers firmly define the electronic properties of the GNRs depending on the width, atomic arrangement at the edges, and even the chemical modifications.1–14 Theoretical and experimental attempts were also made to fabricate atomically controlled heterojunctions of two different GNRs.15–17 With this progress and the recent significant interest in topological insulators,18–20a symmetry-protected topological phase in GNRs was proposed.21It was predicted on the basis of the tight-binding model that the Z 2invariant,22which characterizes the topology of GNRs, depends on how the unit cell is set.21The connection of two unit cells of GNRs with different Z 2invariants produces local- ized junction states in the bandgap.18–20It was also demonstrated that the junction states can act as spin centers and form an antifer-romagnetic (AFM) chain. 12One-dimensional arrays of the junction states were immediately realized as in-gap states in a topologicallydesigned GNR superlattice, which also demonstrated the possibility of designing end (GNR/vacuum interface) states. 23,24This was extended to the flexible design of the Su, Schrieffer, and Heeger(SSH) Hamiltonian,25,26which determines the global topological property of in-gap state bands of the GNRs.27,28These findings present a method for topological band engineering in one- dimensional materials and graphene-based optoelectronics,29spin- tronics, and also quantum computing owing to the expectation thatMajorana fermions will appear at the end of such topologicalGNRs on a superconductor. 30–32In this study, we focus on the energetics and magnetism of topological armchair GNRs (AGNRs) in terms of their periodic length and connected structure, whiletaking into consideration their practical applications. We alsopropose an end atomic structure that does not disturb the globaltopological property of junction state arrays. II. COMPUTATIONAL DETAILS All the density functional theory (DFT) calculations in this study were performed using the OpenMX code, which comprises the use of pseudoatomic orbitals (PAO) as the basis function set. 33–36The exchange-correlation potential was treated with a gen- eralized gradient approximation with or without spin polarization.37 The electron –ion interaction is described by norm-conserving pseudopotentials.38,39The PAOs specified by C7.0-s2p2d1 and H 7 . 0 - s 2 p 1w e r eu s e d ,w h e r eCa n dHa r ea t o m i cs y m b o l s .T h en u m b e rJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 064305 (2021); doi: 10.1063/5.0039777 129, 064305-1 Published under license by AIP Publishing.7.0 represents the cutoff radii of the PAOs in bohrs. For example, s2 indicates the employment of two orbitals for the s component. The energy band occupation is smeared by theFermi distribution function (temperature T= 300 K) to perform stable self-consistent calculations. These methods have been suc-cessfully used to describe the atomic and electronic structures of graphene and GNRs. 11–14,40–44 The atomic structure of uniform N-AGNRs was fully relaxed, where Nrepresents the number of carbon atoms in a row of AGNRs (see Fig. S1 in the supplementary material ). A periodic boundary condition was applied, and 30 kpoints were used for the reciprocal space integration. For geometrical optimization, the con- vergence criterion of the forces acting on the atoms was set as0.1 eV/nm. The number of kpoints and the convergence criterion of the forces were chosen for the numerical error to be less than0.02 meV/atom, which provides the accuracy required for describ- ing the energetics of the connected AGNRs examined in this study. The optimum unit length was determined for each N. The funda- mental features of uniform N-AGNRs used in this study are sum- marized in Table I (see Fig. S2 in the supplementary material , for the band diagrams), along with the Z 2invariants for each type of unit cell21a and b, which are presented in Fig. 1(a) . As shown in Table I , all the N-AGNRs with the optimized structure have a bandgap, despite being gapless in the tight-binding picture.45,46It has been shown that the DFT methods used in this study underes- timate quasiparticle energies; however, the DFT results can qualita- tively reproduce the size dependence of the bandgap and themagnetic states of GNRs presented with a many-electron Green ’s function approach. 46Each N-AGNR has two different Z 2invariants depending on the unit cell, and the N-AGNRs are divided into two categories, N= {5, 7, 9} and {11, 13, 15}, based on their topology. The geometrical optimization of alternating periodic AGNRs was performed while maintaining their total length. When danglingbonds were introduced at the junction by connecting two AGNRsegments or at the end of the AGNRs, the dangling bonds were ter- minated by hydrogen atoms. In the case, where these terminations caused carbon atoms bonded to two hydrogen atoms, i.e., CH 2 groups, it was also considered that the CH 2groups were replaced by hydrogen atoms to reduce the junction formation energy. Thejunction formation energy is required to compare the relative stability of structures with different number of atoms and defined asE J¼E/C0P iEi/C0nC2H4EC2H4/C0nH2EH2, where Eis the total energy for the AGNRs in question; Ei,EC2H4, and EH2are those for each compositional uniform AGNR, C 2H4, and H 2, respectively; and nC2H4and nH2are the number of each molecule determined to offset the difference in the number of atoms. When we found some states in the bandgap of the compositional uniform AGNRs in the calculation without spin polarization, spin-polarized calculationswere also performed with ferromagnetic (FM) and AFM initialconditions. However, the FM states were not obtained or were less stable than the AFM states unless specified otherwise, where we only presented the results for the AFM states. FIG. 1. (a) Two types of unit cells of N-AGNRs. The gray and white spheres represent carbon and hydrogen atoms, respectively. The a- and b-type unit cells are indicated by red (labeled by “a”) and blue (labeled by “b”) rectangles, respectively, using 7-AGNR as an example. Atomic geometry of the junctions:(b) 7b9a, (c) 7a9b, (d) 7a9a, (e) 7b9b, and (f) 7b9b 0. The green (labeled by “b0”) rectangle indicates the b0-type unit cell.TABLE I. Features of N-AGNRs used in this study. Number of atoms, length of the unit cell, bandgap ( Eg), and Z2invariants for each type of unit cell, a and b, are listed. N Number of atoms Length (nm) Eg(eV)Z2 ab 5 14 0.2491 0.40 1 0 7 18 0.2488 1.55 1 09 22 0.2483 0.78 1 011 26 0.2479 0.18 0 113 30 0.2478 0.87 0 1 15 34 0.2478 0.49 0 1Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 064305 (2021); doi: 10.1063/5.0039777 129, 064305-2 Published under license by AIP Publishing.III. RESULTS AND DISCUSSION A. Alternating periodic AGNRs We start with the alternating periodic structures of 7- and 9-AGNRs because both uniform AGNRs have a relatively wide bandgap, as shown in Table I . When junction states exist, they can be more localized at the junctions, which makes the discussion ofthe existence of the junction states clearer. As described above, there are two types of unit cells, a and b, for AGNRs, and we thus have four types of connections, as shown in Figs. 1(b) –1(e). First, we describe the results for 7b( n)9a(n)-AGNRs [Fig. 1(b) ], where two AGNRs can be connected without any dan- gling bond, no extra hydrogen termination is required, and nis the number of repeated unit cells for each AGNR segment. Figure 1(b) presents the optimized atomic structure of the junction for 7b(16)9a(16)-AGNRs without spin polarization, that is, the nonmagnetic(NM) states. We found no remarkable change in the optimizedatomic structure. Figure 2(a) presents the junction formation energy for the NM states, E J(NM), along with that for the AFM states, EJ(AFM). EJ(NM) increases as nincreases and becomes satu- rated at n= 16. In terms of the energy-band structure, two energy bands, which are symmetric with respect to the chemical potential,appear partially or completely in the bandgap of the compositional AGNRs [see Figs. S3(a) and S3(b) in the supplementary material ]. For a small value of n, both the bandgap and width are large, and the wave functions are spread over the entire GNRs. However, for a largevalue of n, the bandgap and width become narrower, and the wave functions are more localized at the junction [see Figs. S3(c) –S3(e) in thesupplementary material ]. These band structures are considered astopological junction states and can be understood via two different hopping parameters between the junction states 23depending on the periodic length. The energies of the lowest unoccupied band of the NM states at the Γpoint are presented in Fig. 2(b) along with those of the AFM states. They are represented by Δ(NM) and Δ(AFM), respectively. Δ(NM) is 0.0071 eV for n= 16 and increases as n decreases. In contrast, Δ(AFM) is nearly constant at 0.07 eV and over- lapsΔ(NM) at n=8 . F o r n< 8, the AFM states were not obtained as per our calculations. This is consistent with the fact that the finite dif- ference of EJ(NM) and EJ(AFM) at a large value of nis almost 0 at n=8 [ Fig. 2(a) ].Figure 2(c) presents the spin-dependent wave func- tions of the highest occupied band at the Γpoint for AFM-7b(16)9a (16)-AGNRs. It is found that each spin state is localized at each junc- tion. The wave functions are spread to a greater extent in 9-AGNRs, wherein the bandgap is narrower than that of 7-AGNRs. FIG. 2. Results for 7b( n)9a(n)-AGNRs. (a) Formation energy. (b) Energies of the lowest unoccupied band with NM and AFM states at the Γpoint. (c) Spin-dependent wave functions of the highest occupied band at the Γpoint for n= 16. Blue and green colors indicate the plus and minus contours, respectively, for the up spin, while the red and yellow colors indicate the plus and minus con-tours, respectively, for the down spin.TABLE II. EJ,Δ(NM), and Δ(AFM) of 7 –9- and 9 –11-AGNRs with b –a, a –b, a –a, b–b, and b –b0connections. The gray-highlighted values indicate that Δis narrower than those of the compositional uniform AGNRs. All the values are in electron volts. AGNRs Connection b –aa –ba –ab –bb –b0 7–9 EJ 0.79 4.16 0.83 1.37 0.97 △(NM) 0.0071 0.000 062 0.45 0.44 0.0073 △(AFM) 0.068 0.17 …… 0.099 9–11 EJ 0.11 4.23 1.07 … 0.35 △(NM) 0.18 0.006 0.068 … 0.18 FIG. 3. Results for 7b( n)9b0(n)-AGNRs. (a) Formation energy. (b) Energies of the lowest unoccupied band with NM and AFM states at the Γpoint. (c) Spin-dependent wave functions of the highest occupied band at the Γpoint for n= 16. Blue and green colors indicate the plus and minus contours, respectively, for the up spin, while the red and yellow colors indicate the plus and minus con-tours, respectively, for the down spin.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 064305 (2021); doi: 10.1063/5.0039777 129, 064305-3 Published under license by AIP Publishing.The combination of 9- and 11-AGNRs is a good counterpart because these AGNRs belong to different topological categories, while 7- and 9-AGNRs have the same topology ( Table I ). It is expected that the two combinations present the opposite resultsregarding the existence of the topological junction states. We foundthat n= 16 for the AGNRs is sufficient for describing the long peri- odic length as observed in Figs. 2(a) and2(b). The E JandΔvalues are listed in Table II . As expected, Δ(NM) of the 9b11a-AGNRs is greater than half the bandgap of 11-AGNRs, which means that9b11a-AGNRs do not comprise topological junction states. Thegreater Δ(NM) than half the bandgap is due to finite size effects. It was also found that E Jof 9b11a-AGNRs is much smaller thanthat of 7b9a-AGNRs, despite the atomic structures being substan- tially equivalent. Furthermore, we examined the combinations 5b7a-AGNRs and 13b15a-AGNRs. They have the same topologi-cal junction states as those of 7b9a-AGNRs. However, we foundthe AFM states only with n≥1 6f o r5 b 7 a - A G N R sa n dn oA F M states for 13b15a-AGNRs, probably because the wave function w a ss p r e a do v e rt h e5 - A G N R so r1 5 - A G N R ss e g m e n t se v e nf o r n=1 6( s e eF i g .S 4i nt h e supplementary material ). We move on to the other connection types. The a –b connection requires two extra hydrogen terminations per junction [ Fig. 2(c) ]. As a result, there are four CH 2groups in the unit cell. As shown inTable II ,EJis very large for both the 7 –9a n d9 –11-AGNRs. FIG. 4. Atomic structure models for AGNRs with ends; (a) 7a(3), (b) 7b(3), and (c) 7b0(3). The gray and white spheres represent carbon and hydro-gen atoms, respectively. The a-, b-, and b 0-type unit cells are indicated by red (labeled by “a”), blue (labeled by “b”), and green (labeled by “b0”) rectan- gles, respectively. TABLE III. Energy (eV) of the states in the bulk bandgap for isolated N-AGNRs with a-, b-, and b0-end structures for n= 20. The energy values smaller than 1.0 × 10−4eV are presented as 0.0 eV . N ab b0 5 −0.022, 0.022 −0.018, −0.0023, −0.0023, 0.024 … 7 0.0, 0.0 −0.0039, −0.0039,0.0039, 0.0039 … 9 0.0, 0.0 −0.0049, −0.0049,0.0049, 0.0048 … 11 −0.050, −0.0023, −0.0023, 0.063 −0.051, −0.0092, −0.0045, 0.0039, 0.0039, 0.068 −0.051, 0.051 13 −0.0054, −0.0054, 0.0051, 0.0058 −0.015, −0.015, 0.0045 0.0045, 0.010, 0.010 −0.00024, 0.00024 15 −0.0079, −0.0078,0.0078, 0.0078 −0.015, −0.015, 0.0034 0.0035, 0.011, 0.011 0.0, 0.0Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 064305 (2021); doi: 10.1063/5.0039777 129, 064305-4 Published under license by AIP Publishing.The atomic structure at the junction is significantly changed. The CH 2groups are deviated from the AGNR plane [see Fig. S5(a) in thesupplementary material ].Δ(NM) of 7a9b-AGNRs is smaller than that of 7b9a-AGNRs by two orders; in contrast, Δ(AFM) of the former is greater than that of the latter. This is consistent with morelocalized wave functions at the junction of 7a9b-AGNRs, especially around CH 2groups [see Fig. S5(b) in the supplementary material ]. It is noteworthy that 9a11b-AGNRs also exhibit a small Δ(NM), as listed in Table II , which means that these states are not simple topo- logical junction states. A larger Δ(NM) of 9a11b-AGNRs than that of 7a9b-AGNRs by two order can be explained by the narrow bandgap of 11-AGNRs. We should say that if the CH 2groups are replaced by hydrogen atoms to reduce the junction formation energy, these a –b structures form b –a connections. The EJvalues of 7a9a- and 7b9b-AGNRs are 0.83 and 1.37 eV, respectively, which are slightly greater than those of 7b9a-AGNRs and 9b11a-AGNRs. This can be attributed to the former ’s atomic structures because they do not have junction states; asshown in Table II , 7a9a- and 7b9b-AGNRs have a large Δof 0.45 and 0.44 eV, respectively. In the case of 7a9a-AGNRs, thehydrogen atoms are deviated from the AGNR plane owing to steric hindrance. 7b9b-AGNRs comprise CH 2groups. In con- trast, 9a11a-AGNRs exhibit topological junction states with asmall Δ(NM) value of 0.068, which is greater than that of 7b9a-AGNRs by one order owing to the small bandgap of 11-AGNRs. If the CH 2groups are replaced by hydrogen atoms in the b–b connection, it becomes a new connection type b –b0,w h i c h is presented in Fig. 1(f) . It can also be said that an eight-atom row exists between the 7b and 9a unit cells. Overall, the formationenergy and the presence or the absence of the topological junctionstates in the case of the b –b 0connection are very similar to those of the b–a connection, as shown in Table II . The results for 7b( n)9b0 (n)-AGNRs are summarized in Fig. 3 . The AFM states were obtained forn≥8 in the same manner as that in the case of the b –a connec- tion. However, the difference between EJ(NM) and EJ(AFM), and Δ (AFM) are greater for the b –b0connection type than those for the b–a connection, which means that the AFM chain can be realized at a higher temperature by using the b –b0connection. B. End (AGNRs/vacuum interface) structures Next, we consider the effect of the end (AGNRs/vacuum inter- face) structures. Prior to the end of the topological AGNRs, weexamined isolated AGNRs with ends, as shown in Fig. 4 , where n= 3; however, we performed calculations for n= 20. Here, b 0(n) denotes b0(1)b( n−2)b0(1). The 5-, 7-, 9-, 11-, 13-, and 15-AGNRs were examined. As shown in Table III , the 5-, 7-, and 9-AGNRs comprise two, four, and no in-gap states for the a-, b-, and b0-end structures, respectively; however, 11-, 13-, and 15-AGNRs comprisefour, six, and two in-gap states for the a-, b-, and b 0-end structures, respectively. We found that the N-AGNRs are divided into two cat- egories according to the number of in-gap states in the same way as those based on the Z 2invariant of the unit cell shown in Table I . There are some states with a large absolute energy value, forexample, −0.050 and 0.63 eV for the a-end structure of 11-AGNRs. These states have antisymmetric wave functions with respect to the center of the ribbon and are hybridized with the bulk band edgestates with the same symmetry, unlike the states with a small abso-lute energy value, for example, −0.0023 eV for the a-end structure of 11-AGNRs (see Fig. S6 in the supplementary material ). FIG. 5. Atomic structure models for topological AGNRs with ends: (a) 9a(2)7b(2)9a(2)-AGNR and (b) 7b0(2)9a(2)7b0(2)-AGNR. The gray and white spheres represent carbon and hydrogen atoms, respectively. The a-, b-, and b0-type unit cells are indicated by red (labeled by “a”), blue (labeled by “b”), and green (labeled by “b0”) rectan- gles, respectively. (c) Spin-dependent wave functions of the highest occupied states for AFM-7b0(20)9a(20)7b0(20)-AGNR. Blue and green colors indicate the plus and minus contours, respectively, for the up spin, while the red and yellow colors indicate the plus and minus contours, respectively, for the down spin.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 064305 (2021); doi: 10.1063/5.0039777 129, 064305-5 Published under license by AIP Publishing.We examined the topological AGNRs with ends derived from 7b9a-AGNRs. Two cases of end structures are considered,9a and 7b 0, as presented in Fig. 5(a) [9a(n)7b(n)9a(n)-AGNR] and Fig. 5(b) [7b0(n)9a(n)7b0(n)-AGNR], where n= 2; however, we per- formed calculations for n= 20. Here, b0(n) denotes b0(1)b( n−1) or b(n−1)b0(1). For the 9a-end structure, we found eight states in the bandgap. The end states have a finite negative energy or zero ener-gies, which results in a non-zero total magnetic moment for the initial AFM conditions (see Fig. S7 in the supplementary material ). However, we obtained only two occupied states in the bandgap forthe 7b 0-end structure. The spin-dependent wave functions are obtained as presented in Fig. 5(c) . Each spin is localized at each junc- tion. The total magnetic moment is zero, and no extra end states are found. We thus confirmed that 9a- and 7b0-end structures provide two and no end states in the bandgap of the compositional uniformAGNRs for topological AGNRs. Finally, we examined 7b 0(2)9b0(8)7b(8)9b0(8)7b(8)9b0(8)7b0 (2)-AGNR presented in Fig. 6(a) to confirm our findings. This model has the shortest periodicity ( n= 8) required for realizing AFM states and the short end structure ( n= 2). We found six occu- pied states in the bandgap of 9-AGNRs, as shown in Fig. 6(b) .E a c h state is localized at each junction, and no extra end states are found. The total magnetic moment is zero. This means that if topo- logical end states exist —for example, the non-trivial phases of theSSH model or magnetic-chain-induced Majorana fermions —they are properly formed at the endmost junctions without the require-ment for extra effort such as long end extention. 23,24,27 IV. CONCLUSION We examined the topological armchair graphene nanoribbons for the case wherein two topologically different armchair graphenenanoribbons were connected. The alternating periodic 7b( n)9a (n)-armchair graphene nanoribbons exhibited antiferromagnetic properties only for n≥8. We found no other combination that pro- vided antiferromagnetic states for a smaller value of n; however, we found that antiferromagnetic-7b9b 0-armchair graphene nanorib- bons were more stable. The end states of the armchair graphene nanoribbons were also examined. We found an end structure thatdid not disturb the global topological property observed at the endof the symmetry-protected junction state array. These resultspresent new possibilities for topological band engineering in one- dimensional materials as well as graphene-based spintronics and quantum computing. SUPPLEMENTARY MATERIAL See the supplementary material for fundamental properties of AGNRs and supplementary data for connected AGNRs. FIG. 6. (a) Optimized atomic structure for the 7b0(2)9b0(8)7b(8)9b0(8)7b(8)9b0(8)7b0(2)-AGNR. The gray and white spheres represent carbon and hydrogen atoms, respec- tively. (b) Spin-dependent wave functions of the occupied in-gap states. Blue and green colors indicate the plus and minus contours, respectively, f or the up spin, while the red and yellow colors indicate the plus and minus contours, respectively, for the down spin.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 064305 (2021); doi: 10.1063/5.0039777 129, 064305-6 Published under license by AIP Publishing.ACKNOWLEDGMENTS This study was supported by CREST JST (No. JPMJCR15F1). We thank the Research Institute for Information Technology,Kyusyu University, Japan for providing access to the Fujitsu PRIMERGY CX2550/CX2560 M4 supercomputer system for per- forming the calculations in this study. DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material . REFERENCES 1J. Cai, P. Ruffieux, R. Jaafar, M. Bieri, T. Braun, S. Blankenburg, M. Muoth, A. P. Seitsonen, M. Saleh, X. Feng, K. Müllen, and R. 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5.0038729.pdf

Appl. Phys. Lett. 118, 100502 (2021); https://doi.org/10.1063/5.0038729 118, 100502 © 2021 Author(s).Quantum dots as potential sources of strongly entangled photons: Perspectives and challenges for applications in quantum networks Cite as: Appl. Phys. Lett. 118, 100502 (2021); https://doi.org/10.1063/5.0038729 Submitted: 25 November 2020 . Accepted: 12 February 2021 . Published Online: 09 March 2021 Christian Schimpf , Marcus Reindl , Francesco Basso Basset , Klaus D. Jöns , Rinaldo Trotta , and Armando Rastelli COLLECTIONS This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN Progress on large-scale superconducting nanowire single-photon detectors Applied Physics Letters 118, 100501 (2021); https://doi.org/10.1063/5.0044057 Progress in quantum-dot single photon sources for quantum information technologies: A broad spectrum overview Applied Physics Reviews 7, 021309 (2020); https://doi.org/10.1063/5.0010193 THz intersubband electroluminescence from n-type Ge/SiGe quantum cascade structures Applied Physics Letters 118, 101101 (2021); https://doi.org/10.1063/5.0041327Quantum dots as potential sources of strongly entangled photons: Perspectives and challenges for applications in quantum networks Cite as: Appl. Phys. Lett. 118, 100502 (2021); doi: 10.1063/5.0038729 Submitted: 25 November 2020 .Accepted: 12 February 2021 . Published Online: 9 March 2021 Christian Schimpf,1,a) Marcus Reindl,1Francesco Basso Basset,2 Klaus D. J€ons,3 Rinaldo Trotta,2 and Armando Rastelli1 AFFILIATIONS 1Institute of Semiconductor and Solid State Physics, Johannes Kepler University, Linz 4040, Austria 2Department of Physics, Sapienza University of Rome, 00185 Rome, Italy 3Department of Physics, Paderborn University, 33098 Paderborn, Germany a)Author to whom correspondence should be addressed: christian.schimpf@jku.at ABSTRACT The generation and long-haul transmission of highly entangled photon pairs is a cornerstone of emerging photonic quantum technologies with key applications such as quantum key distribution and distributed quantum computing. However, a natural limit for the maximum transmission distance is inevitably set by attenuation in the medium. A network of quantum repeaters containing multiple sources of entangled photons would allow overcoming this limit. For this purpose, the requirements on the source’s brightness and the photon pairs’degree of entanglement and indistinguishability are stringent. Despite the impressive progress made so far, a definitive scalable photon sourcefulfilling such requirements is still being sought after. Semiconductor quantum dots excel in this context as sub-Poissonian sources of polari-zation entangled photon pairs. In this work, we present the state-of-the-art set by GaAs based quantum dots and use them as a benchmark to discuss the challenges toward the realization of practical quantum networks. VC2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license ( http:// creativecommons.org/licenses/by/4.0/ ).https://doi.org/10.1063/5.0038729 I. INTRODUCTION After decades of fundamental research, quantum entanglement emerged as a pivotal concept in a variety of fields, such as quantum computation,1-communication,2–4and -metrology.5Am a n i f o l do f quantum systems are being investigated and photons stand out inmany areas due to their robustness against environmental decoherence and their compatibility with existing optical fiber 6and free-space7 infrastructure. Non-local correlations were demonstrated in severalphotonic degrees of freedom such as time-bin, 8,9time-energy,10orbital angular momentum,11polarization,12spin-polarization,13,14or in a combination of them (“hyper-entanglement”15,16). In quantum infor- mation processing the manipulation and measurement of entangled qubits plays a major role. Applications like quantum key distribution (QKD) with entangled qubits4,17,18require high source brightness, high degree of entanglement, transmission through a low-noise quan- tum channel, and finally a straightforward measurement at remote communication partners, all with minimal losses. These prerequisites could be met by polarization entangled photon pairs.19Besides themost prominent sources based on spontaneous parametric downconversion (SPDC),20–22semiconductor quantum dots (QDs)23–28are also capable of generating polarization entangled photon pairs with a fidelity to a maximally entangled state above 0.97.26The probabilistic emission characteristics of SPDC sources prohibit so far a high brightness in combination with a high degreeof entanglement. 29This is not the case for QDs due to their sub-poissionian photon statistics.30Furthermore, in a real-world context, applications like QKD require entanglement to be com-municated over large distances 6,7to be practically relevant. Most transmission channels, like optical fibers, underlie damping, which severely limits the transmission range. This limitation can be alle- viated by exploiting a concept of quantum communication:2The interconnection of multiple light sources in quantum networks2,3 via the realization of a cascaded quantum repeater scheme withentangled photons and quantum memories. 3,31,32In order to reach this goal, properties of the photon sources beyond the maximum entanglement fidelity become relevant, such as the photon Appl. Phys. Lett. 118, 100502 (2021); doi: 10.1063/5.0038729 118, 100502-1 VCAuthor(s) 2021Applied Physics Letters PERSPECTIVE scitation.org/journal/aplindistinguishability.33I nt h i sw o r k ,w ee x a m i n et h ek e yfi g u r e so f merit of entangled photon pairs with an emphasis on the distribu- tion of entanglement in a quantum network. We will start from the state-of-the-art focusing on GaAs QDs. Although the emission wavelength of about 785 nm is currently non-ideal for efficient fiber-based applications, GaAs QDs represent an excellent model system for the here discussed ideas due to their performance. All of the general concepts introduced in the following, however, are also valid for different material systems, such as InGaAs QDs,23,34–37 whose emission wavelength can be extended to the telecom C-band, where the attenuation in silica fibers has a minimum. In the final section, we will outline recent approaches toward the realiza- tion of a viable quantum network. II. POLARIZATION ENTANGLED PHOTON PAIRS FROM QUANTUM DOTS A common scheme to generate entangled photon pairs with semiconductor QDs embedded in photonic structures [see Fig. 1(a) ]i s by resonantly populating the biexciton (XX) state by a two-photon excitation (TPE) process. 38The XX radiatively decays via the biexci- ton-exciton(X)-ground state cascade,39as depicted in Fig. 1(b) . Ideally, the emitted photon pairs are in the maximum entangled Bell state j/þi¼1=ffiffiffi 2p ðjHHiþj VViÞ,w i t h jHiandjVibeing the horizontal and vertical polarization basis states, respectively. The fidelity fj/þiof the real photon pair’s state to j/þiis mostly determined by the fine structure splitting (FSS) Sand lifetime T1;Xof the intermediate X state. In the absence of other dephasing mechanisms, the maximum achiev- able fidelity of an ensemble of photon pairs to a maximum entangled state is given by40fmax j/þi¼1 42/C0gð2Þ 0þ21/C0gð2Þ 0/C16/C17 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þST1;X=/C22h/C0/C12q0 B@1 CA; (1) where gð2Þ 0is the multi-photon emission probability. In the case of GaAs QDs obtained by the Al droplet etching technique,24ah i g h in-plane symmetry25results in average FSS values below 5 leV, while the weak lateral carrier confinement41causes radiative lifetimes of about T1;XX¼120 ps and T1;X¼270 ps. The wavelength of the emit- ted light hereby lies around 780 nm [see Fig. 1(c) ], with the XX and X photons separated by about 2 nm (4 meV), allowing them to be splitby color. In contrast to SPDC-based sources, 42the pair-generation probability and the gð2Þ 0of QDs are decoupled due to the sub- poissonian emission characteristics.39This led to demonstrated values ofgð2Þ 0¼7:5ð16Þ/C210/C05under pulsed TPE,30as illustrated by the corresponding auto-correlation measurement in Fig. 1(d) .Figure 1(e) displays a resulting two-photon density matrix qin the polarization space of the XX and X photons for an as-grown GaAs QD withS/C250:4leV, acquired by full-state tomography 43under pulsed TPE at an excitation rate of R¼80 MHz. The fidelity deduced from this matrix as fj/þi¼h/þjqj/þiis 0.97(1). By utilizing a specifically designed piezo-electric actuator,44capable of restoring the in-plane symmetry and erasing the FSS of the QDs by strain, fidelity values upto 0.978(5) were demonstrated. 26These results suggest that a modest Purcell enhancement of a factor 3 could alleviate remaining dephasing effects and push the fidelity up to 0.99, which would match with thebest results from SPDC sources. 45The minimum time delay 1 =R between the pulses depends on the lifetimes T1;XXandT1;X, allowing for excitation rates up to R/C251 GHz (without Purcell enhancement). This makes GaAs QDs a viable source for applications like QKD with FIG. 1. Compilation of measurements for GaAs QDs. (a) Common sample structure with GaAs QDs in a lambda-cavity sandwiched between distributed Bragg reflecto rs (DBRs). In combination with a solid immersion lens (SIL), this yields an extraction efficiency of about 0.11. (b) Scheme of entangled photon pair gener ation using the resonant two-photon excitation (TPE) process. (c) Emission spectrum of under TPE. (d) Autocorrelation of the XX signal from a GaAs QD excited by TPE and measure d by supercon- ducting nanowire single photon detectors (SNSPDs), with a resulting gð2Þð0Þ¼ 7:5ð16Þ/C2 10/C05. Reproduced with permission from Schweickert et al. , Appl. Phys. Lett. 112, 093106 (2018), Copyright 2018 AIP Publishing. (e) Real and imaginary part of the two-qubit density matrix of the X and XX in the horizontal (H) and verti cal (V) polarization basis. The derived fidelity is fj/þi¼0:97ð1Þ. (f) Two-photon interference visibility from one doubly excited QD with a time delay of 2 ns. (g) Two-photon interference visibility for two remote QDs with a resulting interference visibility of V¼0:51ð5Þ. Reproduced with permission from Reindl et al. , Nano Lett. 17, 4090–4095 (2017). Copyright 2017 Authors, licensed under a Creative Commons Attribution (CC BY) license.Applied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 118, 100502 (2021); doi: 10.1063/5.0038729 118, 100502-2 VCAuthor(s) 2021entangled photons,4,17,18,46where near-unity entanglement fidelity is required in order to reach practical secure key rates. In the case of the widely used InGaAs QDs, presumably the intrinsically longer transi- tion lifetimes currently prevent similarly high degree of entanglementin the absence of time filtering. 6,47 The source brightness of QDs is mostly bound to the extraction efficiency, which is naturally limited in semiconductor structures dueto total internal reflection at the air/semiconductor interface. A simple approach to increase the pair extraction efficiency gð2Þ Efrom less than 10/C04to about 0.01 is to embed the QDs in a lambda cavity defined between two distributed Bragg reflectors and adding a solid immersionlens on top, 48seeFig. 1(a) . A pair extraction efficiency of 0.373(2) has been recently reported for GaAs QDs embedded in antenna structuresconsisting of a semiconductor membrane with a back metal mirror anda top solid immersion lens made of GaP. 49Recently, circular Bragg res- onators (CBRs) have demonstrated values of gð2Þ E¼0:65ð4Þ50and Purcell enhancement up to a factor 11.3.51Although a non-ideal entan- glement fidelity due to the high FSS was reported in Ref. 50, these struc- tures are compatible with the aforementioned strain tuning techniques,which could cancel the FSS to create a bright source of highly entangledphoton pairs, applicable for QKD with key rates potentially in the GHz range. A widely discussed and researched topic is the distribution of entanglement over basically arbitrary distances, for which sourcesoperating at high pair emission rates are especially relevant. Oneapproach is free-space transmission via satellites, where recently a dis-tance of 1120 km was covered. 7 From the practical point of view, it would be desirable to exploit the existing and well-established telecom optical fiber networks.6The obvious effect of fibers on the transmitted light is a uniform damping,which is about 0.2 dB/km for typical fibers in the telecom C-Band at awavelength of 1550 nm wavelength (compared to about 2.5 dB/km for785 nm). When transmitting polarization entangled photons throughoptical fibers, however, also polarization mode dispersion 52(PMD) has to be taken into account. PMD causes the principal states of polari- zation (PSPs) of the photons’ wave packets to drift apart in time,leading to a degradation of the entanglement. The latter is propor-tional to the ratio of the total induced PMD sbetween the two entangled photons and the length of the photon wave packets, givenby 2 T 1. The maximum achievable fidelity to a perfectly entangled state, derived from the 2-qubit density matrix in polarization space,52 is then given by fPMDðsÞ¼1 2þ1 2þs 4T1/C18/C19 e/C0s 2T1: (2) For simplicity, we assume that T1¼minðT1;X;T1;XXÞ,w h i c hr e p r e - sents a worst case scenario. If the two entangled photons experience exactly the opposite relative drift due to a maximum mismatch of the input modes with the PSPs, the PMD from the individual fibers addup to s¼s 1þs2. A typical value for the PMD of a single mode fiber isD¼0:1p s=ffiffiffiffiffiffiffi kmp (e.g., the type “SMF-28e þ”f r o m“ C o r n i n g ” ) , thus with the given lifetimes and a length of l¼200 km for both the X and the XX photon’s fibers, the maximum possible fidelity for T1¼120 ps is still fPMD 2Dffiffi lp/C0/C1 >0:99. For T1¼10 ps, the fPMD degrades to about 0.98. For T1¼1 ps, which is approximately the case for sources based on the SPDC process, the maximum fidelity drops toabout 0.79, unless resorting to lossy measures like frequency filtering53 or conversion to time-bin entanglement.9,54If the input polarization modes are aligned with the PSPs, the total PMD reads s¼s1/C0s2. Therefore, given that s1¼s2, a configuration exists which exactly can- cels out the effect of PMD and preserves the entanglement. This could be achieved by realigning the input modes to the PSPs during opera- tion by polarization controllers.55An additional effect besides PMD in the context of a fiber-based network is chromatic dispersion, which leads to a temporal broadening of the wavefunctions.35,56This lowers the success probability of two-photon interference, but can be coun- tered by the design of the optical setup, as we will discuss in Sec. III. From these considerations, we realize that fiber-based networks with QDs can greatly benefit from the robustness against PMD, espe- cially compared to SPDC sources. However, for emission wavelengths significantly below the telecom bands, as in the case of GaAs QDs and most used InGaAs QDs coherently grown on GaAs substrates, the range remains severely limited. For this purpose, different material sys- tems for QDs emitting in the telecom bands are being developed. An entanglement distribution experiment with an InGaAs QD emitting in the telecom O-band over a distance of about 18 km has already been demonstrated,6while the quality of QDs emitting in the telecom C-band is rapidly approaching that of dots emitting at shorter wave- lengths.34,37,57As an alternative, frequency conversion techniques can be utilized to adapt the emission wavelengths, although at the cost of efficiency.35 III. QUANTUM DOTS IN A QUANTUM REPEATER BASED NETWORK Although transport of highly polarization entangled photons through fibers is possible,58the exponential damping will inevitably lead to an insufficient qubit rate. A possible solution to this problem is the realization of a quantum repeater scheme. The commonly pro- posed approach is based on the DLCZ protocol (and its variants) that relies on spin-photon entanglement.59However, the probabilistic nature of the entangling scheme limits the entanglement creation.60 Although an improved, deterministic version of the spin-photon basedscheme was developed, 61the achieved rates are still modest. An alter- native scheme relies on the use of entangled photon pair sources, like QDs, interfaced with quantum memories capable of receiving and storing entangled states to increase the qubit rate.32This scheme relies on a cascade of entanglement swapping processes62,63among entangled photon pairs emitted by independent emitters. The telepor- tation of the entanglement is enabled by two-photon interference to perform a so-called Bell state measurement (BSM). The success of a BSM strongly depends on the photon indistinguishability, which, in turn, depends on the photon sources and can be experimentally accessed by probing the interference visibility in a Hong-Ou-Mandel (HOM) experiment.56For maximum visibility, the spatiotemporal wave packets of the two photons involved in the BSM have to be iden- tical and pure, i.e., no other physical system should contain informa- tion about the photon’s origin. The latter point plays a crucial role in t h ec a s eo fQ D se x p l o i t i n gt h ed e c a yc a s c a d ef o re n t a n g l e dp h o t o n generation. The XX and X photons are correlated in their decay times,8,64,65which limits the maximum possible indistinguishability for both the XX and X photons according to65Applied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 118, 100502 (2021); doi: 10.1063/5.0038729 118, 100502-3 VCAuthor(s) 2021Vcasc¼1 1þT1;XX=T1;X: (3) Figure 1(f) shows the result of a HOM experiment with two XX photons generated by a GaAs QD, upon excitation with two pulsesseparated by a 2 ns delay. The resulting visibility of 0.69 is typical forboth the XX and X photons under TPE 48and is close to the maximum according to Eq. (3). As a comparison: For single X photons generated by resonant excitation, a visibility of over 0.9 is achieved for the same QDs.65When interfacing two dissimilar QDs, the inherently stochastic nature of the epitaxial growth has to be considered, which primarilyleads to varying emission energies. Further, imperfections in the solid-state environment of the QDs lead to inhomogeneous broadening due to charge noise. 66This results in a jitter of the central emission energy bydE(full width at half maximum) around a mean value typically in the microsecond to millisecond timescale.66The jitter leads to a degra- dation of the indistinguishability described by67 VdE¼/C22hRe FðzÞ½/C138ffiffiffiffiffi 8pp rT1; (4) withr¼dE=2ffiffiffiffiffiffiffiffiffiffi 2l n2p being the standard deviation of the Gaussian distribution of the energy jitter and Re ½FðzÞ/C138being the real part of the Faddeeva function of z¼i/C22h=2pffiffiffi 2p rT1/C0/C1 . Although measurementsunder resonant single-photon excitation reveal that pure dephasing is not fully negligible,65we focus here on the far more dominant energy jitter and do not include pure dephasing in Eq. (4).Figure 1(g) shows the result of a HOM experiment with two X photons from GaAs QDswith dE/C254leV and a resulting visibility of V¼0:51ð5Þ, 68which corresponds, to our knowledge, to the maximum value measured so far for QDs initially prepared into the biexciton state (via phonon- assisted TPE). The center wavelengths were previously matched bytuning the energy of one QD via a monolithic piezo-electric actuator. We will now discuss possible solutions to overcome the two major indistinguishability degrading mechanisms in QDs discussed so far: The partial temporal entanglement in the XX-X decay cascade [Eq. (3)] and frequency jitter [Eq. (4)]. Both effects are influenced by the radiative lifetimes T 1;XXand T1;X, which can be modified by exploiting the Purcell effect in a cavity.50,51Figure 2 illustrates concatenated entanglement swapping processes with a depth of L¼f1;2;3g, i.e., a chain of quantum relays forming the backbone of quantum repeaters. The number of QDs required is 2L, while the range covered scales with 2Ll0,w i t h l0being the total length of both fibers departing from one QD. This example serves as a demonstrationon how the entanglement fidelity evolves over multiple layers of swap- ping operations with photons generated by QDs. Figure 2(a) depicts the final entanglement fidelity as a function of the Purcell factor Pand FIG. 2. Entanglement fidelity in a chain of quantum relays. Simulated entanglement fidelity of the final entangled photon pair in a chain of quantum relays perfo rming entangle- ment swapping operations among pairs of polarization entangled photons emitted by QDs under TPE. The chain depths are L¼f1;2;3g. All QDs are assumed to have an FSS of 0.05 leV. (a) Fidelity as a function of Purcell factor Pand Gaussian energy jitter dEin multiples of the natural X linewidth of dE0¼2.4leV at P¼1. (b) Fidelity with a frequency selective cavity, so that PX¼Pand PXX¼7P. (c) Fidelity as a function of the Lorentzian width of a frequency filter dEfand the Gaussian energy jitter dE(full width at half maximum), for fixed Purcell factors of PX¼2 and PXX¼10.Applied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 118, 100502 (2021); doi: 10.1063/5.0038729 118, 100502-4 VCAuthor(s) 2021the energy jitter dEin multiples of the natural X linewidth of dE0 ¼2.4leV at P¼1, corresponding to T1;X¼270 ps. Values of P>15 are unpractical, as the total relaxation time of the QD then approachesthe typical excitation pulse width of about 10 ps. This primarily leads to an increasing re-excitation probability, 69,70which is detrimental to the indistinguishability and the entanglement. In addition, PMD effects in optical fibers start to become relevant for such short wave packets. For the calculation of the fidelity, we utilize the density matrixformalism for describing one entanglement swapping process with QDs 63with a type of BSM which can detect two Bell states40,71(jwþi andjw/C0i). In order to model a chain of entanglement swapping pro- cesses, the formalism is applied recursively, assuming uncorrelated BSM success probabilities in successive steps. We simultaneously account for varying lifetimes caused by Pa n dad e c r e a s e dB S Ms u c c e s s rate due to dE(see the supplementary material for details). From the simulations, we observe that already for two swapping processes thehomogeneous Purcell enhancement alone cannot recover the entan- glement fidelity sufficiently, as it merely alleviates the impact from inhomogeneous broadening on the indistinguishability, but the visibil- ity degrading effect from the XX-X cascade is still at full force. Figure 2(b) depicts the case for an energy selective cavity, 64which enhances the XX decay rate by a factor of 7 compared to the X, so that PX¼P andPXX¼7P. This approach could strongly increase the BSM success rate and therefore the resulting entanglement fidelity. However, thefinite temporal width of the excitation pulse, whose minimum value is set by the limited spectral separation between X and XX and the neces- sity of suppressing laser stray light, sets a lower limit to the lifetimes— and therefore an upper limit to the Purcell enhancement—in order to limit re-excitation. 69A compromise could be achieved by mild fre- quency filtering of the X photon, as illustrated in Fig. 2(c) .F i l t e r i n g partially erases the temporal information held by the X photon, lead- ing to the same outcome as prolonging the X lifetime and hence decreasing the XX/X lifetime ratio as with the selective Purcell enhancement. In the simulations, we assume a filter with Lorentziantransmission characteristics and a FWHM of dE fand an energy jitter with a FWHM of dEf o rb o t ht h eXa n dt h eX X( s e et h e supplemen- tary material for details). We assume a frequency selective cavity with fixed Purcell enhancement of PX¼2a n d PXX¼10. This asymmetry in enhancement could be achieved by carefully designing the lateral size of the previously introduced CBRs.50As a result of the filtering, the effective lifetime of the X signal increases while simultaneously reducing the impact of the energy jitter. Note that for the here investi-gated values of dEthe interference visibility again drops for dE fvalues below the inhomogeneous broadening dE. In addition, in the presence of a finite FSS, the BSM success rate drops when the filtered linewidth is on the order of the FSS or below.63From the simulations, we can observe that with a low inhomogeneous broadening ( <0:1dE0) and a moderate frequency filtering of about 1 :5dE0one could achieve an entanglement fidelity of approximately 0.93 at L¼2a n d0 . 8 5a t L¼3. A complete repeater scheme requires also quantum memories,72 which can store and retrieve a photonic qubit with high fidelity. To address the noise and bandwidth limitation of quantum memories, two groups invented a cascaded absorption memory scheme, which isintrinsically noise-free. 73,74Furthermore, the possibility to use an off- resonant Raman transition in this cascaded scheme allows for large storage bandwidth, limited mainly by the available control laser power. Currently, the main drawback of these schemes is the limited storagetime, which is determined by the radiative lifetime of the upper state of the cascade (below 100 ns). Another promising approach is to use rare-earth doped crystals as quantum memories,75featuring per- formances that equalize, if not outperform, those of cold atomicensembles 76,77or trapped emitters in terms of efficiency78and coher- ence times.79These memories have shown a full quantum storage pro- tocol with telecom-heralded quantum states of light,80and the first photonic quantum state transfer between nodes of different nature.81 Furthermore, atomic frequency comb quantum memories were thefirst to be successfully interfaced with single photons emitted from aquantum dot. 82 We want to mention at this point that recently an alternative repeater scheme83was proposed, which eliminates the necessity of quantum memories, but instead shifts the challenge toward the realiza-tion of large-scale photonic cluster states. IV. FUTURE OUTLOOK We conclude that bright and nearly on-demand sources of highly entangled photon pairs are on the verge of becoming reality. The ground work has been laid through the development of semiconductor quantum dots (QDs) emitting highly entangled photons, 26of advanced optical cavity structures50,51and technology capable of manipulating the symmetry and emission energy of QDs.44On-chip integration of QDs84and the implementation of electric excitation schemes85can further increase the practicability in emerging quantum technology. The optimal wavelength (about 1550 nm) for transporting entangled photons through fibers is currently determined by the estab-lished telecom fiber infrastructure. Material systems to obtain QDs emitting at this wavelength are under development, 6,34,37,57and exist- ing sources with emission at shorter wavelengths could be adapted byfrequency conversion. 35Recently, a basic GHz-clocked quantum relay with QDs emitting directly in the Telecom-C band was demon- strated.86One of the greatest, yet rewarding challenges is the interfac- ing of dissimilar sources of entangled photons for multi-photonapplications 36and long-haul entanglement distribution6,7in quan- tum networks.2,3The physical limits to the indistinguishability8,65 set by the currently employed cascade for entangled photon pair generation26,39and fluctuations stemming from the solid state environment of QDs25,66pose intricate challenges for the years to c o m e .A sd e m o n s t r a t e di nt h i sw o r k ,t h ea p p l i c a t i o no fs e l e c t i v ePurcell enhancement together with mild frequency filtering could alleviate the limit of indistinguishability of the entangled photon pairs. The different emission energies and radiative life-times of the biexciton (XX) or exciton (X) in QDs could bematched by utilizing strain- 44and electric87degrees of freedom independently. Considering the quantum relay chains depicted in Fig. 2 , three strain degrees of freedom can cancel the fine structure splitting (FSS) and adapt the central energy of the XX or X to thenext neighbor’s. The electric degree of freedom can simulta-neously be used to fine-tune the respective radiative lifetime and therefore the shape of the photonic wave-packet. By repeating this strategy through the whole relay chain for each QD, one couldoptimize the resulting entanglement fidelity of the final photonpair. With these tools at hand, the next leap toward the demonstration of a functional quantum network will be the interconnection of twoApplied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 118, 100502 (2021); doi: 10.1063/5.0038729 118, 100502-5 VCAuthor(s) 2021dissimilar quantum dots via entanglement swapping.62,63Several groups are currently developing devices which merge the concepts of circular Bragg reflectors (CBRs)50,51with the tuning of the in-plane stress tensors44of the QDs. CBRs are ideally suited for this purpose, since they are fabricated on a dielectric-metal structure which can be virtually placed on any other substrate, and in particular, on piezoelec- tric actuators. By carefully designing the dimensions of the CBRs, these devices could provide the necessary magnitude and asymmetry of the Purcell factors for the XX and the X emission for a high photon indistin- guishability, while eliminating remaining FSS and matching the energies of the involved QD’s photons via the three strain degrees of freedom. These concepts are compatible with alternative material systems emitting at telecom wavelength,6,34,37,57which will allow to reach longer distances with optical fibers and possibly to utilize the established telecom fiber network. The next steps could be to interface the photons performing the Bell state measurement with quantum memories72,73,75,78–82and use the resulting entangled photon pairs for quantum key distribution4,17,18 with efficiencies beating the direct transmission through fibers. Fromt h e r eo n ,t h eg o a li st oe x p a n dt h es y s t e mt oac h a i no fm u l t i p l eQ D sa n d implement a quantum repeater scheme 3,31,32in order to enhance the resulting entanglement fidelity and efficiency compared to a repeater-less distribution scheme. SUPPLEMENTARY MATERIAL See the supplementary material for theoretical considerations about the evolution of entangled states in a chain of quantum relays. ACKNOWLEDGMENTS Christian Schimpf is a recipient of a DOC Fellowship of the Austrian Academy of Sciences at the Institute of Semiconductor Physics at Johannes Kepler University, Linz, Austria. This project has received funding from the Austrian Science Fund (FWF): FG 5, P 29603, P 30459, I 4380, I 4320, and I 3762, the Linz Institute of Technology (LIT) and the LIT Secure and Correct Systems Lab funded by the state of Upper Austria and the European Union’s Horizon 2020 research and innovation program under Grant Agreement Nos. 820423 (S2QUIP), 899814 (Qurope), 654384 (ASCENT+), and 679183 (SQPRel). DATA AVAILABILITY Data sharing is not applicable to this article as no new data were created or analyzed in this study. REFERENCES 1J. L. O’Brien, A. Furusawa, and J. Vuc ˇkovic ´, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009). 2N. Gisin and R. Thew, “Quantum Communication,” Nat. Photonics 1, 165–171 (2007). 3H. J. 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5.0043946.pdf

J. Chem. Phys. 154, 144306 (2021); https://doi.org/10.1063/5.0043946 154, 144306 © 2021 Author(s).Understanding the structure of complex multidimensional wave functions. A case study of excited vibrational states of ammonia Cite as: J. Chem. Phys. 154, 144306 (2021); https://doi.org/10.1063/5.0043946 Submitted: 12 January 2021 . Accepted: 23 March 2021 . Published Online: 09 April 2021 Jan Šmydke , and Attila G. Császár COLLECTIONS Paper published as part of the special topic on Quantum Dynamics with ab Initio Potentials ARTICLES YOU MAY BE INTERESTED IN Machine learning meets chemical physics The Journal of Chemical Physics 154, 160401 (2021); https://doi.org/10.1063/5.0051418 Modeling nonadiabatic dynamics with degenerate electronic states, intersystem crossing, and spin separation: A key goal for chemical physics The Journal of Chemical Physics 154, 110901 (2021); https://doi.org/10.1063/5.0039371 Exploring the anharmonic vibrational structure of carbon dioxide trimers The Journal of Chemical Physics 154, 144302 (2021); https://doi.org/10.1063/5.0039793The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Understanding the structure of complex multidimensional wave functions. A case study of excited vibrational states of ammonia Cite as: J. Chem. Phys. 154, 144306 (2021); doi: 10.1063/5.0043946 Submitted: 12 January 2021 •Accepted: 23 March 2021 • Published Online: 9 April 2021 Jan Šmydke1,a) and Attila G. Császár2 AFFILIATIONS 1Department of Radiation and Chemical Physics, Institute of Physics CAS, Na Slovance 1999/2, 18221 Praha 8, Czech Republic 2MTA-ELTE Complex Chemical Systems Research Group and Laboratory of Molecular Structure and Dynamics, Institute of Chemistry, ELTE Eötvös Loránd University, H-1117 Budapest, Pázmány Péter Sétány 1/A, Hungary Note: This paper is part of the JCP Special Topic on Quantum Dynamics with Ab Initio Potentials. a)Author to whom correspondence should be addressed: jan.smydke@gmail.com ABSTRACT Generalization of an earlier reduced-density-matrix-based vibrational assignment algorithm is given, applicable for systems exhibiting both large-amplitude motions, including tunneling, and degenerate vibrational modes. The algorithm developed is used to study the structure of the excited vibrational wave functions of the ammonia molecule,14NH 3. Characterization of the complex dynamics of systems with several degenerate vibrations requires reconsidering the traditional degenerate-mode description given by vibrational angular momentum quantum numbers and switching to a symmetry-based approach that directly predicts state degeneracy and uncovers relations between degenerate modes. Out of the 600 distinct vibrational eigenstates of ammonia obtained by a full-dimensional variational computation, the developed methodology allows for the assignment of about 500 with meaningful labels. This study confirms that vibrationally excited states truly have modal character recognizable up to very high energies even for the non-trivial case of ammonia, a molecule which exhibits a tunneling motion and has two two-dimensional normal modes. The modal characteristics of the excited states and the interplay of the vibrational modes can be easily visualized by the reduced-density matrices, giving an insight into the complex modal behavior directed by symmetry. Published under license by AIP Publishing. https://doi.org/10.1063/5.0043946 .,s I. INTRODUCTION It is a common argument against sophisticated quantum- chemical computations, such as the variational ones, which are widely performed in the fourth age of quantum chemistry,1that they provide a lot of data, some in the form of wave functions, but only limited understanding. To most chemists, an approxi- mate characterization of the (ro)vibrational dynamics of molecular systems requires that certain descriptors are attached to the com- puted wave functions. These descriptors are most often quantum numbers arising from simple models.2Aside from the use of good quantum numbers , the understanding of the internal structure of multidimensional excited-state wave functions requires the use of approximate quantum numbers. When the emphasis is on physicalmeaning and interpretation, the descriptors are commonly associated with the character of a dominant component of the given state expanded in a particular basis set.3For example, the 1s12s1 electronic excited state of a helium atom is distinguished by its domi- nating 1s12s1two-electron full configuration interaction (FCI) basis vector, although this state also has many other non-zero FCI basis contributions. Nevertheless, a general excited state may decompose to more than one dominant basis vector or the composition of the state may even be evenly distributed over the whole basis-set space, not giving a particular vector or a selection of vectors’ significant preference over the others. In such ambiguous cases, the question arises naturally whether one could still associate a given state of interest with a unique physical meaning other than just energy and overall symmetry J. Chem. Phys. 154, 144306 (2021); doi: 10.1063/5.0043946 154, 144306-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp that come naturally from the quantum-chemical computation. One may also ask whether changes in the multidimensional basis could turn more states into those exhibiting a dominant contribution. Such a smart transformation prescription, however, is not known generally. Apart from the state composition, a general and unique charac- teristic of multidimensional excited states of quantum systems is the number of the wave-function nodes in each of the systems’ modes of motion. The given excited state is then associated with a set of modal quantum numbers, regardless of the particular basis set used. This approach proved to be a practical way to label excited rovi- brational states of small molecules.2As we have shown earlier4,5 for vibrational wave functions resulting from sophisticated varia- tional computations, the individual mode excitations tend to have a characteristic and distinguishable density by means of the reduced- density matrices ( vide infra ) that are also easier to handle than a bare multidimensional wave function. Nevertheless, the density, as a function of coordinates, and the position of the nodes may not always match with the underlying modal basis functions. The devi- ation can be significant when several modes interact, and in such cases, the bare basis-set-decomposition method may fail to char- acterize the state. A suitable choice of the modes of motion and the ability to count the number of nodes in each mode of compli- cated multidimensional wave functions are key to the general and practical characterization of excited states. It is, however, not well understood how many states can be unambiguously assigned by this approach. In this paper, we examine vibrational excited states of the ammonia molecule (14NH 3), a system exhibiting nontrivial modal structure. In particular, ammonia is a symmetric-top molecule, it has two twofold-degenerate vibrational modes, and it exhibits a large-amplitude internal motion (tunneling). We demonstrate that by using a simple coordinate system that intuitively mimics the har- monic vibrational modes of ammonia and with the help of reduced- density matrices for reading out the number of modal nodes, we can unambiguously assign hundreds of vibrationally excited states of14NH 3. We compare our results to the recent state-of-the-art ro- vibrational line and energy level list of ammonia6that provides a quick and fully automatic assignment of all states of interest based on the dominant basis-set component, but the assignments may not always be accurate. The rest of this paper is structured as follows. Section II dis- cusses a few theoretical aspects of the vibrational assignment pro- cess together with the treatment of systems possessing degener- ate modes, which is essential for the correct description of the vibrations of ammonia. Section III describes the technical details of variational vibrational wave-function computations. Section IV shows, explains, and discusses the main results of this paper. The most important conclusions of the present study are summarized in Sec. V. II. THEORY A. Multidimensional structure of vibrational states Understanding the dynamical behavior of multidimensional quantum systems may start with a model in which individual degreesof freedom (dof) in the ground state are least coupled or can be considered semi-independent. A well-known example from another field is the Hartree–Fock approximation7of electronic-structure theory, where the individual electrons are moving independently in the averaged field of the other electrons and the nuclei. This model is necessarily far from being perfect; the motion of the electrons is not truly independent as each electron contributes to the self-consistent field by which all the electrons are driven. Nevertheless, the individ- ual one-electron functions, the so-called orbitals, have been playing, for many decades, a crucial role in the qualitative understanding of the structure and reactivity of molecules, and they also form the foundation of correlated wave-function expansions of the ground and of the excited electronic states. Similarly, the harmonic oscillator model8of the vibrations of polyatomic molecules constructs the ground and excited vibrational states by combining independent (the so-called normal) modes of vibrations, as can be seen from the harmonic oscillator energy formula for the ith vibrationally excited state, Ei=∑ mv(i) mεm+ ZPVE, (1) where the index mruns over all vibrational modes, εmandv(i) mare the energy and the excitation quantum number of the mth mode, and ZPVE stands for zero-point vibrational energy. The modes are linear combinations of all the nuclear dofs and are made indepen- dent due to the quadratic nature of the vibrational potential in the harmonic approximation. The decomposition of vibrational states into modes of the harmonic oscillator model is still the de facto stan- dard description of molecular vibrations, and it is generally accepted and used by practitioners of the field. Apart from the anharmonic nature of potential energy sur- faces (PESs),9–11which results in mode coupling and considerable shifting of the harmonic energy levels, when studying higher vibra- tionally excited states or when dealing with semi-rigid systems with several shallow potential energy wells, the strategy of combining the independent normal modes, constrained to a single minimum by definition, quickly becomes inappropriate. In some cases, it is preferable to replace the normal-mode description by a local-mode treatment, for example, for the vibrations of the heavier congeners of water.12For semi-rigid molecules, splitting of energy levels due to tunneling between symmetrically equivalent configurations (such as the two minima of ammonia linked by an umbrella-like nuclear motion) makes the spectra more complex and the normal-mode picture inappropriate, especially for higher excitations. If the model of a normal-mode combination fails quantita- tively and even qualitatively for a general polyatomic system, the question arises whether it is still reasonable and possible to char- acterize highly excited multidimensional wave functions by means of combinations of “modes.” No matter what the modal degrees of freedom are and how strongly coupled they become in various states, the modes should retain a more-or-less intact character for portions of the energy spectrum so that they are clearly distinguish- able and the modal understanding of the excited states remains meaningful. Due to their convenience, modal combinations are routinely used in the qualitative description of vibrational states, although the J. Chem. Phys. 154, 144306 (2021); doi: 10.1063/5.0043946 154, 144306-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp assignment is often cumbersome and tends to become inaccurate as the excitation increases. There are a variety of techniques of vibra- tional assignment. They range from energy decomposition, based on or similar to Eq. (1), finite basis representation (FBR)13and other2 decomposition schemes based on the wave-function overlap with modal basis functions and similar yet conceptually different vibra- tional configuration interaction vector decompositions,14and var- ious expectation-value-based2or even perturbative approaches15,16 to direct inspection of two-dimensional (2D) cuts of vibrational wave functions. The last technique mentioned can provide a valuable visual insight into the nodal structure of vibrations and the interplay of modes. By counting the number of wave-function nodes in each of the vibrational modes, one can directly assign modal quantum numbers that characterize the polyatomic vibrational state. Practi- cally, however, this approach has a limited range of applicability, and it is mainly useful to states with pure bending or stretching char- acter and only to low-lying states.17,18For states excited by more than just a few quanta or states with strongly coupled modes, the nodal information from the 2D plots is mostly not legible.4,19More- over, the number of 2D plots that needs to be inspected rises as (n 2), where nstands for the number of vibrational degrees of free- dom, which is not easy to handle manually even for a few-atomic system. In our recent study about the nuclear dynamics of the vinyl radical,4we came up with an alternative way of visualizing the vibra- tional wave-function modes by means of reduced-density matrices (RDMs) along selected internal coordinates. While this was not the first use of RDM in the context of molecular vibrations, we extended its applicability much further than a simple 2D visual characteriza- tion tool.20Our RDM approach proved to be advantageous over the use of 2D wave-function cuts in several respects. The nodal struc- ture of a particular vibrational mode is far more legible with RDM than with wave-function cuts, and as a result, many more states can be assigned. In contrast to wave-function cuts, RDMs need consid- erably less data generated and inspected. Moreover, RDMs do not depend on a reference vibrational configuration. It also turned out that each modal excitation exhibited a peculiar density pattern that could be used for identifying modal excitations in a semi-automatic way; this has already been successfully applied to the assignment of many vibrational states of the water molecule.5Although the RDMs of particular modal excitations tend to retain their characteristic shapes throughout a large portion of the energy spectrum, heavily coupled modes can deviate from the regular RDM shapes signifi- cantly. Nevertheless, such cases tend to be still well assignable by using a simple logic, and their irregular density shape, then, serves as a specific pattern for similarly coupled modes in other excited states. By using various RDM types, one can decode complicated vibra- tional structures from different contexts, which helps distinguishing states that look similar, as demonstrated on the vibrational states of ammonia during this work. Since the RDM technique proved successful in correctly iden- tifying modal excitations for a considerable number of states, we can conjecture that the view of vibrationally excited states as virtu- ally decomposable to individually excited vibrational modes is truly meaningful for a much higher number of states than suggested by other approaches. We do not imply that there is a clear energy decomposition into modal contributions, but rather that for a given set of vibrational modes, the excited state can be decomposed intoindividual, distinguishable mode excitations with the characteristic nodal structure. In contrast to the method of wave-function overlap with modal basis functions, which can yield incorrect modal quan- tum numbers, the RDM approach allows for a direct visual mode assessment even in cases when the modal density shape strongly deviates from the basis functions. It needs to be stressed that the success of the modal description may heavily depend on the clever definition of the vibrational modes and the underlying coordinate system. In the molecules studied so far, we associated the vibrational modes with intuitively chosen coordinates, expecting particular vibrational motions. Even though they were successful choices, it is not yet clear how to select the most suitable set of coordinates systematically and whether a completely different set of modes/coordinates would lead to equivalent or infe- rior results and would still provide an intuitive physical insight. One question, going beyond the scope of this work, thus, remains open: How far one can reach in the characterization of a vibrational spectrum by using just a single set of vibrational modes? B. Reduced-density matrices and vibrational labeling First, let us briefly summarize the definitions of various types of reduced-density matrices that come handy for inspecting multidi- mensional wave functions. Suppose that we have an N-dimensional vibrational wave function, ∣Ψvib⟩≡Ψ(q1,...,qN). (2) Then, the one-mode reduced-density matrix (1RDM) is defined as Γ1(qi′,qi)=∫dq1...dqi−1dqi+1...dqN ×Ψ∗(q1,...,qi−1,qi′,qi+1,...,qN) ×Ψ(q1,...,qi−1,qi,qi+1,...,qN). (3) We can view the 1RDM as a two-dimensional quantity for a sin- gle vibrational coordinate qi, obtained after integrating out all the remaining vibrational degrees of freedom. Taking only the diag- onal elements of 1RDM, we end up with the diagonal one-mode reduced-density matrix, ΓD 1(qi)≡Γ1(qi,qi)=∫dq1...dqi−1dqi+1...dqN ×Ψ∗(q1,...,qN)Ψ(q1,...,qN), (4) which stands as a true wave-function density for the given coordi- nate qi. Similarly, one can define a two-mode reduced-density matrix (2RDM) as a four-dimensional quantity, Γ2(qi′,qj′,qi,qj) =∫dq1...dqi−1dqi+1...dqj−1dqj+1...dqN ×Ψ∗(q1,...,qi−1,qi′,qi+1,...,qj−1,qj′,qj+1,...,qN) ×Ψ(q1,...,qi−1,qi,qi+1,...,qj−1,qj,qj+1,...,qN). (5) J. Chem. Phys. 154, 144306 (2021); doi: 10.1063/5.0043946 154, 144306-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The diagonal elements of 2RDM can be viewed as a true two- dimensional density, ΓD 2(qi,qj)≡Γ2(qi,qj,qi,qj) =∫dq1...dqi−1dqi+1...dqj−1dqj+1...dqN ×Ψ∗(q1,...,qN)Ψ(q1,...,qN). (6) We suggest to look at this quantity as a convenient resolution of theΓD 1(qi)density in one additional dimension. ΓD 2s greatly improve the understanding of the characteristics of the vibrational states, and they are also necessary for the description of degenerate or otherwise coupled vibrational modes. In the same manner, one can construct even higher-order reduced-density matrices. The high-rank tensor quantities are, how- ever, difficult to visualize, and their storage and manipulation can quickly become intractable. It is, therefore, preferable to restrict the analysis to low-order RDMs, whenever possible. This recommenda- tion was followed in our previous studies.4,5Nevertheless, for the present study, we found it necessary to also use three-dimensional ΓD 3(qi,qj,qk)diagonal densities, ΓD 3(qi,qj,qk)≡Γ3(qi,qj,qk,qi,qj,qk) =∫dq1...dqi−1dqi+1...dqj−1dqj+1... ×dqk−1dqk+1...dqN ×Ψ∗(q1,...,qN)Ψ(q1,...,qN). (7) In our study of the vinyl radical,4we used only the diagonal densitiesΓD 1(qi)andΓD 2(qi,qj), and the assignment of the states was performed exclusively by their visual assessment. It was never- theless a step forward compared to direct wave-function inspection as the two-dimensional wave-function cuts were very complex and extremely confusing, except for the few lowest-energy states. With the help of the densities, which effectively integrate out misleading structural details originating from other vibrational modes, we could clearly count the nodes in any vibrational mode as strongly pro- nounced kinks in the density profiles. Furthermore, we observed that each individual mode excitation had a very characteristic density shape, which did not change in the states of the same modal quan- tum number. Only states with expected strong coupling between particular modes exhibited deviations from the otherwise very regu- lar density pattern. These results were, at the same time, surprising and encouraging for us to pursue the RDM technique further. Not only were we able to assign an order of magnitude more vibrational states with relative ease and confidence than with other methods, but it also suggested that the modal understanding of the vibra- tional structure is a phenomenon valid further than only at the small displacements around local minima and that the same modal char- acter is retained even at higher-energy states in which interactions between modes can take place. In Ref. 5, we further investigated the regular shapes of the excited vibrational modes and suggested a semi-automatic assign- ment procedure based on comparing the modal density shapes by means of overlap integrals. In our study of the water molecule,5about 200 vibrational states were assigned almost automatically, mostly without the need of visually checking the density plots. For water, a molecule with an Abelian molecular-symmetry (MS) group, C2v(M),21and free of vibrational tunneling, we could also make use of a simple build-up principle that helped selecting the appropriate assignment out of several possibilities. For systems exhibiting tun- neling splittings, the ordering of states is less “strict” (it is common that the tunneling splitting is large enough that the + and −states of the same origin enclose several other energy levels in between and there is also the possibility of exchanging the energy order of the + and−states), and one may not generally rely on such a sim- ple automatic approach. In our study on water, we further realized that the full Γ1(qi′,qi)matrix can actually reveal the true nodal structure of a particular mode by visible changes in sign. This is a stronger node indicator than just the variation of diagonal density and can be used complementarily for cases with barely recognizable nodes. In the present study, we are focusing on the ammonia molecule, 14NH 3, which structurally is also a simple but dynamically consid- erably more challenging system than water, as ammonia exhibits tunneling and, at the same time, possesses two doubly degener- ate vibrational modes. The obstacles hindering the assignment of the states arising from tunneling have been explained above. Next, we turn our attention to issues related to excitations that involve degenerate vibrational modes. C. Excitations of a single twofold degenerate vibrational mode Excitations of a twofold degenerate mode are traditionally described by a two-dimensional (2D) isotropic harmonic oscillator (TDIHO) model.21In that model, one transforms the two vibra- tional quantum numbers ( v1v2) (each of which corresponds to one of the two equivalent degenerate normal-coordinate components) into (vl), where vhas the meaning of the total number of quanta in the 2D harmonic oscillator and lis the vibrational angular momen- tum quantum number describing the angular distribution of the 2D wave function. For a given number of quanta v, the wave function is (v+ 1)-fold degenerate, with lvalues ranging v,v−2,...,−v. The TDIHO model perfectly fits a molecule such as acetylene, C2H2.22Nevertheless, considering the high multiplicity ( v+ 1) of the TDIHO associated angular momentum operator irreducible vector spaces, we can see that the group of the TDIHO model is neces- sarily superior to groups of other molecular Hamiltonians, which have mostly singly, doubly, or triply degenerate irreducible rep- resentations (irreps). Hence, the TDIHO-anticipated ( v+ 1)-fold degeneracy breaks for most molecular Hamiltonians and results in a decomposition of the symmetric product of the Erepresentation with itself, denoted as [ E]v,21where the Eirrep belongs to a group of a particular molecular Hamiltonian and not to the group of the TDIHO model. For example, in the C3v(M) molecular-symmetry (MS) group,21the vibrational states having two quanta in the E vibrational mode decompose as [E]2≡[E⊗E]=A1⊕E, (8) and analogously for the D3h(M) group (see Table I). This immedi- ately shows the breaking of the threefold degeneracy expected from J. Chem. Phys. 154, 144306 (2021); doi: 10.1063/5.0043946 154, 144306-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Decomposition of the symmetric product for a few powers of [ E]vunder C3v(M) and [ E′]vand[E′′]vunder the D3h(M) molecular-symmetry groups. v C3v(M): [ E]vD3h(M): [ E′]vD3h(M):[E′′]v 2 A1⊕E A′ 1⊕E′A′ 1⊕E′ 3 A1⊕A2⊕E A′ 1⊕A′ 2⊕E′A′′ 1⊕A′′ 2⊕E′′ 4 A1⊕2E A′ 1⊕2E′A′ 1⊕2E′ 5 A1⊕A2⊕2E A′ 1⊕A′ 2⊕2E′A′′ 1⊕A′′ 2⊕2E′′ 6 2 A1⊕A2⊕2E 2A′ 1⊕A′ 2⊕2E′2A′ 1⊕A′ 2⊕2E′ 7 A1⊕A2⊕3E A′ 1⊕A′ 2⊕3E′A′′ 1⊕A′′ 2⊕3E′′ the TDIHO theory into a single A1and a single Edegenerate level. Therefore, we suggest that it is more appropriate to label single degenerate-mode excitations ( vα) instead of ( v±l), whereαdenotes the appropriate irrep. Thus, for the aforementioned states, the labels would convert as (2±2)→(2E)and(2 0)→(2A1). (9) It is worth noting that this symmetric product decomposition treat- ment of degenerate systems is not confined to degenerate harmonic oscillator models but is valid for degenerate anharmonic oscillators as well. Although the suggested ( vα) labeling scheme may seem merely as a different notation to the ( vl) quantum numbers, it needs to be emphasized that 1. the information about the actual irrep (and degeneracy) is not directly available from the lquantum number in a general case, and 2. in Sec. II D, we explain that for systems with more than one degenerate mode (such as ammonia), labeling the individual modes by ldoes not generally make sense as it could lead to incorrect symmetry and it is also necessary to deal with the total irrep of all the excited degenerate modes, for which the knowledge of the αirreps is inevitable. Because the decomposition of the symmetric product is not as straightforward as that of the ordinary direct product, in Table I, we recall the decomposition for a few powers of [ E]vunder C3v(M) and [E′]vand[E′′]vunder the D3h(M) MS groups. Note that there are a number of publications detailing the use and treatment of symmetric (and antisymmetric) products in theoretical chemistry.21,23–25 D. Excitations of two twofold-degenerate vibrational modes Care must be exercised when two degenerate modes are excited simultaneously. In the ammonia molecule, for example, there are twoE′vibrational modes under the D3h(M) group. Their simulta- neous excitation by v3andv4quanta, respectively, leads to states decomposing as [E′]v3⊗[E′]v4=irreps ∑ iciΓi, (10)TABLE II . Decomposition of simultaneous excitation of the two degenerate vibrational modes of ammonia for both C3v(M) and D3h(M) symmetry models. v3 v4 C3v(M):[E]v3⊗[E]v4D3h(M):[E′]v3⊗[E′]v4 1 1 A1⊕A2⊕E A′ 1⊕A′ 2⊕E′ 1 2 A1⊕A2⊕2E A′ 1⊕A′ 2⊕2E′ 1 3 A1⊕A2⊕3E A′ 1⊕A′ 2⊕3E′ 1 4 2 A1⊕2A2⊕3E 2A′ 1⊕2A′ 2⊕3E′ 1 5 2 A1⊕2A2⊕4E 2A′ 1⊕2A′ 2⊕4E′ 2 2 2 A1⊕A2⊕3E 2A′ 1⊕A′ 2⊕3E′ 2 3 2 A1⊕2A2⊕4E 2A′ 1⊕2A′ 2⊕4E′ 2 4 3 A1⊕2A2⊕5E 3A′ 1⊕2A′ 2⊕5E′ 3 3 3 A1⊕3A2⊕5E 3A′ 1⊕3A′ 2⊕5E′ whereΓistands for the ith irreducible representation of the D3h(M) group. For other systems possessing multiple degenerate dofs under different symmetry groups and different excited combinations of degenerate irreps, the generalization of Eq. (10) is straightforward. For the combination of two (or more) degenerate modes, the result- ing multimode excitation has to be labeled both by the resulting irrep and by all the excitation quantum numbers of individual con- tributing degenerate modes such as ( v3v4α), whereαstands for the particular irrep. In Table II, we provide decompositions of several excited combinations of the two degenerate modes of ammonia in both the C3v(M)- and the D3h(M)-symmetry models. An excitation labeled (1 2 1 E′) would mean a state formed by v3= 1 quantum in oneE′degenerate mode and v4= 2 quanta in the other E′degener- ate mode, making together the E′irrep. Since there are two E′irreps coming from the v3= 1 and v4= 2 excitations, we need to give the E′irrep an additional index, 1 (see Table II). In principle, one could label the individual modes with their symmetric-product labels ( vα), but it remains inevitable to pro- vide the resulting irrep label. Otherwise, the labeling would not be unique. This is apparent from Table III for the v3= 1 and v4= 2 states, where we compare our suggested labeling scheme, which is unique, with the combination of individual mode labels, which is ambiguous. However, one cannot use a particular component of the degenerate irrep for labeling the individual modes as it would not generally result in a pure irrep of the given symmetry group. That would be the same mistake as constructing a two-electron sin- glet spin state as a simple product of the αandβspin functions and ignoring the proper Clebsch–Gordan expansion24that leads TABLE III . An example showing the uniqueness of the newly suggested labeling scheme for excited combinations of degenerate modes, in contrast to the ambigu- ous use of individual degenerate mode labels for the v3= 1 and v4= 2 excitation numbers within the C3v(M) symmetry model. (v3v4α) ( v3α3) (v4α4) (1 2A1) (1 E) (2E) (1 2A2) (1 E) (2E) (1 2 1 E) (1 E) (2A1) (1 2 2 E) (1 E) (2E) J. Chem. Phys. 154, 144306 (2021); doi: 10.1063/5.0043946 154, 144306-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp to the well-known1√ 2(αβ−βα)combination. The appropriate Clebsch–Gordan expansion of the resulting vibrational state in the direct-product space of ν3andν4modes generally reads ∣v3v4Γij⟩=∑ αkβk∣[E′]v3αk⟩∣[E′]v4βk⟩Cij αkβk(v3,v4), (11) where | v3v4Γij⟩is the jth [j= 1...ci; see Eq. (10)] state of Γi irrep formed by the v3-quanta-excited ν3mode and the v4-quanta- excitedν4mode [see Eq. (10)], ∣[E′]v3αk⟩is the kth component of theαth irrep produced from the v3-fold symmetric product of the ν3thE′irrep with itself (note that for one-dimensional αirrep, the index kis just 1), ∣[E′]v4βk⟩is analogous, and Cij αkβk(v3,v4)is the corresponding Clebsch–Gordan coefficient. In contrast to the traditional use of vibrational angular momen- tum quantum number pairs, ( vl), the labeling scheme suggested above directly provides the symmetry and the degeneracy of the resulting states. Moreover, for reasons explained in the previous paragraph and directed by Eq. (11), the labeling with a simple prod- uct of ( v3l3) and ( v4l4) modal quantum numbers, which is common for the vibrational assignment of systems such as ammonia, cannot generally lead to the proper state symmetry. Nevertheless, such a labeling still remains unique, and it is sufficient in that sense. III. COMPUTATIONAL DETAILS A. Internal coordinates and embedding The vibrational motion of the ammonia molecule is parameter- ized, as shown in Figs. 1 and 2. Before shifting the center of nuclear mass into the origin of the coordinate system, the coordinates of the individual atoms read as FIG. 1 . Vibrational parameters r1,r2,r3, andϑof the ammonia molecule with the xzplane shaded. FIG. 2 . Theβ1andβ2vibrational parameters of ammonia shown in the xy projection. x y z N 0 0 0 H1 r1sinϑ 0 r1cosϑ H2r2sinϑcosβ1r2sinϑsinβ1r2cosϑ H3r3sinϑcosβ2−r3sinϑsinβ2r3cosϑ. (12) The actual internal coordinates are symmetry adapted in the radial parameters with respect to permutations of the hydrogen atoms, which gives q1=r1+r2+r3,q2= 2r1−r2−r3,q3 =r2−r3,q4=ϑ,q5=β1, and q6=β2. Such an internal-coordinate system has been chosen to mimic the harmonic oscillator-like vibra- tional modes. The mapping between the internal coordinates and the associated modes is given in Table IV. B. Variational computations Variational computation of the vibrational states of14NH 3 has been performed using our in-house nuclear-motion code GENIUSH,26,27where the abbreviation stands for a general (GE), numerical (N) rovibrational code employing curvilinear inter- nal (I) coordinates and user-specified (US) Hamiltonian (H). Altogether, 600 vibrational states of14NH 3have been com- puted on a direct-product grid utilizing the parameters shown in Table V. The individual coordinates are represented by Hermite- or TABLE IV . Vibrational modes with the associated internal coordinates and symme- tries. Mode Coords. Corresponding normal mode Symmetry ν1 q1 Symmetric stretch A1 ν2 q4 Umbrella motion A1 ν3 q2,q3 Antisymmetric stretch E ν4 q5,q6 Antisymmetric bend E J. Chem. Phys. 154, 144306 (2021); doi: 10.1063/5.0043946 154, 144306-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE V . Parameters of individual internal coordinates in the computational direct-product grid. The radial values are in bohr, and the angles are in degrees. Number of Number of PO-DVR Coord. Min Max Type DVR points PO-DVR points Ref. Conf. q1 4.40 7.55 Hermite 300 10 5.6373 q2 −1.80 2.10 Hermite 300 10 0.0000 q3 −1.20 1.20 Hermite 300 10 0.0000 q4 40 140 Legendre 101 48 90.0 q5 70 179 Hermite 23 ... 120.0 q6 70 179 Hermite 23 ... 120.0 Legendre-type discrete-variable-representation (DVR)28,29grid points, four of them further reduced by the potential-optimized (PO-DVR) procedure with a planar reference molecular configura- tion. The large-scale eigenvalue problem is solved by the iterative Lanczos algorithm.30 The spectroscopic potential NH3-Y201031of Yurchenko and co-workers has been employed during the nuclear-motion compu- tations. The masses of the hydrogen and nitrogen nuclei were set to mH= 1.007 825 u and mN= 14.003 074 u, respectively.IV. RESULTS, ANALYSIS, AND DISCUSSION During this study, altogether, 600 vibrational states of ammonia have been computed and the majority of them assigned. A concise comparison of our results to the recently published CoYuTe6line list is reported in the supplementary material. A selection of states relevant for this discussion is shown in Table VI. The actual com- parison is made only after the first two subsections, which discuss the density structure of the computed excited states with the associated TABLE VI . Comparison of vibrational states from the CoYuTe6list with the results of this work. The energy level ordering is that given by the CoYuTe list. Indices of the states provided by the CoYuTe list and indices of the computed eigenstates of this study are also shown. The excitation energies are given as wavenumbers (cm−1) with respect to the vibrational ground state. The zero-point vibrational energy (ZPVE) computed in this study is 7430.28 cm−1. The labels are [ n1,n2p,n3n4 α], and the CoYuTe labels also include the ( l3l4) quantum numbers (see the text for details). The molecular-symmetry (MS) group used is D3h(M). The two components of the degenerate E′and E′′states of this work are distinguished by + and −subscripts. CoYuTe This work Level Index E Label Index E Label 1 1 0.00 [0, 0+, 0 0 A′ 1(0 0)] 1 0.00 [0, 0+, 0 0 A′ 1] 2 7779 0.79 [0, 0 −, 0 0A′′ 2(0 0)] 2 0.79 [0, 0 −, 0 0A′′ 2] 3 2 932.43 [0, 1+, 0 0 A′ 1(0 0)] 3 932.48 [0, 1+, 0 0 A′ 1] 4 7780 968.12 [0, 1 −, 0 0A′′ 2(0 0)] 4 968.13 [0, 1 −, 0 0A′′ 2] 5 3 1597.48 [0, 2+, 0 0 A′ 1(0 0)] 5 1597.50 [0, 2+, 0 0 A′ 1] 6 3249 1626.27 [0, 0+, 0 1 E′(0 1)] 6 1626.20 [0, 0+, 0 1 E′ +] 7 1626.20 [0, 0+, 0 1 E′ −] 7 9447 1627.37 [0, 0 −, 0 1E′′(0 1)] 8 1627.29 [0, 0 −, 0 1E′′ +] 9 1627.29 [0, 0 −, 0 1E′′ −] 8 7781 1882.18 [0, 2 −, 0 0A′′ 2(0 0)] 10 1882.11 [0, 2 −, 0 0A′′ 2] 10 3250 2540.52 [0, 1+, 0 1 E′(0 1)] 12 2540.40 [0, 1+, 0 1 E′ +] 13 2540.40 [0, 1+, 0 1 E′ −] 14 5 3215.95 [0, 0+, 0 2 A′ 1(0 0)] 19 3215.75 [0, 0+, 0 2 A′ 1] 16 3252 3240.16 [0, 0+, 0 2 E′(0 2)] 21 3240.13 [0, 0+, 0 2 E′ +] 22 3240.14 [0, 0+, 0 2 E′ −] 20 3253 3443.63 [0, 0+, 1 0 E′(1 0)] 27 3443.45 [0, 0+, 1 0 E′ +] 28 3443.54 [0, 0+, 1 0 E′ −] 38 3258 4799.22 [0, 0+, 0 3 E′(0 1)] 55 4799.12 [0, 0+, 0 3 E′ +] 56 4799.12 [0, 0+, 0 3 E′ −] 40 1825 4840.89 [0, 0+, 0 3 A′ 2(0 3)] 60 4842.09 [0, 0+, 0 3 A′ 2] J. Chem. Phys. 154, 144306 (2021); doi: 10.1063/5.0043946 154, 144306-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE VI .(Continued.) CoYuTe This work Level Index E Label Index E Label 41 12 4841.55 [0, 0+, 0 3 A′ 1(0 3)] 59 4841.33 [0, 0+, 0 3 A′ 1] 42 6482 4842.96 [0, 0 −, 0 3A′′ 1(0 3)] 62 4844.01 [0, 0 −, 0 3A′′ 1] 43 7788 4843.36 [0, 0 −, 0 3A′′ 2(0 3)] 61 4842.48 [0, 0 −, 0 3A′′ 2] 47 1826 5052.02 [0, 0+, 1 1 A′ 2(1 1)] 68 5051.17 [0, 0+, 1 1 A′ 2] 48 3260 5052.60 [0, 0+, 1 1 E′(1 1)] 70 5052.36 [0, 0+, 1 1 E′ +] 71 5052.37 [0, 0+, 1 1 E′ −] 52 14 5067.78 [0, 0+, 1 1 A′ 1(1 1)] 74 5069.62 [0, 0+, 1 1 A′ 1] 85 20 6348.84 [0, 2+, 0 3 A′ 1(0 3)] 124 6341.36 [0, 2+, 0 3 A′ 1] 88 21 6356.07 [0, 2+, 0 3 A′ 1(0 3)] 128 6354.33 [0, 0+, 0 4 A′ 1] Note: CoYuTe misassignment; see level 85 101 3274 6608.82 [1, 0+, 1 0 E′(1 0)] 148 6607.21 [1, 0+, 1 0 E′ −] 149 6607.55 [1, 0+, 1 0 E′ +] 103 24 6650.82 [0, 0+, 1 2 A′ 1(1 2)] 152 6647.32 [0, 0+, 1 2 A′ 1] 104 1830 6650.91 [0, 0+, 1 2 A′ 2(1 2)] 154 6650.66 [0, 0+, 1 2 A′ 2] 108 3275 6666.07 [0, 0+, 1 2 E′(1 2)] 156 6666.82 [0, 0+, 1 2 1 E′ +] 157 6666.88 [0, 0+, 1 2 1 E′ −] 109 3276 6677.43 [1, 0+, 1 0 E′(1 0)] 162 6676.34 [0, 0+, 1 2 2 E′ +] 163 6676.60 [0, 0+, 1 2 2 E′ −] Note: CoYuTe misassignment; see level 101 173 37 7860.33 [0, 2+, 0 4 A′ 1(0 0)] 252 7834.07 [0, 2+, 0 4 A′ 1] Note: Noticeably far lower energy in this work; see also level 175 175 3292 7875.62 [0, 2+, 0 4 E′(0 2)] 255 7852.87 [0, 2+, 0 4 1 E′ +] 256 7853.15 [0, 2+, 0 4 1 E′ −] Note: Noticeably far lower energy in this work; see also level 173 195 1836 8135.74 [1, 0+, 1 1 A′ 2(1 1)] 289 8131.49 [1, 0+, 0 3 A′ 2] Note: CoYuTe misassignment; see level 222 200 1837 8174.12 [0, 0+, 1 3 A′ 2(1 1)] 300 8175.03 [0, 0+, 1 3 A′ 2] 215 9496 8261.26 [0, 2 −, 0 4E′′(0 2)] 314 8246.27 [0, 2 −, 0 4 1 E′′ +] 315 8246.56 [0, 2 −, 0 4 1 E′′ −] Note: Noticeably far lower energy in this work 222 1839 8285.96 [0, 0+, 1 3 A′ 2(1 1)] 333 8283.76 [1, 0+, 1 1 A′ 2] Note: CoYuTe misassignment; see level 200 234 3309 8423.94 [1, 4+, 0 1 E′(0 1)] 347 8418.16 [1, 4+, 0 1 E′ −] 348 8418.23 [1, 4+, 0 1 E′ +] 271 9510 8937.31 [1, 4 −, 0 1E′′(0 1)] 413 9013.63 [1, 4 −, 0 1E′′ −] 415 9013.74 [1, 4 −, 0 1E′′ +] Note: Noticeably far higher energy in this work 328 7835 9436.75 [1, 0 −, 0 4A′′ 2(0 0)] 487 9415.74 [0, 0 −, 0 6 1 A′′ 2] Note: CoYuTe misassignment peculiarities of the ammonia molecule (Sec. IV A) and the actual assignment process (Sec. IV B). In this study, we use the following notation for labeling the ammonia vibrational states: [ n1,n2p,n3n4α], where n1,n2,n3,andn4are the number of quanta in the respective vibrational modes (see Table IV), the parity pis either “+” or “ −,” reflecting the inversion symmetry of the molecule, and it is associated with the umbrella-motion-like coordinate q2, whileαis the resulting irrep [in J. Chem. Phys. 154, 144306 (2021); doi: 10.1063/5.0043946 154, 144306-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp theD3h(M) group] of the ( ν3ν4) pair of degenerate modes. Labels in the CoYuTe list are in addition provided with the vibrational angular momentum quantum numbers l3andl4, which we place in parentheses as the last descriptor of the full label. At this point, we need to clarify the distinction between the terms state andenergy level , terms we extensively use in this sec- tion. In this context, the meaning of these terms cannot be freely interchanged. By an energy level, we mean a particular eigenvalue of the Hamiltonian operator, no matter whether it is degenerate or not. However, the states are attributed to particular eigenvectors of the Hamiltonian, that is, one degenerate level corresponds to several states. Since the GENIUSH code that we employ in the computation of the Hamiltonian eigenstates cannot make full use of symmetry, the resulting eigenstates do not have perfectly degenerate numeri- cal values for the eigenenergies. Thus, we provide two energy val- ues, together with two eigenstates, for a single degenerate level. The CoYuTe list, by contrast, provides only one value for each energy level. A. Structure of the vibrational states of ammonia It may seem straightforward to represent the individual vibra- tional modes of14NH 3by the following density matrices: ν1as Γ1(q1′,q1),ν2asΓ1(q4′,q4),ν3asΓD 2(q2,q3), andν4asΓD 2(q5,q6). The first ten states, indeed, are concisely described this way, as shown in Fig. 3, where states are in rows and the columns corre- spond to the various density matrices. For the ground state, state No. 1, the density is concentrated in a simple area around the equilib- rium geometry. There are two equivalent energy minima on the PES of14NH 3related by the inversion symmetry operation and accessed mutually by tunneling via the umbrella-motion-like coordinate q4. This is why both minima are populated in the Γ1(q4′,q4)plot; see the two red areas on the bottom-left to upper-right diagonal. The other two populated areas (on the upper-left to bottom-right diago- nal), representing the cross-terms of the RDM [Eq. (3)], correspond to a product of the wave function in one minimum with the wave function in the other minimum. Comparing plots of the first two states, which differ only in the sign of these two cross-term areas, we can deduce that the ground state has the same wave-function sign in both minima and has thus “+” symmetry with respect to the inversion operation. State No. 2 changes the wave-function sign between the two minima, and it is thus assigned with the “ −” sym- metry label. In states Nos. 3 and 4, we can see a single wave-function node in each of the two minima, suggesting that the ν2mode is singly excited. From the symmetry of the sign distribution in the Γ1(q4′,q4)density plot, we can easily deduce “+” symmetry for state No. 3 and “ −” symmetry for state Nos. 4. State 5 is doubly excited inν2and has “+” symmetry, and its “ −” partner is state No. 10. States Nos. 6 and 7 are degenerate, singly excited in mode ν4, with “+” symmetry. Their complementary “ −” states are Nos. 8 and 9, respectively. One should note that the ΓD 2(q5,q6)densities might more naturally be functions of the q5+q6andq5−q6coordinate combinations. Figure 4 shows another selection of relatively simple states worth commenting on for an improved understanding of higher- energy states. State No. 12 is a combination of a single ν2mode excitation with “+” symmetry and a single ν4mode excitation (one of its degenerate components), as can be deduced by comparing the FIG. 3 . First ten vibrational states of ammonia (rows) depicted for each of its vibra- tional modes (columns) by an appropriate two-dimensional reduced-density matrix. First of the two dimensions corresponds to the vertical axis of the plots, while the other dimension corresponds to the horizontal axis. The stretching coordinates q1−3use atomic units, and the bending coordinates q4−6use degrees. Scale of the densities is shown by the color boxes underneath. J. Chem. Phys. 154, 144306 (2021); doi: 10.1063/5.0043946 154, 144306-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . Selected vibrational states of ammonia (rows) depicted for each of its vibra- tional modes (columns) by an appropriate two-dimensional reduced-density matrix. First of the two dimensions corresponds to the vertical axis of the plots, while the other dimension corresponds to the horizontal axis. The stretching coordinates q1−3use atomic units, and the bending coordinates q4−6use degrees. Scale of the densities is shown by the color boxes underneath.densities of states Nos. 3 and 6. It turns out that for the vast majority of cases, the excitation structure of the states can be easily deduced in the same way by comparing shapes of the individual mode RDMs. States Nos. 19, 21, and 22 correspond to the doubly excited ν4mode. The threefold degeneracy of the TDIHO model is split under the D3h(M) MS group into the A′ 1irrep of state No. 19 and the dou- bly degenerate E′irrep of states Nos. 21 and 22 (see Table I). States Nos. 27 and 28 are degenerate singly excited ν3modes, and one can note that each of the E′irrep components is already well described by the single internal coordinate, in contrast to the excited ν4mode commented above. The four states Nos. 55, 56, 59, and 60 repre- sent the triply excited ν4mode, which is split into the degenerate E′(Nos. 55, 56) and separate A′ 1and A′ 2irreps (Nos. 59 and 60, respectively). In Fig. 5, another type of state is shown, in which both E′degen- erate modes are excited simultaneously. One can immediately note that the density shapes of the two E′modes (ν3andν4) are not sim- ple combinations of densities of only individually excited ν3orν4 modes. The assignment is thus more complicated than what has just been described for states with only one of the two degenerate modes excited. States Nos. 68, 70, 71, and 74 have both the ν3andν4modes singly excited, which leads to state symmetries, E′⊗E′=A′ 1⊕A′ 2⊕E′. (13) Thanks to the observed (computationally obtained) degeneracy of the states, states Nos. 70 and 71 can be assigned with the E′label. Otherwise, the states are hardly distinguishable or assignable just by comparing the two-dimensional densities, which all look almost the same in a given mode. Since all the RDM plots look very similar and they do not resemble any of the E′RDM shapes of the individual singly excited ν3orν4mode patterns (i.e., states Nos. 6, 7, 27, and 28) so that we could assign each of the two modes by one compo- nent of its E′degenerate pair, we can conclude, in accordance with Sec. II D, that every time both E′modes are excited simultaneously, it is necessary to treat both E′modes as a single two-mode-unit with a complex label specifying the distribution of the quanta between the two modes and also the resulting irrep, which needs to be known from the computation or examined explicitly. Thus, the complex two-mode-unit label of all these states has one quantum in mode ν3, one quantum in mode ν4, and the resulting symmetry of the two- mode-unit is A′ 1orA′ 2orE′. A shorthand notation is, for example, (1 1A′ 1). The other six states in Fig. 5 correspond to singly excited ν3and doubly excited ν4modes, which decompose as E′⊗[E′]2=E′⊗(A′ 1⊕E′)=A′ 1⊕A′ 2⊕2E′. (14) Labeling the non-degenerate states Nos. 152 and 154 by mere visual inspection of the depicted densities is not productive, and the explicit knowledge of their irreps is necessary. Looking at the ν3den- sity in the degenerate states (Nos. 156, 157, 162, 163), one cannot fail to note a single excitation pattern similar to the one in states Nos. 27 and 28. The ν4mode in these four states exhibits a clear degenerate doubly excited density pattern, which, however, is very different from the one in states Nos. 21 and 22. This reaffirms the above-mentioned observation that excitations of a single degenerate J. Chem. Phys. 154, 144306 (2021); doi: 10.1063/5.0043946 154, 144306-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . Selected vibrational states of ammonia (rows) depicted for each of its vibra- tional modes (columns) by an appropriate two-dimensional reduced-density matrix. First of the two dimensions corresponds to the vertical axis of the plots, while the other dimension corresponds to the horizontal axis. The stretching coordinates q1−3use atomic units, and the bending coordinates q4−6use degrees. Scale of the densities is shown by the color boxes underneath.mode need a different treatment than combined excitation of two (or generally more) degenerate modes. To successfully distinguish between the highly similar and rather featureless density representations of states possessing sev- eral excited degenerate modes, it is inevitable to resolve the two- dimensional quantities in yet another dimension. Since the only way how to resolve a mode’s RDM structure more deeply means the involvement of another coordinate that is intrinsic to a differ- ent mode, it actually means that the modes are a priori coupled. This, again, confirms the necessity of treating all degenerate modes together and not individually. The new dimension has to be taken from one of the two coordinates describing the other degenerate mode so that the three-dimensional quantity serves to characterize all the coupled modes. How to distinguish the degenerate states Nos. 70 and 71, by resolving the degenerate ν3andν4modes in three- dimensional densities ΓD 3(q5,q2,q3)andΓD 3(q2,q5,q6), is shown in Figs. 6 and 7. Apart from resolving the circular 2D density shapes, which we see in Fig. 5, into more informative 3D structures, one can see why these states are degenerate as their densities symmetri- cally differ only in their orientation. Even more remarkable is that we can see that the modal densities are composed of both E′irrep components, that is, the combination of the ν3densities of states Nos. 27 and 28 and ν4densities of states Nos. 6 and 7. This obser- vation actually confirms the theoretically expected behavior of a pair of excited degenerate modes (see Sec. II D) that cannot be expressed as a single product of respective degenerate irrep basis components but is a combination analogous to the Clebsch–Gordan expansion. Similarly, in Figs. 8 and 9, we resolve states Nos. 152 and 154, respec- tively, which have one quantum in ν3and two quanta in the ν4 modes. By looking at the ΓD 3(q2,q5,q6)density, which corresponds to theν4mode (Fig. 7), and going down the q2coordinate, one can clearly note a transition between density structures recognized in the degenerate states Nos. 21 and 22. Nevertheless, the states Nos. 152 and 154 are not degenerate. As the last example, we resolve the degenerate states Nos. 156 and 157 in Figs. 10 and 11, respectively. One can see there that the 3D density structure is more complex than what might have been deduced from the relatively simple-looking 2D plots of Fig. 5. In summary, to understand the multidimensional structure of vibrational wave functions, one has to start with a wise choice of the coordinate system that mimics well the modes of vibrations. Apart from the symmetry of a particular state’s wave function, which is given by the computation or examined explicitly via projection tech- niques, the structure of the individual modes can be inspected by means of appropriate reduced-density matrices. Each mode’s excita- tion leads to a characteristic RDM pattern that can either be assessed visually or effectively compared between other states by computing RDM overlaps and quickly recognized by pattern matching. This technique is readily applicable to non-degenerate vibrational modes. For a single degenerate mode that is excited, one has to take into account the proper symmetry of the states into which the given excited mode decomposes, as discussed in Sec. II C. In systems with more than one degenerate mode, such as ammonia, one has to treat all the degenerate modes with a single comprehensive label describ- ing both the distribution of quanta among the degenerate modes and also the resulting symmetry of all those coupled degenerate modes, as described in Sec. II D. In order to distinguish the RDMs of the degenerate modes by pattern matching and to correctly assign J. Chem. Phys. 154, 144306 (2021); doi: 10.1063/5.0043946 154, 144306-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 6 . Resolution of the degenerate vibrational modes by a third dimension of an appropriate reduced-density matrix for the selected state. Each row depicts two- dimensional submatrices of the three-dimensional RDMs with the first dimension fixed at a given coordinate value. The stretching coordinates q2−3use atomic units, and the bending coordinates q5−6use degrees. Scale of the densities is shown by the color box underneath. FIG. 7 . Resolution of the degenerate vibrational modes by a third dimension of an appropriate reduced-density matrix for the selected state. Each row depicts two- dimensional submatrices of the three-dimensional RDMs with the first dimension fixed at a given coordinate value. The stretching coordinates q2−3use atomic units, and the bending coordinates q5−6use degrees. Scale of the densities is shown by the color box underneath. J. Chem. Phys. 154, 144306 (2021); doi: 10.1063/5.0043946 154, 144306-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 8 . Resolution of the degenerate vibrational modes by a third dimension of an appropriate reduced-density matrix for the selected state. Each row depicts two- dimensional submatrices of the three-dimensional RDMs with the first dimension fixed at a given coordinate value. The stretching coordinates q2−3use atomic units, and the bending coordinates q5−6use degrees. Scale of the densities is shown by the color box underneath. FIG. 9 . Resolution of the degenerate vibrational modes by a third dimension of an appropriate reduced-density matrix for the selected state. Each row depicts two- dimensional submatrices of the three-dimensional RDMs with the first dimension fixed at a given coordinate value. The stretching coordinates q2−3use atomic units, and the bending coordinates q5−6use degrees. Scale of the densities is shown by the color box underneath. J. Chem. Phys. 154, 144306 (2021); doi: 10.1063/5.0043946 154, 144306-13 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 10 . Resolution of the degenerate vibrational modes by a third dimension of an appropriate reduced-density matrix for the selected state. Each row depicts two- dimensional submatrices of the three-dimensional RDMs with the first dimension fixed at a given coordinate value. The stretching coordinates q2−3use atomic units, and the bending coordinates q5−6use degrees. Scale of the densities is shown by the color box underneath. FIG. 11 . Resolution of the degenerate vibrational modes by a third dimension of an appropriate reduced-density matrix for the selected state. Each row depicts two- dimensional submatrices of the three-dimensional RDMs with the first dimension fixed at a given coordinate value. The stretching coordinates q2−3use atomic units, and the bending coordinates q5−6use degrees. Scale of the densities is shown by the color box underneath. J. Chem. Phys. 154, 144306 (2021); doi: 10.1063/5.0043946 154, 144306-14 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the excited states, each individual (degenerate) mode RDM has to be additionally a function of other coordinates that are specific to all the other degenerate modes. B. The actual assignment process There are three basic assumptions behind RDM-based vibra- tional assignments. First, the chosen coordinate system must mimic the nuclear motion of real vibrational modes. Second, the multimode-excited wave-function density resembles densities of individually excited modes. Third, the densities of individual mode excitations are the characteristic and remain almost constant (i.e., recognizable) throughout a large number of excited states. We note here that the first assumption is mainly a practical prereq- uisite so that one can visually associate a particular mode exci- tation to an appropriate quantum number. In other words, one can use an arbitrary coordinate system, provided that there is a way of associating particular densities with appropriate quantum numbers, which is easily accomplished if coordinates mimicking normal modes are used, for instance. Once a density pattern is known for a particular mode excitation, there is actually no more need for the visual assessment of the density shapes as all is done automatically by pattern matching, even for oddly looking den- sities. To work with the actual densities and assign the states based on them with appropriate labels, we used a set of conve- nient auxiliary codes that made the assignment process relatively straightforward and semi-automatic. The algorithm has four major components. First, each individual state was labeled with symmetry, by computing wave-function characters with respect to the {E,(H2H3),E∗,(H2H3)∗}subgroup of D3h(M). This way the par- ity was recognized and the non-degenerate states were assigned with their D3h(M) irreducible representation labels. To distinguish degenerate states from non-degenerate ones by symmetry, it would also be necessary to evaluate characters of the (H 1H2) permutation operator. This is, however, not straightforward in the chosen coor- dinate system and DVR basis. Fortunately, the degeneracies could mostly be recognized either by energy, when the levels were suf- ficiently isolated from each other, or by visual inspection of the densities, where the two components of a degenerate level formed a characteristic pair. Second, labels of not yet assigned states were estimated by a generalized vibrational build-up principle, similar to what we already used in our study of the water molecule,5but also tak- ing into account the effect of tunneling and that of the degenerate modes. The appropriate code keeps track of all already assigned labels and, for the currently examined state, suggests the logically closest higher excitations accordingly. The basic idea is that for an n-tuplet of quantum numbers, there is a natural ordering between certain excited states. Let us take a quadruplet of quantum num- bers (v1v2v3v4) as an example. Then, the ground state is (0 0 0 0), by definition. Logically, then, there are four descendant states with the labels (1 0 0 0), (0 1 0 0), (0 0 1 0), and (0 0 0 1), one of which has to be the first excited vibrational state. Each excited state thus has to be followed either by one of its natural descendants or by one of the not yet assigned descendants of earlier assigned states. It is also natural that the following ordering of the states, (0 1 0 0) <(0 1 1 0)<(0 1 2 0)<(1 1 2 5), must be correct, even thoughthere may be other states in between them. When a system exhibits vibrational tunneling, which is the case for ammonia, then each state is split into “+” and “ −” components, although the ordering may differ for each state. This makes the general build-up princi- ple complicated as it allows for uneven orderings, such as (0 1 2 0)− <(0 2 2 0)+<(0 1 2 0)+<(0 2 2 0)−, although the principle remains strict. The presence of degenerate modes in ammonia requires inclu- sion of all the irrep labels that come out of their excited-state decompositions (see Secs. II C and II D) into the list of candidate labels for the state under examination. Employing the vibrational build-up principle helps substantially not only to assign complicated highly excited states but also to ensure that no excited-state label is accidentally omitted during the assignment of a large number of states. Third, comparison and recognition of known density pat- terns of individual vibrational modes are performed, similar to that already described in our earlier work.5The similarities between RDMs are quantified automatically by computing their mutual overlaps. The associated code also keeps track of all the already assigned states and stores their modes’ appropriate RDMs as unique patterns corresponding to a particular mode quantum number. All the degenerate modes need to be treated as a single supermode pos- sessing of several RDMs and having a single complex label (see Sec. II D). If there is good match between the RDM of a partic- ular mode and a known pattern (the overlap with one RDM pat- tern is typically high above 0.9, while much less with all the other patterns), then the mode’s label is automatically assigned. Other- wise, the state is marked as 1 with a label not yet known, and it is up to the user to decide, in further steps, about the most feasi- ble label based on hints from the other auxiliary tools or by other reasoning. The last resort is an assignment tool based on the visualiza- tion of the RDMs. Surprisingly, visualization is not needed for the vast majority of the states as the other automatic utility codes do an excellent job, leaving little or no doubt about the labels. Nevertheless, in what follows, we consider the visual inspection of the RDMs as the final decision-making step for all the assignments. In some cases, visual assessment is the only way to decipher skewed poorly converged wave functions when the RDM pattern matching does not provide plausible suggestions. In other cases, by contrast, deciphering the RDM plots can lead to a maze, while the pattern matching suggestions are indisputable. At any rate, the visual inspec- tion of RDMs remains invaluable for revealing the insight into the multidimensional structure of vibrational states. Practically, the assignment process is executed following a five-step pseudocode: 1. Determine the symmetry, including parity, of all the computed vibrational wave functions. 2. For the state currently examined, starting from the ground state, generate all possible label candidates based on the vibrational build-up principle and in accord with the state’s symmetry. 3. For the state currently examined, compare its RDMs with the pool of RDM patterns of states already assigned. Check that the suggested labels also agree with the list of label candidates from step (2). If some of the mode RDMs is not recognized from J. Chem. Phys. 154, 144306 (2021); doi: 10.1063/5.0043946 154, 144306-15 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the known patterns and the list of label candidates suggests a not yet assigned quantum number for that mode, consider including the state RDM into the pattern pool as a new pattern after approval in step (4). 4. Visually check the RDM plots of the currently examined state and make sure that they correspond to the suggested label can- didate from the previous step. If there is no doubt, then do assign the suggested label to the current eigenstate and also update the pool of patterns, if appropriate. Then, take the next eigenstate and go to step (2). If competing label candi- dates remain or if the visual inspection of the only candidate raises doubts, then the current eigenstate is marked as not yet assigned and the procedure continues with the next eigenstate in step (2). 5. Start again at the ground state and reexamine all the states not yet assigned. By now, the list of label candidates generated for each such state by the vibrational build-up principle is much shorter as many labels have already been assigned to different levels. The pool of RDM patterns is also broader than during the first run, which leads to better matches with the appropri- ate modal assignments. Each newly suggested assignment for the state is further examined for consistency of energy gaps between + and −tunneling splitting pairs, completeness of family of states with excited degenerate modes (e.g., no state from the family [0, 0+, 1 2 X], where X∈{A′ 1,A′ 2, 1E′, 2E′} is left unassigned), energy, and RDM similarities with related families of states (e.g., states in families [0, 0+, 1 2 X] and [0, 2+, 1 2 X] are related). If necessary, also inspect the eigenstate by other appropriate RDMs (e.g., three-dimensional) to make sure that the assignment is correct. When there is no doubt, the state is assigned with the suggested label; otherwise, it is left unassigned. To illustrate the algorithm on a concrete example, we describe here how the assignment of the first ten states proceeded (see Table VI and Fig. 3). State No. 1 has + parity, and since there are no visible nodes in its RDM plots, it is assigned as [0, 0+, 0 0 A′ 1], and the RDMs are stored as patterns with appropriate modal labels. State No. 2 has unit overlaps with the stored patterns except for the second mode. By visual inspection, by symmetry, and in accord with the build-up principle, the state is assigned with the label [0, 0−, 0 0 A′′ 2] and the second mode’s RDM is stored as the 0 −pat- tern. State No. 3 also has all RDMs with unit overlap to the stored ground state patterns except for the second mode, which is not yet known. The parity is + , and the build-up principle allows only one such label: [0, 1+, 0 0 A′ 1]. After the clear visual confirmation, the label is assigned to the state and the second mode’s RDM is stored as the 1+ pattern. Similarly, but with the opposite parity, state No. 4 is assigned as [0, 1 −, 0 0 A′′ 2], and the second mode’s RDM is stored as the 1 −pattern. State No. 5 has again not yet recognized RDM for the second mode; otherwise, the other modes have unit overlap with the stored ground state RDMs. From its symmetry and in accord with the build-up principle and the visual assessment, it is labeled [0, 2+, 0 0 A′ 1]. The second mode’s RDM pattern 2+ is again stored. States Nos. 6 and 7 are degenerate with + parity and with the first two modes’ RDMs known from the ground state. There are two possible labels for the degenerate modes suggested bythe build-up principle: either [0, 0+, 1 0 E′] or [0, 0+, 0 1 E′]. By visual inspection and also by mere anticipation that the bending character of the fourth mode results in lower energy than the stretch- ing character of the third mode, the degenerate states are assigned as [0, 0+, 0 1 E′ +] and [0, 0+, 0 1 E′ −], respectively, where we intro- duce the +and−subscripts to distinguish between the two E′states. The RDM patterns 0 1 E′ +and 0 1 E′ −are stored for the combined third and fourth modes since the modes are degenerate and thus need common treatment (see Sec. II D). The pattern matching for states Nos. 8 and 9 yields a direct assignment and labels [0, 0 −, 0 1E′′ +] and [0, 0−, 0 1 E′′ −], respectively, since all the RDM patterns are already known and these labels are in accord with the build-up principle and also with the visual assessment. No new patterns are needed to be stored. For state No. 10, pattern matching suggests [0, X−, 0 0 A′′ 2]. The only allowed label of this kind from the build-up principle and also in accord with the visual inspection of the RDMs is [0, 2−, 0 0 A′′ 2]. The label is thus assigned with the state, and the second mode’s RDM is stored as the 2 −pattern. This way, we were able to reliably provide vibrational labels to about 500 of the lowest 600 eigenstates of14NH 3. C. Comparison with CoYuTe assignments For comparison and an impartial assessment of our results obtained for14NH 3, we chose the recent large ammonia high- temperature ro-vibrational line and energy level list known as the CoYuTe6list, assembled in 2019. There is an even newer list32sup- plementing the CoYuTe list, but it is amended by an earlier set of results of the present study, so the comparison to that list would not be impartial. The much larger number of states in the CoYuTe6list was labeled automatically based on overlaps of the computed wave functions with the FBR basis functions. A comprehensive compar- ison of our results to the vibrational states of CoYuTe with exci- tation energies under 10 000 cm−1is shown in the supplementary material. In Table VI, we selected levels relevant for discussion. The ordering of the energy levels in Table VI is dictated by the CoYuTe list. Most of the energy levels reported by the two studies are as close in energy as a few cm−1. Nevertheless, there are also significant differences, some mentioned below. For the 600 distinct states com- puted during this work, we can distinguish three blocks. The first block covers the lowest 297 energy levels of the CoYuTe list, which means∼440 distinct states (after counting the two components of degenerate levels separately). Within this block, all the states could be smoothly and unambiguously assigned in this work based on symmetry and the semi-automatic RDM pattern-recognition tech- nique. Apart from subtle differences in energy level orderings, where the energy separation of neighboring states is within a fraction of a cm−1(e.g., level Nos. 40 vs 41, Nos. 42 vs 43, Nos. 48 vs 49, Nos. 51 vs 52, and many others), there are also states with unusually large energy differences between the CoYuTe list and the results of this work. Examples include levels No. 173 or No. 175, which differ by more than 20 cm−1between the two studies, while their neighbors are nicely matched within 2–6 cm−1. Similar examples are levels Nos. 215 and 271. Except for level No. 271 out of the four mentioned, this work provides significantly lower energies than CoYuTe. This can be rationalized by the superior Hamiltonian (we employ an exact Hamiltonian) and basis used during our GENIUSH J. Chem. Phys. 154, 144306 (2021); doi: 10.1063/5.0043946 154, 144306-16 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp computations. The odd result is that level No. 271 is predicted to be as much as 76 cm−1above the CoYuTe energy. If we look into the tunneling splittings of the states instead of their absolute ener- gies, the results of this work match the CoYuTe list very well, the exception is again level No. 271. This state is labeled as [1, 4 −, 0 1E′′] and has its tunneling counterpart [1, 4+, 0 1 E′] at level No. 234. The tunneling splitting predicted by CoYuTe is 513 cm−1, which is significantly different from this work’s 595 cm−1. Tun- neling splittings of other states with the same [ ⋅, 4±,...] pattern are about 600 cm−1in both CoYuTe and this work. Therefore, we believe that the CoYuTe energy of 8937.31 cm−1for level No. 271 is incorrect. We identified several misassignments in the CoYuTe list in the first block. For example, CoYuTe uses the same label for levels Nos. 88 and 85, while completely omitting the label [0, 0+, 0 four A′ 1]. Similar label redundance/omitting characterizes the CoYuTe levels Nos. 109 vs 101, Nos. 110 vs 102, Nos. 216 vs 219, Nos. 222 vs 200, and Nos. 223 vs 202. For levels Nos. 139 and 143, the labeling is swapped with respect to results of this work. For lev- els Nos. 195, 196, 201, 203, 204, 205, 212, and 286, the CoYuTe list exhibits a bit more complicated misassignments. Here, we dis- cuss only level No. 195, labeled by CoYuTe as [1, 0+, 1 1 A′ 2(1 1)]. The reasoning employed for the other states can be understood from Table VI. The label of level No. 195 suggested by CoYuTe is assigned during this work for level No. 222. For that level, in turn, CoYuTe suggests the label [0, 0+, 1 3 A′ 2]. However, that has already been assigned to level No. 200 by both CoYuTe and this work. Moreover, CoYuTe does not have the label [1, 0+, 0 3 A′ 2] listed, which is assigned to level No. 195 in our work. This means that the CoYuTe assignment protocol did not succeed to predict these few cases correctly, while our labels remain consistent and complete. The second block of states involves levels Nos. 298–360. Con- trary to the first block, not all the states could be assigned unambigu- ously by the RDM method. States assigned during this work match their CoYuTe counterparts, except level No. 328 whose CoYuTe assignment is odd (the CoYuTe label [1, 0 −, 0 4A′′ 2(0 0)] is missing its tunneling counterstate [1, 0+, 0 4 A′ 1(0 0)], and also the [0, 0 −, 0 6A′′ 2(0 0)] label is missing). The other CoYuTe labels within the second block agree with the RDM-based predictions; nevertheless, due to ambiguities in RDM patterns, such states were left unassigned during this work. In this block, there are still many CoYuTe energy levels that are close to those predicted by this work, but an increasing number of levels differ significantly. In the remaining, third block of states, starting from energy level No. 361, there are only a few states that are reliably assigned by the RDM method. Most of the states could not even be matched by energy and symmetry between the CoYuTe list and the results of this work. Such states are left out from comparison and put at the very end of the table for reference. In order to assign more states reliably, a larger basis set and better convergence of the highly excited wave functions need to be achieved. The auxiliary codes would also need more development in ranking the plausible label candidates by estimating their energy according to earlier assigned levels and tunneling splittings of similarly excited labels. The dif- ferences in symmetry of the remaining unmatched states between the CoYuTe list and the states predicted in this study suggest that at such high energies, the results are sensitive to the qualityof thecomputations, which can lead to a considerable shuffle of states ordering. V. CONCLUSIONS We demonstrated, on the example of the ammonia (14NH 3) molecule, that the use of reduced-density matrices helps to achieve a thorough understanding of the multidimensional structure of non-trivial excited vibrational wave functions. The reduced-density matrix approach is not only limited to using normal coordinates but also particularly suitable for systems described by arbitrary inter- nal coordinates. The vibrational assignment process based on RDMs and internal coordinates had already been employed for the vinyl radical4and the water molecule.5The ammonia molecule used in the present study represents a substantially more complex system as it not only exhibits internal tunneling motion but also possesses two degenerate interacting vibrational modes. To succeed, we had to reformulate the conventional treatment and labeling of systems with several degenerate modes (the method of vibrational angular momenta, which is commonly used to describe ammonia vibrations, does not respect molecular symmetry, cannot predict correct degen- eracy, and prevents full understanding of mutual inter-state rela- tions). The result is a general, unique, and insightful labeling scheme that respects the system’s symmetry. With the RDM algorithm developed, we were able to correctly assign several 100 vibrational states of ammonia with relative ease. Our study also confirms that excited vibrational states tend to have a truly modal structure, i.e., the multidimensional wave function is decomposable to individually excited modes to a good degree of approximation. Reduced-density matrices representing particu- lar excited vibrational modes are very characteristic and can serve as reliable patterns for a quick and semi-automatic state assignment. Moreover, the visual representation of the various RDMs gives an invaluable insight into the physical meaning of the excited modes and their mutual relations. The RDM method presented, together with the reformulated treatment of excited degenerate modes and through the auxiliary codes developed, represents a cutting-edge technique for reveal- ing and understanding the internal structure of excited multidi- mensional vibrational states. The method is applicable to arbi- trary (Abelian or non-Abelian) molecular systems for which excited vibrational wave functions can be practically computed so that the nodal structure significant to a particular vibrational motion can be examined. For large molecules, it may also mean that at least a portion of the PES that is related to a specific vibrational motion is available. The RDM algorithm developed is a significantly improved alternative to basis-set-decomposition-based assignment approaches. Finally, we note that after comparing our results to the large, recently published CoYuTe6list of vibrational states of ammo- nia, we identified several misassignments and inconsistencies in the CoYuTe list. SUPPLEMENTARY MATERIAL See the supplementary material for a complete version of Table VI comparing vibrational states of ammonia from the CoYuTe6list with the results of this work. J. Chem. Phys. 154, 144306 (2021); doi: 10.1063/5.0043946 154, 144306-17 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp ACKNOWLEDGMENTS The work performed in Budapest received support from the NKFIH (Grant No. K119658) and from the ELTE Institutional Excellence Program (Grant No. TKP2020-IKA-05) financed by the Hungarian Ministry of Human Capacities (EMMI). Compu- tational resources used in Prague were supplied by the project “e-Infrastruktura CZ” (e-INFRA LM2018140) provided within the program Projects of Large Research, Development and Innova- tions Infrastructures. The authors are grateful to Professor Sergey Yurchenko for useful discussions and comments. DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material. REFERENCES 1A. G. Császár, C. Fábri, T. Szidarovszky, E. Mátyus, T. Furtenbacher, and G. Czakó, “Fourth age of quantum chemistry: Molecules in motion,” Phys. Chem. Chem. Phys. 14, 1085–1106 (2012). 2E. Mátyus, C. Fábri, T. Szidarovszky, G. Czakó, W. D. Allen, and A. G. Császár, “Assigning quantum labels to variationally computed rotational-vibrational eigen- states of polyatomic molecules,” J. Chem. Phys. 133, 034113 (2010). 3G. Hose and H. S. Taylor, Phys. Rev. Lett. 51, 947 (1983). 4J. Šmydke, C. Fábri, J. Sarka, and A. G. Császár, “Rovibrational quantum dynam- ics of the vinyl radical and its deuterated isotopologues,” Phys. Chem. Chem. Phys. 21, 3453–3472 (2019). 5J. Šmydke and A. G. Császár, “On the use of reduced-density matrices for the semi-automatic assignment of vibrational states,” Mol. Phys. 117, 1682–1693 (2019). 6P. A. Coles, S. N. Yurchenko, and J. Tennyson, “ExoMol molecular line lists–XXXV. A rotation-vibration line list for hot ammonia,” Mon. Not. R. Astron. Soc.490, 4638–4647 (2019). 7C. C. J. Roothaan, “New developments in molecular orbital theory,” Rev. Mod. Phys. 23, 69 (1951). 8E. B. Wilson, Jr., J. C. Decius, and P. C. Cross, Molecular Vibrations (McGraw-Hill, New York, 1955). 9J. N. Murrell, S. Carter, S. C. Farantos, P. Huxley, and A. J. C. Varandas, Molecular Potential Energy Surfaces (Wiley, New York, 1984). 10P. G. Mezey, Potential Energy Hypersurfaces (Elsevier, New York, 1987). 11A. G. Császár, W. D. Allen, Y. Yamaguchi, and H. F. Schaefer, “ Ab initio deter- mination of accurate ground electronic state potential energy hypersurfaces for small molecules,” in Computational Molecular Spectroscopy (Wiley, New York, 2000), pp. 15–68. 12M. S. Child and L. Halonen, “Overtone frequencies and intensities in the local mode picture,” Adv. Chem. Phys. 57, 1–58 (1984).13S. N. Yurchenko, W. Thiel, and P. Jensen, “Theoretical rovibrational energies (trove): A robust numerical approach to the calculation of rovibrational energies for polyatomic molecules,” J. Mol. Spectrosc. 245, 126–140 (2007). 14T. Mathea and G. Rauhut, “Assignment of vibrational states within configura- tion interaction calculations,” J. Chem. Phys. 152, 194112 (2020). 15M. Gruebele, “Intensities and rates in the spectral domain without eigenvec- tors,” J. Chem. Phys. 104, 2453–2456 (1996). 16H.-G. Yu, H. Song, and M. Yang, “A rigorous full-dimensional quantum dynamics study of tunneling splitting of rovibrational states of vinyl radical C 2H3,” J. Chem. Phys. 146, 224307 (2017). 17A. G. Császár, E. Mátyus, T. Szidarovszky, L. Lodi, N. F. Zobov, S. V. Shirin, O. L. Polyansky, and J. Tennyson, “First-principles prediction and partial charac- terization of the vibrational states of water up to dissociation,” J. Quant. Spectrosc. Radiat. Transfer 111, 1043–1064 (2010). 18D. A. Sadovskií, N. G. Fulton, J. R. Henderson, J. Tennyson, and B. I. Zhilinskií, “Nonlinear normal modes and local bending vibrations of H+ 3and D+ 3,” J. Chem. Phys. 99, 906–918 (1993). 19J. Sarka and A. G. Császár, “Interpretation of the vibrational energy level struc- ture of the astructural molecular ion H+ 5and all of its deuterated isotopomers,” J. Chem. Phys. 144, 154309 (2016). 20R. Dawes, X.-G. Wang, A. W. Jasper, and T. Carrington, “Nitrous oxide dimer: A new potential energy surface and rovibrational spectrum of the nonpolar isomer,” J. Chem. Phys. 133, 134304 (2010). 21P. R. Bunker and P. Jensen, Molecular Symmetry and Spectroscopy (NRC, Ottawa, 1998). 22K. Chubb, P. Jensen, and S. Yurchenko, “Symmetry adaptation of the rotation- vibration theory for linear molecules,” Symmetry 10, 137 (2018). 23L. Tisza, Z. Phys. 82, 48 (1933). 24M. Hamermesh, Group Theory and Its Application to Physical Problems (Addison-Wesley, Reading, MA, 1962). 25A. G. Császár, “Anharmonic force field of CO 2,” J. Phys. Chem. 96, 7898–7904 (1992). 26E. Mátyus, G. Czakó, and A. G. Császár, “Toward black-box-type full- and reduced-dimensional variational (ro)vibrational computations,” J. Chem. Phys. 130, 134112 (2009). 27C. Fábri, E. Mátyus, and A. G. Császár, “Rotating full- and reduced- dimensional quantum chemical models of molecules,” J. Chem. Phys. 134, 074105 (2011). 28Z. Bacic and J. C. Light, Annu. Rev. Phys. Chem. 40, 469 (1989). 29J. C. Light and T. Carrington, “Discrete-variable representations and their utilization,” in Advances in Chemical Physics (John Wiley & Sons, Inc., 2007), pp. 263–310. 30C. Lanczos, J. Res. Natl. Bur. Stand. 45, 255 (1950). 31S. N. Yurchenko, R. J. Barber, J. Tennyson, W. Thiel, and P. Jensen, “Towards efficient refinement of molecular potential energy surfaces: Ammonia as a case study,” J. Mol. Spectrosc. 268, 123–129 (2011). 32T. Furtenbacher, P. A. Coles, J. Tennyson, S. N. Yurchenko, S. Yu, B. Drouin, R. Tóbiás, and A. G. Császár, “Empirical rovibrational energy levels of ammo- nia up to 7500 cm−1,” J. Quant. Spectrosc. Radiat. Transfer 251, 107027 (2020). J. Chem. Phys. 154, 144306 (2021); doi: 10.1063/5.0043946 154, 144306-18 Published under license by AIP Publishing

5.0041059.pdf

Appl. Phys. Lett. 118, 082401 (2021); https://doi.org/10.1063/5.0041059 118, 082401 © 2021 Author(s).Observation of electrically detected electron nuclear double resonance in amorphous hydrogenated silicon films Cite as: Appl. Phys. Lett. 118, 082401 (2021); https://doi.org/10.1063/5.0041059 Submitted: 18 December 2020 . Accepted: 09 February 2021 . Published Online: 22 February 2021 Brian R. Manning , James P. Ashton , and Patrick M. Lenahan ARTICLES YOU MAY BE INTERESTED IN Ellipsometric study of the electronic behaviors of titanium-vanadium dioxide (Ti xV1−xO2) films for 0 ≤ x ≤ 1 during semiconductive-to-metallic phase transition Applied Physics Letters 118, 081901 (2021); https://doi.org/10.1063/5.0029279 Impurity band assisted carrier relaxation in Cr doped topological insulator Bi 2Se3 Applied Physics Letters 118, 081103 (2021); https://doi.org/10.1063/5.0039440 Diffusion of charge carriers in pentacene Applied Physics Letters 118, 083301 (2021); https://doi.org/10.1063/5.0026739Observation of electrically detected electron nuclear double resonance in amorphous hydrogenated silicon films Cite as: Appl. Phys. Lett. 118, 082401 (2021); doi: 10.1063/5.0041059 Submitted: 18 December 2020 .Accepted: 9 February 2021 . Published Online: 22 February 2021 Brian R. Manning,a) James P. Ashton, and Patrick M. Lenahan AFFILIATIONS The Pennsylvania State University, University Park, Pennsylvania 16802, USA a)Author to whom correspondence should be addressed: brm5293@psu.edu ABSTRACT We report on the electrical detection of electron nuclear double resonance (EDENDOR) through spin-dependent tunneling transport in an amorphous hydrogenated silicon thin film. EDENDOR offers a many orders of magnitude improvement over classical ENDOR and is exclu-sively sensitive to paramagnetic defects involved in electronic transport. We observe hyperfine interactions with 1H nuclei very close to silicon dangling bond defects. These observations substantially extend recent EDENDOR observations involving silicon vacancy defects and 14N hyperfine interactions with fairly distant nitrogen atoms in 4H-SiC bipolar junction transistors. We have improved the detection scheme utilized in the earlier study by combining magnetic field modulation with RF amplitude modulation; this combination significantly improvesthe operation of the automatic power leveling scheme and the overall sensitivity. Published under license by AIP Publishing. https://doi.org/10.1063/5.0041059 The performance and reliability of solid-state electronic devices are dominated by point defects. Magnetic resonance techniques offerunparalleled analytical power in the identification of paramagneticdefects in semiconductors and insulators. 1The basis for this magnetic resonance detection is electron paramagnetic resonance (EPR).Analysis of the EPR response provides information about the chemicalnature and atomic-scale structure of defects present in the sample. Thesensitivity of conventional EPR is roughly 10 11paramagnetic defects per mT linewidth.2Unfortunately, this number far exceeds the num- ber of performance-limiting defects in technologically relevant nano- scale solid-state devices. The related EPR technique, electron nuclear double resonance (ENDOR), can provide extremely detailed informa-tion about the immediate surroundings of paramagnetic defects byobserving small hyperfine interactions between paramagnetic defectsand nearby magnetic nuclei. 3–6Since the ENDOR response is typically several orders of magnitude smaller than that of conventional EPR, itis even less suitable than conventional EPR for studies of nanoscaledevices. The sensitivity of conventional EPR can be enormouslyimproved via electrically detected magnetic resonance (EDMR).EDMR is an EPR-based technique in which the EPR response isextracted from an electrical measurement that monitors spin-dependent transport. 7–14EDMR sensitivity is typically around seven orders of magnitude better than classical EPR detection.15,16EDMRhas proven to be quite valuable in studies of atomic-scale imperfec- tions in nanoscale solid-state electronics as well as semiconducting and insulating materials.11–14By utilizing EDMR detection, the sensi- tivity of ENDOR may be enhanced by many orders of magnitude withelectrically detected ENDOR (EDENDOR). EDENDOR was firstreported on relatively large volume samples. 6,17Quite high sensitivity EDENDOR was recently reported in measurements on a fully func-tional 4H-SiC bipolar junction transistor. 18,19In this study, we extend the earlier work, all of which involves spin dependent recombination(SDR), to EDEDNOR detected through spin dependent trap assistedtunneling (SDTAT). More importantly, the earlier studies all dealtwith matrix ENDOR. In matrix ENDOR, the response from magnetic nuclei distant from paramagnetic defects is observed in which hyper- fine interactions are not measured. In this work, we extract hyperfineparameters with modest precision. In EPR, the response of paramagnetic defect centers to the simul- taneous application of a quasi-static field and small RF or microwave-frequency field is observed. A difference in energy is caused by theinteractions of the electron magnetic moment with a magnetic field.The application of an oscillating magnetic field with frequency, v, times Planck’s constant, h, equal to this difference in energy induces resonance. For a free electron unperturbed by its local surroundings, the resonance condition is 1 Appl. Phys. Lett. 118, 082401 (2021); doi: 10.1063/5.0041059 118, 082401-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplDE¼hv¼gelBB: (1) Here,DEis the difference in energy between the spin states, geis the Land /C19egv a l u e( ge¼2:002 32… Þ,lBis the Bohr magneton, and Bis the magnitude of the large quasi-static magnetic field. The resonancecondition is altered by a defect’s surroundings. The most commonmechanisms of these alterations are spin–orbit coupling and electron-nuclear hyperfine interactions. Taking into account these factors, theresonance condition is modified and becomes 1,20 DE¼hv¼glBBþX imiAi: (2) Here, gis a second rank tensor and miis the nuclear spin quantum number of the ith nearby magnetic nucleus; Aiis also typically repre- sented by a second rank tensor representing the electron-nuclearhyperfine coupling. However, for the observation reported herein, thehyperfine interactions may be taken to be constants. By measuring themagnetic field at resonance and the frequency v, the physical and chemical nature of atomic scale defects can be identified. The EDMR in this study is detected through SDTAT. 21–23In SDTAT, electrons tunnel from defect to defect. The tunneling is for-bidden between two adjacent defects if unpaired electrons at bothdefects have the same spin quantum number by the Pauli exclusionprinciple. However, if we flip the spin of one of the unpaired electrons,the previously forbidden tunneling event becomes allowed. This isobserved as an increase in current through the device. 24 In classical ENDOR measurements, the EPR response is first measured.25The magnetic field is then held constant at a field that results in EPR microwave absorption. An oscillating magnetic fieldwith the field vector perpendicular to the large applied field is sweptover a frequency range that induces nuclear resonance at a site nearthe defect observed in EPR. When this occurs, the amplitude of theEPR response can change. 25Measuring the EPR amplitude as a func- tion of frequency allows for a measurement of the frequency, vn,a t which nuclear magnetic resonance (NMR) takes place.The ENDOR response is easiest to describe for the case of a defect with electron spin SðÞ¼1 2and a nearby nuclear spin IðÞ¼1 2for a sin- g l ei s o t r o p i cv a l u eo fh y p e r fi n ei n t e r a c t i o n , a. In this case, the ENDOR response condition will be given by vffivn6a 2/C12/C12/C12/C12( t h i si st h ec a s ef o r our study). Somewhat more complex responses can occur. For exam- ple, if I>1 2, the presence of nuclear quadrupole moments changes the ENDOR response frequency conditions to vffia 26vn6Q/C12/C12/C12/C12,w h e r e Q depends upon the electric field gradient at the nucleus and the nuclear quadrupole moment. The EDENDOR spectrometer design used in that study is shown inFig. 1 and is described in detail in a recent paper.19The EDENDOR spectrometer used in this study is somewhat modified from the one described in earlier work. The sample is placed into the microwave cavity and aligned with a single loop antenna. The NMR RF sweep issupplied to this loop via a Fluke-291 arbitrary waveform generator (AWG). In prior EDENDOR experiments, the magnetic field, B 0,w a s fixed and the NMR frequency sweep was 100% amplitude modulated for phase-sensitive detection. In the work reported here, we improve upon the detection sensitivity of the earlier study by using double modulation. We modulated both the B0fi e l da n du t i l i z e da1 0 0 % amplitude modulation of the NMR RF frequency sweep. Double mod-ulation was used in conjunction with an automatic power leveling scheme via a proportional integral differential (PID) controller, which almost completely removes the non-resonant-background that other- wise obscures the EDENDOR spectrum. 17,19 We conducted an EDENDOR experiment on a 10 nm amor- phous hydrogenated silicon thin film (a-Si:H) (99% Si, 1% H) on p-Si (100) wafers with Ti/Al metal contacts with an area of 0 :020 cm2.T h e spin density in these structures is approximately 5 /C21018cm/C03;t h u s , the number of paramagnetic defect sites in the sample is about 1011 spins. The film in this study was deposited via plasma enhanced chem-ical vapor deposition. The device was diced from the wafer via a dia-mond scribe and mounted to a printed circuit board. The electrical FIG. 1. EDMR/EDENDOR spectrometer schematic.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 082401 (2021); doi: 10.1063/5.0041059 118, 082401-2 Published under license by AIP Publishingconnections from the device to the leads were made via wire bond. These measurements were performed at room temperature. TheEDENDOR measurements were taken at the magnetic field at the cen-ter of the EDMR spectrum (zero-crossing), as well as several points oneach side of the center. The representative EDMR spectrum is illus- trated in Fig. 2 . EDENDOR measurements are limited by multiple sources of noise, which influence the EDMR detection such as shot,flicker, and thermal noise. 26Due to the extremely high defect density of this device, the EDMR response is quite strong and these noise sour-ces do not prevent an extremely high EDMR signal-to-noise ratio. TheEDMR response is consistent with SDTAT currents through silicondangling bond sites. 27The center field of the EDMR response is 332:2 m Ta n dt h er e s p o n s ei s /C250:7 mT wide. The measured zero- crossing g¼2:005560:0003 is consistent with silicon dangling bonds, a result widely reported on a-Si:H samples analyzed using EPRtechniques. 12,15,28,29 To observe the EDENDOR response, the magnetic field was modulated at 3350 Hz with an amplitude of 0 :4 mT (approximately half the linewidth). The RF was 100% amplitude modulated at 250 Hz.This modulated response is detected through a virtual lock-in ampli-fier (VLIA) with two demodulation stages. The VLIA was written in LabVIEW 2018 and utilizes two mixing stages in which it is multiplied by a reference sinusoid with an adjustable phase. After each mixingphase, there is a low-pass filter with an adjustable time constant. Thefirst demodulates the field modulated response with a small time con-stant of approximately 0.6 ms. The short first stage time constant isnecessary in order to pass the 250 Hz amplitude modulated responseto the second lock-in. To demonstrate that the response of the system is EDENDOR and not some artifact, we repeated the identical mea- surement but with the field shifted from the EDMR resonance condi-tion of 332 :2 mT to 300 mT, far from the EDMR resonance condition. EDENDOR results of the on- and off-EDMR resonance field areillustrated in Fig. 3 . The signal-to-noise ratio was optimized using adaptive signal averaging. 30–32The extremely large difference between the on- and off-EDMR resonance conditions clearly identifies the upper trace taken at the resonance field of 332 :2 mT as EDENDOR. Note the arrows directed at the peaks of the ENDOR response at 12:58 MHz and 15 :63 MHz. The two peaks correspond to the EDENDOR response of silicon dangling bonds interacting with nearby 1H nuclei. We know that this is so because at a field of 332 :2 mT, the ENDOR response for1H nuclei close to the dangling bond defects should be a two-line spectrum centered around the isolated hydrogennucleus NMR frequency of 14.11 MHz. The center of the two peaks we observe is at ð12:58þ15:68Þ 2MHz¼14:10 MHz, within experimental error of the anticipated value of 14.11 MHz. The asymmetry of the peaks is to be expected and is indicative of the sign of the couplingconstant. 25,33From inspection of the separation of the two1H peaks, we extract a hyperfine coupling of /C253M H z ð60:2M H z Þ.T h i sv a l u ei s consistent within the range of hyperfine couplings reported for1Hi n a-Si:H observed in prior work by Brandt et al.28Aw e a kp e a ki sa p p a r - ently present in our response around 10 MHz. It is possible that this peak corresponds to hyperfine interactions with nearby29Si nuclei. However, due to the asymmetry of the peaks, it is likely we observe the high frequency peak and the low frequency peak is underneath the noise floor of our measurement. Additional verification that the response shown in Fig. 4 is EDENDOR is provided by a plot of the 15 :63 MHz peak amplitude as a function of magnetic field. The peak amplitude should approxi- mately track the field modulated EDMR response.25Figure 4 shows that this is the case. The close correspondence between the EDENDOR amplitude and the EDMR response strongly supports the identification1H FIG. 2. A representative EDMR trace on an a-Si:H sample utilized in EDENDOR measurements. The peak-to-peak linewidth is about 0.7 mT and the zero-crossing g, which is the g-value corresponding to the magnetic field halfway between the top and bottom peaks of the spectrum, is 2.0055, the value generally found for silicondangling bonds in amorphous hydrogenated silicon.FIG. 3. EDENDOR response of the a-Si:H sample. The upper trace (in green) was taken at the magnetic field corresponding to the center of the EDMR response, inthis case 332.2 mT. The lower (black) trace was taken at a field of 300 mT, whichis at a field far off the EDMR resonance.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 082401 (2021); doi: 10.1063/5.0041059 118, 082401-3 Published under license by AIP Publishinginteractions with Si dangling bonds. Most importantly, these observa- tions extend our earlier observations of EDENDOR from nuclei moredistant from the paramagnetic defects. Those earlier observations didnot allow any measure of hyperfine parameters. The EDENDORresults reported here allows the measurement, albeit with modest pre-cision, of 1H-Si hyperfine interactions. In conclusion, we demonstrate EDENDOR detection of1H nuclei interactions nearby Si dangling bond defects in a-Si:H. We extract a1H hyperfine coupling of 3 MHz, consistent with coupling constants corresponding to1H hyperfine interactions in a-Si:H from prior measurements.28These observations extend recent EDENDOR measurements in a 4H-SiC bipolar junction transistor.12,13The results presented are consistent with ENDOR of magnetic nuclei fairly closeto the defect. We believe this approach would be widely applicable toENDOR studies of many semiconducting and insulating materials and solid-state electronic devices. The most important criterion for this approach is the ability to observe EDMR with a reasonable highsignal-to-noise ratio in a system with a significant population ofmagnetic nuclei. See the supplementary material for a picture of the virtual lock-in amplifier’s LabVIEW code (Fig. S1). The picture shows the case fordouble modulation as described within this manuscript. This work was supported by the Air Force Office of Scientific Research under Award No. FA9550-17-1-0242.DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1J. A. Weil, J. R. Bolton, and J. E. Wertz, Electron Paramagnetic Resonance: Elementary Theory and Practical Applications (John Wiley & Sons, Inc., New York, 1994). 2G. R. Eaton, S. S. Eaton, D. P. Barr, and R. T. Weber, Quantitative EPR (Springer Wien New York, 2010). 3L. D. Kispert and L. Piekara-Sady, ENDOR Spectroscopy (Springer Science, New York, 2006). 4A. B. Wolbarst, Rev. Sci. Instrum. 47, 255 (1976). 5B .M .H o f f m a n ,J .M a r t i n s e n ,a n dR .A .V e n t e r s ,J .M a g n .R e s o n . 59, 110 (1984). 6B. Stich, S. Greulich-Weber, and J. M. Spaeth, Appl. Phys. Lett. 68, 1102 (1996). 7C. Boehme and H. Malissa, eMagRes 6, 83 (2017). 8C. J. Cochrane, P. M. Lenahan, and A. J. Lelis, J. Appl. Phys. 109, 014506 (2011). 9D. Kaplan, I. Solomon, and N. F. Mott, J. Phys. Lett. 39, 51 (1978). 10P. M. Lenahan and M. A. Jupina, Colloids Surf. 45, 191 (1990). 11C .J .C o c h r a n e ,P .M .L e n a h a n ,a n dA .J .L e l i s , J. Appl. Phys. 105, 064502 (2009). 12M. J. Mutch, P. M. Lenahan, and S. W. King, J. Appl. Phys. 119, 094102 (2016). 13C .J .C o c h r a n e ,P .M .L e n a h a n ,a n dA .J .L e l i s , Appl. Phys. Lett. 100, 023509 (2012). 14D. J. Meyer, P. M. Lenahan, and A. J. Lelis, Appl. Phys. Lett. 86, 023503 (2005). 15M. Stutzmann, M. S. Brandt, and M. W. Bayerl, J. Non-Cryst. Solids 266-269 ,1 (2000). 16G. Kawachi, C. F. O. Graeff, M. S. Brandt, and M. Stutzmann, Phys. Rev. B 54, 7957 (1996). 17F. Hoehne, L. Dreher, H. Huebl, M. Stutzmann, and M. S. Brandt, Phys. Rev. Lett. 106, 187601 (2011). 18R. J. Waskiewicz, B. R. Manning, D. J. McCrory, and P. M. Lenahan, J. Appl. Phys. 126, 125709 (2019). 19B. R. Manning, R. J. Waskiewicz, D. J. McCrory, and P. M. Lenahan, Rev. Sci. Instrum. 90, 123111 (2019). 20W. Gordy, Theory and Applications of Electron Spin Resonance (John Wiley & Sons, Inc., New York, 1980). 21J. T. Ryan, P. M. Lenahan, A. T. Krishnan, and S. Krishnan, J. Appl. Phys. 108, 064511 (2010). 22C. Boehme, J. Behrends, K. v Maydell, M. Schmidt, and K. Lips, J. Non-Cryst. Solids 352, 1113 (2006). 23D. J. McCrory, P. M. Lenahan, D. M. Nminibapiel, D. Veksler, J. T. Ryan, and J. P. Campbell, IEEE Trans. Nucl. Sci. 65, 1101 (2018). 24J. P. Campbell, P. M. Lenahan, A. T. Krishnan, and S. Krishnan, J. Appl. Phys. 103, 044505 (2008). 25L. Kevan and L. Kispert, Electron Spin Double Resonance Spectroscopy (John Wiley & Sons, Inc., New York, 1994). 26A. Van der Ziel, Noise in Solid State Devices and Circuits (Wiley, 1986). 27M. A. Anders, P. M. Lenahan, C. J. Cochrane, and J. Van Tol, J. Appl. Phys. 124, 215105 (2018). 28M. S. Brandt, M. W. Bayerl, M. Stutzmann, and C. F. O. Graeff, J. Non-Cryst. Solids 227-230 , 343 (1998). 29T. W. Herring, S. Y. Lee, D. R. McCamey, P. C. Taylor, K. Lips, J. Hu, F. Zhu, A. Madan, and C. Boehme, Phys. Rev. B 79, 195205 (2009). 30C. J. Cochrane and P. M. Lenahan, J. Magn. Reson. 195, 17 (2008). 31C. J. Cochrane, Rev. Sci. Instrum. 83, 105108 (2012). 32B. R. Manning, C. J. Cochrane, and P. M. Lenahan, Rev. Sci. Instrum. 91, 033106 (2020). 33G. Feher, C. S. Fuller, and E. A. Gere, Phys. Rev. 107, 1462 (1957).FIG. 4. EDENDOR response of the a-Si:H sample at 15.63 MHz vs magnetic field. Error bars are representative of the amplitude of noise and scale with the numberof scans taken at each field point. The ENDOR points are shifted slightly towardthe lower field to account for a field offset introduced by the software.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 082401 (2021); doi: 10.1063/5.0041059 118, 082401-4 Published under license by AIP Publishing

5.0042759.pdf

J. Chem. Phys. 154, 124110 (2021); https://doi.org/10.1063/5.0042759 154, 124110 © 2021 Author(s).Damped (linear) response theory within the resolution-of-identity coupled cluster singles and approximate doubles (RI-CC2) method Cite as: J. Chem. Phys. 154, 124110 (2021); https://doi.org/10.1063/5.0042759 Submitted: 03 January 2021 . Accepted: 04 March 2021 . Published Online: 23 March 2021 Daniil A. Fedotov , Sonia Coriani , and Christof Hättig ARTICLES YOU MAY BE INTERESTED IN Spin contamination in MP2 and CC2, a surprising issue The Journal of Chemical Physics 154, 131101 (2021); https://doi.org/10.1063/5.0044362 CAS without SCF—Why to use CASCI and where to get the orbitals The Journal of Chemical Physics 154, 090902 (2021); https://doi.org/10.1063/5.0042147 Reliable transition properties from excited-state mean-field calculations The Journal of Chemical Physics 154, 124106 (2021); https://doi.org/10.1063/5.0041233The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Damped (linear) response theory within the resolution-of-identity coupled cluster singles and approximate doubles (RI-CC2) method Cite as: J. Chem. Phys. 154, 124110 (2021); doi: 10.1063/5.0042759 Submitted: 3 January 2021 •Accepted: 4 March 2021 • Published Online: 23 March 2021 Daniil A. Fedotov,1 Sonia Coriani,1,a) and Christof Hättig2,a) AFFILIATIONS 1DTU Chemistry, Technical University of Denmark, Kemitorvet Bldg. 207, DK-2800 Kongens Lyngby, Denmark 2Arbeitsgruppe Quantenchemie, Ruhr-Universität Bochum, D-44780 Bochum, Germany a)Authors to whom correspondence should be addressed: soco@kemi.dtu.dk and christof.haettig@rub.de ABSTRACT An implementation of a complex solver for the solution of the linear equations required to compute the complex response functions of damped response theory is presented for the resolution-of-identity (RI) coupled cluster singles and approximate doubles (CC2) method. The implementation uses a partitioned formulation that avoids the storage of double excitation amplitudes to make it applicable to large molecules. The solver is the keystone element for the development of the damped coupled cluster response formalism for linear and nonlinear effects in resonant frequency regions at the RI-CC2 level of theory. Illustrative results are reported for the one-photon absorption cross section of C60, the electronic circular dichroism of n-helicenes ( n= 5, 6, 7), and the C6dispersion coefficients of a set of selected organic molecules and fullerenes. Published under license by AIP Publishing. https://doi.org/10.1063/5.0042759 .,s I. INTRODUCTION Damped response theory1and the conceptually equivalent complex polarization propagator (CPP) approach2–11are increas- ingly popular frameworks to compute resonance convergent response functions and thereby simulate a variety of spectroscopic effects. They have proven particularly convenient in cases where traditional calculations of explicit excited states (i.e., of individ- ual excitation energies as eigenvalues and corresponding oscillator strengths, also known as “stick” spectra) are impractical due to a large density of states—a prototypical example being the absorp- tion spectrum of large molecules with many equivalent atoms in extended basis sets over a broad frequency range. They allow to specifically target a given spectral region, hereby bypassing the calculation of lower-lying states. They are also advantageous in that they give access to molecular properties in (additional) resonant conditions, as in two-photon absorption (TPA)7,12andresonant inelastic x-ray scattering (RIXS).10,13In addition, the CPP approach can be used to compute, e.g., polarizabilities at imagi- nary frequencies that are needed for the calculation of C6dispersion coefficients.8,14,15 Damped response/CPP frameworks have been successfully implemented at various levels of theory, from Hartree–Fock and time-dependent density functional theory2–4,16to multiconfigura- tional self-consistent field,2,3Algebraic Diagrammatic Construction (ADC),15,17and Coupled-Cluster (CC) theory.8–10,18Extensions to solvated environments (embedding and solvation models)19,20and the relativistic domain21,22have also been presented. Applications to date include linear properties such as one- photon absorption (OPA) and electronic circular dichroism (ECD) in different frequency regions (from UV to x ray),5,6,9,18,23,24C6 dispersion coefficients computed from polarizabilities at imaginary frequencies,8,14,15non-linear effects such as magnetic-field induced circular dichroism (MCD) and nuclear-spin induced circular J. Chem. Phys. 154, 124110 (2021); doi: 10.1063/5.0042759 154, 124110-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp dichroism (NSCD),25–27magneto-chiral dichroism (MChD) and magneto-chiral birefringence (MChB) dispersion,28two-photon absorption in both UV–Vis and x-ray regimes,7,12and resonant inelastic x-ray scattering.10,13,24,29 A keystone element of all damped response/CPP frameworks is the solution of the (linear) response equations for a complex, or damped, frequency.11An implementation of a complex linear response solver within a coupled cluster framework was presented by Kauczor et al.18for the response of the cluster amplitudes and later extended to the response of the Lagrange multipliers by Faber and Coriani,10in both cases using an algorithm that assumes the storage of amplitudes or multipliers for all excitation classes ( vide infra ). Specific strategies to eliminate convergence issues in the x-ray frequency range have been discussed by Faber and Coriani24and by Nanda et al.29Here, we extend the complex solver of Ref. 18 to the case of the resolution-of-identity (RI) coupled cluster sin- gles and approximate doubles (CC2) method as implemented in the Turbomole package,30,31which employs a partitioned formula- tion that avoids the storage of amplitudes and multipliers for double excitation. This is important for large scale applica- tions of CC2, which would otherwise be hampered by I/O and storage demands. As illustrative results, we report the UV–Vis OPA spectra of C 60, the ECD spectra of three helicenes, and the ground-state C6dispersion coefficients of a set of organic molecules previously studied in the literature with other ab initio methods. II. THEORY A. The CC complex linear response function: Definitions and properties of interest In CC damped linear response theory,8–10,18we compute the complex polarizability as ⟨⟨x;y⟩⟩ω+iγ=1 2ˆC±ω{ηxty(ω+iγ)+ηytx(−ω−iγ) +Fty(ω+iγ)tx(−ω−iγ)}, (1) where ˆC±ωis a symmetrization operator, defined as ˆC±ωf(ω) =f(ω)+f∗(−ω). Note that the symmetrization operator only turns the sign of the real frequency ω. In Eq. (1), Fis the matrix of sec- ond derivatives (at zero field strength) of the CC Lagrangian Lwith respect to cluster amplitudes, Fμν=(∂2L ∂tμ∂tν), and the vector ηyis the second derivative of the Lagrangian with respect to cluster ampli- tudes and the field strength εy,ηy μ=(∂2L ∂tμ∂εy). We refer to, e.g., Ref. 32 for the general definitions of vectors and matrices of response functions in CC response theory. The specific CC2 expressions can be found, e.g., in Ref. 33. The solution of the response equations yielding the amplitudes ty(ω+iγ) within the RI-CC2 framework is discussed in Sec. II B. Here, we only note that Fand (for real oper- ators xandy) alsoηyare purely real, while the amplitude responses fulfill the symmetry tx(ω+iγ)∗=tx(ω−iγ). (2)If both operators are real and only diagonal components are con- sidered, the real and imaginary parts of the complex dipole–dipole polarizability in Eq. (1) are R⟨⟨x;x⟩⟩ω+iγ=ηx Rtx R(ω+iγ)+ηx Rtx R(−ω−iγ) +Ftx R(ω+iγ)tx R(−ω−iγ) −Ftx I(−ω−iγ)tx I(ω+iγ), (3) I⟨⟨x;x⟩⟩ω+iγ=ηx Rtx I(ω+iγ)+ηx Rtx I(−ω−iγ) +Ftx I(ω+iγ)tx R(−ω−iγ) +Ftx I(−ω−iγ)tx R(ω+iγ), (4) where we have explicitly split the complex response amplitudes into real and imaginary parts, tx(ω+iγ)=tx R(ω+iγ)+i tx I(ω+iγ). (5) The imaginary part of polarizability can be used to compute, for instance, OPA cross sections, σOPA(ω)∝ωI⟨⟨μα;μα⟩⟩ω+iγ, (6) whereμαis theα-component of the electric dipole operator, and the incident frequency ωis chosen within the specific region of interest, e.g., UV–Vis or x ray. The polarizability dispersion profiles, illustrat- ing the variation of the dipole polarizability over a given frequency range, can conversely be obtained from the real part of the complex dipole polarizability. If one of the two operators in the linear response function, say, X, is purely imaginary, we have tX(ω+iγ)∗=−tX(ω−iγ), (7) and it is the real part of the complex response function that yields the absorption component, R⟨⟨x;X⟩⟩ω+iγ=1 2{ηx RtX R(ω+iγ)−ηx RtX R(−ω−iγ) −ηX Itx I(−ω−iγ)+ηX Itx I(ω+iγ) +Ftx R(−ω−iγ)tX R(ω+iγ)+Ftx R(ω+iγ) ×tX R(−ω−iγ)−Ftx I(−ω−iγ)tX I(ω+iγ) −Ftx I(ω+iγ)tX I(−ω−iγ)}. (8) A prototypical case described by such a response function is the ECD cross section—most often expressed as the difference Δεin extinc- tion coefficients for left and right circularly polarized light—in the length gauge (lg), Δεlg(ω)∝ωR⟨⟨mα;μα⟩⟩ω+iγ, (9) J. Chem. Phys. 154, 124110 (2021); doi: 10.1063/5.0042759 154, 124110-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp whereas the optical rotation dispersion (ORD) profile is given by the imaginary part, σlg ORD(ω)∝ωI⟨⟨mα;μα⟩⟩ω+iγ. (10) We note in passing that, in cases such as ORD and ECD, the sym- metric form of the (complex) polarizability requires solving the complex response equations for both imaginary and real operators. Alternatively, one can resort to the asymmetric form ⟨⟨X;x⟩⟩ω+iγ=1 2ˆC±ω{¯tx(ω+iγ)ξX+ηXtx(ω+iγ)}, (11) which, thus, requires the solution of the CPP equations for the left response multipliers ¯tx(ω+iγ), ¯tx(ω+iγ)[A+(ω+iγ)1]=−ηx−Ftx(ω+iγ), (12) along with those for the response amplitudes tx(ω+iγ), for the real operator x. This allows one to bypass the solution of the response vectors for the imaginary operator. The first-order (com- plex) Lagrange multipliers are also needed for higher-order response and transition properties, such as the previously mentioned two- photon absorption, RIXS, and MCD.10,34 The length-gauge expressions of the optical rotation (OR) ten- sor and of the rotatory strengths within resonant response theory are gauge-origin dependent. Conversely, the velocity-gauge forms, which involve two imaginary operators, namely, the linear momen- tum pαand the magnetic moment mα, are origin independent.35 Within CC theory, the “modified” velocity gauge expression of the OR tensor is typically used,36,37 Gmv αα(ω)=ω−1{⟨⟨pα;mα⟩⟩ω−⟨⟨pα;mα⟩⟩0}, (13) which ensures that the thus-computed OR tensor is zero in the limit of zero frequency, as it should be according to exact theory. We generalize the above expression to obtain the CPP optical rota- tory dispersion and the electronic circular dichroism in the modified velocity gauge, σmv ORD(ω)∝R{⟨⟨pα;mα⟩⟩ω+iγ−⟨⟨pα;mα⟩⟩0} (14) and Δεmv(ω)∝I{⟨⟨pα;mα⟩⟩ω+iγ−⟨⟨pα;mα⟩⟩0}=I⟨⟨pα;mα⟩⟩ω+iγ. (15) Note that the correction to the ECD expression is redundant, since the imaginary part of the (real) response function is zero at the static limit. An alternative choice of CPP expression is to use a lifetime parameter that is scaled with the real frequency, ⟨⟨pα,mα⟩⟩ω(1+iγ)−⟨⟨pα,mα⟩⟩0. (16) This is a slightly different approach than the one typically used with the CPP, again with no correction for ECD. It would have the formaladvantage of conserving the symmetry σORD(−ω) =−σORD(ω). This alternative expression would only be advantageous over the first one in practical applications where ωandγare of similar magnitude, or whenγ>ω, for instance, because one scans with ωthrough 0. If one is interested in computing, for instance, σECD(ω) for the UV–Vis region with γof the order of 0.1 eV, the first expression is to be preferred. This is the case here, so all ECD results presented in the following are obtained according to Eq. (14). Finally, within damped linear response theory, one can also straightforwardly compute the isotropic dipole–dipole polarizability at purely imaginary frequencies, α(iω), by setting the real frequency equal to zero and γ=ωin Eq. (1). From the isotropic averaged polar- izability at imaginary frequency, one can then obtain coefficients to describe the long-range part of London dispersion interactions, e.g., theC6dispersion coefficients,8,14,15 C6=3̵h π∫∞ 0αA(iω)αB(iω)dω, (17) where Aand Blabel the interacting systems. The C6dispersion coefficients can be used, e.g., to compute the long-range dispersion interaction energy between Aand B, also known as the Casimir– Polder potential, according to the simplified expression valid in the van der Waals region,15asΔE(RAB)=−̵h πC6 R6 AB, and to determine long-range dispersion interaction corrections to density functional theory.38–40 B. The complex linear response equations for (RI-)CC2 The properties defined in Sec. II A entail the solution of com- plex response equations to obtain the real and imaginary com- ponents of the response amplitudes tx(ω+iγ) and multipliers ¯tx(ω+iγ), {A−(ω+iγ)1}tx(ω+iγ)=−ξx, (18) ¯tx(ω+iγ){A+(ω+iγ)1}=−ηx−Ftx(ω+iγ), (19) where Ais the CC Jacobian, Aμν=(∂2L ∂¯tμ∂tν), andξx μ=(∂2L ∂¯tμ∂εx).32 We refer once again to, e.g., Refs. 33, 41, and 42 for specific defi- nitions of the CC2 right-hand-side (RHS) vectors ξxandηxand of the matrices AandF. We will in the following concentrate solely on the solution of Eq. (18) within RI-CC2 without storing any double excitation amplitudes, multipliers, or trial vectors. For this, we start from the complex linear response equations in the matrix form of Ref. 18 and explicitly partition them in singles ( S) and doubles ( D) blocks. For ease of notation, we omit in the following the frequency argument on the response amplitudes and write ⎡⎢⎢⎢⎢⎢⎢⎢⎣ASS−ω1SS ASD γ1SS 0 ADS ADD−ω1DD 0 γ1DD −γ1SS 0 A SS−ω1SS ASD 0−γ1DD ADS ADD−ω1DD⎤⎥⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎢⎢⎣tx R,S tx R,D tx I,S tx I,D⎤⎥⎥⎥⎥⎥⎥⎥⎦=−⎡⎢⎢⎢⎢⎢⎢⎢⎣ξx R,S ξx R,D ξx I,S ξx I,D⎤⎥⎥⎥⎥⎥⎥⎥⎦ (20) J. Chem. Phys. 154, 124110 (2021); doi: 10.1063/5.0042759 154, 124110-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp equivalent to the system of equations ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩(ASS−ω1SS)tx R,S+γtx I,S=−ξx R,S−ASDtx R,D, (21a) (ADD−ω1DD)tx R,D+γtx I,D=−ξx R,D−ADStx R,S, (21b) (ASS−ω1SS)tx I,S−γtx R,S=−ξx I,S−ASDtx I,D, (21c) (ADD−ω1DD)tx I,D−γtx R,D=−ξx I,D−ADStx I,S. (21d) Assuming we work with canonical molecular orbitals, the doubles–doubles block ADDof the CC2 Jacobian is diagonal, and so is in this case the doubles–doubles resolvent matrix,41 RDD=−[ADD−(ω+iγ)1DD]−1. We, therefore, define Δ=(ADD−ω1DD), (22) with the diagonal elements Δij ab=(εa−εi+εb−εj−ω). (23) We isolate tx I,Dfrom Eq. (21d) and tx R,Dfrom Eq. (21b), and introduce each resulting expression into the other, to arrive at tx R,D=−Δ γ2+Δ2(ξx R,D+ADStx R,S)+γ γ2+Δ2(ξx I,D+ADStx I,S), (24) tx I,D=−Δ γ2+Δ2(ξx I,D+ADStx I,S)−γ γ2+Δ2(ξx R,D+ADStx R,S). (25) Inserting Eq. (24) into Eq. (21c) and Eq. (25) into Eq. (21c), we finally obtain the effective CC2 CPP linear response equations in the compact matrix form, ⎡⎢⎢⎢⎢⎣Aeff SS(ω,γ)−ω1SS−Γeff SS(ω,γ)+γ1SS Γeff SS(ω,γ)−γ1SSAeff SS(ω,γ)−ω1SS⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎣tx R,S tx I,S⎤⎥⎥⎥⎦=−⎡⎢⎢⎢⎢⎣ξx,eff R,S(ω,γ) ξx,eff I,S(ω,γ)⎤⎥⎥⎥⎥⎦, (26) where Aeff SS(ω,γ)=ASS−ASDΔ Δ2+γ2ADS, (27) Γeff SS(ω,γ)=−ASDγ Δ2+γ2ADS (28) and ξx,eff R,S(ω,γ)=ξx R,S−ASDΔ Δ2+γ2ξx R,D+ASDγ Δ2+γ2ξx I,D, (29) ξx,eff I,S(ω,γ)=ξx I,S−ASDΔ Δ2+γ2ξx I,D−ASDγ Δ2+γ2ξx R,D. (30) Thus, the CPP(-RI)-CC2 building blocks are the same as in the standard linear response case,42,43just with slightly different gener- alized values for the diagonal elements of the resolvent as scalingfactors. These scaling factors are exactly the same as used in the preconditioning step in Ref. 18. III. IMPLEMENTATION A. The iterative CPP solver The general strategy for the implementation of our solver con- sists in working exclusively with real trial vectors, generating two new vectors at each iteration from the real and imaginary parts of the preconditioned residual vectors, and solving the complex linear response equation (27) in the reduced space. In detail, the fundamental steps of the iterative solver are the following: 1.Generation of the start trial vectors by preconditioning the effective RHS vectors: (˜b1)ai=εa−εi−ω (εa−εi−ω)2+γ2ξx,eff R,ai+γ (εa−εi−ω)2+γ2ξx,eff I,ai (31) and (˜b2)ai=εa−εi−ω (εa−εi−ω)2+γ2ξx,eff I,ai−γ (εa−εi−ω)2+γ2ξx,eff R,ai, (32) followed by orthonormalization. 2.Computation of the linearly transformed vectors: σR 1=Aeffb1,σR 2=Aeffb2,σI 1=Γeffb1,σI 2=Γeffb2. (33) 3.Computation of the reduced-space building blocks: Ared ij=bT iσR j,Γred ij=bT iσI j, (34) ξx,red R,i=bT iξx,eff R,ξx,red I,i=bT iξx,eff I, (35) where iand jrun on the number of trial vectors. Note that nred= 2n, where nis the iteration number. 4.Construction and solution of the CPP equation in the reduced space: [Ared−ω1−Γred+γ1 Γred−γ1 Ared−ω1][xR xI]=−[ξx,red R(ω) ξx,red I(ω)]. (36) The CPP reduced equation is solved using standard library solvers to obtain xRandxI. 5.Construction of the solution and residual vectors in the full (singles) space : The solution vectors at iteration nare linear combinations of the trial basis with the reduced space solution vectors as coefficients, tx,(n) R=nred ∑ ixR ibi,tx,(n) I=nred ∑ ixI ibi. (37) They are formally introduced in the effective CPP equation to yield the residual vectors J. Chem. Phys. 154, 124110 (2021); doi: 10.1063/5.0042759 154, 124110-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp R(n) R,S=nred ∑ ixR iσR i,S−ωtx,(n) R,S−nred ∑ ixI iσI i,S+γtx,(n) I,S+ξx,eff R,S, (38) R(n) I,S=nred ∑ ixI iσR i,S−ωtx,(n) I,S+nred ∑ ixR iσI i,S−γtx,(n) R,S+ξx,eff I,S. (39) Solution and residual vectors are, alike the linearly transformed one, stored as vectors that are twice the size of a singles amplitude. 6.Generation of the new trial vectors from the preconditioned residuals: If the residual vectors of step 5 are larger than a preset threshold, new trial vectors are generated and a new iteration is made. In practice, we split the (tentative) trial vector into two vectors, (˜b2n−1)ai=εa−εi−ω (εa−εi−ω)2+γ2⋅R(n) R,ai +γ (εa−εi−ω)2+γ2⋅R(n) I,ai, (40) (˜b2n)ai=εa−εi−ω (εa−εi−ω)2+γ2⋅R(n) I,ai −γ (εa−εi−ω)2+γ2⋅R(n) R,ai, (41) which are normalized and then orthogonalized onto the previ- ous trial vectors. If, after this step, their norm is smaller than a linear-dependence threshold, they are discarded; otherwise, they are normalized once more and added to the set of trial vectors. 7.Extension of the reduced space and iteration until convergence: If the residuals for all equations have decreased below a user- defined threshold, the procedure is stopped, or else the reduced space is extended as in step 3 and steps 4–6 are repeated until convergence. B. Building blocks: The RHS vectors The perturbation operators are in general assumed to be either real or purely imaginary. As a consequence, the (not partitioned) RHS vectors ξxfor the first-order amplitude equations are either real or purely imaginary. In the case of real perturbations (e.g., electric dipole), the effective RHS vectors simplify to ξx,eff R,S(ω,γ)=ξx S−ASDΔ Δ2+γ2ξx D, (42) ξx,eff I,S(ω,γ)=−ASDγ Δ2+γ2ξx D. (43) Within RI-CC2, the doubles elements of the RHS vector ξx Dare com- puted only on the fly and immediately contracted with the elements of the singles–doubles block of the Jacobian matrix ASD, in a loop either over pairs of occupied or over pairs of virtual orbital indices. This entails computing(˜ξx R)ij ab=−Δaibj Δ2 aibj+γ2ξx,ij ab, (44) (˜ξx I)ij ab=−γ Δ2 aibj+γ2ξx,ij ab, (45) where the elements of the unmodified doubles part of the RHS vector are33 ξx,ij ab=ˆPij ab(∑ ctij acˆhx cb−∑ ktik abˆhx kj). (46) Here, tij abare the zero-order double amplitudes and ˆhx pqare the inte- grals of the one-electron operator x, similarity transformed with the exponential function of the single excitation cluster operator (for their definition, see Appendix B). ˆPij abis a symmetrization operator, defined by ˆPpr qsfpq,rs=fpq,rs+frs,pq.33 Then, we contract ˜ξx,ij R,aband ˜ξx,ij I,abwith the elements of the singles–doubles matrix ASD. In general, the contraction of ASDwith a doubles vector bkl cdcan be written as33 ∑ ckdlAai,ckdlbkl cd=+∑ cdk(2bik cd−bik dc)(kdˆ∣ac)−∑ dkl(2bkl ad−bkl da)(ldˆ∣ki) +∑ ck(2bik ac−bik ca)ˆFkc, (47) where ˆFkcis the Fock matrix43and(ldˆ∣ki)are the two-electron inte- grals of the T1-similarity transformed Hamiltonian operator (see their definition in Appendix B). Within RI-CC2, the two-electron integrals are approximated as44–47 ˆ(pq∣rs)=∑ QˆBQ,pqˆBQ,rs, (48) where ˆBQ,pq=∑ P(pqˆ∣P)V−1 2 PQ=∑ μνΛp μpΛh νq∑ P(μν∣P)V−1 2 PQ. (49) In the last equality, ( μν|P) are three-index electron-repulsion integrals (ERIs) for the atomic orbitals μ,νand the auxiliary basis function PandVPQ= (V|P) is a matrix containing as elements the two-index ERIs in the auxiliary basis. The ΛpandΛhmatrices are the T1-transformed molecular orbital coefficients, whose definition is given in Appendix B. With this, the first two terms can be rewritten, e.g., as ∑ cdk(2bik cd−bik dc)(kdˆ∣ac)=∑ Qc(∑ dk(2bik cd−bik dc)ˆBQ,kd)ˆBQ,ac =∑ Qc¯YQ,ciˆBQ,ac, (50) where we introduced the ¯Yintermediate ¯YQ,ai=∑ bj(2bij ab−bij ba)ˆBQ,jb. (51) J. Chem. Phys. 154, 124110 (2021); doi: 10.1063/5.0042759 154, 124110-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp In the case of the CPP RHS vectors, these intermediates become ¯Yx,R Q,ai=∑ bj(2˜ξx,ij R,ab−˜ξx,ij R,ba)ˆBQ,jb, (52) ¯Yx,I Q,ai=∑ bj(2˜ξx,ij I,ab−˜ξx,ij I,ba)ˆBQ,jb, (53) and the real and imaginary parts of the effective singles RHS vectors are computed as ξx,eff R,ai=ξai+∑ ck(2˜ξx,ik R,ac−˜ξx,ik R,ca)ˆFkc+∑ cQ¯Yx,R Q,ciˆBQ,ac−∑ kQ¯Yx,R Q,akˆBQ,ki, (54) ξx,eff I,ai=∑ ck(2˜ξx,ik I,ac−˜ξx,ik I,ca)ˆFkc+∑ cQ¯Yx,I Q,ciˆBQ,ac−∑ kQ¯Yx,I Q,akˆBQ,ki. (55) In case of an imaginary perturbation X(e.g., the magnetic dipole moment or the linear momentum), the effective RHS vector reads ξX,eff R,S(ω,γ)=ASD+γ Δ2+γ2ξX D, (56) ξX,eff I,S(ω,γ)=ξX S+ASD−Δ Δ2+γ2ξX D, (57) that is, ξX,eff R,ai=−∑ ck(˜ξX,ik R,ac−˜ξX,ik R,ca)ˆFkc−∑ cQ¯YX,R Q,ciˆBQ,ac+∑ kQ¯YX,R Q,akˆBQ,ki, (58) ξX,eff I,ai=ξχ ai+∑ ck(2˜ξX,ik I,ac−˜ξX,ik I,ca)ˆFkc+∑ cQ¯YX,I Q,ciˆBQ,ac−∑ kQ¯YX,I Q,akˆBQ,ki. (59) C. Building blocks: The Jacobian transformation To build the reduced-space quantities needed in the CPP solver, we need, for each trial vector b, the result of its transformations with the effective matrices Aeff SS(ω,γ)andΓeff SS(ω,γ), typically referred to asσvectors. To keep the overhead for the CPP small, the transformations with the two matrices are done together. We express the result of the transformation of a singles trial vector bwith the doubles–singles Jacobian matrix ADSas one-index transformed two-electron integrals, ∑ ckAaibj,ckbck=⟨ij ab∣[ˆH,τck]HF⟩bck=(ai¯∣bj), (60) (ai¯∣bj)=ˆPab ij∑ αβγδ(¯Λp αaΛh βi+Λp αa¯Λh βi)Λp γbΛh δj(αβ∣γδ), (61) where ¯Λpand ¯Λhare defined as in Appendix B, with the singles trial vector b1in place of the singles response amplitudes tx 1. Thesefour-index integrals are evaluated on the fly from three-center inter- mediates43[ˆBQ,ai, Eq. (49), and ¯BQ,ai, given in Appendix B] and combined with the energy denominators from Δinto intermediate doubles amplitudes. In other words, for the CPP implementation, the following intermediate doubles amplitudes are built: ˜bR,ij ab=−Δaibj Δ2 aibj+γ2(ai∣bj), (62) ˜bI,ij ab=−γ Δ2 aibj+γ2(ai∣bj). (63) With these, the transformations with Aeff SSandΓeff SScan be expressed as Aeff SS(ω,γ)bS=ASSbS+∑ ckdlAS,ckdl˜bR,kl cd, (64) Γeff SS(ω,γ)bS=∑ ckdlAS,ckdl˜bI,kl cd. (65) The contribution ASSbSis unchanged compared to the standard (non-CPP) solver.41The other contributions are evaluated in a way similar (and partially using the same routines) to the contributions to the effective right-hand sides discussed in Sec. III B, σeff R,ai=∑ ckAeff ai,ck(ω,γ)bck =∑ ckAai,ckbck+∑ ck(2˜bR,ik ac−˜bR,ik ca)ˆFkc +∑ cQ¯YR Q,ciˆBQ,ac−∑ kQ¯YR Q,akˆBQ,ki+∑ ck(2tik ac−tik ca)¯Fkc (66) and σeff I,ai=∑ ckΓeff ai,ck(ω,γ)bck =∑ ck(2˜bI,ik ac−˜bI,ik ca)ˆFkc+∑ cQ¯YI Q,ciˆBQ,ac−∑ kQ¯YI Q,akˆBQ,ki. (67) The real and imaginary ¯Yintermediates are, as in Eq. (51), using the real and imaginary intermediate doubles amplitude trial vectors defined above. D. The first-order perturbed densities Once the real and imaginary response amplitudes have been obtained, we can build the real and imaginary linear response func- tions needed for the properties and spectra discussed in Sec. II A. This entails computing contractions of the (complex) response amplitudes with the ηxvectors and with the Fmatrix. The contributions from the terms of the type ηx⋅tyare formu- lated as contractions of densities and one-electron integrals of the perturbation operator,33,43 ηx⋅ty=∑ pqDη pq(ty)ˆhx pq. (68) J. Chem. Phys. 154, 124110 (2021); doi: 10.1063/5.0042759 154, 124110-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp We do so as, for large systems, we do not want to store the doubles parts ofηxandty. In addition, to recalculate the doubles parts of both vectors for every dot product, i.e., for every pair of perturbations x andy, would require a number of N5-scaling steps that increase with the number of operator pairs. Via densities, on the other hand, the number of N5-scaling steps increases only linearly with the number of operators. The explicit density blocks are43 Dη ij(tx)=−∑ a¯tjatx ai−Xx ij, (69) Dη ia(tx)=Cx ai−∑ ktx akXik−∑ bYbatx bi, (70) Dη ai(tx)=0, (71) Dη ab(tx)=∑ i¯tiatx bi+Yx ba. (72) The real part of Dη(tx) is computed from the real part of txas in the standard response case.43The imaginary part of Dη(tx) is done in the same way using the imaginary part of tx. The contributions to the densities from the singles amplitudes are straightforward to compute since the singles are stored on disk and can be read from the file when needed. Complications arise from the doubles response amplitudes tx,ij ab, as they should also be implemented with O(N2)- scaling memory demands. The expression of the doubles part of the response amplitudes is tx,ij ab=−{ˆPij ab(∑ ctij acˆhx cb−∑ ktik abˆhx kj) +(ai¯∣bj)x}/(εa−εi+εb−εj−ω−iγ). (73) The (complex) tx S-dressed four-index integrals are evaluated within the RI approximation as (ai¯∣bj)x=ˆPij ab⎧⎪⎪⎨⎪⎪⎩∑ Q¯Bx,R Q,aiˆBQ,bj+i∑ Q¯Bx,I Q,aiˆBQ,bj⎫⎪⎪⎬⎪⎪⎭, (74) where the three-center intermediates ¯Bx,Rand ¯Bx,Iare built with, respectively, the real and imaginary parts of the singles amplitudes tx ai, see Appendix B. As described elsewhere,33,48the ground-state double ampli- tudes are evaluated on the fly within the RI approximation and with a numerical Laplace transformation of the denominators, tij ab=−∑QˆBQ,aiˆBQ,bj (εa−εi+εb−εj)≈−∑ m∑ QˆKm Q,aiˆKm Q,bj, (75) with ˆKm Q,ai=ˆBQ,ai√ωmexp{−(εa−εi)θm}, whereθmare the Laplace sampling points and ωmare the weights.33This allows us to do thetransformation with the one-electron integrals for the perturbation operator xat the level of the ˆKintermediates,33 ¯Km,x Q,ai=∑ cˆKm Q,ciˆhx ac−∑ kˆKm Q,akˆhx ki (76) (assuming that xis purely real), so that we can compute the real and the imaginary response double amplitudes on the fly as tx,ij R,ab=ˆPij ab⎧⎪⎪⎨⎪⎪⎩−∑ m∑ Q¯Km,x Q,aiˆKm Q,bj+∑ Q¯Bx,R Q,aiˆBQ,bj⎫⎪⎪⎬⎪⎪⎭ ×−Δaibj Δ2 aibj+γ2−ˆPij ab∑ Q¯Bx,I Q,aiˆBQ,bj⋅−γ Δ2 aibj+γ2, (77) tx,ij I,ab=ˆPij ab⎧⎪⎪⎨⎪⎪⎩−∑ m∑ Q¯Km,x Q,aiˆKm Q,bj+∑ Q¯Bx,R Q,aiˆBQ,bj⎫⎪⎪⎬⎪⎪⎭ ×−γ Δ2 aibj+γ2+ˆPij ab∑ Q¯Bx,I Q,aiˆBQ,bj⋅−Δaibj Δ2 aibj+γ2. (78) The doubles of the first-order response amplitudes are constructed in a loop over pairs of occupied orbitals iandj. In the same loop, the doubles of the ground-state Lagrange multipliers ¯tij abare built. The response amplitudes tx,ij bcare then contracted with the Lagrange multipliers to the intermediates Yx ab=∑ cij¯tij actx,ij bc(79) and Cx ai=∑ bj(2tx,ij ab−tx,ij ba)¯tjb. (80) Then, the same procedure is repeated within a loop over pairs of virtual orbital indices aand b(with occupied and virtual orbitals interchanged) to calculate Xx ik=∑ abk¯tjk abtx,ik ab. (81) The real and imaginary parts for the doubles are computed together to avoid having to compute the doubles multipliers twice, and thus, the real and imaginary parts of Cx,Yx, and Xxare evaluated together. Eventually, the individual blocks of the density Dη(tx) are put together from these intermediates and the singles parts for the response amplitudes and Lagrange multipliers. E. The F-matrix contractions Similar to the evaluation of ηx⋅ty, also the F-matrix contrac- tions are organized such that all O(N5)-scaling steps only depend on one perturbation, and only cheap, low-scaling, steps depend on both amplitude response vectors. The F-matrix contraction is first rewritten as J. Chem. Phys. 154, 124110 (2021); doi: 10.1063/5.0042759 154, 124110-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Ftxty=σx⋅ty, (82) with σx μ=∑ ν=ν1,ν2Fνμtx ν. (83) The singles and doubles blocks of σxare partitioned as summarized in Table I. Algorithm 1 in Appendix B summarizes the main steps in the actual evaluation of the F-matrix contribution to the linear response function. Different from standard response theory, in the CPP case, all intermediates depending on the response amplitudes, i.e., carrying an upper index xory, are complex. The contributions to the real and imaginary parts of the intermediates are evaluated with the real and imaginary parts of tx, respectively, as described for standard response theory in Ref. 33. The explicit evaluation of the doubles blocks is avoided by reformulating the contraction of σx iajbwith ty,ij abas in the following: 1 2∑ ijabσI,x iajbty,ij ab=1 2∑ iajbˆPij ab[¯tia(2ty,ij ab−ty,ij ba)]¯Fx jb=∑ iaCy ai¯Fx ia, (84)1 2ˆPij ab(σG,x iajb+σH,x iajb)ty,ij ab=∑ jbQ[−∑ ck(¯tjctx ckBQ,kb+tx ck¯tkbBQ,jc)]Yy Q,bj =∑ jbQ˘Bx Q,jbYy Q,bj. (85) For the definition of the intermediates, we refer to Appendix B. IV. RESULTS AND DISCUSSION A. Computational details and cost estimates The CPP solver for RI-CC2 has been implemented in a development version of the Turbomole program package.31,49The standard response calculations were performed using previously implemented RI-CC2 functionalities in Turbomole.42,43The exci- tation energies and strengths are given in the supplementary material. The structure of C 60used in the OPA calculations was taken from Ref. 50. It originates from geometry optimization at the level of second-order Møller–Plesset Perturbation (MP2)51theory with TABLE I . Singles and doubles blocks of σx. We refer to Appendix B for further definitions of intermediates. σx ia=σ0,x ia+σF,x ia+σJG,x ia+σJH,x ia +σI,x ia+σJ′,x ia σ0,x ia=2¯Fx ia σF,x ia=∑ cdk¯tki cd(ck¯∣da)x−∑ ckl¯tkl ca(ck¯∣il)x=∑ dQ(˘Yx Q,idˆBQ,da+˘YQ,id¯Bx Q,da)−∑ lQ(˘Yx Q,alˆBQ,il+˘YQ,al¯Bx Q,il) σJG,x ia=−∑ j¯tja¯Fx ij−∑ j¯tja∑ cdk(2tx,jk cd−tx,kj cd)(kd∣ic)=−∑ j¯Ex,2 ij¯tja σJH,x ia=∑ b¯tib¯Fx ba+∑ b¯tib∑ dkl(2tx,kl bd−tx,lk bd)(ld∣ka)=∑ b¯tib¯Ex,1 ba σI,x ia=∑ ckCx ck[2(kc∣ia)−(ic∣ka)]=∑ Q(2∑ ckBQ,ckCx ck)BQ,ia−∑ Qk(∑ cBQ,icCx ck)BQ,ka =∑ Qβ{2mx QCβi−∑ kMx Q,ikCβk}BQ,βa σJ′,x ia=∑ bj¯tjb[2(bj¯∣ia)x−(ij¯∣ba)x] =2∑ Q⎛ ⎝∑ jb¯Bx Q,bj¯tjb⎞ ⎠BQ,ia−∑ Qb⎛ ⎝∑ j¯Bx Q,ij¯tjb⎞ ⎠∑ βΛp βbˆBQ,βa−∑ Qb⎛ ⎝∑ jˆBQ,ij¯tjb⎞ ⎠∑ β¯Λp,x βbˆBQ,βa =2∑ Q˘ix QBQ,ia−∑ Qb⎛ ⎝∑ j¯Bx Q,ij¯tjb⎞ ⎠∑ βΛp βbˆBQ,βa−∑ Qb⎛ ⎝∑ jˆBQ,ij¯tjb⎞ ⎠∑ β¯Λp,x βbˆBQ,βa σx iajb=σI,x iajb+σG,x iajb+σH,x iajb σI,x iajb=2¯tia¯Fx jb−¯tja¯Fx ib σG,x iajb=−∑ck¯tjctx ck[2(kb∣ia)−(ka∣ib)] σH,x iajb=−∑cktx ck¯tka[2(jb∣ic)−(jc∣ib)] J. Chem. Phys. 154, 124110 (2021); doi: 10.1063/5.0042759 154, 124110-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Dunning’s cc-pVTZ basis set.52The structures of the molecules con- sidered for the C6coefficients (alkanes, unsaturated hydrocarbons, aldehydes, and ketones) are also MP2/cc-pVTZ optimized structures from the literature.15The Cartesian coordinates of all molecular sys- tems considered are reported in the supplementary material. The structures of the helicenes37in the ECD calculations are MP2/cc- pVTZ optimized ones. According to the standard convention for helicoidal systems, we used ( −)-5-helicene (M), ( −)-6-helicene (M), and (+)-7-helicene (P) structures. The structures of the fullerenes in theC6calculations are the same B3LYP/cc-pVDZ optimized ones used in Ref. 53. In the calculations of the OPA spectra of C 60, we adopted the aug-cc-pVDZ basis set. The calculations of C6disper- sion coefficients of the fullerenes were carried out using the cc- pVDZ basis set unless otherwise specified. For all other molecules, the aug-cc-pVTZ basis set was used. This also applies to the ECD calculations on the helicenes. The frozen-core approximation was used for the helicenes and in the calculations of the C6coeffi- cients of the selected set of fullerenes. An optimized auxiliary basis set matching the chosen atomic orbital basis was employed in all calculations.54 For comparison with the CPP spectra, the individual oscilla- tor strengths (and energies) were broadened using the Lorentzian function gj(ω)=γ (ω−ωj)2+γ2, (86) with half width at half maximum (HWHM) value γ= 0.004 556 a.u. The frequency steps in the CPP calculations varied between 0.0025 and 0.01 a.u. A cubic spline was used for the interpolation between the computed points to obtain the CPP spectrum. The C6coefficients were obtained according to Eq. (17), with A=B. The integral was evaluated using a Gauss–Legendre inte- gration scheme, with a transformation of variables as suggested in Ref. 55 and followed by a Gauss–Legendre quadrature in the interval −1≤t≤+1. A 12-point scheme was adopted. To conclude this section, a brief comment is in place concern- ing the computational cost of a CPP calculation. The overall costs of response calculations depend, in addition to the system size and computed response properties, on several other factors, e.g., point group symmetry and the number of iterations for convergence. This makes a direct comparison between CPP and non-CPP calculations difficult as these calculations are in general quite different in nature and aim for different properties or, in the case of spectra, for differ- ent cases. However, the largest fraction of the computational time is usually spent for the solution of the response equations, which is dominated by the time needed for the linear transformations of trial vectors with the Jacobian matrix, σi=Abi. In the limit of a large system size N, the operation count for these transformations is dominated by a few (N5)-scaling steps: ●the integrals (aiˆ∣bj)(1 2O2V2X) and(ai¯∣bj)(mO2V2X) ●the intermediates ¯YQ,ci(mO2V2Xfor the real case and 2mO2V2Xfor the complex case) where mis the number of trial vectors, Ois the number of occupied orbitals, Vis the number of virtual orbitals, and Xis the numberof auxiliary basis functions (typically X≈3N) and a system with- out point group symmetry is assumed. The simultaneous transfor- mation of many trial vectors ( m≫1) should thus, in the limit of a large system size for the CPP case, take ≈50% longer than for the (normal) non-CPP case, ≈44% longer for two trial vectors, and ≈40% longer for a single trial vector. Some exemplificative tim- ings for C60are collected in Table IV in Appendix A. The table summarizes the total time per response vector (in min) and the averaged time per vector and per solver iteration when computing the response amplitudes needed to calculate one component of the dipole polarizability of C 60, using either the standard or the CPP solver. Three frequency values ωwere considered: 0.0 a.u. (static case), 0.07 a.u. (far from resonance), and 0.2 a.u. (close to reso- nance). For the imaginary frequency γ, only two values were consid- ered: 0.0 and 4.6 ×10−3a.u. (1000 cm−1). For C 60, the lower scaling steps are, in particular, because D2hsymmetry was used, not yet neg- ligible. For these steps, the extra costs for the CPP case are lower, and we thus observe for C 60overheads for the CPP case of ≈20%. The standard solver is (obviously) more convenient in the static case and at frequencies far from resonance. Close to resonance, the standard solver clearly struggles, and a larger number of iterations are needed for convergence. This effect is smaller with a non-zero gamma. B. One-photon absorption: C 60 The UV spectrum of C 60obtained at the CPP-RI-CC2/aug-cc- pVDZ level (all electrons correlated) is shown in Fig. 1. The spec- trum is compared with the CPP-KS-TDDFT result of Ref. 16. C 60 is a prototypical case where the application of the CPP algorithm is particularly advantageous. The molecular point group symmetry of C60is Ih, and the dipole allowed transitions belong to the T 1uirrep. If the quantum chemistry code used for the spectral calculations only supports Abelian symmetry, the symmetry descent from I hto D 2h implies that the dipole allowed transitions belong to the same irrep (B1u) as several other forbidden excitations. Thus, straightforward calculation of excitation energies and oscillator strengths results in an exceedingly large number of roots with no intensity to be con- verged. Moreover, C 60is an example for a material with a large number of atoms in a similar chemical situation and, thus, a dense spectrum, which makes it costly to compute the spectrum for a given energy or frequency range in the traditional way. C 60already has (at the CC2/aug-cc-pVDZ level) about 500 states below 7 eV (60 of them inB1u). For larger fullerenes, this number will increase, roughly lin- early with the number of C atoms. With the CPP approach, on the other hand, the number of points will not increase with the system size. In the example below, we converged 60 B 1ustates (D 2hpoint group), which covered an energy range up to 6.90 eV, and only obtained four states with non-zero intensity at ∼3.6,∼4.6,∼5.5, and ∼6.4 eV, which are shown in Fig. 1 as red vertical sticks. The spec- trum computed with our CPP-RI-CC2 and the one from a previously reported CPP-B3LYP study (obtained using the pol-Sadlej [10s6p4d| 5s3p2d] basis)16also cover the frequency region up to 7 eV and show four peaks of varying intensity. The intensity of the bands is slightly larger in RI-CC2 compared to CPP-B3LYP. The CPP-B3LYP spec- trum is blue-shifted by ∼0.3 eV with respect to the one obtained with CPP-RI-CC2. J. Chem. Phys. 154, 124110 (2021); doi: 10.1063/5.0042759 154, 124110-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . C 60: RI-CC2/aug-cc-pVDZ UV– Vis OPA spectra from standard linear response (red) and CPP (blue) calcu- lations. The red sticks (RI-CC2/aug-cc- pVDZ/frozen core) are the only excita- tions with non-zero intensity up to 6.9 eV (first 60 B 1uroots in the D 2hpoint group). The CPP spectrum shown as a blue line is a cubic spline of the computed CPP grid points. The dashed green line is the CPP-B3LYP/pol-Sadlej spectrum re-digitized from Ref. 53. C. Electronic circular dichroism: Helicenes Helicenes are prototypical systems that show chiro-optical activity not because of the presence of chiral centers (e.g., asym- metric carbon atoms), but because of the handedness of their helical structure, also known as axial chirality, as the clockwise and coun- terclockwise helices are non-superposable. By convention, a left- handed helix is minus , (−), and labeled M, whereas a right-handed helix is plus, (+), and labeled P. The n-helicenes are also a proto- typical example of overcrowded aromatic chromophores, and the enantiomers possess a strong optical activity,56–60which makes them ideal test systems for our CPP-RI-CC2 computational scheme. ECD spectra of helicenes were theoretically studied before,61–65e.g., in 2000 at the TDDFT level by Furche et al.61and, for 5-helicene and 6- helicene, in 2003 by Köhn62at the CC2 level using the aug-cc-pVDZ basis supplemented with center of mass functions. The standard response spectra in the latter study included the lowest 24 and 20 states, respectively. In 2012, a combined theoretical and experimen- tal study on several helicenes was also presented by Nakai, Mori, and Inoue,65where the computed ECD spectra were obtained at the RI-CC2 level using the TZVPP basis set and 40 excited states. To illustrate the CPP approach, we here extend the RI-CC2 studies of Refs. 62 and 65 by investigating the penta-helicene, hexa-helicene, and hepta-helicene using the larger aug-cc-pVTZ basis set. Experi- mental spectra were re-digitized from the original references and are shown together with the calculated ones. The ECD spectra of ( −)-5-helicene are shown in Fig. 2. By con- verging 40 excited states, we could obtain the standard (broadened) linear response spectrum up to approximately 6.3 eV. Experimen- tal56,60,65and CPP spectra cover the frequency range up to 6.2 and 7.7 eV, respectively. One56,60of the shown experimental spectra was recorded in iso-octane. Note that we re-digitized the experimental spectrum reported in Fig. 2 of Ref. 56. According to the authors,56 this experimental spectrum was taken from the work of Goedickeand Stegemeyer,60even though no image of the spectrum is actu- ally given by Goedicke and Stegemeyer, who only report individual values of Δεat given wavelengths. Spectral data from both articles are presented as a green continuum line and triangles in Fig. 2. We observe small inconsistencies at around 4 and 5.5 eV between the spectrum re-digitized from Ref. 56 and the spectral points taken from Ref. 60 (green triangles). The experimental spectrum from Ref. 65, recorded in 98:2 n-hexane/2-propanol, is shown as a dashed green line. The stick spectrum starts with one positive peak of symmetry A and very low intensity (marked by an arrow). Roughly in the same region, the experiment56,60shows two low intensity positive features (∼50 times weaker than the rest of the spectrum).56,60The CPP and the Lorentzian broadened spectra are practically indistinguishable up to around 5.85 eV, where differences start to emerge, as indi- vidual excitations may be missing in the latter. The computed and experimental spectra have similar features: two negative bands, one at around 4 eV and one just above 5 eV, two positive overlapping bands at around 4.5–4.7 eV, and a feature-rich positive band, start- ing in between 5 and 6 eV, clearly due to a large number of transi- tions. The computed spectra ( in vacuo ) are slightly blue-shifted and of lower intensity compared to the experimental data in iso-octane.60 The ECD spectra for ( −)-6-helicene are presented in Fig. 3. Note that the experimental measurement from Ref. 57 was carried out in methanol on the P structure, so we have reversed its sign when comparing it in Fig. 3 with the spectra computed for the M enantiomer (solid green line). The experimental CD spectrum recorded in acetonitrile from Ref. 65 is also shown as a dashed green line. As the system size increases, it becomes progressively more challenging to converge the standard response spectra. For 6- helicene, the first 20 excited states were explicitly calculated. This, however, only covers the region up to 5.2 eV. The CPP spectrum was computed up to 7.3 eV. The CPP and broadened standard response J. Chem. Phys. 154, 124110 (2021); doi: 10.1063/5.0042759 154, 124110-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . (−)-5-Helicene (M): (frozen- core) RI-CC2/aug-cc-pVTZ ECD spec- tra from resonant linear response and CPP calculations. The individual excita- tion energies and rotatory strengths and their Lorentzian broadened spectrum are reported in red. The blue circles are the CPP points, and the blue line is a cubic spline of the CPP points. The solid green line is the experimental spectrum re-digitized from Ref. 56, which is (sup- posedly) derived from the measurement iniso-octane in Ref. 60 (green triangles). The dashed green line is the experimen- tal spectrum re-digitized from Ref. 65, recorded in 98:2 n-hexane/2-propanol. spectra start to differ at around 5.2 eV. Indeed, the intensity of the strongest positive peak predicted by the CPP spectrum is slightly lower than the one obtained from broadening the individual exci- tation energies and rotatory strengths, probably the effect of the broad negative band, located in between 5.5 eV and 5.7 eV, clearly not present in the broadened spectrum as the corresponding excited states were not computed. All in all, as for 5-helicene, the computed and experimental spectra of 6-helicene have rather similar features: a relatively strong negative peak at around 3.8 eV and two (partly overlapping) posi- tive peaks in between 4.8 and 5.3 eV, followed by a bisignate band in between 5.5 and 6.3 eV. The computed first negative peak at3.9 eV is marginally blue-shifted with respect to the experimen- tal band. The band intensities in the simulated spectrum are only slightly larger than the corresponding ones in the experimental spec- trum recorded in methanol. The lowest-energy band is practically overlapping with the same band from the experimental measure- ment in acetonitrile.65 The ECD spectra for (+)-7-helicene are presented in Fig. 4. The mirror image of the experimental spectrum of ( −)-7-helicene, recorded in ethanol by Brickell et al. ,58is shown as a solid green line in Fig. 4. The experimental spectrum in chloroform reported by Nakai, Mori, and Inoue,65originally taken from the study of Martin and Marchant,66is also shown as a dashed line. The CPP spectrum FIG. 3 . (−)-6-Helicene (M): (frozen- core) RI-CC2/aug-cc-pVTZ ECD spec- tra from standard linear response and CPP calculations. The individual excita- tion energies and rotatory strengths and their Lorentzian broadened spectrum are reported in red. The blue circles are the CPP points, and the blue line is a cubic spline through the CPP points. A mirror image of the experimental spec- trum of (+)-6-helicene (P) from Ref. 57, recorded in methanol, is shown as a solid green line. A dashed green line shows the experimental spectrum recorded in acetonitrile, re-digitized from Ref. 65. J. Chem. Phys. 154, 124110 (2021); doi: 10.1063/5.0042759 154, 124110-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . (+)-7-Helicene (P): (frozen- core) RI-CC2/aug-cc-pVTZ ECD spec- tra from resonant linear response and CPP calculations. The individual excita- tion energies and rotatory strengths and their Lorentzian broadened spectrum are reported in red. The blue circles are the CPP points, and the blue line is a cubic spline through the CPP points. A mirror image of the experimental spec- trum of ( −)-7-helicene (M) from Ref. 58, recorded in ethanol, is shown as a solid green line. The experimental spectrum in chloroform65,66is given as a dashed green line. was obtained up to 6.25 eV and shows a well-separated positive peak between 3 and 3.7 eV; a feature-rich negative band between 3.7 and 5 eV, clearly with contributions from several transitions of different intensity and with maximum at 4.7 eV; and two positive peaks at 5.44 and 5.99 eV, respectively. All peaks are of comparable intensity. Not surprisingly, with the increase in complexity of the system, our ability to solve for the individual excited state energies and rotatory strengths deteriorates. Indeed, for 7-helicene, we only succeeded in converging 12 states, which cover the region up to 4.5 eV, thus only reproducing the first (positive) peak and half of the negative broad band. Despite the different environments, the experimental spectral profiles are, as in the previous two cases, quite similar to the com- puted one, with a broad positive band at lower energy, a structurednegative one in the intermediate region, and two positive bands in the upper frequency region. The experimental intensity of the spec- trum in ethanol is, on the other hand, roughly two times lower, whereas the one in chloroform is more intense, in particular in the intermediate frequency region. D. The C6dispersion coefficients In Table II, we present the C6dispersion coefficients for the dimers of a set of ten organic molecules. In Table III, the results for six different fullerenes are collected. The RI-CC2 values for the C6coefficients of the organic molecules are in line with the results of a previous theoretical study at the ADC(2)/Sadlej-pVTZ level.15The percentage difference TABLE II . RI-CC2/aug-cc-pVTZ C6dispersion coefficients (a.u.) of the dimers of ten organic molecules and comparison with previous theoretical results, obtained at the ADC(2)/Sadlej-pVTZ15and CCSD/Sadlej-pVTZ15levels of theory, and with DOSD results. %ΔCADC(2) 6=100×(CCC2 6−CADC(2) 6)/CADC(2) 6, %ΔCDOSD 6=100×(CCC2 6−CDOSD 6)/CDOSD 6. Molecules RI-CC2 ADC(2)15%ΔCADC(2) 6CCSD15DOSD % ΔCDOSD 6 Acetaldehyde 432.8 434.3 −0.4 407.2 401.8677.7 Acetone 834.0 832.0 0.24 787.4 794.5675.0 Benzene 1874 1926 −2.7 1786 1723688.8 Butane 1285 1263 1.7 1224 1268691.3 Ethane 374.5 365.9 2.4 357.3 381.869−1.9 Ethene 305.9 299.8 2.03 287.3 300.2701.9 Formaldehyde 154.0 157.6 −2.3 144.8 165.267−6.8 Methane 126.3 122.7 2.9 120.7 129.671−2.5 Pentane 1950 1918 1.7 1855 1905692.4 Propane 759.9 745.1 2.0 724.2 768.169−1.1 J. Chem. Phys. 154, 124110 (2021); doi: 10.1063/5.0042759 154, 124110-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE III . RI-CC2 C6dispersion coefficients [a.u. ×10−3] for a set of fullerene dimers and comparison with previous literature results.53,72 RI-CC2/ B3LYP/ CAM-B3LYP/ TD-HF/ Molecule cc-pVDZ pol-Sadlej53pol-Sadlej53pol-Sadlej53DOSD72 C60 97.08a100.8 98.8 100.1 100.3 C70 141.4 143.0 139.8 141.6 C78 179.1 180.0 176.1 178.2 C80 191.3 193.1 189.4 192.5 C82 197.9 199.1 194.8 196.8 C84 208.6 209.8 205.4 207.7 a117.7 ×103(aug-cc-pVDZ); At MP2 geometry, 96.00 ×103(cc-pVDZ); 116.3 ×103(aug-cc-pVDZ); 115.4 ×103(aug-cc- pVTZ). between CC2 and ADC(2) values varies between −2.7% (benzene) and +2.9% (methane). The smallest differences between the two methods are observed for acetaldehyde and acetone. Note, how- ever, that we did not use the same basis set in our CC2 calculations as used in the ADC(2) study. Both RI-CC2 and ADC(2) results are systematically larger than corresponding CCSD/Sadlej-pVTZ results from the literature,15which were obtained using a Lanczos- based implementation of the polarizability at imaginary frequencies9 and a moderate chain length. We also compare our CC2 values to estimates from the literature obtained using the dipole oscillator strength distribution (DOSD) approach.72In the DOSD approach, theC6coefficients are derived from the dipole oscillator strength distributions constructed from theoretical and experimental pho- toabsorption cross sections, combined with constraints provided by the Kuhn–Reiche–Thomas sum rule and molar refractivity data. In this case, the differences in percentage range from ∼−1% (propane) to +9% (benzene).The CC2/cc-pVDZ results for the C6coefficients of the fullerenes, see Table III, are compared to literature results at the CAM-B3LYP, B3LYP, and TD-HF levels of theory,53obtained with the pol-Sadlej basis set. We note that our basis set is on the small side, so our results are probably not fully converged. Indeed, adding one set of augmented functions increased the coefficient for C 60to 117.7×103. At the MP2 geometry used in the OPA calculations, theC6coefficient of C 60changes from 96.00 ×103(cc-pVDZ) to 116.3×103(aug-cc-pVDZ) to 115.4 ×103(aug-cc-pVTZ). A ref- erence value, obtained using the DOSD approach, is available for C60.72As already commented upon in Ref. 53, the TD-HF/pol- Sadlej result is the closest to the DOSD value, but good agreement is probably fortuitous. In Fig. 5, the C6coefficients are plotted as a function of the number Nof carbon atoms in the considered fullerenes. In the inset, a plot of the ratios C6(CN)/C6(C60) vsN/60 is given. Figure 6 reports the base-10 logarithm of C6coefficients as a function of log N. The FIG. 5 .C6coefficients (a.u. ×10−3) of six fullerenes as a function of the number of carbon atoms N, plotted for four dif- ferent electronic structure methods. The inset shows the ratios C6(CN)/C6(C60) vs N/60. Red: RI-CC2; blue: B3LYP; gray: CAMB3LYP; green: TD-HF results. The lines are linear regressions of the C6 points. The regression coefficients for the lines in the main panel are r= 0.9995 (RI-CC2), r= 0.9986 (HF), r= 0.9991 (B3LYP), and r= 0.9989 (CAM-B3LYP). Those of the lines in the inset are r= 0.9995 (RI-CC2), r= 0.9988 (HF), r= 0.9991 (B3LYP), and r= 0.9988 (CAM-B3LYP). J. Chem. Phys. 154, 124110 (2021); doi: 10.1063/5.0042759 154, 124110-13 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 6 . Base-10 logarithm of the C6coef- ficients of six fullerenes as a function of logN, for four different electronic struc- ture methods. Red: RI-CC2 (this work, r= 0.9992); blue: B3LYP ( r= 0.9994); gray: CAMB3LYP ( r= 0.9994); green: TD-HF ( r= 0.9991) results. The inset shows the data without consideration of C80, which slightly improves the linear regression coefficient: r= 0.9998 for HF, r= 0.9998 for B3LYP, r= 0.9998 for CAM-B3LYP, and r= 0.9994 for RI-CC2. latter figure is used to determine the exponent ηat the RI-CC2 level of the ansatz C6∝Nη, as also reported by Kauczor, Norman, and Saidi.53Kauczor, Norman, and Saidi53found that the C6coefficients were non-additive and scaled roughly as N2.2for the three meth- ods they considered. The exponent for the C6power-dependence on Nwas, therefore, much smaller than the values predicted based on a classical-metallic spherical-shell approximation of the fullerenes (≈2.75).73In a later study, some of the same authors74proposed a model based on classical electrodynamics that yielded C6∝N2.8. Our results at the CC2 level, based on small-size fullerenes, give η= 2.3, i.e., only marginally larger than the HF/DFT estimates of Kauczor, Norman, and Saidi. Removing C 80from the series slightly improves the linear regression coefficients but does not significantly change the value of η. V. CONCLUSIONS We have presented an implementation of a damped linear response solver and of the damped linear response function within the resolution-of-identity CC2 method in Turbomole.31The LR- CPP-RI-CC2 approach allows us to directly compute, e.g., ECD and OPA spectra of systems with a high density of excited states, where a standard response approach is hardly or not applica- ble. The combination of the RI approximation with a partitioned formulation that avoids the storage and I/O of four-index two- electron integrals and double excitation amplitudes (employing a Laplace transformation of orbital energy denominators) together with an OpenMP parallelization makes the LR-CPP-RI-CC2 approach applicable to molecular systems as large as fullerenes and helicenes. Examples of application of the approach included the OPA spectra of C 60, the ECD spectra of n-helicenes ( n= 5, 6, 7), andtheC6dispersion coefficients for a sample of organic molecules and fullerenes. The CPP solver for RI-CC2 is also a fundamental stepping stone for the implementation of higher-order response properties in a RI-CC2 CPP framework, e.g., RIXS and MCD, as well as for the extension to excited state properties. SUPPLEMENTARY MATERIAL The following additional information is found in the supple- mentary material: Cartesian coordinates of all studied molecules and raw spectral data (excitation energies and oscillator/rotatory strengths) for C 60, 5-helicene, 6-helicene, and 7-helicene. ACKNOWLEDGMENTS We thank Dr. Thomas Fransson (University of Heidelberg) and Professor Wissam Saidi (University of Pittsburgh) for send- ing us the Cartesian coordinates of the molecular systems consid- ered for the C6coefficients. D.A.F. thanks Dr. Rasmus Faber (DTU) and Dr. Alireza Marefat Khah (RUB) for valuable discussions. D.A.F. and S.C. acknowledge the financial support from the Marie Skłodowska-Curie European Training Network “COSINE- COmputational Spectroscopy In Natural sciences and Engineering” under Grant Agreement No. 765739. S.C. acknowledges Indepen- dent Research Fund Denmark—Natural Sciences, Research Project 2, Grant No. 7014-00258B. C.H. acknowledges the financial support by DFG through Grant No. HA 2588/8. APPENDIX A: TIMINGS Table IV presents computational timings ( t, min) to converge one response amplitude for the calculation of one component of the J. Chem. Phys. 154, 124110 (2021); doi: 10.1063/5.0042759 154, 124110-14 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE IV . Computational timings ( t, min) to converge one response amplitude for the calculation of one component of the dipole polarizability of C 60using the damped and the standard linear response solvers. Niteris the number of iterations and Nvectis the number of response vectors timings are normalized to one response vector. All calculations were performed using Central DTU HPC Cluster, on the node “Lenovo ThinkSystem SD530” with the following configuration: 2 x Intel Xeon Gold 6126 (12 cores, 2.60 GHz); 12 ×192 GB/11 ×384 GB/1 ×768 GB memory; 480 GB SSD. The convergence threshold was 10−6. Solver Total t/vect Niter Nvect t/(iter⋅vect) Standard (ω= 0.00 a.u.) 28.0 10 1 2.80 Standard (ω= 0.07 a.u.) 24.6 14 2 1.76 Standard (ω= 0.20 a.u.) 119.7 74 2 1.62 Damped (ω= 0.00 a.u., γ= 0.0 a.u.) 32.83 9 1 3.65 Damped (ω= 0.07 a.u., γ= 0.0 a.u.) 34.50 13 2 2.65 Damped (ω= 0.20 a.u., γ= 0.0 a.u.) 109.7 54 2 2.03 Damped (ω= 0.20 a.u., γ= 4.60×10−3a.u.) 84.56 43 4 1.97 dipole polarizability of C 60using the damped and the standard linear response solvers. APPENDIX B: ADDITIONAL DEFINITIONS ●T1-similarity-transformed MO coefficients: Λp=C(1−tT 1),Λh=C(1+t1). ●T1-similarity-transformed one-electron integrals: ˆhx pq=∑ αβΛp αpΛh βqhx αβ. ●T1-similarity-transformed two-electron integrals: (pqˆ∣rq)=∑ αβγδΛp αpΛh βqΛp γrΛh δs(αβ∣γδ). ●The elements of ˆFare defined as the elements of the usual Fock matrix, but evaluated with T1-similarity-transformed one-electron and two-electron integrals. ●One-index transformed ˘Λpand ˘Λhmatrices: ˘Λp βi=∑ aΛp βa¯tiaand ˘Λh βa=−∑ iΛh βi¯tia. ●tx-dressed one-index transformed ¯Λp,xand ¯Λh,xmatrices (∗): ¯Λp,x βa=−∑ iCβitx aiand ¯Λh,x βi=∑ aCβatx ai. ●Barred one-electron and three-center and four-center two- electron integrals (∗):¯hx pq=∑ αβ(¯Λp,x αpΛh βq+Λp αp¯Λh,x βq)hαβ, ¯Bx Q,pq=∑ P(pq∣P)V−1 2 PQ =∑ αβ(¯Λp,x αpΛh βq+Λp αp¯Λh,x βq)∑ P(αβ∣P)V−1 2 PQ, ¯(pq∣rs)x=ˆPpr qs∑ αβγδ(¯Λp,x αpΛh βq+Λp αp¯Λh,x βq)Λp γrΛh δs(αβ∣γδ). Here, it is understood that ¯Λp,x αpvanishes if pis an occupied index and ¯Λh,x βqvanishes if qis a virtual index. ●Barred Fock matrices and Eintermediates (∗): ¯Fx ia=∑ ck[2(ia∣kc)−(ic∣ka)]tx ck, ¯Fx ab=−∑ jtx ajˆFjb+∑ ck[2(ab∣kc)−(ac∣kb)]tx ck, ¯Fx ij=+∑ bˆFjbtx bi+∑ ck[2(ij∣kc)−(ic∣kj)]tx ck, ¯Ex,2 ij=¯Fx ij+∑ cdk[2tx,jk cd−tx,jk dc](kd∣ic), ¯Ex,2 ba=¯Fx ba+∑ dkl[2tx,kl bd−tx,kl db](ld∣ka). An asterisk (∗) indicates that the respective intermediates depend linearly on the complex response amplitude txand have been generalized to the CPP case such that their real and imaginary parts are evaluated, respectively, with the real and imaginary parts oftx. J. Chem. Phys. 154, 124110 (2021); doi: 10.1063/5.0042759 154, 124110-15 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp ALGORITHM 1 . Main steps in the evaluation of the F-matrix contraction terms. An asterisk (∗) indicates the terms that have been generalized to the CPP case (complex). •Compute ˘YQ,ck=∑dj¯tjk dcˆBQ,dj(standard code) •Compute ¯Fx ia,¯Ex,1 ab,¯Ex,2 ij,¯Bx Q,bj,¯Bx Q,ij,Cx ck,Yx Q,ai(∗) •Compute dressed integrals ˘Bx Q,jb=−∑ck(¯tjctx ckBQ,kb+tx ck¯tkbBQ,jc)(∗) [Eq. (85)] •Computeσ0,x ia=2¯Fx ia,σJG,x iaandσJH,x ia (∗) •Compute intermediates: –˘Λp βi,˘Λh βaand ˘BQ,ia=∑αβPΛh αa˘Λp βi(αβ∣P)V−1/2 PQ−∑kˆBQ,ik¯tka (standard code) –¯tij ab=(2−ˆPij)ˆPij ab(∑Q˘BQ,aiBQ,jb+¯tiaˆFjb)/(εi−εa+εj−εb)(standard code) –˘Yx Q,ia=∑bj¯tij ab¯Bx Q,jb(∗) –mx Q=∑ckBQ,ckCx ckandMx Q,ik=∑cBQ,icCx ck(∗) –˘ix Q=∑jb¯Bx Q,bj¯tjb(∗) –˘Γx Q,βi=∑Pa(˘Yx P,ia−∑j¯tja¯Bx P,ij)V−1/2 PQΛp βa−∑PkMx P,ikV−1/2 PQΛp βk+ 2∑P(mx P+˘ix P)V−1/2 PQΛp βi(∗) –σIJ′ 12F1,x iα=∑Qβ˘Γx Q,βi(Q∣βα) •ComputeσIJ′ 12F1,x ia=∑ασIJ′ 12F1 iαCαa(σI,x ia+ first term of σF,x ia+ first and second term of σJ′,x ia) (∗) •ComputeσF3,x ia=−∑lQ˘Yx Q,alˆBQ,il(third term of σF,x ia) (∗) •Compute intermediate ˘Γx′′ Q,iβ=∑Pd(˘YP,id−∑jˆBP,ij¯tjd)¯Λp,x βdV−1/2 PQ (∗) •ComputeσF2J′ 3,x iα=∑Qβ˘Γx′′ Q,iβ(Q∣βα)(∗) •ComputeσF2J′ 3,x ia=∑ασF2J3,x iαCαa(second term of σF,x ia+ third term of σJ′,x ia) (∗) •CalculateσF4,x ia=−∑lQ˘YQ,al¯Bx Q,il(∗) •Contract with single amplitudes and add the doubles contributions Ftxty=∑aiσx iaty ai+∑ai¯Fx iaCy ai+∑Q,bj˘Bx Q,jbYy Q,bj(∗) DATA AVAILABILITY The data that support the findings of this study are available within this article and its supplementary material. 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5.0037557.pdf

J. Chem. Phys. 154, 114106 (2021); https://doi.org/10.1063/5.0037557 154, 114106 © 2021 Author(s).A Fock space coupled cluster based probing of the single- and double-ionization profiles for the poly-cyclic aromatic hydrocarbons and conjugated polyenes Cite as: J. Chem. Phys. 154, 114106 (2021); https://doi.org/10.1063/5.0037557 Submitted: 14 November 2020 . Accepted: 26 February 2021 . Published Online: 15 March 2021 Rajat K. Chaudhuri , and Sudip Chattopadhyay ARTICLES YOU MAY BE INTERESTED IN Quantum HF/DFT-embedding algorithms for electronic structure calculations: Scaling up to complex molecular systems The Journal of Chemical Physics 154, 114105 (2021); https://doi.org/10.1063/5.0029536 Modeling nonadiabatic dynamics with degenerate electronic states, intersystem crossing, and spin separation: A key goal for chemical physics The Journal of Chemical Physics 154, 110901 (2021); https://doi.org/10.1063/5.0039371 Equation-of-motion coupled-cluster method with double electron-attaching operators: Theory, implementation, and benchmarks The Journal of Chemical Physics 154, 114115 (2021); https://doi.org/10.1063/5.0041822The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp A Fock space coupled cluster based probing of the single- and double-ionization profiles for the poly-cyclic aromatic hydrocarbons and conjugated polyenes Cite as: J. Chem. Phys. 154, 114106 (2021); doi: 10.1063/5.0037557 Submitted: 14 November 2020 •Accepted: 26 February 2021 • Published Online: 15 March 2021 Rajat K. Chaudhuri1,a) and Sudip Chattopadhyay2,b) AFFILIATIONS 1Indian Institute of Astrophysics, Bangalore 560034, India 2Department of Chemistry, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, India a)Electronic mail: rkchaudh@iiap.res.in b)Author to whom correspondence should be addressed: sudip_chattopadhyay@rediffmail.com ABSTRACT Sequential formation of a poly-cyclic aromatic hydrocarbon (PAH) dication in the H I regions of the interstellar medium (ISM) is pro- posed to be a function of internal energy of the doubly ionized PAHs, which, in turn, is dependent on the single- and double-ionization potentials of the system. This sets a limit on the single- and double-ionization energies of the system(s) that can further undergo sequential absorption of two photons, leading to a dication (PAH+2). Here, we report the single-ionization (I+1) and double-ionization (I+2) energies and the I+2/I+1ratio for some selected PAHs and conjugated polyenes obtained using the Fock space coupled cluster technique, enabling simultaneous consideration of several electronic states of different characters. The I+2to I+1ratio bears a constant ratio, giving allowance to determine I+2from the knowledge of single-ionization (I+1) and vice versa. Our observations are in good agreement with the established literature findings, confirming the reliability of our estimates. The measured single- and double-ionization energies further demonstrate that the sequential formation and fragmentation of a PAH dication in the H I regions of the ISM for systems such as benzene and con- jugated polyenes such as ethylene and butadiene are quite unlikely because I+2–I+1for such system(s) is higher than the available photon energy in the H I regions of the ISM. Present findings may be useful to understand the formation and underlying decay mechanisms of multiply charged ions from PAHs and related compounds that may accentuate the exploration of the phenomenon of high-temperature superconductivity. Published under license by AIP Publishing. https://doi.org/10.1063/5.0037557 .,s I. INTRODUCTION Polycyclic aromatic hydrocarbons (PAHs), molecular systems having several aromatic rings with six carbon atoms in hexagonal (benzene-like) configurations, have been an important domain of exploration for their role as potent agents, leading to the degradation of air quality.1PAHs have been hypothesized to be abundant in the interstellar space as is manifested strongly by their signatures in the infrared spectra from many interstellar sources.2–4The low values of ionization potentials (IPs) of PAHs make them labile to the photo- electric effect upon absorption of light in the far-ultraviolet region.In addition, the planar-structured PAHs permit photo-electrons to easily evade, leading to the observed heating effect of interstellar gases. The possible formation of a PAH dication was first proposed by Leach5in interpreting the infra-red emission bands observed in the H I region of the interstellar medium (ISM), who has also demonstrated that such a formation is a two-photon absorption pro- cess, where the first step is the ionization of the PAH followed by the removal of the second electron through the absorption of the second photon. The minimum energy required for the first step is the ionization potential (I+1) of the PAH and that of second step is I+2–I+1, where I+2is the energy required to remove the second J. Chem. Phys. 154, 114106 (2021); doi: 10.1063/5.0037557 154, 114106-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp electron, i.e., the double ionization potential (DIP) of the system. If Ephis the energy of the incident photon, then the maximum internal energy (E int) of the PAH dication will be Eint=Eph−ΔI, where ΔI is the difference between the double and single ionization potentials of the system, i.e., I+2–I+1. Leach et al.6showed that the fragmenta- tion process and the type of fragmentation in the dication strongly depend on the internal energy E intof the doubly ionized PAH. Therefore, accurate values of single- and double-ionization poten- tials of PAHs are necessary in understanding the fragmentation processes. The single- and double-ionization potentials of PAHs have been measured by various research groups using a wide variety of experimental techniques. For instance, Tobita et al.7measured the single- and double-ionization energies of a series of PAHs (rang- ing from benzene to coronene) using the synchrotron radiation source. They also reported the electron-impact data of anthracene, phenanthrene, and pyrene. Analogous measurements were per- formed by electron-impact8–12and charge-stripping13–15on sev- eral PAHs. The single- and double-ionization energies of fullerenes C60and C 70from photon-impact and charge-stripping experiments were made available by Lichtenberger et al.14,15and Steger et al.16 The C 60photo-double to photo-single ionization ratio show a mod- ulation ranging from threshold to the carbon K-edge that might be related to the particular structure of the C 60systems.17Liter- ature survey shows that the single-ionization potentials measured by photon impact technique agree favorably well (within tenth of an eV or less) with those obtained from electron-impact tech- niques9,10and/or photo-electron spectroscopy measurements. The double-ionization potentials measured through photon-impact also exhibit reasonable agreement with electron-impact and charge- stripping based data, except for the fact that the photon-impact values are consistently lower than those obtained from electron- impact mass spectrometry and charge-stripping experiments. The origin of these differences was attributed to the inherent differ- ence in ascertaining the onset of the appearance potential by Tobita et al.7(i.e., could be of instrumental origin). However, these authors have not ruled out the fact that it could also be due to intrinsic molecular properties and their behavior under electron and photon impact. Compared to the single-ionization potentials, which are known for a large number of PAHs,18reliable theoretical or experimental data of double-ionization potentials for PAHs are very limited. It should be noted that the information of both single- and double- ionization potentials introduces severe impediments in verifying the two-photon model of double-ionization in the H I regions of ISM. Both Tsai and Eland11and Leach et al.6showed that this problem can be mitigated by using the model proposed by Smith19(also by Makov et al.20), who predicted the existence of a simple relationship between successive ionization potentials. Tsai and Eland11reported the double- to single-ionization potential ratio to be 2.69, in which the double-ionization potentials were mainly based on electron- impact data. Using the photon-impact data for the double-ionization potentials, Tobita et al. ,7on the other hand, obtained a value 2.65 ±0.06 for the double- to single-ionization potentials. We also want to mention that recent investigations of photo-double-ionization in aromatic hydrocarbons have shown the appearance of unexpected kinks in the density of doubly charged parent ions to singly charged parent ions, I2+/I+1, inbenzene (deuterated), naphthalene, anthracene, and coronene;21 pentacene;22azulene and phenanthrene;23pyrene,21and tro- pone and cyclo-octatetraene.1Such behavior in the photo-double- ionization spectra of these systems is attributed to Coulomb-pair resonances of πelectrons. However, the origin of these peaks was not well understood, but theoretical attempts to explain this pattern were made.24Here, it is worth noting that the well-known ratio of doubly to singly charged ions of helium can be exploited as a model curve to judge the ratio of other atoms. The close kinship in the ratio curves for different atoms indicates that the same photo-double-ionization mechanisms provide the doubly charged ions, which are the knock- out process and the shake-up process.25However, the ratio of doubly to singly charged parent ions can exhibit additional facets.26Infor- mation about the recent progress in the studies of the interactions of synchrotron radiation with hydrocarbons and aromatic molecu- lar systems leading to doubly charged ions can be found in Ref. 27. Holm et al.28also reported density functional theory estimations of first, second, and double ionization energies for a range of PAHs. From the computational viewpoint, there are various proto- cols [such as Fock-space coupled cluster (FSCC),29,30equation of motion-CC (EOMCC),31,32and related CC-based linear response theory (CCLRT)33–35] by which single- and double-ionization poten- tials can be calculated. In this article, we have employed FSCC technique to estimate them and their ratios for selected poly-cyclic aromatic hydrocarbons (benzene, naphthalene, biphenyl, acenaph- thene, fluorene, and anthracene) and polyenes (ethylene, trans - 1,3-butadiene, trans -1,3,5-hexatriene, trans -1,3,5-octatetraene, and trans- 1,3,5,7,9-decapentaene). Accurate information on the photo- ionization energies of individual PAHs can sometimes furnish a chemical insight (essential for assessments of PAH toxicities and cancer risks) even when large numbers of isomers with ioniza- tion energies that are not easily distinguishable preclude reliable isomer assignments from the photo-ionization efficiency curves.36 The study of properties of spectroscopic interest for these com- pounds is important for describing their electronic characteristics and hence contributing to the design of new materials with selected properties. Reliable information on electronically excited states of neutral and ionized PAHs is necessary in a wide range of appli- cations. Consequently, it is not only useful to accurately compute the structure of electronically excited/ionized states and correspond- ing spectra of individual PAHs but also to investigate their behav- ior with size-extension. An analysis of the present estimates has been done, and an attempt to elucidate agreement between differ- ent experimental methods and other theoretical calculations has also been addressed. In the FSCC method, one can simultaneously access states with a various number of electrons such as the ground state, ionized/electron-attached states, and excited states using the same valence-universal wave operator that makes the theory very attractive (rather the method of choice) for quantum chemistry to compute energy differences of spectroscopic interest for electrons. In recent years, significant efforts have been made to extend the usefulness of the FSCC method to various systems of arbitrary com- plexity.37–44It should be stressed that the double ionization poten- tial (DIP) calculations via the FSCC approach have also been done by Chaudhuri et al.45to be compared with the experimental data obtained by the Auger spectroscopy open shell reference function with the (convenient) closed shell substitute. The DIP-FS approach has already been tested in various contexts.42,46When solving the J. Chem. Phys. 154, 114106 (2021); doi: 10.1063/5.0037557 154, 114106-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FS problem, we have to start from the lowest valence sectors and build the higher sectors step by step, which implies that when we want to compute the DIPs, i.e., the (2,0) sector of the model space, the solutions for the (0,0) and (1,0) sectors must be known. In pass- ing, we want to mention that in Ref. 42, one can find the detailed discussion about the FS realization of the MRCC theory for the DIP process. Works that are related to the implementations of the double ionization potential with the EOMCC47or with the similarity trans- formed EOMCC48are also worth mentioning here. In this context, we want to mention that reliable theoretical benchmark investiga- tions of the UV–Vis absorption spectra of paradigmatic polycyclic aromatic hydrocarbons have also been reported by means of mul- tireference DFT/MRCI, SORCI, and NEVPT2 and single-reference SOS-ADC(2), TD-B3LYP, and TD-CAM-B3LYP methods.49 II. METHODOLOGY: A BRIEF RESUMÉ The underlying principle of single- and double-ionization ener- gies’ computation is the same for all correlated many-body meth- ods and is as follows. For the single-ionization process, the bind- ing energy I+1 αof an electron in the orbital αcan be defined as the energy difference between the (N −1)-electron singly ionized and the N-electron neutral state, i.e., I+1 α=E(N−1)−E(N). (1) The state vector corresponding to the ionized state with a vacancy in the occupied orbital αcan be represented as ∣Φ(1,0) α⟩=aα∣Φ0⟩, (2) where | Φ0⟩is the N-electron Fermi vacuum and a†(a) is the creation (annihilation) operator defined with respect to | Φ0⟩. The unper- turbed reference functions ∣Ψ0(1,0) α⟩corresponding to this ionized state are then constructed through appropriate linear combinations of the spin-adapted functions ∣Φ(1,0) α⟩as ∣Ψ0(1,0) α⟩=d ∑ k=1Ckα∣Φ(1,0) α⟩, (3) where dis the dimension of the unperturbed (N −1) reference state and C kαare the combining coefficients. The ionization energy I+1 α corresponding to the occupied orbital αis corrected up to first order in Coulomb and exchange integrals, which can be expressed as ( vide Koopmans’ theorem50) I+ α=⟨Φ(1,0) α∣H∣Φ(1,0) α⟩−⟨Φ0∣H∣Φ0⟩=−ϵα, (4) where His the electronic Hamiltonian of the system and ϵαis the single-particle of the orbital α. Likewise, the doubly ionized state vec- tor∣Φ(2,0) αβ⟩and the corresponding reference functions ∣Ψ0(2,0) αβ⟩with vacancies in the occupied orbitals αandβcan be expressed as ∣Φ(2,0) αβ⟩=1√ 2[aα↓aβ↑+(−1)Saβ↓aα↑]∣Φ0⟩,α≥β for S=0 (singlet) and α>βfor S=1 (triplet),(5) and ∣Ψ0(2,0) αβ⟩=d ∑ k=1Ck,αβ∣Φ(2,0) αβ⟩, (6)respectively, where C k,αβare the combining coefficients and dis the dimension of the reference state. The double ionization energy I+2 αβ corresponding to the eigenstate ∣Φ(2,0) αβ⟩(assuming k= 1) is corrected up to first-order in Coulomb and exchange integrals, which can be expressed as I+2 αβ=E(N−2)−E(N)=⟨Φ(2,0) αβ∣H∣Φ(2,0) αβ⟩−⟨Φ0∣H∣Φ0⟩ =−ϵα−ϵβ+[⟨αβ∣V∣αβ⟩+(−1)S⟨αβ∣V∣βα⟩], (7) where⟨αβ|V|αβ⟩and⟨αβ|V|βα⟩represent the Coulomb and exchange integrals, respectively. A careful inspection of the first order estimates of single- and double-ionization energies shows that the double- to single-ionization energy ratio Γis of the order of Γ=I+2 αβ I+1α=2−⟨αβ∣V∣αβ⟩+(−1)S⟨αβ∣V∣βα⟩ ϵα, (8) which is greater than 2 as ϵα<0. To obtain more accurate esti- mates of single- and double-ionization energies and their ratio, the terms appearing on the right-hand side of Eqs. (4) and (7) must be replaced by their correlated (or all-order) values. This can be accomplished via various established methods such as Green’s func- tion (GF),51–55symmetry-adapted cluster configuration interaction (SACCI),56EOMCC,31,32,57,58related CCLRT,33–35FSCC,29,30,58–60 and so on. Calculations of double ionization energies can also be performed by a method based on electron propagator theory derived from super-operator theory.55We have already mentioned that the objective of this work is to provide accurate and reliable values for IPs and DIPs of some selected PAHs and conjugated polyenes using the FSCC method with correlation-consistent basis sets. Note that the calculation of doubly ionized states demands proper treat- ments of electron correlation and relaxation effects that can be reached by the FSCC method. The FSCC approach is an explicit fully size-extensive scheme unlike the EOMCC and CCLRT meth- ods, which fails extensively for charge-transfer excitation. More- over, unlike EOMCC/CCLRT, where there is diagonalization of the Hamiltonian in the full active space, the FS scheme depends on an effective Hamiltonian in the model space, and thus, FSCC cluster finding equations often suffer from convergence due to the annoying intruder state problem.61Convergence of the FSCC iter- ations is increased by having a large energy gap (and weak interac- tion) between the two subspaces, PandQ. To avoid this problem, the iterative solution of the Bloch equation of the FSCC scheme can be replaced with the diagonalization of the appropriate matrix by exploiting eigenvalue-independent technique62or the canonical Bloch equation.41,63 The main idea of the FSCC (and other MRCC) method is to find the eigenvalue equation for the effective Hamiltonian operator Heff, Heff∣Ψ0 K⟩=EK∣Ψ0 K⟩, (9) to furnish only few state-to-state energies (we seek) out of the entire spectrum to bypass the diagonalization of the Hoperator in the entire large configurational space. The effective Hamiltonian, H eff, operator can be expressed as Heff=PHΩP, (10) J. Chem. Phys. 154, 114106 (2021); doi: 10.1063/5.0037557 154, 114106-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp where Pstands for the model space projector and Ω is the valence universal wave operator in the Fock-space strategy. They are related in the following manner: PΨK=Ψ0 KandΨK=ΩΨ0 K. The FSCC amplitude finding equations are derived from the Bloch equation, HΩP=ΩHeffP. (11) Exploiting the “valence-universal” form of the wave operator, the Fock-space Bloch equation, mentioned above, for the FSCC model can be written as29,30,58–60 HΩP(k,l)=ΩH(k,l) effP(k,l)⋅ ∀(k,l), (12) where H(k,l) eff=P(k,l)HΩP(k,l), (13) where P(k,l)and H(k,l) effrepresent the model space projector and the effective Hamiltonian for the ( k,l) valence sector, respectively. The Ω operator required to construct H(k,l) effcan be obtained from the fol- lowing Bloch equation which is consecutively resolved in each sector of the Fock space: Q(k,l)HΩP(k,l)=Q(k,l)ΩH(k,l) effP(k,l)∀(k,l), (14) In the FSCC methodology, the model (reference) and the exter- nal (virtual) spaces also inherit the sector-structure. Thus, the model space projector P(k,l)and its orthogonal complement Q(k,l)(projec- tors are mutually exclusive) can be described, respectively, by P(k,l)=∑ α∣Φ(k,l) α⟩⟨Φ(k,l) α∣and Q(k,l)=1−P(k,l). (15) For computational reasons, Ω in FSCC is usually described by the following expression: Ω=exp(T)Ω v=exp(T){exp(˜S(k,l))}, (16) where k(≠0) equals the number of valence holes and l(≠0) equals the number of valence particles and ˜S(k,l)=k ∑ i=0l ∑ j=0S(i,j)[excluding i=j=0], (17) where S(i,j)=S(i,j) 1+S(i,j) 2+⋯. The cluster operators without tilde refer to the particular sector only. Here, it is worth mentioning that Ω v={exp(˜S(k,l))}represents the Fock space cluster operator, defined in a universal manner, for all sectors. The braces represent the normal product form of the expansion where the contraction between the cluster operators ˜Sis not possible in the expansion of Ωv.29This form also ensures the size-extensivity of the computed energies.29,64Note that T[the cluster operator for the (0, 0) sec- tor] is an explicit excitation operator; however, ˜S is always a mixed deexcitation/excitation operator. Pre-multiplying Eq. (12) by exp( −T) and using Eq. (13), we get ¯HΩvP(k,l)=ΩvH(k,l) effP(k,l)∀(k,l)≠(0, 0), (18)where ¯H=exp(−T)H exp(T). (19) If we know the zero-valence operator, one can redefine Eq. (19) as ¯H=exp(−T)HNexp(T) + E gr=̃H + E gr, (20) where H=HN+ E ref. Here, the terms E refand E grcorrespond to the unperturbed and correlated energies of the N-electron reference state, respectively. Following Eq. (20), one can express H(k,l) effas H(k,l) eff=̃H(k,l) eff+ Egr∀(k,l)≠(0, 0). (21) From the above-mentioned definitions, the Bloch equation for energy differences can be redefined as [substituting Eq. (21) into Eq. (18)] ̃HΩvP(k,l)=ΩṽH(k,l) effP(k,l)∀(k,l)≠(0, 0). (22) Here, it should be emphasized that ̃Heffis connected, indicat- ing that the energies yielded by diagonalizing it are size-extensive in nature.29,37,64The aspired wave functions can be constructed by the operation of the valence-universal wave operator on the reference wave functions. A hierarchical structure of the FSCC method (called the subsystem embedding conditions ) leads to the number of amplitude finding equations in each sector being equal to the number of unknown cluster amplitudes in that sector. Here, it is also worth noting that this methodology can be par- ticularly effective in the description of the energy surfaces in the situations when DI-species dissociate smoothly into closed shell fragments. The effective Hamiltonian in the FSCC scheme has a “diagonal” structure with respect to the different valence (i.e., FS) sectors, and two FS-sectors belonging to a common Hilbert space do not mingle even if they have strongly interacting states. The (0, 0)-sector amplitude equations are simply the con- ventional SRCC equations for the ground state [in the present case, solving the coupled-cluster single double (CCSD) equa- tions], which are subsequently used in other Fock space sec- tors. Note that for the (0, 0) sector, we have Ω vP(0,0)=P(0,0) and̃HP(0,0)=0 as̃H(0,0) eff=0. Projection of Eq. (22) onto P(0,0)and Q(0.0), respectively, leads to the standard CC equations for the cluster amplitudes T= S(0,0)and energy for the ground state, P(0,0)(̃H−Egr)P(0,0)=0 and Q(0,0)̃HP(0,0)=0, (23) J. Chem. Phys. 154, 114106 (2021); doi: 10.1063/5.0037557 154, 114106-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp where S(0,0)has a usual cluster definition, S(0,0)=∑ p,asp a{a† pap}+1 4∑ p,q,a,bspq ab{a† pa† qabaa}. (24) The quantities sp aand spq abappearing in Eq. (24) represent cluster amplitudes, and the symbol { ⋯} stands for normal ordering. The indices a,b,. . ., and p,q,. . ., in Eq. (24) represent the occu- pied (hole) and unoccupied (particle) orbitals with respect to the Hartree–Fock reference (or ground) state Φ0, respectively. Likewise, the amplitudes of the cluster operator for the singly ionized state can be obtained by introducing the wave operator Ω, which operating on the model space function ∣Ψ0(1,0) α⟩produces the (N−1)-electron correlated states, ∣Ψ(1,0) α⟩=Ω∣Ψ0(1,0) α⟩, (25) where ∣Ψ0(1,0) α⟩=∑ kCkα∣Φ(1,0) α⟩=∑ kCkαaα∣Φ0⟩, (26) where ˜S(1,0)=S(0,0)+ S(1,0). (27) Here, it should be mentioned that Ckαare the reference (model) space coefficients. Like the S(0,0)operator, limiting the cluster expan- sion to the double components, the (1, 0) sector operator can be expressed as S(1,0)=S(1,0) 1+ S(1,0) 2=∑ i,αsα i{a† αai}+1 2∑ p,α,b,cspα cb{a† pa† αabac}, (28) where the occupied orbitals present (absent) in all (N −1)-electron reference functions are represented by α,β,. . .(i,j,. . .). Substituting Eq. (25) into the Schrödinger equation, we get H{exp(S(0,0)+ S(1,0))}P(1,0) k=Ek{exp(S(0,0)+ S(1,0))}P(1,0) k, (29) and replacing the N-electron Hamiltonian Hin Eq. (29) by ̃H, we get ̃H{exp(S(1,0))}P(1,0) k=(Ek−E0){exp(S(1,0))}P(1,0) k =I+1 k{exp(S(1,0))}P(1,0) k, (30) where I+1 kis the kth state ionization energy of the system. Projection of Eq. (30) onto P(1,0) kandQ(1,0) kyields a set of coupled equations, which has to be solved self-consistently to obtain desired energies and associated cluster amplitudes for the (1,0) valence sector. The final step is the solution for the (2,0) sector. The doubly ionized determinants, as shown in Eq. (5), can be constructed by detaching two electrons from the closed-shell N-electron reference state, and the resulting (N −2)-electron states consist of two active holes and zero active particles. Here, the cluster operator for the (2,0)-sector is ˜S(2,0)=S(0,0)+ S(1,0)+ S(2,0), (31) where the cluster operator S(2,0)must be able to destroy two active holes present in the (2,0) valence space, and S(2,0)=S(2,0) 2=1 4∑ α,β,a,b⟨αβ∣s(1,0) 2∣ab⟩{a† αa† βabaa}, (32)whereα,βdenote the active holes, which are destroyed. Note that the hole orbitals aand bappearing in Eq. (32) cannot be active at the same time. Note that the S(2,0) 1operator does not contribute because it cannot be defined within the DIP framework. Under such a condition, the FSCC approach for the (2,0) valence sector can be expressed as H{exp(S(0,0)+ S(1,0)+ S(2,0))}P(2,0) k =Gk{exp(S(0,0)+ S(1,0)+ S(2,0))}, (33) where Gkis the energy of the (N −2) electron system. Replacing the N-electron Hamiltonian Hin Eq. (33) by ̃H, we have ̃H{exp(S(1,0)+ S(2,0))}P(2,0)=I+2 k{exp(S(1,0)+ S2,0))}P(2,0), (34) where I+2 k=Gk−E0is the double-ionization energy of the system. Projecting Eq. (34) onto P(2,0)andQ(2,0), one can obtain a set of cou- pled equations [which has to be solved self-consistently to obtain the desired energies I+2 kand amplitudes associated with the cluster operator S(2,0)], P(2,0)̃̃H{exp(S(2,0))}P(2,0)=I+2 k (35) and Q(2,0)̃̃H{exp(S(2,0))}P(2,0)⟩=Q(2,0){exp(S(2,0))}P(2,0)I+2 k, (36) respectively. The dressed Hamiltoniañ̃H appearing in the above- mentioned equations can be defined as ̃̃H=[̃H{exp(S(1,0))}] conn =̃H +[̃HS(1,0)] conn+1 2[̃HS(1,0)S(1,0)] conn, (37) where the completely connected terms are denoted by the subscript “conn.” The coupled equations(35) and (36) have to be solved self- consistently to obtain the desired energies I+2 kand amplitudes asso- ciated with the cluster operator S(2,0). Due to the valence universal strategy, the S(0,0)and S(1,0)operators are known from the (0,0) and (1,0) calculations, so the expansion is linear in terms of the unknown S(2,0)operators. The amplitudes associated with the cluster operators S(0,0)and S(1,0)appearing in S(2,0)determining equations (35) and (36) are constants. As long as outer-valence (and double-) ionization is concerned, this method is quite stable and is capable of providing desired results. Although the two-valence sector [say, (2; 0) and (0; 2)] is readily treated, the higher-sector FSCC scheme is still theoretically challenging. This is due to the fact that the higher- sector FSCC equations are rather complex (with unfavorable com- putational scaling) and require significant effort to be put in during the implementation of the scheme. Nonetheless, orbital relaxation effects have become important for higher sectors. The intercon- nection between the Bloch equation based FSCC and the EOMCC method now enables more powerful, unambiguous schemes to be implemented to avoid the complexities of the FSCC approach. It should be reiterated that for the one-valence problem, the FSCC results are identical to those obtained with the EOM-CC scheme. This is no longer true for the sectors that involve the amplitude equa- tions for the products of the one-valence amplitudes. This influences two-valence and higher sectors. J. Chem. Phys. 154, 114106 (2021); doi: 10.1063/5.0037557 154, 114106-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp III. RESULTS AND DISCUSSIONS The FSCC method with single (S) and double (D) excitation schemes is employed to determine the single and double ioniza- tion potentials of PAHs (benzene, naphthalene, biphenyl, acenaph- thene, fluorene, and anthracene) and polyenes (ethylene, trans- 1,3-butadiene, trans- 1,3,5-hexatriene, trans- 1,3,5-octatetraene, and trans- 1,3,5,7,9-decapentaene). Although many investigations on benzene have been performed on single photoionization and disso- ciation employing different experimental techniques, fewer studies have been conducted concerning the doubly charged benzene ions. The photo-double-ionization process in larger molecules consisting of more than six atoms has been investigated only marginally. There are also only a few papers about doubly charged ions of larger hydro- carbons. Here, we want to address the question of how the structure of an aromatic molecule affects the photo-double-ionization pro- cess through theoretical estimates of spectroscopic interest using the FSCC method. The computations of single- and double-ionization energies have been carried out with the cc-pVTZ basis65at the SCF(HF) and MP2 optimized geometries obtained using the cc- pVXZ (X = D and T) basis sets. The FSCC calculations for the sys- tems are also carried with cc-pVDZ and/or cc-pVTZ basis set(s). In this context, we want to mention that the required size of the model space used does not scale with the system size but depends more on the complexity of its nondynamical correlation effect. The selec- tion of the model space is crucial in the FSCC method. Here, we employ complete model space only, which incorporates configura- tions obtained from all possible distributions of the valence electrons in a set of valence orbitals. A protocol permitting construction of the model space by selecting relevant configurations, rather than orbitals, may enhance the convergence behavior of the FS-method. In this work, the highest occupied molecular orbital(s) is (are) chosen to be active, whereas the remaining occupied orbitals are considered to be inactive. This choice of active space is made because the lowest ionization and double ionization are of prime interest in this present context. The deep-lying occupied and high-lying vir- tual orbitals (with single particle energies ±100) are excluded in the post-Hartree–Fock treatment for all the systems as their contribu- tions to the ionization and double-ionization potential are normally negligible.66While the geometry optimizations were carried out with the GAMESS ab initio structure package,67the single- and double- ionization energy calculations were performed with the DIRAC package.68 The reliability of the optimized bond lengths and bond angles of the PAHs and the conjugated polyenes is ascertained by compar- ing them against the available experimental geometry. For instance, MP2/cc-pVDZ estimates the C–C bond lengths in ethylene and ben- zene to be 1.346 and 1.406 Å, which compares well with the experi- mental data of 1.337 and 1.397 Å, respectively. This suggests that the structural parameters computed by the MP2/cc-pVDZ procedure are reliable and can be used for the post-Hartree–Fock calculations. The harmonic vibrational frequencies of the optimized ground state structures for these systems have been calculated using the SCF pro- cedure with the cc-pVDZ basis and are found to be real (positive), confirming the stability of the optimized structures. It should be noted that the non-planar biphenyl molecule is energetically more stable than its planar conformer.69This non-planarity arises due to the fact that the hydrogen atoms at the ortho-positions of bothphenyl rings in the biphenyl system prevent the rotation about the single bond, connecting the two phenyl rings (steric hindrance). This is also supported by the harmonic vibrational frequencies yielded at the SCF/cc-pVDZ level of computation. It is worth noting that the planar conformer renders one of its normal mode vibrational frequencies to be imaginary. Table I demonstrates a comparative account of the first ioniza- tion potential of benzene, naphthalene, biphenyl, acenaphthene, flu- orene, and anthracene obtained using the FSCC method with those observed by other workers in the field. The high-resolution photo- electron spectrum of benzene was reported by Åsbrink et al. ,70where they have detected that the lowest band at 9.24 eV arises due to the ejection of an electron from its highest occupied degenerate molecularπ-orbital. The first ionization potential of deuterium sub- stituted benzene measured by Tobita et al.7using photon-impact technique was found to be 9.22 eV. Using the present FSCC strat- egy with the SD truncation scheme, we have been able to obtain the lowest ionization energy of benzene at 9.26 eV, which is of appre- ciable proximity to the experimentally reported numbers. The single ionization energy of naphthalene has been reported by several exper- imental groups. The photon-impact,7electron-impact, and photo- electron spectroscopic71studies have shown the single ionization energy of naphthalene to be 8.15, 8.24, and 8.12 eV, respectively. The first ionization energy of naphthalene computed using the present FSCCSD/cc-pVDZ method with the HF/cc-pVDZ geometry appears to be 7.98 eV (FSCCSD/cc-pVTZ yields 8.08 eV with the MP2/cc- pVDZ geometry), thus showing a deviation (eV) of 0.17 (0.07), 0.26 (0.16), and 0.14 (0.04), respectively, from the values obtained by the photon-impact,7electron-impact, and photo-electron spec- troscopic71studies. The calculated first adiabatic (IP a) and ver- tical (IP v) ionization energies (eV) using DFT (with the B3LYP functional, combining Beckes’s three parameter non-local hybrid exchange potential in conjunction with the non-local correlation functional) for naphthalene are 7.86 and 7.95,28which agree rather well with the FSCC value. In contrast to this, for the biphenyl system, the FSCC method predicts the lowest ionization energy to be 8.11 eV for the cc-pVDZ basis (8.09 eV for the cc-pVTZ basis) with the MP2 geometry, which accords well with the observed values of 8.16, 8.22, and 8.20 eV, respectively, as obtained from the photon-impact, electron-impact, and photo-electron spectroscopic measurements. Meot-Ner72measured the ionization energy of acenaphthene using the gas-phase ion-equilibrium technique and arrived at a value of 7.74 eV for acenaphthene, which is 0.02 eV higher than the photon- impact value reported by Tobita et al. These numbers are 0.2 eV higher than the theoretical estimates of 7.56 eV (7.58 eV for cc- pVTZ). The present FSCC result for anthracene is in close prox- imity with DFT estimates (IP a= 7.09 eV and IP v= 7.15 eV) of Holm et al.28As can be seen from Table I, the FSCCSD method also underestimates the single-ionization energies of fluorene (7.78 and 7.76 eV with MP2 and HF with cc-pVDZ geometries, respec- tively) and anthracene (7.32 and 7.21 eV with MP2 and HF with cc-pVDZ geometries, respectively). The FSCCSD method yields the maximum deviation from the experimental values of 0.11 and 0.25 eV for fluorene and anthracene, respectively. The FSCC estimates of DIP (related to Auger energy) of benzene, naphthalene, biphenyl, acenaphthene, fluorene, and anthracene are compared against the observed values in Table II. The single-ionization potentials (from theory and experiment) are J. Chem. Phys. 154, 114106 (2021); doi: 10.1063/5.0037557 154, 114106-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Calculated lowest vertical ionization potential (in eV) of poly-cyclic aromatic hydrocarbons. This work Experiment Molecule Point group Basis Optimization level Ionization energy Photon-impact7Others Benzene D 2h cc-pVDZ MP2/cc-pVDZ 9.06 9.22 9.2470 cc-pVTZ MP2/cc-pVDZ 9.26 Naphthalene D 2h cc-pVDZ HF/cc-pVDZ 7.98 8.15 8.1271 cc-pVDZ MP2/cc-pVDZ 7.86 8.2471 cc-pVTZ MP2/cc-pVDZ 8.08 Biphenyl C 2 cc-pVDZ HF/cc-pVDZ 8.09 8.16 8.2071 cc-pVDZ MP2/cc-pVDZ 8.11 8.2271 Biphenyl D 2h cc-pVDZ MP2/cc-pVDZ 7.89 cc-pVTZ MP2/cc-pVDZ 8.02 Acenaphthene C 2v cc-pVDZ HF/cc-pVDZ 7.56 7.72 7.7372 cc-pVDZ MP2/cc-pVDZ 7.45 cc-pVTZ MP2/cc-pVDZ 7.58 Fluorene C 2 cc-pVDZ HF/cc-pVDZ 7.76 7.89 7.9375 cc-pVDZ MP2/cc-pVDZ 7.78 Anthracene D 2h cc-pVDZ HF/cc-pVDZ 7.21 7.46 7.4776 cc-pVDZ MP2/cc-pVDZ 7.32 also listed in this table. It is pertinent to note at this juncture that the observed single- and double-ionization energies of PAHs reported by Tobita et al. suggest that the deuterium substitution hardly has any effect on the ionization (both single and double) energies of PAHs. Thus, we expect that the FSCC estimates of double-ionization energies of benzene, anthracene, etc., should be almost identical to double-ionization energies of deuterium substituted PAHs. As can be seen in Table II, the FSCC method reproduces the double- ionization energies of PAHs listed in Table II within an eV or less, except for benzene. That the FSCC method provides reason- ably accurate estimates of double- and single-ionization is also evi- dent from their ratio, justifying the choice of the present theoreti- cal approach. The FSCC/cc-pVTZ DIP (21.49 eV) for naphthalene agrees very well with the DFT-based calculations (DIP a= 21.05 eV and DIP v= 21.45 eV).28Note that the present FSCC DIP result for anthracene (19.00 eV) is in good harmony with recent theoretical predictions (DIP a= 18.81 eV and DIP v= 19.05 eV) of Holm et al.28 We note that the agreement between present estimates and exper- imental findings generally improves with the increasing basis sets’ size, indicating the usefulness and consistent nature of the FSCC scheme. It is also interesting to note that the geometries obtained at MP2 and HF levels do not alter the computed estimates for the systems treated here. It should be emphasized that for systems such as fluorene, anthracene, and other large molecules considered here, the full-blown FSCCSD calculations with cc-pVTZ and higher basis sets are computationally very demanding since the computational expense of FSCC rises rapidly with the basis set size as that of other CC methods. We must mention that the results presented here are sufficient to demonstrate the aim of the present work along with the usefulness of our FSCCSD code. It is also gratifying that, on the whole, the FSCCSD estimates agree rather well with the results that were independently generated by other authors. This agreement also corroborates the validity of the geometries (obtained by MP2 and HFgradient schemes) that have been employed in computing IP/DIP using the FSCCSD scheme. There is room for further improvement of this work. We hope to address this issue at length in the near future. How unique is the overall character of the double- to single- ionization potential ratio for benzene relative to other aromatic hydrocarbons treated here? In order to understand this query and to be able to make a quantitative comparison between the ratio curves of different PAHs, it is important to realize the energy of the lowest double ionization thresholds so that each ratio curve can be given as a function of excess energy that can be viewed as the difference between photon energy and the particular double ionization threshold. Tsai and Eland11observed that the threshold energy for double ionization energy divided by the single ioniza- tion potential of an aromatic molecule provides an almost constant number of 2.69 ±0.1. With this empirical evidence, one can eas- ily compute the double-ionization threshold from the knowledge of single-ionization energy. Samson25argued that as the ratio val- ues are not small for some of the PAHs, it is more appropriate to use the ratio of doubly charged parent ions to the sum of doubly plus singly charged ones. The effective work function and capac- itive potential ( V) determined using the theoretical and observed data also exhibit the same (see Table III). It is worth mentioning that our computed data corroborate with the observations that are available in the literature. The ratio of double- to single-ionization resulted from FSCCSD estimates of double- and single-ionization turns out to be 2.69 ±0.05 (arrived through a least fitting), which is comparable to the value 2.65 ±0.06 computed using observed energies. Note that UandVdepend on the function of IP and DIP. Since our estimated IPs and DIPs are in agreement with the experi- ment, these two parameters derived from the theoretical value agree well with those calculated using the experimental value (except for acenaphthene). J. Chem. Phys. 154, 114106 (2021); doi: 10.1063/5.0037557 154, 114106-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II . Comparison of the lowest vertical single-ionization potentials (IPs), double-ionization potentials (DIPs), and the ratio of double- to single-ionization potentials ( Γ=I+2 I+1) of PAHs. Energies are given in eV. Experiment DIP Γ TheoryPhoton- Electron- Charge- Photon- Electron- Charge Molecule Basis Geometry IP DIP Γ IP impact impact stripping impact impact stripping Benzene cc-pVDZ MP2/cc-pVDZ 9.06 27.12 2.99 9.22 25.3a26.1b25.4c2.74 2.83 2.75 cc-pVTZ MP2/cc-pVDZ 9.26 27.47 2.97 Naphthalene cc-pVDZ HF/cc-pVDZ 7.98 21.34 2.67 8.15 21.5a22.7b22.7c2.64 2.79 2.79 cc-pVDZ MP2/cc-pVDZ 7.86 21.10 2.68 cc-pVTZ MP2/cc-pVDZ 8.08 21.49 2.66 Biphenyl (C 2) cc-pVDZ HF/cc-pVDZ 8.09 21.14 2.61 8.16 21.3a21.9b2.61 2.68 cc-pVDZ MP2/cc-pVDZ 8.11 21.15 2.61 Biphenyl (D 2h) cc-pVDZ MP2/cc-pVDZ 7.89 20.74 2.63 cc-pVTZ MP2/cc-pVDZ 8.02 20.98 2.62 Acenaphthene cc-pVDZ HF/cc-pVDZ 7.56 20.38 2.70 7.72 21.7a2.81 cc-pVDZ MP2/cc-pVDZ 7.45 20.19 2.71 cc-pVTZ MP2/cc-pVDZ 7.58 20.42 2.69 Fluorene cc-pVDZ HF/cc-pVDZ 7.76 20.56 2.65 7.89 21.0a2.66 cc-pVDZ MP2/cc-pVDZ 7.68 20.79 2.71 Anthracene cc-pVDZ HF/cc-pVDZ 7.21 19.15 2.66 7.46 19.6a21.1a2.63 2.83 cc-pVDZ MP2/cc-pVDZ 7.11 19.00 2.67 aReference 7. bReferences 7–12. cReferences 7, 13, and 77. TABLE III . Effective work function V (in eV) and specific capacitive potential U (eV) for PAHs calculated using the electrostatic model of Smith19from theoretical and experimental (photon-impact) single-ionization potentials (IPs) and double-ionization potentials (DIPs).a Theory Experimentb System Basis Geometry V U V U Benzene cc-pVDZ MP2/cc-pVDZ 4.56 4.50 5.79 3.43 cc-pVTZ MP2/cc-pVDZ 4.79 4.48 Naphthalene cc-pVDZ HF/cc-pVDZ 5.29 2.69 5.55 2.60 cc-pVDZ MP2/cc-pVDZ 5.17 2.69 cc-pVTZ MP2/cc-pVDZ 5.42 2.67 Biphenyl (C 2) cc-pVDZ HF/cc-pVDZ 5.61 2.48 5.67 2.49 cc-pVDZ MP2/cc-pVDZ 5.65 2.47 Biphenyl (D 2h) cc-pVDZ MP2/cc-pVDZ 5.41 2.48 cc-pVTZ MP2/cc-pVDZ 5.55 2.47 Acenaphthene cc-pVDZ HF/cc-pVDZ 4.93 2.63 4.59 3.13 cc-pVDZ MP2/cc-pVDZ 4.81 2.65 cc-pVDZ MP2/cc-pVDZ 4.97 2.72 Fluorene cc-pVDZ HF/cc-pVDZ 5.24 2.52 5.28 2.61 cc-pVDZ MP2/cc-pVDZ 4.97 2.64 Anthracene cc-pVDZ HF/cc-pVDZ 4.85 2.37 5.12 2.34 cc-pVDZ MP2/cc-pVDZ 4.72 2.39 aWn= nV + n2U, where W nis the energy necessary to eject nelectrons from a PAH. bEstimated from photon-impact data. J. Chem. Phys. 154, 114106 (2021); doi: 10.1063/5.0037557 154, 114106-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The experimentally observed and theoretically computed single- and double-ionization energies of conjugated polyenes are displayed in Table IV. The effective work function and spe- cific capacitive potentials computed using the single- and double- ionization energies are also listed in Table IV. In mass spectrum investigation, doubly charged ions have been recorded for several hydrocarbons: ethene, cyclopropane, butene, butadiene, cyclohex- ane, and benzene.8Exploiting electron-impact energies of up to 80 eV, Brehm et al.8observed that the double-ionization threshold is typically 2.8 times the energy of the single-ionization threshold. Both the single- and double-ionization energies yielded by the FSCC method agree well with the experimental values, and this agreement is particularly noticeable for the single-ionization energies. The ratio of double- to single-ionization energies obtained from theoretical estimates of ionization energies is found to be 2.83 ±0.07. This ratio is slightly on the higher side compared to the one obtained for PAHs. Nonetheless, the parameters (the ratio of double- to single- ionization, effective work function, and the specific capacitive poten- tial) deduced from the theoretically determined ionization energies of conjugated polyenes and PAHs display a very similar pattern, which is noteworthy. Present discussion clearly indicates that the FSCC method can be viewed as a viable tool for the investigation and understanding of electronically challenging larger PAHs and conjugated polyenes suffering from (quasi)degeneracy problems, where the single-reference approaches cannot furnish an accurate electronic description. The double- to single-ionization energy ratio is relevant in the context of dication formation of PAHs in the ISM as this can be used in setting the limit on the single-ionization energies of a PAHthat can absorb two photons, leading to a dication. We recall that the internal energy of the PAH E intis related to the incident photon energy E phthrough the following equation: Eint=Eph−ΔI=Eph−(I+2−I+1), (38) where I+1and I+2are the lowest single- and double-ionization ener- gies. Defining the double- to single-ionization ratio Γas I+2/I+1and assuming the incident photon energy to be E ph≤13.6 eV in the H I regions, one can show that the formation of a dication in the H I regions of ISM is possible, provided that the single-ionization energy of the PAH satisfies I+1(Γ−1)≤13.6. (39) Substituting Γ= 2.65±0.05, deduced from the observed single- and double-ionization energies, one finds that the formation and frag- mentation of a PAH dication such as benzene in the H I regions of ISM are very unlikely as the lowest ionization energies of these systems are much higher than the limiting value dictated by the above-mentioned equation. The formation of the ethylene and buta- diene dication in the ISM is also very unlikely for the same reason. Interestingly, Rosi et al.73arrived at a very similar conclusion in their study of the fragmentation pathways of a benzene dication. Using density functional theory, they have shown that systems such as the benzene dication will have a much shorter lifetime than their neutral or cation counterparts. The general overview of the present FSCC procedure yields good estimates for single and double ioniza- tion energies for the group of systems studied here. In order to study the impact of the molecular structure on electron correlations, many TABLE IV . Calculated lowest vertical single- and double-ionization potentials, the ratio of double- to single-ionization potentials, effective work function (V), and specific capacitive potential (U) of polyenes. Parameters derived from observed values are shown in the parentheses. Energies are given in eV. Single-ionization potential Double-ionization potential Ratio System Basis Geometry Theory Experiment Theory Experiment ( Γ) V U Ethylene cc-pVDZ MP2/cc-pVDZ 10.38 10.51a31.79 29.4b, 32.98c3.06 4.87 5.52 cc-pVTZ MP2/cc-pVDZ 10.59 32.17 3.04 5.10 5.50 (2.80) (6.32) (4.19) (3.14) (4.53) (5.98) trans- Butadiene cc-pVDZ MP2/cc-pVDZ 8.91 9.07d26.28 25.9e2.95 4.68 4.23 cc-pVTZ MP2/cc-pVDZ 9.13 26.65 2.92 4.94 4.20 (2.86) (5.19) (3.88) trans- Hexatriene cc-pVDZ MP2/cc-pVDZ 8.08 8.29f23.10 23.3e2.86 4.61 3.47 cc-pVTZ MP2/cc-pVDZ 8.31 23.51 2.83 4.87 3.45 (2.81) (4.93) (3.36) trans- Octatetraene cc-pVDZ MP2/cc-pVDZ 7.75 7.79g21.25 2.74 4.88 2.88 cc-pVTZ MP2/cc-pVDZ 7.99 21.67 2.71 5.15 2.85 trans- Decapentaene cc-pVDZ HF/cc-pVDZ 7.42 19.76 2.66 4.96 2.46 cc-pVTZ HF/cc-pVDZ 7.59 20.07 2.64 5.15 2.45 aReference 79. bReference 80. cReference 81. dReference 82. eReference 83. fReference 84. gReference 85. J. Chem. Phys. 154, 114106 (2021); doi: 10.1063/5.0037557 154, 114106-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp systems are required to be explored. Despite various successes of the method, the FSCC protocol is still awaiting a robust, faithful, and efficient computer implementation, which can illustrate its practical use for chemically interesting problems.74,78 IV. SUMMARY In summary, we have used the coupled cluster (CC) proto- col within the Fock-space (FS) framework (which is suitable for energy difference calculations) to estimate the single- and double- ionization energies of some selected poly-cyclic aromatic hydro- carbons (PAHs) and conjugated polyenes. After converging the FSCC equation, the effective Hamiltonian is diagonalized, yielding directly transition energies (according to the presence of valence holes and/or valence particles). Investigations reported in the lit- erature reflect that the information of IPs and DIPs of the systems studied here is scarce. Generally, the available theoretical studies on these properties of the systems treated are mostly semi-empirical in nature. The estimations of these quantities using the state-of-the- art ab initio methods for these systems are, therefore, highly desir- able. The present FSCC scheme with the cluster operator restricted to one- and two-body contributions is employed in the calcula- tions. The ratio of double- to single ionization energies, effective work functions, and specific capacitive potentials are also evaluated using the theoretically estimated single- and double-ionization ener- gies. We compare our results with the previously published data whenever available. It is encouraging that the ionization energies obtained by the FSCC method, which is rigorously size extensive, furnish values that are in good agreement with the exact values. Parameters such as the ratio of double- to single-ionization energy, the effective work function, and specific capacitive potentials evalu- ated from the theoretically determined ionization energies are found to match quite well with those deduced from the experimentally observed energies. To be more specific, the observed ionization energies estimate the ratio of double- to single-ionization energy of PAHs to be 2.65 ±0.06, which is in accordance with the value of 2.69±0.05 obtained from the present FSCC calculations. The ratio of double- to single-ionization energy for conjugated polyenes is found to be 2.75 ±0.07, which is slightly higher than the pre- dicted value for PAHs. These results support the proposition made by Tobita et al. in the context of the sequential formation of a PAH dication in the H I regions of the interstellar medium. 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5.0034810.pdf

J. Chem. Phys. 154, 094103 (2021); https://doi.org/10.1063/5.0034810 154, 094103 © 2021 Author(s).Energy natural orbitals Cite as: J. Chem. Phys. 154, 094103 (2021); https://doi.org/10.1063/5.0034810 Submitted: 22 October 2020 . Accepted: 11 February 2021 . Published Online: 01 March 2021 Kazuo Takatsuka , and Yasuki Arasaki ARTICLES YOU MAY BE INTERESTED IN CAS without SCF—Why to use CASCI and where to get the orbitals The Journal of Chemical Physics 154, 090902 (2021); https://doi.org/10.1063/5.0042147 Electronic structure software The Journal of Chemical Physics 153, 070401 (2020); https://doi.org/10.1063/5.0023185 From orbitals to observables and back The Journal of Chemical Physics 153, 080901 (2020); https://doi.org/10.1063/5.0018597The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Energy natural orbitals Cite as: J. Chem. Phys. 154, 094103 (2021); doi: 10.1063/5.0034810 Submitted: 22 October 2020 •Accepted: 11 February 2021 • Published Online: 1 March 2021 Kazuo Takatsukaa) and Yasuki Arasakib) AFFILIATIONS Fukui Institute for Fundamental Chemistry, Kyoto University, 606-8103 Kyoto, Japan a)Author to whom correspondence should be addressed: kaztak@fukui.kyoto-u.ac.jp b)E-mail: yasuki.arasaki@fukui.kyoto-u.ac.jp ABSTRACT We propose and numerically demonstrate that highly correlated electronic wavefunctions such as those of configuration interaction, the cluster expansion, and so on, and electron wavepackets superposed thereof can be analyzed in terms of one-electron functions, which we call energy natural orbitals (ENOs). As the name suggests, ENOs are members of the broad family of natural orbitals defined by Löwdin, in that they are eigenfunctions of the energy density operator. One of the major characteristics is that the (orbital) energies of all the ENOs are summed up exactly equal to the total electronic energy of a wavefunction under study. Another outstanding feature is that the population of each ENO varies as the chemical reaction proceeds, keeping the total population constant though. The study of ENOs has been driven by the need for new methods to analyze extremely complicated nonadiabatic electron wavepackets such as those embedded in highly quasi- degenerate excited-state manifolds. Yet, ENOs can be applied to scrutinize many other chemical reactions, ranging from the ordinary concerted reactions, nonadiabatic reactions, and Woodward–Hoffman forbidden reactions, to excited-state reactions. We here present the properties of ENOs and a couple of case studies of numerical realization, one of which is about the mechanism of nonadiabatic electron transfer. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0034810 .,s I. INTRODUCTION It is quite often encountered in quantum chemistry1that very accurate stationary-state electronic wavefunctions, such as those of highly correlated Configuration Interaction (CI) and its variants,2–4the method of cluster expansion,5Complete Active Space Multiconfiguration SCF method6(CASSCF) and various extensions,7Nakatsuji theory,8,9and so on, tend to seem quite different from one another in their profoundly complicated functional forms. In nonadiabatic electron wavepacket dynamics studies,10–14time-dependent states under study are often embed- ded in highly quasi-degenerate electronic state manifolds and rep- resented in the linear combinations of adiabatic or diabatic excited- state wavefunctions. For instance, boron clusters have such a densely packed electronic state manifold in their excited states, each undergoing very frequent nonadiabatic transitions here and there in multiple places of clusters.15–17Under such circumstances, mighty methods are required that can extract clear physical insights without respect to the choice of adiabatic, diabatic, or other representations.We owe the concepts of chemical bonding and reactivity much to classic and fundamental theories such as the valence bond the- ory,18molecular orbital theory,19notion of “orbital interaction” and intermolecular interactions, the Fukui frontier orbital (HOMO– LUMO interaction) theory,20and the Woodward–Hoffmann rule of conservation of orbital symmetry,21and so on. Noting that these theories and concepts have been established in terms of the spe- cific views and approximations, it would be instructive to translate them into a universal representation that is free of dependence on the choice of approximate expressions. This is all the more the case in our age of massive ab initio computation such as large-scale CI calculations. Such an effort will be helpful to extend those theories in a wider chemical context and to examine whether they are valid in high electronic excited states and/or densely quasi-degenerate states. Meanwhile, a very interesting and essential question has been raised by the cutting-edge experiments related to ultrafast elec- tron dynamics within molecules,22the fundamental one of which is “Tomographic imaging of molecular orbitals”23(see Ref. 24 for a recent progress). Molecular orbitals (MOs) themselves are known J. Chem. Phys. 154, 094103 (2021); doi: 10.1063/5.0034810 154, 094103-1 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp not to be invariant quantities with respect to unitary transforma- tion among themselves in a Slater determinant, though the canon- ical MOs are known to be particularly relevant to energetic prop- erties of molecules. Paying the best respect to these revolutionary experiments, we theoreticians are forced to reconsider what kind of physical existence “molecular orbitals” reflect. Although this paper cannot identify presently what the tomographic images really are, it seems quite valuable to deeply consider once again what “electronic orbitals” should be. To approach the above issues, we here propose an idea to extract one-electron functions from an energy density operator. We refer to them as energy natural orbitals (ENOs), since ENO is a fam- ily member of natural orbitals.25In exactly the same sense as in the natural orbital theory, ENOs are invariant with respect to the choice of the functional forms of correlated electronic wavefunctions, for instance, CI, perturbation theories, cluster expansion, or others. As in the standard expansion approach with the use of atomic-orbital- like basis functions, ENOs are generated as many as the number of basis functions. Each ENO is accompanied by an (orbital) energy, and the sum of them exactly amounts to the total electronic energy. Furthermore, and most importantly, it turns out that only a small number of ENOs are enough to characterize the essential features of chemical reactions, as in the HOMO–LUMO theory.20 We define energy natural orbitals in Sec. II and show how to calculate them. To see how the spectral structures and the indi- vidual functional forms of ENOs look, we show two illustrative examples: One is for the adiabatic bond formation of hydrogen molecules in Sec. III, where we revisit the picture of the so-called orbital interaction, which is frequently applied in an explanation of formation of bonding and antibonding orbitals. The other is an application to nonadiabatic electron transfer in LiF in Sec. IV, in which structural beauty and quasi-symmetry are revealed in the course of electron transfer reaction. These examples should serve to clarify the similarity and difference of the ENO picture from those of the existing orbital theories. Selected properties and future per- spective in applications of ENOs are discussed in Appendixes A and B. II. ENERGY NATURAL ORBITAL A. Natural orbitals and generalization Given an electronic density operator ˆρ, natural orbitals λkand occupation numbers nkare defined by Löwdin25as ˆρ∣λk⟩=nk∣λk⟩ (1) and ∑ knk=N, (2) with Nbeing the total number of electrons considered (see a review article of Ref. 26). The density operator is accordingly represented as ˆρ=∑ knk∣λk⟩⟨λk∣. (3) A straightforward extension of the idea of natural orbital is to consider an eigenvalue problem for an arbitrary one-electron operator ˆAmultiplied by ˆρasˆA=1 2(ˆρˆA+ˆAˆρ). (4) Then, one may consider ˆA∣λk⟩=ak∣λk⟩. (5) The possible advantage of this formal extension should depend on the choice of ˆA. B. Energy natural orbitals In the above spirit, we propose the following procedure: Sup- pose that a normalized total electronic wavefunction Ψat a given nuclear configuration is already available, and its total energy is E=⟨Ψ∣ˆH∣Ψ⟩=∫dr1⟨r1∣ˆhˆρ∣r1⟩+∫dr1dr2ˆg12ˆΓ(2)(r1,r2), (6) where ˆhis the sum of one-electron operators (kinetic energy plus electron–nuclei attraction energy) working on the first electronic coordinate r1, and ˆg12being the electron–electron repulsion oper- ator between electrons 1 and 2 at r2. Note that the nuclear repulsion energy is not included here. Ψcan be a highly correlated wavefunc- tion, and the associated first order density operator is normalized toN, while the second-order density operator ˆΓ(2)toN(N−1)/2. Define an electron repulsion density G(r)as ∫dr1dr2g12Γ(2)(r1,r2)≡∫dr1G(r1)=∫dr1⟨r1∣ˆG∣r1⟩, (7) which is utilized in E=Tr[ˆhˆρ+ˆG]=∫dr1⟨r1∣(ˆhˆρ+ˆG)∣r1⟩, (8) where Trindicates the trace operation. Define a one-electron energy operator ˆH(1)such that ˆH(1)=1 2(ˆhˆρ+ˆρˆh)+ˆG. (9) ˆH(1)has been formally symmetrized (Hermitized), and its eigen- values are all real-valued with the eigenfunctions being mutually orthogonal. It holds ⟨Ψ∣ˆH∣Ψ⟩=TrˆH(1)=∫dr1⟨r1∣ˆH(1)∣r1⟩, (10) where∫dr1∣r1⟩⟨r1∣=ˆ1 (identity) with ⟨r′ 1∣r1⟩=δ(r1−r′ 1)is used. Since⟨r1∣ˆH(1)∣r1⟩is an energy density distribution at point r1, we refer to ˆH(1)as the energy density operator. The eigenfunctions ζkand eigenvalues ϵkforˆH(1)are given by ˆH(1)∣ζk⟩=ϵk∣ζk⟩ (11) and ˆH(1)=∑ kϵk∣ζk⟩⟨ζk∣. (12) By definition, the sum of all the eigenvalues is equal to the total energy as J. Chem. Phys. 154, 094103 (2021); doi: 10.1063/5.0034810 154, 094103-2 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp E=TrˆH(1)=∑ kϵk. (13) When the basis set is complete, {ζk}also forms an orthonormal complete set, ⟨ζl∣ζk⟩=δlk (14) and ∑ k∣ζk⟩⟨ζk∣=1. (15) C. Calculations of ENOs Suppose that the energy density operator ˆH(1)is expanded in a complete set of orthonormalized one-particle functions, say {∣i(1)⟩,i=1,. . .,∞}, where the suffix (1) implies that they are func- tions for the first electron (thus distinguishing from another electron later in the electron repulsion integrals). For a complete set, it holds ˆI=∑ i∣i(1)⟩⟨i(1)∣, (16) and the one-electron spin-free density operator is represented as ˆρ=∑ i,j∣j(1)⟩ρji⟨i(1)∣, (17) with ρji=⟨j(1)∣ˆρ∣i(1)⟩. (18) Then, the term arising from one-electron operators in ˆH(1)is ⟨r′ 1∣ˆhˆρ∣r1⟩=∑ j,ij(1)(r′ 1)(∑ khjkρki)i(1)∗(r1), (19) with hjk=⟨j(1)∣ˆh∣k(1)⟩. (20) Likewise, the electron repulsion part is given as ∫dr2⟨r′ 1r2∣ˆg12ˆΓ(2)∣r1r2⟩ =∑ j,ij(1)(r′ 1)(∑ l⟨j(1)l(2)∣ˆg12ˆΓ(2)∣i(1)l(2)⟩)i(1)∗(r1), (21) and each matrix element is further represented as ⟨j(1)l(2)∣ˆg12ˆΓ(2)∣i(1)l(2)⟩ =∑ k,m⟨j(1)l(2)∣ˆg12∣k(1)m(2)⟩⟨k(1)m(2)∣ˆΓ(2)∣i(1)l(2)⟩ =∑ k,mg(j(1)l(2);k(1)m(2))Γ(2)(k(1)m(2);i(1)l(2)), (22) where the resolution of identity ∑ k,m∣k(1)m(2)⟩⟨k(1)m(2)∣=1 (23) has been inserted. The suffix (2) on the molecular orbitals such asm(2)indicates them to be integrated over the second electronic coordinates r2. Thus, we have ˆH(1)=∑ j,i∣j(1)⟩ˆH(1) ji⟨i(1)∣ (24)with H(1) ji=∑ khjkρki+∑ k,mg(j(1)l(2);k(1)m(2)) ×Γ(2)(k(1)m(2);i(1)l(2)). (25) This matrix H(1) jigives a set of ENOs by diagonalization such that ∑ j,i∣j(1)⟩ˆH(1) ji⟨i(1)∣=∑ k∣ζk⟩ϵk⟨ζk∣. (26) The sum of the eigenvalues ϵk, which is the trace of ˆH(1), is TrˆH(1)=∫dr1∑ i,jj(1)(r′ 1)i(1)∗(r1)ˆH(1) ji =∑ i⎛ ⎝∑ khikρki+∑ l∑ k,mg(i(1)l(2);k(1)m(2)) ×Γ(2)(k(1)m(2);i(1)l(2))⎞ ⎠, (27) which is exactly the total electronic energy. In the above realization, which is formally rigorous, it should be a natural practice to use molecular orbitals or variants as {∣i(1)⟩}. Hence, the quality of resultant ENOs and their energies depends on the choice of basis function besides the accuracy of the total wave- function Ψitself. Conversely, for explicitly correlated wavefunctions, in which r12orf12factors are explicitly involved (see Ref. 27 for a recent review), we need another way to evaluate ˆH(1),ζk, andϵk. In what follows, r1is simply replaced with one-electron coordi- nater, asζk(r1)→ζk(r). D. Population of ENOs It, thus, turns out that ∣ζk⟩contributes to the total energy E through the direct sum of ϵk. Its contribution to the density can be estimated by ⟨ζk∣ˆρ∣ζk⟩, (28) which we refer to as the population of ζk.ζktakes a value in the nat- ural interval [0, 2]. If ζkhappens to have a very small population, its energyϵkshould be close to the zero value. That is, the ENO energy ϵkis population-dependent. This is one of the important character- istics of ENOs. We will show this aspect quantitatively in the later examples. One may define a quantity similar to the (normalized) orbital energy, ϵN k≡ϵk/⟨ζk∣ˆρ∣ζk⟩. (29) We have no idea yet if the normalized energy appears to be useful in practice. III. ADIABATIC BOND FORMATION IN ENO REPRESENTATION: HYDROGEN MOLECULE This section and Sec. IV will show a couple of examples of ENOs. We begin with the bond formation of a hydrogen molecule J. Chem. Phys. 154, 094103 (2021); doi: 10.1063/5.0034810 154, 094103-3 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . H2aug-cc-pVDZ full-CI adiabatic ground state potential energy curve V (red, left scale) and the total electronic energy Eealone (black, right scale). from two H atoms in the ground state. This simple example already gives how the standard picture of orbital interaction, which is frequently used in the molecular orbital studies on chemical reac- tions, should be modified. A. Computational level We carried out the full-CI calculation to calculate the ground- state wavefunction of H 2vs the internuclear distance with a rather small basis set, the so-called aug-cc-pVDZ,28which is composed of 18 functions in total including the πcomponents. Consequently, 18 ENOs are generated, which are numbered from the lowest ENO energy to the highest one. All quantum chemistry calculations in this paper are done with the GAMESS quantum chemistry package.29 The red curve in Fig. 1 shows a potential energy curve of the ground state of the H 2molecule. The black curve there indicates the pure total electronic energy with no nuclear repulsion energy being added. The total electronic energy alone is usually not shown in the literature, but it is this value that the sum of the ENO energies is supposed to reproduce. B. Spectral structure A global feature of the ENO spectrum is first shown. Table I lists the energy of all the ENOs at selected internuclear distances, which demonstrates the entire feature of the energy spectrum. It recon- firms that the total sum of ENO energies is exactly equal to the total electronic energy. Table I clearly reveals that among 18 ENOs gen- erated, only a pair of curves dominate, as illustrated in each panel of Fig. 2. One corresponds to the 1 σgorbital (having the lowest energy), while the other to 1 σu(the second lowest) as in the molecular orbital language. Figure 2(a) exhibits the energy ϵkand Fig. 2(b) shows the population ⟨ζk∣ˆρ∣ζk⟩only of these ENOs vs the internuclear distance. The black curve in Fig. 2(a) indicates the total electronic energy Ee, which is exactly equivalent to the sum of all ϵk[the same as the black curve in Fig. 1(a)]. All the other ENOs have nearly zero energy and nearly zero population and are omitted from Fig. 2. Therefore, “zero energy” has an absolute meaning as the origin of scale. It is seen in Table I that the two highest ENOs are of slightly positive energy with very small populations. This positivenessTABLE I . H2ground state ENO energy spectra (in hartree). Basis set aug-cc-pVDZ, full CI. ENO 0.75 (Å) 2.00 (Å) 5.00 (Å) 1 −1.841 072 −1.014 526 −0.556 218 2 −0.019 540a−0.267 520a−0.548 314a 3 −0.004 041b−0.000 852b−0.000 001b 4 −0.004 041b−0.000 852b−0.000 001b 5 −0.001 278 −0.000 257 0c 6 −0.000 150c−0.000 058 0c 7 −0.000 150c−0.000 029a0 8 −0.000 126 −0.000 029c0a 9 −0.000 098a−0.000 029c0a 10 −0.000 021b−0.000 004a0 11 −0.000 021b−0.000 001b0a 12 −0.000 001a−0.000 001b0 13 0c0c0c 14 0c0c0c 15 0 0a0b 16 0a0 0b 17 0.000 004a0a0.000 010a 18 0.000 163 0.000 222 0.000 011 Sum −1.870 379 −1.283 941 −1.104 515 Ee −1.870 379 −1.283 941 −1.104 515 aσu. bπu. cπg, unmarked: σg. appears only in the finite internuclear distance but not in the asymp- totic region. In our experience, the positiveness of a few ENOs in the ground state did not disappear (see also an example in LiF in Sec. IV D), and yet, basis set dependence is seen. Although this issue is not a serious one due to their smallness, we need more survey to comprehend the possible physical implication of this positive- ness. Meanwhile, it is obvious that ENOs of positive energy should have significance in excited states, particularly in those embedded in ionization (continuum) manifold. This aspect will be discussed elsewhere. The two curves in Fig. 1 are very similar in shape to those of occupation number of the corresponding natural orbitals. (Recall though that the natural orbitals do not give energy.) They come to unity in the dissociation limit for the appropriate linear combina- tion of them to reproduce the 1s states of hydrogen atoms. Theory and experience show that only small numbers of natural orbitals have a significant amount of occupation numbers (i.e., close to 2 or 1 for double or single occupancy, respectively) even in highly cor- related electronic wavefunctions. We recall that the natural orbitals having significant large occupation numbers are quite limited even in the very high excited states of the B 12cluster.15–17It has been proven that the number of electronic configurations (configuration state functions) in the CI representation can be minimized with use of natural orbitals.25,26It is anticipated that the ENOs should have similar properties. It is, thus, expected that there are generally two bands in the spectrum of ϵk: J. Chem. Phys. 154, 094103 (2021); doi: 10.1063/5.0034810 154, 094103-4 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . H2ground state ENO (a) energy and (b) population. Numbers 1 and 2 indicate the ENO number, counted from the bottom of the ENO spectrum. ENO numbers 3 through 18 all have very small energy and population and are not shown. The dotted line in (b) marks population 1.0. (1) Core energy band BC, each energy of which is physically meaningful in that ⟨ζk∣ˆρ∣ζk⟩is large. (2) Zero energy band BZ, each energy of which does not repre- sent the essential feature of the wavefunction in that ⟨ζk∣ˆρ∣ζk⟩ is very small. Note, hence, that the energy ϵkshould reflect the information of its norm-contribution to ˆH(1)through ˆρ, or⟨ζk∣ˆρ∣ζk⟩, even ifζkis normalized to unity in Eq. (13). The absolute values of ϵkmust be small for those ζkthat have small ⟨ζk∣ˆρ∣ζk⟩. C. ENOs and RHF MOs in the ground state It is interesting to see how ENOs are geometrically different from or similar to the RHF molecular orbitals. As a starting study, we dare to compare the spatial distribution of the two major ENOs and those of the Hartree–Fock orbitals of 1 σgand 1σu, which are exhib- ited in Fig. 3. Obviously, their functional forms are seen to be similar to each other due to the strong symmetry constraint, although ENOs are found to be of a little more atomic character (higher localiza- tion in atomic regions). This should be a natural reflection that the present CI wavefunction takes better account of electron correlation relative to the RHF wavefunction. The eigenvalues (orbital ener- gies) are of course totally different. We note that in the case where there exists no electron repulsion in Eq. (9), that is, if ˆG=0 and FIG. 3 . Spatial distributions of H 2RHF MOs (in red) and ENOs (in black) at R= 3.0 Å. (a) 1 σgorbitals. (b) 1 σuorbitals. ˆH(1)=1 2(ˆhˆρ+ˆρˆh), the RHF orbitals are also the eigenfunctions of ˆH(1), and they should have one-to-one equivalence with ENOs. In other words, the difference between the Hartree–Fock orbitals and ENOs emerges from the treatment of the electron repulsion oper- ator ˆG. In the case of the ground state of the hydrogen molecule, the deviation of the Hartree–Fock potential from the exact ˆGis rel- atively small, and moreover, the symmetry constraint should lead to their similarity. However, we will show in Sec. IV C treating LiF that RHF orbitals, natural orbitals, and ENOs are generally very different from one another in their functional forms. D. On the picture of orbital interaction The notion of orbital interaction gives the most basic, yet powerful and instructive, picture on how new molecular orbitals are formed from two interacting atomic and/or molecular orbitals. Suppose molecules A and B are forced to come close by varying internuclear distances “parametrically.” The Fukui view is qualita- tively based on a perturbation theory with respect to orbital interac- tions among those initially belonging to the individual molecules, that is, HOMO(A)+LUMO(B) and LUMO(A)+HOMO(B). In the case of hydrogen molecule formation in the ground state, which is the SOMO+SOMO reaction (SOMO; singly occupied molec- ular orbital), we have an orbital interaction to form a doubly occupied 1 σgMO, 1s ( ↑) + 1s ( ↓)→1σg(↑↓) + 1σu(vacant). Of course, this leads to one of the theoretical reasons why the MO picture generally fails in the dissociation limit in the ground state. In ENO picture, the energy profiles of both 1 σgand 1σu ENOs undergo smooth variation [Fig. 2(a)], and the behaviors of their populations [Fig. 2(b)] turn out to be similar to those of the occupation number of the natural orbitals. Hence, the ENO scheme serves as a natural generalization of the picture of orbital interaction. J. Chem. Phys. 154, 094103 (2021); doi: 10.1063/5.0034810 154, 094103-5 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp IV. NONADIABATIC ELECTRON TRANSFER IN ENO REPRESENTATION: LIF A. Possible nonadiabaticity manifesting in ENO The adiabatic theorem due to Born and Fock states is “A quan- tum system remains continuously in its “instantaneous eigenstate” if a perturbation applied on it is slow and small so that there keeps to exist a gap between the eigenvalue and the rest of Hamiltonian’s spectrum.”36This theorem is about the adiabatic change in the total wavefunction and, in turn, suggests a condition for nonadiabatic transition to occur.30–35 Analogously, let us consider a time-dependent problem with respect to the ˆH(1)(t)dynamics, i̵h∂ ∂t∣Ψ(t)⟩=ˆH(1)(t)∣Ψ(t)⟩. (30) We then have instantaneous eigenfunctions and eigenvalues at each time such that ˆH(1)(t)∣ζk(t)⟩=ϵk(t)∣ζk(t)⟩. (31) If the total wavefunction Ψundergoes adiabatic transition, it is highly expected that {ζk}do not have (avoided) crossing either in relevant eigenvalues {ϵk}or populations ⟨ζk∣ˆρ∣ζk⟩. Take the contraposition to the above argument. If we observe a significant (avoided) crossing among ENOs, transition of the parent total wavefunction should not be adiabatic. This notion may be adopted as a guiding principle to understand the behavior of ENOs in nonadiabatic transition to be studied below.B. Global energetic feature of S1 and S2 with avoided crossing of LiF To demonstrate how the ENO works in nonadiabatic chem- istry, we take the well-known avoided crossing between the low- est two adiabatic1Σ+states of the LiF molecule, which we call states S1 and S2. Highly accurate potential curves obtained by state- averaged CASSCF orbital determination followed by multireference configuration interaction, employing basis sets up to the complete basis set limit, have been reported in the literature.37,38Besides, Tóth et al. have very recently discussed photodissociation dynam- ics of LiF in terms of three state descriptions, which includes the 11Πstate in addition to two lowest1Σ+states.39For the demon- stration of ENOs in the study of the avoided crossing between S1 and S2 only, a much simpler description of 2-state averaged, eight electrons in seven orbital CASSCF followed by a single ref- erence singles and doubles configuration interaction (lowest three orbitals frozen) with the small cc-pVDZ basis set,40,41is adequate, except in the very short internuclear distance region. Here, again the GAMESS program has been utilized.29The basis set includes 3 sets of s functions, 2 sets of p functions, and a set of d func- tions on each atom, resulting in 12 σorbitals, 12 πorbitals, and 4 δ orbitals. We first survey the global feature of the electronic energies of the adiabatic ground state S1 and the first excited state S2, as shown in Fig. 4. As seen in Fig. 4(a) and its inset, the two potential energy curves come mutually very close, as though they seemingly crossed. Figure 4(b) shows the nuclear–nuclear repulsion energy Vnnand the pure total electronic energy Eefor the two states. Figures 4(c) and 4(d) magnify the energy scales to explicitly show that the components, the kinetic ( Te), electron–nuclei attraction (Vne), and electron–electron repulsion ( Vee) energies undergo clear FIG. 4 . LiF molecule1Σ+ground (S1) and first excited (S2) states. (a) Adiabatic potential energy curves; V1(V2) for S1 (S2). The inset of (a) shows the mag- nification of the avoided crossing vicin- ity. (b) Electron energy Ee=Te+Vne +Veefor S1 and S2 (scaled to left) and nuclear–nuclear repulsion energy Vnn(scaled to right). The two electron energies are hardly distinguishable in this scale. (c) Virial ratio −Ve/Teand electron kinetic energy Te. (d) Nucleus– electron attraction energy Vne(left scale) and electron–electron repulsion energy Vee(right scale). J. Chem. Phys. 154, 094103 (2021); doi: 10.1063/5.0034810 154, 094103-6 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp “crossing” between the S1 and S2 states. The origin of the crossing of the individual components will be discussed later. C. Comparison with other orbitals We show in Fig. 5 the RHF and CASSCF orbital energies for the ground state to stress the difference of the ENO spectrum (CASSCF orbitals are defined by diagonalizing the respective blocks of the block diagonal Fock matrix for the inactive, active, and virtual orbital subspaces separately42). The CASSCF orbitals are already different from RHF MOs; yet, neither set is seen to exhibit avoided cross- ing in the orbital energies. Since those MOs do not show any token of avoided crossing, the nonadiabatic feature appears only in the relevant electronic configurations in this particular case. We next see in Fig. 6 the difference of the spatial distribution of some relevant orbitals, from fifth to eighth of RHF, CASSCF, natural orbitals (NOs), and ENOs, respectively. NOs and ENOs have been extracted from the S1 CAS-CI wavefunction. In between two inter- nuclear distances chosen at 7.0 Å and 8.0 Å lies the avoided crossing. We here see very clear qualitative difference among the orbitals in contrast to the case of 1 σgand 1σuof the hydrogen molecule, as seen in Fig. 3. D. Electronic structure of electron transfer of LiF 1. ENOs near the avoided crossing Figure 7 displays the spectra of energy and population of the ENOs in the states S1 and S2 vs the internuclear distance. For both FIG. 5 . LiF orbital energies up to 8 σvs the internuclear distance R. (a) RHF MOs. MO 4σis a HOMO. (b) CASSCF orbitals. σorbitals are shown with solid curves; π orbitals are shown with dotted curves. Three lowest orbital energies (1 σ–3σ) have been shifted for plotting within the figure. states, we have plotted the relevant quantities for the lowest six σENOs (1σthrough 6σ) and the 12th (12 σ), which is the high- estσorbital having slightly positive energy. All the other ENOs (except 1π) have energy and population close to zero and are thereby omitted from Fig. 7. ENOs 3 σthrough 6σare the fifth through FIG. 6 . Orbitals five through eight of RHF MOs, CASSCF, state 1 NOs, and state 1 ENOs at (a) R= 7.0 and (b) 8.0 Å. Orange and blue show the surfaces at ±0.025 of the orbital functions. Seen in thexz-plane with LiF along the zaxis with Li at the top and F at the bot- tom of the black lines, which indicate internuclear distances. J. Chem. Phys. 154, 094103 (2021); doi: 10.1063/5.0034810 154, 094103-7 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 7 . LiF (a) state 1 (S1) ENO energy, (b) state 2 (S2) ENO energy, (c) S1 ENO population, and (d) S2 ENO pop- ulation. Only σorbitals are shown. ENO 1σhas low energy and is shown shifted by +55 Ehto include in the figure. The energy of ENOs 7 σto 11σare very close to zero and are not shown. 6 σenergy is slightly negative, while 12 σis slightly positive. Horizontal black dotted lines in (c) and (d) mark population 1.0. ENOs 1 σ and 2σhave population 2.0 and are not shown in (c) and (d). eighth ENOs seen in Fig. 6. The 1 πENOs are located below 3 σin energy. The population spectra shown in Fig. 7 demonstrate that the present nonadiabatic electron transfer is overwhelmingly dominated by two ENOs, the 4 σand 5σ. The population transfer between them reflects the electron transfer within the molecule. The popula- tion dynamics of ENO 12 σappears to make a secondary significant contribution, but only in a limited region. The main feature in the energy spectra of ENOs is that the 4 σ avoids crossing with the 3 σ, and the 5 σdoes with the 6 σas well. (The avoided crossing between 5 σand 6σis not well distinguish- able in the scale of Fig. 7. Yet, the functional variations of them seen in Fig. 8 demonstrate it clearly.) Interestingly, the internuclear dis- tance of the avoided crossing between 3 σand 4σis different from that between 5 σand 6σ. For instance, Fig. 7 (and Fig. 10 as well) demonstrates clearly that the 5 σ/6σavoided crossing on S1 occurs at a place of a little longer Rthan that of 3 σ/4σ. Likewise, the 5 σ/6σ avoided crossing on S2 is located at a place of shorter R. We, thus, see that there are two avoided crossings among ENOs for each adi- abatic total electronic state. This is the first of such an observation to the best of our knowledge. Recall that MOs of RHF and CASSCF happen not to have avoided crossing in markedly contrast to ENOs. We may, therefore, say that ENOs know that the parent total wave- function (here either S1 or S2) undergoes an avoided crossing even if we track only ENOs of one of the adiabatic wavefunctions, S1 or S2. This is one of the consequences suggested by the breakdown of adiabaticity, as discussed in Subsection IV A. A huge energy change is noted for some ENOs in Figs. 7(a) and 7(b); for instance, ENO 4 σin S1 [Fig. 7(a)] drops (becomes stabler) more than 2 hartree by the transition from the distance R= 8 Å to 7 Å, and the steep rise of ENO 4 σis seen in Fig. 7(b) for the S2 state. Such a huge variation of the orbital energy is never seen in the stan- dard molecular orbital theories. This kind of phenomenon happens because the ENOs are population-dependent. For the former case, ENO 4σhas become dramatically stabler because the population FIG. 8 . Spatial distributions of LiF ENOs 3 σ, 4σ, 5σ, and 6σfor states S1 and S2 from R= 6.8 to 8.6 Å in steps of 0.2 Å. Orange (positive) and blue (negative) show surfaces at ±0.025 of the ENO function ζk. Seen in the xz-plane with LiF along the zaxis with Li toward the top and F toward the bottom. The black line indicates the internuclear distance between Li and F. J. Chem. Phys. 154, 094103 (2021); doi: 10.1063/5.0034810 154, 094103-8 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II . Character of primary ENOs. S1 S2 R=6.0aR= 9.0 R= 6.0 R= 9.0 3σ F 2p(2) . . . Li 1s(2) Li 1s(2) . . . F 2p(2) 4σ Li 1s (2) . . . F 2p(1) F 2p(1) . . . Li 1s(2) 5σ F 3d(0) . . . Li 2s(1) Li 2s(1) . . . F 3d(0) 6σ ∼Li 2s(0) . . . F 3d(0) F 3d(0) . . . Li 2s(0) aInternuclear distance in Å. of it increases by as much as almost unity, while ENO 4 σin S2 becomes unstable for the transfer of population to ENO 5 σ. 2. Critical ENOs engaging in the avoided crossing To survey what happens in the relevant ENOs across the avoided crossing, we show in Fig. 8 the spatial distribution of ENOs 3σ, 4σ, 5σ, and 6σof both S1 and S2 states, at selected internu- clear distances. At the top (bottom) of each line segment, the length of which is scaled to the internuclear distance, lies Li (F) atomic nucleus. These orbitals are plotted at the surface value of ±0.025, a little exaggerating the localization of electrons on the atoms. Note that no effect of the variation of ENO population is reflected in these diagrams. To facilitate capturing the characteristics of ENOs of 3 σto 6σ, which play main roles in electron transfer, we summarize them in Table II, in which “F 2p(2)” at the row of the 3 σorbital and the col- umn R= 6.0 Å in S1, for example, reads that this ENO is mainly localized to the 2p orbital (actually 2p σorbital) on the F atom with the population about 2. Again, the avoided crossing exists at about R = 7.5 Å. The character change and the jump of population are clearly captured in this table. The symmetric structure between S1 and S2 is also apparent. Figure 9 shows the variation of ENO 12 σfor a later reference. 3. Further details about the avoided crossing In Figs. 7 and 8, and in Table II, we note a considerably sym- metric structure, which should reflect the electronic structure behind this nonadiabatic electron transfer. To expose it more clearly, we show Fig. 10, in which the energy levels of ENOs from 2 σto 6σas FIG. 9 . Spatial distributions of LiF ENO 12 σfor states 1 and 2 from R= 6.8 to 8.6 Å in steps of 0.2 Å. Orange (positive) and blue (negative) show surfaces at ±0.025 of the ENO function ζk. Seen in the xz-plane with LiF along the zaxis with Li at the top and F at the bottom of the black lines, which indicate the internuclear distance. FIG. 10 . LiF ENO energies across the avoided crossing region for states S1 (color) and S2 (black). (a) ENOs 5 σand 6σ, (b) 3σand 4σ, and (c) 2σand 1π. Note that ENOs in (c) do not undergo avoided crossing in either state. Yet, the counterparts seem to cross each other between S1 and S2. well as 1πare shown for the states S1 and S2 together as a function ofR. A beautiful symmetry between the S1 and S2 states is observed there despite the fact that the molecular field is not symmetric with respect to the ENO 3 σ/4σavoided crossing position. As stated above, the ENO 5σ/6σavoided crossing positions for S1 and for S2 are sepa- rated a little from the central avoided crossing point between 3 σand 4σ. We will discuss this aspect later. To see how the total electronic energy is composed among the kinetic energy Te, electron–nuclei attraction energy Vne, and electron–electron repulsion energy Vee, we further study the energy variation of the representative ENOs. Figure 11 displays these com- ponents of some of the ENO energies. Figures 11(a), 11(c), and 11(e) are for S1, while Figs. 11(b), 11(d), and 11(f) are for S2. Figures 11(a) and 11(b) are for the kinetic energy component, and likewise, Figs. 11(c) and 11(d) are for electron–nuclei attraction, and Figs. 11(e) and 11(f) are for electron–electron repulsion energy. Two aspects are immediately noted in Fig. 11: (1) The energy component variations are dominated almost exclusively by ENOs 3 σand 4σboth for S1 and S2. Note, however, that the sum of Te,Vne, and Veeof 3σ does not vary much before and after the avoided crossing (see Fig. 7). (2) The curves look again very symmetric between S1 and S2 (look J. Chem. Phys. 154, 094103 (2021); doi: 10.1063/5.0034810 154, 094103-9 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 11 . Components of 3 σto 6σENO energies for states S1 [(a), (c), and (e)] and S2 [(b), (d), and (f)]. [(a) and (b)] Electron kinetic energy Te, [(c) and (d)] nucleus– electron attraction energy Vne, and [(e) and (f)] electron–electron repulsion energy Vee. 3σin red dashed, 4 σin red solid, 5 σin solid blue, and 6 σin dashed blue curves. back at Fig. 10). This is another manifestation of the symmetry seen in Figs. 7 and 10. Table II identifies the global characteristics of ENOs 3 σand 4σ as follows: In S1, at the right side ( R= 9 Å), the ENO 4 σis essentially a 2p orbital on the F atom with the population about one electron. We abbreviate this characteristic as F 2p(1) according to Table II. Likewise, ENO 3 σis a 1s orbital on the Li atom with the population 2, which is Li 1s(2). On the other hand, at the left side ( R= 6 Å), the ENO 4σis Li 1s(2) and 3 σis F 2p(2). Thus, the ENO 3 σat the right side directly corresponds to ENO 4 σat the left side. Therefore, each of the energy components for these two are very similar. However, the ENO 4σat the right side [F 2p(1)] and ENO 3 σat the left side [F 2p(2)] are largely different in energy. This is simply because the number of electrons varies between them. As we have seen in Fig. 7, the population of ENO 5 σchanges as much as about one electron. However, the absolute values of the energy of ENO 5 σis so small that the total energy is not affected as much as by the transition from F 2p(1) to F 2p(2). A similar argument can be made for S2. A difference lies in the assignment of the ENOs in that ENO 3 σat the right side is F 2p(2), while at the left side, it is F 2p(1). Thus, the crossing behavior of the energy components observed in Fig. 4 turns out to be compre- hended mainly in terms of the ENOs 3 σand 4σ. It turns out that the 6σin the present (adiabatic) representation, which is, so to say, the lowest “vacant” ENO, does not play a significant role in the present nonadiabatic transition.Incidentally, we clearly observe in Fig. 10 that the πlevels are affected by this nonadiabatic transition taking place in the σ-direction. E. Diabatization of ENO Thus far, we have shown the energy diagram and orbital fea- tures of the ENOs as the first outcome of the ENO study of nonadi- abatic electron transfer. Although the results are robust and already appealing, they appear to be a little complicated when we track the qualitative transition of the electronic states. This is because the energy, spatial distribution, and population of the relevant ENOs can change to a large extent across the avoided crossing. These three varying factors make it somewhat uneasy to comprehend what actually occurs. In order to transform the situation to be more intuitively trans- parent, we introduce the notion of diabatization of the ENO. This is not the usual diabatization of the states between S1 and S2 but that among ENOs given in each state. We here keep the total wavefunc- tions adiabatic as they are. The method of the present approximate ENO diabatization is given in Appendix A. The energy diagram of, thus, diabatized ENOs, indicated with d as d1σ, d2σ, and so on, vs the internuclear distance, is shown in Fig. 12, which is to be compared with Fig. 7. Here, again only the relevant ENOs are mentioned. Note also that the ordering of the diabatized ENOs here is to be made so that it coincides with the adiabatic energy ordering of the ENO in the asymptotic region ( R =∞). It is seen that the adiabatic ENOs 3 σand 4σresult in crossing diabatic ENOs d3 σand d4σ. One of the pair (d3 σfor S1 and d4 σ for S2) now remains almost constant both in energy and population, and the other (d4 σfor S1 and d3 σfor S2) is solely responsible for the dynamics. Likewise, ENOs d5 σand d6σcross, where one of the pair is now constantly zero both in energy and population (d6 σfor S1 and d5σfor S2). The vanishing orbital is excluded from Fig. 12. FIG. 12 . LiF (a) S1 diabatic ENO energy, (b) S2 diabatic ENO energy, (c) S1 dia- batic ENO population, and (d) S2 diabatic ENO population. Only σorbitals are shown. ENO d1 σhas low energy and is shown shifted by +55 hartree to include them in the figure. Horizontal dotted lines in (c) and (d) mark population 1.0. ENOs d1σand d2σhave population 2.0 and are not shown in (c) and (d). See the text for the numbering of diabatized ENOs. J. Chem. Phys. 154, 094103 (2021); doi: 10.1063/5.0034810 154, 094103-10 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The nonadiabatic change in each of the wavefunctions, S1 or S2, is understood as a population transfer between a single pair of diabatic ENOs, (d4σ↔d5σ) in S1 and (d3 σ↔d6σ) in S2. Yet, the tentative but minor roles of d12 σremain in each state. The resultant ENO recombination by the diabatization is dis- played in Fig. 13, which is to be compared with Fig. 8. A sudden change in the ENO is not seen here but continuous changes in ENOs d5σfor S1 and d6 σfor S2 are observed, which do not contribute significantly in energy or population to the total wavefunction. The ENOs responsible for electron transfer within each wavefunction, in particular, those from the adiabatic ENOs 3 σand 4σ, that is, d3 σ and d4σfor both S1 and S2, are all smooth, and their functional patterns are essentially invariant with respect to the internuclear dis- tance. Note that S1 d3 σcorresponds to S2 d4 σ, while S1 d4 σdoes to S2 d3σ. FIG. 13 . Spatial distributions of LiF diabatic ENOs d3 σ, d4σ, d5σ, and d6σfor states S1 and S2 from R= 6.8 to 8.6 Å. Orange (positive) and blue (negative) show surfaces at ±0.025 of the ENO function ζk. Seen in the xz-plane with LiF along the zaxis with Li at the top and F at the bottom of the black lines, which indicate the internuclear distance R. Note that ±ambiguity remains in the phase of the ENO at any R.F. Summary of the mechanism of electron transfer It turns out by diabatization that the mechanism of electron transfer in S1 is predominantly comprehended in terms of d4 σand d5σ. The reaction begins by collision between Li and F atoms in the ground state each. Figure 12 indicates that in the asymptotic region, the populations of S1 d4 σand S1 d5σare both unity. In Fig. 13, ENO S1 d4σis confirmed to be mostly the 2p σfunction mainly localized on the F atom, while S1 d5 σseems to be the 2s function on Li there. This situation lasts up to the region R= 8.0 Å, and Fig. 12(c) tells that one electron is transferred from d5 σto d4σwithin the state, which simply implies that one electron jumps from Li to F at around R= 7.6 Å. Transition in S2 is a little complicated. The reaction begins with d3σfilled with two electrons, which represents an F−state. Therefore, Li actually starts the collision in a cation state Li+. Then, Fig. 12(d) indicates that a small amount of the electrons shift from d3σto d12σ, starting from about R= 7.6 Å, but soon, the contribu- tion from d12 σdisappears and major electron transfer takes place from d3σto d6σas much as about one electron. Meanwhile, Fig. 13 suggests that ENO d6 σofRshorter than 7.8 Å is a covalent bond between Li 2s and F 2p σ. It is known that this covalent bond is not strong enough to compensate the internuclear repulsion. The d12 σ temporally contributing is a covalent bond between Li 2s and F 3p σ, the functional form of which is exactly the same as the ENO “S2 12 σ” of Fig. 9 (see S2 12 σatR= 7.6 Å). Therefore, ENOs S2 d12 σand S2 6σare smoothly connected in the same qualitative feature in their contributing region. The above analyzed mechanism of electron transfer in each adi- abatic state is consistent with what we schematically and intuitively understand as Li + F →Li+F−in S1 and Li++ F−→LiF (nonbonding) in S2. We stress though that such clear views, symmetry, and more have been extracted in an invariant manner by only a small number of ENOs reduced from configuration interaction wavefunctions. V. CONCLUDING REMARKS The notion of energy natural orbitals has been proposed as eigenfunctions of the energy density operator ˆH(1), which is already a function of ˆρ(first order density operator), ˆh(one-electron oper- ators), ˆG(reduced electron–electron repulsion operator), and the total wavefunction Ψ. By definition of the density matrix, both ζk (ENO) and ϵk(energy of ENO) are invariant with respect to the choice of representation of Ψ, exactly in the same sense that the natu- ral orbitals and their occupation numbers are. ENOs are all different from the Hartree–Fock orbitals and their relatives such as CASSCF orbitals. In this regard, the ENO may be regarded as a meta-orbital. The other characteristics of the ENO are summarized as follows: (1) ENO is a one-electron function, namely, an orbital, and each is asso- ciated with an eigenenergy, the sum of all of which is equal to the total electronic energy of the state under study. Hence, electronic force working on nuclei is also decomposed to the individual ENO components (see Appendix B). (2) Although there appear ENOs as many as the number of basis functions, only a small number of them are predominantly responsible for the change of molecular states in reactions, and most of the ENOs behave as spectators. (3) Thus, we may develop a common language to explain the electronic-state mechanisms of the chemical reaction in terms of the small number J. Chem. Phys. 154, 094103 (2021); doi: 10.1063/5.0034810 154, 094103-11 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp of specific ENOs. (4) The populations of those dominating ENOs can vary significantly as the reaction proceeds. Other properties of ENOs, which are conceived to be useful in applications of the analysis of chemical reactions and electron dynamics, are summarized in Appendix B. As the first example of the ENO in the adiabatic reaction, we have shown hydrogen molecule formation from two H atoms. It turns out that the smooth change of ENO populations both in energy and population well modifies the molecular orbital interaction pic- ture for the formation of bonding and anti-bonding orbitals. The second example has been a nonadiabatic electron transfer in LiF. It is rather exciting that much novel insight into the electronic state change associated in such a simple nonadiabatic transition has been attained. Here, again the population transfer between a small num- ber (mostly a pair) of diabatized ENOs turns out to be responsible for the intramolecular electron transfer. ENO is, thus, likely to be a promising means to analyze, on one hand, a very complicated class of nonadiabatic electron wavepacket dynamics10–14in densely quasi-degenerate excited-state manifolds such as those in boron clusters.15–17On the other hand, it is worth- while trying to comprehend the mechanisms of not very celebrated reactions such as the Woodward–Hoffmann symmetry “forbidden” reactions from the view point of nonadiabatic electron dynamics. Our next paper will discuss a relationship between a class of sym- metry forbidden reactions, the concept of transition state, and the pseudo-Jahn–Teller effect from a unified point of view with ENOs. ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI, Grant Nos. JP15H05752 and JP20H00373. APPENDIX A: METHOD OF DIABATIZATION ENOs can be numbered in order of increasing energy (adi- abatic numbering), separately for each irreducible representation. ENOs are determined as eigenfunctions for a single point calcula- tion, and their phases are arbitrary. For approximate diabatization across the internuclear distance, we compute the overlap between ENOs at neighboring points R0andR1in the internuclear distance, Skl(R0,R1)=∫ζk(r;R0)ζl(r;R1)dr, (A1) and set the phases and numbering at R1relative to R0so that the matrix S(R0,R1) is as close to the unit matrix as possible. The deter- mined diabatic numbering of the ENOs are chosen to match the adiabatic numbering at the asymptotic region for facile identifica- tion between the two representations. Thus, only the numbering of ENOs is “diabatized” in the present case. Points where the adiabatic states are significant mixtures of diabatic states (7.34 Å–7.36 Å in S1 and 7.60 Å in S2) are omitted and replaced with interpolated curves in Fig. 12. APPENDIX B: USEFUL PROPERTIES OF ENOs 1. Molecular force and energy flow in molecules Molecular force working on nuclei in the direction of Ra emerges from the individual ENOs as−∑ k∂ϵk ∂Ra, (B1) along with the one from nuclear–nuclear repulsion. The force emerging from the electron distribution ∣ζk(r,t)∣2on the nuclei is ∑ kd dRa(ϵkζ∗ k(r)ζk(r))=∑ k[∣ζk(r)∣2dϵk dRa+ϵkd∣ζk(r)∣2 dRa], (B2) which is immediately reduced to Eq. (B1) when integrated over r. The effect of the electronic fluctuation on the molecular force can be scrutinized with Eq. (B2). In electron wavepacket dynamics13and in the studies using ab initio molecular dynamics, the electronic wavefunctions are often propagated along a path R(t), either classical or nonclassical. Time dependence of ENOs in such cases is reduced to d dtζk(r;R(t))=∑ a˙Ra∂ ∂Raζk(r;R(t)). (B3) Likewise, energy flow within and/or in between molecules, which are placed in laser fields, for example, can be calculated using time-dependent ENOs. The time variation of the total energy is simply dE dt=∑ kdϵk dt=∑ kd dt⟨ζk∣ˆH(1)∣ζk⟩=∑ k⟨ζk∣dˆH(1) dt∣ζk⟩, (B4) whereζkis assumed to be normalized. The spatiotemporal redistri- bution of the energy is given as ∑ kd dt(ϵkζ∗ k(r,t)ζk(r,t))=∑ k[∣ζk(r,t)∣2dϵk dt+ϵkd∣ζk(r,t)∣2 dt]. (B5) All the above quantities and properties will be applied in our analyses of chemical reactions and will be reported in the future. 2. Use of ENO as sampling data The ENO scheme is not directly intended to reproduce accurate wavefunctions but to extract an orbital picture from highly corre- lated wavefunctions. Nevertheless, ENOs can be used rather flexibly as follows: a. Time propagation of electron wavepackets Extending the idea of iterative natural orbital method of Ben- der and Davidson,43Matsuoka and Takatsuka have developed an electron wavepacket propagation method, in which a set of natural orbitals {λi(t)}obtained from a total wavefunction Ψ(t) at time t are substituted into the time evolution equations of motion to attain the next-step wavefunction Ψ(t+Δt) in their studies of molecular ionization.44In this way, one may skip the process of determining Hartree–Fock molecular orbitals to construct the set of configura- tion state functions. It is obvious that a similar algorithm can be well or better adopted by using ENOs. b. Simulation of the potential energy surfaces In order to sample ENOs at many points, we may borrow the idea of the so-called machine learning methodology.45,46Here, the J. Chem. Phys. 154, 094103 (2021); doi: 10.1063/5.0034810 154, 094103-12 © Author(s) 2021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp objects for letting our coherent neural network47learn are the matrix of the one-body Hamiltonian H(1) jiof Eq. (25). At selected initial sampling points of nuclear configuration R, we first calculate the overlap matrix Sfor a given atomic basis. This can be performed almost instantaneously with the well-known analytic expressions of the Gaussian integrals, provided that the Cartesian Gaussian func- tions are adopted. The input set to the neural networks is then {Sab}, aiming at the reproduction of the matrix {H(1) ji}as outputs. One may further want to prepare three sets of the networks, each being for the kinetic energy, electron–nuclei attractive energy, and electron–electron repulsive energy, which are to be summed up to H(1) ji. Once this matrix is simulated at a desired R, ENOs are read- ily estimated by diagonalization, and the resultant eigenvalues are to be summed to the total electronic energy, which in turn gives, together with the nuclear–nuclear repulsion energy, rise to a value of the potential energy surface. The present scheme makes the infor- mation of molecular electronic structures available as an additional outcome, which should serve for chemical analysis. Details will be discussed elsewhere. 3. Sound of molecular electronic states In Eq. (12), we have ˆH(1)(r)=⟨r∣ˆH(1)∣r⟩=∑ kϵk∣ζk(r)∣2, (B6) which immediately indicates that ∣ζk(r)∣2is the spatial distribution of a density responsible for the energy component ϵkin a molecule, where rdenotes the one-electron coordinate. In this sense, ENOs are directly relevant to energetics and dynamics. Let us next consider a frequency arising from each of ENOs, ωk=∣ϵk∣/̵h, (B7) which may be regarded as a frequency of “sound” instead of “light,” and∣ζk(r)∣2is a spatial distribution of the tone from a vibrat- ing media such as a three dimensional musical instrument. Then, ˆH(1)(r)synthesizes a sound in terms of coherent sound compo- nents. Therefore, we can “hear” from which places of a molecule the tone of ωkemerges. As for molecular vibrational motion, the normal modes provide a similar notion. Yet, the ENO must be the first that gives the resolution of electronic sound from a molecule. Furthermore, one can synthesize sounds by coherently summing two total wavefunctions (or wavepackets) such as Ψ1+Ψ2, that is, ∣Ψ1+Ψ2⟩⟨Ψ1+Ψ2∣. Rescaling is of course needed to enable us to physically hear the present “sound” (1 eV corresponds to about 2.41 ×1014Hz). 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Biomicrofluidics 15, 014106 (2021); https://doi.org/10.1063/5.0039087 15, 014106 © 2021 Author(s).Dual-fiber microfluidic chip for multimodal manipulation of single cells Cite as: Biomicrofluidics 15, 014106 (2021); https://doi.org/10.1063/5.0039087 Submitted: 30 November 2020 . Accepted: 05 January 2021 . Published Online: 28 January 2021 Liang Huang , Yongxiang Feng , Fei Liang , Peng Zhao , and Wenhui Wang COLLECTIONS This paper was selected as Featured This paper was selected as Scilight ARTICLES YOU MAY BE INTERESTED IN Misalignment of Optical Fibers Allows Manipulation of Single Cells Scilight 2021 , 051102 (2021); https://doi.org/10.1063/10.0003486 Challenges and opportunities for small volumes delivery into the skin Biomicrofluidics 15, 011301 (2021); https://doi.org/10.1063/5.0030163 A dynamically deformable microfilter for selective separation of specific substances in microfluidics Biomicrofluidics 14, 064113 (2020); https://doi.org/10.1063/5.0025927Dual-fiber microfluidic chip for multimodal manipulation of single cells Cite as: Biomicrofluidics 15, 014106 (2021); doi: 10.1063/5.0039087 View Online Export Citation CrossMar k Submitted: 30 November 2020 · Accepted: 5 January 2021 · Published Online: 28 January 2021 Liang Huang,1,2 Yongxiang Feng,1Fei Liang,1Peng Zhao,1and Wenhui Wang1,a) AFFILIATIONS 1Department of Precision Instrument, State Key Laboratory of Precision Measurement Technology and Instrument, Tsinghua University, Beijing 100084, China 2School of Instrument Science and Opto-Electronics Engineering, Hefei University of Technology, Hefei 230009, China a)Author to whom correspondence should be addressed: wwh@tsinghua.edu.cn ABSTRACT On-chip single-cell manipulation is imperative in cell biology and it is desirable for a microfluidic chip to have multimodal manipulation capability. Here, we embedded two counter-propagating optical fibers into the microfluidic chip and configured their relative position inspace to produce different misalignments. By doing so, we demonstrated multimodal manipulation of single cells, including capture, stretching, translation, orbital revolution, and spin rotation. The rotational manipulation can be in-plane or out-of-plane, providing flexibility and capability to observe the cells from different angles. Based on out-of-plane rotation, we performed a 3D reconstruction of cellmorphology and extracted its five geometric parameters as biophysical features. We envision that this type of microfluidic chip configuredwith dual optical fibers can be helpful in manipulating cells as the upstream process of single-cell analysis. Published under license by AIP Publishing. https://doi.org/10.1063/5.0039087 I. INTRODUCTION Cell research plays an important role in disease diagnosis and investigating cancer metastases. 1,2Single-cell manipulation3,4is imperative in biotechnology to facilitate many potential applications,such as cell injection, 5imaging,6,7positioning,8and electrical proper- ties characterization.9Basic manipulation includes translation and rotation of cells, among many others. Rotational manipulation10–13 has been more challenging than translation. Over the years, different approaches have been practiced for rotational manipulation at thesingle-cell level, 14–16mainly including mechanical, hydrodynamic, optical tweezers, magnetic, acoustic, and electrical means. The mechanical methods17,18use micromanipulation devices to precisely control a probe (e.g., glass pipette) to realize the 3D rotation ofsingle cells. However, they require direct contact with cells and thuseasily damage them. The hydrodynamic methods realize the micro- recirculation of solution through the special microstructure, thus making the cells rotate, but this method relies on complex micro-structures. 19,20The magnetic methods21,22use external magnetic fields to perform 3D cell rotation and require a cell sample pretreat-ment. Acoustic methods 23–25use vibration principle to realize the 3D rotation of single cells, and the vibration modes depend on the microchannel structure.26Electrical methods27–29are basedon the principle that cells can be polarized and moved in the alternating electric field to realize 3D rotation. Electro-thermal and electro-osmotic effects may potentially damage cells. Opticalmethods commonly include optical tweezers 30,31and optical fibers32,33to achieve single-cell rotation. Optical tweezers use a low- power laser beam to exert trapping force and torque onto cells andcause them to rotate. Utilizing multiple optical tweezers can achieve out-of-plane rotation of single cells. 34Compared with traditional optical tweezers, the optical fiber method can realize the rotation ofsingle cells simply by two counter-propagating and misaligned optical fibers. Recently, we used two fibers and four 3D electrodes to achieve single-cell multi-parameter measurements, 35but there are still some limitations in the application of cell study, for example, only in-plane rotation can be achieved. In this paper, we present microfluidic chips configured with two counter-propagating fibers that can achieve multimodal manipulation of single cells. When the two fibers are aligned, single cells can be trapped and stretched by laser-induced forces. Ifthere is a certain offset between the two optical fibers, multimodal manipulation of single cells can be realized, including spin rotation and/or orbital revolution, in-plane or out-of-plane. We successfullyreconstructed the 3D morphology of single cells from a stack ofBiomicrofluidics ARTICLE scitation.org/journal/bmf Biomicrofluidics 15,014106 (2021); doi: 10.1063/5.0039087 15,014106-1 Published under license by AIP Publishing.images from out-of-plane spin rotation and calculated their physical parameters from the 3D image. This 3D imaging techni- que provides a possible alternative to confocal microscopy with noneed for labeling, as a new tool for cell characterization. Thedual-fiber microfluidic chips can efficiently realize single-cellmultimodal manipulation and provide a convenient platform for single-cell research. II. MATERIALS AND METHODS A. Working principle The working principle of multimodal manipulation of a cell is illustrated in Fig. 1 , based on light-induced forces exerted on the cell. Generally, the change of light momentum caused by light irra- diating the cell will induce axial and gradient forces in its axial andnormal direction. 36,37The axial force is generated by the photon hitting the cell along the propagation direction of the light, whichis generally called scattering force; the gradient force is caused by the uneven intensity of the light field, and its direction is perpen- dicular to the direction of light propagation. Visible light is not suitable for manipulating cells because of two reasons. First, the force generated by visible light is one ormore orders of magnitude smaller than the cell weight and fluidic force. Second, heating of the solution via light absorption may cause thermal damage to cells. Thus, near infrared single-modelaser is often used as the light source. The gradient force and axialforce exerted on cells are related to cell volume and the laser wave-length. When the cell radius is much larger than the laser wave- length, which is normally the case, the forces can be computed through a ray-optics approximation. 38,39 The optical field of a non-focused Gaussian beam can be described by w(z)¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þλz nπw2 0/C18/C192s , (1) where wis the beam waist, zis the distance from the fiber end, λis the laser wavelength in vacuum, nis the index of refraction of the solution, and w0is the beam waist as the beam exits the fiber. The optical forces acting on the cell are obtained by applying the relations for the scattering and gradient components exerted by FIG. 1. Working principle of double optical fibers embedded in microfluidic chips for multimodal manipulation of single cells. (a) Optical trap. (b) Optica l stretch. (c) Optical translation. (d) Orbital revolution. (e) In-plane spin rotation. (f ) Out-of-plane spin rotation. Note that P1orP2is the laser power per fiber and dis the cell diameter.Biomicrofluidics ARTICLE scitation.org/journal/bmf Biomicrofluidics 15,014106 (2021); doi: 10.1063/5.0039087 15,014106-2 Published under license by AIP Publishing.each ray38,39via Fs¼nP c/C2[1þRcos 2 θ]/C0T2[cos(2 θ/C02γ)]þRcos 2 θ 1þR2þ2Rcos 2 γ/C26/C27 , (2) FG¼nP c/C2[1þRsin 2θ]/C0T2[sin(2 θ/C02γ)]þRsin 2θ 1þR2þ2Rcos 2 γ/C26/C27 , (3) where Pis the power carried by each ray, cis the light velocity, Rand Tare the Fresnel reflection and transmission coefficients, respectively, at the cell surface at a given incidence angle θ, and γis the refraction angle. Denote the misaligned offset of the two identical counter- propagating optical fibers by L, and the cell motion patterns are categorized by the value of L. When L= 0, the total force on the cell was zero, and the cell was stably trapped. This results in anoptical trap [ Fig. 1(a) ] that drags the cell into the center point. When the trapping force is increased by increasing the laser power on the cell surface such that it is greater than the cellular mechani-cal strength, the cell can be stretched and deformed in situ 40,41 [Fig. 1(b) ].When L≠0, which may be easily the case in reality, cells can exhibit several patterns. If L>d(the cell diameter), cells can be only subject to the scattering force of one optical fiber and be trans-lated toward the other optical fiber [ Fig. 1(c) ]. If L<d, the scatter- ing forces from each laser beam are not collinear and can generatea torque on the trapped cell, causing it to rotate. 42,43Specifically, when d/2 <L<d, the cell does orbital revolution, as shown in Fig. 1(d) ; when 0 < L<d/2, the cell does spin rotation, as shown in Figs. 1(e) and 1(f). Depending on which plane the misalignment occurs, the rotation can be in in-plane rotation [ Fig. 1(e) ]o r out-of-plane rotation [ Fig. 1(f) ]. B. Chip design and fabrication Figure 2(a) shows the sketch of the microfluidic chip. In order to facilitate the integration of the optical fibers into themicrochannel, a 125 μm-thick microchannel was fabricated by pho- tolithography. Constrained by the microchannel geometry, twocounter-propagating optical fibers of 125 μm in diameter were inserted after bonding of the glass substrate and polydimethylsilox- ane (PDMS) microchannel. The optical fibers were carried by a40-nm-resolution 3D micromanipulator (MP285, Sutter, USA) toadjust the spatial position. The y-axis misalignment was confirmed FIG. 2. The microfluidic chip with two counter-propagating optical fibers and the manipulation experimental setup. (a) The sketch of the microfluidic chip with two optical fibers orthogonal to the main channel. (b) The fabrication process of the microfluidic chip. (c) The real picture of the microfluidic chip. (d) Illust ration of the experimental setup.Biomicrofluidics ARTICLE scitation.org/journal/bmf Biomicrofluidics 15,014106 (2021); doi: 10.1063/5.0039087 15,014106-3 Published under license by AIP Publishing.by direct microscopic observation, and the relationship between the optical coupling power and misalignment was recorded and used to adjust the z-axis misalignment when needed. To prevent possiblefluidic leakage through the gap between the round optical fiber andthe square channel, we sealed the outside of the channel with glue.The position of the optical fibers is fixed and non-adjustable after the glue is cured, multiple chips with different optical fiber mis- aligned offsets were made to demonstrate the multimodal manipu-lations of cells. Note here that the optical fibers are intentionallyplaced in the downstream of the main microchannel, which runsthe sample medium and two sheath flows. The sheath flows are used to focus the cells to the centerline of the main channel to ensure that cells enter the effective zone of the optical fibers. In principle, we can obtain all the manipulation modes (Fig. 1 ) of the cell if we can adjust Lin the fly. This can be achieved by adding some on-chip design, which can change the position of one or both of the optical fibers as the proper actuation in associa- tion with external stimulation. For example, adding a pneumaticchannel alongside the fiber-embedded channel may change they-axis misalignment. This is not the focus here, but the concept ofmultimodal manipulations can be demonstrated by current experi- mental settings. Figure 2(b) illustrates the fabrication procedure of the device. (i) The mold of the microchannel was made of SU-8 and fabricatedby soft-lithography technique. The microdevice contains the main flow microchannel and the fiber channel. (ii) PDMS was poured on the SU8 mold to replicate the microchannel. (iii) The solidifiedPDMS was peeled off from the mold. (iv) The PDMS and the glasssubstrate were treated with oxygen plasma and bonded together.The assembled microfluidic device was baked at 120 °C for 1 h to further improve bonding strength ultimately. (v) Two optical fibers were inserted into the fiber microchannel. (vi) Glue was fixed atthe junction of the optical fiber and PDMS microchannel in orderto avoid leakage. The photo of the fabricated device is showninFig. 2(c) . C. Experimental setup Figure 2(d) illustrates the experimental setup. The laser with a wavelength of 980 nm provided by a laser transmitter (VLSS-980-B,Connect Fiber Optics, China) was used in the experiments. Such awavelength is known to be poorly absorbed by water, 44and thus the heat damage to the cells will be minimal. The laser beam is split evenly into two single-mode optical fibers (HI 1060, Corning) via a 1 × 2 fiber coupler (Gould Fiber Optics). Cell solution wasloaded to the microchannel by withdrawing the pump (Legato 200,KD Scientific). The microfluidic chip was placed on an invertedmicroscope (Nikon ECLIPSE Ti-U), and a camera (Nikon DS-Ri1) working at 30 Hz was mounted on the microscope for image capture and video recording. D. Cell preparation Normal human breast cells MCF10A were used in this experi- ment and obtained from School of Medicine of TsinghuaUniversity. The cells were cultured using an incubator (Forma 381, Thermo Scientific, USA) at 37 °C in 5% CO 2.T h ec u l t u r em e d i u m was high-glucose Dulbecco ’s Modified Eagle ’s Medium (DMEM,Life technologies, USA), supplemented with 10% fetal bovine serum (FBS, Life technologies, USA) and 1% penicillin-streptomycin (Life technologies, USA). Cells were rinsed with phosphate buffered saline(PBS, pH 7.4) twice and lifted off by treating with trypsin for 5 min.The cell suspension was washed three times by centrifuging at300gfor 5 min, removing the supernatant with a pipette, and re-suspending the cell pellet in 1 ml PBS buffer supplemented with 1% BSA (Bovine Serum Albumin) to avoid adhesion and shakengently to obtain the uniform cell suspension. To facilitate single-cellloading, the cell suspension was then diluted with additional PBSuntil a density of 1.5 × 10 5cells per ml was achieved. III. RESULTS AND DISCUSSION A. Optical trap, stretch, and translation When the dual fibers are well aligned, cells can be trapped in the center point of two microchannels and stretched in situ .T o estimate the distribution of the optical forces exerted on a singlecell, a finite element simulation (COMSOL Multiphysics 5.5) was performed to show the electric field distribution generated around the dual fibers. For a cell irradiated by an arbitrary monochromaticwave, the time-averaged radiation-induced forces acting on it canbe derived from the nonrelativistic Lorentz force F¼πr 3εmRe[KCM]∇E2þ1 2εmIm[KCM]X lE* l∇Elhi , (4) where εmis the permittivity of the medium, the subscript runs over l=x,y,z, and the superscript * signifies the complex conjugate. According to Eq. (4), the optical field force and the electric field distribution present a functional relationship. In the case that thelinear coefficient cannot be accurately evaluated, the optical field force can be qualitatively estimated by simulating the electric field power. 45The simulation results [ Fig. 3(a) ] show a wonderful sym- metry of the electric fields around the fiber channel center, whichleads to the spatial equilibrium of the cell both horizontally and vertically. Under the condition that the fluidic drag force is less than the trapping force, a cell flowing through the main channelwould be trapped to the center point of the two microchannels,and also the initial position of the cell is not in the middle of themain streamline. The microfluidic device with well-aligned dual optical fibers was used to confirm the trap and stretch manipulation. Accordingto Eq. (1), the diameter of laser spots at the center of the micro- channel is about 12 μm. This size is comparable to the cells we used in experiments, satisfying the purpose of cell manipulations. Referring to the literature, the laser power per fiber was set to be 100 mW, and then the cell medium was flushed through the mainmicrochannel by increasing the flow rate from zero to 100 μm/s. It was found that below 60 μm/s, the cells were nearly 100% trapped by overcoming the fluidic drag force. However, there was the increasing ratio of failure of trapping once the flow rate was increased, probably due to the higher fluidic drag force. The cellcould be trapped even when the cell flowed in with some offsetagainst the middle of the microfluidic streamline but its trajectory fell into the effective zone, which is defined as the overlapping area by the two laser beams. Once it entered the zone by flow, the cellBiomicrofluidics ARTICLE scitation.org/journal/bmf Biomicrofluidics 15,014106 (2021); doi: 10.1063/5.0039087 15,014106-4 Published under license by AIP Publishing.FIG. 3. Cell trap, stretch, and translation. MCF10A cell samples were used in experiments. (a) The simulation results of the electric field in the designated region of the microfluidic channel, when two fibers are aligned well (i.e., L= 0). (b) One cell being trapped (100 mW per fiber) and (c) stretched (400 mW per fiber). (d) Two cells were being trapped (100 mW per fiber) and (e) stretched (400 mW per fiber). (f) The simulation results of the electric fi eld in the designated region of the microfluidic channel, when two fibers are not aligned ( L>d). (g) One cell being landed on the end of the other fiber after translation by one fiber.Biomicrofluidics ARTICLE scitation.org/journal/bmf Biomicrofluidics 15,014106 (2021); doi: 10.1063/5.0039087 15,014106-5 Published under license by AIP Publishing.was dragged with acceleration to the centerline of the aligned optical fibers under the gradient force, and, then, the cell was pushed to the center of the main channel by the scattering axialforce. For the case where the two optical fiber ends were alignedwell with the channel walls, the cell was centered adaptively andtrapped steadily in the main channel. Normally, it took about 2 s for the cell to be trapped from the streamline to the trap site. Figure 3(b) shows the snapshot of the MCF10A cell stably trapped at the center point. Once the cell was stably trapped on site, thelaser power can be increased. Since the scattering force on the cellis proportional to the light intensity, the cell can be stretched by increasing the laser power. Figure 3(c) shows that when the power per fiber was increased to 400 mW, the cell was stretched, and thedeformation was about 27%. This dynamic deformation processcan be made use for characterizing cellular mechanical properties,as one powerful application of the manipulation. 35 Interestingly, the well-aligned dual optical fibers were found to be able to trap multiple cells as well. When they are both in theeffective zone, multiple cells will be trapped and form a cell chainunder the constraints of scattering and gradient forces. Regardless oftheir initial distance, the two cells will eventually converge in the middle of the flow passage and join together to form a cell chain. If the laser power is strong enough, the optical trap can capture threeor more cells. The number of cells or length of the cell chain is posi-tively related to the laser power. Like a single cell, the cell chain can be stretched once the laser power is increased further. Under the action of the scattering force, the long axis direction of the cell chainwas consistent with the direction of light propagation. Figures 3(d) and3(e)show that two MCF10A cells were trapped and stretched as a cell chain. When the optical power was increased by four times to 400 mW, the deformation of the cell chain was 4.5%. Compared to the single cell, the cell chain had six times smaller deformation. Thisobservation indicates that the cell chain has different mechanical andoptical characteristics than single cells. When the misaligned offset is greater than the cell diameter, the cell is only subject to the scattering force of one of the optical fibers and the net force in the horizontal direction translates thecell to the end of the counter-propagating optical fiber. Figure 3(f) shows the simulation results of the electric field distribution for thecase of misaligned optical fibers. The microfluidic device with mis- aligned dual optical fibers ( L>d) was used to demonstrate the translation process. When the optical power is more than 100 mW,the cell can be easily translated and finally land on the end of theother fiber. Since the distribution of flow rate in the microchannel is parabolic and the flow rate near the wall is minimized to 0, the fiber-end-landed cell is unlikely washed away. Under this circum-stance, it is difficult to deform the cell by increasing the opticalpower, because the scattering force is too small at a great distance.Figure 3(g) shows the time-lapsed image of the translation trajec- tory of one MCF10A cell. As the scattering force declined signifi- cantly with the distance, the cell moving speed decreased gradually. B. Orbital revolution and spin rotation Cells were found to rotate with different rotation patterns when the misaligned offset was varying against the cell radius. Generally, orbital revolution and spin rotation were the twopatterns, and the spin rotation was a particular case of the former. Orbital revolution plus spin rotation was observed when d/2 <L<d, and spin rotation was observed when 0 < L<d/2. Figure 4(a) shows the simulation results of the electric field distri- bution of the dual optical fibers for d/2 <L<d.The electric field distribution reveals that the scattering force generated by the dual optical fibers are not collinear. This generates a torque on the cell, causing it to self-spin. In the meantime, the scattering forces in thex axis and the gradient forces in the y axis generated by the twooptical fibers would be location-dependent. The overall effect isalways to generate a centripetal force pointing to the intersection between the two optical fiber cores and the middle streamline along the microchannel. This centripetal force drives the cell toexhibit the orbital revolution. Therefore, the cell has a hybrid rota-tion pattern overlapped by orbital revolution and spin rotation. Figure 4(b) shows the time-lapsed images of the dynamic tra- jectory of one MCF10A cell exhibiting orbital revolution in the horizontal plane, in case that the misaligned offset was 15 μm along the y axis. In the beginning, the cell was positioned at thelight axis of the right fiber. When it was repelled to the left fiber,the cell was decelerated in the x axis direction until the velocity reached 0. Then, it was accelerated by the scatting force generated by the left fiber and moved toward the right fiber. In the y axisdirection, the cell was first accelerated by the gradient force of theright fiber in the y axis direction and then decelerated by the gradient force of the left fiber. When the velocity of the cell in the y-axis direction became 0, it was accelerated by the gradientforce and moved toward the +y-axis direction (Video S1 in thesupplementary material ). The cell would exhibit spin rotation in a dominant manner when the misaligned offset was further shortened. When the mis- aligned offset distance is 0<L<d/2, the cell does spin rotation about the center of the flow channel. The scattering force will gen-erate torque on the cell, causing it to rotate. Because the misalignedoffset of the dual fibers is less than the radius of cells, the gradient force can prevent the cells from shifting and fix it at the center of the flow channel. Figure 4(c) is the simulation results of the electric field distribution of the dual fibers with misaligned offset0<L<d/2. Because the two beams are not collinear, the scattering force acting on the cell still produces a torque on the cell. The cell can be located in the center of the microchannel and does spin rotation. Figure 4(d) shows one MCF10A cell exhibiting spin rota- tion at ∼225°/s under the misaligned fibers acting in the horizontal direction (200 mW per fiber). The rotation speed is nearly propor- tional to the optical power and the direction of spin can be changed by shifting the misaligned offset between the dual fibers (Video S2 in the supplementary material ). C. In-plane rotation and out-of-plane rotation In the above analysis, the misaligned offset occurs in the hori- zontal (y-axis) direction, and the rotation motion of the cell belongs to in-plane rotation; while when the dual fibers are mis-aligned in the vertical ( z-axis) direction, the cell presents out-of-plane rotation. Because the density of cells is close to that of the solution, the buoyancy and gravity can be balanced for simplic- ity. Thus, similar to the in-plane rotation, the different misalignedBiomicrofluidics ARTICLE scitation.org/journal/bmf Biomicrofluidics 15,014106 (2021); doi: 10.1063/5.0039087 15,014106-6 Published under license by AIP Publishing.offset in the vertical direction can also induce orbital revolution and spin rotation of cells. Figure 5 shows a sequence of images for out-of-plane spin rotation of a MCF10A cell. The cell was rotatedat speed of 100°/s (Video S3 in the supplementary material ). The speed was calculated by counting the time and angles for selected feature points on the cell, which was video recorded for several resolutions. Not surprisingly, multiple cells trapped in the center ofthe microchannel can also do out-of-plane rotation (Video S4 inthesupplementary material ). The direction of rotation can be con- trolled by changing the relative spatial position of the two optical fibers. D. 3D reconstruction of cell surface via spin rotation Enabled by out-of-plane rotation, a stack of cell contours can be imaged in several rounds and reconstructed to form the 3Dmorphology of the cell. In the experiment, we used a 40× objective to record the cell contour images. The recorded video clip was con-verted into individual image frames. Each gray-level image of oneround was thresholded into a binary image. The threshold was setto segment sufficiently the cell contour from the surrounding back- ground. Cell contour was extracted from the binary images and the contour points had their 3D coordinates. A standard alpha-shapealgorithm 46,47can be used to reconstruct the 3D morphology from these 3D contour points (Video S5 in the supplementary material ). Then, we extracted the geometric parameters for the cell, like volume, surface area, roughness, and ellipticity from the 3D morphology. Figure 6(a) illustrates the image processing steps for 3D recon- struction of MCF10A cell surface with 241 image frames recorded at the angular displacement of ∼1.5°. To eliminate the noise in contour points, in practice, we extracted contour points from FIG. 4. Rotation patterns of cells when the two fibers are misaligned in space. MCF10A cell sample and 200 mW per fiber were used. The simulation results of the e lectric field (a) and the experimental results of one cell (b) performing in-plane orbital revolution, when d/2<L<dand Loccurs in the y axis. The dotted blue ellipse depicts the trajectory of the cell center. The simulation results of the electric field (c) and the experimental results of one cell (d) performing in-plane spin r otation when 0 < L<d/2 and Loccurs in the y axis. The curved arrow indicates the spin rotation direction. Note that the other two cells out of the effective zone do not rotate at all.Biomicrofluidics ARTICLE scitation.org/journal/bmf Biomicrofluidics 15,014106 (2021); doi: 10.1063/5.0039087 15,014106-7 Published under license by AIP Publishing.several rounds. All these contour points in 3D formed a point cloud, from which the 3D surface was reconstructed consisting ofoptimized triangles. Geometric parameters of the cell were retrieved from the reconstructed 3D morphology to demonstrate thepotential measurement application of 3D imaging. Figure 6(b) illustrates the 3D reconstructed models for ten MCF10A cell samples. The volume and the surface area of MCF10A cells wereaveraged as773.88 fL and 244.78 μm 2, respectively. To further study the geometry of cells, we fitted the 3D morphology cloud points asa tri-axial ellipsoid model, for which ellipticity is denoted by the ratio of the length of the long axis to the length of the short axis, and roughness is denoted by the root mean square of the deviationsbetween the cloud points and the fitting ellipsoid model. The ellip-ticity and roughness of MCF10A cells were averaged as 1.2 and0.25 μm, respectively. Figure 6(c) plotted the distribution of the five parameters after normalization for the ten cell samples. The results reveal the difference between cells, but the set of area and volumeparameters can reflect more differences. Multi-dimensional information can better describe the charac- teristics of cells, but the problem may be redundant characteriza- tion information and difficult visualization in the high-dimensional information. In order to realize the visualization of cell distributionand remove redundant information, principal component analysis(PCA) can be used to reduce the high-dimensional data into twodimensions. To demonstrate this idea, the type-wise values of the five property parameters were processed by PCA, and the result is shown in Fig. 6(d) . It can be seen that the ten samples can be sepa- rated mainly in the x axis, which is the first principal component,and can be further separated in the y axis, which is the second principal component. Actually, the corresponding weights of the FIG. 5. Out-of-plane spin rotation of the MCF10A cell when 0<L<d/2 and L occurs in the z axis. FIG. 6. 3D reconstruction of the cell surface. (a) The image processing steps of 3D reconstruction. A stack of images are captured at an interval of ∼1.5° for the rotation angle. These images are then binarized and the cell contour points in 2D are extracted. All these points are then projected into 3D to generate the cell s urface. (b) The reconstructed 3D surface of ten MCF10A cell samples. (c) The distribution of five geometric parameters for the ten cell samples. Each color represent s one cell sample. (d) PCA results of the five parameters in (c) for the ten cell samples.Biomicrofluidics ARTICLE scitation.org/journal/bmf Biomicrofluidics 15,014106 (2021); doi: 10.1063/5.0039087 15,014106-8 Published under license by AIP Publishing.first two principal components are 95.4% and 4.5%, respectively, which retained almost all the information of the original five- dimensional data. Unlike the original data, the reduced two-dimensional data now have no physical meaning but offer a morestraightforward distribution. This technique can be helpful inpruning the features that characterize single cells or cell types. IV. CONCLUSIONS We achieved multimodal manipulation of single cells on chip by tuning the misaligned offset of dual optical fibers orthogonal tothe main channel. The microchip is capable of not only trappingand stretching single cells but also rotating cells in spin and/or orbital revolution, in a 2D or a 3D manner. When aligned well, the optical fibers can trap and stretch single cells. When misaligned,the optical fibers can cause spin rotation and orbital revolution ofcells in the horizontal (in-plane rotation) and vertical (out-of-planerotation) direction, simply by adjusting the misaligned offset in value and direction. Taking advantage of out-of-plane rotation, the 3D morphology of single cells can be reconstructed, from whichmulti-parameters of physical properties can be calculated. The flex-ible configuration of optical fibers and resultant manipulation pat-terns are expected to be a powerful tool in single-cell analysis. SUPPLEMENTARY MATERIAL See the supplementary material for (1) orbital rotation (Video S1), (2) spin rotation (Video S2), (3) out-of-plane rotation of singlecell (Video S3), (4) out-of-plane rotation of two cells (Video S4),and (5) 3D reconstruction (Video S5). AUTHORS ’CONTRIBUTIONS L.H. and Y.F. contributed equally to this work. ACKNOWLEDGMENTS This work was supported by the NSFC (Nos. 61774095 and 21727813). The authors declare no conflict of interest. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1S. Zhang, Z. Li, and Q. Wei, Nanotechnol. Precis. Eng. 3(1), 32 –42 (2020). 2Y. Ma, K. Middleton, L. You, and Y. Sun, Microsyst. Nanoeng. 4(1), 17104 (2018). 3L. Huang, S. Bian, Y. Cheng, G. Shi, P. Liu, X. Ye, and W. Wang, Biomicrofluidics 11(1), 011501 (2017). 4J. Liu, J. Wen, Z. Zhang, H. Liu, and Y. Sun, Microsyst. Nanoeng. 1(1), 15020 (2015). 5A. Ghanbari, B. Horan, S. Nahavandi, X. 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5.0038560.pdf

J. Chem. Phys. 154, 074303 (2021); https://doi.org/10.1063/5.0038560 154, 074303 © 2021 Author(s).Dipole-bound and valence excited states of AuF anions via resonant photoelectron spectroscopy Cite as: J. Chem. Phys. 154, 074303 (2021); https://doi.org/10.1063/5.0038560 Submitted: 23 November 2020 . Accepted: 21 January 2021 . Published Online: 17 February 2021 Yuzhu Lu , Rulin Tang , Xiaoxi Fu , Hongtao Liu , and Chuangang Ning COLLECTIONS This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN Photodetachment spectroscopy and resonant photoelectron imaging of cryogenically cooled 1-pyrenolate The Journal of Chemical Physics 154, 094308 (2021); https://doi.org/10.1063/5.0043932 Core level photoelectron spectroscopy of heterogeneous reactions at liquid–vapor interfaces: Current status, challenges, and prospects The Journal of Chemical Physics 154, 060901 (2021); https://doi.org/10.1063/5.0036178 Effects of interaction strength of associating groups on linear and star polymer dynamics The Journal of Chemical Physics 154, 074903 (2021); https://doi.org/10.1063/5.0038097The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Dipole-bound and valence excited states of AuF anions via resonant photoelectron spectroscopy Cite as: J. Chem. Phys. 154, 074303 (2021); doi: 10.1063/5.0038560 Submitted: 23 November 2020 •Accepted: 21 January 2021 • Published Online: 17 February 2021 Yuzhu Lu,1Rulin Tang,1Xiaoxi Fu,1 Hongtao Liu,2and Chuangang Ning1,3,a) AFFILIATIONS 1Department of Physics, State Key Laboratory of Low Dimensional Quantum Physics, Tsinghua University, Beijing 10084, China 2Key Laboratory of Interfacial Physics and Technology, Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China 3Collaborative Innovation Center of Quantum Matter, Beijing 100084, China a)Author to whom correspondence should be addressed: ningcg@tsinghua.edu.cn ABSTRACT Gold fluoride is a very unique species. In this work, we reported the resonant photodetachment spectra of cryogenically cooled AuF−via the slow-electron velocity-map imaging method. We determined the electron affinity of AuF to be 17 976(8) cm−1or 2.2287(10) eV. We observed a dipole-bound state with a binding energy of 24(8) cm−1, a valence excited state with a binding energy of 1222(11) cm−1, and a resonant state with an energy of 814(12) cm−1above the photodetachment threshold. An unusual vibrational transition with Δn=−3 was observed in the autodetachment from the dipole-bound state. Moreover, two excited states of neutral AuF were recognized for the first time, located at 13 720(78) cm−1and 16 188(44) cm−1above the AuF ground state. Published under license by AIP Publishing. https://doi.org/10.1063/5.0038560 .,s I. INTRODUCTION Gold fluoride is very unique compared to its congeneric com- pounds MX (M = Cu, Ag, Au and X = F, Cl, Br, I). For example, the number of gold-fluoro compounds known today is still very small in contrast to the vast number of gold compounds containing chlorine, bromine, and iodine.1It was not until 1994 that AuF was unam- biguously identified in the gas phase by Schwarz and co-workers.2 In 1992, Saenger and Sun observed yellow emission bands, which were very likely from AuF.3Some vibrational structures were not resolved due to the limited resolution. An informative experiment by Butler et al.4presented accurate molecular constants of the ground and three excited states of AuF. For the ground state, they reported the vibration frequency ωe= 563.609 04(19) cm−1and the anhar- monicityωeχe= 2.896 284(63) cm−1. The dipole moment of AuF ground state X1Σ+is 4.13(2) D according to the Stark study of Steimle et al. ,5which is large enough ( >2.5 D) to possess a dipole-bound state (DBS).6–11Thus, it is expected that a DBS can be observed in the high-resolution photoelectron spectroscopy ofAuF−. There are many theoretic investigations of neutral AuF.12–21 For example, the work of Guichemerre et al.19made extensive pre- dictions of both ground and excited states of AuF. Recently, the photoelectron spectroscopy of CuF−and AgF−has been reported by the Mabbs group.22,23The electronic structures of CuF−and AgF− are relatively simple. Both CuF−and AgF−have a ground state and a DBS. Above the threshold, some resonance states were observed for both CuF−and AgF−, most of which were described as dipole- stabilized shape resonances.23In this work, we report the resonant photoelectron spectroscopy of a cryogenically cooled AuF anion using the slow-electron velocity-map imaging (SEVI) method.24–28 To the best of our knowledge, no research of AuF−has been reported. II. EXPERIMENT SETUP The experiment was carried out on our SEVI apparatus equipped with a cryogenically controlled ion trap.29,30Our recent J. Chem. Phys. 154, 074303 (2021); doi: 10.1063/5.0038560 154, 074303-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp modifications have enabled us to switch between the SEVI mode and the scan mode so that we can acquire the photoelectron energy spectra in the SEVI mode and observe resonance peaks by scan- ning the wavelength of the photodetachment laser in the scan mode. The AuF−anions were generated by laser ablation of a gold metal disk in the presence of NF 3gas. The NF 3gas was delivered onto the gold target before each ablation laser shot via a pulse valve. The time sequence was optimized to obtain the best AuF−yield. The anions were captured by an octupole radio-frequency (RF) ion trap and cooled through collisions with the buffer gas (20% H 2and 80% He). The ion trap was mounted on the second stage of a liquid helium refrigerator with a tunable temperature in the range of 5 K– 300 K.30–32Then, the anions were ejected out by the pulsed potentials on the end caps of the trap and analyzed using a Wiley–McLaren type time-of-flight (TOF) mass spectrometer.33Strong signals of Au−, AuF−, and AuF 2−were observed, and AuF−was selected via a mass gate. The selected anions were then photodetached with a tunable laser, which crossed the ion beam perpendicularly. In the SEVI mode, the photoelectrons were projected onto a phosphor screen behind a set of microchannel plates and recorded by a charge- coupled device (CCD) camera. The maximum entropy Legendre expanded image reconstruction (MELEXIR) method34was used to reconstruct the 3D photoelectron distribution from the projected images. In the scan mode, electrons and anions were detected by the phosphor screen and recorded using a high-speed oscilloscope.35To obtain the photodetachment spectra of AuF−, the signal light of an optical parametrical oscillator (OPO, 405 nm–709 nm for the sig- nal light, and linewidth ∼5 cm−1) pumped by a Quanta-Ray Lab 190 Nd:YAG laser was used. 100 laser shots were collected for each data point. A tunable dye laser (400 nm–920 nm and linewidth 0.06 cm−1at 625 nm) pumped by a Quanta-Ray Pro 290 Nd:YAG laser (20 Hz and 1000 mJ/pulse at 1064 nm) was employed to mea- sure the electron affinity and investigate the rotational profiles of resonant peaks. The wavelength of the dye laser was measured by a wavelength meter (HighFinesse WS6-600, 0.02 cm−1accuracy). This wavelength meter can also monitor the intensity of each laser pulse. III. RESULTS AND DISCUSSION Figure 1 shows the photoelectron spectra accumulated using the signal light of OPO at room temperature and 15 K. The spectra of Au−were used for the energy calibration. It can be seen that hot bands, i.e., the peaks labeled as 1′→0, 1′→1, and 2′→0, disappeared as the temperature dropped from 300 K to 15 K. These peaks were contributed by the vibrational exited states of AuF−at its electronic ground state X2Σ+. The number with a prime indi- cated the vibrational quantum number of AuF−at the X2Σ+state. The vibrational frequency of AuF−at its electronic ground state X2Σ+was determined to be 405(99) cm−1. To measure the electron affinity of AuF with higher accuracy, the photoelectron spectra were then collected at a photon energy of 18 000.56 cm−1, just above the photodetachment threshold. As a result, the electron affinity of AuF was determined to be 17 976(8) cm−1or 2.2287(10) eV. The uncer- tainty is mainly due to the rotational broadening. To observe the DBS in AuF−, we scanned the photon energy from 16 000 cm−1to 20 900 cm−1and recorded the intensity of photoelectron signals at FIG. 1 . Photoelectron images and spectra of AuF−at 300 K (black line) and 15 K (blue line). The double arrow indicates the polarization of the detachment laser. The vibrational transitions are marked on the top. 15 K. Figure 2 shows the observed resonant structures. The peaks below the photodetachment threshold were contributed by resonant two-photon detachment (R2PD). Clearly, we observed two sets of resonances. The peaks labeled as v0 to v5 are almost evenly spaced, and peaks labeled as d0 to d4 form another nearly equi-spaced set. FIG. 2 . Photodetachment spectrum of cryogenically cooled AuF anions by mea- suring the total electron yields as a function of the laser photon energy across the detachment threshold using the OPO signal light. The curve in red shows the weak peaks multiplied by a factor of 10. The insets show the rotational profiles of resonant peaks v5 and d0 using a dye laser with a narrow linewidth. J. Chem. Phys. 154, 074303 (2021); doi: 10.1063/5.0038560 154, 074303-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The two sets were assigned as the vibrational progressions related to two exited states of AuF−. One is a valence excited state (VES) and the other is a DBS. Peak r, lying 814(12) cm−1above the photode- tachment threshold, is a resonance related to a different state. Peak v0 was assigned to be the vibrational ground state of the VES since no more vibrational peaks were observed when the photon energy was scanned further down. The vibrational frequency of the VES was determined to be 237(8) cm−1. The vibrational frequency of the dipole bound state was determined to be 562(8) cm−1, which is very close to the vibrational frequency 560.712 76(20) cm−1of the ground state of neutral AuF.4Another evidence for the peak assignment is the rotational profiles of the resonant peaks. As shown in the insets of Fig. 2, the rotational profile of v5 is remarkably different from that of d0, which reflects the different Au–F bond lengths between the two states. The rotational profiles were recorded using our dye laser with a narrow linewidth 0.06 cm−1. The energy levels of two excited states were determined to be 16 754(8) cm−1(VES) and 17 952(2) cm−1(DBS) above the AuF−ground state, respectively. Therefore, the binding energy of the DBS was determined to be 24(8) cm−1and 1222(11) cm−1for the VES. To interpret the observed spectra, we conducted multi- reference configuration interaction (MRCI) calculations using the Molpro software package.36The spin–orbit coupling has been included in the calculations. The correlation consistent basis sets aug-cc-pV5Z-PP for Au with ECP60MDF pseudopotentials37and aug-cc-pV5Z for F were used.38The calculations predicted two excited states below the neutral ground state, lying 1.9 eV (22Σ+) and 2.0 eV (12Π3/2) above the AuF−ground state (X2Σ+). Higher states, located 2.4 eV (12Δ5/2) and 3.2 eV (12Π1/2) above the anionic ground state, were also predicted, which were too high to be responsible for the observed valence excited state. The ground state X2Σ+has a dissociation asymptote of Au(2S1/2) + F−(1S0). The excited states 22Σ+and 12Π3/2correspond to the Au−(1S0) + F(2P3/2) asymptote. The calculated excited states 22Σ+and 12Π3/2 might be responsible for the observed VES and the resonance state related to peak r, respectively. Guichemerre and co-workers did not include spin–orbit couplings in their calculations of the potential energy curves of neutral AuF.19Therefore, we calculated the poten- tial energy curves for the ground and excited states of neutral AuF considering spin–orbit couplings. In Fig. 3, the potential energy curves of the related AuF−and AuF states were plotted together. The curves of AuF−DBS, AuF−VES, and AuF−X2Σ+were gen- erated using Morse potentials with the experimentally determined parameters, while the curves for neutral AuF were calculated using the method mentioned above. Since the molecular parameters of neutral AuF have been experimentally determined with high accuracy, the dissociation energy De(AuF, X1Σ+) was estimated to be 3.40 eV using the equation De=̵hωe/4χe.39Then, with EA(AuF) = 2.2287(10) eV determined in the present work and EA(F) = 3.401 1895(25) eV as measured by Blondel et al. ,40the dissociation energy of AuF−can be given by De(AuF−, X2Σ+) =De(AuF, X1Σ+) +EA(AuF) −EA(F) = 2.23 eV, which is above the observed AuF−excited states and very close to EA(AuF). Okabayashi et al.41also gave an estimation of De(AuF, X2Σ+) = 3.01 eV using the molecular constants by fitting the pure rotational spectrum to the Dunham expression. The cor- responding De(AuF−, X2Σ+) = 1.84 eV is below the energy level of AuF−excited states. In addition, based on the reactions producing FIG. 3 . Potential energy curves of the related AuF−and AuF states. The curves of AuF−DBS, AuF−VES, and AuF−X2Σ+are generated using Morse potentials with the experimentally determined parameters, while the curves for the neutral AuF are the calculated results. neutral AuF, Schröder et al.2estimated that the lower bound for De(AuF, X2Σ+) was 3.16 eV and the upper bound was 3.69 eV, which set the bound for De(AuF−, X2Σ+) as [1.99 eV, 2.52 eV]. The reso- nant energies of peaks v0–v5 are in the range [1.99 eV, 2.52 eV]. Therefore, it is likely that some of the vibrational excited AuF− VES are predissociated states. This can explain why the photoelec- tron yield in the range of 18 000 cm−1–18 400 cm−1unexpectedly reduced. If no predissociation occurs, the resonant peaks v6 and v7 are expected to be observed in this range. It should be pointed out that the features in the range of 18 000 cm−1–18 400 cm−1are not likely due to the signal fluctuation since we scanned the region for several times and the spectra showed the same features. A reason- able explanation is that AuF−anions were quickly dissociated into Au and F−and F−cannot be photodetached by the laser due to the very high electron affinity of F atom. The R2PD photoelectron spectra of v5 (VES) and d0 (DBS) are shown in Fig. 4. The spectra are due to the photodetachment from AuF−X2Σ+to the ground state and the excited states of neutral AuF via the intermediated states of AuF−. The AuF excited states [14.0]1, [17.7]1, and [17.8]0+observed by Butler et al.4were recognized in the present work. Besides, two more states [13.7]0−and [16.2]2 were identified, and they are 13 720(78) cm−1and 16 188(44) cm−1above the AuF ground state X1Σ+, respectively. The states are labeled in Hund’s case (c) notion [ T0]Ω, where T0is the energy relative to AuF X1Σ+in 1000 cm−1andΩis the total electronic angular momen- tum about the internuclear axis. The states [13.7]0−and [16.2]2 were predicted by the calculations of Guichemerre et al.19However, they were not observed by Butler et al.4in their search from 553 nm to 800 nm using laser excitation spectroscopy. Figure 5 shows R2PD photoelectron spectra of v1–v5 in comparison with the Franck–Condon simulations. The R2PD pho- toelectron spectrum of v0 is not shown due to the very low signal- to-noise ratio. The Franck–Condon factors were calculated via the J. Chem. Phys. 154, 074303 (2021); doi: 10.1063/5.0038560 154, 074303-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . Resonant two-photon photoelectron spectra and images of AuF−via the valence excited state with the vibration quantum equal to 5 labeled as v5 (a) and the vibrational ground state of the dipole bound state labeled as d0 (b) using the dye laser. The red curves show the weak peaks multiplied by a factor of 5. The excited states of neutral AuF are labeled in Hund’s case (c) notion [ T0]Ω, where T0is the energy relative to AuF X1Σ+in 1000 cm−1andΩis the total electronic angular momentum about the internuclear axis. Morse potential using the method of López et al. ,42and the bond length of AuF−VES was optimized to be 2.16 Å. As shown in Fig. 5, the Franck–Condon simulation can roughly reproduce the major features of R2PD photoelectron spectra. However, significant dis- crepancies exist between the simulations and experimental results. The possible reasons for the discrepancies are as follows: the excited states of AuF−are likely to be predissociated, so the vibrational wavefunction deviates considerably from that of the Morse poten- tial, and the potential energy curves of excited states of neutral AuF also deviate significantly from the Morse potential due to the strong spin–orbit couplings, as illustrated in Fig. 3. Figure 6 shows the resonance-enhanced photoelectron spec- tra at photon energies above the electron affinity. The spectra were labeled with d nindicating an electronic excitation from the ground state of AuF−to a DBS with a vibrational excitation to the nth vibra- tional state simultaneously. The spectra labeled with rare assigned to the resonant state related to 12Π3/2of AuF−tentatively. The energies of these excited states are higher than the electron affin- ity of AuF. Therefore, they quickly autodetached by ejecting an electron due to the vibronic coupling.43Since the extra electron was loosely bounded by the dipole potential, the potential curve of a DBS was almost parallel to that of its neutral core AuF. As a result, there was a propensity rule of the vibrational quantum change Δn=−1 during the autodetachment.44–48It can be seen that the intensity of the peak with a vibration quantum number equal to n−1 was enhanced for each spectrum in Fig. 6 except the spec- trum d3. For d3, both n= 0 and n= 2 were clearly enhanced, which FIG. 5 . Resonant two-photon photoelectron spectra of AuF−via the valence excited state with vibrational quantum numbers equal to 1, 2, 3, 4, and 5 and Franck–Condon simulations. The bars in red denote the Franck–Condon progres- sion corresponding to the AuF [13.7]0−state and the bars in blue denote the Franck–Condon progression corresponding to the AuF [14.0]1 state. was also reflected in the photoelectron angular distributions of n= 0 and n= 2. Both are isotropic. As shown in Fig. 1, the photoelec- tron angular distribution is parallel to the laser polarization for the direct photodetachment from the ground state X2Σ+. And, as shown in Fig. 6, it is isotropic for the autodetachment from a DBS. It should be noted that for n= 0, we have Δn=−3, which is very unusual for the autodetachment from a DBS. The propensity rule is derived based on harmonic approximation,49and violation of this rule has been observed previously due to anharmonic effects.50–54The viola- tion of the propensity rule may also result from the non-negligible correlation effects between the DBS electron and other electrons.55 The interaction between DBS and VES has been discussed in some molecules, such as nitromethane,56,57uracil,58,59nitrobenzene,60and thymine dimer.61Here, the interaction between DBS d3 and a VES may result in the violation of the propensity rule. It should be noted that the lifetimes of these excited states43should be much shorter J. Chem. Phys. 154, 074303 (2021); doi: 10.1063/5.0038560 154, 074303-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 6 . Resonance-enhanced photoelectron spectra and images of AuF−acquired using the OPO signal light. The labels d1 to d4 indicate the resonance via the DBS with vibrational quantum numbers equal to 1, 2, 3, and 4. For d3, the non-resonant photoelectron spectrum collected at a photon energy of 19 564 cm−1is also shown in comparison. The label rindicates a resonance that is tentatively assigned to 1 2Π3/2. The vibrational quantum number of the final AuF (X1Σ+) state is marked on the top. than the duration of the laser pulse ( ∼5 ns) because no two-photon detachment was observed. As a contrast, the lifetime of excited states v0–v5 and d0 should be comparable with the duration of the laser pulse. Otherwise, they cannot be photodetached by a second photon before decaying. IV. CONCLUSION In conclusion, the resonant photodetachment spectra of cryogenically cooled AuF−were obtained via the slow-electron velocity-map imaging (SEVI) method. The electron affinity of AuF was measured as 17 976(8) cm−1or 2.2287(10) eV. Around the photodetachment threshold, a dipole-bound state with a bindingenergy 24(8) cm−1, a valence excited state with a binding energy 1222(11) cm−1, and a resonant state with an energy 814(12) cm−1 above the photodetachment threshold were observed. An unusual vibrational transition was observed in the vibrationally induced autodetachment from the dipole-bound state. Moreover, two excited states of neutral AuF were observed for the first time, located at 13 720(78) cm−1and 16 188(44) cm−1above the AuF ground state. ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (NSFC) (Grant No. 11974199, 91736102) and the National Key R&D Program of China (Grant No. 2018YFA0306504). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1F. Mohr, Gold Bulletin 37, 164 (2004). 2D. Schröder et al. , Angew. Chem. 106, 223 (1994). 3K. L. Saenger and C. P. Sun, Phys. 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5.0045374.pdf

J. Appl. Phys. 129, 125901 (2021); https://doi.org/10.1063/5.0045374 129, 125901 © 2021 Author(s).The ultrahigh pressure stability of silver: An experimental and theoretical study Cite as: J. Appl. Phys. 129, 125901 (2021); https://doi.org/10.1063/5.0045374 Submitted: 26 January 2021 . Accepted: 10 March 2021 . Published Online: 24 March 2021 E. F. O’Bannon , M. J. Lipp , J. S. Smith , Y. Meng , P. Söderlind , D. Young , and Zs. Jenei ARTICLES YOU MAY BE INTERESTED IN Impact response of a tungsten heavy alloy over 23–1100 °C temperature range Journal of Applied Physics 129, 125902 (2021); https://doi.org/10.1063/5.0042939 Thermal oxidation rates and resulting optical constants of Al 0.83 In0.17N films grown on GaN Journal of Applied Physics 129, 125105 (2021); https://doi.org/10.1063/5.0035711 Reconfigurable thin-film transistors based on a parallel array of Si-nanowires Journal of Applied Physics 129, 124504 (2021); https://doi.org/10.1063/5.0036029The ultrahigh pressure stability of silver: An experimental and theoretical study Cite as: J. Appl. Phys. 129, 125901 (2021); doi: 10.1063/5.0045374 View Online Export Citation CrossMar k Submitted: 26 January 2021 · Accepted: 10 March 2021 · Published Online: 24 March 2021 E. F. O ’Bannon,1,a) M. J. Lipp,1 J. S. Smith,2 Y. Meng,2P. Söderlind,1 D. Young,1 and Zs. Jenei1 AFFILIATIONS 1Physics Division, Physical & Life Sciences Directorate, Lawrence Livermore National Laboratory, Livermore, California 94550, USA 2High Pressure Collaborative Access Team, X-Ray Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA a)Author to whom correspondence should be addressed: obannon2@llnl.gov ABSTRACT We measured the atomic volume of Ag in a toroidal diamond anvil cell to a maximum pressure of 416 GPa and calculated the atomic volume and elastic constants of Ag up to 750 and 460 GPa, respectively. Our density functional theory calculations at 0 K utilize an all-elec- tron fully relativistic method and agree well with our volume measurements, particularly at pressures above ∼75 GPa. We corrected our experimental results for non-hydrostaticity using a line shift analysis, and the resulting Vinet equation of state (EOS) parameters arereported. We find that the uniaxial stress sustained by Ag increases linearly up to 4.5 GPa at a pressure of 416 GPa. Our experimental resultsindicate that the fccstructure of Ag remains stable to at least 416 GPa at room temperature. Our theoretical results show that C 44increases as pressure increases and reaches a maximum at ∼100 GPa above which it begins to decrease, a sign that the fccstructure of Ag is becoming unstable, and at V/V0= 0.30, the bccstructure is lower in energy than fcc. Published under license by AIP Publishing. https://doi.org/10.1063/5.0045374 I. INTRODUCTION Silver (Ag) is a group 11 transition metal with an atomic number of 47. Ag has an electron configuration of [Kr]4d105s1and it crystallizes in the face-centered-cubic (fcc) structure at ambient conditions. The group 11 elements Cu, Ag, and Au all have an fcc structure, are chemically inert, and moderately compressible, whichmake them ideal pressure standards for high-pressure diamondanvil cell (DAC) experiments. 1–3Under compression at room temperature, Cu, Ag, and Au are stable in the fccstructure up to at least 153,4150,5,6and 1065 GPa,7,8respectively. However, at high- pressure and temperature, new phases have been reported for allthree of these elements from both static and dynamic compressionexperiments. 9–13Additionally, theoretical studies have predicted a variety of close-packed structures for Ag and Au14–16at high pres- sures, while, surprisingly, Cu has been predicted to be stable in thefccstructure to 100 TPa. 17 Ag has not been commonly used as a pressure marker in static compression experiments, but it does offer many of the same benefits as Au and Cu. Static compression of Ag has been limited to only a few studies.5,6Akahama et al.5compressed Ag at room temperature non-hydrostatically up to 146 GPa andassessed their results for uniaxial stress using a line shiftapproach. 18Dewaele et al.6measured the volume of Ag at room temperature in a helium pressure transmitting medium up to 122 GPa and compared these results to equation of state (EOS)parameters they obtained from various theoretical approaches. Aghas also been studied under shock loading up to pressures as highas 500 GPa 3,19,20and EOS parameters have been published. While the EOS parameters reported from these different studies agreewell on the ambient volume V 0and the isothermal bulk modulus K0, they disagree on the first pressure derivative of the bulk modulus K00. The refined K00reported from static DAC studies are lower by as much as ∼11% than the values obtained from reduced shock wave (RSW) and ultrasonic (US) studies. In this study, the room temperature high-pressure stability of fccAg has been revisited since it has only been investigated under static compression to 150 GPa, and we are motivated by the recentstudies on Ag that have reported a new high-pressure phase, 13 albeit at high temperature. Here, we compressed Ag without a pres-sure transmitting medium to a maximum pressure of 416 GPa andcalculated the volume and elastic constants up to 460 and 750 GPa,respectively. We compare our results with static DAC, US, RSW,and theoretical studies, and discuss the uniaxial stress state of ournon-hydrostatic measurements.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 125901 (2021); doi: 10.1063/5.0045374 129, 125901-1 Published under license by AIP Publishing.II. EXPERIMENTAL DETAILS A. High-pressure x-ray diffraction measurements The Ag sample was a powder with a grain size ranging from 0.5 to 1 μm obtained from Alfa-Aesar with a purity of 99.9%. We measured the ambient atomic volume of Ag and found it to be17.031(2) Å 3in good agreement with previous studies.5,6,21The Au and Cu pressure markers were powders with grain sizes ranging from 0.8 to 1.5 μm and 1.0 to 1.5 μm, respectively, obtained from Alfa Aesar both with a 99.9% purity. Two ultrahigh static pressureexperiments were performed using a LLNL designed membraneDAC 22equipped with toroidal anvils,23experiment 1: Ag with Au and experiment 2: Ag with Cu. Diamonds with toroidal surfaces were fabricated using a standard 16-sided type Ia single beveled diamond anvil with a30μm central culet and a 300 μm bevel diameter at 8.5°. The toroi- dal surfaces of the diamonds are generated by in-house developedsoftware and then translated into a bitmap that is used in the focused ion beam (FIB) instrument to mill the toroidal surfaces onto the beveled diamond anvil surface. The milling is done using9 or 13 nA of 30 keV Ga ions. More details on the fabrication oftoroidal anvils can be found in the study by Jenei et al. 23All the toroidal diamond anvils used in this study had the following parameters: c=9μm, d=3μm, d2=2μm, and rT=1 2μm [Fig. 1(a) ]. Re was used as the gasket in all our experiments, and to obtain “clean ”gasket free diffraction patterns, a hole size of ≥13μm was used in our toroidal experiments [ Fig. 1(b) ]. Obtaining “clean ”diffraction patterns without peaks from the gasket material is crucial to the data analysis. However, this config-uration is a compromise because of the x-ray beam size, whichlikely sacrifices the overall support of the central culet area thatlikely limits our upper achievable pressure. Ag powder was loaded into the gasket hole together with either Au (experiment 1) or Cu (experiment 2) powder for pressureestimation with no pressure transmitting medium. For pressureestimation, we used the Au EOS reported by Yokoo et al. 2and the Cu EOS reported by Fratanduono et al.1Across the pressure range of the experiments reported in this paper, the Cu EOS reported byFratanduono et al.1is indistinguishable from the Cu EOS reported by Kraus et al .24The atomic volume of Ag and the pressure markers was determined from the (111) reflection since it is leastaffected by non-hydrostatic conditions. 25We chose this approach to keep our volume calculation consistent across the entire pressurerange of our experiment since multiple reflections from the sample and pressure markers are not always reliably observed. Angle dispersive x-ray diffraction (XRD) experiments were carried out at the High-Pressure Collaboration Access Team(HPCAT) beamline 16ID-B on either the general purpose or laserheating tables at the Advanced Photon Source at Argonne National Laboratory. Diffraction patterns were collected at room temperature using a monochromatic x-ray beam of ∼30 keV (0.4066 Å) focused to a beam size of 1 μm in the horizontal and 2 μm in the vertical with a tail as large as ∼13μm. Sample positioning and centering was accomplished by the fly scan method. 26Diffraction patterns were collected with a Pilatus 1M-F detector, with exposure times between 4 and 30 s. Detector distance and orientation were cali-brated using a CeO 2standard. To improve the signal-to-noise ratio, we used a multi-channel collimator for data collection above∼100 GPa for some of the experiments. The upstream and down- stream slits of the multi-channel collimator are 50 and 175 mm from the sample position, respectively. The vertical and horizontalopenings are 30° and 60°. The “duty cycle ”is∼10% (only 10% of the detector is illuminated when the slits are stationary) so expo- sure times are ∼10× longer to achieve equivalent counts with the slits removed ( ∼300 s exposure for all diffraction patterns collected above ∼100 GPa). The design is conceptually similar to previous designs 27,28with a larger vertical opening. The two-dimensional diffraction patterns were radially integrated using DIOPTAS29to obtain an intensity curve of the diffraction peaks as a function of the 2 θangle, which were analyzed to obtain peak positions using our in-house analysis codes in OriginPro software package. B. Evaluation of the stress states The lattice parameter is obtained directly from the measured d spacings, and for the cubic system under hydrostatic conditions,the lattice parameter is independent of hklindices. 30Under non- hydrostatic conditions, the measured lattice parameter systemati- cally deviates from the hydrostatic value depending on the hkl indices.18,30The measured lattice parameter am(hkl) is related to the uniaxial stress component, the orientation of the diffractionplane, and the elastic modulus of the sample and is expressed as a m(hkl)¼M0þM1[3(1/C03 sin2θ)Γ(hkl)], (1) where M0¼ap{1þ(αt/3)(1/C03 sin2θ)[S11/C0S12/C0(1/C0α/C01)*(2Gv)/C01]}, (2) M1¼/C0 ap(αtS/3), (3) Γ(hkl)¼(h2k2þk2l2þl2h2)/(h2þk2þl2)2, (4) FIG. 1. (a) Diagram showing the basic parameters of the toroidal design and (b) diagram showing the gasket configuration used in these experiments.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 125901 (2021); doi: 10.1063/5.0045374 129, 125901-2 Published under license by AIP Publishing.S¼S11/C0S12/C0S44/2, (5) with apas the hydrostatic lattice parameter, θis the Bragg angle of the reflection ( hkl),Sijthe single-crystal elastic compliance, and Gv the shear modulus of the polycrystalline aggregate. αdetermines the actual stress of the sample and is assumed to lie between twoextreme values of 0.5 and 1. It can also be verified that for com-monly encountered values of S ijandt,M0≈ap.25,30Thus, assuming M0≈apin Eq. (2), and replacing apwith M0in Eq. (3), the follow- ing relation for the uniaxial stress component tis obtained, t/difference/C03M1/(αM0S): (6) One can obtain tfrom the slope M1and the intercept M0of a plot of am(hkl) vs 3(1 –3 sin2θ)Γ(hkl) (termed the gamma plot) together with αandS. The elastic constants were calculated using density functional theory (DFT), and they were extrapolated from their ambient pres-sure and temperature values following the approach reported byGuinan and Steinberg 31to very high pressure using the following formula: Cij¼C0 ijþdCij dP*P*V V0/C18/C191/3 , (7) where C0 ijis the elastic constant at ambient conditions and dCij/dP is its derivative with respect to pressure. The elastic anisotropyparameter Scan be obtained with Eq. (5), which can be expressed byC ijas S¼1/(C11/C0C12)/C01/(2C44): (8) In the axial diffraction geometry, lattice parameters obtained under positive uniaxial stress are larger than the hydrostatic values,while those obtained under negative uniaxial stress are smaller. 25,30 The so-called hydrostatic values fall in between, and one can calcu- late the hydrostatic value using Eq. (1)and assuming α= 1, which gives the smallest difference between apand am(hkl) by using the following equation:25,30 ap¼am/{1þ(t/3)(1/C03 sin2θ)[S11/C0S12/C03SΓ(hkl)]} :(9) It should be noted that the estimation of tand apis limited by the lack of knowledge of αand that t-values obtained from the analysis of axial geometry data are sensitive to the choice of α. Additionally, uncertainties in extrapolated and/or calculated high-pressure elastic constants will impact the final calculated hydro-static lattice parameter. However, uncertainties in the derivedparameters can be reduced if diffraction from regions of large stress gradients can be avoided. 32The line shift analysis has been used in many studies to compute the EOS that corresponds to hydrostaticcompression. 33–37 III. COMPUTATIONAL DETAILS Density functional theory (DFT) calculations up to 750 GPa utilizing a full-potential linear-muffin-tin orbitals method werecarried out. This method does not rely on any core-electron approximation and is referred to as “all electron. ”This is more accurate than the approach in the previous calculation6and it is fully relativistic, including spin –orbit coupling that was previously ignored.6The setup of the calculations is detailed in our recent assessment of DFT for equation-of-state38and here we use the so-called revised Perdew-Burke-Ernzerhof (PBEsol)39approxima- tion for the electron exchange and correlation functional. Wecompare these new results with previous local density approxima-tion (LDA), generalized gradient approximation (GGA), GGA-PBE, and PBEsol results. For determining the elastic constants, we applied six volume-conserving strains each for the C 0and the C44 moduli then fitting a fourth-order polynomial to the total-energy response and extracting the coefficient for the second-order term.We utilized many k points (3000 –8000) to ensure good resolution of the small energy differences associated with the small strains. IV. RESULTS AND DISCUSSION The maximum pressure achieved in experiment 1 was 416 (3) GPa according to the Yokoo et al. 2Au EOS, and in experiment 2 we achieved a maximum pressure of 392(3) GPa according to the Fratanduono et al.1Cu EOS. The measured volumes and pressures from each experiment and our DFT calculated volumes and DFTcalculated elastic constants are tabulated in the supplementary material (Tables S1 –S4). Selected diffraction patterns from both experiments are shown inFig. 2 , and the measured volumes from experiments 1 and 2 and our new DFT results are shown in Fig. 3(a) . In the diffraction pat- terns, reflections from the sample and pressure marker are doubledup. This is because in these experiments, the culet is smaller thanthe x-ray beam so diffraction from the culet and torus areas can be observed. This doubling up of high- and low-pressure peaks was noted by Jenei et al . 23in their t-DAC and Sakai et al .40in their double-stage DAC papers. Pressure distribution maps of our t-DACwith a 1 μm step size confirm that the low-pressure peaks match the pressure found in the torus of the anvil. The gaps in the pres- sure volume data between ∼20 and ∼180 GPa is primarily due to the steepness of the loading curve, which is due to a large elasticstrain in the diamond tips where a small increase in the loadresults in a large increase in the sample pressure. 8,23There is some overlap between Ag and the chosen pressure markers, but it occurs within or near the pressure gap that originates from compressionstage II. Also as noted by Dewaele et al . 8and from our t-DAC experiments, data collected in this region often appear to have ahigher compressibility than expected, which is likely due to an inhomogeneous stress distribution between the sample and the gasket during this compression stage, and that it is possibly relatedto an insufficient stabilization time before performing the measure-ment. Thus, data collected in this region can be unreliable andwould adversely impact the obtained EOS parameters, and we are confident that the gap in these data does not adversely influence our obtained EOS parameters. In Fig. 3(b) , we show C 11,C12, and C44of Ag as a function of pressure from our DFT calculations and using Eq. (7)the extrapolated ambient isothermal values reported by Biswas et al.41Both the extrapolated and the calculated C11and C12increase approximately linearly, but the extrapolated C44Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 125901 (2021); doi: 10.1063/5.0045374 129, 125901-3 Published under license by AIP Publishing.disagrees with our calculated C44, which increases with pressure up to∼100 GPa and then begins to decrease up to the maximum pres- sure investigated. This softening has been interpreted as a sign of a phase transition, and it has been observed in vanadium both exper-imentally 42,43and theoretically.44,45 For both experiments, the measured pressure volume data have been fitted with a Vinet EOS.46In one fit, we fixed V0to 17.031 Å3(obtained from our ambient measurement) and K0to 101 GPa (the ultrasonically derived isothermal bulk modulusreported by Biswas et al. 41), and in our other fit, we only fixed V0. Notably the correlation between K0and K00is known to be signifi- cant in a simultaneous fit, so we prefer to use the results from the fits where K0is fixed (since this value can be reliably obtained fromother experiments). In our fits where only V0was fixed, we obtain values for K0that are close to the ultrasonically derived value of 101 GPa41for both experiments. The fitted K00values agree well with previous reduced shock wave19,20and ultrasonic studies,41,47 but they are as much as 4% larger than the previous DAC measure- ments.5,6This discrepancy is due to the non-hydrostatic nature of our experiments. In our fits, where both V0and K0were fixed, we obtain values for K00that are still as much as 4% larger than previ- ous DAC studies. The results from the fits to the measured data are shown in the supplementary material (Table S5). Since the EOS parameters were obtained from fits to the mea- sured non-hydrostatic data, we corrected our results for non-hydrostatic conditions using the line shift method 18,25(described in Sec. II B ). We were not able to calculate the uniaxial stress compo- nent tfor every pressure step due to peak overlap between Ag and FIG. 3. (a) The measured experimental results and DFT results from this study. (b) The calculated and extrapolated Cijs for Ag; the ambient Cijs used in the extrapolations are from Ref. 41. FIG. 2. Selected powder x-ray diffraction patterns from experiment 1 (a) and experiment 2 (b) at high pressures. The insets show part of the raw diffractionimages as recorded on the Pilatus 1M-F detector. Note the doubling of the diffraction peaks from the sample and pressure marker, which is described in more detail in the text.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 125901 (2021); doi: 10.1063/5.0045374 129, 125901-4 Published under license by AIP Publishing.the pressure marker and/or not resolving enough reflections for the analysis. We find that for Ag, tincreases approximately linearly with pressure ( Fig. 4 ) and it reaches a maximum of 4.5 GPa at 416 GPa for experiment 1 and 3.2 GPa at 350 GPa for experiment2. These results agree well with Akahama et al.5who also observed an approximately linear increase of tas a function of pressure (Fig. 4 ). A linear extrapolation of the t-values reported by Akahama et al.5to 400 GPa shows that they are slightly larger than the values obtained from our t-DAC experiments. Two possible reasons forthis difference could be that they used a different reference for their ambient elastic constants for Ag, and/or it points to differences in the non-hydrostatic pressurization conditions of conventionalbeveled anvils and toroidal anvils. The approximately linear behav-ior of tas a function of pressure has also been observed in the non- hydrostatic compression of Au to 600 GPa. 8 For our Ag volume corrections, we used uniaxial stress values obtained from linear fits to tas a function of pressure ( Fig. 4 ), the cal- culated Cijsf r o mt h i ss t u d y[ Fig. 2(b) ] (this assumes that the calcu- lated elastic constants capture the true behavior of Ag at highpressure), and assumed α= 1. We used the ambient elastic constants reported in Refs. 41,48,a n d 49for Ag, Au, and Cu. Our corrected volumes are tabulated in Tables S1 –S4 in the supplementary material . W ef i tt h ec o r r e c t e dA gv o l u m ed a t at oaV i n e tE O S 46fixing the same parameters mentioned above. The difference in pres-s u r ef r o mA ua n dC ua sd e t e r m i n e df r o mt h em e a s u r e dl a t t i c e parameter ( a m) and the calculated hydrostatic lattice parameter (ap) are within the uncertainties of the pressure estimation, so we did not apply the hydrostatic correction for the pressuremarkers. We show the results from the fits to the corrected data from both experiments and the EOS parameters from our fit to the calculated volumes in Table I along with reported EOS parameters from previous DAC, ultrasonic, shock, and theoreti-cal studies. The fits to the DFT results were limited to a pressureof 416 GPa, which is equal to the maximum pressure achieved in our t-DAC experiments. TABLE I. EOS parameters from this study (Exp. 1) and previous studies. Bold values were fixed parameters in the fits; we show only the 0 K results from Dewaele et al.6 Reference V0(Å3) K0(GPa) K00 Technique, P range, EOS Experimental results Exp. 1 Ag and Au 17.031(1) 101 6.02(1) XRD, 0 –416 GPa, V Exp. 1 Ag and Au 17.031(1) 103(1) 5.92(4) XRD, 0 –416 GPa, V Exp. 2 Ag and Cu 17.031(1) 101 6.10(1) XRD, 0 –393 GPa, V Exp. 2 Ag and Cu 17.031(1) 104(1) 5.98(3) XRD, 0 –393 GPa, V Akahama et al.5reported 17.057 98 (1.6) 5.47(2) XRD, 0 –146 GPa, V Akahama et al.5re-calibrated 17.057 98 (1.6) 6.05(2) XRD, 0 –163 GPa, V Dewaele et al.617.021 101 5.97(2) XRD, 0 –122 GPa, V Kennedy and Keeler19101 6.19(1) RSW, V Al’tshuler et al.20101 6.15(1) RSW, V Holzapfel et al.317.057 101 6.15 Cal. with RSW and US Biswas et al.41101 6.19 US Barsch and Chang4799.7 6.31 US Theoretical results This Study Ag 16.744 110 6.17 DFTa Dewaele et al.616.046 137 5.87 LDA Dewaele et al.617.907 88 6.13 GGA-PBE Dewaele et al.616.694 116 5.98 GGA-PBEsol aResults from the present DFT all-electron, fully relativistic calculations within the PBEsol approximation. FIG. 4. tfor Ag as a function of pressure for both experiments plotted with the results reported by Akahama et al .5Linear fits to all three experiments are shown with solid lines; shaded regions are the 95% confidence bands from thefits. The results reported in Ref. 5are re-calibrated with the Au EOS reported in Ref. 2, and this is discussed in more detail below.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 125901 (2021); doi: 10.1063/5.0045374 129, 125901-5 Published under license by AIP Publishing.We show the corrected results from both experiments plotted with previous DAC results in Fig. 5 . Notably, the Ag pressure volume data reported by Akahama et al .5are seemingly at odds with the results of Dewaele et al.6and our new corrected results. The non-hydrostatic experiments reported by Akahama et al .5 show that Ag is more compressible than the quasi-hydrostatic He results reported by Dewaele et al.,6which is the opposite of what one would expect under uniaxial compression while collecting data in the axial geometry. Therefore, non-hydrostatic conditionscannot explain this discrepancy, and this behavior is likely an arti-fact related to other experimental parameters or analysis approaches. Moreover, the obtained Ag EOS parameters reported by Akahama et al. 5appear to be aberrant when compared to more recent studies ( Table I ). Perhaps, the next simplest experimental parameter to examine is the Au EOS used for pressure calibration.Akahama et al. 5used an Au EOS reported by Shim et al.50where the fitted EOS parameters appear to disagree with the more recently reported Au EOSs.2,25Indeed, if the results of Akahama et al.5are re-calibrated with these more recent Au EOSs,2,25they plot on top of our corrected results and the ones reported byDewaele et al. 6(Fig. 5 ). Moreover, the fitted EOS parameters agree well with the He results of Dewaele et al.6and our EOS parameters obtained from the corrected data ( Table I ). Thus, it appears that the Au EOS reported by Shim et al.50underestimates pressure by ∼17 GPa at 150 GPa. In the pressure range below ∼150 GPa, our corrected results agree well with the quasi-hydrostatic results of Dewaele et al.6suggesting that our choice of α= 1 is reasonable. We compare all the above EOS parameters in Fig. 6 by plot- ting the difference between the EOS parameters obtained from ourcorrected results from both experiments and the EOS parameters reported from different studies. We find that our results plot in between the RSW and US studies and the previous DAC results 5,6for both experiments. V0and K0for Ag are well constrained, but values reported for K00from DAC studies are consistently lower than the value of K00obtained in RSW or US studies. K00obtained from fitting a Vinet equation to our corrected data from experi- ment 1 is only 0.8% larger than previous DAC studies,5,6while it is as much as 2.8% smaller than RSW3,19,20and as much as 4.7% smaller than US41studies. The K00obtained from fitting a Vinet equation to our corrected data from experiment 2 is 2.2% larger than previous DAC studies,5,6while it is as much as 1.5% smaller than RSW3,19,20and as much as 3.4% smaller than US41studies. The EOS obtained from the corrected data from experiment 1 shows a pressure ∼5 GPa higher at 400 GPa than the extrapolated Ag EOS ’s from Akahama et al.5and Dewaele et al.,6while the dif- ference is 15 GPa higher for the EOS obtained from the corrected FIG. 5. The corrected results from both experiments plotted with the DAC results from Akahama et al .5and Dewaele et al .6Note that the results of Akahama et al.5have been plotted both as reported and as re-calibrated. FIG. 6. The difference between the Ag EOS calibrated with Au as the pressure standard experiment 1 solid black line at 0 on the yaxis (a) and with Cu as the pressure standard experiment 2(b) plotted as the difference between previous Ag EOSs. For DAC (Ref. 5), we used the re-calibrated EOS for comparison.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 125901 (2021); doi: 10.1063/5.0045374 129, 125901-6 Published under license by AIP Publishing.data from experiment 2. The reason for this difference is not obvious; it could be that the stress state in the two experiments was not identical (although the u niaxial stress was similar in both experiments), the pressure gradients could have been dif- ferent, which would influence the calculated “hydrostatic ”lattice parameter, or the choice of EOS for pressure calibration could also be a factor. The volumes obtained from our DFT calculations agree well with our DAC experiments, particularly at higher pressures[Fig. 3(a) ]. The discrepancy at lower pressure is mostly due to thermal expansion as the measurements are performed at room temperature, but even accounting for this, 38the theory underesti- mates the volume due to an imperfect exchange and correlationapproximation in DFT. The DFT obtained EOS parameters deviatefrom the experimentally obtained EOS parameters ( Fig. 6 ) and would overestimate pressure by ∼16 GPa based on experiment 1 and 12 GPa based on experiment 2 at a pressure of ∼350 GPa. This highlights an important point that the development of improvedtheoretical methods that can better estimate the volume of a crystal-line solid at room temperature and pressure is needed. Notably, theGGA-PBEsol calculated Ag EOS reported by Dewaele et al 6shows a similar trend to our newly calculated Ag EOS. V. CONCLUSIONS We have measured the atomic volume of Ag in the t-DAC in two separate experiments using Au and Cu as the pressuremarkers. We find that the uniaxial stress sustained by Ag increaseslinearly as pressure increases reaching a maximum of 4.5 GPa at 416 GPa, which agrees well with the non-hydrostatic measurements of Ag and Au up to 147 GPa reported by Akahama et al. 5The dif- ference in the fitted EOS parameters between the corrected andmeasured Ag results is quite small, but even a small difference canresult in a large overestimation of pressure when using an EOS for pressure estimation in the multi-Mbar regime. Indeed, it is impor- tant to conduct these ultrahigh-pressure static compression experi-ments to quantify the uniaxial stress that common pressuremarkers can sustain to advance our understanding of how the obtained EOS parameters change under these conditions. In our experiments, we observe the fccstructure up to 416 GPa at room temperature. Additionally, our calculated elasticconstants show that C 44softens above ∼100 GPa suggesting that thefccstructure is becoming unstable, and at a V/V0of 0.30, we find that the bccstructure has lower energy than the fccstructure. Although we did not search for the exact transition volume, this isan intriguing result since recently it has been shown experimentallyunder shock compression that Cu, Ag, and Au all transform to abccstructure at high pressure and temperature. 10–13More experi- mental and theoretical studies of these noble metals at high pres- sures and temperatures lower than the temperatures accessed alongthe shock Hugoniot would allow us to constrain the Clapeyronslope of the fcctobcctransition. SUPPLEMENTARY MATERIAL The supplementary material contains tables of the measured and corrected results from both experiments, the calculated pres- sure volumes of Ag from all-electron fully relativistic DFT, DFTcalculated high-pressure elastic constants of Ag, and a table with Vinet fit parameters of the measured data from both experiments. ACKNOWLEDGMENTS We thank C. Reynolds for engineering support at LLNL and C. Stan, D. J. O ’Hara, and T. Shakur for helpful discussions. This work was performed under the auspices of the U.S. Department of Energy (DOE) by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344 and was supported in part bythe LLNL-LDRD program under Project No. 17-ERD-038. We alsogratefully acknowledge support from DOE/NNSA ScienceCampaign-2 and LLNL Science Campaign management. Portions of this work were performed at HPCAT (Sector 16), Advanced Photon Source (APS), Argonne National Laboratory. HPCAT oper-ations are supported by DOE-NNSA ’s Office of Experimental Sciences. The Advanced Photon Source is a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under ContractNo. DE-AC02-06CH11357. DATA AVAILABILITY The data that support the findings of this study are available within the article, its supplementary material , or from the corre- sponding author upon reasonable request. REFERENCES 1D. E. Fratanduono, R. F. Smith, S. J. Ali, D. 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5.0036612.pdf

J. Chem. Phys. 154, 034701 (2021); https://doi.org/10.1063/5.0036612 154, 034701 © 2021 Author(s).Theoretical inspection of the spin-crossover [Fe(tzpy)2(NCS)2] complex on Au(100) surface Cite as: J. Chem. Phys. 154, 034701 (2021); https://doi.org/10.1063/5.0036612 Submitted: 09 November 2020 . Accepted: 24 December 2020 . Published Online: 15 January 2021 Carlos M. Palomino , Rocío Sánchez-de-Armas , and Carmen J. Calzado COLLECTIONS Paper published as part of the special topic on Special Collection in Honor of Women in Chemical Physics and Physical Chemistry ARTICLES YOU MAY BE INTERESTED IN Surface effects on temperature-driven spin crossover in Fe(phen) 2(NCS) 2 The Journal of Chemical Physics 153, 134704 (2020); https://doi.org/10.1063/5.0027641 On the theory of charge transport and entropic effects in solvated molecular junctions The Journal of Chemical Physics 154, 034110 (2021); https://doi.org/10.1063/5.0034782 Relaxation dynamics through a conical intersection: Quantum and quantum–classical studies The Journal of Chemical Physics 154, 034104 (2021); https://doi.org/10.1063/5.0036726The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Theoretical inspection of the spin-crossover [Fe(tzpy) 2(NCS) 2] complex on Au(100) surface Cite as: J. Chem. Phys. 154, 034701 (2021); doi: 10.1063/5.0036612 Submitted: 9 November 2020 •Accepted: 24 December 2020 • Published Online: 15 January 2021 Carlos M. Palomino, Rocío Sánchez-de-Armas, and Carmen J. Calzadoa) AFFILIATIONS Departamento de Química Física, c/Profesor García González, s/n 41012 Sevilla, Spain Note: This paper is part of the JCP Special Collection in Honor of Women in Chemical Physics and Physical Chemistry. a)Author to whom correspondence should be addressed: calzado@us.es ABSTRACT We explore the deposition of the spin-crossover [Fe(tzpy) 2(NCS) 2] complex on the Au(100) surface by means of density functional theory (DFT) based calculations. Two different routes have been employed: low-cost finite cluster-based calculations, where both the Fe complex and the surface are maintained fixed while the molecule approaches the surface; and periodic DFT plane-wave calculations, where the surface is represented by a four-layer slab and both the molecule and surface are relaxed. Our results show that the bridge adsorption site is preferred over the on-top and fourfold hollow ones for both spin states, although they are energetically close. The LS molecule is stabilized by the surface, and the HS–LS energy difference is enhanced by about 15%–25% once deposited. The different Fe ligand field for LS and HS molecules manifests on the composition and energy of the low-lying bands. Our simulated STM images indicate that it is possible to distinguish the spin state of the deposited molecules by tuning the bias voltage of the STM tip. Finally, it should be noted that the use of a reduced size cluster to simulate the Au(100) surface proves to be a low-cost and reliable strategy, providing results in good agreement with those resulting from state-of-the-art periodic calculations for this system. Published under license by AIP Publishing. https://doi.org/10.1063/5.0036612 .,s INTRODUCTION Spin crossover (SCO) molecules belong to the family of bistable molecules, capable of reversible switching from a low-spin (LS) elec- tronic state to a high-spin (HS) electronic state, induced by an external stimulus such as temperature, pressure, or light.1–5Most of these compounds are pseudo-octahedral Fe (II) complexes (d6) where the low and high spin states correspond to S = 0 (t 2g6 eg0) and S = 2 (t 2g4eg2), respectively. Recently, there has been a remarkable interest in these compounds for their potential applica- tions as molecular switches in signal processing, logic data manip- ulation, and information storage.2,3,6–8To be part of a molecu- lar device, the SCO molecule has to be integrated into a circuit and the spin transition must be preserved once the molecule is deposited on the support.9–11The interaction with the support can in fact shift the transition temperatures,12,13allow the coexis- tence of both phases at low temperatures,14–16and even suppress the spin transition,17blocking the molecule in one of the spin states.Two main groups of SCO molecules can be distinguished depending on the strength of the interaction – those weakly interact- ing with the substrate through van der Waals contacts and those with a strong chemical bond with the substrate, mainly through isoth- iocyanate (–NCS) and isoselenocyanate (–NCSe) ligands, where the terminal S and Se atoms act as surface–molecule anchoring groups. It has been reported that [Fe(phen) 2(NCS) 2], in direct con- tact with non-magnetic metallic surfaces18,19as Cu(100), Au(111), and Cu(111), coexists in both the high- and low-spin states but cannot be switched between them. This has been related to the strong interaction between the complex and the substrate, gov- erned by the chemisorption through the isothiocyanate groups. This strong interaction has also been exploited to build single-molecule STM junctions, such as that made up of the spin-crossover Fe(II) complexes [Fe(tzpy) 2(NCX) 2] (X = S or Se, tzpy = 3-(2-pyridyl)- [1,2,3]triazolo[1,5- a]pyridine), on a gold substrate using a magnetic nickel tip,20,21or Fe(terpyridine) 2functionalized with a thiol group on the 4 and 4′′positions of the phenyl rings, bridging two gold electrodes.22 J. Chem. Phys. 154, 034701 (2021); doi: 10.1063/5.0036612 154, 034701-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp In this study, we will focus on the [Fe(tzpy) 2(NCS) 2] complex deposited on the Au(100) surface. The adsorption of SCO com- plexes on this surface has been much less studied than on the other low-index Au-surfaces, but this surface presents the lowest surface energy together with the Au(111) one, and it is the second more abundant in gold nanoparticles.23Indeed, the square symmetry of the Au(100) crystalline surface structure offers a unique packing mode for the deposited molecules. In this [Fe(tzpy) 2(NCS) 2] complex, the Fe(II) metal center lies on an inversion center, and it is coordinated with two tzpy lig- ands occupying the equatorial positions of a compressed octahe- dron and two isothiocyanate ligands on axial positions. The exper- imental data show that it exhibits spin transitions from a low-spin (LS) S = 0 to a high spin (HS) S = 2 state with a critical tem- perature of T 1/2= 108 K for [Fe(tzpy) 2(NCS) 2]⋅2CHCl 3and 118 K for [Fe(tzpy) 2(NCS) 2]⋅H2O.24Additionally, two polymorphs with- out solvent molecules in the lattice have been isolated. Polymorph A exhibits a gradual transition between 180 K and 120 K, while an abrupt transition was observed for polymorph B with transition tem- perature at 102 K.25The enthalpy and entropy changes associated with the spin transition are estimated from calorimetric measure- ments for [Fe(tzpy) 2(NCS) 2]⋅2CHCl 3and [Fe(tzpy) 2(NCS) 2]⋅H2O with values of ΔHexp= 3.67 kJ mol−1and 4.08 kJ mol−1andΔSexp = 34.0 J K−1mol−1and 34.5 J K−1mol−1, respectively.24A quan- titative light-induced excited spin state trapping (LIESST) effect was observed for [Fe(tzpy) 2(NCS) 2]⋅H2O24and the non-solvated polymorphs.25 The aim of this work is to analyze, for the first time, the spin transition process of the [Fe(tzpy) 2(NCS) 2] complex and how it is modified by the deposition on a metal surface. First, geometry opti- mizations of the free molecule are carried out and the HS–LS energy difference is evaluated with different combinations of exchange- correlation functionals and basis sets. Next, the on-surface spin state switching of this complex is analyzed by two approaches. In the first approach, the Au(100) surface is represented by a finite clus- ter, and the total energy of the molecule + cluster entity is computed at different distances for on-top and bridge positions. In both cases, the isothiocyanate S atom acts as an anchoring point. In the sec- ond step, the deposition is inspected by means of periodic density functional theory (DFT) plane-wave based calculations, where both the surface and molecule are optimized. Our results show that the molecule retains the spin crossover properties once adsorbed, theHS–LS energy is enhanced, and the LS state is favored on the surface with respect to the HS one for all the explored binding positions. All these results are relevant for potential applications as a molecular device. Additionally, the use of a reduced size cluster to simulate the Au(100) surface proves to be a low-cost strategy, providing results in good agreement with those resulting from state-of-the-art peri- odic calculations. Finally, the STM images have been simulated for both spin states, and the results indicate that it is possible to distin- guish between both states using the STM bias voltage as a testing probe. DESCRIPTION OF THE SYSTEM AND COMPUTATIONAL DETAILS The main available geometrical parameters for the low- and room temperature structures of this complex are reported in Table I. It should be noted that the inclusion or exclusion of the solvent in the lattice of the SCO complex introduces slight changes in the structure, which induce significant differences in the magnetic prop- erties, as the transition temperatures.25,26We use as reference the [Fe(tzpy) 2(NCS) 2]⋅2CHCl 3crystal for which the x-ray data are avail- able for the room temperature structure and estimates of the transi- tion entropy and enthalpy have been reported.24In the FeN 6core, the axial positions belong to the isothiocyanate ligands (N thio), while the equatorial positions are occupied by the N atoms of the pyridine (Npy) and triazole (N triaz) groups. The theoretical study of this [Fe(tzpy) 2(NCS) 2] complex will be carried out using density functional theory (DFT) based approaches, both at molecular and periodic levels with the Gaussian 09 package27 and VASP code,28–31respectively. Although the reference theoreti- cal approach when dealing with SCO Fe(II) complexes32–37corre- sponds to wavefunction based methods such as CASSCF/CASPT238 and CASSCF/NEVPT2,39they are computationally prohibitive for the study of the deposition on the metallic surface. For this reason, we opt to use the DFT approaches in both scenarios, the isolated molecule and the molecule–surface interaction. As is well known, DFT methods provide accurate structures and vibrational spectra at a reasonable computational cost, but it can be a challenge to correctly describe the energy difference between the two spin states in the SCO complexes. The main difficulty resides on the rearrangements of the occupied orbitals accompanying the SCO TABLE I . Main geometrical parameters for the [Fe(tzpy) 2(NCS) 2] complex at low and high temperatures (LT, HT), from X-Ray data, compared to those resulting from TPPSh/def2-SVP calculations for the HS and LS states. Distance (Å)/angle (deg) Fe–N py Fe–N triaz Fe–N thio Fe–N thio–C(S) LT exp poly A252.021 1.973 1.935 174.98 LS calc 2.016 1.962 1.923 179.98 HT exp poly A252.217 2.181 2.097 174.46 HT exp poly B252.193 2.174 2.107 177.53 HT exp CHCl 3242.204 2.211 2.113 170.75 HS calc 2.251 2.251 2.027 168.64 J. Chem. Phys. 154, 034701 (2021); doi: 10.1063/5.0036612 154, 034701-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp processes, which are exchange-correlation sensitive.40Despite the big effort dedicated to obtain a systematic methodology to study the energy of the spin states in different transition metal com- plexes,41–45there is not a definitive XC functional able to com- pute the relative energy of the LS and HS states, although there exists a certain consensus about the reliability of some function- als. In this study, we first address the evaluation of the HS–LS energy by means of a set of exchange-correlation (XC) function- als and basis sets of different quality. The aim is to establish the best performing XC functional/basis set combination to ana- lyze the spin transition in this Fe(II) complex. The tested XC functionals range from BP8646,47[non-hybrid generalized gradient approximation (GGA)], TPSSh48,49(hybrid meta-GGA, 10% Fock exchange), B3LYP,50and PBE051(20% and 25% Fock, respectively), and the employed basis sets are SDD,52,536-31G(d,p),54–57def2- SVP,58TZVP,59and the combination QZVP58for Fe and TZVP for all non-Fe atoms. The Gaussian 09 package27is employed for this set of calculations. Additionally, for comparison, we have carried out the geometry optimization of the free molecule with the ORCA code60using both TPSSh and rPBE functionals with the def2-SVP basis set. To simulate the deposition on the Au(100) surface, we have performed calculations based on the finite size cluster approach and periodic calculations. In both cases, the hexagonal reconstruction of the Au(100) surface is not taken into account, and all the models refer to the native Au(100) bulk surface. First, a three-layer cluster containing 42 Au atoms (21 + 12 + 9) is used. The interlayer sepa- ration is 2.07 Å, and the Au–Au distance inside each layer is 2.95 Å. The Fe(II) complex is placed on top of the central Au site, with one of the axial −NCS ligands pointing to the surface, or in a bridge posi- tion where the S atom is anchored to two Au atoms. The total energy of the LS and HS solutions has been evaluated at different Au–S dis- tances when the molecule approaches the surface, maintaining fixed its geometry to that of the isolated molecule. The gold atoms are represented with the LANL2DZ basis with the LANL2 relativistic effective core potential.61The Gaussian 09 package27is employed in all these calculations. Finally, the adsorption of the [Fe(tzpy) 2(NCS) 2] complex on the Au(100) surface has been studied within periodic DFT with the VASP (Vienna ab initio simulation package) code28–31 using the revised Perdew–Burke–Ernzerhof (rPBE) functional62and projector-augmented wave (PAW) potentials.63,64The rPBE func- tional is computationally less expensive than the hybrid ones as TPSSh, and it has been proven to provide a good LS–HS balance (much better than other GGA functionals such as PBE) for well- known SCO complexes containing Fe(II) and Fe(III).21,22,43We have tested this functional in a previous study devoted to the deposi- tion of the SCO [Fe((3,5-(CH 3)2Pz) 3BH) 2] complex deposited on the Au(111) surface,65with excellent agreement with wavefunction based methods such as CASSCF/CASPT2 and CASSCF/NEVPT2 and the available experimental data. Compared with previous DFT+U calculations on the same system,66the rPBE functional presents the main advantage of being parameter-free. In fact, LDA+U calculations of the [Fe((3,5-(CH 3)2Pz) 3BH) 2] complex showed that the HS–LS energy difference is dramatically sensitive to the U value, much more than usual, providing results in agreement with the experimental data only for a dramatically narrow window of U values between U = 6.5 eV and 6.6 eV.66Valence electronsare described using a plane-wave basis set with a cutoff of 500 eV, and the Γ-point of the Brillouin zone is used.67The optimized lattice parameters for the Au bulk are a = b = c = 2.97 Å. This calculated value has been used for the (100) surface throughout the present work and maintained fixed during the atomic position relaxation. The Au(100) surface is represented by a slab containing 192 atoms and four layers (23.607 ×17.705 Å2). The atoms of the lowest layer are kept fixed at bulk optimized positions, while the three upper lay- ers as well as the Fe complex have been relaxed. 28 Å of vacuum has been added in the z direction to avoid the interaction between the slabs. The unit cell is big enough to avoid interaction between molecules, as the closest H–H contacts between two molecules along theaandbaxes are found at 8.1 Å and 7.0 Å, respectively. Three starting geometries for geometry optimizations have been used, with one isothiocyanate S atom placed in the top, bridge, or hol- low position. Electronic relaxation has been performed until the change in the total energy between two consecutive steps is smaller than 10−6eV, and the ionic relaxation has been performed until the Hellmann–Feynman forces were lower than 0.025 eV/Å. As we are interested in the different magnetic solutions, the NUPDOWN option is used, which forces the difference between the number of electrons in the up and down spin channels, N α–Nβ, to be equal to 0 (LS) or 4 (HS). Adsorption energies, E ads, were calculated with respect to the isolated complex on a 23.607 ×17.705 ×35.039 Å3box as E ads = E adsorbed_complex −(Eslab+ E complex ). Thus, negative adsorption ener- gies represent bound states. The STM simulations with two differ- ent bias voltages ( −1.5 V and −0.5 V) were carried out using the Tersoff–Hamann approximation.68Constant-height STM images were finally visualized in the p4vasp program using density values of 0.00 e/Å3and 100 e/Å3as low and high boundaries, respectively. Charge density isosurfaces at 0.05 e/bohr3are represented with the VESTA code.69 RESULTS Isolated molecule: Choice of the XC functional and basis sets The Fe(II) complex has been optimized with different basis sets and XC functionals in both spin states and the result- ing geometries compared to the available x-ray data for the [Fe(tzpy) 2(NCS) 2]⋅2CHCl 3complex at a high temperature. For all the considered functionals, the calculated bond distances between the Fe center and the axial ligands, Fe–N thio, are under- estimated, while a slight overestimation of the distances between the metal center and the equatorial ligands Fe–N pyand Fe–N triazis observed. The mean relative errors in Fe–N bond distances are small (1.5%–3.5%) for all the considered functional/basis set combinations (Fig. 1). In fact, the largest deviation comes from the Fe–N–C(S) bond angle (exp 170.7○), underestimated for most of the functionals (mean value for the explored basis sets: 158.7○, 163.6○, and 169.2○ for PBE0, TPSSh, and BP86, respectively) and slightly overestimated for B3LYP (173.4○). The BP86 functional results to be less sensitive to the basis set quality. In fact, it is common to optimize the mod- els employed in benchmark studies42,43at the BP86/def2-SVP basis level. J. Chem. Phys. 154, 034701 (2021); doi: 10.1063/5.0036612 154, 034701-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . Mean relative error (%) for Fe–N bond distances for the different XC functional/basis set explored. Once optimized, the energy difference between the HS and LS state is evaluated for each functional/basis pair (Table II) and compared to the experimental enthalpy change associated with the spin transition, estimated from calorimetric measurements { ΔHexp = 3.67 kJ mol−1and 4.08 kJ mol−1for [Fe(tzpy) 2(NCS) 2]⋅2CHCl 3 and [Fe(tzpy) 2(NCS) 2]⋅H2O, respectively}.24PBE0 favors the HS state over the LS one, providing a qualitatively incorrect HS–LS energy difference (negative instead of a positive value) for all the explored basis sets. Similar results are obtained with B3LYP for most of the chosen basis sets. Both TPSSh and BP86 correctly favor the LS state, with a better agreement with the experimental transition enthalpy for TPSSh, in particular for def2-SVP, TZVP, and the com- bination of QZVP(Fe)+TZVP basis sets. This result is in line with previous benchmark calculations on a set of Fe(II) and Fe(III) SCO complexes.37,41–43The geometrical parameters are of similar quality for these three basis sets, but the dimension of the atomic orbital basis is noticeable smaller for def2-SVP. Hence, we decide to use the TPSSh/def2-SVP combination for the study of the deposition of this Fe(II) complex on the Au(100) surface and refine the energetic TABLE II . Energy difference (in kJ mol−1) between the optimized geometries of the LS and HS states, HS–LS, with different XC functional/basis pairs. The gray cells correspond to qualitatively correct results, and the white cells correspond to those that predict a wrong sign or a large overestimation of the HS–LS difference. XC functional Basis sets B3LYP BP86 PBE0 TPSSh SDD −19.9 68.1 −58.7 −10.1 def2-SVP −29.1 77.8 −54.0 21.5 6-31G(d, p) −19.4 89.9 −44.9 35.9 TZVP 20.9 74.9 −56.2 20.9 QZVP(Fe)+TZVP −30.6 77.2 −56.9 20.3parameters of the deposited molecules by means of single-point cal- culations with higher quality basis sets. The bond distances and angles for LS and HS states obtained with this strategy are collected in Table I. Additionally, we have also evaluated the performance of the rPBE functional for completeness since the periodic calculations in the section titled Adsorption on Au(100) surface will use this functional. We have carried out the geometry optimization of the free molecule with the ORCA code (rPBE is not available in the Gaussian code) using both TPSSh and rPBE functionals with the def2-SVP basis set. TPSSh is included to check the potential deviations due to the different computational code. The HS–LS energy gaps predicted by these two functionals are consistently similar, with a value of 22 kJ/mol for TPSSh (to be compared with the value of 21.5 kJ/mol when using the Gaussian code, Table I) and 19 kJ/mol in the case of rPBE one, in excel- lent agreement with the HS–LS gap obtained for the isolated molecule using periodic plane-wave calculations [20.3 kJ mol−1, see the section titled Adsorption on Au(100) surface]. For the free molecule, the difference in the zero-point energy (ZPE) of the LS and HS states is ZPE(HS)–ZPE(LS) = −10.28 kJ mol−1, evaluated from analytic frequency calculations on the opti- mized geometries at the TPSSh/def2-SVP level. As is well known, the ZPE is more important for the LS state in line with the shorter Fe–N bond distances and then stronger Fe–N bonds. Hence, the zero-point correction favors the HS state in SCO Fe complexes, reducing the HS–LS separation. Thus, the HS–LS separation reduces to 11.2 kJ mol−1for TPSSh/def2-SVP and 10.0 kJ mol−1at the TPSSh/QZVP+TZVP level once the zero-point correction is taken into account, and these values are in reasonable agreement with experimental data. The molecular orbital diagram for the Fe 3d-like orbitals is shown in Fig. 2 for both spin states. The separation between the barycenter of the t 2g-like and e g-like orbitals is 0.2 hartree for LS, while it reduces to 0.1 hartree in the HS state (Fig. 2). Hence, the lig- and field of the Fe center is stronger for the low-temperature struc- ture than the high-temperature one. Consequently, the contribution of the ligands to these orbitals is greater for the LS state than for the HS one, as shown in Fig. 2. Adsorption on Au(100) surface The adsorption of the Fe(II) SCO complex on the Au(100) sur- face has been studied by two different strategies. First, we employ a cluster model for the metal surface and analyze the change in the total energy of the LS and HS states when the molecule approaches the surface, in an on-top position or a bridge position (Fig. 3). In all these calculations, the geometry of the molecule is fixed to the optimized structure of the free molecule on each spin state. For the on-top position, the LS molecule presents the minimum energy at 2.7 Å above the surface, while the optimal isothiocyanate– surface distance for the HS states is 2.6 Å. The LS solution is more stable than the HS one for all the considered distances (Fig. 4), and the bridge position is favored over the on-top one, as observed for the interaction of the isothiocyanate groups of [Fe(phen) 2(NCS) 2] with the Cu(001) surface70and Co/Cu(111) surface.71The bridge sites are found to be also favored for the deposition of alkanethiols on the Au(100)72and Au(111)73,74substrates. J. Chem. Phys. 154, 034701 (2021); doi: 10.1063/5.0036612 154, 034701-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Molecular orbital diagram for the Fe 3d-like orbitals of the LS(left) and HS(right) states. For the HS state, the 3dxy orbital is doubly occupied, while in the LS state, the 3dx2−y2and 3dz2 orbitals are unoccupied. The MO ener- gies are obtained at the TPSSh/def2- SVP level. FIG. 3 . Geometries adopted for the LS and HS molecules deposited on the Au(100) surface in on-top and bridge positions. In the bridge position, the molecule is closer to the surface, the optimal molecule–surface distance is 2.3 Å in both states, cor- responding to a S–Au distance of 2.73 Å. The separation between the LS and HS curves is almost the same for the two considered Au–S coordination contacts, and in both cases, the transition energy FIG. 4 . Total energy (kJ mol−1) of the [Fe(tzpy) 2(NCS) 2] complex, deposited in on- top and bridge sites, in HS and LS states, relative to the LS molecule in the bridge position. All values are at the TPSSh/def2-SVP level.is larger than for the free molecule. Then, the surface induces an enhancement of the relative stability of the LS molecules. In the case of the HS molecule, since the isothiocynate groups are not orthog- onal to the tzpy plane, the on-top curve can be estimated with the −NCS axial group perpendicular to the surface as in the LS molecule (HS top in Fig. 3) or by maintaining the tzpy plane parallel to the surface (HS top parallel in Fig. 3). It is interesting that the interac- tion curves are almost the same (straight cyan and dotted red lines in Fig. 4), regardless of the relative orientation of the molecule with respect to the surface. This indicates that the tzpy ligands are far enough from the surface and do not contribute with any additional interaction with the gold atoms. The adsorption energy can be calculated as E ads= E molec+surf −(Emolec + E surf). Hence, a negative E adsvalue means that the molecule-on-surface is stabilized with respect to the free molecule. For the def2-SVP basis, a large basis set superposition error (BSSE) is observed, mostly due to the Fe complex, described with a basis set of poorer quality than the metal cluster. To reduce the BSSE, single- point calculations have been performed for the optimal interaction distance with basis QZVP for Fe and TZVP for the remaining atoms. Similar results are obtained when using the def2-TVZPP basis. The interaction with the surface favors the LS state, and the interaction energy at the optimal S–Au distance is −26.1 kJ mol−1for the LS state and −18.0 kJ mol−1for the HS state for the on-top position, once the BSSE is corrected by the counterpoise approach.75For the bridge positions, the interactions energies are stronger with values of−53.6 kJ mol−1for the LS state and −44.0 kJ mol−1for the HS one. This spin-dependent adsorption produces an enhancement of the J. Chem. Phys. 154, 034701 (2021); doi: 10.1063/5.0036612 154, 034701-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . The deposition of the SCO complex on Au(100) from periodic cal- culations: on-top (left), bridge (middle), and hollow (right) positions. The LS molecules are represented, similar to the HS ones (Fig. S1). HS–LS energy difference for the deposited molecule (29.1 kJ mol−1 for the on-top position and 30.6 kJ mol−1for the bridge position) with respect to the free molecule (20.3 kJ mol−1) at the same level of calculation. In the second step, we performed periodic DFT calculations of the molecule deposited on three different positions, on-top, hollow, and bridge (Fig. 5). The geometry of the complex and the upper layers of the gold slab have been fully optimized. The purpose is twofold: first to refine the previous calculations where the geom- etry of the molecule was fixed to the optimal structure of the iso- lated molecule for each state and second to test the reliability of the calculations based on finite clusters. Table III shows the relative energy of the molecule on the three explored positions with respect to the most stable one, the bridge position of the LS molecule. We also report the HS–LS energy for each position as well as the adsorption energies, which reflect the relative stability of the deposited molecule with respect to the free one. The complex adsorbs on the three explored sites (negative E ads in all cases), being the bridge position the preferred one for both spin states, in line with the cluster-based results. The relative stability of the different positions is small (between 1 kJ mol−1and 4 kJ mol−1 for on-top with respect to bridge, about 8 kJ mol−1for the difference between hollow and bridge), and then, molecules will be distributed in the different coordination sites on the surface, in particular at room temperature. For the three considered adsorption sites, the HS–LS energy is enhanced about 15%–25% with respect to the free molecule(29.4 kJ mol−1at the same level of calculation). The same effect has been observed for the [Fe((3,5-(CH 3)2Pz) 3BH) 2] complex deposited on the Au(111) surface,65[Fe(phen) 2(NCS) 2] on metallic sub- strates,70,71and [Fe(H 2Bpz 2)2(bipy)] encapsulated in single-walled carbon nanotubes.76Hence, the deposition on the surface impacts the relative stability of the HS and LS phases. Although the calcu- lated HS–LS energy is just a rough estimate of the transition enthalpy since effects such as the zero-point correction or collective effects are not included in our evaluations, the fact that the adsorption energy is spin-dependent suggests that the deposition impacts the SCO prop- erties and could modify the transition temperatures as observed for the [Fe((3,5-(CH 3)2Pz) 3BH) 2] complex supported on the Au(111) surface12and for [Fe(H 2Bpz 2)2(bipy)] encapsulated in single-walled carbon nanotubes.76 The strongest adsorption energies are found for the bridge positions and follow the order bridge >on-top >hollow. Note that, unlike the cluster-based calculations, these adsorption ener- gies also take into account the deformation experienced by the molecule and surface due to the deposition. The adsorption ener- gies are about 50 kJ mol−1, spin-dependent, −59.2 kJ mol−1for LS and −51.4 kJ mol−1for HS for the bridge site, and larger than those calculated for systems where the molecule–substrate deposition is governed by van der Waals interactions, such as [Fe((3,5-(CH 3)2Pz) 3BH) 2] on Au(111)65and [Fe(H 2B(pz) 2)2(bipy)] confined in a single-walled carbon nanotube.76Our estimates are however four times less than the values reported for the chemisorp- tion of [Fe(phen) 2(NCS) 2] on metallic surfaces.70This suggests that TABLE III . Relative energy of the deposited LS and HS molecules (kJ mol−1), HS–LS energy difference, and adsorption energy in on-top, bridge, and hollow positions on the Au(100) surface from periodic calculations. The HS–LS transition energy for the free molecule is 29.4 kJ mol−1at the same level of calculation. Relative energy Adsorption energy Position LS HS HS–LS LS HS On-top 4.4 38.3 33.9 −54.8 −50.4 Bridge 0.00 37.3 37.3 −59.2 −51.4 Hollow 7.9 44.5 36.5 −51.3 −44.2 LS state S–Au distances (Å) HS state S–Au distances (Å) On-top 2.556 2.673 Bridge 2.675 2.644 2.723 2.686 Hollow 2.811 2.848 2.825 2.868 2.884 2.863 2.970 2.989 J. Chem. Phys. 154, 034701 (2021); doi: 10.1063/5.0036612 154, 034701-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the on-surface SCO behavior of the [Fe(tzpy) 2(NCS) 2] complex could be different from that reported for the [Fe(phen) 2(NCS) 2] molecule. The adsorption always favors the LS state (with larger Eads), as has been also observed in our cluster-based calculations. The optimal distance to the surface increases with the number of S–Au contacts (d on-top <dbridge <dhollow ), in reasonable agreement with the minima in Fig. 4, and they are slightly shorter for the LS than the HS molecules. This is in line with a stronger interaction for the LS molecules, and both can be related to the strength of the Fe ligand field that enhances the metal–ligand hybridization in the LS state. Figure 6 (bottom) shows the density of states of the LS and HS molecules deposited on the Au(100) surface, projected on the Fe atom, isothiocyanate, and tzpy ligands. The most significant difference between both states is the hybridization between the Fe and the ligand states, which is larger for the LS state, in particular for the unoccupied bands. This result is in line with the composi- tion of the Fe 3d-like orbitals discussed above (Fig. 2), where the contribution of the ligands to the eg-like orbitals (3dx2−y2and 3dz2) is more important for the LS state than the HS one. In the case of the occupied bands, the main metal–ligand hybridization proceeds through the isothiocyanate groups for the bands close tothe Fermi level, as observed for the 3dxz-like and 3dyz-like orbitals in Fig. 2. Comparing with the projected DOS of the free molecule (Fig. 6 top), the main difference comes from the NCS group clos- est to the surface (NCS1 in Fig. 6), and the states, indistinguishable to those of the NCS2 for the free molecule, strongly differentiate when the molecule is deposited, due to the mixing with the occupied and empty states of the gold surface, in line with the formation of a bond between the terminal S atom and the surface gold atoms. Addi- tionally, the states of the tzpy ligands are slightly shifted to lower energies. Taking into account the role of the NCS groups in the low- lying occupied bands and the fact that they appear at different ener- gies depending on the spin state, it could be possible to distinguish the adsorbed molecules using a STM tip, as observed for [Fe((3,5- (CH 3)2Pz) 3BH) 2] on Au(111).12Our simulated STM images (Fig. 7), based on the DOS resulting from the rPBE calculations, indicate that for a negative bias of −0.5 V, the STM image for HS molecules is completely dark, and only the LS molecules are bright. For a bias of −0.5 V, only the states with energy between the Fermi and −0.5 eV can be probed by the STM tip. In the case of the HS molecule, the occupied states in this range are centered on the Fe site as well as on FIG. 6 . Projected density of states on Fe, isothiocyanate groups (–NCS1 corresponds to the group closest to the surface), and tzpy ligands for the LS (left) and HS (right) states of the free molecule (top) and deposited in a bridge position (bottom). J. Chem. Phys. 154, 034701 (2021); doi: 10.1063/5.0036612 154, 034701-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 7 . Constant-height simulated STM images with two different bias voltages (−0.5 V and −1.5 V) for LS (left) and HS (right) states. The images contain 4 ×4 unit cells. On the right-hand side of each STM image, the charge density isosurface at 0.05 e/bohr3for the corresponding bias voltage is represented. the gold surface, both far away from the STM tip. The charge density isosurface at 0.05 e/bohr3is represented in Fig. 7 (right-hand side) for each spin state and bias voltage. In the case of the HS molecule with a negative bias of −0.5 V, the density is placed on the Fe 3d orbital, in line with the projected DOS in Fig. 6. However, for the LS molecule, the projected DOS shows a significant contribution of the terminal NCS group, and the states spread enough to be probed by the tip, as the charge density isosurface indicates. Then, at this bias voltage, only the LS molecules can be imaged by STM. For negative bias smaller than −1 V, the spots are similar for both states, in line with the main features of the density of states, and the shape of the charge density isosurface is represented in Fig. 7. Then, it is possible to distinguish the spin state of the deposited molecule using the bias voltage of the STM tip as a probe. DISCUSSION AND CONCLUDING REMARKS The on-surface spin state switching of Fe complexes is of rel- evance for their potential applications in molecular electronics and spintronics, information storage, and sensing. We explore for the first time the interaction of [Fe(tzpy) 2(NCS) 2] with the Au(100) sur- face by means of DFT calculations. Our calculations indicate that the molecule can adsorb on the three different sites with similar ener- gies, although the bridge coordination is energetically favored, as observed for other Fe SCO complexes on metallic surfaces as well as alkanethiols. The interaction with the surface is spin-dependent, favors always the LS state, and impacts the SCO properties increasing the HS–LS energy difference. This suggests higher critical tem- peratures than those reported for the bulk material. It should be noted that the adsorption energies are higher than those esti- mated for weakly interacting SCO complexes, such as [Fe((3,5- (CH 3)2Pz) 3BH) 2] on Au(111), but noticeably smaller than those reported for Fe(phen) 2(NCS) 2on Au(111), Cu(100), and Cu(111), whereas in this case, the complex chemisorbs on the metallic substrate.In addition to these results, this study brings information about the reliability of the calculations based on finite clusters as models of the gold surface, confronted with the results provided by state- of-the-art periodic DFT evaluations. Our results show a qualitative agreement between both sets of calculations, and the cluster-based calculations correctly predict the optimal adsorption sites and give interaction energies of the same order than those resulting from the periodic calculations, once the basis set superposition errors are corrected. They also agree that the deposition produces an enhance- ment of the relative stability of the LS state and the spin-dependent hybridization of the 3d-like orbitals due to the different strength of the Fe ligand field on each spin state. The reliability of these cluster- based calculations could be partially ascribed to the fact that the interaction is governed by the terminal isothiocyanate group, and the rest of the molecule has almost no role in the energetics of the deposition. Probably, the scenario would be different for complexes interacting via weak van der Waals contacts where more extended clusters would be necessary to take into account these longer range interactions. Finally, the simulated STM images show that the spin state of the molecule once deposited can be determined using the bias volt- age of the STM tip, and this could be particularly useful at low temperatures, if, as observed for other strongly interacting SCO complexes, the [Fe(tzpy) 2(NCS) 2] molecules coexist in both spin states. SUPPLEMENTARY MATERIAL See the supplementary material for the figures for the optimized HS molecules deposited on Au(100) in on-top, bridge, and hollow positions resulting from periodic calculations. DEDICATION This paper is dedicated to women scientists of all times who made our work possible today. ACKNOWLEDGMENTS The authors acknowledge the financial support by the Min- isterio de Ciencia e Innovación-Agencia Estatal de Investigación (Spain) and FEDER funds through Project Grant No. PGC2018- 101689-B-I00 (MCI/AEI/FEDER, UE). R.S.-d.-A. thanks VPPI-US for the financial support. 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5.0027709.pdf

J. Appl. Phys. 129, 054502 (2021); https://doi.org/10.1063/5.0027709 129, 054502 © 2021 Author(s).Proof-of-concept thermoelectric oxygen sensor exploiting oxygen mobility of GdBaCo2O5+δ Cite as: J. Appl. Phys. 129, 054502 (2021); https://doi.org/10.1063/5.0027709 Submitted: 31 August 2020 . Accepted: 17 January 2021 . Published Online: 03 February 2021 Soumya Biswas , M Madhukuttan , and Vinayak B. Kamble COLLECTIONS Paper published as part of the special topic on Phase-Change Materials: Syntheses, Fundamentals, and Applications ARTICLES YOU MAY BE INTERESTED IN Magnetocaloric and heat capacity studies on NdFe 0.5Mn0.5O3 Journal of Applied Physics 129, 053902 (2021); https://doi.org/10.1063/5.0032627 A method for characterizing near-interface traps in SiC metal–oxide–semiconductor capacitors from conductance–temperature spectroscopy measurements Journal of Applied Physics 129, 054501 (2021); https://doi.org/10.1063/5.0037744 Evidence of improvement in thermoelectric parameters of n-type Bi 2Te3/graphite nanocomposite Journal of Applied Physics 129, 055108 (2021); https://doi.org/10.1063/5.0030745Proof-of-concept thermoelectric oxygen sensor exploiting oxygen mobility of GdBaCo 2O5+δ Cite as: J. Appl. Phys. 129, 054502 (2021); doi: 10.1063/5.0027709 View Online Export Citation CrossMar k Submitted: 31 August 2020 · Accepted: 17 January 2021 · Published Online: 3 February 2021 Soumya Biswas, M Madhukuttan, and Vinayak B. Kamblea) AFFILIATIONS School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram 695551, India Note: This paper is part of the Special Topic on Phase-Change Materials: Syntheses, Fundamentals, and Applications. a)Author to whom correspondence should be addressed: kbvinayak@iisertvm.ac.in ABSTRACT In this paper, we demonstrate a proof-of-concept oxygen sensor based on the thermoelectric principle using polycrystalline GdBaCo 2O5+δ, where 0.45 <δ<0.55 (GBCO). The lattice oxygen in layered double perovskite oxides is highly susceptible to the ambient oxygen partial pressure. The as-synthesized GBCO sample processed in ambient conditions shows a pure orthorhombic phase ( Pmmm space group) and aδ-value close to 0.5 as confirmed by x-ray diffraction Rietveld refinement. The x-ray photoelectron spectroscopy (XPS) shows a signifi- cant Co3+oxidation state in non-octahedral sites in addition to Co3+as well as Co4+in octahedral sites. The insulator-to-metal transition (MIT) is observed at 340 K as seen from resistivity and Seebeck coefficient. The Seebeck coefficient shows a large change of 10 –12μV/K with a time constant of ∼20 s at 300 K, when the gas ambience is changed from 100% oxygen to nitrogen and vice versa. The diffusion of o x y g e ni nt h eG d O δplanes leads to the hole doping, which is a dominant factor for a large change observed in the Seebeck coefficient. This is also evident from the higher fraction of oxidized Co4+as seen from XPS measurements. The interfacial grain boundary in addi- tion to the oxygen diffusion contributes to the change in Seebeck. The change in Seebeck coefficient is minimal in the metallic state dueto an insignificant increase in the carrier concentration, but the response is fairly well and reproducible for stoichiometry δ=0 . 5 ±0.05 below MIT. This principle shall be of significant importance in de signing oxygen sensors operational at room as well as cryogenic temperatures. Published under license by AIP Publishing. https://doi.org/10.1063/5.0027709 I. INTRODUCTION Oxygen sensors are widely exploited for their potential use in domestic, industrial, defense, and space applications. They are uti- lized for monitoring oxygen breathability at high altitudes, main-taining anaerobic conditions in food preservation, maximizingefficiency of the automobile combustion engine, ensuring explo-sive safety limits, and for various other strategic applications. 1–4 The precise design and the versatility of these sensors for a wide oxygen concentration range is technologically a challenging problem.The current generation of oxygen sensors is mostly based on amper-ometric or potentiometric princip les using solid e lectrolytes such as ZrO 2.5The standard operating temperature of these sensors is usually very high, i.e., nearly 400 –500 °C, which hence requires a heater for its operation and increases the power consumption. Inan amperometric mode of operation, a small pinhole (a fewmicrometers in diameter) is used to limit the diffusion of gases and is difficult to achieve. Typically, in a potentiometric mode, acontinuous supply of a reference gas is required for the opera- tion. 1,2,6Thus, there is a demand for a reference free, low temper- ature operating oxygen sensor. In this direction, sensors based on the thermoelectric principle have been explored for sensing atmospheric gases.7–9 Several researchers have reported studies on thermoelectric gas sensors for the detection of gases like H 2, CO, volatile organic com- pounds, etc. Goto et al.10had demonstrated a CO sensor with a double structure of AuPtPd -SnO 2on the hot side and Pt/ α-Al2O3 on the cold side of a micro-machined Thermoelectric (TE) device. Similarly, Park et al.11had designed a thermopile based H 2and O 2 sensor which detects the change in the temperature on the working thermopile due to heat released in the catalytic reaction. The background noise is eliminated by the use of a reference thermopile. The metal catalysts such as Ag and Cu –Bi thermo- couples were deposited on a thermally resistant SU-8 polymer. The reaction between the metal catalyst and oxygen or hydrogenJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 054502 (2021); doi: 10.1063/5.0027709 129, 054502-1 Published under license by AIP Publishing.acts as a heat source for thermoelectric voltage generation. Similarly, several oxide based TE gas sensors are also reported such as Li doped NiO based TE gas sensors for hydrogen gas detection at 50 –100 °C.12BaFe (1−x)−0.01Al0.01TaxO3−δthick film based oxygen sensors utilized co mbined resistive and TE effects for oxygen sensors13operating at 600 –800 °C. In this paper, a simple, proof-of-concept oxygen sensor has been proposed that exploits the sensitivity of thermoemf to the oxygen partial pressure and operates at room temperature and even lower. The response is measured in terms of change in an opencircuit voltage for a constant temperature difference. The response islarge below the metal insulator transition temperature of GBCOwhich is 340 K. In such a double perovskite cobaltate system, the response is generated as a result of mobile lattice oxygen. These cobaltate perovskites are generally complex crystals and are stronglycorrelated systems. 14–18These compounds are of great interest due to their correlated spin and charge degrees of freedom, which is a result of strong correlation among entities like oxygen content (carrier doping), rare earth ion radius, and nonstoichiometry ofcentral B-site ions (charge and spin). 15–18Particularly, in AA0B2O6−δ type structures such as ReBaCo 2O6, the non-stoichiometry of central (B site) Co ions leads to the introduction of various exotic properties like electron –hole asymmetry,19metal –insulator transitions,20mag- netic phase transitions,17oxygen ion mobility, etc. Besides, in ReBaCo 2O6−δdoping at the Co site or varying the oxygen content of the lattice induces either hole doping or electron doping in thesystem. 17,18The oxygen ions incorporated in the system play a vital role in the properties of these layered cobaltates.5,17,18The weak bonding of oxygen in the ReO δlayer enhances the oxygen diffusivity and hence, in turn, increases the mobility of oxygen through thesurface. 21,22This increase in the mobility of oxygen results in a sig- nificant change in the transport properties of these systems. It is pro- posed that the equilibrium δ-value of the system at a given temperature is logarithmic dependent on the oxygen partial pressure(except for δ=0 ) . 17Hence, these double perovskite (AA0B2O5+δ) systems are rich in phases, like antiferromagnetic insulator, ferro- magnetic insulator, ferromagnetic metal, paramagnetic metal, etc.17,23They distinctly show insulator-to-metal transition (MIT) (for intermediate δ-values) near room temperature (340 K for GBCO), wherein there is a small change in the structure within theorthorhombic symmetry. An order –disorder phase transition is observed at relatively high temperatures (723 K for GBCO), due to the structural transformation from an orthorhombic to tetragonalphase. 5,24This order –disorder phase transition is particularly charac- terized by the rearrangement of oxygen ions and their vacancies inthe lattice. This rearrangement results in one-dimensional ordered oxygen vacancies along the aaxis and two-dimensional distributed vacancies in the ReO δplane at low and high temperatures, respec- tively. The ease in the mobility of oxygen causes a change in theelectronic properties of the materials and can be utilized for moni-toring oxygen levels for various applications like oxygen storage or fuel cells, etc. 25The beauty of these layered rare earth cobaltates is that the oxygen concentration in the ReO δplanes can be controlled over a wide range by varying the processing conditions such asannealing. Since the equilibrium δvalue at a given temperature is governed by the size of rare earth ion Re +3,t h eG d+3ion is chosen among the lanthanide series. The δvalue of air synthesized Gdc o b a l t a t es a m p l e si sn e a r0 . 5 ,w h i ch allows the possibility of the electron as well as hole doping. Thus, the crucial role of mobile lattice oxygen in these double perovskite structures needs to bestudied thoroughly for exploring their potential device applica- tions in monitoring oxygen with reliable sensitivity and mayprove to be a straightforward method to measure ambient oxygen in terms of thermoemf. II. EXPERIMENTAL A. Synthesis GdBaCo 2O5+δwas prepared by the solid-state synthesis techni- que. In a typical synthesis, gadolinium oxide (from Sigma Aldrich,Germany) was preheated at 800 °C for 12 h to remove any absorbedvolatilities. The stoichiometric amounts of the rare earth oxides were mixed with BaCO 3(from Sigma Aldrich, Germany) and Co (NO 3)2⋅6H2O (from Sigma Aldrich, Germany) to obtain a yield of 10 g of GdBaCo 2O5+δ. The composition was ground for 2 h in the ethanol medium and heated at 850 °C for 24 h followed by grindingfor 2 h when cooled. Later, it was heated at 1100 °C for 24 h. B. Material characterizations The x-ray diffraction (XRD) was recorded to confirm its crystal structure, phase formation, and to estimate the crystallite size. The high resolution XRD data were recorded in the 2 θrange from 5° to 90° at a step size of 0.02° at room temperature with suf-ficient exposure time on a Philips Panalytical X-pert pro diffrac-tometer equipped with an accelerated detector. The samples wereball milled using, planetary ball milling on a Fritsch system (Model Pulverisette7) for 12 h to reduce the grain size. The samples were pelletized by cold uniaxial pressing and were heated at 1100 °C for24 h. XRD data were recorded for both milled and unmilledsamples. X-ray photoelectron spectroscopy (XPS) measurementswere carried out to verify the stoichiometry in the compound. The electrical resistivity of the sintered pellet was studied using the Van der Pauw method with an in-house built system as a function oftemperature, maintained at a pressure of 10 −2Torr. A Keithley 2401 source meter was used as a current source and a Keithley 2700 digital multimeter was used for measuring the potential drop across the sample using four probes. The system was interfaced to apersonal computer for data acquisition and logging. C. Thermoelectric gas sensing measurements An in-house Seebeck based gas sensing system was developed to carry out the measurements. Figure 1 shows the schematic diagram of the gas sensing system. The samples of size 6 –10 mm diameter and 1 mm thickness were used for the measurement. The sample wasplaced on a heater stage, with a thin alumina substrate below in order to keep some portion of the sample in the hanging/floating condition as shown in Fig. 1 . This configuration builds a small temperature gra- dient which leads to the generation of thermoemf and it is measured.T h es a m p l ei sk e p ta tt e m p e r a t u r e so f3 0 0 –400 K. The sample is equilibrated at different measurement temperatures with a secondary heater to provide a temperature difference of 1 –4K .Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 054502 (2021); doi: 10.1063/5.0027709 129, 054502-2 Published under license by AIP Publishing.For the electrical measurements, the current is passed using Pt wires made into a fine bead, and the voltage is measured in thesimilar way as the measurement for Seebeck has been carried out. The thermocouples are mounted on a spring base to ensure proper pressure at the point of contact. In order to sweep the tem-perature gradient from negative to positive (for Seebeck measure-ments), a secondary heater was placed touching the floating endof the sample. This heater was precisely controlled using a Lakeshore 336 temperature controller. The two thermocouples are used for measuring the temperatures and open circuit voltages aremeasured by crossing two alumel –chromel wires (36 gauge) through four bore alumina ceramics as shown in the top-right of Fig. 1 . This design ensures a precise junction which is in contact with the sample and hence, the cold finger effect is avoided due toa thinner thermocouple junction. 26In addition, the negligible thermal mass of the thermocouple ensures high sensitivity andavoids time delay in attaining thermal equilibrium. The quasisteady state slope method is adopted for the eval- uation of Seebeck coefficient, where the voltages correspondingto different temperature gradients across the ends of the sampleare measured. Seebeck ( S true) of the sample is evaluated from the slope ( Smeasured ) of the graph between voltage and tempera- ture difference including the correction based on the wire mate- rials used, i.e., reference Seebeck coefficient ( Sref) at a given temperature,27,28 Strue¼Smeasured /C0Sref: (1)The value of Sreffor Alumel and Chromel are +18.3 μV/K and −22.7 μV/K,respectively. The system has been first calibrated using a standard Nickel sample, and the data are shown in Fig. 2 . As seen from Fig. 2(a) , the measured Seebeck values agree well with the reported literaturevalues. 29,30The raw data collected, the temperature readings mea- sured using the two thermocouples (hot and cold sides) and thevoltage measured from two different types of leads of the thermo- couples, are shown in Fig. 2(b) . These two voltages have been plotted for corresponding temperature differences and the slope isobtained by fitting a straight line. It can be readily noted fromFig. 2(c) that there is insignificantly small (none) offset voltage such that the line passes through the origin. The slope values obtained have been corrected with the respective Seebeck coeffi-cient values of the positive and negative leads. III. RESULTS AND DISCUSSION A. Material characterizations As shown in Fig. 3(a) , the crystal structure of double perov- s k i t et y p eR B a C o 2O5+δ(RBCO) (where Ris a rare earth element) consists of a sequence of metal oxide layers like CoO 2–BaO– CoO 2–ReO δstacked along the caxis. The δvalue characterizing the oxygen content of the lattice mainly depends on the valencyof Co, which can be 2+, 3+, or 4+. In a regular double perovskite AA 0B2O6lattice, i.e., δ= 1, all Co ions have valency 3+:4+ in the ratio of 50:50. There are only octahedral coordination around FIG. 1. The schematic diagram of (a) the thermoelectric gas sensor system with the cartoon of the thermocouple junction made by crossing the wires shown on the top- right corner and (b) enlarged view of the sample to thermocouple contact with two heaters.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 054502 (2021); doi: 10.1063/5.0027709 129, 054502-3 Published under license by AIP Publishing.Co, and all the CoO 6octahedral (O) sites consist of 50% Co ions have 3+ oxidation state and 50% Co ions have 4+ oxidation state. When δ= 0, there are only square pyramidal (P) coordina- tion around Co, and all the CoO 5pyramidal sites consist of 50%Co ions have 2+ oxidation state and 50% Co ions have 3+ oxida- tion state. However, at δ=0 . 5 ,a l lt h eC oi o n sa r ei nt h e3 +s t a t e , and there is an exact equal contribution from CoO 5square pyra- mids and CoO 6octahedra. This causes the crystal structure to FIG. 2. (a) The measured data for Nickel calibration sample, (b) the raw data of measured temperatures T1,T2, and corresponding voltages measured using positive (V+) and negative (V −) leads of the thermocouple. (c) The typical data of the room temperature Seebeck measurement value from the slope of voltage produced vs tempera- ture gradient and both values after correction from the reference Seebeck of the wires.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 054502 (2021); doi: 10.1063/5.0027709 129, 054502-4 Published under license by AIP Publishing.t r a n s f o r mf r o mt e t r a g o n a l( δ= 0) to orthorhombic ( δ=0 . 5 ) a n d again to tetragonal ( δ= 1) phase as shown in Fig. 3(b) .T h i s results in varying physical properties such as magnetic signa-tures that are affected due to either high spin, low spin, or inter-m e d i a t es p i ns t a t eo ft h eC oi o n . 17,31 The XRD patterns of the synthesized powder with and without ball milling are shown in Fig. 3(c) . All the samples show the pure double perovskites phase. The XRD peaks of milledsamples are considerably broadened due to the smaller crystallitesize. The crystallite size was calculated from the full width at half maxima (FWHM) of the diffraction peaks using the Scherrer formula as given by Eq. (2),t¼ 0:9λffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2/C0β2p cosθ, (2) where ωis the FWHM of the given peak having 2 θas the Bragg angle, βis the FWHM measured for a bulk standard sample in the same x-ray diffractometer. λis the wavelength of the incident x-ray (Cu k α= 0.154 18 nm). The crystallite sizes of the samples before and after ball milling from the XRD patterns are 130 and95 nm, respectively. Thus, the crystallite size ( t) was found to reduce significantly after milling. This size reduction was desired for improving the sinterability of the powders and achieving high FIG. 3. The schematic diagram showing (a) the crystal structure and (b) phase diagram of the GdBaCo 2O5+δ, with average Co oxidation state for δ= 0, 0.5, and 1.0 showing tetragonal, orthorhombic, and tetragonal unit cells, respectively. (c) The x-ray diffraction pattern of as-prepared and ball milled GBCO p owders. (b) Rietveld analy- sis of X-ray diffraction patterns of as prepared GdBaCo 2O5.5withδ= 0.5.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 054502 (2021); doi: 10.1063/5.0027709 129, 054502-5 Published under license by AIP Publishing.density samples, which are necessary for the high electrical con- ductivity of the samples. The x-ray diffraction (XRD) pattern of the GBCO powder sample is shown in Fig. 3 . The Rietveld refinement was carried out on the XRD pattern using Fullprof (version 5.6) software package. Itcan be seen from Fig. 3(d) that the refinement with excellent good- ness of fit (GoF) has been observed, for the orthorhombic ( Pmmm ) structure corresponding to the δ-value of 0.5, GdBaCo 2O5.5. The reduced χ2value was 1.13 as shown in figure which signifies a very close fit. The lattice parameters obtained after refinement werea= 0.387 93 nm, b= 0.783 50 nm, and c= 0.753 49 nm. These values are found to agree well with the reported values in the literature for the orthorhombic phase having δbetween 0.45 and 0.55. 19The small difference between Icaland Iobscould be due to the fact that the delta value may be slightly more than considered 0.5. Even asmall variation in δsuch as 0.501 may result in peak asymmetry and low intensity reflections due to the additional lattice planes as a result of the incorporation of oxygen in the lattice sites. As men-tioned earlier, the double perovskites can have variable latticeoxygen governed by the processing conditions. This large variationin oxygen site occupancy results in the structural phase transitions as a function of oxygen content, precise 0 <δ< 1 values. When it has low oxygen contents, i.e., δ<0:5, the structure is tetragonal, i.e., two of the three crystallographic axes are degenerate, and hence asingle peak is obtained for degenerate ( 0kl)a n d( h0l) reflections for the same value of the lindex. On the other hand, when the δvalue increases and is close to 0.5, the structure becomes orthorhombic(space group Pmmm ) as systematically one oxygen from every alter- nate CoO 6octahedra is missing. Thus, ( 0kl) and ( h0l) reflections are distinct. In fact, the lattice constant of the baxis is doubled as shown in Fig. 3 . However, the second order peak of ( 0 k/2l ) still appears close to ( h0l), and two different Bragg peaks are observed corresponding to ( 0kl) and ( h0l) depending on the difference in the aand blattice parameters. Further increase in δleads the system to the tetragonal phase, and hence ( 0kl)a n d( h0l) again becomes same for the same value of l. In the present study, GBCO showed two splitted peaks (see supplementary material , Fig. S1), and hence, it may be concluded that GBCO is completely orthorhombic, andtheir δvalue is close to 0.5. B. X-ray photoelectron spectroscopy The x-ray photoelectron spectrum (XPS) of the sample was studied before and after the oxygen exposure in Seebeck measure-ments. The core level spectra of Ba 3 d-Co 2 p,G d4 d,and O 1 sare shown in Fig. 4 . Although there are a number of reports in the lit- erature regarding XPS of double perovskites, most of them are focused on the analysis of the spectra qualitatively, and their cor-relation with the electronic transport properties has not been dis-cussed. In this work, an attempt is made to study the spectra indetail in order to establish a correlation with the observed elec- tronic transport properties. The analysis of the XPS spectrum of these systems is rather challenging due to several complexitieslike, variable oxidation state of Co and its 2p line overlap with Ba3d; variation in the oxygen content ( δ-value), and the inherent magnetic moments of certain rare earth elements like Gd. Ideally, the 2p level of Co has the highest photoionization cross section.However, since the binding energies of Co 2 pand Ba 3 dcore levels are overlapping, the next prominent core level spectra, i.e., Co 3 pand Ba 4 dhave been investigated and have been resolved in the spectrum. Figure 4(a) shows the XPS spectrum of overlapped core levels of Co 2 pand Ba 3 dbefore and after the measurements. The several peaks are identified based on the spin –orbit splitting of 3 d(l=2 ) of Ba and 2 p(l= 1) of Co photo-emissions. The Ba 3 d 5/2and 3 d3/2 lines have been assigned based on the observations by Maiti et al., in similar systems32and others.33Maiti et al.32have shown that the binding energy of the Ba 3 dline lies near 778 eV in YBa 2Cu3O7−δ, which is also the oxygen deficient perovskite system, with δ= 0.5 ±0.05. The same observation has been reported by Pramana et al.34Thus, the Ba 3d 5/2line was assigned at this binding energy. The Ba 3 d3/2line has been assigned based on the spin –orbit split- ting energy of the Ba 3d line from the literature.31The binding energy of Co4+is also confirmed from the literature reports.35 The assignment of peaks for Co ions is a challenging task because of the possibility of variable oxidation states and corre-sponding satellites. However, the presence of satellites itself can beeffectively used for fixing the oxidation state of transition metal ions. 36Thus, the two humps seen at 785 and 789 eV (shown by a single broad peak) are ascribed to the satellite peaks of Co3+and Co4+oxidation states, respectively.36Co 2 p3/2is designated at 780 eV by Takubo et al. ,31in the case of Nd and Tb based rare earth cobaltate, this is the most intense and hence abundant oxida- tion state in the structure, i.e., Co3+in the octahedral (O) position forδ=0.5 composition. Thus, the next oxidation state, i.e., 4+ for Co ions (Co4+) is also an octahedral site and should be hardly 1 – 1.5 eV higher than its predecessor, i.e., Co3+(O). Besides, their 2p 1/2 counterparts are included at 795.5 and 797 eV, respectively. In addition to the peaks mentioned above, there is a shoulder seen to the left of the peak, which could neither be attributed toCo 3+(O) nor Co4+(O). Co2+has also been ruled out as it is unlikely to occupy an octahedral site with the 2+ charge in GBCO ( δ<0.5) and cannot be assigned to lower binding energies than Co3+(O). Thus, this has been identified as the Co3+(P) ions occupying a crys- tallographic site, square planar site (marked by P) other than octa-hedral site. The binding energy of photoelectrons ejected from aparticular atom is highly sensitive to its local chemical environment in addition to the nuclear charge. Moreover, it is also found that the binding energies of photoemitted electrons are slightly differentfor the same element, with the same oxidation states but occupiedin different crystallographic sites. 37 The assignment of Co oxidation states is clearly evident after oxygen exposure in Seebeck measurement, i.e., after the sample isexposed to 100% O 2. This exposure has resulted in the diffusion of oxygen into the sub-surface of GBCO. Thus, reduction in oxygenvacancies and increase in cobalt average oxidation state is expected. This indeed can be validated by the shift of the spectrum to higher binding energies as seen from the lower pallet of Fig. 4(a) . This may be due to the increased surface contribution of higher valenceof Co ions, i.e., Co 4+(O). The peak assigned to Co3+(P) is seen to disappear which clearly indicates that when the system is exposed to 100% O 2, the surface oxygen content ( δvalue) increases. Since XPS is a surface sensitive technique and for a given wavelength ofMg Kαradiation, the depth of information is hardly 2 –5 nm;Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 054502 (2021); doi: 10.1063/5.0027709 129, 054502-6 Published under license by AIP Publishing.hence, this technique has been proved to be helpful in fixing the binding energy of Co4+(O). Figure 4(b) shows the XPS spectrum of the 4d core level of Gd. Usually, in the case of d orbital with azimuthal quantumnumber l= 2, the two spin –orbit interactions j 1=l+sand j2=l–s; where s=1/2 as expected for the electrons. This means that two lines of j1= 5/2, j2=3/2 are only expected. Gd3+with 4f7ion expe- riences a strong coulomb as well as exchange interaction between FIG. 4. The x-ray photoelectron spectra of (a) Ba 3 dand Co 2 pas-prepared and after measurement, (b) Gd 4 d,and (c) O 1 score levels.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 054502 (2021); doi: 10.1063/5.0027709 129, 054502-7 Published under license by AIP Publishing.4d and 4f electrons as stated in the literature studies.38,39The two intense lines correspond to9Dand7Dfinal states which show spin–orbit interaction energy of nearly 6 eV. The spin parallel state 9Dshows finer splitting due to spin –spin interaction, whereas the anti-parallel state7Dcould not be resolved. These levels further split into six lines depending on separate mjvalues due to strong 4d–4fcoupling of Gd ions. The broad peak at 154 eV is ascribed to the energy loss feature.38,39 The oxygen 1 sspectrum shows three contributions, which have been identified as the lattice oxygen peak (at 528.5 eV), the peak cor-responding to lattice oxygen vacancies (530.5 eV), and the surface chemisorbed oxygen from atmosphere at 532.5 eV. These accredita- tions are in good agreement with similar systems reported. 31 C. Transport measurements 1. Electrical resistivity The electrical resistivity and Seebeck coefficient of the samples were measured in an in-house built system. The measurement uti-lized the Van der Paw method of four probes using Eq. (3). The system was calibrated using a Ni metal foil as the standard. The current in the range of 10 mA to 1 A is applied, and the voltage drop ( V) is measured as a function of temperature ( T) for all the pellet samples, ρ¼πR ln 2d, (3) where ρis the electrical resistivity, Ris the sheet resistance, and dis the sample thickness. Figure 5 shows the resistivity behavior of the sample in vacuum from 100 to 500 K. It is seen that, at a low temperature range, the resistivity shows a gradual decrease until 330 K, which is followed by a drastic decrease in the value (more than one order ofmagnitude). The resistivity shows a small positive slope beyond350 K. The same sample was measured in two different systems (marked as IISER and INST) to validate the measurement done using an in-house built instrument. The data from Taskin et al. , 18 have also been plotted to compare and match the delta value refer- ence. It is clearly seen from Fig. 5(a) that the data obtained are in agreement with of the data reported by Taskin et al.,18forδ= 0.5. The small deviation in the temperature of transition observed could be attributed to the fact that this study is based on a polycrystallinepellet sample as compared to the single crystal by Taskin et al. Tarancón et al. 25in their report had studied XRD on GBCO and had shown that at 350 K, lattice distortion takes place. The lattice parameter acontracts, while lattice parameters band cshow elongation. This results in lattice distortion and causes the changein the crystal field splitting of CoO 6octahedra, thereby changing the position of t2gand egorbitals relative to each other. This change in the relative position of orbitals shifts non-zero density of states at the fermi level.31The low binding energy region of the XPS spectrum which essentially depicts the valance band is showninFig. 6 recorded at room temperature (300 K). This shows a near flat region at E F, zero binding energy. Since this is close to the MIT (350 K), there may be small non-zero electron occupancy even at room temperature. According to the literature, this transition occursfor compositions with the δvalue in the range of 0.45 <δ<0.65. The resistivity of the compositions with the δvalue outside this range shows an exponential decrease with respect to the temperatureand behaves like a normal insulator. The signature of MIT is quite evident from the variation of Seebeck coefficient with respect to temperature as shown in Fig. 5(b) . The Seebeck coefficient is fairly large in the insulating state and decreases gradually as the resistivity decreases. The Seebeck coefficient(S) shows a significant change in slope and has very small value (1μV/K) indicating its metallic behavior beyond MIT. The first deriv- ative of Swith respect to the temperature shows distinct minima, which is a signature of MIT around 345 K. The thermoemf behavior of this system has been studied thoroughly and the effect of deduc-tion of oxygen content in the lattice has been compared by corrobo-rating the results from the literature. The detailed study conducted by Taskin et al. 17,18h a dr e v e a l e dt h em a x i m u mt e m p e r a t u r eo fM I T obtained for δ= 0.5 and the decrease in temperature was observed in FIG. 5. (a) The variation of electrical resistivity of the GBCO sample measured in two different systems and compared with that of the reported data. (b) TheSeebeck coefficient measurement on the same sample across the metal insula- tor transition marked by the first derivative of Seebeck with temperature.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 054502 (2021); doi: 10.1063/5.0027709 129, 054502-8 Published under license by AIP Publishing.t h ec a s eo f δdeviation. Moreover, the behavior of Seebeck is found to be the same for single crystals and polycrystalline bulk samples. It isalso evident that the Seebeck values obtained in this study match well withδof 0.497. It is well known from the literature that Co 3+in octahedral coordination has a low spin (LS, t6 2ge0 g, S = 0) state, whereas in square pyramidal coordination, same Co3+has intermediate spin (IS, t5 2ge1 g, S = 1). On the other hand, when the system is doped with electrons, i.e., Co2+, it occupies square pyramidal coordination and has high spin (IS, t5 2ge2 g, S = 3/2). Doping with holes results in Co4+which occupies octahedral coordination has low spin ( t5 2ge0 g, S = 1/2). Although the change in δupon incorporation or removal of oxygen is symmetric, at about δ= 0.5, the transport properties are not symmetric.17–19Thus, the system becomes conducting for hole doping (i.e., δ>0.5) and is insulating for electron doping (δ<0.5). This is due to the fact that the transport of HS electrons Co2+is suppressed due to spin blockade19in the background of Co3+which has LS. This leads to the asymmetry observed in the Seebeck coefficient of the system in spite of divergence at 0.5δ-value. The Seebeck coefficient for δ= 0.5 composition is negative at low temperatures (<250 K), which becomes large positive evenwith a small change in δ, i.e., 0.501. The δ-values in Fig. 7(b) show a good correlation as per the arguments. When the sample ambience is changed from oxygen rich to oxygen lean conditions, the Seebeck value shows a drastic changeas compared to its resistivity, which is nearly the same across the δ value. In this study, the Seebeck coefficient has been measured by changing the atmosphere from N 2to O 2and vice versa. This enables the observation of the change in the potential difference fora fixed temperature gradient across the hot and cold ends of thesample. The system is calibrated using the Ni metal sample in the given temperature range. Nickel calibration data of Seebeck mea- surements and their data analysis have been discussed in Fig. 2 .It is seen from Fig. 7(a) that the Seebeck coefficient in the oxygen atmosphere is lower than that in the nitrogen atmosphere.This is due to the diffusion of oxygen into the lattice, which changes the oxidation state of some of Co 2+ions to Co3+ions (i.e., the coordination changes from square pyramidal to octahedralcoordination), thereby increasing the conductivity. However, thischange is conspicuous only for high Seebeck values as the resolu-tion of measurement may be a limiting factor to observe such changes in low values of Seebeck coefficient. It is clearly evident that the ambient cycling at room temperature changes the oxygenstoichiometry. The sample was ground in a mortar and pestle and annealed in ambient oxygen at 1000 ° Cfor 2 h and naturally cooled to room temperature in order to estimate the change in Seebeck uponambient cycling for samples with high initial δ-values, the sample so obtained showed a large Seebeck value as shown in Fig. 7(b) , which confirmed the value of δto be greater than 0.5 (upon com- parison with data in the literature 17). The nominal MIT tempera- ture value is shown by the vertical dashed line in Fig. 7(b) . The Seebeck measurement data for samples which showed consistent results are shown in Fig. 8 . As mentioned earlier, the as synthesized sample ( δ= 0.5) showed a small change in oxygen stoi- chiometry ( δ<0.5) as manifested in Seebeck after the first ambient cycling at room temperature. However, the resulting oxygen stoichi-ometry was fairly consistent until it was re-annealed at high temper-atures to give a larger δ(>0.5) value. Figures 8(a) and8(c)show the response of Seebeck for δ>0.5 sample at 300 K. The similar trend is observed for samples with smaller Seebeck ( δ<0.5) near room tem- perature, 333 K and is shown in Fig. 8(c) . The response is also mea- sured at 353 K which is beyond its MIT, i.e., in the metallic state. Itis seen from Fig. 8(a) that the change in the Seebeck response obtained at 300 K is significant, whereas a change in Seebeck response obtained at 353 K [ Fig. 8(d) ] is negligible. A large change FIG. 6. The valance band region of x-ray photoelectron spectra showing the crystal filled energy levels of Co ions at room temperature (300 K). The inset cartoon on the topleft shows the change in electron occupancy at the fermilevel ( E F) when the system is heated beyond 350 K, CoO 6octahedra undergoes a lattice distortion changing the density of states at the EFfrom near zero (insulating) to non-zero (metallic) behavior.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 054502 (2021); doi: 10.1063/5.0027709 129, 054502-9 Published under license by AIP Publishing.in Seebeck has been observed (10 –15μV/K) for the insulating state, whereas in the metallic state, there is no significant change. The corresponding response in resistivity are shown in Figs. 8(b) , 8(d),8(f),a n d 8(h). The resistivity of GBCO at room temperature (300 K) is very low (a few milli Ωcm), thus it becomes difficult to note the change in the voltage produced. Increasing the current foran observable potential drop produces Joule heating. The response in the resistivity is very low unlike Seebeck coefficient. It can be seen from Figs. 8(b) and8(d) that the response of δ>0.5 sample at 300 K is as low as 7% for 100% O 2and is 1% for 20% O2.I nt h e case of sample with δ<0.5, the response is barely visible in noise as seen in Fig. 8(f) . Efforts were also made to measure the response of the sample in the metallic state, i.e., at 353 K which is beyond MIT,but no reasonable change was observed and was submerged in noise due to low resistance in the metallic state and resolution limit of the measurement system. GBCO is further known to exhibit order –dis- order transition at about 720 K. This transition is characterized bythe transformation of ordered occupancy of oxygen vacancies torandom (disordered) occupancy of oxygen vacancies within the lattice. Thus, the lattice structure undergoes a low temperature ordered to high temperature disordered phase transition. No notice-able change in resistance was observed at 723 K as shown in Fig. 8(h) . In RBCO double perovskite systems, the change in resistance is onlyappreciable post order –disorder transitions, wherein the disorder in oxygen vacancy aids the oxygen diffusion resulting in measurable resistance change. 40 The response of the Seebeck coefficient for fractional oxygen atmosphere such as ambient 20% O2and balance 80% N2to that of 100% N2is also measured and is shown in Fig. 8(b) , wherein a distinct sharp rise and fall are seen for change in atmosphere. The chamber was first evacuated and then filled with 100% N2gas and subsequently 20% O2was introduced before the measurement. Similarly, the change was recorded for 10% O2. Notably, the change in Seebeck for a given concentration of O2was practically the same irrespective of the initial Seebeck value. This means that the change in the Seebeck coefficient was found to be indepen-dent of the initial δvalue except for conditions, where Sis really low. This change in S(ΔS) was plotted against the given O 2con- centration in percentage. The data were found to obey the power law of semiconductor gas sensors41as shown in Fig. 9 .T h i s implies that the response of the material is scaled as a power ofconcentration of the gas. It is known that almost all the thermoelectric response based gas sensor devices exhibit power law dependence. 7,10Usually, the response is measured in the change in resistance, unlike change inSeebeck in this case. Thus, the power-law dependence is argued tohave a similar origin in thermoelectric response as chemoresistiveresponse. In either case, the transport of electrons across the sample with or without a temperature gradient is affected by the energy barriers at the grain boundaries because of surface bandbending due to the adsorption of gases. The governing equation of the law is stated as ΔS¼αC β, (4) where αis a scaling prefactor and βis the exponent which is gov- erned by the gas –solid interaction. In this case, the interaction is diffusion of oxygen through the abplane which leads to the change in the Seebeck coefficient of the system. This change may be indi- rectly attributed to the change in carrier concentration ( n). Taskin et al.19had mentioned in his work that even a small change in the δvalue of 0.001, i.e., from 0.5 to 0.501, a large change in the carrier concentration of nearly 1019cm−3is observed. The Seebeck coeffi- cient depends mainly on three quantities, viz., carrier concentration (n), temperature ( T), and effective mass ( m*), which is the slope of the density of states at EF. The relation is shown below, S¼8π2k2 B 3eh2m*Tπ 3nhi2/3 : (5) FIG. 7. (a) The variation of Seebeck coefficient of the GBCO sample measured in two different atmospheres, i.e., O2andN2. (b) The Seebeck coefficient mea- surement on the same sample across the metal insulator transition before andafter annealing differing in δ-values.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 054502 (2021); doi: 10.1063/5.0027709 129, 054502-10 Published under license by AIP Publishing.Thus, if the temperature of sample is held constant, the two quantities, i.e., nand m* can affect the value of Seebeck. It is unlikely that m* would change drastically and lead to a large change in Seebeck. Moreover, when oxygen is diffused inside thelattice and occupies the vacant sites along the ordered oxygenvacancies in the abplane, excess holes are generated in the system, thereby increasing the conductivity. This change is clearly evident inSvariation at lower temperatures. 17When δ-values drop below 0.5, the Seebeck value is found to diverge with a low positive valueforδ>0.7. A large positive value for 0 .7>δ>0.5 and large nega- tive value for 0.5 >δ>0.45 and moderately large negative value for δ<0.45 is reported. It is found that the Seebeck coefficient of the sample shows different trends and is found to mismatch for δ= 0.5 after the first (virgin) test as seen in Fig. 7(b) . However, this trend is found to be reproducible ( δ<0.5) at room temperature. The average oxygen content in the lattice ( /C22δ) has been found to show an exponential dependence with time, and the time constant ( t)i s dependent on the temperature 17as per Eq. (6),/C22δ¼δ1/C0[δ1/C0δ0]e(/C0t/τ), (6) where δ∞is the oxygen content at equilibrium ( t=∞)a n d δ0is the initial oxygen content ( t= 0). This time constant is governed by the diffusion coefficient of oxygen into the lattice. Taskin et al.17had concluded from their detailed kinetic studies that, as the temperature is increased, the diffusivity of oxygen increases, and hence at higher temperatures, the oxygen content ( δ) is lower at a given oxygen partial pressure (Po 2). However, once the sample is annealed at higher temperatures for a long duration (several hours) and cooledslowly to room temperature in a constant Po 2, the lattice oxygen content is increased. Thus, the sample synthesized in atmospheric Po2∼10−1bar shows a high δvalue of near 0.5. The data obtained are fitted with an exponential function according to Eq. (6)and the values of time constants tis estimated for response as well as recov- ery. A small time constant of 10 s is obtained which is remarkably fast for a bulk diffusion phenomenon. FIG. 8. The response of GdBaCo 2O5+δto oxygen gas in Seebeck coefficient and resistivity (a), (b) 100% N 2to 100%O 2, (c) and (d) for 100% N 2to 20% O 2at 300 K tem- perature for δ>0.5. (e) and (f) 100% N 2to 100% O 2response at 333 K for δ<0.5; and (g) Seebeck response in the metallic state at 355 K (h) the resistivity response at 723 K, i.e., near the order disorder phase transition temperature.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 054502 (2021); doi: 10.1063/5.0027709 129, 054502-11 Published under license by AIP Publishing.The kinetics of the response and recovery times are governed by the diffusion coefficient of oxygen at a given temperature whichdecides the time constant as shown in Eq. (7), 42 D¼2L2 π2τ, (7) where L2is the area of surface available through which the diffu- sion takes place and tis the time constant of diffusion for rise and recovery. The diffusivity ( D) of oxygen in GBCO is 3 /C210/C08cm2/s at 250 ° C.17The diffusion is a thermally activated process, hence the diffusivity is expected to be lower at room temperature. However, using this value and typical surface area of the sample in this study, i.e., 1 cm radius, of time constant as nearly 10+7s is obtained which is much larger than that found in this study.One of the possible reasons could be the smaller crystallite size(100 nm) in the present case as compared to the reported value. 17Moreover, in this case since the time constant is mea- sured using transport measurements with the thermocouples incontact with the surface, an instantaneous change in the surfacevoltage is obtained which is found to saturate quickly. Hence,the response measured in this study is dominated by the surface interaction and is fast as compared to the reports by Taskin et al ., 17,18and by Hossein-Babaei et al .7The thickness of the sample (either in pellets or thin film) is not found to affect theresponse time. The idea of a viable thermoelectric gas sensor was first pro- posed by Rettig and Moos 43who demonstrated oxygen sensing abilities of SrTi 0.6Fe0.4O3using thermoelectric principles at 750 ° C. Thus, thin films of GBCO on a microheater platform may be pre-pared with an inherent design to maintain a small (calibrated) tem- perature gradient and open circuit voltage maybe measured. Higher the temperature gradient would facilitate a higher resolution withan increase in oxygen concentration. A similar TE based sensor has been demonstrated by Masoumi et al. , 8using ZnO for volatile organic compound (VOC) sensing. However, higher operation tem-perature (400 ° C) required and limited selectivity among VOCs is observed for these sensors. Nonetheless, similar power-law depen-dence was observed. The primary mechanism identified for change in Seebeck upon exposure to VOCs was the change in the inter- grain barrier height due to carrier injection or trapping because ofgas–surface interaction, 7unlike dominant bulk oxygen diffusion in the present study. The probable mechanism of oxygen sensing due to enhanced oxygen diffusion is as follows: GdBaCo 2O5+δis a system where the stoichiometry of oxygen and the transport properties are strongly correlated. Here, thethermoemf and electrical conductivity show a varied behavioreven for a small change in the oxygen content of the lattice, which is denoted by a variable δ.A l t h o u g hm a n yo t h e ro x i d e s show such oxygen deficient behavior, the possible δvalue for those oxides is a very small fraction. On the other hand, here inthis system, it could be as large as one in six oxygen ions in aunit cell. This significant mobility of the oxygen ions at room temperature allows the diffusion of oxygen in and out of the lattice depending on the partial pressure of oxygen in theambient. This results in a marked change in the Seebeck coeffi-cient of the GBCO lattice. Hence, it could be effectively used for monitoring oxygen levels due to its reversible change in Seebeck coefficient. Hossein-Babaei et al. 7in their work thoroughly discussed t h et h e r m o p o w e rv a r i a t i o nd u et ot h ea t m o s p h e r ec h a n g e ,which is dominated by the grain boundaries in oxides like ZnO. The grain boundaries contribute to the enhanced Seebeck coef- ficient in polycrystalline nanograined samples. Hence, the highSeebeck value are observed in polycrystalline samples thansingle crystals. In Fig. 10(a) , the schematic view is shown, depicting the polycrystalline pellet sample with abundant grain boundaries, while Fig. 10(b) shows the incorporation of O 2 within the vacant sites of the lattice ( ReO 2planes). This results in relatively conducting surface and sub-surface regions includ-ing the grain boundaries. The porous nature of the pellet samplesaids in the diffusion of oxygen in the bulk. This diffusion of oxygen is also probable at room temperature and bulk diffusion has a dominant contribution compared to surface chemisorption.Thus, the oxygen diffusion attains saturation at a given tempera-ture. However, in the other oxides, the oxygen diffusion at room temperature (300 K) is negligible, and the bulk diffusion takes place at high temperatures (>600 K). 43,44 When a temperature gradient is applied, a charge carrier density gradient is created. This drives a diffusion current from high to lowcarrier densities (high to low temperatures). Here, GBCO is a p-type semiconductor, since its δ-value is higher than 0.5. The thermoemf is developed across the sample due to the diffusion of carriers, andan electric field is created that drives a reverse drift current into thesample. Here, the conductivity of the sample plays a major role indeciding the properties of the sample. If the sample is highly con- ducting (metallic), built-in thermoemf is counteracted by the spon- taneous back flow of carriers. Since this system usually has highthermal conductivity, hence, the temperature equilibrium is reached FIG. 9. The change in the Seebeck coefficient ( ΔS) for different O 2concentra- tions in % with power law fit.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 054502 (2021); doi: 10.1063/5.0027709 129, 054502-12 Published under license by AIP Publishing.quickly, thereby canceling the diffusion current. However, for insu- lating samples, the temperature gradient is maintained due to low thermal conductivity. Furthermore, due to the low electrical conduc- tivity, the drift current is also comparatively small. At equilibrium,the temperature difference thus results in a balance between the thermal diffusion current ( JDiffusion ) and drift current ( JDrift)d u e to the induced potential difference on either sides as shown in Figs. 10(c) and10(d) and is represented by Eqs. (8)and(9), FIG. 10. (a) Schematic cartoon of two thermocouples placed on a polycrystal-line GdBaCo 2O5+δsample having a tem- perature gradient of ΔT, (b) the inclusion of oxygen in the lattice at vacant sites.(c) The conduction and valance bandswith hole occupancy for a given temper- ature gradient, resulting in thermal diffu- sion current ( J TH). (d) The voltages developed positive (+ve) and negative(−ve) due to temperature gradient and resulting electric field driven drift current (J D). These two currents balance each other at equilibrium. (e) The zoomed inview of energy band bending at the grain boundary, acting as the hole dense region in the given oxygen ambience.(f) The same grain boundary regionenergy barrier changes upon exposure to the high oxygen ambience. The c a r r i e rf l o ws h o w ni n( x) in front is stopped by the lattice, leading to energyfiltering. (g) The typical variation of delta at room temperature Seebeck ( S) and electrical conductivity ( σ) in vicinity of δ= 0.5. The lines show change in Sand σfor a given change in δ.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 054502 (2021); doi: 10.1063/5.0027709 129, 054502-13 Published under license by AIP Publishing.Jtoial¼JDrift/C0JDiffusion , (8) Jtotal¼σE/C0AΔT, (9) where Eis the electric field and Ais the thermal diffusion dependent cofactor which is related to the Seebeck coefficient of the material.Thus, in addition to grain boundaries, the Seebeck coefficient inGBCO significantly changes due to the lattice incorporation ofoxygen in the sub-surface at a temperatures as low as 300 K. When this sample is placed in an O 2atmosphere, the diffusion of oxygen into the bulk of the grains from the surface takes place.Thus, there exists a graded doping profile from surface to the bulk.The surface gets readily saturated with diffused oxygen that resultsinto a relatively higher delta value at the grain boundaries as com- pared to the bulk. This leads to a conducting shell and a relatively less conducting core as shown in Fig. 10 . Subsequently, high oxygen partial pressure results in the widening of surface conducting shells.On the other hand, in the N 2atmosphere, the core is wider and interface is narrower than the O2atmosphere. Therefore, in the N2 atmosphere, the narrower grain boundary region with E Fcloser to the valence band traps the majority holes. Thus, the holes with suit-able energy can only participate in the transport phenomenon. Thiscondition is similar to the electrons trapped in a potential well, where charges at the continuum can only participate in the trans- port. The enhancement of O 2partial pressure in the ambient leads to its incorporation deeper into the bulk. This widens the grainboundaries, as a result, some of the trapped holes become mobile,which, in turn, increases the overall electrical conductivity of the solid. This is due to the surface conductance (along the grain bound- aries) σ Sbecomes higher than the bulk conductance σB(which varies in an opposite way than Seebeck coefficients, i.e., SS<SB). Overall, it increases the amount of diffusion current and reduces theSeebeck coefficient in a high O 2atmosphere as shown in Figs. 10(e) and 10(f) . The variation of the Seebeck coefficient and electrical conductivity in the vicinity of δ=0.5 has been reported in the literature.17–19The qualitative nature of Sandσdependence on the δ value is shown in Fig. 10(g) at 300 K. It may be noted that, for a small change in δ, a large change in S is observed, whereas the elec- trical conductivity shows a negligible change. The experimentally observed reduction in the resistance of the sample is much less ascompared to the enhancement in the Seebeck coefficient values asshown in Fig. 10(g) . The change in the resistance may even be of the order of the noise level of the measuring instrument. Therefore, it is imperative to monitor the change in the Seebeck coefficient due to the change in atmosphere than the change in resistance. IV. CONCLUSION The polycrystalline bulk ceramic sample of GdBaCo 2O5.5 (GBCO) prepared by the solid state route was used to demonstrate the thermoelectric principle based oxygen sensor which operates at room temperature. The as prepared GBCO sample showed an optimum oxygen content ( δ∼0.5) resulting in the demonstration of metal insulator transition close to room temperature (at 340 K)and an oxygen ambient dependent transport properties. The Seebeck coefficient shows a step change at MIT and also exhibits a large change with change in oxygen partial pressure. Thischange is due to the selective hole doping as a result of oxygen movement in ReO δplanes of the double perovskite lattice vis-a-vis with grain boundary barrier modulation. This phenomenon wasexploited to show that the reproducibility of Seebeck coefficientwith the oxygen ambience and hence can be utilized for fabricat-ing the oxygen sensor. The smaller time constants (20 s) obtained at room temperature depict a surface sensitive nature of measure- ment method adopted in this study. Thus, this method of measur-ing open circuit voltage for a small temperature difference has anedge over the existing oxide based oxygen sensors which work onpotentiometric or amperometric principle, which requires high temperature of operation (500 ° C) in addition to the requirement of a reference oxygen level. SUPPLEMENTARY MATERIAL See the supplementary material for Rietveld refinement of both, as prepared and ball milled samples, the raw XPS data com-parison and the temperature dependence of resistivity analysis. ACKNOWLEDGMENTS The authors are thankful for the funding revived from the Science and Engineering Research Board (SERB), Government ofIndia (Grant No. EEQ/2018/000769). The authors are thankful toDr. Chandan Bera of INST Mohali for electrical conductivity mea-surements at low temperatures, Professor Arun M Umarji for his guidance about the system of double perovskites, and Mr. Andrews P. Alex for the XPS data recording and Dr. Geetika Srivastava formanuscript correction. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1R. Ramamoorthy, P. Dutta, and S. Akbar, J. Mater. Sci. 38, 4271 (2003). 2R. Moos, N. Izu, F. Rettig, S. Reiß, W. Shin, and I. Matsubara, Sensors 11, 3439 (2011). 3T. Liu, X. Zhang, L. Yuan, and J. Yu, Solid State Ionics 283, 91 (2015). 4S. Akbar, P. Dutta, and C. Lee, Int. J. Appl. Ceram. Technol. 3, 302 (2006). 5W. Liu, C. Yang, X. Wu, H. Gao, and Z. 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A 2, 9919 (2014).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 054502 (2021); doi: 10.1063/5.0027709 129, 054502-15 Published under license by AIP Publishing.

9.0000122.pdf

AIP Advances 11, 025313 (2021); https://doi.org/10.1063/9.0000122 11, 025313 © 2021 Author(s).Size dependence of charge order and magnetism in Sm0.35Ca0.65MnO3 Cite as: AIP Advances 11, 025313 (2021); https://doi.org/10.1063/9.0000122 Submitted: 22 October 2020 . Accepted: 18 January 2021 . Published Online: 08 February 2021 Lora Rita Goveas , K. S. Bhagyashree , K. N. Anuradha , and S. V. Bhat COLLECTIONS Paper published as part of the special topic on 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials and 65th Annual Conference on Magnetism and Magnetic Materials ARTICLES YOU MAY BE INTERESTED IN Magnetic and electron paramagnetic resonance studies of Ln 0.5Ca0.5MnO 3 (Ln = Pr, Bi) manganite AIP Advances 11, 015144 (2021); https://doi.org/10.1063/9.0000172 The impact of solvent tan on the magnetic characteristics of nanostructured NiZn-ferrite film deposited by microwave-assisted solvothermal technique AIP Advances 11, 025003 (2021); https://doi.org/10.1063/9.0000190 Critical magnetic behavior in [Ag 8/Co0.5]x64, [Ag 8/Co1]x32 and [Ag 16/Co1]x32 epitaxial multilayers AIP Advances 11, 025220 (2021); https://doi.org/10.1063/9.0000086AIP Advances ARTICLE scitation.org/journal/adv Size dependence of charge order and magnetism in Sm 0.35Ca0.65MnO 3 Cite as: AIP Advances 11, 025313 (2021); doi: 10.1063/9.0000122 Presented: 5 November 2020 •Submitted: 22 October 2020 • Accepted: 18 January 2021 •Published Online: 8 February 2021 Lora Rita Goveas,1,a)K. S. Bhagyashree,2K. N. Anuradha,3and S. V. Bhat2 AFFILIATIONS 1Department of Physics, St. Joseph’s College (Autonomous), Bangalore 560027, India 2Department of Physics, Indian Institute of Science, Bangalore 560012, India 3Department of Physics, Dr. Ambedkar Institute of Technology, Bangalore 560056, India Note: This paper was presented at the 65th Annual Conference on Magnetism and Magnetic Materials. a)Author to whom correspondence should be addressed: loragoveas@gmail.com ABSTRACT We report a systematic tracking of consequences of size decrease to nanoscale for charge order (CO) and magnetic properties of electron doped manganite Sm 0.35Ca0.65MnO 3by magnetization measurements. The bulk form of this system is charge ordered below 270 K and anti- ferromagnetic (AFM) below 130 K. The bulk sample and nanoparticles of various sizes (mean diameter ∼15, 30, 90 nm) were synthesized by sol-gel technique. Our studies show that the robust CO in the bulk gets weakened by size reduction and the nanoparticles exhibit ferromag- netic (FM) ordering. Magnetization at high temperatures, in the paramagnetic region, reflecting the behaviour of the most part of the samples arising due to FM fluctuations caused by double exchange interaction is found to decrease as the particle size reduces. However, at low tem- perature the trend of FM magnetization as a function of the size is found to be reversed. This result is understood in terms of the dominance of surface effects where uncompensated bonds and an increase in the charge density at the surface layers lead to weak ferromagnetism which increases with decreasing size. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/9.0000122 INTRODUCTION Over the last few decades the doped perovskite manganites RE1-xAxMnO 3(RE, trivalent rare-earth ion; A, divalent alkaline earth ion) have fascinated scientific community due to the variety of phe- nomena like colossal magnetoresistance (CMR), charge order (CO), orbital order (OO) and phase separation(PS) exhibited by them.1–6 The intimate mutual coupling of many degrees of freedom namely charge, spin, lattice and orbital in these strongly correlated systems produce huge responses to small perturbations. Currently the size induced effects in manganites like emergence of ferromagnetism (FM), suppression of CO, exchange bias (EB) effect, spin glass (SG) transition, magnetocaloric effect, training effect and memory effect have resulted in paving the way for many technological applications making the research in this field much exciting.7–10 CO in rare earth manganites is an interesting manifestation occurring due to interactions between phonons and charge carriers, resulting in localization of charge carriers at particular locations inthe lattice below a certain characteristic temperature (T CO). Predom- inance of Coulomb interactions over the kinetic energy of the charge carriers is the driving force of this phenomenon. The CO transi- tion is indicated by an increase in resistivity, which is due to gradual freezing of the e gelectrons at the Mn3+sites. A peak in magnetiza- tion is observed at T COoccurring due to the FM fluctuations present for T>TCOmaking way for AFM fluctuations below T CO. CO state is sensitive to perturbations like magnetic field,11pressure,12exposure to X-ray photons, structural and geometrical changes.13The sup- pression of CO and emergence of ferromagnetism consequent to the reduction of particle size was first reported by Rao et al.14Since then different groups have studied various systems and demonstrated that the CO phase could be suppressed by size reduction.15–17 Though the melting of CO and emergence of ferromagnetism has been the focus of a huge number of studies the mechanism responsible for such size effects are topic of intense debate. The phe- nomenological model proposed by Dong et al. , based on surface separation infers an increase in FM shell thickness with decrease AIP Advances 11, 025313 (2021); doi: 10.1063/9.0000122 11, 025313-1 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv in grain size due to the relaxation of superexchange interaction on the surface layers.18Monte-Carlo simulation study on the CE-type CO/AFM phase has shown that an increase in the charge density at the surface layer due to unscreened Coulomb interactions results in suppressing the CO state leading to FM tendency.19‘Core-shell model’ put forward by Zhang et al. ,20explains that the anoma- lous behaviour of destruction of CO/AFM and emergence of FM is arising because of the uncompensated surface spins. However, Aliyu et al. ,21find no evidence for a shell in their high resolu- tion transmission electron microscopic study and conclude that excess of Mn3+over Mn4+in a single surface layer can cause the observed ferromagnetism. A recent theoretical study done using dynamical mean-field theory and density-functional theory showing that the structural changes caused by reduction in size is account- able for the size induced effects,22has been the subject of some controversy.23 It is well established that size reduction to nanoscale results in disappearance of charge ordered state and appearance of a FM phase at low temperature in half doped manganites.15,24–26Electron spin resonance (ESR) measurements on Nd 0.5Ca0.5MnO 3by Zhou et al. ,16shows that though the magnetic measurements indicate the complete disappearance of CO in nanoparticles(size - 40nm), the temperature dependence of g-factor of ESR and linewidth clearly display the distinctive characteristic of the CO pointing towards the presence of short range charge order even though the long range charge order has disappeared. The phase diagrams of doped manganites show that T COis maximum not for the composition x =0.5, but around x =0.6 the CO is found to be more robust.27When the size of RE and A ions in doped manganites become smaller as in the case of Sm and Ca, the GdFeO 3-type distortion becomes larger which favours the CO state to stabilize resulting in higher T CO. The Sm 1-xCaxMnO 3man- ganites have shown promising applications in the field of infrared emissivity, microwave absorption28,29and thermoelectric effect. In particular, Sm 0.35Ca0.65MnO 3(SCMO), the subject of the present investigation, has been found to have the highest emissivity contrast at 280 K corresponding to its T CO.30,31 EXPERIMENTAL DETAILS Sol-gel method was followed for the synthesis of samples. Sto- ichiometric amounts of high purity ( ≥99.9%) Sm 2O3, CaCO 3and MnCO 3were first converted into nitrates and then subjected to hydrolysis and condensation followed by polymerisation. The pre- cursor powder obtained was separated into numerous parts and each portion was annealed separately at temperatures 550○C, 650○C and 850○C for about 6 hours. Heating the precursor at 1200○C for 24 hours resulted in micrometer sized particles which we designate as bulk particles. Structural investigations were car- ried out by X-ray diffraction (XRD) using BRUKER D8 ADVANCE X-ray diffractometer with monochromatic radiation source Cu-K α (λ=1.54056 Å). The XRD patterns were analysed with Rietveld method using the software GSAS. Using transmission electron microscopy (TEM) and scanning electron microscopy (SEM) the particle sizes and morphology were found. The compositional exam- ination was done by energy dispersive X-ray analysis (EDAX). Oxygen stoichiometry was measured by iodometric titration. Super- conducting quantum interference device (SQUID) magnetometer at FIG. 1. Rietveld plots for various sizes indicated in the panel. The experimental data points are indicated by black solid dots, the calculated by red solid line and difference patterns by blue solid lines. The Bragg positions of the reflections are indicated by vertical lines below the patterns. a field of 0.01 T was used to carry out magnetization measurements in the temperature range 10-300 K. RESULTS AND DISCUSSION The XRD patterns of all synthesised particles obtained at room temperature are given in Figure 1 which confirm single phase and crystalline nature with orthorhombic crystal structure. The decrease in the full width at half maximum (FWHM) of the diffraction peak with increase in annealing temperature indicates growth in particle size. Volume (Å3) of unit cell calculated from the XRD profile fitting using the software GSAS are 222.78, 220.28, 219.76 and 219.23 respectively for 15nm, 30 nm, 90 nm and bulk parti- cles. The shrinkage of volume due to size reduction is observed in Pr0.5Ca0.5MnO 3,15,32La0.4Ca0.6MnO 3,17La0.9Ca0.1MnO 333whereas increase in cell volume with reduced size has been reported in Nd 0.5Ca0.5MnO 324and Sm 0.5Ca0.5MnO 3.34The magnetic behaviour of the nano particles in all the above-mentioned systems are found to be similar though the structural changes observed are different. The TEM images (not shown) of nanosamples reveal that the average particle size is about 15 nm, 30 nm and 90 nm for 550○C, 650○C and 850○C sintered sample respectively. The sam- ples are henceforth named as 15 nm, 30 nm, 90 nm and Bulk in the increasing order of their size. The SEM image (not shown) of Bulk showed that the grains are of a few micrometres in size. The cationic composition was confirmed using EDAX. The oxygen contents estimated by iodometric titration for bulk particles were determined to be 3.021 ±0.015. Temperature dependence of field cooled (FC) and zero field- cooled (ZFC) magnetizations (M) measured at H =100 Oe, between 10 K and 300 K for 15 nm, 30 nm, 90 nm and Bulk particles is shown in Figure 2. The bulk sample shows a clear CO peak at temperature 270 K and AFM transition at 130 K (Fig. 2 (b)) which is consistent with the observations reported earlier.27For the nanosamples, as the size of the particle reduces the CO transition peak is found to be of decreased intensity, broadened and shifted towards the lower tem- perature indicating that the CO phase is weakened in the nanosized AIP Advances 11, 025313 (2021); doi: 10.1063/9.0000122 11, 025313-2 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 2. (a) Temperature dependence of ZFC and FC magnetization at H =100 Oe (solid fills for FC and hollow for ZFC). (b) FC curves at high temperature(c) Temperature variation of inverse dc susceptibility. samples. For 15 nm the broad CO peak is found to have completely disappeared. While this could be an evidence for the absence of long- range charge order, short range CO can still be present and local probe techniques like electron paramagnetic resonance can help ver- ify that possibility. The AFM transition peak is visibly absent in the case of all nanosamples. All the nanosamples show increase in magnetization with lowering temperature with a rapid rise below 100 K reflecting presence of FM behaviour at low temperatures. The FC and ZFC curves bifurcate around 100 K; ZFC exhibiting a peak shows a blocking temperature (T B) which reduces with size of the particle. The irreversibility between FC and ZFC, the non-saturation of FC are an indication of strong anisotropy and glassy nature of spins in nanoparticle.34 The expanded view of the FC curves in the temperature range 300 - 90 K is shown as inset (b) to Figure 2. It is interesting to note from Figure 2(a) and (b) that closer to room temperature, as par- ticle size decreases magnetization also decreases whereas at lower temperature as the particle size decreases magnetization increases. This reversal in the order of magnitude of magnetisation could be attributed to different behaviours of the spins on the surface and in the core of the particles. The magnetization at room tempera- ture is due to FM fluctuations occurring in the paramagnetic region of the entire sample. As the temperature is reduced and the CO sets in, FM fluctuations of paramagnetic region begin to decrease giving way to AFM fluctuations resulting in a decrease in magnetiza- tion in the core. When the size of the particle decreases the core, size lessens eventually reducing the magnetization. But when the temper- ature is reduced further the magnetization in nanosamples increases due to the ordering of uncompensated spins at the surface of the nanoparticles. As particle size reduces surface effect rules over the core, consequently magnetization increases in the nanoparticles.30 We provide more quantitative understanding of this behaviour next. Figure 2(c) shows the plots of inverse susceptibility 1/ χvs tem- perature for the bulk as well as the three nanosized particle samples. It is seen that for each nano sample the plot contains two separate linear regions: the high temperature (HT) range, ( ∼300 – 250 K) and the low temperature (LT) ( ∼90 – 130 K). The two regions areTABLE I. CW-parameters obtained by fitting CW-law for low temperature and high temperature regimes. Sample C HT(emu-K/g-Oe) oHT(K) C LT(emu-K/g-Oe) oLT(K) 15 nm 0.0250 ±0.0003 17 ±4 0.0068 ±0.0002 92 ±7 30 nm 0.0243 ±0.0004 31 ±5 0.0060 ±0.0002 95 ±3 90 nm 0.0248 ±0.0005 90 ±5 0.0046 ±0.0002 94 ±7 Bulk 0.0247 ±0.0005 111 ±6 separated by a transition region which is characterized by an ini- tial upward change in 1/ χsignalling development and dominance of AF correlations followed by a downward change indicating the development of FM correlations which eventually culminate in FM transitions. This behaviour is quite different from that of the bulk sample which has only the HT linear region and it does not undergo any FM transition. The data in the linear regions of all the samples are fitted to the Curie-Weiss law χ=C/(T – Θ) (Fig. 2 (c)) and the Curie constant C ( =(NA/3k B)μ2 effwhere N Ais the Avogadro num- ber, k Bis the Boltzmann constant and μeffis the effective magnetic moment per ion) and the paramagnetic Curie temperature Θ(which is a measure of the strength of the inter-spin coupling,35in the present case, strength of the ferromagnetic exchange interactions) are extracted and are presented in Table I. The fitting errors are also indicated for each value of C HT, CLT,ΘHTandΘLT. The following points are to be noted: (a) ΘHTdecreases with decreasing particle size implying that in the high temperature region the Zener double exchange mediated FM correlations weaken as the size is reduced. (b) In contrast, ΘLTis independent of the nanopar- ticle size. This implies that the strength of the FM correlations in the surface layer is independent of the nano particle size in as much as it is believed that the origin of the ferromagnetism in nanopar- ticles is in the uncompensated spins on the surface shell of the nanoparticles.19This interesting result, being reported for the first time to the best of our knowledge, requires further experimental and theoretical confirmation and understanding. (c) C HTand C LTalso exhibit contrasting behaviour in that whereas C HTis seen to be inde- pendent of particle size, C LTincreases with decreasing size. The latter behaviour could be understood as a consequence of the fact that as the size decreases, the relative contribution of the surface shell to the measured magnetization increases. The field dependent magnetization studies done at different temperatures for bulk and nanoparticles are shown in Figures 3(a) at 50 K and Figures 3(b) at 280 K. The bulk shows a linear variation at both the temperatures, confirming the absence of a ferromag- netic phase. The formation of ferromagnetism at low temperature is evident from the hysteresis loop at 50 K in the field dependent mag- netization of the nanosamples (15 nm and 30 nm). Magnetization does not saturate even up to 5 T magnetic fields for all the nanopar- ticles showing the presence of residual AFM in the particles. In our system the fraction of Mn3+and Mn4+ions is 0.35: 0.65 and if all the spins were to be ferromagnetically aligned, the value of μeffwould be 6.321 μBper ion. The spontaneous magnetization M Sobtained by the linear extrapolation of the high field magnetization above 2.5 T to H =0 is found to be 6.5817 emu/g ( μeff=0.2141 μBper ion ∼3.4% of the ideal value) and 7.997 emu/g ( μeff=0.2603 μBper ion ∼4.1% of the ideal value) for 30 nm and 15 nm respectively. This AIP Advances 11, 025313 (2021); doi: 10.1063/9.0000122 11, 025313-3 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 3. Field dependence of magnetization of (a) at T =50 K and (b) T =150 K. indicates that only a very small fraction of the sample is ferromag- netic, consistent with the understanding of the presence of only surface ferromagnetism in the nanoparticles. CONCLUSION Nanoparticles of average diameter 15 nm, 30 nm, and 90 nm were synthesized by adopting sol-gel technique and their proper- ties were compared with bulk samples. Structural analysis shows that there is an increase in unit cell volume with decrease in particle size. The magnetization curve of bulk shows a clear CO peak at temper- ature 270 K and AFM transition at 130 K. Magnetization studies of the nanosamples, indicate that as the size of the particle reduces the CO transition peak is found to be broadened and shifted towards the lower temperature indicating that the CO phase is weakened in the nanosized samples. As the particle size is reduced to 15 nm, the broad CO peak is found to have completely disappeared indicating that long range CO is no longer present though, at this stage the pres- ence of short-range charge ordering cannot be ruled out. Our studies are consistent with the suppression of the CO and the appearance of the weak ferromagnetism in the nanoparticles arising from the dominance of the surface effects. ACKNOWLEDGMENTS K.N.A. acknowledges Technical Education Quality Improve- ment. Programme III (Grant No. TEQIP III), NPIU/SPIU and Dr. Ambedkar Institute of Technology, Bangalore, India for financial support. S.V.B. thanks The National Academy of Sciences, India for financial support. DATA AVAILABILITY The data that support the findings as well as the TEM and SEM micrographs of this study are available from the corresponding author upon reasonable request.REFERENCES 1J. M. D. Coey, M. Viret, and S. von Molnár, Adv. Phys. 48, 167 (1999). 2T. Hotta, A. L. Malvezzi, and E. Dagotto, Phys. Rev. B 62, 9432 (2000). 3A. Arulraj, P. N. Santhosh, R. S. Gopalan, A. Guha, A. K. Raychaudhuri, N. Kumar, and C. N. R. Rao, J. Phys. Condens. Matter 10, 8497 (1998). 4L. Gorkov and V. Kresin, Phys. Rep. 400, 149 (2004). 5Y. Tokura, Rep. Prog. Phys. 69, 797 (2006). 6S. S. Rao and S. V. Bhat, J. Phys. Condens. Matter 21, 196005 (2009). 7T. Zhang, G. Li, T. Qian, J. F. Qu, X. Q. Xiang, and X. G. Li, J. Appl. Phys. 100, 094324 (2006). 8A. Sawa, T. Fujii, M. Kawasaki, and Y. Tokura, Appl. Phys. Lett. 88, 232112 (2006). 9M. Aparnadevi and R. Mahendiran, J. Appl. Phys. 113, 013911 (2013). 10S. N. Jammalamadaka, S. S. Rao, S. V. Bhat, J. Vanacken, and V. V. 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5.0040804.pdf

J. Appl. Phys. 129, 145104 (2021); https://doi.org/10.1063/5.0040804 129, 145104 © 2021 Author(s).Acoustic flatbands in phononic crystal defect lattices Cite as: J. Appl. Phys. 129, 145104 (2021); https://doi.org/10.1063/5.0040804 Submitted: 16 December 2020 . Accepted: 19 March 2021 . Published Online: 08 April 2021 Tian-Xue Ma , Quan-Shui Fan , Chuanzeng Zhang , and Yue-Sheng Wang ARTICLES YOU MAY BE INTERESTED IN Origami inspired phononic structure with metamaterial inclusions for tunable angular wave steering Journal of Applied Physics 129, 145103 (2021); https://doi.org/10.1063/5.0041503 Experimental evidence of a hiding zone in a density-near-zero acoustic metamaterial Journal of Applied Physics 129, 145101 (2021); https://doi.org/10.1063/5.0042383 Progress and perspectives on phononic crystals Journal of Applied Physics 129, 160901 (2021); https://doi.org/10.1063/5.0042337Acoustic flatbands in phononic crystal defect lattices Cite as: J. Appl. Phys. 129, 145104 (2021); doi: 10.1063/5.0040804 View Online Export Citation CrossMar k Submitted: 16 December 2020 · Accepted: 19 March 2021 · Published Online: 8 April 2021 Tian-Xue Ma,1 Quan-Shui Fan,2 Chuanzeng Zhang,1,a) and Yue-Sheng Wang2,3,a) AFFILIATIONS 1Department of Civil Engineering, University of Siegen, Siegen D-57076, Germany 2Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, People ’s Republic of China 3School of Mechanical Engineering, Tianjin University, Tianjin 300350, People ’s Republic of China a)Authors to whom correspondence should be addressed: c.zhang@uni-siegen.de andyswang@tju.edu.cn ABSTRACT In this paper, we investigate the acoustic flatbands (FBs) in phononic crystal (PnC) defect lattices. The defects are introduced into a PnC composed of periodic rigid rods in the air background. Since the acoustic energy is highly confined inside the PnC defects, the interactionbetween the defects can be described by the tight-binding model. We construct the PnC defects in two bipartite lattices, namely, the stub and Lieb lattices. The acoustic FBs can be observed for both of the lattices. Moreover, the acoustic FBs are protected by the chiral symmetry. That is, the FBs can be preserved even though the hopping strengths between the neighboring defects are perturbed. The proposed PnCdefect lattices provide a feasible platform for the study of acoustic FB systems and topological insulators. Published under license by AIP Publishing. https://doi.org/10.1063/5.0040804 I. INTRODUCTION Recently, flatband (FB) systems have attracted considerable attention in condensed matter physics. 1–3An FB is a dispersionless energy band that spreads over the entire Brillouin zone. The FBsystems were first proposed to study the ferromagnetism in multi- band Hubbard models. 4–6Over the years, the FBs have been inves- tigated in a variety of physical systems, such as ultracold atoms inoptical lattices, 7,8photonic waveguide networks,9–11synthetic atomic lattices,12–14metamaterials,15,16and photonic crystals.17–19 Typically, the FBs are found in various lattice geometries due to the destructive interference, such as the stub,20Dice,21kagome,12and Lieb9,10lattices. The FB systems have emerged as promising platforms for the investigation of many-body physics. Within pho-tonics, the systems with an FB have been utilized for enhancing light–matter interaction by slow light 22,23and achieving non- diffractive transmission of optical images,9,10to name a few. Phononic crystals (PnCs), the acoustic counterpart of photonic crystals, are artificial acoustic materials with a periodic modulation of their material properties.24,25The PnCs can be designed to achieve unprecedented wave phenomena, including bandgaps, nega- tive refraction, acoustic rectification, and sound collimation.26–28In particular, the PnCs can provide controllable platforms to studyphysical phenomena that are difficult to realize in electronic systems,such as topological phase. In the last decade, acoustic topological phases have been observed in one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D) PnC systems. 29,30 Defects in phononic or photonic crystals are of great interest due to their abilities of confining or guiding waves and desirable in various applications, such as wave filters,31–33waveguides,34,35and sensors.36,37 Additionally, the interaction between defects offers new opportunities to control wave transport. For example, coupled resonator waveguidescomposed of the PnC point defects can guide waves along complexroutes. 38,39More recently, photonic topological states have been observed in the photonic crystal defects with special lattice configura- tions.40,41In this work, we construct the stub and Lieb lattices using the PnC defects. As the interaction between the PnC defects can bedescribed by the tight-binding (TB) model, the acoustic FBs are realizedin the PnC defect lattices. For the stub and Lieb lattices, the acoustic FBs are topologically protected, sho wing robustness to the perturbation of hopping strength between the neighboring PnC defects. II. BAND STRUCTURES OF PHONONIC CRYSTAL DEFECT LATTICES A. Stub lattice One of the quasi-1D FB lattices is the stub lattice, as shown in Fig. 1(a) . The unit-cell of the stub lattice [the dashed square inJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 145104 (2021); doi: 10.1063/5.0040804 129, 145104-1 Published under license by AIP Publishing.Fig. 1(a) ] consists of three sublattices, namely, A, B, and C. In this paper, the PnC defects are used to realize the FBs in the acousticregime. It is well known that such defects can act as resonators(cavities) where the acoustic field is highly confined. Hence, the PnC defects can be considered the acoustic analogs to the atomic orbitals of crystals. Different from the periodicity of the unit-cell,the defects can also be arranged periodically that form a variety ofPnC defect lattices. For the stub lattice, the super-cell of the PnCmodel is illustrated in Fig. 1(b) . Note that here, we use the term super-cell to distinguish it from the unit-cell of the PnC [the blue dashed square in Fig. 1(b) ]. The PnC contains periodic steel rods embedding in the air background, where the lattice constant of theunit-cell and the rod radius are a 0¼25 mm and r¼10 mm, respectively. A complete bandgap can be obtained by the consid- ered PnC (see Appendix A ). Here, it should be pointed out that the complete bandgap is necessary for the point defect modes. Weremove the steel rods from the perfect PnC to form the defects andthen construct the stub lattice [see Fig. 1(b) ]. The distances between the two neighboring defects are denoted by d 1,d2, and d3, where d1þd2¼awith abeing the super-cell lattice constant. Forthe super-cell shown in Fig. 1(b) , these distances are set as d1¼d2¼d3¼4a0. Under the TB approximation, the Hamiltonian in the momen- tum space for this system can be written as ^HkðÞ ¼f0 γ1þγ2e/C0ikaγ3 γ1þγ2eikaf0 0 γ3 0 f00 @1 A, (1) where krepresents the wave vector, f0the resonant frequency of the PnC defects, and γi(i¼1, 2, 3) the hopping strength. It is pointed out here that for simplicity, only the nearest-neighbor (NN) inter- actions between the PnC defects are taken into account, as depicted byγ1,γ2, and γ3[the black solid lines in Fig. 1(a) ]. Notably, the NN hopping strength depends on the distance between the NNPnC defects. For the case in Fig. 1(b) , the hopping strengths are identical; i.e., γ 1¼γ2¼γ3. The detailed parameters for the TB model are given in Appendix B . The sound propagation in this system satisfies the pressure- wave equation ∇2þω2=c2ðÞ p¼0, where p,c, and ωare the sound pressure, the sound speed in the air, and the angular frequency,respectively. In addition to the TB model, we solve the pressure- wave equation by using the finite element method (FEM). In this paper, the commercial software COMSOL Multiphysics is utilizedto perform the numerical simulations. Because of the large contrastbetween the impedances of steel and air, the steel rods are regardedas rigid bodies. The material parameters of the air background are the sound speed c¼343 m/s and the mass density ρ¼1:21 kg =m 3. The band structure of the PnC defect lattice is plotted in Fig. 1(c) , where the results obtained by the TB model and the FEM are denoted by the blue solid lines and the red dots, respectively. One FB and two dispersive bands appear in the band structure.Additionally, it is seen that the analytical results are in good agree-ment with the numerical ones, demonstrating that the TB model isapplicable to the PnC defect lattice. However, one difference between the TB and numerical results should be noticed. The TB model shows an FB that is completely flat, while the numericalresults show that this band is actually a nearly flat one. For the TBmodel, the dispersion relations can be derived from Eq. (1), fkðÞ ¼ f 0orfkðÞ ¼ f0+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ12þγ22þγ32þ2γ1γ2coskaðÞp :(2) Thus, there is an energy band fkðÞ ¼ f0independent of the wave vector k, implying that this band is completely flat in the momen- tum space. It is noted here that the (nearly) flatbands may appear in a PnC with a single point defect [e.g., Fig. 9(b) ]. In such a case, the point defect does not interact with the other point defects, andthus, the flat bands correspond to the localized modes in thebandgap. However, the formation mechanism of the acoustic FB in this work is different from that in the PnC with a single point defect. When the distance between the point defects is not suffi-ciently far, they can interact with the others via the evanescent cou-pling. 42Herein, by tuning the hopping strengths (e.g., using special lattice structures), the acoustic FB is generated due to the destruc- tive interference of these point defects.11,19,20We indicate that in FIG. 1. (a) Schematic of the TB model of the 1D stub lattice, where the red, green, and yellow circles denote sublattice sites A, B, and C, respectively. (b)Schematic of the PnC defects in the stub lattice, where the size of the super-cell is 8a 0/C211a0. (c) Band structure of the PnC defect lattice, where the blue solid lines and the red dots represent the results calculated by the TB model and theFEM, respectively. (d) Pressure field distributions of the acoustic FB at points Γ (k¼0) and X ( k¼π=a). The corresponding eigenvectors obtained by the TB model are shown in the insets.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 145104 (2021); doi: 10.1063/5.0040804 129, 145104-2 Published under license by AIP Publishing.the TB approximation, only the NN hopping between the PnC defects is considered. However, in the realistic PnC structure, the interactions do not only occur between the NN defects but alsobetween farther defects, such as the next-nearest-neighbor (NNN)hopping [ γ 0inFig. 1(a) ]. The interactions beyond the NN hopping lead to the FB slightly dispersive or the so-called nearly flat. Here, we have to emphasize that compared to the NN hopping, the other hopping strengths are quite weak. The details about the NNNhopping can be found in Appendix C . Moreover, from Eq. (2), one can see that the existence of the FB is independent of the hopping strengths ( γ 1,γ2,γ3) as well. This FB is a “topologically protected ”one occurring in bipartite systems.2,43For such kind of FBs, the FB is protected by the chiral symmetry, and the hopping strength values have no effects on theFB. We will plot the acoustic band structures for the systems withchanged hopping strengths in Sec. III. For the FB of the stub lattice, the wave amplitude on sublattice site A is zero. 20It is necessary to confirm whether the same phe- nomenon can be found in the present acoustic system. To this end,the acoustic fields of the eigenmodes at the Brillouin zone center(point Γ) and boundary (point X) are illustrated in Fig. 1(d) .I n addition, the eigenvectors calculated by the TB model are given in the insets. One can observe that the acoustic fields are confined inthe PnC defects on sublattice sites B and C. Meanwhile, the acous-tic fields corresponding to sublattice A are almost zero. The eigen- modes obtained from the real acoustic and theoretical models are consistent. Next, the frequency response of a finite-size PnC defect lattice is investigated to examine the existence of the acoustic FB, asshown in Fig. 2(a) . For the sake of simplicity, we use a plane waveas the incident wave. The top and bottom boundaries are consid- ered sound-hard walls, while the absorbing boundary is placed at the right end of the structure. Two detection points 1 and 2 are setin the PnC defects, which correspond to sublattice sites A and B,respectively. The FB as well as the other two dispersive bands canbe identified from the transmission spectra [ Fig. 2(b) ]. It is clearly seen that the amplitude at point 2 is much larger than that at point 1. Furthermore, from the pressure distribution, one can confirmthat the acoustic waves concentrate on sublattice sites B and C andsignificantly decay on site A. Considering the experimental aspect, we note that the disper- sion relation of the acoustic FB can be measured by the transmis- sion test using the finite-sized structure shown in Fig. 2(a) . The pressure fields at each defect center along the green dashed lineshown in Fig. 2(a) are collected and recorded. The collected results are then represented in the wave number domain by performing a spatial Fourier transform. 44,45By sweeping the exciting frequency in the concerning frequency window, the dispersion relations of thePnC defect lattice can be obtained. The dispersion relationsobtained by the transmission calculations are plotted in Fig. 3 , where the bright and dark colors correspond to the high and low Fourier amplitudes, respectively. Additionally, the dispersion rela- tions calculated within the super-cell are also depicted; see the cyandots in Fig. 3 . The FB as well as the two dispersive bands can be clearly identified from the band structure, meaning that the disper- sion relations of the system can be measured in a finite-sized PnC structure, which can be easily conducted in the experiment. B. Lieb lattice In this subsection, the PnC defects are arranged in the Lieb lattice, known as a typical 2D FB lattice. The schematic diagram of the TB model of the Lieb lattice is illustrated in Fig. 4(a) , where the NN hoppings between different sublattices are denoted by γ 1/C0γ4 (the black solid lines). For the Lieb lattice, the unit-cell [the dashed FIG. 2. (a) Schematic of the frequency response calculation of the PnC defect lattice. (b) Bulk transmission spectra for the PnC defect lattice, where the red and blue solid lines are the results obtained at detection points 1 and 2, respec-tively. (c) Pressure field distribution of the PnC defect lattice at f¼7904 Hz. FIG. 3. Dispersion relations of the PnC defect lattice obtained by the transmis- sion test shown in Fig. 2(a) .Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 145104 (2021); doi: 10.1063/5.0040804 129, 145104-3 Published under license by AIP Publishing.square in Fig. 4(a) ] also contains three sublattices (A, B, and C). The scheme of the PnC defect lattice is illustrated in Fig. 4(b) . The geometrical parameters a0and rare the same as those in Sec. II A. The distances between the neighboring defects are denoted by d1, d2,d3, and d4, where d1þd2¼aand d3þd4¼awith abeing the lattice constant of the super-cell. For the model shown in Fig. 4(b) , we have d1¼d2¼d3¼d4¼4a0. For the Lieb lattice, the Hamiltonian in the momentum space reads ^HkðÞ ¼f0 γ1þγ2e/C0ikxaγ3þγ4e/C0ikya γ1þγ2eikxaf0 0 γ3þγ4eikya0 f00 @1 A, (3)where k¼(kx,ky) is the wave vector, f0is the resonant frequency, and γi(i¼1, 2, 3, 4) is the NN hopping strength. According to Eq.(3), the dispersion relations of the system can be obtained as fkðÞ ¼ f0orfkðÞ ¼f0+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ12þγ22þγ32þγ42þ2γ1γ2cos(kxa)þ2γ3γ4cos(kya)q : (4) The band structure of the defect Lieb lattice is shown in Fig. 4(c) , where the wave vector kvaries along the boundary of the first irre- ducible Brillouin zone Γ—X—M—Γ. The bands calculated by the full-wave simulation are plotted by the red dots. Meanwhile, theresults based on the TB model are given as a comparison (the bluesolid lines). From the band structure, one can see an FB intersect- ing two dispersive bands at point M of the Brillouin zone boun- dary, where the two dispersive bands form a Dirac cone. Based onthe TB model, the FB fkðÞ ¼ f 0is independent of the wave vector kmeaning that this band is completely flat over the entire Brillouin zone. However, for the real acoustic system, the hoppings beyond the NN one cause the FB slightly bent [the red dots in Fig. 4(c) ], which is similar to the stub lattice. Additionally, the 3D band struc-ture is plotted in Fig. 4(d) . One can clearly see that this FB spreads over the whole Brillouin zone and touches the other two dispersivebands at point M. It is noteworthy that the FB fkðÞ ¼ f 0in the Lieb lattice is also protected by the chiral symmetry and indepen- dent of the hopping strengths.2,43 In the Lieb lattice, the wave function of the FB is localized on sublattice sites B and C and vanishes on site A.9,10We plot t h ea c o u s t i ce i g e n f i e l d sa tt h eB r i l l o u i nz o n ec e n t e r( p o i n t Γ) and boundary (point M) in Fig. 4(e) . It is found that the acoustic energy is confined only in the PnC defects on sublat-tice sites B and C. The eigenvectors calculated by the TBmodel are shown in the insets, which agree with the full-wave simulation results. Figure 5(a) illustrates the scheme for calculating the bulk transmission in the PnC defect lattice. In the simulations, a planewave as the source transmits from the left to the right side. Weassume that the structure is infinite along the vertical direction, and then the periodic boundary conditions are applied on the top and bottom boundaries. In the spectra, the transmission peaks forthe FB and the first dispersive band are apparent, where the peakfor the FB appears in a narrow frequency window. However, thesecond dispersive band can hardly be excited by the incident wave because the plane wave source does not match its modal pattern. The pressure field distribution of the acoustic FB is plotted inFig. 5(c) . The field distribution exhibits the typical pattern of an FB: the field amplitude is large on sites B and C and rapidlydecays on site A. Similar to the stub lattice, the dispersion relations of the acoustic Lieb lattice along the ΓX direction are obtained by the transmission test [ Fig. 5(a) ]; see Fig. 6 . The FB and the lower dis- persive band can be clearly observed, while the upper dispersive band can hardly be identified from the transmission calculation. This is because that the upper dispersive band can hardly be FIG. 4. (a) Schematic of the TB model of the Lieb lattice, where the red, green, and yellow circles denote sublattice sites A, B, and C, respectively. (b) Schematic of the PnC defects in the Lieb lattice, where the size of the super-cell is 8a0/C28a0. (c) Band structure of the PnC defect lattice, where the blue solid lines and the red dots represent the results calculated by the TB model and theFEM, respectively. The high-symmetry points follow the usual conventions where Γ¼(0,0),X¼(π=a,0), and M ¼(π=a,π=a). (d) The 3D band struc- ture of the PnC defect lattice obtained by the FEM. (e) Pressure field distribu-tions of the acoustic FB at points Γand M. The corresponding eigenvectors obtained by the TB model are also shown in the insets.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 145104 (2021); doi: 10.1063/5.0040804 129, 145104-4 Published under license by AIP Publishing.excited by the incident plane wave incident, as we have already mentioned previously. For both lattices, the realization of the acoustic FBs is based on the PnC defect mode. Therefore, the acoustic FBs in these PnC defect lattices can be tuned by modifying the basic defect mode(seeAppendix D ). III. ROBUSTNESS OF ACOUSTIC FLATBANDS AGAINST PERTURBATION OF HOPPING STRENGTHS The acoustic FBs in the stub and Lieb lattices are independent of the NN hopping strength [see Eqs. (2)and(4)]. In other words,they are the topologically protected FBs due to the chiral symme- try. 2,21,43,46The corresponding discussions on the chiral symmetry of the PnC defect lattices are given in the supplementary material . For the PnC defect lattices, the hopping strength can be tuned by FIG. 5. (a) Schematic of the frequency response calculation of the PnC defect lattice. (b) Bulk transmission spectra for the PnC defect lattice, where the red and blue solid lines are the results obtained at detection points 1 and 2, respec-tively. (c) Pressure field distribution of the PnC defect lattice at f¼7901 Hz. FIG. 6. Dispersion relations of the PnC defect lattice obtained by the transmis- sion test shown in Fig. 5(a) , where the acoustic waves propagate along the ΓX direction. FIG. 7. (a) Schematic of the PnC defects in the stub lattice, where d1¼3a0, d2¼5a0, and d3¼4a0. (b) Band structure of the PnC defect lattice, where the blue solid lines and the red dots represent the results for the models shownin (a) and Fig. 1(b) , respectively. (c) Pressure field distribution of the acoustic FB at point Γ(k¼0). FIG. 8. (a) Schematic of the PnC defects in the Lieb lattice, where d1¼d4¼ 3a0and d2¼d3¼5a0. (b) Band structure of the PnC defect lattice, where the blue solid lines and the red dots represent the results for the models shown in(a) and Fig. 4(b) , respectively. (c) Pressure field distribution of the acoustic FB at point Γ(k x¼ky¼0).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 145104 (2021); doi: 10.1063/5.0040804 129, 145104-5 Published under license by AIP Publishing.changing the distances between the two neighboring PnC defects. Then, we modify the distances between the two neighboring defects in Figs. 1(b) and 4(b), leading to the perturbation of the hopping strengths. The modified PnC models are depicted inFigs. 7(a) and 8(a) for the stub and Lieb lattices, respectively. We calculate the band structures by the FEM; see Figs. 7(b) and8(b). The results with unperturbed hopping strengths are also illustrated for comparison. For both lattices, the acoustic FBs keep almostunchanged under the perturbation in the hopping strength, whilethe other two dispersive bands shift obviously. Especially for theLieb lattice, it is seen that the Dirac cone at point M opens when the perturbation in the hopping strength is introduced. In addition, the calculated eigenfields at the Brillouin zone center (point Γ)a r e plotted. One can see that the acoustic waves are confined on sublat-tice sites B and C, showing similar modal patterns as the unper-turbed ones. IV. CONCLUSIONS Inspired by the FB lattices in electronic and photonic systems, we design the PnC defects in the stub and Lieb latticesand realize the FBs in acoustic systems. The acoustic waves in the PnC defect lattices can be well described by the TB model. For the stub and Lieb lattices, the NN hopping between the PnC defectsdetermines the appearance of the acoustic FBs. The NNNhopping shows weak effects on the FBs but induces the FBs bentslightly. Moreover, the results show that the acoustic FBs are insensitive to the perturbation of the hopping strengths. We point out, in particular, that the acoustic FB can be utilized for logicoperations. 20The PnC point defects can enhance the amplitude of the incident waves and hence be designed as piezoelectric energyharvesters. 47For the PnC defect lattices, all three bands (one FB and two dispersive bands) can be potentially exploited in the broadband energy harvesting. Besides, a further interesting appli-cation of the acoustic FB is the ultra-sensitive detection of smallnearby scatterers. 17 We note that the acoustic FBs could be realized by properly setting the PnC defects in other forms of lattices, ranging from 2Dto 3D lattices. Such a strategy is not limited to acoustic systems andcan be extended to other physical systems, such as elastic and pho-tonic systems. Furthermore, as the bands can be engineered by tuning the hopping strengths and lattice geometries, it is interesting to investigate the topological characteristics of different PnC defectlattices. To date, a variety of PnC devices have been proposed andvalidated. It is expected that the defect lattices and other functionalcomponents can be integrated into the same PnC platform accord- ing to practical requirements, opening up new avenues for design- ing multi-functional acoustic devices. SUPPLEMENTARY MATERIAL See the supplementary material for chiral symmetry of the PnC stub and Lieb lattices and the effective Hamiltonian near the Dirac cone for the PnC Lieb lattice.ACKNOWLEDGMENTS This work was supported by the German Research Foundation (DFG, ZH 15/27-1) and the Joint Sino-GermanResearch Project (Grant No. GZ 1355). Y.-S. Wang is also grateful to the support by the Major Program of the National Science Foundation of China (No. 11991031) and the Innovative ResearchGroup of NSFC (No. 12021002). APPENDIX A: BAND STRUCTURES OF THE UNIT-CELL AND THE SUPER-CELL WITH A POINT DEFECT The band structure of the PnC unit-cell calculated by the FEM is shown in Fig. 9(a) . A complete bandgap is clearly observed, which ranges from 6.87 to 8.84 kHz. It is noteworthy that the com- plete bandgap is the prerequisite for generating the point defectmodes. Then, the band structure of the PnC super-cell containingone point defect is plotted in Fig. 9(b) , where the size of the super- cell is 9 a/C29a. Considering Bloch ’s boundary condition, the dis- tance between the adjacent point defects is 9 a, and hence, the inter- action between these point defects is negligible. It is seen that onedefect mode emerges inside the complete bandgap, which isemployed to construct the acoustic FB systems. The pressure distri-bution of the defect mode at point Γ[Fig. 9(d) ] shows that the acoustic waves are confined in the defect region. The modal pattern of the defect mode exhibits the s-type symmetry. 41Besides, notice FIG. 9. Band structures of the PnC unit-cell (a) and the super-cell with a point defect (b). (c) The first irreducible Brillouin zone of the square lattice. (d)Pressure field distribution of the defect state at point Γ(k x¼ky¼0).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 145104 (2021); doi: 10.1063/5.0040804 129, 145104-6 Published under license by AIP Publishing.that the acoustic fields of the defect lattices [ Figs. 1(d) and4(e)]a r e similar to that shown in Fig. 9(d) . This confirms that the PnC defect lattices in this work are originated from the point defectmode. APPENDIX B: FITTING PARAMETERS OF THE TIGHT-BINDING MODEL The parameters used in the TB model are estimated by fitting the full-wave simulation results to the TB model. We start with the simple PnC coupled resonator waveguide, 42as shown in Fig. 10(a) . The lattice constant of the super-cell is a¼4a0, which is the same as the distance between two neighboring defects. This system canbe described by the TB model, 48 i@An @t¼f0AnþγAn/C01þγAnþ1, (B1) where Andenotes the amplitude of the defect mode within the nth super-cell, f0is the resonant frequency of the PnC defects, and γis the hopping strength. Considering the time-harmonic behavior ofthe wave fields and the periodicity of the system, we can derive thedispersion relation analytically as fkðÞ ¼ f 0þ2γcoskaðÞ. The full- wave simulation is performed by the FEM. The Floquet and peri- odic boundary conditions are applied to the boundaries of the computational domain along the horizontal and vertical directions,respectively. The finite element results are fitted to the analyticalexpression, and then the corresponding parameters (i.e., f 0andγ) can be obtained. In this system, the resonant frequency and hopping strength are set as f0¼7906 Hz and γ¼/C030:01 Hz, respectively. The band structure of the coupled resonator waveguideis shown in Fig. 10(b) . One can see that the energy band calculated by the full-wave simulations can be well predicted by theTB model.APPENDIX C: EFFECT OF THE NEXT-NEAREST-NEIGHBOR HOPPING In this section, the effect of the NNN hopping in the TB model is discussed, for example, γ 0inFig. 1(a) . First, let us consider the stub lattice case. For the sake of simplicity, the NN and NNN hopping strengths in the stub lattice are denoted by γand γ0, respectively. In the TB model, the Hamiltonian of this system canbe obtained as follows: ^HkðÞ ¼f 0 γþγe/C0ikaγ γþγeikaf0 γ0þγ0eika γγ0þγ0e/C0ikaf00 @1 A: (C1) Figure 11(a) shows the band structure of the stub defect lattice [the model in Fig. 1(b) ], where the red dots and the black dashed lines correspond to the results calculated by the FEM and the TB model with the NNN hopping, respectively. As a comparison, the TB results without the NNN hopping are plotted as well (the bluesolid lines). It is observed that if the NNN hopping is considered inthe TB model, the middle band (FB) is no longer completely flatbut slightly bent. By comparison with the FB, the NNN hopping does not show a significant effect on the other two dispersive bands. Therefore, it is inferred that the flatness of the FB isdependent on the NNN hopping. For the Lieb lattice, the bandstructure is illustrated in Fig. 11(b) , where similar phenomena can also be found. It is noteworthy that compared to the NN hopping (γ¼/C030:01 Hz), the NNN hopping strength ( γ 0¼3:45 Hz) is quite weak. Therefore, the NN hopping between the PnC defectsplays a dominant role in the TB model. FIG. 10. (a) Schematic of the PnC coupled resonator waveguide. (b) Band structure of the coupled resonator waveguide, where the blue solid line and thered dots are the results calculated by the TB model and the FEM, respectively. FIG. 11. Band structures of the PnC defect lattices: (a) stub and (b) Lieb lat- tices. The red dots and the blue solid and black dashed lines represent theresults calculated by the FEM and the TB model with and without the NNN hopping, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 145104 (2021); doi: 10.1063/5.0040804 129, 145104-7 Published under license by AIP Publishing.APPENDIX D: TUNING THE FLATBANDS VIA THE DEFECT MODE The acoustic FBs originate from the PnC defect mode, imply- ing that the frequency of the FBs could be tuned by modifying thedefect mode. An easy way to tune the PnC defect mode is changing the radius of the steel rod on the defect site, 49,50as shown in the inset of Fig. 12(a) . For the PnC considered in this work, the defect mode frequency increases gradually as the defect rod radius rd increases. We change the defect rod radius and plot the band struc- tures obtained by the FEM for the stub [ Fig. 12(a) ] and Lieb [Fig. 12(b) ] lattices. As we change rdfrom 0 :1a0to 0 :3a0, the FBs keep the flatness while they shift to higher frequencies. Theseresults show that the FBs can be effectively tuned by changing therelated defect mode. 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Phys. 84, 3026 –3030 (1998).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 145104 (2021); doi: 10.1063/5.0040804 129, 145104-9 Published under license by AIP Publishing.

5.0033181.pdf

J. Chem. Phys. 154, 044307 (2021); https://doi.org/10.1063/5.0033181 154, 044307 © 2021 Author(s).A SA-CASSCF and MS-CASPT2 study on the electronic structure of nitrosobenzene and its relation to its dissociation dynamics Cite as: J. Chem. Phys. 154, 044307 (2021); https://doi.org/10.1063/5.0033181 Submitted: 14 October 2020 . Accepted: 11 January 2021 . Published Online: 28 January 2021 Juan Soto , Daniel Peláez , and Juan C. Otero ARTICLES YOU MAY BE INTERESTED IN Modern quantum chemistry with [Open]Molcas The Journal of Chemical Physics 152, 214117 (2020); https://doi.org/10.1063/5.0004835 The ORCA quantum chemistry program package The Journal of Chemical Physics 152, 224108 (2020); https://doi.org/10.1063/5.0004608 Accelerating coupled cluster calculations with nonlinear dynamics and supervised machine learning The Journal of Chemical Physics 154, 044110 (2021); https://doi.org/10.1063/5.0037090The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp A SA-CASSCF and MS-CASPT2 study on the electronic structure of nitrosobenzene and its relation to its dissociation dynamics Cite as: J. Chem. Phys. 154, 044307 (2021); doi: 10.1063/5.0033181 Submitted: 14 October 2020 •Accepted: 11 January 2021 • Published Online: 28 January 2021 Juan Soto,a) Daniel Peláez,2 and Juan C. Otero1 AFFILIATIONS 1Department of Physical Chemistry, Faculty of Science, University of Málaga, Andalucía Tech., E-29071 Málaga, Spain 2Institut des Sciences Moléculaires d’Orsay (ISMO) - UMR 8214, Université Paris-Saclay, 91405 Orsay Cedex, France a)Author to whom correspondence should be addressed: soto@uma.es ABSTRACT The photodissociation channels of nitrosobenzene (PhNO) induced by a 255 nm photolytic wavelength have been studied using the complete active space self-consistent method and the multistate second-order multiconfigurational perturbation theory. It is found that there exists a triplet route for photodissociation of the molecule. The reaction mechanism consists of a complex cascade of nonadiabatic electronic tran- sitions involving triple and double conical intersections as well as intersystem crossing. Several of the relevant states (S 2, S4, and S 5states) correspond to double excitations. It is worth noting that the last step of the photodissociation implies an internal conversion process. The experimentally observed velocity pattern of the NO fragment is a signature of such a conical intersection. Published under license by AIP Publishing. https://doi.org/10.1063/5.0033181 .,s I. INTRODUCTION Nitrosobenzene (C 6H5NO) is the smallest member of the aro- matic C-nitroso family of compounds. This class of compounds presents a rich chemistry1as they are involved in many interesting processes in biological processes, organic synthesis, decomposition of energetic materials, or combustion chemistry.2–5The underlying reason for such a varied chemistry lies in its rich electronic structure, which has motivated its extensive study.6From an experimental per- spective, its photodissociation dynamics has been studied at various excitation wavelengths using different experimental techniques (see Table I).5–14 (1) The UV absorption spectrum of the molecule in the vapor phase shows a weak absorption band7,15with a maximum near13 333 cm−1(1.65 eV) together with three strong bands with maxima at 34 500 (4.28), 37 000 (4.59), and 46 500 cm−1(5.77 eV).12These features have been assigned to excitations from the ground state to the first four singlet excited states of the molecule. With respect to the photoproducts, there is consensus in that the only possible reac- tion channel is the Ph–NO bond breaking, i.e., C 6H5NO→C6H5 + NO, independently of the excitation energy. Only at the shortest wavelength used in the experiments5(193 nm), an additional reac- tion product is observed, benzyne (C 6H4), which arises from the subsequent decomposition of the phenyl radical (C 6H5) generated in the first dissociation step into C 6H4and the H atom.5A remark- able feature in this system is the large energy gap between the S 1 state and its upper neighboring states despite of which no fluores- cence from S nto S 1has been observed.6–9This is a relevant aspect since it provides a first hint on the photobehavior of nitrosobenzene: its photodissociation takes place on the S 0or S 1states. Experiments suggest that after excitation to S n(n≥2), the molecule undergoes internal conversion to the S 1state, in accordance with the absence of observed fluorescence.6–8Furthermore, velocity mapping ion imaging data6–9suggest that the nascent NO is generated with a propeller-like motion. Indeed, there is direct evidence that NO does J. Chem. Phys. 154, 044307 (2021); doi: 10.1063/5.0033181 154, 044307-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Observed products and detection techniques in the photodissociation of nitrosobenzene. Detection λexc/nm; [eV] Observed products techniqueaReference 248; [5.00] C 6H5+ NO MII 5 193; [6.42] C 6H5+ NO//C 6H4+ H MII 5 305; [4.07] C 6H5+ NO REMPI/VIP 6 255; [4.86] C 6H5+ NO LIF 7 290; [4.28] C 6H5+ NO REMPI/VIP 8 and 12 266; [4.66] C 6H5+ NO REMPI/VIP 8 and 12 225; [5.51] C 6H5+ NO REMPI 11 266; [4.66] C 6H5+ NO TOFMS 13 266; [4.66] C 6H5+ NO LIF 14 aMII: Multimass ion imaging; REMPI: resonance enhanced multiphoton ionization; VIP: velocity-mapped ion imaging; LIF: laser induced fluorescence; and TOFMS: time of flight mass spectrometry. have a preferred parallel arrangement of the velocity and angular momentum vectors ( v||j) in the recoil trajectory, which, in turn, supports the hypothesis that deactivation of molecules occurs via internal conversion.6–9 Concerning theoretical studies, to the best of our knowl- edge, nitrosobenzene has not been much studied in contrast to its aliphatic C-nitroso (R–NO) and nitrite (RO–NO) counterparts.16–21 In this respect, multiconfiguration self-consistent field (MCSCF) approaches are arguably the most appropriate quantum chemical approximations to understand the chemistry of these strongly cor- related systems.22–24This issue is especially relevant in the case of nitrosobenzene where double excitations, multiple coupled excited states, and surface crossings are involved. The objective of this work is to precisely disentangle the electronic structure and photodisso- ciation dynamics of the title molecule, and for this, we will make use of second-order perturbation theory on a MCSCF reference wavefunction. II. METHODS OF CALCULATION Extended relativistic basis sets of the atomic natural orbital (ANO)-type, the so-called ANO-RCC basis sets,25,26have been used throughout this work by applying the (C,N,O)[4 s3p2d1f]/(H) [3s2p1d] contraction scheme. The complete active space self- consistent field (CASSCF)27–33and the multi-state second-order per- turbation (MS-CASPT2)34,35methods have been applied as imple- mented in the MOLCAS 8.4 program.36,37MS-CASPT2 energies have been calculated for the CASSCF-optimized geometries. To avoid the inclusion of intruder states in such calculations, an imag- inary shift set to 0.1 has been applied. Equally, the IPEA empiri- cal correction has been fixed at 0.25 in all the calculations. When CASSCF has been applied with the state average approximation, the notation SA n-CASSCF has been used, where ndenotes the number of states of a given symmetry included in the calculation. Mini- mum energy crossing points of the same spin multiplicity (i.e., con- ical intersections) have been optimized with the algorithm38imple- mented in MOLCAS. The analysis of molecular geometries and molecular orbitals has been performed with the programs Gabe- dit39and Molden,40while the analysis of vibrational normal modesand vectors defining the branching space (energy difference and derivative coupling) has been performed with the program Mac- Molplt.41Spin–orbit coupling constants have been computed with a spin–orbit Fock-type Hamiltonian.42–44For the sake of clarity, two conventions have been used to label the electronic states: (i) spectroscopic and (ii) energetic ordering (S 0, S1,. . .; T 0, T 1,. . .). Construction of potential energy curves (PECs) has been done with our linear interpolation method45–51using the full space of non- redundant internal coordinates. The latter provide an accurate 1D- representation of the potential energy surface in the space spanned by a given set of internal coordinates. The PECs for the dissociation are built as follows: First, a common set of 3 N-6 internal coordi- nates is defined for the target geometries, the reactant ( R1), and the dissociation fragments separated by a physically reasonable distance (R2). Our experience shows that a separation distance of 4.7 Å of the dissociative bond (C–N distance for dissociation of nitrosoben- zene into phenyl and NO) is enough to reach the asymptotic limit of the PEC of interest. Second, the difference between R2andR1yields an interpolation vector ( ΔR) that connects reactants and products. Third, ΔRis divided by n(an entire number at the choice of the user). Each of the divisions constitutes what we will call a step. In consequence, each step mcorresponds to a nuclear configuration given by Rm=R1+ (m/n)ΔR. Since we are using internal coor- dinates (internuclear distances, valence bond, and dihedral angles), we cannot give a unique unit for the reaction coordinate. In what follows, we will indicate arbitrary units . Linear interpolations in internal coordinates present two favorable characteristics: (i) they are less demanding computationally than a scan with relaxation of geometry and (ii) all the points along the interpolation vector (reac- tion coordinate) are necessarily in a straight line. That is not true, however, for scanning of the potential energy surfaces with geometry relaxation. III. RESULTS AND DISCUSSION A. Electronic structure of nitrosobenzene and excited states As is well known, the choice of different orbital subspaces lead- ing to the definition of a CASSCF wavefunction is crucial to give a correct answer to the problem under study. Otherwise, the con- clusions will be unfortunately erroneous.52–54In spite of methods for the automated selection of the active space such as the novel AVAS,55this task requires some chemical intuition combined with a trial and error process to select the appropriate active orbitals.22,23 This is particularly true when the process under scrutiny requires the sampling of quite different regions of the configuration space (e.g., asymptotic channels where several states converge). Concerning the title molecule, after consideration of the targeted processes, we have chosen an active space consisting of 18 electrons distributed in 15 orbitals, hereafter represented as (18e, 15o; Fig. 1). As a first stage, we have calculated the vertical excitation energies of the molecule at the MS-CASPT2 level using a SA6-CASSCF(18e, 15o) reference wavefunction. Provided that the geometry of the electronic ground state of nitrosobenzene has been experimentally determined to be planar ( Cssymmetry),56,57these SA-CASSCF calculations have been performed under such a symmetry point group, unless otherwise indicated. Geometries have also been enforced to remain C s. The J. Chem. Phys. 154, 044307 (2021); doi: 10.1063/5.0033181 154, 044307-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . SA6-CASSCF/ANO-RCC natural orbitals included in the active space (18e, 15o) of nitrosobenzene. Mean occupation numbers are given in square brackets. results of such vertical excitations are given in Table II together with the experimental vacuum ultraviolet absorption maxima registered in the vapor phase.7,58Overall, we observe quite a satisfactory agree- ment between the present results and experiments, both in energies and intensities. Of course, it should be understood that this compar- ison is not straightforward, given that the computed transitions are not corrected by the zero-point energy of the two states involved in the transition. As a first important result, we are able to reproduce the large gap ( >2.5 eV) responsible for the absence of fluorescence from S nto S 1. Moreover, the calculated values of oscillator strengths and electronic transitions are consistent with the experimental observation that photolysis at 255 nm (4.86 eV) produces the largest absolute yield of NO fragments,7that is, the estimated oscillator strength of the S 6state yields the highest calculated value among all the studied states. Finally, we remark that S 2(11A′→21A′), S 4(11A′ →21A′′), and S 5(11A′→31A′′) states possess a double-excitation character.B. Excitation of nitrosobenzene at 255 nm: Photodissociation into phenyl radical and nitric oxide In the following, we will present and discuss our theoreti- cal results regarding the photolysis of nitrosobenzene and compare them to the different experimental results available.5–14As a starting point to analyze the photodissociation dynamics of this molecule, we consider the potential energy curves depicted in Fig. SI of the supplementary material. These linear interpolations, computed at the MS-CASPT2 level of theory with a SA6-CASSCF(16e, 14o) ref- erence wavefunction (C ssymmetry), depict the dissociation of the parent molecule into a phenyl radical and nitric oxide on the lowest- lying singlet and triplet potential energy surfaces. It should be high- lighted that for their computation, we have omitted the 2 s(O) orbital from the active space. The initial geometry of the interpolations cor- responds to the ground state of nitrosobenzene [ M0, Fig. 2(a)] and the end point to the dissociated fragments in their respective ground states (phenyl and nitric oxide). As a proof of the accuracy of the method, we have calculated the enthalpy of the dissociation reac- tion at 0 K (using the CASSCF frequencies of all species for the ZPE correction and the electronic energy at the end point of the interpolation), and its magnitude amounts to 2.32 eV, a value that compares quite well with the enthalpy of the reaction at 0 K from Active Thermochemical Tables59–61(2.33±0.02 eV). As a first remark, we observe that population of the S n(n = 2–5) excited states cannot lead to direct NO elimination since the dissociation limit of each of these singlet states lies well above the respective vertical excitation energy (Table SI). We also observe that the large energy gap between the S 1state and the upper excited states is well preserved along the dissociation path (Fig. SI). The calcu- lated energy difference between the minima on the first two singlet excited states ( M1(S1) and M2(S2) in Figs. 2(b) and 2(c), respec- tively) is∼1.5 eV. Given the absence of S n→S1fluorescence as well as the energetic hindrance to direct NO dissociation inferred from the potential energy curves of Fig. SI, we have searched for mini- mum energy crossing surfaces (conical intersections and intersystem crossings). As guess geometries, we have used the structures in the vicinity of the crossings revealed by our interpolations. The result- ing points are presented in Fig. 2. In view of these results, curves, relative energies, and critical points, we can propose a mechanism to explain the photodissociation of nitrosobenzene after excitation up to S 6(4.86 eV: 255 nm). This process occurs via an electronic cascade of nonadiabatic transitions. The correlation diagram of the states of the reaction is shown in Fig. 3. First, the system decays from S 6to S 4through a nonadiabatic transition involving a multi- state intersection62–64(of states 6, 5, and 4).65Second, the S 4state leads to the S 3surface by means of another nonradiative mecha- nism via an S 4/S3conical intersection [ CI1(S4/S3), Fig. 2(d)]. Third, the S 3state decays to S 2via a third conical intersection [ CI2(S3/S2), Fig. 2(e)]. As mentioned before, there is a very large energy gap between the S 2and S 1states. Thus, we have searched for an S 2/S1 minimum energy crossing point that would provide the sought internal conversion without fluorescence emission and eventually to the formation of NO and phenyl on the S 1surface. After an exhaus- tive search, we have found such a conical intersection [ CI3(S2/S1), Fig. 2(f)]. However, the minimum energy crossing point of CI3 is even higher in energy (4.1 eV) than CI1. Therefore, we think that this channel is not very probable. Hence, to explain the dissociation J. Chem. Phys. 154, 044307 (2021); doi: 10.1063/5.0033181 154, 044307-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II . MS-CASPT2 vertical excitation energies of the lowest-lying singlet and triplet states of nitrosobenzene ( Cs).a VTbΔE (eV)cΔE (eV)dAbMeAbMffOSCgCharacterhWi 11A′′1.67 1.60 1.63∗4.15×10−4(nσ)1(π∗ NO)177 21A′4.27 4.18 4.20 4.22 4.26 ×10−3(nσ)0(π∗ NO)269 31A′4.45 4.37 4.59 4.58 1.67 ×10−2(π2)1(π∗ NO)151 21A′′4.52 4.38 1.40 ×10−3(nσ)1(π3)1(π∗ NO)229 (nσ)1(π∗ 3)128 31A′′4.69 4.49 8.04 ×10−5(nσ)1(π2)1(π∗ NO)219 (nσ)1(π∗ 2)135 41A′4.80 4.62 4.75 4.73 2.06 ×10−1(π3)1(π∗ NO)175 41A′′5.63 5.32 . . . (nσ)1(π∗ NO)2(π2)1(π∗ NO)1(π∗ 2)137 51A′′6.17 5.96 5.88 5.83 1.99 ×10−3(πσ)1(π∗ NO)146 51A′6.23 6.35 6.39∗5.13×10−2(π3)1(π∗ 2)123 (π2)1(π∗ 3)122 61A′′6.62 6.85 2.27 ×10−4(nσ)1(π3)1(π∗ NO)1(π∗ 2)126 (nσ)1(π2)1(π∗ NO)1(π∗ 3)116 61A′7.13 6.85 7.12∗1.85×10−2(π1)1(π∗ NO)127 aReference wave function: CsSA6-CASSCF(18e, 15o)/ANO-RCC; IPEA = 0.25. Imaginary shift = 0.1. bVertical transition from the S 0(11A′) state. cCASSCF optimized geometry ( Cs). dMP2/def2-TZVPP optimized geometry ( Cs). eUV absorption maxima in the vapor phase from Ref. 58;∗represents those in n-hepatane. fUV absorption maxima in the vapor phase from Ref. 7. gOscillator strength. hMS-CASPT2 main electronic configurations of the excited states referred to the ground state configuration (see text). iWeight of the configuration in %. Only contributions greater than 15% are included [reference configuration: (2s(O))2(nσ)2(π1)2(π2)2(π3)2(πσ)2(σNO)2(σCN)2(σNO)2||(π∗ 1)0(π∗ 2)0(π∗ 3)0(σ∗ NO)0(σ∗ CN)0(σ∗ NO)0]. of the molecule after excitation to S n(2≤n≤6), another path is necessary. The only one that we have found involves a surface cross- ing between states of different multiplicities, a mechanism common to another family of N-containing aromatic compounds, the phenyl azide derivatives.66–68Once the S 2state is populated, it experiences an S 2(1A′)/T 2(3A′′) intersystem crossing, whose minimum energy crossing point is displayed in Fig. 2(g) [ ISC1 (S2/T2), with the spin– orbit coupling constant of 23.6 cm−1]. Afterward, the T 2(3A′′) state decays to the T 1(3A′) state via a triplet/triplet internal conversion whose minimum energy crossing point is depicted in Fig. 2(h) [ CI4 (T2/T1)]. At this point, it is important to note that the derivative coupling vector of CI4 [Fig. 2(h)] involves a wagging motion of the system with the opposite phase for nitrogen and oxygen in combi- nation with a torsional motion of the NO moiety. In Fig. SIV, we compare the torsional normal mode of M0 with the derivative cou- pling vector of CI4. Thus, we have searched for another singular point with a non-planar arrangement of the molecule on the T 1sur- face and what we have found is a T 1/T0conical intersection [ CI5 (T1/T0), Fig. 2(i)] whose molecular configuration shows a dihedral angle between the NO moiety and the aromatic ring of ∼115○. The SA4-CASSCF energy profiles that connect CI3 with CI4 are plot- ted in Fig. SVa. Therefore, in accordance with this result, it can be proposed that once the lowest triplet state T 0is populated, the sys- tem has sufficient accumulated energy to dissociate into phenyl and NO because this surface is reached via a cascade of non-radiative transitions. Figure SVb represents the potential energy curve of thedissociation process starting at the geometry of the minimum energy triplet state T 0[M3, Fig. 2(j)]. Given that the molecular configura- tion of CI5 shows a dihedral angle between the NO moiety and the aromatic ring of ∼115○and the minimum energy geometry of the electronic state T 0is planar [ M3, Fig. 2(j)], the CC–NO torsional mode is highly excited and will transform into rotational motion of the salient fragment (NO) at the dissociation limit, in accordance with the conclusion obtained by Ke βler and co-authors.7It must be remarked that ISC1 is very close to CI2 both geometrically and energetically [the distance (in Cartesian coordinates) between the geometries of ISC1 and CI2 is 0.04 Å, and the energy difference between the two crossings is 0.16 eV]. This fact has been shown to increase the efficiency of the intersystem crossing as recently dis- cussed by us66–68and reported by the old masters .69–71To conclude this section, for the sake of clarity, it must be noted that all the singular points given in Fig. 2 are out of the interpolation domain presented in Fig. SI. Figure 3 shows the correlation diagram for the states involved in the photodissociation route. C. Excitation of nitrosobenzene at 193 nm: Formation of benzyne from phenyl radical Benzyne (C 6H4) is a product observed only when nitrosoben- zene is irradiated at 193 nm (6.42 eV). This compound is generated from the phenyl radical (C 6H5) by elimination of atomic hydrogen. It must be noted that the formation of benzyne does not require the J. Chem. Phys. 154, 044307 (2021); doi: 10.1063/5.0033181 154, 044307-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . CASSCF/ANO-RCC geometries of the critical points in the photodisso- ciation of nitrosobenzene. (a) S0mini- mum, (b) S1minimum A′′, (c) S2A′ minimum, (d) CI1(S4/S3) conical inter- section, (e) CI2(S3/S2) conical intersec- tion, (f) CI3(S2/S1) conical intersection, (g)ISC1 (S2/T2) intersystem crossing, (h)CI4(T2/T1) conical intersection, (i) CI5(T1/T0) conical intersection, and (j) M3(T0) minimum A′′. Arrows in main figures: gradient difference vector; in small figures: derivative coupling vec- tors. Cartesian and internal coordinates are given in the supplementary material. Relative energies (in eV) are displayed in brackets. absorption of an extra photon by the phenyl radical. The available internal energy of the latter, after dissociation from NO, suffices for this.5Curiously, although the dissociation–rearrangement channel C6H5NO→C6H4+ HNO is also energetically accessible, after both excitations at 248 nm and 193 nm, no such products were observed by Tseng et al.5Figure 4 presents the potential energy curves corresponding to the lowest-lying doublet states leading to the dissociation of the phenyl radical into benzyne and atomic hydrogen. The interpola- tion curves depicted in Fig. 4 start at the geometry of the mini- mum energy of the radical [CASSCF(7e, 7o)/ANO-RCC], formed in its doublet ground state (12A1:C2v), and end at the (11A1:C2v) FIG. 3 . Diagram of correlation of the states involved in the photodissociation of nitrosobenzene. MS-CASPT2/SA7- CASSCF relative energies in eV are given in parentheses. J. Chem. Phys. 154, 044307 (2021); doi: 10.1063/5.0033181 154, 044307-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . MS-CASPT2/ANO-RCC potential energy curves of the low-lying doublet states of the phenyl radical leading to dissociation into benzyne and atomic hydro- gen. Reference wave function: SA6-CASSCF(7e, 7o). A′states (solid lines); A′′ states (dotted lines). minimum energy of benzyne, with the dissociating hydrogen sep- arated by ∼6.7 Å of the aromatic ring. The computed asymptotic limit for the dissociation of C 6H5into benzyne and atomic hydro- gen in the ground state (12A′in Fig. 4) amounts to 3.76 eV. Again, after correcting by the zero-point energy of phenyl and benzyne (by using their respective CASSCF frequencies), we obtain a reaction enthalpy of 3.44 eV, a value that compares quite well with the value at 0 K obtained from Active Thermochemical Tables59–61[3.45(7) ±0.01 eV]. This limit is easily accessible for C 6H5, given that it accumulates around 95% of the available energy (from the excit- ing photon, 6.42 eV) in its vibrational degrees of freedom, that is, 4.56 eV, after subtraction of translational energy.5Concerning the possibility of dissociating on the excited states, two features must be remarked: (i) the next dissociation limit (22A′in Fig. 4) is computed almost at the same energy than the excitation of the source, while the other states are even higher in energy than the excitation line; and (ii) almost all the dissociation limits do not reach an asymptotic behavior, that is, they are not dissociative with respect to the for- mation of benzyne. Therefore, it must be concluded that benzyne is formed from the benzyl radical on its lowest potential energy surface (12A1). D. Photodissociation of nitrosobenzene into phenylnitrene and atomic oxygen In this section, we tackle the study of the elimination of atomic oxygen from nitrosobenzene with the concomitant formation of phenylnitrene. Figure 5 displays the adiabatic Cspotential energy curves of the low-lying singlet and triplet states for such a reac- tion. According to the spectroscopic convention, the hierarchy of the energy levels of an atom is configuration-term-level-state . The configuration of an atom, that is, the orbitals that are occupied by its electrons, is split by electrostatic interactions between electrons into terms ; due to spin–orbit coupling, these terms are separated into levels ; and finally, the interaction of electrons with an external field uncouples these levels into states . Given that in the calculations of FIG. 5 . MS-CASPT2/ANO-RCC potential energy curves leading to the dissociation of nitrosobenzene into phenylnitrene and atomic oxygen. Reference wave function: SA5-CASSCF(16e, 14o). A′states (solid lines); A′′(dotted lines). Singlet (blue lines); triplet (red lines). this reaction (Fig. 5), the spin–orbit Hamiltonian is not included, we will consider a configuration-term-state hierarchy. In this case, the role of the external field over the electrons of each fragment (oxy- gen and phenylnitrene) is played by the other fragment. Thus, we would observe at the dissociation limit of nitrosobenzene the fol- lowing almost degenerate states arising from the oxygen atom: three triplet states arising from the O(3P) term, five singlet states from the O(1D) term, and one singlet state from the O(1S) term. However, as the ground state of phenylnitrene is a triplet66(Table SII), the rela- tionships of the states at such dissociation limit are somewhat more cumbersome. For example, the six lowest energy states are almost asymptotically degenerate because they come from the triplet states of oxygen and the ground state of phenylnitrene, respectively. Concerning the energetics of the reaction of oxygen extrusion, it must be noted that its lower dissociation limit, despite accessible, is considerably higher in energy ( ΔE∼2.4 eV) than the analogous dissociation limit for the nitric oxide formation channel (Fig. SI). From our point of view, this explains only partially why this reaction channel is not observed experimentally. The most important factor, in our opinion, is the driving force of the cascade of conical inter- sections after excitation at 255 nm, which guides the system after photon excitation to dissociate into NO and phenyl. IV. CONCLUSIONS In this work, we propose a reaction mechanism for photodis- sociation of nitrosobenzene into phenyl and NO that occurs via two different routes, singlet or triplet. In essence, it consists in an elec- tronic cascade of nonadiabatic transitions (triple and double conical intersections plus intersystem crossing) that starts at the S 6state and ends at the T 0state. Our work explains the experimental observations, that is, prod- ucts and the dynamical properties of the products: (1) The estimated oscillator strength of the S 6state yields the higher calculated value among all the studied states, which J. Chem. Phys. 154, 044307 (2021); doi: 10.1063/5.0033181 154, 044307-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp is in accordance with the experimental observation that pho- tolysis at 255 nm produced the largest absolute yield of NO fragments,7provided that there is only one dissociation channel. (2) The proposed cascade of nonadiabatic transitions, which dominates the process, is in accordance with the conclusion obtained by Ke βleret al. ,7that is, the NO elimination occurs after fast internal conversion (or intersystem crossing) to the triplet state T 0that induces a highly excited torsional state. (3) The origin of the torques producing the propeller-like tra- jectory6–9of NO is a signature of the deactivation pro- cess associated with the CI5(T 1/T0) conical intersection [Fig. 2(i)]. SUPPLEMENTARY MATERIAL See the supplementary material for energetics of dissociation limits, Cartesian coordinates of all the critical points, and vertical excitation energies of phenylnitrene. ACKNOWLEDGMENTS This work was supported by the Spanish Ministerio de Economía, Industria y Competividad (Grant No. CTQ2015-65816- R) and Project Nos. UMA18-FEDER-JA-049 and P18-RT-4592 of Junta de Andalucía and FEDER funds. The authors thank Rafael Larrosa and Darío Guerrero for the technical support in running the calculations and the SCBI (Supercomputer and Bioinformatics) of the University of Málaga for computer and software resources. D.P. gratefully acknowledges computational support from Andrei Borissov (ISMO). DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material. REFERENCES 1H. Yamamoto and N. Momiyama, Chem. Commun. 2005 , 3514. 2P. Zuman and B. Shah, Chem. Rev. 94, 1621 (1994). 3J. Chen, J. Xiong, Y. Song, Y. Yu, and L. Wu, Appl. Surf. Sci. 440, 1269 (2018). 4D. B. Galloway, J. A. Bartz, L. G. Huey, and F. F. Crim, J. Chem. Phys. 98, 2107 (1993). 5C.-M. Tseng, Y. M. Choi, C.-L. Huang, C.-K. Ni, Y. T. Lee, and M. C. Lin, J. Phys. Chem. A 108, 7928 (2004). 6J. A. Bartz, S. C. Everhart, and J. I. Cline, J. Chem. Phys. 132, 074310 (2010). 7A. Kessler, A. Slenczka, R. Seiler, and B. Dick, Phys. Chem. Chem. 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Chem. Phys. 154, 044307 (2021); doi: 10.1063/5.0033181 154, 044307-8 Published under license by AIP Publishing

5.0024559.pdf

J. Chem. Phys. 154, 054102 (2021); https://doi.org/10.1063/5.0024559 154, 054102 © 2021 Author(s).Effect of varying the TD-lc-DFTB range- separation parameter on charge and energy transfer in a model pentacene/ buckminsterfullerene heterojunction Cite as: J. Chem. Phys. 154, 054102 (2021); https://doi.org/10.1063/5.0024559 Submitted: 10 August 2020 . Accepted: 01 January 2021 . Published Online: 02 February 2021 Ala Aldin M. H. M. Darghouth , Mark E. Casida , Xi Zhu (朱熹) , Bhaarathi Natarajan , Haibin Su (蘇海斌) , Alexander Humeniuk , Evgenii Titov , Xincheng Miao (缪昕丞) , and Roland Mitrić COLLECTIONS Paper published as part of the special topic on Excitons: Energetics and Spatio-temporal Dynamics ARTICLES YOU MAY BE INTERESTED IN Excitonic and charge transfer interactions in tetracene stacked and T-shaped dimers The Journal of Chemical Physics 154, 044306 (2021); https://doi.org/10.1063/5.0033272 DFTB+, a software package for efficient approximate density functional theory based atomistic simulations The Journal of Chemical Physics 152, 124101 (2020); https://doi.org/10.1063/1.5143190 -machine learning for potential energy surfaces: A PIP approach to bring a DFT-based PES to CCSD(T) level of theory The Journal of Chemical Physics 154, 051102 (2021); https://doi.org/10.1063/5.0038301The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Effect of varying the TD-lc-DFTB range-separation parameter on charge and energy transfer in a model pentacene/buckminsterfullerene heterojunction Cite as: J. Chem. Phys. 154, 054102 (2021); doi: 10.1063/5.0024559 Submitted: 10 August 2020 •Accepted: 1 January 2021 • Published Online: 2 February 2021 Ala Aldin M. H. M. Darghouth,1,a)Mark E. Casida,2,b) Xi Zhu ( 朱熹),3,c)Bhaarathi Natarajan,3 Haibin Su ( 蘇海斌),3,d) Alexander Humeniuk,4 Evgenii Titov,4,e) Xincheng Miao ( 缪昕丞),4,f) and Roland Mitri ´c4,g) AFFILIATIONS 1Department of Chemistry, College of Science, University of Mosul, Mosul, Iraq 2Laboratoire de Spectrométrie, Interactions et Chimie Théorique (SITh), Département de Chimie Moléculaire (DCM), Institut de Chimie Moléculaire de Grenoble (ICMG), Université Grenoble-Alpes, 301 rue de la Chimie, CS 40700, 38058 Grenoble Cedex 9, France 3Institute of Advanced Studies, Nanyang Technological University, 60 Nanyang View, 639673, Singapore 4Institut für Physikalische und Theoretische Chemie, Julius-Maximilians-Universität Würzburg, Emil-Fischer-Straße 42, D-97074 Würzburg, Germany Note: This paper is part of the JCP Special Topic on Excitons: Energetics and Spatio-Temporal Dynamics. a)E-mail: aladarghouth@uomosul.edu.iq b)Author to whom correspondence should be addressed: mark.casida@univ-grenoble-alpes.fr c)Present address: The Chinese University of Hong Kong, Shenzhen, No. 2001 Longxiang Blvd., Longgang District, Shenzhen, Guangdong 518172, China. d)Present address: Department of Chemistry, The Hong Kong University of Science and Technology, Hong Kong, China. E-mail: haibinsu@ust.hk e)E-mail: evgenii.v.titov@gmail.com f)E-mail: xincheng.miao@uni-wuerzburg.de g)E-mail: roland.mitric@uni-wuerzburg.de ABSTRACT Atomistic modeling of energy and charge transfer at the heterojunction of organic solar cells is an active field with many remaining outstand- ing questions owing, in part, to the difficulties in performing reliable photodynamics calculations on very large systems. One approach to being able to overcome these difficulties is to design and apply an appropriate simplified method. Density-functional tight binding (DFTB) has become a popular form of approximate density-functional theory based on a minimal valence basis set and neglect of all but two center integrals. We report the results of our tests of a recent long-range correction (lc) [A. Humeniuk and R. Mitri ´c, J. Chem. Phys. 143, 134120 (2015)] for time-dependent (TD) lc-DFTB by carrying out TD-lc-DFTB fewest switches surface hopping calculations of energy and charge transfer times using the relatively new DFTBABY [A. Humeniuk and R. Mitri ´c, Comput. Phys. Commun. 221, 174 (2017)] program. An advantage of this method is the ability to run enough trajectories to get meaningful ensemble averages. Our interest in the present work is less in determining exact energy and charge transfer rates than in understanding how the results of these calculations vary with the value of the range-separation parameter ( Rlc= 1/μ) for a model organic solar cell heterojunction consisting of a gas-phase van der Waals complex P/Fmade up of a single pentacene ( P) molecule together with a single buckminsterfullerene ( F) molecule. The default value of Rlc= 3.03 a0 is found to be much too small as neither energy nor charge transfer is observed until Rlc≈10a0. Tests at a single geometry show that the best agreement with high-quality ab initio spectra is obtained in the limit of no lc (i.e., very large Rlc). A plot of energy and charge transfer rates as a function of Rlcis provided, which suggests that a value of Rlc≈15a0yields the typical literature (condensed-phase) charge transfer J. Chem. Phys. 154, 054102 (2021); doi: 10.1063/5.0024559 154, 054102-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp time of about 100 fs. However, energy and charge transfer times become as high as ∼300 fs for Rlc≈25a0. A closer examination of the charge transfer process P∗/F→P+/F−shows that the initial electron transfer is accompanied by a partial delocalization of the Phole onto F, which then relocalizes back onto P, consistent with a polaron-like picture in which the nuclei relax to stabilize the resultant redistribution of charges. Published under license by AIP Publishing. https://doi.org/10.1063/5.0024559 .,s I. INTRODUCTION Charge transfer (CT) and energy transfer (ET) are major top- ics in physical chemistry/chemical physics1that enter into diverse problems such as photobiology2and organic electronics.3Time and distance scales for CT and ET may differ over many orders of mag- nitude for different systems, necessitating the development and use of different theoretical models. Here, we are interested in a method able to treat phenomena occurring on time scales of the order of 100 fs, such as CT occurring at the heterojunction of an organic solar cell. In general, proper atomistic modeling of such a system involves nonadiabatic photodynamics where electronic states are delocalized over several molecules. The number of atoms involved and the need to calculate ensemble-averaged properties mean that efficiency is important. At the same time, the methods used must be realistic enough to give at least a qualitatively correct description of the phe- nomena involved. In the present article, we consider the use of time- dependent (TD)4–8long-range-corrected (lc)9–12density-functional tight binding (DFTB)13–16fewest switches surface hopping (FSSH) photodynamics17,18for atomistic calculations of CT and ET times in model organic solar cell heterojunctions. The particular version of TD-lc-DFTB used in this paper is that implemented in the relatively recent program DFTBABY.9,19,20(Recent applications of DFTBABY include Refs. 21–24.) Unlike conventional TD-DFTB, TD-lc-DFTB contains a range-separation parameter Rlc= 1/μ, which has a large effect on CT energetics and so can also be expected to have a large effect on CT times. It also turns out to have a large effect on ET times. It is, thus, important to understand how these times vary with Rlcin order to be able to make rational choices for particular prob- lems. Here, we investigate how the choice of Rlcaffects CT and ET in a gas-phase model of a P/Fjunction ( P= pentacene and F= buck- minsterfullerene; see Fig. 1). It is to be emphasized that our interest at this time is not in obtaining quantitatively correct estimates of ET and CT times but rather is in understanding which values of Rlclead to qualitatively correct behavior. Indeed, our model is most likely too small to be the basis for a quantitatively correct com- parison with ET and CT times at the heterojunction in a real P/F solar cell (Fig. 2). Not only would a quantitatively correct compar- ison require, at a minimum, the inclusion of bulk dielectric effects, which are not included here and should account for explicit delo- calization of excitations over more than just two molecules, but a quantitatively correct comparison also needs to take into account that the well-known preferred geometry of Pis perpendicular, rather than parallel, to the Finterface. However, the calculations that we are reporting could, in principle, be compared with potential molec- ular beam experiments on our small gas-phase complex, although to the best of our knowledge such experiments have yet to be car- ried out. Nevertheless, the results of our calculations are useful before applications to larger, more realistic models are carried outand may already provide some new insight into this well-studied system. Indeed, although P/Fsolar cells may be the most heavily stud- ied and best characterized organic solar cells, both at the experi- mental level and at the theoretical level, the fact that this solar cell continues to be studied is also a good indication that plenty of things remain to be understood. Table I shows the well-accepted six-step model for the physics of organic solar cells. To this, we could add that singlet fission also occurs in P/Fsolar cells,29although we shall not discuss singlet fission any further in the present work. Our study concerns step (iii) of the six-step model. Tables II and III provide experimental and theoretical times for this CT step obtained via FIG. 1 . The model system used in this work to study the behavior of TD-lc-DFTB calculations as Rlcis varied. Some very rough dimensions have been given for comparison with Rlc. Bottom row: Buckminsterfullerene ( F), roughly 7 Å (13.2 a0) in diameter. Middle row: Pentacene ( P), roughly about 14 Å (26.4 a0) long. Top row: two views of the same P/Fvan der Waals complex, the distance of the clos- est approach between the two molecules is roughly 3.0 Å (5.7 a0). Note that all these dimensions are subject to change during geometry optimizations and during dynamics calculations. J. Chem. Phys. 154, 054102 (2021); doi: 10.1063/5.0024559 154, 054102-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Cartoon of an idealized Tang-type25P/Fsolar cell. ITO is a common abbreviation for indium titanium oxide, a transparent conducting oxide. (A realistic working P/FTang-type solar cell is described in Ref. 26.) various methods for a few organic solar cells, including but not limited to P/F. We have also included results from the present work (PW) in these tables. All results (not just our own) should be taken with a grain of salt as they are subject to interpretational and methodolog- ical limitations. We have already emphasized that our own model is too simple to be expected to be quantitative. Rather, our model should be regarded as a necessary step in testing methodology, which could be extended to a more realistic model on the basis of what TABLE I . The generally accepted model for organic heterojunction solar cells.28 Six-step model (i) Exciton formation via photon absorption (ii) Excition diffusion to the heterojunctiona (iii) Exciton dissociation into closely bound charge-transfer (CT) states at the heterojunctionb (iv) Dissociation of these CT states into charge-separated (CS) states composed of free mobile chargesc (v) Charge transport away from the heterojunctiond (vi) Charge collection at the electrodes aDiffusion length ≤about 10 nm.25 bCT time∼100 fs (Table III). cOver a time scale on the order of a picosecond and over a distance scale on the order of a nanometer.27 dThis is a mesoscopic to macroscopic step with time scales from nanoseconds to milliseconds and distance scales on the order of millimeters.TABLE II . This is a sampling of some organic heterojunction coherent CT times col- lected from the literature. Except for P/F, the reader is referred to cited references for the definitions of the various acronyms used to denote the donor and acceptors. Our own results have also been included in the table with the reference “PW” (present work). Donor/acceptor CT time Reference Expt. P3HT/PCBM 25 fsa30 p-DTS(FBTTh 2)2/PC 71BM <40 fsb38 P3HT/PCBM 20 fsc39 Theory 4T/F 25 fsd30 P/F 25 fse40 P/F ∼15 fsfPW aHigh time-resolution pump–probe spectroscopy and time-dependent density func- tional theory (DFT). bTransient absorption spectroscopy. cTwo-dimensional electronic spectroscopy. dEhrenfest TDLDA calculations. eTD-DFT FSSH. fCIS/AM1. we are learning from calculations on the present, more restricted, model. The first thing to note about Tables II and III is the presence of two different CT times. Coherent CT (Table II) refers to rapid charge transfer back and forth between PandFand may go by additional names, such as Rabi or Stückelberg oscillations, depend- ing on how the oscillations are explained. Incoherent CT (Table III) refers to a longer time scale non-oscillatory charge transfer. Caution is in order when studying the literature as sometimes the distinc- tion between coherent and incoherent CT does not seem to be very clear. Here, we are primarily only interested in incoherent CT. One reason for this is that the FSSH method is most likely too crude to be able to provide a realistic description of the coherent periodic localization/delocalization of wavefunctions30–33needed to describe coherent CT,34although the FSSH method can give at least a partial description of phenomena such as Stückelberg oscillations18,35and the localization/delocalization of flickering polarons.36,37The table shows that literature numbers for incoherent CT times vary from as low as about 40 fs to as high as about 700 fs with the most common estimates being around 100 fs. We will also find an incoherent CT time of this order of magnitude from our TD-lc-DFTB calculations with an appropriate choice of Rlc. Of the many theoretical studies of the P/Fsystem reported in the literature, a few stand out as overlapping in one way or another with the present work. The group of Brédas have a series of papers where they have established the van der Waals complex of a single Pmolecule with a single Fmolecule as one of their model systems of choice.40,45,47,48(See also Ref. 49.) They considered many orien- tations of one molecule relative to the other molecule. The orien- tation used in the present work was somewhat arbitrarily chosen because of its relation to the known thermal electrocyclic reaction of the two molecules whose product is shown in Fig. 3.50,51[Although J. Chem. Phys. 154, 054102 (2021); doi: 10.1063/5.0024559 154, 054102-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE III . This is a sampling of some organic heterojunction incoherent CT times collected from the literature. Except for P/F, the reader is referred to cited references for the definitions of the various acronyms used to denote the donor and acceptors. Our own results have also been included in the table with the reference “PW” (present work). Donor/Acceptor CT time Reference Expt. P3HT/PCBM <100 fsa41 APFO3/PCBM 200 fsb42 MDMO-PPV/PC 70BM ≤100 fsc43 PCPDTBT/PC 70BM ≤100 fsc43 p-DTS(FBTTh 2)2/PC 71BM 82 fs 38 P/F 110 fsd44 Theory 4T/F 97 fse30 P/F 100 fsf45 P/F 714 fsg46 P/F 40 fsh29 P/F 164 fsiPW P/F 79 fsjPW aTransient absorption spectroscopy and quasi-steady-state photoinduced absorption spectroscopy. bTransient absorption spectroscopy. cInfrared pump–probe spectroscopy. dFemtosecond time-resolved two-photon photoemission spectroscopy. eEhrenfest TDLDA calculations. fFigure 7 of the cited reference, Marcus theory, dimer parallel bond-alignment geometry 2A,1BP 1u⊗1AC60 g,di= 3.5 Å, fastest decay time for geometries with dint≤2.0 Å. gFrom Table I of the cited reference, Marcus theory, bulk singlet CT, quoted result is for the aligned structure without interfacial effects. hTD-DFT FSSH (simplified scheme). iCIS/AM1 long-time CT time. A fit for all times 0–500 fs produces a CT time of 39 fs. jTD-lc-DFTB FSSH Rlc= 15 a0. FIG. 3 . Electrocyclic addition product.no indication of such a reaction was found in our TD-lc-DFTB photochemical simulations, we did see it in a DFTB thermal simulation. See the supplementary material.] It is essentially the same as geometry 2Ain the Marcus theory study of CT and charge recombination times in Ref. 45. In Ref. 48, they tuned the ωB97X-D functional52for describing charge transfer in this system. (Note that the range-separation parameter is called ωin this functional rather thanμ. In fact, both notations are used in the literature.) Inverting the resultant value ω=0.137 a−1 0gives Rlc= 7.30 a0, which is about equal to the radius of F, is about a quarter the length of P, and is a little longer than the closest distance between PandF(Fig. 1). Three years later, they carried out TD- ωB97X-D FSSH calculations to gain insight into coherent CT times.40(See the comments above about the problems with FSSH modeling of coherent CT times.) Note that only seven trajectories were included in these calculations, probably because these calculations rapidly become very time and resource intensive. The papers from the Brédas group include environmental effects directly through the use of a dielectric cavity and indirectly in their tuning of the range-separation parameter, but results from gas-phase calculations were also reported either in the articles or in the associated supplementary material. Another notable study comes from the group of Prezhdo.29 This is an approximate periodic TD-DFT FSSH calculation of a P/Finterface in which the repeated unit cell consisted of two P and one F. This is not a true TD-DFT FSSH calculation but rather involves some simplifications: Precomputed ground state trajecto- ries are used instead of true FSSH trajectories, excluding applica- tions where photochemical reactions occur, which are different from those in the ground state. In addition, the nonadiabatic couplings (NACs) are calculated assuming excited-state wave functions of a single determinantal form53,54rather than using the correct multide- terminantal form.55Excitation energies are approximated as orbital energy differences.54These approximations are one way to increase the efficiency of an approximate TD-DFT FSSH calculation. The TD-lc-DFTB FSSH approach explored in this paper represents a dif- ferent way to increase the efficiency of approximate TD-DFT FSSH calculations and the one which we believe is more faithful to the spirit of TD-DFT FSSH calculations. The remainder of this paper is organized as follows: Sec- tion II reviews those aspects of TD-lc-DFTB FSSH theory, which are important for this paper, and explains how we define ET and CT. Computational details are provided in Sec. III. Results are presented in Sec. IV. Section V contains our concluding discussion. II. THEORETICAL METHODS The purpose of this section is to describe the methodology in this paper, which might be considered to be either novel or at least nonstandard. A. TD-lc-DFTB We now explain the need for and difficulty in formulating TD-lc-DFTB. The explanation assumes a certain familiarity with DFTB and TD-DFTB, but Appendix A provides a minimal review, which should make this section comprehensible for those new to the field. It is now well-established that TD-DFT with conventional J. Chem. Phys. 154, 054102 (2021); doi: 10.1063/5.0024559 154, 054102-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp density-functional approximations works best for localized low- energy excitations without too much CT. On the other hand, CT excitations may be underestimated by one or two electron volts.56,57 The best solution to date for fixing the problem of underestimated CT excitation energies is to use TD-lc-DFT.58–63In long-range cor- rected (lc) functionals, the electron repulsion is separated into a short-range (sr) part and a long range (lr) part, 1 r12=(1 r12) sr+(1 r12) lr, (1 r12) sr=erfc(μr12) r12, (1 r12) lr=erf(μr12) r12,(2.1) where erf is the error function and erfc is the complementary error function and μ= 1/Rlcis the range-separation parameter. The sr part is described with DFT using special functionals based on only the form of the short-range part of the electron repulsion, and the lr part is described using the appropriate long-range exact exchange. Work on lc-DFTB began with work by Niehaus and Della Sala10 but has become more active in the last five years.9,11,12,19Three years ago, it was extended to TD-lc-DFTB FSSH.19 The TD-lc-DFTB scheme9,19used in the present work is based on the simplest type of range-separated functional, namely, the lc scheme of Iikura et al.64which was later generalized to TD- lc-DFT by Tawada et al.58In this scheme, the range separation is applied only to exchange so that the new exchange–correlation energy is Elc xc(μ)=Esr,GGA x(μ)+Elr,HF x(μ)+Ec. (2.2) Now, suppose we want to do this in the context of DFTB. A minor point is that DFTB is based on certain separability assump- tions, which are rigorous for local and semi-local density func- tionals but are lost when including Hartree–Fock exchange. This is compensated by long experience gained with approximating Hartree–Fock exchange in the context of traditional semi-empirical quantum chemistry techniques.65However, there is also a major problem, which is the need (at least in principle) for extensive reparameterization each time a new density-functional approx- imation is considered. A priori lc-DFTB should involve three terms, E(μ)=EBS(μ)+Erep(μ)+Ecoul(μ). (2.3) This is avoided in the method of Humeniuk and Mitri ´c (used in the present article) where the long-range part of the Coulomb interaction Elr,HF x is added without reparametrization. Furthermore, Humeniuk and Mitri ´c neglect the lr contribution to the BS energy on the grounds that the zero-order system “already accounts for all interactions between electrons in theneutral atoms,”19which means that we are actually using an expres- sion of the form Elc-DFTB(μ)=EBS+Erep+Ecoul(μ). (2.4) While it often seems to be true that a good value of μto use in this formulation of TD-lc-DFTB is close to that found to be good to use in TD-lc-DFT, we must be aware that this need not always be the case. It should be clear from the above-mentioned very rapid review that the largest differences between TD-lc-DFTB and TD-lc-DFT are in the lc part. Developing lc-DFTB is still an active area of research, and much testing of the new methodology is needed. The present work on ET and CT in our P/Fvan der Waals complex investigates one particularly challenging case. B. TD-lc-DFTB FSSH Appendix B contains a very brief review of the TD-DFT FSSH method. This allows us to emphasize what is either particularly important or what is new in TD-lc-DFTB FSSH used here. We use the version of TD-lc-DFTB FSSH implemented in the DFTBABY program.19,20This solves the full TD-lc-DFTB equation but calcu- lates the non-adiabatic coupling (NAC) elements in the same way as Tapavicza et al.66did using Casida’s ansatz.67Three additions were added to the DFTBABY code specifically for the project reported here. 1. Choice of initial state In order to be as consistent as possible with the conventional model of an organic solar cell, we do not start with an adiabatic wavefunction as is usually done in FSSH. Instead, we imagine the formation of the lowest excited state of P, which is known to be the singlet spin projected highest occupied molecular orbital (HOMO) →lowest unoccupied molecular orbital (LUMO) of P.22We sim- ulate the arrival of this P∗excited state at our model P/F“het- erojunction” by projecting the molecular P∗excited state onto the van der Waals complex to form a P∗/Fdiabatic state whose wave- function is then projected onto the basis of P/Fadiabatic excited states to obtain the initial wavefunction as a superposition of elec- tronic states. The initial adiabatic surface is selected stochastically with the probability equal to the modulus squared of the corre- sponding component in the electronic wavevector. Note that this was only done for our TD-lc-DFTB FSSH calculations and not for the CIS/AM1 FSSH calculations. In the latter case, it was judged that the initial adiabatic excited state was close enough to a pure P∗/F excitation. 2. Incorporation of a decoherence correction We will be using Casida’s ansatz wavefunction to obtain a quan- titative measure of ET and CT. Before doing this, we need to first discuss the issue of how observables should be calculated in the FSSH method. It turns out that this is less trivial than might at first J. Chem. Phys. 154, 054102 (2021); doi: 10.1063/5.0024559 154, 054102-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp be believed because the FSSH method is constantly switching back and forth between a quantum description of the evolution of the electronic wave function and the classical propagation of nuclei on adiabatic potential energy surfaces. The classical part of the algo- rithm represents a measurement of the state of the quantum sys- tem, which collapses the quantum wave packet in the sense of fix- ing the adiabatic electronic state. On the other hand, the quantum part of the calculation ignores this collapse and continues on with the time-dependent propagation of the wave function in the field of the moving nuclei. This raises the question of whether observ- ables should be calculated from the classical populations of the different adiabatic states or quantum mechanically from the time- dependent wave function. Following the work of Landry, Falk, and Subotnik,68we call the first method the “surface method,” while the second method is called the “electronic wavevector method.” It was recognized early on that the surface and electronic wavevec- tor methods gave different expectation values and that the surface method usually agreed better with the results of fully quantum cal- culations. This may be a little surprising as the basic assumption of Tully’s FSSH is that the two methods should agree in the limit of an ensemble average over a very large number of trajectories. It turns out that the primary reason for the discrepancy is that the FSSH method is overcoherent. Put simply, there is a problem because the electronic wave function can have nonnegligible probability den- sity for finding the state on different adiabatic potential energy surfaces even once the classical trajectories have left the region of significant surface hopping probability. The result is the appear- ance of artifactual Stückelberg-like oscillations,18improper scaling of electron transfer rates in Marcus theory,69,70and disagreement between the surface and electronic wavevector methods for calcu- lating observables.68These problems are solved by the inclusion of proper decoherence corrections. Reference 71 contains an excellent review of decoherence corrections in trajectory-based photodynam- ics modeling. For the present study, an energy-based decoherence correction72,73has been implemented in DFTBABY. Our method of calculating ET and CT requires access to an electronic wavefunc- tion. This wavefunction could be an adiabatic wavefunction (surface method) or the time-dependent wavefunction (wavevector method). We usually use the wavevector method. However, as we shall see, the decoherence correction makes this choice of wavefunction largely irrelevant. 3. Particle–hole definition of charge and energy transfer Defining ET and CT is actually less trivial than might at first be assumed. We take a very direct approach here by looking at the charges due to excited electrons (particles) and holes on each frag- ment molecule. However, it should be noted that this is not the only way to define ET and CT. For example, Ref. 22 treats ET and CT in stacks of ethylene and pentacene molecules using a generaliza- tion of Kasha’s exciton model.74It is shown there that CT excitations may be identified whose energy is underestimated by TD-DFT and by TD-DFTB but corrected by TD-lc-DFT and TD-lc-DFTB even though no actual charge is transferred. ET and CT in the sense of Kasha’s exciton model could be treated using the methodology developed by Plasser and Lischka.75,76However, we have preferred in the present work to use the simpler, more direct, definition of ETand CT in terms of the transfer (or nontransfer) of particle and hole charges between fragments. Appendix C explains how to calculate four non-negative numbers—namely, the population qP hof holes on P, the population qP pof excited electrons (“particles”) on P, the population qF hof holes onF, and the population qF pof excited electrons (“particles”) on F. These could be gathered into a single matrix, q=[qP h qP p qF h qF p], (2.5) if so desired, although this would be more for esthetic than for prac- tical reasons. These will allow us to quantify ET and CT. Note that the symbol qhas been used, which usually refers to charge, but that these charges are all non-negative: qP h≥0,qP p≥0,qF h≥0, and qF p≥0. Nevertheless, by conservation of charge, we have that qF h+qP h=qF p+qP p=1. (2.6) It is easy to see that the amount of charge transferred from pentacene to buckminsterfullerene ( P→F) is CT=qP h−qP p=qF p−qF h. (2.7) [Note that the two different ways to define CT are equivalent because of Eq. (2.6).] With this definition, CT = +1 for P+/F−and CT =−1 for P−/F+. It is also possible for a neutral excitation to be localized either onPor on For partially on both at the same time. In order to quan- tify the amount of neutral excitation transferred from pentacene to buckminsterfullerene, we define ET=(qF h+qF p)−1=1−(qP h+qP p) =qF h+qF p 2−qP h+qP p 2. (2.8) Then, ET = −1 for P∗/Fand ET = +1 for P/F∗. Note that CT and ET are independent in the sense that these two parameters were created from the four components of the charge matrix qby making use of the two dependence relations given in Eq. (2.6). III. COMPUTATIONAL DETAILS Four different programs were used to carry out the calculations reported in this paper: (A) GAUSSIAN versions 0977and 1678were used to construct start geometries and to carry out some single point spectra calculations. (B) TURBOMOLE79versions 6.580and 7.081 were used to carry out second-order coupled cluster82(CC2) and second-order algebraic diagrammatic construction [ADC(2), also known as strict ADC(2) or ADC(2)-s, Eq. (53c) of Ref. 83] single point spectra calculations. (C) DFTBABY19was used to carry out Tully-type TD-lc-DFTB/classical trajectory surface hopping calcu- lations. The energies, gradients, and nonadiabatic couplings pro- duced by the (D) MNDO2005 program84were used as input for a local program (FIELD_HOPPING) to carry out AM1/CIS FSSH dynamics. J. Chem. Phys. 154, 054102 (2021); doi: 10.1063/5.0024559 154, 054102-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp A. GAUSSIAN GAUSSIAN was used to generate start geometries and for some single-point spectra calculations. Start geometries for the indi- vidual PandFmolecules were obtained by gas-phase optimiza- tion of initial crystal geometries taken from the Crystallography Open Database (COD)85–87and then optimized at the B3LYP/6- 31G(d,p) level—that is, with the B3LYP functional (i.e., Becke’s B3P functional88with Perdew’s correlation generalized gradient approximation (GGA) replaced with the Lee–Yang–Parr GGA89 without further optimization90)88,91using the 6-31G(d,p) basis set.92,93 The initial geometry for the P/Fcomplex was generated as follows: A first geometry, called A, was obtained from the min- imum of the potential energy curve for the unrelaxed molecules as a function of the intermolecular distance using the orientation shown in the upper left-hand corner of Fig. 1. These curves were calculated at the CAM-B3LYP-D3/6-31G(d,p) level—that is, the range-separated CAM-B3LYP94was supplemented with Grimme’s semi-empirical van der Waals correction.95A second geometry, called C, was then obtained by completely relaxing geometry Aat the same level of theory. We only report spectra calculated at geom- etryCin the present work. (Spectra calculated with geometry A were found to be very similar to those calculated with geometry C.) The ( x,y,z)-coordinates of geometry Cmay be found in the supplementary material. B. TURBOMOLE TURBOMOLE was used to carry out single point spectra calcu- lations at the CC2 and ADC(2) levels of theory. All these calculations were carried out using the cc-pVDZ basis set.96The resolution-of- the-identity (RI) approximation was employed with the standard TURBOMOLE cc-pVDZ-RI auxiliary basis set.97,98Tight conver- gence criteria were employed ( thrdiis = 4 ,thrpreopt = 5 in $excitations ). Oscillator strengths have been calculated using the length gauge ( operators = diplen ). The freeze option was used to freeze out the core orbitals from the active space in our CC2 and ADC(2) calculations. Besides CC2 and ADC(2) calculations, two variants were also tried using default parameters. These are spin-component scaled99,100(SCS) CC2 and ADC(2) ( cos = 1.2 css = 0.33333 ) and scaled opposite-spin (SOS)100,101CC2 and ADC(2) ( SOS: cos = 1.3 ). In particular, Lischka and co-workers found SOS-ADC(2) to be an improvement over ADC(2) for charge- transfer complexes.102 C. DFTBABY DFTBABY was used to carry out the initial calculation of the absorption spectrum and was used to carry out mixed TD-lc-DFTB/classical trajectory surface hopping using the FSSH algorithm. DFTB uses an underlying atomic basis set whose definition requires atomic calculations on both the free atoms and for atoms confined within a quadratic potential. These calculations were car- ried out with the PBE functional.103,104The atomic confinement radius used for hydrogen was r0(H) = 1.757 a0. The confine- ment radius for carbon used in the present work differs from ther0(C) = 2.657 a0default in DFTBABY. Its purpose in the DFTB parameter set is to describe the compressed nature of atoms in molecules and solids relative to free atoms. It does, however, vary between implementations of DFTB. For example, the default in DFTBABY is smaller than that used in DFTB+ with the 3ob-3-1 parameter set ( rwf(s,p) = 3.3 a0in Table 1 of Ref. 105). The usual choices are meant for describing intra molecular interac- tions but may be problematic when inter molecular interactions are important such as in the present investigation of intermolecular charge- and energy-transfer. This is why a different value of the con- finement radius of carbon was used here. In particular, the chosen value r0(C) = 4.309 a0has been optimized to give roughly the same electronic density in the P/Fintermolecular region as that found in DFT calculations (Fig. 4). A full active space was used to calculate the absorption spec- trum at a fixed geometry. The dynamics calculations are more involved. For each value of Rlc, the ground state ( S0) geometry was FIG. 4 . Comparison of the DFTB density in the intermolecular region calculated with DFTBABY using two different confinement radii for carbon against the density calculated using DFT with the PBE functional and the cc-pVDZ basis. Here, rcov(C) = 1.436 a0, sor0(C) = 1.85×rcov(C) = 2.657 a0is the DFTBABY default, while r0(C) = 1.85×rcov(C) = 4.309 a0is the value used in the present work. J. Chem. Phys. 154, 054102 (2021); doi: 10.1063/5.0024559 154, 054102-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp reoptimized starting from geometry C. A thermal ensemble was then generated for this geometry by choosing velocities randomly with a 300 K Maxwell-Boltzmann distribution. The electronic wavevec- tor was initialized by projecting the diabatic P∗/Fexcited state onto the adiabatic basis, and the initial adiabatic surface for the classical propagation of the nuclei was drawn randomly, as described above in Sec. II B 1. FSSH calculations were then carried out at constant energy without the use of a thermostat. Calculations were carried out with a nuclear time step of 0.1 fs. All trajectories were at least 500 fs long. The active space used in the FSSH calculations was restricted to 40 occupied orbitals and 40 unoccupied orbitals. At least ten excited states, in addition to the ground state, have been followed during each trajectory calculation. (For Rlc= 20 a0and Rlc= 25 a0, 20 excited states were followed. This increased number of excited states was needed only for these values of the range-separation parameter as it was found that the P∗/Fdiabatic state had significant amplitude for some of the higher excited states.) Excited state gradients were calculated analytically (Appendix B of Ref. 19). Adiabatic energies and scalar non-adiabatic couplings were interpolated linearly when integrating the electronic Schrödinger equation between nuclear time steps. As explained in Sec. II B 2, a decoherence correction was added to DFTBABY for the present work. Hops from a lower to a higher state were rejected if the kinetic energy was less than the energy gap between the states so as not to violate the principle of conservation of energy. Velocities were uniformly scaled after an allowed hop so that the total (kinetic plus potential) energy was conserved. Trivial crossings are barely avoided crossings whose adiabatic couplings appear as a spatial δ-function in the nonadiabatic cou- pling.106These are particularly likely to arise for weakly interacting molecules and become increasingly likely for larger systems as the density of excited states increases. If not dealt with in an appropriate manner, they can give rise to numerical problems at crossings where trajectories that should have followed a diabatic surface, thereby maintaining the nature of the electronic state, instead remain on the same adiabatic surface, resulting in an artificially high probability of changing the nature of the electronic state. Failure to account for trivial crossings could conceivably, for example, lead to an artificially high probability of long distance charge transfer.106The problem of trivial crossings is minimized in our DFTBABY calculations by using the locally diabatic formalism.107,108Unlike some previous imple- mentations of DFTB, which may not have been suitable for some dynamics calculations because they were not parameterized for all interatomic distances, DFTBABY is well adapted for dynamics cal- culations as it is parameterized for every interatomic distances from zero to infinity. In a few of our calculations, DFTBABY was used to run ground state dynamics at 300 K without the lc option. [Although just an aside to the present work, we note that the electrocyclic addition product (Fig. 3) was found after a few hundred fs in some of our DFTB + D3 thermal dynamics calculations.] D. MNDO2005 MNDO2005 was used to generate energies, gradients, and nonadiabatic couplings, which were then used together with FIELD_HOPPING, a local program in the Würzburg group, to do CIS/AM1 photochemical dynamics. FIELD_HOPPING containsthe decoherence correction [Eq. (17) of Ref. 73 with C= 1 Ha]. AM1 is described in Ref. 109, and its CIS variant is described in Ref. 110. A description of the method for calculating nonadiabatic coupling elements is given in Ref. 111. Our AM1/CIS calculations used an active space of 12 occupied and 12 virtual orbitals. The initial state preparation began with ground-state molecular dynam- ics at the PBE/def2-SVP level with TURBOMOLE 6.5 interfaced with METAFALCON.112Equilibration to 300 K used the Berend- sen thermostat and a time constant of 100 fs. The ground-state dynamics was propagated with a time step of 0.5 fs for 20 ps. The temperature was found to be stable after 5 ps. After that time, 100 initial configurations were selected equally spaced in time for starting the FSSH dynamics. Excitation was to the lowest lying excited state in the first 19 excited states having greater than 60% P∗/Fcharacter. As was the case with the DFTBABY calculations reported in this paper, our MNDO2005 FSSH calculations were also carried out at constant energy without the use of a ther- mostat. The locally diabatic formalism was not available for our CIS/AM1 FSSH calculations. Instead, each nuclear time step was divided into 2500 time steps for the integration of the electronic wave function in order to reduce the possibility of trivial crossing problems. IV. RESULTS Although physics suggests a rough physical value for the range- separation parameter Rlc= 1/μto be used in TD-lc-DFT, this param- eter is often optimized, in practice, in order to fit known physical quantities from the experiment or from the results of a more exact theory. As the long-range correction (lc) in TD-lc-DFTB affects only some of the terms that the lc affects in TD-lc-DFT, there is no clear reason why the same range-separation parameter should be used in the two theories. Thus, a separate investigation of the effect of vary- ingRlcis needed for TD-lc-DFTB. Our results are presented in this section. Note that optimization by fitting to the experiment is not really available to us here for two reasons. The first is that our particular physical system P/F, consisting of a van der Waals complex of pen- tacene ( P) and of buckminsterfullerene ( F), is unlikely to exist in nature, except possibly in outer space. Second, although we do not doubt that P/Fmight be made, for example, in a molecular beam, it would most likely involve a larger mixture of configurations than we consider here. As will be seen below, we can compare against the results of high-quality ab initio calculations of spectra at a single geometry and that is done below to find the range of values of Rlc, which appear to give the most physical results. Along the way, we also explore how some simpler theories that resemble TD-lc-DFTB do when compared to high-quality ab initio results. TD-lc-DFTB FSSH calculations of charge transfer (CT) and energy transfer (ET) are carried out specifically to see how these properties vary with Rlc. This provides additional information, which supports and extends what we learned by looking at spectra calculated at only a single geometry. However, before doing any of this, we find it useful to review the importance of having good statistics and what types of FSSH results we get from another semi-empirical theory, namely, CIS/AM1. This J. Chem. Phys. 154, 054102 (2021); doi: 10.1063/5.0024559 154, 054102-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp will form a different type of comparison against which to judge TD-lc-DFTB FSSH calculations. A. Importance of statistics It is worth emphasizing the importance of ensemble averag- ing and hence of good statistics. It has been pointed out that, “the dynamics of individual FSSH trajectories are not physical, and it is unclear how many trajectories in the swarm must be averaged before observables can be calculated.”71While this statement seems overly pessimistic in the sense that individual FSSH trajectories are often thought to reflect different possible photochemical pathways, it is also true that the most rigorous interpretation of FSSH observables requires taking ensemble averages over many trajectories and that great care must be taken to obtain the necessary statistics. We have tried TD-DFT FSSH calculations and have found them to be quite resource intensive compared to semi-empirical FSSH calculations for our problem. However, we mentioned one previ- ous TD-DFT FSSH study of the P/Fsystem.40As the objective of that study was to study coherent CT and the primary subject of the present study is incoherent CT, the procedure followed was rather different. Calculations in Ref. 40 were started in the lowest singlet state. This state has dominant P+/F−character. Only seven trajectories were carried out, presumably because these are compu- tationally resource intensive to do. The charge on Fwas studied as a function of time. The excited electron (particle) always began onFbut then moved back and forth between PandFwith an apparently regular frequency. The results were deemed consistent with previous theoretical38,39and experimental29,30work, indicating the presence of charge oscillations occurring with periods of some 20–25 fs. Now, let us take a different point of view and look at ensemble averages rather than individual trajectories. When studying coherent CT, ensemble averaging has the possible disadvantage of averaging away at least some of the coherence, but we are interested in inco- herent CT whose study demands ensemble averaging. As the authors of Ref. 40 provided graphs of the charge on Ffor all seven trajec- tories, we may take their data and use it to illustrate the effect of ensemble averaging. Chemical intuition might have suggested that the electron should remain on F, but Fig. 5 is consistent with the above-mentioned physical picture of the charge moving back and forth between FandP. As the oscillations are not the same for all seven trajectories, the averaging process leads to the picture of the decay of the initial P+/F−charge separation, with, on average, the electron spending half of its time on Pand half of its time on F. Of course, the statistics from averaging over only seven trajectories are far from satisfactory if our goal is to calculate an ensemble property. It is, for example, impossible to extract ensemble-based relaxation times from these data. Semi-empirical calculations are much less resource intensive but are also more approximate than DFT calculations. However, since we can carry-out many more semi-empirical than DFT cal- culations with the same computer resources, we are able to cal- culate ensemble averages over a larger number of trajectories with semi-empirical than with DFT calculations. Ensemble averages over larger numbers of trajectories have the significant advantage of hav- ing less statistical noise. We illustrate this concept by showing the ensemble averages from our own CIS/AM1 FSSH calculations. Our FIG. 5 . Fraction of an electron on FinP/Fstudied in Ref. 40: All seven trajectories and their ensemble average. minimum objective is to calculate a CT relaxation time. As it hap- pens, out of 100 initial trajectories, all but four ran to completion to that ensemble averages are reported over 96 trajectories. As discussed in Sec. II, there are two main ways to calculate expectation values in the FSSH method. Tully’s FSSH method was designed so that the fraction of running trajectories for any given state (method 1, surfaces68) should equal the diagonal element of the electronic density matrix for that state (method 2, electronic wavevector68), provided that averaging occurs over a sufficiently large number of trajectories. Unfortunately, this does not happen for the original FSSH algorithm with a reasonable number of trajecto- ries. As shown in Fig. 6, the inclusion of the decoherence correction of Ref. 73 brings CT and ET observables calculated with the two dif- ferent methods into nearly perfect agreement. It is now possible to fit the CT curve to either a single exponential or a linear combination FIG. 6 . CIS/AM1 FSSH calculations: “current state” is the same as the surface method; “superposition” is the same as the wavevector method. J. Chem. Phys. 154, 054102 (2021); doi: 10.1063/5.0024559 154, 054102-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp of two exponentials and extract P∗/F→P+/F−relaxation times. A single exponential gives a CT time of τCT= 1/kCT= 164 fs. A fit with a linear combination of two exponentials gives two relaxation times, a fast oneτfast CT≈16 fs and a slow one τslow CT≈91 fs. It is very interesting to convert these ET and CT plots into particle and hole charge plots, as in Fig. 7. Small oscillations reminis- cent of the coherent charge oscillations of Ref. 40 or of Stückelberg oscillations18,35appears with a period of about 15 fs (PW value in Table II). However, more interesting to us is that we observe that the transfer of the excited electron (particle) from PtoFinduces apartial delocalization of the hole from PtoFconsistent with the idea that the pairing of opposite charges should be energeti- cally favorable. In the present case, this process only lasts about 5 fs before the hole relocalizes back onto P. Such a mechanism is con- sistent with the solid-state concept of a polaron, namely, a charge defect, which is localized by a lattice deformation. Recently, FSSH calculations have revealed the possibility of exciton hopping via a flickering polaron mechanism where excitations localized on one molecule “hop” by first becoming delocalized over several molecules and then relocalizing again on another molecule.36,37OurP/Fsys- tem is too small to see flickering polarons, but we do seem to be seeing delocalization of the particle–hole state followed by relocal- ization of the hole on the molecule where the excitation was first formed. The reader is reminded that a complicated process is taking place, which requires nuclear motion in order for there to be nonadi- abatic hopping between different electronic states. Nevertheless, the particle and hole dynamics suggest that qualitative (perhaps, even semi-quantitative) results may be obtained from the very simple “textbook” model, P∗/FkET/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne→ P/F∗kCT/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne→ P+/F−. (4.1) The well-known solution is[P∗/F]=e−kETt, [P/F∗]=kET kCT−kET(e−kETt−e−kCTt), [P+/F−]=1 +kETe−kCTt−kCTe−kETt kCT−kET(4.2) assuming that [P∗/F]0=1. Since qp=qP p=[P∗/F], qh=qP h=[P∗/F]+[P+/F−],(4.3) then qp=e−kETt, qh=1 +kET kCT−kET(e−kCTt−e−kETt).(4.4) The easiest way to find values for kETand for kCTis to first obtain kETfrom a fit of qpand then to adjust kCTuntil the fit of qhlooks reasonable. This was done for 0 <t<50 fs and then simply applied at longer times. CT and ET may then be obtained from CT=qh−qp, ET=1−(qh+qp).(4.5) The result of the fit is shown in Fig. 7. The fit is by no means quan- titative, but it is at least qualitative (if not semi-quantitative). This is FIG. 7 . CIS/AM1 FSSH calculations of particle and hole charges along with state populations. J. Chem. Phys. 154, 054102 (2021); doi: 10.1063/5.0024559 154, 054102-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp even more impressive when it is realized that the fit was done using only the data from the first 50 fs and then simply applied to the entire 500 fs run. The resultant CT time is τCT= 1/kCT= 20 fs and τET= 1/kET= 8.3 fs. It is interesting to note how different this is from the above-mentioned τCT= 164 fs extracted from the same data but using a different kinetics model. However, the 20 fs CT time does resembleτfast CT≈16 fs obtained from the double exponential fit. This simply means that great care must be taken when specifying relax- ation times to also specify the fitting method. Finally, we note that it is possible to obtain even better fits with a more elaborate kinetic model (supplementary material). B. Importance of spectra As emphasized above, our choice of a semi-empirical method is governed by the need to make a compromise between accuracy and being able to carry out a large number of trajectories. We would therefore like to benchmark TD-lc-DFTB for our system against high-quality ab initio methods in order to establish an “exact” result against which to compare. At the same time, we would also like to establish two other benchmarks: The first is TD-DFT as TD-DFTB is supposed to be designed to behave like TD-DFT, rather than ab initio methods. The second is the semi-empirical CIS/AM1 method as TD-DFTB shares many aspects in common with semi- empirical theories. As the ab initio methods considered here are fairly resource intensive, this subsection is restricted to calculations at a single geometry (geometry C). Since our initial model geometry is a purely theoretical con- struct, there are no experimental results against which to com- pare. Nevertheless, we can make a common sense crude estimate of the position of the lowest CT excitation using the following formula: ΔECT≈IP−AF−e2 R, (4.6) where IP= 6.61 eV is the experimental gas phase ionization potential ofP,113AF= 3.22 eV is the experimental gas phase electron affin- ity of F,114and R= 6.26 Å. This gives an estimated CT energy of ΔECT= 1.09 eV. It is only a rough estimate because the third term is the Coulomb attraction between point charges, which is unlikely to be an accurate reflection of the charge distributions in Pand in F. An attempt to improve on this approximation by calculating the repulsion energy between the LUMO on Fand the HOMO on P increased the estimated CT energy to ΔECT= 1.35 eV. (See also Ref. 115 for more information about experimental and calculated ionization potentials and electron affinities for molecules of interest for organic solar cells.) As both FandPareπ-systems, it is useful to take a brief look at the results of simple Hückel molecular orbital theory. The frontier orbitals are shown in Fig. 8. In this crude picture, the Pb2gHOMO is well above the FhuHOMO, but the Pb3uLUMO is quasi-degenerate with the triply degenerate Ft1uLUMO. Thus, we should expect two types of low-lying valence excitations, namely, P∗/Fand three quasi- degenerate P+/F−states. It is impossible from this level of theory to predict which of these two types of excitations should be the lowest in energy. FIG. 8 . Frontier molecular orbitals for Pand for Ffrom simple Hückel molecular orbital theory. Energies are in Hückel units (i.e., the energy is α+βE, where both αandβare negative numbers). Symmetry assignments are for the point groups of the isolated molecules. Tables IV and V show the results of ADC(2) and CC2 calcu- lations and their spin-scaled alternatives [SCS-ADC(2), SCS-CC2, SOS-ADC(2), and SOS-CC2]. These latter might be important because ADC(2) and CC2 are only second-order methods and should not be considered to be the ultimate truth. Indeed, recent work has shown that improved results are obtained via spin scal- ing. In particular, the SOS-ADC(2) method has been claimed to be an improvement over ADC(2) calculations for describing CT excitations.102It should also be pointed out that these are heavy calculations. Calculating 20 states at the SCS-ADC(2) and SOS- ADC(2) levels took 13 days using 20 central processing units (CPUs) on our machines. CC2 calculations are even more resource intensive. Calculating 10 states at the SCS-CC2 and SOS-CC2 lev- els each took around 20 days using 20 CPUs on our machines. Thus, these methods are notsuitable for on-the-fly photodynamics simulations. Note that the assignments given in Tables IV and V and else- where in the present work are only given as a brief indication of the nature of the state. For reasons of space and brevity, we can- not give a detailed analysis of these assignments, which therefore should be considered as only very approximate and subject to sub- jective bias. Nevertheless, we do see the expected three P+/F−states lying lower in energy than the expected P∗/Fstate. In the case of our high-quality ab initio calculations, a transition density matrix analysis is given in the supplementary material. We have noticed that adding spin-scaling increases the P/F∗character relative to the P+/F−character in the first three excited states. Another important observation from these ab initio calcula- tions is that there is a dense manifold of singlet states (many of P/F∗character), which is well separated from the ground state. The precise ordering of these excited states can depend on both the par- ticular method used and small changes in the geometry of our P/F system. As semi-empirical methods are only intended to describe the valence orbitals and make use of a minimal-basis, we cannot expect to be able to describe all these excited states with TD-lc-DFTB J. Chem. Phys. 154, 054102 (2021); doi: 10.1063/5.0024559 154, 054102-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE IV . Characterization of P/Fexcited states using the CC2 ab initio methods with the cc-pVDZ basis set for model geometry C. In all cases, 10 excited states have been requested. Oscillator strengths near or larger than 0.01 have been italic. State ΔE(eV) f Character CC2 S10 2.01 0.0005 P/F∗ S9 2.00 0.0000 P/F∗ S8 1.98 0.0000 P/F∗ S7 1.90 0.0000 P/F∗ S6 1.90 0.0002 P/F∗ S5 1.88 0.0000 P/F∗ S4 1.87 0.0000 P/F∗ S3 1.70 0.0355 P+/F− S2 1.68 0.0097 P+/F− S1 1.67 0.0001 P+/F− State ΔE(eV) f Character ΔE(eV) f Character SCS-CC2 SOS-CC2 S10 2.37 0.0000 P/F∗2.54 0.0000 P/F∗ S9 2.35 0.0000 P/F∗2.53 0.0000 P/F∗ S8 2.34 0.0000 P/F∗2.51 0.0000 P/F∗ S7 2.25 0.0000 P/F∗2.46 0.0001 P+/F−a S6 2.24 0.0020 P/F∗2.45 0.0089 P+/F− S5 2.23 0.0001 P/F∗2.41 0.0246 P+/F− S4 2.21 0.0000 P/F∗2.39 0.0044 P/F∗ S3 2.17 0.0380 P+/F−2.38 0.0000 P/F∗ S2 2.14 0.0001 P+/F−2.36 0.0001 P/F∗ S1 2.14 0.0011 P+/F−2.35 0.0000 P/F∗ aThis state is actually about 50/50 P+/F−andP/F∗according to the transition density matrix analysis shown in the supplementary material. although we should be able to describe the most important valence excitations. Let us return once more to the results of the ab initio calcu- lations shown in Tables IV and V. The lowest states are seen to be P+/F−CT states with some admixture of P/F∗for some methods and are somewhat higher than our estimated P+/F−CT energy of 1.35 eV [1.70 eV for ADC(2), 1.67 eV for CC2, 2.15 eV for SCS-ADC(2), 2.14 eV for SCS-CC2, 2.36 eV for SOS-ADC(2), and 2.35 eV for SOS-CC2]. As illustrated in Fig. 9, CC2 and ADC(2) calculations without spin-scaling give very similar results. As previously men- tioned (Sec. III B), spin-scaling is thought to improve CT energies. In this case, we see that spin-scaling leads to blue-shifting the energies. Figure 10 shows spectra calculated with ADC(2) with and without spin-scaling over a larger energy range. In the present case, we see that spin-scaling increases the energy of all states but increases the energy of CT states more than of the energy of a localized excitation onP. This is pretty much exactly what is usually seen when an lc functional is used to improve CT energies in TD-lc-DFT, although we do not understand why such a qualitatively similar observation should also hold for spin-corrected ADC(2) and CC2.TABLE V . Characterization of P/Fexcited states using post-Hartree–Fock ADC(2) ab initio methods with cc-pVDZ for a model geometry C. In all cases, 20 excited states have been requested. Oscillator strengths near or larger than 0.01 have been italic. State ΔE(eV) f Character ADC(2) S20 2.92 0.0000 P+/F− S19 2.40 0.0415 P∗/F S18 2.15 0.0002 P/F∗ S17 2.14 0.0001 P/F∗ S16 2.13 0.0002 P/F∗ S15 2.11 0.0003 P/F∗ S14 2.10 0.0001 P/F∗ S13 2.10 0.0002 P/F∗ S12 2.10 0.0001 P/F∗ S11 2.08 0.0000 P/F∗ S10 2.07 0.0003 P/F∗ S9 2.05 0.0000 P/F∗ S8 2.04 0.0000 P/F∗ S7 1.96 0.0000 P/F∗ S6 1.96 0.0001 P/F∗ S5 1.94 0.0000 P/F∗ S4 1.93 0.0000 P/F∗ S3 1.73 0.0179 P+/F− S2 1.71 0.0072 P+/F− S1 1.70 0.0001 P+/F− State ΔE(eV) f Character ΔE(eV) f Character SCS-ADC(2) SOS-ADC(2) S20 3.34 0.0027 P∗/F 3.32 0.0029 P∗/F S19 2.62 0.0477 P∗/F 2.73 0.0409 P∗/F S18 2.53 0.0004 P/F∗2.72 0.0008 P/F∗ S17 2.53 0.0004 P/F∗2.72 0.0042 P/F∗ S16 2.51 0.0004 P/F∗2.70 0.0005 P/F∗ S15 2.49 0.0013 P/F∗2.68 0.0072 P/F∗ S14 2.48 0.0000 P/F∗2.67 0.0000 P/F∗ S13 2.43 0.0001 P/F∗2.59 0.0003 P/F∗ S12 2.43 0.0011 P/F∗2.58 0.0032 P/F∗ S11 2.41 0.0007 P/F∗2.58 0.0007 P/F∗ S10 2.41 0.0000 P/F∗2.57 0.0000 P/F∗ S9 2.39 0.0000 P/F∗2.55 0.0000 P/F∗ S8 2.38 0.0000 P/F∗2.54 0.0000 P/F∗ S7 2.29 0.0000 P/F∗2.47 0.0002 P/F∗ S6 2.28 0.0009 P/F∗2.46 0.0049 P/F∗ S5 2.27 0.0000 P/F∗2.43 0.0013 P/F∗ S4 2.25 0.0000 P/F∗2.41 0.0000 P/F∗ S3 2.18 0.0318 P+/F−2.40 0.0337 P+/F− S2 2.16 0.0002 P+/F−2.37 0.0002 P+/F−a S1 2.15 0.0027 P+/F−2.36 0.0001 P+/F−a aThis state is actually about 50/50 P+/F−andP/F∗according to the transition density matrix analysis shown in the supplementary material. J. Chem. Phys. 154, 054102 (2021); doi: 10.1063/5.0024559 154, 054102-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 9 . Comparison of calculated spectra at geometry CGaussian convoluted with a 0.1 eV FWHM: ADC(2), CC2, SCS-ADC(2), and SCS-CC2. All peaks correspond to the formation of the P+/F−state. So which should we consider as “exact”? As there is less expe- rience with the spin-scaled methods, we might be conservative and focus on the ADC(2) and CC2 results. However, in the present con- text, the differences between the ab initio calculations with and with- out spin-scaling are much less important than the simple result that there is an intense P+/F−CT peak, which is lower in energy than a still more intense peak corresponding to a local excitation on P (P∗/F). This is the “exact” behavior that we should hope to match with a semi-empirical method. FIG. 10 . Comparison of calculated ADC(2), SCS-ADC(2), and SOS-ADC(2) spec- tra Gaussian convoluted with a 0.1 eV FWHM at geometry Cover an expanded spectral range. The double-headed arrow indicates the energy difference between the lower P+/F−CT peak and the upper P∗/Fpeak.As “DFTB inherits the faults of DFT as well as some of its own,”116it is interesting to first see how well DFT works before examining DFTB. We carried out TD-B3LYP/6-31G(d,p), TD- CAM-B3LYP/6-31G(d,p), TD-HF/6-31G(d,p), and CIS/6-31G(d,p) calculations of the spectra of our P/Fcomplex. B3LYP is a well- known global hybrid functional with about 20% exact exchange. CAM-B3LYP adds a range-separated hybrid on top of the B3LYP global hybrid, thus increasing the percentage of exact exchange at long-range. Although pre-dating Kohn–Sham DFT, Hartree–Fock (HF) may be considered as an extreme density functional with 100% exact exchange and no correlation. Configuration interac- tion singles (CISs) are TD-HF in the Tamm–Dancoff approxima- tion. TD-DFT is known to underestimate CT excitations by as much as 1–2 eV. Range-separated functionals are known to cor- rect this problem at the cost of slightly increasing local excita- tion energies. Figure 11 shows that TD-B3LYP gives a qualita- tively correct spectrum with the P+/F−CT lower than the P∗/F locally excited state, but the intensities are wrong. The P∗/Fpeak should be more intense than the P+/F−peak, but, instead, it is the P+/F−peak that is more intense. As expected, the range-separated TD-CAM-B3LYP functional increases the energy of the P+/F−CT peak relative to that of the P∗/Fpeak but incorrectly makes them quasi-degenerate. Figure 12 shows that TD-HF and CIS calcula- tions are qualitatively similar but that the P+/F−CT peak is now higher in energy than the peak for the P∗/Flocally excited state. On the other hand, it is reassuring that the relative intensities of the two peaks are close to what we expected from our high-quality calculations. Finally, before turning to TD-lc-DFTB spectra, let us look at semi-empirical CIS/AM1 spectra. Note that this method has been parameterized using experimental data. As such, although the struc- ture of the calculations resembles a CIS calculation, the CIS/AM1 calculation actually interpolates (or even extrapolates) experimental FIG. 11 . Comparison of TD-B3LYP and TD-CAM-B3LYP spectra, Gaussian con- voluted with a 0.1 eV FWHM at geometry C. The TD-B3LYP peak at around 1.5 eV corresponds to the P+/F−CT state, and the peak at around 2.0 eV corresponds to the local excitation P∗/F. In contrast, these two peaks have merged into a single peak at about 2.4 eV in the TD-CAM-B3YP spectrum. J. Chem. Phys. 154, 054102 (2021); doi: 10.1063/5.0024559 154, 054102-13 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 12 . Comparison of calculated CIS/6-31G(d,p) and TD-HF/6-31G(d,p) spec- tra Gaussian convoluted with a 0.1 eV FWHM at geometry C. The lower peaks (at around 2.6 eV for TD-HF and at around 3.0 eV for CIS) correspond to the local excitation P∗/F, while the higher peaks at around 3.6 eV for the two types of calculations correspond to the P+/F−CT state. data by including implicit electron correlation effects in its parame- terization. As shown in Fig. 13, the CIS/AM1 spectrum is very dif- ferent from the CIS spectrum in that AM1/CIS correctly places the P+/F−CT peak at lower energy than the P∗/Fpeak, while CIS places theP+/F−CT peak at higher energy than the local P∗/Fpeak. More- over, the CIS/AM1 spectrum, although shifted to higher energy than the ADC(2) spectrum, is in qualitative agreement with the ADC(2) calculations so far as the P+/F−CT peak lies at lower energy than theP∗/Flocal excitation peak and the energy differences between FIG. 13 . CIS/AM1 spectrum, Gaussian convoluted with a 0.1 eV FWHM at geom- etryC. The lower peak ( ∼2.4 eV) corresponds to the P+/F−CT state, while the upper peak ( ∼3.0 eV) corresponds to the local P∗/Fexcitation.theP+/F−andP∗/Fpeaks are about the same in the CIS/AM1 and ADC(2) calculations. We explain this by the implicit inclusion of correlation effects in CIS/AM1 via fitting of parameters to exper- imental values. On the other hand, the P∗/Fpeak is much more intense compared to the P+/F−peak in the CIS/AM1 spectrum than is the case in the ADC(2) spectrum. We now turn to TD-DFTB with and without lc. Figure 14 shows the TD-DFTB spectrum without lc and the TD-lc-DFTB spectrum with the default value of Rlc= 3.03 a0. It is immediately obvious that the TD-DFTB spectrum is at least qualitatively correct with a lower energy P+/F−CT peak and a higher energy P∗/Fpeak. The posi- tion of the P∗/Fpeak is even in good agreement with that of the corresponding ADC(2) P∗/Fpeak, although we note that this is the unexpected consequence of using the new r0(C) = 4.309 a0confine- ment radius. The older r0(C) = 2.657 a0confinement radius results in a TD-DFTB spectrum with the P∗/Fpeak at about 1.9 eV and a larger energy difference between the P∗/FandP+/F−peaks (∼0.4 eV as opposed to ∼0.2 eV with the new confinement radius). This high- lights how sensitive this particular calculation is to the choice of confinement radius. Figure 14 also shows that the TD-lc-DFTB spec- trum with the default Rlc= 3.03 a0is qualitatively incorrect as the lowest energy peak corresponds to the P∗/Fstate irrespective of the choice of confinement radius. We conclude that something more profound is going on. This is why we now turn to how the spectra of TD-lc-DFTB calculations vary as a function of Rlc= 1/μ. Figure 15 shows the results of TD-lc-DFTB calculations for several values of Rlc. There is one excitation, identified as a local pentacene excitation, P/F→P∗/F, which has significantly greater oscillator strength than the other excitations. It is useful to split the graph into regions: I. Above Rlc≳12.5 a0, this excited state is quasidegenerate with a locally excited buckminsterfullerene state P/F∗and FIG. 14 . TD-DFTB ( Rlc=∞) and TD-lc-DFTB with the default value of Rlc= 3.03 a0Gaussian convoluted with a 0.1 eV FWHM at geometry C. The TD-DFTB cal- culation is qualitatively correct with a lower energy P+/F−CT peak ( ∼2.1 eV) and a higher energy P∗/Fpeak ( ∼2.3 eV). The TD-lc-DFTB calculation only shows the P∗/Fpeak ( ∼2.7 eV). J. Chem. Phys. 154, 054102 (2021); doi: 10.1063/5.0024559 154, 054102-14 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 15 . TD-lc-DFTB P/F excited-state energies as a function of Rlc: upper, oscillator strengths; lower, excitation energies. the two states can mix. There are also three quasidegenerate P+/F−CT excited states below the quasidegenerate P∗/Fand P/F∗states. The absorption spectrum in this region is qual- itatively similar to what is seen in the high-quality ab initio calculations. II. The region ∼12.5 a0>Rlc≳10a0is a crossing region where all the states become quasidegenerate and considerable mixing occurs. III. Below Rlc<10a0, the P+/F−states move to higher energies and the two lower states have local excitation character with theP∗/Fstate being lower than the P/F∗state. Inverse CT states of type P−/F+are higher energy states not shown on this graph. The absorption spectrum in this region is qualitatively similar to what is seen in the TD-HF and CIS calculations. Note that this makes good sense as Rlcis (roughly speaking) the cut-off beyond which exact (i.e., Hartree–Fock) exchange is used. Thus, a smaller value of Rlccorresponds to a more Hartree–Fock-like calculation. On the basis of this picture, we might expect CT above Rlc>12.5 a0and little or no CT below Rlc<10a0. We also tried one more idea that we call μ-scanning. This con- sists of finding the value of Rlc, which gives the best agreement with some other high-quality calculation. In our case, we found that Rlc≈15a0allowed us to reproduce the spectra reported in Ref. 40 at their geometry using the older r 0(C)confinement radius. We have nottried to repeat the procedure with the new value of the confinement radius. Finally, it is interesting to speculate how these results might change when the model is extended to several molecules and envi- ronmental effects are included via a dielectric cavity according to the methods described in Ref. 106. CT excitation energies are known to undergo significant solvent shifts because a polar (or polarizable) environment will act to stabilize the CT state, hence reducing the CT excitation energy. In contrast, local excitation energies are expected to be much less sensitive to environmental effects. In fact, this is an old experimental test used to distinguish between CT and local exci- tations in molecular spectra. Assuming, then, that the CT curves in Fig. 15 shift down while the local excitation states remain roughly constant suggests that the critical crossover region should be shifted from Rlc≈11a0to lower values of Rlc. In fact, a reduction of only a couple of tenths of an eV would be enough to shift the critical value of Rlcto the value Rlc= 7.30 a0used in the papers from Brédas’ group. C. Charge and energy transfer as a function of Rlc We now wish to see how P/Fcharge and energy transfer vary in TD-lc-DFTB FSSH calculations as a function of Rlc. This will give us some idea of how robust this model is with respect to Rlc and will provide an estimate of ET and CT times as a function of Rlc. The basic procedure is already explained in Secs. II and III. Ensemble averages were over at least 50 trajectories for all values ofRlcexcept for Rlc= 15 a0where the average is over 100 tra- jectories. Results are shown in Fig. 16. Note that the range of Rlc covers regions I, II, and III defined by the spectroscopic analysis of Sec. IV B. Thus, we may expect substantial variation in the physics of CT and ET. From Fig. 15, we see that the initial excited state for Rlc= 5a0 isS1, which has P∗/Fcharacter. There are no low-lying CT excited states, so no CT is expected. Furthermore, P/F∗lies above S1so that ET also may not take place. The top row of Fig. 16 shows that neither CT nor ET takes place and that the system remains in S1for the entire 500 fs run. From the upper part of Fig. 15 or from the second row of Fig. 16, we see that the initial excited state is S1forRlc= 10 a0. There are several quasi-degenerate states near S1so that some mixing may occur with other states, making it difficult to anticipate how it will decay. The second row of Fig. 16 shows that CT is negligible but that P∗/F→P/F∗ET is important. As S1remains by far the dominant state, ET must be explained by variations in the geometry, which leads to changes in the nature of S1. From Fig. 15, we see that the initial excited state for Rlc= 15 a0 isS4, which has P∗/Fcharacter. There are three lower lying states of P+/F−CT character so that we may expect to see CT dynamics. This is confirmed by the third row of Fig. 16, which shows rapid pop- ulation of P+/F−states. Interestingly, Fig. 16 also shows significant ET. Examination of Fig. 15 might lead to the expectation that the CT and ET dynamics for Rlc= 20 a0and Rlc= 25 a0should be sim- ilar to that for Rlc= 15 a0. However, the last two rows of Fig. 16 show that this is not the case. Instead, ET dominates over CT for short times, and CT dominates over ET for longer times. As we shall see, this switch-over may be explained by the idea of that electron J. Chem. Phys. 154, 054102 (2021); doi: 10.1063/5.0024559 154, 054102-15 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 16 . CT, ET, particle/hole populations, and state populations for different values of the long-range parameter. (particle) transfer from PtoFbegins by dragging the hole along with it, but then, nuclear motion kicks in and restores the hole on P. This is similar to the mechanism of polaron formation in solids where the lattice distorts to create localized charge defects. This “polaron formation” is easiest to see after recasting our results in terms of the particle qp=qP pand hole qh=qP hpopula- tions on P. The results are also shown in Fig. 16. Note that there is a smooth trend in going from the top to the bottom of Fig. 16. AsRlcincreases, the qpcurve, which is always decreasing, changes its form from convex down to a nearly exponential decay. Also asRlcincreases, the qhcurve, which is initially convex down (for Rlc= 10 a0), straightens out (for Rlc= 15 a0) and then becomes increasingly sharply concave upward when Rlc= 25 a0. This grad- ual and seemingly smooth change is a little deceptive because, as we have seen with the spectra, the underlying physics changes quite a bit with the value of Rlc. Let us take a closer look. For Rlc= 10 a0(second row of Fig. 16), we see that the particle and hole move together from PtoF(P∗/F→ P/F∗). No significant CT happens, but the ET is almost exponential. ForRlc= 15 a0(third row of Fig. 16), the hole transfer has slowed so that it is nearly linear. On the other hand, the particle transfer has become nearly exponential. For Rlc= 20 a0(fourth row of Fig. 16), the particle transfer is slowing and the hole, which initially left Ptogo to F, starts to return to Pafter some time. This phenomenon is even more marked for Rlc= 25 a0(last row of Fig. 16). Figure 16 shows that the overall trends can be captured very qualitatively in the case of Rlc= 10 a0andRlc= 15 a0and much more quantitatively for Rlc= 20 a0and Rlc= 25 a0using the kinetic model shown in Eq. (4.1) (see also the supplementary material). This allows us to summarize CT and ET times in a single figure (Fig. 17). Using the three points on the right to extrapolate linearly backwards, then somewhere around Rlc= 12±2a0, both the ET and CT times go to zero. This corresponds to our earlier conclusion that Rlc≈10a0 is the lower limit of region I. At the value obtained by our μ-scan (Rlc= 15 a0),τCT= 79 fs, in qualitative agreement with the CIS/AM1 value ofτCT= 164 fs obtained from the single exponential fit or the τslow CT≈91 fs obtained from the double exponential fit. However, there is another kinetics model, which provides a significantly better fit for the Rlc= 15 a0case. It is two indepen- dent processes: hole migration from PtoF(kh= 1/τh) and particle migration from PtoF(kp= 1/τp). The corresponding rate laws are then qh=e−t/τh, qp=e−t/τp,(4.7) J. Chem. Phys. 154, 054102 (2021); doi: 10.1063/5.0024559 154, 054102-16 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 17 . CTτCT= 1/kCTand ETτET= 1/kETtimes for different values of Rlc obtained using the kinetic model shown in Eq. (4.1). which is the solution of the kinetic problem, kp kh P∗/F/leftr⫯g⊸tl⫯ne→P+/F−/leftr⫯g⊸tl⫯ne→P/F∗ (4.8) when kp≫kh(τh≫τp). A good fit (up to about 700 fs) is obtained withτp= 111 fs and τh= 1380 fs, as shown in Fig. 18. We see that neither ET nor CT occurs when Rlc<10a0, but that both ET and CT occur when Rlc>10a0. ET and CT relaxation times are on the order of 100–300 fs in this range with values of around 100 fs being our preferred best estimate based on our μ-scan value of Rlc= 15 a0, consistent with the CT relaxation time obtained from CIS/AM1. At Rlc= 20 a0and Rlc= 25 a0, the observed kinet- ics is in semi-quantitative agreement with a mechanism where CT follows as a second step after an initial ET step, while at Rlc= 15 a0, best agreement is found with a kinetic model involving rapid particle transfer and slow hole transfer. In terms of the “polaron formation” picture, rapid electron transfer is too rapid to delocalize the hole very much, while slower electron transfer gives the hole time to delocalize partially off of Pbefore relocalizing back onto Pagain. FIG. 18 . ET and CT curves for the Rlc= 15 a0fit with Eq. (4.7).V. CONCLUSION Our interest in TD-lc-DFTB is motivated by potential applica- tions in the field of organic electronics. The present work may be regarded as a continuation of our other work in this field.22,115,117–119 In particular, this is the third in a series of papers22,115aimed at exploring the use of density-functional tight-binding (DFTB) for investigating ET and CT for molecules and assemblies of molecules typical of organic electronics. The choice of TD-lc-DFTB is governed, in part, by practical- ity. FSSH photodynamics calculations easily become very resource intensive, so computational efficiency is important even when deal- ing with only moderately large systems. DFTB has recently gained immense popularity as demonstrated by DFTB options in increas- ingly many major quantum chemistry codes. This is because DFTB is designed to behave like DFT, the dominant workhorse for rou- tine quantum chemistry calculations these days, but DFTB has better scalability than DFT with respect to the number of atoms in our sys- tem. DFTB accomplishes this by borrowing approximation methods from the toolbox of semi-empirical methods and adding some of its own. We emphasize that technically DFTB is an approximate form of DFT rather than a semi-empirical theory because it is parameter- ized to fit DFT rather than parameterized using experimental data. However, many of the approximations made in DFTB are inherited from earlier work on semi-empirical methods. This is even more true once a long-range correction is introduced because of the need to include Hartree–Fock exchange for which older semi-empirical methodology is well-developed. The conventional six-step model for the physics of organic solar cells was presented in the Introduction. We would like to explore phenomena that happen on the scale of hundreds of femtoseconds. From Table I, it is mainly step (iii) CT. However, we might also get some insight into the initial physics of longer steps such as (ii) exciton diffusion and (iv) charge separation. We know from our knowledge of DFT that we will need a dispersion correction and that we should use an lc functional, especially for TD-DFT calcu- lations of CT excitations. Not every computer code has the DFTB version of all of these options and can carry out FSSH calculations. However, the DFTBABY TD-lc-DFTB computer code19that we used here is one of the few codes to have all of these options (see also Ref. 120). Our objective has been to test the TD-lc-DFTB FSSH method for its description of CT at a model heterojunction of an organic solar cell. For this purpose, we chose a particularly well-studied sys- tem. The pentacene ( P)/buckminsterfullerene ( F) solar cell is not the most efficient of all the organic solar cells, but it is probably the most studied organic solar cell both experimentally and theo- retically. It should probably be emphasized that, while a good deal is already known about this solar cell, it is also frequently used as a model system for a deeper investigation of new or older insuffi- ciently understood phenomena. Thus, we should not discount the possibility of learning something new about this system. Our testing of TD-lc-DFT focused on a bimolecular model of a P/Fheterojunc- tion with an eye to going to yet larger systems with several Pand several Fmolecules. Given the reliance of DFTB on semi-empirical technology, we might ask what we should expect to obtain from TD-lc-DFTB. The J. Chem. Phys. 154, 054102 (2021); doi: 10.1063/5.0024559 154, 054102-17 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp answer to the question depends, in part, on whether it is possible to use a more sophisticated and rigorous method for the applica- tion of interest. If the answer is “yes,” then semi-empirical theories have a well-established place as tools for building understanding by showing what phenomena follow from simple models. If the answer is “no,” then semi-empirical theories provide a way to extend the more sophisticated and rigorous method beyond its normal range of applicability. DFT does this with more resource intensive ab initio quantum-chemistry methods, and DFTB does this with DFT. Nev- ertheless, given the number of approximations that are made, our emphasis should be on qualitative phenomena and on trends. Here, we have emphasized trends in ET and CT times as a function of the range-separation parameter Rlc. Although DFTBABY seemed the aptest starting point for our study, we still found it necessary to add three improvements. The first improvement is the choice of the initial state. In photochemical applications, the initial state is usually thought of as the excitation of a wave packet to the Franck–Condon point of one of the adiabatic potential energy surface of one of the excited electronic states. This would be appropriate if we believed that the excitation occurred at the heterojunction of the organic solar cell. However, the conven- tional model prescribes that the exciton is most probably formed away from the heterojunction and diffuses to the heterojunction. We have, therefore, modified the program to model the arrival of theP∗exciton at the interface by the projection of the P∗excita- tion onto the P∗/Fmodel system. Note that, while this may be a rare choice of FSSH initial condition, it is also completely in line with applications anticipated by Tully who allows an arbitrary initial state. The second improvement has been the implementation of the decoherence correction of Ref. 73. This solves (or at least reduces the effects of) several known problems caused by overcoherence in FSSH.18,68–70In particular, we have shown by explicit CIS/AM1 FSSH calculations that the surface and the electronic wavevector methods become essentially indistinguishable when it comes to cal- culating ensemble-averaged electronic expectation values. This is important for the third improvement. This third improvement is the implementation of a way to auto- matically calculate the charge of excited electrons (particles) and holes on each of the fragments. This gives us a direct definition of CT and ET. We note that it is not the only definition as another definition is possible on the basis of Kasha’s exciton model,22,74,75 which can account for the exchange of charge between neighbor- ing molecules even when no net charge transfer occurs. However, our direct approach seems better adapted to the needs of organic electronics where the primary concern is the movement of real net physical charges. Our P/Fmodel is already beginning to be rather large (96 atoms) for studying CT and ET by the mixed quantum/classical trajectory-based fewest-switches surface hopping (FSSH) method, which requires many thousands of electronic structure calcula- tions. Nevertheless, both the CIS/AM1 FSSH and TD-lc-DFTB FSSH methods showed themselves to be suitable for meeting this chal- lenge. Thus, we were able to calculate ensemble averages for up to 500–800 fs with reasonably good ensemble statistics over 50–100 tra- jectories. This may be compared with the two previous FSSH studies that we know for P/Fsystems: The TD- ωB97X-D FSSH study from the Brédas group40used a more accurate electronic structure methodthan the one used here but was limited to 100 fs and seven tra- jectories. The periodic TD-DFT FSSH calculations by the Prezhdo group29treated a larger, more realistic, heterojunction model but made several severe approximations, including the assumption that the excited-state trajectories follow the same paths as ground-state thermal trajectories and several uses of the independent particle approximation. The present study represents a compromise between the accuracy of the electronic structure method and the need to run enough trajectories to get reasonably good ensemble averages, while still maintaining the basic structure of a rigorous TD-DFT FSSH calculation. As we have emphasized, lc-DFTB is an active area of methods development. As it stands, the structure of the TD-lc-DFTB differs enough from that of TD-lc-DFT that, while Rlcis often similar for the two methods, it need not always be similar. Indeed, the present work shows that a much larger value of Rlcis needed in our TD-lc-DFTB calculations than might have been expected based on experience with TD-lc-DFT. Examination of the calculated spectra shows that small values of Rlcbehave like TD-HF or CIS in that the P+/F−peak is higher in energy than the P∗/Fpeak. Increasing Rlcbeyond 10 a0 leads to the lowering of the energy of the P+/F−peak to below that of theP∗/Fpeak, in agreement with the spectra of high-quality ab initio calculations. A simple chemical kinetics model allowed us to extract ET and CT times as a function of Rlc, hence anticipating that the choice of Rlcmight be tuned to achieve ET and CT times obtained from high-quality ab initio FSSH calculations. Unfortunately, such calculations are much too resource intensive to be practical at the present time. Nevertheless, we did carry out CIS/AM1 FSSH cal- culations, a method whose parameters are fit to experiment, whose spectra show the same qualitatively correct ordering of P+/F−and P∗/Fpeaks as in the high-quality ab initio calculations. TD-lc-DFTB FSSH CT times were found to be similar to CIS/AM1 FSSH times for Rlc= 15 a0. The fact that we can calculate not only net charges but also electron (particle) and hole charges allowed us to make a remark- able observation of an unexpected yet physically reasonable phe- nomenon. During the initial P∗→Ftransfer of an electron, the positively charged hole also starts to delocalize, following the nega- tively charged electron. This delocalization does not last long before the hole relocalizes back onto P, presumably because of electronic and nuclear relaxation effects. As such, this resembles polaron for- mation in solids. It may also be thought of as being like the flicker- ing polaron phenomenon recently seen in some FSSH calculations on solids where excitations hop from one molecule to the next by delocalization followed by relocalization on a different molecule,36,37 except that here the hole delocalizes off of Pand then relocalizes back onto P. This phenomenon only lasts about 5 fs in our CIS/AM1 calculations but is present. It is also present in our TD-lc-DFTB calculations and is of longer duration with the “polaron formation time” increasing as Rlcincreases. Looking forward to the future, we think that the experi- ence gained through the present work should allow us to design TD-lc-DFTB FSSH in silico experiments for larger cluster models of organic heterojunctions. A next local step would be to include environmental effects106in our calculations. We speculate on the basis of the results given in the present work that this would lead to the use of reduced values with the TD-lc-DFTB range-separation parameter, possibly bringing it more in line with literature values J. Chem. Phys. 154, 054102 (2021); doi: 10.1063/5.0024559 154, 054102-18 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp used in TD-DFT calculations with range-separated hybrids. Because DFTB is more efficient than DFT, we think that the present approach can be developed into a useful approach for realistic modeling of charge transfer at organic heterojunctions and also as a tool to investigate at least the initial steps of exciton diffusion and charge separation. SUPPLEMENTAL MATERIAL See the supplementary material for the following: C 60+ pen- tacene electrocyclic addition reaction; the ( x,y,z) coordinates for geometry Cof our 96 atom F/Pvan der Waals complexes; transition density matrix analysis of some of our ab initio calculations; and the results of a more elaborate kinetics model: AkAB /leftr⫯g⊸tl⫯ne→ ←/leftr⫯g⊸tl⫯ne kBABkBC /leftr⫯g⊸tl⫯ne→ ←/leftr⫯g⊸tl⫯ne kBCC, (5.1) withA=P∗/F,B=P/F∗, and C=P+/F−. ACKNOWLEDGMENTS This work was supported, in part, by the French National Research Agency ( Agence Nationale de la Recherche , ANR) ORGA- VOLT (ORGAnic solar cell VOLTage) (Project No. ANR-12- MONU-0014-02). Ala Aldin M. H. M. Darghouth acknowledges a Franco-Iraqi Ph.D. scholarship administered via the French agency Campus France during the period of his doctoral research and sup- port from the University of Mosul for funding that allowed him to continue working on this project after taking up his current posi- tion in Iraq. The authors wish to acknowledge the support from Grenoble Alps University’s ICMG ( Institut de Chimie Moléculaire de Grenoble ) Chemistry Nanobio Platform PCECIC ( Plateau du Centre d’Expérimentation et de Calcul Intensif en Chimie ) on which this work was performed. Pierre Girard is gratefully acknowledged for his help and support using this platform. Alexander Humeniuk and Roland Mitri ´c acknowledge financial support within the European Research Council (ERC) Consolidator Grant DYNAMO (Grant No. 646737). Some of the early work on this project was performed in Singapore where it was supported, in part, by the Society of Inter- disciplinary Research (SOIRÉE). Mark E. Casida and Ala Aldin M. H. M. Darghouth would also like to acknowledge a useful trip to Würzburg funded by the German GRK 2112 Project “Biradicals,” as well as useful discussions with Dr. Mathias Rapacioli and Dr. Hemanadhan Myneni. Ala Aldin M. H. M. Darghouth acknowledges having followed an advanced course taught by Professor Mario Bar- batti on “Theoretical Aspects of Organic Femtochemistry” together with a tutorial on the NEWTON-X program. We are grateful to Mario Barbatti for his extensive comments on an early version of this manuscript. Felix Plasser is acknowledged for having drawn our attention to Ref. 75. Mark E. Casida acknowledges an inspiring dis- cussion with Lucia Reining regarding the place of semi-empirical methods in the theorist’s toolbox. We gratefully acknowledge help- ful correspondence with Oleg Prezhdo, Mario Barbatti, and SergeiTretiak, who pointed out references useful for addressing key points raised by the reviewers. The authors declare no conflict of interest. APPENDIX A: BRIEF REVIEW OF DFTB AND TD-DFTB This appendix contains a very concise review of DFTB and of TD-DFTB as an aid for understanding the still-rather-new imple- mentation of TD-lc-DFTB used in this work.9We give a brief review of DFTB14,15,121and TD-DFTB4–8and of their relation to DFT122–124 and TD-DFT.125–127DFT and TD-DFT are now so well established that it seems that little needs to be said about them. DFTB and TD- DFTB are more recent but would still be familiar to experts. Here, we will concentrate on a concise review of just enough of the basic methodology to be able to explain why it is difficult to introduce lc into TD-lc-DFTB and what compromises have been made in the current methodology. In both DFT and DFTB, the total energy is expressed as E=EBS+Erep, (A1) where the band-structure term is the occupation-number weighted sum of Kohn–Sham orbital energies, EBS=∑ iniϵi, (A2) and the repulsion potential term in DFT is Erep=EH[ρ]+Exc[ρ]−∫vxc[ρ](⃗r)d⃗r+Vn,n, (A3) whereρis the electron density, Hrefers to the Hartree energy expres- sion, xcstands for exchange–correlation terms, and Vn,nstands for the nuclear repulsion terms. DFTB makes use of several ideas from semi-empirical quantum chemistry as well as some of its own. Like semi-empirical methods, DFTB assumes a minimal basis descrip- tion of the valence electrons, and all the other electrons are treated implicitly as part of ionic cores. In the original form of DFTB,13 the total charge density was simply assumed to be the sum of unperturbed atomic charge densities, ρ0=∑Iρ0 I, and the Hartree plus xc potential is assumed separable, vHxc[ρ] =∑IvHxc[ρI]. These two approximations are used, together with neglect of any three- center terms, to construct the matrix of the Kohn–Sham operator. Solving the matrix form of the DFTB Kohn–Sham equation then gives the orbital energies and hence the band-structure term in the total energy. The repulsion potential is assumed to be the sum of pairwise interatomic repulsion potentials, Erep=∑I<JVI,J(RI,J). In principle, a new set of pairwise potentials is needed for each new approximate density-functional. In practice, generating these new pairwise potentials for each new density functional is impractical. Modern self-consistent charge (SCC) DFTB adds a Coulomb term, Ecoul=1 2∬δρ(⃗r1)(fH(⃗r1,⃗r2)+fxc(⃗r1,⃗r2))δρ(⃗r2)d⃗r1d⃗r2, (A4) J. Chem. Phys. 154, 054102 (2021); doi: 10.1063/5.0024559 154, 054102-19 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp to the energy expression of Eq. (A1) to account for the fact that the charge density is not simply the sum of the unperturbed charge densities but rather ρ(⃗r)=ρ0(⃗r)+δρ(⃗r). (A5) Here, fHand fxcare the second functional derivatives with respect toρof the classical Coulomb (Hartree) repulsion and Exc, respec- tively. Mulliken’s integral approximation and a monopole expansion of products of molecular orbitals in terms of atom-centered s-type functions gI(⃗r), then, allow δρto be replaced by Mulliken charges, and the essential Coulomb integral in Eq. (A4) becomes Ecoul=1 2∑ I,JΔqIγI,JΔqJ, (A6) where γI,J=∬ gI(⃗r1)(fH(⃗r1,⃗r2)+fxc(⃗r1,⃗r2))gJ(⃗r2)d⃗r1d⃗r2 (A7) andΔqIare Mulliken charge differences. This has a couple of inter- esting consequences. The first is that variational minimization of the three term energy expression results in a Kohn–Sham orbital equation, which must now be solved self-consistently because the Kohn–Sham operator depends on the Mulliken charges, which are themselves calculated from the molecular orbital coefficients (hence, SCC-DFTB). The second interesting consequence is that it is now possible to set up and solve the TD-DFTB analog4of Casida’s equation,67 [A B B∗A∗](⃗XI ⃗YI)=ωI[1 0 0−1](⃗XI ⃗YI), (A8) where Aiaσ,jbτ=δi,jδa,bδσ,τ(ϵaσ−ϵiσ)+Kiaσ,jbτ, Biaσ,jbτ=Kiaσ,bjτ.(A9) Here,ωIis the electronic excitation energy and (⃗XI,⃗YI)are used for calculating spectra oscillator strengths and nonadiabatic coupling elements. In TD-DFTB, Kiaσ,jbτ=∑ I,Jqia Iγσ,τ I,Jqjb I, (A10) where qia Iare Mulliken transition charges. APPENDIX B: BRIEF REVIEW OF TD-DFT FSSH This appendix provides a very brief review of how TD-DFT is usually combined with Tully’s molecular dynamics with quan- tum transitions (nowadays, often simply referred to as FSSH).17,18 As FSSH is now widely known, it is not our intention toreview that method here. Instead, the reader is referred to recent reviews.106,128 The first implementation of TD-DFT FSSH was due to Tapavicza, Tavernelli, and Röthlisberger in 200766in a development version of the CPMD (Car-Parrinello Molecular Dynamics) code. They proposed that the nonadiabatic coupling be calculated using Casida’s ansatz ,which was originally intended as an aid for assigning TD-DFT excited states.67Specifically, an excited-state wave function ΨI=∑ i,a,σΦaσ iσCiaσ (B1) made up of singly excited determinants Φaσ iσ(corresponding to the iσ→aσexcitation) is postulated, and it is argued that CI iaσ=√ ϵaσ−ϵiσ ωIFI iaσ, (B2) where ⃗FI∝(A−B)−1/2(⃗XI+⃗YI) (B3) is renormalized so that ⃗F† I⃗FI=1. (B4) Then, Eqs. (B1) and (B2) are combined with dI,J(R(t+Δ 2))=1 2Δ[⟨ΨI(R(t))∣ΨJ(R(t+Δ)⟩ −⟨ΨI(R(t+Δ)∣ΨJ(R(t)⟩] (B5) to obtain the nonadiabatic coupling (NAC) elements. This leads to a linear combination of overlap terms between two Slater deter- minants at different times, which is evaluated using the observa- tion that the overlap of two Slater determinants is the determinant of overlap integrals.129Their implementation was followed by an application to the photochemical ring opening of oxirane,130which showed that the nonexistence of a proper conical intersection in conventional TD-DFT131was not a serious practical problem for TD-DFT FSSH. TD-DFT FSSH has also been implemented in a version of TURBOMOL capable of calculating nonadiabatic cou- pling elements analytically,132and this was applied early to study the photochemistry of vitamin-D.133 APPENDIX C: ALGORITHM FOR CALCULATING pAND hCHARGES The particle density matrix γpand the hole density matrix γhare easy to calculate using second quantization. We will use the common molecular orbital (MO) index convention, a,b,c,...,g,h ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ unoccupiedi,j,k,l,m,n ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ occupiedo,p,q,...,x,y,z ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ free. (C1) J. Chem. Phys. 154, 054102 (2021); doi: 10.1063/5.0024559 154, 054102-20 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The physical vacuum |0 ⟩is taken to be the ground state single deter- minant. Casida’s ansatz takes the form of a single configuration interaction (CIS) wave function,67 ∣I⟩=∑ i,aa†i∣0⟩Ci,a. (C2) Specifically, Ci,a=N(Xi,a+Yi,a), (C3) and the normalization constant Nis chosen so that ∑ i,a∣Ci,a∣2=1. (C4) The hole density matrix in the MO representation is γh k,l=⟨I∣lk†∣I⟩=∑ aC∗ l,aCk,a, γh=CC†,(C5) and the particle density matrix in the MO representation is γp c,d=⟨I∣d†c∣I⟩=∑ iC∗ i,dCi,c, γp=C†C.(C6) Finally, we must calculate the particle and hole charges on each fragment. This is done in the usual way using Mulliken charges. Atom-centered basis functions are labeled with lower case Greek indices. The overlap matrix is Sμ,ν=⟨μ∣ν⟩. 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5.0028874.pdf

J. Phys. Chem. Ref. Data 50, 023102 (2021); https://doi.org/10.1063/5.0028874 50, 023102Rate Constants for Abstraction of H from the Fluoromethanes by H, O, F, and OH Cite as: J. Phys. Chem. Ref. Data 50, 023102 (2021); https://doi.org/10.1063/5.0028874 Submitted: 07 September 2020 . Accepted: 25 February 2021 . Published Online: 14 April 2021 Donald R. Burgess , and Jeffrey A. Manion COLLECTIONS This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN A method of flight path and chirp pattern reconstruction for multiple flying bats Acoustics Research Letters Online 6, 257 (2005); https://doi.org/10.1121/1.2046567 Solution of Multiscale Partial Differential Equations Using Wavelets Computers in Physics 12, 548 (1998); https://doi.org/10.1063/1.168739 Chinese Abstracts Chinese Journal of Chemical Physics 34, i (2021); https://doi.org/10.1063/1674-0068/34/01/ cabsRate Constants for Abstraction of H from the Fluoromethanes by H, O, F, and OH Cite as: J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 Submitted: 7 September 2020 •Accepted: 25 February 2021 • Published Online: 14 April 2021 Donald R. Burgess Jr.a) and Jeffrey A. Manion AFFILIATIONS Chemical Sciences Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA a)Author to whom correspondence should be addressed: dburgess@nist.gov ABSTRACT In this work, we compiled and critically evaluated rate constants from the literature for abstraction of H from the homologous series consisting of thefluoromethanes (CH 3F, CH 2F2, and CHF 3) and methane (CH 4) by the radicals H atom, O atom, OH, and F atom. These reactions have the form RH + X →R + HX. Rate expressions for these reactions are provided over a wide range of temperatures (300 –1800 K). Expanded uncertainty factors f(2σ) are provided at both low and high temperatures. We attempted to provide rate constants that were self-consistent within the series —evaluating the system, not just individual reactions. For many of the reactions, the rate constants in the literature are available only over a limited temperature range (or there are no reliable measurements). In these cases, we predicted the rate constants in a self-consistent manner employing relative rates for other reactions in the homologous series using empirical structure –activity relationships, used empirical correlations between rate constants at room temperature and activation energies at high temperatures, and used relative rates derived from ab initio quantum chemical calculations to assist in rate constant predictions. ©2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved. https://doi.org/10.1063/5.0028874 Key words: chemical kinetics; critical evaluation; hydro fluorocarbons; reaction mechanism. CONTENTS 1. Introduction and Background ................ 3 1.1. Hydro fluorocarbon chemistry ............. 3 1.2. Rationale for evaluation and estimation methodology . . . ..................... 3 1.3. Some general characteristics of the reactions and rate expression .......................... 4 2. Overview of Available Experimental Data and Analysis . 4 2.1. Availability of data .................... 4 2.2. Data analysis and least-squares fit with additional constraints ......................... 5 2.3. Network of related reactions .............. 7 2.4. Overview of classes of reactions ............ 8 2.4.1. Estimation and interpolation .......... 8 3. Thermochemical Data ..................... 9 4. Fluoromethanes + H →Fluoromethyls + H 2....... 9 4.1. CH 4+H→CH 3+H 2................. 1 0 4.2. CH 3F+H→CH 2F+H 2and CH 2F2+H→CHF 2 +H 2............................. 1 0 4.3. CHF 3+H→CF3+H 2................. 1 2 4.3.1. Overview . ..................... 1 24.3.2. Rate constants at high temperatures ..... 1 3 4.3.3. Rate constants at low temperatures ...... 1 4 4.3.4. Quantum chemical calculations for CHF 3 +H ......................... 1 6 4.3.5. Discussion of systematic trends in fluorome- thanes + H rate expressions . . . ....... 1 7 4.3.6. Equilibrium constant for CHF 3+H↔CF3 +H 2......................... 1 8 5. Fluoromethanes + O →Fluoromethyls + OH ...... 1 9 5.1. Overview . . . ....................... 1 9 5.2. CH 4+O→CH 3+O H................. 1 9 5.3. CH 3F+O→CH 2F+O H ............... 2 2 5.4. CH 2F2+O→CHF 2+O H ............... 2 2 5.5. CHF 3+O→CF3+O H ................ 2 2 6. Fluoromethanes + OH →Fluoromethyls + H 2O ..... 2 4 6.1. Overview . . . ....................... 2 4 6.2. CH 4+O H→CH 3+H 2O ............... 2 4 6.3. (CH 3F, CH 2F2, CHF 3)+O H→(CH 2F, CHF 2,C F 3) +H 2O............................ 2 5 7. Fluoromethanes + F →Fluoromethyls + HF ....... 2 7 7.1. Overview . . . ....................... 2 7 J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-1 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jpr7.2. Evaluation procedure ................... 2 9 7.3. Discussion of fluorine impact on rate constants . . 31 8. Discussion ............................. 3 3 8.1. Overview .......................... 3 3 8.2. Correlations for rate constant Afactors . . ..... 3 6 8.3. Activation energies .................... 3 7 8.4. Tunneling rates . ..................... 3 7 9. Summary ............................. 3 9 10. Notation Used .......................... 4 0 11. Supplementary Material .................... 4 1 12. Data Availability ......................... 4 1 13. References ............................. 4 1 List of Tables 1. Counts of available data for the fluoromethanes + X →fluoromethyls + HX reactions, where X /equalsH, O, O H ,F............................. 5 2. Standard enthalpies of formation ΔfH°(298.15 K) and C–H BDEs used in this work. ................. 9 3. Recommended rate expressions for H abstraction from the fluoromethanes by the H atom and enthalpies of reaction 10 4. CH 4+H→CH 3+H 2rate expressions .......... 1 1 5. CH 3F+H→CH 2F+H 2rate expressions ......... 1 3 6. CH 2F2+H→CHF 2+H 2rate expressions . . . ..... 1 3 7. CHF 3+H→CF3+H 2rate expressions .......... 1 5 8. CF3+H 2→CHF 3+ H rate expressions (reverse direction to CHF 3+H→CF3+H 2) .................. 1 6 9. Equilibrium constants for CHF 3+H↔CF3+H 2.... 1 7 10. CF3+C F 3→C2F6rate constants .............. 1 7 11. Recommended rate expressions for H abstraction from the fluoromethanes by O atoms .................. 1 9 12. CH 4+O→CH 3+ OH rate expressions .......... 2 0 13. CH 3F+O→CH 2F + OH rate expressions . . . ..... 2 3 14. CH 2F2+O→CHF 2+ OH rate expressions . . ..... 2 3 15. CHF 3+O→CF3+ OH rate expressions ......... 2 4 16. Recommended rate expressions for H abstraction from the fluoromethanes by OH radicals ................ 2 5 17. CH 4+O H→CH 3+H 2O rate expressions . . . ..... 2 6 18. CH 3F+O H→CH 2F+H 2O rate expressions . ..... 2 9 19. CH 2F2+O H→CHF 2+H 2O rate expressions . ..... 3 0 20. CHF 3+O H→CF3+H 2O rate expressions . . ..... 3 1 21. Recommended rate expressions for H abstraction from the fluoromethanes by F atoms .................. 3 2 22. CH 4+F→CH 3+ HF rate expressions .......... 3 3 23. CH 3F+F→CH 2F + HF rate expressions ......... 3 4 24. CH 2F2+F→CHF 2+ HF rate expressions . . . ..... 3 4 25. CHF 3+F→CF3+ HF rate expressions .......... 3 5 26. Recommended rate expressions for H abstraction from the fluoromethanes by H, O, OH, and F ............. 3 9 List of Figures 1. Potential energy as a function of reaction coordinate for R –H+X→R + XH bimolecular abstraction r e a c t i o n............................ 42. CH 4+H→CH 3+H 2experimental data and fitted rate expression ............................. 1 2 3. CH 4+H→CH 3+H 2residuals for rate data excluded from fit ................................... 1 2 4. CH 3F+H→CH 2F+H 2and CH 2F2+H→CHF 2+H 2 experimental data and rate expressions ........... 1 4 5. Rate constants for CHF 3+H→CF3+H 2........ 1 6 6. CHF 3+H→CF3+H 2.................... 1 6 7. Fluoromethanes + H rates (per H atom) at 300 K vs barriers at (1200 –1800) K ................... 1 8 8. Fluoromethanes + H rate constants (per H atom) vs B D E s .............................. 1 8 9. CH 4+O→CH 3+ OH rate constants and recommended rate expression . . . ....................... 2 1 10. Residuals for CH 4+O→CH 3+ OH rate constants excluded from fit......................... 2 1 11. Residuals for CH 4+O→CH 3+ OH rate constants from quantum chemical calculations ................ 2 1 12. Rates of reaction for H abstraction by O from CH 3F, CH 2F2, and CHF 3........................ 2 2 13. CH 4+O H→CH 3+H 2O rate constants and recom- mended rate expression ..................... 2 7 14. CH 4+O H→CH 3+H 2O rate constant residuals relative to the recommended rate expression ............ 2 7 15. CH 4+O H→CH 3+H 2O rate constant residuals for excluded data relative to the recommended rate expression ............................. 2 8 16. CH 3F+O H→CH 2F+H 2O................. 3 0 17. CH 2F2+O H→CHF 2+H 2O ................ 3 0 18. CHF 3+O H→CF3+H 2O .................. 3 2 19. Fluoromethanes + OH →fluoromethyls + H 2O ..... 3 2 20. Fluoromethanes + F →fluoromethyls + HF rate c o n s t a n t s............................ 3 5 21. Normalized Afactors (per H atom) (on a logarithmic scale) at high temperatures as a function of the number of H atoms for the different reactants H, O, O H ,a n dF........................... 3 5 22. Evans –Polanyi55plot of activation energy Eaat high tem- peratures (1200 –1800 K) as a function of heat of reaction ΔrH................................. 3 6 23. Activation energies (open circles) for hydrogen abstraction by H, O, OH, and F, and TS bond distances ( filled circles) vsfluorine substitution in the fluoromethane series . . . 37 24. Contribution of tunneling to the rate constants for CH 4+O H→CH 3+H 2O................. 3 7 25. Correlation between activation energies Eaat low temper- atures (300 K) and activation energies Eaat high temper- atures (1200 –1800 K) ...................... 3 8 26. Normalized (per H atom) rate constant k(cm3mol−1s−1) at 300 K as a function of activation energy Ea(kJ mol−1) for H abstractions in the fluoromethane series . . ....... 3 8 27. The compensation effect correlation of the temperature coefficient nwith energy coef ficient Ein expressions k/equalsATnexp(−E/RT) for H abstractions in the fluoro- m e t h a n es e r i e s .......................... 3 9 J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-2 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jpr1. Introduction and Background 1.1. Hydro fluorocarbon chemistry In this work, we have compiled, evaluated, and recommended rate constants for a homologous series of reactions for the abstraction of H from the fluoromethanes (and methane) by the radicals H atom, O atom, F atom, and OH. Reactions for methane are included as benchmarks for each series. These series of reactions can be described by the set R –H+X→R + HX, CH 4+X→CH 3+HX, CH 3F+X→CH 2F+HX, CH 2F2+X→CHF 2+HX, CHF 3+X→CF3+HX, where X /equals(H, O, OH, F) and HX /equals(H2, OH, H 2O, and HF). These reactions involving the fluoromethanes are an important set of reference reactions for larger hydro fluorocarbons (HFCs), particularly the fluoroethanes used as refrigerants. They can also be used as references of hydro fluoroole fins (HFOs), hydro fluoroethers (HFEs), and brominated HFCs (used in a variety of applications as refrigerants, fire suppressants, blowing agents, and cleaning solvents). Uses of these chemical classes of compounds are regulated because of their global warming potential (GWP), and uses of the brominated compounds are regulated because of their ozone depletion potential(ODP). 1,2 A number of common refrigerant agents that are used are difluoromethane (CH 2F2, R32), 1,1-di fluoroethane (CH 3CHF 2,R - 152a), 1,1,1-tri fluoroethane (CH 3CF3, R-143a), 1,1,2-tri fluoroethane (CH 2FCHF 2, R-134a), penta fluoroethane (CHF 2CF3, R-125), 1,1,1,3,3,3-hexa fluoropropane (CF 3CH 2CF3, HFC-236fa), 1,1,1,2,3,3,3-hepta fluoropropane (CF 3CHFCF 3, HFC-227ea), 2,3,3,3-tetra fluoropropene (CF 3CF/equalsCH 2, HFO-1234yf), penta- fluoroethyl methyl ether (CF 3CF2OCH 3, HFE-245mc), hepta- fluoropropyl methyl ether (CF 3CF2CF2OCH 3, HFE-7000), and 2- bromotri fluoropropene (CF 3CBr/equalsCH 2, 2-BTP). The above agents are often used in various blends to achieve specific physicochemical properties optimized for speci fic applica- tions. The physicochemical properties include such properties as critical temperature, enthalpy of vaporization, thermal conductivity,and vapor density. Additional property considerations are toxicity andflammability. Flammability of the components and blends is a significant safety concern in the context of engineering and regulatory needs. In the absence of individual empirical testing of all possible formulations and conditions, a daunting if not impossible task, predictive detailed kinetic models provide a valuable and more-rapidscreening tool. Such flammability models require accurate knowledge of the kinetic properties at high temperatures. Kinetic information at lower temperatures relevant to the troposphere and stratosphere islikewise necessary to understand and minimize the GWPs and ODPs of the formulations. The temperatures considered in the present evaluation range from near-ambient to those relevant to combustion. In general, there are more kinetic studies at the lower temperatures and a focus of thiswork is extrapolation of the rate constants to the higher temperatures necessary for flammability models of refrigerants. Indeed, it was the absence of reliable self-consistent data at higher temperatures that was a primary motivation for undertaking this work. The reactionsdescribed pertain to fluorinated C 1species. Such species will naturally arise at some level during the breakdown of larger fluorinated compounds at high temperatures. The results will thus apply to combustion models of most refrigerants. Despite our primary mo- tivation being the kinetics at high temperatures, we have critically considered data from lower temperatures and our recommendations are also applicable to atmospheric chemistry. As noted, self-consistent rate constants for these reactions comprise a valuable training set that can subsequently be used to, e.g., validate future quantum chemical calculations or structure –activity rules used to predict the reactivity of more complex agents including the fluoroethanes,3,4 fluoropropanes,5,6fluoroalkenes,7–12andfluoroethers.13–18 1.2. Rationale for evaluation and estimation methodology When evaluating chemical kinetic data, it is often found that the quality of the available information is highly variable. Ideally, the reactions under consideration will all have been studied by several researchers using different techniques over a wide range of tem- peratures and conditions. This is rarely the case, however. More often, one is faced with sparse, incomplete data of variable quality, obtained with methods ranging from crude estimates to highly sophisticated techniques. Even if a reliable measurement exists, each method spans a limited temperature range and the data will have to be ex- trapolated to cover all conditions of interest. If reactions are con-sidered individually, such realities can lead to large uncertainties in the rate parameters. Fortunately, the experimental information can usually be placed in a broader context that will considerably constrain the possibilities. In the present work, we do this in several ways. For example, transition state (TS) theory allows one to relate kinetic parameters and thermodynamic properties. If one considers a series of related reactions, one usually finds that barrier heights and TS entropies (related to pre-exponential factors) vary in a regular way with properties such as reaction enthalpy or electronic characteristics of substituents. Such behavior has been known for many decades and forms the basis of thermochemical kinetics and many predictive structure –activity relationships (SARs). More recently, high-level quantum chemical calculations, if available, provide improved methods of quantifying behavior. Although quantum calculations may not provide quantitatively exact rate constants, they usually capture trends with high accuracy. In the present evaluations, we seek to establish networks that, for related reactions, bring together all available sources of information, including experimental measure- ments of absolute and relative rates, empirical relationships, and quantum chemical calculations. While the reader is referred to the evaluations for details, the general principle of our approach is to first establish within each reaction set the cases that have the best in- formation and experimental measurements and then to use these data to set absolute rates and bounding behavior. The less-well-studied reactions are then considered in the context of broader information that establishes rate trends. This yields a self-consistent set of rec- ommended values that we believe are the best presently available. J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-3 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprIn any evaluation, it is necessa ry to select, among often-con flicting data, the results that are to be preferred and used to derive the final recommendation. A dif fic u l t yi nt h es t u d yo fk i n e t i c s ,a si sw e l lk n o w n ,i s that the most important errors are usually not random but rather sys-tematic in nature. Furthermore, the sources of the systematic errors vary with the technique and are often unknown; indeed, had they been un- derstood, the researchers would have accounted for them. There are various ways to deal with this problem. Ideally, one would assign to all determinations a correct statistica l weighting that accounts for the sys- tematic errors. However, the unknown nature of the errors makes this difficult to do in a consistent manner; we ighting assignments thus easily become ad hoc justifications rather than statistically relevant assessments. In the present work, we have adopted what we believe is a more transparent approach of using only what we have deemed “preferred data”in deriving our recommended fits. The results not used directly in thefits remain valuable, however, and have been considered in our as- signment of the overall uncertainties. For clarity, the tables split the results into “preferred data ”and “excluded data. ”Assignment to the latter category is not meant to necessarily imply that we have dismissed these data or consider them to be unimportant. Details of the data selection process for speci fic reactions are found in the individual evaluations. 1.3. Some general characteristics of the reactions and rate expression All of the reactions considered in this evaluation are direct bimolecular reactions that are abstraction reactions that have the form R –H+X→R + XH, where the reactants are the stable (full valence) molecules R –H/equals(CH 4,C H 3F, CH 2F2, CHF 3) and the radicals X /equals(H, O, OH, F) with the products being the corresponding radicals R /equals(CH 3,C H 2F, CHF 2,C F 3) and the stable molecules HX/equals(H2, OH, H 2O, HF). This reaction type can also be written in the form R –H+X →[R–H–X]‡→R + XH where the intermediate entity is the TS [R–H–X]‡corresponding to the highest point on the potential energy surface (PES) that connects reactants and products along the reaction coordinate. The overall reaction is usually exothermic, and the products are at lower energies than the reactants, leading to the generic PES illustrated in Fig. 1 . Note that the reactions of CH 4and CHF 3with H are slightly exothermic. This moves the products above the reactants in energy, but does not change the general shape of the potential energy curves. The bimolecular reaction rate can be written as r/equalsk[RH][X]/equals−d[RH]/dt, where kis the rate constant and [RH] and [X] are the concentrations of the reactants. Throughout this paper, we use the units mol, kJ, cm3, s, and K. Consequently, the concentrations [M] are given in mol cm−3, the rate of reaction is given in mol cm−3s−1, and the rate constants k are given in cm3mol−1s−1. Rate constants k(T) are normally well-described over moderate temperature ranges by the simple two-parameter Arrhenius ex- pression k/equalsAe−Ea/RT,w h e r e Ais a pre-exponential factor (cm3mol−1 s−1),Eais the activation energy (kJ mol−1),Tis temperature (K), and Ris the gas constant (8.314 J mol−1K−1). For the broad range of temperatures considered in the present evaluation, a more accurate representation ofk(T) is given by the three-parameter extended Arrhenius expression(k/equalsAT ne−E/RT), where nis an additional temperature exponent co- efficient and Eis an exponential temperature coef ficient (it is not an “activation energy ”Ea). The three-parameter expression allows one to better account for factors that lead to curvature in standard Arrhenius plots of log kvs 1/T, such as tunneling or temperature-dependent changes in the TS properties. There is no required relationship between corre-sponding Aand Evalues in fits using the two- and three-parameter expressions; in addition, the values of A, E, andnare highly correlated in the extended expression and the fits do not necessarily relate in a straightforward way to thermodynamic properties in the context of TStheory. When considering a limited su bset of temperatures, we therefore sometimes use the simpler form k/equalsAexp(−E a/RT) in order to highlight relationships involving AorE. The simpler rate expression k(A,Ea)i sa l s o useful when comparing rate expressio ns in a homologous reaction series or optimizing the agreement of mod els with a series of observables. 2. Overview of Available Experimental Data and Analysis 2.1. Availability of data As mentioned above, there are no experimental determinations of rate constants for all of the H abstraction reactions over a wide range of temperatures (200 –2000 K). The rate constants for H ab- stractions by OH at lower temperatures (200 –400 K) are pertinent to atmospheric chemistry and have been extensively measured (cited later). Rate constants at higher temperatures (1200 –2000 K) are pertinent to combustion chemistry and flammability. Because of their importance in hydrocarbon combustion, the reactions of H, O, and OH have seen much study. F atoms are not present in most fuels, however, and little-to-no work has been done on their reactions at high temperatures. The reactivity of F atoms with HFCs is nonetheless important in the flammability of the neat refrigerants, where the absence of hydrocarbon fuel makes conditions relatively hydrogen- poor and fluorine-rich. InTable 1 , we brie fly characterize the available data for the considered reactions. More details are provided in the subsequent corresponding reviews, which include comprehensive lists of the FIG. 1. Potential energy as a function of reaction coordinate for R –H+X→R+X H bimolecular abstraction reaction. J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-4 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprmeasurements and calculations. In many cases, the data are sparse. By “High T ”inTable 1 , we mean moderate to high temperatures, roughly (500–2000) K, that is, substantially above the near-room-temperature measurements, which are denoted “Low T ”and pertain to ( <300–500) K. The case counts of Table 1 include only the “preferred experimental data”; there may be additional data not speci fically used in the fits, but that was considered in the broader assessments of overall uncer- tainties (see Sec. 8for each reaction). In a few instances, the counts include rate expressions for H abstractions by X that are derived from the reverse reaction (i.e., R + HX →R–H + X) using thermodynamics and detailed balance. For rate constants derived from computations, we include counts for all values without regard to perceived quality or accuracy. We note that in a number of cases, the rate constants from quantum chemical calculations differed substantially (an order of magnitude or more) from the recommended rate constants (and each other); see, for example, CHF 3+H→CF3+H 2. More typically, however, they differed by factors of (1.3 –2.5) and (2 –4) at high and low temperatures, respectively, and outside our assigned uncertainties. Our recommended rate expressions are derived (for the most part) from fits to the available data. Some data were excluded from the fits (as mentioned above) for several reasons since they would in- troduce bias to the recommended rate expressions because (a) the data were simply revised measurements made by the same research group and method; (b) the rate constants were substantially outside our recommended uncertainties; (c) the data were derived frommodels where the values were dependent upon assumptions that were not reported, ill-de fined, or more generally uncertain; and (d) the data showed incorrect temperature dependencies (differing substantially from the recommended values) and, consequently, would inappro-priately skew the temperature dependence. In all cases, the recom- mended uncertainties that we provide take into account these outliers using appropriate weighting, and in all cases, we provide the devi- ations of the excluded data from the recommended values. 2.2. Data analysis and least-squares fit with additional constraints The problem at hand is to develop self-consistent fits to rate data in a situation where the experimental data are often sparse, of variable quality, possibly con flicting, and not available over the full temper- ature range of interest. In a few cases, there are no experimental data. In the above circumstances, a simple least-squares fit to individual datasets would yield a multitude of possible parameters with equal statistical uncertainties in the fits. In addition, the derived Arrhenius parameters ( A,n,E) would be unphysical: the resultant rate ex- pression might accurately represent the rate constants in the limited temperature ranges but would be inaccurate at all other temperatures. We therefore apply additional information that will further constrain the kinetic parameters such as A-factors and activation energies. In particular, all the considered reactions are simple abstractions and each reaction set is comprised of a homologous series having a similar generic PES. It is expected that the functional similarities will lead to regular trends in the kinetic parameters. A key point is that there are usually good experimental data for (CH 4+ X) and (CHF 3+ X), the two reactions that are expected to bound the high and low rate constants in each homologous series. Some form of interpolation is therefore expected to yield reliable results for the other reactions. The solution to this problem of fitting datasets where the data are sparse is straightforward: a least-squares fit to the data with additional constraints —an optimization that fits the experimental data while ensuring that deviations from objective functions involving other properties are minimized. This numerical analysis method is widely used in all fields of engineering in solving practical problems.19In our case for rates of reactions, given the scarcity of the data, a model employing just an extended Arrhenius expression is insuf ficient to find solutions that can be extrapolated with any certainty. Fortu- nately, one can relatively easily construct a more expansive modelemploying the concept of SARs, which is widely used in chemistry (and other fields). 20–27The reactions studied here are a set of ho- mologous reactions where the PESs for the reactions are functionally the same and, hence, SARs should work relatively well. This concept goes by many names and has been implemented in different ways in thefield of chemical kinetics including SARs,17,28 –30valid kinetic parameter range analysis,31group additivity kinetics,32–35rate con- stant rules,36reaction classes,37–46hierarchical classes,47isodesmic reactions for TSs,48,49graph theory for reactions,50,51and even re- action symbolic computing,52deep learning of activation energies,53 and genetic algorithm-based method for kinetics.54 For each homologous series, the following assumptions were included as constraints in developing our fits: 1. log(A/nH)∼nH. The log of the normalized Afactors (per H atom) at high temperatures (we chose 1200 –1800 K) should vary roughlyTABLE 1. Counts of available data for the fluoromethanes + X →fluoromethyls + HX reactions, where X /equalsH, O, OH, F. Only reliable data ( <3σ) are included in the counts. “High T ” denotes data at intermediate to high temperatures (500 –2000) K, while “Low T ”denotes near room temperature measurements ( <300 –500) K. “Calc. ”denotes data from quantum chemical calculations. The last column contains estimated expanded uncertainty factors f(2σ)i nt h e experimental measurements at high and low temperatures, with the low-temperature valuesgiven in parentheses Species Expt. High T Expt. Low T Calc. f R–H+H→R+H 2 CH 4 9 1 5 1.4 (1.6) CH 3F 1 0 3 1.5 (1.8) CH 2F2 1 0 3 1.6 (1.8) CHF 3 3 1 7 1.1 (2) R–H+O→R+O H CH 4 5 2 12 1.18 (1.5) CH 3F 0 9 5 1.4 (2) CH 2F2 0 0 2 1.4 (2) CHF 3 2 0 1 1.7 (2.3) R–H+O H→R+H 2O CH 4 4 8 5 1.12 (1.05) CH 3F 0 9 9 1.5 (1.22) CH 2F2 0 8 6 1.5 (1.17) CHF 3 1 7 6 1.3 (1.22) R–H+F→R+H F CH 4 0 9 6 1.35 (1.23) CH 3F 0 7 1 1.35 (1.24) CH 2F2 0 7 1 1.35 (1.35) CHF 3 1 10 3 1.35 (1.20) J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-5 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprlinearly with the number of H atoms. The simplest assumption would be to make the normalized A-factors identical, but this is not the case for a number of reasons —the partition functions will change with increasing fluorine substitution, the PES will change due to polar- izability changes, and for the reaction with OH, the degree of hin- drance in the OH rotor in the TS will change. Consequently, the reaction coordinate will vary in a regular way across a homologousseries of reactions, which will slightly impact the TS entropies andhence the A-factors. This criterion is applied at high temperatures to avoid interference from tunneling. As will be illustrated and dis- cussed in Sec. 8, the uncertainties in the optimized Afactors of about f/equals1.1±0.1 are substantially less than the assigned expanded un- certainty factors (2 σ) of about f/equals1.4±0.2, and thus, the normalized Afactors are highly correlated with the number of H atoms. 2.E a∼ΔrH. The activation energy Eaat high temperatures should vary roughly linearly with the heat of reaction ΔrH.This is an expression of the well-known Evans –Polanyi relationship.40,55 –58We have qualified this to apply at high temperatures because tunneling is expected to cause the activation energy (an empirical quantity) to deviate at low temperatures from the relevant thermodynamic en- ergy barrier, whereas tunneling plays only a minor role at high temperatures (see Sec. 8.4for a further discussion of tunneling). One thus expects a better correlation at high temperatures. No particular assumptions were made about the slope in this correlation, but the value is approximately determined by the behavior and data for the bounding reactions. As will be illustrated and discussed in Sec. 8, the uncertainties in the optimized activation energies Eaof about (3 –7)% (excluding a few outliers) that correspond to an expanded uncer- tainty factor f(2σ) of about (1.2 –1.3) for the rate constants at high temperatures are less than the assigned expanded uncertainty factors f(2σ) of about f/equals1.4±0.2, and thus, the optimized activation energies Eaare well correlated with the heats of reactions. 3.n∼Ea. The temperature coef ficient nshould vary roughly linearly with the activation energy Eaat high temperatures. This re flects the well-known kinetic compensation effect between the Arrhenius parameters [ Aexp(−Ea/RT)/equals(A′Tn) exp( −E/RT)]: a rate expression in order to go through the same rate constants would require a higher Afactor along with a higher activation energy Eaand similarly would require a higher temperature coef ficient nalong with a higher ex- ponential energy coef ficient E.The extended Arrhenius format accounts for an upward curvature in Arrhenius plots over an ex- tended temperature range. Such a curvature is partly related to temperature-dependent changes in the thermodynamic properties of the TS, but tunneling also signi ficantly increases the rate over the base non-tunneling value at near-ambient temperatures. The larger the barrier, the greater the relative contribution of tunneling at lower temperatures, leading to a greater curvature and a larger required n. In deriving our rate constant fits, we found that only modest changes innwere required for each reaction partner (H, O, OH, F) and that assuming a roughly linear dependence on Eaat high temperatures gave good results. As will be illustrated and discussed in Sec. 8,w e estimate expanded uncertainty factors f(2σ) of about f/equals1.25 for rate constants (based on a derived empirical uncertainty of about 10% in the temperature coef ficient n) using the correlation between nandEa, which is less than our assigned uncertainty factors of f/equals1.4±0.2, demonstrating that the kinetic compensation effect is a good assumption.4.ΔEa∼ΔEa(ab initio ). The relative activation energies at high temperatures for the different reactions in each set of reactions (R–H+X→R + HX) should roughly scale with those from ab initio calculations. In this work, we employed the rate expressions fromMatsugi and Shiina 59who carried out high-level calculations for (X/equalsH, O, OH). We do not expect the computations to be quan- titatively exact but do consider the relative rate constants to be goodinitial guesses. We do not provide a graphical characterization of thiscorrelation but do brie fly discuss it in each section of homologous reactions. 5.E a(Tlow)∼Ea(Thigh). There is an exact relationship between Eaat high and low temperatures that can be analytically derived from an extended Arrhenius expression, Ea(Thigh)−Ea(Tlow)/equalsnR(Thigh−Tlow), where Ris the gas constant and nis the temperature coef ficient. Thus, the uncertainty in Eaat low temperatures relative to that at high temperatures should be on the order of the uncertainty in thetemperature coef ficient nor about 10% (see above). We illustrate and discuss this correlation in Sec. 8. We estimate expanded uncertainty factors f(2σ) of about f/equals1.25 for rate constants (based on a derived empirical uncertainty of about 10% in the temperature coef ficient n) using the correlation between nand E a, which is less than our assigned uncertainty factors of f/equals1.4±0.2. This demonstrates clearly that the kinetic compensation effect is a good assumption with regardtoE aat high and low temperatures. 6. log(k300)∼Ea. Since there are correlations (see above) between the normalized Afactors (per H atom) and the number of H atoms, between the temperature coef ficients n, and between the activation energies Eaat high and low temperatures, there will also be a de- pendent correlation between the rate constants at low temperatures(k 300) and the activation energies Eaat high temperatures. The log of the rate constant at room temperature (300 K) should vary roughlylinearly with the activation energy E a. For these reactions where an H atom is being abstracted, the rate constants at low temperatures aredominated by tunneling and the tunneling rate is, to first order, a direct function of the height of the barrier through which the H atomtunnels. It is also a function of the width and asymmetry in thebarrier, but since we are considering a homologous set of reactionshaving PESs that are functionally the same, to first order, these will also scale roughly linearly. This correlation is illustrated and dis-cussed in Sec. 8, although the uncertainty factor is on the order of f/equals2.0—this is re flected in our assigned uncertainties at low tem- peratures (provided in tables in Secs. 4–7). We note that because of high electronegativity of F, there will be some deviations from these linear correlations. We point out these in- stances in the discussion in Sec. 8. This appears to be a complicated model requiring computer optimization, but in practical terms, it is not, and we optimized the parameters manually without much dif ficulty. Not all of the 3 Arrhenius parameters ( A,n,E) for all of the 16 different reactions, RH + X→R+H X( R H /equals{CH 4,C H 3F, CH 2F2,C H F 3}; X/equals{H, O, OH, F}), need to be simultaneously optimized. First, only nand Eare inde- pendent parameters with Abeing a dependent parameter determined through best agreement (least-squares fit) with the experimental data. We note that Eis essentially a first-order correction ( “slope ”)t ot h er a t e expression, while nis a second-order correction ( “curvature ”). Second, one only needs to optimize the parameters within each homologous series (X /equals{H, O, OH, or F}). Finally, the rate parameters for the J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-6 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprbounding cases RH /equals{CH 4and CHF 3) are largely determined by experimental data that are for the most part available over wide temperature ranges, and thus, the parameters nand Efor the rate expressions involving the intermediate species CH 3Fa n dC H 2F2can be estimated fairly well through simple interpolation. The optimization procedure was quite standard and fairly simple. First, initial guesses for the rate constants at high temperatures (1200 –1800 K) were generated where the relative Afactors and relative activation energies Eawere based on those from the ab initio calculations of Matsugi and Shiina59benchmarked to AandEaof the bounding parameters (from fits to experimental data) for CH 4and CHF 3. Second, initial guesses for the rate constants at low temper- atures ( k300) were generated through interpolation of log( k300)v sEaat high temperatures from the bounding cases of CH 4and CHF 3and the initial estimate of Eaat high temperatures for CH 3Fo rC H 2F2. These estimated data at high and low temperatures plus the available ex- perimental data were then fit to an extended Arrhenius expression to determine an initial guess for the parameters A,n, and E. This can be termed a “tight ”optimization since the rate expression is forced to agree with the initial guesses including the parametric constraints from SARs. In contrast, a “loose ”optimization is where the fit does not consider the parametric constraints. The initial tight optimization was then iteratively made “looser ” to generate more relaxed fits by adjusting the rate constants at high temperatures and/or at low temperatures and/or by adjusting the independent parameters nand E. This process was then manually iterated a few times until a fully relaxed fit was obtained —that is, the data were fit to a least-squares regression to an extended Arrhenius expression while the deviations from the structure –activity correla- tions were also minimized. This is the same procedure as a formal (computer) optimization, which would, of course, find the “true” global minimum but would likely produce fits with uncertainties statistically similar to our recommended uncertainties. The figures and discussion in Sec. 8show that the different parameters related to the 6 constraints above are well-correlated —the uncertainties in the parameters result in uncertainties in the rate constants that are less than our assigned uncertainties. There are two caveats to the additional constraints. First, fluorine is highly electronegative and can change the PES in a non-linear fashion, especially at high fluorine substitution (i.e., CHF 3). Second, abstraction by OH has a hindered rotor in the TS. Consequently, the effective barrier to the hindered rotation will change signi ficantly with addition of highly electronegative fluorine atoms and will become both sterically hindered and strongly temperature-dependent. In- stances of these exceptions will be discussed in Sec. 8. In each section for the homologous reactions RH + X →R+H X (X/equalsH, O, OH, F), we provide tables and discussion regarding the goodness of these fits including statistically derived uncertainty factors f,Δn, andΔE. In Sec. 8, we also provide tables and discussion regarding the goodness of these fits, along with figures showing how well the parameters A, n, and Eand the rate constants at low tem- peratures ( k300) are correlated with the SAR constraints. 2.3. Network of related reactions As discussed above, this reference series of reactions can be used to aid in determination of the reactivity of more complex agents. There are many agents where rate constants have not been measured,have only been measured in a restricted temperature range, or have been measured just at low temperatures but not at high temperatures (or vice versa). Rate constants are often (and best) determined bymeasuring the rate of one reaction relative to another reaction withwell-established absolute rate constants. For some agents, reactivities(rate constants) have not been measured, and one can estimate fairlywell the reactivities by interpolating using structure –activity relations (and energy relationships) such as the relative number of hydrogen and/or fluorine atoms or the relative bond strengths (equivalently, relative heats of reactions). In addition, high-level ab initio quantum chemical calculations can be used as an aid in determining relativerate constants and their temperature dependencies. Although thequantum calculations may not be completely accurate, they do very well in predicting relative rates. For example, if TS calculations based on quantum chemical calculations predict that the magnitude of the relative rates of two reactions at high temperatures differs by about a factor of 6, then the actual (experimental) relative rates likely differ on the order of (4.5–8). Our recommended rate expressions show differences with theab initio calculated relative values on the order of f/equals1.3. Similarly, if quantum chemical calculations predict a relative barrier of about 1.25, the actual difference in barriers is likely on the order of (1.20 –1.30). In short, the quantum chemical calculations often have modest differences when compared to experimentally derived rate constants, but these differences are largely systematic and hence lead to fairly precise relative rates. Within a homologous series of reactions, the activation energies often correlate well with bond strengths (or equivalently, heats of reaction). This correlation is often termed Evans –Polanyi relation- ship. 55For example, in the reactions considered here, a change in bond strength of 10 kJ mol−1results in a change of about 3.6 kJ mol−1 (±6%) in the activation energy. See figures and discussion about this correlation in Sec. 8. Since all of these reactions involve H atom transfer, the rates of reaction at low temperatures are dominated by quantum chemicaltunneling. There is a strong correlation between tunneling rates andbarriers to reaction, and the tunneling rate is, to first order, a direct function of the height of the barrier through which the H atom tunnels (it is also a function of the width and asymmetry of the barrier). Consequently, both experimental data and quantum chemical cal-culations for other similar reactions can be used to assist in predictingrelative tunneling rates using this correlation. At low temperatures, the rates of these H abstraction reactions are dominated by tunneling. These tunneling rate constants can be predicted (scaled) using ab initio barriers and curvature in the energy potentials. Although the tunneling calculations may not be accurate, there will be systematic differences with the actual (experimental) rate constants. For example, if the effective barrier, or activation energy (E a), at high temperatures (e.g., 1500 –1800 K) between two ho- mologous reactions changes by about 10 kJ mol−1, then the effective activation energy at low temperatures (e.g., 300 K) might differ on the order of 8.5 kJ mol−1(±15%). The tunneling roughly scales with the barrier because the shape of the energy potential along the reaction coordinate changes only slightly. See figures and discussion about this correlation in Sec. 8. In short, the accuracy in the determination of absolute rate constants over a range of temperatures for all H abstraction reactions J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-7 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprinvolving a set of HFCs can be signi ficantly increased by utilizing the more precise relative rates in a network of self-consistent reactions using correlations based on SARs and assigning uncertainties based on bracketing of the relative rates. 2.4. Overview of classes of reactions The evaluations are divided into four sections, one for each reaction class, R –H+X→R + HX (X /equalsH, O, OH, F). These sections are then followed by a discussion section that provides an overview of this critical evaluation and then analyzes the trends in Afactors, energies EaorE(temperature dependencies), and tunneling with fluorine substitution, as well as the trends with regard to the attacking radical (H, O, OH, F). Within each section, an overview of the available data and evaluations is first provided for the homologous series with increasing fluorine substitution (CH 4,C H 3F, CH 2F2, CHF 3) and this includes a table of recommended rate expressions for the series. These overviews are then followed by subsections for each of the compounds containing tables of available data from the lit- erature (reviews, experimental, and theoretical), figures with the data and our fits, and a discussion of the evaluation for each individual reaction. In all cases, we have done our best to provide a good un- certainty estimate, 2 σ(type B, a level of con fidence of approximately 95%).60We note that uncertainties for the rate constants are often not provided in the literature for both experimental and (especially) theoretical values. In addition, in cases where uncertainties are provided, they are generally simply precision estimates and do not include systematic uncertainties, and the coverage (1 σor 2σ) is often not speci fied. We note that a column “Method ”is provided in the tables. This is a brief characterization of the methods used to determine the rateconstants. The acronyms and abbreviations used for “Methods ”are more fully speci fied in Sec. 10. We now provide a brief overview for each of the four classes of reactions R –H+X→R + HX (X /equalsH, O, OH, F). For all of these abstraction reactions, note that the available ab initio calculations, our (recommended) parameterization of the rate constants, and all available experimental data for abstraction reactions involving H, O, F, and OH suggest that the difference in barriers between abstraction reactions for CH 3F and CH 2F2are on the order of (2 –3) kJ mol−1and that the differences in rate constants at low and high temperatures are on the order of f/equals(1.5–2.5). Consequently, the relative rate of an abstraction reaction from one fluoromethane can be estimated rel- atively well from the same reaction involving another fluoromethane because of relatively small differences. R–H+H→R+H 2. For the reactions involving H abstraction by H atoms from the fluoromethanes, there are reliable data for CH 4and CHF 3at both high and low temperatures and usually a single measurement each at intermediate temperatures for CH 3F and CH 2F2. There are possible con flicting data (about a factor of f/equals1.8) at high temperatures for the reaction CHF 3+H→CF3+H 2.W e address this in our evaluation. In all cases, our recommended rate expressions agree with the available (reliable) experimental data (within uncertainties) at the midpoint of their limited temperature ranges. The temperature dependencies of the recommended rate expressions were determined by benchmarking the experimental data while considering trends in the series for the rate parameters A,n, and Eand relative values from ab initio calculations.R–H+O→R+O H . For the reactions involving H abstraction by O atoms from the fluoromethanes, there are reliable data for CH 4 over a wide range of temperatures and rate constants for CHF 3at intermediate and high temperatures, but none at room temperaturefor CHF 3. There are possible con flicting data at high temperatures for the CHF 3+ O reaction, which we address in our evaluation. For the reaction of O atoms with CH 3F, there is a single measurement at high temperatures. In that same work, the reaction of O atoms with CHF 3 was determined (as well as the reaction with CH 4)—providing an excellent set of relative values that can be used. There are no reliable measurements for the reaction of the O atom with CH 2F2at any temperature. We provide a recommended rate expression for CH 2F2 considering trends within the homologous series in the rate pa-rameters A,n, and Ealong with relative values from ab initio calculations. R–H+O H→R+H 2O.There are extensive measurements for the reaction of OH with the fluoromethanes (and other HFCs) near room temperature (above and below) because of the importance of these reactions in atmospheric chemistry that impact global warming. Reliable rate constants for the reaction of OH with both methane and CHF 3are available at high temperatures, but none for CH 3Fo rC H 2F2. The recommended rate constants at high temperatures for these two compounds were determined by considering trends within the ho- mologous series in the rate parameters A,n, and Ealong with relative values from ab initio calculations. R–H+F→R + HF. There are a few reliable rate constants that have been measured for the reaction of F atoms with methane and all of the fluoromethanes at low (near room temperature) and inter- mediate temperatures, but unfortunately no measurements at high temperatures where the rate constants are important in the com- bustion of refrigerants. Rate constants for the reactions involving CH 4 and CHF 3have been measured from about 200 K up to about (450–550) K, while rate constants involving CH 3F and CH 2F2have only been measured in the range (200 –300) K. Unfortunately, many of the researchers who carried out ab initio calculations for these reactions did not report rate constants at higher temperatures where the rate constants are not known —they instead focused on rate constants at lower temperatures where the rate constants are well-established . 2.4.1. Estimation and interpolation Where rate constants for R –H+X→R + HX in the series of reactions are unavailable, or only available over a limited temperature range, we loosely utilized the high-level ab initio quantum chemical calculations of Matsugi and Shiina59to guide the estimation and interpolation of rate constants in the different homologous series of reactions. (We did not repeat these calculations at other levels of theory because we believe their results are adequate as relative values.) Unfortunately, for some reactions, they did not provide their original calculations but instead provided “adjusted rate expressions ”to agree with low-temperature measurements where tunneling is important. Because the exact adjustments are uncertain, we used the trends to guide our fits, but did not take the relative rates to be exact. For abstractions involving OH, we utilized the relative rate constants from the ab initio quantum calculations of Schwartz et al.,61 rather than those of Matsugi and Shiina, because the latter reported “adjusted ”rate constants for these reactions instead of their original J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-8 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprvalues. Based on our inspection of these ab initio calculations and also considering them relative to the available experimental data, we believe that the relative rate constants from the ab initio calculations with regard to their magnitude ( Afactor), temperature dependence (E), and curvature including tunneling ( Tn) can be used fairly well to aid as interpolation and extrapolation tools when viewed in context of SARs, the relative number of hydrogen and fluorine atoms, and relative heats of reactions in these homologous series of reactions. Quantifying this, we believe that the uncertainty in this parametric estimation is on the order of the experimental uncertainties. The network of self-consistent rate constants based on all the experimental data in conjunction with parametric constraints leads to relatively accurate rate constants —certainly within less than a factor of f/equals2.0 where little data are available but with uncertainties on the order of a factor of f/equals(1.3–1.5) where more data are available. 3. Thermochemical Data InTable 2 , we provide standard enthalpies of formation for reactant and product species involved in H abstraction from the fluoromethanes b yt h er a d i c a l sH ,O ,F ,a n dO H .T h ee n t h a l p i e so ff o r m a t i o np r o v i d e da r e single high-quality values taken f rom original sources. We did not use evaluated sources that were weighted av erages of different values yielding yet another value that is largely statistically indistinguishable from the single high-quality value. In some ca ses, we provide values from evaluated sources where the original sources were simply corrected using updated supplementary values (see the supplementary material ). We also provide C–H bond dissociation ene rgies (BDEs) for the fluoromethanes. These values are used in tables and fig u r e si nt h i sp a p e r . 4. Fluoromethanes +H→Fluoromethyls +H2 In this section, we compile and evaluate rate constants for re- actions involving the abstraction of H from C –H bonds by H atoms from methane and the fluoromethanes. Data are included from bothexperimental determinations and quantum chemical calculations. Based on considering all of the data, we provide expanded uncertainty factors f(2σ) that are generally a factor of about f/equals(1.4–1.5) over the entire range. Table 3 lists our recommended rate expressions for abstraction of Hf r o m fluoromethanes by H atoms. Preferred values for each reaction are subsequently discussed in Sec. 4with each individual reaction. Two sets of rate expressions are provided in Table 3 .T h e first set uses the three-parameter extended Arrhenius format ( k/equalsATne−E/RT) to specify values at temperatures of (300 –1800) K. The second set uses the simpler two-parameter Arrhenius expression ( k/equalsAe−E/RT) to describe the rate constants over a narrower range of temperatures more relevant to combustion (1200 –1800 K). The latter expressions make it easier to inspect the parameters of related reactions for compatibility and are useful when adjusting rate parameters in models to optimize agreement with experimental observables such as ignition delays and burning velocities. Both the extended and simple rate expressions can be used at temperatures above 1800 K; however, validation data at such tem- peratures are limited and the reliability is not well-tested. We note that there exist in the literature many “recommended ”rate expressions (usually based on ab initio calculations) that give (300 –2500) K as the range. We think values at temperatures above 1800 K should be treated with caution. When placed in a standard Arrhenius format, the pre- exponential Afactors that describe H abstraction from the fluoro- methanes by H at high temperatures (1200 –1800 K) for CH 2F2,C H 3F, and CH 4, relative to that for CHF 3(1.0) are 1.92, 2.85, and 3.84, respectively, and are very similar to the reaction path degeneracies (the number of hydrogen atoms) of 2, 3, and 4. These ratios are about (10–15)% larger than the ratio of Afactors from the rate expressions in the quantum chemical work of Matsugi and Shiina,59which were 1.80, 2.51, and 2.99, respectively. The effective barriers Eaat high temperatures (1200 –1800 K) for the recommended rate expressions TABLE 2. Standard enthalpies of formation ΔfH°(298.15 K) and C –H BDEs used in this work. Units for enthalpies and uncertainties are kJ mol−1. Expanded absolute uncertainties U(2σ) as reported are provided. We also provide our estimates of the uncertainties (in parentheses) Species ΔfH°(298 K) U Reference BDE(C –H) CH 4 −74.55 0.15 04RUS/PIN62439.1 ±0.3 CH 3F −235.55 0.70 (1.2) 18GAN/KAL63423.2 ±1.4 CH 2F2 −451.66 0.68 (1.2) 18GAN/KAL63426.2 ±1.4 CHF 3 −697.45 0.65 (1.4) 18GAN/KAL63446.4 ±1.7 CH 3 146.55 0.25 09BOD/JOH64 93BLU/CHE65 CH 2F −30.39 0.50 (0.8) 18GAN/KAL63 CHF 2 −243.45 0.51 (0.8) 18GAN/KAL63 CF3 −469.06 0.52 (1.0) 18GAN/KAL63 H2 0.0 0.0 By de finition H2O −241.831 0.026 13RUS66 HF −272.72 0.05 06HU/HEP67 H 217.9979 <0.0001 04ZHA/CHE68 O 249.229 0.002 13RUS/FEL69 OH 37.51 0.03 13BOY/KOS70 F 79.46 0.05 (0.10) 05YAN/HAO71,72 J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-9 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprfor CH 4,C H 3F, CH 2F2, and CHF 3are 6.0%, 3.7%, 4.2%, and 4.1% lower, respectively, than those derived from the Matsugi and Shiina rate expressions. There is relatively good agreement between our recommendations and the computed values, in terms of both absolute Afactors and the relative scaling of Afactors and effective barriers E; this provides a degree of validation that the rate expressions are self- consistent and accurate within the indicated uncertainties. This will be discussed further later in this section. InTable 3 , we have assigned an expanded uncertainty factor f(2σ) to each of the rate constants at both high and low temperatures (the latter in parentheses). For the CH 4+ H and CHF 3+ H reactions, the uncertainties are based on scatter in the experimental mea- surements. We have, however, increased this uncertainty slightly for CHF 3+ H. This accommodates the relatively imprecise measurement by Takahashi et al.73that is at odds with the recommended rate expression; it likewise accounts for uncertainties in the equilibrium constants for CHF 3+H→CF3+H 2and CF 3+C F 3→C2F6that were used to derive the rate constants at low temperatures. The deviations between the recommended rate expressions and the experimental data for CH 3F + H and CH 2F2are about f/equals1.2 and f/equals1.45, re- spectively. However, we believe that higher uncertainties of f/equals1.5 and f/equals1.6, respectively, are appropriate, given uncertainties in the ex- perimental measurements and our estimation methods. Uncertainties for these reactions at low temperatures are assigned in accord with that estimated for CH 4+ H, with a higher uncertainty for CHF 3+H because of the limited measurements. 4.1. CH 4+H→CH3+H2 InTable 4 , we present rate constants complied from the liter- ature for CH 4+H→CH 3+H 2. This is one of the most studied reactions in chemical kinetics. Table 4 does not include all mea- surements but is rather a substantial and representative set of the data considered to be most reliable. The recommended rate expression is based on our fit to these data. The first set in this table is comprised of evaluated values, the second set is the data used in our fits, the third set is data excluded from our fits, and the fourth set is from ab initio quantum chemical calculations. Data were excluded from the fits if the magnitude of the rate constants differed substantially (well outside our recommended uncertainty), the temperature dependence wasinconsistent with the well-established temperature dependence by many other measurements (and calculations), and/or they were re- vised values from prior measurements (duplicates). We note that our fit is statistically equivalent to the 2001 fit by Sutherland et al. ,74an often-cited evaluation based on their measurements at high tem- peratures along with their analysis of measurements by others in- cluding at intermediate and low temperatures. We utilized our rate expression in this work instead because (1) we have included more data in our fit, particularly at low temperatures, and (2) our rate expression yields an effective rate expression at high temperatures (1200 –1800 K) that has an Afactor that is self-consistent with A factors we find for the fluoromethanes (see Table 3 and the quantum chemical calculations of Matsugi and Shiina59). Our recommended rate expression differs from that of Sutherland et al. in that it agrees slightly better with the high-temperature experimental data of both Sutherland et al. and Baeck et al.75,76and also with the low- temperature experimental data of Marquaire et al.77 Figure 2 presents, from Table 4 , the experimental data deemed reliable, the results of the ab initio calculations of Matsugi and Shiina, and the recommended fit. Note that the plotted experimental data points are smoothed values obtained from the reported rate ex- pressions rather than the primary measurements. We estimate the uncertainty in the rate constant to be a factor of about f/equals1.4 at high temperatures (1200 –1800) K and about f/equals1.6 at low temperatures (300–450) K based on the scatter in the experimental data and the different statistically similar fits that could be obtained. The rec- ommended uncertainty factor includes consideration —with appro- priate weighting —of rate constants excluded from the fit. Those data excluded from the fit deviated from the recommended value by generally factors of about 2 –3 (or more) and most had temperature dependencies that were signi ficantly different than derived from the recommended values —on the order of ΔE/equals(10–20) kJ mol−1. The residuals between the fitted expression and the data excluded from the fit are provided in Fig. 3 [note in ln( k) units]. 4.2. CH 3F+H→CH2F+H2and CH 2F2+H→CHF 2+H2 Rate expressions for abstraction of H from CH 3F and CH 2F2by H atoms that we compiled from the literature are provided in Tables 5 and 6. There is only one reliable measurement for the H atomTABLE 3. Recommended rate expressions for H abstraction from the fluoromethanes by the H atom and enthalpies of reaction. k(T)/equalsATnexp(−E/RT).k(T), temperature-dependent rate constant. A, pre-exponential factor. n, temperature coef ficient. E, energy coef ficient. f(2σ), expanded uncertainty factors. ΔrH298, standard enthalpy of reaction at 298 K. Units: k (cm3mol−1s−1),E(kJ mol−1),T(K). Uncertainty factors fare at high(low) temperatures. Uncertainty factors in parentheses “(f)”are at low temperatures (300 –500) K. A/ACHF3 is the ratio of the pre-exponential to that for CHF 3. Reaction An E f T log10(k300) ΔrH298 CH 4+H→CH 3+H 2 9.0131052.41±0.31 38.21 ±2.13 1.4(1.6) 300 –1950 5.28 3.1 CH 3F+H→CH 2F+H 2 2.1331052.56±0.42 31.98 ±2.34 1.5(1.8) 300 –1800 6.11 −12.8 CH 2F2+H→CHF 2+H 2 8.2931042.63±0.45 30.65 ±2.41 1.6(1.8) 300 –1800 6.09 −9.8 CHF 3+H→CF3+H 2 2.8931042.95±0.28 38.61 ±2.28 1.1(2.0) 300 –1800 4.05 10.4 A/ACHF3 CH 4+H→CH 3+H 2 4.993101468.10 1.4 1200 –1800 3.84 CH 3F+H→CH 2F+H 2 3.713101463.06 1.5 1200 –1800 2.85 CH 2F2+H→CHF 2+H 2 2.493101462.53 1.6 1200 –1800 1.92 CHF 3+H→CF3+H 2 1.303101472.45 1.5 1200 –1800 1.00 J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-10 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprabstraction reaction for CH 3F, but none for CH 2F2. The rate mea- surement for CH 3F by Westenberg and DeHaas97is at intermediate temperatures (750 –900) K. Note that in that work, the decay of CH 3F was measured (products were not identi fied), and the authors in- correctly assumed that the F atom was being abstracted by analogy with the other halogenated methanes CH 3Cl and CH 3Br where the halogen (Cl or Br) is abstracted.97,98C–F bonds are much stronger (about 480 kJ mol−1) than C –Cl (about 350 kJ mol−1) and C –Br (about 300 kJ mol−1) bonds; consequently, for CH 3F, it is the H atom rather than the F atom that is being abstracted. For more discussion of abstraction of halogens (C –X) by the H atom, see the work by Manion and Tsang.99 There is also a reasonable rate constant reported in the literature for CH 2F2+H→CHF 2+H 2by Pritchard and Perona,100who measured the rate of the reaction in the reverse direction CHF 2+H 2→CH 2F2+ H. This measurement, however, was determined relative to the rate of combination CHF 2+ CHF 2, which is pressure- and temperature-dependent, and deriving the rate constant involved modeling of a set of reactions and some assumptions. We display the rate constants derived from these measurements but did not use them in determining our recommended rate expression (although they roughly agree). Note that we computed the equilibrium constant (from available thermochemical data) in the temperature range of this measurement (500 –635 K), Keq/equals42.2 * exp( −46.9/ T), and utilized this to determine the “forward ”rates from the measured “reverse ” rates. The equilibrium constant was computed using Burcat ’s ther- mochemical polynomials101with an adjustment to make them compatible with the enthalpies of formation given in Table 2 . There are also several unreliable measurements for these two reactions by Parsamyan and Azatyan,102Parsamyan andTABLE 4. CH 4+H→CH 3+H 2rate expressions. k(T)/equalsATnexp(−E/RT).k(T), temperature-dependent rate constant. A, pre-exponential factor. n, temperature coef ficient. E, energy coefficient. f(2σ), expanded uncertainty factors. Units: k(cm3mol−1s−1),E(kJ mol−1),T(K). Uncertainty factors fare at high temperatures. Uncertainty factors fin parentheses “(f)” are our estimates at low temperatures (300 –500) K. Uncertainty factors in square brackets “[f]”are the deviations from our recommended values. See Sec. 10for de finitions of acronyms and abbreviations in the Method column An E f T Method Reference Evaluations 9.0131052.41 38.21 1.4(1.6) 300 –1950 Recommended This work 4.383101467.37 1.4 1200 –1800 Recommended, high T fit This work 4.0831033.16 36.63 [1.5] 348 –1950 Review 01SUT/SU74 3.8631062.11 32.43 [2(5)] 400 –1800 Review 91RAB/SUT78 6.313101352.71 2.0[3] 350 –2000 Review 67DIX/WIL79 Experimental (preferred data) 1.773101457.65 1.2 913 –1697 Fwd/rev rxn, shock, photol, H abs 01SUT/SU74 1.543101457.15 1.35 748 –1054 Dischg flow, reson fluor 01BRY/SLA80 3.103101463.19 [1.2] 1310 –1820 Rev rxn, shock, CH3 UV abs 95BAE/SHI175 4.403101465.19 [1.2] 1250 –1950 Rev rxn, shock, CH3 UV abs 95BAE/SHI276 3.213101133.92 [1.3] 348 –421 Dischg flow, ESR 94MAR/DAS77 1.073101453.55 1.5 897 –1730 Flash photol, shock, H abs 91RAB/SUT78 1.823101455.12 1.8 640 –818 Dischg flow, GC 79SEP/MAR81 3.20310121.2 1600 Flame, MS 73PEE/MAH82 6.253101348.56 1.3 474 –732 Dischg flow, ESR 70KUR/HOL83 4.00310101.5 900 Flame, MS 67DIX/WIL79 3.313101463.18 [1.4] 673 –753 Ignition LIB 62GOR/NAL84 Experimental (excluded data) 1.0231071.2[5] 298 Dischg fast flow, ESR 86JON/MA85 7.233101463.0 [2.5] 1700 –2300 Model, flow, H reson abs 75ROT/JUS86 1.243101449.80 [3] 298 –753 Rev rxn, static, thermal, GC 71BAK/BAL87 6.93101349.37 1.09[1.7] 426 –747 Flow, H dischg, H ESR 69KUR/TIM88 1.213101344.32 1.7[2.5] 843 –933 Thermal, ESR 67AZA89 3.443101335.59 [70] 298 –398 Ultrasonic, GC 66LAW/FIR90 6.033101130.93 2.4 500 –787 Fast flow, dischg, GC 64JAM/BRO91 2.003101448.14 1.7[2.5] 1200 –1800 Model, flame 61FEN/JON92 1.003101018.62 [3] 372 –436 Thermal, H heat combin 54BER/LER93 Theoretical 4.5031052.57 41.65 [1.2(1.8)] 300 –2000 CBS-QB3 Eckart 14MAT/SHI59 2.823101472.84 [1.2] 1200 –1800 CBS-QB3 Eckart, this work, High T fit 14MAT/SHI59 2.5931042.93 38.30 [1.2(1.8)] 300 –2500 CC/pVQZ Eckart 97KON/KRA94 1.0731013.78 30.76 [2.0(2.0)] 298 –2500 G2MP2 97BER/EHL95 4.25310−86.51 15.47 [1.5(4.0)] 200 –1000 MP4 CVT/SCT 99MAI/DUN96 J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-11 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprNalbandyan,103Hart and Grunfelder,104and Aders et al.105The re- ported rates are an order of magnitude (or more) different from that which is reasonable, and in some cases, in the papers, there are in- consistencies between the rate constants given in the tables and the reported rate expressions. In the evaluation by Baulch et al. ,106they corrected the determination of the rate constant for CH 3F+Hb y Parsamyan and Azatyan using correct values for relative reactions. The revised rate constant is in good agreement with that from Westenberg and DeHaas,97but we do not include it in the fit because of the many uncertainties in how it was derived. There are also several rate expressions for these reactions based onab initio quantum calculations, and these are provided in Tables 4 –7. The rate constants from the work of Matsugi and Shiina,59however, are the only ones that are in relatively good agreement with the available experimental data. InFig. 4 ,w ep r o v i d et h ee x p e r i m e n t a lr a t ec o n s t a n t sf o r CH 3F+Ha n dC H 2F2+ H that we think are relatively reliable, along with our recommended rate expressions and those from the quantum chemical calculations of Matsugi and Shiina. Note that in thisfigure, the curves for CH 2F2have been shifted up by a factor of 10 (1 log 10unit) for clarity because they overlapped with the curves for CH 3F. The actual rate constants for CH 2F2+ H are about 92% and 59% of the rate constants for CH 3F at 300 K and 1800 K, respectively. Our recommended rate expressions for CH 3F + H and CH 2F2+H given in Tables 5 and6and displayed in Fig. 4 were based on trial-and- error iterative parametric fits. The constraints were these: (1) The rate expression for CH 3F+H→CH 2F+H 2agrees with the midpoint of the single measurement for this reaction; (2) when combined with the data for CH 4and CHF 3,t h e Afactors and activation energies Eaat high temperatures (1200 –1800) K should vary smoothly across the series CH 4,C H 3F, CH 2F2,C H F 3; and (3) the effective Aand Eaat high temperatures should roughly scale with those from the quantum chemical calculations of Matsugi and Shiina.59In short, we attempted to make the rate expressions self-consistent with other available data. Determined first in our procedure were the rate expressions for CH 4and CHF 3because these comprise the upper and lower rate bounds and both reactions have rate measurements extending from atmospheric to combustion temperatures; these bounding values were then used to help establish intermediate rates for the less-well-studied compounds CH 3F and CH 2F2. 4.3. CHF 3+H→CF3+H2 4.3.1. Overview We have compiled all available rate constants for CHF 3+H →CF3+H 2from the literature. This includes both experimental and computational work and also rate constants for the reverse reaction CF 3+H 2→CHF 3+ H. The forward rates in these cases were derived using the equilibrium constant for the reaction (see discussion below regarding computation of the equilibrium con- stant). The forward rates are given in Table 7 , and the reverse rates are given in Table 8 . Our recommended rate expression is based on an unweighted fit to rate constants from four measurements at high temperatures (1000 –1700 K) and one measurement at low temperatures (375 –450 K). (See Table 7 and discussion below for details.) Other values were excluded in the fit because of their large deviation from others or lack reliability. The recommended rate expression along with almost all of the experimental data (included or excluded from the fit) is given in Fig. 5 . Based on all of the considerations, we assign an expanded un- certainty factor f(2σ)o f f/equals1.1 for rate constants for our recom- mended rate expression at high temperatures and f/equals2 at low temperatures. In this section, we first provide the data for CHF 3+H→CF3+H 2 from the literature and the reverse reaction CF 3+H 2→CHF 3+Hi n Tables 7 and8, respectively, and rate constants in Fig. 5 .T h i si sf o l l o w e d by discussions of the data at both high temperatures and low tem- peratures. We then compare rate constants for CHF 3+H→CF3+H 2 FIG. 2. CH 4+H→CH 3+H 2experimental data and fitted rate expression. Data: Evaluation 01SUT/SU,74Quantum 14MAT/SHI,59other experimental data as given inTable 4 .k(cm3mol−1s−1). Note that other quantum chemical data rate expressions are not provided in this figure. Note that data points are not individual measurements but rather smoothed values obtained from the reported rateexpression. Legend: This work (rec) recommendation (red line), 14MAT/SHI59 (dotted line), 01SUT/SU74(rec, their recommended expression, dashed line), 01SUT/SU74(experimental, red dots), 01BRY/SLA80(yellow squares), 95BAE/ SHI175,76(green dots), 94MAR/DAS77(blue dots), 91RAB/SUT78(brown dashes), 79SEP/MAR81(black dashes), 73PEE/MAH82(large brown dot), 70KUR/HOL83 (blue circles), 67DIX/WIL79(large green dot), and 62GOR/NAL84(blue squares). FIG. 3. CH 4+H→CH3+H 2residuals for rate data excluded from fit. Note that not all of the excluded data are shown here because they either deviated much more than shown here or were revised measurements (duplicates). J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-12 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprcomputed using quantum chemical methods —these data are provided inTable 7 and in Fig. 6 . We also provide here a discussion of systematic trends in rate constants for the entire homologous series fluoromethanes + H →fluoromethyls + H 2. It is shown that there appears to be strong relationships between the rate constant at room temperature ( k300)a n d the activation energy Eaat high temperatures (1200 –1800 K) and between the rate constant at high temperatures ( k1500) and C –H BDEs. We present these empirical correlations here because data for thepresent reaction CHF 3+H→CF3+H 2are the most reliable, and consequently, these systematic trends can be used for other fluoro- methanes + H reactions employing CHF 3+H→CF3+H 2as the benchmark reaction. Finally, at the end of this section, we also provide ancillary data in order to derive some of the rate expressions for CHF 3+H→CF3+H 2 from the reverse reaction CF 3+H 2→CHF 3+H . Table 9 providesequilibrium constants for CHF 3+H→CF3+H 2, and Table 10 provides rate constants for the reference reaction CF 3+C F 3→C2F6. 4.3.2. Rate constants at high temperatures The three measurements included in the fit at high temperatures consist of two sets measured in the forward direction by Hrani- savljevic and Michael107and Richter et al.111,112and one set measured in the reverse direction by Hranisavljevic and Michael107and Berces et al.117Although the rate constants from Richter et al. agree well with the other three sets of data, they are from flame measurements (and estimated flame temperatures) and are based on a model that required rate constants for abstraction of H by O atoms and OH radicals, as well as other reactions. The rate constants reported by Berces et al.117 agree similarly well with the other rate expressions but were also derived using a complex model. Consequently, there is much more uncertainty in these data and the apparent good agreement may beTABLE 6. CH2F2+H→CHF 2+H 2rate expressions. k(T)/equalsATnexp(−E/RT).k(T), temperature-dependent rate constant. A, pre-exponential factor. n, temperature coef ficient. E, energy coef ficient. f(2σ), expanded uncertainty factors. Units: k(cm3mol−1s−1),E(kJ mol−1),T(K). The uncertainty fin parentheses is at low temperatures (300 –500) K An E f T Method Reference Evaluations 8.2931042.63 30.65 1.6(1.8) 300 –1800 Recommended This work 2.493101462.54 1.6 1200 –1800 Recommended, high T fit This work Experimental (preferred data) 2.723101455.26 1.3[1.5] 500 –635 This work, from rev rxn 69PRI/PER100 Theoretical 3.1331042.83 30.93 [1.7(1.7)] 200 –1800 CBS-QB3 Eckart 14MAT/SHI59 5.92310−65.74 7.59 [2.5(5)] 200 –1000 MP4 CVT/SCT 99MAI/DUN96 4.52310−24.39 18.44 1.5(2.0)] 300 –2500 G2MP2 97BER/EHL95TABLE 5. CH3F+H→CH2F+H 2rate expressions. k(T)/equalsATnexp(−E/RT).k(T), temperature-dependent rate constant. A, pre-exponential factor. n, temperature coef ficient. E, energy coef ficient. f(2σ), expanded uncertainty factors. Units: k(cm3mol−1s−1),E(kJ mol−1),T(K). The uncertainty factor is at high temperatures. The uncertainty fin parentheses is at low temperatures (300 –500) K An E f T Method Reference Evaluations 2.1331052.56 31.98 1.5(1.8) 300 –1800 Recommended This work 3.713101463.06 1.5 1200 –1800 Recommended, high T fit This work 2.33101341.2 1.5 600 –1000 Review 81BAU/DUX106 Experimental (preferred data) 1.803101339.33 1.2 750 –900 Dischg flow, ESR 75WES/DEH97 Experimental (excluded data) 6.303101321.78 1.1[10] 298 –652 Dischg flow, ESR 75ADE/PAN105 6.303101334.27 1.7[5] 870 –1088 Model, rel rate CH3Br, flame 74HAR/GRU104 1.38310126.80 1.3[5] 858 –933 Static 67PAR/AZA102 Theoretical 7.8331042.76 32.00 [1.6(1.2)] 200 –1800 CBS-QB3 Eckert 14MAT/SHI59 2.13310−65.94 8.44 [2.5(5)] 200 –1000 MP4 CVT/SCT 99MAI/DUN96 5.57310−24.40 18.95 [1.5(2.0)] 300 –2500 G2MP2 97BER/EHL95 J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-13 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprcoincidental —as a result, we excluded these two sets of data from our fit to the rate constants. There are four additional rate constants at high temperatures reported by Skinner and Ringrose,120,121Hidaka et al. ,115Fargash et al. ,119and Westbrook116that were excluded from the fits because they differed by about an order of magnitude and the determinations had some signi ficant uncertainties due to the lack of reported details and the use of complex model reaction systems to derive the reported rate expressions. There is an additional 1997 measurement of this reaction (in the forward direction) at high temperatures by Takahashi et al.73that we did not weigh as strongly as the 1998 measurements of Hranisavljevicand Michael 107even though we generally find both groups to provide reliable data and the methods used were similar. In agreement with Hranisavljevic and Michael, Takahashi et al. found much smaller rate constants than most early investigators (see Fig. 5 ); nonetheless, their values are about 1.8 times larger than those of Hranisavljevic and Michael and lie outside of the experimental uncertainties of each. In addition, the barrier of Takahashi et al. is about 20 kJ mol−1lower than that of Hranisavljevic and Michael, a discrepancy larger than ex- pected. In considering the work of Takahashi et al. , we note several issues: the H atom absorption coef ficients did not follow a linear Beer–Lambert law; the H atom absorbance had to be corrected by subtracting absorbance by CHF 3; the relatively high concentration of CHF 3could promote secondary chemistry; the decay traces were noisy in comparison to those of Hranisavljevic and Michael; the rate constants were derived using a model that required rate constants for the decomposition of the precursor C 2H5I (used to produce the H atoms) as well as rate constants for the reaction of H atoms with the decomposition product CHF 2. We also find that the data of Takahashi et al. are inconsistent with the trends in Arrhenius parameters A,n, andEobserved for the other reactions in this series and, further, with the scaled rate parameters from the ab initio calculations of Matsugi and Shiina. The effective Afactor and activation energy Eaat high temperatures (1200 –1800 K) were higher than what one might expect, and the temperature coef ficient ( Tn) was much lower ( flatter curve) than one would expect, n/equals(2.1–2.2) vs n/equals(2.5–3.0). For all thesereasons, we favor the results of Hranisavljevic and Michael and did not directly use the data of Takahashi et al. in deriving our fit. In the work of Hranisavljevic and Michael, the H atom decay profiles showed precise single exponential decays with little noise (2%–5% uncertainty). The derived rate constants, however, showed significantly more variability —on the order of (20 –30)% ( k/equals1) likely due to uncertainties in determining the temperatures and/or con- centrations. Inspection of a scatter plot of the rate data relative to a bestfit rate expression (for both forward and reverse reactions) seems to show that the deviations at the higher pressure measurements (∼16 bar) were signi ficantly larger and a signi ficant proportion of data points were outside 3 sigma deviations of the mean. Eliminating just 9 outlier points of the 54 total data points signi ficantly reduces the uncertainty ( k/equals1) from about (25 –30)% to about (13 –18)%. Note that Hranisavljevic and Michael measured the rate con- stants for this reaction in both the forward and reverse directions. As with the other “reverse ”rate constants, we employed the equilibrium constant to derive forward rates. The good agreement between the rate constants measured by Hranisavljevic and Michael in both di- rections is another con firmation of the validity of their data since these are essentially independent measurements. We compared the magnitude of the forward and reverse rate constants in the tem- perature range of 1100 –1600 K and found that the equilibrium constant derived from the relative rates was within about a factor of f/equals1.35 of the more accurate equilibrium constant computed from thermodynamic properties —a very good agreement for thermody- namic data derived from kinetics measurements. When we examine the self-consistency of the recommended rate expression, we find that it is consistent with the effective rate ex- pression at high temperatures (1200 –1800 K) for the root reaction in this system CH 4+H→CH 3+H 2, the effective Afactor being about 3.4 times less (the ratio of hydrogens is 4:1), and the effective barrier Ea is 5.1 kJ mol−1higher, consistent with the difference of the BDEs of 5.3 kJ mol−1. This fit is also consistent with slightly scaled rate pa- rameters from the ab initio work of Matsugi and Shiina.59 Based on all of the considerations discussed, we assign an ex- panded uncertainty factor f(2σ)o ff/equals1.1 for rate constants for our recommended rate expression at high temperatures (1500 –1800 K)—there are five measurements that agree within this uncertainty (although two may be coincidental since they were derived using complex mechanisms). 4.3.3. Rate constants at low temperatures As indicated in Tables 7 and8, rate constants for CHF 3+H ⇌CF3+H 2at lower temperatures have been measured in both the forward and reverse directions by several groups. However, some of these reported values are based on reworking earlier values or are based on estimates using BDEs. After consideration, we think the rate constants from the work of Ayscough et al.113and Kibby et al.118are most reliable. These two determinations measured the rate constant of CF 3with H 2relative to the rate of recombination of CF 3to form C2F6. To convert these measurements to rate constants for CHF 3+H →CF3+H 2, we must know the equilibrium constant, Keq/equalskfwd/krev, and the rate constant for the reference reaction CF 3+C F 3→C2F6. There are two main issues to address here in order to derive rate constants for the reaction in the forward direction. First, the equi- librium constant must be calculated: Keq/equalskfwd/krev. We have done FIG. 4. CH 3F+H→CH 2F+H 2and CH 2F2+H→CHF 2+H 2experimental data and rate expressions. The CH 2F2curves and points have been displaced upward by a factor of 10 for clarity. Rate constants k(cm3mol−1s−1). Note that data points shown simply represent the range of values and are not actual measured datapoints. Legend: This work recommended (red lines), 14MAT/SHI59(quantum, dotted lines), 69PRI/PER100(blue squares), and 75WES/DEH97(green circles). J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-14 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprthis, and the equilibrium constants for CHF 3+H→CF3+H 2and the reference reaction CF 3+C F 3→C2F6are derived, shown, and dis- cussed later in this section in Tables 9 and10, respectively. The biggest uncertainty here is the heat of reaction derived from the enthalpies of formation of CHF 3and CF 3. The uncertainty in the heat of reaction (at 298 K) is about 2.6 kJ mol−1(seeTable 2 )—this translates into an uncertainty of about a factor of 2 at low temperatures. Second, Ayscough et al. and Kibby et al. measured a relative rate constant and both workers employed a fixed equilibrium constant Keq(400 K) for the reference reaction CF 3+C F 3→C2F6as determined by Ayscough et al. and did not consider the possible pressure dependence of this unimolecular reaction. Later determinations129–131showed that the value of Ayscough et al. was low by about a factor of 2.8. In addition, the theoretical work of Cobos et al.131showed that there was a small tem- perature dependence —it varied by about a factor of 1.15 over thetemperature range of the work by Ayscough et al. (375–447 K) and about a factor of 2.1 over the temperature range of Kibby et al. (333 K –870 K). Fortunately, Cobos et al. showed that there was no pressure de- pendence under the cond itions used by Ayscough et al. Rate constants for the recombination of CF 3(CF 3+C F 3→C2F6) are given in Table 10 . At low temperatures, we utilized the corrected rate constants from Ayscough et al. in our fits. The reverse reaction was determined using the updated rate expression for the relative reference reaction of recombination of CF 3. The forward rate constants were then derived using the equilibrium constant for the reaction CHF 3+H→CF3+H 2 (seeTable 10 and the discussion below). Although the rate constants from Kibby et al. are similar, we did not use them in the fits for several reasons: (1) signi ficant uncertainties in both the equilibrium con- stants and the reference reaction; (2) the temperature dependence of the data being systematically weaker than that for Ayscough et al. andTABLE 7. CHF 3+H→CF3+H 2rate expressions. k(T)/equalsATnexp(−E/RT).k(T), temperature-dependent rate constant. A, pre-exponential factor. n, temperature coef ficient. E, energy coef ficient. f(2σ), expanded uncertainty factors. Units: A(cm3mol−1s−1),E(kJ mol−1),T(K). The uncertainty fin parentheses is at low temperatures (300 –500) K. See Fig. 5 showing an Arrhenius plot of the experimental rate expressions provided in this table, along with the recommended rate expression from a fit to the data An E f T Method Reference Evaluations 2.8931032.95 38.61 1.1(2) 300 –1800 Recommended This work 1.303101472.45 1.1 1200 –1800 Recommended, high T fit This work 2.1431013.62 37.77 [1.1(1.3)] 300 –1673 Rev rxn, Fit w/ k300/equals2.0310498HRA/MIC107 3.203101246.89 [2] 350 –600 Review 67AMP/WHI108 2.893101338.07 [5] 219 –447 Review 64PRI/FOO109 1.053101316.5 [5] 300 –700 Review 72KON110 3.393101430.4 [5] 336 –1300 Review 72KON110 Experimental (preferred data) 3.693101361.22 [1.2] 1111 –1550 Shock, H reson abs 98HRA/MIC107 1.783101478.00 [1.1] 1168 –1673 Rev rxn, shock, H reson abs 98HRA/MIC107 1.163101473.16 [1.1] 1050 –1350 Model, flame, MS 94RIC/VAN111,112 8.933101252.47 2.3[1.1] 375 –447 Static, this work rev rxn 55AYS/POL113 Experimental (excluded data) 9.543101364.60 1.36[2] 1100 –1350 Model, shock, H reson abs 97TAK/YAM73 3.763101354.99 1.26[2] 1100 –1350 Model, shock, H reson abs 96YAM/TAK114 2.753101458.05 [2] 1000 –1600 Model, shock, IR, this work rev rxn 93HID/NAK115 5.013101220.92 [5] 1400 –1800 Model, flame, MS 83WES116 7.083101247.84 1.26[1.1] 1029 –1132 Shock, this work rev rxn 72BER/MAR117 1.223101353.68 [3] 333 –550 This work rev rxn 68KIB/WES118 4.203101461.69 [10] 738 –854 This work rev rxn 68FAR/MOI119 5.003101220.95 [20] 970 –1300 Model, shock, IR 65SKI/RIN120,121 n/a Rev rxn, rel rate 56PRI/PRI122 Theoretical 3.6731042.74 42.32 [2.0(1.2)] 200 –1800 CBS-QB3 Eckart 14MAT/SHI59 2.823101475.58 2.0 1200 –1800 This work, fit high T 14MAT/SHI59 1.7331052.5 48.30 [1.3(5)] 200 –2000 CCSD/pVTZ, this work, extracted pts from figures, fit 200 –2000 K13SHA/CLA123 1.18310140 72.40 [1.3] 1000 –1600 CCSD/pVTZ, this work extracted pts from figures, fit 1000 –1600 K13SHA/CLA123 8.3731062.08 53.20 [1.5(5)] 300 –2000 G3B3, TST Wigner 07ZHA/LIN124 7.8331062.05 51.46 [1.3(5)] 300 –2000 CCSD/6-311G 04LOU/GON125 6.80310−65.26 21.32 [1.1(1.3)] 300 –1700 G2, adjusted E, Eckart 98HRA/MIC107 9.0031033.00 38.91 [4] 300 –2000 BAC-MP4 97BUR/ZAC126 8.6731013.38 37.53 [1.3(2)] 300 –2000 CH 97BER/EHL95 J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-15 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprour recommended expression; and (3) other possible systematic errors that are introduced because of the much wider temperature range in their measurements. Based on the various considerations discussed above, we assign expanded uncertainty factors f(2σ)o ff/equals2 for rate constants for our recommended rate expression at low temperatures (300 –500 K) —this encompasses the difference in the measurements of Ayscouigh et al. and Kibby et al.4.3.4. Quantum chemical calculations for CHF 3+H InFig. 6 , we present rate expressions from quantum chemical calculations for the reaction CHF 3+H→CF3+H 2. There are large differences between the different calculations that used different quantum chemical methods. These calculations also used different methods for predicting the dependence of the rate constant (increase) on tunneling at low temperatures. At high temperatures (1200 –1800 K), the spread in rate constants from the different quantum chemical calculations is about a factor of 6, while at low temperatures (300 –400 K), the spread is very large —about a factor of 60. Compared to the rate expression recommended in this work, the effective rate expressions at high temperatures from the qua ntum chemical calculations have pre-exponential Afactors mostly about (1.3 –1.9) times higher,TABLE 8. CF3+H 2→CHF 3+ H rate expressions (reverse direction to CHF 3+H→CF3+H 2).k(T)/equalsATnexp(−E/RT).k(T), temperature-dependent rate constant. A, pre- exponential factor. n, temperature coef ficient. E, energy coef ficient. f(2σ), expanded uncertainty factors. Units: k(cm3mol−1s−1),E(kJ mol−1),T(K). The uncertainty fin parentheses is at low temperatures (300 –500) K An E f T Method Reference Evaluations 7.49310−13.70 27.29 [1.1(1.3)] 300 –1673 Fit w/ k300/equals2.0310498HRA/MIC107 Experimental (preferred data) 1.543101371.08 [1.3] 1168 –1673 Shock, H reson abs 98HRA/MIC107 1.133101368.05 [1.3] 1111 –1673 Shock, fit to k1, k-1 98HRA/MIC107 7.193101139.74 2.3 375 –447 Photol, UV abs 55AYS/POL113 Experimental (excluded data) 2.23101350.21 [3.0] 1000 –1600 Model, shock, IR 93HID/NAK115 5.013101138.91 [2.0] 1029 –1132 Model, static, GC 72BER/MAR117 4.903101115.82 [2.5] 1400 –1800 Model, flame 83WES116 8.913101139.74 [2.5] 350 –600 Review 78ART/BEL127 2.453101131.68 [3.5] 350 –600 Review 75ART/DON128 9.333101140.84 [3.5] 333 –870 Rel rate, photol, MS 68KIB/WES118 2.523101351.47 [3.5] 738 –854 photol, IR abs 68FAR/MOI119 Theoretical 1.26310−44.88 10.01 [2.5(6)] 200 –1000 MP4/pVTZ CVT/SCT 99MAI/DUN96 1.263101125.86 [3(6)] 200 –1000 MP4/pVTZ CVT/SCT 99MAI/DUN96 6.80310−65.26 21.32 [1.1(1.3)] 300 –1673 Quant, G2, Eckart, adjusted E 98HRA/MIC107 2.543101131.68 [4] 350 –600 Theory BEBO 75ART/DON128 FIG. 5. Rate constants for CHF 3+H→CF3+H 2. Open circles and open squares are high-temperature and low-temperature values, respectively, used in fit. All other rate constants were excluded (see Table 7 and text for details). Rate constants k (cm3mol−1s−1). Note that data points shown simply represent the range of values and are not actual measured data points. Legend: Recommended rates (red line);68FAR/MOI119(green triangles); 93HID/NAK115(blue squares); 65SKI/RIN120,121 (black asterisks); 83WES116(blue dots); 97TAK/YAM73(red dots); and 98HRA/ MIC107and 94RIC/VAN111,112(blue circles). FIG. 6. CHF 3+H→CF3+H 2. Rates from quantum chemical calculations. Rate constants k(cm3mol−1s−1). Deviations are on the order of factors of f/equals2.5 and f/equals7.5 at high and low temperatures, respectively. 97BUR/ZAC,12697BER/EHL,95 04LOU/GON,12507ZHA/LIN,12413SHA/CLA,123and 14MAT/SHI.59 J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-16 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprcommensurate with temp erature dependencies Ethat are roughly (2.5–3.2) kJ mol−1higher (about 4%). The rate expression from the work of Matsugi and Shiina59has the best agreement (within about 12%) with the experimental rate constants at low temperatures (not shown in Fig. 6 for clarity). This agreement, however, is because they adjusted the overall reaction barrier to match the low-temperature rate constants. This adjustment came “at a price ”—their rate expression differs from our recom- mended value (based on high-temperature measurements) by a factor of 2. In the work of Matsugi and Shiina, they employed the CBS-QB3 quantum chemical method132using optimized structures with a high- quality DFT (density functional theory) method ωB97X-D133and corrected for tunneling using asymmetric Eckart potentials.134They computed rate expressions for the abstraction of H from methane and thefluoromethanes (CH 4,C H 3F, CH 2F2, and CHF 3) not just by H but also by the radicals O atom and OH. There is relatively good agreement between these computed rate expressions and the limitedexperimental data. Based on our analysis, however, there are modest systematic differences between the computed and experimentally derived rate constants. This is “good news ”in the sense that the modest systematic differences enable using the relative values of thecomputed rates to help evaluate and predict rate expressions where experimental data are limited, not available, or in disagreement. For example, the effective pre-exponential Afactors that describe the computed rate expressions for the fluoromethanes + H reactions at high temperatures (1200 –1800 K) for CH 2F2,C H 3F, and CH 4relative to that for CHF 3are 1.8, 2.5, and 3.0, respectively, similar to (75% –90%) of the reaction path degeneracy (i.e., the number of hydrogen atoms) of 2, 3, and 4. This can provide a framework for generating self-consistent rate expressions among the fluoro- methanes. The “bad news ”is that any calculation that adjusts the overall reaction barrier to “hit”low-temperature data provides a bias and lowers the accuracy of the rate constants at high temperatures. The proper procedure would be to adjust the parameters for the tunneling potential parameters. 4.3.5. Discussion of systematic trends influoromethanes +H rate expressions The tunneling (enhanced rate constants) at low temperatures is a function of a number of factors: barrier heights, barrier widths, and asymmetry in the potentials. For a homologous series of reactions where the potential energy reaction surfaces will be similar (but scaled), there should be a rough correlation between the tunneling rate constants (at low temperatures) and the barrier height. We have considered the barrier heights as characterized by the temperature dependence (effective activation energy) at high temperatures (1200 –1800 K) compared to the rate constants at low temperatures (300 K). We considered both the experimentally derived recom- mended values from this work and those from the ab initio quantum chemical calculations by Matsugi and Shiina.59These comparisons are shown in Fig. 7 . Note that in this figure, the rate constants are corrected for the number of hydrogen atoms (reaction path degen- eracy). In Fig. 7 , for CH 4,C H 3F, and CH 2F2, there is a strong cor- relation (for both the experimental and ab initio values) between the rate constant at low temperatures ( k300) and the effective activation energy ( Ea) at high temperatures. We calculated that the decrease in the rate constant at low temperatures with the barrier at highTABLE 9. Equilibrium constants for CHF 3+H↔CF3+H 2. See text for details T(K) Keq 300 0.282 400 0.751 500 1.393 600 2.123 700 2.870 800 3.584 900 4.238 1000 4.820 1100 5.324 1200 5.753 1300 6.114 1400 6.412 1500 6.656 1600 6.852 1700 7.006 1800 7.125 1900 7.213 2000 7.275 TABLE 10. CF3+C F 3→C2F6rate constants. k(T)/equalsATnexp(−E/RT).k(T), temperature-dependent rate constant. A, pre- exponential factor. n, temperature coef ficient. E, energy coef ficient. Units: k(cm3mol−1s−1),E(kJ mol−1),T(K). See text for discussion of uncertainties An E T Method Reference 8.33310100.77 0.00 300 –600 Scaled 10COB/CRO This work 9.69310100.77 0.00 300 –2000 10COB/CRO131 6.7331012300 From weighted average This work 8.4431012400 From weighted average This work 6.6231012300 08SKO/KHR130 7.8331012300 10COB/CRO131 5.9631012300 70OGA/CAR141 9.7731012400 10COB/CRO131 7.6531012400 08SKO/KHR130 2.3431013400 56AYS137 J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-17 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprtemperatures is consistent with a change in the effective barrier at low temperatures that is about 1.7 of the change in the effective barrier at higher temperatures (suggestive of the barrier becoming propor- tionally wider). Likely, this effective relative decrease in the tunneling rate is due to the turning point in the reaction moving along the reaction coordinate —a higher barrier likely results in a later barrier (more product-like), a relatively wider barrier, and less tunneling possible. The rate constant for CHF 3+H→CF3+H 2, however, is not consistent with the trend for the other reactions, with the high- temperature barrier being about 3 kJ mol−1higher than that would be predicted (about 1 kJ mol−1higher for the ab initio value). We can also consider this in the context of an Evans –Polanyi type relationship55where the barrier to reaction in a homologous series of reactions should roughly scale with the heat of reaction or, equivalently, with the BDE. This correlation is represented in Fig. 8 where we plotted the rate constants at 300 and 1500 K vs the C –H BDEs for this series. Note that the rate constants are normalized to the number of H atoms (reaction path degeneracy). We see that the rate constants at both temperatures vary only slightly, except for CHF 3 where the rate constant is much lower at 300 K. These relative dif-ferences are consistent with a barrier for CHF 3that is about (5 –6) kJ mol−1higher than one might expect based on trends in BDEs.This anomaly for CHF 3can be likely explained because of the extreme electronegativity of fluorine. In CH 4, the carbon is somewhat electronegative because of the electron-donating H atoms. With addition of fluorine, which is highly electron-withdrawing, the carbon becomes increasingly electropositive. In the case of CHF 3, the carbon is highly electropositive and the C –H bond involved in the reaction has a semi-ionic nature. Consequently, for CH 4,C H 3F, and CH 2F2, the C –H bond breaking is largely covalent in nature. However, for CHF 3, there is an additional ionic contribution. The trends we observe suggest that the additional contribution to the barrier may be on the order of about (3 –6) kJ mol−1—not an unreasonable amount. We would like to note here that we could generate fitted rate expressions with substantially different coef ficients ( A,n,E) but that were essentially statistically identical, because the experimental rate constants are only available in limited temperature ranges. We found differences that ranged from about factors of 1.3 –1.5 for A, tem- perature coef ficients n(Tn) that ranged as much as a factor of 1.5, and barriers Ethat ranged from 2 to 3 kJ mol−1. In our fitting procedure, however, we selected intermediate values that were best consistent with trends in the parameters A,n, and Eand the rate constants at low and high temperatures. 4.3.6. Equilibrium constant for CHF 3+H↔CF3+H2 Equilibrium constants for this reaction at selected temperatures are given in Table 9 and were computed using Burcat ’s thermo- chemical polynomials101with a slight modi fication. Standard en- thalpies of formation of CHF 3and CF 3were updated to −695.0 kJ mol−1and−467.6 kJ mol−1, respectively. These revised enthalpies of formation are based on the iPEPICO measurements and thermo- chemical network analysis of Harvey et al.135and the high-level ab initio calculations by Cobos et al.136Burcat ’s thermochemical polynomials utilized −693.2 kJ mol−1and−467.4 kJ mol−1for the enthalpies of formation of CHF 3and CF 3, respectively. The updated heat of reaction (at 298 K) is more endothermic by 1.6 kJ mol−1(9.5 kJ mol−1vs 7.9 kJ mol−1). The resultant equilibrium constants are then about 0.52, 0.68, 0.83, and 0.90 times smaller at 300 K, 500 K, 1000 K, and 1800 K, respectively. The biggest impact is at low temperatures. The overall uncertainty in the heat of reaction is about 2.6 kJ mol−1, based on the uncertainties in the enthalpies of formation for CHF 3and CF 3(see Table 2 ). Thus, the equilibrium constants computed using either set of enthalpies of formation are within the experimental uncertainties. The equilibrium constants given in Table 9 can be described between 300 and 2000 K to within 1.0% by the following equation: Keq(T)/equalsATnexp(−E/RT )exp(−T/T 1), where A/equals5.11310−2,n/equals0.90, E/equals8.02 kJ mol−1, and T1/equals1425 K. On a practical level, however, the uncertainty in the equilibrium constant at low temperatures is on the order of a factor of f/equals2.0. Coupled with the uncertainty in the measurement of the rate of the reverse reaction CF 3+H 2→CHF 3+ H (estimated to be about a factor off/equals1.5), the calculated rate of the forward reaction CHF 3+H →CF3+H 2has an uncertainty on the order of a factor of about f/equals2.5. We estimated the uncertainty in the reverse reaction by comparing the measurements of Ayscough137and Berces et al.117 We note that the equilibrium constants computed using K/equalsk1/k−1from the rate expressions provided by Hranisavljevic and FIG. 7. Fluoromethanes + H rates (per H atom) at 300 K vs barriers at (1200 –1800) K. Rate constants k(cm3mol−1s−1).Eafor CHF 3is about 3 kJ mol−1higher than the trend for other fluoromethanes. Filled circles and open circles are derived from experimental data and quantum chemical data (14MAT/SHI59), respectively. FIG. 8. Fluoromethanes + H rate constants (per H atom) vs BDEs. Rate constants k (cm3mol−1s−1). Open circles and squares were derived assuming the barrier for CHF 3+ H is 5.5 kJ mol−1lower than predicted using the trend in BDEs. J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-18 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprMichael107do not agree with the equilibrium constants we provide above nor those derived using the JANAF tables.138–140The equilibrium constants derived from their expressions decrease from K/equals7.8 to K/equals5.0 from 1000 K to 1600 K, while those provided above increase from K/equals5.8 to K/equals7.7, as one might expect. They indicated that they used equilibrium constants derived from the JANAF tables. We were unable to identify the source of this discrepancy (it is on the order of about 6 kJ mol−1). 5. Fluoromethanes +O→Fluoromethyls +OH 5.1. Overview In this section, we compile and evaluate rate constants from the literature and provide recommended rate expressions for reactions involving the abstraction of H from C –H bonds by O atoms from the fluoromethanes (and methane). The data for the fluoromethanes + O reactions are extremely limited. There are no experimental rate constants (except for CH 4) near room temperature. At medium to high temperatures (about 600 –1600 K), there are two sets of relatively reliable rate constants for CHF 3, one reliable measurement for CH 3F, and none for CH 2F2. There have been a number of rate constants for these series of reactions derived using ab initio quantum chemical methods that include tunneling. The work by Matsugi and Shiina59 using the CBS-QB3 method for energies and Eckart potentials fortunneling is the best work employing quantum chemical calculations for the fluoromethanes + O reactions. There is somewhat of a dis- agreement (a factor of 1.8 –2.2) between the two intermediate to high temperature (600 –1300 K) measurements for CHF 3+ O, both by reliable groups. InTable 11 , we provide recommended rate expressions for abstraction of H from the fluoromethanes (and methane) by O atoms. This includes rate expressions (with three parameters A,n, and E) that are good over a wide temperature range (300 K –1800 K), as well as simpler rate expressions (with two parameters Aand E) to describe the rate constants at temperatures relevant to combustion (1200 –1800 K). Also included in this table are relative Afactors at high temperatures. We see that the relative Afactors for the CHF 3, CH 2F2,C H 3F, and CH 4reactions from the recommended rate ex- pressions (1.0, 1.89, 2.75, and 4.05) are roughly proportional to the number of hydrogen atoms in the molecules (reaction pathdegeneracy). The TS calculations from the ab initio quantum chemical work of Matsugi and Shiina also yielded relative Afactors (1.0, 1.55, 2.66, and 4.11) that are also roughly proportional to re- action path degeneracy. This analysis provides a measure of the self- consistency (estimated 20% –30%) of these recommended rate ex- pressions at high temperatures. The assigned expanded uncertainty factors f(2σ) for CH 4+O were determined from the scatter in the experimental measurements and the recommended rate expression. Although the measurements extend down to nearly 400 K, these are limited data, and consequently, a higher uncertainty is assigned. The uncertainty factors at high temperatures for CH 3F + O and CHF 3+ O were assigned based on experimental uncertainties from the work of Miyoshi et al.142and of Fernandez and Fontijn143and agreement between the two measurements for the CHF 3+ O reaction. At low temperatures, the uncertainty factors were increased to account for the uncertainty in tunneling based on the comparison of the cal- culated tunneling from Matsugi and Shiina59and that from the recommended expressions. The uncertainty factors for CH 2F2+O were assigned by analogy to CH 3F + O, which has nearly identical rate constants. 5.2. CH 4+O→CH3+OH We have compiled rate constants from the literature for CH 4+O →CH 3+O H —this is the benchmark reaction for analogous reac- tions for the fluoromethanes. Our recommended rate expression is based on a fit to six sets of measurements that range from about 400 to 1900 K. This recommended rate expression is largely statistically equivalent to a number of recommended rate expressions for this reaction in the literature. We, however, utilize our recommended rate expression for self-consistency in evaluating the other reactions (the fluoromethanes). Table 12 contains the rate expressions from the literature along with our recommended value. The rate expression from Miyoshi et al.142was not included in the fit since two other measurements by this group were included. Figure 9 shows the fit (solid curve) to the experimental rate constants (various symbols). For comparison, in Fig. 9 , we also show the recommended rate expression of Sutherland et al.144and the quantum calculations of Matsugi and Shiina.59Figure 10 shows the deviations of the data not included in theTABLE 11. Recommended rate expressions for H abstraction from the fluoromethanes by O atoms. k(T)/equalsATnexp(−E/RT).k(T), temperature-dependent rate constant. A, pre-exponential factor. n, temperature coef ficient. E, energy coef ficient. f(2σ), expanded uncertainty factors. ΔrH298, standard enthalpy of reaction at 298 K. Units: k(cm3mol−1s−1),E(kJ mol−1),T(K). A/ACHF3 is the ratio of each pre-exponential Ato that for CHF 3. Uncertainty factors fin parentheses “(f)”are at low temperatures (300 –500) K. Reaction An E f T log10(k300) ΔrH298 CH 4+O→CH 3+ OH 6.63 31062.16±0.24 31.42 ±1.91 1.18(1.5) 300 –1800 6.70 9.4 CH 3F+O→CH 2F + OH 1.66 31062.28±0.39 26.34 ±2.44 1.4(2.0) 300 –1800 7.28 −6.6 CH 2F2+O→CHF 2+ OH 4.10 31052.40±0.41 27.19 ±2.52 1.4(2.0) 300 –1800 6.72 −3.5 CHF 3+O→CF3+ OH 2.70 31042.65±0.58 42.17 ±3.52 1.7(2.3) 300 –1800 3.65 16.7 A/ACHF3 CH 4+O→CH 3+ OH 3.98 3101457.55 1.18 1200 –1800 4.26 CH 3F+O→CH 2F + OH 2.70 3101453.92 1.4 1200 –1800 2.89 CH 2F2+O→CHF 2+ OH 1.86 3101456.32 1.4 1200 –1800 1.99 CHF 3+O→CF3+ OH 9.33 3101374.27 1.7 1200 –1800 1.00 J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-19 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprTABLE 12. CH4+O→CH3+ OH rate expressions. k(T)/equalsATnexp(−E/RT).k(T), temperature-dependent rate constant. A, pre-exponential factor. n, temperature coef ficient. E, energy coef ficient. f(2σ), expanded uncertainty factors. Units: k(cm3mol−1s−1),E(kJ mol−1),T(K). Letters in the uncertainty factor fcolumn refer to quantum chemical data presented inFig. 11 . See Sec. 10for de finitions of acronyms and abbreviations in the Method column An E f T Method Reference Evaluations 6.6331062.16 31.42 1.18(1.5) 300 –1840 Recommended This work 3.983101457.55 1.18 1200 –1800 Recommended, high T fit This work 6.9231081.56 35.50 [1.4] 400 –2250 Review 86SUT/MIC144 4.9031062.2 31.76 [1.3] 420 –1520 Review 86COH/WES154 1.6231062.3 29.68 [1.3] 300 –1800 Review 86COH155 3.16310120.50 43.07 [1.5(2.5)] 475 –2250 Review 81KLE/TAN156 1.1431072.08 31.93 [1.2] 300 –2200 Review 79ROT/JUS157 2.13101337.83 1.3 350 –1000 Review 73HER/HUI158 2.193101337.08 [1.6] 300 –1000 Review 69HER/HUI159 Experimental (preferred data) 2.833101454.13 1.52 980 –1529 Shock, flash photol, reson abs 94MIY/TSU160 5.013101458.28 1.55 980 –1520 Shock, flash photol, reson abs 93MIY/OHM142 6.753101464.44 1.86 1340 –1840 Shock, flash photol, reson abs 92OHM/YOS161 1.6131081.75 34.26 2.0 763 –1760 Shock, flash photol, reson abs 87SUT/KLE162 5.5531052.40 24.44 2.0 763 –1760 Shock, flash photol, reson abs 86SUT/MIC144 5.2431052.50 29.67 2.4 420 –1670 Fast flow,flash photol, reson fluor 86FEL/MAD163 1.213101445.15 1.7 474 –1160 Dischg flow, reson fluor 81KLE/TAN156 7.603101234.18 [1.3] 474 –520 Dischg flow, reson fluor 81KLE/TAN156 1.1431072.08 31.93 2.6 525 –1250 Flow, flash photol, reson fluor 79FEL/FON164 2.03101338.50 1.12[1.7] 400 –900 Fast dischg flow, ESR 69WES/DEH145 1.233101336.03 [1.3] 363 –605 Re fit 69WES/DEH This work Experimental (excluded data) 4.03105[10] 293 Static, photol 73FAL/HOA165 1.93101230.51 1.5[5] 350 –1000 Revised 65CAD/ALL 73HER/HUI158 1.263107[3] 300 Fast dischg flow, ESR 69WES/DEH145 4.03108[100] 300 Fast dischg flow, ESR 68FRO166 4.03101441.82 [10] 375 –576 Stirred reactor, dischg, MS 67WON/POT167 7.03101232.18 1.29[5] 450 –600 Dischg flow, ESR 67BRO/THR168 3.3031091.5 295 Dischg flow, O emiss 65CAD/ALL169 2.33101327.6 1.5 295 –533 Dischg flow, O emiss 65CAD/ALL169 7.13101230.51 1.5[10] 295 –533 Dischg flow, oxygen emiss 65CAD/ALL169 2.003101228.85 1.5[3] 353 –580 Stirred reactor, MW dischg, MS 63WON/POT170 Theoretical 5.64310−45.20 12.97 [3] x 200 –1000 Quantum PES, our fit to data in tables 16ZHA/WAN171 1.8731033.24 24.86 [1.6] a 300 –1800 CBS-QB3 Eckart 14MAT/SHI59 8.683101464.06 [1.6] 1200 –1800 This work; high T fit 14MAT/SHI59 6.7831023.27 23.15 [1.3] b 300 –2500 Quant PES; our fit to data in tables 14GON/COR147,148 2.09310 4.01 18.65 [1.3] c 200 –2500 Quant PES; our fit to data in tables 05TRO/GAR149 5.6231042.70 28.06 [1.7] d 200 –1000 Quant PES; our fit to data in tables 02HUA/MAN151 8.1231042.60 26.32 [1.7] e 200 –1250 Quant PES 00YU/NYM152 3.6331062.17 30.09 [1.3] f 200 –2500 Quant PES 00ESP/GAR146 2.1631091.32 38.43 [3.5] x 300 –1000 Quant PES; our fit to data in figures 99CLA172 3.393101451.8 [2.0] x 300 –2500 Quant PES 98COR/ESP173 5.3831013.65 22.95 [1.3] g 300 –2000 Quant PES 90GON/MCD150 2.1331062.21 27.11 [1.3] h 500 –2500 BEBO theory 83MIC/KEI153 5.1231062.00 26.94 [1.4] i 300 –2500 Semi-empirical 78SHA174 1.033101438.58 [4] x 300 –1000 BEBO Theory 68MAY/SCH175,176 J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-20 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprfit from the recommended rate expression, and Fig. 11 shows the deviations for the quantum calculations. The rate constants at intermediate to high temperatures (500–1800 K) are well established (within about 20%). At low temperatures (300 –500 K), there is a single fairly reliable measure- ment by Westenberg and Dehaas.145There are, however, some in- consistencies in the data. First, the barrier from the reported rate expression (38 kJ mol−1) is substantially lower than the effective temperature dependencies (44 kJ mol−1) of measurements at inter- mediate temperatures (600 –900 K) by other workers. Second, the reported rate expression is inconsistent with the individual rate constant data presented in the paper. In our fits, we utilized an in- termediate temperature subset of the reported data (4 points from 363 K to 605 K) and ignored the lowest (297 K) and highest (904 K) data points —and our fit goes through roughly the midpoint (450 K) of these data.Figure 11 shows the deviations of the rate constants Δln(k) for quantum calculations from the recommended rate expression. Note that several sets of rate constants are not included in this figure because they deviated by large amounts (these are denoted in Table 12 with “x”in the uncertainty factor column). Note also that identi fi- cation of the rate expressions with the labels “a”through “i”is also denoted in this table. Overall, the differences between the rate constants from quantum calculations and the recommended values are on the order of a factor of f/equals1.7. The rate constants from the quantum calculations are on the average about 20% lower at high temperatures and about 10% higher at low temperatures compared to the recommended rate constants based on experimental data. The rate expression fromquantum chemical calculations that is closest to the recommended expression is that of Espinosa-Garc ´ıa and Garc ´ıa-Bern´ aldez 146—the temperature coef ficients nare virtually identical (2.17 vs 2.16), the temperature dependencies differ by under 1.5 kJ mol−1, and the rate constants at 300 K differ by just 1.2%. The rate constants at high temperatures from the calculations of Gonz´ alez-Lavado et al. ,147,148Troya and Garcia-Molina,149and Gonzalez et al.150agree satisfactorily (within about 25%) with the experimental values. The temperature dependencies at low tem- peratures from the quantum calculations agree satisfactorily (within about 2 kJ mol−1) with that derived from the recommended rate expression in the work of Gonzalez-Lavado et al. ,147,148Hurarte- Larranga and Manthe,151Yu and Nyman,152and Michael et al. ,153 although the absolute rate constants deviate by factors of f/equals(1.3–2.0). Other key things to note about the ab initio calculations are that the rate expression at high temperatures (1200 –1800 K) from the work of Matsugi and Shiina has an Afactor about 2.2 times higher than the fitted recommended value and a temperature dependence E that is about 11% higher. They adjusted their barrier so that their rate expression agreed with the low-temperature data, which then affects the rate constants at high temperatures. This is an incorrect proce- dure, and the tunneling parameters should have been adjusted. Other observations are that at 300 K, the predicted tunneling rate by Yu and Nyman is about 15% higher than that derived from the recommended rate expression, and the tunneling rate from Matsugi and Shiina is FIG. 9. CH 4+O→CH3+ OH rate constants and recommended rate expression. Rate constants k(cm3mol−1s−1). Legend: Red line (recommended), dotted line (14MAT/SHI59), dashed line (86SUT/MIC rec, their recommendation), black trian- gles (94MIY/TSU160), black circles (93MIY/OHM142), yellow circles (92OHM/ YOS161), blue circles (87SUT/KLE162), green circles (86SUT/MIC144), red dots (86FEL/MAD163), brown dashes (81KLE/TAN156), black dashes (79FEL/FON164), brown dots (69WES/DEH145), and red circles (69WES/DEH145tw) were derived from our re fitting of the original data. FIG. 10. Residuals for CH 4+O→CH3+ OH rate constants excluded from fit. Rate constants k(cm3mol−1s−1). Residuals in ln( k) units. FIG. 11. Residuals for CH 4+O→CH 3+ OH rate constants from quantum chemical calculations. Rate constants k(cm3mol−1s−1). Residuals in ln( k) units. See Table 12 for legend ( “a”through “i”labels). J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-21 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprabout 1.9 times higher. We will employ the latter difference to assist in estimating k(300 K) for the other fluoromethane + O reactions. 5.3. CH 3F+O→CH2F+OH There is one set of reliable experimental data for this reaction from the shock tube work of Miyoshi et al.142in the temperature range (920–1570) K. There are other reported rate constants, but they differ from the recommended and self-consistent rate expression by as much as an order of magnitude. In the work of Miyoshi et al. , the authors measured rate constants for the abstraction of H by O atoms from CH 4,C H 3F, and CHF 3(as well as other molecules). Conse- quently, these three sets of measurements provide a good measure of the relative rates of these reactions. The rate constants for the homologous reaction CH 4+ O in the work of Miyoshi et al. are slightly different (about 15% higher) than the rate constants measured in other work for this reaction by the same group160,161and slightly different from the rate expression recommended in this work. The most recent work of Miyoshi et al.142 reanalyzed/corrected the rate expression for CH 4+ O. Thus, we have also corrected the reported rate constants for the reaction CH 3F+O →CH 2F + OH by Miyoshi et al. by 15% (decreased) to account for this difference. There are no measured rate constants for this reaction at low temperatures (i.e., near room temperature). In order to provide a rate expression valid over a range of temperatures (300 –1800 K), we needed to fit the high-temperature data along with providing an estimate for the rate constant at low temperatures. We estimated k(300 K) for CH 3F (and for CH 2F2and CHF 3) by extrapolating the rate constants at high temperatures with a fixedk(300 K) for CH 4. [Note that the high-temperature rate constants for CH 2F2were es- timated by analogy to CH 3F + O (see discussion for that reaction below).] We bracketed k(300 K) for the fluoromethanes that provided a range of reasonable parameters for the fitted rateexpressions using the systematic differences between the calculated tunneling rates from Matsugi and Shiina59and the fitted values. We then used intermediate values from the bracketing of k(300 K) for each fluoromethane. The procedure we used was iterative and sys- tematic but manual (not computer optimization). [Note that by “reasonable parameters, ”we mean different sets of parameters ( A, n,E) that gave statistically similar fits and were constrained by the cor- relations that we discuss in Sec. 8. For example, a temperature co- efficient of n/equals(2–3) is reasonable, while n/equals(1 or 4) is not.] The resultant fit to the experimental rate constants of Miyoshi et al. and the estimated k(300 K) from bracketing are shown in Fig. 12 . Rate expressions from the literature are given in Table 13 . The fitted rate expression not only agrees very well with the rate constants at high temperatures but also displays the same effective activation energy (temperature dependence) at high temperatures. From in- spection of the raw data of Miyoshi et al., we estimated an uncertainty factor of f/equals1.4 at high temperatures. Considering the uncertainties inherent in the bracketing methodology to estimate a rate constant at low temperatures (along with the uncertainties in the rate constants at high temperatures), we estimate an uncertainty factor of f/equals2.0 for k(300 K). 5.4. CH 2F2+O→CHF 2+OH There are no reliable experimental measurements for this re- action. The reported values by Parsamyan and Nalbandyan103are off by an order of magnitude from what would be reasonable. In order to provide a rate expression, we utilized estimated rate constants at both high and low temperatures. An initial guess for the rate constants at high temperatures was estimated using the recommended rate ex- pression for CH 3F + O and then scaling the rate constants by the ratio of the constants from the ab initio calculations of Matsugi and Shiina.59An initial guess for the low-temperature k(300 K) rate constant was estimated by bracketing reasonable values for thetemperature coef ficient n(2.2–2.5)—considering the trend in tem- perature coef ficients nfor the other molecules CH 4,C H 3F, and CHF 3, as well as considering the derived activation energy at low temper- atures to be about 60% of the derived activation energy at high temperatures. See Sec. 8forfigures and discussion regarding these correlations. These parameters were manually adjusted to obtain a best relaxed fit while maintaining the relative rate constants for the reactions involving CH 3F and CH 2F2from the ab initio calculations of Matsugi and Shiina. The resultant estimated rate constants are shown inFig. 12 , and the rate expressions are provided in Table 14 . As before, we estimated expanded uncertainty factors f(2σ)o ff/equals1.5 and f/equals2.0 at high and low temperatures, respectively. Note that the rate constants in Fig. 12 for CH 2F2+ O are dis- placed upward by an order of magnitude (one log 10unit) for clarity. The absolute values for the CH 2F2+ O rate constants are about 1.8 times lower at high temperatures and about 2.9 times lower at low temperatures than those for CH 3F+O . 5.5. CHF 3+O→CF3+OH There are two relatively reliable measurements of the reaction of O with tri fluoromethane (CHF 3) by Fernandez and Fontijn143and Miyoshi et al.142They are, however, in disagreement by a factor of about (1.8 –2.2) and outside their respective expanded uncertainty FIG. 12. Rates of reaction for H abstraction by O from CH 3F, CH 2F2, and CHF 3. Rate constants k(cm3mol−1s−1). Circles and squares are experimental rate constants. Open squares (01FER/FON143),filled circles (93MIY/OHM142), and filled squares (adjusted 93MIY/OHM142). Solid curves are recommended rate expressions. Dotted curves are from quantum calculations of Matsugi and Shiina.59The rate constants for CH 2F2, where no experimental data are available, were estimated by interpo- lating the rate expressions using the rate constants calculated by Matsugi and Shiinaas a relative measure, as well as trends in A,n, and Ein this homologous series. J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-22 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprfactor f(2σ)o ff/equals1.3 and f/equals1.5. Both measurements were made by reliable researchers who have obtained essentially the same rate constants (within <10%) for the reference reaction CH 4+O →CH 3+ OH, as well as having made reliable measurements for other reactions. The limited data for this reaction are given in Table 15 . The rate constants from the literature and our recommended rate expression are presented in Fig. 12 along with the rate expression from the ab initio calculations of Matsugi and Shiina.59The other values in the table not considered are in signi ficant disagreement with the ex- pression recommended here. We have examined the measurement s of this reaction by Fernandez and Fontijn and by Miyoshi et al. and see no obvious ap r i o r i reason to consider one or the other to be in error. In order to discriminate between the two measurements, we utilized the ab initio calculations of Matsugiand Shiina combined with parametric fits to the data and also self- consistency with the other rate expressions. We found that utilizing the rate constants of Miyoshi et al. yielded fitted rate expressions with A factors at high temperatures (1200 –1800 K) that closely scaled with the A factors derived from the TS/quantu m chemical calculations of Matsugi and Shiina and that these rate express ions yielded exponential temper- ature coef ficients ( Tn) that were more reasonable, n/equals(2.0–2.5), and relaxed (not a forced fit ) .O nt h eo t h e rh a n d ,t h ea c t i v a t i o ne n e r g yi nt h e temperature range of interest from the fitted curve was consistent with the measured temperature dependence o f Fernandez and Fontijn, while the data of Miyoshi et al. showed a higher activation than the trend from the fitted curve. Consequently, we used both of these sets of data to determine the recommended rate expression with the data of Fernandez and Fontijn and Miyoshi et al. a factor of about 1.35 lower and 1.65 higher at the midpoint of their respective tempe rature ranges than the recommendedTABLE 13. CH3F+O→CH2F + OH rate expressions. k(T)/equalsATnexp(−E/RT).k(T), temperature-dependent rate constant. A, pre-exponential factor. n, temperature coef ficient. E, energy coef ficient. f(2σ), expanded uncertainty factors. Units: k(cm3mol−1s−1),E(kJ mol−1),T(K) An E f T Method Reference Evaluations 1.6631062.28 27.19 1.4(2.0) 300 –1800 Recommended This work 2.703101453.92 1.4 1200 –1800 Recommended, high T fit This work Experimental (preferred data) 1.483101447.47 1.1 920 –1570 Adjusted 93MIY/OHM142This work Experimental (excluded data) 1.703101447.47 1.3 920 –1570 Shock, laser photol, O reson abs 93MIY/OHM142 7.833101240.57 [7] 858 –933 Static 67PAR/AZA102 Theoretical 3.0131033.13 20.45 [1.7(3.0)] 300 –1800 CBS-QB3 Eckart 14MAT/SHI59 5.763101458.44 [1.7] 1200 –1800 This work, high T fit 14MAT/SHI59 n/a 51.25 298 G2M 01KRE/SEY177 7.53106298 G2M 01KRE/SEY177 2.54310−34.80 10.08 200 –3000 G2MP2 99WAN/HOU178 TABLE 14. CH 2F2+O→CHF 2+ OH rate expressions. k(T)/equalsATnexp(−E/RT).k(T), temperature-dependent rate constant. A, pre-exponential factor. n, temperature coef ficient. E, energy coef ficient. f(2σ), expanded uncertainty factors. Units: k(cm3mol−1s−1),E(kJ mol−1),T(K) An E f T Method Reference Evaluations 4.1031052.49 27.19 1.4(2) 300 –1800 Recommended This work 1.863101456.32 1.4 1200 –1800 Recommended, high T fit This work Experimental (preferred data) NONE Experimental (excluded data)2.65310 1236.83 1.14[10] 873 –953 Static 68PAR/NAL103 Theoretical 1.2031033.17 22.61 [1.8] 300 –1800 CBS-QB3 Eckart 14MAT/SHI59 3.283101461.08 [1.6(1.13)] 1200 –1800 This work, high T fit 14MAT/SHI59 n/a 54.48 298 G2M 01KRE/SEY177 2.73106298 G2M 01KRE/SEY177 2.24310−45.03 9.30 200 –3000 G2MP2 99WAN/HOU178 J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-23 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprrate expression, respectively. We no te that in the small temperature range (940–956 K) where the two sets of data overlap, they differ by (20 –30)%, d i v e r g i n ga th i g h e ra n dl o w e rt e m p e r a t u r e s . In the fitting procedure, while utilizing the experimental high- temperature data, it was necessary to estimate k(300 K) for CHF 3.T o do this, we used a bracketing procedure (as discussed earlier) based on extrapolating the rate constants from high temperature, using k(300 K) for CH 4, and utilizing the differences between calculated tunneling rates by Matsugi and Shiina and that obtained through fits of the experimental data. This was not a rigorous procedure but rather (manually) iteratively adjusting k(300 K) to obtain a range of rea- sonable fit parameters. See Secs. 2and8for discussion and figures regarding this optimization procedure where the experimental data arefit subject to constraints involving correlations due to SARs. In summary, the recommended rate expression is based on the rate constants of Fernandez and Fontijn and Miyoshi et al. at high temper- atures and an estimated k(300 K). Based on the uncertainties in the experimental data and the uncertainty in the parametric fitt ot h ed a t a ,w e estimate an expanded uncertainty factor f(2σ) of about a factor f/equals1.7 to the rate constant at high temperatures and about f/equals2.3 at 300 K. 6. Fluoromethanes +OH→Fluoromethyls +H2O 6.1. Overview All of the fluoromethanes and methane have many reliable measurements for the abstraction of H by OH radicals at low tem- peratures, both below and above room temperatures (250 –500 K). The extent of the number of these measurements is because the lifetimes of these molecules in the upper atmosphere are important to radiative forcing and global warming.1High-temperature measure- ments (1000 –2000 K) related to the combustion of these molecules have been made only for CH 4and CHF 3, but not for CH 3Fo rC H 2F2. However, the rates of these two latter reactions at low temperatures (up to about 500 K) are due to tunneling and the curvature in the rate expressions (characterized by the temperature exponent Tn)i sa constraint on the rate constant at high temperatures.The recommended rate expressions for the reaction of OH with methane and the fluoromethanes are collected in Table 16 . The relative Afactors at high temperatures for abstraction by OH do not follow the reaction path degeneracy (number of hydrogen atoms) like they roughly do for abstraction by H and O atoms —1.0, 3.0, 4.3, and 10.9 vs 1, 2, 3, and 4 for CHF 3,C H 2F2,C H 3F, and CH 4, respectively. This is likely because the TS has a hindered rotor; this is discussed in more detail in the discussion near the end of this paper. 6.2. CH 4+OH→CH3+H2O InTable 17 , we provide rate expressions compiled from the literature for the reaction CH 4+O H→CH 3+H 2O. These include evaluations/review, experimental measurements, and quantum chemical calculations. We fit the experimental data from 9 different measurements to provide a recommended rate expression with ex- panded uncertainty factors f(2σ) of about f/equals1.15 at higher tem- peratures (1000 –2000 K) and f/equals1.07 at low temperatures (200 –500 K), respectively. Although we assign an uncertainty of 7% at low temperatures, we note that at (300 –400) K Bryukov et al. ,181Bonard et al. ,13Dunlop and Tully,182Gierczak et al. ,183 Vaghjiani and Ravishankara,184Amedro et al. ,185and Overend et al.186all agree to within about 3% of the recommended values. For more discussion of the low-temperature data for this reaction, see Burkholder et al.187The recommended rate expression by Burkholder et al. agrees with our recommended rate expression (and the experimental data) to wi thin about 5% in the temperature range (200 –350) K but diverges rapidly above about 400 K. The data excluded from our fit deviated from the recommended expression by factors of f/equals(1.2–5.0), and almost all of these data had rate constants that were higher than the recommended values. In addition, many of these had temperature dependencies that were inconsistent with the recommended rate expression. In general, thedata included had residuals on the order of (3 –13)%, while the data excluded had residuals of about (20 –70)%. Figure 13 shows the ex- perimental rate data along with the recommended rate expression.TABLE 15. CHF 3+O→CF3+ OH rate expressions. k(T)/equalsATnexp(−E/RT).k(T), temperature-dependent rate constant. A, pre-exponential factor. n, temperature coef ficient. E, energy coef ficient. f(2σ), expanded uncertainty factor. Units: k(cm3mol−1s−1),E(kJ mol−1),T(K) An E f T Method Reference Evaluations 2.7031042.65 42.17 1.7(2.3) 300 –1800 Recommended CHN This work 9.333101374.27 1.7 1200 –1800 Recommended, high T fit CHN This work 3.073101479.28 630 –1330 Review 01FER/FON143 Experimental (preferred data) 4.573101481.80 1.6 956 –1328 Shock, laser photol, O reson abs 93MIY/OHM142 3.983101481.80 [1.65] 956 –1328 Adjusted 93MIY/OHM This work 1.513101360.16 [1.35] 630 –940 Fast flow, VUV flash photol, O reson abs 01FER/FON143 Experimental (excluded data) 1.503101124.94 1.25[5] 500 –750 Flow, photol, LIF 97MED/FLE179 1.103101213.30 [7] 920 –1150 Model, flame 94RIC/VAN111 4.883101245.15 [6] 298 –600 Plasma excit, MS 77JOU/POU180 Theoretical 1.51310236.58 [1.2(2)] 300 –1800 CBS-QB3 Eckart 14MAT/SHI59 2.113101477.48 [1.2] 1200 –1800 High T fit 14MAT/SHI59 J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-24 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprFigures 14 and15show the deviations of the included and excluded data, respectively, in ln( k) units from the recommended expression. There have been many precise measurements at lower temperatures for this reaction but only one at high temperatures by Srinivasan et al.188This is re flected in an uncertainty at high temperatures that is about 2 times higher than at low temperatures (15% vs 7%). We note that our fit is statistically similar to the recommended rate expression (not their experimental one) by Srinivasan et al. (seeTable 17 ) based on their fit to available data with their rate expression being about 5% and 14% higher at low and high temperatures, respectively. Our fiti s slightly different because we consider other data and use different weightings. Note that we excluded the data from flame measurements. 6.3. (CH 3F, CH 2F2, CHF 3)+OH→(CH 2F, CHF 2,C F 3)+H2O We can estimate (in a relative fashion) the rate constants for CH 3F + OH and CH 2F2+ OH at high temperatures (1200 –1800 K) utilizing the low-temperature measurements and extrapolating and bracketing considering the relative activation energies from ab initio calculations (benchmarked to that for CH 4+ OH and CHF 3+ OH), considering a range of temperature exponents by analogy to CH 4and CHF 3, and considering that the Afactors should vary (roughly) uniformly from that for CH 4to that for CHF 3. Using the low-temperature data and extrapolating based on a range of temperature exponents ( n/equals1.95±0.10), we find that the activation energies for CH 3F + OH and CH 2F2+ OH should be about 6.8±0.8 kJ mol−1and 5.5 ±0.8 kJ mol−1lower, respectively, than that for CHF 3+ OH. Using the relative activation energies from TSs computed in the quantum chemistry work of Schwartz et al.61and Matsugi and Shiina,59the activation energies for CH 3F + OH and CH 2F2+ OH should both be about 6.9 ±0.3 kJ mol−1lower than that for CHF 3+ OH. There is some uncertainty in the relative activation energies from the work of Matsugui and Shiina because they arbi- trarily adjusted the barriers (at 0 K) in order to make their tunneling calculations agree with the experimental rate constants at low tem- peratures (and do not report the unadjusted rate constants) —hence, we increase the uncertainty and estimate the activation energies for CH 3F + OH and CH 2F2+ OH as 6.9 ±0.3 kJ mol−1lower than that for CHF 3. Concurrent with the lower activation energies, the rate con- stants ( k) at high temperature should be about 6.2 ±0.3 and 5.1 ±0.3 times higher for CH 3F + OH and CH 2F2+ OH compared toCHF 3+ OH, and Afactors should be about 3.9 ±0.3 and 2.9 ±0.3 times higher, respectively. Our recommended rate expressions for CH 3F + OH and CH 2F2+OH are based on bracketing the rate parameters A,n, and E using the recommended rate expressions for CH 4+O Ha n dC H F 3+O H as benchmarks and interpolating using the ab initio calculations as a measure. The uncertainties provided in square brackets “[]”for the rate constants in Table 18 are based on deviations for the ex- perimental measurements at low and high temperatures. We have assigned uncertainty factors of f/equals1.4 for both CH 3F + OH and CH 2F2+ OH based on our bracketing estimate. This corresponds to an uncertainty in the activation energy of 4.2 kJ mol−1at 1500 K. Tables 17 –19list the rate constants compiled from the literature for the reactions of OH with CH 3F, CH 2F2, and CHF 3, and Figs. 15 –17 present the recommended rate constants along with the selected data for these three reactions. Figure 19 is a summary comparing the temperature dependence of the recommended rate constants with reactions involving CH 3F and CH 2F2being similar, while CHF 3+O H has a much stronger temperature dependence. Table 18 provides rate constants from the literature for CH 3F+O H→CH 2F+H 2O, and Fig. 16 compares the experimental rate constants to our recommended rate expression. The only ex- perimental rate constant data that exists for this reaction CH 3F+O H→CH 2F+H 2O is at low temperatures between about 240 and 500 K. The scatter in the experimental data is about a factor of f/equals1.3. In this range, our recommended rate expression agrees with the recommended values from the NASA/JPL Kinetics Panel Eval- uation187to within about (6 –8)% (see the dashed line in Fig. 16 ). We assign an uncertainty to our rate expression in this range of f/equals1.2. Table 19 provides rate constants from the literature for CH 2F2+O H→CHF 2+H 2O, and Fig. 17 compares the experimental rate constants to our recommended rate expression. There are no experimental data above about 500 K. The majority of the data are between about 220 and 400 K. There are two sets of data that extended slightly above 400 K (420 and 490 K). The agreement between the experimental data at low temperatures is good, and we assign an uncertainty factor f/equals1.17. Because of the uncertainty in extrapolating the rate constants to high temperatures, we assign an uncertainty factor f/equals1.5 in this range. The recommended rate expressions provided here for the re- actions of OH with CH 3F and CH 2F2are based on fits to theTABLE 16. Recommended rate expressions for H abstraction from the fluoromethanes by OH radicals. k(T)/equalsATnexp(−E/RT).k(T), temperature-dependent rate constant. A, pre-exponential factor. n, temperature coef ficient. E, energy coef ficient. f(2σ), expanded uncertainty factor. ΔrH298, standard enthalpy of reaction at 298 K. Units: k(cm3mol−1s−1),E(kJ mol−1),T(K).A/ACHF3 is the ratio of each pre-exponential Ato that for CHF 3. Reaction An Ef T log10(k300) ΔrH298 CH 4+O H→CH 3+H 2O 6.19 31052.23±0.09 9.84 ±1.54 1.12(1.05) 200 –2025 9.62 −58.2 CH 3F+O H→CH 2F+H 2O 1.06 31062.04±0.26 5.70 ±1.93 1.5(1.22) 240 –1800 10.08 −74.2 CH 2F2+O H→CHF 2+H 2O 3.29 31061.86±0.24 7.52 ±1.90 1.5(1.17) 220 –1800 9.82 −71.1 CHF 3+O H→CF3+H 2O 1.20 31061.85±0.19 13.71 ±1.78 1.3(1.22) 250 –1800 8.27 −51.1 A/ACHF3 CH 4+O H→CH 3+H 2O 6.12 3101335.33 1.12 1200 –1800 10.9 CH 3F+O H→CH 2F+H 2O 2.40 3101330.44 1.5 1200 –1800 4.26 CH 2F2+O H→CHF 2+H 2O 1.68 3101330.09 1.5 1200 –1800 2.98 CHF 3+O H→CF3+H 2O 5.64 3101236.15 1.2 1200 –1800 1.00 J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-25 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprTABLE 17. CH 4+O H→CH3+H 2O rate expressions. k(T)/equalsATnexp(−E/RT).k(T), temperature-dependent rate constant. A, pre-exponential factor. n, temperature coef ficient. E, energy coef ficient. Expanded uncertainty factor f(2σ). Units: k(cm3mol−1s−1),E(kJ mol−1),T(K). See Sec. 10for de finitions of acronyms and abbreviations in the Method column An E f T Method Reference Evaluations 8.6831052.18 9.98 1.15(1.07) 200 –2025 Recommend This work 6.123101335.33 1.25 1200 –1800 Recommend, high T fit This work 1.69310100.67 13.09 1.1[1.2] 243 –480 Review 15BUR/SAN187 1.483101214.76 1.1[1.2] 243 –480 Review, our fit 15BUR/SAN187 1.0031062.18 10.24 1.2 195 –2025 Review 05SRI/SU188 2.4631033.04 7.65 1.05[1.12] 195 –1234 Review 02BON/DAE13 1.5731071.83 11.64 1.41[1.23] 250 –2500 Review 92BAU/COB189 1.9331052.40 8.81 1.4[1.1] 300 –2500 Review 86TSA/HAM190 5.7231061.96 11.04 [1.17] 268 –1512 Review 86FEL/MAD163 1.9031052.40 8.81 2.0[1.08] 240 –3000 Review 83COH/WES191 1.5531062.13 10.26 [1.13] 250 –2000 Review 78ERN/WAG192 Experimental (preferred data) 3.6131091.03[1.06] 296 Flow, laser photo, LIF 12AME/MIY185 5.703101334.37 1.17 840 –2025 Shock, OH abs 05SRI/SU188 2.3031052.38 9.45 1.05[1.03] 298 –1009 Laser photol, OH LIF 04BRY/KNY181 3.4031033.01 7.97 1.09[1.03] 295 –668 Laser photol, OH LIF 02BON/DAE13 1.1231042.82 8.21 1.06[1.03] 196 –420 Laser photol, OH LIF 97GIE/TAL183 1.543101214.63 1.67[1.1] 233 –343 Laser photol, OH LIF 94MEL/TET193 5.8331042.58 8.98 1.05[1.03] 293 –800 Laser photol, OH LIF 93DUN/TUL182 1.713101214.72 [1.06] 378 –422 Dischg flow, OH EPR LIB 92LAN/LEB194 2.413101216.13 1.4[1.06] 278 –378 Dischg flow, OH res fluor 92FIN/EZE195 9.5931032.84 8.13 1.12[1.03] 223 –420 Laser photol, OH LIF 91VAG/RAV184 1.81310121.07 1240 Flow, laser pyrolysis, LIF 85SMI/FAI196 4.6131091.08[1.14] 300 Flash photol, reson fluor 81HUS/PLA197 4.2031091.01[1.09] 296 Flash photol, reson fluor 80SWO/HOC198 4.5831091.12 298 Rel rate, flow, chemilum 76COX/DER199 3.9231091.04 295 Flash photo, UV abs 75OVE/PAR186 Experimental (excluded data) 1.70310121.19[1.3] 1072 –1139 Shock, OH abs 12HON/DAV200 5.823101111.81 [1.25] 178 –298 Laser photol, OH LIF 93SHA/SMI201 1.6231012[1.5] 1030 Flame, GC, model 92YET/DRY202 2.60310121.19[1.5] 1200 Shock, OH abs 89BOT/COH203 1.5531071.83 11.64 1.16[1.3] 298 –1510 Fast flow,flash photol, reson fluor 85MAD/FEL204 2.233101321.2 1.25[1.7] 340 –1250 Flow disch, OH res fluor 84JON/MUL205 1.5531062.13 10.23 1.1[1.2] 403 –696 Photol, GC 83BAU/CRA206 3.373101216.38 1.27 278 –473 Flow disch, OH res fluor 82JEO/KAU207 7.9631061.92 11.31 1.13[1.3] 298 –1020 Flash photo, reson fluor 80TUL/RAV208 2.5310121.3 1300 Shock, flash photol, OH reson abs 78ERN/WAG192 3.4731033.08 8.4 1.2[1.6] 300 –900 Flash photo, reson fluor 76ZEL/STE209 5.7331091.15[1.4] 296 Dischg flow, LMR 76HOW/EVE210 2.333101215.46 [1.5] 300 –700 Flash photo, reson fluor 75STE/ZEL1211 2.83101215.48 [1.3] 300 –480 Flash photo, reson fluor 75STE/ZEL2211 2.313101215.3 1.05[1.22] 290 –440 Flow disch, OH res fluor 74MAR/KAU212 1.423101214.22 1.09[1.16] 240 –373 Flash photo, reson fluor 74DAV/FIS213 3.603101325.11 1.2[1.5] 1100 –1900 Model, flame, MS LIB 73PEE/MAH82 2.523101320.95 1.38[2.3] 298 –753 Rev rxn, photol, GC 71BAK/BAL87 3.973101215.8 1.25[1.7] 301 –492 Flash photo, OH abs 70GRE214 6.531091.23[1.5] 300 Dischg flow, OH ESR 67WIL/WES215 5.013101320.92 [4] 298 –423 Static, flash photo, OH abs 67HOR/NOR216 3.00310121.33[1.4] 1280 Flame, MS, model 67DIX/WIL79 1.433101427.19 [4] 372 –1340 Flame, model, MS LIB 63FRI217 J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-26 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprexperimental data at low temperatu res and extrapolation to higher temperatures using parametric corre lations that we developed (see the discussion in Sec. 8). The rate constants at high temperatures for these two reactions were interpolated between those for CH 4and for CHF 3(ex- perimental data exist for both of these reactions at high temperatures). The interpolation was determined by considering the relative Afactors and activation energies from the qu antum chemical study of Matsugi and Shiina,59trends in Afactors with fluorine substitution, and correlations between the rate constants at room temperature and the activation en- ergies at high temperatures for t his and other homologous series ( fluo- romethanes + X →fluoromethyls + HX). This procedure provides a relatively self-consistent set of rate expressions. We assign uncertainties of f/equals1.5 for these two rate constants at high temperatures by considering a range of parameters that a re consistent with our empirical correlations.Table 20 lists rate constants compiled from the literature for CHF 3+O H→CF3+H 2O, and Fig. 18 shows the recommended rate expression along the experimental rate constants used in the fits. This reaction, unlike those for CH 3F and CH 2F2, has rate data at high temperatures. There are also a fair amount of reliable rate constants at low temperatures —six that are individual points at room temperature and four with temperature dependencies. Because of the extent of the data and good agreement, we assign expanded uncertainty factor f(2σ)f/equals1.3 and f/equals1.22 at high and low temperatures, respectively. Finally, Fig. 19 compares the recommended rate constants for the reaction of OH with the three fluoromethanes, showing that the rate constant for CHF 3+ OH has a much stronger temperature depen- dence than those for CH 3F and CH 2F2. 7. Fluoromethanes +F→Fluoromethyls +HF 7.1. Overview In this section, we compile rate constants from both experi- mental and quantum chemical studies for the reaction of F atoms withTABLE 17. (Continued. ) An E f T Method Reference 3.53101437.66 1.7[5] 1200 –1800 Flame, model, MS 61FEN/JON92 Theoretical 6.6231042.56 9.15 [1.4(1.08)] 300 –2000 CBS-QB3 Eckart 14MAT/SHI59 n/a 24.4 200 –1000 CBS-RAD 09BLO/HOL218 n/a 15.1 0 B3LYP/6 –311 0 K barrier 02KOR219 4.6431052.30 11.47 [1.3] 200 –1500 CCSD VTST 01MAS/GON220 n/a 16.6 298 MP2/6-31 01ELT221 n/a 19.7 280 –420 G2 99KOR/KAW222 8.3731023.21 6.59 [1.5] 200 –2500 G2 Eckart 98SCH/MAR61 n/a 10.0 0 B3LYP/6 –311 0 K barrier 96JUR223 FIG. 13. CH4+O H→CH3+H 2O rate constants and recommended rate expression. Rate constants k(cm3mol−1s−1). Legend: Black dots (05SRI/SU188), black circles (04BRY/KNY181), green squares (02BON/DAE13), black triangles (97GIE/TAL183), blue circles (94MEL/TET193), blue dots (93DUN/TUL182), green dots (91VAG/ RAV184), red squares (92LAN/LEB194), yellow dots (92FIN/EZE195), black X ’s (15BUR/SAN187evaluation), red plus (92YET/DRY202), blue X (67DIX/WIL79), green plus (85SMI/FAI196), yellow dash (89BOT/COH203), brown circle (12HON/DAV200), and red triangle (12AME/MIY185). Note that only data within about 3 σof the recommended rate expression are included in this figure. Note also that all of the experimental data below room temperature are represented by the recommended rate expression from the evaluation by Burkholder et al.187(using black X ’s). FIG. 14. CH 4+O H→CH 3+H 2O rate constant residuals relative to the recommended rate expression. Rate constants k(cm3mol−1s−1). Legend: Black dots (05SRI/SU188), black circles (04BRY/KNY181), green squares (02BON/DAE13), black triangles (97GIE/TAL183), blue circles (94MEL/TET193), blue dots (93DUN/ TUL182), green dots (91VAG/RAV184), brown squares (92LAN/LEB194), yellow dots (92FIN/EZE195), black X ’s with line (15BUR/SAN187evaluation), blue asterisks (92FIN/EZE195), red X (81HUS/PLA197), blue square (76COX/DER199), red dia- mond (80SWO/HOC198), blue triangle (75OVE/PAR186), green plus (85SMI/FAI196), red triangle (12AME/MIY185). J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-27 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprthefluoromethanes (and methane) that abstract H atoms. Based on our analysis and fitting procedures, we recommend rate expressions for these reactions over a wide range of temperatures. Experimental rate constants are only available at lower temperatures (about 200–550 K). We extend these rate constants to higher temperatures employing quantum chemical calculations (done by others) as frameworks and empirical correlations (see discussion later). A summary of the recommended rate expressions is given in Table 21 . Individual rate constants and expressions for H abstraction by F atoms from the fluoromethanes are given in the following tables: CH 4 (Table 22 ), CH 3F(Table 23 ), CH 2F2(Table 24 ), and CHF 3(Table 25 ). Figure 20 provides an Arrhenius plot of the experimental data for these four compounds along with the recommended rate expressions. The rate expressions are represented by k(T)/equalsATnexp(−E/RT), where A,n, and E(pre-exponential factor, temperature coef ficient, and energy coef ficient, respectively) are given in the tables. Expanded uncertainty factors f(2σ) are also provided. The uncertainty factors are as provided in the original works. Uncertainty factors that we assigned are 2-sigma coverage. The range of temperatures for the rate expressions are given, as is a short description of the method used to determine the rate constants. Along with the experimental and computed rate constants, we provide recommended rate expressions for these reactions over a range of temperatures. Two sets of rec- ommended expressions are provided: the first are extended Arrhenius expressions with A,n, and Ethat are good from about 200 to 2000 K and the second are simple Arrhenius expressions with just Aand E that are good at high temperatures from about 1200 to 1800 K — relevant to combustion chemistry. We also provide a rate constant for each reaction at room temperature (298 K) based on a weighted average of the experimental measurements —although we would recommend values at 298 K from our rate expressions. Unfortunately, there are no measurements at high temperatures for H abstraction from the fluoromethanes by F atoms where this chemistry is very important in describing the decomposition of the agents. The flammability of the neat agents in combustion processesin air is extremely dependent upon the hydrogen content. For neat agents, where there is no hydrocarbon fuel present, the flames become hydrogen-poor and fluorine-rich. The only rate measurements for these reactions are at low temperatures. For CH 4and CHF 3, rate constants are available in the range of about 200 –550 K, while for CH 3F and CH 2F2, it is a much more limited range just below room temperature (from about 200 to 300 K). For each of the reactions,there are multiple determinations of the rate constant at roomtemperature (300 K) but only a few temperature-dependent mea-surements. However, these temperature-dependent measurementsare deemed reliable. A summary of the rate constants for this series of reactions is provided in Fig. 20 . The many single room-temperature measure- ments are not shown for clarity. There also are a few temperature-dependent measurements that are not shown —they are inconsistent with the other measurements and the recommended rate expressions. The experimental data are represented by points (circles and squares). The solid lines represent the recommended rate expression. In ad-dition, dashed lines represent the variational transition state theorycalculations by Wang et al. 247,248using a PES determined from ab initio quantum chemical calculations using coupled cluster theory. Other rate expressions from ab initio calculations are not shown in Fig. 20 for clarity. We now discuss in more detail the experimental data and the recommended rate expressions for this homologous series ofreactions. There are many studies measuring the rate of H abstraction by F atoms at room temperatures. The experimental data for temperature-dependent rate constants, however, are more limited. Persky hasmeasured the rate constants at below room temperature (189 –298 K) for CH 3F and CH 2F2and at below and above room temperature (184–406 K) for CH 4.249–253Wagner et al.254and Beiderhase et al.255 have measured the rate constants at below room temperature (220–312 K) for CH 4and CH 3F, respectively. There have been three measurements from below room temperature to slightly highertemperatures ( >400 K) of the rate constants for CHF 3+F→CF3+H F by Louis and Sawerysyn,256Maricq and Szente,257and Clyne et al.258 Wang and coworkers have used quantum chemical methods to compute rate expressions for the reaction of F atoms with methaneand all of the fluoromethanes. 247,248,259Other work using quantum chemical methods is listed in the tables. Wefitted our rate expressions to the low-temperature data and then utilized empirical trends in the derived Afactors at high tem- peratures, activation energies vs heats of reaction, and temperaturecoefficients T nto extrapolate the rate expressions to higher tem- peratures in a self-consistent manner (see Sec. 8about this param- eterization). We also considered the predicted temperaturedependence from quantum calculations. We benchmarked our rate expression for CH 4+ F to the ex- perimental data of Persky and found that it agrees with our rec-ommended values to within about 20%. Note that Persky revised hisexpression in 2008 from that reported in 2006 using an updatedexpression for the rate constant of H 2+ F (relative rate constants were measured in this work).249,250We note that the temperature de- pendence at low temperatures for the data of Persky is weaker thanthat from our recommended expression (we calculated effectiveactivation energies of 1.8 kJ mol −1vs 2.5 kJ mol−1), and the FIG. 15. CH 4+O H→CH 3+H 2O rate constant residuals for excluded data relative to the recommended rate expression. Legend: Black dots (86FEL/MAD163), black circles (83BAU/CRA206), black open triangles (82JEO/KAU207), green squares (80TUL/RAV208), blue circles (76ZEL/STE209), blue dots (75STE/ZEL1211), green dots (75STE/ZEL2211), red squares (73PEE/MAH82), yellow filled triangles (71BAK/ BAL87), brown filled diamonds (70GRE214), brown open diamonds (67HOR/ NOR216), blue plus signs (63FRI217), and red asterisks (61FEN/JON92). Note that not all of the excluded data provided in Table 17 are included here. J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-28 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprtemperature dependence from Foon and Reid is even weaker (1.2 kJ mol−1). Given this uncertainty in the temperature dependence of the rate expression, we assign an uncertainty of a factor of f/equals1.4 at high temperatures. Our recommended rate expressions for reaction of F atoms with CH 3F and CH 2F2agree well with the data of Persky to within 2% and 4%, respectively. We excluded the data of Beiderhase for CH 3F be- cause they had a temperature dependence (activation energy) that wasinconsistent with the temperature dependence (5.0 kJ mol −1vs 3.5 kJ mol−1) from our self-consistent rate expressions developed for the fluoromethanes + F homologous series. We benchmarked the rate constants for CHF 3at low temperatures with the data of Louis and Sawerysyn and Maricq and Szente with our recommended values andfound that they agree to within 6% and 11%, respectively. We ex-cluded the higher temperature data by Clyne et al. because they had a weak temperature dependence inconsistent with our self-consistentrate expressions, resulting in a rate constant that was about 40% lower at the highest temperature (667 K). 7.2. Evaluation procedure The recommended rate expressions were determined by fitting the low-temperature experimental data and the scaled rate constants at high temperatures (1200 K –1800 K) from the work of Wang et al. 247 The procedure was to firstfit the ab initio rate expressions at T/equals (1200 –1800) K to determine AandE(simple Arrhenius expression). The values of Aand Ewere then (iteratively) adjusted to arrive at scaled values that were best self-consistent with fits to the experi- mental data and the scaled rate constants. This procedure worked well for CHF 3+ F and CH 2F2+F—where the ab initio andfitted curves are very similar in curvature. For CH 3F + F, however, there was some tension in the fitting procedure —where we obtained values for the temperature coef ficient ( n) that ranged from about 1.1 to 1.4. WeTABLE 18. CH3F+O H→CH2F+H 2O rate expressions. k(T)/equalsATnexp(−E/RT).k(T), temperature-dependent rate constant. A, pre-exponential factor. n, temperature coef ficient. E, energy coef ficient. f(2σ), expanded uncertainty factor. Units: k(cm3mol−1s−1),E(kJ mol−1),T(K). Uncertainty factors when reported in the literature are given in the table. In addition, we provide deviations (uncertainty factors f) from our recommended rate constants in square “[f]”brackets with the uncertainty factor at low temperatures given in parentheses “(f).” An E f T Method Reference Evaluations 1.0631062.04 5.70 1.5(1.22) 240 –1800 Recommend This work 2.403101330.44 1.5 1200 –1800 Recommend, high T fit This work 1.323101211.64 1.1[1.1] 243 –480 Review 15BUR/SAN187 Experimental (preferred data) 2.653101213.8 [1.2] 308 –398 Photol, FTIR 96DEM224 1.323101212.06 [1.15] 298 –363 Photol, FTIR 95HSU/DEM225 1.053101210.81 [1.1] 243 –373 Laser photol, LIF 93SCH/TAL226 4.933101215.71 1.08[1.4] 292 –480 Dischg flow, reson fluor 82JEO/KAU207 Experimental (preferred data, 298 K) 1.1631010[1.3] 298 Ultrasonic, UV abs 98KOW/JOW227 9.0431091.07[1.3] 298 Photol, FTIR 93WAL/HUR228 1.03310101.14[1.2] 298 Pulse radiolysis, reson abs 88BER/HAN229 1.31310101.08[1.1] 297 Flash photo, reson abs 79NIP/SIN230 9.6431091.22[1.3] 296 Dischg flow, LMR 76HOW/EVE210 Experimental (excluded data, all duplicates) 1.0131010[1.2] 298 Photol, FTIR 96DEM224 1.0231010[1.15] 298 Photol, FTIR 95HSU/DEM225 1.3431010[1.1] 298 Laser photol, LIF 93SCH/TAL226 8.643109[1.4] 298 Dischg flow, reson fluor 82JEO/KAU207 Theoretical 1.9331052.37 6.15 [2(1.2)] 300 –2000 CBS-QB3 Eckart 14MAT/SHI59 4.0331010[3] 300 DFT KKMLYP/6 –311 Wigner 12PET/HAR231 1.9131062.15 8.92 200 –700 CC/cbs CVT/SCT, our fit 08MAR/GRU232 4.1231042.60 5.72 200 –1500 mPW1PW91/6 –31 CVT/ μOMT 06ALS/SWA233 n/a 9.6 0 B3LYP/6 –311 0 K barrier 02KOR219 2.2131062.03 7.66 200 –1000 MP2/6 –31 CVT/ μOMT, our fit 01LIE/YOU234 n/a 21.34 298 CC/pVTZ 01LIE/YOU234 1.143101213.18 [2.5] 250 –400 MP4/6-311G Wigner 00LOU/GON235 2.2831023.35 3.06 [4(1.3)] 300 –2000 G2 Eckart 98SCH/MAR61 4.4431034.78 −4.52 200 –1000 CVT/ μOMT 98ESP/COI236 n/a 0.4 0 B3LYP/6 –311 0 K barrier 96JUR223 J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-29 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprTABLE 19. CH 2F2+O H→CHF 2+H 2O rate expressions. k(T)/equalsATnexp(−E/RT).k(T), temperature-dependent rate constant. A, pre-exponential factor. n, temperature coef ficient. E, energy coef ficient. f(2σ), expanded uncertainty factor. Units: k(cm3mol−1s−1),E(kJ mol−1),T(K) An E f T Method Reference Evaluations 3.2931061.86 7.52 1.5(1.17) 220 –1800 Recommend This work 1.683101330.09 1.5 1500 –1800 Recommend, high T fit This work 1.023101212.47 1.2 220 –492 Review 15BUR/SAN187 Experimental (preferred data) 1.083101212.89 1.07 297 –383 Photol, FTIR 95HSU/DEM225 9.463101112.22 1.13 220 –380 Flash photol, LIF 91TAL/MEL237 2.653101214.72 1.08 250 –492 Dischg flow, reson fluor 82JEO/KAU207 3.893101216.63 1.74 293 –429 Dischg, reson fluor 79CLY/HOL238 Experimental (preferred data, 298 K) 6.0531091.03 298 Dischg flow, reson fluor LIB 00SZI/DOB239 5.3031091.16 298 Pulse radiolysis, reson abs 88BER/HAN229 7.0531091.12 297 Flash photol, reson abs 79NIP/SIN230 4.7031091.15 296 Dischg flow, LMR 76HOW/EVE210 Experimental (excluded data, 298 K, some duplicates) 9.893109[1.6] 298 Ultrasonics, UV abs 98KOW/JOW227 5.9431091.07 298 Photol, FTIR 95HSU/DEM225 6.8231091.13 298 Flash photol, LIF 91TAL/MEL237 6.973109[1.17] 298 Dischg flow, reson fluor 82JEO/KAU207 4.7331091.74[1.2] 298 Dischg, reson fluor 79CLY/HOL238 Theoretical 1.3231052.35 6.40 300 –2000 CBS-QB3 Eckart 14MAT/SHI59 3.7731010300 DFT KKMLYP/6 –311 Wigner 12PET/HAR231 n/a 24.4 298 CBS-RAD 09BLO/HOL218 2.3831042.86 10.06 200 –1500 mPW1PW91/6 –31 CVT/ μOMT 07ALB/SWAa240 n/a 8.8 0 B3LYP/6 –311 0 K barrier 02KOR219 n/a 16.8 298 MP2/6-31 01ELT221 3.253101213.39 250 –400 MP4/6-311G Wigner 00LOU/GON235 3.33310−55.43 −5.13 210 –500 CCSD/pVTZ VTST/LCT 01GON/LIU241 2.6431023.27 3.98 300 –2000 G2 Eckart 98SCH/MAR61 n/a 0.4 0 B3LYP/6 –311 0 K barrier 96JUR223 n/a 9.2 298 MP2/3-21 93BOT/POG242 FIG. 17. CH 2F2+O H→CHF 2+H 2O. Rate constants k(cm3mol−1s−1). Red solid line (this work recommended), blue dots (91TAL/MEL237), green squares (95HSU/ DEM225), black X ’s (79CLY/HOL238), and red circles (82JEO/KAU207). FIG. 16. CH3F+O H→CH2F+H 2O. Rate constants k(cm3mol−1s−1). Red solid line (this work recommended), large black dot (98KOW/JOW227), open blue circles (96DEM224), green squares (95HSU/DEM225), large blue triangle (93WAL/HUR228), red circles (93SCH/TAL226), blue dots (82JEO/KAU207), large blue X (79NIP/ SIN230), large yellow dot (76HOW/EVE210), and dashed line (15BUR/SAN187 recommended). J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-30 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprselected an intermediate value of n/equals1.25. As can be seen in Fig. 20 , the calculated rate curve for CH 3F + F shows much more curvature than thefitted curve. We speculate that the additional curvature is due to the inadequacy of variational transition-state theory to represent acomplicated and multi-dimensional quantum dynamical PES (seediscussion below). This is evident in the many quantum dynamics studies for the CH 4+ F reaction (see Table 22 ), which disagree substantially on the degree of curvature —these studies found that the rate constants were extremely dependent upon fine details of the PESs used to determine the rates of reaction. The recommended rate expression for the reaction CH 4+Fi s taken from the quantum dynamics study of Chu et al.260This rate expression agrees with the experimental data at low temperatures within the experimental uncertainties. Note that all of the other quantum calculations for CH 4provided in Table 22 are also consistent with the experimental data. We chose the rate expression from Chu et al. for several reasons. It is one of the higher level calculations using a re fined PES. The temperature coef ficient ( n/equals0.91) from thisstudy was an intermediate value compared to other computations (n/equals0.53–1.57). In addition, we found that the temperature coef ficient (n) for the reaction series CH 4,C H 3F, CH 2F2,a n dC H F 3varied smoothly (with a slight quadratic curvature) as a function of log 10(k298) where k298is the rate constant for each reaction at 298 K. In summary, we compiled rate expressions for H abstraction by F atoms from the fluoromethanes (and methane) and provide rec- ommended rate expressions over a range of temperatures. These rate expressions agree with the experimental data at low temperatures, appear to be self-consistent with one another, and are generally consistent with rate expressions from quantum calculations. 7.3. Discussion of fluorine impact on rate constants If we consider the rate parameters in Table 21 ,w efirst see that the effective Afactors at high temperatures do not scale with the number of hydrogen atoms as found for abstraction by other radicals. In contrast, the effective activation energies at high temperatures scaleTABLE 20. CHF 3+O H→CF3+H 2O rate expressions. k(T)/equalsATnexp(−E/RT).k(T), temperature-dependent rate constant. A, pre-exponential factor. n, temperature coef ficient. E, energy coef ficient. f(2σ), expanded uncertainty factor. Units: k(cm3mol−1s−1),E(kJ mol−1),T(K) An E f T Method Reference Evaluations 1.2031061.85 13.71 1.3(1.22) 250 –1800 Recommend This work 5.643101236.15 1.3 1200 –1800 Recommend, high T fit This work 3.673101118.79 chk 253 –500 Review 15BUR/SAN187 Experimental (preferred data) 5.843101236.56 1.22 995 –1663 Shock CH 07SRI/SU243 2.893101118.13 1.27 253 –343 Rel rate, static, photol, GC 03CHE/KUT244 3.853101119.54 [1.1] 298 –383 Photol, FTIR 95HSU/DEM225 4.183101119.12 [1.2] 298 –500 Laser photol, LIF 93SCH/TAL226 1.823101224.20 1.1 387 –480 Dischg flow, reson fluor 82JEO/KAU207 4.0310111.25 1350 Shock, flash photol, OH reson abs 78ERN/WAG192 0.19 1300 Rel rate CH4+OH, shock, flash photol, OH reson abs76BRA/CAP245 Experimental (excluded data, some duplicates at 298 K) 1.9231081.27 298 Rel rate, static, photol, GC 03CHE/KUT244 6.623101119.12 [2] 298 –750 Photol, LIF 97MED/FLE179 2.953108[1.8] 298 –750 Photol, LIF 97MED/FLE179 1.453108[1.10] 298 Photol, FTIR 95HSU/DEM225 1.863108[1.16] 298 Laser photol, LIF 93SCH/TAL226 1.3931091.17[9] 298 Pulse radiolysis, reson abs 88BER/HAN229 1.0431081.1 298 Dischg flow, reson fluor 82JEO/KAU207 2.131081.5 297 Flash photo, reson abs 79NIP/SIN230 1.2131082 296 Dischg flow, LMR 76HOW/EVE210 Theoretical 1.3931032.83 11.31 300 –2000 CBS-QB3 Eckart 14MAT/SHI59 5.943107300 DFT KKMLYP/6 –311 Wigner 12PET/HAR231 4.2531033.22 9.96 298 –2500 G3B3 07ZHA/LIN124 4.6031032.87 14.70 200 –1500 mPW1PW91/6 –31 CVT/ μOMT 07ALB/SWAb246 n/a 29.6 298 MP2/6-31 01ELT221 1.203101222.82 250 –400 MP4/6-311G Wigner 00LOU/GON235 6.93310 3.66 8.86 300 –2000 G2 Eckart 98SCH/MAR61 n/a 5.9 0 B3LYP/6 –311 0 K barrier 96JUR223 J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-31 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprroughly with the C –H BDEs (see Table 2 ):Eaand BDEs: CH 3F <CH 2F2<CH 4<CHF 3. In contrast, we see that both the curvatures as expressed by the temperature coef ficients n(Tn) and the rate constants at low temperatures scale well with the number of fluorine atoms in the molecule. We interpret this as a clear indication of semi- ionic contributions to the rates of these reactions in this extraordinary class of reactions involving the reaction of F atoms (highly electro- negative) with fluorinated methanes (electropositive carbon atom). These are nearly barrierless reactions, and thus, the strong curvature in the rate constants cannot be due to tunneling. The strong overall temperature dependence would be indicative of the turning point from products to reactants moving along the reaction coordinate and likely related to the charge on the central carbon atom. At low temperatures (low collision energies), semi-ionic forces may domi- nate the rate of reaction (related to the charge on the carbon atom). On the other hand, at high temperatures (high collision energies), the covalent nature of the bonding is dominant (evident by the barrier scaling with BDEs). Much of the following discussion is based on that in earlier work by others.247,248,259,260,268 –271A more complete and accurate dis- cussion of these quantum dynamical effects for F atom abstraction of H atoms can be found in the quantum chemical works referenced in the rate expression tables for these molecules (the quantum dy- namical studies of the CH 4+F→CH 3+ HF provide the most information). All of these reactions CX 3H+F→CX3+ HF display a large change in rate constants with temperature and signi ficant curvature to the Arrhenius rate expressions. These effects need to be explained because these reactions are barrierless reactions (or nearly) that are not classical abstraction reactions. Classical abstraction reactions essentially involve the intersection of two covalent bond-breaking PESs (e.g., C –H and H –F in the forward and reverse directions, respectively). In contrast, these reactions proceed essentially only on attractive potential surfaces with very early turning points. The high electronegativity of the F atom results in a more ionic and less covalent character to the bond breaking with polarization of the H –F bond and a charge separation. The largest temperature dependence isobserved with the reaction CHF 3+ F, where electron withdrawal by the three fluorine atoms on CHF 3leads to a much stronger C –H bond. FIG. 18. CHF 3+O H→CF3+H 2O. Rate constants k(cm3mol−1s−1). Red solid line (this work recommended), blue dots (07SRI/SU243), blue triangles (95HSU/ DEM225), green squares (03CHE/KUT244), black X ’s (93SCH/TAL226), and red circles (82JEO/KAU207). FIG. 19. Fluoromethanes + OH →fluoromethyls + H 2O. Recommended rate constants k(cm3mol−1s−1). See Figs. 15 –17for rate data and fits for individual reactions. TABLE 21. Recommended rate expressions for H abstraction from the fluoromethanes by F atoms. k(T)/equalsATnexp(−E/RT).k(T), temperature-dependent rate constant. A, pre- exponential factor. n, temperature coef ficient. E, energy coef ficient. f(2σ), expanded uncertainty factor. Units: k(cm3mol−1s−1),E(kJ mol−1),T(K).A/ACHF3 is the ratio of the pre- exponential to that for CHF 3.ΔrH298, standard enthalpy of reaction at 298 K Reaction AnE f T log10(k300) ΔrH298 CH 4+F→CH 3+ HF 2.93 310110.91±0.21 0.79 ±1.51 1.35(1.23) 180 –1800 13.58 −131.1 CH 3F+F→CH 2F + HF 1.91 310101.25±0.29 0.81 ±1.49 1.35(1.24) 180 –1800 13.24 −147.0 CH 2F2+F→CHF 2+ HF 2.05 31081.79±0.25 1.30 ±1.65 1.35(1.35) 180 –1800 12.52 −144.0 CHF 3+F→CF3+ HF 4.57 31042.77±0.20 3.00 ±1.47 1.35(1.20) 180 –1800 11.00 −123.8 A/ACHF3 CH 4+F→CH 3+ HF 5.81 3101411.95 1.35 1200 –1800 1.42 CH 3F+F→CH 2F + HF 6.86 310146.87 1.35 1200 –1800 1.68 CH 2F2+F→CHF 2+ HF 5.54 310149.77 1.35 1200 –1800 1.35 CHF 3+F→CF3+ HF 4.09 3101415.54 1.35 1200 –1800 1.00 J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-32 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprThe strong temperature dependence of the rates of these r e a c t i o n si sd u et oan u m b e ro ff a c t o r s .Q u a n t u mc h e m i c a lc a l - culations reveal the existence of van der Waals adducts X 3C–H–F in the entrance channel (reactants) due to long-range forces. Thequantum dynamical studies of these systems (supported by ex- perimental evidence) suggest that rovibrational excitations cou- pled to the electronic PES result in reactive resonances thatenhance the reactivity. Although there are essentially no “barriers ” to these reactions, the turning point moves along the reaction pathas a function of energy, leading to an entropic temperature de-pendence to the reaction rate. This effect requires the use ofvariational transition state the ory or quantum dynamical studies to accurately predict reactivity i n these systems. In addition, the ground state of the F atom is split into two states 2P3/2and2P1/2 with degeneracies of 4 and 2, resp ectively, and the higher state2P1/2 is separated by a spin –orbit splitting of 404.10 cm−1(4.83 kJ mol−1). This leads to a change in effective electronic degeneracy (and rate constant) with temperature. In addition, the quantumdynamics studies suggest that the 2P1/2state may have a signi fi- cantly higher reaction rate.In summary, all of these factors contribute to the large change in rate constants as a function of temperature —even though these are barrierless reactions. 8. Discussion 8.1. Overview In this work, we complied and evaluated rate constants for the abstraction of H from the fluoromethane homologous series (CH 4,CH 3F, CH 2F2,C H F 3)b yt h er a d i c a l sHa t o m ,Oa t o m ,F atom, and OH and provided self-consistent recommended rate expressions for these reactions from room temperature to com- bustion temperatures (300 –1800 K). There were limited experi- mental temperature-dependent rate constants for some of the reactions. In these cases, we pre dicted the rate constants using relative rates employing a combination of rate constants from ab initio calculations and self- consistent parametric fits considering all the reactions as a set of homologous reactions. In this work, the rate expressions we provided are consistent with all the reliable experimental data within reported uncertainties. Other rateTABLE 22. CH 4+F→CH 3+ HF rate expressions. k(T)/equalsATnexp(−E/RT).k(T), temperature-dependent rate constant. A, pre-exponential factor. n, temperature coef ficient. E, energy coef ficient. f(2σ), expanded uncertainty factor. Units: k(cm3mol−1s−1),E(kJ mol−1),T(K). See Sec. 10for de finitions of acronyms and abbreviations in the Method column. An E f T Method Reference Evaluations 2.93310110.91 0.79 1.35(1.23) 180 –1800 Recommend. Selected 09CHU/HAN260This work 5.813101411.95 1.4 1200 –1800 Recommend, high T fit This work 7.71310131.79 1.12 184 –406 Review 06PER252 4.09310131.2 298 Review 93WAL/HUR228 Experimental (preferred data) 7.71310131.79 1.12 184 –406 Corrected 96PER24998PER250 Experimental (preferred data, 298 K) 3.78310131.26 298 Average This work 3.73310131.1 298 Rel rate, dischg flow, MS 98PER250 2.83310131.2 298 Rel rate, photol, FTIR 95MOO/SMI261 3.19310131.1 294 Rel rate, photol, FTIR 94MOO/SMI262 3.97310131.1 298 Fast flow, MS 89WOR/HEY263 4.0031013[1.03] 298 Pulse photol, UV detect 88PAG/MUN264 3.44310131.05 298 IR photodissoc, HF emission 82FAS/NOG265 4.52310131.2 298 Dischg flow, atomic reson 78CLY/NIP266 4.3031013[1.1] 298 Laser photol, HF emission 72KOM/WAN267 3.49310131.15 298 Rel rate, flow, GC 71FOO/REI253 4.74310131.5[1.3] 298 Fast flow, MS 71WAG/WAR254 Experimental (excluded data) 9.88310132.20 1.1 184 –406 Rel rate, dischg flow, MS 96PER249 3.3310144.81 [1.5] 250 –312 Fast flow, MS 71WAG/WAR254 Theoretical 2.6331091.57 −1.74 [1.07] 184 –404 Quant dynamics, PES, CC/pVTZ 13WAN/CZA259 2.93310110.91 0.79 selected 180 –400 Quant dynamics, PES, CC/pVTZ 09CHU/HAN260 2.00310120.53 0.58 [1.5] 180 –500 Quant dynamics, PES, CC/pVTZ 07ESP/BRA268 1.2731091.70 −1.58 [1.5] 200 –2000 CC/pVTZ, VTST 05ROB/MAC269 5.68310110.71 −0.29 [1.5] 180 –500 Quant dynamics, PES, CC/pVTZ 05RAN/NAV270 7.0731091.30 −1.92 [2.0] 200 –500 Quant dynamics, PES, CC/pVTZ 04TRO/MIL271 J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-33 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprTABLE 23. CH 3F+F→CH 2F + HF rate expressions. k(T)/equalsATnexp(−E/RT).k(T), temperature-dependent rate constant. A, pre-exponential factor. n, temperature coef ficient. E, energy coef ficient. f(2σ), expanded uncertainty factor. Units: k(cm3mol−1s−1),E(kJ mol−1),T(K) An E f T Method Reference Evaluations 1.91310101.25 0.81 1.35(1.24) 190 –1800 Recommend This work 6.86310146.87 1.25 1200 –1800 Recommend, high T fit This work Experimental (preferred data) 6.20310133.24 1.15 189 –298 Flow, dischg, MS 03PER251 2.4231072.4 0 1.1[1.2] 220 –300 Flow, dischg, MS 95BEI/HAC255 Experimental (preferred data, 298 K) 1.85310131.24 298 Weighted average This work 1.66310131.1 298 Flow, dischg, MS 03PER251 2.06310131.1 298 Flow, dischg, MS 95BEI/HAC255 1.69310131.2 298 Rel rate, photol, FTIR 95MOO/SMI261 2.23310131.2 295 Rel rate, laser photol, FTIR 93WAL/HUR228 1.33310131.06 300 Rel rate, electron beam, IR 83MAN/SET272 1.8231013[1.1] 298 Rel rate, flow, dischg, HF emission 77SMI/SET273 2.1631013[1.1] 298 Rel rate, H2+F →HF + H 75MAN/GRA274 Experimental (excluded data) 4.00310121.23[5] 298 Pulse radiolysis, UV detect 98KOW/JOW227 2.4131091.75[1.3] 298 Electron beam, ESR, CH 86JON/MA85 5.3031013[2.8] 298 Rel rate, fast flow, chemilum 73POL/JON275 Theoretical 3.2431081.75 −0.91 180 –1800 CC/6 –311 VTST 05WAN/LIU247 6.28310148.60 1200 –1800 High T fit to 05WAN/LIU247This work TABLE 24. CH2F2+F→CHF 2+ HF rate expressions. k(T)/equalsATnexp(−E/RT).k(T), temperature-dependent rate constant. A, pre-exponential factor. n, temperature coef ficient. E, energy coef ficient. f(2σ), expanded uncertainty factor. Units: k(cm3mol−1s−1),E(kJ mol−1),T(K) An E f T Method Reference Evaluations 2.0531081.79 1.30 1.35(1.35) 180 –1800 Recommend This work 5.54310149.77 1.35 1200 –1800 Recommend, high T fit This work Experimental (preferred data) 2.23310134.78 1.15 182 –298 Flow, dischg, MS 03PER181251 Experimental (preferred data, 298 K) 3.03310121.35 298 Weighted average This work 3.25310121.1 298 Flow, dischg, MS CH 03PER181251 2.83310121.2 298 Rel rate, photol, FTIR 95MOO/SMI261 2.59310121.2 295 Rel rate, laser photol, FTIR 93WAL/HUR228 3.44310121.05[1.1] 298 Electron beam, dischg flow, LIF 85CLY/HOD276 2.34310121.33 300 Rel rate, electron beam, IR 83MAN/SET272 4.0631012[1.3] 298 Rel rate H2 + F /equalsHF + H 75MAN/GRA274 Experimental (excluded data) 1.26310121.47[2.5] 298 Pulse radiolysis, UV detect 98KOW/JOW227 5.9031012[1.9] 298 Rel rate, laser photol, FTIR 92NIE/ELL82277 1.1031013[3.5] 298 Rel rate, fast flow, chemilum 73POL/JON275 Theoretical 1.5431072.1 0.13 [1.4] 180 –1800 CC/6 –311 VTST 08WAN/LIU248 5.383101410.74 [1.2] 1500 –1800 High T fit to 05WAN/LIU247This work J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-34 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprexpressions could be provided that are statistically equivalent, but our intent in this work was to provide well-behaved rate expressions allowing for accurate interpolation and extrapolation.TABLE 25. CHF 3+F→CF3+ HF rate expressions. k(T)/equalsATnexp(−E/RT).k(T), temperature-dependent rate constant. A, pre-exponential factor. n, temperature coef ficient. E, energy coef ficient. f(2σ), expanded uncertainty factor. Units: k(cm3mol−1s−1),E(kJ mol−1),T(K) An E f T Method Reference Review 4.5731042.77 3.00 1.35(1.2] 210 –1800 Recommend This work 4.093101415.54 1.40 1200 –1800 Recommend, high T fit This work Experimental (preferred data) 1.263101312.22 1.5 297 –421 Dischg flow, MS 98LOU/SAW256 2.17310127.90 1.55 210 –353 Rel rate, flash photol, UV abs 93MAR/SZE257 6.383101210.00 [1.4] 301 –667 Dischg flow, MS 73CLY/MCK258 Experimental (preferred data, 298 K) 8.49310101.20 298 Weighted average This work 8.13310101.2 298 Rel rate, photol, FTIR 95MOO/SMI261 7.23310101.35 294 Photol, chemilum 94MOO/SMI262 8.43310101.3 295 Rel rate, laser photol, FTIR 93WAL/HUR228 7.2331010[1.2] 298 Rel rate CH 92NIE/ELL1009278 7.2331010[1.2] 298 Rel rate, flash photol, UV abs CH 92MAR/SZE279 9.0331010[1.05] 298 Dischg flow LIB 83CLY/HOD280 8.9931010[1.05] 298 Pulse radiolysis, ESR 76GOL/SCH281 Experimental (excluded data, 298 K, some duplicates) 9.08310101.5 298 Dischg flow, MS 98LOU/SAW256 4.88310121.2[60] 298 Pulse radiolysis, UV detect 98KOW/JOW227 7.2331010[1.2] 298 Rel rate, flash photol, UV abs CH 92MAR/SZE279 9.0331010[1.05] 298 Dischg flow LIB 83CLY/HOD280 4.3531011[5] 298 Rel rate, H2+F /equalsHF + H 75MAN/GRA274 1.9031011[2] 298 Rel rate, fast flow, chemilum 73POL/JON275 1.1331011[1.3] 298 Dischg flow, MS 73CLY/MCK258 Theoretical 7.1131042.69 4.14 180 –1800 CC/6 –311 VTST 08WAN/LIU248 3.283101415.63 1200 –1800 High T fit to 08WAN/LIU248This work n/a G2MP2, found no barrier 00OKA/TOM282 n/a 16.1 MP2/6 –311 barrier 99LOU/RAY283 FIG. 20. Fluoromethanes + F →fluoromethyls + HF rate constants. k(cm3mol−1s−1). Red solid lines represent recommended rate expressions. Dashed lines represent thecalculations by Wang et al. (05WAN/LIU,24708WAN/LIU248). Experimental data: CH 4 circles (06PER252), CH 3F circles (03PER251), squares (95BEI/HAC255), CH 2F2circles (03PER251), CHF 3circles (93MAR/SZE257),filled circles (98LOU/SAW256), and squares (73CLY/MCK258). FIG. 21. Normalized Afactors (per H atom) (on a logarithmic scale) at high temperatures as a function of the number of H atoms for the different reactants H, O, OH, and F. A(cm3mol−1s−1). J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-35 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprWe now discuss the systematic trends we have observed and provide a basis for our claim that the rate expressions are self- consistent. 8.2. Correlations for rate constant Afactors It was found that the relative Afactors for abstractions by H and O atoms for rate constants at high temperatures (1200 –1800 K) roughly scaled (within about 30%) with the number of hydrogens in the molecule (reaction path degeneracy), while the relative A factors for abstractions by OH and F changed upon fluorine substitution. This included both Afactors derived from our rate expressions and from TSs derived from reliable ab initio quantum chemical calcu- lations. The Afactors, of course, should not scale exactly with the number of hydrogens for a number of reasons: (a) TSs will move along the reaction coordinates in a homologous series, changing the shapes of the PESs; (b) the addition of heavy F atoms to replace light H atoms will impact the enthalpic differences between the TS and the reactants; and (c) the large electronegativity of additional F atoms will, of course, change the nature of the TSs. Figure 21 shows the normalized A factors (per H atom) at high temperatures for all of the H abstraction reactions from the fluoromethanes by H, O, OH, and F. We see that for abstraction by H and O, the normalized Afactors are very nearly independent of the degree of fluorination (within about 10% –20%). This is less than the assigned uncertainties of f/equals(1.3–1.5) for the rate constants. The independence of the normalized Afactors is unsur- prising since abstractions are simple reactions —bond breaking concurrent with bond formation, relatively tight (closer) TSs, and only slight changes expected in the structure and bond frequencies involving spectator atoms (those not involved in the reaction). In contrast, we see that the normalized Afactors for H abstraction by OH decrease by a factor of about (2.5 –3) from CH 4to CHF 3, and the opposite trend is observed for abstraction by F atoms, where the normalized Afactors increase by about a factor of (2.5 –3) from CH 4to CHF 3. The deviations of the normalized Afactors (per H atom) from the linear correlations illustrated in Fig. 21 are about factors of 1.01, 1.12, 1.20, and 1.08 (1%, 12%, 20%, and 8%) for abstractions by H, O, OH, and F. Thus, there is a high degree of correlation between the A factors and the number of H (or F) atoms. Although these deviations (10±10)% are the optimized a posteriori derived factors, nevertheless, they are signi ficantly less than the assigned uncertainty factors that are on the order of (40 ±20)%. The absolute values of the normalized Afactors for the series of reactions CH 4+ (O, H, F) only differ by a small amount, about (1.031014, 1.2531014, 1.531014)c m3mol−1s−1, respectively, while theAfactor for CH 4+ OH is about (6 –8) times less at about 1.5 31013 cm3mol−1s−1, suggesting a much tighter TS. The decrease in the nor- malized Afactor for abstractions by OH with fluorine substitution from CH 4to CHF 3is a valid systematic trend since the rate constants for these two reactions at high temperatur es are relatively well-known ( f/equals1.3). The change in the normalized Afactors for these reactions with OH is likely a consequence of several factors invol ving interaction between the elec- tropositive O –H bond and the very electronegative C –F bonds in the TS. The hindered rotor in the TS involving the OH with a barrier that will depend upon the number of flu o r i n ea t o m si nt h em o l e c u l e s ,a n d consequently a factor of about 1/2 Rin the entropy or a factor of 1.65 in the Afactor, might be expected considering that the OH in the CH 4+O HT Sis signi ficantly less constrained (essentia lly a free rotor) compared to the OH in CHF 3+ OH, which is much more constrained (hindered rotor) and closer to a vibration. In addition, both the magnitude and anhar- monicity of the bending normal modes involving the OH bond will be significantly impacted by the electrostatics of any O –H/C–Fi n t e r a c t i o n s . The consequences of these factors are that the Afactors do not scale roughly with the number of hydrogens like for abstractions by H and O atoms; instead, the ratios are about (1. 0, 3.0, 4.3, 10.9), indicating that the TSs become increasingly more constrained with fluorine substitution. There is also an apparent increase in the normalized Afactor for thefluoromethanes + F abstractions upon fluorine substitution, changing by about a factor of (2.5 –3.0) from CH 4to CHF 3. This is a consequence of the semi-ionic nature of this class of reactionsinvolving a highly electronegative F atom with molecules where the charge distribution drastically changes depending upon the number offluorine substitutions. This is a complicated series of reactions. They are very nearly barrierless reactions (see the references in Tables 21–24), and the temperature dependences of the rate constants are due to the turning point for the reactions moving along the reaction coordinate with temperature (energy), due to quantum dynamical effects, and due to the semi-ionic nature of these reactions. The “TS” for CHF 3+ F is earlier and looser (higher entropy), while that for CH 4+ F is later and tighter (closer). Such changes in the rate constants are to be expected because of the large changes in the electronegativity of the central carbon atom with fluorine substitution impacting the PES involving the attacking highly electronegative F atom. A much different type of barrier is to be expected for CHF 3+ F with a large positive charge on the carbon atom, compared to that for CH 4+Fw h e r et h e r ei sa substantial negative charge on the carbon atom. CHF 3has a large dipole moment (positive pointing toward the reaction site) along the reaction coordinate due to the electronegative fluorine sub- stitutions, and consequently, a long-range (loose) semi-ionic interaction with the attacking electronegative F atom is avail- able. In contrast, H atoms are electropositive, and the effective dipole moment along the reaction coordinate for the CH 4reaction is pointing away from the reaction site. Somewhat equivalently, thefluorine substitutions in CHF 3make the central carbon atom FIG. 22. Evans –Polanyi55plot of activation energy Eaat high temperatures (1200 –1800 K) as a function of heat of reaction ΔrH. The correlation is approx- imately Ea/equals55.79 + 0.356 ΔrH. The barriers for H and F atoms are about (8 –11) kJ mol−1and (4 –5) kJ mol−1higher, respectively, than the trend for O atoms and OH radicals. The barrier for CHF 3+O→CF3+ OH is about 10 kJ mol−1higher than the trend for other abstractions by O atoms. See discussion in text. J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-36 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprhighly electropositive, which makes the H atom being abstracted less electropositive than in CH 4, and consequently will have more electron density to react with the attacking F atom (and a higher rate constant). 8.3. Activation energies InFig. 22 ,w ep r o v i d ea nE v a n s –Polanyi type relationship55 showing a correlation between the activation energy Eaat high temperatures (1200 –1800 K) determined from the recommended rate expressions and the exp erimental heats of reaction ΔrH.T h e reactions involving F, OH, O, and H with the fluoromethanes follow the relation Ea/equals55.79 + 0.356 ΔrH(kJ mol−1)w i t ha n expanded uncertainty factor f(2σ) of about 2.1 kJ mol−1in the barrier with a “correction ”of–(8–11) kJ mol−1and –(4–5) kJ mol−1for H and F atoms, respectively. The uncertainty of about 2.1 kJ mol−1corresponds to uncerta inties of only about (10 –30)%, orf/equals(1.1–1.3) in the rate constants at 1500 K, and well within our assigned uncertainties fo rt h e s er e a c t i o n so fa b o u t f/equals1.4±0.2. The barriers for abstraction b y H and F atoms are likely higher because of electronegativity effects. The other reactive species (O atoms and OH radical) are electronegative, while H atoms are more electropositive and F atom s are highly electronegative. Consequently, the interaction between the attacking H and F atoms and the H atom being abstracted will be less attractive (more repulsive) than for the reaction involving O atoms and OH radical. There is signi ficant uncertainty in the barriers for ab- straction by F atoms at high temperatures (which were derived using correlations) since there are no experimental rate constants for any of the fluoromethanes (and methane) above about 450 K. Undoubtedly, there are some differences in the attractiveness ofthe abstraction potentials for these electronegative species; however, with uncertainties on the order of (2 –5) kJ mol −1and over a large range of heats of reaction (about 100 kJ mol−1), any effect is likely masked. There are undoubtedly competing factors related to polarization of the reactant molecule and polarization effects in the TS due to substitution of highly electronegative fluorine atoms. The barrier for CHF 3+O→CF3+ OH is about 10 kJ mol−1 higher than the trend for other abstractions by O atoms. CHF 3is a special case —with three fluorine atoms withdrawing electrons, the electron density on the carbon atom is small. This low electron density on the carbon atom in CHF 3is evident by considering the radical product CF 3in this reaction, which is nearly sp3hybridized with near tetrahedral angles —we calculate (B3LYP/6-31G**) that it has a principal moment of inertia of about 91.1 amu ˚A2compared to 90.0 amu ˚A2for CHF 3. The combination of this electronegativity effect with the triplet oxygen atom (biradical) likely affects the polarization of electrons during the reaction. Substitution of H atoms by F atoms in methane results in a withdrawal of electron density from the carbon, thus lowering the bond order in the remaining C –H bonds and making them weaker. In general, lowering the C –H bond strength will decrease the activation energy for the corresponding H abstraction. However, although the withdrawal of electron density by fluorine weakens the remaining C–H bonds, it also makes the starting molecule more polarized and therefore less able to transfer electron density that would contribute to the formation of a new bond in an abstraction reaction (i.e., in the TS).This latter effect tends to increase the activation energy for ab- straction. The first is a static effect in the reactant molecule, while the latter is a dynamic effect in the TS. InFig. 23 , we see the competition between these two effects and see a correlation between the C –H bond distance in the TS and the activation energy (which is correlated with bond strengths/ heats of reaction; see Fig. 22 ). Shorter C –H bonds indicate more bond energy and thus a more stabilized lower-energy TS. The data suggest that the bond weakening effect dominates for the first two fluorine substitutions (CH 4to CH 3Ft oC H 2F2) but that this is overshadowed by electronic eff ects in the extremely polarized CHF 3. The activation energies in Fig. 23 are from our recom- mended rate expressions at high temperatures (1200 –1500 K), while the C –H bond distances are from the ab initio calculations of Matsugi and Shiina.59They did not calculate abstraction by fluorine, and consequently, those bond distances are not con- tained in the figure. 8.4. Tunneling rates These abstraction reactions involve H atom transfer, and con- sequently, the rates of reaction at low temperatures are dominated by FIG. 23. Activation energies (open circles) for hydrogen abstraction by H, O, OH, and F, and TS bond distances ( filled circles) vs fluorine substitution in the fluoromethane series. The lines are only to guide the eyes. Activation energies are from this work and bond distances are from Matsugi and Shiina.59The four points for each curve are left-to-right for CH 4,C H 3F, CH 2F2, and CHF 3, respectively. FIG. 24. Contribution of tunneling to the rate constants for CH 4+O H→CH 3+H 2O. J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-37 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprquantum chemical tunneling. The tunneling rate is, to first order, a function of the height of the barrier through which the H atom tunnels (to second order, it is also a function of the width and asymmetry of the barrier). It is well beyond the subject of this chemical kinetic evaluation to go into any detail on the subject of tunneling. There are many theories, treatments, PESs, and computer programs utilized in this field in- cluding variational transition state theory,284semiclassical TS the- ory,285,286large curvature tunneling approximation,287zero curvature tunneling,288centrifugal dominant small curvature semiclassical tunneling,289multidimensional tunneling,290–292Wigner potentials,293–295Eckart barriers,296,297unsymmetrical Eckart bar- riers,134Polyrate,298,299and TheRate.300 A generic example of a potential energy curve and barrier was shown in Fig. 1 . Here, we present a complementary figure ( Fig. 24 )t o illustrate the contribution of tunneling to these types of H abstraction reactions for the case of CH 4+O H→CH 3+H 2O. In this figure, the solid curve is the total rate constant from our recommended rate expression, while the dashed and dotted lines are the contributions from the classical PES and from quantum chemical tunneling, re- spectively. The classical rate is represented by a rate expression considering an activated complex in TS theory,301,302which corre- sponds to an extended Arrhenius expression with n/equals1 that as- ymptotes to the total rate constant at high temperatures. Here, we derived the tunneling contribution by simply subtracting the classical rate from the total rate. In Fig. 24 , we see that the contribution of tunneling extends over a wide temperature range, that the rate constant is dominated by tunneling up to about 500 K, that it becomes equal to the classical rate at about 800 K, that the tunneling rate maximizes at about 1200 K, and that it does not finally drop to 10% of the classical rate until above about 1600 K. Figure 25 shows a strong correlation between Eaat high (1200 K –1800 K) and low temperatures (300 K). These were derived from our recommended rate expressions. There should be a strong correlation since it can be derived analytically from an extended Arrhenius expression, Ea(Thigh)−Ea(Tlow)/equalsnR(Thigh−Tlow), whereRis the gas constant and nis the temperature coef ficient. For example, using a high temperature of 1440 K (1/ Taverage of 1200 K and 1800 K) and a low temperature of 300 K, one calculates the difference between Eaat high and low temperatures to be about n*9.5 kJ/mol from this simple equation. Using the four data for OH abstractions in Fig. 25 , one calculates on the average Ea(high) and Ea(low) to be about 32.8 kJ mol−1and 14.2 kJ mol−1or a difference of about 18.7 kJ mol−1. Thus, nshould be about 1.97, in excellent agreement with the roughly (1.9–2.0) for OH abstractions in Fig. 27 . Similarly, using the three lowest data for abstractions by O (excluding CHF 3), one calculates on the average Ea(high) and Ea(low) to be about 56.6 kJ mol−1and 35.0 kJ mol−1, respectively, or a difference of about 21.6 kJ mol−1. Thus, nshould be about 2.27, in excellent agreement with the roughly (2.3–2.4) in Fig. 27 . The exact values here are not important —other than to demonstrate that the activation energies Eaat high and low temperatures and the temperature coef ficients nare all well- correlated and thus are valid for use in our least-squares optimiza- tion with additional constraints. Wefind uncertainties in Eaat low temperatures to be about (3–4)% for reactions involving H and O atoms and about (6 –8)% for OH where the barriers are about half as much. From these data, we estimate uncertainty factors of about f/equals(1.4–2.0) for these rate constants, which are consistent with our assigned uncertainty factors of about f/equals(1.2–2.3). Although barriers for H, O, and OH follow this trend very well, those for F atoms do not. These reactions involving F atoms have very low barriers, and the interactions are much less covalent and much more ionic in nature and thus do not have similar PESs to those for H, O, and OH. A reaction with a large barrier will have much less tunneling. This is illustrated in Fig. 26 , which shows the reaction rates (entirely due to tunneling) at low temperatures k(300 K) derived from our recommended rate expressions, on a log 10scale, as a function of the effective barrier Eaat high temperatures. The values for k300and Ea were all derived from our recommended rate expressions. The tunneling rates k(300 K) were normalized to the number of H atoms (reaction path degeneracy). These effective barriers were derived from the recommended rate expressions for reaction of all of the fluoro- methanes (and methane) with H, O, F, and OH over the range FIG. 25. Correlation between activation energies Eaat low temperatures (300 K) and activation energies Eaat high temperatures (1200 –1800 K). Ea(kJ mol−1). Black squares are reactions involving H atoms, red dots are reactions involving O atoms,blue triangles are reactions involving OH radicals, and green circles are reactions involving F atoms. The activation energies were derived from our recommended rate expressions. FIG. 26. Normalized (per H atom) rate constant k(cm3mol−1s−1) at 300 K as a function of activation energy Ea(kJ mol−1) for H abstractions in the fluoromethane series. Correlation between rate constants with tunneling and activation energies athigh temperatures (1200 –1800 K). The rates at 300 K and the activation energies at high temperatures were derived from our recommended rate expressions. J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-38 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jpr1200 –1800 K. We see that abstraction by F atoms has the smallest barriers (near 10 kJ mol−1) and largest rate constants, while ab- straction by H atoms has the largest barriers (near 65 kJ mol−1) and smallest rate constants. Abstraction by OH (near 30 kJ mol−1) and by O (near 60 kJ mol−1) have intermediate barriers and rate constants. The correlation in Fig. 26 corresponds to ( k300/nH)/equals4.7031013 exp(−0.306 Ea), or the changes in the rate constants (per H atom) at 300 K scale as about 31% of the activation energies at high tem- peratures (1200 –1800 K). The deviations of k300from this correlation with Eaare relatively large —about a factor of f/equals3.5. However, we found that if we applied a correction of +(5 –7) kJ mol−1toEafor the reactions involving CHF 3 (an outlier, see Figs. 7 ,22, and 23), the uncertainty factor for this correlation was on the order of f/equals2.0, not an unreasonable value (consistent with our assigned uncertainties). On the other hand, this isactually a dependent correlation since there are relatively strong correlations (see Sec. 8elsewhere) between the Afactors, the number of H atoms, the temperature coef ficients n, and the activation energies Eaat high and low temperatures. Thus, the correlation between k300 and Eaat high temperatures should also be satisfactory. In any case, Fig. 25 illustrates that there is a correlation between the rate constant at low temperatures and the activation energy at high temperatures. A consequence of the correlation between the enhanced rate constant at low temperatures due to tunneling and the activation energy at high temperatures means that there should also be a correlation between the temperature coef ficient nand the activation energy Ea—this is shown in Fig. 27 , where the temperature coef ficient nranges from about 2.6 ±0.3 to 2.4 ±0.2 to 2.0 ±0.2 for H, O, and OH reactions, respectively, or uncertainties in nof about (10 ±2)%. Note that the temperature coef ficient nfor F atom reactions does not follow this correlation because of compensation with the very low energy coefficients E. Considering the compensation effect between nandEa, we estimate expanded uncertainty factors f(2σ) of about f/equals1.25 for rate constants optimized utilizing the correlation shown in Fig. 27 , well within our assigned uncertainty factors of about f/equals1.4±0.2. Consequently, we can conclude that recommended values of nandEa are fully self-consistent. 9. Summary We have complied and critically evaluated rate constants from the literature for abstraction of H from the fluoromethanes (CH 3F, CH 2F2, CHF 3) and methane (CH 4) by the radicals H atom, O atom, OH, and F atom. We provide about 350 determinations of rate constants abstracted from the literature for these 16 reactions from just over 200 references. Based on our evaluation, we provide self- consistent recommended rate expressions for these reactions over a wide range of temperatures (about 300 –1800 K). We provide expanded uncertainty factors f(2σ) for abstractions by H atoms, O FIG. 27. The compensation effect correlation of the temperature coef ficient nwith energy coef ficient Ein expressions k/equalsATnexp(−E/RT) for H abstractions in the fluoromethane series. TABLE 26. Recommended rate expressions for H abstraction from the fluoromethanes by H, O, OH, and F. k(T)/equalsATnexp(−E/RT).k(T), temperature-dependent rate constant. A, pre-exponential factor. n, temperature coef ficient. E, energy coef ficient. The expanded uncertainty factor f(2σ) is at high temperatures with the uncertainty factor at low temperatures (300 –500) K given in parentheses. k300is the rate constant at 300 K. Units: k(cm3mol−1s−1),E(kJ mol−1),T(K). Reaction An Ef T log10(k300) CH 4+H→CH 3+H 2 9.0131052.41±0.31 38.21 ±2.13 1.4(1.6) 300 –1950 5.28 CH 3F+H→CH 2F+H 2 2.1331052.56±0.42 31.98 ±2.34 1.5(1.8) 300 –1800 6.11 CH 2F2+H→CHF 2+H 2 8.2931042.63±0.45 30.65 ±2.41 1.6(1.8) 300 –1800 6.09 CHF 3+H→CF3+H 2 2.8931042.95±0.28 38.61 ±2.28 1.1(2.0) 300 –1800 4.05 CH 4+O→CH 3+ OH 6.63 31062.16±0.24 31.42 ±1.91 1.18(1.5) 300 –1800 6.70 CH 3F+O→CH 2F + OH 1.66 31062.28±0.39 26.34 ±2.44 1.4(2.0) 300 –1800 7.28 CH 2F2+O→CHF 2+ OH 4.10 31052.40±0.41 27.19 ±2.52 1.4(2.0) 300 –1800 6.72 CHF 3+O→CF3+ OH 2.70 31042.65±0.58 42.17 ±3.52 1.7(2.3) 300 –1800 3.65 CH 4+O H→CH 3+H 2O 6.19 31052.23±0.09 9.84 ±1.54 1.12(1.05) 200 –2025 9.62 CH 3F+O H→CH 2F+H 2O 1.06 31062.04±0.26 5.70 ±1.93 1.5(1.22) 240 –1800 10.08 CH 2F2+O H→CHF 2+H 2O 3.29 31061.86±0.24 7.52 ±1.90 1.5(1.17) 220 –1800 9.82 CHF 3+O H→CF3+H 2O 1.20 31061.85±0.19 13.71 ±1.78 1.3(1.22) 250 –1800 8.27 CH 4+F→CH 3+ HF 2.93 310110.91±0.21 0.79 ±1.51 1.35(1.23) 180 –1800 13.58 CH 3F+F→CH 2F + HF 1.91 310101.25±0.29 0.81 ±1.49 1.35(1.24) 180 –1800 13.24 CH 2F2+F→CHF 2+ HF 2.05 31081.79±0.25 1.30 ±1.65 1.35(1.35) 180 –1800 12.52 CHF 3+F→CF3+ HF 4.57 31042.77±0.20 3.00 ±1.47 1.35(1.20) 180 –1800 11.00 J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-39 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jpratoms, OH radicals, and F atoms at higher temperatures (1200 –1800 K) that range about (1.4 –1.6), (1.2 –1.5), (1.2 –1.5), and (1.3 –1.5), respectively. Uncertainty factors for these reactions at lower tem- peratures (300 –500 K) are also provided —and are slightly lower (for OH) or slightly higher (for H, O, and F atoms). For a number of reactions, the reported rate constants in the literature are available only over a limited temperature range, or there are no reliable measurements. In these cases, we predicted the rate constants and their temperature dependencies in a self-consistent manner employing relative rates based on the rate constants and expressions that are established for other reactions in the homologous series. In these cases, we utilized empirical SARs correlating changes in rate constants with characteristics of the reaction, used empirical correlations between the rate constants at room temperature and the activation energy at high temperatures, and also used relative rates derived from ab initio quantum chemical calculations to guide (but not force) the rate constant predictions. We found that the relative A factors generally scaled with the reaction path degeneracies (number of H atoms) with some systematic differences due to (a) changes in the PESs from changes in polarizabilities due to substitution by the highly electronegative F atoms and (b) changes from free-rotor to hindered- rotor TSs for the reactions involving OH radicals. We also found that there was a strong correlation between the activation energy Eaat high temperatures where the rate constants can be considered at the classical limit and the activation energy Eaat low temperatures where the rate constants are dominated by quantum mechanical tunneling. A summary of the evaluated rate constants that are recom- mended in this study and the uncertainties in the parameters for these reactions is provided in Table 26 . 10. Notation Used Notation Description Acronyms EPR Electron paramagnetic resonance spectroscopy ESR Electron spin resonance spectroscopy FTIR Fourier transform infrared spectroscopy GC Gas chromatography IR Infrared spectroscopy LIF Laser induced fluorescence LMR Laser magnetic resonance spectroscopy MS Mass spectrometry MW Microwave excitation UV Ultraviolet light VUV Vacuum ultraviolet light Abbreviations abs absorption chemilum chemiluminescence detect detection dischg discharge(Continued. ) Notation Description emiss emission spectroscopy fastflow fast flow reactor flame flame measurements flow flow reactor fluor fluorescence fwd/rev rxn forward/reverse reaction heat recomb heat of recombination High T fit High-temperature fit ignition ignition measurement model complex chemical kinetic model photodissoc photodissociation photol photolysis plasma excit plasma excitation pulse photol pulse photolysis Recommended Recommended value Rel rate Relative rate reson resonance rev rxn reverse reaction Review Evaluation based on literature review shock shock tube measurement static static reactor stirred stirred reactor thermal thermal reactor ultrasonic ultrasonic chemical kinetic method Theoretical methods BEBO Bond energy bond order theory Semi-empirical Semi-empirical theoretical method TST Transition state theory VTST Variational transition state theoryQuantum chemical methods BAC-MP4 BAC-MP4 quantum chemical theory CBS-QB3 CBS-QB3 composite quantum chemical method CBS-RAD CBS-RAD composite quantum chemical method CC Coupled cluster quantum chemical theory CCSD CC theory with single and double excitations CVT Canonical variational transition state theory DFT Density functional theory quantum chemical methods Eckart Eckart tunneling approximation G2 G2 composite quantum chemical method G2MP2 G2MP2 composite quantum chemical method G3B3 G3B3 composite quantum chemical method MP2 MP2 quantum chemical theory MP4 MP4 quantum chemical theory J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-40 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jpr11. Supplementary Material See the supplementary material for __. all the tables in the paper in Excel format. 12. Data Availability The data that support the findings of this study are available within the article. 13. References 1V. Masson-Delmotte, P. Zhai, H.-O. P¨ ortner, D. Roberts, J. Skea, P. R. Shukla, A. Pirani, W. Moufouma-Okia, C. P´ ean, R. Pidcock, S. Connors, J. B. R. Matthews, Y. Chen, X. Zhou, M. I. Gomis, E. Lonnoy, T. Maycock, M. Tignor, and T. Water field, IPCC, 2018: Global warming of 1.5°C. 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Eyring, “The activated complex and the absolute rate of chemical reactions, ” Chem. Rev. 17, 65 (1935). 302H. Eyring, “The activated complex in chemical reactions, ”J. Chem. Phys. 3, 107 (1935). J. Phys. Chem. Ref. Data 50,023102 (2021); doi: 10.1063/5.0028874 50,023102-47 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jpr

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Chin. J. Chem. Phys. 34, 61 (2021); https://doi.org/10.1063/1674-0068/cjcp2102026 34, 61 © 2021 Chinese Physical Society.Advanced techniques for quantum-state specific reaction dynamics of gas phase metal atoms Cite as: Chin. J. Chem. Phys. 34, 61 (2021); https://doi.org/10.1063/1674-0068/cjcp2102026 Submitted: 03 February 2021 . Accepted: 18 February 2021 . Published Online: 10 March 2021 Ang Xu , Yu-jie Ma , Dong Yan , Fang-fang Li , Jia-xing Liu , and Feng-yan Wang COLLECTIONS Paper published as part of the special topic on Special Issue of “New Advanced Experimental Techniques on Chemical Physics” ARTICLES YOU MAY BE INTERESTED IN Photodissociation dynamics of CS 2 near 204 nm: The S(3PJ)+CS(X1Σ+) channels Chinese Journal of Chemical Physics 34, 95 (2021); https://doi.org/10.1063/1674-0068/ cjcp2010183 A three-dimensional velocity-map imaging setup designed for crossed ion-molecule scattering studies Chinese Journal of Chemical Physics 34, 71 (2021); https://doi.org/10.1063/1674-0068/ cjcp2012219 Interaction energy prediction of organic molecules using deep tensor neural network Chinese Journal of Chemical Physics 34, 112 (2021); https://doi.org/10.1063/1674-0068/ cjcp2009163CHINESE JOURNAL OF CHEMICAL PHYSICS VOLUME 34, NUMBER 1 FEBRUARY 27, 2021 REVIEW Advanced Techniques for Quantum-State Specic Reaction Dynamics of Gas Phase Metal Atomsy Ang Xu ;Yu-jie Ma ;Dong Yan ;Fang-fang Li ;Jia-xing Liu ;Feng-yan Wang Department of Chemistry and Shanghai Key Laboratory of Molecular Catalysis and Innovative Materials, Collaborative Innovation Center of Chemistry for Energy Materials (iChEM), Fudan University, Shanghai 200433, China (Dated: Received on February 3, 2021; Accepted on February 18, 2021) One of the themes of modern molec- ular reaction dynamics is to characterize ele- mentary chemical reactions from \quantum state to quantum state", and the study of molec- ular reaction dynamics in ex- cited states can help test the validity of modern chemical theories and provide methods to control chemical reactions. The subject of this review is to describe the recent experimental techniques used to study the reaction dynamics of metal atoms in the gas phase. Through these techniques, information such as the internal energy distribution and angular distribution of the nascent products or the three-dimensional stereodynamic reactivity can be obtained. In addition, by preparing metal atoms with specic excited electronic states or orbital arrangements, information about the reactivity of the electronic states enriches the relevant understanding of the electron transfer mechanism in metal reaction dynamics. Key words: Time-sliced ion velocity map imaging, Crossed molecular beams, Laser abla- tion, Metal atom reaction dynamics, Stereodynamics I. INTRODUCTION With the development of molecular beam and various detection techniques, especially with the emergence and development of lasers since the 1960s, the experimen- tal studies of elementary reactions prepared in specic †Part of special topic on \the New Advanced Experimental Tech- niques on Chemical Physics". ∗Author to whom correspondence should be addressed. E-mail: fengyanwang@fudan.edu.cnquantum states have provided a great deal of quantita- tive and qualitative information for the understanding and control of chemical reactions [1{17]. Recent stud- ies using infrared optical parameter oscillator/amplier system have shown that vibration excitation has a much more complex eect on reactions than just considering energy factors [5{7, 15, 16]. The eects of electronic states on chemical reactions are more complicated due to the possibility of nonadiabatic transitions between potential energy surfaces (PESs). The subject of the present review is about the techniques used in the gas- DOI:10.1063/1674-0068/cjcp2102026 61 c⃝2021 Chinese Physical Society62 Chin. J. Chem. Phys., Vol. 34, No. 1 Ang Xu et al. phase metal atom reactions. These reactions, such as, oxidation reactions of metal atoms, are usually fast due to the crossing between ionic and covalent potential energy surfaces, so they are called harpoon \electron" mechanism, and therefore are suitable for technique de- velopment [18, 19]. The study on gas-phase metal reactions began in the 1930s with ame chemistry. Polanyi and coworkers studied the atomic diusion ame of reactions between free metal atoms and halogen-containing molecules, suggesting that alkali metal atoms react quickly with many halogen species [20]. However, ame technology cannot separate the homogeneous elementary reactions from the reactions of particles formed in the gas phase, nor can it distinguish between the reactions from dier- ent electronic states. The systematic kinetic studies of gas-phase metal reactions were carried out in the fast- owing or pseudo-static reactors. The technique used vaporization, photolysis, or discharge methods to pro- duce gas phase metal atoms in a ow of buer gas. The metal atomic beam enters the reaction chamber, which contains a calibrated number density of the second re- actant gas of interest at low pressure. By changing the pressure, the single and double collisions can be distin- guished. The energy distribution of reactants or prod- ucts can be observed through chemical luminescence or laser-induced uorescence (LIF). By comparing mea- surements at dierent temperatures or pressures, infor- mation about the reaction coecients of metal atoms in dierent electronic states can be provided. However, since direct reactions and those involving long-lived in- termediates also depend on temperatures and can con- tribute over a wide temperature range, such distinctions are not that obvious in bulk experiments. Moreover, the eect of vibration excitation on the high temperature reactions has to be considered. Crossed molecular beams (CMB) studies provide some of the most detailed quantitative experimental results for the single collision dynamics of elementary metal reactions with the well-controlled collision en- ergy and precise measurement of the velocity and angu- lar distribution of products [1, 21{23]. In 1955, Taylor and Datz used a CMB device consisting of a nickel fur- nace containing molten potassium, which was used as a source of potassium vapor, and of a detector using a surface ionization gauge with a tungsten and a platinum alloy lament to study the collision reaction mechanism between K and HBr [24]. In the 1960s, Herschbach led FIG. 1 The schematic diagram of the crossed beam experi- mental setup for metal atom reaction studies. This gure is modied with permission from Ref.[27] c⃝AIP Publishing. the \alkali era" by studying reaction dynamics between alkali metals and various halides through CMB exper- iments [25]. The reactions of alkali metal atoms with halogen species were suggested to proceed the nona- diabatic transition or crossing from covalent to ionic potential energy surfaces (harpoon \electron" mecha- nism), which is the main feature of distinguishing metal reactions from non-metallic reactions. Until 1969, Lee and coworkers introduced a universal CMB apparatus equipped with a skimmer between the beam source and the reaction chamber, drastically reducing the back- ground, marking that the studies of CMB entered into the \chemical era" [26]. The schematic diagram of the crossed beam ex- perimental setup for metal atom reaction studies is shown in FIG. 1. The apparatus basically includes two source chambers, for producing two atomic or molec- ular beams, and a main chamber for reaction and de- tection system [27]. The technical advantages of CMB enable the reaction studies in a state-to-state manner, including the preparation of electronically excited state in the metal atom reactant, the orbital alignment ex- periments relative to the collision direction, and the measurement of the energy and angular distribution of their reaction products. The dierential cross sec- tions from crossed beam studies can distinguish between direct reactions and non-direct reaction with longer- lived intermediates. Using the crossed molecular beams and polarized laser excitation method, Lee and co- workers studied the reactions of the ground and elec- tronically excited state Ba (1S,1P,3P,1D) reaction with diatomic and polyatomic molecules [28{33]. For DOI:10.1063/1674-0068/cjcp2102026 c⃝2021 Chinese Physical SocietyChin. J. Chem. Phys., Vol. 34, No. 1 Quantum-State Specic Reaction Dynamics of Gas Phase Metal Atoms 63 FIG. 2 Schematic designs for a laser vaporization source (a) \cutaway" and (b) \oset". Supersonic metal atomic beam is generated by laser vaporization of metal rod and free expansion design without gas ow channel which has been employed to obtain a good quality of metal atomic beam. This gure is modied with permission from Ref.[34] c⃝AIP Publishing. example, Ba(1S)+H 2O led to dominate formation of BaO+H 2, but Ba(1D)+H 2O was primarily channeled into BaOH+H. To date, we have seen that almost all experiments with orbital-aligned atom reactions have involved alkaline earth metal atoms, mainly Ca and Ba atoms. II. DEVELOPMENT OF A SUPERSONIC METAL ATOMIC BEAM The development of metal atomic beams has been go- ing for years accompanied with the advances in metal- containing cluster beams [34]. The gas-phase metal ef- fusive sources are obtained by heating the desired metal above its boiling point in a high temperature oven sys- tem (temperature can be up to 2800 K) [35]. In 1981, Smalley and co-workers introduced supersonic metal cluster beams with greater intensity by using pulsed laser vaporization method [36]. In their design, a pulsed Nd:YAG laser at 532 nm was focused on the target metal rod to vaporize the metal [36, 37]. To prevent the rod from being perforated by the high-power laser, the rod was kept rotating and moving continuously through a screw drive mechanism. The short vaporization laser pulses (6 ns) can only vaporize a tiny amount of metal rod. This method is preferred because there is no need to heat the supersonic nozzle source and beams of even extremely refractory metals can be easily generated. The pulsed laser vaporization design was initially adopted by Costes et al. [38] in 1987 for the genera- tion of supersonic metal atomic beam in crossed-beam experiments and in 1999 was modied to a \free abla- tion condition" without gas ow channel [39], and then followed by other groups [40, 41]. Duncan in 2012 has systemically reviewed various designs of laser vaporiza-tion [34]. As indicated in Ref.[34], a \cutaway" or \o- set" source is preferred to produce more metal atoms instead of clusters. As shown FIG. 2, the \cutaway" or \oset" source without gas ow channel, proposed by Duncan [34], provides a good quality supersonic beam of single metal atoms instead of clusters. The super- sonic metal atomic beam is produced by using a pulsed supersonic nozzle source for collisional cooling of hot plasma formed by laser ablation. There are many fac- tors aecting the quality of supersonic atomic beam, in- cluding carrier gas species, backing pressure, temporal duration of the carrier gas pulse, timing between the gas pulse and vaporization laser, laser ablation wavelength, laser ablation energy, laser focus length, the distance from metal rod to the pulse valve etc. The main dier- ence between these two designs is that the metal rod is oset out of the carrier gas line in the latter design, as shown in FIG. 2(b). III. DEVELOPMENT OF DETECTION TECHNIQUE Earlier studies on gas-phase metal atom reactions mainly used spectroscopic detection methods, including laser-induced uorescence (LIF) or chemiluminescence detection methods, and the consumption of transition metal atoms in the reactant can be monitored by LIF to determine the reaction rate constant [35, 38, 42{56]. Spectra, such as chemiluminescence spectroscopy, LIF spectroscopy and resonance enhanced multiphoton ion- ization (REMPI) spectroscopy can provide information about the energy distribution of products at high or low energy levels. However, these spectroscopic meth- ods need to know the energy level information of the upper and lower states of molecules or atoms. Chemi- luminescence or LIF methods can only be used when DOI:10.1063/1674-0068/cjcp2102026 c⃝2021 Chinese Physical Society64 Chin. J. Chem. Phys., Vol. 34, No. 1 Ang Xu et al. the detected molecules or atoms emit uorescence. For example, in the laboratories of Costes or Honma, the detection technique relies or has relied on LIF to analyse the internal energy distributions of the metal atom reaction product and obtain the threshold energy through the excitation function [39, 40, 55, 57]. In the Al atom oxidation reaction Al+O 2!AlO+O at the col- lision energy of 12.2 kJ/mol, the internal energy state distribution of AlO is determined by the LIF spectrum of AlO product through the transition of B2+2+. The derived rotation and vibration distributions show more than that statistically expected in low vibrational and rotational levels [58]. REMPI spectroscopy combined with mass spectrom- eter technique shows strong advantages in obtain- ing spectral information of specic species. However, REMPI is usually a multiphoton absorption process and so higher laser energy is required. For example, the (1+1) REMPI spectrum of the AlO product produced from the oxidation reaction of the Al atom at a colli- sion energy of 12.2 kJ/mol was obtained through the D2+2+transition, which gives the AlO rovibra- tional distribution similar to LIF [59]. The use of mass spectrometers with electron impact ionizer and quadrupole mass lter to detect neutral products has facilitated the studies of a very wide range of reactions. The velocity and angular distributions of the product can be obtained by rotating the mass spectrometer detector in the beam plane, and the in- ternal energy distribution of the product can be ob- tained by spectroscopy method [35, 38, 60{64]. The an- gular distribution in the center of mass frame can give clear information about the reaction mechanism. For example, forward scattering usually indicates a strip- ping reaction, backward scattering usually indicates a rebound reaction, and for long-lived collision complex formation, the forward scattering and backward scatter- ing are symmetrically distributed. Generally, the more information provided by experimental techniques, the more we understand the dynamics and mechanisms of elementary chemical reactions. An important research system of metal atom reaction dynamics is to study the reaction dynamics of methane CH activation by transition metal. In view of the im- portance of transition metal selective C H activation of methane in the conversion of abundant but inert nat- ural gas into more useful products, many related stud- ies have been conducted [65{67]. Blomberg theoreti-cally studied the atoms in the three transition rows and found that the rhodium atom has the lowest CH inser- tion reaction barrier, which was observed in the matrix separation experiment [68{72]. In the reactor at room temperature, no reaction between metal and methane was observed by kinetic techniques [48, 51, 73]. Using rotatable source crossed molecular beams and mass spectrometer apparatus [60], Davis group stud- ied the reaction of electronically excited Mo ( a5S2 s1d5) with methane [74, 75]. By optically pumping the Mo atomic beam on the5P3 7S3transition at 345.7 nm, the metastable Mo(5S2) was prepared by the transition followed by an allowed radiative decay to the desired metastable5S2state lying at 10768 cm1 (30.8 kcal/mol). In their experiment, they used the pulsed 157 nm radiation from a F 2excimer laser in the ultrahigh vacuum region of the mass spectrometer for ionization of products MoCH 2to improve detection sensitivity. For the ground state Mo ( a7S3s1d5), it has been calculated that the potential barrier for insertion into CH 4is as high as 37.8 kcal/mol [48], and for the metastable low spin Mo(a5S2s1d5), the experimen- tally measured barrier is as small as 2.1 kcal/mol. Thus, the low spin state Mo(a5S2s1d5), which correlates with the ground state insertion intermediate HMoCH 3, reacts eciently with methane. In other words, for a given total energy of electronic and translational form, it is found that electronic energy is highly eective in promoting the reaction of Mo+CH 4whereas collisional energy is ineective [75]. However, the chemical reac- tion dynamics of some transition metal ions (such as cobalt ion, nickel ion and copper ion) show that the electronic excited state is less reactive than the ground state [76]. The above results indicate that the electronic structure is correlated with the reactivity of transition metal atoms. Currently ion imaging technique has been used in the study of molecular reaction dynamics [77{79]. The de- velopment history of ion imaging technique began in 1987, when Chandler and Houston [80] rst introduced ion imaging into the study of CH 3I photodissociation dynamics with two-dimensional position sensitive de- tector composed of microchannel plates and phosphor screen. In 1997, Eppink and Parker [81] improved the velocity resolution in ion imaging by removing grids in ion optics. It is called velocity map imaging (VMI), that is, ions with the same initial velocity are focused to the same point on the detector regardless of the initial DOI:10.1063/1674-0068/cjcp2102026 c⃝2021 Chinese Physical SocietyChin. J. Chem. Phys., Vol. 34, No. 1 Quantum-State Specic Reaction Dynamics of Gas Phase Metal Atoms 65 FIG. 3 (a) Crossed-beam and time-sliced ion velocity imaging apparatus for studying metal atomic reaction dynamics, (b) raw slice images of rotational-state selected AlO products formed from the reaction of Al+O 2→AlO+O at the collision energy of 6.07 kJ/mol, (c) corresponding speed distributions of AlO products from the raw slice images of (b). The velocity image mode is sensitive to almost zero speed near the center of the mass coordinate. This gure is based on Ref.[85] c⃝The Royal Society of Chemistry and Ref.[41] c⃝Chinese Physical Society . position. In 2001, Kitsopoulos group [77] introduced a novel method called slicing imaging, which extended the ight time of Newton sphere to several hundreds of nanoseconds and extracted the center part for slices. The technical advantage of slice imaging is that the in- verse Abel transformation is no longer required and the speed and angular distribution of products can be di- rectly obtained from the slice image. In 2003, Liu group [78] developed a weak electronic eld DC time-sliced ion velocity imaging method and applied it to crossed molecular beams research. Since 2011, Honma group and Wang group have applied time-sliced velocity map imaging technique to the studies of metal atom reac- tions [82{86]. Using the symmetry of the speed and angular distributions in the image, the position of the center of the Newton sphere or the origin of the center- of-mass coordinate can be easily determined, and so the velocity distribution and angular distribution in the center of mass coordinate system can be directly ob- tained. Honma et al. used velocity map imaging technique for the studies of the metal reaction systems that were stud- ied using LIF method [59, 82, 87, 88]. The molecular beam experimental device includes a supersonic metal atomic beam, which is produced by laser vaporization of a metal rod, and the other supersonic molecular beam which crosses the metal atomic beam at the center of the ion optics. They used an OPO/OPA system formultiphoton ionization of the neutral products that are formed in the reactions. They observed the angular dis- tributions of AlO products from the oxidation reaction of Al atoms. All angular distributions show forward and backward peaks, and the forward peak is more pro- nounced than the backward peak for low internal energy states. The backward peak intensity becomes compara- ble to the forward peak intensity for the high internal energy states. These results were compared with pre- vious spectroscopic studies and added new insights. It is believed that the oxidation reaction of Al atoms pro- ceeds via an intermediate and the lifetime of the inter- mediate is comparable to or shorter than its rotational period. Wang et al. improved the energy resolution of the experimental setup for imaging reaction dynamics of metal atoms. FIG. 3 shows the crossed beams and imag- ing setup with a combination of laser ablation source, which carries out a full state-to-state resolution reaction dynamics of the Al atoms by measuring the speed and angular distribution of products at dierent rotational quantum states. In the velocity images, the dierent rotational states of the AlO products and the contri- butions of the spin-orbit split Al (2P3=2;1=2) reactants with an energy dierence of 112 cm1are distinguished. By accurately measuring the rotational state and veloc- ity distribution of the AlO products, the derived impact parameter at 2.5 A is consistent with the electron trans- fer distance predicted by the harpoon \electron" model DOI:10.1063/1674-0068/cjcp2102026 c⃝2021 Chinese Physical Society66 Chin. J. Chem. Phys., Vol. 34, No. 1 Ang Xu et al. RC=ke2/(I.E.E.A.), where RCis the predicted elec- tron transfer distance, kis Coulomb constant, I.E. rep- resents the ionization energy of metal atoms and E.A. represents the electron anity of the oxidant molecule [41, 85]. Combining crossed-beam velocity map imaging and laser state selective excitation method, Honma group studied the oxidation reactions of the gas-phase tita- nium atoms in electronically excited state with oxy- gen molecules at the collision energy of 14.3 kJ/mol [88]. Metastable excited Ti ( a5FJ) was generated by an optical pumping method and the reaction products were detected by single photon-ionization followed by the time-sliced ion velocity imaging detection. The an- gular distributions with forward-backward peaks from the images of TiO products conrmed the mechanism proposed by the previous chemiluminescence study, and the reaction proceeds through a long-lived intermediate complex. We can see that the electronic excited state reactivity of metal atoms can be achieved in many technologies, such as photodissociation studies or matrix isolation infrared spectroscopy of the complex formed between the metal and the reactant [90]. Van der Waals com- plexes can be formed in supersonic expansion or in a matrix at low temperatures from reactants in electron- ically excited states. Recently, Ming-fei Zhou group [91] using matrix isolation infrared spectroscopy re- ported the isolation and spectroscopic characterization of eight-coordinate carbonyl complexes M(CO) 8(where M=Ca, Sr, or Ba) in a low-temperature neon matrix (4 K). In the complexes, the alkaline earth metal atoms M (M=Ca, Sr, and Ba) are electronically excited to (n1)d states with ns0(n1)d2valence electronic con- guration, which allows M(d )!(CO) 8backdonation and behaves like transition metal chemistry. IV. STEREO DYNAMICS TECHNIQUE The preparation of the quantum state involves the control of the reagent approach geometry, which is also called stereodynamic control [92]. So far, the experi- mental information on orientation or alignment eects mainly comes from crossed molecular beams experi- ments, where the steric geometry of one reactant with respect to the relative collisional direction is usually achieved by applying an external electric eld, such as a strong uniform electric eld or a hexapole [93{104]. Another important method is use of lasers to prepareone of the reagents whose molecular framework or elec- tronic orbitals preferentially pointing to a certain di- rection in the center of mass coordinate and study the eect of that alignment or orientation on chemical reac- tivity. Non-spherical atomic orbitals with orbital angu- lar momentum quantum number l>0 (p, d and higher l orbitals) can be controlled by absorbing linearly polar- ized light. The change of the polarization direction of the laser will change the direction of the rotational an- gular momentum vector in the upper state through the interaction with the transition dipole moments. The stereodynamic studies on metal atom reactions provide rich information on the three-dimensional geometries of reaction pathways. In a beam-gas scattering geometry for open-shell reagents, Rettner and Zare [105, 106] studied the reac- tions of electronically excited calcium atoms Ca (3s3p 1P) with various halogen containing compounds using linearly polarized laser. The dierent alignments of the p orbits of the Ca atoms evolve into the ps or pp or- bitals of the CaCl product, resulting in the products in two electronic states, CaCl ( B2+) and CaCl ( A2). Ding et al. using beam-gas and chemiluminescence ex- periment further investigated the eect of laser-aligned Ca (1P) on the vibrational distributions of CaI products when colliding with CH 3I molecules [107]. The results showed that product vibrational distributions are sensi- tive with the reactant orbital alignment by the impact parameter. Lee and his colleagues [31, 33, 108, 109] comprehen- sively studied the dierent alignments of barium p or- bital Ba (1P1) in crossed beams experiments. As seen in FIG.4 (a), (c) and (d), the excitation laser prop- agated perpendicularly to the plane of the molecular and atomic beams, allowing the p orbital to rotate in the collision plane. Alternatively, the laser propagated antiparallelly to the molecular beam, allowing the p or- bital to rotate out of the collision plane. The dierent dependence of out-of-plane and in-plane orbital align- ment observed in the reaction of Br 2indicates that the reaction is dominated by the large impact parameter collisions. In the collisions of Ba (1P1) with NO 2, for production of Ba+from Ba+NO 2!Ba++NO 2, the symmetry of the orbital does not seem to be impor- tant and the orbital is directed toward the molecule for collisions with large impact parameter. However, for the BaO+from Ba+NO 2!BaO+NO channel, the alignment eects show a high sensitivity to the electron DOI:10.1063/1674-0068/cjcp2102026 c⃝2021 Chinese Physical SocietyChin. J. Chem. Phys., Vol. 34, No. 1 Quantum-State Specic Reaction Dynamics of Gas Phase Metal Atoms 67 FIG. 4 Schematic views of laser aligned experiments in (a) universal crossed beams apparatus, and (b) imaging crossed beam apparatus. Illustration of alignment geometry for (c) in-plane and (d) out-of-plane rotation of the Ba (1P1) orbital, (e) (, ) geometries for the vibrationally excited C −H bond alignment of methane. In (c), the laser was aligned perpendicular to the plane of the crossed molecular and atomic beams, allowing the p orbital to rotate in the collision plane. In (d), the laser was aligned antiparallel to the molecular beam, allowing the p orbital to rotate out of the collision plane. In (e), the laser was aligned in the plane of the crossed beams and at least three independent ( , ) geometries were required for stereo control experiment: the vector kis the relative velocity of the reactants, dened as the zaxis; k′is the direction of product recoil velocity, and is the product scattering angle dened by kandk′in the imaging plane. The ight time axis is the y axis, which is dened by kxk′. The spatial direction of the laser polarization vector Ein the scattering frame is specied by the polar angle and the azimuth angle ( , ). When the Evector points to the x,y, orzaxis, the reaction system under study can obtain three independent stereo reaction geometries. This gure is modied with permission from Ref.[33] and Ref.[110] c⃝AIP Publishing. transfer or the nonadiabatic transitions between the po- tential energy surfaces. Recently, a full stereo picture for the reaction of rovi- brationally excited methane with chlorine atoms have been shown in detail by using time-sliced ion velocity map imaging, rotatable crossed molecular beams and linearly polarized infrared laser [7, 8, 97, 110{112]. As shown in FIG. 4 (b) and (e), by actively controlling the collisional geometry through controlling the polar- ization direction of the laser, the alignment-dependent slice images are obtained and the image given by dif- ferent collision geometry is dierent. By combining the images of dierent collision geometry, the three- dimensional reactive structure information can be ob- tained. The time-sliced ion velocity imaging has shown signicant technical advantage in obtaining the polar- ized velocity and angular distributions in polarizationscattering experiments. V. OUTLOOK In summary, to explore the reaction mechanism of the elementary reactions, it is necessary to obtain the infor- mation of the angular distribution, velocity distribution and internal energy distribution in experiments to help theoretical calculations of the potential energy surface and reaction trajectory. The current experimental tech- niques for metal atom reaction studies include laser in- duced uorescence, chemiluminescence, mass spectrom- etry, translational energy distribution and ion velocity imaging. The time-sliced ion velocity imaging technique has shown great advantages in stereo reaction studies. The electronic conguration of metal atoms especially transition metals is relatively complicated. However, the chemical reaction dynamics of these low-energic DOI:10.1063/1674-0068/cjcp2102026 c⃝2021 Chinese Physical Society68 Chin. J. Chem. Phys., Vol. 34, No. 1 Ang Xu et al. electronic excited states atoms is very important for un- derstanding the activation reaction mechanism of tran- sition metals. In the past, experimental studies on the preparation of specic electronic excited states of metal atoms and the detection of internal energy states and spatial scattering distribution of reaction products on the reactivity of bare metal atoms have provided rich information about atomic reactions, which plays an im- portant role in the study of metal clusters and the ac- tivation of solid transition metals. The combination of the latest experimental methods and techniques will help us better understand the eects of electronic spin, orbital symmetry and spin-orbital coupling on the reaction dynamics, and nd eective methods to control chemical reactions, and nally pro- mote the future development of reaction dynamics re- search. Especially recently, with the development of laser technique and related stereodynamics methods [2{ 4, 7, 8, 10{15, 113], including the emergence of infrared and ultraviolet OPO/OPA sources, and the construc- tion of high-brightness ultraviolet free electron laser light sources and infrared free electron laser sources, we have better experimental techniques and conditions in the study of excited state molecular reaction dynamics, and can provide sucient details for state selective reac- tion dynamics and photochemistry to construct chemi- cal reaction models. VI. ACKNOWLEDGMENTS The work was supported by the National Natu- ral Science Foundation of China (No.21673047 and No.22073019), the Shanghai Key Laboratory Founda- tion of Molecular Catalysis and Innovative Materials, and the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning. [1]R. D. Levine, Molecular Reaction Dynamics , Cam- bridge: Cambridge University Press, (2005). 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9.0000154.pdf

AIP Advances 11, 025314 (2021); https://doi.org/10.1063/9.0000154 11, 025314 © 2021 Author(s).Formation of quasi-free-standing graphene on SiC(0001) through intercalation of erbium Cite as: AIP Advances 11, 025314 (2021); https://doi.org/10.1063/9.0000154 Submitted: 21 October 2020 . Accepted: 05 January 2021 . Published Online: 08 February 2021 P. D. Bentley , T. W. Bird , A. P. J. Graham , O. Fossberg , S. P. Tear , and A. Pratt COLLECTIONS Paper published as part of the special topic on 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials and 65th Annual Conference on Magnetism and Magnetic Materials ARTICLES YOU MAY BE INTERESTED IN Fabrication of a single layer graphene by copper intercalation on a SiC(0001) surface Applied Physics Letters 104, 053115 (2014); https://doi.org/10.1063/1.4864155 Surface intercalation of gold underneath a graphene monolayer on SiC(0001) studied by scanning tunneling microscopy and spectroscopy Applied Physics Letters 94, 263115 (2009); https://doi.org/10.1063/1.3168502 The quasi-free-standing nature of graphene on H-saturated SiC(0001) Applied Physics Letters 99, 122106 (2011); https://doi.org/10.1063/1.3643034AIP Advances ARTICLE scitation.org/journal/adv Formation of quasi-free-standing graphene on SiC(0001 ) through intercalation of erbium Cite as: AIP Advances 11, 025314 (2021); doi: 10.1063/9.0000154 Presented: 4 November 2020 •Submitted: 21 October 2020 • Accepted: 5 January 2021 •Published Online: 8 February 2021 P. D. Bentley, T. W. Bird, A. P. J. Graham, O. Fossberg, S. P. Tear, and A. Pratta) AFFILIATIONS Department of Physics, University of York, Heslington West, York YO10 5DD, United Kingdom Note: This paper was presented at the 65th Annual Conference on Magnetism and Magnetic Materials. a)Author to whom correspondence should be addressed: andrew.pratt@york.ac.uk ABSTRACT Activation of the carbon buffer layer on 4H- and 6H-SiC substrates using elements with high magnetic moments may lead to novel graphene/SiC-based spintronic devices. In this work, we use a variety of surface analysis techniques to explore the intercalation of Er under- neath the buffer layer showing evidence for the associated formation of quasi-free-standing graphene (QFSG). A combined analysis of low energy electron diffraction (LEED), atomic force microscopy (AFM), X-ray and ultraviolet photoemission spectroscopy (XPS and UPS), and metastable de-excitation spectroscopy (MDS) data reveals that annealing at temperatures up to 1073 K leads to deposited Er clustering at the surface. The data suggest that intercalation of Er occurs at 1273 K leading to the breaking of back-bonds between the carbon buffer layer and the underlying SiC substrate and the formation of QFSG. Further annealing at 1473 K does not lead to the desorption of Er atoms but does result in further graphitization of the surface. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/9.0000154 .,s I. INTRODUCTION Graphene’s large spin relaxation length at room temperature continues to attract significant attention in terms of potential appli- cations in spintronics.1Thermal decomposition of silicon carbide provides a route towards high-quality epitaxial graphene,2how- ever is limited for the 4H and 6H-SiC Si polar faces due to sub- surface Si back-bonds. These lead to the first carbon layer that forms having the same structure as graphene but acting as a buffer layer that displays no linear dispersion at the K-point in reciprocal space.3Adatoms such as H, Ca, Mn, Li, Au, Fe and Co4–10have all been used to break these back-bonds, resulting in the formation of “quasi-free-standing graphene” (QFSG),4with the additional bene- fit of potentially enhancing the magnetic properties of this graphene layer.10 Specifically, tailoring graphene for spintronic applications gen- erally involves enhancing its intrinsically weak spin-orbit cou- pling (SOC),1either by adatom decoration11or through proxim- ity effects.12Since the strength of SOC scales with atomic number (Z), there has been increasing research activity related to interca- lation/adatom decoration of heavier elements under/on grapheneincluding Au8and the lanthanide series.13–16In addition to form- ing QFSG, it is proposed that the high magnetic moment associated with the rare earth metals will lead to enhanced SOC following inter- calation providing a route to magnetic storage and other spintronic devices.13For example, one such study has shown Eu to interca- late, form a periodic array of atoms underneath the buffer layer and be protected from oxidation for months after being removed from ultra-high vacuum (UHV).15To date, Eu, Dy, Yb, and Gd have all been studied,14–16although Er intercalation has not yet been reported, despite it having one of the highest magnetic moments (7μB) of all the lanthanides. Additionally, the mechanism of rare- earth interaction with the buffer layer and how intercalation leads to QFSG is not fully understood. In this work, we explore the intercalation of Er underneath the carbon buffer layer that forms on 6H-SiC(0001) substrates using a variety of surface analysis techniques. In particular, we make use of the extreme surface sensitivity associated with metastable de- excitation spectroscopy (MDS), which we apply here to the study of rare-earth intercalation for the first time.17,18The large de-excitation cross-section of helium atoms in the long-lived metastable 23S state (19.82 eV) provides access to the outermost electronic and AIP Advances 11, 025314 (2021); doi: 10.1063/9.0000154 11, 025314-1 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv magnetic structure of a surface providing complementary informa- tion to, for example, ultraviolet photoemission spectroscopy (UPS). Together with low energy electron diffraction (LEED), atomic force microscopy (AFM), and X-ray photoemission spectroscopy (XPS) results, the UPS and MDS data reveal that Er intercalates beneath the buffer layer, likely breaking back-bonds to form QFSG and thereby providing a potential system for future studies of SOC enhancement. II. EXPERIMENTAL TECHNIQUES Wafers of n-type vanadium- and nitrogen-doped 6H-SiC were purchased from Semiconductor Wafer Inc., samples cut, and mounted onto Scienta Omicron GmbH direct-current heating plates with the (0001) silicon-rich face up. Electron spectroscopy mea- surements were performed in a UHV system consisting of a prepa- ration chamber and an analysis chamber with base pressures of <5×10−10mbar and <1×10−10mbar respectively. All spectra were obtained using a Scienta Omicron GmbH EA 125 hemispher- ical energy analyzer. A Scienta Omicron GmbH VUV HIS 13 cold cathode discharge source was used to generate He I αUV pho- tons ( hf=21.22 eV). Metastable helium 23Satoms for performing MDS were produced using a bespoke liquid-nitrogen-cooled direct- current discharge source.19AlKα1(1486.7 ±0.1 eV) monochro- mated X-rays were generated using a Scienta Omicron GmbH XM1000 MKII Mono X-ray source. All LEED and LEED I/V mea- surements were performed in a separate UHV system with a base pressure of <1×10−10mbar. LEED images were obtained using a Scienta Omicron GmbH SPECTALEED instrument equipped with a LaB 6filament. LEED I/V measurements were performed using a combination of Neptune and kSA 400 software, and all images weretaken using a kSA K-30FW camera. AFM images were obtained on a Bruker BioScope Resolve instrument operating in tapping mode. A buffer layer of carbon was prepared by annealing each SiC sam- ple at 1473 K for 30 minutes, with temperature measured using a Raytek Corp infrared pyrometer. Erbium was deposited using bespoke evaporation sources consisting of erbium lumps mounted in a tantalum boat with deposition rate calibrated using a quartz crystal microbalance.20 III. RESULTS A. Low energy electron diffraction and LEED I/V Following preparation of the buffer layer, several SiC substrates were covered with 0.5 monolayer (ML) of erbium with the samples at room temperature during growth ( ρ= 8.795 gcm−3, thickness of an Er ML τ= 2.36 Å).21Following deposition, the (6√ 3×6√ 3) R30○LEED pattern associated with the buffer layer (Figure 1(a)) dis- appeared at all energies (Figure 1(b)) implying the formation of a disordered Er layer atop the buffer layer. By annealing for 5 min- utes in increasing 100 K steps, the 6√ 3 LEED pattern was eventually recovered at 1073 K, with a similar LEED I/V curve to that prior to deposition (Figure 1(e)). Partial restoration of the 6√ 3 period- icity is consistent with a proportion of the buffer layer again being exposed suggesting that annealing at 1073 K either results in Er des- orbing from the surface, or alternatively, Er clustering at the surface. The latter is supported by Vaknin et al. who reported Eu to form clusters on the buffer layer over a similar temperature range.15To investigate this further, we performed AFM on buffer layer sam- ples exposed to a coverage of 0.5 ML of Er at room temperature with several resulting images shown in Figure 2. At lower magni- FIG. 1 . LEED images taken at 126 eV of (a) the buffer layer and (b) after a 0.5 monolayer deposition of erbium with the 6H-SiC sample at room temperature. Further annealing of the sample was then performed in 100 K steps. After annealing at (c) 1073 K and 1173 K (not shown), the graphene spot (dashed circle) and subsidiary spots of the 6√ 3 buffer layer LEED pattern remain but are increasingly diminished. (d) At 1273 K, subsidiary spots associated with the buffer layer have almost disappeared but the graphene spot remains. (e) The evolution of the LEED pattern is reflected in the corresponding LEED I/V spectra of the graphene spot with a pronounced change evident at 1173 K and above. AIP Advances 11, 025314 (2021); doi: 10.1063/9.0000154 11, 025314-2 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 2 . Atomic force microscopy (AFM) images of a sample of 0.5 ML of Er deposited at room temperature on to a carbon buffer layer prepared on a 6H- SiC(0001) substrate. Clear terraces are observed in the lower magnification image (a) along with bright spots which are shown in greater detail in (b). fication (Figure 2(a)), clear terraces of the substrate can be seen which have a typical width of ∼500 nm and height of ∼0.5 nm. Step- edges between terraces appear to be decorated by bright spots which can be seen in more detail in Figure 2(b). Individual atoms are not resolvable in these images but the location and prevalence of these bright spots support Er clustering at disordered step edges of the SiC substrate. Annealing at 1173 K and above results in further changes to the LEED pattern and associated LEED I/V spectra as seen in Figure 1(d) and (e). The subsidiary 6√ 3 spots that surround the graphene spot (dashed circle) are heavily suppressed although the graphene spot itself remains visible. These changes are further reflected in the LEED I/V data where the spectrum after annealing at 1173 K is sig- nificantly different from the buffer layer and 1073 K anneal spectra. The most notable changes in the spectra occur in the 70-110 eV energy range where characteristic Bragg peaks indicate the forma- tion of buffer-layer graphene. In particular, the reduction of the peak at∼74 eV for the 1173 K anneal to a shoulder-like feature for higher temperatures indicates the evolution of the buffer layer to mono- layer or bilayer graphene.22Changes in the LEED I/V spectra at theselow energies are indicative of a structural change close to the sur- face (within the first few atomic sublayers) based on the information depth of electrons within this energy range. The LEED and LEED I/V data suggest that erbium has either desorbed at 1173 K and above, clustered in a manner so that the buffer layer becomes partially covered, or intercalated underneath the buffer layer. Intercalation of Eu beneath the buffer layer leading to the breaking of Si-C back-bonds following annealing has previ- ously been reported.15A similar mechanism of intercalation possibly explains the LEED data for Er presented here. Spots in the LEED pat- tern associated with the 6√ 3 periodicity of the buffer layer mostly disappear after annealing at 1273 K but the graphene spots remain suggesting the formation of QFSG. With a 0.5 ML deposition of Er, it is possible that not all C-Si back-bonds have been replaced by Er-Si bonds so that some regions of buffer-layer graphene remain. Based on this picture, a simplistic schematic of the suggested Er intercalation process is presented in Figure 3. Annealing at 1473 K does not result in further substantial changes in the LEED I/V spectrum suggesting that Er remains even after heating to such elevated temperatures.22In contrast, Vaknin et al. observed Eu to “deintercalate” and desorb from the surface at 1473 K although this only followed “prolonged annealing”16which results in further graphitization of the sample. As our experiments only involved annealing for 5 minutes at each temperature, it is likely that erbium remained after the 1473 K anneal. It should also be noted that erbium diffusion into the bulk is a possibil- ity, as has been observed for intercalation experiments with other metals.9 FIG. 3 . Schematic representation of the Er intercalation process showing (a) depo- sition of Er on to a carbon buffer layer formed on a 6H-SiC(0001) substrate, (b) Er migration to defects and step edges after annealing at ∼1073 K resulting in cluster formation, and (c) Er intercalation taking place only after annealing above ∼1273 K. AIP Advances 11, 025314 (2021); doi: 10.1063/9.0000154 11, 025314-3 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv B. X-ray photoemission spectroscopy To help elucidate the behaviour of Er following deposition on the buffer layer, XPS was performed with the results presented in Figures 4 and 5. The peak at ∼284.5-285.5 eV in the C 1 sbuffer layer spectrum (Figure 4) arises due to emission from C atoms in this layer, approximately 30% of which are back-bonded to underly- ing Si atoms at the surface of the substrate.23At∼283.3 eV, emission is due to C atoms in the SiC substrate itself. No significant changes to the C 1 sspectrum are observed following deposition of 0.5 ML of Er with the sample at room temperature or following annealing at 1073 K, consistent with erbium forming clusters at the surface rather than intercalating. Annealing to 1273 K, however, does result in a very different C 1 sspectrum, with the clear appearance of two peaks in Figure 4 centered at approximately 284.6 eV (i) and 282.6 eV (ii). At an anneal temperature of only 1273 K, such changes can- not be attributed to further graphitization of the surface3,4and so are most likely a result of Er strongly interacting with C atoms either in the buffer layer or at the substrate surface. Peak (i) is assigned to C 1 semission from the C-C environment present in the QFSG layer in line with previous studies for hydrogen intercalation.4The lower energy peak (ii) is at a similar energy to the modified SiC’ peak observed for QFSG/bilayer graphene observed by Chung et al.24 Those authors suggested the shift in the peak position occurred as a result of transfer doping from Eu intercalation underneath the buffer layer although other changes in the C environment of the SiC substrate due to intercalation could also contribute to the shift. FIG. 4 . Normalized XPS spectra of the C 1 score state for the buffer layer, following deposition of 0.5 ML of Er with the substrate at room temperature, and following subsequent annealing at 1073, 1273 and 1473 K. FIG. 5 . Normalized XPS spectra of the Er 4 dcore state for the buffer layer, follow- ing deposition of 0.5 ML of Er with the substrate at room temperature, and following subsequent annealing at 1073, 1273 and 1473 K. Figure 5 shows the Er 4 dcore level XPS spectra that correspond to the sample preparation conditions as for the C 1 sdata presented in Figure 4. The complex multiplet structure of the Er 4 dstate makes detailed analysis of the Er bonding environment difficult but the strong peak at a binding energy of ∼167-169 eV is characteristic of both metallic Er and Er-Si bonds.25The broadening of this peak and subtle shifts in its position as the anneal temperature increases sug- gest a change in Er bond formation with Er-C bonds typically giving rise to stronger emission at around 170 eV.26 Altogether, the XPS results support the occurrence of Er inter- calation beneath the buffer layer when annealing at high enough temperatures, likely leading to the breaking of Si-C back-bonds in similarity with Eu intercalation.15,24Erbium, as for other interca- lated species such as Dy and Au, most likely intercalates through defects in the buffer layer which would lead to the formation of Er- C bonds that may also contribute to the carbide C 1 speak.8,13Note that the presence of Er even after annealing the sample to 1473 K suggests that Er has not desorbed from the sample to any significant degree. C. Ultraviolet photoemission spectroscopy and metastable de-excitation spectroscopy Figures 6 and 7 show respective UPS and MDS spectra corre- sponding to each step of sample preparation and annealing from the same sample used to obtain the above XPS data. After the Fermi edge at (i) in Figure 6, states (ii) and (iii) at energies of 1.7 and 3 eV AIP Advances 11, 025314 (2021); doi: 10.1063/9.0000154 11, 025314-4 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 6 . Normalized UPS spectra of the buffer layer before and after deposition of 0.5 ML of erbium and following subsequent annealing cycles at increasing temperatures. are features intrinsic to the buffer layer with (ii) arising from the back-bonding state and (iii) due to π-band electrons at the Mpoint in reciprocal space.27These latter states appear in the MDS spectra of the buffer layer, Figure 7, as a shoulder at approximately 3 eV from the cutoff point, marked (i). Feature (ii) in the buffer layer MDS spectrum at 11.5 eV is associated with πhybridization at the Γpoint in reciprocal space.28Both of these states are broad as a result of the dominant de-excitation channel for this surface being the two-step process of resonance ionization followed by Auger neutralization.18 Deposition of 0.5 ML of erbium leads to significant changes in the UPS and MDS spectra, with features associated with the buffer layer replaced by the formation of a broad peak in both spectra cen- tered around 6.2 - 6.6 eV from the cutoff point. Previous analysis of erbium thin films on homoepitaxial diamond showed that an increase in emission around 5 eV from the Fermi level is a result of Er 4 fstates although it is possible that some emission in this region also occurs due to 2 pstates of oxygen atoms adsorbed on the deposited Er.29Annealing the sample at 1073 K results in peak (iv) of the UPS spectrum shifting to a lower binding energy and the par- tial reappearance of peaks (ii) and (iii), again indicating clustering of Er. Interestingly, these buffer layer features again reduce on anneal- ing to 1273 K which provides further evidence that Er intercalation, and the associated breaking of the C-Si back bonds that give rise to these features, has taken place. Stronger emission at deeper binding FIG. 7 . Normalized MDS spectra of the buffer layer before and after deposition of 0.5 ML of erbium and following subsequent annealing cycles at increasing temperatures. energies is possibly a result of the formation of Er-C at defects in the buffer layer, as discussed above. After annealing at 1473 K, the UPS spectrum shows stronger emission at the Fermi level and a new peak at∼2.2 eV indicating graphitization of the sample.4The new feature appears at a similar energy to emission observed for ErSi 2arising as a result of the flat part of a band crossing the Γpoint in reciprocal space due to an Er atom on a buckled Si top layer site.30The forma- tion of Er-Si bonds necessarily means the breaking of back-bonds between C in the buffer layer and Si in the substrate again suggesting the formation of QFSG. In the MDS spectra, annealing at 1073 K leads to the main peak (iii) moving to a higher kinetic energy suggesting clustering of the deposited erbium atoms. Although both XPS and UPS indicate inter- calation of Er atoms occurs after further annealing at 1273 K, the MDS spectrum for this temperature is similar to that at 1073 K. Due to the extreme surface sensitivity of MDS, this indicates that some Er has remained at the surface or is present in relatively large quantities at defected regions of the buffer layer which act as intercalation path- ways. Upon annealing to 1473 K, the features in the MDS spectrum evolve to be similar to those for π-band emission from a graphite surface.18Together with the UPS data, the MDS results support the picture that Er intercalation occurs following deposition and anneal- ing at sufficient temperatures, in this case 1273 K. Er intercalation leads to bonding with substrate Si atoms and the breaking of C-Si back-bonds leading to the formation of QFSG. AIP Advances 11, 025314 (2021); doi: 10.1063/9.0000154 11, 025314-5 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv IV. CONCLUSIONS In this work, we present results from a variety of surface anal- ysis techniques supporting the intercalation of erbium underneath a carbon buffer layer formed on a 6H-SiC(0001) substrate. Upon annealing at temperatures of 1273 K, LEED, XPS, UPS, and MDS all provide evidence that Er intercalation leads to the breaking of sub- surface C-Si back-bonds and the formation of quasi-free-standing graphene. Deposition at room temperature initially leads to disor- dered Er coverage of the buffer layer which then progresses to Er island formation at anneal temperatures below that needed for inter- calation. Intercalation likely proceeds through defects in the buffer layer leading to the formation of Er-C bonds. Further investiga- tion, including with complementary techniques such as Raman spec- troscopy, will help further elucidate the mechanisms of Er intercala- tion and the associated changes in the carbon bonding environment. The results presented here suggest an additional way of potentially activating buffer layer graphene using elements with high magnetic moments, possibly providing options for novel graphene/SiC-based spintronic devices. ACKNOWLEDGMENTS The authors acknowledge the Engineering and Physical Sciences Research Council (EPSRC) for funding this project (EP/N509802/1). The AFM work was carried out in the York JEOL Nanocentre at the University of York. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1A. Avsar, H. Ochoa, F. Guinea, B. Özyilmaz, B. J. van Wees, and I. J. Vera-Marun, “Colloquium: Spintronics in graphene and other two-dimensional materials,” Rev. Mod. Phys. 92, 021003 (2020). 2D. M. Pakdehi, K. Pierz, S. Wundrack, J. Aprojanz, T. T. N. Nguyen, T. Dziomba, F. Hohls, A. Bakin, R. Stosch, C. Tegenkamp, F. J. Ahlers, and H. W. 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5.0034396.pdf

Appl. Phys. Lett. 118, 052411 (2021); https://doi.org/10.1063/5.0034396 118, 052411 © 2021 Author(s).A frustrated bimeronium: Static structure and dynamics Cite as: Appl. Phys. Lett. 118, 052411 (2021); https://doi.org/10.1063/5.0034396 Submitted: 21 October 2020 . Accepted: 20 January 2021 . Published Online: 03 February 2021 Xichao Zhang , Jing Xia , Motohiko Ezawa , Oleg A. Tretiakov , Hung T. Diep , Guoping Zhao , Xiaoxi Liu, and Yan Zhou COLLECTIONS Paper published as part of the special topic on Mesoscopic Magnetic Systems: From Fundamental Properties to Devices ARTICLES YOU MAY BE INTERESTED IN Perspective: Magnetic skyrmions—Overview of recent progress in an active research field Journal of Applied Physics 124, 240901 (2018); https://doi.org/10.1063/1.5048972 Current-driven skyrmionium in a frustrated magnetic system Applied Physics Letters 117, 012403 (2020); https://doi.org/10.1063/5.0012706 The design and verification of MuMax3 AIP Advances 4, 107133 (2014); https://doi.org/10.1063/1.4899186A frustrated bimeronium: Static structure and dynamics Cite as: Appl. Phys. Lett. 118, 052411 (2021); doi: 10.1063/5.0034396 Submitted: 21 October 2020 .Accepted: 20 January 2021 . Published Online: 3 February 2021 Xichao Zhang,1 Jing Xia,2 Motohiko Ezawa,3,a) Oleg A. Tretiakov,4,5 Hung T. Diep,6 Guoping Zhao,2 Xiaoxi Liu,7 and Yan Zhou1,a) AFFILIATIONS 1School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong 518172, China 2College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, China 3Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Tokyo 113-8656, Japan 4School of Physics, The University of New South Wales, Sydney 2052, Australia 5National University of Science and Technology “MISiS,” Moscow 119049, Russia 6Laboratoire de Physique Th /C19eorique et Mod /C19elisation, Universit /C19e de Cergy-Pontoise, 95302 Cergy-Pontoise Cedex, France 7Department of Electrical and Computer Engineering, Shinshu University, 4-17-1 Wakasato, Nagano 380-8553, Japan Note: This paper is part of the APL Special Collection on Mesoscopic Magnetic Systems: From Fundamental Properties to Devices. a)Authors to whom correspondence should be addressed: ezawa@ap.t.u-tokyo.ac.jp andzhouyan@cuhk.edu.cn ABSTRACT We show a topological spin texture called “bimeronium” in magnets with in-plane magnetization. It is a topological counterpart of skyrmio- nium in perpendicularly magnetized magnets and can be seen as a combination of two bimerons with opposite topological charges. We report the static structure and spin-orbit-torque-induced dynamics of an isolated bimeronium in a magnetic monolayer with frustratedexchange interactions. We study the anisotropy and magnetic field dependences of a static bimeronium. We also explore the bimeroniumdynamics driven by the damping-like spin-orbit torque. We find that the bimeronium shows steady rotation when the spin polarizationdirection is parallel to the easy axis. Moreover, we demonstrate the annihilation of the bimeronium when the spin polarization direction is perpendicular to the easy axis. Our results are useful for understanding the fundamental properties of bimeronium structures and may offer an approach to build bimeronium-based spintronic devices. Published under license by AIP Publishing. https://doi.org/10.1063/5.0034396 Topological magnetism and spin frustration are important and hot topics in the fields of magnetism and spintronics. 1–13The link between topological magnetism and spin frustration lies in the fact that many topological spin textures can be stabilized in frustrated spinsystems. 14–37For example, the magnetic skyrmion is an exemplary topological spin texture,1,2which can be regarded as a quasi-particle and shows intriguing dynamics.3–13 The magnetism and spintronics community has focused on sky- rmions stabilized by the Dzyaloshinskii–Moriya (DM) interaction,38–44 however, recent progress in the field revealed that skyrmions and other topological spin textures can be found in a different system, where topological spin textures are stabilized by exchange frustration.14–37 Typical frustrated topological spin textures include the skyrmion,14–36 the skyrmionium37,45,46(i.e., target skyrmion47and 2 p-skyrmion48), and the bimeron20,34,36(i.e., asymmetric skyrmion49and meron pair50). Indeed, skyrmioniums and bimerons can also be stabilized bythe DM interaction.34,51–56In principle, all of these particle-like topo- logical spin textures can be used to carry information,4,5,7–11,13and thus are promising for building future information storage and logic computing devices.4,5,7–11,13 In this Letter, we report that the topological counterpart of sky- rmioniums, which is called the bimeronium [Fig. 1(a) ], can be stabi- lized in an in-plane magnetic system with competing exchange interactions. We study the static structure of an isolated bimeronium with a topological charge of zero at different anisotropies and magneticfields. We also investigate the dynamics of an isolated bimeronium induced by the damping-like spin-orbit torque (SOT). Our results sug- gest that the frustrated bimeronium could be used as a special buildingblock for spintronic applications, however, it cannot move like the frustrated bimeron 36due to its complex and non-circular symmetric spin structure that could be annihilated by the SOT under certainconditions. Appl. Phys. Lett. 118, 052411 (2021); doi: 10.1063/5.0034396 118, 052411-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplOur simulated system is a J1-J2-J3classical Heisenberg model on a simple monolayer square lattice,12,16,21,30,36,37,57where the nearest- neighbor (NN) exchange interaction J1is ferromagnetic (FM), while the next-NN (NNN) J2and next-NNN (NNNN) J3exchange interac- tions are antiferromagnetic (AFM). The Hamiltonian Hincludes the FM and AFM exchange interactions, in-plane easy-axis magneticanisotropy ( K), and applied magnetic field ( B), given as H¼/C0 J 1X <i;j>mi/C1mj/C0J2X /C28i;j/C29mi/C1mj /C0J3X ni;jomi/C1mj/C0KX iðmx iÞ2/C0X iB/C1mi; (1) where mirepresents the normalized spin at the site i,jmij¼1.hi;ji, hhi;jii,a n dhhhi;jiiirun over all the NN, NNN, and NNNN sites in the magnetic monolayer, respectively. Kis the easy-axis magnetic anisotropy constant, and the easy axis direction is aligned along the x axis.Bis the applied magnetic field. The spin dynamics is simulated by using the Object Oriented MicroMagnetic Framework (OOMMF)58with our extension modules for the J1-J2-J3classical Heisenberg model.16,21,30,36,37We also use the OOMMF conjugate gradient minimizer for obtaining relaxed spinstructures, which is a method that locates local minima in the energysurface through direct minimization techniques. 58The spin dynamics is governed by the Landau–Lifshitz–Gilbert (LLG) equation58 dm dt¼/C0c0m/C2heffþam/C2dm dt/C18/C19 ; (2) where jmj¼1 represents the normalized spin, heff¼/C01 l0MS/C1dH dmis the effective field, tis the time, ais the Gilbert damping parameter, c0 is the absolute value of the gyromagnetic ratio, and MSis the satura- tion magnetization. For the spin dynamics driven by the SOT, we consider the damping-like SOT sdexpressed as4,9,11,59sd¼u aðm/C2p/C2mÞ,w h e r eu¼j ðc0/C22h=l0eÞj /C1 ðjhSH=2MSÞi st h es p i nt o r q u ec o e f fi c i e n t . /C22his the reduced Planck constant, eis the electron charge, l0is the vacuum permeability constant, ais the thickness of the FM monolayer (i.e., the lattice constant here), jis the applied current density, hSHis the spin Hall angle. pdenotes the spin polarization direction. sdis added to the right-hand side of Eq. (2)when the damping-like SOT is turned on. In this work, we define the topological charge Qin the contin- uum limit by the formula3,11Q¼1 4pÐmðrÞ/C1ð@xmðrÞ/C2@ymðrÞÞd2r. We parametrize the bimeronium [ Fig. 1(b) ]a n db i m e r o n[ Figs. 1(c) and1(d)]a smðrÞ¼mðh;/Þ¼ð cosh;sinhsin/;/C0sinhcos/Þ,a n d we parametrize the skyrmionium [ Fig. 1(e) ] and skyrmion [ Figs. 1(f) and 1(g)]a smðrÞ¼mðh;/Þ¼ð sinhcos/;sinhsin/;coshÞ.W e define /¼Qvuþg,w h e r e uis the azimuthal angle in the y-zplane (0/C20u<2p). For the bimeron and skyrmion, we assume that h rotates pfor spins from the texture center to the texture edge.51For the bimeronium and skyrmionium, we assume that hrotates 2 pfor spins from the texture center to the texture edge.51Hence, Qv¼1 2pÞ Cd/is the vorticity and gis the helicity defined mod 2 p. Note that g¼0i si d e n t i c a lt o g¼2p. The default simulation parameters are:16,21,30,36,37J1¼30 meV, J2¼/C00:8( i nu n i t so f J1¼1),J3¼/C00:9 (in units of J1¼1), K¼0.02 (in units of J1=a3¼1),B¼0( i nu n i t so f J1=a3MS¼1), a¼0:3,c0¼2:211/C2105mA–1s/C01,hSH¼0:2, and MS¼580 kA m/C01. The lattice constant is a¼0.4 nm (i.e., the mesh size is 0 :4/C20:4 /C20:4n m3). We have simulated the metastability diagram showing that the frustrated bimeronium can be a metastable state for a wide range of J2and J3(see the supplementary material ). The minimum required value of J3for stabilizing bimeroniums decreases with increasing J2. We first study the static structures and properties of a relaxed iso- lated bimeronium in the magnetic monolayer with competing exchange interactions and in-plane easy-axis anisotropy, where we setJ 2¼/C00:8;J3¼/C00:9,K¼0.02, and B¼0.Figure 1 shows the top views of relaxed compact bimeronium and bimeron structures. For FIG. 1. (a) Schematic illustration of a bimeronium with Q¼0. The arrows represent the spin directions. The out-of-plane spin component ( mz) is color coded. (b) Top view of a relaxed bimeronium with Q¼0. (c) Top view of a relaxed bimeron with Q¼/C0 1. (d) Top view of a relaxed bimeron with Q¼þ 1. (e) Top view of a relaxed skyrmionium with Q¼0. (f) Top view of a relaxed skyrmion with Q¼/C0 1. (g) Top view of a relaxed skyrmion with Q¼þ 1. Here, J2¼/C0 0:8;J3¼/C0 0:9,K¼0.02, and B¼0. For (b)–(d), the easy axis is aligned along the xdirection. For (e)–(g), the easy axis is aligned along the zdirection. The displayed area is 10 /C210 nm2.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 052411 (2021); doi: 10.1063/5.0034396 118, 052411-2 Published under license by AIP Publishingthe purpose of comparison, we also show the relaxed solutions of com- pact skyrmionium and skyrmion obtained with the same parameters but an easy-axis aligned along the zaxis. The given skyrmionium with Q¼0[Fig. 1(e) ] consists of an inner skyrmion with Q¼/C01[Fig. 1(f) ] and an outer skyrmion with Q¼þ1[Fig. 1(g) ]. Similarly, the corre- sponding bimeronium with Q¼0[Fig. 1(b) ] exists as a combination of an inner bimeron with Q¼/C0 1[Fig. 1(c) ] and an outer bimeron with Q¼þ1[Fig. 1(d) ]. Namely, the bimeronium in in-plane magne- tized magnets can be seen as a topological counterpart of the skyrmio- nium in out-of-plane magnetized magnets. The total energy as well as different energy contributions for a relaxed bimeronium are shown in Fig. 2(a) . It shows that the competi- tion among the FM NN, AFM NNN, and AFM NNNN exchange interactions is considerable. The magnetic anisotropy energy is posi- tive, which means that larger anisotropy constant could raise the total energy of a bimeronium and may reduce its stability. A controllable degree of freedom of topological spin textures in frustrated magnetic systems is the helicity g.15,16,21,30,36,37It is found that the energy [ Fig. 2(b)] and spin components [ Fig. 2(c) ] of the bimeronium are indepen- dent of its helicity. We note that although the bimeronium helicity can freely vary between 0 and 2 p, the orientation of the background spins outside the bimeronium is aligned with the easy axis orientation, i.e., thexd i r e c t i o ni nt h i sw o r k .As the magnetic anisotropy can be adjusted experimentally, we study the bimeronium structure for different anisotropy constants K, as shown in Fig. 3 . By increasing Kfrom 0 to 0.095, while keeping the easy-axis orientation aligned along the xdirection, the size of relaxed bimeronium decreases obviously. The spin component mxreduces with increasing K,w h i l e myand mzdo not depend on K[Fig. 3(a) ]. The total energy [ Fig. 3(b) ], anisotropy energy [ Fig. 3(c) ], AFM NNN exchange energy [ Fig. 3(e) ], and AFM NNNN exchange energy [ Fig. 3(f)] increase with K, while the FM NN exchange energy [ Fig. 3(d) ] decreases with increasing K.W h e n Kis larger than a certain threshold (i.e., 0.1 in this work), the bimeronium structure becomes unstable and cannot exist in the system. We also study the effect of an external in-plane magnetic field on the bimeronium structure (see the supplementary material ). As the bimeronium size is related to its spin component along the easy-axis orientation, which is mxin this work [ Fig. 3(a) ], we apply a magnetic field along the xdirection with a strength of Bx. Within a reasonable range of Bxthat does not destroy the bimeronium (i.e., in this work Bx=1000¼/C00:2/C240:8i nu n i t so f J1=a3MS¼1), it is found that the bimeronium size is insensitive to Bx.T h es p i nc o m p o n e n t mxincreases with Bx,w h i l e myand mzare independent of Bx.T h et o t a le n e r g y , anisotropy energy, and FM NN exchange energy are proportional to Bx, while the AFM NNN and NNNN exchange energies are inversely FIG. 2. (a) Different energy contributions for a relaxed bimeronium with g¼0. The energies are given in units of J1¼1. (b) Total energy of a relaxed bimeronium as a function ofg. (c) Spin components of a relaxed bimeronium as functions of g. Top views of relaxed bimeroniums with (d) g¼0, (e) g¼p=2, (f)g¼p, and (g) g¼3p=2 are given. Here, J2¼/C0 0:8;J3¼/C0 0:9,K¼0.02, and B¼0. The arrows represent the spin directions. The out-of-plane spin component ( mz) is color coded. The displayed area is 10/C210 nm2.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 052411 (2021); doi: 10.1063/5.0034396 118, 052411-3 Published under license by AIP Publishingproportional to Bx. Note that a larger external magnetic field may lead to the deformation and annihilation of the bimeronium texture. As shown in Fig. 4 , we continue by investigating the dynamics of an isolated bimeronium driven by the damping-like SOT sd.W e assume that sdis generated by the spin Hall effect in a heavy-metal substrate layer underneath the magnetic monolayer.4,5,7–9,11,60We first consider that the direction of spin polarization is aligned along the easy-axis direction, i.e., p¼þ^x. A recent report36numerically dem- onstrates that a frustrated bimeron shows SOT-induced selfrotationwhen the spin polarization direction is parallel to the easy-axis direc-tion. Similarly, we find that the bimeronium also shows steady selfro-tation induced by the damping-like SOT when p¼þ^x,w h i c hc a nb e described by a theoretical approach (see the supplementary material ). The rotation period decreases with increasing driving current density[Fig. 4(a) ] and the rotation frequency is proportional to the driving current density [ Fig. 4(b) ]. At a given current density, the inner and outer consisting bimerons of the bimeronium rotate in an identical manner, and the rotation frequency of the whole structure decreases with increasing damping parameter a. For the rotating bimeronium, its total energy [ Fig. 4(c) ] and spin components [ Fig. 4(d) ] are inde- pendent of time when the steady rotation state is reached. Note thatthe rotating bimeronium is a dynamic stable object, which will not be annihilated even at a high speed rotation as the inner and outerconsisting bimerons rotate in an identical frequency and direction. On the other hand, due to the non-circular symmetric out-of-plane spinstructure of the bimerionium, its rotation can be observed by imaging the out-of-plane cores, while the selfrotation of skyrmions and sky- rmioniums can only be observed by imaging their in-plane spin com-ponents. Thus, we point out that it is possible to observe rotatingbimeroniums by using the Kerr microscope system. The bimeronium rotation depends on both the internal structure of the bimeronium as well as the spin polarization direction. As showninFig. 4(e) , when the bimeronium consists of an inner bimeron with Q¼/C01a n da no u t e rb i m e r o nw i t h Q¼þ1, it shows counterclock- wise rotation driven by the damping-like SOT with p¼þ^x.I ft h ei n i - tial bimeronium structure consists of an inner bimeron with Q¼þ1 a n da no u t e rb i m e r o nw i t h Q¼/C01, it may show clockwise rotation driven by the damping-like SOT with p¼/C0^x(see the supplementary material ). As reported in Ref. 36, a frustrated bimeron could show SOT- induced translational motion when the spin polarization direction is perpendicular to the easy-axis direction. However, we find that thefrustrated bimeronium cannot be driven into steady motion when thespin polarization direction is perpendicular to the easy-axis direction,i.e.,p¼6^y. Instead, the damping-like SOT leads to the deformation and annihilation of the bimeronium structure, as shown in Fig. 5 .T o FIG. 3. (a) Spin components as functions of K. A bimeronium with g¼0 is relaxed at the center of a monolayer with J2¼/C0 0:8;J3¼/C0 0:9, and B¼0. (b) Total energy ETotal as a function of K. The energies are given in units of J1¼1. (c) Anisotropy energy EKas a function of K. (d) NN exchange energy ENNas a function of K. (e) NNN exchange energy ENNN as a function of K. (f) NNNN exchange energy ENNNN as a function of K. Top views of relaxed bimeroniums with g¼0 at (g) K¼0.01, (h) K¼0.02, (i) K¼0.04, and (j) K¼0.08 are given. The arrows represent the spin directions. The out-of-plane spin component ( mz) is color coded. The displayed area is 12 :4/C212:4n m2.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 052411 (2021); doi: 10.1063/5.0034396 118, 052411-4 Published under license by AIP Publishingbe more specific, the damping-like SOT leads to different rotation behaviors of the inner and outer consisting bimerons of the bimero- nium, and the bimeronium structure is twisted upon the application of the SOT. When the out-of-plane cores of the inner and outer con- sisting bimerons merge together, the bimeronium is thus destroyed and annihilated. The total energy [ Fig. 5(a) ] and spin component mx [Fig. 5(b) ] decrease significantly during the SOT-induced annihilation of the bimeronium. The spin components myandmzalso show certain fluctuations during the annihilation process. Note that the initial bimeronium structure in Fig. 5 consists of an inner bimeron with Q¼/C01 and an outer bimeron with Q¼þ1. For the bimeronium structure consisting of an inner bimeron with Q¼þ1 and an outerbimeron with Q¼/C01, it also shows deformation and annihilation w h e nt h ed a m p i n g - l i k eS O Tw i t h p¼6^yis applied (see the supple- mentary material ). In conclusion, we have studied the static structures and SOT- induced dynamics of an isolated bimeronium in a frustrated magneticmonolayer with competing FM and AFM exchange interactions. Note that the small FM NN and large AFM NNN exchange interactions could be realized in low-dimensional compound Pb 2VO(PO 4)2with a frustrated square lattice.57The bimeronium structure carries a topo- logical charge of Q¼0b u ti tc a nb er e g a r d e da sac o m b i n a t i o no ft w o bimerons with opposite topological charges. Namely, it may consist ofan inner bimeron with Q¼/C01 and an outer bimeron with Q¼þ1, FIG. 4. (a) Bimeronium rotation period Tas a function of driving current density jfor different damping parameters a. The spin polarization direction p¼þ^x. (b) Bimeronium rotation frequency fas a function of jfor different a. (c) Time-dependent total energy of a typical rotating bimeronium. (d) Time-dependent spin components of a typical rotating bimeronium. (e) Top views of a typical rotating bimeronium at selected times. The arrows represent the spin directions. The out-of-plane spin compon ent (mz) is color coded. Here, J2¼/C0 0:8;J3¼/C0 0:9,K¼0.02, and B¼0. The initial bimeronium state consists of an inner bimeron with Q¼/C0 1 and an outer bimeron with Q¼þ 1.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 052411 (2021); doi: 10.1063/5.0034396 118, 052411-5 Published under license by AIP Publishingand it may also consist of an inner bimeron with Q¼þ1 and an outer bimeron with Q¼/C01. It is found that the frustrated bimeronium energy is independent of its helicity in the in-plane magnetized system, however, the size and energy of a bimeronium are subject to the easy-axis magnetic anisot- ropy. A larger anisotropy will lead to a smaller compact bimeroniumwith higher total energy. Indeed, extremely large anisotropy may resultin the instability of the bimeronium structure. On the other hand, the bimeronium energy can be subtly adjusted by an external in-plane magnetic field, however, the bimeronium size is insensitive to the mag-netic field. In this work, we have also numerically demonstrated that the bimeronium can be driven into steady rotation by the damping-like SOT, where the spin polarization direction is parallel to the easy-axis direction. It is found that the rotational dynamics depends on both theinternal bimeronium structure and the spin polarization direction. In particular, when the spin polarization direction is perpendicular to the easy-axis, the bimeronium is annihilated by the damping-like SOT. We point out that the rotational feature of a bimeronium may be used for building a multi-state memory device based on bimeronium-hosting nanodots, 61where a bimeronium with variable helicity values in a unit nanodot can be used to store different information. The current-controlled rotation of a bimeronium could also be useful for future spin-wave applications, where arrays of bimeroniums serve asreconfigurable spin wave guides. We believe our results are important for understanding the frustrated bimeronium structures, and can pro- vide guidelines for the design of spintronic devices based onbimeroniums.See the supplementary material for additional simulation results. AUTHORS’ CONTRIBUTIONS X.Z. and J.X. contributed equally to this work. This work was primarily supported by the National Natural Science Foundation of China (Grant No. 12004320). X.Z. also acknowledges the support by the Guangdong Basic and AppliedBasic Research Foundation (Grant No. 2019A1515110713). M.E.acknowledges the support by the Grants-in-Aid for ScientificResearch from JSPS KAKENHI (Grant Nos. JP18H03676 andJP17K05490) and also the support by CREST, JST (Grant Nos.JPMJCR16F1 and JPMJCR20T2). O.A.T. acknowledges the supportby the Australian Research Council (Grant No. DP200101027), theCooperative Research Project Program at the Research Institute ofElectrical Communication, Tohoku University (Japan), and by theMinistry of Science and Technology Higher Education of theRussian Federation in the framework of Increase CompetitivenessProgram of NUST MISiS (No. K2–2019-006), implemented by agovernmental decree dated 16th of March 2013, N 211. G.Z.acknowledges the support by the National Natural ScienceFoundation of China (Grant Nos. 51772004, 51771127, and51571126). X.L. acknowledges the support by the Grants-in-Aid forScientific Research from JSPS KAKENHI (Grant No. JP20F20363).Y.Z. acknowledges the support by the Guangdong Special Support Project (Grant No. 2019BT02X030), the Shenzhen Peacock Group Plan (Grant No. KQTD20180413181702403), the Pearl RiverRecruitment Program of Talents (Grant No. 2017GC010293), and FIG. 5. (a) Total energy of a bimeronium as a function of time during its annihilation induced by the SOT. The spin polarization direction p¼þ^y. (b) Spin components of a bimeronium as a function of time during its annihilation induced by the SOT. (c) Top views of the bimeronium annihilation at selected times. The arrows represent the spin directions. The out-of-plane spin component ( mz) is color coded. Here, J2¼/C0 0:8;J3¼/C0 0:9,K¼0.02, and B¼0. The initial bimeronium state consists of an inner bimeron withQ¼/C0 1 and an outer bimeron with Q¼þ 1.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. 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5.0031419.pdf

J. Math. Phys. 62, 032204 (2021); https://doi.org/10.1063/5.0031419 62, 032204 © 2021 Author(s).Generalized Pauli constraints in large systems: The Pauli principle dominates Cite as: J. Math. Phys. 62, 032204 (2021); https://doi.org/10.1063/5.0031419 Submitted: 30 September 2020 . Accepted: 26 February 2021 . Published Online: 19 March 2021 Robin Reuvers ARTICLES YOU MAY BE INTERESTED IN Infinite wall in the fractional quantum mechanics Journal of Mathematical Physics 62, 032104 (2021); https://doi.org/10.1063/5.0026816 Unitary, continuum, stationary perturbation theory for the radial Schrödinger equation Journal of Mathematical Physics 62, 032105 (2021); https://doi.org/10.1063/5.0023630 Particle–hole symmetries in condensed matter Journal of Mathematical Physics 62, 021101 (2021); https://doi.org/10.1063/5.0035358Journal of Mathematical PhysicsARTICLE scitation.org/journal/jmp Generalized Pauli constraints in large systems: The Pauli principle dominates Cite as: J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 Submitted: 30 September 2020 •Accepted: 26 February 2021 • Published Online: 19 March 2021 •Publisher error corrected: 19 March 2021 Robin Reuversa) AFFILIATIONS DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom and QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark a)Author to whom correspondence should be addressed: rr@math.ku.dk ABSTRACT Lately, there has been a renewed interest in fermionic one-body reduced density matrices and their restrictions beyond the Pauli princi- ple. These restrictions are usually quantified using the polytope of allowed, ordered eigenvalues of such matrices. Here, we prove that this polytope’s volume rapidly approaches the volume predicted by the Pauli principle as the dimension of the one-body space grows and that additional corrections, caused by generalized Pauli constraints, are of much lower order unless the number of fermions is small. Indeed, we argue that the generalized constraints are most restrictive in (effective) few-fermion settings with low Hilbert space dimension. Published under license by AIP Publishing. https://doi.org/10.1063/5.0031419 I. INTRODUCTION Fermionic quantum states are antisymmetric: their N-body space is a wedge product ∧NHofNcopies of the one-body space H. In particular, this implies the Pauli principle. Of course, antisymmetry is a more restrictive property, and it is a long-standing problem to find out just how restrictive it is.10,11For example, it is unknown what k-particle reduced density matrices γΨ k∶=(N k)Trk+1...N[∣Ψ⟩⟨Ψ∣] (1) can arise from pure states ∣Ψ⟩∈∧NH⊂⊗NH. This is particularly relevant for k=2 since such knowledge would provide significant computational advantages. In this paper, we focus on the simpler case k=1. The set of interest is {γΨ 1∣∣Ψ⟩∈∧NH,∥∣Ψ⟩∥=1}. (2) EachγΨ 1is diagonalizable: it has eigenvalues and eigenvectors, but the latter can easily be changed with a unitary transformation. Indeed, the set is closed under such transformations: it is entirely defined by the allowed eigenvalues of γΨ 1. ForH=Cdwith N≤d, this information amounts to Fd,N∶={(λ1,...,λd)∈Rd∣λ1≥⋅⋅⋅≥λdeigenvalues of γΨ 1for∣Ψ⟩∈∧NCd,∥∣Ψ⟩∥=1}. (3) This is a convex polytope in Rd,1,28,29and it can be determined numerically for small Nand d.1Less is known about higher Nand d, and that is the focus of this paper. We are motivated by the ongoing attempts to use knowledge about Fd,Nin physics and chemistry.3,4,14,18–21 J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 62, 032204-1 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp To start the investigation, let us check how Fd,Nrelates to an important physical fact: the Pauli principle. The Pauli principle says that the expectation value of any particle number operator ni∶=a† iaiin a normalized fermionic state ∣Ψ⟩is bounded by 1, ⟨ni⟩Ψ=⟨Ψ∣a† iai∣Ψ⟩=⟨Ψ∣𝟙−aia† i∣Ψ⟩=1−∥a† i∣Ψ⟩∥2≤1, (4) but this is equivalent to saying that the eigenvalues of γΨ 1are all bounded by 1. After all, an annihilation operator aiacts as√ N(⟨ui∣⊗𝟙)on N-fermion states such as ∣Ψ⟩, for some one-particle state ∣ui⟩, and ⟨ui∣γΨ 1∣ui⟩=NTr1...N[(∣ui⟩⟨ui∣⊗𝟙)∣Ψ⟩⟨Ψ∣]=⟨Ψ∣a† iai∣Ψ⟩=⟨ni⟩≤1. (5) Hence, we know that λ1≤1 for points in Fd,N. Since Fd,Nis a convex polytope,1,28,29it is completely defined by inequalities involving λi. These are known as generalized Pauli constraints1,22and have the general shape c1λ1+⋅⋅⋅+ cdλd≤b (6) forci,b∈R. Are these as valuable as the Pauli inequality λ1≤1? To investigate this from a purely mathematical viewpoint, define Pd,N∶={(λ1,...,λd)∈Rd∣1≥λ1≥⋅⋅⋅≥λd≥0 andλ1+⋅⋅⋅+λd=N}. (7) This is the crudest approximation to Fd,Nwe can make, and it uses only the Pauli inequality and the normalization condition λ1+⋅⋅⋅+λd=N. Does this give a good approximation to Fd,N? For low Nand d, certainly not. Example 1. The N =2case has been understood since 1961,25,26and the N =3, d=6cases have been understood since 1972.5The relevant sets are Fd,2={(λ1,...,λd)∣1≥λ1≥⋅⋅⋅≥λd≥0,∑ iλi=2,λ2i−1=λ2i,λd=0 ifdis odd}, F6,3={(λ1,...,λ6)∣1≥λ1≥⋅⋅⋅≥λ6≥0,∑ iλi=3,λi=1−λ7−i,λ4≤λ5+λ6}. In 2008, an algorithm was devised to calculate general F d,N.1The resulting polytopes for low N and d do not resemble P d,N, but more so than in the instances mentioned above. Clearly, the difference between Fd,Nand Pd,Nis huge in these cases. Does this remain true when dincreases or when both Nand d increase? One way to measure this is by comparing volumes. Although volume does not carry any physical information, it is a useful way to investigate the two polytopes. Indeed, in line with what is suggested by the explicit results for small Nand d,1we will show that Fd,Nand Pd,N quickly have similar volume as dincreases and explain why the polytopes are mostly alike. This paper is divided into three parts. We discuss theorems about volume in Sec. II, important insights from the proof in Sec. III, and the proof itself in Sec. IV. II. THEOREMS ABOUT VOLUME A. Comparing the volumes of Fd,NandPd,N Recall that we want to compare Fd,N={(λ1,...,λd)∈Rd∣λ1≥⋅⋅⋅≥λdeigenvalues of γΨ 1for∣Ψ⟩∈∧NCd,∥∣Ψ⟩∥=1}, Pd,N={(λ1,...,λd)∈Rd∣1≥λ1≥⋅⋅⋅≥λd≥0 andλ1+⋅⋅⋅+λd=N}.(8) Note that Vold−1(Fd,N)=Vold−1(Fd,d−N)by particle–hole duality and similar for Pd,N. J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 62, 032204-2 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp Our first theorem gives the volume’s limiting behavior. Theorem 2 (Limit behavior). Let N≥8be fixed. Then, lim d→∞Vold−1(Fd,N) Vold−1(Pd,N)=1. Alternatively, for a fixed filling ratio r ∈(0, 1), lim d→∞Vold−1(Fd,⌊rd⌋) Vold−1(Pd,⌊rd⌋)=1. This theorem is a corollary of the following estimates, which are proved in Sec. IV. Theorem 3 (Quantitative estimate). Let8≤N≤d/2be fixed. Then, if d is large enough to guarantee d (N−1 N)d−1≤1, 1≥Vold−1(Fd,N) Vold−1(Pd,N)≥1−dN 1−d(N−1 N)d−1⎛ ⎝min[1 2(N+7),√ 32N] N⎞ ⎠d−1 . In addition, for integers d and N =rd≥20for some r∈(0, 1/2), 1≥Vold−1(Fd,rd) Vold−1(Pd,rd)≥1−1 rr+1/2(1−r)3/2−r(8 rr+1/2(1−r)1−r1√ d)d−1 . Remarks. 1. Volume is used here as a way to compare Fd,Nand Pd,N—it does not carry any physical information. We do argue that insights from the proof allow us to draw some conclusions. These are discussed in Sec. III. 2. Although these estimates show that convergence occurs rapidly, we can obtain better estimates for low Nand d. Remark 6 discusses this; Fig. 1 illustrates the result. 3. The ratios mentioned above concern the effect of the generalized Pauli constraints in excess of the Pauli principle. It is useful to compare this to the effect that the Pauli principle itself has on the bosonic analog of Fd,N. We discuss this in Subsection II B. FIG. 1. A contour plot demonstrating the lower bound to Vold−1(Fd,N)/Vold−1(Pd,N)obtained in Remark 6. The blue part corresponds to N>d, which is not allowed. The convergence happens as orange turns to yellow, and it occurs extremely rapidly if N≥80. Numerical simulations inspired by Ref. 8 (now see Ref. 9) suggest that convergence should actually happen more quickly in the region 8 ≤N≤80 so that the yellow region extends a long way toward the contour that forms a triangle in the orange region, but this cannot be demonstrated with our method. Similarly, we have no bound for N,d−N≤8, but numerics also suggests rapid convergence in dfor 4≤N,d−N≤8. J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 62, 032204-3 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp B. Comparing with the effect of the Pauli principle Define the bosonic polytope Bd,N∶={(λ1,...,λd)∈Rd∣λ1≥⋅⋅⋅≥λd≥0 andλ1+⋅⋅⋅+λd=N}, (9) which is Pd,Nwithout the Pauli condition. It is well known that this set is physically correct for N≥2: it is equal to {(λ1,...,λd)∈Rd∣λ1≥⋅⋅⋅≥λdeigenvalues of γΨ 1for∣Ψ⟩∈⊗N SYMCd,∥∣Ψ⟩∥=1}. (10) Indeed, the discrepancy between “naive” Pd,Nand correct, more complicated Fd,Nis a purely fermionic phenomenon. Nevertheless, it is useful to compare Bd,Nand Pd,Nsince the Pauli principle cuts Bd,Ndown to Pd,N, after which the generalized constraints cut Pd,Ndown to Fd,N. It seems reasonable to compare the volumes lost in these two steps, as it suggests something about the impact of the generalized constraints compared to that of the Pauli principle. Let us stress again that this is the main motivation behind this work: volume itself is not important, but it is used here as a tool to investigate the structure of these polytopes. To make a comparison, we first need information about the difference between Pd,Nand Bd,N. This is proved at the end of Sec. IV D. Proposition 4 (Volume loss due to Pauli). For1≤N≤d, 1−d(N−1 N)d−1 ≤Vold−1(Pd,N) Vold−1(Bd,N)≤1−(N−1 N)d−1 . This immediately implies two things: First, for fixed Nand large d, the effect of the Pauli principle on volume is negligible, and second, for a fixed ratio r=N/d, the Pauli principle has a non-negligible effect on volume. As shown in Sec. II A, the generalized constraints have a negligible effect in all cases. Using Theorem 3 and Proposition 4, a quantitative comparison can be made. To do this, note that Vold−1(Pd,N/Fd,N) Vold−1(Bd,N/Pd,N)=1−Vold−1(Fd,N) Vold−1(Pd,N) Vold−1(Bd,N) Vold−1(Pd,N)−1(11) so that we obtain expressions such as the ones in Theorem 3. Qualitatively, nothing changes, except that Nd−1gets replaced by (N−1)d−1 in the denominator of the first estimate. This says that the volume effect of the generalized Pauli constraints is much smaller than that of the Pauli principle. The qualitative conclusions are listed below. Corollary 5 (Comparing to Pauli). Let N≥10be fixed. Then, lim d→∞Vold−1(Pd,N) Vold−1(Bd,N)=1, lim d→∞Vold−1(Pd,N/Fd,N) Vold−1(Bd,N/Pd,N)=0. In addition, for a fixed filling ratio r ∈(0, 1), lim sup d→∞Vold−1(Pd,⌊rd⌋) Vold−1(Bd,⌊rd⌋)≤1−e−1/r, lim d→∞Vold−1(Pd,⌊rd⌋/Fd,⌊rd⌋) Vold−1(Bd,⌊rd⌋/Pd,rd)=0. III. INSIGHTS FROM THE PROOF A. Proof strategy 1. Pd,Nis a polytope. We first determine which of its extreme points lie in Fd,N. As discussed in Subsection III B, this turns out to be the vast majority as Nand dincrease. However, these do not yet capture a volume. 2. To deal with this, replace extreme points outside Fd,Nby one or more intermediate points that do lie in Fd,N. This captures part of the volume of Pd,Nby convexity, and this volume must be contained in Fd,N. In particular, we verify in Sec. IV C that for integers 1≤m≤N−7 and t=N−m+1 N−m+9, Ad,N,m,t∶={(λ1,...,λd)∈Pd,N∣λm≤t}⊂Fd,N. (12) See Fig. 2 for an illustration. J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 62, 032204-4 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp FIG. 2. Suppose that the triangle is Pd,N. It has three extreme points. Imagine that one (blue) is in Fd,N, whereas two (red) are not, and that we can verify that the black points are in Fd,N. This means that Fd,Ncontains the blue set, say, Ad,N,m,t, and hence, Vold−1(Ad,N,m,t)≤Vold−1(Fd,N)≤Vold−1(Pd,N). 3. The above implies Vold−1(Fd,N) Vold−1(Pd,N)≥Vold−1(Ad,N,m,t) Vold−1(Pd,N). (13) We estimate these volumes in Sec. IV D and prove Theorems 2 and 3. Note that mis a variable that can be optimized. Remark 6. The volumes of Pd,Nand Ad,N,m,tcan also be calculated explicitly; see Proposition 7 and Appendix A, respectively. This is the sharpest estimate our method can give, and it demonstrates how quickly Vold−1(Fd,N)/Vold−1(Pd,N)converges to 1 already for low Nand d (see Fig. 1). B. Extreme points of Pd,N We now discuss which extreme points of Pd,Nare also in Fd,N. This provides important clues as to why and when these two polytopes resemble each other. We start by indexing the extreme points of Pd,N. Proposition 7 (Properties of Pd,N).The extreme points of P d,Nare the Slater point (1,..., 1 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ N, 0,..., 0 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ d−N)∈Rd(14) and the N (d−N)distinct points (1,..., 1 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ i,N−i d−i−j,...,N−i d−i−j ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ d−i−j, 0,..., 0 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ j)∈Rd(0≤i≤N−1, 0≤j≤d−N−1). (15) The polytope has (d−1)-dimensional volume, Vold−1(Pd,N)=1 d!√ d (d−1)!N−1 ∑ k=0(−1)k(d k)(N−k)d−1. (16) This is proved at the end of Sec. IV A. For now, note that the extreme points of Pd,Nare completely defined by the fact that they have i entries that are 1 and jentries that are 0. As we discuss in Sec. IV B, the ones correspond to a Slater determinant that can be split off from the remainder of the state; the zeros can be ignored as unoccupied dimensions. This gives the following observation. J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 62, 032204-5 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp Proposition 8. For 0≤i≤N−1, 0≤j≤d−N−1, (1,..., 1 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ i,N−i d−i−j,...,N−i d−i−j ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ d−i−j, 0,..., 0 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ j)∈Fd,N ⇐⇒(N−i d−i−j,...,N−i d−i−j ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ d−i−j)∈Fd−i−j,N−i. States with this latter eigenvalue structure have been studied before. Definition 9 (Completely entangled or fermionic LME states). A normalized state ∣Ψ⟩in∧NCdis Locally Maximally Entangled (LME),6,7 alternatively, completely entangled,1if its one-body reduced density matrix satisfies γΨ 1=NTr2...N[∣Ψ⟩⟨Ψ∣]=N d⋅𝟙d. (17) These states form a subset VN,d LME⊂∧NCd. It turns out that fermionic LME states exist for almost all Nand d. Theorem 10 (Altunbulak and Klyachko1). Fermionic LME states exist unless d≥2,N=1; dodd, N=2; dodd, N=d−2; d≥2,N=d−1. Table I illustrates this. Note that particle–hole symmetry is present because γΨholes 1=𝟙d−γΨparticles 1for particle–hole duals ∣Ψparticles⟩∈∧NCd and∣Ψholes⟩∈∧d−NCdso that the LME property is preserved. Remark 11. Although it is not needed in this paper, the dimension of VN,d LME/SU(d)can be computed with techniques from Refs. 2, 6, 7, 12, and 13. For completeness, we include a theorem in Appendix B. From Theorem 10 and Proposition 8, we can now tell which extreme points (15) of Pd,Nare in Fd,N: each extreme point is indexed by (i,j)and corresponds to a different box of Table I. Table II illustrates this for d=11,N=5. This observation leads to the following conclusion: as dgrows, more and more extreme points of Pd,Ncorrespond to blue boxes in Table I—that is, they are in Fd,N. The points that Fd,Ndoes not reach effectively correspond to N=1, 2, d−1,d−2. Note that these points have “few-body” character. As mentioned in Sec. III A, we will have to approach these problematic points to capture a large volume. That is, we will seek points in Fd,N that are fairly close to the problematic points. Lemma 21 shows which (suboptimal) points we use. It is interesting to note that these again have TABLE I. Existence of LME states in ∧NCd. No LME states exist for (d≥2,N=1),(dodd, N=2)and their particle–hole duals (d≥2,N=d−1),(dodd, N=d−2). 1 2 3 4 5 6 7 8 9 10 11 12dN 0 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ 1 ✓ × × × × × × × × × × × 2 ✓ × ✓ × ✓ × ✓ × ✓ × ✓ 3 ✓ × × ✓ ✓ ✓ ✓ ✓ ✓ ✓ 4 ✓ × ✓ ✓ ✓ ✓ ✓ ✓ ✓ 5 ✓ × × ✓ ✓ ✓ ✓ ✓ 6 ✓ × ✓ ✓ ✓ ✓ ✓ 7 ✓ × × ✓ ✓ ✓ 8 ✓ × ✓ ✓ ✓ 9 ✓ × × ✓ 10 ✓ × ✓ 11 ✓ × 12 ✓ J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 62, 032204-6 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp TABLE II. The 30 extreme points (15) of P11,5that are not the Slater point can be associated with the filled boxes. Each extreme point is indexed by (i,j), with i=number of ones and j=number of zeros. The red points are not in F11,5. 1 2 3 4 5 6 7 8 9 10 11 12dN 0 1 (4, 5) ( 4, 4) ( 4, 3) ( 4, 2) ( 4, 1) ( 4, 0) 2 (3, 5) ( 3, 4) ( 3, 3) ( 3, 2) ( 3, 1) ( 3, 0) 3 (2, 5) ( 2, 4) ( 2, 3) ( 2, 2) ( 2, 1) ( 2, 0) 4 (1, 5) ( 1, 4) ( 1, 3) ( 1, 2) ( 1, 1) ( 1, 0) 5 (0, 5) ( 0, 4) ( 0, 3) ( 0, 2) ( 0, 1) ( 0, 0) 6 few-body characteristics, in the sense that they consist of a Slater determinant and two constituent parts that correspond to N=3, 4, 5 states or their particle–hole duals. All this supports the idea that the problematic parts of Pd,Nsomehow relate to few-body states—antisymmetry is most restrictive at low particle numbers, and the non-trivial Pauli constraints quantify this. The following remark makes this a little more precise. Remark 12. In Sec. IV C, we show that for 1 ≤m≤N−7 and t=N−m+1 N−m+9, Ad,N,m,t∶={(λ1,...,λd)∈Pd,N∣λm≤t}⊂Fd,N. (18) This means that any point on a non-trivial boundary of Fd,Nneeds to have λm≥N−m+1 N−m+9for 1≤m≤N−7. For example, for N=1000, this implies that λ209≥0.99 andλ609≥0.98. For large N, this shows that a state on a non-trivial boundary of Fd,Nhas a dominant Slater determi- nantal part. Based on numerics (inspired by Ref. 8; now see Ref. 9), we expect that sharper bounds can be found, which could mean that even states with N=O(100)have an approximate Slater determinantal part if they lie on a non-trivial boundary of Fd,N. C. Discussion and outlook Many suggested that applications of the Pauli constraints involve non-trivial boundaries of Fd,N(e.g., Refs. 3, 14, and 18–21). This paper provides some guidance on where these boundaries are and clarifies which extreme points of Pd,Ncannot be reached. As discussed above, the problematic extreme points relate to one and two-particle (or hole) states, and the non-trivial boundaries seem to be in the neighborhood of these points. Of course, volume convergence does not mean that the Pauli constraints cannot play a role in nature. Effective few-fermion states appear in atoms and Cooper pairs; many ground states involve correlated pockets with only a few electrons. In general, it remains unclear whether near-Slater determinant ground states of many-electron systems have the tendency to lie close to non-trivial boundaries of Fd,N. To decide if this is the case, it would be good to study more specific examples, notably ones with more electrons than those considered in Refs. 22–24. Another open problem is the implication of our results for Reduced Density Matrix Functional Theory (RDMFT). Since physical systems often involve spin, let us add a final remark about the spin-dependent polytopes discussed in Ref. 1. The analysis and methods used here extend easily to that case, with similar conclusions. IV. ESTIMATES AND PROOFS A. Geometry of Bd,N,Pd,N It will be convenient to gather some facts about polytopes before we start. Definition 13. A convex polytope is an intersection of a finite number of half-spaces. It can therefore be characterized as the points (x1,...,xd)∈Rdthat satisfy a finite system of linear inequalities Ax ≤b, A :Rd→Rk, or A11x1+⋅⋅⋅+ A1dxd≤b1 ⋮ Ak1x1+⋅⋅⋅+ Akdxd≤bk.(19) J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 62, 032204-7 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp An extreme point of a set Pis a point x∈Pthat cannot be written as a convex combination of two points in Pthat are distinct from x. It is that standard fact that the extreme points of Pcan be characterized with Eq. (19). Lemma 14. Given a polytope P defined by k equations (19), the extreme points of P are those points in P that satisfy d linearly independent equations with equality. Proof. We study the two inclusions separately. 1. Assume that k≥dand that a point x∈Psatisfies dlinearly independent equations of (19) with equality. Also suppose that x=μy+(1 −μ)zfory,z∈P,μ∈[0, 1]. Gather the satisfied equations in ˜A:Rd→Rdand ˜b∈Rd. Since ˜Ay,˜Az≤˜band ˜Ax=μ˜Ay+(1−μ)˜Az=˜b, we have ˜Ay=˜Az=˜b, but such a system of dlinearly independent equations can have at most one solution, so x=y=z. 2. Suppose a point x∈Pdoes not satisfy dlinearly independent equations of (19) with equality. We want to prove that it can be written asx=μy+(1−μ)z, with y,z∈Pdistinct from x. We will do this by finding v∈Rdandϵ>0 such that A(x+ϵv),A(x−ϵv)≤bso that x+ϵv,x−ϵv∈P. The existence of such vis obvious if ker (A)≠{0}, so assume that ker (A)={0}, which implies that k≥dand rank(A)=d. By our assumption, there are at least k−d+1 equations in (19) that are strict inequalities, and the corresponding basis vectors define a (k−d+1)-dimensional subspace of Rk. If we can find v∈Rdsuch that Avlies completely in that subspace, there exists ϵ>0 such that A(x+ϵv),A(x−ϵv)≤b. However, such vexists since we have assumed that the image of Aisd-dimensional and so intersects any (k−d+1)-dimensional subspace of Rkin some non-zero point. ◻ The convex polytopes we will study are all closed and bounded. In this case, the Krein–Milman theorem says that they are, in fact, the convex hull of their extreme points. The minimal such d-dimensional object is a d-dimensional simplex—a convex hull of d+1 linearly independent points. The bosonic polytope Bd,Nis a simplex. For completeness, we discuss it before turning to Pd,N. Proposition 15 (Properties of Bd,N).The extreme points of B d,N(9)are (N, 0,..., 0 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ d−1) (N 2,N 2, 0,..., 0 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ d−2) ⋮ (N d,...,N d),(20) and so, B d,Nis a(d−1)-dimensional simplex. It has volume Vol d−1(Bd,N)=√ d d!1 (d−1)!Nd−1. (21) Proof. According to Lemma 14, any extreme point has to satisfy dlinearly independent defining equations with equality. There are d inequalities λ1≥λ2≥⋅⋅⋅≥λd≥0 and one equality λ1+⋅⋅⋅+λd=N. Hence, an extreme point is obtained when we ignore one inequality from the list, solve the system of equations, and find that the solution lies in Bd,N. This gives the dextreme points (20). To calculate the volume, note that the set Bunord d,N∶={(λ1,...,λd)∈Rd∣λi≥0 andλ1+⋅⋅⋅+λd=N} (22) can be split into d! pieces of equal volume based on the ordering of the λi, one of which is Bd,N. Hence, Vold−1(Bd,N)=1 d!Vold−1(Bunord d,N). (23) Also note that Bunord d,Nhasdlinearly independent extreme points (N, 0,..., 0),...,(0,..., 0,N), and so, it is a (d−1)-dimensional simplex. In fact, it is a regular simplex since all its points are equally spaced. If we add the origin (0,..., 0), we obtain a d-dimensional simplex whose J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 62, 032204-8 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp FIG. 3. Bosonic polytopes B2,NforN=λ1+λ2=1, 2, 3 (bold). The unordered polytopes Bunord 2,Nared!=2 times as large and include the dashed continuations. Bunord 2,Nhave length√ 2N. One way to calculate this is to note that the lightly shaded region (drawn for N=2) has area N2/2. This increases to (N+√ 2ϵ)2/2 if we move a distance ϵ in the normal direction (1, 1)/√ 2. Hence, the derivative in ϵatϵ=0, which is the surface length, is√ 2N. volume is easily calculated to be Nd/d! using the standard volume formula for cones (base height /dimension). As illustrated in Fig. 3, the (d−1)-dimensional volume of Bunord d,Ncan then be found by traveling distance ϵin the normal direction (1,..., 1)/√ dand noting that now λ1+⋅⋅⋅λd=N+√ dϵso that the new volume is (N+√ dϵ)d/d!. Taking a derivative with respect to ϵand combining this with (23) give the volume of Bd,N.27◻ Proof of Proposition 7. Properties of P d,N. Using Lemma 14 again, this time there are d+1 inequalities 1 ≥λ1≥λ2≥⋅⋅⋅≥λd≥0 and one equalityλ1+⋅⋅⋅+λd=N. Ignoring two inequalities from the list, solving the resulting set of dequations, and checking whether solutions lie inPd,Nresult in (15). The extreme points are completely defined by the number of λiequal to 1 ( ≤N) and 0 (≤d−N). To calculate the volume, we use the same method as for Bd,N. To deal with the ordering, define Punord d,N∶={(λ1,...,λd)∈Rd∣1≥λi≥0 andλ1+⋅⋅⋅+λd=N}. (24) As illustrated in Fig. 4, the bound λi≤1 complicates the volume of this object. It is now convenient to use the Irwin–Hall distribution of probability theory. For uniformly distributed i.i.d. random variables and x∈R, PXi∼U(0,1)[X1+⋅⋅⋅+ Xd≤x]=1 d!⌊x⌋ ∑ k=0(−1)k(d k)(x−k)d. (25) Note that this is exactly the volume of the convex set generated by Punord d,Nand the origin (0,..., 0)(see Ref. 16 for a review). Hence, as mentioned before, Vold−1(Punord d,N)=∂ϵPXi∼U(0,1)[X1+⋅⋅⋅+ Xd≤N+√ dϵ]∣ ϵ=0 =√ d (d−1)!N ∑ k=0(−1)k(d k)(N−k)d−1.(26) Vold−1(Pd,N)acquires an extra 1 /d! because of ordering. ◻ B. Fermionic states: Proof of Proposition 8 To prove this proposition, we need to review some properties of fermionic states. First of all, an N-fermion Slater determinant built from orthonormal ∣u1⟩,...,∣uN⟩∈Cdis defined as J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 62, 032204-9 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp FIG. 4. The Pauli polytope P2,1(bold). The unordered polytope Punord 2,1isd!=2 times as large and includes the dashed continuation. The method to determine the volume still applies, and we still have λ1+λ2=N+√ 2ϵupon moving distance ϵin the normal direction. Naturally, the area of the shaded region is subject to the Pauli bound λi≤1. ∣u1∧⋅⋅⋅∧ uN⟩∶=1√ N!∑ σ∈SNsgn(σ)∣uσ(1)⊗⋅⋅⋅⊗ uσ(N)⟩. (27) This satisfies antisymmetry under permutations σ∈SNor ∣u1∧⋅⋅⋅∧ uN⟩=sgn(σ)∣uσ(1)∧⋅⋅⋅∧ uσ(N)⟩. (28) For an orthonormal basis ∣u1⟩,...,∣ud⟩ofCd, the(d N)Slater determinants built from that basis are an orthonormal basis of ∧NCd. For a state ∣Φ⟩∈∧NCdwhose expansion in this basis does not involve Slater determinants containing ∣ui⟩, we define ∣ui∧Φ⟩∧N+1Cdby linearity. For example, ∣u1⟩∧(1√ 2∣u2∧u3⟩+1√ 2∣u4∧u5⟩)=1√ 2∣u1∧u2∧u3⟩+1√ 2∣u1∧u4∧u5⟩. (29) We will extend the definition of ∧somewhat in Lemma 17. To do this, define the projection onto ∧NCd⊂⊗NCdby ΠN A∶=1 N!∑ σ∈SNsgn(σ)Uσ, (30) where Uσis the permutation operator corresponding to σ. Comparing this to the definition of a Slater determinant (27), we note ∣v1∧⋅⋅⋅∧ vN⟩=√ N!ΠN A∣v1⊗⋅⋅⋅⊗ vN⟩. (31) Finally, recall that the annihilation operator aicorresponding to ∣ui⟩acts as ai=√ N(⟨ui∣⊗𝟙):∧NCd→∧N−1Cd. (32) This implies that ai∣uj1∧⋅⋅⋅∧ ujk∧ui∧ujk+1∧⋅⋅⋅∧ ujN−1⟩=(−1)k∣uj1∧⋅⋅⋅∧ ujN−1⟩ (33) J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 62, 032204-10 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp and also that aigives 0 on Slaters that do not contain ∣ui⟩. Consequently, splitting an N-fermion state ∣Ψ⟩=∣Ψ1⟩+∣Ψ2⟩into a part ∣Ψ1⟩containing Slaters without ∣ui⟩and a part ∣Ψ2⟩=∣ui∧Φ⟩, we obtain ai∣Ψ⟩=∣Φ⟩. (34) We are now ready to prove Proposition 8 with the two following lemmas. Lemma 16. Let ∣Ψ⟩∈∧NCdbe normalized with a one-body reduced density matrix γΨ 1that has ordered eigenvalues (λΨ 1,...,λΨ d)and corresponding eigenbasis ∣u1⟩,...,∣ud⟩. 1. IfλΨ d=0, then∣Ψ⟩can be expanded in Slaters not containing ∣ud⟩and, hence, be embedded in ∧NCd−1. 2. IfλΨ 1=1, then∣Ψ⟩=∣u1∧Φ⟩and∣Φ⟩∈∧N−1Cd−1can be expanded in Slaters not containing ∣u1⟩. Proof. 1. Using (32), we find that the norm of the part of ∣Ψ⟩that contains ∣ud⟩is ∥ad∣Ψ⟩∥2=⟨ud∣γΨ 1∣ud⟩=0, (35) so according to (34), no Slater in ∣Ψ⟩contains ∣ud⟩. 2. Similarly, the norm of the part of ∣Ψ⟩containing ∣u1⟩is ∥a1∣Ψ⟩∥2=⟨u1∣γΨ 1∣u1⟩=1, (36) so all Slaters in this basis contain ∣u1⟩, and we can write ∣Ψ⟩=∣u1∧Φ⟩and a1∣Ψ⟩=∣Φ⟩. ◻ Lemma 17. For i =1, 2, suppose that ∣Ψi⟩∈∧NiCdihas one-body reduced density matrix γΨi 1with eigenvalues λ(i) j,1≤j≤di,and corresponding eigenvectors ∣u(i) j⟩such that ∣u(1) j⟩and∣u(2) j′⟩are mutually orthogonal for all j ,j′. Then, extend (29) by ∣Ψ1∧Ψ2⟩∶=√ (N1+N2 N1)ΠN1+N2 A∣Ψ1⊗Ψ2⟩∈∧N1+N2Cd1+d2. (37) This state is normalized, and its one-body reduced density matrix is γΨ1 1+γΨ2 1with eigenvalues {λ(i) j∣1≤j≤di,i=1, 2}. Proof. Using (31) and the projection property ΠN1+N2 A(ΠN1 A⊗ΠN2 A)=ΠN1+N2 A , it is easy to see that for v1...v N1+N2orthonormal, √ (N1+N2 N1)ΠN1+N2∣v1∧⋅⋅⋅∧ vN1⊗vN1+1∧⋅⋅⋅∧ vN1+N2⟩=∣v1∧⋅⋅⋅∧ vN1+N2⟩ (38) is normalized. By linearity, this directly extends to ∣Ψ1⊗Ψ2⟩. To show the eigenvalue property, denote sets of N1distinct vectors {u(1) j1,...,u(1) jN1}bySand their corresponding (ordered) Slater determinant by ∣S⟩. Then, ∣Ψ1⟩=∑ ScS∣S⟩ (39) for suitable coefficients cSwith∑S∣cS∣2=1. By (32) and (33), this implies that λ(1) jδjj′=⟨u(1) j∣γΨ1 1∣u(1) j′⟩=⟨Ψ1∣(a(1) j′)†a(1) j∣Ψ1⟩ =∑ S∋j,S′∋j′sgn(u(1) j,S)sgn(u(1) j′,S′)cS′cS⟨S′/u(1) j′∣S/u(1) j⟩,(40) J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 62, 032204-11 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp where sgn (u(1) j,S)is the sign of the permutation that reorders the elements of Sfrom increasing to u(1) jfirst and then increasing. Note that the inner product is 1 if S/u(1) j=S′/u(1) j′and 0 otherwise. Adopting a similar notation for ∣Ψ2⟩with an index T, we find that ∣Ψ1∧Ψ2⟩=∑ S,TcScT∣S∪T⟩, (41) noting that S∩T=∅for all S,T. It is then easy to see that cross terms ⟨u(1) j∣γΨ1∧Ψ2 1∣u(2) j′⟩=⟨Ψ1∧Ψ2∣(a(2) j′)†a(1) j∣Ψ1∧Ψ2⟩=0 (42) by orthogonality. In addition, ⟨u(1) j∣γΨ1∧Ψ2 1∣u(1) j′⟩=⟨Ψ1∧Ψ2∣(a(1) j′)†a(1) j∣Ψ1∧Ψ2⟩ =∑ S,S′,Tsgn(u(1) j,S)sgn(u(1) j′,S′)cS′cS∣cT∣2⟨S′/u(1) j′∪T∣S/u(1) j∪T⟩=λ(1) jδjj′,(43) soλ(1) jis indeed an eigenvalue of γΨ1∧Ψ2 1. The same argument applies to λ(2) j. ◻ C. Proving Ad,N,m,t⊂Fd,Nfor certain m,t Recall that Ad,N,m,twas defined for integers 1 ≤m≤dand t∈[0, 1]as Ad,N,m,t∶={(λ1,...,λd)∈Rd∣1≥λ1≥⋅⋅⋅≥λd≥0 andλm≤tandd ∑ i=1λi=N}. (44) Proposition 18. Let 8≤N≤d/2, m≤N−7,and t=N−m+1 N−m+9. Then, A d,N,m,t⊂Fd,N. Note that Ad,N,m,tis a polytope. Our goal is to show that all its extreme points are contained in Fd,N. Recall from Lemma 14 that the extreme points of a polytope in Rdsatisfy dof the polytope’s defining equations. Since Ad,N,m,tisPd,Nconstrained by λm≤t, its extreme points come in two types: ●Extreme points of P d,N(45) satisfyingλm≤t.These satisfy d−1 equations of 1 ≥λ1≥⋅⋅⋅≥λd≥0 with equality and ∑iλi=N. ●Extreme points with λm=t. In addition, these satisfy d−2 equations of 1 ≥λ1≥⋅⋅⋅≥λd≥0 with equality, as well as ∑iλi=N. We need to check that both types are contained in Fd,N. For the first type, recall the extreme points of Pd,Nfrom Proposition 7. These were the Slater point (1,..., 1, 0,..., 0)and (1,..., 1 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ i,N−i d−i−j,...,N−i d−i−j ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ d−i−j, 0,..., 0 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ j)∈Rd(0≤i≤N−1, 0≤j≤d−N−1). (45) Also recall from Sec. III B that these are definitely in Fd,Nunless N−i=1, 2 or N−i=d−i−j−2,d−i−j−1. The following lemma now shows that the condition λm≤texcludes these problematic cases. Lemma 19. Let 8≤N≤d/2, m≤N−7,and t=N−m+1 N−m+9. Then, of all the extreme points of P d,N, only those in (45) with 0≤i≤m−1and 0≤j≤d−i−N−i tare contained in A d,N,m,t. In particular, points with i >N−8and j>d−N−8lie outside of A d,N,m,t. Proof. Since t<1 and m<N, the Slater point and all points with i≥mare excluded. The points with i≤m−1 haveλm=N−i d−i−jsince d−j≥N+1. These points are only included ifN−i d−i−j≤t, which gives the equation for j. The final remark follows from i≤m−1≤N−8 and j≤d−i−N−i t≤d−m+1−N−m+1 t=d−N−8. ◻ J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 62, 032204-12 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp It is now clear that Ad,N,m,t’s extreme points of the first type are in Fd,N, but we are not ready for a Proof of Proposition 18. We also need to study extreme points of the second type, namely, those with λm=t. We will actually ignore this defining property and focus on the fact that they satisfy d−2 equations of 1 ≥λ1≥⋅⋅⋅≥λd≥0 with equality instead. Lemma 20. Let (λ1,...,λd)∈Pd,N, and assume that at least d −2equations of 1≥λ1≥⋅⋅⋅≥λd≥0are satisfied with equality. Then, (λ1,...,λd)can be written as a convex combination of (at most) two extreme points of P d,Nthat satisfy the same d −2equations with equality. Proof. Note that the d−2 equalities, together with ∑iλi=N, define a one-dimensional subspace of Rd. Our point must lie in the inter- section of Pd,Nwith this subspace, which is a bounded convex set with at most two extreme points. It is defined by the above-mentioned d−1 equalities, together with the two remaining inequalities of 1 ≥λ1≥⋅⋅⋅≥λd≥0. According to Lemma 14, its extreme points satisfy one of those inequalities with equality and, hence, dof the defining equations of Pd,N: they are extreme points of Pd,N. ◻ This says that we will not miss out on any extreme points of Ad,N,m0,t0if we restrict attention to line segments between extreme points of Pd,Nthat share d−2 equalities of 1 ≥λ1≥⋅⋅⋅≥λd≥0. Fortunately, we know that many such line segments are completely contained in Fd,N, simply because their defining extreme points are, according to the analysis from Sec. III B. We will need more information for line segments whose endpoints are not both contained in Fd,N. It turns out that the following lemma provides this, as will be explained in the Proof of Proposition 18. Lemma 21. Let 8≤N≤d/2, m≤N−7,and t=N−m+1 N−m+9. Consider an extreme point (45) of P d,Nindexed by (i,j),with 0≤i≤m−1and 0≤j≤d−i−N−i t. Then, the following points are contained in F d,N: (1,..., 1 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ i,N−i−4 N−i−1,...,N−i−4 N−i−1 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ N−i−1,4 d−N−j+1,...,4 d−N−j+1 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ d−N−j+1, 0,..., 0 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ j) (1,..., 1 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ i,N−i−5 N−i−2,...,N−i−5 N−i−2 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ N−i−2,5 d−N−j+2,...,5 d−N−j+2 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ d−N−j+2, 0,..., 0 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ j) (1,..., 1 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ i,N−i−3 N−i+1,...,N−i−3 N−i+1 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ N−i+1,3 d−N−j−1,...,3 d−N−j−1 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ d−N−j−1, 0,..., 0 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ j) (1,..., 1 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ i,N−i−3 N−i+2,...,N−i−3 N−i+2 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ N−i+2,3 d−N−j−2,...,3 d−N−j−2 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ d−N−j−2, 0,..., 0 ⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫m⟩⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ j).(46) These points lie on the line segments between the extreme point of P d,Nindexed by (i,j)and those indexed by (N−1,j),(N−2,j),(i,d−N−1), and(i,d−N−2),respectively. Any extreme points of A d,N,m,ton these line segments are contained in F d,N. Proof. We only discuss the first point of (46). The others can be treated in a similar way. To prove that the point is contained in Fd,N, we concatenate LME states using Lemma 17. To start, take an i-dimensional subspace of Cdand construct a Slater determinant ∣Ψ1⟩. We then pick an (N−i−1)-dimensional subspace of remaining Cd−iand construct an LME state∣Ψ2⟩ofN−i−4 particles, which exists by Theorem 10 since N−i−4≥4. Finally, we pick an (d−N−j+1)-dimensional subspace of remaining Cd−N+1and construct an LME state ∣Ψ3⟩of four particles, which exists since d−N−j+1≥9. Lemma 17 then says that ∣Ψ1∧Ψ2 ∧Ψ3⟩∈∧NCdwith the desired (ordered) eigenvalue vector. Hence, the point is in Fd,N. Now, consider the statement about the line segment. It is easy to see that the three points are on a line. Their order is also simple to check, for instance, in the first case by verifyingN−i−4 N−i−1∈[N−i d−i−j, 1]usingN−i d−i−j≤N−i N−i+8≤N−i−4 N−i−1since j≤d−N−8 as used in Lemma 19. For the final statement, note that the extreme point of Pd,Nindexed by (i,j)is in Fd,Nand that it has λm=N−i d−i−j≤tby Lemma 19. The first point of (46) is also in Fd,N, but it has λm=N−i−4 N−i−1≥N−m−3 N−m≥N−m+1 N−m+9=tby our assumptions. Since λmis strictly increasing on the line segment between (i,j)and(N−1,j), this means that all points with λm≤ton that line segment are in Fd,N, but then so must any extreme points of Ad,N,m,tbe. ◻ J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 62, 032204-13 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp We are now ready to prove that Ad,N,m,t⊂Fd,Nwhen m≤N−7 and t=N−m+1 N−m+9. Proof of Proposition 18. Recall that we wanted to show that all extreme points of Ad,N,m,tare in Fd,N. We identified two types of extreme points below Proposition 18: points that are also extreme points of Pd,Nand points that are not, but satisfy λm=t. Lemma 19 says that points of the first type are all contained in Fd,N. For points of the second type, Lemma 20 proves that we can restrict our attention to line segments between certain pairs of extreme points of Pd,N. In many cases, such line segments are entirely in Fd,Nsince their endpoints are according to Theorem 10 and Proposition 8. Which pairs of extreme points are left to check? We claim that the pairs addressed in Lemma 21 suffice. Why? By our reasoning just now, any remaining pairs must contain a member that corresponds to one of the problematic extreme points of Theorem 10, which leaves only the cases i=N−1,N−2 or j=d−N−1,d−N−2. These points are outside Ad,N,m,tby Lemma 19, so the other member of the pair should be inside—the interpolation could not contain an extreme point of Ad,N,m,totherwise. In addition, Lemma 20 says that it suffices to consider two points that have d−2 equalities of 1 ≥λ1≥⋅⋅⋅≥λd≥0 in common. Each point (45) satisfies d−1 of these with equality, so it is easy to see that only pairs (i,j),(i′,j′)with i=i′orj=j′qualify—see Table III for the position of such pairs in the LME table. All these considerations reduce our efforts to exactly the pairs discussed in Lemma 21. That lemma also showed that any extreme points ofAd,N,m,ton the corresponding line segments are in Fd,Nso that all extreme points of Ad,N,m,tare, and indeed, the set itself is. ◻ D. Volume estimates The important conclusion from Proposition 18 is that for certain mand t, Vold−1(Fd,N) Vold−1(Pd,N)≥Vold−1(Ad,N,m,t) Vold−1(Pd,N). (47) We now start estimating this ratio. First note that we can remove the ordering by adding a factor 1 /d! to both volumes and replacing λm≤tbyλ[m]≤t, where the latter denotes the mth largest entry of the tuple (λ1,...,λd). This implies that Vold−1(Ad,N,m,t) Vold−1(Pd,N)=Vold−1({(λ1,...,λd)∈Rd∣λi∈[0, 1]andλ[m]≤tand∑d i=1λi=N}) Vold−1({(λ1,...,λd)∈Rd∣λi∈[0, 1]and∑d i=1λi=N}) =1−Vold−1({(λ1,...,λd)∈Rd∣λi∈[0, 1]andλ[m]>tand∑d i=1λi=N}) Vold−1({(λ1,...,λd)∈Rd∣λi∈[0, 1]and∑d i=1λi=N}) ≥1−Vold−1({(λ1,...,λd)∈Rd∣λ[m]>tand∑d i=1λi=N}) Vold−1({(λ1,...,λd)∈Rd∣λi∈[0, 1]and∑d i=1λi=N}) ≥1−(d m)Vold−1({(λ1,...,λd)∈Rd∣λ1,...,λm>tand∑d i=1λi=N}) Vold−1({(λ1,...,λd)∈Rd∣λi∈[0, 1]and∑d i=1λi=N}),(48) where we use permutation invariance in the last step (see Fig. 5 for a geometric example). TABLE III. Illustration of the Proof of Proposition 18, following the ∧5C11example of Table II. Consider pairs involving the point (1, 1). According to Lemma 20, it suffices to consider the bold points because we can connect (1, 1)to these points by a line that satisfies d−2 of 1≥λ1≥⋅⋅⋅≥λdwith equality. Of the bold points, only the line segments connecting to the red points are not automatically contained in Fd,Nby Theorem 10, so these require the additional work from Lemma 21. Note, though, that this illustration is not perfect: the case ∧5C11is not actually covered by the main theorem—this example just explains these considerations. 1 2 3 4 5 6 7 8 9 10 11 12dN 0 1 (4, 5) ( 4, 4) ( 4, 3) ( 4, 2) ( 4, 1) ( 4, 0) 2 (3, 5) ( 3, 4) ( 3, 3) ( 3, 2) ( 3, 1) ( 3, 0) 3 (2, 5) ( 2, 4) ( 2, 3) ( 2, 2) ( 2, 1) ( 2, 0) 4 (1, 5) ( 1, 4) ( 1, 3) ( 1, 2) ( 1, 1) ( 1, 0) 5 (0, 5) ( 0, 4) ( 0, 3) ( 0, 2) ( 0, 1) ( 0, 0) 6 J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 62, 032204-14 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp FIG. 5. This image illustrates (48) for d=3 and N=2. Starting with the image at the top, the large triangle represents B3,2, containing P3,2as a smaller triangle. The area defined byλ[2]≤tis indicated in gray in the first two images. We then proceed through the steps of (48) image by image. These drawings were kindly contributed by an anonymous referee. In Proposition 7, we showed that the volume in the denominator is equal to √ d∂yPXi∼U(0,1)[d ∑ i=0Xi≤y]∣ N=√ d1 (d−1)!⌊N⌋ ∑ k=0(−1)k(d k)(N−k)d−1, (49) which is the probability density function of the Irwin–Hall distribution. To give a lower bound on (48), we need a lower bound on this quantity, but for fixed Nand d→∞that amounts to large deviations estimate. The only exception is N=d/2, so we aim to reduce to that case by proving the following lemma. J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 62, 032204-15 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp Lemma 22. Assuming that d ≥3, the quantity √ d∂yPXi∼U(0,1)[d ∑ i=0Xi≤y]∣ xx−(d−1)(50) is continuously differentiable and monotone decreasing in x ∈(0,∞). Proof. Since the derivative mentioned above is the probability density function of a sum of i.i.d. uniform random variables, it is easy to confirm with induction that it is a repeated convolution of U(0, 1)-density 𝟙[0,1]. That is, ∂yPXi∼U(0,1)[d ∑ i=0Xi≤y]∣ x=𝟙∗(d−1) [0,1](x)=1 (d−1)!⌊x⌋ ∑ k=0(−1)k(d k)(x−k)d−1. (51) This gives continuous differentiability in x∈(0,∞)for (50) as long as d≥3. For monotonicity, we use induction. Starting from d=3, 𝟙∗2 [0,1](x) x2=1 2𝟙[0,∞)(x)−3 2(1−1 x)2 𝟙[1,∞)(x)+3 2(1−2 x)2 𝟙[2,∞)(x)−1 2(1−3 x)2 𝟙[3,∞)(x) (52) is indeed monotone decreasing on (0,∞). Now, assume that the statement is true for some d≥3, and consider the derivative of (50) for d+1, x∈(0,∞), √ dxd∂y𝟙∗d [0,1](x)−dxd−1𝟙∗d [0,1](x) x2d. (53) We claim that this is negative on (0,∞). Note that induction tells us this is the case for dand so for x∈(0,∞), (d−1)𝟙∗(d−1) [0,1](x)≥x∂y𝟙∗(d−1) [0,1](x). (54) Adding 𝟙∗(d−1) [0,1](x)and convoluting with 𝟙[0,1]give d𝟙∗d [0,1](x)≥∫x x−1𝟙∗(d−1) [0,1](s)+s∂y𝟙∗(d−1) [0,1](s)ds=∫x x−1∂y(y𝟙∗(d−1) [0,1])(s)ds =x𝟙∗(d−1) [0,1](x)−(x−1)𝟙∗(d−1) [0,1](x−1) ≥x(𝟙∗(d−1) [0,1](x)−𝟙∗(d−1) [0,1](x−1))=x∂y𝟙∗d [0,1](x),(55) where we used the explicit form of (51) in the last step and its positivity in the one before. Hence, (50) is monotone decreasing on (0,∞).◻ This allows us to estimate the volume of Pd,N. Proposition 23. For d ≥7and N≤d/2, d! Vold−1(Pd,N)=Vold−1({(λ1,...,λd)∈Rd∣λi∈[0, 1]andd ∑ i=1λi=N}) ≥√ d 2(2N d)d−1 .(56) Proof. According to Lemma 22 and (51), this quantity is lower bounded by √ dNd−1∂yPXi∼U(0,1)[d ∑ i=0Xi≤y]∣ d/2(d 2)−(d−1) =√ d(2N d)d−1 𝟙∗(d−1) [0,1](d/2). (57) J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 62, 032204-16 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp According to the last equality of (55) and (51), 𝟙∗(d−1) [0,1](d/2)=∫d/2 −∞[𝟙∗(d−2) [0,1](s)−𝟙∗(d−2) [0,1](s−1)]ds=PXi∼U(0,1)[d−1 ∑ i=0Xi∈[d 2−1,d 2]]. (58) By Chebyshev’s inequality, this is PXi∼U(0,1)[ ∣d−1 ∑ i=0Xi−d−1 2∣≤1 2]≥1−3 d−1≥1 2(59) ford≥7. ◻ Having dealt with the denominator of (48), it remains to calculate the numerator. Proposition 24. Let m ≥0be an integer and t ∈R. Assuming that N ≥mt, Vold−1({(λ1,...,λd)∈Rd∣λ1,...,λm>tandd ∑ i=1λi=N})=√ d1 (d−1)!(N−mt)d−1. (60) Proof. In close analogy with (49), the volume mentioned above is equal to √ d(2N)d∂yPXi∼U(0,2N)[X1,...,Xm>tandd ∑ i=0Xi≤y]∣ N, (61) where the value 2 Nwas chosen as a convenient number bigger than N. Note that (2N)darises as the volume of [0, 2N]d. Since this is again a probability density, its value is similar to (51), namely, √ d(2N)d(1 (2N)m𝟙∗(m−1) [t,2N]∗1 (2N)d−m𝟙∗(d−m−1) [0,2N])(N)=√ d(𝟙∗(m−1) [t,2N]∗𝟙∗(d−m−1) [0,2N])(N). (62) To show that this is indeed (60), we use induction on d,m→d+1,m+1 to prove a slightly more general claim, namely, that for x≤2N, (𝟙∗(m−1) [t,2N]∗𝟙∗(d−m−1) [0,2N])(x)=1 (d−1)!(x−mt)d−1𝟙[mt,∞](x). (63) The base case— m=0 for any d—is covered by the above analysis and (49). We now assume that the formula is true for d,mand note that 𝟙[t,2N]∗(𝟙∗(m−2) [t,2N]∗𝟙∗(d−m−1) [0,2N])(x)=1 (d−2)!∫∞ −∞(s−mt)d−2𝟙[mt,∞](s)𝟙[t,2N](x−s)ds =1 (d−2)!∫max(mt,x−t) mt(s−mt)d−2ds,(64) which proves the claim. ◻ As a final ingredient, we prove Proposition 4 stated in Sec. II B. Proof of Proposition 4. We use techniques mentioned before. Similar to (48), we obtain Vold−1(Pd,N) Vold−1(Bd,N)=Vold−1({(λ1,...,λd)∈Rd∣λ[1]≤1 and∑d i=1λi=N}) Vold−1({(λ1,...,λd)∈Rd∣∑d i=1λi=N}) =1−Vold−1({(λ1,...,λd)∈Rd∣λ[1]>1 and∑d i=1λi=N}) Vold−1({(λ1,...,λd)∈Rd∣∑d i=1λi=N}) ≥1−dVold−1({(λ1,...,λd)∈Rd∣λ1>1 and∑d i=1λi=N}) d! Vold−1(Bd,N),(65) J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 62, 032204-17 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp so the lower bound follows from Proposition 15 and Proposition 24. For the upper bound, start again from the middle line of (65) and useλ1>1/Leftr⫯g⊸tl⫯ne⇒λ[1]>1 and Proposition 24. ◻ The main result now follows by combining the results mentioned above. Proof of Theorems 2 and 3. Theorem 2 follows directly from the bounds of Theorem 3. These can be derived as follows. 1. Fix 8≤N≤d/2, and recall that we previously obtained Proposition 18 and (48). Combining this with Proposition 24 and the lower bound of Proposition 4 gives Vold−1(Fd,N) Vold−1(Pd,N)≥Vold−1(Ad,N,m,t) Vold−1(Pd,N)≥1−(d m)1 1−d(N−1 N)d−1(N−mt N)d−1 ≥1−dN 1−d(N−1 N)d−1(N−mt N)d−1(66) form≤N−7 and t=N−m+1 N−m+9. To obtain a good estimate for both low and high N, we use two different m. The first is simply m=N−7, which gives N−mt =1 2(N+7). The second is m=N+9−⌈√ 8√ N+9⌉, for which it can be verified that N−mt≤√ 32N. This bound is not allowed if N+9−⌈√ 8√ N+9⌉≥N−7, but in this case, min [1 2(N+7),√ 32N]=1 2(N+7). This proves the estimate. 2. For N=rd≥20, we again use not only (48) and Proposition 24 but also Proposition 23. This gives Vold−1(Fd,N) Vold−1(Pd,N)≥Vold−1(Ad,N,m,t) Vold−1(Pd,N)≥1−2(d N)(1 2r)d−11 (d−1)!(N−mt)d−1. (67) We choose m=N+9−⌈√ 8√ N+9⌉as before and use N−mt≤√ 32N=√ 32rd. For the factorials, we use Stirling’s formula, 1 (d−1)!≤1√ 2π√ d−1(e d−1)d−1 (68) and (d N)=(d rd)=d! (rd)!((1−r)d)!≤edd+1/2e−d 2π(rd)rd+1/2e−rd((1−r)d)(1−r)d+1/2e−(1−r)d =e 2π√ d1 rr+1/2(1−r)3/2−r(1 rr(1−r)1−r)d−1 .(69) Since d≥2N≥40, all this gives Vold−1(Fd,rd) Vold−1(Pd,rd)≥1−2e (2π)3/2√ d(d−1)1 rr+1/2(1−r)3/2−r(ed 2(d−1)√ 32 rr+1/2(1−r)1−r1√ d)d−1 ≥1−1 rr+1/2(1−r)3/2−r(8 rr+1/2(1−r)1−r1√ d)d−1 .(70) ◻ ACKNOWLEDGMENTS This work was supported by The Royal Society through a Newton International Fellowship, by Darwin College Cambridge through a Schlumberger Research Fellowship, and by the Villum Centre of Excellence QMATH (University of Copenhagen). I thank Jan Philip Solovej for asking me about the volume of these polytopes and for his hospitality at QMATH, where this work was initiated. I also thank Michael Walter for discussions about numerical and physical aspects of the problem and Christian Schilling for comments on the initial draft of this paper. Figure 5 was kindly suggested and contributed by an anonymous referee. J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 62, 032204-18 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp APPENDIX A: EXACT VOLUME OF Ad,N,m,t As discussed in Remark 6, the following calculation gives the sharpest estimate our method can produce, but it is not used in the proof of the main theorems. First, recall that Ad,N,m,twas defined for integers 1 ≤m≤d,N∈R, and t∈[0, 1]as Ad,N,m,t∶={(λ1,...,λd)∈Rd∣1≥λ1≥⋅⋅⋅≥λd≥0 andλm≤tandd ∑ l=1λl=N}. Theorem 25. Let X 1,...,Xd∼U(0, 1)i.i.d. and x ∈R. For 1≤m≤d, let X (d+1−m)be the(d+1−m)th order statistic, that is, the (d +1−m)th smallest value, which means that it is the mth largest value. Then, for t ∈[0, 1], P[X(d+1−m)≤t∩d ∑ l=1Xl≤x]=1 d!min(m−1,⌊x⌋) ∑ i=0(−1)i(d i)(x−i)d +1 d!min(m−1,⌊x⌋) ∑ i=0⌊x−i t⌋ ∑ k=m−i(−1)k+i(d k+i)(k+i i)⎛ ⎝m−i−1 ∑ j=0(−1)j(k j)⎞ ⎠(x−kt−i)d.(A1) Compared to Proposition 7, this gives the volume Vold−1(Ad,N,m,t)=√ d d!∂xP[X(d+1−m)≤t∩d ∑ l=1Xl≤x]∣ x=N. (A2) Remark 26. When differentiated in x, this probability relates to the order statistics of a bunch of uniform random variables with constraint ∑lXl=x. Such order statistics are most likely well-known, but we were unable to find a suitable reference. Note that by permutation invariance, we have P[X(d+1−m)≤t∩d ∑ l=1Xl≤x] =m−1 ∑ j=0(d j)P[X1,...,Xj>t,Xj+1,...,Xd≤t∩d ∑ l=0Xl≤x].(A3) We compute the latter probabilities separately. Lemma 27. Let X 1,...,Xd∼U(0, 1)i.i.d. and x ∈R. For 0≤j≤d, t∈[0, 1], P[X1,...,Xj>t,Xj+1,...,Xd≤t∩d ∑ l=0Xl≤x] =1 d!j ∑ i=0(−1)i(j i)⌊x−i t⌋−(j−i) ∑ k=0(−1)k(d−j k)(x−(k+j−i)t−i)d.(A4) Note that this is 0 if j ≥⌊x t⌋+1. Proof. We use induction on j−1,d−1 to j,d, that is, we add a random variable and assume that it is bigger than t. The base case has j=0 and general dor P[X1,...,Xd≤t∩d ∑ l=0Xl≤x]=1 d!⌊x t⌋ ∑ k=0(−1)k(d k)(x−kt)d. (A5) J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 62, 032204-19 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp This can be verified by seeing that this probability is equal to P[d ∑ l=0Xl≤x∣X1,...,Xd≤t]P[X1,...,Xd≤t]=P[d ∑ l=0Xl t≤x t∣X1,...,Xd≤t]td(A6) and using (25). For the induction step, we integrate over Xj=s∈[t, 1]. This gives P[X1,...,Xj>t,Xj+1,...,Xd≤t∩d ∑ l=0Xl≤x] =∫1 tP[X1,...,Xj−1>t,Xj+1,...,Xd≤t∩d ∑ l=0Xl≤x−s]ds =1 (d−1)!j−1 ∑ i=0(−1)i(j−1 i)∫1 t⌊x−s−i t⌋−(j−1−i) ∑ k=0(−1)k(d−j k)(x−s−(k+j−1−i)t−i)d−1ds.(A7) Note that for all terms k≤⌊x−i−1 t⌋−(j−i−1), the integral is over the entire range [t, 1], but that for k≥⌊x−i−1 t⌋−(j−i−1)+1, it is only over[t,x−i−(k+j−i−1)t]. This interval is empty if k≥⌊x−i t⌋−(j−i)+1, and so, (A7) is equal to 1 d!j−1 ∑ i=0(−1)i(j−1 i)⎡⎢⎢⎢⎢⎣⌊x−i t⌋−(j−i) ∑ k=0(−1)k(d−j k)(x−(k+j−i)t−i)d −⌊x−i−1 t⌋−(j−i−1) ∑ k=0(−1)k(d−j k)(x−(k+j−i−1)t−(i+1))d⎤⎥⎥⎥⎥⎦ =1 d!j ∑ i=0(−1)i[(j−1 i)+(j−1 i−1)]⌊x−i t⌋−(j−i) ∑ k=0(−1)k(d−j k)(x−(k+j−i)t−i)d,(A8) which is the desired result. ◻ Proof of Theorem 25. Rewriting Lemma 27 slightly and checking which terms are clearly 0, we obtain that (A3) is equal to 1 d!min(m−1,⌊x t⌋) ∑ j=0(d j)min(j,⌊x⌋) ∑ i=0(j i)⌊x−i t⌋ ∑ k=j−i(−1)k−j(d−j k−(j−i))(x−kt−i)d. (A9) A careful exchange of the sums gives 1 d!min(m−1,⌊x⌋) ∑ i=0min(m−1,⌊x t⌋) ∑ j=i⌊x−i t⌋ ∑ k=j−i(−1)k−j(d j)(j i)(d−j k−(j−i))(x−kt−i)d. (A10) We then use (d j)(j i)(d−j k−(j−i))=(d k+i)(k+i i)(k j−i)and a second exchange of sums to obtain 1 d!min(m−1,⌊x⌋) ∑ i=0⌊x−i t⌋ ∑ k=0(−1)k(d k+i)(k+i i)⎛ ⎜ ⎝min(m−1,⌊x t⌋,i+k) ∑ j=i(−1)j(k j−i)⎞ ⎟ ⎠(x−kt−i)d. (A11) J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 62, 032204-20 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp Note that k≤⌊x−i t⌋implies k≤⌊x t⌋−i, and so, the part between the big brackets equals (−1)imin(m−i−1,k) ∑ j=0(−1)j(k j), (A12) which is (−1)iifk=0 and 0 if k≤m−i−1 so that k=0 gives rise to the first term of (A1) and k≥m−igives rise to the second term. ◻ APPENDIX B: DIMENSION OF VN,d LME/SU(d) For completeness, we extend the results of Refs. 6 and 7 (and the predating qubit case15) to fermions by calculating the dimension of VN,d LME/SU(d)(Definition 9). These dimensions are not otherwise used in this paper. Theorem 28. Given that SU(d)acts as A∈SU(d)↦A⊗⋅⋅⋅⊗ A on VN,d LME(Definition 9), dim(VN,d LME/SU(d))=⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩0 if N=0,N=d −1 if d≥2and N=1,N=d−1 0 if d≥2is even and N=2,N=d−2 −1 if d≥2is odd and N =2,N=d−2 ≥0 if d=6,N=3 ≥0 if d=7and N=3,N=4 ≥0 if d=8and N=3,N=5 (d N)−d2if d=8,N=4or d≥9and 3≤N≤d−3. Here, dimension −1indicates that VN,d LME/SU(d)=∅, whereas dimension 0 indicates that it is a point. The orange results only indicate the existence of LME states. Proof. N=0,N=d This case is trivial since there is only one normalized state, and it satisfies (17). d≥2and N=1,N=d−1 A one-body pure state has eigenvalues (1, 0,..., 0), so it cannot be LME for d≥2. The other case is identical by particle–hole duality. d≥2even and N =2,N=d−2 For any state ∣Ψ⟩∈∧2Cd, there are numbers c1≥⋅⋅⋅≥c⌊d/2⌋≥0 and an orthonormal basis ∣u1⟩,...,∣ud⟩such that25,26 ∣Ψ⟩=⌊d/2⌋ ∑ j=1cj∣u2j−1∧u2j⟩. (B1) To obtain an LME state for deven, we need c1=⋅⋅⋅=cd/2=√ 2/d. It is then the choice of the basis that defines the LME state, but this can be changed with K=SU(d), so dim(VN,d LME/K)=0. Particle–hole duality gives the same for N=d−2. d≥2odd and N =2,N=d−2 Ifdis odd, the general form (B1) rules out the existence of LME states. d=6and N=3 The following state is LME: 1√ 2(∣u1∧u2∧u3⟩+∣u4∧u5∧u6⟩). (B2) d=7and N=3,N=4 The following state is LME: 1√ 7(∣u1∧u2∧u3⟩+∣u1∧u4∧u5⟩+∣u1∧u6∧u7⟩ +∣u2∧u4∧u6⟩+∣u2∧u5∧u7⟩+∣u3∧u4∧u7⟩+∣u3∧u5∧u6⟩).(B3) d=8and N=3,N=5 J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 62, 032204-21 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp The following state is LME: 1√ 8(∣u1∧u2∧u3⟩+∣u1∧u4∧u5⟩+∣u1∧u6∧u7⟩+∣u2∧u4∧u6⟩ +∣u2∧u5∧u8⟩+∣u3∧u5∧u7⟩+∣u3∧u6∧u8⟩+∣u4∧u7∧u8⟩).(B4) d=8,N=4or d≥9and 3≤N≤d−3 We rely on Ref. 7. To comply with the notation, set V∶=∧NCd. The groups K∶=SU(d)and G∶=SL(d)act on V symmetrically, that is, A/leftfootl⫯ne→A⊗⋅⋅⋅⊗ A. For the Lie algebras, this defines a representation a∈sl(d)/leftfootl⫯ne→ a⊗𝟙⊗⋅⋅⋅⊗ 𝟙+⋅⋅⋅+ 𝟙⊗⋅⋅⋅⊗ 𝟙⊗a. The moment map μ:P(V)→sl(d)∗can then be written in terms of the one-body reduced density matrix, μ(∣Ψ⟩)(a)=⟨Ψ∣a⊗𝟙⊗⋅⋅⋅⊗ 𝟙+⋅⋅⋅+ 𝟙⊗⋅⋅⋅⊗ 𝟙⊗a∣Ψ⟩=Tr[aγΨ 1], (B5) by antisymmetry of ∣Ψ⟩. The moment map maps to 0 if and only if γΨ 1is proportional to the identity, but this happens if and only if ∣Ψ⟩is LME. Hence, VN,d LME/K=μ−1(0)/K. We now simply apply the steps from Ref. 7. The recipe is as follows: a) Ifρ:G→GL(V)is a representation of a complex reductive group Gand Vhas a norm that is invariant under a maximal compact subgroup KofG, the Kempf–Ness theorem13applies and we have μ−1(0)/K≃P(V)//G, (B6) which is the geometric invariant theory quotient of the projective space P(V). b) The dimension of this quotient P(V)//Gis then derived in Ref. 7 using two facts. The first is that, under the additional assumption that the representation ρis finite-dimensional, there exists a “generic” stabilizer group S17such that dim (P(V)//G)=dim(V)−dim(G) +dim(S)−1. This Sis defined to be a closed subgroup of Gsuch that there exists an open dense subset U⊂Vwith the property that for every x∈U, the stabilizer Gxatxis conjugate to S. c) All that remains is to determine the dimension of S. This is done with the work of Élashvili,12which says that assuming that Gis semisimple and ρis irreducible, we should check whether l(ρ∣H)≥1 for every (non−trivial)simple normal subgroup HofG. (B7) Given a faithful finite-dimensional representation ρ:H→GL(V)of a simple complex linear algebraic group H, this index is defined as2 l(ρ)∶=Tr[ρ∗(a)2] Tr[ad(a)2], (B8) where a∈Lie(H),ρ∗is the representation of Lie (H)associated with ρ, and ad is the adjoint representation of Lie (H). This is inde- pendent of the choice of aas long as Tr [ad(a)2]≠0. If the criterion (B7) holds, Élashvili12provides us with the dimension dim (S), allowing for a calculation of dim (P(V)//G). It is easy to check that all the required assumptions hold and the recipe can be applied. We just need to verify (B7) for SL (d). To calculate the index, take a∈sl(d)to be a=diag(μ1,...,μd)with Tr[a]=∑iμi=0. As shown in example 3.4 in Ref. 2, it is easy to calculate Trsl(d)[ad(a)2]=∑ i≠j(μi−μj)2=∑ i≠jμ2 i+μ2 j−2μiμj =2(d−1)Tr[a2]−2(Tr[a]2−Tr[a2])=2dTr[a2].(B9) For Tr[ρ∗(a)2]=Tr∧NCd[(a⊗𝟙⊗⋅⋅⋅⊗ 𝟙+⋅⋅⋅+ 𝟙⊗⋅⋅⋅⊗ 𝟙⊗a)2], (B10) J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 62, 032204-22 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp we use a basis of Slater determinants (27) built from the eigenvectors ∣u1⟩,...,∣ud⟩ofa. A single Slater determinant contributes terms of the form ⟨ui1∧⋅⋅⋅∧ uiN∣a2⊗𝟙⊗⋅⋅⋅⊗ 𝟙∣ui1∧⋅⋅⋅∧ uiN⟩=1 N∑ 1≤k≤Nμ2 ik, (B11) and similarly, ⟨ui1∧⋅⋅⋅∧ uiN∣a⊗a⊗𝟙⊗⋅⋅⋅⊗ 𝟙∣ui1∧⋅⋅⋅∧ uiN⟩=1 (N 2)∑ 1≤k<k′≤Nμikμik′. (B12) Noting that the contribution μ2 ikis obtained from the (d−1 N−1)Slaters that contain ∣uik⟩and each contribution μikμik′is obtained from the (d−2 N−2) Slaters that contain both ∣uik⟩and∣uik′⟩, we find that (B10) becomes Tr[ρ∗(a)2]=(d−1 N−1)∑ iμ2 i+2(d−2 N−2)∑ 1≤i<j≤dμiμj =[(d−1 N−1)−(d−2 N−2)]Tr[a2]+(d−2 N−2)Tr[a]2=(d−2 N−1)Tr[a2].(B13) Therefore, the index of the representation A⊗⋅⋅⋅⊗ Aof SL(d)on∧NCdis l(ρ)=1 2d(d−2 N−1). (B14) We check l(ρ)=5/4 for d=8,N=4. For d≥9 and 3≤N≤d−3, note that 1 2d(d−2 N−1)≥1 2d(d−2 2)=d2−5d+6 4d≥7 6, (B15) where the first inequality is obvious from the properties of binomial coefficients and the second can easily be derived by noting that the derivative in d≥9 is positive. Since (B7) holds, Ref. 12 says that the connected component S0ofSis trivial, but then dim (S)=dim(S0)=0 and dim (P(V)//G) =dim(V)−dim(G)−1=(d N)−d2. ◻ DATA AVAILABILITY Data sharing is not applicable to this article as no new data were created or analyzed in this study. REFERENCES 1M. Altunbulak and A. Klyachko, “The Pauli principle revisited,” Commun. Math. Phys. 282, 287–322 (2008). 2E. M. Andreev, E. Vinberg, and A. G. Élashvili, “Orbits of greatest dimension in semi-simple linear Lie groups,” Funct. Anal. Appl. 1, 257–261 (1967). 3C. L. Benavides-Riveros, J. M. Gracia-Bondía, and M. Springborg, “Quasipinning and entanglement in the lithium isoelectronic series,” Phys. Rev. A 88, 022508 (2013). 4C. L. Benavides-Riveros and M. A. L. Marques, “Static correlated functionals for reduced density matrix functional theory,” Eur. Phys. J. B 91, 133 (2018). 5R. E. Borland and K. Dennis, “The conditions on the one-matrix for three-body fermion wavefunctions with one-rank equal to six,” J. Phys. B: At. Mol. Phys. 5, 7 (1972). 6J. Bryan, S. Leutheusser, Z. Reichstein, and M. Van Raamsdonk, “Locally maximally entangled states of multipart quantum systems,” Quantum 3, 115 (2019). 7J. Bryan, Z. Reichstein, and M. Van Raamsdonk, “Existence of locally maximally entangled quantum states via geometric invariant theory,” Ann. Henri Poincare 19, 2491–2511 (2018). 8P. Bürgisser, C. Franks, A. Garg, R. Oliveira, M. Walter, and A. Wigderson, “Efficient algorithms for tensor scaling, quantum marginals, and moment polytopes,” in 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, 2018), pp. 883–897. 9P. Bürgisser, C. Franks, A. Garg, R. Oliveira, M. Walter, and A. Wigderson, “Towards a theory of non-commutative optimization: Geodesic first and second order methods for moment maps and polytopes,” in 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, 2019), pp. 845–861. 10A. J. Coleman, “Structure of fermion density matrices,” Rev. Mod. Phys. 35, 668 (1963). 11A. J. Coleman and V. I. Yukalov, Reduced Density Matrices: Coulson’s Challenge (Springer Science & Business Media, 2000), Vol. 72. 12A. G. Élashvili, “Stationary subalgebras of points of the common state for irreducible linear Lie groups,” Funct. Anal. Appl. 6, 139–148 (1972). 13G. Kempf and L. Ness, “The length of vectors in representation spaces,” Algebraic Geom. 233–243 (1979). J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 62, 032204-23 Published under license by AIP PublishingJournal of Mathematical PhysicsARTICLE scitation.org/journal/jmp 14A. A. Klyachko, “The Pauli exclusion principle and beyond,” arXiv:0904.2009 (2009). 15T. Maci ˛ a ˙zek, M. Oszmaniec, and A. Sawicki, “How many invariant polynomials are needed to decide local unitary equivalence of qubit states?,” J. Math. Phys. 54, 092201 (2013). 16J. E. Marengo, D. L. Farnsworth, and L. Stefanic, “A geometric derivation of the Irwin-Hall distribution,” Int. J. Math. Math. Sci. 2017 , 3571419. 17R. W. Richardson, “Principal orbit types for algebraic transformation spaces in characteristic zero,” Inventiones Math. 16, 6–14 (1972). 18C. Schilling, “Hubbard model: Pinning of occupation numbers and role of symmetries,” Phys. Rev. B 92, 155149 (2015). 19C. Schilling, “Quasipinning and its relevance for N-fermion quantum states,” Phys. Rev. A 91, 022105 (2015). 20C. Schilling, M. Altunbulak, S. Knecht, A. Lopes, J. D. Whitfield, M. Christandl, D. Gross, and M. Reiher, “Generalized Pauli constraints in small atoms,” Phys. Rev. A 97, 052503 (2018). 21C. Schilling, C. L. Benavides-Riveros, A. Lopes, T. Maci ˛ a ˙zek, and A. Sawicki, “Implications of pinned occupation numbers for natural orbital expansions: I. Generalizing the concept of active spaces,” New J. Phys. 22, 023001 (2020). 22C. Schilling, D. Gross, and M. Christandl, “Pinning of fermionic occupation numbers,” Phys. Rev. Lett. 110, 040404 (2013). 23F. Tennie, D. Ebler, V. Vedral, and C. Schilling, “Pinning of fermionic occupation numbers: General concepts and one spatial dimension,” Phys. Rev. A 93, 042126 (2016). 24F. Tennie, V. Vedral, and C. Schilling, “Influence of the fermionic exchange symmetry beyond Pauli’s exclusion principle,” Phys. Rev. A 95, 022336 (2017). 25C. N. Yang, “Concept of off-diagonal long-range order and the quantum phases of liquid He and of superconductors,” Rev. Mod. Phys. 34, 694 (1962). 26D. C. Youla, “A normal form for a matrix under the unitary congruence group,” Can. J. Math. 13, 694–704 (1961). 27A more reliable, but less intuitive way to calculate the volume is to use the parameterization (λ1,. . .,λd−1,N−λ1− ⋅ ⋅ ⋅ −λd−1), where√ darises as the determinant of the Jacobian. 28V. Guillemin and S. Sternberg, “Convexity properties of the moment mapping,” Inventiones mathematicae 67, 491–513 (1982). 29L. Ness and D. Mumford, “A stratification of the null cone via the moment map,” American Journal of Mathematics 106, 1281–1329 (1984). J. Math. Phys. 62, 032204 (2021); doi: 10.1063/5.0031419 62, 032204-24 Published under license by AIP Publishing

5.0043791.pdf

J. Chem. Phys. 154, 135102 (2021); https://doi.org/10.1063/5.0043791 154, 135102 © 2021 Author(s).Toward photoswitchable electronic pre- resonance stimulated Raman probes Cite as: J. Chem. Phys. 154, 135102 (2021); https://doi.org/10.1063/5.0043791 Submitted: 11 January 2021 . Accepted: 15 March 2021 . Published Online: 02 April 2021 Dongkwan Lee , Chenxi Qian , Haomin Wang , Lei Li , Kun Miao , Jiajun Du , Daria M. Shcherbakova , Vladislav V. Verkhusha , Lihong V. Wang , and Lu Wei COLLECTIONS Paper published as part of the special topic on 2021 JCP Emerging Investigators Special Collection This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN Combining the best of two worlds: Stimulated Raman excited fluorescence The Journal of Chemical Physics 153, 210901 (2020); https://doi.org/10.1063/5.0030204 Gas-phase studies of chemically synthesized Au and Ag clusters The Journal of Chemical Physics 154, 140901 (2021); https://doi.org/10.1063/5.0041812 Perspective: Coherent Raman scattering microscopy, the future is bright APL Photonics 3, 090901 (2018); https://doi.org/10.1063/1.5040101The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Toward photoswitchable electronic pre-resonance stimulated Raman probes Cite as: J. Chem. Phys. 154, 135102 (2021); doi: 10.1063/5.0043791 Submitted: 11 January 2021 •Accepted: 15 March 2021 • Published Online: 2 April 2021 Dongkwan Lee,1 Chenxi Qian,1Haomin Wang,1 Lei Li,2Kun Miao,1Jiajun Du,1Daria M. Shcherbakova,3 Vladislav V. Verkhusha,3,4 Lihong V. Wang,2 and Lu Wei1,a) AFFILIATIONS 1Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA 2Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125, USA 3Department of Anatomy and Structural Biology, and Gruss-Lipper Biophotonics Center, Albert Einstein College of Medicine, Bronx, New York 10461, USA 4Medicum, Faculty of Medicine, University of Helsinki, Helsinki, Finland Note: This paper is part of the 2021 JCP Emerging Investigators Special Collection. a)Author to whom correspondence should be addressed: lwei@caltech.edu ABSTRACT Reversibly photoswitchable probes allow for a wide variety of optical imaging applications. In particular, photoswitchable fluorescent probes have significantly facilitated the development of super-resolution microscopy. Recently, stimulated Raman scattering (SRS) imaging, a sen- sitive and chemical-specific optical microscopy, has proven to be a powerful live-cell imaging strategy. Driven by the advances of newly developed Raman probes, in particular the pre-resonance enhanced narrow-band vibrational probes, electronic pre-resonance SRS (epr-SRS) has achieved super-multiplex imaging with sensitivity down to 250 nM and multiplexity up to 24 colors. However, despite the high demand, photoswitchable Raman probes have yet to be developed. Here, we propose a general strategy for devising photoswitchable epr-SRS probes. Toward this goal, we exploit the molecular electronic and vibrational coupling, in which we switch the electronic states of the molecules to four different states to turn their ground-state epr-SRS signals on and off. First, we showed that inducing transitions to both the electronic excited state and triplet state can effectively diminish the SRS peaks. Second, we revealed that the epr-SRS signals can be effectively switched off in red-absorbing organic molecules through light-facilitated transitions to a reduced state. Third, we identified that photoswitchable pro- teins with near-infrared photoswitchable absorbance, whose states are modulable with their electronic resonances detunable toward and away from the pump photon energy, can function as the photoswitchable epr-SRS probes with desirable sensitivity ( <1μM) and low photofa- tigue (>40 cycles). These photophysical characterizations and proof-of-concept demonstrations should advance the development of novel photoswitchable Raman probes and open up the unexplored Raman imaging capabilities. Published under license by AIP Publishing. https://doi.org/10.1063/5.0043791 .,s I. INTRODUCTION Photoswitchable probes are molecules whose signals can be turned on and off reversibly upon irradiation of light. The devel- opment of such optical-highlighter probes could greatly expand the range of questions that can be investigated, particularly in biology. For example, the emergence of photoswitchable fluorophores has allowed unique imaging of protein dynamics in cells, sensing of subcellular environment, and optical data writing and storage.1–5By precisely activating and deactivating fluorescence in space and time, these probes have also largely facilitated the development of the ground-breaking super-resolution microscopy, pushing the spa- tial resolution of optical imaging to tens of nanometers.6–8Promi- nently, utilizing the non-fluorescent states (i.e., the OFF state) of photoswitchable fluorescent proteins, which have a much longer lifetime (>ms) compared to the fluorescence lifetime ( ∼ns) of excited states, RESOLFT (reversible saturable optical fluorescence transitions) has addressed the high-power photo-damage issues in J. Chem. Phys. 154, 135102 (2021); doi: 10.1063/5.0043791 154, 135102-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp STED (stimulated emission depletion) microscopy and allows an eight-order-of-magnitude smaller illumination intensity than STED for super-resolution live-cell imaging.9–13 In addition to fluorescence microscopy, Raman imaging, which targets the vibrational transitions of chemical bonds, has shown its promises to be a powerful biomedical imaging modality that offers complementary information when interrogating biological systems. In particular, stimulated Raman scattering (SRS) microscopy, which harnesses the stimulated emission amplification principle, could accelerate the vibrational transitions by 108times compared to spon- taneous Raman [Fig. 1(a)]. Overcoming the low sensitivity issue in conventional spontaneous Raman imaging, SRS has achieved sub- cellular imaging with speed up to the video rate.14,15It allows detec- tion of endogenous biomolecules in a label-free fashion and also offers bioorthogonal chemical imaging of small metabolites and drugs in live cells, tissues, and animals with tiny Raman tags.16–18 However, no photoswitchable Raman probes have been reported so far. Recently, by bringing the pump photon energy close to, but still slightly detuned away from, the electronic absorption maximum of the red-absorbing dyes, electronic pre-resonance SRS (epr-SRS) has been invented, enhancing the sensitivity of Raman imaging down to 250 nM. Such a sensitivity level is close to that offered by typical confocal fluorescence microscopy.19,20Compared to conventional non-resonance SRS [Fig. 1(a)], epr-SRS obtains an up to 105-fold signal boost while keeping the electronic-resonance-related back- ground to a minimum level [Fig. 1(b)]. With a newly developed and synthesized probe palette, epr-SRS has enabled optical super- multiplex imaging for up to simultaneous 24-color visualization of biological targets.19These probes, which incorporate narrow-band and isotope edited nitrile ( ∼11 cm−1) or alkyne ( ∼14 cm−1) moi- eties to their conjugation systems, share similar electronic absorp- tion peaks, but show distinctly separated Raman bands in the desired cell-silent Raman spectral region (1700–2700 cm−1).19–21 Herein, we explore and develop the photoswitchable epr-SRS probes. Since epr-SRS probes manifest strong coupling between elec- tronic and vibrational transitions, we use light-induced transitions from one electronic state to another as a general strategy to switchepr-SRS signals on and off. Specifically, we adopt an additional exci- tation beam to induce electronic state transitions of the molecules [Fig. 1(c), green arrow]. As the excitation beam depletes the ON state population to the OFF state, the SRS laser pair [Fig. 1(d), the Stokes beam fixed at 1031.2 nm and the pump beam tunable around 830–880 nm for epr-SRS imaging] is utilized to probe the depleted ON state epr-SRS vibrational modes. In principle, the epr- SRS signals could possibly be switched off and on with and with- out the excitation from the additional laser, respectively [Figs. 1(c) and 1(d)]. One envisioned application with such photoswitchable molecules is super-multiplex (i.e., >10 plex) super-resolution imag- ing, which has remained as a highly challenging but long sought- after goal.22Since epr-SRS probes all share similar absorption peaks, only a single doughnut depletion laser would be required to switch off the periphery signals and leave the spectrally-separated epr- SRS signals in the center (Fig. S1). As a comparison, if STED or RESOLFT were to achieve this goal of super-multiplex super- resolution, an additional pair of excitation and depletion beams is required for each extra color.9–13This is highly challenging due to two main reasons. First, the added laser lines and optics would largely increase the complexity for precise instrumentation align- ment. Second, the existing color-barrier in fluorescence (i.e., the spectral overlap) would typically limit the number of possible colors to 3–5.23–25 Guided by the rationales above, we study the photophysics of different molecular electronic states to evaluate whether they can serve as an OFF state for the ground-state epr-SRS excita- tions. We first investigated the excited state and the triplet state to implement the ground-state depletion photoswitching strategy. We found that transitions to the first excited state and triplet state result in vanishing epr-SRS peaks but also induce large electronic background. Guided by the Albrecht A-term pre-resonance approx- imation equation (see the supplementary material, Scheme 1), we then revealed that a thiol-promoted long-lived dark state of organic dyes and an absorption-detuned transition state of near-infrared (NIR) proteins could effectively eliminate the epr-SRS signals and serve as the OFF states for cyclic photoswitching. We envision this work to motivate further research in developing and optimizing the FIG. 1 . Principle and design of photoswitchable electronic pre-resonance SRS (epr-SRS) via electronic state transition. [(a)–(c)] Energy diagrams of SRS (a), epr-SRS (b), and the proposed electronic-state modulated epr-SRS with an additional electronic excitation laser (green) (c). (d) Experimental scheme of the electronic-state modulated epr-SRS by introducing a third excitation beam (green) to a conventional SRS microscope. EOM, electro-optic modulator; REF, reference; and X, in-phase X-output of the lock-in amplifier. J. Chem. Phys. 154, 135102 (2021); doi: 10.1063/5.0043791 154, 135102-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp photoswitchable epr-SRS probes and to expand the capabilities of optical imaging. II. RESULTS A. Photoswitching by ground-state-depletion epr-SRS First, we explored the possibility of harnessing electronic excited states as the OFF state for epr-SRS signals [Fig. 2(a)]. It is known that bond properties (e.g., force constants, bond orders) change between the ground and excited states.26Recent studies have also shown that Raman spectra of molecules in the excited state exhibit shifted peaks compared to those of the ground state for certain vibrational modes in the molecules.27,28We hence rea- soned that shifting the population to the excited state could likely deplete the ground-state Raman signals. To test this for epr-SRS, we adopted Rhodamine 800 (Rh800), a near-infrared-absorbing dye peaked around 680 nm with high and well-characterized epr-SRS signals [the structure shown in Figs. 2(b) and 2(c)].19–21For elec- tronic excitation, we integrated and aligned a 660 nm continuous wave (CW) laser into the SRS system for steady state excitation [Fig. 1(d)]. We planned to excite Rh800 by the 660 nm excitation beam and simultaneously probe the ground-state epr-SRS signals by the SRS beams [Fig. 2(a), OFF state, dashed laser lines for pump and Stokes]. We would then compare the resulting epr-SRS spectra with and without 660 nm excitation for signal suppression analysis. To ensure over 80% excitation of the population from the ground state to the excited state, we applied up to 34 mW of the 660 nm excitation beam (Table S1). Since excited-state Raman peaks are possibly shifted from those of the ground state, we expected to observe a decrease in epr-SRS signals for electronic pre-resonance enhanced peaks from Rh800 upon 660 nm excitation. We, indeed, observed a gradual decrease in the epr-SRS peaks for both the double bond [Fig. 2(b) and Fig. S2(a)] and triple bond [Fig. 2(c) and Fig. S2(b)] of Rh800 withincreasing excitation beam powers. However, we simultaneously detected a large increase in the broad background [Figs. 2(b) and 2(c)]. The increase in background shows strong resemblance to the SRS spectra in the rigorous resonance regime.19,20,29We, hence, rea- soned that the observed background increase should originate from the reduced energy gap between the first electronic excited state (S1) and the second electronic excited state (S 2) compared to that between the ground state (S 0) and the first electronic excited state (S1) [Fig. 2(a), OFF state]. epr-SRS excitations for the excited-state Rh800 would, therefore, invoke high rigorous-electronic-resonance- involved background [Fig. 2(a), OFF state, solid laser lines for pump and Stokes]. In addition to the peak shift that can induce peak decrease at the original epr-SRS frequency channel as we initially hypothesized, there are a few additional possible factors that may likely underlie the decrease in the epr-SRS signals upon 660 nm excitation even without a peak shift. One possibility is the population competition between the epr-SRS excitation and the invoked rigorous-electronic-resonance-involved four-wave mix- ing processes.19Second, since the frequency-independent K term in the Albrecht A equation includes a quadratic dependence on the oscillation strength of the molecular absorption (i.e., σabs), a smallerσabsfor the S 1–S2transition compared to that of the S 0–S1 transition may also lead to a much-lowered excited-state epr-SRS peaks. The invoked high electronic background from the excited state would complicate the analysis for imaging applications. In princi- ple, utilizing a frequency-modulation SRS scheme, which subtracts SRS signals between on-resonance and off-resonance frequencies in real-time30,31instead of the intensity-modulation SRS scheme in our setup, should resolve this issue. Nonetheless, it is still desirable to have a high signal-to-background ratio for straightforward imag- ing interpretations. Since the background is potentially induced by the S 1–S2transition, we then aimed at investigating whether the triplet state (T 1) could serve as an OFF state for epr-SRS signals with a decreased background [Fig. 3(a)]. To increase the T 1popu- lation, we added potassium iodide (KI) to the dye solution, which is known to accelerate intersystem crossing by the heavy atom induced FIG. 2 . Photoswitching of epr-SRS signals via transition to the electronic excited state. (a) Energy diagrams of the proposed ON (left, the green shade indicates the ground state as the ON state) and OFF (right, the gray shade indicates the excited state as the OFF state) states for epr-SRS signals. (b) epr-SRS spectra of the conjugated double bond mode (i.e., highlighted red in the molecular structure) of Rh800 at 0, 2, 8, 20, and 34 mW of 660 nm excitation beam irradiation. (c) epr-SRS spectra of the triple bond mode (red colored in the molecular structure) of Rh800 at 0, 2, 8, 20, and 34 mW of 660 nm excitation beam irradiation. J. Chem. Phys. 154, 135102 (2021); doi: 10.1063/5.0043791 154, 135102-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Photoswitching of epr-SRS signals via transition to the triplet state. (a) Energy diagrams of the proposed ON (left, the green shade indicates the ground state as the ON state) and OFF (right, the gray shade indicates the triplet state as the OFF state) states for epr-SRS signals. (b) epr-SRS spectra of the double-bond mode (red-colored in the molecular structure) of Rh800 at 0, 0.4, 2, 8, and 20 mW of 660 nm excitation beam power in the presence of potassium iodide (KI). (c) epr-SRS spectra of the triple-bond mode (red colored in the molecular structure) of Rh800 at 0, 0.4, 4, 12, and 34 mW of 660 nm excitation beam power in the presence of (KI). spin–orbit coupling.32–34The increase in triplet state population was confirmed by fluorescence intensity measurements (Fig. S3). Simi- lar to the excited state SRS spectra, the double bond [Fig. 3(b) and Fig. S4(a)] and triple bond [Fig. 3(c) and Fig. S4(b)] peaks disap- peared when the Rh800 molecules were further shifted to the triplet state in the presence of KI. Surprisingly, while we still detected a large background increase for the triple-bond peaks, we observed a large negative background signal across the double-bond frequency range. These negative signals indicate an increase in pump photons, since we detected the stimulated Raman loss (i.e., the pump pho- ton loss) as SRS signals (see the Methods section). Although a complete understanding of the molecular pathways would require further studies, a plausible reason for such an increase in pump photons is the population depletion in the presence of Stokes pho- tons due to the excitation competition for the T 1–Tntransitions. Since the pump beam wavelength for the triple-bond excitation (i.e., 838 nm) is further blue shifted from that for the double bond (i.e., 880 nm) and from the Stokes beam (i.e., 1031.2 nm), it is possible that 838 nm light falls out of the T 1–Tntransition range, and hence do not induce a significant negative background. Here, we success- fully demonstrated that both the excited state and the triplet state could effectively deprive the epr-SRS signals for both the double and the triple bond of the Rh800 molecules. However, both induced negative and positive electronic backgrounds, which could intro- duce artifacts in analyzing the epr-SRS images. In addition, inducing transitions to the excited and triplet state would lead to increased photobleaching. We hence continued to explore two other molecular states as the effective OFF state. B. Photoswitching by modulating the epr-SRS enhancement: Organic dyes The third electronic state we aimed at exploiting was the long- lived reversible dark state ( ∼100 ms to s in an oxygenated environ- ment). This photo-reduced state in the presence of electron donors for organic fluorophores has been heavily explored in STORM and d-STORM super-resolution fluorescence microscopy. For example, oxazine and rhodamine dyes are known to form semi-reduced rad- icals (F⋅) or leuco (FH) structures in buffers containing primarythiols (RSH) upon irradiations to the triplet state (3F) [Fig. 4(a), gray box, RS-indicates the thiolate anions].35Similarly, cyanine dyes have also been shown to form a cyanine-thiol adduct under similar excitation and buffer conditions.36A photophysical change associ- ated with this photochemical reduction is the diminishment of the electronic absorption peaks. Since epr-SRS signals strongly depend on the oscillation strength of the molecular absorption (see the supplementary material, Scheme 1, parameter K),20,37the disappear- ance of absorption peaks in these photo-reduced states indicates that they would be ideal candidates to serve as the OFF state for epr-SRS excitations [Fig. 4(a) OFF state, gray box]. We tested this hypothesis with ATTO680, a red-absorbing oxazine dye that falls into the desired electronic pre-resonance exci- tation region under our SRS laser excitation and was reported to undergo light-induced transitions to the above-mentioned long- lived dark state [Figs. 4(a) and 4(b)].35Exciting ATTO680 solu- tions containing primary thiol β-mercaptoethylamine (MEA) with a 660 nm excitation beam could indeed transform the color of the solutions into transparent [Fig. 4(c) vs Fig. 4(d), cuvette in the inset, before and after 660 nm illumination]. Subsequent absorption measurement confirmed the disappearance of the corresponding absorption peak for ATTO680 [Fig. 4(c) vs Fig. 4(d)]. The rem- nant absorption after irradiation [Fig. 4(d)] was due to a layer of unconverted molecules at the interface between the solution and the airspace of the cuvette. After gentle shaking of the cuvette to facilitate dissolution of oxygen in the headspace, both the color and the absorbance peaks of the same solution were fully recovered [Fig. 4(e)], which indicates that the molecules are relaxed back to the ground state (1F0). These results confirm that the absorption peaks of ATTO680 can be switched on and off, as reported.38 We next examined whether the same excitation and oxida- tion steps can switch the epr-SRS signals on and off. We followed the same excitation procedure but contained the sample solution in a glass chamber typically used for SRS measurement [Fig. 5(a)]. Indeed, we successfully observed reversibly switchable epr-SRS sig- nals targeting the double bond peak of ATTO680 at 1661 cm−1 [Fig. 5(b), solid line, red]. The concurrent switching of the fluores- cence over multiple cycles was also observed [Fig. 5(b), solid line, blue]. The reduced epr-SRS signals could reach as low as to 5% of the original epr-SRS intensity. Control experiments in the absence J. Chem. Phys. 154, 135102 (2021); doi: 10.1063/5.0043791 154, 135102-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . Proposed strategy to photo- switch epr-SRS signals via transition to the absorption diminished long-lived dark state. (a) Energy diagrams of the pro- posed ON (left, the green shade indi- cates the ground state as the ON state) and OFF (right, the gray shade indi- cates the long-lived dark state as the OFF state) states.1F0, singlet ground state;1F1, singlet excited state;3F, triplet state; F⋅, semireduced radical state; FH, fully reduced leuco state; ISC, intersys- tem crossing; red, reduction; and RS−, thiolate anions. (b) Molecular structure of ATTO680. [(c)–(e)] Absorption spectra of ATTO680 before (c) and after (d) irra- diating with a 660 nm excitation beam, and after agitating the cuvette to facilitate oxidation (e). Insets show the images of solution color change in the same cuvette. of MEA showed no effect of such photoswitching [Fig. 5(b), dotted line, red for epr-SRS and blue for fluorescence signals]. Together, these data confirm that shifting the molecules between the ground state and the photo-reduced dark state could serve as an effective strategy to photoswitch epr-SRS signals. Although we have demonstrated the recovery of epr-SRS sig- nals by mechanically accelerating the oxidation through shaking or pipetting the solutions, it is more appealing to utilize light to recover SRS signals for the precise control of the activation kinetics. As the oxidation of semi-reduced radicals is known to be accelerated by irradiation of UV light,35we asked whether we could turn the epr- SRS signals on from the dark state by illumination with a 405 nm laser instead. We observed that the 405 nm laser irradiation could increase the epr-SRS signals by 1.7 times compared to that from the OFF state [Fig. 5(c), green vs pink]. Fluorescence signals also showed a similar level of recovery (Fig. S5). We note that such a recovery was not observed in the absence of the 405 nm activation [Fig. 5(d)], implying that the recovery was not caused by other processes such as diffusion. Going beyond the solution characterization, we further con- firmed this photoswitching effect in epr-SRS imaging. 5-ethynyl-2′- deoxyuridine (EdU) was incorporated into newly synthesized DNA of dividing HeLa cells and was then click-labeled by ATTO680 azide. The labeled cells immersed in a MEA-containing buffer were subsequently imaged by epr-SRS, targeting the 1661 cm−1peak of ATTO680 [Fig. 5(e)]. After sequential irradiation of the excitation beam and the 405 nm beam, the epr-SRS signals from ATTO680 (1661 cm−1) were first decreased to 50% and then recovered back to 70% of the original signals [Figs. 5(e)–5(h)]. The limited deple- tion of the SRS signals in cell samples compared to solutions is likely due to the restricted transport of the thiolate anions in cells. We note that the limited recovery of the SRS signals after 405 nm irradiation [Figs. 5(c) and 5(h), green] should be due to the fact that ATTO680 has a high electron affinity and accepts another electron to form aleuco dye (FH) [Fig. 4(a)].38As this leuco dye is more stable than the semi-reduced radical (F⋅), the oxidation is not easily facilitated by the 405 nm laser. Screening other rhodamine and oxazine dyes with a stable dark-state in the semi-reduced radical form should further increase the activation efficiency. However, such currently known structures (e.g., ATTO532) mostly fall outside the desired epr-SRS excitation regime (640–790 nm). Shifting the wavelength of SRS lasers to the bluer region, as recently reported, should facilitate screening of better performing photoswitchable epr-SRS probes.39,40 C. Photoswitching by modulating the epr-SRS enhancement: Photoswitchable proteins Based on the pre-resonance Raman approximation,19,20,37 epr-SRS signals are nonlinearly dependent on another photo- physical parameter, the detuning between the pump photon energy and the molecular electronic resonance [see supplementary material, Scheme 1, parameter1 (ω2 0−ω2 pump)4]. The epr-SRS signals would decrease over 105-fold when the pump laser energy ( ωpump) is detuned away from the molecular electronic transition energy (ω0).20Therefore, molecules with switchable electronic resonances closer to and further away from the pump laser energy could also serve as ON and OFF epr-SRS states, respectively, with a decent ON-to-OFF ratio [Fig. 6(a)].41We tested a recently engineered truncated version of a reversibly switchable far-red absorb- ing soluble bacterial phytochrome photoreceptor (BphP) from Deinococcus radiodurans , DrBphP-PCM [Fig. 6(b)].42The absorb- ing core of DrBphP-PCM is composed of a photosensory core mod- ule (PCM), which is shared by all BphPs, and a covalently attached biliverdin IXa chromophore, which is the enzymatic product of heme catabolism and present in all mammalian cells. Biliverdin undergoes reversible cis–trans isomerization when irradiated with different wavelengths of light, causing BphP transitions between two J. Chem. Phys. 154, 135102 (2021); doi: 10.1063/5.0043791 154, 135102-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . Photoswitching of epr-SRS signals via transition to the long-lived dark state. (a) A photo of ATTO680 solution containing 0.5M MEA at pH 9.5 in an SRS imaging chamber before (left) and after (middle) excitation beam irradiation, and after oxidation (right). (b) Reversible switching of epr-SRS (red) and fluorescence (blue) signals for multiple cycles of irradiation and oxidation (solid line). No switching was observed in the absence of MEA for both epr-SRS (red) and fluorescence (blue) (dashed line). (c) epr-SRS signals before (black) and after (magenta) excitation beam, and after 405 nm activation (green) irradiation. (d) epr-SRS signals of ATTO680 solutions without 405 nm activation. [(e)–(g)] epr-SRS images of ATTO680-click-labeled DNA in HeLa cells before irradiation (e), after excitation beam irradiation (f), and after 405 nm laser irradiation (g). (h) Quantification of epr-SRS signals from the arrowed cell in (e)–(g). Scale bars, 10 μM. In (c) and (d), statistical significance was determined by the unpaired two tailed ttest. ns, not significant (p >0.05),∗p<0.05. Data are shown as mean ±standard deviation (n = 3 replicates for each group). absorbing states, Pr and Pfr [Fig. 6(b), cis–trans isomerization high- lighted in the blue circle].42With purified DrBphP-PCM solutions, we first confirmed their reversibly switchable absorptions, peaked at 750 nm [Fig. 6(c), magenta, the Pfr state in Fig. 6(b)] and 700 nm [Fig. 6(c), black, the Pr state in Fig. 6(b)], upon illumination with 690 and 780 nm, respectively.We next quantified its epr-SRS signal magnitude and reversibil- ity. As the absorption peaks of DrBphP-PCM fall within the desired epr-SRS excitation regime,19,20DrBphP-PCM in its epr-SRS ON state (i.e., the Pfr state) shows an around 300 times signal magni- tude to that of EdU, adopting the recent RIE (the relative intensity to EdU) quantification metrics [Fig. 6(d)].43Such a signal size is J. Chem. Phys. 154, 135102 (2021); doi: 10.1063/5.0043791 154, 135102-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 6 . Photoswitching of the purified near-infrared absorbing DrBphP-PCM protein. (a) Energy diagrams of the proposed ON (left, green shaded) and OFF (right, gray shaded) states. (b) Cis–trans configuration change of the biliverdin chromophore in DrBphP-PCM upon irradiation. (c) Absorption spectra of DrBphP-PCM in the Pr (black) and Pfr (magenta) conformation states. (d) Relative epr-SRS signals of DrBphP-PCM in the Pfr (1615 cm−1, magenta) and Pr (both 1628 cm−1, black; and 1615 cm−1, brown) states compared to the standard SRS signal of EdU (2124 cm−1, green). (e) epr-SRS spectra of DrBphP-PCM in the Pr (black) and Pfr (magenta) states. (f) Cycles of reversible photoswitching of DrBphP-PCM fluorescence, observed in the Pr state. (g) Cycles of reversible photoswitching of DrBphP-PCM epr-SRS signal at 1615 cm−1, the ON state (Pfr state) on-resonance channel. (h) Photoswitching of DrBphP-PCM at 1615 cm−1for over 40 cycles demonstrates no detectable photofatigue. equivalent to a detection sensitivity below 1 μM for this probe.44 When switched to the epr-SRS OFF state (i.e., the Pr state) by 780 nm laser, DrBphP-PCM indeed presents a lowered epr-SRS signal for the double bond mode with a Raman peak shifted from1615 to 1628 cm−1[Fig. 6(e), magenta and black, respectively]. This peak shift is resulted from the change in cis–trans conformation of the double bond between ring C and ring D of the biliverdin chro- mophore [Fig. 6(b)].45Detecting the absolute intensity changes at J. Chem. Phys. 154, 135102 (2021); doi: 10.1063/5.0043791 154, 135102-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the 1615 cm−1channel yielded an about three-fold signal decrease between the epr-SRS ON and OFF states [Figs. 6(d) and 6(e)]. The residual 33% of epr-SRS signals in the OFF state [Figs. 6(d) and 6(e)] originates from a combination of the remaining epr-SRS enhance- ment from the OFF-state Raman peak and a slight electronic back- ground by the pump laser from the relatively broad absorption bands of DrBphP-PCM [Fig. 6(c)]. After the initial characterization of DrBphP-PCM, we tested the robustness of the photo-switching for epr-SRS signals as the resistance to switching fatigue is an important photophysical param- eter in reversibly switchable probes. We first monitored the cycles of reversibility for the DrBphP-PCM fluorescence signals using the sequence of alternating 60 and 30 s illuminations by the 780 nm [Fig. 6(f), yellow] and 640 nm [Fig. 6(f), green] lasers. The 780 nm laser switches the protein to the Pr state exhibiting a weak fluorescence peak at 720 nm,46whereas the 640 nm laser serves as both the readout laser and the deactivation laser that shifts the protein back to the Pfr non-fluorescent state. We note that the ON and OFF states for fluorescence signals are reversed from those of epr-SRS signals as quantum yields of the Pr and Pfr states are 2.9% and 0%, respectively.46Our observed fluorescence depletion and recovery are similar to what were reported previously [Fig. 6(f)].42,46 We then probed the reversibility of the epr-SRS signals. Here, a 780 nm laser was adopted as the deactivation laser, a 690 nm laser was used as the activation laser, and the SRS beams were used as the signal readout laser. Figure 6(g) shows the laser sequence and the corresponding epr-SRS ON and OFF intensity. Similar to that for fluorescence, clear photo-switching of epr-SRS signals was observed over multiple cycles [Fig. 6(g)], whereas the epr-SRS sig- nal levels remained unchanged in the absence of the 690 nm and 780 nm lasers [Fig. S6(a)]. In a separate control experiment, we irradiated the protein solution with Stokes and pump beams for 30 s and allowed the molecules to diffuse and replenish for 60 s [Fig S6(b)]. However, no SRS recovery was observed [Fig. S6(b), dotted line]. In contrast, when the protein sample was irradiated by a 690 nm activation laser, the SRS signal immediately increased back to the original intensity [Fig. S6(b), green arrow]. This result indicates that the role of diffusion is minimal and shows that the recovery of SRS signals is not caused by diffusion of the ON molecules into the focal volume, but from activation via the 690 nm light. Interestingly, we observed a decrease in the epr-SRS signals when the DrBphP-PCM solution was irradiated with SRS beams [Fig. 6(g), black arrow]. We attribute this switching-off effect to pump beam excitation (around 884 nm), which could excite at the very tail of the absorption peak of the Pfr state [Fig. 6(c), magenta]. We note that this decrease is not due to photobleaching, as epr-SRS signals always recovered back to 100% with 690 nm beam activa- tion [Fig. 6(g)]. We further extended the epr-SRS switchable cycles to more than 40 cycles with minimum photobleaching, demonstrat- ing the robustness of the DrBphP-PCM protein as a photoswitchable epr-SRS probe [Fig. 6(h)]. We reasoned that the minimal photo- bleaching observed here should likely be due to two reasons. First, the competing switching-off pathway from the SRS beams may have likely helped in reducing the potential photobleaching kinetics from either the one-photon or the two-photon excitation by the SRS lasers. Second, the adopted picosecond SRS beams should contributemuch less to the higher-order multi-photon excitation induced photobleaching, since their peak power is much smaller compared to that of the femtosecond lasers. III. DISCUSSION This work presents a series of photophysical characterizations and proof-of-principle observations toward a new type of opti- cal imaging probes, i.e., the photoswitchable epr-Raman probes. We explored the possibilities of four electronic states to serve as the effective OFF state for the epr-SRS signals. In our first two strategies, we induced transitions to the excited state and the triplet states, which led to successful reduction of the epr-SRS peaks. However, robust frequency-modulation SRS techniques are required to remove the large electronic background (both the posi- tive and negative ones) for further applications.30,31Guided by the Albrecht A-term pre-resonance approximation equation (see the supplementary material, Scheme 1), we subsequently explored another two states, the long-lived dark state with a diminished absorption peak from organic dyes and a tunable absorption tran- sition state with modulable detuning to the pump photon energy from photoswitchable proteins. We proved that all four states together with the ground state could serve as the ideal candidates for reversibly photo-switching the epr-SRS signals upon further optimizations through additional engineering and designing of the photoswitchable Raman probes. As we indicated above, for red-absorbing organic molecules, extensive screening of rhodamine and oxazine dyes with a stable dark-state in the semi-reduced radical form should help improve the activation efficiency with 405 nm laser. In addition, cyanine dyes with a similar cyanine-thiol adduct may also offer new opportu- nities. In parallel with the dye screening, doubling the frequencies of the SRS lasers as recently demonstrated would offer the match- ing excitation region for molecules across the entire visible absorp- tion range, vastly expanding the pools for photo-switchable epr-SRS probes.39,40 We have also demonstrated that the DrBphP-PCM protein shows high promise toward generating a new category of photo- switchable Raman proteins. Further protein engineering efforts for obtaining larger dynamic ranges of the ON-to-OFF signal ratios are required. This would offer multiplexable epr-SRS peaks and allow genetical encodability for future cell imaging applications. First, slightly blue shifting the absorption peak (e.g., for 30–50 nm) would minimize the electronic background. This would lead to a clearer separation of Raman peaks between the epr-SRS ON and OFF states for easier multiplexing. Second, larger shift in absorption peaks between the epr-SRS ON and OFF states should also con- tribute to enhancing the dynamic ranges of the ON-to-OFF ratios. Third, mutagenesis of the amino acid residues around the biliverdin binding pocket may shift its double bond vibrational frequency by changing the interacting environment and, hence, creating more colors.47Fourth, incorporation of a nitrile bond to the conjuga- tion system of the biliverdin would significantly help expand the epr-SRS color palette owing to the features of the narrow-band and editable nitrile bonds that are ideal for multiplexing. Fifth, the superior property of photoswitchable Raman protein-based probes is their genetic encodability, which is critical for live cell imaging J. Chem. Phys. 154, 135102 (2021); doi: 10.1063/5.0043791 154, 135102-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp and is not offered by organic dyes. Moreover, it is worth noting that whereas photoswitching of organic dyes frequently requires UV light, which is phototoxic for cells, the Raman protein probes derived from BphPs use non-cytotoxic photoswitching far-red and near-infrared light, which penetrates much deeper in biological tissues, thus enabling intravital imaging.48Further engineering of distinct BphP-based Raman probes, along with their different intracellular targeting, will allow super-multiplex epr-SRS imaging in live cells. Ultimately, with successful invention of a new category of photoswitchable epr-SRS probes, super-multiplex super-resolution optical imaging may be implemented, as we rationalized above. Adopting a doughnut setup similar to RESOLFT but with only one additional switching laser, super-multiplex imaging could be brought into the super-resolution regime and offer a valuable new addition to the toolbox of optical imaging in investigating biological activities and functions. SUPPLEMENTARY MATERIAL See the supplementary material for experimental methods, scheme, supporting figures, and tables. ACKNOWLEDGMENTS This work was supported by the grants from the National Institutes of Health, Grant No. DP2 GM140919 (to L.W.) and Grant No. R35 GM122567 (to V.V.), and by the start-up fund from the California Institute of Technology (to L.W.). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1R. Ando, H. Mizuno, and A. Miyawaki, “Regulated fast nucleocytoplasmic shut- tling observed by reversible protein highlighting,” Science 306(5700), 1370–1373 (2004). 2G. U. Nienhaus, K. Nienhaus, A. 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Rev. Sci. Instrum. 92, 041501 (2021); https://doi.org/10.1063/5.0039730 92, 041501 © 2021 Author(s).Rotational energy harvesting systems using piezoelectric materials: A review Cite as: Rev. Sci. Instrum. 92, 041501 (2021); https://doi.org/10.1063/5.0039730 Submitted: 06 December 2020 . Accepted: 05 March 2021 . Published Online: 02 April 2021 Zhe Wang , Lipeng He , Xiangfeng Gu , Shuo Yang , Shicheng Wang , Pingkai Wang , and Guanggming Cheng ARTICLES YOU MAY BE INTERESTED IN A comprehensive review on piezoelectric energy harvesting technology: Materials, mechanisms, and applications Applied Physics Reviews 5, 041306 (2018); https://doi.org/10.1063/1.5074184 Entangled photon-pair sources based on three-wave mixing in bulk crystals Review of Scientific Instruments 92, 041101 (2021); https://doi.org/10.1063/5.0023103 Energy harvesting from low frequency applications using piezoelectric materials Applied Physics Reviews 1, 041301 (2014); https://doi.org/10.1063/1.4900845Review of Scientific InstrumentsREVIEW scitation.org/journal/rsi Rotational energy harvesting systems using piezoelectric materials: A review Cite as: Rev. Sci. Instrum. 92, 041501 (2021); doi: 10.1063/5.0039730 Submitted: 6 December 2020 •Accepted: 5 March 2021 • Published Online: 2 April 2021 Zhe Wang,1,a) Lipeng He,1,b) Xiangfeng Gu,1,c)Shuo Yang,1,d)Shicheng Wang,1,e)Pingkai Wang,1,b) and Guanggming Cheng2,f) AFFILIATIONS 1School of Mechatronic Engineering, Changchun University of Technology, Changchun, Jilin 130012, China 2Institute of Precision Machinery, Zhejiang Normal University, Jinhua 130022, China a)zhewang0926@163.com b)Authors to whom correspondence should be addressed: lipeng_he@126.com and wangpingkai@ccut.edu.cn c)xiangfeng_gu@163.com d)yangshuo88088@163.com e)shicheng_wang126@126.com f)cgm123@163.com ABSTRACT In the past few decades, rotary energy harvesting has received more and more attention and made great progress. The energy harvesting device aims to collect environmental energy around electronic equipment and convert it into usable electrical energy, developing self-powered equipment that does not require replaceable power supplies. This paper provides a holistic review of energy harvesting techniques from rotary motion using piezoelectric materials. It introduces the basic principles of piezoelectric energy harvesting, the vibrational modes of piezoelectric elements, and the materials of piezoelectric elements. There are four types of rotational energy harvesting technologies: inertial excitation, contact execution, magnetic coupling, and hybrid systems. An overview of each technology is made, and then, a detailed analysis is carried out. Different types of rotating energy harvesting technologies are compared, and the advantages and disadvantages of each technology are analyzed. Finally, this paper discusses the future direction and goals of improving energy harvesting technology. This Review will help researchers understand piezoelectric energy harvesting to effectively convert rotational energy into electrical energy. Published under license by AIP Publishing. https://doi.org/10.1063/5.0039730 I. INTRODUCTION With the development of micro-electromechanical system technology, various radio devices are developing at a high speed toward miniaturization, intelligence, multi-function, and low-power consumption. Most radio equipment requires an uninterrupted operation, so energy harvesting becomes a major issue for the long-term normal operation of radio equipment.1,2The conven- tional battery power supply has the disadvantages of limited service life, chemical pollution, and large volume. The energy harvester collects various forms of energy from the surrounding environment to replace traditional batteries or extend the life of tra- ditional batteries to power microelectronic devices.3,4The energy present in the surrounding environment includes solar energy,thermal energy, magnetic field energy, nuclear energy, and vibra- tion energy. Vibration energy is the most widely distributed green energy in the environment, and energy harvesting from these envi- ronments is beneficial. Vibration energy can be obtained from sea wave fluctuations, heart movement, train operation, mechanical equipment operation, and car driving vibration. The conversion of vibration energy into electrical energy is not only sustainable, energy-saving, and environmentally friendly but also enables various self-powered functions. The collection and utilization of green and renewable vibration energy have become a research hotspot widely concerned.5,6 The energy harvesting device can convert useless vibration energy in the environment into electric energy that can be reused. The main methods of vibration energy conversion include the Rev. Sci. Instrum. 92, 041501 (2021); doi: 10.1063/5.0039730 92, 041501-1 Published under license by AIP PublishingReview of Scientific InstrumentsREVIEW scitation.org/journal/rsi electromagnetic,7,8electrostatic,9,10triboelectric,11,12and piezoelec- tric mechanism.13,14The electromagnetic generator system has the characteristics of stable operation and high current. The output power density is low when subjected to low-frequency excitation. The electrostatic generator system outputs a high voltage in a small size. The conditions for generating high power density are the need for a small gap and starting voltage, leading to dielectric degrada- tion and system complexity. The friction generator system has a high power density, but the current density is low and the charge accumulation is difficult. Among the various techniques of energy harvesting, the piezo- electric energy harvester (PEH) has multiple advantages over other solutions, including smaller size, high output voltage, simple mech- anism, highly sensitive to the applied strain, and high power den- sity.15,16In the past ten years, the Web of Science thesis database included up to 3968 articles on the subject of piezoelectric energy collection. As shown in Fig. 1, the annual global collection of the lit- erature on the theme of piezoelectric energy harvesting has increased from 138 in 2010 to 730 in 2019, which indicates that the scholar’s attention to piezoelectric energy harvesting technology has increased dramatically. Vibration energy harvesting may prolong the service life of electronic devices and provide energy for inaccessible electronic devices or special electronic devices that require expensive main- tenance. The method of collecting environmental vibration energy through the piezoelectric mechanism has been widely used, for example, in the biomedical field, portable electronic products, navi- gation systems, environmental monitoring, and many other fields. Several review papers have been published on the technology of piezoelectric energy collection performance;17–22however, this article will only focus on the rotational energy collection technol- ogy. To meet the self-powered sensor detection of rotating machin- ery systems, rotating piezoelectric generators of various structures have been proposed.23–28Rotational motion is common in natural environments or mechanical equipment. The piezoelectric trans- ducer collects the energy in the rotating motion, and its structure is FIG. 1. Number of peer-reviewed papers during 2010–2020 as a function of publication year. simple, the output voltage is high, and the power density is high. Fu and Yeatman29introduced a review of rotational energy harvesting. The electromagnetic, piezoelectric resonant, and piezoelectric non- resonant rotating energy harvesters were reviewed and compared based on the device size and operating frequency. This article differs in the technical classification and expands the previous review by including the excitation form not previously mentioned or analyzed in detail. In this paper, an overview of energy harvesting is provided, with the main focus on the usage of piezoelectric harvesters in energy harvesting in the rotational motion. This Review presents a FIG. 2. Overview of the classification of PEH systems in rotational motion. Rev. Sci. Instrum. 92, 041501 (2021); doi: 10.1063/5.0039730 92, 041501-2 Published under license by AIP PublishingReview of Scientific InstrumentsREVIEW scitation.org/journal/rsi FIG. 3. PZT electronic movement trend diagram after force: (a) no external force, (b) tension, and (c) pressure. classification of the current techniques that collect energy through rotational motion. There is a lot of literature on energy collection through rotational motion; we show classification in Fig. 2. This classification method shows energy harvesting devices that collect rotational energy under different excitation conditions. An overview of each technology is made, and then, a detailed analysis is carried out. The examples of the current research illustrate each technology and summarize the advantages and disadvantages of each technol- ogy. Finally, the challenges based on the existing methods and future requirements of energy harvesting are discussed. II. PEH PRINCIPLES AND MATERIALS A. Principles of PEH Piezoelectric energy harvesting utilizes environmental vibra- tion to induce the deformation of piezoelectric structures. As shown in Fig. 3, the piezoelectric ceramic material itself does not generate electricity when it receives no external force. Once subjected to an external force, the original state of the piezoelectric ceramic material is changed, generating a potential difference between its two ends. The positive and negative charge inside the piezoelectric material is separated, thereby generating a polarization voltage, which will drive the free charge on the polar plate to flow directionally, and output electrical energy, converting the useless mechanical energy from the environment into usable electrical energy. It is usually designed tomatch the natural frequency of PEH with the external environment’s excitation frequency. PEH is composed of a mechanical vibration system, energy conversion equipment, a power management cir- cuit, and energy storage. First, the vibration energy is converted into mechanical energy through the structure in the external envi- ronment. Then, the mechanical energy is converted into electrical energy by using the piezoelectric element and finally stored in the electrical form. B. Vibration mode of piezoelectric element Generally, piezoelectric energy harvesting devices made of piezoelectric materials have three vibration modes, namely, d 33, d31, and d 15, as shown in Fig. 4. When a force is applied to the piezo- electric sheet in the longitudinal direction, the piezoelectric element generates strain in the longitudinal direction at this time. The voltage in the piezoelectric sheet refers to the voltage when the piezoelectric sheet tries to return to a non-resistive state without deformation. The vibration mode generated in this process is called d 33mode. When a stress is applied to the piezoelectric sheet in the lateral direc- tion, strain is generated in the lateral or longitudinal direction of the piezoelectric element at this time. This vibration mode is called d31mode. As for the d 15mode, its principle is essentially the same as that of the d 33and d 31modes, except that the direction of the stress it receives is the stress of rotating shear, not longitudinal or transverse. The d 15mode has the largest value of the piezoelectric FIG. 4. Illustration of the d 31, d33, and d 15operation modes. Rev. Sci. Instrum. 92, 041501 (2021); doi: 10.1063/5.0039730 92, 041501-3 Published under license by AIP PublishingReview of Scientific InstrumentsREVIEW scitation.org/journal/rsi coefficient component. Therefore, theoretically, a higher electrome- chanical conversion efficiency can be achieved. However, since the d15mode requires piezoelectric materials to obtain shear stress and it is, therefore, difficult to obtain shear stress in the actual applica- tion environment, the more commonly used piezoelectric conver- sion modes are d 33and d 31modes. In the d 31piezoelectric mode, the electrode of the piezoelectric material is a flat electrode, the externally applied stress and the mechanical deformation caused by it are all along direction 1, and the polarization direction is direction 3, so the generated voltage is also direction 3. Thus, the direction of the voltage and electric field generated by the piezo- electric material and the direction of the applied stress and strain are perpendicular to each other. In the d 33piezoelectric mode, the electrode of the piezoelectric material is a surface interdigital elec- trode. The externally applied stress and the mechanical deformation caused by this are all along direction 3, and the polarization direc- tion is direction 3, so the voltage generated is also the direction 3. That is, the voltage and electric field generated by the piezoelectric material is consistent with the direction of the applied stress and strain. For the piezoelectric cantilever beam energy harvester, the d31mode completes the collection of charges in the direction per- pendicular to the stress and strain, while the d 33mode completes the collection of charges in the direction parallel to the stress and strain. C. Materials of piezoelectric element Piezoelectric materials are functional materials that realize the interconversion of mechanical and electrical energy and are elec- tronic materials that are sensitive to electricity, light, and heat and are widely used in transducers. Since the emergence of piezoelec- tric materials in the twentieth century, they have gradually become an important part of the materials field. Piezoelectric materials are mainly divided into three categories: inorganic piezoelectric materi- als, organic piezoelectric materials, and piezoelectric composites. 1. Inorganic piezoelectric materials Inorganic piezoelectric materials are divided into piezoelec- tric crystals and piezoelectric ceramics. Piezoelectric crystals refer to piezoelectric single crystals, while piezoelectric ceramics refer to piezoelectric polycrystals, in general. Piezoelectric crystals generally refer to piezoelectric single crystals, which are crystals grown in a long-range order according to the crystal space dot matrix. This kind of crystal structure has no center of symmetry, so it has piezoelec- tricity. Piezoelectric ceramics are chemical raw materials for mixing, molding, and high-temperature sintering, and there is a separation between the solid-phase reaction and the sintering process to obtain the dimensional grains’ irregular collection of polycrystals. Many piezoelectric materials have been developed over the past century, but the most common piezoelectric material is perovskite lead zirconate titanate, a polycrystalline monolithic piezoelectric ceramic called PZT, usually doped with niobium or lanthanum to form soft and hard piezoelectric materials, respectively. The performance of PZT can be adjusted in a wide range by the preparation process and the change in doping. Since the sintering temperature of PZT is generally 1200–1300○C and the volatilization temperature of lead oxide is about 850○C, lead oxide is more likely to volatilize at such high temperature, and the volatilization of leadoxide will directly affect the performance of ceramics, on the one hand, and bring great harm to the environment, on the other hand, so the research direction in recent years has gradually shifted to reducing the sintering temperature while ensuring that the sintering speed is not too fast Therefore, in recent years, the research direction has shifted to lowering the sintering temperature while ensuring that the sintering rate is not too fast so that the grain size remains fine. By adding low melting point sintering aids, such as Li 2CO 3, Bi2O3, and MnO 2, to PZT ceramics, the sintering temperature can be reduced by about 200–300○C. The nanosized ultrafine powder can be prepared to control the sintering temperature below the volatilization temperature of lead oxide; the sintering temperature of PZT can also be reduced by 150–200○C by hot pressing sintering, which is beneficial to reduce the volatilization of lead oxide. There are three main methods to prepare high power and high conversion efficiency PZT piezoelectric ceramics: modification by doping with manganese elements, YMnO 3, and other elements; adding the fourth group element to the multi-system piezoelectric ceramics to develop a new material system; and preparing ultrafine powder by the chemical method by adding low temperature co- sintering additives and using hot press forming sintering to explore a new preparation process. 2. Organic piezoelectric materials Although piezoelectric ceramics are relatively inexpensive and provide good coupling, they are fragile and have high densities. Given the increasing use of piezoelectric ceramics in microelec- tromechanical systems (MEMS), PZT films have been developed to achieve flexibility using small-scale production. Organic materials, also known as piezoelectric polymers, such as vinylidene fluoride (PVDF) and other organic piezoelectric film materials represented by others are used. These materials and their material flexibility, low density, low impedance, and high voltage constant are advantages for the world’s attention, and the development is very rapid. How- ever, the simple PVDF dielectric constant is generally low, so it is difficult to improve the dielectric properties, and although there are a few polymers with high dielectric constants, their application is hindered by the complexity of their manufacturing process. There- fore, the preparation of polymer-based dielectric materials with high dielectric constants and good mechanical properties is an important way to solve the above-mentioned problems by using the respec- tive advantages of each single-phase material through the compos- ite effect of materials. Piezoelectric ceramic films are mainly made by using various deposition techniques to produce piezoelectric films of various sizes and thicknesses. The deposition techniques are the screen printing method, sol-gel method, chemical vapor deposition method, magnetron sputtering method, etc. Vapor-phase methods are suitable for preparing nanoscale piezoelectric films, mainly physical vapor deposition and chemical vapor deposition. Among them, the sputtering method is the most commonly used method. The chemical vapor deposition method can precisely con- trol the chemical composition of reaction products, and doping is convenient. 3. Piezoelectric composites Piezoelectric composites are composed of sheet, rod, bar, or powdered piezoelectric materials embedded in an organic polymer matrix material. The resulting composites have improved strength Rev. Sci. Instrum. 92, 041501 (2021); doi: 10.1063/5.0039730 92, 041501-4 Published under license by AIP PublishingReview of Scientific InstrumentsREVIEW scitation.org/journal/rsi and flexibility, as well as improved robustness, due to the polymer matrix protecting the fragile ceramics. Research in this area has led to the widespread use of piezoelectric devices using piezoelectric particles (0–3 piezoelectric composites) and piezoelectric fibers (1–3 piezoelectric complex composites). Early development of 0–3 piezoelectric composites focused on the development of material properties, often referred to as “piezoelectric coatings.” These materials offer unprecedented ease of use for the desired surface and can be quickly applied as dis- crete pastes over large surface areas by spraying or using a squeegee or brush. 1–3 piezoelectric fiber composites retain the stiffness and bandwidth benefits of pure piezoelectric ceramic materials, but because of the use of fiber structures, piezoelectric composites add new advantages such as anisotropy, increased damage resistance and static tensile strength, and the possibility of extending the applica- tion range to non-planar applications. The methods used to pre- pare composite piezoelectric materials are alignment-casting and cut-and-cast methods. III. RESEARCH STATUS OF PIEZOELECTRIC ROTATIONAL ENERGY HARVESTING DEVICE The rotation frequency of the motion source in the natu- ral environment is random and changeable and uncontrollable. The natural frequency of the piezoelectric energy harvesting device matches the rotation frequency, and when it is the resonance fre- quency, the output voltage value is high, and the energy conversion efficiency is high. When the rotation frequency shifts from the nat- ural frequency of the piezoelectric energy collection device, the gen- erated voltage drops sharply and the conversion efficiency is low. The analytical solution for the natural frequency of the piezoelec- tric energy harvester was derived from the parameter design process, which could specify a structure approaching resonance at any wheel rotating frequency. Scholars optimized the structure parameters of the energy harvester by designing them to achieve effective energy harvesting in a wide range of rotation frequencies. In recent decades, many studies have been conducted to develop knowledge, and new technologies have been introduced in the field of extracting energy from rotational motion. This Review presents a classification of the current techniques that collect energy through rotational motion. There is a lot of literature on energy collection through rotational motion; we recommend classification. The techniques for collecting energy in rotation fall into four cate- gories: inertial excitation, contact execution, magnetic coupling, and hybrid systems. A. Inertial excitation Inertial excitation refers to the action of centrifugal force in the rotating motion to drive the piezoelectric element to swing and gen- erate a charge. According to the configuration mode, it is divided into four types: outward, inward, vertical, and other configurations. 1. Outward configuration Gu and Livermore30,31reported on a passively tuned collec- tor that bends and deforms the radial piezoelectric cantilever beam caused by centrifugal force, thereby passively adjusting the reso- nance frequency of the beam. Khameneifar et al.32,33presented arotating flexible piezoelectric beam, which can be used to collect energy in a rotating motion. The structure consists of a cantilever beam with cutting-edge mass and piezoelectric ceramics connected along the beam, and the cantilever beam is mounted on a rotating shaft. Due to the gravity applied to the tip mass as the hub rotates, mechanical vibration energy is induced in the flexible piezoelectric beam and converted into electrical energy. Hsu et al.34presented a self-tuning piezoelectric cantilever beam to collect energy from rotational motion. The piezoelectric beam is installed on the rotat- ing shaft in the radial direction, and the tensile stress caused by the centrifugal force effectively deforms the beam, thereby passively tun- ing the resonance frequency. Kim35introduced a PEH that collects electrical energy from torsional vibrations caused by internal com- bustion engines. The piezoelectric cantilever beam structure is on the top surface of the rotating shaft and has a power of about 14 μW under a vibration torque of 30 Nm. Yang and Zhang36–38proposed the use of one or more thin beams attached to the end of the cantilever beam instead of using a tip-mass energy harvester. By adjusting the rotation angle of the connected thin beam, the resonance frequency of the energy collec- tor can be effectively adjusted. It aims to study the influence of the joint beam system on the performance of the piezoelectric energy collector. On this basis, an energy collector composed of three piezo- electric cantilever beams with attached beams is designed, as illus- trated in Fig. 5(a). The rotating hinged beam is utilized to adjust the operating frequency of the PEH. By utilizing two piezoelectric can- tilever beams aligned perpendicular to each other, two resonance peaks are obtained. In addition, by adjusting the rotation angle of these piezoelectric beams, the operating frequency bandwidth has been expanded by 70%–100%. 2. Inward configuration Guan and Liao39presented a PEH for rotational motion appli- cations. As the basic frame rotates, the piezoelectric elements in the energy collector repeatedly change to produce electrical energy. The constrained frame can limit the swing range of the piezoelectric ele- ment for protection and can use the collision to change the equiv- alent rigidity of the system to obtain the piecewise linear restoring force. The output power is 83.5–825 μW at a rotation frequency of 7–13.5 Hz. Wang et al.40designed a device that collects vibrational energy caused by friction. The feasibility of energy collection through friction-induced vibration is confirmed, and the effects of speed and load on system dynamics and energy collection are verified. The device is able to convert normal and tangential vibration energy into electrical energy. Ruiet al.41–43proposed a method for designing a PEH using centrifugal force for rotation. The schematic diagram of the PEH is illustrated in Fig. 5(b). The acrylic turntable is fitted to the rotat- ing shaft in the radial direction, and the piezoelectric cantilever beam base is fixed outward. During rotation, the stress caused by centrifugal force reduces the stiffness of the beam. The resonance frequency of the energy harvester can be adjusted by changing the distance between the mass and the center of rotation. On this basis, the energy harvesting system introduces a limiter, which decreases the amplitude of the piezoelectric element and extends the life of the harvester. The collision between the limiter and the harvester causes shock excitation, and the frequency band will become wider. Finally, Rev. Sci. Instrum. 92, 041501 (2021); doi: 10.1063/5.0039730 92, 041501-5 Published under license by AIP PublishingReview of Scientific InstrumentsREVIEW scitation.org/journal/rsi FIG. 5. The PEH is installed on the rotating hub in different installation methods. (a) Outward configuration. Reproduced with permission from L. Yang and H. Zhang., “A wide-band piezoelectric energy harvester with adjustable frequency through rotating the angle of the jointed beam,” Ferroelectrics 520(1), 237 (2017). Copyright 2017 Taylor & Francis. (b) Inward configuration. Reproduced from Rui et al. , “A design method for low-frequency rotational piezoelectric energy harvesting in micro-applications,” Microsyst. Technol. 26(3), 981 (2019). Copyright 2019 Springer Nature. (c) Perpendicular configuration. Reproduced with permission from Ramírez et al. , “Energy harvesting for autonomous thermal sensing using a linked E-shape multi-beam piezoelectric device in a low frequency rotational motion,” Mech. Syst. Signal Process. 133, 106267 (2019). Copyright 2019 Elsevier. (d) Other configuration. Reproduced with permission from Wang et al. , “Resonant frequency self-tunable piezoelectric cantilevers for energy harvesting and disturbing torque absorbing,” Sens. Actuators, A 285, 25 (2019). Copyright 2019. Elsevier. the designed piezoelectric energy collector is installed in the wheel spokes. Su et al.44assessed the influence of the installation direction of the rotating PEH on its performance and compared the outward, inward, and oblique directions. The installation direction of the PEH affects the axial component of the centrifugal force operating in the beam. Both the installation radius and the tilt angle can adjust the resonance frequency of the PEH to be adapted to the environment and high output power. 3. Perpendicular configuration Febbo et al.45,46provided a one-dimensional finite element capable of modeling a three-dimensional rotating energy harvest- ing device. Two structures were tested: a cantilever beam with a free end and two beams with the fixed end on the same side. The free ends of the two beams are connected by a spring. The results of the finite element model show good accuracy relative to the experimen- tal results and Abaqus simulation. Afterward, improvements were made, placing the fixed ends of two beams on opposite sides. The energy harvesting device produces the output power in the range of 26–105 μW. Machado et al.47introduced a flexible stop to limit the max- imum displacement and maintain the structural integrity of the beam. The energy harvesting device includes two beams, two weights connected by a linear spring, and a single-sided spring stop.Ramírez et al.48,49designed a multi-beam PEH for rotary motion, as shown in Fig. 5(c). It composes of two E-shaped multi-beam sys- tems and a rigid beam. The effects of hub distance, speed, centrifugal force, softening effect, and resistance on output voltage and power are studied. 4. Other configuration Sadeqi et al.50designed a piezoelectric rotating energy harvest- ing device including a coupled spring-mass system connected to a PZT beam. By changing the tip mass and spring stiffness, the reso- nance frequency can be adjusted. Wang et al.51,52proposed a PEH with a trapezoidal cantilever, as shown in Fig. 5(d). Fix the weight- ing block at the position of the cantilever and adjust the moment of inertia of the root of the trapezoidal cantilever by centrifugal force to ensure that the natural frequency coincides with the wheel. When the speed of the wheels is 240–1200 rpm, the output power of the energy harvesting device is about 65–110 μW. 5. Summary of inertial excitation Inertial excitation is divided into four types: outward configura- tion, inward configuration, vertical configuration, and other config- urations. In both the outward configuration and the inward config- uration, piezoelectric beams are parallel to the plane of rotation. The outward configuration means that the fixed end of the piezoelectric Rev. Sci. Instrum. 92, 041501 (2021); doi: 10.1063/5.0039730 92, 041501-6 Published under license by AIP PublishingReview of Scientific InstrumentsREVIEW scitation.org/journal/rsi cantilever beam is at the center of rotation and the free end of the piezoelectric beam points to the outside of the rotation plane. The inward configuration means that the fixed end of the piezoelectric cantilever beam is offset from the center of rotation and the free end of the piezoelectric beam points to the center of rotation. The verti- cal configuration means that the piezoelectric beam is perpendicular to the plane of rotation. Other configurations refer to configurations other than these three. A summary of the literature review on inertial excitations is listed in Table I. The piezoelectric beam is installed on the rotating shaft. During the rotation, the tensile stress generated by the centrifugal force dur- ing the rotating movement makes the beam hard, thereby passively tuning the resonance frequency to follow the driving frequency of the rotation. The tensile stress generated by centrifugal force in the radially oriented piezoelectric cantilever beam will passively adjust the resonance frequency during the rotation. The centrifugal force in the beam changes the stiffness and resonance frequency of the beam. The centrifugal force generated by the rotation will change the resonance frequency of the flexible drive beam and the frequency response of the harvester. Periodic excitation is generated by the gravity action of the mass of the free end in rotation. The installation distance and tilt angle can be used to adjust the resonant frequency of the system to match the excitation frequency. By changing the load mass and/or the mechanical characteristics of the spring mass system, the natural frequency of the system can be easily adjusted.By carefully selecting the load quality, the harvester proposed in this study can be easily pre-adjusted to generate voltage in a wider fre- quency range. The piezoelectric energy harvester has the ability to self-adjust its resonance frequency in a rotating environment. B. Contact execution The principle of contact excitation is that the low-frequency vibrator captures the vibration energy from the external vibration source, transmits it to the high-frequency vibrator through mechan- ical collision, converts it into the vibration energy of the high- frequency vibrator, and then turns into electrical energy output. There are two types according to the position of the excited piezo- electric element: plucking excitation and impact excitation. The for- mer means that the mechanical body is in contact with the edge of the piezoelectric element, and the latter means that the mechanical body is in contact with the center of the piezoelectric element. 1. Plucking excitation Pozzi et al.53,54proposed a PEH that can be worn on the knee joint. During the plucking process, the piezoelectric cantilever beam will be pulled by the deflection paddles, and the double- pressure cantilever beam will vibrate freely at its resonance fre- quency, thereby generating the highest efficiency of electrical energy. Priya55reported a theoretical model based on the bending beam TABLE I. Summary of literature review of inertial excitation. References Mechanism MaterialResonant frequency (Hz) Power Voltage Load resistance Gu and Livermore30 Outward configurationPZT 13.2 13.5 Gu and Livermore31PZT/PVDF 15.2 123 μW 23.2 V 220 k Ω Khameneifar et al.32PZT 1.2 7.7 μW 40 k Ω Khameneifar et al.33PZT/PVDF 6.4 mW/147 μW 40 k Ω/600 kΩ Hsuet al.34PZT-5A 1.6 mW 400 k Ω Kim et al.35PZT 93 14 μW 4 V Yang and Zhang36PMN-PT Yang and Zhang37PZT 2 60 V Yang and Zhang38PZT 16 17.5 V Guan and Liao39 Inward configurationPZT 7–13.5 83.5–825 μW Wang et al.40PZT-5A 316 30 k Ω Ruiet al.41MFC 8.7 200 k Ω Ruiet al.42MFC 8 82–103 μW 220 k Ω Ruiet al.43MFC 7.1 Suet al.44MFC Ramírez et al.45 Perpendicular configurationPZT-5A 5/6 2.2 μW/5.5 μW 0.15 V/0.225 V 10 k Ω Febbo et al.46PZT-5A 2.54 104.74 μW 4.438 V 10 k Ω Machado et al.47PZT-5A 2.3 920 μW 1.03 V 0.84 M Ω Ramírez et al.48MFC 3 442.25 μW 6.5 V 1000 k Ω Ramírez et al.49MFC 3 233 μW 14 V/9 V 100 k Ω/400 kΩ Sadeqi et al.50 Other configurationPZT 55–60 3 V Wang et al.51PVDF 70–140 μW 50 V 3.3 M Ω Wang et al.52PVDF 130 65–110 μW 600 k Ω Rev. Sci. Instrum. 92, 041501 (2021); doi: 10.1063/5.0039730 92, 041501-7 Published under license by AIP PublishingReview of Scientific InstrumentsREVIEW scitation.org/journal/rsi theory of bimorphs and equivalent circuits of capacitors for the com- putation of the generated electric power using piezoelectric bimorph transducers. This model was verified by comparing the calculated results with the measured ones on a prototype of a piezoelectric windmill consisting of ten piezoelectric bimorphs arranged in a cir- cular fashion. The schematic diagram of the piezoelectric windmill is shown in Fig. 6(a). Janphuang et al.56presented configuration to harvest energy from a rotating gear using piezoelectric system har- vesters. The piezoelectric cantilever beam is plucked by the teeth of the rotating gear, which may eventually be driven by the oscillating mass to generate electrical energy. A fan-structure PEH was proposed and tested to collect wind energy.57The collision induction PEH consists of a stator and a rotor and a circular array of four cantilevers. The periodic impact of the rotor blades on the free end of the cantilever is used to make the can- tilever oscillate and generate electrical energy. Fu and Liao58devel- oped a comprehensive model of plucking PEH. The model predicts the response of the PEH removal under different removal speeds and overlapping lengths. Jung et al.59,60proposed alternating resis- tive impedance matching for an impact-type microwind PEH. ThePEH consists of piezoelectric cantilevers and small wind turbines. The wind turbine hits the free end of the cantilever and vibrates to produce electrical energy. The wind-induced energy harvester utilizing a piezoelectric bimorph with a pulling strip was designed to collect wind energy.61 Both the rotation speed and the installing angle play crucial roles in determining the generated power capacity of the energy harvester. Tang et al.62proposed a PEH using a wind vane to convert multi- directional wind energy into electrical energy. The multi-directional wind collecting module keeps the piezoelectric element in the most windward area. The piezoelectric element is deformed by the impact of the rod and converts wind energy into electrical energy. The stor- age module stores electrical energy in a capacitor to provide electri- cal energy to the wireless sensor. Kim et al.63reported a propeller- based underwater PEH consisting of a propeller, hitting sticks, and a piezoelectric module, as shown in Fig. 6(b). The hitting sticks spin with a rotating axis connected to a propeller rotated by water flow and hit a piezoelectric cantilever beam. The running water rotates the propeller of the system to run four hitting sticks, which exert force to the piezoelectric module at a frequency of 24.5 Hz. FIG. 6. (a) Schematic structure of the energy harvester. Reproduced with permission from S. Priya, “Modeling of electric energy harvesting using piezoelectric windmill,” Appl. Phys. Lett. 87(18), 184101 (2005). Copyright 2005 AIP Publishing LLC. (b) Architecture of the multidirectional wind energy harvester. Reproduced with permission from Kim et al. , “Propeller-based underwater piezoelectric energy harvesting system for an autonomous IoT sensor system,” J. Korean Phys. Soc. 76(3), 251–256 (2020). Copyright 2020 Springer Nature. (c) Schematic diagram of the piezoelectric windmill. Reproduced with permission from Yang et al. , “Rotational piezoelectric wind energy harvesting using impact-induced resonance,” Appl. Phys. Lett. 105(5), 053901 (2014). Copyright 2014 AIP Publishing LLC. (d) Schematic diagram of the gull-wing structure piezoelectric rotating energy harvester. Reproduced with permission from Yang et al. , “A gullwing-structured piezoelectric rotational energy harvester for low frequency energy scavenging,” Appl. Phys. Lett. 115(6), 063901 (2019). Copyright 2019 AIP Publishing LLC. Rev. Sci. Instrum. 92, 041501 (2021); doi: 10.1063/5.0039730 92, 041501-8 Published under license by AIP PublishingReview of Scientific InstrumentsREVIEW scitation.org/journal/rsi 2. Impact excitation Makki and Pop-Iliev64designed a tire pressure measurement system sensor powered by on-the-go piezoelectric energy harvest- ing. Varela and Sierra65designed a PEH mounted on the inner layer of a pneumatic tire for providing enough power for microelectronic devices required for monitoring intelligent tires. Roundy and Tola66 used the characteristics of the eccentric pendulum and bistable non- linearities to improve the operating bandwidth of the system. The restoring force acting on the system at low speeds produces nonlin- ear bistable oscillations, while at high speeds, it turns into rigid oscil- lators. Yang et al.67proposed collision-induced resonance to excite the vibration of the piezoelectric cantilever and obtain the optimal deformation. As shown in Fig. 6(c), the device is rotated by the wind, and the ball hits the piezoelectric cantilever beam fixed on the cir- cumference of the inner surface of the rotating fan, and the cantilever beam vibrates to generate electricity. A rotational PEH for efficiently harvesting wind energy is developed.68The piezoelectric beam gen- erates electricity using the impact-induced vibration. As shown in Fig. 6(d), a PEH with a gull-wing structure composed of two typical nonlinear buckling bridges is designed.69The energy harvester can generate a peak open-circuit voltage of 20 V at a rotation frequency of 7.8 Hz. Micek et al.70,71presented a contact technique of stress monitoring in the rotating shaft. The Macro-Fiber Composite (MFC) patch is directly attached to the surface of the rotating shaft and generates a certain amount of electrical energy, which is sufficient to supply power only to the radio transmitter. da Silveira and Daniel72used piezoelectric mate- rials to collect the vibration energy of tilting pad journal bearings and provide power for condition monitoring sensors. The energy generated by the collector is up to 2.25 mW, which is enough to power the wireless sensor without compromising the stability of the bearing. 3. Summary of contact execution There are two types of contact excitation. Plucking excitation and impact excitation have their own characteristics. Toggle exci- tation means that the protrusion of the rotating object toggles the piezoelectric beam to vibrate or there is a protrusion plate around the rotating object, and the piezoelectric beam rotates to strike the plate. Impacting excitation means that an object hits the piezoelec- tric beam during its rotation. The operation principle is to pluck the piezoelectric bimorphs with plectra so that they produce electrical energy during the ensuing mechanical vibrations. By deflecting the piezoelectric bimorph based on the paddle and then quickly releasing the piezoelectric bimorph to vibrate unim- peded, the frequency upconversion based on the mechanical toggle is obtained. In the following oscillation cycle, part of the mechani- cal energy is converted into electrical energy. During the excitation process, the piezoelectric bimorph will be deflected by the pad- dle; when released due to loss of contact, the bimorph will vibrate freely at its resonance frequency, thereby generating the highest effi- ciency of electrical energy. It is best not to excite bimorphs at the same time, and the number of picks and bimorphs must be rela- tively prime. Steel paddles can greatly increase the energy produced by the energy harvester, thereby making plucking more powerful. The impact force is introduced by forming a cantilever polygon of piezoelectric bimorphs fixed on the circumference of the innersurface of the rotating fan. The impact point is carefully selected to use the first bending mode as much as possible to maximize the harvesting efficiency. The impact will excite resonance under any operating conditions. In addition, using shock-induced resonance to effectively excite the natural vibration mode of the piezoelectric cantilever can obtain the best deformation, which will facilitate the conversion of mechanical/electrical energy. It is found that if the impact frequency exceeds the critical value, the output power of the harvester decreases. A summary of the literature review on contact execution is listed in Table II. C. Magnetic coupling Magnetic coupling is a non-contact transmission method, which can convert the rotational movement in the external environ- ment into piezoelectric beam vibration. According to the structure and excitation principle, the magnetic coupling piezoelectric rotary energy harvesters that have been developed can be divided into two categories: a single cantilever and an array of cantilevers. 1. Single cantilever Manla et al.73proposed a method based on the non-contact PEH. This method deforms the piezoelectric beam to generate elec- tric energy due to centripetal force and magnetic force. Luong and Goo74introduced a PEH that uses the magnetic force of a small windmill to excite piezoelectric elements to vibrate. The wind forces the rotor to rotate, and the rotor rotates to generate an interactive magnetic force, thereby exciting the piezoelectric element to vibrate. The deformation of the piezoelectric element enables it to generate electric power. To transport the electric power from stationary to rotating equipment, a vibration-based electromagnetic microgenerator has been presented.75The piezoelectric disk is bonded near the fixed end to obtain the maximum deformation of the piezoelectric mate- rial, and the voltage output is much higher than other positions. Pillatsch et al.76,77proposed an energy harvesting technique that introduces an inertial device. Piezoelectric cantilever beams generate electrical energy by vibrating magnetically. The device can operate in a wide-frequency range and different directions. A nonlinear rotating PEH is proposed by adding a nonlin- ear magnetic force.78The distance between the two magnets has a great influence on the nonlinear rotating energy harvester. Nonlin- ear rotational energy harvesters have wider resonance bandwidth and lower resonance frequency than linear rotational energy har- vesters. Chen et al.79presented a Melnikov-theory-based method to explore the broadband mechanism and necessary conditions of the nonlinear rotating piezoelectric vibration energy harvesting system. Cheng et al.80presented an improved P-SSHI circuit with control- lable optimal voltage by using a voltage control strategy between the storage capacitor and the electric load. As shown in Fig. 7(a), a rotary piezoelectric frequency up- converting energy harvester was investigated.81Due to the magnetic interaction, the piezoelectric cantilever is excited to swing intermit- tently. When the angular velocity of the detection mass increases linearly, several peak voltages will appear in the signal output by the piezoelectric beam. In order to improve the performance of PEH, auxiliary components are added to the rotary PZT frequency upcon- version energy harvester.82By appropriately adjusting the magnet Rev. Sci. Instrum. 92, 041501 (2021); doi: 10.1063/5.0039730 92, 041501-9 Published under license by AIP PublishingReview of Scientific InstrumentsREVIEW scitation.org/journal/rsi TABLE II. Summary of literature review of contact execution. References Mechanism Material Resonant frequency (Hz) Power Voltage (V) Load resistance Pozzi and Zhu53 Plucking excitationPZT-5H 320 5–7 mW 24 1 M Ω Pozzi et al.54PZT-5H 30 mW Priya et al.55PZT 10 5.75 mW 6.7 k Ω Janphuang et al.56PZT 3.2 12 μW 7.8 K Ω Baiet al.57PZT 0.27 mW 100 k Ω Fu and Liao58PZT Jung et al.60PZT 305 μW 2.7 30 k Ω Chen et al.59PZT 1.31 1.71 mW 200 k Ω Zhou et al.61PZT-4 140 1 mW 10 100 k Ω Tang et al.62PVDF 5.3 46.2 mW 15.5 Kim et al.63PZT 24.5 17 mW 50 10.8 k Ω Makki and Pop-Iliev64 Impact excitationPZT 6.5 mW 10 42 k Ω Varela and Sierra65PVDF 0.8 W 27.5 1 k Ω Roundy and Tola66PZT 10 μW Yang et al.67PZT 200 613 μW 20 k Ω Zhang et al.68PVDF 82 2566.4 μW 160.2 Yang et al.69PZT 7.8 0.4 mW 20 30 k Ω Grzybek and Micek70MFC 20 7.44 μW 56.17 k Ω Micek and Grzybek71MFC da Silveira and Daniel72PZT 43 2.25 mW separation distance in the auxiliary component, the generated power can be increased. Zhang et al.83analyzed the effect of stochastic resonance on the efficiency of PEH. When the wheel rotates at a variable speed, PEH is stimulated by environmental noise, and gravity acts as a peri- odic modulation force, which will stimulate the stochastic resonance of the piezoelectric cantilever. Under a certain vehicle speed range, stochastic resonance can effectively optimize the performance of the energy harvester. Due to the centrifugal effect caused by the rotation, the restoring force of the piezoelectric cantilever may increase.84 When the rotation frequency exceeds the identifiable boundary fre- quency, the system will be converted to a monostable barrier system. The bistable system can provide enough kinetic energy so that the cantilever maintains its high-energy orbital oscillation. As shown in Fig. 7(b), a blade rotates on a shaft carrying a magnet and sweeps the tip of the piezoelectric layer, causing a serial buckling effect result- ing in energy generation.85Xieet al.86,87proposed a rotating energy collector that uses piezoelectric bistable bending beams to collect low-speed rotating energy. The harvester consists of a piezoelectric bending beam with a central magnet and a pair of rotating mag- nets with opposite poles mounted on a rotating body. In addition, a double-attracting magnet is used to overcome the suppression phe- nomenon at higher frequencies. Wu et al.88proposed and developed PEH excited by magnetic force for rotating machinery applications. For the piezoelectric energy harvesting system, different directions and configurations of magnetic force, attractive force, and repulsive force are designed and optimized. When the excitation frequency is close to the natural frequency of the piezoelectric cantilever and the duty cycle is relatively large, the alternating power of attrac- tion and repulsion can achieve a maximum power of 1.23 mW.Mei et al.89investigated the performance of a tri-stable PEH in rotational motion. 2. Cantilever array Karami et al.90proposed a compact wind collector in which the rotation of the blade causes the piezoelectric beam to vibrate. Through the interaction of the permanent magnet at the end of the piezoelectric beam and the permanent magnet rotating with the blade, a bistable piezoelectric cantilever beam is promoted. Ramezanpour et al.91proposed a device consisting of a rotating proof mass and eight piezoelectric bimorph beams. The magnet mounted on the rotating pendulum interacts with the tip magnet of the piezoelectric beam. When the pendulum passes through the piezoelectric beam, the free vibration of the piezoelectric beam gen- erates energy. A compact and low-profile energy harvester designed to be worn on the outside of the knee joint is presented.92The har- vester was installed on a knee simulator, and the measured rectified electric energy during walking and running was about 53 mW and 72 mW, respectively. A new contactless piezoelectric wind energy harvester has been proposed as a battery charger,93given in Fig. 7(c). Due to elec- tromechanical damping, high-frequency mechanical vibration, and electromagnetic nonlinearity, a combined system with multiple lay- ers can give a complex response. Since the three layers have three different natural frequencies, each layer can produce a larger ampli- tude close to its natural frequency, widening the frequency band- width of the structure. Fang et al.94proposed a music-box-like extended rotational plucking energy harvester with multiple piezo- electric cantilevers. The plucking frequency is the same as the rota- tion frequency, thereby expanding the operating frequency range Rev. Sci. Instrum. 92, 041501 (2021); doi: 10.1063/5.0039730 92, 041501-10 Published under license by AIP PublishingReview of Scientific InstrumentsREVIEW scitation.org/journal/rsi FIG. 7. (a) Schematic diagram of a nonlinear PEH. Reproduced with permission from Ramezanpour et al. , “Electromechanical behavior of a pendulum-based piezoelectric frequency up-converting energy harvester,” J. Sound Vib. 370, 280–305 (2016). Copyright 2016 Springer Nature. (b) New wind PEH introducing magnetic force. Reproduced with permission from Çelik et al. , “Experimental and theoretical explorations on the buckling piezoelectric layer under magnetic excitation,” J. Electron. Mater. 46(7), 4003–4016 (2017). Copyright 2017 Springer Nature. (c) Schematic diagram of a nonlinear array PEH. Reproduced with permission from Bouzelata et al. , “Mitigation of high harmonicity and design of a battery charger for a new piezoelectric wind energy harvester,” Sens. Actuators, A 273, 72–83 (2018). Copyright 2018 Elsevier. (d) Prototype of wind PEH. Reproduced with permission from Na et al. , “Wind energy harvesting from a magnetically coupled piezoelectric bimorph cantilever array based on a dynamic magneto-piezo-elastic structure,” Appl. Energy 264, 114710 (2020). Copyright 2020 Elsevier. and expanding the power bandwidth. Rashidi et al.95proposed a technique for energy harvesting using multiple magnetic actuators and piezoelectric beams. One end of the piezoelectric beam is fixed on the central axis of the fixed wheel, and a small magnet is bonded to the free end. A small magnet is bonded to the concentric wheel fixed on the working shaft. When the working shaft drives the mov- ing wheel, the magnets on the moving surface attract the magnets on the piezoelectric beam so that the piezoelectric cantilever beam vibrates to generate electrical energy. Zhao et al.96proposed a rotating piezoelectric wind energy col- lection method using magnetic coupling and force amplification mechanism. Through the symmetrical reverse magnetic arrange- ment, the resistance torque has been greatly reduced, while the effective force has been maximized. It can produce high and rel- atively stable open-circuit voltage from low wind speed to high wind speed. Na et al.97proposed a wind energy collector based on a magneto-piezo-elastic structure, as shown in Fig. 7(d). The device consists of a power generating stator, wind rotor, and tur- bine blades. The permanent magnets on the free end mass of the vertically arranged piezoelectric cantilever interact with the rotating permanent magnets fixed on the rotor top plate to generate electrical energy.3. Summary of magnetic coupling The magnetic coupling mechanism refers to the nonlinear mag- netic force that is attached to the end of the piezoelectric beam and interacts with the fixed magnet. By changing the distance between the fixed magnet and the end magnet along the cantilever beam axis, hard spring and soft spring systems can be realized, respec- tively. The resonance frequency of the piezoelectric energy harvester is adjusted by the interaction force between the magnets. The non- linearity caused by the magnetic force also plays an important role in broadening the frequency band and improving the energy capture efficiency. The magnetically coupled bistable piezoelectric energy harvester can produce large-scale, wide-frequency periodic or non- periodic vibration under weak excitation, thereby obtaining large structural deformation in a non-resonant state and improving power generation efficiency. The energy harvester integrating multiple can- tilever beam arrays can provide continuous wide bandwidth and realize multi-peak energy harvesting. Multimodal energy harvesting technology uses cantilever arrays of different sizes. By adjusting the number and size of the cantilevers as needed, this technology can better control the final frequency range of the harvester. Therefore, the output power is the sum of the output power provided by each Rev. Sci. Instrum. 92, 041501 (2021); doi: 10.1063/5.0039730 92, 041501-11 Published under license by AIP PublishingReview of Scientific InstrumentsREVIEW scitation.org/journal/rsi cantilever beam. A summary of the literature review on magnetic coupling is listed in Table III. D. Hybrid systems Most energy collection uses a single power generation princi- ple, which has the defects of the contracted working frequency band, low energy collection efficiency and low output energy density, and insufficient collection of energy. The trend of using hybrid energy to increase the production energy of a single power generation sys- tem is increasing. The hybrid scheme effectively combines two or more generator mechanisms to improve energy collection efficiency and output voltage. The hybrid energy harvesting system combines piezoelectric, electromagnetic, and triboelectric conversion meth- ods. By applying these hybrid solutions to rotate environments, the performance of the energy harvester can be further improved. 1. Piezoelectric and electromagnetic hybrid system Hybrid energy harvesting systems usually combine piezoelec- tric and electromagnetic mechanisms for energy harvesting. Cooley and Chai98studied the energy collected from the translational vibra- tion of a rotating host system. The energy harvesting device includes mass detection, elastic structure, electromagnetic, and piezoelectric energy harvesting elements. The vibration and rotation of the host system work together, and there are multiple resonance points in the energy harvesting device. Rotary vibration energy harvesters usually have two resonances: one for fixed speed and one for vari- able speed. The electromagnetic energy harvesting device harvests the maximum average power at a specific excitation frequency. Thepiezoelectric energy harvesting device has two local average power maxima. Zhao et al.99proposed a hybrid piezoelectric and electromag- netic wind power generation device incorporating magnetic cou- pling and force amplification mechanism, as shown in Fig. 8(a). With a symmetrical opposite magnetic arrangement, the reluctance torque is significantly reduced and the effective magnetic force is increased. The magnetic force can be further amplified by the curved structure and applied to the piezoelectric layer more uniformly. This waterproof hybrid system wind energy generator can not only obtain higher output power but also has great flexibility in practical appli- cations. Vargas and Tinoco100proposed an energy harvesting device using piezoelectric induction technology. The device consists of a voice coil motor extracted from a hard drive. The dynamic excita- tion in the system is generated by the pendulum motion of a rigid- flexible hybrid arm that generates energy by rotating and cutting electromagnetic coils and piezoelectric cantilever beam strain. Lal- lart and Lombardi101proposed a hybrid nonlinear interface combin- ing both piezoelectric and electromagnetic effects for energy harvest- ing purposes. To overcome these main drawbacks when considering hybrid piezoelectric–electromagnetic energy harvesters, this paper proposes a hybrid energy harvesting electrical interface based on the coupling of these two energy conversion mechanisms. This produces an immediate enhancement of the final energy that can be extracted without adding significant complexity to the systems in terms of electrical components and space. Shi et al.102proposed a multi-directional vibration piezoelectric–electromagnetic composite energy harvester. The interaction between the magnet on the cantilever and the magnet on TABLE III. Summary of literature review of magnetic coupling. References Mechanism MaterialResonant frequency (Hz) Power Voltage Load resistance Luong and Goo74 Single cantileverPZT 2 mW 17.5 V 100 k Ω Manla et al.73PZT-5A 0.2 μW–3.5 μW Yanet al.75PZT-5H 9.2 23.3 mW 185.1 mV Pillatsch et al.76PZT 2 5.2 μW 2.5 V 120 k Ω Pillatsch et al.77PZT 2 43 μW 150 k Ω Chen et al.79PZT 200 mW Zhang et al.83PZT 5–9.2 0.081 mW 4.5 V Celik et al.85PZT 4.76 6 mW 6 V 10 M Ω Xieet al.86PZT 1–14 6.91–48.01 μW 7.22 V 110 k Ω Xieet al.87PZT 6 1450 μW 230 k Ω Wuet al.88PZT 24–34.1 1.23 mW 7.17 V 40 k Ω Zhang et al.84PZT 61 μW 6 V 150 k Ω Meiet al.89PZT 4–73 26–105 μW 18 V Ramezanpour et al.91 Cantilever arrayPZT 3 0.93 mW 40 V 2 M Ω Pozzi92PZT-5H 72 mW 3.3 V 74.9 k Ω Bouzelata et al.93PZT 13 40 V Fang et al.94PZT 19.5 mW 55 k Ω Rashidi et al.95PZT 3.9 15 V Zhao et al.96PZT-5H 10 mW 23 V Naet al.97PZT-5H 24.95 mW 30 k Ω Rev. Sci. Instrum. 92, 041501 (2021); doi: 10.1063/5.0039730 92, 041501-12 Published under license by AIP PublishingReview of Scientific InstrumentsREVIEW scitation.org/journal/rsi FIG. 8. (a) Piezoelectric and electromagnetic hybrid system. Reproduced with permission from Zhao et al. , “A water-proof magnetically coupled piezoelectric-electromagnetic hybrid wind energy harvester,” Appl. Energy 239, 735–746 (2019). Copyright 2019 Elsevier. (b) Piezoelectric and triboelectric hybrid system. Reproduced with permission from Zhao et al. , “Hybrid piezo/triboelectric nanogenerator for highly efficient and stable rotation energy harvesting,” Nano Energy 57, 440–449 (2019). Copyright 2019 Elsevier. (c) Piezoelectric, electromagnetic, and triboelectric hybrid system. Reproduced with permission from Rahman et al. , “Natural wind-driven ultra-compact and highly efficient hybridized nanogenerator for self-sustained wireless environmental monitoring system,” Nano Energy 57, 256–268 (2019). Copyright 2019 Elsevier. the rotating mass drives the piezoelectric cantilever beam to vibrate. Through the combination of electromagnetic power generation and piezoelectric power generation, more vibration energy can be harvested. By forming a force–electricity–magnetic multi-field nonlinear coupling system, a higher operating frequency band can be achieved. 2. Piezoelectric and triboelectric hybrid system Chung et al.103proposed a handheld gyroscope energy collec- tor. The gyro generator is composed of a flywheel triboelectric nano- generators (TENG) and a sleeve generator. The casing generator can generate output through triboelectric and piezoelectric effects. During operation, electrical energy is generated by the movement of equipment caused by rotation, vibration, and centrifugal force. Zhao et al.104developed a piezoelectric and triboelectric hybrid sys- tem for efficiently collecting mechanical rotational energy. As shown in Fig. 8(b), the composite piezoelectric and triboelectric devices are mounted on the upright posts of the frame as a “stator,” while the rotatable shaft with an acrylic blade acts as a “rotor.” Three wind cups were added to the top of the rotatable shaft to collect wind energy. When subjected to the action of a “rotor” wiper blade, an equivalent contact separation process and external strain are applied to the hybrid device. The conversion of rotating mechanical energy into electrical energy is achieved.3. Piezoelectric, electromagnetic, and triboelectric hybrid system Koh et al.105proposed a 3D active inertial sensor for multi- dimensional energy harvesting and rotational inertial detection. It is composed of magnetic buckyballs encapsulated in a 3D printed spherical shell. The inner wall of the shell is made of multiple lay- ers of PTFE, PVDF, and Al films, and the outside is wound with electromagnetic coils. It can collect energy from various energy sources through piezoelectric, electromagnetic, and triboelectric hybrid mechanisms. Ma et al.106proposed a hybrid friction elec- tromagnetic voltage electric hybrid energy harvester based on rota- tional motion. Three different generator components are integrated through the shell and the acrylic cylinder to form a hybrid energy collector. The hybrid energy collector is mounted on the turntable to achieve rotation. With the aid of a rectifier circuit and a capac- itor, the rotation speed is 45 rpm, and the hybrid energy harvester can supply power to the temperature and humidity meter and light up 111 blue LEDs. Rahman et al.107proposed a small windmill hybrid nanogenerator based on three conversion mechanisms of tri- boelectric nanogenerator, piezoelectric nanogenerator, and electro- magnetic generator. As shown in Fig. 8(c), the wind rotation system is impacted by the wind, frictional electric nano-generator, piezo- electric nano-generator, and electromagnetic generator that gen- erate electricity at the same time. Developed power management Rev. Sci. Instrum. 92, 041501 (2021); doi: 10.1063/5.0039730 92, 041501-13 Published under license by AIP PublishingReview of Scientific InstrumentsREVIEW scitation.org/journal/rsi TABLE IV. Summary of literature review of the rotation-based hybrid power generator. References Mechanism MaterialsResonant frequency (Hz) Power Voltage Currents Load resistance Zhao et al.99 Piezoelectric andPZT/copper coil36 1724.1/ 1239.0 μW17.4/0.53 V 300/400 Ω Vargas and Tinoco100 triboelectricPZT/copper coil0–15 Shiet al.102PZT/copper coil42 1.31 mW 100 mV 100/80 Ω Chung et al.103 Piezoelectric andP(VDF- TrFE)/PTFE200 0.581 mW 90 V 11 μA 20 M Ω Zhao et al.104 triboelectricP(VDF- TrFE) and PET/Al and PTFE/Au69 10.88 mW 210 V 395 μA 1 and 10 M Ω Koh et al.105 Piezoelectric andPVDF and PTFE/Al and copper coil3 0.12/0.19, 0.22/0.72, and 13.8/22.4 nW1.2/1.7 V, 3.7/5.7 V, and 8.6/10.8 mV116/132 nA, 90/150 nA, and 40/37.5 μA10 MΩ, 12/15 MΩ, and 20Ω Maet al.106 triboelectric andPZT and PTFE and copper coil0.75 6.37, 712.3, and 30.9 μW51 and 250 V 21 mA 50 M Ω, 20 MΩ, and 200Ω Rahman et al.107electromagnetic PVDF and PET/Cu PTFE/Al and copper coil65, 360, and 23.2 V1.38, 1.67, and 268.6 mW135μA, 128 μA, and 87 mA330 kΩ, 10 MΩ, and 180Ω circuits; wireless sensor units with temperature, humidity, and pres- sure sensors; and smartphone applications are used to study their effectiveness. 4. Summary of hybrid systems The output voltage of the piezoelectric energy harvester is higher, but the output current is lower, and its output impedance is capacitive. The output voltage of the electromagnetic energy har- vester is lower, but the output current is higher, and its output impedance is inductive. Electromagnetic energy harvesting can pro- vide enough current as of the best current source with suitable out- put resistance. Triboelectric energy harvesters have the advantages of high generation voltage, high power density, lightweight, small size, flexibility and shape flexibility, and compatibility. Compared with a single energy harvesting mechanism, the hybrid energy har- vester can achieve both high voltage and high current at the same time, which can increase the efficiency of energy harvesting. Hybrid energy harvesting mechanisms can provide high output voltage and current as well as large average power and power density. Hybrid energy harvesters combine the advantages of multiple energy har- vesters for greater utility and greater flexibility. Table IV lists a literature review of hybrid generators based on rotation. IV. COMPARISON OF PERFORMANCE This section compares energy harvesting techniques based on different excitation methods. Energy harvesting techniques based onrotational motion are classified as inertial excitation, contact excita- tion, magnetic coupling, and hybrid systems, each of which has its advantages and disadvantages. Table V evaluates different energy harvesting technologies based on the categories presented here, highlighting the advantages and disadvantages of each technology. Rotational motion results in centrifugal force, which causes the axial load on the beam and alters the resonant frequency of the sys- tem. The centrifugal force and inertial force are used to drive the piezoelectric element to vibrate in the rotary motion, which is suit- able for the low-speed state. In a high-speed rotating environment, the inertial force generated by the movement is relatively large, which can easily break the piezoelectric element. Contact excitation is performed by a rotating object touch- ing a piezoelectric beam, which converts the rotational motion into vibration and further into electrical energy. Rotational motion is converted into vibration by means of contact, which is simple, not limited by the layout, does not need to consider the influence of the magnetic field, and can increase the excitation frequency through multiple contact points. However, there is more energy loss and the device is easier to wear and damage in mechanical conversion through contact. Magnetic coupling technology is a method to increase the power output of the vibration energy collector while increasing the working range of the device. This has been done by the introduc- tion of nonlinear magnetic stiffness through levitated magnet sys- tems. Systems with bistable nonlinearities are more applicable than monostable nonlinearities because they are better suited to random Rev. Sci. Instrum. 92, 041501 (2021); doi: 10.1063/5.0039730 92, 041501-14 Published under license by AIP PublishingReview of Scientific InstrumentsREVIEW scitation.org/journal/rsi TABLE V. Summary of the presented techniques. Method Advantages and disadvantages Inertial excitationThere is a higher charge output in low-frequency environment. Not suitable for high-speed rotation environment. Contact executionThe piezoelectric ceramic sheet deforms greatly, and the generated charge is high. There is energy loss, and piezoelectric elements are more susceptible to wear and damage. Magnetic couplingThe working range increases while increasing the power output of the vibration energy collector. The introduction of magnets can easily cause magnetic pollution. Hybrid systemsIncreased power harvested; slight bandwidth enhancement More complex circuit design; sometimes results in high damping vibration, which is realistic for the actual operating conditions of the equipment. Magnetic tuning can be used to increase the band- width of the device by applying magnetic interactions with differ- ent gap pitches to effectively change the natural frequency of the system. However, the presence of magnetic field after the introduc- tion of magnets interferes with the surrounding environment. The force transmission process is prone to hysteresis. The efficiency is relatively low compared to contact excitation. Hybrid schemes show a slight increase in operating frequency compared to single systems; however, the power output of such a design is improved using hybrid systems. Hybrid solutions combine multiple energy harvesting technologies to expand the operating fre- quency range of energy harvesting devices and improve energy con- version efficiency. However, this design requires all devices to be in an optimal condition to operate effectively, and the circuit design becomes more complex. V. FUTURE DIRECTIONS FOR ENERGY HARVESTING IN ROTATIONAL MOTION Although people have conducted a lot of research on rota- tional energy collection methods and structures based on various mechanisms, there are still several problems to be solved, such as improving the collection efficiency, power density, coupling per- formance, integration performance, expanding the frequency range, and reducing costs. Many designs are only in the experimental stage, and the energy collected is not enough to maintain the stable oper- ation of electronic devices. In view of the above-mentioned prob- lems, major improvements can be made to the existing or proposed energy harvesting models or novel harvesting methods to achieve significantly higher output power requirements. From the research progress of rotational energy collection technology based on various mechanisms, the following aspects are worthy of study. A. Research on new rotational energy collection structure 1. Development of different shapes of piezoelectric elements Rectangular piezoelectric elements are widely used for piezo- electric energy harvesting due to their simple design and ease of fabrication. However, the main disadvantage of using this shape is the very low average strain of the piezoelectric element. The strain is improved by developing new designs of piezoelectric elementshapes. Different shapes of piezoelectric elements may introduce multi-peak energy capture and extend the frequency bandwidth system. 2. Development of frequency-controllable and wide-band rotational energy collection structure The rotational frequency of the motion source in the environ- ment varies. The study of the piezoelectric energy harvesting device with controllable frequency and high-frequency band is done so that it can be set to match the rotational frequency of the motion source in the environment and make the piezoelectric energy harvesting device in a resonant state to improve the efficiency of energy capture. 3. Development of miniaturized rotational energy collection structure Research on miniaturization and integration of rotating energy harvesting structures is carried out. The design of micro- and nano- scale rotational energy harvesting devices is made to facilitate the integration and mass production of MEMS. 4. Develop a rotating energy harvesting structure for specific application As theoretical studies of energy harvesting devices mature, they will eventually lead to practical engineering applications. Based on the specific application context, the broadband piezoelectric energy harvesters are optimized to meet the application requirements by combining the characteristics of the motion source and the load. B. Development of new high-strength multifunctional materials Conventional piezoelectric materials have the disadvantages of low energy conversion efficiency, low energy density, poor cou- pling performance, and low strength. Therefore, it is important to improve and research new preparation processes to prepare piezo- electric materials with high electromechanical conversion perfor- mance, new composite materials with high magnetoelectric coupling performance, and other new functional materials to improve power generation performance. 1. Lowering the sintering temperature of piezoelectric materials The preparation methods to reduce the sintering temperature of piezoelectric materials to less lead volatilization are carried out. Rev. Sci. Instrum. 92, 041501 (2021); doi: 10.1063/5.0039730 92, 041501-15 Published under license by AIP PublishingReview of Scientific InstrumentsREVIEW scitation.org/journal/rsi The sol-gel method, hot pressing method, ultra-fine powder prepa- ration method, and the addition of the co-solvent method are used to reduce the sintering temperature of piezoelectric materials while trying to increase their piezoelectric constants. 2. Development of ternary and multi-system piezoelectric ceramics Piezoelectric ceramics are constantly being improved and per- fected. Based on lead zirconate titanate, ternary and quaternary piezoelectric ceramics with multiple elements have also been devel- oped. The multi-system piezoelectric ceramics can make up for the defects of the low-system ceramics with single performance and have the advantages of comprehensive piezoelectric, electromechanical, and mechanical properties, and the applications are more extensive. 3. Development of new piezoelectric composite materials The currently used piezoelectric ceramics have many draw- backs, which limit the wide application. In contrast, new synthetic piezoelectric composites and piezoelectric fiber materials have supe- rior electrical performance parameters. Piezoelectric composites have the following trends to develop connection types: according to the distribution state of piezoelectric ceramics and polymer in the composite, piezoelectric composites can be divided into ten connec- tion types. They are noted as 0–0, 0–1, 0–2, 0–3, 1–1, 1–2, 1–3, 2–2, 2–3, and 3–3 types, respectively. It is generally agreed that the first word represents the piezoelectric phase and the second number rep- resents the non-piezoelectric phase. The processing of piezoelectric composites is continuously improved to enhance the stability, com- patibility, and service life of piezoelectric materials, improve energy harvesting efficiency, and reduce production costs. 4. Development of high Curie temperature piezoelectric ceramics Atomic energy, energy, aerospace, metallurgy, petrochemical industry, and many other industrial and scientific research sectors urgently need to be able to work at higher temperatures of electronic devices. The Tc of PZT-based piezoelectric ceramics is generally 300–360○C, which cannot meet the needs of some applications, and the development of some piezoelectric ceramics with excellent high Curie temperature Tc has become a hotspot for research. C. Research on low-power and high-efficiency energy harvesting and circuits Most of the current work is focused on the study of energy har- vesting structures, while relatively little research has been done on energy recovery circuits. 1. Development of more efficient energy capturing circuits The AC power generated by the voltage energy harvesting device is easily influenced by the vibration source and its own struc- ture. It is difficult to supply power to the load directly. In order to provide stable and continuous power to the load, it is necessary to design and develop energy harvesting circuits with higher energy conversion and harvesting efficiency.2. Reduced energy consumption in energy harvesting circuits The energy captured by piezoelectric energy traps is small, typi- cally in the milliwatt range. There is a loss when the energy collected by the energy harvesting structure passes through the harvesting cir- cuit. Therefore, it is beneficial to study the low energy consumption and high-efficiency energy harvesting circuit to improve the output performance of the whole harvesting structure. Low energy con- sumption storage circuits and high-energy density energy storage elements are developed to improve the storage efficiency and make the captured energy more functional for the load. 3. Development of a post-processing circuit based on the cooperative energy capture of multiple piezoelectric oscillators For a device structure with multiple piezoelectric elements, multiple full-bridge rectifier circuits need to be connected, which not only increases the energy consumption of the circuit itself but also increases the complexity of the circuit. Researchers have improved and optimized the dual synchronous switching inductor circuit. The researchers proposed a post-processing circuit applicable to mul- tiple piezoelectric oscillators to coordinate energy capture, solving the problem of difficult processing of coupled electrical signals and realizing synchronous collection and storage of electrical energy. D. Research on the new mechanisms, new theory, and new technology of compound energy harvesting Researchers at home and abroad have proposed a variety of rotational energy recovery methods: thermoelectric conversion, photovoltaic conversion, electromagnetic conversion, electrostatic conversion, piezoelectric conversion, magnetostrictive conversion, magnetoelectric conversion, and friction nanopower generation. Hybrid energy harvesting technology can recover from the short- comings of a single energy harvesting mechanism. The prior art usually focuses on a certain energy harvesting mechanism, while ignoring other forms of energy harvesting. Regular in-depth discus- sion and exploration of new mechanisms and methods of hybrid energy collection technology are carried out. VI. CONCLUSION In recent years, scholars have made many research studies on the collection of rotational energy. Piezoelectric materials can be used to convert rotational energy into electrical energy. This article reviews the piezoelectric energy harvesting technology in rotational motion. It introduces the basic principles of piezoelec- tric energy harvesting, the vibrational modes of piezoelectric ele- ments, and the materials of piezoelectric elements. This Review presents a classification of the current techniques that collect energy through rotational motion. There is a lot of literature on collecting energy through rotational motion, and our classification includes the four categories: inertial excitation, contact execution, magnetic coupling, and hybrid systems. This classification method shows energy harvesting devices that collect rotational energy under dif- ferent excitation conditions. An overview of each technology is made, and then, a detailed analysis is carried out. The examples of the current research illustrate each technology and summarize the Rev. Sci. Instrum. 92, 041501 (2021); doi: 10.1063/5.0039730 92, 041501-16 Published under license by AIP PublishingReview of Scientific InstrumentsREVIEW scitation.org/journal/rsi advantages and disadvantages of each technology. Finally, this paper discusses the future direction and goal of improving energy harvest- ing technology. ACKNOWLEDGMENTS This work was supported by the NSFC of China (Grant No. 51805489), the National Study Abroad Fund of China (Grant No. 202008220173), and the Project of Jilin Provincial Education Department: Research on River Hydrological Monitoring Device Based on ADCP Sensor under Grant jijiaokehezi (2014) No. 140. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1S. Bahrami, M. H. Amini, M. Shafie-khah, and J. P. S. Catalao, “A decentralized electricity market scheme enabling demand response deployment,” IEEE Trans. Power Syst. 33(4), 4218–4227 (2018). 2S. 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5.0038883.pdf

J. Appl. Phys. 129, 093103 (2021); https://doi.org/10.1063/5.0038883 129, 093103 © 2021 Author(s).From single-particle-like to interaction- mediated plasmonic resonances in graphene nanoantennas Cite as: J. Appl. Phys. 129, 093103 (2021); https://doi.org/10.1063/5.0038883 Submitted: 27 November 2020 . Accepted: 12 February 2021 . Published Online: 04 March 2021 Marvin M. Müller , Miriam Kosik , Marta Pelc , Garnett W. Bryant , Andrés Ayuela , Carsten Rockstuhl , and Karolina Słowik COLLECTIONS Paper published as part of the special topic on Plasmonics: Enabling Functionalities with Novel Materials ARTICLES YOU MAY BE INTERESTED IN Coherent diffraction rings induced by thermal–mechanical effect of a flexible Dirac semimetallic composite structure Journal of Applied Physics 129, 093102 (2021); https://doi.org/10.1063/5.0035647 Modal polarization analysis using Fourier-rectangular function transform in a cylindrical plasma Journal of Applied Physics 129, 093301 (2021); https://doi.org/10.1063/5.0037352 Tunable low-frequency and broadband acoustic metamaterial absorber Journal of Applied Physics 129, 094502 (2021); https://doi.org/10.1063/5.0038940From single-particle-like to interaction-mediated plasmonic resonances in graphene nanoantennas Cite as: J. Appl. Phys. 129, 093103 (2021); doi: 10.1063/5.0038883 View Online Export Citation CrossMar k Submitted: 27 November 2020 · Accepted: 12 February 2021 · Published Online: 4 March 2021 Marvin M. Müller,1,a) Miriam Kosik,2,b) Marta Pelc,2,3,4 Garnett W. Bryant,5,6 Andrés Ayuela,3,4 Carsten Rockstuhl,1,7 and Karolina S łowik2 AFFILIATIONS 1Institute of Theoretical Solid State Physics, Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany 2Institute of Physics, Nicolaus Copernicus University in Toru ń, Grudziadzka 5, 87-100 Toru ń, Poland 3Donostia International Physics Center (DIPC), Paseo Manuel Lardizabal 4, 20018 Donostia-San Sebastián, Spain 4Centro de Física de Materiales, CFM-MPC CSIC-UPV/EHU, Paseo Manuel Lardizabal 5, 20018 Donostia-San Sebastián, Spain 5Joint Quantum Institute, University of Maryland and National Institute of Standards and Technology, College Park, Maryland 20742, USA 6Nanoscale Device Characterization Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA 7Institute of Nanotechnology, Karlsruhe Institute of Technology (KIT), 76021 Karlsruhe, Germany Note: This paper is part of the Special Topic on Plasmonics: Enabling Functionalities with Novel Materials. a)Author to whom correspondence should be addressed: marvin.mueller@kit.edu b)Electronic mail: mkosik@doktorant.umk.pl ABSTRACT Plasmonic nanostructures attract tremendous attention as they confine el ectromagnetic fields well below the di ffraction limit while simultaneou sly sustaining extreme local field enhancements. To fully exploit these prope rties, the identification and classi fication of resonances in such nanos truc- tures is crucial. Recently, a novel figure of merit for resonance classification has been proposed [Müller et al., J. Phys. Chem. C 124, 24331 –24343 (2020)] and its applicability was demonstrated mostly to toy model sys tems. This novel measure, the energy-based plasmonicity index (EPI), characterizes the nature of resonances in molecular nanostructures. The EPI distinguishes between either a single-particle-like or a plasmonic nature of resonances based on the energy space coherence dynamics of the excitation. To advance the further development of this newly established measure, we present here its exemplary application to chara cterize the resonances of graphene nanoantennas. In particular, we focus on resonances in a doped nanoantenna. The structure is of interest, as a c onsideration of the electron dynamics in real space might suggest a plasmonic nature of selected resonances in the low doping limit but our ana lysis reveals the opposite. We find that in the undoped and moderately doped nanoantenna, the EPI classifies all emerging resonances as predominan tly single-particle-like, and only after doping the structure heavily , the EPI observes plasmonic response. Published under license by AIP Publishing. https://doi.org/10.1063/5.0038883 I. INTRODUCTION The field of plasmonics experienced huge interest in the past two decades2–6with a recent shift of focus toward related quantum pro- cesses and applications at the nanoscale.7–9The possibility of confin- ing and enhancing electromagnetic fields5,10–12attracts attention not only for fundamental research reasons but also due to many potentialapplications in plasmonic sensing, 13–15photodetection,16–19medi- cine,20,21optical metamaterials,22–24and single photon sources.25,26Graphene supports intrinsic tunable plasmons and, therefore, is a well-suited platform for exploring and exploiting plasmonic phenomena.22,27–29Recent progress in nanostructure fabrication allows us to produce graphene flakes consisting of only a fewhundred atoms 30,31that support a plasmonic response at near infrared frequencies. Since a classical description based on the Drude model fails to properly predict the properties of metallic nanoantennas with a size below 10 nm,32–34more accurate attemptsJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 093103 (2021); doi: 10.1063/5.0038883 129, 093103-1 Published under license by AIP Publishing.to model such systems should account for fine details at atomistic scale and are mostly based on quantum mechanical methods. Among these, one can find density functional theory (DFT),35–38 the tight binding (TB) model,39–43or quantum fluid dynamics.44,45 Graphene nanoantennas support both single-particle-like res- onances and plasmonic ones, which leads to the important question how to identify the nature of resonances in such nano- structures.46,47This complex issue has received substantial interest in the past few years and several works have been devoted toaddressing it. The collective charge density oscillation in real spaceis typically considered as the smoking gun to classify a specific res- onance as plasmonic in nature. However, as will be shown below, the real space analysis cannot be the sole basis for decisions on thenature of the resonance. Therefore, it continues to be a major chal-lenge in the field of plasmonics to decide whether a specific excita-tion is plasmonic or not. Associated with that characterization is, of course, the question how to actually define a plasmonic excitation in nanoscaled systems. This contribution addresses these questionsand aims to provide an answer. Early studies found that in graphene nanoantennas, a single extra electron from doping can switch on infrared plasmons that were absent from the structure before doping and that adding further electrons causes a significant shift in plasmon frequency. 33 Studying how the spectral position of a resonance changes with theaddition of doping electrons is one of the clues that can serve as a guide to determine its character. Another approach to this problem relies on scaling of the elec- tron –electron interaction strength in simulations to find how the frequency of a given resonance depends on this variation. 35,48,49In this approach, the excitations that show little dependence on the scaling parameter are classified as single-particle like, while those which blue shift considerably with the increase of the Coulombinteraction strength are deemed plasmonic. Bryant and Townsend have used real space and real-time time-dependent DFT to examine jellium spheres. They were able to distinguish two different types of behavior that the occupations of electronic states in energy basis of a given resonance can show. Thefirst one was called “sloshing. ”It is a pattern of oscillatory move- ment between shells above and below the Fermi energy. Thesecond was called “inversion, ”which is associated with a continu- ous transition of electrons from occupied to unoccupied states. 46,50 The ratio of “sloshing ”and ”inversion ”that is linked to a given res- onance can be used as a clue to determine its nature. Otherattempts to classify resonances were based on a bi-local auto- correlation function 45or a model in which the electrons were con- fined to a potential box.51 Among the numerous studies conducted on this topic, a line of research focused on the construction of a universal figure ofmerit for resonance classification. In 2011, Yasuike et al. introduced the ”collectivity index ”(CI) to characterize plasmonic behavior by the quantification of how many single-particle transitions contrib-ute to an excitation. 52In 2017, Fitzgerald et al. refined this approach and proposed to build an index as the product of the CIand the dipole strength of an excitation. 53In 2016, Bursi et al. pro- posed the plasmonicity index (PI) to characterize and quantify plasmonic behavior based on how much the induced potential in ananostructure deviates from a neutral case. 36Shortly afterward,Zhang et al. defined a dimensionless, but unnormalized metric called generalized plasmonicity index (GPI) to distinguish plas- mons from single-particle-like excitations based on a similaraspect. 47Both of these measures can be determined using the real space charge distribution of the structure ’s resonances as the only input. However, as we outline below, there exist resonances of single-particle-like nature that reveal strong dipolar character even innon-interacting systems. On the other hand, Pines and Bohm state in their pioneering work in 1952 that charge oscillations mayhave individual and collective components, where the latter emergeonly in systems with long-range electron –electron interaction. 54 As a contribution to resolve this issue, recently, a new figure of merit for resonance classification in quantum mechanicallydescribed nanostructures has been proposed. 1The energy-based plasmonicity index (EPI) is a normalized and dimensionlessmeasure for characterizing the nature of resonances in nanostruc- tures. It does not rely on charge carrier oscillation patterns on the nanostructure and, hence, cannot be determined by the analysis ofatomic site population dynamics. Unlike the PI and the GPI, theEPI probes the manifestation of the resonance directly in energyspace. It quantifies if the existence of a given resonance can be explained predominantly by the system ’s energy landscape or if electronic interaction energy, e.g., Coulomb energy, needs to betaken into account to properly determine its spectral position. Thedefinition of the EPI is based on the coherences of the system ’s density operator and the single-particle energies. While the EPI has been applied so far mostly to toy model systems, it remains anopen question how this measure can be used to explain and under-stand resonances sustained in structures of practical relevance.Here, we concentrate on the study of an armchair-edged graphene nanoantenna with triangular shape that is of relevance in the context of nonlinear frequency conversion processes such as higherharmonic generation, for instance. 42 The paper is organized as follows. In Sec. II, we shortly intro- duce the basics of the model and the EPI measure. Further, we present a thorough analysis of a few chosen resonances in a gra- phene nanoantenna doped with various numbers of electrons. Tofully understand the nature of each resonance, we analyze thedependence of the absorption spectra on doping with and withoutCoulomb interaction taken into account, the dependence of spectral positions and absolute strengths of the nanoantenna ’s resonances on Coulomb interaction scaling, their real space charge distribu-tion, and energy space fingerprints. Then, we compute and discussthe EPI for the shown resonances. As a conclusion, we find that in the undoped and moderately doped nanoantenna, the EPI classifies all emerging resonances as predominantly single-particle-like. Onlyin the 20-fold heavily doped nanoantenna, the EPI observes trulyplasmonic response. II. METHOD Our general modeling framework is based on previous work by Cox and Garcia de Abajo. 42We model the graphene nanostruc- tures relying on the tight binding (TB) approach55in the nearest- neighbor approximation. The coupling to an externally applied laser field is considered in the quasistatic limit and dipolar approxi- mation. To describe the Coulomb interaction between electrons atJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 093103 (2021); doi: 10.1063/5.0038883 129, 093103-2 Published under license by AIP Publishing.different carbon sites land l0, we employ a Coulomb interaction matrix vλ ll0¼λvll0, where vll0values are based on Ref. 56for the onsite-, nearest neighbor-, and next-to-nearest neighbors. For theatomic sites that are further away from each other, we use the usual1=rpower law. The parameter λ[[0, 1] is used to continuously vary the Coulomb interaction from being completely turned off (λ¼0) to being fully taken into account ( λ¼1). To describe the dynamics of the system, we first construct the ground state densitymatrix from the TB-Hamiltonian eigenstates of the nanoantennaaccording to the Aufbau principle. Then, we evolve this state with amaster equation that contains the influence of an external optical illumination as well as a phenomenological damping term that accounts for the dissipative processes in the nanoantenna. Thismethod allows us to find the optical response of graphene nanoan-tennas in terms of the time-dependent polarization function of thesystem 1,42and the absorption spectrum related to its Fourier trans- form. Different resonances observed in the absorption spectrum can then be related to either single-particle-like or collective effects,and, in general, both physical mechanisms contribute to a givenresonance and determine its spectral position. It is the main chal-lenge to find out which resonance is caused by which effect. The figure of merit we will use to quantify the contribution of collective effects to a specific resonance is the energy-based plasmo-nicity index (EPI). It is based on the stationary density matrix ρ ωof a nanoantenna subject to continuous wave (CW) illumination at the resonance frequency ω. To define the EPI, we introduce an aux- iliary quantity ~ρωrelated to ρωas follows: ~ρω jj0¼jρω jj0j jEj/C0Ej0j/C0/C22hωþiϵ/C12/C12/C12/C122, (1) where the elements ρω jj0of the density matrix are given in the basis of TB-Hamiltonian eigenstates. The parameter ϵ¼0:05 eV in the denominator prevents a divergence if the incident illumination fre- quency ωhappens to be perfectly resonant to the transition fre- quency between the electronic energy states Ejand Ej0. Since information on electronic transitions that contribute to the density- matrix dynamics is already contained in its off-diagonal elements (coherences), we deplete the diagonal of the density matrix ρω jj!0 before it enters the definition of ~ρin Eq. (1). Please note again, that in this way, no information about the occupations of energy statesenters the definition of the EPI. Having introduced ~ρ, we define the EPI according to EPI(ω)¼1/C0h~ρ ω,ρωi[[0, 1], (2) where we use a scalar product of two matrices aand b, which is defined as ha,bi:¼P mnjamnbmnj (P mnjamnj2/C1P mnjbmnj2)1=2[[0, 1] : (3) For a resonance that comprises the single-particle-like transition from state jjitojj0i, the density operator ’s coherence element ρω jj0is non-zero. In the definition of ~ρω jj0, these elements get enhanced in case the excitation energy /C22hωmatches the energy difference of thesingle-particle states, jEj/C0Ej0j. On the other hand, if coherence ele- ments in the density operator ρωexist, which cannot be related to the excitation energy, they get suppressed in ~ρω. Consequently, if a resonance is predominantly composed of single-particle-liketransitions, we find ~ρ ω/C25k/C1ρωwith a constant k, furthermore h~ρω,ρωi/C251 and, therefore, EPI( ω)/C250, which renders the reso- nance single-particle-like. On the other hand, if a resonance is com- prised predominantly by coherences that cannot be associated withthe excitation energy (which is the case in plasmonic resonances), wefind ~ρ ω/C250, because most of the coherence elements in ρωget sup- pressed; furthermore, h~ρω,ρωi/C250 and, therefore, EPI( ω)/C251. With that, we get an explicit normalized measure of whether a resonance is single-particle-like or plasmonic. More details concerning the EPI,especially graphical illustrations of the density operator coherenceelements for the single-particle-like and plasmonic resonances, and athorough analysis of the measure, as well as a complete description of the used methodology, can be found in the literature. 1 III. RESONANCE ANALYSIS In this section, we examine the absorption spectrum of a tri- angular armchair-edged graphene nanoantenna consisting of N¼270 atoms. The eigenenergies of the non-interacting system that are located near the Fermi energy of an undoped flakeE F= 0 eV are shown in Fig. 1(a) . We label the eigenstates corre- sponding to different energy levels with Greek letters. Note thatsome of these might be degenerate, e.g., in Fig. 1(a) , the letter α denotes the 133rd and 134th eigenstates. In the absorption spectrum of the considered nanostructure, one can identify a few resonances that co rrespond to single-particle-like transitions. This can be seen particularly well in Fig. 1(b) ,w h e r et h e Coulomb interaction and, therefore, collective interaction-mediated processes are turned off. Some prominent resonances are present, e.g., at 1.12 eV (associated with the α/C0!α 0transition), at 2 eV ( δ/C0!β0 transition) and at 2.2 eV ( δ/C0!δ0transition). Moreover, after doping the system with one electron, which populates the α0state and enables transitions from it, three new resonances in the absorption spectrum appear in the energy range between 0.2 eV and 0.5 eV. These clearlycorrespond to the α 0/C0!β0,α0/C0!γ0,a n d α0/C0!δ0transitions. Moreover, we observe that the resonance associated with the α/C0!α0 transition vanishes after doping wit h four additional electrons, since theα0states are fully occupied and cannot serve as acceptors for any transition any more. When the Coulomb interaction is taken into account [ Fig. 1(c) ], the character of some resonances is modified. In particular, someexcitations blueshift with increasing number of doping electrons. If they appear only when the interaction between a collection of elec- trons is allowed, we expect them to originate from electron –electron interaction. We will denote the fundamental mode as P 1and the higher-order mode as P2[Fig. 1(c) ]. Apart from these resonances, we find resonances of a different type. They are mostly of a single- particle character, but are dressed (superscript d)b yi n t e r a c t i o n energy, as we argue below. These can still be attributed to a transitionbetween a single pair of eigenstates, e.g., the resonance denoted asE d α/C0!α0inFig. 1(c) . While plasmonic resonances are, in general, known to blueshift with increasing charge carrier density, this specific example of a dressed single-particle-like resonance appears to redshiftJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 093103 (2021); doi: 10.1063/5.0038883 129, 093103-3 Published under license by AIP Publishing.as additional electrons are introduced into the system. Interestingly, another example of a dressed resonance, the one associated with theα 0/C0!δ0transition, behaves differently and blueshifts. From this observation and from the similar charge carrier distribution of these two resonances in real space (see Fig. 4 later in the article), we con- clude that the shifting behavior of a resonance as a function of chargecarrier density cannot be used to unambiguously determine thenature of said resonance. For a closer inspection, we employ the scaling approach 48,49 and smoothly scale the Coulomb interaction strength by a parameter λ[0, 1½/C138 . A look at how the absorption spectrum changes with λ shows that in the undoped structure, there is a continuous transitionfrom the E α/C0!α0resonance at 1.12 eV for λ¼0 to the dressed Ed α/C0!α0 resonance at 1.4 eV for λ¼1[Fig. 2(a) ]. Therefore, we can argue that the Ed α/C0!α0resonance in Fig. 1(c) exists also without Coulomb interaction and that it is dominated by the single-particle compo-nent; hence, it is not predominantly plasmonic. Interestingly, thesame resonance in the case of the twofold doped nanostructure (d¼2) behaves similarly [ Fig. 2(b) ], which suggests that it is also more of a single-particle-like nature. A qualitative difference appearsin the scaling of the most prominent resonances in the absorptionspectra in the case of 10 or 20 doping electrons [ Figs. 2(c) and2(d)]. A close look at the absolute strength of these resonances highlights the difference between the undoped/twofold doped and the 10-/ 20-fold doped cases even more ( Fig. 3 ). While the first pair of reso- nances ( d¼0, 2) decreases in strength with increasing value of Coulomb interaction, the latter two behave differently. In the struc- ture with the strongest doping ( d¼20), the relation is exactly oppo- site and the resonance becomes stronger with growing Coulombinteraction. The resonance in the tenfold doped structure is an inter- mediate case that grows with λto a certain point and then decreases. From this, we deduce that there is a qualitative difference betweenthe resonance P 1in the 20-fold doped nanoantenna and the reso- nance Ed α/C0!α0in the twofold doped structure. A. Real space dynamics Another important feature of a resonance is its real space dynamics. We compare the resonances in a twofold doped and in a FIG. 1. (a) Jab łonski diagram of the armchair triangle consisting of 270 atoms. Only the neighborhood of the energy gap between the highest occupied molecular orbita l (HOMO) and the lowest unoccupied molecular orbital (LUMO) is shown, which determines the properties of resonances discussed in the main text. (b) Abs orption cross section of the graphene armchair triangle consisting of 270 atoms as a function of doping charge. Coulomb interaction is turned off ( λ¼0). (c) Absorption cross section of the graphene armchair triangle consisting of 270 atoms as a function of doping charge. The Coulomb interaction is fully taken into account ( λ¼1). FIG. 2. Absorption spectra of the triangular 270-atom graphene nanoantenna as functions of the Coulomb scaling parameter λplotted for four levels of doping: (a) no doping d¼0, (b) d¼2, (c) d¼10, and (d) d¼20 doping electrons.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 093103 (2021); doi: 10.1063/5.0038883 129, 093103-4 Published under license by AIP Publishing.20-fold doped nanoantenna. In Fig. 4 , the results for a structure with two doping electrons are shown. The real space distribution ofthe induced charge at resonances corresponding to the α 0/C0!δ0(at 0.55 eV for λ¼0 and 0.72 eV for λ¼1) and α/C0!α0(at 1.12 eV forλ¼0 and 1.36 eV for λ¼1) transitions shows dipolar charac- ter [ Figs. 4(a) ,4(b),4(d), and 4(e)]. This dipolar character can cause some of the real space based plasmonicity metrics to qualifythese resonances as plasmonic. The GPI, for instance, exhibits peaks both for the Ed α0/C0!δ0resonance around 0.72 eV and the Ed α/C0!α0 resonance around 1.4 eV of strengths 3 and 5.5, respectively, for the very same structure in the twofold doped case.47Thus, in the sense of the GPI, they are both classified plasmonic. The EPI, however, yields results of 0.05 for the resonance at 0.55 eV and 0.02 for the resonance at 1.12 eV when Coulomb inter- action is turned off and gives only slightly higher values of0.13 and 0.23 for the respective resonances when the Coulombinteraction is included, e.g., for the E d α0/C0!δ0and Ed α/C0!α0resonances. Consequently, they are both classified predominantly single-particle-like. For comparison, in Figs. 4(c) and 4(f),w e present real space induced charge patterns for the 2.98 eV and3.06 eV resonances, which are typical single-particle-like excitations(which barely change their spectral position both with doping andwith Coulomb interaction scaling). The distributions of induced charge for the resonances P 1and P2in the same nanoantenna doped with 20 electrons also show regular patterns (see Fig. 5 ). These resonances, however, exhibit much higher EPIs of 0.49 and0.78, respectively, which is in accordance with the qualitative dis-cussion of Fig. 3 . The EPI takes a comparably low value for the P 1 resonance because a prominent single-particle-like transition is located at the same spectral position that also contributes to theexcitation. From Fig. 4 , we conclude that the presence of electron – electron interaction is apparently not a necessary prerequisite for long-range structured (e.g., dipolar) charge oscillation in real space. Since the existence of plasmons, however, is linked to long-rangeelectron –electron interaction energy, 54we prefer to define a FIG. 3. Strength of the resonances around 1 eV in Fig. 2 as a function of the Coulomb scaling parameter λplotted for four levels of doping ( d= 0, 2, 10, 20). σmax absstands for the peak value of the absorption cross section at resonance. FIG. 4. Comparison of the induced charge distribution with Coulomb interaction turned off [ λ= 0, upper row, subfigures (a), (b), and (c)] and taken into account [ λ=1 , lower row, subfigures (d), (e), and (f)]. Snapshots present the real space induced charge distribution in the triangular 270-atom graphene nanoante nna with two doping electrons under vertically polarized CW illumination. Snapshots were taken at the time of maximum dipole moment. We present three different resonan ces: the α0!δ0 resonance [subfigures (a) and (d)], the α!α0resonance [subfigures (b) and (e)], and the pronounced resonance around /C22hω/C253 eV [subfigures (c) and (f )]. The corre- sponding values of the EPI are shown in the upper-left corners of the subfigures.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 093103 (2021); doi: 10.1063/5.0038883 129, 093103-5 Published under license by AIP Publishing.plasmon at the nanoscale not by charge occupation characteristics in real space, but rather by coherence considerations in energyspace. From the charge occupation patterns in real space only, onecan hardly tell apart bare single-particle-like resonances [ Fig. 4(b) ] from dressed single-particle-like resonances [ Fig. 4(e) ] and plas- monic resonances [ Fig. 5(a) ]. B. Energy space dynamics Finally, to get a systematic picture of the population dynamics in the nanoantenna under consideration, we examine the energyspace dynamics as well. In Fig. 6 , we depict the difference of theoccupation of the eigenstates located near the Fermi energy with respect to the initial state of the system as a function of time. Weshow the last five optical cycles of the simulation period both fornon-interacting ( λ¼0) and for interacting ( λ¼1) electrons. Without the Coulomb interaction, one can recognize pairs or groups of states among which charge transfer appears. When theCoulomb interaction is taken into account, the oscillatory move-ment of occupation emerges on top. This “sloshing ”effect appears both in the twofold and 20-fold doped nanoantenna, but it is more pronounced in the latter one. The close relation between the subfigures in Fig. 6 correspond- ing to the case of non-integrating electrons (left) and interacting FIG. 5. Snapshots present the real space induced charge distribution in the triangular 270-atom graphene nanoantenna with 20 doping electrons under vertic ally polarized CW illumination. Snapshots were taken at the time of maximum dipole moment. We present (a) the P1mode [cf. Fig. 2(d) ], (b) the P2mode, and (c) the higher energy mode at /C22hω¼3:05 eV. The corresponding values of the EPI are shown in the upper-left corners of the subfigures. The Coulomb interaction is taken into account. FIG. 6. Population difference ρjj(t)/C0ρ0 jjof the energy states of the triangular 270-atom graphene nanoantenna doped with two electrons [upper row, subfigures (a), (b), and (c)]and 20 electrons [lower row, subfigures (d), (e), and (f)] under CW illumination for the last five optical cycles of the simulation period with respect to the ground state ρ0. The illumination frequencies coincide with those in Figs. 4 and 5and represent the (a) α0!δ0, (b)α!α0, (c) /C22hω/C253 eV, (d) P1, (e) P2, and (f) /C22hω/C253e V modes. The left panel in each subfigure shows results for non-interacting electrons ( λ¼0), whereas the right panel shows data for interacting electrons ( λ¼1).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 093103 (2021); doi: 10.1063/5.0038883 129, 093103-6 Published under license by AIP Publishing.electrons (right) reveals that the investigated resonances share a sig- nificant component of single-particle-like transitions. Population transfer between the pairs of states that contribute to a transition inthe non-interacting case, is present also in the interacting case. Ontop of that, the sloshing-type [ Figs. 6(b) and 6(d)] or even the inversion-type [ Figs. 6(c) and 6(f)] population dynamics in the interacting case may engage additional states, building up the complex collective interaction-mediated response. The EPI is ameasure of such collective influence and naturally its larger valuescorrespond to larger differences between the dynamics in the caseof non-interacting and interacting electrons. C. The EPI in the classical limit In Ref. 57, the transition between the classical and quantum regimes in graphene nanotriangle plasmonics has been reported indetail. Classically, the intra-band dynamics of conduction bandelectrons in graphene can be described in the local Drude model by an ac surface conductivity 58σ(ω)/EF=(ωþi=τ), where EF denotes the Fermi energy and τis an inelastic relaxation time. From that, one can derive that the resonance frequency of the fun-damental plasmonic mode scales as ω 0/ffiffiffiffiffiffiffiffiffiffi EF=Lp , where Lis a characteristic length of the nanoantenna. In the quantum mechani- cal structures at hand in this paper, we identify the Fermi energyE Fwith the energy of the HOMO and Lwith the triangle side- length. In Fig. 7 , we show the EPIs of the lowest-energy dipolar mode for graphene triangles with various sidelengths, but constant Fermi energy EF/C250:75 eV. For a triangle with sidelength L/C255(6, 7) nm, this translates to 8 (12,16) doping electrons. We can observe that the EPI of the lowest-energy dipolar mode isincreasing for larger flake sizes and, therefore, the more plasmonicthe resonance gets. For small flakes, this low-energy dipolar mode is intrinsically still associated to a single-particle transition (see also Fig. 4 ). For larger flakes, in the classical limit (EPI !1), the energy level diagram becomes continuous and single-particle transitionscannot occur any more.IV. CONCLUSIONS AND SUMMARY We have inspected selected resonances in a triangular gra- phene nanoantenna to distinguish those of predominantlysingle-particle-like from those of collective character. Based on the fact that the resonances corresponding to the transitions α/C0!α 0 and α0/C0!δ0are clearly visible in the spectrum of an undoped/ twofold doped structure when the Coulomb interaction is turnedoff, we claim that they are predominantly of a single-particle-likenature. Even though their real space charge distributions show dipolar patterns, the EPI does not classify these resonances as plas- monic, like the GPI, for instance, does. This classification is sup-ported by the fact that the real space charge distribution pattern isalso dipolar for non-interacting electrons. Moreover, said resonan-ces exhibit a qualitatively different dependence on the Coulomb scaling parameter λthan the resonances in 10-/20-fold doped structures, which appear to be plasmonic. Their energy patternsshow a stronger “sloshing ”behavior in the strongly doped structure as compared to a weak presence of this effect in the twofold dopednanoantenna. This behavior arises due to Coulomb interactions among electrons, whose influence on their dynamics is significantly stronger in the case of collective plasmonic resonances. We want toemphasize that there is no inconsistency when the GPI and EPIclassify certain resonances differently, but simply another definitionof a plasmon at the nanoscale. The GPI relies on the Coulomb energy of a given charge distribution associated to an excitation; the EPI relies on the impact of long-range electron –electron inter- action on the spectral position of a given resonance. In the classicalmacroscopic system, these two characteristics go hand in hand, but not necessarily in nanostructures. The EPI takes a low value for α/C0!α 0andα0/C0!δ0resonan- ces and significantly higher values for the P1and P2resonances, which is in agreement with the predictions that those pairs of reso-nances are of different nature. This is a qualitative difference beyond analyzing the real space induced charge patterns that requires a careful look at both the absorption spectra of the nano-structure and the coherence dynamics in energy basis. These conclusions ultimately raise the question of whether one can find plasmons in the sense of the EPI, i.e., originating from long-range electron –electron interactions in weakly or moderately doped nanoantennas at all. Our research rather suggests that inthese structures, resonances are predominantly single-particle-likeand only moderately dressed by interaction effects. To obtain trulyplasmonic resonances in the sense of the EPI, which, in the first place, emerge due to long-range electron –electron interaction, one needs heavy doping. ACKNOWLEDGMENTS M.M.M. acknowledges financial support through the Research Travel Grant by the Karlsruhe House of Young Scientists (KHYS).M.M.M. and C.R. acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) (Project No. 378579271) within Project RO 3640/8-1 and from theVolkswagenStiftung. M.M.M. is grateful for the support of theToru ńAstrophysics/Physics Summer Program TAPS 2019 and the PROM (Project No. PPI/PRO/2018/1/00016/U/001) by the Polish National Agency for Academic Exchange. M.M.M., M.K., and K.S. FIG. 7. EPI of the most prominent low-energy dipolar mode for graphene trian- gles of different sidelengths between L¼3 nm (168 atoms, quantum regime) and L¼9 nm (1386 atoms, classical limit).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 093103 (2021); doi: 10.1063/5.0038883 129, 093103-7 Published under license by AIP Publishing.acknowledge the hospitality of the Donostia International Physics Center. M.K. acknowledges financial support from the National Science Centre, Poland (Grant No. 2016/23/D/ST2/02064). M.P.and A.A. acknowledge financial support by Project No.PID2019-105488GB-I00 of the Spanish Ministry of Science andInnovation, and the Gobierno Vasco UPV/EHU (Project No. IT1246-19). K.S. acknowledges support from the National Science Centre, Poland (Project No. 2016/23/G/ST3/04045), the EuropeanCommission from the NRG-STORAGE project (GA 870114) andH2020-FET OPEN project MIRACLE (GA 964450). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1M. M. Müller, M. Kosik, M. Pelc, G. W. Bryant, A. Ayuela, C. Rockstuhl, and K. 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9.0000142.pdf

AIP Advances 11, 025122 (2021); https://doi.org/10.1063/9.0000142 11, 025122 © 2021 Author(s).Study of canonical spin glass behavior in Co doped LaMnO3 Cite as: AIP Advances 11, 025122 (2021); https://doi.org/10.1063/9.0000142 Submitted: 15 October 2020 . Accepted: 18 January 2021 . Published Online: 11 February 2021 Farooq H. Bhat , Ghazala A. Khan , Gitansh Kataria , Ravi Kumar , Deshdeep Sahdev , and Manzoor A. Malik COLLECTIONS Paper published as part of the special topic on 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials and 65th Annual Conference on Magnetism and Magnetic Materials ARTICLES YOU MAY BE INTERESTED IN Investigation of coherence properties of white light emission of tungsten lamp additionally excited with laser radiation AIP Advances 11, 025119 (2021); https://doi.org/10.1063/5.0038219 Structural, magneto transport and magnetic properties of Ruddlesden–Popper La2-2x Sr1+2x Mn2O7 (0.42≤x≤0.52) layered manganites AIP Advances 11, 025331 (2021); https://doi.org/10.1063/9.0000109 Some optical soliton solutions to the perturbed nonlinear Schrödinger equation by modified Khater method AIP Advances 11, 025130 (2021); https://doi.org/10.1063/5.0038671AIP Advances ARTICLE scitation.org/journal/adv Study of canonical spin glass behavior in Co doped LaMnO 3 Cite as: AIP Advances 11, 025122 (2021); doi: 10.1063/9.0000142 Presented: 5 November 2020 •Submitted: 15 October 2020 • Accepted: 18 January 2021 •Published Online: 11 February 2021 Farooq H. Bhat,1,2,a) Ghazala A. Khan,3Gitansh Kataria,4Ravi Kumar,5Deshdeep Sahdev,4 and Manzoor A. Malik2 AFFILIATIONS 1Department of Physics, Islamic University of Science and Technology, Awantipora, J & K 192122, India 2Department of Physics, University of Kashmir, Srinagar, Jammu and Kashmir 190006, India 3Department of Physics, SP College, Cluster University Srinagar, Srinagar, J & K 190001, India 4Quazar Technologies Private Limited, 105, Village, Begumpur, New Delhi 110017, India 5Department of Materials Science and Engineering, NIT, Hamirpur, Hamirpur, HP 177005, India Note: This paper was presented at the 65th Annual Conference on Magnetism and Magnetic Materials. a)Author to whom correspondence should be addressed: fhbhat@gmail.com ABSTRACT Temperature-dependent magnetic and a.c.susceptibility measurements were done on single phase polycrystalline LaMn 1-yCoyO3(y=0.1, 0.4) samples. The field cooled and zero field cooled magnetic measurements performed indicate the presence of spin glass state which is established using a.c.susceptibility measurements. The a.c.susceptibility data analyzed using Arrhenius and Vogel–Fulcher (VF) law reveal the presence of canonical spin glass. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/9.0000142 I. INTRODUCTION The perovskite manganites RMnO 3(R=La, Nd, etc.) have been the subject of exhaustive research because of competing interactions among magnetic, electronic, and structural phases. The doping of a cation like Ca at a La site or of Co on a Mn site brings out different properties (e.g., ferro/anti-ferromagnetism, metal–insulator transi- tion, and colossal magnetoresistance1) in them, besides properties involving spin, orbital and charge ordering.2 The doping of manganese (Mn) by cobalt (Co) stimulates ferro- magnetism (FM) in LaMn 1-xCoxO3.3–7This was explained through superexchange (SE) interactions among Mn3+ions,3positive SE interactions among Mn4+and Co2+ions4,5and double exchange (DE) mechanisms involving Mn3+/Mn4+.8The competing positive and negative SE interactions among different Mn and Co charge states were also cited as the cause of FM.7,9The Mn NMR stud- ies done on LaMn 0.5Co0.5O3indicated the ordering of Co2+–Mn4+ ions.10–13The findings on charge distribution are contentious too, as some studies8,9,14–16have shown the charge distribution ofMn4+-Co2+. Joy and co-workers17–19found that LaMn 0.5Co0.5O3 exists in two FM phases, representing various spin and valence and spin states of Mn and Co. The lower T C(150 K) phase contains Co2+and Mn4+, while a higher T C(225 K) phase has high spin (HS) Mn3+and low spin (LS) Co3+ions. However, Burnus et al. ,20using XAS measurements, found a low T Cphase having LS Co3+ions and Mn3+/Mn4+ions. The high T Cphase was found to contain Co2+ and Mn4+ions. Viswanathan et al.21attributed the difference in Tcof two distinct LaMn 0.5Co0.5O3FM phases to the extent of dis- tortion present in them. Autret et al.22found that cobalt enters as Co2+at lower concentrations, gradually shifting to Co3+in higher concentrations consequently leading to mixed Co2+/Co3+valency. The manganites also show spin-glass (SG) behaviour;23–28 which may arise due to disorder in the system. Recently, Manna et al.29observed a re-entrant glass state in LaMn 0.5Co0.5O3sin- gle crystal due to competing FM and antiferromagnetic (AFM) interactions caused by high B-site disorder. We doped Co at a Mn site in LaMnO 3to study the magnetic properties. The temperature-dependent magnetic measurements AIP Advances 11, 025122 (2021); doi: 10.1063/9.0000142 11, 025122-1 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv done in ZFC and FC mode at 0.01T applied field, exhibit the occurrence of a SG state in the samples LaMn 1-yCoyO3with Co concentration y =0.1 and y =0.4. To authenticate these findings frequency-dependent a.c.susceptibility measurements were made at different frequencies of 21 Hz, 1101 Hz and1701 Hz in the presence of the applied field of 0.0004 T and analyzed using Arrhenius and Vogel–Fulcher (VF) law. II. EXPERIMENT Single-phase LaMn 1-yCoyO3(y=0.1 and 0.4) samples were pre- pared using a solid state reaction method. The oxides of highly pure La2O3,MnO 2, and Co 3O4were mixed in stoichiometric amounts and ground in an agate mortar until a homogenous form was achieved. This homogenous mixture was sintered at 1200○C for 20 h after calcination. The pellets of 10 mm diameter were made from sintered powder and further sintered at 1250○C for 24 h. An x-ray diffraction (XRD) pattern of the samples was obtained at room temperature by a Bruker D8 diffractometer with Cu-K αradi- ation. The temperature-dependent magnetization measurements were done by using a commercial 7T SQUID-vibrating sample mag- netometer of Quantum Design and a.c.susceptibility measurements were made in an a.cfield of 0.0004T at 21 Hz, 1101 Hz, and 1701Hz using a commercial Physical Quantity Measurement System. III. RESULTS AND DISCUSSIONS A. X-ray diffraction The X-ray diffraction (XRD) spectra of powdered LaMn 1-yCoyO3(y=0.1 and 0.4) samples reveal a single-phase perovskite structure. The XRD pattern (Fig. 1) is indexed using powder X software to an orthorhombic structure with Pbnm as a space group. The decrease of unit cell parameters afrom 5.4291 to 5.4151 Å, bfrom 5.7207 to 5.6805 Å, and cfrom 7.7489 to 7.6591 Å, decreases the overall unit cell volume from 240.667 to 235.600Å. The XRD pattern displays a peak shift towards higher 2 θvalues as y increases indicating the lattice contraction due to Co substi- tution at the Mn site resulting in the decrease of overall unit cell volume. This reduction may be recognized due to smaller ionic radii of Co as compared to that of Mn. The intensity of the peak indexed at 2θ∼72.6○in y=0.1 sample goes on decreasing as Co doping is increased; hence, is not indexed in y =0.4 sample. This has also been witnessed earlier.30 B. Magnetization and ac susceptibility The temperature variation of magnetization for LaMn 1-yCoyO3 (y=0.1 and 0.4) samples during zero-field cooled (ZFC) and field cooled (FC) measurements is presented in Fig. 2. In the y =0.1 sam- ple the paramagnetic (PM) to FM transition takes place at Curie temperature (T c)∼170 K, while as in y =0.4 double FM transition is observed at T c2=151 K, T c1=207 K. It is seen that the order- ing temperature rises with an increase in Co concentration. This indicates the enhancement of FM interactions. The double mag- netic transition in y =0.4 can be explained due to mixed valency of Mn and Co ions, which earlier study18proposed as the reason FIG. 1. XRD pattern of LaMn 1-yCoyO3(y=0.1 and 0.4) samples. for the samples heated in the temperature range of 973-1573 K. The present samples have been prepared at 1523 K; in y =0.4 Co3+starts contributing along with Co2+ions in addition to the appreciable contribution of Mn4+. The anomaly at T c1may occur because of FIG. 2. Variation of magnetic moment in LaMn 1-yCoyO3(y=0.1 and 0.4) samples with temperature in a field of 0.01 T. The inset figure shows plot of temperature dependent inverse magnetic susceptibility for the same samples. AIP Advances 11, 025122 (2021); doi: 10.1063/9.0000142 11, 025122-2 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv Mn4+and Co2+ordering and consequently, the Co2+-O-Mn4+SE FM interaction and the second anomaly at T c2points to the placing of LS Co3+as well as HS Co3+.31In addition to SE, FM interaction mixed valence of Mn (Mn3+/Mn4+) introduces double exchange FM interaction too. The noteworthy information on valence/spin states is determined by fitting the inverse magnetic susceptibility to Curie Weiss lawχ=C T−θ, shown in the inset of Fig. 2. The μeffobtained from the best fit is 5.10 μBand 4.30μBfor y =0.1 and y =0.4 respectively. The calculated value of paramagnetic moment μcalis 4.77μBand 4.21μBfor y =0.1 and y =0.4 respectively. The theo- retical moments were calculated by taking into account the charge distribution Mn3+/Mn4+-Co2+and Mn3+/Mn4+-Co2+/Co3+in y=0.1 and y =0.4 samples respectively. The signatures of irreversibility between ZFC and FC curves, cusp in the ZFC curve, and rising FC magnetization are characteris- tics of spin glass state. To explore it a.c.susceptibility measurements were done in an a.c. field of 0.0004 T for both the samples at the frequencies of 21, 1101, and 1701 Hz. The temperature dependence of the real part of the a.c.susceptibility ( χ′) for both the samples is presented in Fig. 3. It is seen that the peak position in χ′, corre- sponding to Tf, moves towards higher temperatures with decreasing magnitude as the frequency increases. This is a strong signature of the existence of SG. FIG. 3. Temperature dependence of real part of a.c. susceptibility for LaMn 0.9Co0.1O3(y=0.1) and LaMn 0.6Co0.4O3(y=0.4).To differentiate between various SG systems, the parameter called the relative shift in freezing temperature ( δTf) per decade of frequency is a convenient tool and is defined as32,33 δTf=ΔTf TfΔ(log10ω), (1) where ΔTf=(Tf)υ1−(Tf)υ2 and Δlog10ω=log10(2πυ1) −log10(2πυ2). We obtain δTf=0.009 for LaMn 0.9Co0.1O3andδTf=0.0052 for LaMn 0.6Co0.4O3samples which agree with the reported values for canonical SG (CSG) systems.32,34 To further validate the above findings a fit of a.c. susceptibility data to the Arrhenius law32for thermal activation is done and shown in Fig. 4. This law is appropriate for noninteracting magnetic entities and is written as τ=τ0exp(Ea kBTf). (2) Here,τ0is the relaxation time of a single spin–flip of the fluctuating entities,Ea kBTfis the average activation energy of the relaxation bar- rier,ωis the driving frequency of a.c. susceptibility measurement, andTfits peak. The activation energy essentially gauges the energy barrier parting the metastable states and the Arrhenius law explains the timescale to surmount the energy barriers by the activation process.33By plotting ln(f) versus 1 /Tfthe values of τ0obtained are τ0∼10-114, 10-195s and of the barrier height Ea/kB=43438, 95058 K FIG. 4. Plot of ln(f) versus 1/T f, solid line shows best fit using Arrhenius law for LaMn 0.9Co0.1O3(y=0.1) and LaMn 0.6Co0.4O3(y=0.4). AIP Advances 11, 025122 (2021); doi: 10.1063/9.0000142 11, 025122-3 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv for LaMn 0.9Co0.1O3and LaMn 0.6Co0.4O3respectively. These values are highly unrealistic; the possible reason for such highly unrealistic values is small changes in Tf. Therefore, the possibility of the exis- tence of a superparamagnet, which obeys the Arrhenius law, is ruled out; hence the absence of non-interacting dynamics. The phenomenological Vogel–Fulcher (VF) law is another dynamical scaling law, which deals with the interacting spins. This law defines the frequency-dependent Tfas32,33 τ=τ0exp(Ea kB(Tf−T0)), (3) where T0is the empirical VF temperature, having a value between 0 K and T f. It is linked to the interaction strength between the dynamic entities. The plot of ln τversus 1 /(Tf−T0)is dis- played in Fig. 5, fit to Eq. (3) with T0=155 K and 202 K for LaMn 0.9Co0.1O3and LaMn 0.6Co0.4O3, respectively. The value of τ0 obtained from the intercept of the linear fit is of the order of ∼10-13s for both the samples which are typically those of canon- ical spin glass (CSG) and are approximately equal to the spin–flip time of atomic magnetic moments ( ∼10−13s).35The slope Ea/kB of the linear fit has values of ∼319 and ∼337 for LaMn 0.9Co0.1O3 and LaMn 0.6Co0.4O3respectively. As per Tholence criterion, δTTh =(Tf−T0)/Tf,36the value of δTThobtained for LaMn 0.9Co0.1O3 and LaMn 0.6Co0.4O3is∼0.088 and ∼0.064 respectively, which is less than the values reported for cluster glass (CG) by one order of mag- nitude.32These values are comparable to those obtained for a CSG system like CuMn.37Further, the ratio of Ea/kBand T0indicates FIG. 5. Plot of lnτvs 1/(T f-T0), solid line represent the best fit using VF law for LaMn 0.9Co0.1O3(y=0.1) and LaMn 0.6Co0.4O3(y=0.4).the strength of coupling among the interacting entities.37The val- ues of Ea/kBT0calculated for LaMn 0.9Co0.1O3and LaMn 0.6Co0.4are ∼2 and ∼1.66 respectively, which lies in the range those of CSG systems.37 IV. CONCLUSIONS In the current study, we report the existence of canonical spin glass state in polycrystalline LaMn 0.9Co0.1O3and LaMn 0.6Co0.4O3 samples. These samples were prepared in a single-phase using the solid state reaction method, having the orthorhombic crystal struc- ture and Pbnm as a space group. The occurrence of a canonical spin glass state is confirmed through the a.c.susceptibility measurements and its analysis using Arrhenius law and Vogel–Fulcher law. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1R. von Helmolt, J. Wecker, B. 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Magn. Magn. Mater. 210, 63 (2000). 24I. O. Troyanchuk, A. P. Sazonov, H. Szymczak, D. M. Többens, and H. Gamari-Seale, J. Exp. Theor. Phys. 99, 363 (2004). AIP Advances 11, 025122 (2021); doi: 10.1063/9.0000142 11, 025122-4 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv 25X. L. Wang, M. James, J. Horvat, and S. X. Dou, Supercond. Sci. Technol. 15, 427 (2002). 26R. Laiho, E. Lähderanta, J. Salminen et al. , Phys. Rev. B 63, 094405 (2001). 27J.-W. Cai, C. Wang, B.-G. Shen, J.-G. Zhao, and W.-S. Zhan, Appl. Phys. Lett. 71, 1727 (1997). 28J. Dho, W. S. Kim, and N. H. Hur, Phys. Rev. Lett. 89, 027202 (2002). 29K. Manna, S. Elizabeth, and P. S. Anil Kumar, J. Appl. Phys. 119, 043906 (2016). 30G. Pecchi, C. Campos, O. Peña, and L. E. Cadus, J. Mol. Catal A: Chemical 282, 158–166 (2008).31Y. Q. Lin, X. M. Chen et al. , J. Am. Ceram. Soc. 94(3), 782 (2011). 32J. A. Mydosh, Spin Glasses: An Experimental Introduction (Taylor & Francis, 1993). 33P. Bag, P. R. Baral, and R. Nath, Phys. Rev. B 98, 144436 (2018). 34N. Khan, A. Midya, P. Mandal, and D. Prabhakaran, Appl. Phys. 113, 183909 (2013). 35A. Malinowski, V. L. Bezusyy, R. Minikayev, P. Dziawa, Y. Syryanyy, and M. Sawicki, Phys. Rev. B 84, 024409 (2011). 36J. Souletie and J. L. Tholence, Phys. Rev. B 32, 516(R) (1985). 37S. Shtrikman and E. P. Wohlfarth, Phys. Lett. A 85, 467 (1981). AIP Advances 11, 025122 (2021); doi: 10.1063/9.0000142 11, 025122-5 © Author(s) 2021

5.0040973.pdf

J. Chem. Phys. 154, 084107 (2021); https://doi.org/10.1063/5.0040973 154, 084107 © 2021 Author(s).Analysis of the kinetic energy functional in the generalized gradient approximation Cite as: J. Chem. Phys. 154, 084107 (2021); https://doi.org/10.1063/5.0040973 Submitted: 18 December 2020 . Accepted: 31 January 2021 . Published Online: 23 February 2021 Héctor I. Francisco , Javier Carmona-Espíndola , and José L. Gázquez ARTICLES YOU MAY BE INTERESTED IN The Devil’s Triangle of Kohn–Sham density functional theory and excited states The Journal of Chemical Physics 154, 074106 (2021); https://doi.org/10.1063/5.0035446 Implementation of Perdew–Zunger self-interaction correction in real space using Fermi– Löwdin orbitals The Journal of Chemical Physics 154, 084112 (2021); https://doi.org/10.1063/5.0031341 r2SCAN-D4: Dispersion corrected meta-generalized gradient approximation for general chemical applications The Journal of Chemical Physics 154, 061101 (2021); https://doi.org/10.1063/5.0041008The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Analysis of the kinetic energy functional in the generalized gradient approximation Cite as: J. Chem. Phys. 154, 084107 (2021); doi: 10.1063/5.0040973 Submitted: 18 December 2020 •Accepted: 31 January 2021 • Published Online: 23 February 2021 Héctor I. Francisco,1,a) Javier Carmona-Espíndola,2 and José L. Gázquez1,a) AFFILIATIONS 1Departamento de Química, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, México, Ciudad de México 09340, Mexico 2Departamento de Química, CONACYT-Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, Ciudad de México 09340, Mexico a)Authors to whom correspondence should be addressed: hifr@xanum.uam.mx and jlgm@xanum.uam.mx ABSTRACT A new density functional for the total kinetic energy in the generalized gradient approximation is developed through an enhancement factor that leads to the correct behavior in the limits when the reduced density gradient tends to 0 and to infinity and by making use of the con- joint conjecture for the interpolation between these two limits, through the incorporation, in the intermediate region of constraints that are associated with the exchange energy functional. The resulting functional leads to a reasonable description of the kinetic energies of atoms and molecules when it is used in combination with Hartree–Fock densities. Additionally, in order to improve the behavior of the kinetic energy density, a new enhancement factor for the Pauli kinetic energy is proposed by incorporating the correct behavior into the limits when the reduced density gradient tends to 0 and to infinity, together with the positivity condition, and imposing through the interpolation function that the sum of its integral over the whole space and the Weiszacker energy must be equal to the value obtained with the enhancement factor developed for the total kinetic energy. Published under license by AIP Publishing. https://doi.org/10.1063/5.0040973 .,s I. INTRODUCTION The interest to express the total energy of a many electron system as a functional of the electron density, ρ(r), goes back to the early days of quantum mechanics. In a first attempt, Thomas1 and Fermi2,3approximated the kinetic energy (KE) by introduc- ing the local density approximation (LDA) in the KE per parti- cle of an homogeneous electron gas. Shortly after, using the same approach, Dirac4derived the LDA for the exchange energy, and later, von Weizsacker5introduced a gradient correction to the KE functional. With the advent of density functional theory (DFT),6 particularly through the Kohn–Sham(KS)7approach, the interest focused mainly on the development of density functionals for the exchange–correlation energy. In this theoretical framework, the KE is expressed in terms of the KS orbitals, TS[ρ]=∫drtorb S(r)=∫dr1 2N ∑ i=1∣∇φi(r)∣2, (1)while the electronic density is given by ρ(r)=N ∑ i=1∣φi(r)∣2. (2) However, despite the fact that the Kohn–Sham method has become a very important tool for the study of the electronic structure of atoms, molecules, and solids,8–17the introduction of orbitals to describe the noninteracting kinetic energy, TS, implies a large increment in the computational effort, with respect to the one associated with an energy functional that only depends on the electron density. In this context, it is certainly desirable to develop accurate representations ofTSin terms of ρ(r) to recover what is nowadays called orbital-free density functional theory.18–21Research in this area has concentrated mainly on two types of approximations, namely, the nonlocal two- point21–38and the semilocal one-point39–61functionals. Although the former have achieved, in general, through constraint satis- faction combined with the appropriate parameterization, the best J. Chem. Phys. 154, 084107 (2021); doi: 10.1063/5.0040973 154, 084107-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp performance so far, some of the recent approximations60,61of the latter have been found to lead to similar results with a significantly lower computational effort. The semilocal one-point expressions have, in general, as a starting point, the Thomas–Fermi LDA followed by the gradi- ent expansion approximation (GEA) for the inhomogeneous elec- tron gas.6,8,9,62–66However, because of the exponential behavior of the electronic density in the asymptotic limit, far away from the nuclei,67–70the sixth and higher order terms diverge for finite systems,71and from the perspective of considering the gradient expansion to derive the Euler–Lagrange equation for the ground- state electronic density, one finds that, for the same reason, the func- tional derivative of the fourth and higher order terms diverges. A similar situation occurs for the GEA of the inhomogeneous electron gas for the case of the exchange energy,72–77where the fourth and higher order terms and the functional derivative of the second and higher order terms diverge. Thus, in order to avoid this problem, one makes use of the methodology of the generalized gradient approxi- mation (GGA),41originally developed for the exchange energy func- tional.78–82The GGA for the kinetic energy is also motivated by the conjoint gradient correction approach.39 Now, a relevant aspect to take into account in the derivation of KE functionals is concerned with the behavior of the kinetic energy density. Several studies83–86have revealed that, in general, although many of the one-point approximations may lead to a rea- sonable estimate of the KE when they are evaluated with Hartree– Fock electron densities, the analysis through other integrals intended to measure the local behavior with the same Hartree–Fock den- sities shows that there are large discrepancies with respect to an orbital-based expression like the one given in Eq. (1). Although the KE density is not uniquely defined because one can always add another function to the original integrand that integrates to 0, one certainly expects that the proposed one-point expression could lead to a satisfactory description at the local (KE density) and global (KE) levels. In fact, there has been an increased attention in the local level because it constitutes a fundamental ingredient to simplify meta-GGA exchange–correlation functionals.17,84–102 The objective of the present work is to perform an analysis of the development of the GGA for the exchange energy to con- jecture about the plausibility of expressing the kinetic energy in the form of a GGA. Then, we will make use of several constraints associated with the kinetic energy, when it is expressed in the form of a GGA, to propose a non-empirical one-point functional that provides good estimates of the energy when used in combination with Hartree–Fock densities. In addition, we will make use of the proposed expression to derive from it a system-dependent GGA functional for the Pauli kinetic energy, which, when added to the Weizsacker contribution, provides an improved description of the KE density that by construction leads to the same KE values of the original GGA. II. TOTAL KINETIC ENERGY One-point KE functionals based on the GGA are usually writ- ten in the form TGGA S[ρ]=∫drtGGA S(r)=∫drtTF(r)FGGA T(s), (3)where tGGA S(r)is the local KE, tTF(r)=3 10(3π2)2/3ρ5/3(r) (4) is the local Thomas–Fermi kinetic energy, and FT(s) is the enhance- ment factor, a function of the reduced gradient of the density, namely, s=∣∇ρ(r)∣ 2(3π2)1/3ρ4/3(r). (5) The concept of a generalized gradient approximation was first proposed by Perdew and collaborators78–82for the exchange energy functional. Their starting point was the expression for the exchange energy written in terms of the exchange hole density at point r+R about an electron at point r,ρX(r,r+R), namely, EX[ρ]=1 2∫drρ(r)∫dRρX(r,r+R)/R. (6) The exact hole fulfills the following constraints: ρX(r,r)=−ρ(r)/2, (7) ρX(r,r+R)≤0, (8) and ∫dRρX(r,r+R)=−1. (9) The LDA for the exchange energy derived by Dirac satisfies the three conditions. However, the gradient expansion approximation for densities that show a small variation over the space does not fulfill the conditions given by Eqs. (8) and (9). With this situation in mind, Perdew and collaborators introduced real space cutoffs in the second order gradient expansion of the exchange hole density to impose these two constraints and found that the final expression for the exchange energy could be written in the form EGGA X[ρ]=∫drεLDA X(r)FGGA X(s), (10) where εLDA X(r)=−3 4(3/π)1/3ρ4/3(r)is the local exchange energy. In the original work,78the enhancement factor FGGA X(s)was determined numerically, but later on, analytic expressions were used to fit as best as possible the numerical GGA values for different inter- vals of the reduced density gradient, taking into account the limits when s→0 and s→∞. For the first limit, one should recover the second order GEA, that is, FGGA X(s)/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne→ s→01 +μXs2, (11) where the value of the coefficient μXmay be fixed through different constraints, while for the large slimit, one may also consider differ- ent constraints, which lead to different expressions of FGGA X(s). This J. Chem. Phys. 154, 084107 (2021); doi: 10.1063/5.0040973 154, 084107-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp is the origin of the PW8679and the PW9180exchange energy func- tionals. The analytical fit for PW86 took into account almost all the numerical GGA values, together with the value found by Sham103 for the coefficient in Eq. (11), μSham X=7/81, and the numerical large s limit, which is given by82,104 FGGA X(s)/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne→ s→∞0.8862 s2/5, (12) to within 1%. On the other hand, the analytical fit of PW91 lies very close to the numerical GGA in the interval 0.2 <s<3, which cor- responds to the physically important region105–107for the exchange energy, it leads to μGEA X=10/81, which is the accepted value for the coefficient of the second order GEA,75,76instead of the value obtained by Sham, and it reaches a maximum value of 1.641 that leads to a tighter local bound80than the original Lieb–Oxford bound,108and for large s, it decays as s−1/2in order to approximately satisfy the non-uniform scaling constraint,109 lim s→∞s1/2FGGA X(s)<∞, (13) which implies that FGGA X(s)/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne→ s→∞CXs−1/2, (14) where CXis a constant. From these two experiences, the PW86 and the PW91 exchange energy functionals, it was recognized that the GGA in a non- empirical approach could be simplified through an analytical expres- sion, with two parameters that could be fixed to satisfy the constraint given by Eq. (11), with a specific criterion to fix the value of the coefficient μX, and by imposing one of the several constraints that one can consider for the large slimit. The analytical form in itself, together with the parameter values obtained as described, should lead to a reasonable proximity to the numerical GGA values in the interval 0.2 <s<3. This is, in part, the origin of the well- known exchange energy functional of Perdew, Burke, and Ernzerhof (PBE)110in which μPBE X=0.219 51 to preserve the LDA description of the linear response uniform gas, which is known to be quite accurate, and the large slimit is given by FPBE X(s)/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne→ s→∞1.804 (15) so that FX(s)≤1.804 for any value of sto satisfy the local Lieb–Oxford bound. If the gradient expansion of the exchange hole density is slightly modified by introducing a factor that damps some of the oscillations, one is led to a numerical GGA that is described quite well by the PBE enhancement factor81in the interval 0 <s<3. Since in the case of the exchange energy functional there are other important constraints related to the large sbehavior, this procedure has been recently followed to derive GGA functionals that lead to a rather good prediction of thermodynamic, kinetic, and response properties.111–114 An important consequence of the real space cutoffs is that the numerical GGA or its analytical approximations lead to an enhance- ment factor that when used in combination with Eq. (10) does notdiverge for finite systems, and the same situation applies to the functional derivative of the exchange energy when used with the analytical approximations to FGGA X(s). In the case of the KE, its expression in terms of the exchange hole density is given by115 TS[ρ]=TW[ρ]+1 4∫dr∇2ρ(r)+1 2∫dr∇2 RρX(r,r+R)∣R=0, (16) where TW[ρ]=∫drtW(r)=1 8∫dr∣∇ρ(r)∣2 ρ(r)=∫drtTF(r)FW(s), (17) with FW(s)=(5/3)s2(18) being the von Weizsacker term.5Thus, one can see, according to the third term in the right-hand side of Eq. (16), that since the deriva- tives of ρX(r,r+R) are evaluated at R= 0, the real space cutoffs on the exchange hole density do not have any effect on the KE density. However, in this case, one may still consider a generalized gradient approximation through the use of an interpolating func- tion between the limits when s→0 and s→∞. This approach can be seen as a convergent resummation of the KE gradient expansion for slowly varying densities that only involves the electronic density and its gradient, that eliminates all higher order derivatives of ρ(r), and that leads to a well-behaved functional and functional derivative for finite systems. Furthermore, the GGA form of the KE functional expressed in Eq. (3) satisfies the same spin scaling relations of the GGA exchange energy functional that allows one to convert density functionals into spin-density functionals; it is size consistent, and by imposing the constraint FGGA T(0)=1, it recovers the electron gas limit.10 Thus, following a non-empirical approach, one can propose an analytic function for the enhancement factor FGGA T(s)in Eq. (3), in which its form and the parameters are fixed to fulfill some additional conditions that are known to be obeyed by the exact KE functional. Therefore, when s→0, the enhancement factor should adopt the form of the second order KE gradient expansion for slowly varying densities, that is, FGGA T(s)/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne→ s→01 +μTs2. (19) The value of the coefficient μTmay be fixed through the GEA,6,9,62–66 leading to a value of μGEA T=5/27, or by making use of the asymptotic expansion of the semiclassical neutral atom,50,116,117which implies μMGE T=0.238 89. On the other hand, when s→∞, the KE density becomes equal to the Weizsacker term given in Eq. (18)9,118–120so that FGGA T(s)/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne→ s→∞5 3s2. (20) J. Chem. Phys. 154, 084107 (2021); doi: 10.1063/5.0040973 154, 084107-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp However, since for the enhancement factor for the exchange energy, FGGA X(s), one has several constraints associated with the large slimit that lead to different behaviors such as in the case of Eqs. (12) and (14), one can analyze the situation for the KE for which it has been proven that40,121 lim λ→0TS[ρx λ]<∞, (21) where ρx λ(x,y,z)=λρ(λx,y,z)is a one-dimensional nonuniformly scaled density. Thus, if one makes use of the GGA expression given in Eq. (3) and noting that when λ→0 for this scaling s∝λ−1/3, then Eq. (21) leads to lim s→∞s−2FGGA T(s)<∞, (22) which implies that FGGA T(s)/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne→ s→∞CTs2, (23) where CTis a constant. In addition, one can analyze the constraint that arises from the functional derivative in order to obtain the appropriate asymp- totic behavior when r→∞in the Euler–Lagrange equation that minimizes the total energy subject to the constraint related to the fact that the electronic density must integrate over the whole space to the specified number of electrons. In the case of the exchange energy, one may consider several asymptotic forms for FGGA X(s)in Eq. (10) whose functional derivative leads to the correct asymp- totic behavior of the exchange potential.112,114,122–124Nevertheless, in the case of FGGA T(s), the correct asymptotic form is also the one given in Eq. (20) because the functional derivative of the Weiz- sacker term [Eqs. (17) and (18)] introduces the term in the Euler– Lagrange equation that leads to the asymptotic form required70to obtain the exact asymptotic behavior of the electron density67–69 when r→∞. In summary, it seems that while in the case of the exchange energy FGGA X(s)adopts different forms for different constraints asso- ciated with the limit s→∞, in the case of the KE, FGGA T(s)adopts the same form, given in Eq. (20), for all the different constraints considered. The fact that the leading term when s→0 and when s→∞is proportional to s2indicates that a plausible GGA for the KE could be just the second order gradient expansion, that is,FGGA T(s)≈1 +(5/27)s2, as conjectured by Perdew41some time ago. However, for all the reasons given above, it is important to pro- pose an enhancement factor that recovers the appropriate coefficient value of the s2term when s→0 and when s→∞. This situation implies that one needs to introduce an interpolation between these two limits. Some time ago, Ernzerhof proposed45a simple enhance- ment factor that was designed to recover the gradient expansion up to fourth order and to lead to the value of (5/3) s2in the limit s →∞, but it did not have any information on the possible behav- ior of FGGA T(s)in the interval 0.2 <s<3, which corresponds to the physically important region. With respect to the last point, one may consider the conjoint gradient correction to the kinetic and exchange energy functionals,39which establishes that the enhancement factor could be approxi- mately the same for the two quantities. Thus, through this pro- cedure, one can make use of an analytic expression for FGGA X(s) to approximate FGGA T(s)but by replacing the values of the param- eters that characterize the exchange energy by the ones corre- sponding to the KE. In fact, this has been done empirically and non-empirically.83,86,125,126In the empirical approach, one, gener- ally, makes use of Hartree–Fock electron densities of a given test set of atoms, and the values of some of the parameters are fixed by minimizing the mean absolute deviation of the calculated KE with the approximate functional with respect to the Hartree–Fock KE.39,43,44,47In the non-empirical approach, the values of the param- eters are fixed by imposing constraints that are known to be satisfied by the exact KE functional.50,51,53,87,89 In this context, the analytical form of the PBE enhancement fac- tor for the exchange energy functional110has been proposed as an approximation to FGGA T(s), that is, FPBE T(s)=1 +κ−κ 1 +μs2/κ. (24) Tran and Wesolowski,47following an empirical approach, fixed the values of the parameters μandκto reproduce the KE of several noble gas atoms, and this functional is named PBEK. On the other hand, Constantin et al. ,50following a non-empirical approach, set the value ofμby making use of the asymptotic expansion of the semiclassical neutral atom,50,116,117which implies μMGE T=0.238 89, and they kept the value of κ= 0.804, which corresponds to the value used in PBE to satisfy the local Lieb–Oxford bound [Eq. (15)] for the exchange energy, and this functional is named APBEK. They also considered other slight variations to fix the values of the parameters.51 In the present work, we propose a new non-empirical GGA kinetic energy functional that retains, in part, the analytical form of the PBE exchange enhancement factor that satisfies the limit when s→0 [Eq. (19)] with the value of μthat comes from the asymptotic expansion of the semiclassical neutral atom or from the GEA but with two important changes. One refers to the value of κ, which as mentioned in PBE takes a value of 0.804 that comes from the Lieb– Oxford bound. However, we will make use of the tighter bound of 0.641 established by Perdew80,82,109, which is the one used in PW91. The other change refers to the limit when s→∞that in the func- tional proposed here fulfills the condition given by Eq. (20) and starts to dominate the enhancement factor at around s≈3, which is where the asymptotic region starts to manifest. With all these characteristics, the KE enhancement factor proposed is given by the expression FWPBEK T(s)=(1 +κT−κT 1 +μTs2/κT)+(1 1 +e−A(s−s0))5 3s2, (25) where κT= 0.641, μMGE T=0.238 89, A= 3.0, and s0= 4.0, and the name WPBEK is intended to indicate that the enhancement factor has the PBE form parameterized for the KE, with an additional term that asymptotically leads to the Weizsacker KE density. In Fig. 1, one can see a plot of this enhancement factor together with the plot of the first term, which is the one that dominates in the region around J. Chem. Phys. 154, 084107 (2021); doi: 10.1063/5.0040973 154, 084107-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . Plot of the enhancement factor and its components. The blue curve cor- responds to the first term in the right-hand side of Eq. (25), the red curve corre- sponds to the second term, and the green curve corresponds to the sum of both terms. 0≤s≤2, and the plot of the second term, which is the one that dominates in the region around 3 ≤s<∞. Thus, through this approach, we are proposing an expression forFGGA T(s)that satisfies the limits for the KE density when s→0 and when s→∞, through an interpolation function that resem- bles PW91 FGGA X(s)in the interval around 0.2 <s<3, which, as mentioned before, is where the analytical fit of PW91 lies very close to the numerical GGA of the exchange energy. The motivation behind this analysis comes from the assumption that the similarity between the kinetic and the exchange enhancement factors estab- lished in the conjointness conjecture could occur mainly in the inter- val around 0.2 <s<3, which is in between the s→0 and s→∞ limits. Therefore, this way, one may consider that the effects of the real space cutoffs introduced in the exchange energy functional to satisfy Eqs. (8) and (9) are reflected in the KE functional through aninterpolation function based on the form of the GGA exchange enhancement factor. In order to analyze the performance of the KE functional obtained by substituting Eq. (25) into Eq. (3), we have consid- ered three different test sets. The first one corresponds to the first eighteen atoms of the periodic table (called A18) to study the behavior of systems with a relatively small number of electrons, and the second is composed of the noble gas atoms, from He to Rn (called NG), to compare the results of systems with a small, medium, and large number of electrons, while the third one corre- sponds to a group of 30 molecules (called M30) that covers different types of chemical bonding and also a wide range in the number of electrons. All the calculations reported in this work were done with a modified version of NWChem 6.5.127The Hartree–Fock energies and densities were determined using the universal Gaussian basis set128in the case of the atomic systems and the def2-TZVPP basis set129,130for the molecular systems. In the case of the atomization energies, the atoms present in each molecule were also calculated with the def2-TZVPP basis set for consistency. The Hartree–Fock densities thus obtained were used in all the calculations. In Table I, we present the results obtained for the total kinetic energy TWPBEK S for the test set formed by the noble gas atoms and compare it with other GGA functionals and with KE functionals that also include a dependence on the Laplacian of the density (GEA463 and PC88), while in Table II, we present the mean absolute devi- ation (MAD) with respect to the Hartree–Fock KE for the three test sets for a larger group of functionals. In the case of the A18 test set, we include the results obtained with one and two-point KE functionals based on the weighted density approximation.35One can see that the non-empirical GGA functional proposed in this work is the one that provides the best description of the KE in the three test sets. Particularly, the results obtained in the NG set show that the functional WPBEK is also appropriately normed. In the supplementary material, one can see the values for all the individ- ual systems that conform the three test sets, for all the functionals reported. Additionally, we have determined the kinetic energy contribu- tion to the atomization energies of the G3 set131(223 molecules) for the same list of functionals considered in Table I. The results are given in Table III, where one can see that present day approxima- tions are still rather far from the chemical accuracy required to study this property. However, one can note that the functional WPBEK is TABLE I . Non-interacting kinetic energy of the noble gas atoms for several functionals calculated with Hartree–Fock electronic densities (a.u.). NG HF GEA262GEA463LC9443PC88APBEK50WPBEK He 2.862 2.878 2.9622 2.8683 2.9931 2.8659 2.9137 Ne 128.547 127.829 129.7581 128.5486 129.3132 128.7259 128.4247 Ar 526.818 524.223 530.4233 527.9850 530.6598 528.6283 527.5483 Kr 2 752.055 2 733.064 2 757.0756 2 753.9753 2 761.2507 2 756.0790 2 752.2036 Xe 7 232.138 7 183.789 7 237.4623 7 237.2947 7 249.6999 7 240.8511 7 232.6655 Rn 21 866.768 21 725.448 21 859.1300 21 877.3026 21 896.1013 21 882.2970 21 863.0648 MAD 35.331 3.817 3.131 10.138 5.043 0.881 J. Chem. Phys. 154, 084107 (2021); doi: 10.1063/5.0040973 154, 084107-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II . MAD with respect to Hartree–Fock kinetic energy for several functionals for the three test sets considered in this work (a.u.). Functionals A18 GN M30 GGA GEA2620.781 35.331 1.830 PW86K440.323 11.831 0.787 PBEK470.316 2.079 0.779 APBEK500.551 5.043 1.382 LC94430.363 3.131 0.915 E450.514 22.829 1.530 WPBEK 0.246 0.881 0.598 GGA + Laplacian GEA4631.527 3.817 4.062 TFLreg850.569 56.866 1.607 GEA4reg881.774 5.950 4.902 PCreg881.216 10.138 3.792 WDA 1-point355.446 2-point351.387 the one that leads to the lowest MAD for the total molecular KE and for the KE contribution to the atomization energy. Thus, we have seen that the total KE can be reasonably determined through this functional when it is combined with the Hartree–Fock density, and one should also analyze other quantities to assess its performance for the description of the KE density, tWPBEK S(r)=tTF(r)FWPBEK T(s). (26) In this direction, in the present work, we have considered three indicators. One of them is related to the total KE,83,84namely, σ=∫dr∣torb S(r)−tapprox S(r)∣ TS, (27) TABLE III . MAD with respect to Hartree–Fock kinetic energy (KE) and kinetic energy contribution to the atomization energy (KE-AE) for several functionals for the G3 set of 223 molecules (a.u.). Functionals KE KE-AE GEA2 1.596 0.171 GEA4 4.052 0.347 LC94 0.702 0.179 PC 3.247 0.534 APBEK 1.086 0.213 WPBEK 0.523 0.152while the other one is related to the α(r) function, which is used in the development of meta-GGA exchange–correlation energy func- tionals in Kohn–Sham. This function is given by132 αorb(r)=torb S(r)−tW(r) tTF(r), (28) and the quality measure based on this quantity is expressed as84,85 Δα=1 N∫drρ(r)∣αorb(r)−αapprox(r)∣ =1 N∫drρ(r)∣torb S(r)−tapprox S(r)∣ tTF(r), (29) where Nis the number of electrons in the system. However, since σtends to over emphasize the core region where tStakes the largest values and αtends to over emphasize the asymp- totic region where α→∞, Mejia-Rodriguez and Trickey proposed a third indicator84in which the radial integration in the previous equation is restricted to a small sphere with a radius of 4 bohrs that surrounds the system, leading to Δnear α=4π Nnear∫4 0dr r2ρ(r)∣αorb(r)−αapprox(r)∣, (30) where Nnearis the number of electrons inside the sphere. Clearly, the lower the values of σ,Δα, and Δnear α, the better the agreement with torb S(r)of the kinetic energy density tapprox S(r). In Table IV, one can see the average values of σ,Δα, and Δnear α, for each test set, for the group of functionals considered. It is observed that in the cases corresponding to GGA functionals, the values are larger than those for the functionals that include a contribution of the Laplacian of the electronic density. It is impor- tant to recall that in the gradient expansion of the kinetic energy for slowly varying densities, the second order term of the enhance- ment factor, which corresponds to Eq. (19), has the additional term8,9bTq, where qcorresponds to the reduced density Laplacian, q=∇2ρ(r)/[4(3π2)2/3ρ5/3(r)], and the coefficient bTmay also be fixed through the GEA, leading to a value of bGEA T=20/9, or through other procedures.85,90,133,134In this context, the enhancement fac- tor goes beyond the GGA because it becomes a function of the reduced density gradient and of the reduced density Laplacian, FT(s, q). Note that when this Laplacian term, bTq, is multiplied by the local Thomas–Fermi kinetic energy given in Eq. (4), its integral over the whole space, for finite systems, is equal to 0 so that its addition to the enhancement factor does not contribute to the total KE nor to the KE potential.85Nevertheless, its presence is important for the descrip- tion of the local KE. However, the Laplacian of the density diverges at the nuclei and has very large negative values in the regions close to the nuclei, which leads to negative values of the KE density, in con- tradiction with the properties of torb S(r), which according to Eq. (1) must be positive and finite everywhere. Thus, in order to fulfill these requirements, one can regularize the functional by enforcing the Weizsacker lower bound, that is, Freg T(s,q)=max(FT(s,q),FW(s)), at every point.88,135All the functionals presented in Table IV with the Laplacian term have been regularized. In addition, in TFLopt and in J. Chem. Phys. 154, 084107 (2021); doi: 10.1063/5.0040973 154, 084107-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE IV . Average σ,Δα, andΔnear αvalues for several kinetic energy functionals for the three test sets considered in this work (a.u.). σ Δα Δnear α Functionals A18 GN M30 A18 GN M30 A18 GN M30 GGA APBEK500.533 0.432 0.527 1.778 1.839 1.733 1.080 1.348 1.012 LC94430.532 0.432 0.527 1.763 1.824 1.723 1.071 1.335 1.009 WPBEK 0.530 0.429 0.527 0.924 0.956 1.158 0.843 0.925 0.983 GGA + Laplacian TFLreg850.075 0.073 0.085 0.173 0.186 0.230 0.111 0.132 0.186 TFLopt840.043 0.047 0.057 0.165 0.171 0.211 0.089 0.104 0.158 GEA4reg880.201 0.167 0.195 0.390 0.429 0.357 0.230 0.342 0.225 PCreg880.060 0.073 0.069 0.189 0.310 0.177 0.365 0.397 0.389 PCopt840.047 0.068 0.051 0.330 0.367 0.294 0.170 0.280 0.164 PCopt, the coefficients of the s2andqterms have been optimized84 to minimize the value of (Δα+Δnear α). One can see that indeed there is a significative reduction in the values of σ,Δα, and Δnear αso that the presence of the Laplacian leads to a substantial improvement of the KE density. However, this regularization has two important con- sequences. On the one hand, the functional becomes system depen- dent because the values at the points where FT(s,q) is replaced by FW(s) are specific for each particular system. On the other hand, the integral of the Laplacian over the region that is left after the regu- larization is not equal to 0 so that the presence of this additional term in the KE density leads, in this case, to a different value of the total KE than the one obtained when the Laplacian term is absent. The analysis of Tables II and IV indicates that the presence of the regularized Laplacian term improves the KE density, but, at least for the functionals reported, it does not lead to a good description of the KE. Therefore, if we add the term bTqin Eq. (25), there is a reduc- tion in the value of σ, but the local kinetic energy becomes negative near the nuclei, and if one carries out the regularization to eliminate the negative values, one loses the level of accuracy achieved for the total KE with Eq. (25), for the test sets reported in Table II. III. PAULI KINETIC ENERGY As it was previously mentioned, the KE density is not uniquely defined. In fact, there are two commonly used expressions in terms of the KS orbitals, the one given by Eq. (1) and TS[ρ]=∫drtorbL S(r)=∫drN ∑ i=1φ∗ i(r)(−1 2∇2)φi(r). (31) Although both KE densities integrate to the same value, they are different at the local level,8 torb S(r)=torbL S(r)+1 4∇2ρ(r). (32)It is important to mention that due to the positive definite nature of torb S(r), in contrast with torbL S(r), which can be negative, it is usually preferred to describe the local KE.90,134 From the perspective of the development of the KE functional of electronic density, this positive definite behavior at the local level has been related to the description of TS[ρ] in terms of two contributions,136 TS[ρ]=∫drtS(r)=TW[ρ]+Tθ[ρ]=∫dr(tW(r)+tθ(r)), (33) where TW[ρ] is the Weizsacker term already defined in Eqs. (17) and (18) and Tθ[ρ] is the Pauli kinetic energy. The presence of the Weizsacker term is very important because it is equal to the exact TS[ρ] for one-electron and for two-electron singlets, and its func- tional derivative with respect to ρ(r) leads to an Euler equation that together with the Pauli potential,70,136,137which is given by vθ(r) =δTθ[ρ]/δ ρ(r), and the Kohn-Sham exchange-correlation, Coulombic, and external potentials is an exact equation for the square root of the electronic density. Additionally, from the local viewpoint, tW(r), by itself, provides a very good description of torb S(r) in the regions close to the atomic nuclei,138and the presence of δ TW[ρ]/δ ρ(r) leads to the correct behavior of ρ(r) at large distances from the nuclei.70 On the other hand, the Pauli kinetic energy contains all the effects of the antisymmetry requirement. Since the Weizsacker con- tribution is a lower bound to the total KS kinetic energy, then Tθ[ρ] ≥0. Furthermore, it can be shown that136 tθ(r)≥0∀r, (34) tθ(r)/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne→ ∣r∣→∞0, (35) and vθ(r)=δTθ[ρ] δ ρ(r)≥0∀r. (36) J. Chem. Phys. 154, 084107 (2021); doi: 10.1063/5.0040973 154, 084107-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Thus, because tW(r)≥0 for all values of r, the total KE density related to Eq. (33) satisfies the condition tS(r)=tW(r)+tθ(r)≥0∀r (37) in agreement with the positive definite nature of torb S(r)defined in Eq. (1). In order to satisfy the constraints given by Eqs. (34) and (35) when the Pauli KE is expressed through the GGA expression, namely, TGGA θ[ρ]=∫drtTF(r)FGGA θ(s), (38) the enhancement factor, FGGA θ(s), must fulfill the following condi- tions:90 FGGA θ(s)≥0∀s (39) and FGGA θ(s)/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne→ s→∞0. (40) In addition, from the integrands in the right-hand side of Eq. (3), in the third equality of Eq. (33), and in Eq. (38), combined with Eqs. (37) and (19), one finds that FGGA θ(s)/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne→ s→01 +(μT−5/3)s2. (41) According to Eqs. (33) and (37), the enhancement factors of the total and the Pauli kinetic energies are related by FGGA T(s)=5 3s2+FGGA θ(s). (42) However, if one considers that the enhancement factor of the Pauli KE could be obtained, according to this relationship, by making use of the approximation of the total KE enhancement factor given in Eq. (25), FWPBEK T(s), one would be led to an expression for FGGA θ(s) that does not fulfill the positivity condition indicated in Eq. (39); see Fig. 2. Thus, although the approximate FWPBEK T(s)satisfies several important constraints and, as we have seen, leads to a reasonable description of the total KE, when used in combination with Hartree– Fock densities, it does not mean that the KE energy density that generates could properly describe the local features that character- ize the Weizsacker and the Pauli KE densities separately. That is, the Weizsacker term by itself is a very important local component to obtain a rather good description of the KE density near to the nuclei and far away, when r→∞. On the other hand, the positivity of the Pauli term by itself is also important because it guarantees that when used through the decomposition of the total KE density given by Eq. (37), the latter will be positive or equal to 0 at every point of space. In this context, it seems that an alternative approach could be based on directly performing an interpolation between the lim- its given by Eqs. (41) and (40), making use of a function of the reduced density gradient that fulfills the positivity condition given by FIG. 2 . Plot of the enhancement factor as a function of the reduced density gra- dient. The green curve corresponds to FWPBEK T(s)given by Eq. (25), and the red curve corresponds to [FWPBEK T(s)−(5/3)s2]. Eq. (39). However, in order to retain the same accuracy in the total KE calculated with Hartree–Fock densities than the one obtained with Eq. (25), one can propose an expression for the Pauli enhance- ment factor that satisfies Eq. (39) and that reduces to the values expressed by Eqs. (41) and (40) in the limits s→0 and s→∞, respec- tively, with an additional parameter that is fixed so that the Pauli KE obtained through this expression, plus the Weizsacker contribution, leads to the same values as those obtained through FWPBEK T(s). This way, the local behavior of the KE density is improved making use of the global information contained in the parameter. A simple expression that may be used to achieve these objec- tives may be constructed as follows. Through the analysis of the Pauli KE for the atoms of the first row of the periodic table, Li to Ne, and the noble gases atoms, Ne to Rn, obtained by subtracting from the Hartree–Fock KE the Weizsacker contribution, one finds that in order to satisfy Eqs. (41) and (40), the Pauli enhancement factor for the light atoms must decay to 0 very fast, while as one advances from Ne to Rn, it must decay gradually more slowly. Thus, to cover both regimes, one may consider the combination of two functions, which satisfy both Eqs. (41) and (40), namely, Fθ1(s)=eαs2 (1 +b1s4)2andFθ2(s)=eαs2 (1 +b2s4)2, (43) where α=μT−5/3 and the parameters b1andb2are fixed so that Fθ1(s) and Fθ2(s) decay to 0 fast and slowly, respectively, as the reduced density gradient increases from 0 to infinity. This way, the Pauli enhancement factor may be written in the form FWPBEK θ(s)=AθFθ1+(1−Aθ)Fθ2, (44) J. Chem. Phys. 154, 084107 (2021); doi: 10.1063/5.0040973 154, 084107-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Plot of the Pauli enhancement factor FWPBEK θgiven by Eq. (44) as a function of the reduced density gradient for several atoms. where Aθand its complement (1 −Aθ) represent the fraction of the fast and slow decay functions, respectively. Accordingly, if one defines the quantities I1=∫drtTF(r)Fθ1(s)andI2=∫drtTF(r)Fθ2(s), (45) one can set the values of b1and b2so that I1and I2are smaller and greater than the Pauli KE of the Li and the Rn atoms, respec- tively. Through this procedure, one finds that one can set b1= 50 and b2= 0.5. Once Fθ1(s) and Fθ2(s) are known, one can make use of the Pauli KE associated with the enhancement factor proposed in this work, namely, TWPBEK θ=TWPBEK S−TW, (46)where the quantity TWPBEK S is obtained by substituting Eq. (25) into Eq. (3) to determine the value of Aθmaking use of Eqs. (38) and (43)–(46) through the following expression: Aθ=I2−TWPBEK θ I2−I1. (47) Since by construction, TWPBEK θ<I2andI1<I2, then 0<Aθ<1, as expected. This way, as in the case of regularized Laplacian dependent functionals, the value of Aθand, therefore, the enhancement factor expressed in Eq. (44) are system dependent. Note that through this procedure, the local KE tWPBEK S,θ(r)=tW(r)+tWPBEK θ(r)=tW(r)+tTF(r)FWPBEK θ(s)(48) is different, at the local level, from tWPBEK S(r), Eq. (26), although, by construction, both local functions integrate to the same total KE, that is, if the integral over tWPBEK S,θ(r)is denoted as TWPBEK S,θ , then TWPBEK S=TWPBEK S,θ . In Fig. 3, we present a plot of the Pauli enhancement factor for Li and the noble gases Ne through Rn. One can observe that the KE criterion imposed seems to indicate that the relevant region for FGGA θ(s)occurs in the interval 0 ≤s≤1.5, and although it seems that the changes are small as the number of electrons increases, the total KE is very sensible with respect to these changes. In Table V, we report the average values of σ,Δα, and Δnear α, for each test set, for the functional corresponding to the local KE given by Eq. (48) and compare it with other GGA Pauli KE function- als. In first instance, one can see that there is an improvement with respect to the values obtained through tWPBEK S(r), Eq. (26), indicat- ing that while retaining the performance in the description of total KE energies, through the present procedure, one can improve the local behavior by imposing the constraints associated with the Pauli KE. The present Pauli functional also gives a slightly better descrip- tion in comparison with other GGA Pauli KE functionals and gets closer to the performance of functionals that include a contribution of the Laplacian of the electronic density. However, the latter lead to significantly larger errors in the total KE. Again, in the supplemen- tary material, one can see the values of all the individual systems that conform the three test sets for all the functionals reported in this section. In Fig. 4, we present a plot of FT(r) =tS(r)/tTF(r) as a func- tion of the distance from the nucleus in the Ne atom when tS(r) TABLE V . Average σ,Δα, andΔnear αvalues for several GGA Pauli kinetic energy functionals for the three test sets considered in this work (a.u.). σ Δα Δnear α Functionals A18 GN M30 A18 GN M30 A18 GN M30 VT84K530.726 0.558 0.716 0.463 0.499 0.674 0.458 0.487 0.631 LKT600.579 0.477 0.578 0.394 0.440 0.612 0.389 0.428 0.570 PGS960.466 0.400 0.478 0.357 0.417 0.608 0.351 0.404 0.566 WPBEK- θ 0.274 0.297 0.337 0.307 0.384 0.611 0.300 0.371 0.570 J. Chem. Phys. 154, 084107 (2021); doi: 10.1063/5.0040973 154, 084107-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . Plot of the enhancement factor FT(r) as a function of the distance from the nucleus for the Ne atom. In the case of HF, FT(r)=torb S(r)/tTF(r), for WPBEK (Pauli), FT(r) = (5/3) s2(r) +Fθ(r), with Fθ(r) given by Eq. (44), and for TFLreg, FT(r) = 1 + (20/9) q(r) with regularization. is the exact one determined with the Hartree–Fock orbitals, when tS(r) is replaced by84,85tTFLreg S(r)=tTF(r)[1 +(20/9)q], and when it is replaced by tWPBEK S,θ(r). One can see that the latter, despite not including the Laplacian term, reproduces the shape of the exact function reasonably well. Finally, it is important to mention that we have determined the Pauli potential, vθ(r) =δTθ[ρ]/δ ρ(r), through the enhancement fac- tor given in Eq. (44), for the noble gas atoms Ne to Rn, to confirm that it is a non-negative function, in agreement with Eq. (36). IV. CONCLUDING REMARKS In the present work, we have analyzed the total and the local KE in the generalized gradient approximation. For the case of the total KE, we have found that the enhancement factor may be developed by imposing the correct behavior in the limits when s→0 and when s→∞and making use of the conjoint conjecture for the interpo- lation between these two limits since following this procedure in which all the parameters are fixed non-empirically either through the satisfaction of some of the known KE constraints or, in the intermediate region, through constraints that are associated with the exchange energy functional, we have derived an enhancement factor that leads to a rather good description of the KE when Hartree–Fock densities are used. However, it has been found that through this procedure, one does not arrive at a good description of the KE density, unless one goes beyond the GGA level, by incorporating terms that depend on the Laplacian of the density. Thus, one can see that at the GGA level, it is more convenient to express the total KE density as thesum of the Weizsacker and the Pauli KE densities. In this con- text, the enhancement factor of the latter can also be developed by imposing the correct behavior in the limits when s→0 and when s→∞, together with the positivity condition, but by making use, in this case, of the information contained in the total KE for the interpolation between these two limits. This way, one improves the KE density without losing the accuracy achieved in the total KE formulation. Finally, it is important to note that in the present work, we have restricted the development of the total and the Pauli KE functionals to the GGA level. However, one could follow the same procedure with functionals that depend on the reduced density gradient and Laplacian in order to improve, particularly, the local behavior of the KE density. It is important to mention that the ultimate goal in orbital- free DFT should be to develop a KE functional that provides a good description of the KE and the KE density and that, at the same time, entails a functional derivative that leads to an Euler–Lagrange equa- tion able to produce an accurate electronic density. In this context, the approach adopted in this work may be considered as a useful intermediate step in that direction. That is, we have derived in a first step a KE functional that when used in combination with densities of Hartree–Fock or orbital Kohn–Sham quality leads to a reasonable description of the KE. Then, in a second step, the KE thus obtained has been used to derive a KE density that is given by the explicit sum of the Weizsacker and the Pauli terms. Thus, while the latter is approximated, the former is a fundamental component whose func- tional derivative in the Euler–Lagrange equation is the key to obtain an accurate electronic density close to and far away from the nuclei. Therefore, it seems that future development could be oriented by imposing more constraints associated with the KE functional deriva- tive139and its relationship with the KE energy through the virial theorem140,141in the second step, together with the improvement of the KE functional derived in the first step. SUPPLEMENTAL MATERIAL See the supplementary material for the values of all the indi- vidual systems that conform the three test sets for all the functionals and quantities reported in the Tables. ACKNOWLEDGMENTS We thank the Laboratorio de Supercómputo y Visualización of Universidad Autónoma Metropolitana-Iztapalapa and the Labora- torio Nacional de Cómputo de Alto Desempeño (LANCAD) for the use of their facilities. 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Chaos 31, 023133 (2021); https://doi.org/10.1063/5.0011622 31, 023133 © 2021 Author(s).Nonlinear excitations of magnetosonic solitary waves and their chaotic behavior in spin-polarized degenerate quantum magnetoplasma Cite as: Chaos 31, 023133 (2021); https://doi.org/10.1063/5.0011622 Submitted: 23 April 2020 . Accepted: 01 February 2021 . Published Online: 19 February 2021 Zakia Rahim , Muhammad Adnan , and Anisa Qamar ARTICLES YOU MAY BE INTERESTED IN Rogue waves for the fourth-order nonlinear Schrödinger equation on the periodic background Chaos: An Interdisciplinary Journal of Nonlinear Science 31, 023129 (2021); https:// doi.org/10.1063/5.0030072 Solitons in complex systems of chiral fields with Kuramoto interactions Chaos: An Interdisciplinary Journal of Nonlinear Science 31, 023138 (2021); https:// doi.org/10.1063/5.0039991 Multistability for nonlinear acoustic-gravity waves in a rotating atmosphere Chaos: An Interdisciplinary Journal of Nonlinear Science 31, 023108 (2021); https:// doi.org/10.1063/5.0020319Chaos ARTICLE scitation.org/journal/cha Nonlinear excitations of magnetosonic solitary wavesandtheirchaoticbehaviorinspin-polarized degenerate quantum magnetoplasma Cite as: Chaos 31, 023133 (2021); doi: 10.1063/5.0011622 Submitted:23April2020 ·Accepted:1February2021 · PublishedOnline:19February2021View Online Export Citation CrossMark Zakia Rahim,1Muhammad Adnan,2,a) and Anisa Qamar1 AFFILIATIONS 1DepartmentofPhysics,UniversityofPeshawar,Peshawar25000, Pakistan 2DepartmentofPhysics,KohatUniversityofScience&Technology( KUST),Kohat26000,Pakistan a)Author to whom correspondence should be addressed: adnan_physiks@yahoo.com ABSTRACT The quantum hydrodynamic model is used to study the nonlinear propagation of small amplitude magnetosonic solitons and their chaotic motions in quantum plasma with degenerate inertialess spin-up electrons, spin-down electrons, and classical inertial ions. Spin effects are considered via spin pressure and macroscopic spin magnetization current, whereas the exchange effects are considered via adiabatic local density approximation. By applying the reductive perturbation method, the Korteweg–de Vries type equation is derived for small ampli- tude magnetosonic solitary waves. We present the numerical predictions about the conservative system’s total energy in spin-polarized and usual electron–ion plasma and observed low energy in spin-polarized plasma. We also observe numerically that the soliton characteristics are significantly affected by different plasma parameters such as soliton phase velocity increases by increasing quantum statistics, magnetization energy, exchange effects, and spin polarization density ratio. Moreover, it is independent of the quantum diffraction effects. We have analyzed the dynamic system numerically and found that the magnetosonic solitary wave amplitude and width are getting larger as the quantum statis- tics and spin magnetization energy increase, whereas their amplitude and width decrease with increasing spin concentration. The wave width increases for high values of quantum statistic and exchange effects, while their amplitude remains constant. Most importantly, in the presence of external periodic perturbations, the periodic solitonic behavior is transformed to quasiperiodic and chaotic oscillations. It is found that a weakly chaotic system is transformed to heavy chaos by a small variation in plasma parameters of the perturbed spin magnetosonic solitary waves. The work presented is related to studying collective phenomena related to magnetosonic solitary waves, vital in dense astrophysical environments such as pulsar magnetosphere and neutron stars. Published under license by AIP Publishing. https://doi.org/10.1063/5.0011622 In recent times, considerable interest is devoted to spin-polarized plasma featuring electron spin orientation mismatch. Here, we focused on the nonlinear localized solution as well as the chaotic trajectories associated with magnetosonic waves. The nonlinear Korteweg–de Vries type equation accounts for the evolution of small but finite amplitude magnetosonic waves. The collective modes pile up to weak chaos and eventually to fully chaotic based on the configurational sensitivity. We found that the total energy of the spin-polarized system is high compared to the usual electron–ion quantum plasma.I. INTRODUCTION In recent times, quantum plasmas have been extensively stud- ied due to its potential applications in dense astrophysical environ- ments (such as white dwarf, neutron stars, magnetars)1in the labo- ratory, for instance, in the laser–matter interaction experiments2,3 and in the nano-scale electromechanical systems.4–6Other recent applications include the expansion of a quantum electron gas into a vacuum,7the self-consistent dynamics of Fermi gases,8quan- tum plasma echoes,9quantum plasma instabilities,10and quantum Penrose diagrams.11Such a cold, dense plasma system can be of Chaos31,023133(2021);doi:10.1063/5.0011622 31,023133-1 PublishedunderlicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha considerable importance when the thermal de Broglie wavelength is comparable or equal to the inter-particle distance1and quantum effects such as diffraction, statistical Fermi pressure, exchange inter- action, and spin play a significant role in the dynamics of collective modes.8,12,13The exchange interaction and spin effects also come into play in a plasma medium when the difference between two spin energy states is greater than the thermal energy, whereas the higher dense Debye sphere does not affect the importance of spin.14 In the past few decades, spin quantum plasma has been rapidly becoming an emerging interest in the scientific community. The quantum hydrodynamical (QHD) equations for a single spin-half particle were first derived and analyzed by Takabayasi.12Later on, Kuz’menkov et al.13,15studied the effects of electron spin on plasma dynamics and derived many-particle QHD equations, which con- sist of the Euler equation, the continuity, the energy balance, and the magnetic moment evolution equations for spin-half quantum plasma. The model has been used in the excitations of collective modes in spin-half plasma, such as spin waves, theory relating to angular momentum, and spin-related forces.16,17The spin waves were discussed in Ref. 18by using the spinor Bose condensation. A set of separate spin evolution (SSE) QHD equations have recently derived from the Pauli equation, which treats the spin-up and spin- down electrons as two different fluids. Here, the Fermi pressure comes from the difference in the population of spin-up and spin- down state electrons in the presence of an external magnetic field, which revealed a new mode called spin electron acoustic waves (SEAW).19The spin polarization properties of plasmas and gases are essential in the optical orientation of atoms on the electrical conductivity, in the intensity of radiation in plasma,20and the emer- gence of new research areas such as spintronics.21One way to create such spin-polarized plasma is to use a ferromagnetic cathode in a glow discharge.22,23Electrons’ interactions can also be significant in cold, dense plasma systems due to electron spin-half effects. The electron’s interaction can be separated into the Hartree term (leads from the electrostatic potential of total electron density),24and the exchange interaction term appeared because of the anti-symmetric electron wave-functions.25Some electronic devices with increasing minimization degrees lead to a high value of electron density for which the exchange effects may no longer be negligible. The effect can be expressed in QHD as a complicated time-dependent electron density function via adiabatic local density approximation such as Vxc=0.985 n1/3 ee2 ε/bracketleftbigg 1+0.034 n1/3 eaBln/parenleftbig 1+18.376 n1/3 eaB/parenrightbig/bracketrightbigg , (1) where aB=ε/planckover2pi12/mee2being the Bohr radius where εis the dielectric permeability constant of the medium.26,27 The classical magnetohydrodynamic (MHD) theory is extended to magnetized quantum plasmas and a developed quantum mag- netohydrodynamic (QMHD) model leading to different types of localized mode shocks, soliton, vortices, and double layers. These nonlinear modes arise due to different plasmas, namely, disper- sion, nonlinearity, and dissipation. Including the spin magnetization effects, Brodin and Marklund28extended the QMHD model and studied the propagation properties of nonlinear waves in quantum magnetoplasma. They also discussed that the plasma characteristicsmight be change by the spin properties of the electron and the ferro- magnetic behavior of plasma in low temperature and higher dense systems.29Marklund et al.30investigated the effects of quantum diffraction and spin on magnetosonic solitons in non-degenerate quantum plasma. The two dimensional nonlinear fast and slow spin magnetosonic solitary waves (SMSWs) with resistive effects in spin-half quantum plasma were studied in the framework of small amplitude perturbation.42Recently, dust magnetosonic waves with spin and EC effects have been studied by Maroof et al.43and found that the dust concentration and EC significantly modify the dispersive properties of excitations. Moreover, multidimensional magnetosonic monotonic and oscillatory shock waves in dissipa- tive quantum plasma with EC and other quantum effects were also studied.44Andreev36considered the SSE in the QMHD model with a Coulomb exchange interaction between the spin-down elec- trons and showed a spin electron acoustic soliton influenced by concentration, spin polarization, and exchange interaction. While investigating the nonlinear dynamics in quantum plas- mas, another astonishing property of the nonlinear systems is the chaotic oscillations with a small number of degrees of freedom com- pared to periodicity (solitonic) in a system with a large number of degrees of freedom.45The interesting study of periodic, quasiperi- odic, and chaotic behaviors in the presence of external force per- turbation is a growing research area in a nonlinear plasma system. A nonlinear equation such as the Korteweg–de Vries (KdV) lead- ing from many physical fields is a completely integrable system.46 The chaotic behavior of perturbed systems such as sine-Gordon, nonlinear Schrödinger, and KdV have been studied numerically.47,48 However, some perturbations in many physical processes convert the integrable (solitonic) system to a chaotic dynamics.49,50In our recent study, we have explored the chaotic behavior of dust acous- tic waves in Thomas–Fermi plasma and showed the effects of dust concentration and external force on the transition of periodic behav- ior to aperiodic behavior.51However, the quasiperiodic and chaotic motions of the magnetosonic waves are not reported in the liter- ature. Our work investigates the total energy of the conservative system related to the KdV equation and focuses on the dramatic changes of magnetosonic solitonic structures to chaotic motion with SSE and exchange effects. The purpose of the present study is to consider quantum effects with exchange, external force effects, and developed model to the nonlinear magnetosonic solitary waves (MSWs) in spin-polarized plasma, particularly to the soliton formation. One of the interest- ing features of this investigation is applying the exchange term to the two-fluid model of electrons. An electron with spin-up and elec- trons with spin-down states will be treated separately. We also show that the exchange effects play a dominating role in the formation of MSWs. Moreover, we also study that the external periodic force is responsible for transitioning periodic behavior to quasiperiodic and chaotic behaviors. This work is organized as follows: we present the developed model equations for the ion-scale electromagnetic excitations, prop- agating perpendicular to the external magnetic field in Sec. II. The KdV equation is derived with the help of the reductive perturbation technique in Sec. III. Section IVis devoted to numerical solutions of the KdV equation and the conservative system’s total energy. The effects of the external periodic force on the integrable system Chaos31,023133(2021);doi:10.1063/5.0011622 31,023133-2 PublishedunderlicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha will be discussed in Sec. V. Finally, we summarize our results in Sec.VI. II. MODEL EQUATIONS Here, we present a model for nonlinear quantum electron–ion magnetoplasma excitations having classical ions and separated spin-up and spin-down states non-relativistic degenerate electrons. We develop the magnetohydrodynamic equation with electrons’ exchange potential by writing down the usual one fluid equations for ions and electrons with spin-up and spin-down states separately. To study the magnetosonic spin soliton in low-frequency regimes (in comparison with the electron gyro-frequency), the electrons’ iner- tia and displacement current in Faraday’s law will be ignored. In the Cartesian coordinate system, the external magnetic field and spin magnetization are assumed to be along the z-direction and can be expressed as− →B=B(x,t)ˆzand− →M=M(x,t)ˆz, respectively, where ˆz is the unit vector along the z-coordinate. The multi-fluid QMHD model is considered in low-frequency perturbations and neglects the ion quantum effects due to its large inertia. The ion dynamic equations are given by14,29 midvi dt=eE+e(vi×B), (2) ∂ni ∂t+∇.(nivi)=0. (3) The basic set of equations for inertialess spin-up and spin- down electrons can be expressed as 0= −nse(E+vs×B)−∇Ps+FQs, (4) ∂ns ∂t+∇.(nsvs)=0, (5) dSs dt=2µe /planckover2pi1Ss×B. (6) The subscript s/parenleftbig =e↑,e↓/parenrightbig denotes the electron with spin-up and spin-down states, respectively. In the above equation, d/dt =∂/∂t+(vi.∇)is the hydrodynamic derivative, vi(e)[=vi(e)(x,t)ˆx] is the ion (electron) velocity along the x-direction, and nj(mj) being the jth particle density (mass) where j=i,e↑,e↓. Consid- ering the spin-1/2 plasma, one may include the spin polariza- tion effect of the degenerate electron gas to the pressure35,55as Pe=/braceleftBig ϑ3D(3π2)2/3/planckover2pi12n5/3 e/bracerightBig /(5me), where ϑ3D=[(1+κ)5/3 +(1−κ)5/3]/2 is the coefficient of spin polarization of the elec- tron gas, spin polarization κ=3µBB0/2εFe, withµBbeing the Bohr magneton,εFethe electron Fermi energy, and /planckover2pi1the Planck con- stant. However, for unpolarized electrons, equation of pressure can be written as Pe= {(3π2)2/3/planckover2pi12n5/3 e}/(5me). The partial pressure caused by the evolution of each spin-up and spin-down electron isPe↑= {(6π2)2/3/planckover2pi12n5/3 e↑}/(5me)andPe↓= {(6π2)2/3/planckover2pi12n5/3 e↓}/(5me), respectively. The relation of the electron concentration with the coefficient of spin polarization is presented as ne=ne↑+ne↓, wherene↑=(1+κ)n0/2 and ne↓=(1−κ)n0/2. The quantum force FQs including the exchange potential30is given by FQs=ns/planckover2pi12/Gamma1D 2me∇/parenleftbigg∇2√ns√ns/parenrightbigg −µens 2/planckover2pi1∇(Ss.B)+ns∇Vxcs, (7) where the first term represents the quantum Bohm potential and the second term represents the spin evolution due to the spin- 1/2 effect of electron, where the +(−)sign denotes the spin-up (spin-down) states. Furthermore, µe=|µB|=e/planckover2pi1/2meis the Bohr magneton and TFe=(3π2n0)2/3/planckover2pi12/(2mekB)is the electron Fermi temperature measured in unit of energy with the Boltzmann con- stant kB. The pre-factor /Gamma1Dis the gradient correction to the Bohm potential in local density approximation that depends upon the wavenumber, frequency, and dimensionality. In strongly degener- ate plasmas for low-frequency excitations, /Gamma1D= −1/3 is chosen in one dimension.31–34The second term describes the action of the z-component of the magnetic field on the z-component of the parti- cle magnetic moment. In Eq. (5), the spin-flip is neglected because of the smaller temporal variation in the magnetic field compared to the inverse electron cyclotron frequency. We also assume the dynam- ics on a time scale greater than the inverse cyclotron frequency and smaller than the inverse spin transition frequency. For such kind of plasma, the spin vector Ssfrom spin evolution [Eq. (6)] reduces to Ss×B=0 and gives Ss= ∓(/planckover2pi1/2)ˆB. The spin vector Ss= ∓(/planckover2pi1/2)ˆB since spin-up and spin-down electrons have oppositely directed spin projection on the z-direction and, therefore, have opposite signs.14,28,39–41The last term is associated with the exchange force, which strongly depends on the spin polarization index (κ)of all electrons. Following the SSE-QMHD model35,36where the exchange term only involves the interaction of a spin-down electron, i.e., Vxe↓=0.985 e2ξ3Dn1/3 e↓ (8) for a three-dimensional partially spin-polarized electron gas dis- tributionξ3D=(1+κ)4/3−(1−κ)4/3. In this regime, the wave function is anti-symmetric on the space variable and symmetric on the spin variable relative to the permutation of spin-down elec- tron results to the same sign of an exchange interaction. How- ever, spin-up electrons give no contribution to the momentum equation.35,36Moreover, for high density and µBB0/Te/lessmuch1 envi- ronments, the exchange potential is dominated by the correlation potential effects. The polarization dependence of spin-down elec- tron exchange potential is more pronounced than the spin-up elec- tron exchange potential.37,38The Maxwell’s equations, i.e., Ampere’s law in a magnetized medium can take the form, given as ∇×B=µ0/parenleftbig Jp+Jm/parenrightbig , (9) where Jpis the plasma current density given as Jp=enivi−ene↑ve↑−ene↓ve↓, (10) andJm=∇×Msis the magnetization current density of the elec- tron with magnetization Ms= ±µensˆB. The displacement current in Eq. (7)is neglected because of the low value compared to J=Jp+Jmin a conducting medium (plasma). The Faraday’s law Chaos31,023133(2021);doi:10.1063/5.0011622 31,023133-3 PublishedunderlicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha takes the form ∇ ×E= −∂B ∂t, (11) ∇ ·B=0, ∇ ·E=4π(eni−ene↑−ene↓).(12) The quasi-neutrality condition is assumed ni≈n0≈ne↑+ne↓. The SSE description of electrons with different z-projections of spins fol- lows from the structure of the Pauli equation. The Pauli equation is a spinor equation and a short representation of two coupled equations. These are equations for spin-up and spin-down state electrons, and the spin z-projection (Sz)is related to the dif- ference between the concentrations of electrons. The difference Sz=ne↑−ne↓is caused by the external magnetic field. Since the electrons are negatively charged, their spins have a preferable direc- tion opposite to the external magnetic field. The spin polarization κis directly related to the effective magnetic field containing the external magnetic field and internal fields existing in magnetically ordered materials and caused by the exchange interaction with bound electrons of an incomplete d or f shell. Hence, the SSE- QMHD model gives a more accurate account to treat spin-up and spin-down electrons separately.35,36The above set of three fluid QMHD momentum equations for ions and spin-up and spin-down electrons with the help of Eq. (9)may be written in a normalized form as42,52,53 ni/bracketleftbigg∂vi ∂t+(vi.∇)vi/bracketrightbigg =(∇ ×B)×B−/parenleftbig 2δ↑/parenrightbig5/3β 6n2/3 e↑∇ne↑−/parenleftbig 2δ↓/parenrightbig5/3β 6n2/3 e↓∇ne↓ +H2 eδ↑/Gamma1D 2ne↑∇/parenleftbigg∇2√ne↑√ne↑/parenrightbigg +H2 eδ↓/Gamma1D 2ne↓∇/parenleftbigg∇2√ne↓√ne↓/parenrightbigg −(∇ ×Ms)×B+ε2 0βB∇B+γ1ξ3Dδ4/3 ↓∇n1/3 e↓. (13) Normalized equation of continuity for a spin-down state electron by using Eqs. (2),(5), and (10)may be written as ∂ne↓ ∂t+∇./bracketleftbigg ne↓/parenleftbiggB×(vi×B) B2/parenrightbigg/bracketrightbigg −∇./bracketleftbigg ne↓/parenleftbiggB×(∇×B)×B niB2+B×(∇×M)×B niB2/parenrightbigg /bracketrightbigg =0; (14) with the help of an ion momentum equation along with Eq. (10), the normalized magnetic induction equation can be obtained from Faraday’s law [Eq. (11)] as ∂B ∂t=∇×(vi×B). (15) It is important to mention here that in the standard QMHD model, we have neglected the Hall term and the anisotropic part of pres- sure such that the ion gyro frequency (/Omega1i)is larger than the wave frequency, i.e., /Omega1i/greatermuchω, whereas in the Hall–QMHD model, the ion gyrofrequency may be comparable to the wave frequency(/Omega1i∼ω).28,30,54Equation (12)for spin magnetosonic waves may be written in a normalized form as ni=δ↑ne↑+δ↓ne↓. (16) The number densities njare normalized by their respective equilib- rium densities n0j; other parameters are as follows: r/prime=r/Omega1i VA,v/prime i=vi VA,B/prime=B B0,M/prime s=µ0Ms B0,t/prime=/Omega1it, where/Omega1i=eB0/miis the ion gyrofrequency, the Alfven speed isVA=B0/√µ0n0mi, plasma beta β=c2 qs/V2 A=(2µ0n0iεFe)/B2 0 measures the quantum statistical effects with cqs=√2εFe/mibeing the quantum ion sound speed, µ0is the permeability of the free space with the Fermi energy of degenerate electron gas given asεFe=(3π2n0)2/3/planckover2pi12/(2me), and the dimensionless parameter He=/planckover2pi1/Omega1i//parenleftbig√memiV2 A/parenrightbig appears due to collective electron tunnel- ing through Bohm potential. The normalized magnetization energy Ms=ε0βtanh(α)ˆB=ε2 0βBwithε0=µBB0/Te; here, ne↑−ne↓ =n0tanh(α)≈n0α, where the Brillion function tanh (α)≈αfor µBB0/Te/lessmuch1 for spin-1/2 electron with α/parenleftBig =µBB Te/parenrightBig being the Zeeman energy normalized by Fermi temperature.14,29The term γ1= {0.985 e2(n0)4/3}/(miV2 A)is a dimensionless exchange inter- action plasma parameter, which shows that the exchange term is inversely proportional to the background magnetic field B0. Further- more,δ↑=n0e↑/n0i=(1+κ)/2,δ↓=n0e↓/n0i=(1−κ)/2 where the density polarization ratio κis given by κ=n0e↑−n0e↓ n0e↑+n0e↓,κ∈[0, 1]. (17) The value of κ=0 corresponds to the spin unpolarized system, κ=1 for fully polarized, whereas 0 ≺κ≺1 shows the partially spin-polarized system. The unperturbed non-zero total concentra- tion of the spin-up and spin-down electrons is ne0=n0e↑+n0e↓, whereas the difference of the concentrations of the spin-up and spin-down electrons ∇ne0=n0e↑−n0e↓is caused by the external magnetic field B0. III. DERIVATION OF THE KDV EQUATION We study the nonlinear magnetosonic soliton structures in a planar geometry with separated spin-up and spin-down electrons and non-degenerate ions. Here, the plane of MSWs (i.e., along x- axes) is propagating perpendicular to the external magnetic field, which is along z-axes. We consider dynamic, positively charged ions and electrons (spin-up and spin-down states separately) with exchange and other quantum effects. To obtain a spin-dependent soliton solution, we apply the reductive perturbation technique.56 Accordingly, the space and time variables can be stretched as ζ=/epsilon11/2(x−vt), τ=/epsilon13/2t,(18) where the small parameter /epsilon1measures the strength of nonlinear- ity and the MSW velocity is denoted by v. We also expand the Chaos31,023133(2021);doi:10.1063/5.0011622 31,023133-4 PublishedunderlicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha hydrodynamic dependent variables in terms of /epsilon1as nj=1+/epsilon1nj1+/epsilon12nj2+ · · ·· · · .., B=1+/epsilon1B1+/epsilon12B2+ · · ·· · · .., vi=0+/epsilon1vi1+/epsilon12vi1+/epsilon13vi2+ · · · .(19) Using the stretching (18) and expansions (19) in Eqs. (3) and(13)–(16)and collecting the lowest order terms in equal power of∼/epsilon13/2, we may get the solution of first order quantities in terms of magnetic field intensity as follows: ni1(ζ,τ)=B1(ζ,τ), vi1=vB1(ζ,τ), n1e↑(ζ,τ)=B1(ζ,τ), n1e↓(ζ,τ)=B1(ζ,τ), where wave speed vcan be calculated with the help of the above set of the solution as v2=1−2ε2 0β+(1+κ)5/3β 6+(1−κ)5/3β 6 −7γ1ξ3D 9/bracketleftBigg/parenleftbigg1+κ 2/parenrightbigg4/3 +/parenleftbigg1−κ 2/parenrightbigg4/3/bracketrightBigg . (20) Equation (20)exclusively shows that the phase velocity depends on the spin polarization index κ, the plasma beta β, Zeeman energy ε0, and exchange term γ1and is independent of quantum diffrac- tion effect He. Equation (20)also shows that as the values of spin polarization (means spin-up and spin-down electrons remain inde- pendent species) increase in the presence of quantum effects, the phase velocity gets smaller values and then approaches to larger values asκ→1 (can also be seen in Fig. 1 ). Moreover, as the values ofβ,ε0and exchange term γ1increase, the phase velocity FIG. 1.Phase velocity against κfor various values of (a) plasma beta for ε0=1.2,γ1=0.7,(b)spinmagnetizationenergy ε0forβ=0.2,γ1=0.7,and (c)exchangeterm γ1withβ=0.15,ε0=1.6.decreases with κas depicted in Figs. 1(a) –1(c). However, no change has been observed in the phase velocity for κ=0 or 1 by changing the exchange values, as presented in Fig. 1(c) . The next higher-order terms in ∼/epsilon15/2provide a set of equa- tions in terms of second-order quantities, and with the help of first order quantities, we obtain after long calculations; then, by integrating, we have ∂B1 ∂τ+AB1∂B1 ∂ζ+D∂3B1 ∂ζ3=0. (21) The coefficients of Eq. (21)are the nonlinear Aand dispersion D. Based on our plasma configuration, the coefficients acquired the following mathematical forms: A=1 2v/bracketleftbigg 3−6ε2 0β+4(1+κ)5/3β 9+4(1−κ)5/3β 9/bracketrightbigg −7γ1ξ3D 18v/bracketleftBigg/parenleftbigg1+κ 2/parenrightbigg4/3 +/parenleftbigg1−κ 2/parenrightbigg4/3/bracketrightBigg , (22) D=H2 e 24v. (23) Equation (21) represents the compressive or rarefactive solitary wave magnetic induction equation. One may also note that the non- linear coefficient is independent of the quantum diffraction effect, where the dispersion coefficient depends on the diffraction and other quantum effects. In order to get a real-valued secant hyper- bolic (or bell) type solution of Eq. (21), the nonlinear coefficient may be negative or positive, and the dispersion coefficient must be positive. The dispersion coefficient expressed by Eq. (23)has a neg- ative value. Hence, we solve the system (21)numerically in Sec. IV to obtain a soliton type solution. Soliton, like the solution of pos- itive and negative polarities, depends on the product of nonlinear and dispersion coefficients. Therefore, the coefficients AandDare responsible for such a situation that depend on β,κ,ε0,He, andγ1 such as (i) when AD>0 (positive polarity), the KdV solution gives the compressive MSWs (CMSWs). (ii) They also represent the rar- efactive MSWs (RMSWs) if AD<0; i.e., the soliton amplitude is observed negative (polarity). IV. CONSERVATION OF THE KDV EQUATION Equation (21) presented the KdV equation with SSE and exchange effects propagating in the reference frame of ζ+τ. One can also study the conservation of the stationary soliton from the well-known Korteweg–de Vries (KdV) equation by using the trans- formationη=ζ+u0τ, where u0is the speed of spin magnetosonic traveling waves. Using the notation ψ(η)=B1(ζ,τ)into Eq. (21) and integrating once with appropriate boundary conditions as η→ ∞ ,ψ→1,dψ/dη→0, Eq. (21)reduces to Dd2ψ dη2+A 2/parenleftbig ψ2−1/parenrightbig +u0(ψ−1)=0. (24) The above equation can also be expressed as d2ψ dη2+d/Gamma1 dψ=0, (25) Chaos31,023133(2021);doi:10.1063/5.0011622 31,023133-5 PublishedunderlicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha FIG. 2.The 3D graph of the energy function represented by system (24)for (left)spin-polarizedplasma(i.e., κ=0.1)and(right)nearlyelectron–ionplasma κ=0.9,where He=0.1,ε0=1.8,β=0.16,u0=0.2,andγ1=0.2. where/Gamma1is the corresponding potential energy of the system given by /Gamma1=1 6D(Aψ+3u0)ψ2−1 2D(A+2u0)ψ, whereas the total energy of Eq. (25)may be written as /Xi1/parenleftbig ψ,˙ψ/parenrightbig =˙ψ2 2+1 6D(Aψ+3u0)ψ2−1 2D(A+2u0)ψ. (26) The plots of energy function represented by Eq. (26)in the ψ−˙ψplane are presented in Fig. 2 for different spin polarization density ratios. We have observed lower energy values for κ=0.1, i.e., more electrons having in spin-up states and others are in spin- down states (see the left of Fig. 2 ), where large values of the energy function for nearly electron-ion plasma are shown in the right of Fig. 2 . To study periodic behavior numerically, Eq. (24)may be written in a dynamical form as dψ dη=Z, dZ dη= −u0 D(ψ−1)−A 2D/parenleftbig ψ2−1/parenrightbig .(27) System (27)shows a traveling solitary wave solution and will be ana- FIG. 3.The numerical solution of system (27)for different spin polarization densityratio(a)showstheplotof ψagainstspacecoordinate ηand(b)thecor- responding phase portrait in the Z–ψplane with(ψ,Z)=/parenleftbig 0.2,−1×10−6/parenrightbig , whereasothervaluesare He=0.05,ε0=1.8,β=0.16,γ1=0.1,u0=0.1.FIG.4.Thenumericalsolutionofsystem (27)fordifferentplasmabeta(a)shows theplotofψagainstspacecoordinate ηand(b)thecorrespondingphaseportrait intheZ–ψplanewithκ=0.7,whereasothervaluesarethesameasin Fig.3. lyzed numerically for different plasma parameters such as quantum diffraction He, plasma beta β, the energy of the spin-up and spin- down electrons characterized by the spin pressure parameters via ε0, exchange dependent term γ1, and spin polarization density ratio κ.Figure 3 shows a series of solitons for spin-polarized and nor- mal electron-ion plasma, respectively. Soliton amplitude and width show smaller values in nearly electron–ion plasma (i.e., κ=0.9) than for low spin index plasma (i.e., for κ=0.1) as depicted in Fig. 3(a) . Accordingly, we have also shown the phase portrait in Fig. 3(b) , which reveals different orbit centers for both types of plasma. One may also illustrate that the nonlinear coefficient and soliton speed have increasing values with increasing κ, where the dispersion decreases with increasing κ. We have also noted a strong influence of plasma beta on soliton characteristics, as depicted in Fig. 4 . For a small increase in βproduces a large increase in the soliton amplitude and width, as shown in Fig. 4(a) and their cor- responding phase portrait in Fig. 4(b) . The plasma beta increases (decreases) the dispersion (nonlinear) coefficient and also increases the MSW speed in spin-polarized plasma via the exchange effects. FIG.5.Thenumericalsolutionofsystem (27)fordifferentvaluesof ε0(a)shows theplotofψagainstspacecoordinate ηand(b)thecorrespondingphaseportrait in theZ–ψplane withβ=0.16,ε0=1.6, whereas other values are the same asinFig.4. Chaos31,023133(2021);doi:10.1063/5.0011622 31,023133-6 PublishedunderlicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha FIG.6.Thenumericalsolutionofsystem (27)fordifferentvaluesof He(a)shows the plot ofψagainst the space coordinate ηand (b) the corresponding phase portraitinthe Z–ψplanewithβ=0.15,whereasothervaluesarethesameas inFig.5(b). The influence of spin magnetic energy on the soliton solution is pre- sented in Fig. 5 , which shows that larger values of ε0enhance the soliton amplitude and width [see Fig. 5(a) ]. The phase portrait in Fig. 5(c) shows that the center of periodic oscillations is the same for different values of ε0. One may also notice that higher magnetic energy decreases the nonlinearity, increases the plasma medium’s dispersion, and decreases solitary wave speed. The Bohm potential, which only affects the dispersion coefficient, also influences the soli- ton properties. Higher values of Heincrease the soliton width where the amplitude remains constant, as shown in Fig. 6(a) . However, the nonlinearity and magnetosonic soliton speed are independent of the quantum diffraction effects. Hence, we have observed in the phase portrait of Fig. 6(b) that the center of periodic oscillations is the same for different values of He. In our study, we have also mentioned the effects of exchange potential on solitary waves. The larger exchange term increases the nonlinearity and dispersion in the medium, which leads to a lower speed of MSWs. Figure 7(a) shows that as the values of γ1increase, the width of periodic oscil- lations decreases with increasing amplitude. The phase portrait in FIG. 7.The numerical solution of system (27)for different values of exchange term(a)showstheplotof ψagainstspacecoordinate ηand(b)thecorresponding phase portrait in the Z–ψplane with boundary value (Z,ψ)=/parenleftbig 0.1,−10−6/parenrightbig , whereasothervaluesarethesameasin Fig.6(b).Fig. 7(b) shows that different centers of periodic oscillations are found for different exchange values. V. CHAOTIC BEHAVIOR OF THE PERTURBED SYSTEM In this section, we explore whether a system (21)can show chaotic or quasiperiodic behavior in the presence of external peri- odic force perturbation. We discussed in Sec. IVthat system (21) shows periodic (solitary) behavior in the absence of an external force. By considering the effects of an external force, the conservative system (21)may be presented as47,57 dψ dη=Z, dZ dη= −u0 D(ψ−1)−A 2D/parenleftbig ψ2−1/parenrightbig +f0cos(ωη),(28) where f0andωare the strength and frequency of the external peri- odic force, which is a root cause of the chaotic motions. Such kind of periodic force maybe exists in plasma in different forms such asf0cos(ωη),f0sin(ωη), and f0exp(ωη).58,59To explore the effects of different plasma variables on the transition of periodic oscilla- tions to aperiodic motions, we need to employ different numerical techniques such as (i) phase portrait analysis, (ii) time series analy- sis, (iii) Poincaré mapping, and (iv) Lyapunov exponent.45,60,61The phase portrait is a geometric representation of a dynamical system in the phase space, which gives information on whether aperiodic oscillations exist for a certain number of chosen parameters. It con- sists of plots in the form of limit cycles, attractor, and repellor, which strongly depend on the boundary conditions. Figures 8 –15(a) and15(b) presented plots in the (Z−ψ)-plane for different plasma parameters and show that chaotic behavior affects by quantum FIG. 8.The numerical solution of the perturbed system (28)is shown via theZ–ψplane forκ=0.1,ε0=1.84,He=0.1,γ1=0.2,u0=0.9, f0=0.72,ω=15.8 and different values of βsuch that (a) β=0.14, (b) β=0.16,and(c)thecorrespondingLyapunovexponentswithbou ndaryvalues (ψ,Z)=/parenleftbig 1,10−6/parenrightbig andβ=0.16. Chaos31,023133(2021);doi:10.1063/5.0011622 31,023133-7 PublishedunderlicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha FIG. 9.The numerical solution of the perturbed system (28)is shown via the Z–ψplane forκ=0.1,β=0.2,He=0.1,γ1=0.2,u0=0.9, f0=0.72,ω=15.8 and different values of ε0such that (a) ε0=1.52, (b) ε0=1.88,and(c)thecorrespondingLyapunovexponentswithbou ndaryvalues (ψ,Z)=/parenleftbig 1.0003,1 ×10−6/parenrightbig andε0=1.88. FIG. 10. The numerical solution of the perturbed system (28)is shown via theZ–ψplane forκ=0.1,β=0.2,ε0=1.88,γ1=0.2,u0=0.9, f0=0.1,ω=9.4 and different values of Hesuch that (a) He=0.1, (b) He=0.2,and(c)thecorrespondingLyapunovexponentswithboun daryvalues (ψ,Z)=/parenleftbig 1.003,1 ×10−6/parenrightbig andHe=0.1.FIG. 11.The numerical solution of the perturbed system (28)is shown via theZ–ψplane forκ=0.1,β=0.2,ε0=1.6,He=0.1,u0=0.9,f0=1.2, ω=13 and different values of exchange term such that (a) γ1=0.58, (b) γ1=0.74,and(c)thecorrespondingLyapunovexponentswithbou ndaryvalues (ψ,Z)=/parenleftbig 1.001,1 ×10−6/parenrightbig andγ1=0.74. FIG. 12. The numerical solution of the perturbed system (28)is shown via theZ–ψplane forγ1=0.74,β=0.2,ε0=1.6,He=0.1,u0=0.9, f0=0.4,ω=46 and different values of κsuch that (a)κ=0.1, (b)κ=0.9, and (c) the corresponding Lyapunov exponents with boundary values(ψ,Z) =/parenleftbig 1,1×10−6/parenrightbig andκ=0.9. Chaos31,023133(2021);doi:10.1063/5.0011622 31,023133-8 PublishedunderlicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha FIG. 13. The numerical solution of the perturbed system (28)is shown forγ1=0.5,β=0.2,κ=0.1,ε0=1.84,He=0.1,u0=0.9,f0=1.2, ω=46anddifferentvaluesofboundaryconditionsinthe Z–ψplanesuchthat (a)(ψ,Z)=(0.35,0),(b)(ψ,Z)=(0.96,−0.01),and(c)thecorresponding Lyapunovexponentswithboundaryvalues (ψ,Z)=(0.96,−0.01). FIG. 14. The numerical solution of the perturbed system (28)is shown via theZ–ψplane forγ1=0.1,κ=0.1,β=0.2,ε0=1.75,He=0.1, u0=0.8,f0=0.2 and different values of external force frequency ωsuch that (a)ω=2, (b)ω=8, and (c) the corresponding Lyapunov exponents with boundaryvalues (ψ,Z)=(0.52,−0.03)andω=8.FIG. 15. The numerical solution of the perturbed system (28)is shown viaZ–ψplane for γ1=0.1,κ=0.1,β=0.2,ε0=1.75,He=0.1, u0=0.8,f0=0.2,ω=10 and different values space coordinate (η) such that (a)η=0–0.04, (b)η=0–0.4, and (c) the corresponding Lyapunov exponentswithboundaryvalues (ψ,Z)=(0.41,0)andη=0–0.4. statistics, diffraction, magnetic energy, exchange, spin density ratio, driving frequency, and boundary values. These plots reveal that the plasma parameters play a key role in the transition of periodic oscil- lations to weak chaos and then developed chaos in the presence of external periodic perturbation. The plots also predicted that the amplitude of aperiodic oscillations changes from small values to large values by small variations in plasma variables and vice versa. Figure 8(a) shows that for small β(=0.148), weak chaotic behav- ior of Eq. (28)is observed, while a smaller change in βcan notice heavy, chaotic oscillations of enormous amplitude (=0.16)as seen inFig. 8(b) . Moreover, the positive largest Lyapunov exponents in Fig. 8(c) prove the strong aperiodic oscillations with large ampli- tudes and widths. Similarly, the low value of Zeeman energy shows quasiperiodic and weak chaotic oscillations, while strong chaotic uncorrelated vibrations of a sudden rise in amplitude for higher ε0(=1.8)can be seen in Figs. 9(a) and9(b), respectively. The cor- responding positive Lyapunov exponents in Fig. 9(c) prove the aperiodic nature of the system (28). The aperiodic random oscil- lations of large amplitude and width are observed in Figs. 10(a) and10(b) for different values of Bohm potential (He), and their corresponding Lyapunov exponent is expressed in Fig. 10(c) . The chaotic behavior of system (28)is also affected by the exchange term represented by Fig. 11 . The system can be transferred from quasiperiodic to weak chaotic and heavy chaotic oscillations of a large amplitude by varying the interaction value. For different spin polarization values (κ=0.1, 0.9), chaotic oscillations are noticed in Figs. 12(a) and12(b) ; moreover, different heavily developed chaos of a large amplitude and width for spin-polarized plasma with SSE can be seen in Fig. 12(b) , and accordingly, the Lyapunov Exponent Chaos31,023133(2021);doi:10.1063/5.0011622 31,023133-9 PublishedunderlicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha is presented in Fig. 12(c) . The chaotic motion is strongly affected by the boundary values; that is, a small change in initial values at η=0, (Z,ψ)=(0, 0.35)to(−0.01, 0.96)produces a drastic change in the behavior of aperiodic oscillations shown by Figs. 13(a) and13(b) and the Lyapunov exponent is plotted in (c). The figure also shows that the initial values transform the system from quasiperiodic to chaotic behaviors. The external periodic driving frequency (ω=2, 8)affects the amplitude and width of aperiodic oscillations and their Lya- punov exponent is represented by Fig. 14 . The traveling space coordinate represented by ηin system (28)brings a drastic change in the chaotic oscillations as depicted in Fig. 15 , as the values of space coordinate slightly change from η=0→0.04 toη=0→ 0.4, uncorrelated random oscillations of a higher amplitude and width are observed, whereas the related Lyapunov exponent is plot- ted in Fig. 15(c) . The Lyapunov exponents determine a quantitative measure of the divergence (or convergence) of given nearby tra- jectories for a dynamical system. If we assume a set of aperiodic oscillations for short time scales of given initial values in the phase space, the effects of dynamics will be distorted, which stretched along with some directions and contracted along with others, as seen inFigs. 8(c) –15(c). The large values of Lyapunov exponents prove the most unstable direction of flow. The value of Lyapunov expo- nents is zero for periodic oscillation; for quasiperiodic oscillations, some values are zero and others are negative. On the other hand, the chaotic motions have positive values (at least one value). As the graphs show that some negative values correspond to quasiperiodic behavior, zero value corresponds to periodic, while the highest pos- itive values correspond to the chaotic behavior of system (28). In all these figures, some plots show heavily chaotic behavior, while some are showing weak chaotic motions. VI. CONCLUSIONS In this study, the propagation of MSWs and their chaotic motion in a weakly coupled spin-polarized plasma comprising heav- ier ion and inertialess degenerate electrons with spin-up and spin- down states separately is investigated numerically. The main interest here is to study the effects of spin polarization numerically κ, quan- tum diffraction He, plasma beta β, the energy of the spin-up and spin-down electrons characterized by the spin pressure parameters viaε0, and exchange effects γ1on periodic and aperiodic oscil- lations. To show also the experimental significance of our work, we choose the dense astrophysical environments such as neutron stars, magnetars, and massive white dwarf where number densities are about n0≈1030−1035m−3, temperatures are T∼105−107K, and magnetic fields B0=109−1014G.62,63The numerical data for such a type of system have the value of He≤1,βhas some finite value≤1, whereas the spin energy and exchange terms get higher values asε0/greaterorsimilar1,γ1/greaterorsimilar0.1 and spin polarization κhaving a value between zero and one. Based on the above discussion, we used the reductive perturbation method to derive the KdV equation, which results in solitary waves produced due to the balance of the non- linearity and dispersion in the medium. We have explored that the conservative system’s total energy is smaller in spin-polarized plasma than the normal electron–ion plasma. We have also noted that with increasing spin index values, the soliton amplitude andwidth become smaller. However, with increasing quantum statis- tics and spin magnetization energy values, the soliton amplitude and width are increasing. In contrast, the higher values of quan- tum diffraction affect the periodic wave width increases, whereas their amplitude remains constant. The higher values of exchange term(γ1)and the solitary magnetosonic wave width decrease repre- sented by Eq. (27)where their amplitude increases. Moreover, peri- odic (solitonic) oscillations also change with changing the described parameters represented by phase portraits. External periodic forcing term is the key to quasiperiodic and chaotic motions in the conser- vative dynamical system (27). The numerical analysis shows that the periodic oscillations change to heavily aperiodic oscillations by vary- ing the plasma parameters. Figures 8 –15illustrate the transition of solitonic behavior to quasiperiodic and chaotic motions by chang- ing the values of quantum diffraction He, plasma beta β, the energy of the spin-up and spin-down electrons characterized by the spin pressure parameters via ε0, spin polarization density ratio κ, interac- tion effectγ1, boundary conditions, the external force periodic force frequency, and the space coordinate. The positive largest Lyapunov exponents show the chaotic oscillations of the dynamical system (28). 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5.0043658.pdf

J. Chem. Phys. 154, 104304 (2021); https://doi.org/10.1063/5.0043658 154, 104304 © 2021 Author(s).Rainbow scattering in rotationally inelastic collisions of HCl and H2 Cite as: J. Chem. Phys. 154, 104304 (2021); https://doi.org/10.1063/5.0043658 Submitted: 10 January 2021 . Accepted: 12 February 2021 . Published Online: 08 March 2021 Masato Morita , Junxiang Zuo , Hua Guo , and Naduvalath Balakrishnan ARTICLES YOU MAY BE INTERESTED IN Shape resonance determined from angular distribution in D 2 (v = 2, j = 2) + He D2 (v = 2, j = 0) + He cold scattering The Journal of Chemical Physics 154, 104309 (2021); https://doi.org/10.1063/5.0045087 Direct time delay computation applied to the O + O 2 exchange reaction at low energy: Lifetime spectrum of species The Journal of Chemical Physics 154, 104303 (2021); https://doi.org/10.1063/5.0040717 Classical molecular dynamics The Journal of Chemical Physics 154, 100401 (2021); https://doi.org/10.1063/5.0045455The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Rainbow scattering in rotationally inelastic collisions of HCl and H 2 Cite as: J. Chem. Phys. 154, 104304 (2021); doi: 10.1063/5.0043658 Submitted: 10 January 2021 •Accepted: 12 February 2021 • Published Online: 8 March 2021 Masato Morita,1,a)Junxiang Zuo,2 Hua Guo,2,b) and Naduvalath Balakrishnan1,c) AFFILIATIONS 1Department of Chemistry and Biochemistry, University of Nevada, Las Vegas, Las Vegas, Nevada 89154, USA 2Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, New Mexico 87131, USA a)Electronic mail: masatomorita2013@gmail.com b)Electronic mail: hguo@unm.edu c)Author to whom correspondence should be addressed: naduvala@unlv.nevada.edu ABSTRACT We examine rotational transitions of HCl in collisions with H 2by carrying out quantum mechanical close-coupling and quasi-classical tra- jectory (QCT) calculations on a recently developed globally accurate full-dimensional ab initio potential energy surface for the H 3Cl system. Signatures of rainbow scattering in rotationally inelastic collisions are found in the state resolved integral and differential cross sections as functions of the impact parameter (initial orbital angular momentum) and final rotational quantum number. We show the coexistence of dis- tinct dynamical regimes for the HCl rotational transition driven by the short-range repulsive and long-range attractive forces whose relative importance depends on the collision energy and final rotational state, suggesting that the classification of rainbow scattering into rotational andl-type rainbows is effective for H 2+ HCl collisions. While the QCT method satisfactorily predicts the overall behavior of the rotationally inelastic cross sections, its capability to accurately describe signatures of rainbow scattering appears to be limited for the present system. Published under license by AIP Publishing. https://doi.org/10.1063/5.0043658 .,s I. INTRODUCTION Molecular collision and chemical reaction outcomes are strongly influenced by initial internal states of reactants, collision energy, kinematics, stereodynamics, intermolecular forces, external fields, etc. Identifying the collisional parameters that control the collision outcomes has received much attention in theoretical and experimental studies.1–5The Wigner threshold law,6Breit–Wigner and Fano–Feshbach profiles for resonances,7–10and the Langevin model11,12are well-known examples of useful concepts to character- ize the physical origin and behavior of cross sections as a function of collision energy. In the rapidly evolving field of ultracold atoms and molecules, zero-energy Feshbach resonances as functions of external magnetic and/or electric fields play a central role in the control of the collision outcome.13–15Cross sections and rate coefficients for bar- rierless reactions and inelastic collisions are controlled by the values of long-range coefficients such as C4andC6in capture models from the ultracold to the Langevin energy regime.1,16 A long sought goal in molecular dynamics is to predict param- eter dependence of scattering properties from limited information.An extreme example is the J-shifting approximation17for predict- ing the behavior of reaction probabilities for non-zero values of the total angular momentum quantum number Jfrom that of J= 0. While this approach has found some success for activated reactions, it does not work well for complex-forming reactions and inelastic scattering. More recently, novel methods based on machine learning are actively being developed to interpolate cross sections and rate coefficients as functions of energy and temperature.5,18–22 Rainbow scattering, leading to characteristic parameter depen- dence of the scattering properties, may occur in systems with isotropic potentials and multiple partial waves.1,2,4,23–28Rainbow fea- tures arise from an interplay between the short-range repulsive and long-range attractive forces. In classical mechanics, a characteris- tic feature of rainbow scattering is a singularity (divergence and discontinuity) in the elastic differential cross section (DCS) at the maximum negative deflection angle χ(b) with the impact parameter b=brsatisfying (dχ(b)/db)br=0, where the relation between the deflection angle and scattering angle is | χ(b)| =θ.1–3Semiclassically, rainbow signature in the DCS is an oscillatory quantum interfer- ence pattern associated with a confluence of two paths scattered into J. Chem. Phys. 154, 104304 (2021); doi: 10.1063/5.0043658 154, 104304-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the sameθwith different values of bor orbital angular momentum quantum number l(≈bk, where kis the wave vector) due to the con- dition(dχ(b)/db)br=0 or(dχ(l)/dl)lr=0.1,2,4,29The sensitivity of the rainbow signature in the DCS to small changes in the interaction potential can be used as a delicate probe to calibrate the interaction potential.1,2,24,26 Rainbow scattering in the DCSs for rotationally inelastic ( j→j′) collisions in atom + molecule and ion + molecule systems has been the subject of both experimental and theoretical investigations.4,24–28 Signatures of rainbow scattering can be characterized by the exci- tation function j′(l,γ) and the deflection function χ(l,γ), whereγ is the molecular orientation angle against the initial relative veloc- ity vector for the collision although orientation or alignment of the initial molecular state is not a requirement for observing rain- bow scattering. Classically, a singularity in the DCS appears when the determinant D(l,γ) = (∂χ/∂l)γ(∂j′/∂γ)l−(∂χ/∂γ)l(∂j′/∂l)γof the Jacobian of the dynamical mapping between ( l,γ) and (χ,j′) becomes zero. Semiclassically, the quantum interference pattern is observed when “adjacent” trajectories with slightly different initial values of landγgive rise to the same final jandχ. For impulsive collisions, D(l,γ) may be well approximated by the diagonal terms as D(l,γ) = (∂χ/∂l)γ(∂j′/∂γ)l, suggesting that the singularity is classified according to whether ( ∂χ/∂l)γor (∂j′/∂γ)lvanishes.24–28,30When (∂χ/∂l)γ= 0, the rainbow is called an l-type (or potential or impact parameter) rainbow, similar to that observed in elastic scattering. Another type is the rotational (or orientation or stereodynamic) rainbow, which can occur due to the inner repulsive region even in the absence of a potential well. In contrast to elastic scattering, rotationally inelastic scattering requires anisotropy in the potential to change the rotational level. While the anisotropy of the poten- tial is usually large enough at the short-range to cause rotational transitions (provided energetically allowed), the anisotropy in the long-range attractive part is not necessarily strong enough, particu- larly for neutral species. Thus, the l-type rainbow is generally promi- nent only for small changes in rotational quantum numbers. Indeed, many of the early studies have focused on rotational rainbow scat- tering. Aoiz and co-workers have examined the role of long-range attractive forces in the l-type rainbow in a series of studies of rota- tionally inelastic atom + diatomic molecule collisions: Ar + NO, Cl + H 2, Ne + NO, Xe + NO, and Kr + NO.31–34We note that there is no definitive criterion to classify the rainbows into the two types in advance.26,28,30 Recent experimental progress in high-resolution measure- ments of state-to-state DCSs and stereodynamic control of rota- tionally inelastic collisions may facilitate observation of rainbows with fewer averaging effects.34–39Indeed, rainbow structures have been observed in the state-to-state DCS of rotationally inelastic col- lisions for more complex systems such as He + H 2O,40H2+ NO,41 D2+ NO,41,42and CH 4+ NO.43Furthermore, signatures of reac- tive rainbows have been observed in reactions F + CH 3D→CH 2D + HF and F + CH 3D→CH 3+ DF by Liu and co-workers.44,45On the other hand, full-dimensional quantum scattering calculations for exploring rotational rainbows are rare for molecule + molecule collisions. Recently, within a reduced four-dimensional (4D) treat- ment, detailed quantum scattering calculations were reported for the DCS in H 2+ NO collisions.41In addition, even for atom + diatomic molecule collisions, full-dimensional quantum scattering calculations are scant for open-shell molecules regardless of theappearance of rainbow scattering except for very recent reports on Ar + NO and H + NO collisions.46 In this paper, we report explicit six-dimensional (6D) quan- tum mechanical scattering calculations and quasi-classical trajectory (QCT) calculations of integral and differential cross sections for rotational excitation of HCl by para -H2at collision energies ranging from Ec= 100 cm−1to 6000 cm−1. Quantum calculations of rota- tional relaxations in HCl by H 2within a 4D rigid rotor model were previously reported by Lanza et al.47In our recent work, we have reported a 6D potential energy surface (PES) for the H 2+ HCl sys- tem48as well as 6D quantum calculations of rotational transitions in cold collisions with H 2with an emphasis on stereodynamic control at around 1 K.49Unlike rare gas-molecule collisions, the interaction potential between HCl and H 2is much more anisotropic, making it a more compelling case for rainbow scattering in rotationally inelastic molecule + molecule collisions. Whether the lightness of the colli- sion partners and the high asymmetry of the masses of Cl and H may preclude rainbow scattering in this system is an open question. While the experimental DCS for the rotational excitation of HCl by heavier colliders, e.g., rare gas atoms (Ne, Ar, and Kr), N 2, and CH 4, was reported by Chandler and co-workers, the energy range and the number of final rotational states explored are limited.50,51 II. THEORY AND COMPUTATION A. Potential energy surface The computations are performed using a recently developed globally accurate 6D ab initio PES for the electronic ground state of the H 3Cl system.48Briefly, the PES was fitted using a permutation- invariant polynomial neural network (PIP-NN) technique52,53based on a grid of interaction energies calculated with the CCSD(T)- F12b54/aug-cc-pVQZ55method. The long-range part of the interac- tion potential between H 2and HCl includes an accurate description of the electrostatic and dispersion interactions. The well depth of the interaction potential at the global minimum is −215.5 cm−1cor- responding to a T-shaped geometry with the H 2molecule on the H side of the HCl molecule. A second minimum with a depth of −102.6 cm−1exists for another T-shaped structure with H 2on the Cl side of HCl. These potential wells are significantly deeper than that between two H 2molecules ( −35 cm−1). At the global minimum, the distance between the centers of mass of H 2and HCl is R= 3.55 Å. In our previous work,48we have compared key features of this PES with the 4D PES of Lanza et al.47and confirmed its accuracy for pure rotational transitions in HCl induced by H 2. B. Quantum scattering calculation We numerically solve the time-independent Schrödinger equa- tion for H 2+ HCl collisions based on the close-coupling (CC) method56in the space-fixed (SF) coordinate frame within the total angular momentum representation for the angular degrees of free- dom as implemented in a modified version of the TwoBC code.57,58 The computational details including relevant basis set parameters are discussed in our previous publications,48,49and thus, we provide only a brief outline of the scattering formalism. The Hamiltonian for the relative motion of two diatomic molecules (1Σ) in a set of Jacobi vectors ( r1,r2,R) in the SF J. Chem. Phys. 154, 104304 (2021); doi: 10.1063/5.0043658 154, 104304-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp coordinate frame is given by (̵h= 1 hereafter) ˆH=−1 2μRd2 dR2R+ˆl2 2μR2+ˆHasym +Uint(R,r1,r2), (1) whereμis the reduced mass of the collision partners, Ris the magni- tude of the vector Rjoining the centers of mass of the two molecules, ˆlis the orbital angular momentum operator for the relative motion, Uint(R,r1,r2) is the interaction potential, and ˆHasym is the asymp- totic Hamiltonian expressed as the sum of the Hamiltonians of the separated molecules, ˆHasym=ˆh1(r1)+ˆh2(r2). The Hamiltonian for each of the diatomic molecule, ˆhi(ri)(i= 1, 2), is ˆhi(ri)=−1 2μirid2 dr2 iri+ˆji2 2μiri2+Vi(ri), (2) whereμiis the reduced mass, ˆjiis the rotational angular momentum operator, and Vi(ri) is the potential energy function of the diatom. The PES of the tetra-atomic system, V(R,r1,r2), is expressed as V(R,r1,r2)=V1(r1)+V2(r2)+Uint(R,r1,r2). (3) Since the total angular momentum Jof the collision complex, its projection Monto the SF z-axis, and the inversion parity εI =(−1)j1+j2+lare conserved, the total wavefunction for given values ofJ,M, andεImay be expanded as ΨJMεI=1 R∑ v j lFJMεI vjl(R)ϕj1 v1(r1) r1χj2 v2(r2) r2∣JMεI(lj12(j1j2))⟩, (4) where v=v1v2and j=j1j2j12collectively denote the vibra- tional and rotational quantum numbers, respectively, FJMεI vjl(R)are the radial expansion coefficients in R,ϕj1 v1(r1)/r1andχj2 v2(r2)/r2 denote the radial parts of the rovibrational eigenfunctions of the diatomic molecules with the Hamiltonian ˆhi(ri)in Eq. (2), and |JMεI(lj12(j1j2))⟩denotes the eigenstates of the total angular momen- tum for a given Mand parityεI. The coefficients FJMεI vjl(R)satisfy the CC equations obtained by substituting Eq. (4) into the Schrödinger equation, [1 2μd2 dR2−l(l+ 1) 2μR2+Ec]FJMεI vjl(R) =∑ v′j′l′FJMεI v′j′l′(R)∫∞ 0∫∞ 0⟨JMεI(lj12(j1j2)∣ϕj1 v1(r1)χj2 v2 ×(r2)Uint(R,r1,r2)ϕj′ 1 v′ 1(r1)χj′ 2 v′ 2(r2) ×∣JMεI(l′j′ 12(j′ 1j′ 2)⟩dr1dr2. (5) For a given total energy E, the collision energy Ec=E−Ev1j1−Ev2j2, where Eviji(i=1, 2)denotes the initial ro-vibrational energies for the diatomic molecules. Note that the CC equations in Eq. (5) are independent of M. In the TwoBC code, the matrix elements of the interaction potential given in the right-hand side of Eq. (5) are evaluated by expanding the angular dependence ( ˆR,ˆr1,ˆr2) of the interac- tion potential in a triple series of spherical harmonics.59A Gauss– Hermite quadrature is used to evaluate the integrals over r1and r2of the expansion coefficients of the interaction potential withthe vibrational wave functions ϕ(r1) andχ(r2). The explicit form of the matrix elements and relevant details are given in Ref. 58. The radial propagation of the CC equations is carried out by Johnson’s log-derivative propagation method and the S-matrix is obtained by applying asymptotic boundary conditions at Rmax= 50.0 Å at each energy. The S-matrix carries the relevant information to compute the state-resolved differential and integral cross sections. In the following, we omit the symbol MandεIfor simplicity. The state-to-state DCS may be written in terms of the scattering amplitude qin the helicity representation as60,61 dσα→α′ dΩ=1 (2j1+ 1)(2j2+ 1)∑ k1,k2,k′ 1,k′ 2∣qα,k1,k2→α′,k′ 1,k′ 2∣2, (6) whereα≡v1j1v2j2andα′≡v′ 1j′ 1v′ 2j′ 2refer to the initial and final com- bined molecular rovibrational states, respectively, dΩ is the infinites- imal solid angle, and the quantum number k/k′is the helicity com- ponent of the initial/final molecular rotation angular momentum j/j′. The contributions of all possible degenerate k/k′components inj/j′are summed over in the right-hand side of Eq. (6). The scattering amplitude qis given by60 qα,k1,k2→α′,k′ 1,k′ 2=1 2kα∑ J(2J+ 1)∑ j12,j′ 12,l,l′il−l′+1 ×TJ αj12l,α′j′ 12l′(E)dJ k12,k′ 12(θ) ×⟨j′ 12k′ 12J−k′ 12∣l′0⟩⟨j12k12J−k12∣l0⟩ ×⟨j′ 1k′ 1j′ 2k′ 2∣j′ 12k′ 12⟩⟨j1k1j2k2∣j12k12⟩, (7) where kαis the wave vector for the incident channel, the T-matrix is given in terms of the S-matrix as TJ(E) = 1 −SJ(E), the projec- tion quantum numbers k12=k1+k2andk′ 12=k′ 1+k′ 2,dJ k12,k′ 12(θ)is the Wigner’s reduced rotation matrix, and the bracket ⟨|⟩denotes a Clebsch–Gordan coefficient. The scattering amplitude includes energy dependence through kαand the T-matrix and angle depen- dence through dJ k12,k′ 12(θ). By taking the integral of the DCS [Eq. (6)] overθ(0 toπ) andϕ(0 to 2π), one obtains the state-to-state integral cross section (ICS) as48,60,61 σα→α′(E)=π (2j1+ 1)(2j2+ 1)k2α∑ j12j′ 12,ll′J(2J+ 1) ×∣TJ αj12l,α′j′ 12l′(E)∣2. (8) C. Quasi-classical trajectory method The QCT calculations are carried out using the VENUS pro- gram package.62Batches of 5 ×105trajectories were calculated at each collision energy on the 6D PES. The maximum impact param- eters, bmax, were determined to be 8 Å after carrying out a small set of trajectories with trial values. All trajectories in QCT calcu- lations were started from a diatom–diatom distance of 15 Å and ended when the products or reactants reach a separation of 15.5 Å. The propagation time step was chosen to be 0.1 fs, which guarantees the energy conservation of all trajectories better than 0.01 kcal/mol (∼3.5 cm−1). J. Chem. Phys. 154, 104304 (2021); doi: 10.1063/5.0043658 154, 104304-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The state-to-state ICS corresponding to Eq. (8) is calculated by σα→α′(E)=πb2 maxNα→α′ Ntot,α, (9) where Nα→α′denotes the number of trajectories, which result in a final combined molecular rovibrational state α′with an initial com- bined molecular state α. The values of the quantum numbers for the final rovibrational states were rounded to the nearest integers except for final state j′= 0 and 1. To avoid the “leak” of final rotational states from j′= 0 to j′= 1 and unphysically large populations in j′= 1, the real number j′in intervals [0, 1.0] and [1.0, 1.5] was assigned to j′= 0 and j′= 1, respectively.31,32The number of total trajectories with the initial combined molecular state αis denoted as Ntot,α. In the present study, the initial combined molecular state αis composed of the rovibrational ground states of HCl and H 2. The values of the quantum numbers for the final rovibrational states were rounded to the nearest integers. The DCS is computed by dσα→α′(E) dΩ=σα→α′(E)Pα→α′(θ) 2πsinθ, (10) whereθis the scattering angle, Pα→α′(θ) is the normalized probabil- ity obtained by Pα→α′(θ)=θ′<θ+Δθ ∑ θ′≥θ−ΔθNα→α′(θ′) Nα→α′, (11) where Nα→α′(θ′) is the number of trajectories in a bin from θ−Δθ toθ+Δθ. In the computations, Δθis taken to be 1.0○. III. RESULTS We consider excitations of HCl from the rovibrational ground state ( v= 0,j= 0) to the rotationally excited state ( v′= 0,j′= 1) in col- lisions with para -H2(v= 0,j= 0), namely, HCl ( v= 0,j= 0) + H 2(v = 0,j= 0)→HCl ( v′= 0,j′= 1) + H 2(v′= 0,j′= 0), at collision ener- gies ranging from Ec= 100 cm−1to 6000 cm−1. At these collision energies, the vibrationally inelastic scattering is either energeticallydisallowed or very weak even when energy is sufficiently large, and we will focus on pure rotationally inelastic transitions of HCl. For the same reason, in subsequent discussions, we will suppress vibrational quantum numbers of both molecules. While low energy rotational transitions of HCl by H 2exhibit isolated shape resonances,48,49the resonances vanish as the collision energy is increased. Indeed, the left panel of Fig. 1 shows ICS as a function of the collision energy, which depicts a smooth behavior due to contributions from many partial waves. However, when the ICS is examined as a function of l,l′, orJ, we find multiple peaks and associated minima that dip to almost zero as the collision energy is increased. In the higher energy region ( Ec>500 cm−1) where many incoming and outgoing partial waves contribute, the plots of ICS vs l,l′, and Jall look very similar. In particular, since both HCl and H 2are initially in their rotational ground states ( j= 0), l=Jis satisfied. For subsequent analysis and discussions of the impact parameter dependence of the cross sec- tions, we show the ICS as a function of the incoming partial wave lin the right panels of Fig. 1 at selected values of collision ener- gies, Ec= 100 cm−1, 500 cm−1, 1000 cm−1, 2000 cm−1, 3000 cm−1, 4000 cm−1, 5000 cm−1, and 6000 cm−1. At each of these collision energies, the partial ICS summed over all integer lyields the total ICS shown in the left panel. For energies between Ec= 500 cm−1and 3000 cm−1, the partial ICS vs lexhibits a double-peak structure and a third peak begins to appear in the lower lregion ( l<25) for Ec >3000 cm−1. In what follows, we unambiguously attribute the peak at higher lto an l-type rainbow. The peak positions and minima shift to higher values of las the collision energy is increased. Also, the intensity of the peak at higher lvalues decreases monotonically with collision energy, indicating its diminishing importance compared to the dominant peak at lower l. More detailed analysis of the partial wave contributions and relation with elastic scattering are given in the supplementary material. To gain more insight into the multi-peak structure of the partial ICS with respect to land to make a comparison with QCT results, we examine the impact parameter ( b) dependence of the ICS plotted in Fig. 1. In the classical limit, bis given in terms of lasb=l/√ 2μEc. While the conversion between land bis not unique due to the quantum nature of ˆl, the non-uniqueness does not cause any issue in the following discussion since many partial waves contribute. The FIG. 1 . Integral cross section for HCl ( v= 0, j= 0) + H 2(j= 0)→HCl ( v′= 0, j′= 1) + H 2(j′= 0) rotationally inelastic collisions at energies ranging from Ec= 100 cm−1 up to 6000 cm−1(left panel). Partial cross section decomposition with respect to the incoming partial wave lis displayed in the right panels at Ec= 100 cm−1, 500 cm−1, 1000 cm−1, 2000 cm−1, 3000 cm−1, 4000 cm−1, 5000 cm−1, and 6000 cm−1. Results are from quantum mechanical close-coupling calculations, and thus, lis an integer. J. Chem. Phys. 154, 104304 (2021); doi: 10.1063/5.0043658 154, 104304-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp bdependence of the partial ICS obtained from the results in Fig. 1 is shown in Fig. 2(a) at selected collision energies. It is clearly seen that the peaks and minima of the partial ICS depend weakly on the collision energy, and thus, we observe a minimum at around b= 3 Å and maximum on both sides of it regardless of the energy. Strictly, as the collision energy increases, the location of the minimum also increases from b= 2.86 Å at Ec= 500 cm−1tob= 3.13 Å at Ec = 5000 cm−1. At the lowest collision energy of Ec= 100 cm−1, the ICS is dominated by the peak at the right of the minimum. The relative importance of the peak at lower bgrows with the collision energy, and eventually, the low- bpeak becomes dominant for collision ener- gies above 3000 cm−1. Furthermore, at the highest collision energy (Ec= 5000 cm−1), the low- bpeak is split into two distinct peaks with a minimum at around b= 1.5 Å. The multi-peak structure, similar to Fig. 2(a), is also observed for the non-zero initial rotational level of HCl and for DCl + H 2collisions (see the supplementary material). Generally, low impact parameters lead to head-on collisions and backward scattering, while high- bapproaches favor glancing collisions and forward scattering. Thus, Fig. 2(a) reveals that there are two competing collision mechanisms that correlate with low- b FIG. 2 . Partial rotational inelastic cross section as a function of the impact param- eterbfor HCl ( v= 0,j= 0) to ( v′= 0,j′= 1) in collisions with para-H2(j= 0). (a) CC and (b) QCT. The partial cross section at each energy is normalized to the peak value for Ec= 100 cm−1(black curve) for the convenience of plotting.and high- bcentered around b∼3 Å. Such clear separation of dynam- ical regimes by a minimum that becomes almost zero in the partial ICS or opacity function has been reported for ion + molecule and atom + molecule systems.23,31–33,63,64If the classification of rainbows into rotational and l-type rainbows is valid for H 2+ HCl collisions, then the high- bpeak (∼3.8 Å) corresponds to an l-type rainbow.31–33 The relative dominance of the high- bpeak at lower collision energies in Fig. 2(a) implies that the scattering is dominated by long-range force, and thus, the energy dependence of the high- bpeak is con- sistent with the character of an l-type rainbow. The rainbow impact parameter brfor the potential ( l-type rainbow) is roughly approx- imated by the location of the potential minimum.1As stated in Sec. II A, the minimum of the potential corresponds to R= 3.55 Å, and thus, the position of the high- bpeak can be reasonably approximated as an l-type rainbow. Before delving further into the analysis of the high- bpeak, we consider the complementary QCT results depicted in Fig. 2(b). Overall, the behavior of the QCT results as a function of bexhibits similar energy dependence to that shown in Fig. 2(a). However, the clear modal structure obtained with the CC calculation is not repro- duced by QCT. We can identify that the high- bpeaks are located at b= 5.0 Å and b= 4.5 Å at Ec= 500 cm−1(blue) and 1000 cm−1 (green), respectively, with the associated shallow dips at around b = 4.3 Å and b= 3.9 Å. Also, we see shoulders around b= 3.5 Å at higher energies ( Ec>3000 cm−1). These might be a manifestation of the high- bpeak ( l-type rainbow) in the QCT results. From the resulting trajectories, it is possible to directly confirm the occurrence ofl-type rainbows from the correlation between the scattering angle and the impact parameter, as shown in Fig. 3. At Ec= 500 cm−1and 1000 cm−1, we see a clear signature of an l-type rainbow with a max- imum at around br= 4.2 Å, where a subset of trajectories give rise to the same scattering angle, θr∼10○to 20○. The fact that the same scattering angles are reached by trajectories with different impact FIG. 3 . Deflection functions (| χ(b)| =θ) for HCl ( v= 0,j= 0) + H 2(j= 0)→HCl ( v′ = 0,j′= 1) + H 2(j′= 0) collisions. J. Chem. Phys. 154, 104304 (2021); doi: 10.1063/5.0043658 154, 104304-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp parameters implies that in quantum mechanics, these different scat- tering paths could constructively or destructively interfere, leading to the rainbow signature. We also see the glory impact parameter at around bg= 3.6 Å arising from a balance between the repulsive and attractive forces resulting in a scattering angle of ∼0○. At higher energies, the maximum arising from the rainbow is not clear due to the small magnitude of θabove bg, as shown in Figs. 3(c) and 3(d). While we see a clear signature of an l-type rainbow in the deflection functions in Fig. 3, it is worth noting the discrepancies between CC and QCT results in panels (a) vs (b) in Fig. 2, includ- ing the values of glory and rainbow impact parameters extracted from Fig. 3. For rotational excitations in Ar + NO collisions, Aoiz et al.31demonstrated excellent agreement of the opacity functions derived from quantum mechanical (QM) and QCT calculations, including the locations of minimum and maximum. Similar good agreement between QM and QCT results in the partial ICS or opac- ity functions were reported for Xe + NO and Cl + H 2.32,33In these systems, there is a region of b(∼bg) in which only very few trajec- tories are observed, leading to the discontinuity in the distribution of trajectories. The discontinuity is the origin of the minimum that becomes almost zero in the partial ICS or opacity function in their QCT results. On the contrary, we do not observe such discontinuity around bgin our trajectory results (Fig. 3). The limitation of QCT calculation to reproduce QM result was also reported for Kr + NO.34 However, in this case, the quantum results correspond to specific Λ-doublet transition ( f→f) in NO, while the QCT treatment used a closed-shell approximation for the NO molecule. An explicit com- parison between QM and QCT results would require a sum over the two Λ-doublet transitions as well as the two spin–orbit states. For Ar + HF, a classical mechanical treatment did not reproduce a minimum structure in the opacity function obtained with a semi- classical treatment that includes the interference effect.63Strictly, the origin of rainbow signature is quantum interference, while the existence of the rainbows is conveniently explained by the singulari- ties observed in classical mechanics. Thus, the observed discrepancy between QM and QCT may not be altogether surprising. Further systematic studies are required to isolate systems for which classi- cal and quasi-classical calculations are not effective in reproducing specific rainbow signatures. Next, we examine whether the nature of the high- bpeak observed in the CC calculation [Fig. 2(a)] is consistent with an l- type rainbow. Since l-type rainbows for rotational excitation arise from anisotropy in the long-range attractive part of the potential, it would become difficult to observe l-type rainbows in the rotational transition processes that accompany large changes in rotational lev- els.32To illustrate this, we show results for rotational excitations to higher rotational states at Ec= 1000 cm−1(Fig. 4). Compared to | Δj| = 1 (green), we see a gradual suppression of contributions from ∼b >3 Å with the increase in | Δj|. In particular, for | Δj| = 4 (black), the contribution from high bvanishes. This result combined with the discussions of the energy dependence of the ICS in Fig. 2(a) and the QCT trajectories distribution in Fig. 3 support the conclusion that the high- bpeak observed in the | Δj| = 1 cross sections is an l-type rainbow. We have confirmed that long-range attractive forces give rise to a modest well in the potential and lead to the high- bpeak (l-type rainbow) in the partial ICS [Fig. 2(a)]. On the other hand, the dynamics at low- bis significantly influenced by the inner FIG. 4 . Partial rotational inelastic cross section as a function of the impact parame- terbfor rotational excitations out of the ground state HCl ( v= 0,j= 0) to ( v′= 0,j′) (j′= 1–5) in collisions with para-H2(j= 0) from CC calculations at Ec= 1000 cm−1. repulsive region of the interaction potential, and thus, the peaks observed at low- bcould be ascribed to a rotational rainbow. One typical way to examine the occurrence of the rotational rainbow is to explore the DCS and its final state j′dependence for a given scat- tering angle,23,27,65,66and thus, we move on to the analysis of the DCS. In Fig. 5, we show the energy dependence of the DCS for j = 0 to j′= 1 transition in HCl from CC (a) and QCT (b) cal- culations. As discussed earlier, high- band low- bcollisions corre- spond to small (forward) and large (backward) scattering angles ( θ), respectively. Therefore, there exists a correspondence between Figs. 2 and 5, although different values of bcan result in the same scattering angle θeven in the classical limit. Indeed, in Fig. 5(a), we observe two broad peaks in the DCS above θ= 30○corresponding to the two peaks in the low- bregion in Fig. 2(a) above Ec= 3000 cm−1. As the collision energy is decreased below Ec= 3000 cm−1, the two broad peaks completely vanish at Ec= 500 cm−1, consistent with the energy dependence of the low- bpeak in Fig. 2(a). We can also observe signatures of the l-type rainbow at small angle scattering in the DCS. Below the inflection point, the cross section increases rapidly with a decrease in θ. The inflection point is an indicator of the region of the l-type rainbow. Indeed, the energy dependence of the position of the inflection point is consis- tent with the minimum that separates the low- band high- bregions in Fig. 2(a). However, as we can see, the diffraction pattern makes it difficult to identify the positions of the maxima of the l-type rainbow and its secondary (supernumerary) rainbows. Figure 5(b) indicates the capability of QCT in describing the behavior of DCS. Similar to the discussion of Fig. 2, the DCS from QCT fails to reproduce the specific details of the quantum DCS although the overall behavior is qualitatively reproduced. The broad peaks in the large θregion or the detailed structures in the low- b region in Fig. 2(b) are not reproduced in the QCT results. On the other hand, we see a change in the behavior of the DCSs at lower θcorresponding to the inflection points in Fig. 5(a), the boundary between low- band high- bregions, while the signature of the l-type rainbow is not evident or correct. Features of l-type rainbows in clas- sical mechanics are expected to appear as an abrupt change in the cross section separating the region of θrinto the bright side ( θ<θr) J. Chem. Phys. 154, 104304 (2021); doi: 10.1063/5.0043658 154, 104304-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . Differential cross section as a function of the scattering angle for HCl ( v = 0,j= 0) to ( v′= 0,j′= 1) in collisions with para-H2(j= 0). (a) CC and (b) QCT. and dark side ( θ>θr).28,31We find a slight change in the behavior of the small peaks at very low θ(<5○) at higher collision energies in Fig. 5(b). To further examine the rainbow signatures in the DCS from quantum calculations, in the left panel of Fig. 6, we show the DCS atEc= 3000 cm−1for rotational excitations out of the ground state of HCl to all energetically accessible rotational states ( v′= 0,j′= 1 –14) in collisions with para -H2. The behavior and magnitude of DCS at smallθare drastically changed at j′= 5 (black curve) due to the disappearance of the l-type rainbow, which is consistent with the dis- cussion of the partial ICS in Fig. 4. We observe an overall decrease in trend of the cross sections with the increase in j′. However, it is pos- sible to find some exception to this trend, suggesting more nuanced nature of the j′dependence of the DCS in identifying features of rotational rainbows. In particular, for j′≥5, we observe a gradual shift of the broader peaks to the higher θregion with the increase inj′. This trend is consistent with signatures of a rotational rain- bow.3,28The right panels of Fig. 6 depict DCS as a function of j′forθ = 60○, 120○, and 180○. The results for θ= 120○and 180○mainly reveal the j′dependence of the broader peaks observed in the DCS related to the low- bpeak in the partial ICSs. The appearance of mul- tiple peaks with respect to j′seems to be consistent with the signa- tures of rotational rainbows. However, the distributions of the peaks FIG. 6 . Left panel: state-to-state differential cross section as a function of the scat- tering angle for rotational excitations out of the ground state HCl ( v= 0, j= 0) to (v′= 0,j′) (j′= 1–14) in collisions with para-H2(j= 0) from CC calculations at Ec = 3000 cm−1. Right panels: differential cross section as a function of final rotational state j′for selected values of the scattering angle. are significantly different from typical j′-distributions for heteronu- clear target molecules that feature multiple peaks with two distinct maxima.67–69In particular, the j′distributions in Fig. 6 are charac- terized by a peak at low j′(∼j′= 2) and the absence of an intense peak at high j′. Very similar signatures were reported in atom + molecule systems with a large asymmetric mass for the diatomic molecule, He + HF65and Li++ BeF .70The overall trend is robust with respect to the collision energy. Indeed, in Fig. 4, we can see that the excitation toj′= 2 results in the largest (total) cross section at 1000 cm−1(see also the supplementary material). IV. CONCLUSION In this paper, we performed full-dimensional quantum mechanical scattering calculations using the close-coupling formal- ism for rotationally inelastic scattering of HCl ( v= 0, j= 0) in collisions with ground state para -H2on a globally accurate full- dimensional ab initio potential energy surface. Our results for rota- tional excitations of HCl ( j= 0→j′) at collision energies ranging from 100 cm−1to 6000 cm−1show signatures of rainbow scattering in both state-to-state partial integral and differential cross sections. For the j= 0→j′= 1 excitation, signatures of l-type rainbow are observed in the impact parameter dependence of the partial integral cross sections as a pronounced peak in the higher impact parameter region. This peak is clearly separated from a peak at a lower bwith a minimum where the cross section becomes almost zero at around b= 3.0 Å. Features of the high- bpeak are examined in terms of the collision energy and final rotational quantum numbers, confirming the appearance of the l-type rainbow. In addition, we demonstrated that the l-type rainbow does not occur for high rotational excita- tions of HCl since the anisotropy of the long-range attractive force is not sufficiently strong to generate the required torque to excite high rotational states. An analysis of the classical deflection function J. Chem. Phys. 154, 104304 (2021); doi: 10.1063/5.0043658 154, 104304-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp from quasi-classical trajectory calculations also qualitatively sup- ports the presence of an l-type rainbow. These analyses indicate the coexistence of distinctive dynamical regimes for HCl rotational tran- sition driven by the short-range repulsive and long-range attractive forces whose relative importance depends on the collision energy and final rotational states. We have also identified the characteristic multi-peak structure in the final rotational state dependence of the differential cross section that is consistent with the previously stud- ied atom + heteronuclear molecule systems with a large asymmetric mass for the molecules (composed of light and heavy atoms). SUPPLEMENTARY MATERIAL See the supplementary material associated with this article for a comparison of cross sections from QCT and quantum mechanical calculations, partial-wave resolved elastic and rotationally inelastic cross sections, rainbow effects in DCl + H 2collisions, the effect of initial rotational excitation of the HCl molecule, and inelastic transition involving rotational excitation of the H 2molecule. ACKNOWLEDGMENTS This work was supported, in part, by the ARO MURI [Grant No. W911NF-19-1-0283 (H.G. and N.B.)] and the NSF [Grant No. PHY-1806334 (N.B.)]. The calculations at UNM were performed at the Center for Advanced Research Computing (CARC). DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material. REFERENCES 1R. Levine, Molecular Reaction Dynamics (Cambridge University Press, 2009). 2M. Child, Molecular Collision Theory , Dover Books on Chemistry Series (Dover Publications, 1996). 3R. Levine and R. Bernstein, Molecular Reaction Dynamics and Chemical Reactiv- ity(Oxford University Press, 1987). 4R. Schinke and J. M. Bowman, “Rotational rainbows in atom-diatom scatter- ing,” in Molecular Collision Dynamics , edited by J. M. Bowman (Springer, Berlin, Heidelberg, 1983), pp. 61–115. 5A. Jasinski, J. Montaner, R. C. Forrey, B. H. Yang, P. C. Stancil, N. Balakrishnan, J. Dai, R. A. Vargas-Hernández, and R. V. 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5.0032732.pdf

AIP Advances 11, 025205 (2021); https://doi.org/10.1063/5.0032732 11, 025205 © 2021 Author(s).Spin–orbit torque induced magnetization switching for an ultrathin MnGa/Co2MnSi bilayer Cite as: AIP Advances 11, 025205 (2021); https://doi.org/10.1063/5.0032732 Submitted: 10 October 2020 . Accepted: 03 January 2021 . Published Online: 01 February 2021 Kohey Jono , Fumiaki Shimohashi , Michihiko Yamanouchi , and Tetsuya Uemura COLLECTIONS Paper published as part of the special topic on Chemical Physics , Energy , Fluids and Plasmas , Materials Science and Mathematical Physics ARTICLES YOU MAY BE INTERESTED IN Tunable spin–orbit torque efficiency in in-plane and perpendicular magnetized [Pt/Co] n multilayer Applied Physics Letters 118, 042405 (2021); https://doi.org/10.1063/5.0034917 Origin of spin–orbit torque in single-layer CoFeB investigated via in-plane harmonic Hall measurements AIP Advances 11, 025033 (2021); https://doi.org/10.1063/5.0035845 Enhancement of low-frequency spin-orbit-torque ferromagnetic resonance signals by frequency tuning observed in Pt/Py, Pt/Co, and Pt/Fe bilayers AIP Advances 11, 025206 (2021); https://doi.org/10.1063/9.0000066AIP Advances ARTICLE scitation.org/journal/adv Spin–orbit torque induced magnetization switching for an ultrathin MnGa/Co 2MnSi bilayer Cite as: AIP Advances 11, 025205 (2021); doi: 10.1063/5.0032732 Submitted: 10 October 2020 •Accepted: 3 January 2021 • Published Online: 1 February 2021 Kohey Jono,1Fumiaki Shimohashi,1Michihiko Yamanouchi,1,2and Tetsuya Uemura1,a) AFFILIATIONS 1Graduate School of Information Science and Technology, Hokkaido University, Sapporo 060-0814, Japan 2Research Institute for Electronic Science, Hokkaido University, Sapporo 001-0020, Japan a)Author to whom correspondence should be addressed: uemura@ist.hokudai.ac.jp ABSTRACT We investigated spin–orbit torque (SOT) induced magnetization switching and SOT efficiency for Mn 1.8Ga1.0(MnGa) single layers and MnGa/Co 2MnSi (CMS) bilayers. Magnetization measurements showed that ultrathin MnGa and CMS were antiferromagnetically coupled to each other with clear perpendicular magnetization. SOT-induced magnetization switching was observed for both MnGa/CMS/Ta and MnGa/Ta stacks, and the switching current was reduced by a half in the MnGa/CMS/Ta stack. Examination of SOT acting on the domain walls revealed that the effective magnetic field originating from the SOT was approximately five times stronger in the MnGa/CMS/Ta stack than in the MnGa/Ta stack. These results indicate that the MnGa/CMS bilayer structure is effective in enhancing the efficiency of SOT generation. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0032732 Spin–orbit torque (SOT) induced magnetization switching has attracted much interest as an efficient writing technology for next- generation magnetic random access memory.1–3Mn xGa1.0(MnGa) is a promising ferromagnetic electrode for perpendicular magnetic tunnel junctions (p-MTJs) that use SOT-induced magnetization switching because it exhibits relatively large perpendicular mag- netic anisotropy (PMA)4and low saturation magnetization.5,6There have been several reports on SOT-induced magnetization switching for an ultrathin MnGa single layer.7–11Moreover, a relatively high spin polarization of 88%, which enables highly sensitive detection of the magnetization direction in MTJs, has been predicted.12How- ever, MnGa-based MTJs with an MgO barrier showed a low tunnel magnetoresistance (TMR) ratio (23% at 10 K13), probably due to the large lattice mismatch ( ∼7%) between MnGa and MgO. Recent studies have demonstrated that the TMR ratio can be enhanced by inserting an ultrathin ferromagnetic material, such as Fe or Co,14 which exhibits high tunneling spin polarization in conjunction with MgO(001) tunnel barriers. Cobalt-based Heusler alloys (Co 2YZ, where Y is usually a tran- sition metal and Z is a main group element) are also candidatesfor such an insertion layer because of the half-metallic nature theoretically predicted for many of these alloys. Within the Co 2YZ family, Co 2MnSi (CMS) is one of the materials most exten- sively applied to spintronic devices, including MTJs15–19and giant magnetoresistance devices,20–24and also used for spin injec- tion into semiconductors.25–29This is because of its theoreti- cally predicted half-metallic nature with a large half-metal gap of 0.81 eV for its minority-spin band,30high Curie temperature of 985 K,31and experimentally observed low damping constant.32,33 We have demonstrated TMR ratios of up to 1995% at 4.2 K and up to 354% at 290 K in CMS-based MTJs with an MgO barrier,19indicating that CMS has high spin polarization. Thus, a MnGa/CMS bilayer is a promising ferromagnetic electrode structure for p-MTJs with a high TMR ratio and high thermal stability. Previous studies have demonstrated that MnGa and CMS are antiferromagnetically coupled to each other,34,35and a p-MTJ with MnGa/CMS electrodes has been reported.36Moreover, SOT-induced magnetization switching for a MnGa/CMS bilayer near the magnetic moment compensation point of 270 K has AIP Advances 11, 025205 (2021); doi: 10.1063/5.0032732 11, 025205-1 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv been demonstrated.35However, the effect of CMS on the SOT characteristics has not been fully clarified. The purpose of this study was to quantitatively investigate the effect of CMS in MnGa/CMS bilayers on the SOT characteristics. To do this, we fabricated MnGa/Ta and MnGa/CMS/Ta stacks grown on the same substrate and compared the SOT-switching characteristics and SOT-induced effective magnetic field. A layer structure [Fig. 1(a)] consisting of (from the sub- strate side) a MgO buffer (10)/NiAl buffer (5)/MnGa (2 or 3)/CMS (1)/Ta (5)/MgO cap (2) was deposited on a MgO(001) single- crystalline substrate (numbers in parentheses are nominal thick- nesses in nanometers). The CMS layer was deposited on an area equal to half the substrate surface area by using a slide shutter in order to precisely evaluate the effect of the CMS. The MnGa (CMS) layer was deposited at room temperature and annealed at 250○C (400○C); epitaxial growth was confirmed by reflection high-energy electron diffraction. Finally, a 5-nm-thick Ta was deposited as the SOT spin source. Although the Ta layer should be placed under the MnGa/CMS bilayer in the MTJ structure, it was placed on top of the MnGa/CMS bilayer in this study because of the ease of fabrication. The compositions of NiAl and MnGa were Ni 1.1Al1and Mn 1.8Ga1, respectively, from inductively cou- pled plasma optical emission spectroscopy measurements. The layer structure was processed into Hall devices with a 5- μm-wide chan- nel by photolithography and Ar ion milling to investigate the SOT-switching characteristics. The effective magnetic field origi- nating from the SOT in the MnGa(3)/CMS/Ta and MnGa(3)/Ta devices was also evaluated by examining the SOT acting on the domain walls (DWs).11,37A schematic of the measurement setup with a definition of the Cartesian coordinate system is shown in Fig. 1(b). Figures 2(a) and 2(b) show the total magnetic moment per area at room temperature as a function of the out-of-plane magnetic field μ0Hz(μ0is the permeability of vacuum) for (a) a 2-nm-thick MnGa layer with and without CMS and (b) a 3-nm-thick MnGa layer with and without CMS, obtained by vibrating sample mag- netometer (VSM) measurement. All curves show clear PMA characteristics, including that for the 2-nm-thick MnGa with CMS, indicating that the CMS layer was also perpendicularly magnetized FIG. 1. Schematic of (a) the layer structure and (b) a typical Hall bar device with a 5-μm wide channel and a pair of Hall probes, together with a definition of the Cartesian coordinate system. The transverse resistance Ryxwas measured with an in-plane dc current Iunder application of external magnetic fields: in-plane magnetic field ( Hx) and out-of-plane magnetic field ( Hz). FIG. 2. (a) and (b) Total magnetic moment per area at room temperature as a function of μ0Hzfor (a) a 2-nm-thick MnGa layer with and without CMS and (b) a 3-nm-thick MnGa layer with and without CMS, obtained by vibrating sample mag- netometer measurement. The blue and red arrows in the inset of (a) indicate the magnetization direction of MnGa and CMS, respectively. The magnetization curves were obtained by subtracting background signals arising from the MgO substrate and NiAl buffer layer. The step structure at Hz=0 appeared in the magnetiza- tion curve for a 2-nm-thick MnGa single layer that probably comes from some artifacts in subtracting the background signal. (c) and (d) Transverse resistance Ryxas a function of μ0Hzfor (c) a 2-nm-thick MnGa layer with and without CMS and (d) a 3-nm-thick MnGa layer with and without CMS. The value of Ryxfor a 2-nm-thick MnGa layer with CMS was five-times magnified and was plotted [the red curve in (c)]. due to the interfacial exchange coupling with MnGa. The total sat- uration magnetic moment per area msis given by Ms(MnGa)⋅d(MnGa) for a MnGa single layer and by Ms(MnGa)⋅d(MnGa)±Ms(CMS)⋅d(CMS) for a MnGa/CMS bilayer, where Ms(MnGa)andd(MnGa)are, respec- tively, the saturation magnetization and thickness of MnGa, Ms(CMS) and d(CMS)are those of CMS, and +(−) corresponds to ferromag- netic (antiferromagnetic) coupling between MnGa and CMS. As shown in Figs. 2(a) and 2(b), the magnetic moment for a 2-nm-thick MnGa layer was increased by adding CMS, while that for a 3-nm- thick MnGa layer was decreased by adding CMS. Such a complicated behavior cannot be explained by the deterioration of magnetic prop- erties. On the other hand, if we assume that MnGa with MS(MnGa) ≈0.2 T and CMS with MS(CMS)≈1.0 T are antiferromagnetically coupled to each other, the observed saturation magnetic moment for four kinds of samples shown in Figs. 2(a) and 2(b) can be well explained. This antiferromagnetic coupling of MnGa and CMS is consistent with the findings in previous studies.34,35Moreover, it is supported by the results of anomalous Hall effect measurements AIP Advances 11, 025205 (2021); doi: 10.1063/5.0032732 11, 025205-2 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 3. (a) and (b) Ryxas a function of pulse current Ipfor a MnGa(2) single layer for (a) μ0Hx=+0.05 T and (b) μ0Hx=−0.05 T. (c) and (d) Ryxas a function of Ip for a MnGa(2)/CMS(1) bilayer for (c) μ0Hx=+0.05 T and (d) μ0Hx=−0.05 T. shown later. The value of Ms(MnGa)is comparable to that in previous work,6while that of Ms(CMS)corresponds to 76% of the theoretical Slater–Pauling value, possibly due to the existence of a magnetically dead layer and/or relatively low annealing temperature of 400○C for the CMS. Figures 2(c) and 2(d) show the transverse resistance Ryxas a function of μ0Hzfor (c) a 2-nm-thick MnGa layer with and without CMS and (d) a 3-nm-thick MnGa layer with and with- out CMS. The Ryxwas measured with dc current I=1 mA. The hysteresis loops for the MnGa single layers were similar to the magnetization curves, indicating that Ryxfor MnGa single layers are dominated by the anomalous Hall signal arising from MnGa. The polarity of the hysteresis loops for MnGa/CMS bilayers, on the other hand, was reversed with respect to those of the mag- netization curves. It was confirmed in the other experiment that the anomalous Hall coefficients for both a MnGa single layer and a CMS single layer are positive in our coordinate system. Thus, the change of polarity of the hysteresis loops for the MnGa/CMS bilayers suggests that either MnGa or CMS is magnetized antipar- allel to the external field. This result also supports the existence of the antiferromagnetic coupling between MnGa and CMS. Because Ms(CMS)⋅d(CMS)>Ms(MnGa)⋅d(MnGa)in the bilayer system investi- gated in this study, the CMS (MnGa) layer was magnetized parallel (antiparallel) to the external magnetic field. Thus, we conclude that Ryxfor MnGa/CMS bilayers are also dominated by the anomalous Hall signal arising from MnGa, and the magnetization direction ofMnGa could be detected from the values of Ryxfor all the samples investigated. Next, we describe the current-induced magnetization switching for a MnGa(2,3) single layer and a MnGa(2,3)/CMS(1) bilayer. After the magnetization direction was aligned by applying μ0Hz=−0.5 T or 0.5 T, a current pulse Iphaving a duration of 100 μs was applied to the channel with a varying amplitude [from −40 (−50) mA to +40 (+50) mA for a 2 (3)-nm-thick MnGa layer with and without CMS], where a positive (negative) Ipis defined as current flowing in the +x(−x) direction. For deterministic switching, an in-plane magnetic field of μ0Hx=±0.05 T was applied along the xaxis. Figures 3(a) and 3(b) show Ryxas a function of Ipfor a MnGa(2) single layer for (a) μ0Hx=+0.05 T and (b) μ0Hx=−0.05 T. The Ryxwas measured with Iset to 100 μA after the application of each Ip. Clear magnetization switching with current was observed. When the directions of Ipand Hxwere parallel (antiparallel), the magnetization switched from −z(+z) to+z(−z). This relationship agrees with that expected from the switching of perpendicular mag- netization by Slonczewski-like SOT originating from the spin Hall effect (SHE) in the Ta layer.2,3 Figures 3(c) and 3(d) show Ryxas a function of Ipfor a MnGa(2)/CMS(1) bilayer for (c) μ0Hx=+0.05 T and (d) μ0Hx =−0.05 T. The polarity of the hysteresis loops for a MnGa/CMS bilayer is opposite that for a MnGa single layer because the magne- tization configuration of MnGa in the bilayer is opposite that in the single layer, as shown in Figs. 2(a) and 2(c). Interestingly, the switch- ing current for the bilayer was approximately half of that for the single layer, indicating that combining CMS with MnGa is effective in reducing the switching current. Clear SOT-induced magnetiza- tion switching was also observed in a 3-nm-thick MnGa layer with CMS at Ip=±40 mA ( not shown ), whereas it was not observed in a 3-nm-thick MnGa single layer at ∣Ip∣≤50 mA. Table I summarizes the switching current density ( JC) in the Ta layer, the saturation magnetic moment per area ( ms), and the anisotropic magnetic field ( HK) for each sample. The values of HK were estimated by fitting the RyxvsHxcurves with the simple rela- tion given by Ryx=Ryx0[1 – ( Hx/HK)2]1/2, where Ryx0isRyxatHx=0. It should be noted that the values of HKfor MnGa single layers are approximated ones because the fitting range of Hxis four or five times smaller than the estimated values of HK. The JCwas calcu- lated from the ratio of the sheet resistances of each layer. According to the macro-spin model for SOT-induced switching of perpendicu- lar magnetization,38JCis proportional to ms⋅HK. Since the value of ms⋅HKwas reduced by inserting the CMS layer, the reduction in switching current can be explained by the reduction in ms⋅HK. The value of HKwas significantly reduced. However, since too much TABLE I. Summary of switching current density ( JC) in the Ta layer, the saturation magnetic moment per area ( ms), and the anisotropic magnetic field ( HK) for each sample. MnGa(2)/CMS MnGa(2) MnGa(3)/CMS MnGa(3) JC(A/m2) 2.36 ×10114.58×10114.64×1011No switching ms(T nm) 0.7 0.35 0.35 0.66 μ0HK(T) 0.8 11 2 8 AIP Advances 11, 025205 (2021); doi: 10.1063/5.0032732 11, 025205-3 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv reduction in HKgenerally degrades the thermal stability of magne- tization, further optimization is necessary to balance low switching current and high thermal stability. Since the SOT-induced switching for a 3-nm-thick MnGa sin- gle layer was not obtained, we evaluated the strength of the SOT- induced effective magnetic field ( μ0Heff) for a 3-nm-thick MnGa layer with and without CMS by measuring the shift in the out-of- plane hysteresis loops under in-plane field Hx. Figures 4(a) and 4(b) show the normalized Ryxfor a MnGa(3)/CMS bilayer as a function ofμ0Hzwith dc current I=±12 mA under (a) μ0Hx=+0.45 T and (b)μ0Hx=−0.45 T. When Iwas parallel (antiparallel) to Hx, the center of the hysteresis loop was shifted in the negative (positive) μ0Hz-axis direction. Moreover, the shift amount μ0Hshiftwas pro- portional to the dc current, as shown in Fig. 4(c). The origin of this shift is explained by the effective field of the SOT acting on the DW as follows.37The magnetization reversal for the MnGa single layer and MnGa/CMS bilayer with μ0Hzwas accompanied by DW prop- agation since the size of the Hall bar is relatively large. Therefore, positive application of in-plane μ0Hxaligns the internal magnetiza- tion of the DWs in the x-direction, and the effective field originates from the SOT acting on the DW points in the z-direction, which induces a shift of the hysteresis loop. Thus, the shift amount of the hysteresis loop gives μ0Heff. We evaluated μ0Heffin this manner atI=±12 mA for various values of Hxand calculated the slope μ0Heff/I, which corresponds to the efficiency of Heffgeneration, at each Hx. Figure 4(d) shows μ0Heff/Ias a function of μ0Hxfor a MnGa(3) single layer and a MnGa(3)/CMS(1) bilayer. The value of μ0Heff/I was approximately five times larger for the MnGa(3)/CMS(1) bilayer than for the MnGa(3) single layer at μ0Hx=0.5 T. Saturation of FIG. 4. (a) and (b) Normalized Ryxfor a MnGa(3)/CMS bilayer as a func- tion of μ0Hzwith dc current I=±12 mA under (a) μ0Hx=+0.45 T and (b) μ0Hx=−0.45 T. (c) The shift amount of hysteresis loops μ0Hshiftas a function of current Ifor a MnGa(3)/CMS bilayer. (d) μ0Heff/Ias a function of μ0Hx for a MnGa(3) single layer and a MnGa(3)/CMS(1) bilayer, where Heffis an effec- tive field of the SOT acting on the domain wall, and μ0Heff/Icorresponds to the efficiency of generating SOT.μ0Heff/Iwith respect to μ0Hxwas observed for the MnGa(3)/CMS(1) bilayer, which may be related to realignment of the internal magne- tization in the DWs, as reported previously.11Assuming that Heff is induced by the SOT acting on the DWs, the saturated μ0Heffper current density JTacan be expressed as ∣μ0Heff/JTa∣=(π/2)(μ0θSHeff ̵h/2ems), where θSHeff,̵h, and eare the effective spin Hall angle for the Ta layer, the Dirac constant, and the elementary charge, respectively.39We assumed that the value of ∣μ0Heff/I∣atμ0Hx =0.5 T is equal to the saturated ∣μ0Heff/I∣and estimated the satu- rated∣μ0Heff/JTa∣values for the MnGa(3)/CMS(1) bilayer and the MnGa(3) single layer by using the sheet resistance of each layer. Substituting the values of ∣μ0Heff/JTa∣atμ0Hx=0.5 T into the expres- sion, we calculated the ∣θSHeff∣values for the MnGa(3)/CMS(1) bilayer and the MnGa(3) single layer to be 0.03 and 0.006, respec- tively. These results indicate that a MnGa/CMS bilayer is effec- tive in enhancing the efficiency of generating SOT. However, the physical origins for the enhancement of ∣θSHeff∣obtained in the MnGa/CMS bilayer are unclear at present, and further studies are necessary. In summary, we experimentally found that MnGa and CMS are antiferromagnetically coupled to each other with clear perpendicular magnetic anisotropy. The current needed for SOT-induced magne- tization switching was reduced by half, and the effective SOT field was approximately five times higher in a MnGa/CMS bilayer than in a MnGa single layer. These results indicate that a MnGa/CMS bilayer structure is effective in enhancing the efficiency of generat- ing SOT, mainly due to the reduction in the magnetization and/or anisotropic field and the enhancement of the effective spin Hall angle by inserting a CMS layer. This work was supported, in part, by the Japan Society for the Promotion of Science KAKENHI (Grant Nos. 20H02174 and 20H02598), the Center for Spintronics Research Network, and the Hitachi Metals-Materials Science Foundation. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq, A. Schuhl, S. Pizzini, J. Vogel, and P. Gambardella, Nat. Mater 9, 230 (2010). 2L. Liu, C. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336, 555 (2012). 3M. Cubukcu, O. Boulle, M. Drouard, K. Garello, C. O. Avci, I. M. Miron, J. Langer, B. Ocker, P. Gambardella, and G. Gaudin, Appl. 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5.0036404.pdf

J. Chem. Phys. 154, 064102 (2021); https://doi.org/10.1063/5.0036404 154, 064102 © 2021 Author(s).Resolution of the identity approximation applied to PNOF correlation calculations Cite as: J. Chem. Phys. 154, 064102 (2021); https://doi.org/10.1063/5.0036404 Submitted: 04 November 2020 . Accepted: 15 January 2021 . Published Online: 10 February 2021 Juan Felipe Huan Lew-Yee , Mario Piris , and Jorge M. del Campo ARTICLES YOU MAY BE INTERESTED IN r2SCAN-3c: A “Swiss army knife” composite electronic-structure method The Journal of Chemical Physics 154, 064103 (2021); https://doi.org/10.1063/5.0040021 r2SCAN-D4: Dispersion corrected meta-generalized gradient approximation for general chemical applications The Journal of Chemical Physics 154, 061101 (2021); https://doi.org/10.1063/5.0041008 Analytic evaluation of energy first derivatives for spin–orbit coupled-cluster singles and doubles augmented with noniterative triples method: General formulation and an implementation for first-order properties The Journal of Chemical Physics 154, 064110 (2021); https://doi.org/10.1063/5.0038779The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Resolution of the identity approximation applied to PNOF correlation calculations Cite as: J. Chem. Phys. 154, 064102 (2021); doi: 10.1063/5.0036404 Submitted: 4 November 2020 •Accepted: 15 January 2021 • Published Online: 10 February 2021 Juan Felipe Huan Lew-Yee,1 Mario Piris,2,a) and Jorge M. del Campo1,a) AFFILIATIONS 1Departamento de Física y Química Teórica, Facultad de Química, Universidad Nacional Autónoma de México, Mexico City C.P. 04510, Mexico 2Donostia International Physics Center (DIPC), 20018 Donostia, Euskadi, Spain; Euskal Herriko Unibertsitatea (UPV/EHU), PK 1072, 20080 Donostia, Euskadi, Spain; and Basque Foundation for Science (IKERBASQUE), 48009 Bilbao, Euskadi, Spain a)Authors to whom correspondence should be addressed: mario.piris@ehu.eus and jmdelc@unam.mx ABSTRACT In this work, the required algebra to employ the resolution of the identity approximation within the Piris Natural Orbital Functional (PNOF) is developed, leading to an implementation named DoNOF-RI. The arithmetic scaling is reduced from fifth-order to fourth-order, and the memory scaling is reduced from fourth-order to third-order, allowing significant computational time savings. After the DoNOF- RI calculation has fully converged, a restart with four-center electron repulsion integrals can be performed to remove the effect of the auxiliary basis set incompleteness, quickly converging to the exact result. The proposed approach has been tested on cycloalkanes and other molecules of general interest to study the numerical results, as well as the speed-ups achieved by PNOF7-RI when compared with PNOF7. Published under license by AIP Publishing. https://doi.org/10.1063/5.0036404 .,s I. INTRODUCTION Recently,1an open-source implementation of natural orbital functional (NOF) based methods has been made available to the scientific community. The associated computer program DoNOF is designed to solve the energy minimization problem of an approx- imate NOF, which describes the ground-state of an N-electron system in terms of the natural orbitals (NOs) and their occu- pation numbers (ONs). Approximate NOFs have been demon- strated2to be more accurate than density functionals for highly multi-configurational systems and scale better with the number of basis functions than correlated wave-function methods. A detailed account of the state of the art of the NOF-based methods can be found elsewhere.3–7 A route8for the construction of an approximate NOF involves the employment of necessary N-representability conditions9for the two-particle reduced density matrix (2RDM) reconstructed in terms of the one-particle reduced density matrix (1RDM). Appropriate 2RDM reconstructions have led to different implementations known in the literature as PNOFi (i = 1–7).10–17This family of functionalsprovide an efficient way of including dynamic and static correla- tion with chemical accuracy in many cases.18,19It has recently been shown20,21that PNOF7 is an efficient method for strongly correlated electrons in one and two dimensions. In addition, the use of pertur- bative corrections allows us to improve the dynamic correlation in order to achieve a complete method to describe electron correlated systems.22,23 In the current implementation, the DoNOF computer code needs to transform the atomic orbital (AO) electron repulsion inte- grals (AO-ERIs) into molecular orbital (MO) electron repulsion integrals (MO-ERIs) in order to evaluate the Coulomb and exchange integrals required in PNOF. The optimization process involves searching for ONs, which requires the computation of Coulomb and exchange matrices in the MO representation, and for NOs, which requires computing Coulomb and exchange matrices in the AO rep- resentation for each MO. These procedures have an overall fifth- order arithmetic scaling factor. While this scaling factor is lower compared to other procedures such as those based on configuration interaction and coupled cluster approaches, there is still room for improvement. J. Chem. Phys. 154, 064102 (2021); doi: 10.1063/5.0036404 154, 064102-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Resolution of the identity (RI), also known as density fit- ting,24–26approximates the product of basis functions as a linear combination of auxiliary basis sets.27It usually reduces the arith- metic and memory scaling factors and produces intermediate easy- to-handle arrays, as has been reported in other methodologies28–39 such as RI-MP2,34,40–44DF-MP2,45DF-MP2.5,46,47DF-MP3,46,47 DF-LCCD,48DF-CCSD,33,49,50and DF-CCSD(T).50,51In particular, the use of the RI approximation in v2RDM-CASSCF calculations52,53 has been shown, leading to energy expressions and handling of the MO-ERIs in the optimization procedure different from those nec- essary in the PNOF family of functionals. Applying the RI approx- imation in PNOF correlation calculations allows faster calculations, decreasing the arithmetic scale factor of the integral transformation of AO-ERIs to MO-ERIs from fifth-order to fourth-order, as shown in this work. This paper is structured as follows: In Sec. II, the elemental theory of PNOF formulation is shown and the use of the RI approx- imation in the ON and NO optimization process is analyzed. In Sec. III, the details about the implementation are given. In Sec. IV, the time savings due to the use of the RI approximation as well as the energy results in the standard cycloalkane test set up to nine car- bon atoms are presented, and other relevant molecules such as oxa- zole, borazine, coumarin, cyanuric chloride, benzene, thiepine, and thieno[2,3-b]thiophene are also presented. Finally, the conclusions are given in Sec. V. II. THEORY The ground-state electronic energy of an approximate NOF is given by the expression E=2∑ pnpHpp+∑ pqrsD[np,nq,nr,ns](pq∣rs), (1) where Hppdenotes the one-electron matrix elements of the kinetic energy and outer potential operators, ( pq|rs) are the MO-ERIs in chemists’ notation, and D[np,nq,nr,ns] represents the reconstructed 2RDM from the ONs {np}. Restrictions on the ONs in the range 0≤np≤1 represent the necessary and sufficient conditions for ensemble N-representability of the 1RDM under the normalization condition, 2 ∑pnp= N. It is worth noting that any explicit dependence of Don the NOs {ϕp}themselves is neglected. Accordingly, NOs are the MOs that diagonalize the 1RDM of an approximate ground-state energy, so it is more appropriate to speak of a NOF rather than a functional of 1RDM due to the explicit dependence on the 2RDM.54 It is clear that the construction of an N-representable func- tional given by Eq. (1) is related to the N-representability problem ofD. Using its ensemble N-representability conditions to generate a reconstruction functional leads to PNOF.8This particular recon- struction is based on the introduction of two auxiliary matrices Δ andΠexpressed in terms of the ONs to reconstruct the cumu- lant part of the 2RDM.55For the sake of simplicity, let us address only singlet states in this work. The generalization of our results to spin-multiplet states23is straightforward. Consequently, the energy expression of Eq. (1) becomesE=2∑ pnpHpp+∑ qpΠqpLpq +∑ qp(nqnp−Δqp)(2Jpq−Kpq), (2) where Jpq,Kpq, and Lpqare the Coulomb, exchange, and exchange- time-inversion integrals.56Note that Lpq=Kpqfor real MOs, as developed in this work. Therefore, only two-index JpqandKpqinte- grals are necessary due to our approximation for the 2RDM. Appro- priate forms of matrices ΔandΠlead to different implementations known as PNOFi (i = 1–7). Remarkable is the case of PNOF5, which turned out to be strictly pure N-representable.57 In the current implementation, minimization of the energy E[{np},{ϕp}]is performed under orthonormality requirement for real NOs, whereas ONs conform to the ensemble N-representability conditions. The solution is established by optimizing the functional of Eq. (2) with respect to the ONs and the NOs separately.58 In DoNOF,1the Coulomb integrals are built according to the following equation: Jpq=∑ μνPp μνJq μν =∑ μCμp∑ νCνp∑ σCσq∑ λCλq(μν∣σλ), (3) where the indices μ,ν,σ, andλlabel AOs of dimension Nband (μν|σλ) is an AO-ERI. Hence, Jqis the Coulomb matrix in the AO basis for the MO ϕqandPpis computed by means of the MO coefficient matrix, C, as Pp μν=CμpCνp. (4) Similarly, the exchange integrals are defined as Kpq=∑ μσPp μσKq μσ =∑ μCμp∑ σCσp∑ νCνq∑ λCλq(μν∣σλ), (5) where Kqis the exchange matrix in the AO basis for the MO ϕq. From Eqs. (3) to (5), we observe that the four-index transforma- tion of the ERIs generally scales as N5 b. In the occupancy optimiza- tion, this operation is carried out once for fixed orbitals; however, in the orbital optimization, it is necessary to perform this trans- formation every time orbitals change, which is a time-consuming process. It is worth noting that the last members of the PNOF family, namely, PNOF5–PNOF7, use electron-pairing constraints.7Until now, only these NOFs can provide the correct number of electrons in the fragments after a homolytic dissociation.19,59Moreover, the con- strained nonlinear programming problem for the ONs can be trans- formed into an unconstrained optimization with the correspond- ing saving of computational time. In the case of electron-pairing approaches, we can additionally reduce the number of orbitals in calculations and use just orbitals in the pairing scheme, which we will represent as NΩ(NΩ≤Nb). From now on, we will focus on the electron-pairing-based PNOFs. J. Chem. Phys. 154, 064102 (2021); doi: 10.1063/5.0036404 154, 064102-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Algorithm used to compute JandKin the occupancy optimization and JqandKqin the orbital optimization. Scaling Step Operation Memory Arithmetic Common 0 Evaluation of ( μν|σλ) N4 b N4 b 1 Pp μν=CμpCνp N2 bNΩ N2 bNΩ Jpq 2 Jq μν=∑σλPq σλ(μν∣σλ) N2 bNΩ N4 bNΩ 3 Jpq=∑μνPp μνJq μν N2 Ω N2 bN2 Ω Kpq 2 Kq μσ=∑νλPq νλ(μν∣σλ) N2 bNΩ N4 bNΩ 3 Kpq=∑μσPp μσKq μσ N2 Ω N2 bN2 Ω In Table I, we show the conventional algorithm used to com- pute the Coulomb ( J) and exchange ( K) integrals in the MO representation and the Coulomb ( Jq) and exchange ( Kq) matrices in the AO representation for each orbital ϕq. In the last columns, the memory and arithmetic scaling of the steps are reported. We see that the evaluation of the AO-ERIs ( μν|σλ), labeled step zero, has an arithmetic scaling of N4 b. In the current implementation, they are evaluated and stored at the beginning; consequently, this step does not contribute significantly to the computational time. However, its storage represents the highest memory demand with a memory scaling of N4 b. The first step corresponds to the evaluation of Pmatrix, as shown in Eq. (4), which has low arithmetic and memory scaling fac- tors of N2 bNΩ. The second step corresponds to the evaluation of Jq andKqmatrices for each MO in the AO basis. This is the bottleneck of the current implementation with an arithmetic scaling factor of N4 bNΩand memory scaling of N2 bNΩ. Finally, in the third step, Jand Kintegrals in the MO representation are computed with an arith- metic scaling factor of N2 bN2 Ω. The memory scaling of this step is N2 Ω, which is not significant compared to the other steps. As mentioned above, energy minimization is made up of two independent optimization procedures: an outer one that involves the optimization of the ONs for fixed orbitals and an inner one that involves the optimization of the NOs for fixed occupancies, as shown in Fig. 1. Both optimizations are iterative procedures in which many inner iterations are performed for each outer iteration until con- vergence. In Subsections II A and II B, the introduction of the RI approximation in each optimization procedure applied to PNOFi (i = 5–7) is analyzed. For further reference, to emphasize the spe- cific functional used, the calculations within this approach will be labeled PNOFi-RI (i = 5–7), while the global implementation will be named DoNOF-RI. A. Occupancy optimization with RI In DoNOF,1bounds on {np}are imposed automatically by expressing the ONs through new auxiliary variables {γp}. In this way, the constrained minimization problem with respect to ONs for a fixed set of NOs is transformed into an unconstrained minimiza- tion problem with respect to auxiliary γ-variables.Since the orbitals do not change, JandKcan be computed once and stored along the occupancy optimization process of an outer iteration. The RI approximation can be used to reduce the arithmetic scaling factors of JandKintegrals. In this approximation, the four- center AO-ERI, ( μν|σλ), is expressed using three-center ERIs, ( μν|k), and two-center ERIs, ( k|l), according to the following equation: (μν∣σλ)=∑ k(μν∣k)∑ lG−1 kl(l∣σλ), (6) where kandlrepresent functions of the auxiliary basis of dimension NauxandGis a metric matrix defined as Gkl= (k|l). In a symmetric approach, G−1/2would be computed through eigenvalue decom- position or singular value decomposition and multiplied by the three-center AO-ERIs; however, the metric matrix may be numer- ically ill conditioned,38having small or even negative eigenvalues. Although this problem might be surpassed truncating eigenvalues below a certain tolerance, the overall process is slow and may affect FIG. 1 . General scheme of the energy optimization. A guess for ONs and NOs is considered, and then, an iterative procedure composed of two independent opti- mizations with respect to ONs and NOs, respectively, is performed. For a more detailed description, see Ref. 1. J. Chem. Phys. 154, 064102 (2021); doi: 10.1063/5.0036404 154, 064102-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the numerical stability. Recently, a modified Cholesky decompo- sition has been applied to factorize the metric matrix and correct the numerical problems if required.38,39In this approach, the metric matrix is expressed as60 G=PLDLTPT, (7) where Pis a permutation matrix, Lis a lower triangular matrix, and Dis a block diagonal matrix with blocks of dimensions 1 ×1 and 2×2.61The eigenvalue spectrum of the Dmatrix is analyzed block by block to correct negative and very small eigenvalues, giving a cor- rected matrix, ˜D.62In PNOF correlation calculations, a symmetric approach results convenient; thus, the Gmatrix is expressed as G=PL˜D1/2˜D1/2LTPT. (8) The process of decomposing the Dmatrix in its eigenvectors and eigenvalues is fast due to the small dimension of its blocks. Once the eigenvalues have been corrected, its square root can be evaluated directly. Then, a btensor is found by solving the following linear equation system: PL˜D1/2bT=(k∣μν). (9) Using RI, the Coulomb and exchange integrals can be expressed as Jpq=∑ lbl ppbl qq, (10) Kpq=∑ lbl pqbl pq, (11) where the change of indices in bdenotes contractions from AOs ( μ, ν) to MOs ( p,q) according tobl pν=∑ μCμpbl μν, (12) bl pq=∑ νCνqbl pν. (13) An equivalent btensor is employed in RI implementations that useG−1/2; particularly, the equations are similar to those used in RI- MP234,40–44to build other MO-ERIs. The memory and arithmetic scaling factors of Eqs. (6)–(13) with the RI approximation are shown in Table II. The zero step cor- responds to the evaluation of the ( μν|k) AO-ERIs, and the first step corresponds to solve the linear equation system for the btensor with a memory scaling factor of N2 bNauxand an arithmetic scaling fac- tor of N2 bN2 aux. Assuming that enough memory is available to store thebtensor in the AO representation, this step can be performed only once at the beginning of the calculation; hence, although the first step has the largest memory scaling, it does not pose a problem through the iterative process. The second step is the contraction of an index of the btensor from AO to MO with the memory scaling of NbNauxNΩand the arithmetic scaling of N2 bNauxNΩ, being the most demanding step per outer iteration; in the third step, the remaining atomic orbital is contracted with the arithmetic scaling of NbNauxN2 Ω and the memory scaling of NauxN2 Ω, respectively. Finally, in step four, thebtensor is used to build the Coulomb and exchange integrals with the arithmetic scaling of NauxN2 Ωand the memory scaling of N2 Ω. The overall procedure has a fourth-order arithmetic scaling of N2 bNauxNΩ. B. Orbital optimization with RI In the inner optimization procedure of the current imple- mentation (see Fig. 1), the energy minimization is performed with respect to real MOs under the requirement of orthonormality and considering a fixed set of ONs. In general, an approximate NOF is not invariant with respect to an orthogonal transformation of the orbitals. Consequently, orbital optimization cannot be reduced to a pseudo-eigenvalue problem like in the Hartree–Fock approxima- tion. TABLE II . Algorithm used to compute JandKin the occupancy optimization with RI.a Scaling Step Operation Memory Arithmetic Common 0 Evaluation of ( μν|k) N2 bNaux N2 bNaux 1 Solve PL˜D1/2bT=(k∣μν) N2 bNaux N2 bN2 aux 2 bl pν=∑μCμpbl μν NbNauxNΩ N2 bNauxNΩ 3 bl pq=∑νCνqbl pν NauxN2 Ω NbNauxN2 Ω Jpq 4 Jpq=∑lbl ppbl qq N2 Ω NauxN2 Ω Kpq 4 Kpq=∑lbl pqbl pq N2 Ω NauxN2 Ω aFormal memory scaling is shown. However, to optimize memory usage, the contraction of btensor for JandK(steps 2, 3, and 4) is carried out simultaneously for each lsuch that the dimension of the auxiliary basis does not affect the memory scaling. J. Chem. Phys. 154, 064102 (2021); doi: 10.1063/5.0036404 154, 064102-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE III . Algorithm used to compute JqandKqin the orbital optimization with RI.a Scaling Step Operation Memory Arithmetic Common 0 Evaluation of ( μν|k) N2 bNaux N2 bNaux 1 Solve PL˜D1/2bT=(k∣μν) N2 bNaux N2 bN2 aux 2 bl qν=∑μCμqbl μν NbNauxNΩ N2 bNauxNΩ Jq μν 3 bl qq=∑νCνqbl qν NauxNΩ NbNauxNΩ 4 Jq μν=∑lbl qqbl μν N2 bNΩ N2 bNauxNΩ Kq μν 3 Kq μν=∑lbl qμbl qν N2 bNΩ N2 bNauxNΩ aFormal memory scaling is shown. However, to optimize memory usage, the contraction of btensor for Jq(steps 2, 3, and 4) andKq(steps 2 and 3) is carried out simultaneously for each lsuch that the dimension of the auxiliary basis does not affect the memory scaling. In DoNOF,1the optimal NOs are obtained by iterative diago- nalizations of a symmetric matrix Fλdetermined by the Lagrange multipliers {λpq}associated with the orthonormality conditions. A remarkable advantage of this procedure is that the orthonormality constraints are automatically satisfied. Unfortunately, the diagonal elements cannot be determined from the symmetry property of λ, so this procedure does not provide a generalized Fockian in the conven- tional sense. Nevertheless, {Fλ pp}may be determined with the help of an Aufbau principle.58 Thus, the orbital optimization requires to calculate {λpq}in each step of the inner iterations in order to determine the symmet- ric matrix Fλ. Since orbitals change in each step, JqandKqmust be recomputed in each inner iteration. Many inner iterations are per- formed per outer iteration, so the computation of these matrices in the orbital optimization is the most important contribution to the computational time of the present algorithm. The RI approximation can also be applied in this case using the procedure shown in Table III. The zero and first steps evaluate the (μν|k) AO-ERIs and the btensor in the AO basis, both are com- mon steps shared with the occupancy optimization and performed at the beginning of the calculation. In the second step, an index of the btensor is contracted from AO to MO with the arithmetic scaling of N2 bNauxNΩ. In the third step of the Coulomb procedure, an additional contraction is performed for the btensor. Finally, in the last steps of both the Coulomb and exchange procedures, the intermediate tensors are multiplied to compute JqandKq. The algo- rithm reduces the arithmetic scaling factor of orbital optimization to fourth-order ( N2 bNauxNΩ) as in the previous case. Hence, an over- all reduction in the arithmetic scaling factor from fifth-order to fourth-order and of the memory scaling factor from fourth-order to third-order is achieved due to the RI approximation. III. COMPUTATIONAL DETAILS The proposed PNOFi-RI (i = 5–7) algorithm was implemented in a modified version of the DoNOF software1using Cartesian FIG. 2 . Analysis of occupancy (top panel) and orbital optimizations (bottom panel) for PNOF7 and PNOF7-RI computing time using aug-cc-pVDZ/GEN-A2∗. The achieved speed-up is presented over each pair of bars. (a) Outer occupancy optimization cycle. (b) Outer orbital optimization cycle. J. Chem. Phys. 154, 064102 (2021); doi: 10.1063/5.0036404 154, 064102-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE IV . Comparison of the energies (hartree) obtained with PNOF7 and PNOF7-RI using aug-cc-pVDZ/GEN-A2∗for the cycloalkane test. Mean difference: 2.2 ×10−4. Molecule EPNOF 7 ΔEPNOF 7−RIa Cyclopropane (C 3H6)−117.228 991 1.5 ×10−4 Cyclobutane (C 4H8)−156.328 758 1.9 ×10−4 Cyclopentane (C 5H10)−195.449 913 2.5 ×10−4 Cyclohexane (C 6H12)−234.549 938 2.2 ×10−4 Cycloheptane (C 7H14)−273.630 436 2.3 ×10−4 Cyclooctane (C 8H16)−312.714 209 2.4 ×10−4 Cyclononane (C 9H18)−351.799 073 2.9 ×10−4 aPositive differences mean that the PNOF7-RI energy is above the PNOF7 energy. Gaussian basis functions and MPI parallelization, leading to a new implementation labeled DoNOF-RI. We assume that there is enough memory available to compute at the beginning all the required AO-ERIs as well as the bten- sor on the atomic basis and store them for use in the calculation. Operations of optimization procedures correspond only to arith- metic manipulations and not to AO-ERI evaluations. Four-center AO-ERIs, ( μν|σλ), have been screened to discard those lower than 10−9. This approach has been taken to reduce the arithmetic scaling when four-center ERIs are used.24,63–65All results shown in this arti- cle were calculated using 24 threads of an Intel Xeon Gold 5118 CPU. The basis sets were taken from the basis set exchange66–68website (www.basissetexchange.org). IV. RESULTS Single point energy calculations were performed to study the numerical stability and speed-up achieved with the DoNOF- RI implementation. The structures were optimized with the Psi4 software69using M06-2X70and aug-cc-pVDZ/aug-cc-pVDZ-jkfit71 basis sets. Initial auxiliary variables {γ0 p}corresponding to a Fermi– Dirac distribution of {n0 p}were employed. For NOs, the guess MOs were taken from a Hartree–Fock calculation.Figure 2 presents the computational times of an outer itera- tion for occupancy optimization (top panel) as well as for orbital optimization (bottom panel) from cyclopropane to cyclononane employing the aug-cc-pVDZ basis set72,73and GEN-A2∗auxiliary basis set,74–76which generates auxiliary basis functions according to the basis set. In both plots, the blue bars represent the elapsed time obtained with PNOF7 and the yellow bars correspond to the com- puted time with PNOF7-RI; the speed-up achieved by PNOF7-RI with respect to PNOF7 is presented over each pair of bars. The differ- ent sizes of the blue bars compared to the yellow bars make evident the different arithmetic scaling factors between PNOF7 and PNOF7- RI. For the smallest cycloalkane tested, C 3H6, an outer iteration of PNOF7-RI is 12 times faster than the equivalent iteration in PNOF7; on the other hand, for the largest cycloalkane tested, C 9H18, PNOF7- RI is 83 and 37 times faster for occupancy and orbital optimiza- tion, respectively. Speed-ups for occupancy and orbital optimization behave according to the described arithmetic scaling factors since the final steps of the integral evaluation for the orbital optimiza- tion shown in Table III have slightly higher arithmetic scaling fac- tors than the final steps of the integral evaluation in the occupancy optimization described in Table II. Although a significant reduction in computational time has been achieved, it is important to analyze the numerical impact of the RI approximation applied to PNOF7 on the final energy values. For this purpose, the NOs and ONs of the converged PNOF7-RI calcula- tion have been used to restart the calculation using four-center ERIs, namely, a PNOF7 calculation. The results are presented in Table IV, where the PNOF7 energy and PNOF7-RI energy difference for each cycloalkane is tabulated. It can be seen that PNOF7-RI allows achiev- ing a general accuracy between three and four decimal places with a mean difference of 2.2 ×10−4hartree. In all cases, a restart of the PNOF7-RI calculation is converged to the PNOF7 energy in at most two outer iterations, allowing for a PNOF7 result in a reduced amount of time. The described restarting procedure using cc-pVTZ/GEN-A2∗ basis sets for molecules of general interest has been performed. The results are shown in Table V, where the PNOF7 energy is shown with the corresponding deviation of the PNOF7-RI result. The min- imum error of 6.7 ×10−4corresponds to the benzene molecule, and the maximum error of 1.7 ×10−3corresponds to the coumarin TABLE V . Comparison of the energies (hartree) obtained with PNOF7 and PNOF7-RI using cc-pVTZ/GEN-A2∗for molecules of general interest. Mean difference: 3.1 ×10−3. Molecule EPNOF 7 ΔEPNOF 7−RIaSpeed-upb Oxazole (C 3H3NO) −244.980 370 8.8 ×10−423 Borazine (B 3H3N3) −241.487 944 7.0 ×10−419 Coumarin (C 9H6O2) −494.724 761 1.7 ×10−319 Cyanuric chloride (C 3Cl3N3)−1655.966 373 8.0 ×10−323 Benzene (C 6H6) −231.058 747 6.7 ×10−428 Thiepine (C 6H6S) −628.585 882 2.4 ×10−337 Thieno[2,3-b]thiophene (C 6H4S2) −1239.953 451 7.1 ×10−327 aPositive differences mean that the PNOF7-RI energy is above the PNOF7 energy. bGlobal speed-up per outer iteration. J. Chem. Phys. 154, 064102 (2021); doi: 10.1063/5.0036404 154, 064102-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp molecule. The global times of an outer iteration of PNOF7-RI and PNOF7 were compared, and the result can be seen in the column labeled speed-up, where it is shown that PNOF7-RI is 37 times faster than PNOF7 for the case of thiepine, as well as important speed-ups for the other cases. Overall, the results prove that DoNOF-RI allows us to compute medium size molecules of general interest. V. CONCLUSIONS The resolution of the identity approximation has proved to be significant to decrease the arithmetic and memory scaling factors of the PNOFi (i = 5–7) functionals, leading to the DoNOF-RI imple- mentation. The generality of the algorithm proposed here makes it applicable to all approximate natural orbital functionals known so far. While having an acceptable deviation of the final energy value, the solution for the natural orbitals and occupation numbers can be used as a start guess for a regular PNOF calculation with conver- gence in few iterations. Consequently, DoNOF-RI provides a way of reaching accurate results in a reduced amount of time, allowing PNOFi (i = 5–7) functionals to be used to study systems of general interest. ACKNOWLEDGMENTS J.F.H.L-Y. with CVU No. 867718 gratefully thanks CONA- CyT for the Ph.D. scholarship. J.MdelC acknowledges funding from CONACyT Project CB-2016-282791 and PAPIIT-IN114418 and computing resources from the LANCAD-UNAM-DGTIC- 270 project. M.P. acknowledges the financial support from MCIU/AEI/FEDER, UE (Grant No. PGC2018-097529-B-100), and Eusko Jaurlaritza (Ref. No. IT1254-19). DATA AVAILABILITY The data that support the findings of this study are available within the article. REFERENCES 1M. Piris and I. Mitxelena, “DoNOF: An open-source implementation of natural- orbital-functional-based methods for quantum chemistry,” Comput. Phys. Com- mun. 259, 107651 (2021). 2I. Mitxelena, M. Piris, and J. M. 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5.0040208.pdf

Matter Radiat. Extremes 6, 024201 (2021); https://doi.org/10.1063/5.0040208 6, 024201 © 2021 Author(s).Pulsed-field nuclear magnetic resonance: Status and prospects Cite as: Matter Radiat. Extremes 6, 024201 (2021); https://doi.org/10.1063/5.0040208 Submitted: 11 December 2020 . Accepted: 31 January 2021 . Published Online: 25 February 2021 Qinying Liu , Shiyu Liu , Yongkang Luo , and Xiaotao Han COLLECTIONS This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN On the study of hydrodynamic instabilities in the presence of background magnetic fields in high-energy-density plasmas Matter and Radiation at Extremes 6, 026904 (2021); https://doi.org/10.1063/5.0025374 Studies of laser-plasma interaction physics with low-density targets for direct-drive inertial confinement fusion on the Shenguang III prototype Matter and Radiation at Extremes 6, 025902 (2021); https://doi.org/10.1063/5.0023006 Recent progress in atomic and molecular physics for controlled fusion and astrophysics Matter and Radiation at Extremes 6, 023002 (2021); https://doi.org/10.1063/5.0045218Pulsed- field nuclear magnetic resonance: Status and prospects Cite as: Matter Radiat. Extremes 6,024201 (2021); doi: 10.1063/5.0040208 Submitted: 11 December 2020 Accepted: 31 January 2021 Published Online: 25 February 2021 Qinying Liu,1,2 Shiyu Liu,1 Yongkang Luo,1 and Xiaotao Han1,2,a) AFFILIATIONS 1Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan 430074, China 2State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan 430074, China a)Author to whom correspondence should be addressed: xthan@mail.hust.edu.cn ABSTRACT High-magnetic- field nuclear magnetic resonance (NMR) has manifested itself as an indispensable tool in modern scienti fic research in the fields of physics, chemistry, materials science, biology, and medicine, among others, owing to its great advantages in both measurement sensitivity and quantum controllability. At present, the use of pulsed fields is the only controllable and nondestructive way to generate high magnetic fields of up to 100 T. NMR combined with pulsed fields is therefore considered to have immense potential for application in multiple scienti fic and technical disciplines. Irrespective of the paramount technical challenges, including short duration of the pulsed fields, unstable plateaus, and poor field homogeneity and reproducibility, great progress has been made in a number of pulsed- field laboratories in Germany, France, and Japan. In this paper, we brie fly review the status of the pulsed- field NMR technique, as well as its applications in multiple disciplines. We also discuss future trends with regard to the upgrading of pulsed- field NMR. © 2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http:// creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0040208 I. INTRODUCTION Since Rabi invented the magnetic resonance method to study the nuclear magnetism of gaseous atoms in 1944, research achievements related to nuclear magnetic resonance (NMR) have won five Nobel Prizes for work in areas including nuclear spin, spectral transformation, and magnetic resonance imaging (MRI): two in Physics, two in Chemistry, and one in Medicine. The essential characteristic of NMR is that it opens the door to direct study of the magnetic moments of nucleons, and canmeasure the magnetic properties of an atomic nucleus whose mass, in the case of hydrogen, is just 1840 times the mass of an electron. Techniques based on NMR are therefore widely used as research tools in several areas of physics, chemistry, materials science, and biomedicine. Much effort has been dedicated to the development of NMR techniques, with several breakthroughs occurring in the past few decades that have made NMR one of the most widely used experimental methods. In the past, there was a general consensus that it was very dif ficult to perform NMR experiments in an unsteady magnetic field, especially a pulsed field. However, with the development of NMR technology, many amazing nuclear magnetic phenomena have been found in unstable magnetic fields such as the high magnetic fields provided by hybrid magnets. 1–3The discovery of such field-inducedexotic behavior under extreme conditions has motivated attempts to further increase the background field intensity and explore ways to apply NMR techniques in higher pulsed magnetic fields, bringing pulsed-magnetic- field NMR (PF-NMR) to center stage in recent years. The use of PF-NMR promises to be of great value in many ways, including the following. First, PF-NMR should allow improvements in the detection signal-to-noise ratio (SNR). Although NMR has precise site-selectionproperties, its inherent low signal strength has always been a noto-rious disadvantage. With the increasing demand for high-throughput and multidimensional studies, in order to improve the SNR, it is necessary to perform multiple signal sampling, which results inmassive amounts of data, and the time required has become a major constraint on applications. 4The SNR of an NMR spectrogram has a power-law dependence on the magnetic field intensity: SNR}γf3/2}γ5/2B3/2, (1) where γis the gyromagnetic ratio, fis the resonance frequency, and B is the intensity of the magnetic field. Therefore, to improve the SNR, the background magnetic field strength needs to be increased. However, permanent magnets, superconducting magnets, and hybrid Matter Radiat. Extremes 6,024201 (2021); doi: 10.1063/5.0040208 6,024201-1 © Author(s) 2021Matter and Radiation at ExtremesREVIEW scitation.org/journal/mremagnets all have a critical temperature for operation and are all subject to deformation by electromagnetic forces. So far, the maxi- mum magnetic field strength produced by superconducting magnets can only reach 32.35 T,5and that of hybrid magnets can only reach 45 T.6,7 For example, in the biological field, in order to ensure an ac- ceptable SNR in existing MRI studies of macromolecules, multiplecollections of data must be superimposed. The acquisition time can be reduced by the addition of a paramagnetic reagent as co-solute to shorten the relaxation time, by increasing the scanning speed, or by optimizing the filter performance to allow estimation of multiple samples at the same time; alternatively, the sample concentration can be increased to achieve high-throughput multidimensional NMR measurements. 8–11Nevertheless, many problems remain, such as limited space for the arrangement of the probe terminals12and the large amounts of data that are acquired.13Much work has been devoted to these problems, but they have not been tackled at root. High- field NMR provides a fundamental approach to solving these problems. Increasing the background magnetic field will directly expand the range of energies corresponding to each energy level and enhance the resonance signal. In biological applications, on the one hand, a higher background field signi ficantly enhances the SNR of the resonance signal, which reduces the time required for an experiment thus allowing study of the structure and dynamical changes of bio- logical macromolecules with molecular weights up to several meg- adaltons. On the other hand, the Larmor resonance frequency becomes higher in an ultrahigh magnetic field, leading to higher resolution. More subtle chemical environmental changes can be distinguished, and the capability of NMR to recognize various groups of biological macromolecules can be increased. Second, PF-NMR should allow the exploration of the peculiar physical properties of special systems that are far from the normal state under ultrahigh field strengths. NMR provides an effective method to directly detect the electron energy density, which de fines the properties of materials. Therefore, it is a very important tool for exploring new phases and phenomena driven by magnetic fields. Moreover, a large number of experiments have shown that when theexternal magnetic field reaches a certain strength, some special phenomena that do not appear at low fields can be observed, such as the Wigner crystal state in low-dimensional quantum systems, 14 suppression of superconductivity,15,16the de Haas –van Alphen (dHvA) effect, the Shubnikov –de Haas (SdH) effect17in two- dimensional ultrathin materials,18topological materials,19and complex magnetic materials.20,21Where appropriate, PF-NMR can provide microscopic information at the nucleon level, as well as having great potential for the study of exotic field-induced effects in special systems.22 For example, in solid state physics, steady high-magnetic- field NMR at 33.5 T revealed the existence of a charge density wave (CDW) phase state in underdoped p/equals0.108 and 0.12 samples of the well- known high-temperature superconductor YBa 2Cu3O7.23This proved the existence of orderly charges in high- Tcsuperconductors and indicated that the ordered state at the optimally doped quantum critical point may be this kind of CDW. However, no direct evidence could be obtained, because sample experiments close to the optimal doping must be carried out at higher magnetic fields, exceeding the design limit of known steady high-magnetic- field techniques.Since the inherent SNR of the detection method is directly proportional to the 3/2 power of the background field strength, an increase in this strength can greatly improve the detection accuracy as well as signi ficantly shortening the detection time. Furthermore, a high magnetic field can directly affect the electronic state and quantization of matter and signi ficantly change the electronic structure, thus leading to special properties that would not appear under normal circumstances. This provides further motivation for the development of NMR techniques in higher magnetic fields, especially unstable pulsed fields. The remainder of this paper is organized as follows. The current state of development of PF-NMR is described in Sec. IIwith regard to the pulsed magnets, high-frequency spectrometers, and signal pro- cessing strategies that are available. In Sec. III, prospective research applications of NMR in extremely strong magnetic fields are de- scribed. Then, in Sec. IV, considering these potential scenarios, the key technical problems faced in attempts to realize high- field and high-frequency NMR are analyzed. In Sec. V, a perspective for future developments of PF-NMR is presented. Finally, the paper is con- cluded in Sec. VI. II. DEVELOPMENTAL STATUS In an NMR experiment, Zeeman splitting is induced by applying a background magnetic field to nuclear spins, resulting in energy separation between sublevels. When the magnetic field is constant, according to the Boltzmann distribution law, the system will reach a certain thermal equilibrium. The population difference be- tween the upper and lower spins is determined by the Boltzmann factor. At this time, the spectrometer emits a radio-frequency (RF) signal to cause the low-energy nucleus to transition to a high-energy level, with the original population distributions being destroyed. After removal of the RF field, the high-energy nucleus then spontaneously returns to the low-energy level, and the system re-establishes thermal equilibrium, i.e., relaxation occurs. In solids containing unpaired electrons, the relaxation stems from fieldfluctuations caused by pulsed electron moments. This can be regarded as an interactive electron –nuclear spin reversal scattering process, which keeps the angular momentum unchanged. The free induction decay (FID) signal is recorded by an LCcoil around the sample. Through or- thogonal frequency conversion, spectral transformation, and other signal processing methods, we can obtain nuclear spin information such as the Knight/chemical shift and relaxation time,24with the aim of analyzing the crystal structure and electronic state of the sample. To transplant the NMR method to higher magnetic fields, especially unstable strong fields such as pulsed fields, we need to ensure field stability, develop NMR spectrometers suitable for high fields and high frequency, and establish new algorithms for dynamic analysis of FID signals. A. Early origin In recent years, several high- field laboratories have carried out research on high- field NMR. The National High Magnetic Field Laboratory (NHMFL) in Florida started work very early on. In 2000, Murali ’s team pointed out that in order to take advantage of NMR in high magnetic fields, a magnetic field with high intensity, good spatial homogeneity, and temporal stability was needed. They demonstrated that in high- field NMR experiments, transient instability of steady- Matter Radiat. Extremes 6,024201 (2021); doi: 10.1063/5.0040208 6,024201-2 © Author(s) 2021Matter and Radiation at ExtremesREVIEW scitation.org/journal/mrestate magnets is mainly caused by ripples in the power supply and changes in the temperature of the cooling system. They therefore designed a de-ripple feedback coil and cooling system with an au- tomatic correction mode that allowed them to obtain a 1.7 ppm linewidth in 2D NMR at 24 T. A liquid sample was then spin-manipulated by the intermolecular zero-quantum-coherence (iZQC) method to reduce the in fluence of poor spatial homogeneity and temporal stability of the magnetic field on the spectrum, and they were thereby able to obtain the first high-resolution NMR spectrum up to 1 GHz.25 In 2002, NHMFL showed that phase noise results from the phase changes of continuously acquired NMR signals caused by an unstable external magnetic field, which leads to serious FID signal distortion during the signal averaging process.1With a reduced sample volume and the use of magic-angle spinning (MAS), NMR experiments on alumina mixtures were carried out for different field strengths, and the resonance spectra of a solid27Al sample were analyzed under a high field generated by a 40 T hybrid magnet (an 11 T super- conducting magnet combined with a 29 T resistive magnet). As shown inFig. 1 , the spectral resolution gradually improved with increasing external magnetic field, and the resonance site was more clearly identi fied, thus solving the problem of spectral line aliasing of samples with strong quadrupole coupling in lower magnet fields. Further studies in the following years have shown that the NMR spectral resolution of resistive –superconducting hybrid magnets can be im- proved by ferromagnetic shimming, MAS, and heteronuclear phase correction, and the linewidth achieved under ideal conditions can be close to that achievable with superconducting magnets.26Other approaches, such as the use of pulse sequences that are not sensitive to magnetic fieldfluctuations to excite the sample, or feedback control systems that actively compensate for the shimming field to stabilize the magnetic flux, can signi ficantly reduce the phase difference causedby magnetic fieldfluctuations between spin-echo signals.27These methods are expected to narrow the aliasing spectral peaks, allowing the observation of NMR phenomenon that cannot be detected with high resolution in low fields. At about the same time, in 2003, the Nijmegen High-Field Magnet Laboratory obtained FID spectral lines of27Al in a 24 T steady magnetic field by means of techniques including the use of a shimming insert, feedback of the power supply, and phase reference deconvolution.28The result indicated a clear nuclear quadrupole resonance (NQR). It is worth noting that the although the quadrupole interaction provides important structural information, it impairs spectral resolution and widens the central linewidth, and some nuclei will over flow outside the detection edge. This situation is called “NMR invisibility, ”which means that these nuclei in these special positions can only be observed under a higher magnetic field. Therefore, the results are consistent with NHMFL ’s inference, namely, that a high- field physical environment provides unique advantages for the NMR study of half-integer quadrupole nuclei (such as27Al,23Na, and17O). In 2002, the National Institute for Materials Science (NIMS) conducted an NMR experiment in a 21.6 T superconducting magnetic field. In this study, although the magnetic field intensity was not very high, the problem of inhomogeneous broadening of spectral lines had already appeared. It was demonstrated in Ref. 29that this problem can be alleviated by adding shimming coils, and FID results with a res- olution of up to 4 Hz were obtained at a resonance frequency of 920 MHz. Although these pilot studies did not directly involve NMR experiments in a pulsed high magnetic field, they all re flected the trend of NMR experiments toward higher field strengths. B. Development track 1. Budding technology In 2003, the High Magnetic Field Laboratory Dresden (HLD) group proposed the concept of PF-NMR and conducted systematic experimental research,30,31including a feasibility analysis of magnetic field homogeneity, spectral resolution, SNR, and other factors. They carried out NMR experiments on63Cu under 12 T and 33 T pulsed high magnetic fields, and selected a time window near the peak to trigger the RF pulse at a fixed point, obtaining the results shown in Fig. 2 .32Although there is a large ripple and the linewidth is not ideal, this was the first time that a useful FID signal had been collected in a pulsed field, and it marked the germination of PF-NMR technology. Some disadvantages were also noted, such as the strong time de- pendence of the pulsed field, the far lower spatial homogeneity than that available with commercial NMR superconducting magnetic fields, and the long cooling time. The experiments were optimized by reducing the sample volume to meet the requirements of higher magnetic field intensity, by broadening the instantaneous bandwidth of the RF signal to expand the scanning range, and by extending the pulse flattening time to ensure full polarization of the nuclear spin system. The HLD group realized NMR of2D at a 58 T magnetic field in 2004,33and proposed a method of isotope comparison in the same year.34First, the1H(γ/equals42.5774) resonance was excited in a steady- state field, and then2D(γ/equals6.5359) was calibrated with gyromagnetic ratio multiples in the pulsed field. Taking advantage of the constant spin magnetic ratio of isotopes, the uncertainty caused by different electronic environments among elements was eliminated, and the FIG. 1.27Al MAS-NMR spectra from 14 T to 40 T. Reprinted with permission from Gan et al. , J. Am. Chem. Soc. 124, 5634 (2002). Copyright 2002 American Chemical Society. Matter Radiat. Extremes 6,024201 (2021); doi: 10.1063/5.0040208 6,024201-3 © Author(s) 2021Matter and Radiation at ExtremesREVIEW scitation.org/journal/mreelements with low gyromagnetic ratio under a high pulsed field were measured. Then, by reducing the sample volume to adapt to thepoorly homogeneous field, increasing the power ampli fier magni fi- cation to narrow the bandwidth, and collecting multiple FID to reduce the signal uncertainty, the 1H NMR spectrum was observed at fre- quencies of 1.3 GHz,352 GHz,36and 2.4 GHz.37NMR signal ac- quisition of elements with high gyromagnetic ratio was completed under a pulsed magnetic field of up to nearly 50 T, as shown in Fig. 3 . Although the SNR is low in this stage, this is still strong evidence that NMR detection can be realized in a pulsed magnetic field. These studies have indicated that PF-NMR represents a new stage in the development of NMR. It is signi ficant not principally because of its ability to improve SNR and resolution, since measures such as increased flux, sample concentration, number of time averageaccumulations, and scanning speed can already do this. Rather, the field-induced collective electronic behavior that appears in ultrahigh magnetic fields, leading to exotic phase transitions and states of matter, can only be observed in a pulsed high magnetic field, and it is this that constitutes the unparalleled advantage of pulsed- field NMR over steady- field NMR. 2. Further studies The years 2007 –2016 represented the peak of PF-NMR devel- opment. Research teams from various institutions continuously optimized the relevant techniques. On the one hand, the background field conditions were improved through means such as spectrometer upgrades and magnet optimization, while methods such as magnetic field time-dependent interlocking FID phase, frequency-domain deconvolution, and signal normalization averaging were incorpo- rated into algorithms to improve the quality of spectral analysis. In terms of spectrometer development, in 2009, the Zheng team at Okayama University, using a home-built phase-coherent NMR spectrometer, obtained a59Co NMR shift38that was consistent with the 8 T steady-state field results in Ref. 39and a peak width that was roughly equivalent to the spectral width including the spectral line due to the NQR effect, as presented in Fig. 4 . Even though, compared with results obtained under a steady field, the quality of the spectral lines is not high and the NQR effect is not very obvious, this was the first time that nuclear spin detection excited by a spin-echo pulse sequence was realized in a pulsed field and was of great signi ficance for the development of PF-NMR. In 2012, the HLD group built a complete spectrometer system suitable for high- field, high-frequency NMR experiments under pulsed fields. The upper-level computer program was written in LabVIEW, and the lower-level program was embedded in the NI system.40As shown in Fig. 5 , the bandwidth range of the spectrometer could theoretically reach 500 kHz –2.7 GHz. Based on this spectrometer, a spin-echo pulse sequence was used to solve the dephasing problem in an inhomogeneous magnetic field. The transverse relaxation time T2of the nuclear spin system under a FIG. 2. (a) Background pulsed magnetic field. (b)63Cu FID at 12 T; (c) Fourier transform of63Cu FID at 33 T. Reprinted with permission from Haas et al. , J. Magn. Magn. Mater. 272-276 , e1623 (2004). Copyright 2004 Elsevier. FIG. 3.1H FID under a 50 T pulsed high magnetic field. Reprinted with permission from Haas et al. , Solid State Nucl. Magn. Reson. 28, 64 (2005). Copyright 2005 Elsevier. Matter Radiat. Extremes 6,024201 (2021); doi: 10.1063/5.0040208 6,024201-4 © Author(s) 2021Matter and Radiation at ExtremesREVIEW scitation.org/journal/mrepulsed field was obtained for the first time, indicating that it is feasible to observe dynamic relaxation processes of NMR in ultrahigh pulsed fields. It has been speculated that the spin coherence time is long enough to allow further NMR experiments such as crystal structure analysis and electronic state capture.In 2011, the group at the Laboratoire National des Champs Magn´ etiques Intenses (LNCMI) used a spin-coherent NMR spec- trometer similar to that described in Refs. 38and41to realize solid state measurements in PF-NMR experiments, observing1H and93Nb spectral lines at 30.4 T.42Two years later, this research group pre- sented the detailed structure of the PF-NMR spectrometer, as shown inFig. 6 . The main difference from the HLD group ’s spectrometer is that a dual-channel Marconi 2024 RF signal generator was used, and a Lake Shore Model 340 temperature control system was con figured at the sample end. Using this system, 48.8 T NMR experiments on YBCO were carried out, and the main and satellite peaks of63Cu and 65Cu were observed.43This was the first time that NMR studies were performed on high-temperature superconductors in a pulsed high magnetic field. As far as magnet development is concerned, the HLD group completed the design, construction, and testing of a PF-NMR ex- perimental magnet. On the basis of 70 T/100 ms, they planned to build a 60 T/1000 ms long pulsed magnet.44With this goal, in 2012, they successfully built a 60 T/1500 ms long pulse magnet that stayed for about 70 ms at the maximum field level ( Bmax±1 T).45The magnet structure is composed of a modular high-power capacitor bank and special copper alloy wire mixed with Zylon –Stycast composite re- inforcement, which allowed NMR experiments to be performed at 52.2 T with a long pulsed flat top. In 2016, the LNCMI group analyzed the spatial homogeneity of the magnetic field in a PF-NMR experiment.46It is generally accepted that the main reason for decreased homogeneity of high- field pulses is deformation of the magnet geometry rather than noise. By changing the winding direction of the local magnet coil and reducing the sample volume and center hole diameter of the magnet, the LNCMI study achieved a high degree of homogeneity of 10 ppm at 12 T over a sample volume of 2 mm3–3m m3in the central part of the magnet. Although this does not represent much of an advantage over superconducting magnets, the same method was used to achieve a spatial homogeneity of 33 ppm at 47 T, which improved the spatial distribution of magnetic flux at the level of pulsed high magnetic field intensity, as shown in Fig. 7 . This further widens the scope for further development of NMR technology in pulsed field environments. In terms of signal processing strategy, in addition to common problems such as radio-frequency interference (RFI) noise,47,48the FIG. 5. Scheme of the pulsed NMR spectrometer at HLD. Reprinted with permission from Meier et al. , Rev. Sci. Instrum. 83, 083113 (2012). Copyright 2012 AIP Publishing LLC. FIG. 4. (a)59Co NMR spectra under a steady field. Reprinted with permission from Kawasaki et al. , Phys. Rev. B 79, 220514 (2009). Copyright 2009 American Physical Society. (b)59Co NMR spectra under a pulsed field. Reprinted with permission from Zheng et al. , J. Phys. Soc. Jpn. 78, 095001 (2009). Copyright 2009 The Physical Society of Japan. Matter Radiat. Extremes 6,024201 (2021); doi: 10.1063/5.0040208 6,024201-5 © Author(s) 2021Matter and Radiation at ExtremesREVIEW scitation.org/journal/mremost important issue in PF-NMR is phase correction of FID signals modulated by the rapidly changing pulsed field. In 2007, the NIMS group published Ref. 49on how to use deconvolution to calculate the NMR signal in a time-dependent field to achieve high resolution. In this method, the pick-up coil is used to monitor the real-timemagnetic field strength, and the magnetic fieldfluctuations are di- vided into two parts: recognizable and unrecognizable. The unrec- ognizable part mainly refers to the initial offset or slow fluctuations with a period longer than a few milliseconds. Such fluctuations cannot be detected by the pick-up coil in an instantaneous measurement. The constant value is determined by adjusting all the FID spectral peaks collected in one discharge to the average position. Figure 8(a) shows thefluctuating curve obtained from a synthesis of five tests. After the initial fluctuations have been corrected, the induced voltage of the pick-up coil is double-integrated to obtain the phase offset, thereby realizing an interlock of magnetic field and phase. In this way, the FID signal correction process of a MAS-NMR experiment can be com- pleted under the high field provided by a 30 T hybrid magnet. Two years later, the NIMS group further proposed a reference signal deconvolution compensation method for phase reconstruction in response to the large magnetic fieldfluctuations that occur in NMR experiments on liquid samples. The NMR signals of1H and2D were measured synchronously in the fluctuating field, and1H phase compensation was performed with the2D picked signal as a reference with high resolution.50 The LNCMI group used a similar method for obtaining the induced voltage in the additional pickup coil to estimate the phase offset. They performed deconvolution correction on the measured signal according to the feedback voltage value, and obtained an FID FIG. 6. Scheme of the pulsed NMR spectrometer at LNCMI. Reprinted with permission from Stork et al. , J. Magn. Reson. 234, 30 (2013). Copyright 2013 Elsevier. FIG. 7. Magnetic field homogeneity of the pulsed magnet at LNCMI at 12.5 T and 47 T. Reprinted with permission from Orlova et al. , J. Magn. Reson. 268, 82 (2016). Copyright 2016 Elsevier. FIG. 8. (a) Initial value of the induced electromotive force measured synchronously with the FID signal. Reprinted with permission from Iijima et al. , J. Magn. Reson. 184, 258 (2007). Copyright 2007 Elsevier. (b)1H spectrum before and after deconvo- lution. Reprinted with permission from Stork et al. , J. Magn. Reson. 234, 30 (2013). Copyright 2013 Elsevier. Matter Radiat. Extremes 6,024201 (2021); doi: 10.1063/5.0040208 6,024201-6 © Author(s) 2021Matter and Radiation at ExtremesREVIEW scitation.org/journal/mrespectral line consistent with the steady field.43The result is shown in Fig. 8(b) . In 2011, the HLD group published a signal averaging algorithm for NMR experimental data in a pulsed field.51Byfitting FID signals, the time-dependent B(t) of the magnetic field was obtained, and the initial phase was then inverted by B(t). The phase change of the FID signal was interlocked with the change in the magnetic field through the resonance gyromagnetic ratio to give the spectral analysis results for the FID signal. This method enables the correlation experiment to carry out signal averaging in a pulsed field similar to the way in which this is done in a steady state, which goes a long way toward solving the problems of rapid signal change and poor repeatability of the pulsed field. The resonance spectrum of 25/10 FID signals in 25/5 ms under a 7.7 T –7.8 T/61.7 T –62.2 T pulsed field was obtained, as shown in Fig. 9 . This means that Knight/chemical shifts in a pulsed magnetic field can be measured, which allows nuclear magnetic electronic states to be studied and spin magnetization information to be obtained in high- field systems in a variety of applications, including biological, chemical, and condensed matter physics. This is an important achievement in the development of PF-NMR. It is noteworthy that the above problems, such as phase cor- rection and signal averaging, are all accomplished by B(T) decon- volution, because they are only applied in a single-pulse scenario. However, in actual experiments, in order to extend high- field de- tection results to newly discovered field-induced systems, it is still necessary to measure the signal average value under multiple pulses. At the same time, judging from the current development of magnet technology, no matter which facility produces a pulsed high magnetic field, repeatability is limited, and the discharge mode means that it is impossible to achieve an accurate setting value for each peak magnetic field, and therefore the waveform in a certain region at the top cannot be completely reproduced. From this point of view, in 2016, the HLD team further proposed a method to demodulate the magnetic field intensity with the phase of the strong nuclear spin signal as the reference value, and they used the magnetic field strength to modulate the weak nuclear spin signal in turn. The NMR shifts of27Al and LindeA zeolite were detected under a maximum magnetic field of 58 T,52as shown in Fig. 10(a) , thereby setting a precedent for high- field NMR multinuclear detection. In addition, the longitudinal relaxation time T1is a very im- portant measurement objective in conventional NMR experiments. However, it is very dif ficult to measure T1in a pulsed field, because it ranges between milliseconds and seconds, and the peak plateau time of the pulsed magnetic field is shorter than this. The HLD team presented a method for measuring T1in a fast relaxation system by using adiabatic inversion in the pulsed field. The measurement process is shown in Fig. 10(b) . The T1values of 29 T/302 K aluminum powder and 58 T/308 K liquid gallium were successfully measured, with errors that were within the allowable fluctuation range. In the same year, the HLD team further considered nuclear quadrupole energy other than Zeeman energy,53and introduced Bmax as a normalization factor in the deconvolution average method in the signal processing. This method was applied to the spin dimer system SrCu 2(BO 3)2to observe the NMR phenomenon of the11Bs y s t e ma t 54 T. An overview of the procedure is shown in Fig. 11 . The result shows that the problem of spectral line broadening caused by inhomogeneity, instability, and poor repeatability of the magnetic field has largely been solved. Compared with the previous single-pulse phase correction, a more feasible high- field NMR signal processing method under multi- pulse experiments has been proposed, which improves the quality of the spectrum and gives hyper fine interaction information at the micro- scopic electronic state level that cannot be obtained under low- field conditions. This provides a demonstration of the analysis of nuclear spin NMR spectra in a pulsed high magnetic field. In general, many effective signal processing methods are available that allow higher-quality FID spectral lines to be obtained in NMR detection under unsteady magnetic fields. The methods described in this paper are listed in Table I . Up to now, PF-NMR technology has achieved the four most important objectives of observation in experiments: the Knight/ chemical shift, the NQR effect, the longitudinal relaxation time T1, and the transverse relaxation time T2. FIG. 9. Spectrogram of the FID signal of a single-pulse peak segment: (a) 7 T; (b) 62 T. Reprinted with permission from Meier et al. , J. Magn. Reson. 210, 1 (2011). Copyright 2011 Elsevier. Matter Radiat. Extremes 6,024201 (2021); doi: 10.1063/5.0040208 6,024201-7 © Author(s) 2021Matter and Radiation at ExtremesREVIEW scitation.org/journal/mreWith the development of high- field NMR over the last 20 years (Table II ), it has been demonstrated that NMR experiments are quite feasible in pulsed magnetic fields. Research teams from Germany, France, and Japan have cooperated with the high-magnetic- field laboratories and have made outstanding achievements in related research. Although there have not been many papers published in this area to date, and the quality of NMR spectral analysis in pulsed magnetic fields remains far lower than that in steady fields, the huge development potential of PF-NMR provides a great incentive for high-magnetic- field research centers worldwide to focus more on this area and gradually move from exploration to application.From an interdisciplinary point of view, high-magnetic- field technology and NMR have a strong correlation. If a high- field and high-frequency NMR technology using flat-top pulsed magnetic fields (FTPMFs) can be developed on existing experimental plat- forms, and background fields with stability and homogeneity com- parable to those of a steady-state magnetic field can be realized, it should be possible to use PF-NMR in a wider range of applications in biology, medicine, and solid state physics. This will not only promote the development of pulsed high-magnetic- field technology, but will also have long-term signi ficance for innovation in NMR detection methods. FIG. 10. (a) FID signals of Linde A (weak, left) and27Al (strong, right) under a 55.7 T pulsed field. (b) Adiabatic reversal experiment for measuring T1in a pulsed field. Reprinted with permission from Kohlrautz et al. , J. Magn. Reson. 263, 1 (2016). Copyright 2016 Elsevier. FIG. 11. Overview of procedure for reconstruction of broad spectra in a pulsed magnetic field using the normalized deconvolution method. For more details, see Ref. 53. Reprinted with permission from Kohlrautz et al. , J. Magn. Reson. 271, 52 (2016). Copyright 2016 Elsevier. Matter Radiat. Extremes 6,024201 (2021); doi: 10.1063/5.0040208 6,024201-8 © Author(s) 2021Matter and Radiation at ExtremesREVIEW scitation.org/journal/mreIII. RESEARCH PROSPECTS There have been a number of proposals for practical applications of PF-NMR, with the ability to carry out NMR experiments under higher magnetic fields offering new research opportunities in many areas. A. Bio-macromolecular dynamics As one of the main methods for structural analysis of biological macromolecules, NMR has been widely used to reveal the relationship between structure and function of proteins and nucleic acids. Compared with other detection methods, NMR can capture the instantaneous dynamic structure of biological macromolecules, with the molecular characterization being closer to that in physiological states, and it can provide a better re flection of the relationships between structure, dynamics, and function. For example, although interactions between proteins are weak, they are fundamental to cell signal transduction and many other important cellular processes. Taking advantage of the nuclear Overhauser effect (NOE) between molecules, it is possible to use NMR spectroscopy to detect ultraweak interactions between proteins and determine their skeletal structure54 Recent years have seen the successful capture of the protein complex with the currently known weakest interaction strength (dissociation constant Kdup to 25 mM) using paramagnetic NMR technique, together with a structural analysis at atomic resolution, ascan be seen in Fig. 12 .55A new type of rigid paramagnetic probe with unpaired electrons was developed to label the protein, and the subtle changes in protein structure were then successfully observed by NMR, with a detection spatial resolution reaching 1 ˚A.56Moreover, protein site-speci fic labeling based on unnatural amino acids and a label-free NMR method has also been promoted, greatly expanding the scope ofapplication of magnetic resonance techniques. 57–59 The SNR is the key parameter determining NMR detection sensitivity. Because the interval between nuclear spin energy levels is very small (the smallest among almost all types of absorption spectra), the energies and sensitivity of NMR are very low (e.g., smaller than those in electron spin resonance60,61by a factor of 1000 or more). In recent years, methods such as dynamic nuclear polarization (DNP) have been proposed to excite electron nuclear double resonance with the aim of solving this problem, but it is dif ficult and expensive to manufacture high-power millimeter-wave microwave sources. Therefore, the most direct way is still to improve the SNR based on existing NMR experiments. One method is to increase the number of particles distributed in an energy level by increasing the concentration of the sample. However, the sample concentration is usually limited by natural abundance and extraction and separation techniques. Another convenient approach is to increase the magnetic field in- tensity to widen the energy level interval, thereby obtaining a greater energy level difference and a higher Larmor frequency of atomicTABLE I. NMR signal processing strategies in unstable magnetic fields. Facility Reference MagnetaAcquisition of B(t) Phase correction NIMS 49 Hybrid Pick-up coil Deconvolution averaged HLD 51 Resistive L-M algorithm Deconvolution averaged HLD 53 Resistive Phase demodulation Normalized deconvolution LNCMI 43 Resistive Pick-up coil Deconvolution aThe devices that generate unstable high magnetic fields are generally divided into hybrid magnets and resistive magnets. The former produce magnetic field pro files closer to those of permanent magnets or superconducting magnets with a lower intensity, while the latter produce pulsed magnetic fields with greater fluctuation but higher intensity. TABLE II. Research status of NMR in unsteady high magnetic fields worldwide. Facility Year ReferenceBmax (T)Resonance fre- quency (MHz)Temperature (K)Rprobe (mm) RF sequenceTarget nucleus (object) HLD 2003 31 12 140 300 3π 2(0.5μs)63Cu (shift) 2003 31 33 360 300 2π 2(0.5μs)63Cu (shift) 2004 33 58 375 300 3π 2(0.5μs)2D (shift) 2005 37 56 2400 300 6 <π 2(0.3μs)1H (shift) 2012 40 62 400 ... 6π 2(1μs)−τ(150μs)−π(2μs)2D(T2) 2016 52 58 600 308 16 ≪π 269Ga (shift, T1) 2016 53 54 740 2 ...π 2(0.2μs)11B (shift, NQR) Okayama university2010 41 48 495 ... ...π 2(2.5μs)−τ(20μs)−π(5μs)59Co (shift, NQR) LNCMI 2011 42 30 300 80 ...π 2(4.7μs)−τ(20μs)−π(9.4μs)93Nb (shift) 2013 43 47 300 2.5 ...π 2(0.8μs)−τ(3.9μs)−π(1.6μs)63Cu/65Cu (shift, NQR) Matter Radiat. Extremes 6,024201 (2021); doi: 10.1063/5.0040208 6,024201-9 © Author(s) 2021Matter and Radiation at ExtremesREVIEW scitation.org/journal/mremagnetic moment precession. This enhances the useful signal in the NMR experiment and signi ficantly improves the SNR. At present, the maximum available pulsed field intensity has reached 80 T –100 T, generated at high-magnetic- field research in- stitutions, the maximum flat-top magnetic field has reached 64 T, and the platform period has reached tens of milliseconds or even longer. Collecting nuclear magnetic data under ultrahigh magnetic fields can greatly shorten the experimental time and signi ficantly improve the acquisition ef ficiency of massive multidimensional NMR data. In addition, the structural changes of certain biological macromolecules with speci fic biological functions are dynamic. A single sample can be detected with different magnetic field strengths under a pulsed magnetic field, and dynamic information about the protein structure can then be inferred from the magnetic resonance characteristics (relaxation, chemical shift, chemical exchange, etc.). A large amount of NMR information can be obtained through rapid sampling under different magnetic field intensities, which is helpful for analyzing the dynamic characteristics of protein structures in the process of their function. Clearly, PF-NMR can provide a high-precision and high-resolution method for analyzing structural information on biological macromolecules, thereby helping to solve a number of important basic biological problems. B. Condensed matter physics In condensed matter physics, NMR is often used to study the interaction between nuclear systems and a magnetic field, the in- teraction between nuclei and the outside environment, and the re- laxation of nuclear systems. Owing to the hyper fine interaction between the electronic and nuclear moments, the electronic system can be directly detected through the nucleus, thereby providing valuable information on many different phenomena, 7with the ad- vantages of high spatial resolution and flexible element selection. NMR experiments can be used to observe the characteristics of some unconventional materials, including physical properties such as the Knight shift, NQR, and relaxation time, in order to detect spin density waves (SDWs), electron nematic order, superconducting energy gap changes, etc. The extreme conditions generated by a pulsed field may produce some field-induced effects that cannot be observed in lowfields, and PF-NMR is a powerful tool for detecting these abnormal phenomena. 1. Unconventional superconductors It has been found that field-induced effects are very common in unconventional superconductors. For example, in a high field, the inner and outer regions of the unconventional superconductor vortex core with nodes in the gap can be clearly distinguished, and spin diffusion and vortex vibrations can be suppressed, making obser- vations much easier and more conclusive than in low fields.62Further studies have proved that a high magnetic field can not only suppress the superconducting state,63but also induce some exotic super- conducting states.64,65Both conventional and unconventional su- perconductors have unique nuclear spin magnetization and energy gap properties,66,67which can be detected by NMR, and the crystal structure of topological superconductors can also be studied using the Knight shift.68 Despite the growing interest in the study of superconductivity in pulsed high magnetic fields, there has been a lack of detection techniques that are suitable for use with such pulsed fields. For ex- ample, in a low-dimensional unconventional superconductor, when the upper critical magnetic fieldHpdetermined by the Pauli splitting effect is higher than the upper critical magnetic field generated by the orbital effect, an unconventional phase state of Cooper pairs with nonzero total momentum and a spatially nonuniform order pa- rameter can be induced by fields higher than Hp. This leads to the appearance of a normally conducting region in the super- conductor,69–71which is called the Fulde –Ferrell –Larkin – Ovchinnikov (FFLO) phase. This phase was originally discovered when a suf ficiently strong magnetic field parallel to the conducting plane was applied to a quasi-two-dimensional superconductor. Under this strong in-plane magnetic field, as a result of the Zeeman effect, the Fermi surface will be split. Furthermore, some electrons are polarized, and so the Cooper pair whose central momentum is zero is destroyed in this case, and a new type of pairing state becomes stable. The Cooper pair with finite nonzero momentum will bring about spontaneous spatial symmetry breaking and periodic modulation. Although some NMR experiments have provided good evidence for the FFLO state in organic superconductors,72–74the existence of this state is still controversial, and there is an urgent need for more phase transition information from ultrahigh-magnetic- field experiments. Heat capacity and PF-NMR experiments with a magnetic field strength greater than Hpshould be able to facilitate observation of the FFLO phase in unconventional superconductors. It is evident that PF-NMR has better performance than NMR with steady-state fields in some respects. The LNCMI team carried out 48.8 T PF-NMR experiments on YBCO, the second class of high- temperature superconductors, and observed the spectral lines of63Cu and65Cu,43comparing these with their previous NMR results for Cu under a steady magnetic field.23Under the high field, the observed gyromagnetic ratio and satellite line position were consistent with those under the low field, and a more obvious spectral splitting (at 554 MHz) could be obtained, as shown in Fig. 13 . Although the SNR was not ideal, it was still clear that the PF-NMR technique was su- perior to steady- field NMR for observing atomic hyper fine interac- tions. More importantly, the application of NMR with a pulsed high magnetic field will make it possible to study the unconventional FIG. 12. (a) EIIAGlctitration results with an 800 MHz NMR spectrometer. (b) Surface mapped by residues with chemical shift perturbations >3 Hz. Reprinted with permission from Xing et al. , Angew. Chem., Int. Ed. 53, 1 (2014). Copyright 2014 John Wiley and Sons. Matter Radiat. Extremes 6,024201 (2021); doi: 10.1063/5.0040208 6,024201-10 © Author(s) 2021Matter and Radiation at ExtremesREVIEW scitation.org/journal/mreconducting state of high-temperature superconducting samples at temperatures and doping regimes where NMR experiments could not previously be performed. 2. Magnetic materials In recent years, there have been extensive studies of phase transition mechanisms in quantum magnetic systems, ranging from quantum spin liquids with topological order75to decon fined quan- tum critical points disallowed by Landau –Ginzburg –Wilson symmetry-breaking theory,76first-order phase transition with con- tinuous symmetry,77and Bose –Einstein condensation.78With the emergence of these new phenomena, NMR has become an important tool to study the magnetic structure of the associated systems. Take ferromagnetic and antiferromagnetic phase transitions as examples. If a ferromagnetic phase transition occurs in a system, since the total magnetic field to which each nuclear spin is exposed is the sum of the external magnetic field and the nucleus ’s own internal field, then a large NMR shift will be generated; if a antiferromagnetic phase transition occurs in the system, then the total magnetic field on half of the atoms is the sum of the external and internal fields, while that on the other half of the atoms is the difference between the external and internal fields, and consequently the spectral peaks will split. On the basis of these characteristics, it is possible to infer the magnetic structure of a material based on NMR spectral information. Fur- thermore, in higher- field NMR experiments, some magnetic mate- rials can exhibit different phase states from those in a low field, which provides a new perspective for related research. For example, in a 2009 NHMFL publication, it was shown that the heavy fermion antiferromagnetic material CeIn 3had an abnormal skin depth at 45 T.79Ten years later, the LNCMI team conducted PF- NMR experiments on CeIn 3again. Within the acceptable range of resolution, the obvious changes in the NMR spectrum at different temperatures con firmed the existence of a magnetic phase transition from the paramagnetic state to the antiferromagnetic state. However, Fig. 14 shows that at the same temperature, there is almost no changein the NMR spectral peak under three different magnetic field in- tensities: a steady 10.7 T field, a 36.4 T hybrid magnetic field, and a 52 T pulsed high magnetic field.80This indicates that the abnormal phenomenon exhibited by115In at 45 T cannot be attributed simply to the change in magnetic structure or to distortion of the crystal structure and the charge density distribution, but rather it is possible that hyper fine coupling changes the properties of the115In electric field gradient. An investigation of the well-known frustrated magnetic material SrCu 2(BO 3)2again illustrates the unique advantages of PF-NMR. As a quasi-two-dimensional spin system, SrCu 2(BO 3)2is the prototype of a material with a highly symmetric and frustrated Shastry –Sutherland Hamiltonian. Its average magnetization shows a strong background field dependence, as shown in Fig. 15(a) .81–84In 2016, the HLD team studied its NMR behavior in a pulsed magnetic field. A conventional quadrupole spectrum was found at 54 T/119 K,53but when the temperature dropped to 2 K, further splitting peaks [blue curve in Fig. 15(b) ] were observed, which was similar to the behavior of the material in a 41 T steady magnetic field [black curve in Fig. 15(b) ]. This result is consistent with the magnetic superlattice in Fig. 15(c) , namely, the spin superstructure for the plateau phase shows the triplet dimers mentioned in Ref. 81. Compared with the NMR experiment under a 41 T steady field, the use of the 54 T pulsed field signi ficantly FIG. 14. (a) CeIn 3NMR spectra (56 T) at different temperatures. (b) CeIn 3NMR spectra (1.5 K) at different magnetic field intensities. Reprinted with permission from Tokunaga et al. , Phys. Rev. B 99, 085142 (2019). Copyright 2019 American Physical Society. FIG. 13. Resonance spectra of YBa 2Cu3Oxin a pulsed 47 T field. The black solid line is the sum of several experiments (2.5 K). Reprinted with permission from Stork et al. , J. Magn. Reson. 234, 30 (2013). Copyright 2013 Elsevier. Matter Radiat. Extremes 6,024201 (2021); doi: 10.1063/5.0040208 6,024201-11 © Author(s) 2021Matter and Radiation at ExtremesREVIEW scitation.org/journal/mrereduces the experimental energy consumption, improves the ex- perimental ef ficiency, and broadens the available magnetic field range. With regard to the 2/5 and 1/2 magnetization plateaus mentioned in Refs. 85and86, further research under higher pulsed magnetic fields (e.g., >75 T) is warranted. 3. Nematic materials Nematicity here refers to a liquid crystal phase in an electronic state that is similar to the nematic state in liquid crystals. The pre- ferred orientation formed by nematic electrons destroys the rotational symmetry of the crystal, exhibiting short-range order and long-range disorder. In an NMR experiment, the spin fluctuation state is inferred from the observed Knight shift and the temperature dependence of 1/(T1T). When the background field intensity is high, a field-induced nematic phase transition may appear in some strongly correlatedsystems, which is of great signi ficance for both theoretical and experimental research. Take LiCuVO 4as an example. Kazuhiro Nawa ’s research group at Kyoto University has studied the NMR behavior of7Li and51Vi n 4T–10 T steady magnetic fields. They observed that the energy gap of 51V above 10 T was suppressed, re flecting a phase transition from a spiral spin form to a spin density wave (SDW) state at low field.87They further studied the change in the energy gap of51V at a 45 T steady field provided by a hybrid magnet and pointed out that51V may continue to change from the SDW state to a spin nematic phase in the magnetic field range 40.5 T –41.4 T ( H/bardblc).88The LNCMI group used a pulsed high-magnetic- field technique to further increase the background field intensity and obtained a PF-NMR spectrum of LiCuVO 4at 56 T, as shown in Fig. 16 .89The result shows that the phase transition from a SDW state to a magnetic saturation state occurs in the range 42.41 T –43.55 T with increasing magnetic field intensity ( H/bardblc). In this magnetic field range, the normalized spin polarization Sz/Ssat zis linear, while the internal magnetic fieldΔHint remains constant. The system exhibits an obvious spin nematic phase. This experimental result is slightly different from the nematic ap- pearance range obtained by Nawa ’s group, which may be caused by different defect concentrations in the sample. In recent research,90Yoshimitsu Kohama ’s team at the Uni- versity of Tokyo has studied the spin phase sequence of LiCuVO 4by NMR. In experiments below 27.5 T, with a steady magnetic field, the fitted nematic phase NMR line shape was found to be close to the saturated state, while in experiments above 28 T, with a 33 T pulsed magnetic field, the speci fic heat and magnetocaloric effect (MCE) were measured. The magnetic order detected in this study once again provided evidence for a nematic phase state of spin 1/2 in the magnetic frustration lattice from a thermodynamic point of view, further illustrating the reliability of results obtained by means of PF-NMR.89 In contrast to normal materials, nematic materials do not have the rotational symmetry of crystals and retain time reversal sym- metry, which is different from the traditional magnetic sequence. However, in the case of LiCuVO 4, this exotic phenomenon cannot be observed in the lower magnetic fields provided by ordinary super- conducting magnets or permanent magnets, while the use of hybrid FIG. 15. (a) Curve of average magnetization of SrCu 2(BO 3)2vs magnetic intensity.81–84(b) NMR spectra of11B under pulsed 54 T (blue) and steady 41 T (black) magnetic fields (2 K). (c) Magnetic superlattice in the 1/3 magnetization plateau. Reprinted with permission from Kohlrautz et al. , J. Magn. Reson. 271,5 2 (2016). Copyright 2016 Elsevier. FIG. 16. Field dependence of the normalized spin polarization Sz/Ssat zand distri- bution widths of the internal magnetic fieldΔHintobtained from the51V PF-NMR spectra in LiCuVO 4(H/bardblc). Reprinted with permission from Orlova et al. , Phys. Rev. Lett. 118, 247201 (2017). Copyright 2017 American Physical Society. Matter Radiat. Extremes 6,024201 (2021); doi: 10.1063/5.0040208 6,024201-12 © Author(s) 2021Matter and Radiation at ExtremesREVIEW scitation.org/journal/mremagnets is restricted by the power supply and cryogenic system. However, by using PF-NMR methods, it is easy to obtain excellent results for such materials. IV. BOTTLENECK PROBLEMS At present, the development of nuclear magnetic resonance technology is focused mainly on magnets, spectrometers, and probes. However, to facilitate the further development of PF-NMR in par- ticular, it is also necessary to consider three particular issues. A. Flat-top pulsed magnetic fields High magnetic fields used in NMR can be divided into two types according to their duration: steady fields and pulsed fields. The in- tensity of a steady field is constant and persistent, while that of a pulsed field changes greatly and lasts for a short time.91,92The three main types of steady strong magnetic field are those generated by permanent, superconducting, and hybrid magnets, respectively, while pulsed high magnetic fields are usually generated by resistive magnets. Owing to their characteristics of high stability and long duration, steady high magnetic fields are widely used in NMR. Because of their zero resistance, superconducting materials have the apparent merits of low heat loss and uniform conduction current and are commonly used to generate steady magnetic fields. Therefore, commercial NMR spectrometers based on superconducting magnets are in widespread use in medicine,93the study of biological macromolecules,54,94,95solid state physics,23,39,87,96and other research on organic and inorganic materials.97Nevertheless, superconducting materials are limited by their critical magnetic field and critical current, and so the range of magnetic fields that can be reached is subject to severe constraints. As mentioned above, the continuous development of NMR techniques has provided an incentive for developing ways to achieve further increases in background field intensity. The requirements of some experimental investigations have already exceeded the design limits of available steady high-magnetic- field technology. At the same time, however, pulsed magnets can generate fields of more than 50 T, and thus provide a route to obtaining insight into material properties in the regime up to 100 T. This is manifestly the most direct way to meet the magnetic field intensity requirements for ultrahigh- field NMR research. A stable and homogeneous magnetic field is a basic requirement for an NMR experiment. Most commercial solid state NMR equip- ment requires that the homogeneity of the steady magnetic field be better than 10 ppm over mm/DSV and that the stability be better than 10 ppm/h. For this reason, various shimming methods have been proposed, such as field-frequency locking, additional coil compen- sation, and installation of flux stabilizers.108However, these methods are limited by the aperture and pulse power of the pulse magnet, and so they cannot be simply copied for the PF-NMR technique. In the case of a pulsed magnetic field, the discharge method determines the inherent time dependence and spatial distribution of the magnetic field. The temporal stability and spatial uniformity of pulsed magnets are usually as high as several thousand ppm or even tens of thousands of ppm near the peak, and the Larmor resonance frequency is directly proportional to the magnetic field strength. Under these circum- stances, any instability of the magnetic field will cause violent fluc- tuations of the resonance frequency, which is re flected in the spectrum of the FID signal, leading to inhomogeneous broadening ofspectral peaks and, in severe cases, to aliasing. This will make the NMR displacement blurred, resulting in the measured relaxation time being much smaller than the actual value. Until recently, the range of materials that could be detected by PF-NMR was quite limited.7For systems with long relaxation times, the millisecond-level duration of the pulsed magnetic field is too short to cover a complete relaxation process, thus failing to meet the basic conditions for NMR detection. Even though the peak width of a robust spin correlation system measured in a high magnetic field is usually more than 10 kHz –100 kHz, which greatly tempers the demands on the homogeneity and stability of the background field, it is still necessary to control the resolution within a few hundreds of ppm to achieve acceptable resolution. Moreover, the repeatability of pulsed strong magnetic fields is poor, and their accuracy of restoration is not suf ficient to accumulate signals from multiple experiments. In this case, the NMR detection method almost completely loses its advantages, and pulsed magn ets were therefore long considered unsuitable for NMR.109 Aflat-top pulsed magnetic field (FTPMF) refers to a kind of magnetic field that stabilizes the pulsed magnetic field at the crest within a certain period of time. It is able to form a pro file similar to a steady magnetic field in the platform segment. Table III lists the current technical advances with respect to the use of FTPMFs in large facilities.45,98 –107FTPMFs combine the advantages of the high sta- bility of steady magnetic fields and the high field strengths of pulsed magnetic fields, and thus provide a new basis for NMR experiments that cannot be performed using steady magnetic fields with low field strength. Since 2000, research institutions in Germany, Japan, France, and elsewhere have successively carri ed out NMR studies with pulsed high magnetic fields.1H,63Cu/59Co, and63Cu/65Cu NMR spectra have been observed at 56 T, 55 T, and 48.8 T respectively ( Table II ), and the highest RF frequency has reached 2.4 GHz. However, the use of pulsed magnetic fields in these experiments has encountered problems such as short flat- top duration and insuf ficient stability (existing PF-NMR experiments have stabilities not less than a few hundred ppm), resulting in failure to meet relaxation time conditions, br oadening of FID signal linewidth, baseline distortion, and phase error. These cause dif fic u l t i e si nN M R spectral analysis, leading to poor s ignal quality. As mentioned above, HLD and LNCMI have published deconvolution algorithms to solve theproblem of insuf ficient stability, and have th ereby improved the reso- lution of FID spectral lines, but this is only part of signal post-processing, and does not improve the stability of the background magnetic field itself. It is worth noting that, in addition to stability, the homogeneity and repeated additivity of the background magnetic field are important concerns even for traditional NMR, but there have been few studies of these issues in the context of PF-NMR. All of these things show that the current state of PF-NMR technology is far from mature and further developments are needed. A key task to allow high- field, high-frequency NMR experiments to be carried out is the development of an FTPMF system with high stability, homogeneity, and repeatability. This will involve, among other things, optimization of the magnet structure to improve spatial homogeneity at the sample position, the development of power supplies that can provide high-stability, ripple-free, and high-current excitation to generate strong magnetic fields with long flat-top du- ration, and satisfaction of requirements on relaxation time and NMR signal acquisition without loss of quantum controllability of the strong magnetic field. Matter Radiat. Extremes 6,024201 (2021); doi: 10.1063/5.0040208 6,024201-13 © Author(s) 2021Matter and Radiation at ExtremesREVIEW scitation.org/journal/mreB. High-frequency spectrometer The spectrometer is at the core of any NMR experiment. Figure 17 shows the overall structure of a current commercial spectrometer that is suitable for the working environment of a steady- state low-intensity magnetic field. In a pulsed high magnetic field, for example, when the magnetic field intensity is higher than 60 T, even a material with a low spin ratio will excite a resonance frequency of hundreds of megahertz, while for a material with high spin magnetic ratio, the resonance frequency may reach several gigahertz. In recentyears, more powerful (and expensive) NMR spectrometers have appeared on the market. However, owing to limitations on the strength of the background field, the conventional frequency band is usually between 10 MHz and 400 MHz, so there is no need for NMR spectrometers with upper limits at the gigahertz level. By contrast, forNMR in pulsed high magnetic fields, there is an urgent requirement for NMR spectrometers suitable for a high- field and high-frequency environment. First of all, to operate at high field strength when observing nuclei with a high spin ratio, the NMR spectrometer must support a resonance frequency much higher than the common resonance frequency in NMR experiments (e.g., the Larmor frequency of 1Ha t 60 T will reach 2.6 GHz). Second, the temporal stability and spatial homogeneity of the pulsed magnetic field must be considered. On the one hand, because of the inherent time dependence of the magnetic field, the resonance frequency generated by each RF signal is un- known, so the NMR frequency will fluctuate continuously. On the other hand, because the experimental system is in the background of an ultrahigh magnetic field, these fluctuations will be ampli fied several times, and therefore the receiver needs a real-time bandwidth of at least tens of megahertz. Third, the phase of the collected FID signal is different, and direct averaging is not allowed, and so all data pointsmust be stored in memory, waiting for subsequent correction pro- cessing. Therefore, compared with a steady-state NMR experiment, PF-NMR will generate a large amount of data in a short time, which imposes more stringent requirements on the storage capacity of the spectrometer. Besides, the ultrahigh magnetic field enlarges the Larmor frequency range, meaning in the RF power ampli fier must maintain high power and small attenuation over a wide frequencyrange. These strict requirements on the power ampli fier will sig- nificantly increase the investment cost of establishing an NMR spectrometer. Finally, since the holding time of a pulsed high magnetic field is very short, the sequence structure of the NMR spectrometer and the magnetic field generator must be set through a reasonable timing system to achieve precise digital control, so as to ensure the orderly calling of working modules such as the magnetic field trigger, RF trigger, and duplexer switching. The construction of a PF-NMR spectrometer is a systematic project involving both software and hardware aspects. The hardware part includes a pulse sequence programming module, an RF exci- tation module, an ampli fier module, an acquisition module, and an RF analysis module. The software part includes timing control and data analysis programs. The design goal is to have a broadband low-noise RF circuit, a fast receiver recovery time, accurate pulse programming, continuous data acquisition, and automatic data analysis functions, such that the high- field NMR spectrometer system will be universal and easy to replicate. This will allow a variety of researchers to conduct series of NMR studies under extremely high magnetic fields.TABLE III. Progress in FTPMF research worldwide. Facility References Year Power supplyBmax (T)Duration (ms)Stability (ppm) Advantage Limitation UvA, NLD 98and99 1985 Grid recti fier 40 150 ... First appeared Deleterious effects on the power gird NHMFL, USA 100 1996 650 MJ generator 58.5 140 ... High Bmaxand long durationHigh-power generator TU Wien, AUT101 2004 Grid recti fier 40 100 ... Long duration Uncertain stability HLD, GER 45 2012 50 MJ capacitor bank 55.2 70 18 000 High Bmaxand long durationHeavy device (1200 kg) and long cooling time (8 h) WHMFC, CHN102 2012 185 MJ generator 50 100 5 000 Run smoothly High power ripple WHMFC, CHN103 2014 900 3200 Ah battery bank25 200 300 High Bmax High PWM ripple ISSP, JPN 104 2015 900 kJ capacitor bank 60.64 2 85 High stability Short duration WHMFC, CHN105 2020 1400 3200 Ah battery bank23.37 100 65 High stability Low Bmax WHMFC, CHN106and107 2020 12 MJ capacitor bank 65 10 3 000 Low energy consumption Open-loop control system Matter Radiat. Extremes 6,024201 (2021); doi: 10.1063/5.0040208 6,024201-14 © Author(s) 2021Matter and Radiation at ExtremesREVIEW scitation.org/journal/mreC. Probe In any NMR experiment, the probe carrying the sample is a key part. With the increasing application of NMR to investigate materials in extreme environments such as high pressures and strong magneticfields, 110there is a need to develop probes with a wide tuning range, accurate matching performance, good robustness, controllable strain, and coverage of the entire frequency range required by the nuclear resonance in the corresponding magnetic field.7,111Such probes should be able to provide samples with reliable signal transmissionconditions and a stable loading environment. The space available for scienti fic experiments is usually located only at the center of the magnet aperture, 112as shown in Fig. 18 . Therefore, the diameter of the NMR probe for pulsed field is limited to a very small range. Taking a FTPMF of 60 T/2 ms/30 ppm as an example, the diameter of the aperture channel through which the probe can enter and exit is only12 mm,104which is much smaller than the typical diameter of the probes used in ordinary steady magnetic fields. Apart from the aperture limitations, a PF-NMR probe will also be affected by factors such as sample positioning, an inhomogeneous magnetic field distribution, electromagnetic noise, structural stresses, and temperature variations. Since the distribution of a pulsed high magnetic field is not as controllable as that of a steady field, it is dif ficult to accurately position the sample in the aperture, and the rapidly changing magnetic field may cause eddy currents in the LCcoil of the probe, which generate a locally asymmetric magnetic field. This leads to distortion of the magnetic field in the aperture, thereby increasing the uneven broadening of the resonance signal. Worse still, strong elec- tromagnetic noise and structural stress will be excited at the moment of discharge, which imposes stringent requirements with regard to electromagnetic shielding and structural stability of the probe. Finally, given that the pulsed magnetic field is usually immersed in liquid FIG. 17. Structure of a traditional NMR spectrometer. Matter Radiat. Extremes 6,024201 (2021); doi: 10.1063/5.0040208 6,024201-15 © Author(s) 2021Matter and Radiation at ExtremesREVIEW scitation.org/journal/mrenitrogen, for some materials that require a high-temperature environment, a high-sensitivity temperature controller must be in- stalled near the probe to ensure the normal operation of the experiment. The ideal PF-NMR probe should be small and delicate. In terms of mechanical structure, positioning of a samples in the Zdirection should be adjustable at the operating end of the sample rod, and a flippable platform should be available for rotat ion angle experiments and to enable NMR experiments with all crystal axes. By adding a digital feedback loopto control the linear movement and rot ation of the sample table, greater positioning accuracy (angular accuracy <0.1 °112and displacement ac- curacy <1μm40) can be achieved. It is helpful to be able to quickly position the sample in the region with the most uniform magnetic field distribution and thereby overcome th e problem of spatial inhomogeneity of the field to a certain extent. To overcome the problem of low magnetic field repeatability, an electrical circuit with flexible tuning and a matching coil is placed around the sample, and the circuit is tuned by the feedback control system to achieve resonanc e conditions and obtain a higher quality factor Q. Meanwhile, impedance matching is imposed to suppress reflected waves. For this part of the sys tem, it is important to have mutual shielding and heat insulation between circuit elements and wires. A pick- up coil should be wound around the sample to measure the electromotive f o r c ei n d u c e db yt h em a g n e t i c field and thereby monitor the time- dependent intensity. For experime nts with temperature changes, a thermocouple auto-veri fication system is also needed. The instantaneous release of the pulsed high current will cause strong mechanical vibrations. The LCresonant circuit needs to be fixed on a replaceable board to provide mechanical stability for the circuit while also facilitating sample packaging. At the same time, all the connecting terminals between the probe and the control system and the opening of the movable rod should be sealed to prevent leakage of helium gas. Generally speaking, the mechanical structure of the NMR probe depends on the geometry of the pulsed magnet aperture, and theresonance parameters depend on the working frequency band of the NMR spectrometer. Installation of samples and adjustment of probes are usually the last critical steps in the preparation of an NMR ex- periment. At present, RF microcoils are widely used, owing to their good integration capability, inherent sensitivity, high excitation/ resonance frequency, and wide receiving bandwidth compared with earlier types.113,114However, because of the limited number of labo- ratories that have the facilities to study matter in extreme pulsed highmagnetic fields, such NMR probes have not been extensively studied in the ultrahigh-frequency range. Therefore, the design of appropriate probes with good mechanical and electrical properties is one of the key tasks in the further development of PF-NMR. V. PERSPECTIVES AND FUTURE DIRECTIONS As evidenced here, PF-NMR is expected to open up a new perspective for high- field physical property measurement. Especially in the life sciences and condensed matter physics, high magnetic fields are helpful for manipulating and studying the nuclear spin and electronic state properties of strongly correlated electronic materials, thereby providing more information on the complex behavior of these challenging systems. The development of the PF-NMR technique over the past 20 years shows that it is feasible to carry out NMR experiments in pulsed high magnetic fields. However, there are a number of issues that remain. (1) The advantages of NMR at high magnetic field strength should be stressed within the scienti fic community. In steady low- field NMR, in order to improve the SNR and optimize signal processing, various methods have already been proposed, and it is not dif ficult to obtain the same resolution as at high fields. Therefore, it is necessary to understand that the main motivation for developing high- field NMR is to establish ways to observe systems in which special field-induced FIG. 18. NMR detection environment in a pulsed magnetic field. Matter Radiat. Extremes 6,024201 (2021); doi: 10.1063/5.0040208 6,024201-16 © Author(s) 2021Matter and Radiation at ExtremesREVIEW scitation.org/journal/mreeffects occur, which represents the principal advantage of PF-NMR. Taking into account the large fluctuations of the unsteady pulsed field, under the premise of not sacri ficing the strength of the magnetic field, it is necessary to optimize the magnet structure and to use a power supply topology and power supply method that provide high stability and ripple-free high-current excitation to generate a ho- mogeneously distributed high-intensity magnetic field within a certain duration. It is necessary to meet not only the criterion of an acceptable NMR relaxation time, but also the acquisition conditions for most nuclear magnetic signals. In addition, the quantum con- trollability of the high magnetic field must be maintained. (2) NMR devices that are highly adaptable to pulsed high-magnetic- field environments must be made available. Spectrometer designs should be suitable for high- field and high-frequency NMR experiments, with these tailored spectrometers having good electromagnetic shielding performance and timing coordination with the pulsed magnetic field crest. For the probe, the in fluence of sample volume and resonant coil volume on RF pulse width must be considered, as well as pulse field probe tuning and impedance matching. A comprehensive structural optimization scheme should be developed to allow temperature control, monitoring, and precise positioning of samples. Moreover, the time dependence and repetition error of the magnetic field cannot be ignored. An RF communication loop with high-speed broadband, large-capacity signal transmission and acquisition should be set up. The use of appropriate optimization algorithms and control system strategies should solve the problems of NMR signal phase error and baseline distortion. (3) The feasibility of observing four key parameters of nuclear magnetic phenomena in the extreme environment of pulsed fields should be explored: Knight/chemical shift, nuclear quadrupole shift, longi- tudinal relaxation time T1, and transverse relaxation time T2. For thefirst two, improvements in the SNR should be taken as the core task, and the holding time of the pulsed magnetic field should be prolonged to make the nuclear spin as polarized as possible and increase the population difference between different energy levels, thereby giving a high-quality resonance spectrum. With regard to T1andT2, owing to the limited flat-top time that can be achieved with a pulsed magnetic field, it is best to choose a material system with a shorter relaxation time, or to shorten the relaxation time by doping with appropriate reagents. Furthermore, a stable magnetic field can be applied to pre-polarize the nuclear spin to obtain an initial magnetization similar to that in the steady state. VI. CONCLUSIONS PF-NMR has both the measurement sensitivity of conven- tional NMR and the quantum controllability of a high magnetic field. It has the potential to become an important NMR technique and to play signi ficant roles in physics, materials science, chemistry, biomedicine, and other disciplines. Typical applications include but are not limited to the following: (1) It can effectively shorten the NMR detection time of bio-macromo lecules, improve the detection sensitivity, and provide a high SNR, high-resolution, and high- throughput detection method, all owing structural determination and the study of dynamic chang es of macromolecules such as proteins and nucleic acids. (2) Using the time-varying characteristics of the magnetic field strength at the rising and falling edges of each pulse of the pulsed magnetic field, it is possible to rapidly analyze the nuclearrelaxation characteristics of bio-macromolecules and other key nuclear magnetic parameters under different magnetic field intensities, thus significantly improving the sensitivity of nuclear magnetic detection such that the lower limit of detection of sample concentrationapproaches or even reaches the single-molecule level. (3) The Knight shift and electric field gradient changes can be measured in differently doped samples of high-temperature superconductors, the evolution of spin/charge-ordered states can be explored, it can be determined whether a quantum critical point really exists, and the mechanism(s) of high-temperature superconductivity can be revealed. (4) In strongly correlated systems such as heavy fermion materials and quantum spin liquids, various quantum phase transitions can be observed in an ultrahigh magnetic field. NMR can then be used to measure the line splitting, longitudinal relaxation rate, and transverse relaxation rate caused by magnetic interaction. It is also possible to study the order parameters and fluctuations of various quantum states and to explore the general laws of quantum phase transitions. Other mechanisms affected by magnetic fields, such as structural phase transitions, nematic phase transitions, hidden states, magnetic phase transitions, and re-entrant superconductivity, could also be investigated by NMR methods under extremely high magnetic fields. PF-NMR has been around for nearly two decades. Owing to technical bottlenecks such as poor magnetic field stability and the low quality of FID signal spectral analysis under such harsh conditions, research has lingered at the exploratory stage. However, once the technical problems of PF-NMR have been overcome and its detection accuracy and reliability are comparable to those of NMR in a steady field, this method should provide an important route to signi ficant discoveries about the properties of matter under extreme conditions in condensed matter physics, materials science, chemistry, bio- medicine, and other fields. Particularly important future avenues of research concern im- provements in pulsed magnet technology, spectrometer upgrades, and probe structure optimization. Such efforts are expected to in- crease the ef ficiency and widen the range of application of NMR under high fields. Although this work is far from complete, and steady- field NMR will remain the mainstream for some time, PF-NMR still has theprospect of providing a powerful detection method for new phe- nomena in advanced research. ACKNOWLEDGMENTS We would like to thank Professor Chun Tang (Peking Uni- versity) for help with this manuscript. This work was supported by the National Key Research and Development Program of China (Grany No. 2016YFA0401703) and the National Natural Science Foundation of China (Grany No. 51821005). DATA AVAILABILITY Data sharing is not applicable to this article as no new data were created or analyzed in this study. REFERENCES 1Z. Gan, P. Gor ’kov, A. S. T. Cross, and D. 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5.0033283.pdf

AIP Advances 11, 015142 (2021); https://doi.org/10.1063/5.0033283 11, 015142 © 2021 Author(s).Perpendicular magnetic anisotropy and its electrical control in FeNiB ultrathin films Cite as: AIP Advances 11, 015142 (2021); https://doi.org/10.1063/5.0033283 Submitted: 15 October 2020 . Accepted: 29 December 2020 . Published Online: 27 January 2021 Tatsuya Yamamoto , Takayuki Nozaki , Kay Yakushiji , Shingo Tamaru , Hitoshi Kubota , Akio Fukushima , and Shinji Yuasa COLLECTIONS Paper published as part of the special topic on Chemical Physics , Energy , Fluids and Plasmas , Materials Science and Mathematical Physics ARTICLES YOU MAY BE INTERESTED IN Exceeding 400% tunnel magnetoresistance at room temperature in epitaxial Fe/MgO/ Fe(001) spin-valve-type magnetic tunnel junctions Applied Physics Letters 118, 042411 (2021); https://doi.org/10.1063/5.0037972 Voltage-controlled magnetic anisotropy in an ultrathin Ir-doped Fe layer with a CoFe termination layer APL Materials 8, 011108 (2020); https://doi.org/10.1063/1.5132626 Implementation of complete Boolean logic functions in single spin–orbit torque device AIP Advances 11, 015045 (2021); https://doi.org/10.1063/5.0030016AIP Advances ARTICLE scitation.org/journal/adv Perpendicular magnetic anisotropy and its electrical control in FeNiB ultrathin films Cite as: AIP Advances 11, 015142 (2021); doi: 10.1063/5.0033283 Submitted: 15 October 2020 •Accepted: 29 December 2020 • Published Online: 27 January 2021 Tatsuya Yamamoto,a) Takayuki Nozaki, Kay Yakushiji, Shingo Tamaru, Hitoshi Kubota, Akio Fukushima, and Shinji Yuasa AFFILIATIONS National Institute of Advanced Industrial Science and Technology (AIST), Research Center for Emerging Computing Technologies, Tsukuba, Ibaraki 305-8568, Japan a)Author to whom correspondence should be addressed: yamamoto-t@aist.go.jp ABSTRACT We study the perpendicular magnetic anisotropy in (Fe 100−xNix)80B20(FeNiB) films with various Ni contents. Perpendicularly magnetized films are achieved when the Ni content is in the range of 30 at. %–70 at. %. An effective perpendicular magnetic anisotropy (PMA) constant of 1.1 ×105J/m3is achieved for the (Fe 50Ni50)80B20film. We also fabricate magnetic tunnel junction devices containing FeNiB films, and electrical measurements show that a tunneling magnetoresistance ratio of more than 20% can be achieved for devices having an orthogonal magnetization configuration. The PMA of the FeNiB film clearly changes by varying the bias voltage applied along the FeNiB/MgO junction, and a voltage-controlled magnetic anisotropy (VCMA) efficiency of over 30 fJ/Vm is demonstrated. From systematic investigations, there is no clear correlation between PMA and VCMA efficiency in the FeNiB/MgO junction. These experimental results should facilitate the development of energy-efficient magnetic random-access memory. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0033283 Ultrathin ferromagnetic films possessing large perpendicular magnetic anisotropy (PMA) are key elements in magnetic random- access memory (MRAM). The memory bits in MRAM, called mag- netic tunnel junctions (MTJs), consist of a tunneling barrier sand- wiched between two ferromagnetic layers; one has a fixed magneti- zation and the other has a “free” magnetization whose polarity can be switched by external fields. The information stored in MRAM is nonvolatile, which is guaranteed by the energy barrier KeV, where Keis the effective uniaxial magnetic anisotropy constant and Vis the volume of the free layer. This nonvolatility allows the standby power of MRAM to be minimized; however, there is growing demand for reducing the energy required to switch the free-layer magne- tization. A breakthrough approach for energy-efficient switching involves using the spin-transfer torque (STT),1,2which originates from the transfer of quantum-mechanical angular momentum from spin-polarized conduction electrons to the magnetization in the free layer. In fact, the combination of perpendicularly magnetized MTJs and the STT has enabled the scaling down of MTJ size while reduc- ing the switching energy lower than 1 pJ/bit.3,4Further substantialreduction in the switching energy can be achieved by using the voltage-controlled magnetic anisotropy (VCMA) effect at the fer- romagnetic/dielectric interface5–7in MTJs. The VCMA effect on Fe/MgO and related materials systems can essentially be explained by the sum of selective electron–hole doping in the d-electron orbitals and induction of magnetic dipole moment.8–10Since these purely electronic processes do not require electrical current through the MTJ, the VCMA effect enables high-speed manipulation of mag- netization with minimal energy consumption.11–16Recent studies have shown that the switching energy can be reduced to below 10 fJ/bit by using the VCMA effect.17,18 Although a larger Keis necessary to improve the nonvolatility of MRAM irrespective of the technique used to switch the magne- tization, an MTJ has to also possess a large tunneling magnetore- sistance (TMR) as well as high STT efficiency and/or high VCMA efficiency ( ξ). Therefore, a broader choice of materials is necessary to improve the performance of MRAM. However, MTJs possess- ing both a large Keand large TMR are currently limited to those involving single crystalline Fe(Co)/MgO,19,20Co2FeAl/MgO,21or AIP Advances 11, 015142 (2021); doi: 10.1063/5.0033283 11, 015142-1 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv polycrystalline (Co 100−xFex)80B20(CoFeB)/MgO junctions.22There- fore, in this paper, we report the magnetic and electric properties of (Fe 100−xNix)80B20(FeNiB)/MgO junctions prepared on a Ta buffer layer. Fe–Ni alloys are known to exhibit unique magnetic properties depending on the composition ratio,23whereas details on the mag- netic properties of FeNiB ultrathin films are yet to be investigated. Very recently, Su et al. demonstrated a large Keas well as large TMR over 100% in Mo/FeNiB/MgO junctions with Fe-rich FeNiB layers (Fe 68Ni14B18).24We fabricated FeNiB films with various Ni contents (x) including Ni-rich regions ( x>50%) and systematically investi- gated how the magnetic properties are affected by x. We also exam- ined the TMR and VCMA effect on microfabricated MTJ devices containing an FeNiB free layer to discuss the compatibility of FeNiB films with the MRAM application. Multilayered films of Ta (5)/FeNiB ( tn)/MgO (2.1)/Ru (3)/ Ta (2)/Ru (5) and Ta (5)/FeNiB (1.1)/MgO (2.1)/CoFeB (5)/Ru (5)/Pt (2) (values in parentheses in nm) were prepared on thermally oxidized Si substrates with a Ta/Ru electrode using a sputtering sys- tem (Cannon ANELVA EC7800). The former were used for the characterization of magnetic properties, and the latter were used for electrical measurements after microfabrication. The Fe–Ni com- position ratios in the FeNiB films was controlled by co-sputtering Fe80B20and Ni 80B20alloy targets. All films were deposited at room temperature and ex situ annealed in vacuum in the temperature range 200○C–300○C for 1 h. The magnetic properties of the films were investigated using a vibrating sample magnetometer. Micro- fabrication of MTJ devices was carried out using photolithography, Ar-ion etching, and sputtering. These MTJ devices had an elliptical shape with dimensions of 1840 ×840 nm2, and TMR curves were measured under the application of an in-plane magnetic field along the long-axis of the devices. Figure 1 shows the magnetization curves of the 1.1-nm-thick FeNiB films with various Ni contents x, where the black and red curves were measured under the application of in-plane ( H//) and out-of-plane ( H/⊙◇⊞) magnetic fields, respectively. The films were annealed at 250○C prior to measurement. In contrast to a previ- ous study using a Mo buffer layer,24our Fe-rich films prepared on a Ta buffer layer do not exhibit a perpendicular easy axis, as shown in Fig. 1(a). When the Ni content xis increased to 30 at. %, FIG. 1. Magnetization curves of 1.1-nm-thick FeNiB films with various Ni contents. Black and red curves were measured under application of in-plane ( H//) and out- of-plane ( H/⊙◇⊞) magnetic fields, respectively.the easy axis turns perpendicular to the film plane, and the anisotropy field increases by further increasing xto 50 at. %. This is an interesting result in which the partial substitution of Fe with Ni in FeNiB thin film enhances PMA. However, when xis increased to 70 at. %, the saturation field decreases, which is associated with a notice- able reduction in saturation magnetization ( Ms). In fact, for x≥75 at. %, Msbecomes vanishingly small, and the magnetization curves measured under H//andH/⊙◇⊞become almost identical, suggesting the transition from the ferromagnetic to paramagnetic states. To provide quantitative discussions on the values of Msand Ke, we characterized magnetic properties as a function of FeNiB film thickness. In Fig. 2(a), magnetic moment ( Ms×V) is displayed as a function of nominal thickness ( tn). Here, x=50 at. %, and the annealing temperature is 250○C. For 0.9 nm ≤tn≤1.5 nm, Ms×V increases linearly with tn, and the linear fit to the experimental data shown with the broken line gives the values of Msand the thickness of the magnetically dead layer ( td); the slope and the intercept to the horizontal axis give u0Ms=1.13 T and td=0.34 nm, respectively. Note that tdis comparable to that of the Ta/CoFeB/MgO multilay- ered films.25In the following discussions, td=0.34 nm is used to estimate Ke,Ms, and ξ. Figure 2(b) shows the effective thickness ( te) dependence of Ke×te, where te=tn−td. Positive Ke×te, i.e., per- pendicularly magnetized films, were obtained for 0.36 nm ≤te≤0.86 nm. From the linear fitting to the experimental data shown with the broken line, the interfacial ( Ki) and bulk ( Kb) contributions to PMA were estimated to be 0.4 ×10−3J/m2and 6.6 ×104J/m3, respectively. Figures 3(a) and 3(b) show Msand Keas functions of x, respectively. We prepared three series of 1.1-nm-thick FeNiB films annealed at 200○C, 250○C, and 300○C. For 10 at. % ≤x≤60 at. %,Msalmost linearly decreases as xincreases. In other words, our FeNiB films do not exhibit deviation in Msfrom the Slater–Pauling curve in the Invar region, as has been observed in Fe 100−xNix(30 at. %<x<40 at. %) thin films.26Msdecreases rather quickly for x>60 at. %, and all the FeNiB films become paramagnetic for x>75 at. %, regardless of the annealing temperature. More pronounced FIG. 2. (a) Magnetic moment as a function of nominal FeNiB layer thickness ( tn) and (b) Ke×teas a function of effective FeNiB layer thickness ( te). Broken lines correspond to linear fits to experimental data used for estimating values of the dead layer ( td) and interfacial PMA constant ( Ki). AIP Advances 11, 015142 (2021); doi: 10.1063/5.0033283 11, 015142-2 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 3. (a) Saturation magnetization ( Ms) and (b) the effective uniaxial magnetic anisotropy constant ( Ke) as a function of Ni content. The broken line in (a) is a guide for the naked eye. effects of Ni substitution and annealing are observed for Ke. Regard- less of the annealing temperature, the Fe-rich ( x<20 at. %) films exhibit in-plane magnetization. The slight increase in Keat the higher annealing temperature in this region can be attributed to the improved crystallinity of the FeNiB layer. Keis clearly enhanced by increasing x, and the largest Keof 1.1 ×105J/m3is obtained for the (Fe 50Ni50)80B20film annealed at 250○C. The dependence of Keonx shown in Fig. 3(b) proves that the PMA in FeNiB films is enhanced by the moderate substitution of Fe with Ni, regardless of the reduc- tion in Ms. However, the appearance of Kemaximum at x=50 at. % is a curious result since there has been no report that pro- vides evidence of the existence of PMA at the Ni-(Mg)O interface. Improved corrosion resistance and/or formation of nanocrystalline FeNi ordered alloys27–29may be related to enhanced PMA, but the crucial factor has not yet been revealed. Nevertheless, we stress that the current Ta/FeNiB/MgO junction combines a linear tunability of Msand PMA in a broad range of x. Figure 4(a) shows TMR curves of the microfabricated MTJ devices along with schematic illustrations of the corresponding mag- netization configurations. These curves were measured at a bias voltage ( Vb) of 0.05 V under the application of an in-plane mag- netic field. We restricted the Ni content xin the FeNiB layer from 30 at. % to 60 at. % to provide a sufficiently large PMA. Owing to the PMA, the magnetization in the FeNiB layer points perpendic- ular to the film plane at H=0 T. By applying the in-plane H, the magnetization in the FeNiB layer tilts from the perpendicular axis and finally points the direction of H. Since the CoFeB layer on top of the MgO layer has an in-plane magnetization, these TMR curves can be interpreted as representing the magnetization process of the FeNiB layer. As expected from the dependence of Keonx [Fig. 3(b)], the MTJ device whose FeNiB layer has a Ni content of 50 at. % shows a largest saturation field. On the other hand, the TMR ratio decreases with the increase in x. Figure 4(b) summarizes the dependence of the TMR ratio as a function of x. The orthogonally magnetized MTJ devices have reasonably large TMR ratios of over 20% for 30 at. % ≤x≤40 at. %, showing that TMR ratios as large as 40% can be attained if parallel-antiparallel magnetization configu- rations are achieved in perpendicularly magnetized MTJs. Although the TMR ratios of current FeNiB/MgO/CoFeB MTJs are lower than FIG. 4. (a) TMR curves of microfabricated MTJ devices along with schematic illus- trations of corresponding magnetization configurations, (b) the TMR ratio as a function of Ni content, (c) normalized magnetization curves obtained with vary- ing bias voltage ( Vb); the inset shows Vbdependence of Ke×te. The broken line is linear fit to experimental data, and (d) VCMA efficiency ( ξ) as a function of Ni content. typical CoFeB/MgO/CoFeB MTJs,25the TMR ratio would be readily enhanced by inserting an ultrathin CoFe(B) layer between the FeNiB and MgO layers.30 Finally, we discuss the VCMA effect on the FeNiB/MgO junc- tions. Figure 4(c) shows Vbdependence of normalized magnetiza- tion ( M/Ms) curves obtained from the TMR measurements. A posi- tive (negative) Vbis defined as a voltage polarity in which electrons accumulate (deplete) at the FeNiB/MgO interface. The saturation field of the FeNiB layer decreases (increases) due to the applica- tion of positive (negative) Vb. A linear dependence of Ke×teon Vbis obtained, as shown in the inset of Fig. 4(c). From the slope of linear dependence, we estimated the VCMA efficiency ξ, and the obtained values are summarized in Fig. 4(d). The ξvaries from 10 fJ/Vm to 40 fJ/Vm as xvaries, which is comparable or even higher than that reported for CoFeB films.25Since the current FeNiB films do not contain heavy elements with large spin–orbit coupling as well as proximity-induced magnetization, the VCMA effect originating from the induction of dipole moment can be ignored.10Therefore, the observed variation in ξas a function of xcan be attributed to the change in the efficiency of selective electron–hole doping in the d- electron orbitals.8–10From the fact that ξdecreases with the increase inx, we argue that the Fe atoms in the FeNiB layer play an important role in the VCMA effect. A similar reduction in ξhas recently been reported for epitaxial Fe/MgO junctions with an ultrathin Ni inser- tion layer.31It is noteworthy that the dependence of ξonxis distinct from the dependence of Keonx[Fig. 3(b)], which demonstrates that the key factors for the large Keand ξdiffer. The shallow dependence ofξonxin the range of 30 at. % ≤x≤40 at. % would be useful in tuning MsandKewithout changing ξ. To summarize, we studied the magnetic properties of FeNiB/MgO films with various Ni contents x. Perpendicularly AIP Advances 11, 015142 (2021); doi: 10.1063/5.0033283 11, 015142-3 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv magnetized FeNiB films were obtained for a wide range of xinclud- ing Ni-rich region ( x>50 at. %). The largest Keof 1.1 ×105J/m3was obtained for the (Fe 50Ni50)80B20film annealed at 250○C. The electri- cal measurement of orthogonally magnetized MTJ devices revealed reasonably large TMR ratios of over 20%, showing that TMR ratios as large as 40% can be attained if parallel-antiparallel magnetization configurations are achieved in perpendicularly magnetized MTJs. In addition, the FeNiB/MgO junctions exhibited ξof over 30 fJ/Vm by the moderate substitution of Fe with Ni in the FeNiB layer. From systematic investigations, we also demonstrated that there is no clear correlation between ξand Kein the FeNiB/MgO junc- tions. These findings will be useful for designing MTJs for MRAM applications. The authors thank T. Nozaki, M. Konoto, A. Sugihara, S. Tsunegi, Y. Hibino, M. Endo, H. Ohmori, Y. Higo, Y. Kageyama, and M. Hosomi for their fruitful discussions, and E. Usuda and M. Toyoda for assisting with the experiments. This work was based on the results obtained from a project, Grant No. JPNP16007, commis- sioned by the New Energy and Industrial Technology Development Organization (NEDO), Japan. The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1L. Berger, “Emission of spin waves by a magnetic multilayer traversed by a current,” Phys. Rev. B 54, 9353 (1996). 2J. C. Slonczewski, “Current-driven excitation of magnetic multilayers,” J. Magn. Magn. Mater. 159, L1 (1996). 3H. Zhao, A. Lyle, Y. Zhang, P. 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5.0033540.pdf

J. Chem. Phys. 154, 054108 (2021); https://doi.org/10.1063/5.0033540 154, 054108 © 2021 Author(s).Transition states, reaction paths, and thermochemistry using the nuclear– electronic orbital analytic Hessian Cite as: J. Chem. Phys. 154, 054108 (2021); https://doi.org/10.1063/5.0033540 Submitted: 16 October 2020 . Accepted: 10 January 2021 . Published Online: 03 February 2021 Patrick E. Schneider , Zhen Tao , Fabijan Pavošević , Evgeny Epifanovsky , Xintian Feng , and Sharon Hammes-Schiffer COLLECTIONS This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN -machine learning for potential energy surfaces: A PIP approach to bring a DFT-based PES to CCSD(T) level of theory The Journal of Chemical Physics 154, 051102 (2021); https://doi.org/10.1063/5.0038301 Nuclear–electronic orbital Ehrenfest dynamics The Journal of Chemical Physics 153, 224111 (2020); https://doi.org/10.1063/5.0031019 Alchemical transformations for concerted hydration free energy estimation with explicit solvation The Journal of Chemical Physics 154, 054103 (2021); https://doi.org/10.1063/5.0036944The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Transition states, reaction paths, and thermochemistry using the nuclear–electronic orbital analytic Hessian Cite as: J. Chem. Phys. 154, 054108 (2021); doi: 10.1063/5.0033540 Submitted: 16 October 2020 •Accepted: 10 January 2021 • Published Online: 3 February 2021 Patrick E. Schneider,1 Zhen Tao,1 Fabijan Pavoševi ´c,1 Evgeny Epifanovsky,2 Xintian Feng,2 and Sharon Hammes-Schiffer1,a) AFFILIATIONS 1Department of Chemistry, Yale University, New Haven, Connecticut 06520, USA 2Q-Chem, Inc., 6601 Owens Drive, Suite 105, Pleasanton, California 94588, USA a)Author to whom correspondence should be addressed: sharon.hammes-schiffer@yale.edu ABSTRACT The nuclear–electronic orbital (NEO) method is a multicomponent quantum chemistry theory that describes electronic and nuclear quan- tum effects simultaneously while avoiding the Born–Oppenheimer approximation for certain nuclei. Typically specified hydrogen nuclei are treated quantum mechanically at the same level as the electrons, and the NEO potential energy surface depends on the classical nuclear coordinates. This approach includes nuclear quantum effects such as zero-point energy and nuclear delocalization directly into the potential energy surface. An extended NEO potential energy surface depending on the expectation values of the quantum nuclei incorporates coupling between the quantum and classical nuclei. Herein, theoretical methodology is developed to optimize and characterize stationary points on the standard or extended NEO potential energy surface, to generate the NEO minimum energy path from a transition state down to the corre- sponding reactant and product, and to compute thermochemical properties. For this purpose, the analytic coordinate Hessian is developed and implemented at the NEO Hartree–Fock level of theory. These NEO Hessians are used to study the S N2 reaction of ClCH 3Cl−and the hydride transfer of C 4H9+. For each system, analysis of the single imaginary mode at the transition state and the intrinsic reaction coordinate along the minimum energy path identifies the dominant nuclear motions driving the chemical reaction. Visualization of the electronic and protonic orbitals along the minimum energy path illustrates the coupled electronic and protonic motions beyond the Born–Oppenheimer approximation. This work provides the foundation for applying the NEO approach at various correlated levels of theory to a wide range of chemical reactions. Published under license by AIP Publishing. https://doi.org/10.1063/5.0033540 .,s I. INTRODUCTION To incorporate nuclear quantum effects during the optimiza- tion of geometries and the generation of minimum energy paths (MEPs) for chemical systems in a computationally tractable man- ner, approaches beyond conventional quantum chemistry methods must be developed. One such approach is multicomponent quan- tum chemistry theory, where more than one type of particle is treated quantum mechanically at the same level using molecular orbital techniques.1–7The nuclear–electronic orbital (NEO) multi- component theory4,8treats specified nuclei, typically key protons, and all electrons quantum mechanically. This approach removes theBorn–Oppenheimer separation between the electrons and the quan- tum nuclei but invokes a Born–Oppenheimer separation between the quantum portion of the system (i.e., the specified quantum nuclei and electrons) and the remaining nuclei, which are typically denoted “classical” for notational convenience. Within the NEO framework, the potential energy surface (PES) and the associated coordinate NEO Hessian depend on only the classical nuclei. The NEO PES inherently includes the quantum mechanical effects associated with the specified quantum nuclei, such as zero-point energy and nuclear delocalization. The NEO opti- mized geometries are stationary points on the NEO PES. These optimized geometries can be characterized by a vibrational analysis J. Chem. Phys. 154, 054108 (2021); doi: 10.1063/5.0033540 154, 054108-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp based on diagonalization of the NEO Hessian.9In this manner, the stationary points can be identified as minima or transition states (i.e., first-order saddle points). Moreover, the imaginary mode associated with a NEO transition state can be analyzed to elucidate the domi- nant motions contributing to the chemical reaction. Related to this analysis, the minimum energy path (MEP) from a NEO transition state down to the reactant and product states can also be computed on the NEO PES. The nuclear motions associated with the imaginary mode at the NEO transition state and the intrinsic reaction coordinate along the MEP provide insight into the dominant motions of the classi- cal nuclei that drive the chemical reaction. This framework is par- ticularly relevant to hydrogen transfer (i.e., proton, hydride, and proton-coupled electron transfer), where the transferring hydrogen nucleus is treated quantum mechanically.10–12In this case, the NEO MEP elucidates the classical nuclear motions that drive hydrogen transfer, analogous to conventional electronic structure calculations probing the nuclear motions that drive electron transfer.13–15Due to the simultaneous quantum mechanical treatment of electrons and protons, analysis of the electronic and nuclear orbitals or densi- ties along the MEP can provide mechanistic information, such as whether the electron and proton transfer synchronously or asyn- chronously in a proton-coupled electron transfer reaction.16Despite the conceptual simplicity and utility, NEO MEPs have not previously been implemented. Although these types of analyses of the NEO PES provide use- ful insights, the calculation of full molecular vibrational frequencies requires inclusion of the coupling between the classical and quan- tum nuclei,17,18and a complete understanding of the chemical reac- tion path should also include this coupling. For this purpose, an extended NEO PES can be defined to include the expectation val- ues of the quantum nuclei. All stationary points on the NEO PES are also stationary points on the extended NEO PES because of the vari- ational principle, as discussed elsewhere.17Previously, we developed the methodology to compute the extended NEO Hessian, which is defined in terms of second derivatives with respect to the classical nuclear coordinates and expectation values of the quantum nuclei on the extended NEO PES.17,18The stationary points on the extended NEO PES can be characterized as minima or transition states by a vibrational analysis based on diagonalization of the extended NEO Hessian. Moreover, analysis of the imaginary mode at the transition state on the extended NEO PES provides insights into the role of the quantum protons in the chemical reaction. This extended Hessian approach has been shown to produce accurate molecular vibrational frequencies that incorporate significant anharmonic effects17,18 but has not previously been used for the analysis of transition states. The goal of this paper is to develop the methodology to char- acterize stationary points on the NEO PES and extended NEO PES, generate the NEO MEP, and perform thermochemistry calculations. All of these objectives require the NEO coordinate Hessian. Previ- ous work has utilized either a semi-numerical or fully numerical implementation for calculations of the NEO Hessian.9However, as the field of multicomponent quantum chemistry expands and larger systems are studied, an analytic implementation of the NEO Hessian is necessary. Herein, we develop and implement analytic Hessians at the NEO-HF level of theory. We also use these ana- lytic NEO Hessians to characterize stationary points, generate MEPs,and calculate thermochemical properties of two chemical systems: the self-exchange S N2 reaction of ClCH 3Cl−and the intramolec- ular hydride shift in C 4H9+.This work provides the foundation for future extensions to compute NEO Hessians at correlated levels of theory, such as NEO-coupled cluster singles and doubles (NEO-CCSD),19NEO orbital optimized second-order perturba- tion theory (NEO-OOMP2),20and NEO density functional the- ory (NEO-DFT).21–25Moreover, the results and analyses highlight the fundamental conceptual insights obtained by computing MEPs within the NEO framework. II. THEORY A. General definition of the NEO Hessian The Hamiltonian in the NEO framework is given by HNEO=−1 2Ne ∑ i∇2 i−Ne ∑ iNc ∑ AZA ∣re i−rc A∣+Ne ∑ i>j1 ∣re i−re j∣ −1 2mpNp ∑ i′∇2 i′+Np ∑ i′Nc ∑ AZA ∣rp i′−rc A∣+Np ∑ i′>j′1 ∣rp i′−rp j′∣ −Ne ∑ iNp ∑ i′1 ∣re i−rp i′∣+Nc ∑ A>BZAZB ∣rc A−rc B∣, (1) where electronic coordinates reare indicated by lower-case indices iand j, quantum nuclear coordinates rpare indicated by primed versions of these indices, and the remaining classical nuclear coor- dinates rcare indicated by capital indices AandB. Moreover, Ne, Np, and Ncdenote the number of electrons, quantum nuclei, and classical nuclei, respectively. The proton mass is denoted by mp. The Hamiltonian in Eq. (1) includes the standard electronic terms (i.e., the kinetic energy, interaction of the electrons with the clas- sical nuclei, and electron–electron repulsion terms), the analogous quantum proton terms, the electron-proton attraction terms, and the classical nuclear repulsion term. For ease of presentation, the quantum nuclei are presumed to be protons, but this approach is generalizable for any type of particle and multiple types of particles. The NEO energy is computed as the expectation value of the NEO Hamiltonian in Eq. (1) with respect to the mixed nuclear–electronic wave function, which is typically expressed in terms of electronic and nuclear orbitals that are expanded in electronic and nuclear basis sets. Within this framework, the NEO PES depends on only the posi- tions of the classical nuclei, represented by the collective coordinate rc. However, in practice, the quantum protons are typically associ- ated with electronic and nuclear basis functions with centers repre- sented by the collective coordinate rb. These basis function centers must be optimized for any given geometry of the classical nuclei to satisfy the variational principle. Thus, the energy for any geometry of the classical nuclei is variationally optimized with respect to the nuclear basis function center positions and the electronic and pro- tonic orbital coefficients. In this case, the NEO PES can be expressed asENEO(rc,rb(rc)). According to the variational procedure within the NEO framework, the equality ∂ENEO ∂rb=0 (2) J. Chem. Phys. 154, 054108 (2021); doi: 10.1063/5.0033540 154, 054108-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp is satisfied for any point on the NEO PES.9The dimensionality of the NEO PES will be lower than that of the conventional PES for the same system because certain nuclei are treated quantum mechani- cally. Similarly, the corresponding coordinate Hessian matrices will also be of lower dimensionality. As shown in previous work,17,18the NEO gradient and Hessian elements on the NEO PES can be expressed as dENEO drc=(∂ENEO ∂rc) rb+(∂ENEO ∂rb) rcdrb drc=(∂ENEO ∂rc) rb(3) and d2ENEO drc 2=(∂2ENEO ∂rc 2) rb−∂2ENEO ∂rc∂rb(∂2ENEO ∂rb 2)−1 rc∂2ENEO ∂rb∂rc. (4) Equation (4) shows that the Hessian elements require the calcula- tion of not only the partial second derivatives of the NEO energy with respect to the classical coordinates but also the pure and mixed derivatives with respect to the basis function centers. If Ncis the number of classical nuclei and Nbis the number of nuclear basis function centers, then∂2ENEO ∂rc2corresponds to a 3 Nc×3Ncmatrix, ∂2ENEO ∂rb2corresponds to a 3 Nb×3Nbmatrix, and∂2ENEO ∂rc∂rbcorresponds to a 3 Nc×3Nbmatrix. Given that each quantum nucleus has at least one unique nuclear basis function center, the calculation of the NEO Hessian will require the calculation of at least as many second deriva- tives of the energy as required for the calculation of the conventional Hessian. These terms are combined as expressed in Eq. (4) to yield the final NEO Hessian elements. B. NEO-HF theory, gradients, and Hessians The equations for the analytic NEO-HF energy gradients were presented previously in Ref. 9. However, as the analytic formulation of the Hessian extends that theory to a higher-order derivative, it is constructive to reproduce these expressions here. For the purposes of this paper, the theory is presented for closed-shell electronic and high-spin protonic configurations, but the extension to open-shell electronic configurations is straightforward. Moreover, the NEO- HF framework presented herein serves as the foundation for other multicomponent methods such as NEO-CCSD,19NEO-OOMP2,20 and NEO-DFT.21–25Analogous to the conventional formulation of HF theory, NEO-HF4begins with a wave function ansatz of the form Ψtot(re,rp)=Φe 0(re)Φp 0(rp), (5) which is a product of electronic and nuclear Slater determinants, Φe 0(re)andΦp 0(rp), composed of electronic and nuclear orbitals, respectively. The electronic and nuclear spatial orbitals are expressed as linear combinations of Gaussian basis functions, ψe p(re)=∑ μce μpφe μ(re), (6) ψp P(rp)=∑ μ′cp μ′Pφp μ′(rp). (7) In this notation, μ,ν,σ,. . .are the electronic Gaussian basis func- tion indices and the primed equivalents μ′,ν′,σ′,. . .are the nuclearGaussian basis function indices. Hereafter, the indices i,j,k,. . . denote occupied orbitals, a,b,c,. . .denote virtual orbitals, and p, q,r,. . .denote general orbitals for the electrons; the upper case variants are used for the quantum nuclear indices. The NEO-HF Roothaan equations, which will be discussed below, are solved using a self-consistent field (SCF) approach to yield the orbital coefficients. The total closed-shell NEO-HF energy can be written in the form ENEO -HF =∑ μνPe μνHe μν+1 2∑ μνPe μνGe μν−1 2∑ μνPe μνGep μν +∑ μ′ν′Pp μ′ν′Hp μ′ν′+1 2∑ μ′ν′Pp μ′ν′Gp μ′ν′−1 2∑ μ′ν′Pp μ′ν′Gpe μ′ν′+Vnuc, (8) where Vnucis the Coulomb repulsion between the classical nuclei, the electronic and nuclear density matrices are defined as Pe μν=2Ne/2 ∑ ice μice* νi, (9) Pp μ′ν′=Np ∑ Icp μ′Icp* ν′I, (10) and the integrals in the atomic orbital basis are given by He μν=∫dre 1φe* μ(re 1)he(re 1)φe ν(re 1), (11) Hp μ′ν′=∫drp 1φp* μ′(rp 1)hp(rp 1)φp ν′(rp 1), (12) he(re 1)=−1 2∇2 1−Nc ∑ AZA ∣re 1−rc A∣, (13) hp(rp 1)=−1 2mp∇2 1+Nc ∑ AZA ∣rp 1−rc A∣, (14) Ge μν=∑ λσPe λσ[(μν∣σλ)−1 2(μσ∣νλ)], (15) Gp μ′ν′=∑ λ′σ′Pp λ′σ′[(μ′ν′∣σ′λ′)−(μ′σ′∣ν′λ′)], (16) Gep μν=∑ λ′σ′Pp λ′σ′(μν∣σ′λ′), (17) Gpe μ′ν′=∑ λσPe λσ(μ′ν′∣σλ), (18) (μν∣σλ)=∫dre 1∫dre 2φe* μ(re 1)φe ν(re 1)1 ∣re 1−re 2∣φe* σ(re 2)φe λ(re 2), (19) with analogous definitions to Eq. (19) for the pure nuclear and mixed two-particle integrals. J. Chem. Phys. 154, 054108 (2021); doi: 10.1063/5.0033540 154, 054108-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp In the NEO-HF method, a set of Roothaan equations is solved variationally to yield a single-configuration mixed nuclear– electronic wave function.4Specifically, the equations for the elec- trons and quantum protons are given by FeCe=SeCeεe, (20) FpCp=SpCpεp, (21) where orbital coefficients and energies are contained in the Cand εmatrices, respectively, the elements of the basis function overlap matrices are given by Se μν=∫dre 1φe* μ(re 1)φe ν(re 1), (22) Sp μ′ν′=∫drp 1φp* μ′(rp 1)φp ν′(rp 1), (23) and the elements of the Fock matrices are defined as Fe μν=He μν+Ge μν−Gep μν, (24) Fp μ′ν′=Hp μ′ν′+Gp μ′ν′−Gpe μ′ν′. (25) Notably, the electronic and nuclear Roothaan equations are cou- pled and influence each other through the electron-proton Coulomb terms. Furthermore, the solutions of these equations are subject to the orthonormality constraints, Ce†SeCe=I, (26) Cp†SpCp=I. (27) As in the conventional electronic case,26the application of the first derivative to the total energy given in Eq. (8) with respect to ageometric perturbation xand subsequent simplification to remove the density response terms yields an expression for the energy gradient, ∂ENEO−HF ∂x=∑ μνPe μν(He μν)x+1 2∑ μνPe μν(Ge μν)x−1 2∑ μνPe μν(Gep μν)x −∑ μνWe μν(Se μν)x+∑ μ′ν′Pp μ′ν′(Hp μ′ν′)x +1 2∑ μ′ν′Pp μ′ν′(Gp μ′ν′)x −1 2∑ μ′ν′Pp μ′ν′(Gpe μ′ν′)x −∑ μ′ν′Wp μ′ν′(Sp μ′ν′)x +(Vnuc)x. (28) In this notation, terms such as (He μν)xare the geometric deriva- tive integrals, corresponding to the partial derivative of the integral in parentheses with respect to the geometric perturbation x[i.e., (He μν)x≡∂He μν ∂x]. The perturbation xcan be either a classical nuclear coordinate or a nuclear basis function center coordinate. The elec- tronic and nuclear energy-weighted density matrices introduced in the above expression are defined as We μν=2Ne/2 ∑ ice μice* νiεe i, (29) Wp μ′ν′=Np ∑ Icp μ′Icp* ν′Iεp I. (30) The calculation of the closed-shell NEO-HF gradient from Eq. (28) is straightforward. Most integral codes already include the analytic evaluation of the first-order electronic derivative integrals and, with minor modification, can provide the quantum nuclear analogs. In order to calculate∂2ENEO ∂rc2,∂2ENEO ∂rb2, and∂2ENEO ∂rc∂rb, which are needed to compute the NEO Hessian, another derivative of Eq. (28) with respect to a second geometric perturbation, y, is taken, producing ∂2ENEO - HF ∂x∂y=∑ μνPe μν(He μν)xy+1 2∑ μνPe μν(Ge μν)xy−1 2∑ μνPe μν(Gep μν)xy−∑ μνWe μν(Se μν)xy +∑ μ′ν′Pp μ′ν′(Hp μ′ν′)xy +1 2∑ μ′ν′Pp μ′ν′(Gp μ′ν′)xy −1 2∑ μ′ν′Pp μ′ν′(Gpe μ′ν′)xy −∑ μ′ν′Wp μ′ν′(Sp μ′ν′)xy +(Vnuc)xy +∑ μν(Pe μν)y(He μν)x+1 2∑ μν(Pe μν)y(Ge μν)x−1 2∑ μν(Pe μν)y(Gep μν)x−∑ μν(We μν)y(Se μν)x +∑ μ′ν′(Pp μ′ν′)y (Hp μ′ν′)x +1 2∑ μ′ν′(Pp μ′ν′)y (Gp μ′ν′)x −1 2∑ μ′ν′(Pp μ′ν′)y (Gpe μ′ν′)x −∑ μ′ν′(Wp μ′ν′)y (Sp μ′ν′)x . (31) The analytic evaluation of the electronic and nuclear terms that contain second-order derivative integrals has already been derived and implemented for Gaussian basis sets.27The response of the SCF solution quantities to geometric perturbations is manifested in the density and energy-weighted density matrix response terms(Pe μν)y,(Pp μ′ν′)y ,(We μν)y, and(Wp μ′ν′)y . These quantities can be computed with coupled-perturbed NEO-HF equations analogous to those derived and implemented for conventional electronic HF theory.26,28,29The formulation for the solution of these equations is presented in the supplementary material. J. Chem. Phys. 154, 054108 (2021); doi: 10.1063/5.0033540 154, 054108-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp C. Transition states and minimum energy paths Two of the most popular uses of coordinate Hessians in con- ventional quantum chemistry are the optimization and characteri- zation of transition state geometries and the generation of MEPs. In the NEO formalism, these structures and paths are computed on a lower-dimensional PES, which incorporates the zero-point energy, anharmonicity, and nuclear delocalization of the quantum nuclei. Analogous to the conventional electronic structure case, the transi- tion state on the NEO PES is defined to be a stationary point with a single imaginary vibrational frequency (i.e., a first-order saddle point). For this reason, any algorithm that optimizes conventional transition state geometries based on gradient and Hessian informa- tion30,31can be used to find a NEO transition state with the NEO gradient and Hessian as input. Also analogous to the conventional electronic structure case,32 the NEO MEP begins at the transition state and is guided by steepest descent steps with subsequent corrective steps down to the reactant and product optimized geometries.33,34Analytic Hessians are useful for MEP calculations because the energy gradient of the transition state is zero, and the imaginary normal mode motion associated with the negative eigenvalue of the Hessian is used for the first step of the reaction path. An additional technical feature arises for the NEO MEP generation because the NEO energy must be optimized with respect to the basis function centers and the orbital coefficients. As a result, after every steepest descent step and corrective step of the NEO MEP algorithm, the nuclear basis function center positions are optimized. III. RESULTS AND DISCUSSION A. Implementation and model systems The theory presented in Sec. II was implemented in a devel- opment branch of the Q-Chem 5.3 quantum chemistry software.35 This implementation built upon the existing NEO-HF framework in the release version of the software, and existing conventional electronic machinery was repurposed for the calculation of quan- tum nuclear derivative integrals, optimization of NEO-HF transi- tion states, and determination of NEO-HF MEPs. The CP-NEO-HF equations are solved using the conjugate gradient algorithm for solu- tions of inhomogeneous linear systems.36The analytic Hessian was compared with numerical and semi-numerical Hessian calculations to validate the implementation. These validation tests, as well asmore details on the implementation of the NEO analytic Hessian in Q-Chem, are provided in the supplementary material. All NEO- HF calculations herein utilized the 6-31G∗∗electronic basis set37 with the PB4-D nuclear basis set for the quantum protons,38and all conventional calculations were performed at the HF/6-31G∗∗ level of theory. The PB4-D basis set is composed of four s-type, three p-type, and two d-type primitive Gaussians (4 s3p2d). For these applications, each quantum proton was represented by a sin- gle basis function center for both the electronic and nuclear basis functions, which were assumed to move together. The optimization of these nuclear basis function centers utilizes the energy gradients given in Eq. (28). Although the NEO-HF approach lacks electron– electron and electron–proton correlation, this implementation illustrates the important fundamental concepts underlying NEO Hessians, transition states, and MEPs. Extensions to NEO-DFT, which includes these correlation effects,21–25are currently underway. To illustrate the properties of transition states and MEPs at the NEO-HF level, two qualitatively different systems were studied. The self-exchange S N2 reaction of ClCH 3Cl−(Fig. 1) provides an exam- ple of a transition state in which motion of the protons does not dominate the MEP. This system is intended to demonstrate the cal- culation of NEO transition states and reaction paths with multiple quantum protons, and it is expected to be qualitatively similar to the conventional analog because the quantum nuclei are predomi- nantly spectators in the overall motion along the reaction path. For an example in which the quantum proton motion is expected to play a dominant role along the reaction path, the intramolecular hydride shift (i.e., carbocation rearrangement) between the middle two carbon atoms of the C 4H9+species (Fig. 1) was studied. In this application, only the transferring hydrogen was treated quantum mechanically to simplify the analysis. B. Transition states The transition states of both systems were found using the ana- lytic Hessian and gradients in conjunction with an existing quasi- Newton-like transition state search algorithm30with the NEO-HF and conventional HF methods. In this algorithm, the analytic Hes- sian is used at the first step. The conventional transition state of C4H9+was found to correspond to a single-well proton poten- tial (i.e., the three-dimensional proton potential energy surface computed at the transition state geometry with all other nuclei FIG. 1 . Schematic depiction of the reac- tant (left), transition state (middle), and product (right) for the ClCH 3Cl−(top row) and C 4H9+(bottom row) reactions. J. Chem. Phys. 154, 054108 (2021); doi: 10.1063/5.0033540 154, 054108-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Depictions of the imaginary mode at the transition state for the (a) ClCH 3Cl−and (b) C 4H9+systems calcu- lated using the NEO-HF and HF meth- ods. In the NEO cases, the quantum protons are excluded from the depic- tion, as their motion is not explicitly described by the normal mode. The transition state geometry is represented in gray, while positive and negative displacements of the nuclei along the imaginary mode are depicted with red and blue, respectively. Multimedia view: https://doi.org/10.1063/5.0033540.1 fixed exhibited a single minimum, as shown in Fig. S1). This feature allows the use of a single determinant NEO-HF description along the entire MEP, rather than requiring multireference methods.39The transition states were verified to be first-order saddle points because the gradients on the NEO PES were zero within the specified toler- ance (3×10−5hartree/bohr) and the vibrational analysis performed with the NEO Hessian produced only a single imaginary frequency. The imaginary modes and associated frequencies for the ClCH 3Cl− and C 4H9+transition states are depicted in Fig. 2 (Multimedia view). The NEO and conventional calculations for the ClCH 3Cl−system produce nearly identical imaginary frequencies of 412 cm−1and 414 cm−1, respectively. Even when the proton motion dominates the normal mode in the C 4H9+system, these imaginary frequencies differ only by 5 cm−1. The visual comparisons shown in Fig. 2 (Multimedia view) demonstrate that the motions of the classical nuclei in the NEO-HF imaginary mode mirror the motions of the same atoms in the conventional HF imaginary mode. To further analyze the differences between the conventional and NEO imaginary normal mode vec- tors, we calculated the dot product between these vectors, as given in Table I. As the dimensionalities of the normal mode vectors are different in the two cases, we excluded the normal mode elements in the conventional HF vector associated with the quantum protons in the NEO-HF calculation. We also computed this dot product after renormalizing the HF normal mode vector when these elements are excluded. As the protons of ClCH 3Cl−do not contribute signifi- cantly to the conventional HF imaginary mode (i.e., these coordi- nates correspond to 6% of this normal mode vector), its dot product with the NEO-HF mode is nearly unity before renormalization and is 1.000 after renormalization. In contrast, the conventional C 4H9+ imaginary mode is dominated by the transferring proton motion (i.e., these coordinates correspond to 74% of this normal mode vec- tor), and its dot product with the NEO-HF mode is only 0.495. How- ever, after renormalization, this dot product increases to 0.973, indi- cating that the motions of the classical nuclei are nearly identical inthe NEO-HF and conventional HF imaginary modes for this system as well. These findings are consistent with the visual observation that the motions of the classical nuclei in the NEO imaginary mode are the same as the corresponding motions in the conventional descrip- tion even though the conventional mode is dominated by proton motion. The Cartesian coordinates associated with each imaginary normal mode are presented in the supplementary material, Table SIV. The description of the imaginary mode obtained by diagonal- izing the NEO Hessian does not include the motions of the quan- tum protons, which respond instantaneously to the motions of the classical nuclei due to the Born–Oppenheimer separation between the classical and quantum nuclei. This instantaneous response is TABLE I . NEO-HF, conventional HF, and NEO-HF(V) vibrational frequencies associ- ated with the imaginary mode at the optimized transition states for the ClCH 3Cl−and C4H9+systems. All frequencies are given in wavenumbers and are imaginary. ClCH 3Cl−C4H9+ NEO-HF frequency 412 170 HF frequency 414 165 NEO-HF(V) frequency 410 118 HF and NEO-HF dot producta0.968 0.495 HF and NEO-HF dot product 1.000 0.973 after renormalizationb HF and NEO-HF(V) dot productc0.998 0.996 aThe dot product between the HF and NEO-HF normal mode vectors excludes the normal mode elements corresponding to the coordinates of the quantum protons. bThe dot product after renormalization entails renormalization of the HF vector after the normal mode elements corresponding to the coordinates of the quantum protons are excluded. cThe dot product between the HF and NEO-HF(V) normal mode vectors includes all elements. J. Chem. Phys. 154, 054108 (2021); doi: 10.1063/5.0033540 154, 054108-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp analogous to the response of the electrons to the nuclei in con- ventional electronic structure calculations. In general, however, the molecular vibrational modes are mixtures of all nuclear coordi- nates, and a description of the full molecular vibrational frequen- cies requires coupling between the classical and quantum nuclei. To enable the calculation of molecular vibrational frequencies com- posed of both classical and quantum nuclei, we developed the NEO- DFT(V) method.17,18This approach involves the diagonalization of an extended NEO Hessian that is defined in terms of the second derivatives of the NEO energy with respect to the expectation val- ues of the quantum nuclear coordinates and the classical nuclear coordinates. The extended portion of this Hessian is constructed with input from the proton vibrational excitations computed using the NEO time-dependent DFT (NEO-TDDFT) method.40,41 The NEO-DFT(V) approach has been shown to produce accu- rate molecular vibrational frequencies that incorporate the signifi- cant anharmonic effects associated with the quantum protons. The NEO-HF(V) method is the NEO-HF analog of the NEO-DFT(V) method. We performed NEO-HF(V) calculations for the transition state geometry of C 4H9+to compute the full molecular vibrational frequencies. Computational details for the NEO-HF(V) calcula- tions are provided in the supplementary material. The optimized NEO-HF geometry is a stationary point in the extended NEO space corresponding to the classical nuclear coordinates and the expecta- tion value of the quantum proton position coordinate because the derivatives of the NEO energy with respect to both types of coordi- nates are zero in this optimized geometry (see Ref. 17 for relevant derivation). Diagonalization of the extended Hessian produced a single imaginary frequency of 118 cm−1, indicating that this geom- etry is a first-order saddle point in the extended NEO space. The resulting imaginary mode is nearly indistinguishable from the con- ventional HF mode, as depicted in Fig. 3. To quantify this similarity, the dot product of the NEO-HF(V) and conventional HF imaginary normal mode vectors is 0.996. A similar NEO-HF(V) analysis was performed for the ClCH 3Cl−system (Table I). In addition to the imaginary mode, the majority of other nor- mal modes from the NEO-HF(V) calculation are nearly indistin- guishable from their conventional counterparts at the C 4H9+tran- sition state geometry. The root-mean-square deviation between the NEO-HF(V) and conventional HF normal mode frequencies is 58 cm−1, with only six modes exhibiting differences greater than 10 cm−1. These six modes, which include the imaginary mode, correspond to vibrations with significant contributions from the quantum proton. The quantum mechanical treatment of the pro- ton in the NEO framework incorporates anharmonic effects that are absent in the conventional HF harmonic normal mode analy- sis. Thus, the NEO-HF(V) frequencies associated with the quantumproton motion are expected to be more accurate than their conven- tional counterparts. This anharmonicity is most likely responsible for the significantly lower imaginary normal mode frequency pro- duced by NEO-HF(V) than by conventional HF for C 4H9+(Table I). The enhanced accuracy in the frequencies due to the inclusion of anharmonic effects has been shown previously when comparing NEO-DFT(V) calculations with second-order vibrational perturba- tion theory42calculations for a series of molecules.17,18The com- plete results for this comparison are given in the supplementary material, Table SVI. Furthermore, the supplementary material also describes an additional test of the transition state search algorithm for an asymmetric variant of the hydride transfer reaction shown in Fig. 1. C. Minimum energy paths Starting from the transition state structures, we calculated the MEPs for both processes studied, as depicted in Fig. 4. As expected, the multicomponent and conventional MEPs for ClCH 3Cl−are vir- tually identical, with barrier heights that differ by only 0.23 kcal/mol. The C 4H9+reaction barrier is significantly lower than the ClCH 3Cl− barrier. Moreover, the difference in the barrier heights computed using the conventional electronic and NEO methods is slightly larger (0.79 kcal/mol) for the C 4H9+system, presumably because the transferring proton is contributing more to the intrinsic reaction coordinate. The main origin of the slightly lower barrier for the NEO MEP compared with the conventional electronic MEP is that the NEO PES includes the zero-point energy of the quantum proton. To benchmark this effect, we performed Fourier Grid Hamiltonian (FGH) calculations43,44to compute the three-dimensional proton vibrational wave functions and zero-point energies at the conven- tional optimized transition state and reactant geometries of C 4H9+. Computational details for the FGH calculations are available in the supplementary material. The proton zero-point energy at the tran- sition state (5.79 kcal/mol) was found to be smaller than that at the reactant state (6.94 kcal/mol). Thus, including the zero-point energy of the quantum proton with the FGH method lowers the barrier height by 1.15 kcal/mol. This trend is similar to the lower- ing of the C 4H9+barrier height by 0.79 kcal/mol for the NEO-HF method compared with the conventional HF method. The quan- titative differences are most likely due to the lack of electron- proton correlation in the NEO-HF calculations, and the agreement is expected to be better with the NEO-DFT method21–25or with a higher level wave function-based method such as NEO-OOMP2 or NEO-CCSD.19,20 The MEP algorithm implemented in this study33,34uses the imaginary normal mode at the transition state as well as the energy FIG. 3 . Depictions of imaginary vibra- tional normal modes at the transition state geometry of C 4H9+computed using the NEO-HF(V) (teal arrows, left) and conventional HF (orange arrows, center) methods, as well as an overlay of the two depictions (right). J. Chem. Phys. 154, 054108 (2021); doi: 10.1063/5.0033540 154, 054108-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . Minimum energy paths for ClCH 3Cl−(left) and C 4H9+(right) calculated using NEO-HF/6-31G∗∗/PB4-D (blue triangles) and conventional HF/6-31G∗∗(red circles). The intrinsic reaction coordinate is calculated in Cartesian coordinates. gradients at points along the MEP to determine the pathway. Infor- mation from the Hessian is used only at the transition state. Steep- est descent steps are taken using energy gradients at points already determined to be on the pathway, and gradient bisector corrective steps or backup steps may be executed based on the local infor- mation of the surface. These backup steps can cause non-uniform spacing of points along the MEP, as observed in the conventional HF and NEO-HF C 4H9+MEPs shown in Fig. 4. We have found that this behavior is typically more prevalent when the imaginary normal mode frequency is relatively low and the gradients along the MEP are relatively small. The points along the MEP will often become more evenly spaced when a smaller steepest descent step size is used, although small gradients along the MEP may require a sufficiently large step size for numerical reasons. Some of these tech- nical issues may be alleviated by other algorithms, such as a Hessian based predictor-corrector integrator,45which utilizes information from the coordinate Hessian at each subsequently determined point along the MEP. In the NEO framework, the intrinsic reaction coordinate is composed of only the classical nuclei. Analysis of the geometries along the MEP for a hydrogen atom transfer reaction, where thetransferring hydrogen nucleus is treated quantum mechanically, provides insights into the classical nuclear reorganization that drives hydrogen transfer. This perspective is analogous to the nuclear reor- ganization that induces electron transfer in conventional calcula- tions of electron transfer reactions. For the C 4H9+system, analy- sis of the geometries along the NEO MEP illustrates that one of the two central carbon atoms adopts a tetrahedral geometry, while the other remains planar with movement along the reaction path in either direction from the transition state, as depicted in Fig. 5 (Multimedia view). Thus, the dominant reorganization that facili- tates the movement of the quantum proton toward either carbon is the tetrahedral-to-planar (i.e., sp3to sp2) rearrangement around the carbon atoms. An advantage of the NEO approach is that the electronic and nuclear orbitals are computed simultaneously on equal footing. Figure 5 (Multimedia view) also presents a visualization of the pro- tonic orbital and the reactive electronic orbital along the MEP for the C4H9+hydride transfer reaction. The localization of the electronic wave function to produce intrinsic bond orbitals and the visual- ization of these orbitals were performed using the IboView soft- ware developed by Knizia and co-workers.46,47The protonic orbital FIG. 5 . Depictions of protonic and elec- tronic orbitals along the MEP for C 4H9+. The protonic orbital is represented as the purple isosurface, and the reac- tive electronic intrinsic bond orbital is represented as blue (positive) and red (negative). The same isovalue is used for the protonic and electronic orbitals, indicating that the proton is signifi- cantly more localized. Multimedia view: https://doi.org/10.1063/5.0033540.2 J. Chem. Phys. 154, 054108 (2021); doi: 10.1063/5.0033540 154, 054108-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp appears to be quite localized when using the same isovalue for both the electronic and protonic orbitals, mainly due to the significant mass disparity. When a much smaller isovalue is used, the protonic orbital displays the expected behavior of greater delocalization in the transition state region of the MEP compared with the reactant and product state regions. Moreover, the protonic orbital is expected to become more delocalized upon inclusion of electron-proton corre- lation with an approach such as NEO-DFT.8,23Our analysis based on Fig. 5 shows that the quantum proton moves between the two carbon atoms concertedly with the shifting of the electronic orbital as one C–H bond breaks and another C–H bond forms. This type of simultaneous visualization of quantum nuclear and electronic orbitals during a chemical reaction provides insights beyond the previous conventional electronic structure studies with this soft- ware. In particular, this analysis will be able to classify proton- coupled electron transfer reactions as synchronous or asynchronous mechanisms.16 D. Thermochemistry The analytic NEO Hessian also enables the calculation of ther- mochemical properties. As discussed above, the zero-point energies of the quantum nuclei are already included in the NEO PES. The zero-point energies of the other nuclei can be included by diag- onalization of the NEO Hessian, analogous to the conventional electronic structure vibrational analysis, although this approach neglects the coupling between the classical and quantum nuclei in the vibrational modes. Similarly, the normal modes obtained from the NEO Hessian can be used to compute the entropic contribu- tions to produce free energies, although these entropic contributions do not include the contributions from the quantum nuclei or the vibrational coupling between the classical and quantum nuclei. To account for the entropic contributions associated with all nuclei, including the vibrational coupling among all of them, the entropic contributions for both classical and quantum nuclei can be com- puted using the extended NEO Hessian within the NEO-HF(V) framework. Importantly, both the zero-point energies inherent to the NEO PES and the entropic contributions resulting from a NEO-HF(V) calculation include the anharmonicity of the quantum nuclei. Table II shows the values of these various quantities for the two systems studied herein. The temperature used for inclusion of the vibrational entropy was 298.15 K. For the ClCH 3Cl−system, the barrier decreases when accounting for zero-point energy for both the NEO and conventional methods, and the addition of the respective vibrational entropy contributions increases each barrier by∼1.7 kcal/mol–1.8 kcal/mol. For the C 4H9+system, the zero- point energy contribution to the barrier of the hydride transfer pro- cess is nearly equal and opposite ( ±0.5 kcal/mol) for the HF and NEO-HF methods, yielding barrier heights closer in value. The vibrational entropy contributions are very similar for the HF and NEO-HF methods (0.8 kcal/mol–0.9 kcal/mol). After addition of both zero-point energy and vibrational entropy contributions, the barrier heights are the same within 0.5 kcal/mol for the ClCH 3Cl− system and within 0.1 kcal/mol for the C 4H9+system. These results illustrate several general concepts relevant to ther- mochemistry calculations within the NEO approach. The zero-pointTABLE II . Barrier heights (kcal/mol) for ClCH 3Cl−and C 4H9+reactions calculated using NEO-HF and HF methods including zero-point energy ( ΔZPE) and vibrational entropy ( TΔS) contributions. ClCH 3Cl−C4H9+ ΔENEO-HF 14.01 2.06 ΔENEO-HF +ΔZPE NEO-HF 13.62 2.51 ΔENEO-HF +ΔZPE NEO-HF−TΔSNEO-HFa15.41 3.32 ΔENEO-HF +ΔZPE NEO-HF−TΔSNEO-HF(V)b15.47 3.32 ΔEHF 13.78 2.85 ΔEHF+ΔZPE HF 13.23 2.35 ΔEHF+ΔZPE HF−TΔSHF 14.96 3.24 aThe vibrational entropy was computed with the NEO Hessian and, therefore, does not include contributions from the quantum nuclei. bThe vibrational entropy was computed with the extended NEO Hessian using the NEO- HF(V) method and, therefore, includes contributions from both quantum and classical nuclei. energy and vibrational entropy corrections computed with the vibra- tional modes obtained from the NEO Hessian are not expected to be the same as those obtained from the conventional Hessian because the number of modes is different. However, inclusion of the zero-point energy corrections from these Hessians for both con- ventional and NEO calculations is expected to lead to more simi- lar energy differences (i.e., barrier heights) because the NEO PES already includes the zero-point energies associated with the quan- tum nuclei. However, these energies will not be identical because the NEO approach includes the anharmonic effects associated with the quantum nuclei, as well as the impact of the nuclear quantum effects on the geometry optimization. Additionally, the vibrational entropy contributions of the quantum nuclei can be included by obtaining the vibrational modes from the extended NEO Hessian within the NEO-HF(V) framework. The vibrational entropy contributions to the barriers of the ClCH 3Cl−and C 4H9+systems computed in this manner are 1.86 kcal/mol and 0.82 kcal/mol, respectively, compared with the conventional HF values of 1.73 kcal/mol and 0.89 kcal/mol. The slight differences between NEO-HF(V) and conventional HF vibrational entropy contributions are attributable to the anharmonic effects of the quantum nuclei included in the NEO-HF(V) approach. The neglect of correlation effects and limitations of the basis sets may also contribute to these differences. IV. CONCLUSIONS As multicomponent methods gain traction within the quantum chemistry community, the development of tools to enable the mul- ticomponent study of chemical reactions becomes essential. Herein, we developed and implemented the NEO-HF analytic Hessian in the Q-Chem software package. We used this NEO Hessian to charac- terize stationary points as minima or transition states, analyze the imaginary mode at transition states, generate and analyze MEPs, and calculate thermochemical properties within the NEO framework. An advantage of the NEO approach is that the nuclear quantum effects, such as zero-point energy and nuclear delocalization, of specified nuclei are inherently included in the PES. Thus, these nuclear quantum effects are included during the optimizations of J. Chem. Phys. 154, 054108 (2021); doi: 10.1063/5.0033540 154, 054108-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp geometries and generation of MEPs rather than added subsequently as corrections. Analysis of the intrinsic reaction coordinate illus- trates the dominant nuclear motions that drive the chemical reac- tion. Moreover, analysis of the electronic and nuclear orbitals along the MEP highlights the coupled motions of the electrons and quan- tum protons beyond the Born–Oppenheimer approximation. An extended NEO PES that depends on the classical nuclear coordi- nates and the expectation values of the quantum nuclear coordinates allows the inclusion of coupling between the classical and quantum nuclei. In principle, the NEO framework can be used to treat all nuclei quantum mechanically. However, such a treatment requires the elimination of the overall rotations to allow the calculation of meaningful vibrational excitations.48,49Moreover, within the NEO- DFT framework, this treatment also requires the development of electron–nucleus correlation functionals for each type of nucleus. We have developed electron-proton correlation functionals that have been shown to provide accurate proton densities and energies within the NEO-DFT framework.23–25The development of more general electron–nucleus correlation functionals for other nuclei is an even more challenging task. On the other hand, the treatment of only specified protons quantum mechanically enables the inves- tigation of a wide range of chemical processes, particularly those involving hydrogen transfer. The extension of the NEO-HF analytic Hessian to the NEO- DFT analytic Hessian as a means of incorporating electron-proton correlation is currently underway. The description of hydrogen tun- neling requires a multireference method such as NEO multistate DFT (NEO-MSDFT), which has been shown to produce bilobal, delocalized proton vibrational wave functions and accurate hydro- gen tunneling splittings for systems such as malonaldehyde.50The combination of the NEO MEP methods presented herein with NEO- MSDFT will enable the incorporation of hydrogen tunneling effects into the reaction paths. Correlation effects could also be included in geometry optimizations and MEPs with wave function meth- ods such as the NEO-CASSCF,4NEO-OOMP2, and NEO-CCSD19,20 methods. This work provides the foundation for all of these future directions. SUPPLEMENTARY MATERIAL See the supplementary material for the derivation of the coupled-perturbed NEO-HF equations, a comparison of semi- numerical and analytic Hessian elements, the proton potential for the C 4H9+transition state, a comparison of conventional and NEO hydrogen motion along the MEP, imaginary normal mode vec- tors for transition states, a comparison of NEO-HF(V) and HF frequencies, and additional computational details. ACKNOWLEDGMENTS The authors thank Dr. Tanner Culpitt, Dr. Qi Yu, Ben- jamin Rousseau, Dr. Saswata Roy, Professor John Tully, and Dr. Kurt Brorsen for useful discussions. This work was supported by the National Science Foundation (Grant No. CHE-1954348). P.E.S. was supported by a National Science Foundation Graduate Research Fellowship (Grant No. DGE-1752134).The authors declare the following competing financial interest: E.E. is a part-owner of Q-Chem, Inc. 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5.0042316.pdf

Appl. Phys. Lett. 118, 072106 (2021); https://doi.org/10.1063/5.0042316 118, 072106 © 2021 Author(s).Prospects for hole doping in dilute-anion III- nitrides Cite as: Appl. Phys. Lett. 118, 072106 (2021); https://doi.org/10.1063/5.0042316 Submitted: 06 January 2021 . Accepted: 15 January 2021 . Published Online: 19 February 2021 Justin C. Goodrich , Chee-Keong Tan , Damir Borovac , and Nelson Tansu ARTICLES YOU MAY BE INTERESTED IN Suppression of the regrowth interface leakage current in AlGaN/GaN HEMTs by unactivated Mg doped GaN layer Applied Physics Letters 118, 072103 (2021); https://doi.org/10.1063/5.0034584 Split Ga vacancies in n-type and semi-insulating -Ga2O3 single crystals Applied Physics Letters 118, 072104 (2021); https://doi.org/10.1063/5.0033930 Impurity band conduction in Si-doped β-Ga2O3 films Applied Physics Letters 118, 072105 (2021); https://doi.org/10.1063/5.0031481Prospects for hole doping in dilute-anion III-nitrides Cite as: Appl. Phys. Lett. 118, 072106 (2021); doi: 10.1063/5.0042316 Submitted: 6 January 2021 .Accepted: 15 January 2021 . Published Online: 19 February 2021 Justin C. Goodrich,1,a) Chee-Keong Tan,2 Damir Borovac,1 and Nelson Tansu1,3,a) AFFILIATIONS 1Center for Photonics and Nanoelectronics, Department of Electrical and Computer Engineering, Lehigh University, Bethlehem, Pennsylvania 18015, USA 2Department of Electrical and Computer Engineering, Clarkson University, Potsdam, New York 13699, USA 3School of Electrical and Electronic Engineering, The University of Adelaide, Adelaide, SA 5005, Australia a)Authors to whom correspondence should be addressed: jcg213@lehigh.edu andnelson.tansu@adelaide.edu.au ABSTRACT Efficient p-type doping of III-nitride materials is notoriously difficult due to their large bandgaps, intrinsic n-type doping, and the large ionization energy of acceptors. Specifically, aluminum-containing nitrides such as AlN and AlGaN have demonstrated low p-type conductiv- ity, which increases device resistances and reduces carrier injection in optoelectronic applications. Dilute-anion III-nitride materials are apromising solution for addressing this issue and increasing the activation efficiency of p-type dopants. The upward movement of the valencebands in these materials reduces the ionization energy of the dopants, allowing for enhanced p-type conductivity in comparison to theconventional nitrides. Incorporation of a dilute-arsenic impurity into AlN is hypothesized to significantly reduce the ionization energy of Mg-acceptors from 500 meV to 286 meV, allowing for a two-order magnitude increase in activation efficiency in 6.25%-As AlNAs over that of AlN. Published under license by AIP Publishing. https://doi.org/10.1063/5.0042316 Group III-nitride semiconductors, which primarily consist of AlN, GaN, InN, and their alloys AlGaN, InGaN, and AlIn(Ga)N, forma technologically important class of materials for high-efficiency light-emitting and power devices. 1–16These materials possess a wide range of direct bandgaps, from 6.2 eV in AlN down to 0.7 eV in InN, allow-ing for a broad spectral coverage over the entire visible spectrum and into the deep UV. 17In particular, the aluminum-containing nitrides show great potential for UV emitter and detector applications down to200 nm. In order to realize such applications, the growth of high-quality device grade material is necessary. Although tremendousprogress has been made in the development of these semiconductormaterials and implementing them into functional devices, achievinghigh p-type conductivity has proven to be difficult. 18–23III-Nitride semiconductors are intrinsically n-type and notoriously difficult to achieve efficient p-doping due to their large bandgaps. Aluminum containing nitrides, such as AlN and AlGaN, are particularly challeng-ing due to the high ionization energy of acceptors and strong compen-sation from intrinsic defects. 24–34There is a need to improve the hole concentration in such materials in order to continue to progress in thedevelopment of highly efficient devices, especially those operating inthe UV regime.Previous works have presented several methods to attempt to enhance the p-type current conduction in GaN, AlN, and AlGaNalloys. Some of the proposed techniques include exploiting the intrin-sic polarization effect, 29utilizing a tunnel junction design,30co-doping methods,31,32the use of AlGaN/GaN superlattices,33and nanowire structures.34While many of these methods enhance the p-type conductivity over that in conventional bulk materials, there is stillsignificant work to be done in achieving high-quality p-type III-nitride epilayers. In contrast to the conventional III-nitride materials, the class of dilute-anion III-nitride semiconductors is relatively unexplored but shows promise in extending the capabilities of the material system. 35–46 Incorporation of dilute amounts of arsenic, phosphor, and antimonite into GaN and AlN has been shown to drastically reduce theirbandgaps 35–42along with providing other benefits,43–45such as a reduced Auger coefficient,46in comparison with the host materials. G a N A si sc o n s i d e r e da na l t e r n a t i v et ot h eI n G a Na l l o y ,w h e r e b yt h esame bandgap can be achieved with significantly less arsenic incorpora-tion in comparison to the indium required. 35,37–40Similarly, AlNAs may be used as an alternative candidate for AlGaN.43In theory, adding a dilute amount of a group V element (i.e., P, As, Sb, and Bi) into a III- Appl. Phys. Lett. 118, 072106 (2021); doi: 10.1063/5.0042316 118, 072106-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplnitride materials creates a dilute-anion III-nitride alloy with a signifi- cant upward movement of valence bands.35–44Recent work by Tan and co-workers47indicates a strong possibility of achieving p-type dop- ing using dilute-anion III-nitride materials, if the upward motion of the valence band to reduce the ionization energy of acceptors can be achieved. However, there have been no quantifiable results and consid-erable work is still required to solidify the concept. The uncertainty of the p-type doping characteristics of these dilute-anion III-nitride materials is intriguing and requires the utmost attention for clarifica- tion. A thorough investigation into the enhancement possible through dilute-anion incorporation in III-nitride material will revolutionize state-of-the-art device design and kick start the next generation in III-nitride based devices and relevant architecture. In this work, we explore how dilute-anion incorporation in AlN affects its p-type hole concentrations. Density functional theory (DFT) is used to calculate the band structures and activation energies of magnesium and beryllium acceptors dilute-As AlNAs alloys. We pre- dict that alloying a small amount (6.25%) arsenic into AlNAs raises the valence band edge, thereby significantly reducing the ionization energy of magnesium acceptors in the lattice, allowing for up to a two order of magnitude higher p-type conductivity than that in AlN.AlNAs:Mg is also predicted to have superior activation efficiency than similar AlGaN alloys and comparable to that of GaN:Mg. Our work indicates that the dilute incorporation of anions into III-nitride semi- conductors reveals a promising pathway for achieving high p-type conductivity. The use of density functional theory (DFT)-calculated band structures was employed to analyze the effect that a dilute-anion incor- poration will have on p-doped III-nitrides. Specifically, in our study, we analyzed the GaN:Mg, AlN:Mg, AlNAs:Mg, AlN:Be, and AlNAs:Be material systems. Incorporation of other dilute-anion elements, such as phosphorous and antimonite, will be the subject of future studies. Due to similarities seen in the bandgap reductions in dilute-As GaNAs, dilute-P GaNP, and dilute-Sb GaNSb, 35–40we expect that the trends elucidated in our analysis should hold to other dilute-anion incorporation. We performed our DFT analysis using the MedeA VASP software.48The supercell approach was used to build appropriate crys- tal structures for the band structure calculations. Figure 1 depicts a constructed AlNAs alloy crystal structure with a 2 /C22/C22 supercell consisting of a total of 32 atoms. The 32-atom model of the AlNAs crystal consists of 16 Al atoms, 15 N atoms, and 1 substituted As atom,corresponding to 6.25% As in the dilute-As alloy. Larger supercells, such as a 72-atom model for 2.78%-As AlNAs and a 128-atom model for 1.56%-As AlNAs, were also employed with the purpose of analyz- ing the trend of Mg ionization energy as a function of arsenic content. The external stress was set to 0 Pa with the energy convergence tolerance set to 1 /C210 /C01eV/atom. The Gamma-centered Monkhorst– Pack grid and high symmetry k-points were used for the band struc- ture calculations. Spin–orbit coupling was excluded in our calculation because of its insignificant effect in wideband gap III-nitride semicon- ductors. Additional calculation details have been discussed in our previous literature.37–45Due to the known bandgap error of LDA cal- culations, the scissor operator method was employed to correct the band edge positions.49 InFig. 2 , we plot the DFT-calculated band structures for AlN and 6.25%-As AlNAs. The theory indicates that incorporation of thedilute anion in the AlN crystal significantly modifies the band struc- ture from that of AlN, perturbing both the band edge positions and effective masses. 6.25%-As AlNAs has a bandgap of 3.87 eV, a drastic reduction from that of AlN. In order to determine how incorporationof the dilute anion will impact p-type conductivity of the resultantmaterial system, we calculate the ionization energies for the selectedIII-nitride material compositions and dopants. The ionization energyof a dopant reveals the percentage of dopants that will contribute car-riers as a function of temperature. To determine the ionization energy, we follow the treatment from the study by Van de Walle and co-workers. 50The formulation of the equations for the case of AlN:Mg is presented, EfMg Al0/C2/C3 ¼EtotMg Al0/C2/C3 /C0EtotAlN;bulk ½/C138 /C0lMgþlAl/C0Ecorr; (2.1) EfMg Al/C0½/C138 ¼EtotMg Al0/C2/C3 /C0EtotAlN;bulk ½/C138 /C0lMgþlAl /C0½EFþEvþDVM g GaðÞ /C138; (2.2) EA¼EfMg Al/C0½/C138 EF¼0 ðÞ /C0 EfMg Al0/C2/C3 : (2.3) In these equations, Etot[AlN] is the total energy derived from a supercell calculation with one impurity (Mg) in the cell andE tot[AlN,bulk] is the total energy for the equivalent supercell contain- ing only bulk AlN. The ionization energy EAis found from the differ- ence in these two terms. lrepresents the chemical potential of the atomic species. EFand Evrefer to the Fermi level and valence band maximum, respectively. EcorrandDVare correction terms that align the reference potential in the impurity supercell to that of the bulkmaterial. The calculations of the impurity level for the selected materialsystems are presented in Fig. 3 . AlN:Mg has an ionization energy of /C24500 meV, which signifi- cantly limits the activation of the magnesium dopants. However, theincorporation of 6.25%-As into the lattice drastically reduces thisenergy to 286 meV, close to the value obtained for GaN:Mg of260 meV. Ionization energies of 421 meV and 397 meV were alsofound for 1.56%-As incorporation (E g¼5.35 eV) and 2.78%-As incor- poration (E g¼5.08 eV), respectively. Such a large movement of the ionization energy encourages further exploration of the potential of FIG. 1. Illustrative 2 /C22/C22 supercell for AlN 0.9375 As0.0625 . The cell contains a total of 32 atoms, with one nitrogen being replaced with an arsenic. This replacementcorresponds to 6.25% arsenic incorporation.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 072106 (2021); doi: 10.1063/5.0042316 118, 072106-2 Published under license by AIP Publishingthis material for achieving high p-type conductivity. Although this method slightly overestimates the activation energy for GaN:Mg from experimental values, the trends elucidated in this study clearly illus- trate the potential improvements made possible by the inclusion of thed i l u t ea n i o ni nA l N A s : M g . An upward motion of the valence band is expected to contribute to the reduction in ionization energy. For the purpose of further inves- tigating the relationship between AlN and the dilute-anion AlNAs alloy and to identify the mechanism by which the reduction of ioniza-tion occurs, we have calculated the natural band alignment betweenAlN and AlN 0.9375As0.0625. Our approach is to use the potential lineup method.41,51–56In this method, the determination of the valence band offset (VBO) between the materials is performed by summing a bandstructure term ( DE v) and an electrostatic term ( DV), VBO ¼DEvþDV: (2.4) The band structure term is the difference in the VBM of the bulk materials, and the electrostatic term is the difference between themacroscopic average electrostatic potential of the materials in slab configurations. The conduction band offset (CBO) is then found byadding the bandgaps onto the calculated valence band positions, CBO ¼E g;AlN/C0Eg;AlNAs /C0VBO : (2.5) We plot our results for AlN and 6.25%-As AlNAs in Fig. 4 .A l N and AlNAs have a type-I natural band alignment. Our analysisindicates the upward movement of the valence band when arsenic isintroduced into the AlN material. Interestingly, the addition of arsenicalso leads to large downward movement of the conduction band, which contrasts with the some of the other dilute-anion III-nitride materials such as dilute-As GaNAs. Further investigations will beconducted to understand the primary cause behind this phenomenon,along with the impact of other dilute-anion incorporation in AlN.FIG. 2. Band dispersions of (a) AlN and (b) AlN 0.9375 As0.0625 from the LDA DFT calculations. Note the significant changes in the band edges and effective masses, resulting from the arsenic incorporation into the AlN lattice. FIG. 3. The ionization energy E Aof acceptors, in eV, as calculated from DFT for AlN:Be, 6.25%-As AlN:Be, AlN:Mg, 1.56%-As, 2.78%-As, and 6.25%-As AlNAs:Mg,and GaN:Mg.FIG. 4. Natural (type-1) band alignment of AlN and AlN 0.9375 As0.0625 as calculated from the potential lineup method. Ionization energies ( EA) for magnesium dopants are included.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 072106 (2021); doi: 10.1063/5.0042316 118, 072106-3 Published under license by AIP PublishingWhile a relatively small valence band offset was found, with most of the reduction in the bandgap coming from a downward movement of the conduction band, the upward movement of the valence band is expected to contribute to the reduction of ionization energy of magne-sium in the AlNAs alloy and lead to higher activation of acceptors.The band-anticrossing model, when applied to III-nitride materialssuch as GaNP and GaNAs, 40predicts a large VBO in contrast to the result here, which indicates that a significant amount of bandgapreduction is in a CBO. To date, no known work has been published dedicated to predicting or determining the band alignment of dilute- anion aluminum-containing nitrides or developing a comprehensivek/C1pmodel of their band structures. It is our belief that a VBO of /C240.5 eV is substantial enough to account for the reduction of the ioni- zation energy predicted although a more comprehensive analysis ofthe band alignments of this material is required. Experimental investi-gation of these alloys is necessary to refine the impact that the dilute-anion incorporation will have on the band structures. The DFT calculations of AlNAs:Mg have revealed that the dilute incorporation of the group III anion can offer a reduction in both thebandgap energy of the material and the activation energy of the mag-nesium dopants for achieving a p-type material. These results indicatethe possibility of tuning the activation energy with the arsenic incorpo-ration into the lattice. The results from the calculations are summa- rized in Table I . To determine the effect that the changes in the band structure and ionization energy will have on the activation efficiency of accept-ors and resulting hole concentrations in the material, we determinethe hole concentration p (1/cm 3) assuming the theoretical expression of the charge neutrality in semiconductors,24 ppþND ðÞ NA/C0ND/C0p¼NV gexp /C0EA kT/C18/C19 ; (3.1) where pis the hole concentration (1/cm3),NDis the donor concentra- tion (1/cm3),NAis the acceptor concentration (1/cm3), and NVis the effective density of states (1/cm3), given by NV¼22pm/C3 hkT/C0/C1 3=2 h3; (3.2) where m/C3 hi st h ee f f e c t i v em a s so fh o l e s , gis the electron degeneracy (g¼2 for nitride semiconductors), EAis the acceptor ionization energy (eV), and Tis the temperature (K). The Boltzmann constant kand the Planck constant hare also used. In our calculations, we assume a doping concentration of NA ¼[Mg] ¼1/C210201/cm3and a background of ND¼1/C210161/cm3. The ionization energies EAfor GaN:Mg, AlNAs:Mg, and AlN:Mg weredetermined from the DFT calculation as described in Sec. II. The bandgap of AlN 0.9375As0.0625 is 3.87 eV. Using Eg,GaN¼3.4 eV, Eg,AlN¼6.2 eV, and a bowing parameter of b¼0.94 eV,17the compa- rable AlGaN composition for a bandgap of 3.87 eV is found to be Al0.23Ga0.77Nf r o m Eg;AlxGa1/C0xN¼Eg;GaN1/C0x ðÞ þEg;AlNxðÞ/C0bxðÞ1/C0x ðÞ :(3.3) The ionization energy and effective masses for this AlGaN alloy were determined using a linear interpolation between GaN and AlN. InTable II , we present the parameters used (the ionization energy EAand the hole effective mass mh/C3) in and the results (the hole concentration pand the activation efficiency p/N A) from our calcula- tion at a temperature of T¼300 K ( Table II ). From our results, one can see a very substantial increase in activa- tion efficiency when alloying dilute As into AlN, due to a significantreduction in the acceptor ionization energy. Arsenic incorporation modifies the valence band structure by raising the valence band maximum (VBM), reducing the thermal energy required to activatethe dopants. AlN 0.9375As0.0625 is predicted to have a hole concentration similar to GaN and significantly higher than that of the comparable Al0.77Ga0.23N alloy, indicating that incorporation of arsenic into aluminum-containing nitrides can significantly improve the hole con-centration. The hole concentrations as a function of temperature fromthe charge neutrality equation are plotted in Fig. 5 . The temperature dependence of the hole concentration shows that the improvement that 6.25%-AlNAs:Mg has over AlN:Mg and the comparable 23%-AlGaN:Mg alloy is clear. AlNAs:Mg can achieve a doping profile several orders of magnitude larger than AlN:Mg, overa wide range of temperatures, owing to the incorporation of a minuteconcentration of arsenic into the crystal. In summary, our work indicates that incorporation of dilute anions into III-nitride materials provides an alternative pathway to drastically increasing the activation efficiency of acceptors. An upward movement of the valence band reduces the ionization energy of thedopants, significantly increasing the number, which may contribute tototal conductivity. These alloys show promise in achieving hole con-centrations on the order of 10 17/cm3in large bandgap materials, indi- cating their potential to alleviate many of the challenges presented in and revolutionize the p-type doping of III-nitrides. Although incorpo-ration of the arsenic anion into AlN reduces the bandgap of the mate-rial along with the acceptor ionization energy, the ratio of the change in E Avs E gappears greater in AlNAs than in AlGaN. We believe that the reduction will be technologically relevant, as it improves the abilityto p-type dope materials with bandgaps between GaN (3.4 eV) and TABLE I. AlNAs bandgap and Mg activation energy of dilute-As AlNAs, as a function of arsenic incorporation. Eg(eV) AlNAs As content AlNAs E A(eV) 6.20 0% 0.500 5.35 1.56% 0.4225.08 2.78% 0.3973.87 6.25% 0.286TABLE II. Results from the charge neutrality equation at T ¼300 K for GaN:Mg, 6.25%-As AlNAs:Mg, 23%-Al AlGaN:Mg, and AlN:Mg. AlNAs:Mg show that they pre-dicted to possess a significant improvement in activation efficiency ( p/N A) over the conventional III-nitride materials. Material EA(eV) m/C3 h(me) p(1/cm3) p/N A GaN 0.260 0.8 1.91 /C210170.19% AlN 0.9375 As0.0625 0.286 3.0 2.74 /C210170.27% Al0.23Ga0.77N 0.315 1.2 4.29 /C210160.04% AlN 0.500 2.5 2.21 /C210150.002%Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 072106 (2021); doi: 10.1063/5.0042316 118, 072106-4 Published under license by AIP PublishingAlN (6.2 eV). Furthermore, dilute-anion incorporation could be poten- tially employed with other techniques to improving p-type conductivity,such as co-doping methods, 31,32superlattices,33and nanostructures34 to achieve even greater p-doping in large bandgap III-nitrides.Experimental realization of this effect and integration into III-nitride-based devices will bring about a significant advancement in the field. Our work focused on obtaining the band structures of various p-type III-nitrides, including GaN:Mg, AlN:Mg, AlNAs:Mg, AlN:Be,and AlNAs:Be, via a first-principles approach using the local density approximation. The formation energies of point defects representing the acceptor dopants were obtained and used to determine the ioniza-tion energy for the various material systems. Analysis of ionizationenergy via the charge neutrality condition reveals that the incorpora-tion of dilute arsenic into the host material system significantly changes the doping profile. 6.25%-As AlNAs is found to have the potential to achieve hole concentrations two orders of magnitudehigher than AlN and over three times higher than a comparable 23%-Al AlGaN. Further investigation is required into the impact that dilute-phosphorus and other dilute-anion III-nitrides will have on p-type doping, but it is expected that a similar trend will be observeddue to the reduction in the bandgap brought about by dilute-anionincorporation. This result is promising for achieving high-quality, large-bandgap p-type layers, which is essential for the development of UV laser diodes, LEDs, and power transistors. 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J. Chem. Phys. 154, 014104 (2021); https://doi.org/10.1063/5.0026849 154, 014104 © 2021 Author(s).Toward chemical accuracy at low computational cost: Density-functional theory with σ-functionals for the correlation energy Cite as: J. Chem. Phys. 154, 014104 (2021); https://doi.org/10.1063/5.0026849 Submitted: 25 August 2020 . Accepted: 13 December 2020 . Published Online: 05 January 2021 Egor Trushin , Adrian Thierbach , and Andreas Görling ARTICLES YOU MAY BE INTERESTED IN r2SCAN-3c: A “Swiss army knife” composite electronic-structure method The Journal of Chemical Physics 154, 064103 (2021); https://doi.org/10.1063/5.0040021 Electronic structure software The Journal of Chemical Physics 153, 070401 (2020); https://doi.org/10.1063/5.0023185 r2SCAN-D4: Dispersion corrected meta-generalized gradient approximation for general chemical applications The Journal of Chemical Physics 154, 061101 (2021); https://doi.org/10.1063/5.0041008The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Toward chemical accuracy at low computational cost: Density-functional theory with σ-functionals for the correlation energy Cite as: J. Chem. Phys. 154, 014104 (2021); doi: 10.1063/5.0026849 Submitted: 25 August 2020 •Accepted: 13 December 2020 • Published Online: 5 January 2021 Egor Trushin, Adrian Thierbach, and Andreas Görlinga) AFFILIATIONS Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstr. 3, D-91058 Erlangen, Germany a)Author to whom correspondence should be addressed: andreas.goerling@fau.de ABSTRACT We introduce new functionals for the Kohn–Sham correlation energy that are based on the adiabatic-connection fluctuation-dissipation (ACFD) theorem and are named σ-functionals. Like in the well-established direct random phase approximation (dRPA), σ-functionals require as input exclusively eigenvalues σof the frequency-dependent KS response function. In the new functionals, functions of σ replace the σ-dependent dRPA expression in the coupling-constant and frequency integrations contained in the ACFD theorem. We optimizeσ-functionals with the help of reference sets for atomization, reaction, transition state, and non-covalent interaction energies. The optimized functionals are to be used in a post-self-consistent way using orbitals and eigenvalues from conventional Kohn–Sham calculations employing the exchange–correlation functional of Perdew, Burke, and Ernzerhof. The accuracy of the presented approach is much higher than that of dRPA methods and is comparable to that of high-level wave function methods. Reaction and transition state energies from σ-functionals exhibit accuracies close to 1 kcal/mol and thus approach chemical accuracy. For the 10 966 reac- tions of the W4-11RE reference set, the mean absolute deviation is 1.25 kcal/mol compared to 3.21 kcal/mol in the dRPA case. Non- covalent binding energies are accurate to a few tenths of a kcal/mol. The presented approach is highly efficient, and the post-self- consistent calculation of the total energy requires less computational time than a density-functional calculation with a hybrid functional and thus can be easily carried out routinely. σ-Functionals can be implemented in any existing dRPA code with negligible programming effort. Published under license by AIP Publishing. https://doi.org/10.1063/5.0026849 .,s I. INTRODUCTION Density-functional methods based on the adiabatic-connection fluctuation-dissipation (ACFD) theorem1,2for the Kohn–Sham (KS) correlation energy represent a new branch in density-functional the- ory (DFT) that has attracted considerable interest in recent years.3–91 The simplest and computationally most efficient ACFD method results from the direct random phase approximation; see Refs. 31, 36, 39, and 92 for a review of direct random phase approximation (dRPA) methods in chemistry and physics. The dRPA is imple- mented in many electronic structure codes but, nevertheless, has not become a widely used method so far. The reason is that the ratio of accuracy to computational effort of the dRPA is not sub- stantially better than that of well-established DFT and wave functionmethods. In this work, we present new ACFD functionals for the KS correlation energy that are technically closely related to the dRPA and therefore can be easily implemented in existing dRPA codes and have the same computational demands. The new functionals are significantly more accurate, e.g., for reaction energies by almost a factor of three, and thus are attractive alternatives and supplements of existing methods with the perspective to turn into widely used approaches. In ACFD methods, all parts of the total electronic energy except the correlation energy are calculated exactly. This includes the exchange energy, which is calculated exactly from the orbitals like in the Hartree–Fock (HF) method. The KS and the HF exchange energy have the same form in terms of orbitals and differ in val- ues only because the orbitals differ in the two cases. For a closed J. Chem. Phys. 154, 014104 (2021); doi: 10.1063/5.0026849 154, 014104-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp shell system, the total energy is given by the energy expectation value of the KS determinant, i.e., an expression identical to that of the HF total energy in terms of orbitals, plus the ACFD cor- relation energy, which, in practice, depends on KS orbitals and eigenvalues. Thus, in ACFD methods, the electron density is not required in the evaluation of the total energy because all parts of the latter are calculated from KS orbitals and their eigenval- ues. Nevertheless, ACFD methods are KS and thus DFT methods because KS orbitals and eigenvalues are functionals of the electron density, and thus, any energy expression in terms of KS orbitals and eigenvalues, implicitly, is a density functional. The KS for- malism of DFT, indeed, is crucial for the formal basis of ACFD methods. In ACFD methods, the KS correlation energy is calculated with the help of dynamic density–density (potential–density) response functions for purely imaginary frequencies. Besides the KS response function, which is known in terms of KS orbitals and eigenval- ues, response functions for interacting electron systems are needed, which are not known exactly. The latter are usually calculated within time-dependent DFT (TDDFT)93–95in the linear response regime. To that end, the exchange–correlation kernel, the frequency- dependent functional derivative of the exchange–correlation poten- tial with respect to the electron density, is needed. ACFD methods differ by the way the exchange–correlation kernel is handled. Taking into account only the known frequency-independent Hartree ker- nel and neglecting the exchange–correlation kernel amounts to the dRPA.31,36,39,92 The dRPA typically is used in a post-SCF (Self-Consistent Field) manner. This means KS orbitals and eigenvalues are calcu- lated by a conventional DFT method, e.g., within the generalized gradient approximation (GGA).96–98In this case, dRPA methods have the advantage that they are computationally highly efficient, in particular if handling of two-electron (four-index) integrals is avoided by density-fitting in the calculation of the Coulomb and exchange energies. The calculation of the dRPA total energy is com- putationally more demanding than a GGA calculation but about as or even somewhat less expensive than a HF or hybrid DFT cal- culation in typical cases. Calculating the dRPA total energy scales with N4with the system size N, which is higher than the scaling of GGA calculations and equal to the formal scaling of HF and hybrid DFT calculations. Actual computational timings show that the calculation of the dRPA total energy is somewhat faster than HF and hybrid calculations even for larger molecules; see below and Ref. 64. Therefore, a calculation of the dRPA energy can be easily carried out routinely in cases that are accessible by hybrid DFT methods. This means, compared to the computational effort of a GGA calculation, there is some extra expense in computa- tional effort required for a dRPA total energy calculation, but it is moderate. In contrast to conventional DFT methods, the dRPA method can describe non-covalent interactions quite well.4,5,22,35Reac- tion energies typically are of a quality comparable to that of conventional DFT methods, e.g., of the hybrid type.36,64 dRPA atomization or ionization energies, on the other hand, are very poor.36This is related to the basic shortcoming that the dRPA is plagued by unphysical self-interactions. Because of these shortcomings and because non-covalent interactions can be taken into account as well by Møller–Plesset perturbationtheory or semi-empirical corrections,99,100dRPA methods have not become widely used mainstream quantum chemistry approaches. Various attempts to improve the dRPA have been investigated in recent years, e.g., by supplementing it by approximate exchange or exchange–correlation kernels.65–91So far, this has not led to approaches that are generally more accurate than the dRPA and simultaneously retain the dRPA computational simplicity and effi- ciency. Another strategy is to systematically go beyond the dRPA by taking into account the exact exchange kernel and by approximating the correlation kernel by a power series approximation in terms of the sum of the Hartree and the exchange kernel.29–31,37,51,53,63While this solves the self-interaction problem and generally improves the accuracy, this goes along with a higher computational effort and complexity. Here, we propose a new type of ACFD methods that require as input quantities exclusively the eigenvalues σμof the KS response matrix, such as the dRPA method, and therefore are as efficient as the latter. Technically, the new methods are closely related to the dRPA method. Within the dRPA, a specific expression con- taining the eigenvalues σμenters the coupling-constant and fre- quency integrations leading to the dRPA correlation energy. This expression is replaced by a function of the eigenvalues σμthat is optimized with the help of reference sets of atomization, reaction, transition state, and non-covalent binding energies. The function- als resulting from such optimizations are named σ-functionals due to the crucial role the function of the eigenvalues σμplays. While σ-functionals are technically closely related to the dRPA, their for- mal background is somewhat different. They are inspired by and emerge from ACFD methods that employ a power series expan- sion for the correlation kernel in terms of the KS response func- tion and the sum of the Hartree and the exchange kernel.53,63For σ-functionals, the exchange kernel is omitted in such power series expansions and the complete exchange–correlation kernel is repre- sented approximately by a power series in terms of the KS response function and the Hartree kernel. If such a power series converges, it defines a function of the eigenvalues σμ. In practical applica- tions, this function can be represented in various ways. In this work, among others, we present a representation involving cubic splines. The computationally most expensive step of dRPA methods and of methods relying on σ-functionals is the construction of the KS response function. The subsequent calculation of the correla- tion energy by integrating over functions of the eigenvalues σμof the KS response function or over the corresponding dRPA expres- sion does not significantly contribute to the computational demand. This explains the similar computational requirement of dRPA and σ-functional methods. In Sec. II on theory, σ-functionals and their formal background are introduced. In Sec. III on the optimization of σ-functionals, the specific ansatz for the required function of the eigenvalues of the KS response matrix is presented and the optimization proce- dure and its outcome are discussed. In Sec. IV, the performance of theσ-functional is discussed and compared to that of various DFT and wave function methods. Section V provides information on computational timings, and Sec. VI draws conclusions. Three appendices present generalizations for spin-polarized systems, list parameters of presented σ-functionals, and briefly discuss an ansatz J. Chem. Phys. 154, 014104 (2021); doi: 10.1063/5.0026849 154, 014104-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp forσ-functionals that is an alternative to the one presented in the main text. II. THEORY A. Direct random phase approximation and σ-functionals The ACFD theorem1,2 Ec=−1 2π∫∞ 0dω∫1 0dα∫drdr′1 ∣r−r′∣ ×[χα(iω,r,r′)−χ0(iω,r,r′)] (1) is an exact expression for the KS correlation energy that reads Ec=−1 2π∫∞ 0dω∫1 0dαTr[[Xα(iω)−X0(iω)]FH] (2) in a matrix representation. Equation (1) contains two density– density (potential–density) response functions, the KS response functionχ0and the response function χαfor an “electron” system with the electron–electron interaction scaled by the coupling con- stant 0≤α≤1. The path from α= 0 toα= 1 constitutes the adiabatic connection between the KS model system and the physical electron system and, according to the Hohenberg–Kohn theorem, is uniquely defined by the requirement that the ground state electron density is fixed to that of the physical electron system for all values of α. Equation (2) contains matrix representations X0,Xα, and FHof the response functions and the Coulomb interaction in a resolution of the identity (RI) basis. The KS response function χ0is given by χ0(iω,r,r′)=4∑ i∑ a(ϵi−ϵa) (ϵi−ϵa)2+ω2φi(r)φa(r)φa(r′)φi(r′)(3) in terms of occupied and unoccupied KS orbitals φi(r) andφa(r), respectively, and the corresponding eigenvalues ϵiandϵa. The orbitalsφi(r) andφa(r) are real-valued spatial orbitals, and spin is taken into account by appropriate prefactors. Throughout Sec. II on theory, we will consider non-spin-polarized molecules. The generalization to the spin-polarized case is straightforward and sketched in Appendix A. Generally, indices iand jrefer to occu- pied and indices aand brefer to unoccupied orbitals, while orbitals with indices sortcan be occupied or unoccupied. Unless noted otherwise, summations always run over the full range of an index. The elements X0,μνof the matrix X0representing the KS response function χ0are constructed with respect to the Coulomb norm and are given by X0,μν(iω)=∫dr1dr2dr3dr4fμ(r1)χ0(iω,r2,r3)fν(r4) ∣r1−r2∣∣r3−r4∣ =∑ i∑ aDia,μλiaDia,ν (4) withDia,μ=(φiφa∣fμ)=∫drdr′φi(r′)φa(r′)fμ(r) ∣r−r′∣(5) and with λia=(ϵi−ϵa) (ϵi−ϵa)2+ω2. (6) The RI basis functions fμare linear combinations of Gaussian functions from RI basis sets from the literature that are orthonor- malized with respect to the Coulomb norm, i.e., δμν=∫drdr′fμ(r)fν(r′) ∣r−r′∣=FH,μν. (7) See Ref. 37 for details on the construction of the RI basis sets. For RI basis functions orthonormalized according to Eq. (7), the matrix FHrepresenting the Coulomb interaction in the ACFD the- orem (2) is a unit matrix, i.e., FH=1. For clarity, the matrix FH is nevertheless included explicitly in Eq. (2) and some equations below. The matrix Xαrepresenting the interacting response functions χαis given by TDDFT in the linear response regime as Xα(iω)=[1−X0(iω)Fα Hxc(iω)]−1X0(iω), (8) with the matrix Fα Hxcrepresenting the sum of the Hartree and the exchange–correlation kernel. The latter is the frequency- and coupling-constant-dependent functional derivative of the exchange–correlation potential with respect to the electron density. Inserting Eq. (8) into Eq. (2) yields the expression Ec=−1 2π∫∞ 0dω∫1 0dαTr{[[1−X0(iω)Fα Hxc(iω)]−1−1]X0(iω)FH} (9) for the KS correlation energy. The frequency-independent Hartree kernel, of course, is known. The frequency-dependent exchange kernel is acces- sible via an integral equation.63,101–103The correlation ker- nel is unknown and therefore needs to be approximated. The way the exchange–correlation kernel is treated distin- guishes different ACFD methods. Simply omitting the exchange– correlation kernel amounts to the dRPA, the most widely imple- mented ACFD method. In this case, Eq. (9) turns into the expression EdRPA c=−1 2π∫∞ 0dω∫1 0dαTr{[[1−αX0(iω)FH]−1−1]X0(iω)FH} (10) for the dRPA correlation energy. With the spectral representation −X0(iω)=V(ω)σ(ω)VT(ω) (11) for the negative of the KS response matrix and with FH=1, Eq. (10) for the dRPA simplifies to J. Chem. Phys. 154, 014104 (2021); doi: 10.1063/5.0026849 154, 014104-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp EdRPA c=−1 2π∫∞ 0dω∫1 0dαTr{[−(1+ασ(ω))−1+1]σ(ω)} =−1 2π∫∞ 0dω∫1 0dαTr{hdRPA(ασ(ω))σ(ω)} (12) with the function hdRPA(ασ)=−1 1 +ασ+ 1. (13) In Eq. (12), the spectral representation of the negative −X0of the KS response matrix X0was introduced. Within the dRPA, one could use the spectral representation of X0itself as well. In this case, the sign in front of the diagonal matrix σwould change in the following formu- las. In more advanced ACFD methods, however, the square root of −X0is required. Because X0is negative semi-definite, −X0is positive semi-definite and its square root can be taken. To be consistent with previous works on advanced ACFD methods,29–31,37,51,53,63we here chose to use the spectral representation of −X0. The integration over the coupling constant αcan be carried out analytically to obtain EdRPA c=−1 2π∫∞ 0dωTr{−ln[1+σ(ω)]+σ(ω)} =−1 2π∫∞ 0dωTr{HdRPA(σ(ω))} (14) with the function HdRPA(σ)=∫1 0dαhdRPA(ασ)σ =−ln[1 +σ]+σ. (15) Theσ-functionals introduced in the present work are obtained by replacing the function hdRPAby a function hσfobtained from opti- mizing reference sets of atomization, reaction, transition state, and non-covalent binding energies, as described below. The function Hσf is defined by Hσf(σ)=∫1 0dαhσf(ασ)σ (16) in analogy to the dRPA case [Eq. (15)]. This means, σ-functionals yield the KS correlation energy as Eσf c=−1 2π∫∞ 0dω∫1 0dαTr{hσf(ασ(ω))σ(ω)} =−1 2π∫∞ 0dωTr{Hσf(σ(ω))}. (17) B. Background of σ-functionals In order to elucidate the formal background of σ-functionals for the KS correlation energy, we consider expansions of the sum of Hartree plus exchange–correlation kernel with respect to the cou- pling constant α.53,63Within the matrix representation introduced in Sec. II A, such an expansion reads as Fα Hxc=αFHx+α2F(2) c+α3F(3) c+⋯, (18)with FHxdenoting the sum of the Hartree plus the exchange kernel andF(n) cdenoting the contribution of order nto the correlation ker- nel. Various quantities appearing in Eq. (18) and in the following equations are frequency dependent. For notational simplicity, this frequency dependence is not shown. The leading contribution to Fα Hxc, i.e., the sum FHxof Hartree and exchange kernels, obeys the equation63,101–103 FHx=X−1 0X(1)X−1 0, (19) which contains the inverse of the KS response matrix X0and the first order contribution X(1)to the response matrix Xαin the coupling strength expansion Xα=X0+αX(1)+α2X(2)+α3X(3)+⋯. (20) In expansion (20), the KS response matrix constitutes the zeroth order term, i.e., X0=X(0). For the first order contribution, X(1)per- turbation theory yields an explicit expression in terms of KS orbitals and eigenvalues and matrix elements of the KS exchange poten- tial.101–103(Note that in Refs. 63 and 101–103, the first order con- tribution X(1)is denoted HHxin matrix representation and hHxin real space representation or Hxand hxif only the exchange con- tribution to X(1)is considered.) Construction of the matrix X(1)is somewhat involved and requires a computational effort that scales with N5with the system size N.37,63This is higher than the N4scal- ing of the construction of the KS response matrix X0, which is the computationally most expensive step in dRPA methods. In exact exchange RPA methods, alternatively called ACFD[Hx] methods, X(1)is constructed.37,53,63In the present work, we aim at a computa- tionally highly efficient method and therefore avoid the construction of the matrix X(1). In passing, we note that, in principle, the higher order contribu- tions X(n)to the response matrix Xαare accessible by perturbation theory as well. However, as usual in perturbation theory, the number of terms and the computational effort to construct them explodes and seems unmanageable in practice. At this point, the question arises whether it is possible to truncate expansion (20) after the first two terms and to use the resulting approximation for Xαdirectly in the ACFD theorem [Eq. (2)] for the KS correlation energy. In this way, neither the Hartree nor the exchange–correlation kernel would be needed. Technically, this is indeed possible.63,104However, the resulting correlation energies are quite poor.104 Next, we consider the higher order contributions F(n) cto the kernel. They obey equations of the form F(2)=−X−1 0X(1)X−1 0X(1)X−1 0+X−1 0X(2)X−1 0 (21) and F(3)=X−1 0X(1)X−1 0X(1)X−1 0X(1)X−1 0−X−1 0X(1)X−1 0X(3)X−1 0 −X−1 0X(3)X−1 0X(1)X−1 0+X−1 0X(3)X−1 0 (22) and analogous expressions for the higher orders.63We now follow Ref. 63 and omit in the equations for the contributions F(n) call those J. Chem. Phys. 154, 014104 (2021); doi: 10.1063/5.0026849 154, 014104-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp terms that contain quantities X(n)with n≥2 and in turn renormalize the coefficients in front of the remaining terms. The kernel Fα Hxcthen is approximated by the expansion Fα Hxc≈αX−1 0X(1)X−1 0+X(−1) 0∞ ∑ n=2(−)n−1βnαn[X(1)X−1 0]n , (23) which can be rearranged to Fα Hxc≈αFHx+FHx∞ ∑ n=2(−)n−1βnαn[X0FHx]n−1(24) if Eq. (19) for FHxis taken into account. The renormalization coef- ficientsβnhave to compensate approximately for the neglect of the terms containing X(n)with n≥2. This approximation was named power series approximation in Ref. 53. In order to construct σ-functionals, we make one more approx- imation. We neglect the exchange contribution in the kernel FHxand keep only the Hartree contribution FH. This yields Fα Hxc≈αFH+FH∞ ∑ n=2(−)n−1˜βnαn[X0FH]n−1. (25) The tilde above the renormalization coefficients ˜βnsymbolizes that inσ-functionals, the renormalization coefficients not only shall com- pensate to some extent the neglect of terms containing X(n)with n≥2 but additionally the neglect of the exchange contribution to FHx. If, like in the present case, the RI basis set to represent the Hartree kernel is orthonormalized with respect to the Coulomb norm such that FHturns in to a unit matrix [see Eq. (7)], then expansion (25) assumes the form Fα Hxc≈α1+∞ ∑ n=2(−)n−1˜βnαnXn−1 0 ≈V[α1+∞ ∑ n=2˜βnαnσn−1]VT. (26) By inserting expansion (26) for Fα Hxcinto the ACFD theorem (9), we obtain the σ-functional for the KS correlation energy, Eσf c=−1 2π∫∞ 0dω∫1 0dαTr⎧⎪⎪⎨⎪⎪⎩⎡⎢⎢⎢⎢⎣(−)⎡⎢⎢⎢⎢⎣1+ασ(ω) +∞ ∑ n=2˜βnαnσ(ω)n−1⎤⎥⎥⎥⎥⎦−1 +1⎤⎥⎥⎥⎥⎦σ(iω)⎫⎪⎪⎬⎪⎪⎭ =−1 2π∫∞ 0dω∫1 0dαTr{hσf(ασ(ω))σ(ω)} =−1 2π∫∞ 0dωTr{Hσf(σ(ω))} (27) with hσf(ασ(ω))=−[1+ασ(ω)+∞ ∑ n=2˜βnαnσ(ω)n−1]−1 +1 (28)and with Hσfdefined according to Eq. (16). With the last two lines of Eq. (27), which are identical to those in defining Eq. (17) of σ-functionals, we arrived at the latter. From Eq. (26), for the kernel Fα Hxc, the spectral representation of the KS response function (11), the definition (28) of the function hσf, and Eq. (8) for the response function Xαfollow that the differ- ence Xα(iω)−X0(iω) of response matrices occurring in the ACFD theorem (2) is given by Xα(iω)−X0(iω)=Vhσf(ασ(ω))σ(ω)VT(29) and that Xα(iω)=V[hσf(ασ(ω))−1]σ(ω)VT(30) equals the response matrix Xα(iω). For purely imaginary arguments iω, both the KS response matrix Xand the exact response matrix Xα are negative semi-definite. If the response matrix Xαis approximated in the way presented here, then it is negative semi-definite only if the function hσfobeys the condition hσf≤1. (31) III. OPTIMIZATION OF σ-FUNCTIONALS A. Ansatz for the function hσf In Eq. (28), the function hσfis defined by a series expansion. This assumes that the expansion converges. The argument in expan- sion (28) is ασ, i.e., the product of the coupling constant αwith the diagonal matrix σcontaining the eigenvalues σμof the negative −X0 of the KS response matrix X0. The definition of hσfthus implies that expansion (28) converges for all possible values ασμ. The lat- ter obey the condition 0 ≤ασμbecause 0 ≤α≤1 and 0≤σμ. The eigenvalues σμare non-negative because the KS response matrix X0 is negative semi-definite, and thus, −X0is positive semi-definite. In the following, we will suppress the subscript μand simply consider the function hσffor arguments ασwithσrepresenting any of the eigenvalues σμ. There are various conceivable strategies to optimize the func- tion hσf. One possibility would be to truncate expansion (28) at some order nand to optimize the coefficients ˜βnup to this order. An analogous approach was followed in Ref. 53 for the power series approximation of the correlation kernel in terms of the sum FHxof Hartree and exchange kernels. Here, we pursue a different strategy and decompose the function hσfaccording to hσf(ασ)=hdRPA(ασ)+h(ασ) =[−1 1 +ασ+ 1]+h(ασ) (32) in the dRPA σ-function hdRPA[Eq. (13)] and a remainder hthat we represent by cubic splines.105If expansion (28) converges for all its arguments, then hσfis differentiable arbitrarily often. A function given in terms of cubic splines is only twice differentiable. This, J. Chem. Phys. 154, 014104 (2021); doi: 10.1063/5.0026849 154, 014104-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp however, is sufficient for all practical purposes, even if analytic energy gradients of second order were to be calculated. On the other hand, cubic splines are highly flexible and the integration over the coupling constant can be carried out analytically (see below). Analternative ansatz for the function hσfand its performance is briefly discussed in Appendix C. Our representation of the function hin terms of cubic splines reads as h(x)=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩c1,1x forx∈I1 c0,m+c1,m(x−xm)+c2,m(x−xm)2+c3,m(x−xm)3forx∈Imwith 2≤m≤M−1 c0,M forx∈IM(33) with the argument x=ασand intervals Imdefined as Im={[xm,xm+1] for 1≤m≤M−1 [xm,∞] form=M(34) and with x1= 0 and the other abscissae xmchosen before the opti- mization. By M, the number of intervals, which is equal to the num- ber of abscissae xm, is denoted, and the cp,mare the coefficients of the cubic polynomial in the interval m. The first y-value correspond- ing to x1= 0 is fixed to zero, i.e., y1=h(x1) =h(0) = 0, and the other y-values ym=h(xm) are the quantities to be optimized. Fixing h(0) = 0 means that the complete σ-function hσf=hdRPA+h is zero for x1= 0, i.e., hσf(0) = 0, because hdRPA(0) = 0. The condition hσf(0) = 0 needs to be obeyed because for a coupling constantα= 0, implying ασ=x= 0, the contribution to the correlation energy in the coupling strength integration has to bezero. This contribution to the correlation energy needs to be zero becauseα= 0 corresponds to the KS model system of noninter- acting “electrons” that are not correlated by definition. Further- more, only if hσf(0) = 0 holds true, the correlation energy can converge with the size of the RI basis set determining the dimen- sion of the KS response matrix X0. With an increasing RI basis set,X0has more and more eigenvalues σμapproaching zero that, however, do not contribute to the correlation energy in the limit σμ→0 if hσf(0) = 0. In the first interval, the function h(x) is a linear function with slope y2/x2=c1,1. In the intervals Imwith 2 ≤m≤M−1, the function h(x) is given by cubic splines determined with the boundary condition that the derivative at x2equals y2/x2 and that the derivative at xMequals zero. In the last interval IM, the function h(x) is constant equaling yM=c0,M. In the present work, in all optimizations, yMis fixed to zero, i.e., yM=c0,M= 0. The coupling strength integration of the function hyields the function H, H(σ)=⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩1 2c1,1σ2forσ∈I1 1 2c1,1x2 1+c0,1(σ−x1)+1 2c1,1(σ−x1)2+1 3c2,1(σ−x1)3+1 4c3,1(σ−x1)4forσ∈I2 1 2c1,1x2 1+∑m−1 i=2[c0,iΔxi+1 2c1,iΔx2 i+1 3c2,iΔx3 i+1 4c3,iΔx4 i] +c0,m(σ−xm)+1 2c1,m(σ−xm)2+1 3c2,m(σ−xm)3+1 4c3,m(σ−xm)4 forσ∈Imwith 3≤m≤M−1 1 2c1,1x2 1+∑M−1 i=2[c0,iΔxi+1 2c1,iΔx2 i+1 3c2,iΔx3 i+1 4c3,iΔx4 i]+c0,M(σ−xM) forσ∈IM(35) withΔxi=xi+1−xi. The function Hσfof Eq. (16) then is given by Hσf(σ)=HdRPA(σ)+H(σ) =−ln(1 +σ)+σ+H(σ). (36) The function Hσfhas to be evaluated instead of HdRPAif the KS cor- relation energy is calculated by a σ-functional instead of the dRPA functional. B. Computational details Data for the optimization of σ-functionals were calculated with a local version of Molpro-2015.106,107Computational timings werecarried out with the recent Molpro-2020.1 development version. The latter version was furthermore used to calculate data from the INV24 reference set. Throughout, density fitting108–116was employed in the evaluation of the two-electron (four-index) integrals for the Coulomb as well as the exchange energy, with the S22 reference set being the only exception. For the S22 set, a correction for the basis set superposition error seemed appropriate despite the quite large basis sets that were employed (see below). For technical reasons, we could not use density fitting together with dummy atoms required for the correction of the basis set superposition error. The input orbitals and eigenvalues for the post-SCF σ-functional method presented here were obtained in all cases by a GGA calculation employing the exchange–correlation functional of J. Chem. Phys. 154, 014104 (2021); doi: 10.1063/5.0026849 154, 014104-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Perdew, Burke, and Ernzerhof (PBE).117The SCF convergence cri- terion for the energy was 10−10Hartree, and the density matrix was converged to 10−8a.u. In some cases, the SCF calcula- tions only converged to slightly lower but still sufficiently tight values. Except for the reference sets ALKDBE10 und HEAVYSB11, aug-cc-pwCVQZ atomic orbital basis sets118were used. For density- fitting, i.e., the treatment of the two-electron integrals, aug-cc- pwCV5Z/mp2fit119,120basis sets were employed. The RI basis sets to represent the Coulomb kernel and the KS response function were aug-cc-pwCVQZ/mp2fit basis sets.119,120The reference sets ALKDBE10 und HEAVYSB11 contain heavier elements and met- als. For some of them, the above basis sets are not available. For the reference sets ALKDBE10 und HEAVYSB11, we therefore useddef2-QZVPP121,122atomic basis sets, def2-QZVPP/jkfit123basis sets for density fitting, and def2-AQZVPP/mp2fit119,124basis sets as RI basis sets to represent the Coulomb kernel and the KS response function. The numerical frequency integration required in the ACFD theorem was performed with a Gauss–Legendre quadrature scheme with 50 quadrature points. The nodes ˜ωiand the weights ˜γiof the Gauss–Legendre quadrature for the interval [ −1, 1] were mapped on the interval [0, ∞] to obtain the actually used nodes ωi=w01 +˜ωi 1−˜ωi(37) and weights TABLE I . Reference sets used for optimizing and evaluating σ-functionals. Atomization energies for small systems W4-11aAtomization energies ALKBDE10 Dissociation energies of group-1 and -2 diatomics HEAVYSB11 Dissociation energies of heavy-element compounds Reaction energies of small molecules W4-11REaReaction energies of all A + B →C + D and B →C + D reactions contained in W4-11 BH76RC Reaction energies of the BH76 set FH51 Reaction energies of various (in-)organic systems Reaction energies for large systems and isomerization reactions ISO34 Isomerization energies of small- and medium-sized organic molecules Barrier heights BH76 Barrier heights of hydrogen transfer, heavy atom transfer, nucleophilic substitution, unimolecular, and association reactions BHDIV10 Diverse reaction barrier heights PX13 Proton-exchange barriers in H 2O, NH 3, and HF clusters WCPT18 Proton-transfer barriers in uncatalyzed and water-catalyzed reactions INV24bInversion/racemization barrier heights Intramolecular dispersion interactions ICONF Relative energies in conformers of inorganic systems ACONF Relative energies of alkane conformers Intermolecular non-covalent interactions S22cBinding energies of non-covalently bound dimers Difficult cases G21EA Adiabatic electron affinities G21IP Adiabatic ionization potentials SIE4x4 Self-interaction-error related problems aAtomization and reaction energies containing C 2are excluded because the PBE calculation for the KS orbitals and eigenvalues did not converge to an electronic structure obeying the aufbau principle. bReaction 9 is excluded because the PBE calculation for the KS orbitals and eigenvalues for the required transition state containing PF 3did not converge properly. cOnly dimers 1–4, 8, 9, and 16 are included and were calculated without density-fitting in the evaluation of Coulomb and exchange energy. Other dimers are excluded because, due to technical limitations, a correction of the basis set superposition error and density-fitting could not be combined; see text for details. J. Chem. Phys. 154, 014104 (2021); doi: 10.1063/5.0026849 154, 014104-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp γi=˜γi2w0 (1−˜ωi)2(38) withw0= 2.5. The optimization of the coefficients of the cubic spline repre- sentation of the σ-function fσfwas carried out using the Broyden– Fletcher–Goldfarb–Shanno (BFGS) method125as implemented in theSciPy library.126 C. Optimization of function hσf As data basis for the optimization and evaluation of σ-functionals, we use the reference sets listed and briefly described in Table I. With the exception of the set W4-11RE of reaction ener- gies all reference sets were taken from the GMTKN55 database.127 The W4-11RE reaction energies were calculated from the W4-11 atomization energy set128using the AutoRE script129that gener- ates the reaction energies of all reactions of the type A + B →C + D and B →C + D. Atomization and reaction energies contain- ing the C 2molecule had to be excluded because the PBE calcu- lation of C 2did not converge to an electronic state obeying the aufbau principle. In the W4-11RE reference set, then remained a total of 10 966 reactions energies determined from the atom- ization energies of 139 molecules. Similarly, for the INV24 refer- ence set130of inversion/racemization barrier heights, reaction 9 was excluded because the PBE calculation of the corresponding transi- tion state containing a PF 3molecule could not be converged prop- erly. From the non-covalent dimer binding energies of the S22 ref- erence set,127,131only dimers 1–4, 8, 9, and 16 were considered. The remaining dimers were excluded because we could not com- bine density fitting of the two-electron (four-index) integrals with the use of dummy atoms required for correcting the basis set super- position error in the Molpro version we used in the optimization ofσ-functionals. We therefore had to evaluate the Coulomb and exchange energy without density-fitting, which renders the calcula- tions considerably more expensive computationally. Basis set super- position errors are small for the employed atomic orbital basis sets and do not need to be corrected, except for the comparably small non-covalent dimer binding energies of the S22 reference set. We optimized σ-functions for various setups, distinguished by the number of abscissae in the cubic spline representation of the σ-function, the included reference sets, and the relative weights of the reference sets. In Table I, the results of three representa- tive setups are shown, named parameterizations P1, P2, and P3. Parameterization P1 is the one that underlies the comparison with other DFT and wave function methods in Sec. IV. Table VII of Appendix B lists the chosen x-values and the y-values result- ing from the optimizations, which together determine the cubic splines and thus the σ-function. Figure 1 displays the resulting σ-functions hσf. For parameterization P1 exclusively, the mean absolute error (MAE) of the W4-11RE reaction energies was optimized. Never- theless, the MAEs of all other reference sets in Table II improved substantially over the dRPA values with the exception of the atom- ization energies of the HEAVYSB11 set and the proton-exchange barriers of the PX13 set. Often an improvement by a factor of two or more could be achieved. Striking is the dramatic improvement FIG. 1 .σ-Functions hσf(ασ) of parameterizations P1, P2, and P3 compared to the corresponding function hdRPA(ασ) of the dRPA. The σ-function of parameterization P1 for values of ασ<3 is almost identical to that of parameterization P2 and therefore barely visible. of the W4-11 atomization energies from an MAE of 18.78 kcal/mol in the dRPA case to just 4.72 kcal/mol for the σ-functional. This means the poor description of atomization energies by dRPA methods is not observed for the σ-functionals presented here. Reaction energies (reference sets W4-11RE, BH76RC, FH51, and ISO34) and barrier heights (reference sets B76, BHDIV10, PX13, WCPT18, and INV24) obtained with the σ-functional P1 reach chemical accuracy of 1 kcal/mol or exhibit MAEs just moder- ately above with the MAE of 1.46 kcal/mol for the reaction energy set BH76RC being largest. The MAEs of the atomization ener- gies (reference sets W4-11, ALKBDE10, and HEAVYSB11) around 5 kcal/mol are competitive as well. Non-covalent binding energies (reference sets ICONF, ACONF, and S22) are described quite well already by dRPA, and the largest dRPA MAE of 0.76 kcal/mol results for the considered dimers of the S22 set. With the σ- functional, the accuracy of non-covalent binding energies is increas- ing further. In particular, the S22 MAE reduces to 0.19 kcal/mol. The fact that an optimization with respect to one reference set, W4-11RE, improves the accuracy for almost all other sets, often drastically, suggests that the resulting σ-functional is suitable for general application. J. Chem. Phys. 154, 014104 (2021); doi: 10.1063/5.0026849 154, 014104-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II . MAEs in kcal/mol for various reference sets for PBE-DFT, the dRPA method, and three differently parameterized σ-functional methods. The parameterizations are designated P1, P2, and P3. The columns labeled “weight” show the weight of the reference set in the optimization of the σ-function. The spline representation of parameterization P3 contains a larger number of x- and y-values than the one of parameterizations P1 and P2 (see text and Appendix B for details). PBEaRPAσ-functional (P1) σ-functional (P2) σ-functional (P3) MAE MAE Weight MAE Weight MAE Weight MAE W4-11b15.09 18.78 0 4.72 0.1 4.69 0.1 4.41 ALKBDE10 6.21 11.90 0 4.68 0.1 4.59 0.1 4.15 HEAVYSB11 4.58 4.62 0 5.15 0.1 5.12 0.1 5.19 W4-11REb5.95 3.21 1 1.25 1 1.27 1 1.26 BH76RC 4.09 2.44 0 1.46 1 1.46 1 1.54 FH51 3.40 1.93 0 0.84 1 0.83 1 0.83 ISO34 1.80 0.78 0 0.37 1 0.36 1 0.34 BH76 9.15 1.66 0 1.26 1 1.28 1 1.33 BHDIV10 8.23 1.57 0 0.72 1 0.71 1 0.74 PX13 11.54 0.56 0 0.74 1 0.72 1 0.72 WCPT18 8.61 1.06 0 0.28 1 0.25 1 0.26 INV24c2.67 0.80 0 0.48 0 0.46 0 0.49 ICONF 0.43 0.28 0 0.13 1 0.14 1 0.13 ACONF 0.61 0.05 0 0.08 1 0.08 1 0.10 S22d0.54 0.76 0 0.19 0 0.20 0 0.22 aFrom Refs. 127 and 129. bAtomization and reaction energies containing C 2are excluded because the PBE calculation for the KS orbitals and eigenvalues did not converge to an electronic structure obeying the aufbau principle. cReaction 9 is excluded because the PBE calculation for the KS orbitals and eigenvalues for the required transition state containing PF3did not converge properly. dOnly dimers 1–4, 8, 9, and 16 are included; see text and Table I for details. In parameterization P2, we included 13 reference sets instead of just the W4-11RE reference set. From the 13 reference sets, ten were taken into account with equal weights of 1, while the three atomization sets were considered with a weight of 0.1. The weight- ing of the reference sets was carried out at the level of their MAEs that are added up with the weights listed in Table II to give the tar- get to be minimized in the optimization for the parameterization. This means that in parameterization P2, the 10 966 reactions of ref- erence set W4-11RE together have only the same weight as, e.g., the 13 proton-exchange barriers in reference set PX13. Thus, param- eterizations P1 and P2 effectively use a highly different database. Nevertheless, only marginal changes in the MAEs are observed. The fact that even a drastic change in the optimization target leads to only negligible or small changes in the performance of the σ-functional is crucial because it is a strong hint that transferability of σ-functionals from the reference they are optimized for to other sets of molecules is given. The last parameterization P3 shown in Table II is identical to parameterization P2 as far as reference sets and their weights are concerned but differs in the number of x-values chosen for the spline representation, 19 instead of 13. This means the spline representation is more flexible. The resulting MAEs again are differing little from those of the previous parameterizations P1 and P2. Parameterizations P1, P2, and P3 have a similar quality. P1 and P2 require less x-values in the spline representation thanP3. Moreover, parameterization P1 depends exclusively on the highly reliable W4-11RE reference set. We therefore have chosen parameterization P1 for the comparison with other methods in Sec. IV. Theσ-functions hσfcorresponding to three parameterizations P1, P2, and P3 are compared to the corresponding function hdRPA of the dRPA in Fig. 1. For values ασ>2, the functions hσffor all three parameterizations are quite similar and quite close to the dRPA function hdRPA. In the region ασ<2, theσ-functions hσfdiffer distinctively from the dRPA function and show a feature consist- ing of a maximum followed by a minimum in the region around 0.1<ασ<0.5. Theσ-functions of P1 and P2 are very close to each other, whereas the one of P3 is slightly different with the feature being a bit more pronounced and a weak second maximum follow- ing the first one. Nevertheless, the MAEs of P3 on the one hand and P1 and P2 on the other hand are almost identical for most reference sets (see above). Only for the atomization reference sets W4-11 and ALKBDE10, parameterization P3 exhibits a bit smaller MAEs. Limits of the applicability of the σ-functionals presented here become evident in Table III, which displays the MAEs of refer- ence sets G21EA, G21IP, and SIE4x4 for electron affinities, ioniza- tion potentials, and self-interaction problems. The MAEs of elec- tron affinities and ionization potentials roughly double when going from the dRPA to σ-functionals, while the large MAE of the dRPA J. Chem. Phys. 154, 014104 (2021); doi: 10.1063/5.0026849 154, 014104-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE III . MAEs in kcal/mol for reference sets G21EA, G21IP, and SIE4x4 resulting from the regular dRPA and σ-functionals of parameterizations P1, P2, and P3. σ-functional σ-functional σ-functional dRPA (P1) (P2) (P3) G21EA 5.63 10.86 10.88 11.04 G21IP 6.56 14.34 14.34 14.72 SIE4x4 20.99 21.60 21.33 21.57 for the reference set SIE4x4 remains roughly as is. Here, it shows that we optimized exclusively energy differences when constructing theσ-functional presented here but did not take into account total electronic energies. These therefore worsened for the σ-functionals presented here. That is, the overestimation of the magnitude of the correlation energy in the dRPA leading to very low total electronic energies is even more pronounced for the σ-functionals. The total electronic energy is an extensive quantity that grows in magnitude with the electron number. The underestimation of the total elec- tronic energy therefore increases with the electron number. For pro- cesses such as ionization or electron attachment that change the elec- tron number, this introduces errors. An analysis of this problem and possible remedies can be found in Refs. 89 and 90. Table III shows that the difficulties in calculating ionization potentials and electron affinities are more pronounced in the σ-functionals presented here than in the dRPA. Both the dRPA and σ-functionals neglect the exchange kernel that cancels self-interactions and thus would remedy the problem of a systematic underestimation of total electronic energies. In σ- functionals, the role of the exchange kernel can be modeled to some extent by the σ-function. For the σ-functionals presented here, this did not happen in a way that total electronic energies improved. This is not surprising because neither total electronic energies nor electron affinities or ionization potentials were taken into account in the optimizations of the σ-functions presented here. We tried to include total electronic energies, electron affinities, and ioniza- tion potentials in optimizations of σ-functions. It is possible to improve total electron energies, electron affinities, and ionization potentials in this way albeit at the price of poorer accuracies for most other properties, in particular, reaction energies and barrier heights. Because total electronic energies are of little importance, in practice, we opted to improve relative instead of total energies, which works well for most properties of interest as shown above. Electron affinities and ionization potentials, however, are beyond the scope of the σ-functionals presented here, and similarly, we cannot expect to improve on systems such as those in the SIE4x4 reference set. We note that the situation may be different if other orbitals and eigenvalues are selected as basis for σ-functionals than the PBE orbitals and eigenvalues chosen here. We currently investigate this option.132 IV. COMPARISON OF σ-FUNCTIONALS WITH OTHER ELECTRONIC STRUCTURE METHODS In this section, analyzing the performance of σ-functionals we concentrate on the σ-functional of parameterization P1. Table IVTABLE IV . Comparison of MAEs in kcal/mol for various reference sets resulting from theσ-functional method presented here with the dRPA method, the density-functional methods PBE, SCAN, and B3LYP, and the double-hybrid method PWPB95. PBE, SCAN, B3LYP, and PWPB9 data are taken from Ref. 127, unless noted otherwise. σ-functional dRPA PBE SCAN B3LYP PWPB95 W4-11a4.72 18.78 15.09 4.01 4.22 3.27 ALKBDE10 4.68 11.90 6.21 19.21 4.57 3.26 HEAVYSB11 5.15 4.62 4.58 7.17 7.91 3.81 W4-11REa1.25 3.21 5.95b... 3.78b2.06b BH76RC 1.46 2.44 4.09 3.38 2.38 1.14 FH51 0.84 1.93 3.40 2.58 3.97 1.47 ISO34 0.37 0.78 1.80 1.35 2.31 0.90 BH76 1.26 1.66 9.15 7.66 4.94 1.62 BHDIV10 0.72 1.57 8.23 6.50 2.76 1.33 PX13 0.74 0.56 11.54 8.25 3.54 1.19 WCPT18 0.28 1.06 8.61 6.04 1.11 0.93 INV24c0.48 0.80 2.67 1.20 1.87 0.99 ICONF 0.13 0.28 0.43 0.32 0.58 0.35 ACONF 0.08 0.05 0.61 0.32 0.96 0.06 S22d0.19 0.76 0.54 0.80 1.99 0.91 G21EA 10.86 5.63 3.43 3.64 1.91 1.71 G21IP 14.34 6.56 3.85 4.69 3.55 1.94 SIE4x4 21.60 20.99 23.44 17.91 17.63 9.93 aAtomization and reaction energies containing C 2are excluded because the PBE calcu- lation for the KS orbitals and eigenvalues did not converge to an electronic structure obeying the aufbau principle. bFrom Ref. 129. cReaction 9 is excluded because the PBE calculation for the KS orbitals and eigenvalues for the required transition state containing PF 3did not converge properly. dOnly dimers 1–4, 8, 9, and 16 are included; see text and Table I for details. compares the MAEs of this σ-functional with the dRPA method, with the density-functional methods PBE, SCAN, and B3LYP, and with the double-hybrid method PWPB95. For atomization energies (reference sets W4-11, ALKBDE10, and HEAVYSB11), the double-hybrid method PWPB95 is most accurate. The σ-functional method exhibits MAEs that are higher by about 1.5 kcal/mol. The density-functional methods PBE, SCAN, and B3LYP show quite nonuniform MAEs that in some cases are very high. For reaction energies and barrier heights (second and third groups of reference sets in Table IV), the σ-functional method clearly outperforms all other methods. The double-hybrid method PWPB95 is second best with, however, clearly higher MAEs in all cases except the reference set BH76RC. The density-functional methods PBE, SCAN, and B3LYP are considerably less accurate for reaction energies and barrier heights and far away from chem- ical accuracy. For non-covalent interaction energies (reference sets ICONF, ACONF, and S22) again the σ-functional method is most accurate followed by the double-hybrid method PWPB95. For elec- tron affinities and ionization potentials (reference sets G21EA and G21IP), the σ-functional shows the poorest results of all considered methods. This is not surprising because electron affinities and ion- ization potentials are outside the scope of the presented σ-functional due to its construction, as discussed in Sec. III C. Similarly, the J. Chem. Phys. 154, 014104 (2021); doi: 10.1063/5.0026849 154, 014104-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp σ-functional MAE of the reference set SIE4x4 considering self- interaction related problems is the second highest after the one of PBE. Table V compares the performance of the σ-functional method presented here, the dRPA, and the wave function methods MP2 (Møller–Plesset perturbation theory of second order), CCSD (coupled-cluster singles doubles), MP4 (Møller–Plesset perturba- tion theory of fourth order), and CCSD(T) (coupled-cluster singles doubles with perturbative triples). For atomization energies (refer- ence set W4-11), MP4 exhibits the lowest MAE followed by the σ- functional method and CCSD(T), which perform similar. MP2 and CCSD show distinctively higher MAEs. For barrier heights (refer- ence set BH76), the best results are obtained with the σ-functional closely followed by MP4. MP2 and CCSD exhibit about twice as large MAEs beyond 2 kcal/mol and therefore clearly do not reach chemical accuracy. The BH76 reference values are calculated within the W2 scheme133that essentially amounts to CCSD(T) in the com- plete basis set limit. Therefore, a comparison with CCSD(T) val- ues does not make sense. The best reaction energies (reference set WE-11RE) are obtained by CCSD(T) closely followed by the σ-functional method. Both methods approach chemical accuracy. MP2 and MP4 exhibit distinctively larger MAEs far from chemical accuracy. Next, we analyze in more detail the errors of the presented σ-functional method. To that end, the upper panel of Fig. 2 dis- plays the distribution of errors of the 10 966 reactions taken into account in the W4-11RE energy reference set for the dRPA and theσ-functional. Figure 2 shows that the errors are symmetrically distributed both for the dRPA and for the σ-functional. The distribu- tion of errors for the σ-functional is much more narrow. In particu- lar, there are almost no errors with a magnitude beyond 5 kcal/mol– 6 kcal/mol. The error distribution thus improves in a very favorable way when going from the dRPA to the σ-functional. In the lower panel of Fig. 2, a corresponding distribution of errors is shown for the 115 reaction energies of the reference sets BH76RC, FH51, and ISO34. Again, the σ-functional exhibits a distribution of errors more narrow than that of the dRPA. Due to the much smaller number of reactions, the error distribution is less smooth than in the W4-11RE case. In Table VI, MAEs of the dRPA and the σ-functional for sub- sets of the W4-11RE reference set for reaction energies are listed. If only reactions between molecules containing exclusively first row TABLE V . Comparison of MAEs in kcal/mol of the σ-functional method presented here, the dRPA, and several wave function methods for the reference sets W4-11, BH76, and W4-11RE. σ-functional dRPA MP2 CCSD MP4 CCSD(T) W4-11a4.72 18.78 8.17b11.52b2.89,c2.10b5.92,c3.23b BH76 1.26 1.66 2.68b2.34b1.31b W4-11REa1.25 3.21 5.37c2.89c1.08c aAtomization and reaction energies containing C 2are excluded because the PBE calcu- lation for the KS orbitals and eigenvalues did not converge to an electronic structure obeying the aufbau principle. bFrom Ref. 134. cFrom Ref. 129. FIG. 2 . Distribution of the errors of the W4-11RE reference set (upper plot) and BH76RC, FH51, and ISO34 reference sets (bottom plot) for reaction energies for the dRPA and the σ-functional with parameterization P1. Reaction energies containing C 2are excluded because the PBE calculation for the KS orbitals and eigenvalues did not converge to an electronic structure obeying the aufbau prin- ciple. The remaining W4-11RE reference set contains 10 966 reaction energies of 139 molecules. The reference sets BH76RC, FH51, and ISO34 together contain 115 reactions. TABLE VI . MAEs in kcal/mol of σ-functional and dRPA for subsets of the W4-11RE reference set for reaction energies. Reaction energies containing C 2are excluded because the PBE calculation for the KS orbitals and eigenvalues did not converge to an electronic structure obeying the aufbau principle. The remaining W4-11RE reference set contains 10 966 reaction energies. No. of No. of σ-functional dRPA molecules reactions All 1.25 3.21 139 10 966 1stRow 1.20 3.22 98 9 135 1stRow-noMR 1.15 2.93 88 7 647 1stRow-noSpin 0.97 3.24 73 4 061 1stRow-noMR-noSpin 0.90 2.84 67 3 993 noMR-noSpin 0.93 2.82 96 3 983 noMR 1.19 2.91 124 8 868 noSpin 1.00 3.21 104 4 532 J. Chem. Phys. 154, 014104 (2021); doi: 10.1063/5.0026849 154, 014104-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Atomization energies calculated with the dRPA and with the σ-functional with parameterization P1 vs W4-11, ALKBDE10, and HEAVYSB11 reference energies. The C 2molecule was excluded from the W4-11 set because the PBE calculation for the KS orbitals and eigen- values did not converge to an electronic structure obeying the aufbau principle. elements (in addition to hydrogen) are taken into account (row labeled 1stRow), this has an almost negligible effect on the MAE both for the dRPA and for the σ-functional. This means the reac- tions with molecules containing heavier elements are described with almost the same accuracy as those containing only molecules with light elements. If only molecules without multireference charac- ter are considered (rows labeled noMR and 1stRow-noMR), the MAEs of the σ-functional are improved only very little. This sug- gests that a treatment of reactions containing molecules with some multi-reference character poses no problems for σ-functionals. The dRPA MAEs react a bit stronger on the absence of systems with multireference character. If only reactions between non-spin- polarized molecules are taken into account (rows labeled 1stRow- noSpin and no Spin), then the σ-functional MAE is substantially reduced by about 0.25 kcal/mol both for the comparison “1stRow” vs “1stRow-noSpin” and “All” vs “noSpin.” In the dRPA case, on the other hand, it has little effect whether or not spin-polarized systems are considered. In the cases “1stRow-noMR-noSpin” and “noMR-noSpin,” the σ-functional MAE lies below 1 kcal/mol, and in the case “noSpin,” the MAE is 1 kcal/mol. This means that for the important case of reactions between non-spin-polarized molecules, the presented σ-functional method reaches chemical accuracy. In Fig. 3, dRPA and σ-functional atomization energies are plot- ted against the reference values from the W4-11, ALKBDE10, and HEAVYSB11 sets. The dRPA atomization energies are systemat- ically very large with a deviation from the reference values that increases with the atomization energy, i.e., with increasing system size. Theσ-functional atomization energies do not exhibit a clear systematic over- or underestimation of the reference values but, for the most part, are scattered around the reference values. Further- more, a significant increase of the errors with the magnitude of the atomization energy is not observed. V. COMPUTATIONAL TIMINGS In Fig. 4, computational timings for the post-SCF calcula- tion of the total energy in σ-functional methods are compared to HF calculations. Not only the evaluation of the correlation energy by the σ-functional but that of the complete post-SCFtotal energy calculation is compared to a complete self-consistent HF calculation typically consisting of 10–15 SCF iterations. The total energy evaluations of the σ-functional method and the self- consistent HF calculations all employ density-fitting for the two- electron (four-index) integrals. The HF calculations serve as ref- erence for an SCF method. PBE calculations that are used in the present work to generate KS orbitals and eigenvalues are consider- ably faster than HF calculations. Hybrid DFT calculations, on the other hand, are computationally slightly more expensive than HF calculations. Systems consisting of 1–8 benzene molecules are considered in order to analyze the behavior of the computational time with increasing system size. Two setups, using basis sets of triple zeta FIG. 4 . Computational timings for the evaluations of the total energy in σ-functional methods compared to the computational time for Hartree–Fock calculations serv- ing as examples for self-consistent electronic structure methods. Both in the Hartree–Fock calculations and in the total energy evaluations with the σ-functional method, density-fitting for the two-electron (four-index) integrals was employed. Calculations using basis sets of triple zeta quality (VTZ) and of quadruple zeta quality (VQZ) are displayed. Computational timings for systems consisting of an increasing number of benzene molecules are shown. Calculations were carried out on four Intel Broadwell nodes with 512 GB memory and 28 cores per node using an hierarchical parallelization with four MPI processes (one per node) and up to 28 OMP threads per node; see text for details. J. Chem. Phys. 154, 014104 (2021); doi: 10.1063/5.0026849 154, 014104-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp (VTZ) and of quadruple zeta (VQZ) quality, are considered; see Sec. III B for further details on basis sets. In this work, basis sets of VQZ quality were used in order to not impair the parameteriza- tion ofσ-functionals by issues of basis set insufficiencies. Timings for VTZ, however, may be of interest because VTZ basis sets are often employed in practice. The calculations were carried out on four Intel Broadwell nodes with 512 GB memory and 28 cores per node. The code is hierar- chically parallelized with one MPI (Message Passing Interface) pro- cess per node and up to 28 OMP (Open Multi-Processing) threads per node. The OMP parallelization in the HF as well as in the σ-functional code is quite basic, leaving ample room for further improvement. Figure 4 shows that the evaluation of the post-SCF total energy in aσ-functional method is computationally distinctively less expen- sive than a HF calculation for all considered system sizes and for VTZ as well as VQZ basis sets. The ratio of the computational times might change somewhat if parts of the involved codes are optimized. However, this will not change the basic message that the post-SCF use ofσ-functionals requires a computational effort similar or lower to that of HF or hybrid DFT methods and thus can be carried out routinely for systems of practical relevance. We note, furthermore, that the computational time of σ-functional and dRPA calculations is almost identical. VI. CONCLUDING REMARKS We introduced a new type of functionals for the KS corre- lation energy. The functionals are named σ-functionals because they contain a function of the eigenvalues σof the KS response function. Technically, the new functionals are closely related to the dRPA. An existing dRPA program can be converted into a program for σ-functionals with minimal programming effort. Indeed, in order to turn a dRPA into a σ-functional program, it is only necessary to replace the evaluation of the function ln(1 +σ) +σin dRPA programs via the intrinsic logarithm func- tion by a call to a subroutine for evaluating the σ-function intro- duced here. The formal background of σ-functionals is power series expansions of response functions and exchange–correlation kernels.53,63 Theσ-functionals presented here are parameterized for a post- SCF use based on input orbitals and eigenvalues from a preced- ing PBE calculation. The reaction and transition state energies calculated with the presented σ-functional approach chemical accu- racy. Non-covalent interaction energies are described highly accu- rate, and even atomization energies exhibit an accuracy like that of CCSD(T) methods. In contrast to high-level wave function meth- ods,σ-functional methods are computationally quite efficient. The required computational time for a total energy calculation with σ-functionals is lower than that of hybrid DFT calculations. This meansσ-functionals can be evaluated routinely after a conventional DFT calculation in order to boost the accuracy and to approach chemical accuracy. This makes the new functionals interesting for routine applications. The parameterization of the σ-functional depends on the input orbitals and eigenvalues. This means that there is chance to improve the accuracy even further by changing from PBEinput orbitals and eigenvalues to those from other DFT meth- ods. We presently investigate this point, and first results indicate that substantial improvements in accuracy are possible along this route.132 Processes that change the electron number, i.e., ionization potentials or electron affinities, are outside the scope of the spe- cificσ-functionals presented here. Whether other parameterizations and/or other input orbitals and eigenvalues may enable an accurate description of these quantities together with reaction and transition state energies remains to be investigated. Other desirable future work includes the implementation of analytic energy gradients of σ-functionals in order to be able to carry out geometry optimizations efficiently. A construction of σ- functionals for self-consistent use would be interesting in order to explore further the scope of this type of functionals. Com- putationally, self-consistent σ-functional methods are consider- ably more expensive than non-self-consistent ones, even though they exhibit the same scaling with the system size as the post- SCFσ-functional method. However, the prefactor of the compu- tational time is distinctively higher for self-consistent σ-functional methods. ACKNOWLEDGMENTS This work was funded by the German Science Foundation (DFG) through SFB 953 Synthetic Carbon Allotropes (Project No. 182849149). APPENDIX A: σ-FUNCTIONALS FOR SPIN-POLARIZED ELECTRONIC SYSTEMS If a spin-polarized calculation is carried out to provide the KS orbitals and eigenvalues entering the calculation of the KS corre- lation energy by σ-functionals, then Eq. (3) for the KS response function assumes the form χ0(iω,r,r′)=2∑ τ=α,β∑ i∑ a(ϵi,τ−ϵa,τ) (ϵi,τ−ϵa,τ)2+ω2 ×φi,τ(r)φa,τ(r)φa,τ(r′)φi,τ(r′), (A1) withφi,τandφa,τdenoting occupied and unoccupied spatial spin orbitals with spin τ=α,βandϵi,τandϵa,τdesignating the cor- responding eigenvalues. The elements of the KS response matrix then are calculated by Eq. (4) analogously to the non-spin-polarized case. The definition and computational evaluation of σ-functionals is identical to the non-spin-polarized case once the KS response matrix is constructed. All contributions to the total energy aside from the correlation energy are calculated exactly as in a spin- polarized HF calculation, of course, using KS instead of HF orbitals. APPENDIX B: PARAMETERS OF THE SPLINE REPRESENTATION OF σ-FUNCTIONS Table VII lists the x- and y-values determining the spline rep- resentation of the σ-functions of the three parameterizations P1, P2, and P3 discussed in the main text. The x-values were chosen, and the y-values are the result of the optimizations described in Sec. III C. J. Chem. Phys. 154, 014104 (2021); doi: 10.1063/5.0026849 154, 014104-13 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE VII . Parameters, x- and y-values, for the spline representation of the σ-functions of the three parameterizations P1, P2, and P3 discussed in the main text. P1 P2 P3 xi yi xi yi xi yi 0 0 0 0 0 0 10–4−0.000 058 857 2 10–4−0.000 058 427 0 10–5−0.000 007 096 3 10–3−0.000 664 498 3 10–3−0.000 662 974 0 10–4−0.000 048 363 0 10–2−0.011 805 792 0 10–2−0.011 817 243 2 10–7/2−0.000 244 944 1 10–3/2−0.015 708 445 1 10–3/2−0.015 738 416 8 10–3−0.000 624 961 2 10–10.176 098 505 0 10–10.176 099 888 0 10–5/2−0.001 155 930 3 10–3/40.098 145 514 7 10–3/40.098 112 504 1 10–2−0.011 726 142 8 10–1/2−0.036 283 244 9 10–1/2−0.036 218 578 6 10–5/3−0.010 005 946 2 10–1/4−0.042 471 783 7 10–1/4−0.044 400 137 4 10–4/3−0.008 544 536 1 1 −0.049 440 747 7 1 −0.052 408 361 5 10–10.210 686 485 0 101/3−0.009 490 347 2 101/3−0.002 815 746 9 10–4/50.118 075 663 2 102/30.004 773 528 5 102/3−0.021 971 711 3 10–3/50.015 483 958 7 10 0 10 0 10–2/5−0.062 286 840 8 10–1/5−0.044 018 530 9 1 −0.055 200 940 0 101/4−0.021 452 753 8 101/20.013 375 549 1 103/4−0.015 590 463 4 10 0 APPENDIX C: ALTERNATIVE ANSATZ FOR FUNCTION hσf Besides the ansatz for the σ-function hσfpresented in Sec. III A, we tested an alternative form for hσfgiven by hσf(x)=−1 1 +x+c1x2ec2x+c3x3ec4x+c5x2 c6+x2+ 1 (C1) with x=ασas before. The alternative ansatz contains six parameters ciand represents the complete σ-function hσf, not just a correction hto the corresponding dRPA function hdRPAas in the ansatz of Sec. III A. The integration over the coupling constant αcannot be carried out analytically for this alternative form of hσf. This means both the frequency and the coupling strength integration have to be carried out numerically to obtain the correlation energy Ec=−1 2π∫dωσ(ω)∫dαhσf(ασ). (C2) We determined the coefficients ciby optimizing the W4- 11RE reaction energy test in analogy to parameterization P1 of the ansatz for hσfin Sec. III A and obtained c1= 37.507 342 15, c2=−11.452 833 85, c3=−20.731 478 78, c4=−4.684 671 05, c5= 0.421 598 55, c6= 9.296 015 24. Table VIII compares results with the new ansatz for hσfwith those of parameterization P1 of the ansatz presented in Sec. III A and with the dRPA. The new ansatz yields results that are similar and only slightly inferior to those ofTABLE VIII . MAEs in kcal/mol for various reference sets for the dRPA method and twoσ-functional methods employing different forms of σ-functions hσf. Column A1 displays results for the alternative ansatz for hσfdiscussed in this appendix, while column P1 lists the results for parameterization P1 of the ansatz for hσfpresented in Sec. III A. dRPA A1 P1 W4-11a18.78 5.24 4.72 ALKBDE10 11.90 5.02 4.68 HEAVYSB11 4.62 5.06 5.15 W4-11REa3.21 1.39 1.25 BH76RC 2.44 1.59 1.46 FH51 1.93 0.90 0.84 ISO34 0.78 0.37 0.37 BH76 1.66 1.35 1.26 BHDIV10 1.57 0.65 0.72 PX13 0.56 0.94 0.74 WCPT18 1.06 0.24 0.28 INV24b0.80 0.48 0.48 ICONF 0.28 0.15 0.13 ACONF 0.05 0.06 0.08 S22c0.76 0.33 0.19 aAtomization and reaction energies containing C 2are excluded because the PBE calcu- lation for the KS orbitals and eigenvalues did not converge to an electronic structure obeying the aufbau principle. bReaction 9 is excluded because the PBE calculation for the KS orbitals and eigenvalues for the required transition state containing PF 3did not converge properly. cOnly dimers 1–4, 8, 9, and 16 are included; see text and Table I for details. J. Chem. Phys. 154, 014104 (2021); doi: 10.1063/5.0026849 154, 014104-14 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 .σ-Function hσf(ασ) of the alternative ansatz A1 presented in this appendix compared to that of parameterization P1 from the main text and to the correspond- ing function hdRPA(ασ) of the dRPA. parameterization P1 for most of the considered test sets and that are therefore again drastically better than the dRPA results. In Fig. 5, theσ-function hσfof the new ansatz is compared to that of param- eterization P1 and to the corresponding dRPA function. The two σ-functions are clearly different, but both exhibit a feature in the region around 0.1 <ασ<0.5 that is absent in the corresponding dRPA function. DATA AVAILABILITY The data that support the findings of this study are avail- able within the article and from the corresponding author upon reasonable request. REFERENCES 1D. C. Langreth and J. P. 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5.0035640.pdf

Appl. Phys. Lett. 118, 042406 (2021); https://doi.org/10.1063/5.0035640 118, 042406 © 2021 Author(s).Interfacial chemical states and recoverable spin pumping in YIG/Pt Cite as: Appl. Phys. Lett. 118, 042406 (2021); https://doi.org/10.1063/5.0035640 Submitted: 30 October 2020 . Accepted: 10 January 2021 . Published Online: 26 January 2021 Mingming Li , Dainan Zhang , Lichuan Jin , Bo Liu , Zhiyong Zhong , Xiaoli Tang , Hao Meng , Qinghui Yang , Lei Zhang , and Huaiwu Zhang ARTICLES YOU MAY BE INTERESTED IN Tunable spin–orbit torque efficiency in in-plane and perpendicular magnetized [Pt/Co] n multilayer Applied Physics Letters 118, 042405 (2021); https://doi.org/10.1063/5.0034917 Spin-memory loss induced by bulk spin–orbit coupling at ferromagnet/heavy-metal interfaces Applied Physics Letters 118, 042408 (2021); https://doi.org/10.1063/5.0039088 Large damping-like spin–orbit torque and perpendicular magnetization switching in sputtered WTe x films Applied Physics Letters 118, 042401 (2021); https://doi.org/10.1063/5.0035681Interfacial chemical states and recoverable spin pumping in YIG/Pt Cite as: Appl. Phys. Lett. 118, 042406 (2021); doi: 10.1063/5.0035640 Submitted: 30 October 2020 .Accepted: 10 January 2021 . Published Online: 26 January 2021 Mingming Li,1 Dainan Zhang,1Lichuan Jin,1,a) BoLiu,2,a)Zhiyong Zhong,1 Xiaoli Tang,1Hao Meng,2 Qinghui Yang,1 LeiZhang,1and Huaiwu Zhang1 AFFILIATIONS 1State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, People’s Republic of China 2Zhejiang Hikstor Technology Co., Ltd., Hangzhou, 310000, People’s Republic of China a)Authors to whom correspondence should be addressed: lichuanj@uestc.edu.cn and liubo@hikstor.com ABSTRACT Ion etching is an essential step in the processing of spintronic devices. In this work, we investigated the role of argon ion (Arþ) bombardment in the spin pumping and inverse spin Hall effect in the Y 3Fe5O12/Pt (YIG/Pt) heterostructure. The inverse spin Hall voltage is found to reduce by an order of two when the argon ion bombardment is employed on the YIG surface before the deposition of the Pt layer.This giant inhibition of spin injection efficiency is undesirable. In this work, we propose an experimental technique for its recovery via a chemical route. The interface property and chemical state were identified by transmission electron microscopy, Raman spectroscopy, and x-ray photoelectron spectroscopy. We found that the argon ion bombardment on the YIG surface leads to an increase in the ratio of Fe 2þ ions in the YIG/Pt interface region. Moreover, the interface magnetic moment reduces in the presence of Fe2þions, which resulted in the decrease in spin injection efficiency. A strong oxidizing solution (a mixture of concentrated H 2SO4and 30% H 2O2of 1:1 volume ratio) was used to recover the valence of iron and subsequently the interface magnetic moment. Our results are helpful for the understanding of the importance of interface properties and the optimization of spintronic device processing technology. Published under license by AIP Publishing. https://doi.org/10.1063/5.0035640 The conversion between spin currents and charge currents is fun- damental to the development of devices based on information carriedby the electron spins. 1Pure spin current, associated with angular momentum, avoids the Joule heating, circuit capacitance, and electro-migration in charge current. 2The common methods to generate spin current include spin Hall effect (SHE),3,4spin Seebeck effect (SSE),5,6 spin pumping (SP),7,8and nonlocal spin valves.9Af e r r o m a g n e t i c (FM)/normal metal (NM) heterostructure is employed to investigatethe spin pumping and inverse spin Hall effect. Yttrium iron garnetY 3Fe5O12(YIG) thin films are widely studied ferromagnetic material due to their low magnetization damping. The normal metal as a spindetector refers to a material with strong spin–orbit coupling, such asparamagnetic Pt, 10,11b-W,12and Ta,13ferromagnetic Py,14antiferro- magnetic IrMn,15,16and semiconductors.17,18The spin mixing con- ductance g"#is used to quantitatively describe the spin injection efficiency at the FM/NM interface.19 The properties of the interface region play a key role in spin cur- rent flow across the FM/NM interface. Several techniques have beenused to improve the interface condition, such as argon ion (Ar þ)bombardment. However, experimental studies have shown different results; argon ion (Arþ) bombardment could enhance spin pumping or even reduce spin pumping. For example, in the YIG/Au/Fe hetero-structure, Burrowes et al. reported that Ar þion beam etching together with the heating in 400/C14C is capable of increasing the spin mixing conductance by a factor of 5 compared to the untreated interface.Then, a prolonged etching time leads to the appearance of metallic Fe,which results in the reduction of spin injection. 20Qiu et al. found that Arþion bombardment leads to the appearance of an amorphous YIG layer at the YIG/Pt interface, which results in the reduction of longitu-dinal SSE (LSSE) voltage. 21Ion etching is an essential step in the processing of spintronic devices, and this possible giant inhibition ofspin injection efficiency is undesirable. Therefore, it is crucial to figureout a method to recover interfacial spin injection efficiency. It is alsoimportant to clearly understand the interfacial structure, magnetism,and chemical constituents in spin injection. In this Letter, we studied the effects of argon ion (Ar þ) bombard- ment on the spin pumping and inverse spin Hall effect in the YIG/Ptheterostructure. The inverse spin Hall voltage was reduced by an order Appl. Phys. Lett. 118, 042406 (2021); doi: 10.1063/5.0035640 118, 042406-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplof two when the argon ion bombardment was employed on the YIG surface before the deposition of the Pt layer. The interfacial properties of YIG/Pt heterostructures are investigated using high-resolution transmission electron microscopy (HRTEM), Raman spectroscopy, and x-ray photoelectron spectroscopy (XPS). We found that the argon ion bombardment on the YIG surface leads to the increase in the ratio of Fe2þions and the subsequent decrease in the interface magnetic moment in the YIG/Pt interface region. Moreover, we propose a chemical technique to recover interface magnetic moment and thespin injection. The YIG (420 nm)/Pt bilayers were fabricated by liquid epitaxy (LPE) and magnetron sputtering on (111)-oriented, single-crystalline Gd 3Ga5O12(GGG) substrates. The YIG film was grown at 1278 K, and t h eg r o w t hr a t ew a s0 . 1 9 lm/min. The argon ion bombardment was employed on the YIG surface before the deposition of the Pt layer. The acceleration voltage of the ion beam was set at 500 V. Subsequently,the Pt layers were deposited by DC magnetron sputtering and the base vacuum is 3.5 /C210 /C05Pa. A mixture of concentrated H 2SO4and 30% H2O2of 1:1 volume ratio was used in chemical recovery processing. T h eY I Gs a m p l e sw e r ei m m e r s e di nt h i ss o l u t i o nf o r3 0 m i na f t e r argon ion bombardment for recovery of the interfacial spin injection. One may concern that the oxidizing solution would corrode the YIG film rather than recovering the interfacial states. So we fabricated a half masked samples to demonstrate that the oxidizing solution would not corrode the YIG film (see the supplementary material ). The crystal texture of YIG/Pt samples was studied by using x-ray diffraction and high-resolution transmission electron microscopy (HRTEM). Atomic force microscopy (AFM) was used to study the surface roughness. The spin pumping and inverse spin Hall effect were measured using abroadband ferromagnetic resonance system. First of all, we investigated the crystal texture of YIG/Pt samples with an Ar þion bombardment for 600 s before the deposition of Pt layers. The ion beam was kept perpendicular to the sample surface. The high-resolution XRD pattern is shown in Fig. 1(a) . Two distinct diffraction peaks correspond to Pt (111) and YIG and GGG (444), respectively. The inset shows a zoomed image of the diffraction peaknear 51 /C14. The diffraction peak of YIG is very close to the GGG sub- strate, which confirms the epitaxial growth of the YIG film.22The three-dimensional AFM images of the surface of YIG after Arþion bombardment are found to be flat and smooth as shown in Fig. 1(b) . The root mean square surface roughness is 0.46 nm. The atomic-scale microstructure of the YIG/Pt interface is obtained by HRTEM imag- ing. As shown in Fig. 1(c) , the interface is atomically sharp and the for- mation of the amorphous layer is invisible. The selected area electron diffraction (SAED) pattern of the interface area is shown in Fig. 1(d) . The polycrystalline diffraction ring came from the Pt layer, while the single crystal lattice is rooted in the YIG layer. As shown in Figs. 1(e) and1(f), the elemental maps reveal a sharp interface between the Pt and YIG film and there is no obvious atomic diffusion. The compari- son of AFM and TEM images of YIG and YIG/Pt with and without Arþetching could be found in the supplementary material . The above results confirm that Arþion bombardment on the YIG surface has no impact on the interface structure of YIG/Pt samples. So the decrease in interfacial spin injection efficiency (described later) may come from other factors. A series of YIG/Pt (10 nm) samples treated with different Arþ ion bombardment times were prepared, and the Pt layer was sputterdeposited at the same time. The spin pumping effect was studied with the help of FMR in a swept in-plane magnetic field. During the FMRmeasurement, samples were capped on the coplanar waveguide. Themicrowave frequency used is in the range of 2–15 GHz, and the copla-nar waveguide signal linewidth is 500 lm. The absorption derivative spectra of YIG/Pt (10 nm) samples with different interfacial treatments at 5 GHz are shown in Fig. 2(a) . The peak to peak linewidth ( DH P-P) of the YIG is markedly increased after the capping of the Pt layer,which indicated the generation of spin current. Furthermore, thebroadening of the linewidth diminished gradually when the Ar þion bombardment time enhanced and recovered after the surface treat- ment. Also, the intensity of the absorption derivative spectrum showsa similar trend. The energy of YIG magnetization precession trans-ferred to the Pt layers as spin currents and gave rise to the damping a. A higher damping constant means higher absorption power. The over- all magnetic properties of YIG/Pt samples with different interface FIG. 1. The interfacial properties of YIG (420 nm)/Pt samples with an Arþion bom- bardment for 600 s before the deposition of Pt layers. (a) High-resolution XRD pat- tern of the YIG/Pt sample. (b) Three-dimensional AFM image of the YIG (420 nm) sample after Arþion bombardment. (c) High-resolution transmission electron micro- scope (HRTEM) images of the YIG/Pt interface. (d) The selected area electron dif-fraction (SAED) pattern of the interface area. (e) and (f) show the elemental mappings of the Fe and Pt elements.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 042406 (2021); doi: 10.1063/5.0035640 118, 042406-2 Published under license by AIP Publishingtreatment are obtained from the relationship between the extracted resonance field ( HR) and the microwave frequency as shown in Fig. 2(b) , which is described by the Kittel law,23,24 f¼cjjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi HRþHk ðÞ HRþHkþ4pMeff ðÞp : (1) Here, Hkis the in-plane anisotropy field and Meffis the effective saturation magnetization of the YIG layer. cjj¼2:8G H z =kOe is the gyromagnetic ratio. The measured in-plane anisotropy field and effec-tive saturation magnetization were found to be 1.2 Oe and 1790 Gs,respectively, which remain almost constant. The properties of theinterface in YIG/Pt structures are of great importance. To identify the damping constant and the interfacial spin injection efficiency of the samples, we investigated the FMR linewidth ( DH¼ffiffiffi 3p DH p-p)a sa function of microwave frequency ( f) for a bare YIG and YIG/Pt samples having different surface treatment. The results are shown inFig. 2(c) . The effective Gilbert damping a effis obtained by25,26 DH¼2aeff cjjx 2pþDH0; (2) where x/2p¼fis the microwave frequency, which is in the range of 2–15 GHz, and DH0is the zero-frequency offset arising from long- range magnetic inhomogeneities. The damping constants for bare YIG with different treatments are given in the supplementary material ,a n dwe found that different surface treatments had no effect on the damping constants of bare YIG. The spin mixing conductance g"#is used to quantitatively describe the spin injection efficiency at the FM/NM interface. Here, we calculated the effective spin mixing conductance g"# effusing27,28 g"# eff¼4pMsdYIG glBaYIG =Pt/C0aYIG ðÞ ; (3) where 4 pMs,dYIG,g,a n d lBare the saturation magnetization, the YIG thickness, the Lande factor, and the Bohr magneton, respectively. The spin backflow in our structures is negligible as the thickness of the Ptlayer is much larger than the spin diffusion length. Replacing our experimental results in Eqs. (2)and(3), we calculated the values of a eff and g"# effas shown in Fig. 2(d) . The effective damping constants aeffare found to be around 2.87 /C210/C04for YIG/Pt (10 nm) without interface treatment and 1.82 /C210/C04after Arþion bombardment for 150 s. Even smaller values are obtained when the Arþion bombardment time was 600 s (1.54 /C210/C04). When we employed a surface recovery process before the deposition of the Pt layer, the damping constant recovers to 2.71 /C210/C04. Furthermore, the effective spin mixing con- ductance g"# effexhibited a similar trend, which indicated that Arþion bombardment on the YIG surface restricted the spin injection inthe interface region of the YIG/Pt heterostructure, and the recovery process could effectively restore the spin pumping effect. FIG. 2. FMR absorption derivative spectrum of YIG/Pt (10 nm) samples with different interfacial treatments at 5 GHz. (b) Hras a function of ffor YIG/Pt (10 nm) samples. (c) DH as a function of ffor YIG/Pt (10 nm) samples. (d) Extracted effective Gilbert damping aeffand effective spin mixing conductance g"# efffor YIG/Pt (10 nm) samples.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 042406 (2021); doi: 10.1063/5.0035640 118, 042406-3 Published under license by AIP PublishingTo further explain the spin injection phenomena at the YIG/Pt interface, we measured the inverse spin Hall effect of several samples. Compared to the results of spin pumping described above, the changes in inverse spin Hall voltage caused by Arþion bombardment and sub- sequent chemical recovery were even more dramatic. The schematic ofthe inverse spin Hall effect measurement is shown in Fig. 3(a) .T h e static field His applied perpendicular to the sample and the rf field h rf in parallel for measuring the inverse spin Hall voltage (V ISHE).hHis the angle between V ISHEand the static field H. The magnetization M of YIG precesses around Hand pumps a pure spin current into the Pt layer. This pure spin current converted into a charge current because of the inverse spin Hall effect in the Pt layer. The measured inversespin Hall voltage can be expressed as follows: 29,30 VISHE¼RhSHwkSDtanhdN 2kSD/C18/C19 jeff s; (4) jeff s¼ex 2pg"# effPsin2h: (5) Here hSH,kSD,a n d jeff sare the spin Hall angle, spin diffusion length, and effective spin current density, respectively. Moreover, R,dN, and ware the resistance, the thickness, and the width of the Pt layer [Fig. 3(a) ].Pis the ellipticity correction factor raised from the elliptic- ity of the magnetization precession. Figure 3(b) shows the raw data and the Lorentz fit result of the obtained inverse spin Hall voltage ofthe YIG/Pt sample, for which the YIG surface was bombarded by the Arþion for 600 s before the deposition of the Pt layer. It is obvious that the voltage is a symmetric Lorentzian signal around the resonance field. To exclude the influence of the heat effect, we changed the signof the applied field. The sign of the measured voltage changed, how-ever, keeping the intensity same. Thus, we determined that the mea- sured voltage originates from the inverse spin Hall effect. We measured the inverse spin Hall effect and extracted the inverse spin Hall voltage for the samples etched with different Ar þion e t c ht i m e sa ss h o w ni n Figs. 3(c) and3(d).W h e nA rþion etch was employed on the YIG/Pt interface, V ISHE dropped by nearly two orders of magnitude compared to the untreated sample. In our mea-surements, the inverse spin Hall voltage for the untreated YIG/Pt(10 nm) sample was found to be 12.4 lV, which reduced to 0.18 lV after an Ar þion etching of 600s. The spin injection almost disap- peared. It should be pointed out that the changes in the inverse spinHall voltage did not match the changes in the spin mixing conduc-tance before. So the calculated damping constant and effective spin mixing conductance of Ar þetched samples may contain some interfa- cial scattering component, such as electron-magnon scattering causedby the transition from Fe 3þto Fe2þeven Fe0.A sm e n t i o n e db e f o r e , this giant inhibition of spin injection efficiency is undesirable and we propose a chemical route to recover interfacial spin injection effi- ciency. YIG samples after the Arþion etch for 600 s were immersed in FIG. 3. (a) Schematic of the inverse spin Hall effect measurement. (b) Inverse spin Hall voltage of the YIG/Pt(10 nm) sample, for which the YIG surface was bomb arded by the Arþion for 600 s before the deposition of the Pt layer. (c) Inverse spin Hall voltage as a function of H-H Rfor the YIG/Pt(10 nm) samples with different interface treatments. (d) Extracted inverse spin Hall voltage for the YIG/Pt(10 nm) samples.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 042406 (2021); doi: 10.1063/5.0035640 118, 042406-4 Published under license by AIP Publishinga mixture of concentrated H 2SO4and 30% H 2O2of 1:1 volume ratio for 30 min. Then, Pt layers were deposited on the recovered YIG sam-ples. The inverse spin Hall voltage was found to be 11.6 lV. Strong oxidant treatment helped in recovering the spin injection efficiency in the interface region, and the potential changes in the chemical state caught our attention. It is important to understand the spin injection efficiency reduc- tion due to Ar þion etching and subsequent recovery by strongoxidant treatment. To clarify the transformation of the interfacial chemical state during the spin injection efficiency reduced by Arþion etch and then recovered by strong oxidant treatment, we investigated the Raman spectrum and x-ray photoelectron spectroscopy (XPS) ofthe YIG/Pt (2 nm) samples. Raman spectroscopy and XPS have highsurface sensitivity, nondestructivity, and the ability to obtain chemical state information, which makes them powerful tools for surface analysis. However, the effective attenuation ranges are 4–6 nm in FIG. 4. (a) Raman spectra of YIG/Pt (2 nm) samples with different Arþion etch times and recovery treatments. (b) Extracted intensities of Raman modes at 229 cm/C01264 cm/C01, and 339 cm/C01. (c) The Fe 2p XPS spectra of Arþion etched and recovered YIG/Pt (2 nm) samples. (d) and (e) Peak decomposition of Fe 2p XPS spectra for the Arþion etch of 600s and interface chemical recovered YIG/Pt (2 nm) samples. (f) Fe3þ/Fe2þratio as a function of different interface treatments.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 042406 (2021); doi: 10.1063/5.0035640 118, 042406-5 Published under license by AIP PublishingRaman spectroscopy and XPS. So we can investigate the interface chemistry of YIG/Pt bilayers. The Raman spectra of YIG/Pt (2 nm) samples with different Arþion etch times and recovery treatments are shown in Fig. 4(a) . The Raman shift was found to be in the range from 100 to 800 cm/C01. There are thirteen Raman modes in this range. It should be pointed out that these Raman modes are all associated withFe 3þions as Fe2þions do not show any Raman modes.31,32Thus, one can determine the content of Fe3þions in the interface region by com- paring the intensity of Raman modes. We extracted the intensities of Raman modes at 229 cm/C01, 264 cm/C01, and 339 cm/C01as shown in Fig. 4(b) . The intensities of these modes kept falling when the Arþion etch time is increased and recovered after the chemical recovery pro- cess. We believe that the Fe3þion content has the same variation (fall and then rise) trend. To further investigate the changes in the Fe ion valence state in the interface region, the XPS measurement was employed on the same YIG/Pt (2 nm) samples. The Fe 2p XPS spectra of Arþion etched and recovered samples are shown in Fig. 4(c) . The Fe 2p core level splits into 2p 1/2and 2p 3/2components. The binding energies range from 737 eV to 700 eV. Here, Fe 2p 3/2is considered for the determination of the valence state and its content. The binding energy of the Fe3þion (711.4 eV) is higher than that of the Fe2þion (709.3 eV).33The Fe 2p3/2peak moves toward lower binding energy with the increase in the Arþion etch time and returns after the chemical process, which means ac h a n g ei nt h eF e3þ/Fe2þr a t i o .T h eF e2 p 3/2band was divided into two subbands corresponding to the Fe3þion and Fe2þion by Lorentzian-Gaussian curve fitting, and the peak area ratio was used for quantitative determination of the Fe3þ/Fe2þratio as shown in Figs. 4(d) and4(e).W ee x t r a c t e dt h eF e3þ/Fe2þratio as shown in Fig. 4(f) . Arþion etches lead to the decrease in the Fe3þ/Fe2þratio and then recovery after the chemical process. Thus, it may be concluded that the decrease in the Fe3þion content at the interface is the origin of the decrease in spin injection efficiency. As we all know, the magnetic moment of YIG originates from the local Fe3þion.34The magnetic moment of the Fe3þion (5 lB)i ss t r o n g e rt h a nt h a to ft h eF e2þion (4lB). A split-off peak at 700 eV as shown in Fig. 4(d) may correspond to the metallic state of Fe, which disappears after chemical recovery as shown in Fig. 4(e) . This will not affect our conclusion as the peak area of this peak is relatively small compared to that of the iron ion and Fe0 holds a lower magnetic moment of 2.2 lB. The decreasing Fe3þcon- centration means that the interfacial magnetic moment will reduce. As predicted by theoretical calculations, the spin injection efficiency was correlated with interfacial magnetization.35Arþion etches changed the Fe3þion into the Fe2þion even Fe0and reduced the interfacial magnetic moment. Strong oxidant treatment would reverse this pro- cess and recover the spin injection efficiency. In summary, we have systematically studied the effects of argon ion (Arþ) bombardment on the spin pumping and inverse spin Hall effect in YIG/Pt heterostructures. The interfacial chemical state was investigated by Raman spectroscopy and XPS. Arþion bombardment leads to the conversion of the Fe3þion into the Fe2þ ion and reduces the spin injection efficiency. Moreover, we have developed a chemical method to recover interface magnetic moment and spin injection efficiency. Our findings are helpful forthe optimization of spintronic device processing technology and the understanding of the importance of interface properties in spin injection.See the supplementary material for the details about (i) whether the oxidizing solution will corrode the YIG surface, (ii) the comparisonof AFM and TEM images of YIG/Pt with and without Ar þetching, and (iii) the damping constants for bare YIG with different surfacetreatments. This work was supported by the National Key Research and Development Plan under Grant No. 2016YFA0300801, the NationalNatural Science Foundation of China under Grant Nos. 51702042,61734002, 61571079, and 51672007, the National Key ScientificInstrument and Equipment Development, under Project No.51827802, and the Sichuan Science and Technology Support Projectunder Grant Nos. 2017JY0002 and 2017GZ0111. DATA AVAILABILITY The data that support the findings of this study are available within this article and its supplementary material . 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5.0034232.pdf

Appl. Phys. Lett. 118, 042404 (2021); https://doi.org/10.1063/5.0034232 118, 042404 © 2021 Author(s).Magnetocrystalline anisotropy correlated negative anisotropic magnetoresistance in epitaxial Fe30Co70 thin films Cite as: Appl. Phys. Lett. 118, 042404 (2021); https://doi.org/10.1063/5.0034232 Submitted: 21 October 2020 . Accepted: 12 January 2021 . Published Online: 26 January 2021 Yu Miao , Dezheng Yang , Lei Jia , Xiaolin Li , Shuanglong Yang , Cunxu Gao , and Desheng Xue ARTICLES YOU MAY BE INTERESTED IN Tunable spin–orbit torque efficiency in in-plane and perpendicular magnetized [Pt/Co] n multilayer Applied Physics Letters 118, 042405 (2021); https://doi.org/10.1063/5.0034917 Exceeding 400% tunnel magnetoresistance at room temperature in epitaxial Fe/MgO/Fe(001) spin-valve-type magnetic tunnel junctions Applied Physics Letters 118, 042411 (2021); https://doi.org/10.1063/5.0037972 Interfacial chemical states and recoverable spin pumping in YIG/Pt Applied Physics Letters 118, 042406 (2021); https://doi.org/10.1063/5.0035640Magnetocrystalline anisotropy correlated negative anisotropic magnetoresistance in epitaxial Fe30Co70thin films Cite as: Appl. Phys. Lett. 118, 042404 (2021); doi: 10.1063/5.0034232 Submitted: 21 October 2020 .Accepted: 12 January 2021 . Published Online: 26 January 2021 YuMiao, Dezheng Yang, LeiJia,Xiaolin Li,Shuanglong Yang, Cunxu Gao,a) and Desheng Xueb) AFFILIATIONS Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education, Lanzhou University, 730000 Lanzhou, People’s Republic of China a)Author to whom correspondence should be addressed: gaocunx@lzu.edu.cn b)xueds@lzu.edu.cn ABSTRACT We report on the magnetoresistance in different crystallographic directions of epitaxial ferromagnetic Fe 30Co70thin films with magnetization rotated in the film plane. A negative single crystal anisotropic magnetoresistance (SCAMR) is found when the current is alongthe easy magnetization axis [110], and the SCAMR can be tuned to the conventional positive one when the current flows along the hardmagnetization axis [100]. This finding is explained comprehensively by a magnetocrystalline anisotropy (MCA) symmetry-adapted modelexpanded along the easy magnetization direction, with which the SCAMR can be represented as a MCA-independent conventional term cos 2u Mand a series of MCA-dependent terms cos 2 nuA(n/C211). The results show that the MCA-dependent twofold term contributes to the negative SCAMR, which cannot be used as a fingerprint of the half-metallicity. Our finding provides an approach to understand and designthe magnetoresistance with ferromagnets by MCA. Published under license by AIP Publishing. https://doi.org/10.1063/5.0034232 Since the discovery of longitudinal and transverse magnetization effects on the resistance of ferromagnetic materials founded in a rect- angular polycrystalline sheet of Fe and Ni by Lord Kelvin in 1856, 1the twofold symmetry characteristic of the anisotropic magnetoresistance(AMR) in polycrystalline ferromagnets is established. 2Conventionally, the resistivity qJin the current density Jdirection of the polycrystalline ferromagnets is represented as3 qJ/C0q? q?¼qk/C0q? q?cos2uM; (1) where qkðq?Þis the resistivity when the magnetization Mis parallel (perpendicular) to JanduMis the angle of Mfrom the Jdirection. In the polycrystalline ferromagnets, most of the metals and alloys exhibita positive AMR with q k/C0q?>0,3–5and a negative AMR was also observed in several 3 dmetals and alloys containing Cr (Mo, Si)6or Ir,7 Co2MnAl 1/C0xSixHeusler alloys,8and textured Fe 4N metallic compounds.9The anisotropy effect of the current direction- independent AMR is a consequence of s–dscattering with spin–orbit interaction (SOI),10–12which is called conventional AMR. Based on the two-current model proposed by Campbell–Fert–Pomeroy,13,14theeffect of spin-related s–dscattering on AMR was supposed.15–19The sign of conventional AMR and its reversal can be reasonably under- stood with spin-polarized conduction states and localized dstates with SOI. That is, when the dominant s–dscattering process was s"! d#(s"! d")o rs#! d"(s#! d#), the sign tended to be positive (negative).20 Magnetoresistance of single crystal Fe in the longitudinal mag- netic field was studied decades later.21A crystallographic direction- dependent AMR resistance was observed in bulk22,23and thin films of traditional ferromagnetic metals,24–28ferrimagnetic oxides,29and anti- ferromagnetic material CuMnAs,30and of half-metallic alloys,31 oxides,32and nitrides,33as well as magnetic semiconductors.34,35Two significant characteristics of the single crystal AMR (SCAMR), whichis beyond the cos 2 u Mpolycrystalline symmetry, are revealed clearly. One is higher order terms occur in addition to the twofold term, and the other is a phase-shift of the AMR terms exists in different crystallo-graphic directions. A symmetry-adapted phenomenological theory of the AMR on current and magnetization orientation with respect to the crystal axes was given by D €oring. 36T h eS C A M Rc a nb ed i v i d e di n t oa crystalline-independent term and a series of crystalline-dependent Appl. Phys. Lett. 118, 042404 (2021); doi: 10.1063/5.0034232 118, 042404-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplterms according to the symmetry of crystal.35For the cubic structure, the longitudinal SCAMR can be represented as27,37 qJ¼C0 0þC0 2cos 2aMcos 2aJþS0 2sin 2aMsin 2aJ þC0 4cos 4aMþ/C1/C1/C1 ; (2) where C0 i(i¼0;2;4;…) and S0 2are the coefficients related to the crystal and aJ(aM) is the angle between J(M) and the reference crystal axis. The SCAMR is frequently used to successfully describe the AMR of the single crystal,38,39polycrystal,40as well as textured 3 d-ferromag- netic layers in sandwich.41 However, no bridge between s–dscattering with SOI and the symmetry-adapted phenomenological theory indicates that an insightinto the nature of crystalline-dependent terms is still a challenge. For adefined SCAMR experimental result, the sign and phase of thecrystalline-dependent terms can be differently described by thephenomenological theory with different reference crystal axes. 42On the contrary, if the reference crystal axis along the current directionwas selected, the phase-shift curve of the SCAMR 25,26cannot be fitted well by the phenomenological theory with one set of coefficients. The fact indicates that the selection of the reference crystal axis should belimited rather than free. From this point, only the correct referencecrystal axis selected in the expansion of the phenomenological theory,the conventional AMR resulted from the magnetization can be derivedfrom the SCAMR in which the twofold symmetry term comes fromboth crystal-dependent and crystal-independent terms. Experimentally,the SCAMR with the same crystal structure revealed that the crystalline-dependent terms are related to the temperature 43–46and composition.47Both the sign and symmetry changes of the SCAMR in Co2MnGa,48GaMnAs34and Ni xFe4/C0xN33,49indicate that the reference crystal axis should be related to the anisotropic properties. Recently, theinterface spin–orbit fields significantly change the symmetry ofSCAMR in quasi-two-dimensional Fe/GaAs. 50Considering that the SCAMR comes from s–dscattering with SOI (Refs. 10–12 and20), and the magnetocrystalline anisotropy (MCA) originates from the SOI (Ref. 51), the SCAMR can be understood based on the symmetry of MCA rather than the crystalline symmetry. If so, the negative SCAMRcan be observed in a certain crystallographic direction of ferromagneticmetal alloy, which has a well-known positive traditional AMR. Here, we report the SCAMR of the epitaxial single crystal in dif- ferent crystallographic directions of the body-centered cubic (bcc)Fe 30Co70thin films on MgO(001). It is found that the sign of SCAMR can be changed from negative to positive through the phase-shift grad-ually when the current is applied from the [110] to [100] direction inFe 30Co70. Considering the MCA-dependent resistivity tensor, the SCAMR can be divided into a MCA-independent term and a series of MCA-dependent terms. The former corresponds to the traditional AMR with a positive cos 2 uMsymmetry and that in the later terms contribute to the negative AMR. The MCA-related properties are con-sistent with the results of ferromagnetic resonance (FMR). Theseresults provide ideas for researching and regulating positive and nega-tive AMR in single crystalline magnetic metals under consideringMCA. High quality epitaxial Fe 30Co70thin films were fabricated and monitored by the in situ reflection high energy electron diffraction pattern ( supplementary material I). A bcc structure was characterized by high resolution x-ray diffraction, and a lattice parameter of 2.851 A ˚is calculated by the diffraction peak at 32 :5/C14of the Fe 30Co70(002) plane ( supplementary material II). Hysteresis loops were measured at room temperature by a vibrating sample magnetometer. The easy magnetization axis (EA) and hard magnetization axis (HA) along[110] and [100] directions are determined by the remanence of about1.0 and 0.7, respectively. Then, the thin films were patterned into a designed current-direction-dependent Hall bar by photolithography and ion beam etching techniques for electrical transport measure-ments. All of the Hall bars along the different crystallographic direc-tions have a transverse width of 50 lm and a longitudinal length of 1000lm. Both SCAMR and FMR were performed at room tempera- ture by a physical property measurement system (PPMS) equipped with a motorized sample rotator and coplanar waveguide FMR mea-surement system, respectively. In-plane hysteresis loops are shown in Fig. 1 . The loop measured in the [110] direction has a reduced remanence of about 1.0, which indicates that the EA is along [110]. With the decreasing field, the loop measured in the [100] direction (hard axis) decreases almost linearlyand has a reduced remanence of about 0.7, which is consistent withthe in-plane coherent rotation magnetization process under cubicMCA. 26With analyzing the loop in the [100] direction,52the anisot- ropy field of about 50 mT can be obtained. Figure 2(a) shows the schematic diagram of the SCAMR mea- surement, where the current flows along the longitudinal directionof each Hall bar. The angles of magnetization Mand the EA with respect to the current density Jareu ManduEA, respectively. uM¼b/C0a,w h e r e aandbare the angles of magnetization Mand the current Jwith respect to the applied magnetic field H, respec- tively. Figure 2(b) shows the longitudinal resistivity qJwith uEA¼0/C14;15/C14;30/C14,a n d4 5/C14measured as a function of bin the film plane under an applied magnetic field of 6 T. The significant phase-shift appears in the qJ/C24bcurves when the direction of the EA changes from uEA¼45/C14touEA¼0/C14.I ts h o w sat y p i c a lp o s i - tive SCAMR as Jk[100] at uEA¼45/C14, which is the characteristic of traditional 3 dmagnetic metal. However, the negative SCAMR asJk[110] at uEA¼0/C14occurs. Those curves with phase-shift obviously cannot be described by the conventional AMR as shownin Eq. (1). FIG. 1. Hysteresis loops with the applied magnetic field in [110] (black line) and [100] (red line) crystallographic directions.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 042404 (2021); doi: 10.1063/5.0034232 118, 042404-2 Published under license by AIP PublishingWhen the EA is selected as the reference crystal axis, the SCAMR in the (001) crystal plane of the bcc structure by the same process ofthe phenomenological theory 35,36,42can be represented as qJðuM;uEAÞ¼q0þq2cos 2uMþqC 2cos 2ð2uEAþuMÞ þqC 4cos 4ðuEAþuMÞþ/C1/C1/C1 ; (3) where q0¼C0 0;q2¼ðC0 2þS0 2Þ=2;qC 2¼ðC0 2/C0S0 2Þ=2;qC 4¼C0 4 compared with Eq. (2).q2,qC 2;qC 4are expressed as the noncrystalline AMR, crossed noncrystalline/crystalline, and cubic crystalline coeffi- cients (Refs. 35and53), respectively. It clearly shows that the second term is MCA-independent, which represents the conventional AMR.The terms higher than the second term are MCA-dependent, whichreveal the main difference of SCAMR from the conventional AMRand will disappear as the MCA is averaged out in polycrystalline. Certainly, the resistivity q Jwith uEA¼0/C14;15/C14;30/C14,a n d4 5/C14 shown in Fig. 2(b) can be well-fitted with Eq. (3)when the applied magnetic field is high enough to satisfy uM¼b,w h e r e q2¼4:185 /C210/C02lXcm and qC 2¼/C07:337/C210/C02lXcm. The fitting process is similar to other reference crystal axes.3,37,42However, three aspects are significant when we select the EA as the reference crystal axisrather than the arbitrary crystallographic direction. First, it is reason- able in mathematics. If we regard the reference crystal axis as the cur-rent direction, Eq. (2) will be q JðuM;0/C14Þ¼C00þC20cos2uM þC40cos4uMþ/C1/C1/C1 . This means there cannot be any phase-shift when Jis applied along the reference axis, which is not the case as shown in Fig. 2(b) . Second, it is more reasonable in physics. The coefficients of Eq.(3)obtained from any one of curves shown in Fig. 2(b) can be used to fit others well. The last is the selection of the EA as the refer-ence axis constructs the connection between the phenomenologicaltheory and the SOI. Equation (3)is obtained by the expansion of q ijð^MÞ, which is quite similar to that of the free energy density of MCA.51A tt h es a m et i m e ,t h eS O Ii st h ei m p o r t a n ti s s u eo ft h e MCA.51 Actually, the validity of the expansion of qijð^MÞrather than qijð^HÞis not sure because most of the measurements are performed in a high applied magnetic field, which satisfy MkH.I fa nu n s a t u r a t e d field is applied, the direction of Mrather than Hchanges. It is reason- able to represent the variation of SCAMR by the expansion of qijð^MÞ rather than qijð^HÞ, which has been used to determine the magnetic anisotropy.54,55Figure 3(a) shows the results of unsaturated SCAMR measured at 100 mT under different current directions relative to theEA. The significant deviation from the saturated result shown in Fig. 2(b)reveals the validity of our suggestion. When we try to fit the experimental results in Fig. 3(a) with Eq. (3), it is necessary to obtain the angle of u M. Considering the Zeeman energy and cubic MCA energy, under the coherent rotation model, thetotal free energy density of the system can be expressed as F¼/C0l 0MsHcosaþ1 4Ksin22ðuEAþb/C0aÞ; (4) where l0is the vacuum permeability, Msis the saturation magnetiza- tion, and Kis the cubic MCA constant. uM¼b/C0acan be derived by @F=@a¼0 with small angle approximation uMub/C0sin 4ðuEAþbÞ 4hþcos 4ðuEAþbÞ ½/C138; (5) where h¼H=HKand HK¼2K=l0Msi st h ee f f e c t i v ec u b i cM C A field. This approach has been successfully used to analyze the conven-tional AMR. 52,54,55 Figures 3(b) and3(c)show the fitting results by Eq. (3)combined with Eq. (5)when Jis along [110] and [100] directions, respectively. The 100 mT magnetic field ensures the validity of Eq. (5), under which theHis larger than twice of the anisotropy field of about 50 mT of the Fe30Co70thin films and the magnetization process is coherent rotation rather than domain-wall motion or nucleation.52The purple curves in Figs. 3(b) and3(c)represent the second term in Eq. (3), and the blue and green curves represent the third and fourth terms, respectively.Adopting the parameters q 2¼4:185/C210/C02lXcm and qC 2¼/C07:337 /C210/C02lXcm under 6 T, it can be seen that the experimental data are fitted by the total results well. At the same time, qC 4¼4:460 /C210/C03lXcm and HK¼38 mT along the EA are obtained. It shows that the magnitude of fourfold symmetry term qC 4is almost one order of smaller than that of the twofold symmetry terms q2andqC 2. Therefore, the fourfold symmetry of SCAMR is not obvious, in gen-eral, unless the two terms of the twofold symmetry are approxi-mately equal and signs are opposite. In the Fe 30Co70films, the MCA-dependent twofold symmetry term qC 2is larger than q2, which FIG. 2. (a) Measurement schematic of the SCAMR. (b) SCAMR as a function of b in the film plane under 6 T in different current directions with respect to the EA. The dots are the experimental data. The solid lines are the fitted curves by Eq. (3).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 042404 (2021); doi: 10.1063/5.0034232 118, 042404-3 Published under license by AIP Publishingultimately dominates the phase-shift of SCAMR and leads to a nega- tive SCAMR in the [110] crystallographic direction. This also causesthe SCAMR ratio to have an order of magnitude difference in differ-ent crystallographic directions as reported in other systems, 26,31,32,56 but there is no negative SCAMR and significant phase-shift reportedin ferromagnetic metals like Fe 26and Fe 100/C0xCox(x/C2065).56 In order to verify the correctness of the effective MCA field obtained by fitting the SCAMR, Fig. 4 shows the frequency conversionFMR results with the applied magnetic field along the EA [110] direc- tion. The resonance field Hrcan be obtained from the curves with dif- ferent frequencies. It is known that a more general expression of theresonance frequency derived by Baselgia et al. is 57 /C18x c/C192 ¼1 M2 s/C20@2F @h21 sin2h@2F @u2þcosh sinh@F @u/C18/C19 /C01 sinh@2F @h@u/C0cosh sin2h@F @u/C18/C19 /C21 ; (6) where xis the resonance frequency, cis the gyromagnetic ratio, andhanduare the polar angle and azimuthal angle with respect to the EA direction, respectively. With h¼90/C14andu¼0/C14,t h e resonance frequency of a thin film with the EA along the xdirec- tion in the film plane can be obtained as an improved Kittel for- mula x¼l0cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðMsþHKþHrÞðHKþHrÞp .F o rF e 30Co70films, HK¼40 mT at the EA is worked out by fitting the resonance field dependence of resonance frequency. This result corresponds tothat obtained from the unsaturated SCAMR and reveals the rea-sonableness of Eq. (3)where the EA is the reference crystal axis. At this point, we rewrite the SCAMR as Dq J¼qJðuM;uEAÞ/C0qJð0;uEAÞ: (7) In Eq. (7), the influence of crystalline resistance and device difference is ruled out as shown in Fig. 5(a) , where the SCAMR of magnetization and the MCA is highlighted. It clearly shows the significant phase- shift of SCAMR and the sign change from negative AMR along [110] to positive AMR along [100]. Figure 5(b) shows the SCAMR at uEA¼30/C14under different applied magnetic fields. It is found that there is almost no change with the increase in the field from 1 T to 9 T,which indicates further that the influence of the applied magnetic fieldon our results can be ignored. Correspondingly, with temperaturedecreasing, the H KandqC 2increase with a similar trend, which further proves the correlation between MCA and SCAMR ( supplementary material III). To conclude, we investigated the SCAMR of Fe 30Co70thin films on MgO(001) with current applied along different crystallographic FIG. 3. (a) SCAMR as a function of bunder 100 mT in different current directions with respect to the easy axis. The fitting results by Eq. (3)at (b) uEA¼0/C14and (c) uEA¼45/C14. The open triangles are the experimental data. The purple solid lines represent MCA-independent SCAMR, while the blue and green solid lines represent twofold and fourfold symmetry MCA-dependent AMR, respectively. FIG. 4. The frequency conversion FMR spectra with the magnetic field applied along [110] crystallographic directions.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 042404 (2021); doi: 10.1063/5.0034232 118, 042404-4 Published under license by AIP Publishingdirections. The SCAMR accompanied a huge phase-shift until its sign changes from negative to positive with the direction of the currentapplied from [110] to [100]. Under the MCA symmetry-adapted model, a more general SCAMR expression is derived by using the EA reference axis and is composed of a MCA-independent conventionalterm cos 2 u Mand a series of MCA-dependent terms cos 2 nuA (n/C211), which describes the experimental results well under whatever the high or low magnetic field. We also verified the correctness of the SCAMR, in which the MCA-dependent twofold term contributes the negative AMR, which cannot be used as a fingerprint of the half-metallicity. This MCA symmetry-adapted SCAMR can provide anapproach for the design and research of the magnetoresistance in thespin electronic devices, and the influence of symmetry-adapted MCA on the electrical transport properties can also be extended to analyze various Hall effects. See the supplementary material for the sample growth, structural characterization, and the temperature dependence of the SCAMR ofthe Fe 30Co70films. This work was supported by the National Natural Science Foundation of China (Grant Nos. 91963201, 11674143, and11674141); 111 Project (B20063); and PCSIRT (Grant No. IRT 16R35).DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material . REFERENCES 1W. Thomson, Proc. R. Soc. 8, 546 (1857). 2E. Englert, Ann. 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5.0024516.pdf

Appl. Phys. Lett. 118, 052404 (2021); https://doi.org/10.1063/5.0024516 118, 052404 © 2021 Author(s).Modulating the transport property of flexible La0.67Ca0.33MnO3 thin film by mechanical bending Cite as: Appl. Phys. Lett. 118, 052404 (2021); https://doi.org/10.1063/5.0024516 Submitted: 10 August 2020 . Accepted: 15 January 2021 . Published Online: 01 February 2021 Wentao Hua , Lu Lu , Lvkang Shen , Jing Jin , He Wang , Ming Liu , Chunrui Ma , and Chun-Lin Jia ARTICLES YOU MAY BE INTERESTED IN Electrically tunable inverse spin Hall effect in SrIrO 3/Pb(Mg 1/3Nb2/3)0.7 Ti0.3O3 heterostructures through interface strain coupling Applied Physics Letters 118, 052904 (2021); https://doi.org/10.1063/5.0027125 Strain-controlled electrical and magnetic properties of SrRuO 3 thin films with Sr 3Al2O6 buffer layers Applied Physics Letters 118, 072407 (2021); https://doi.org/10.1063/5.0038588 Coexistence of different magnetic ordering in thin films of SrMnO 3 studied by spin transport Applied Physics Letters 118, 052407 (2021); https://doi.org/10.1063/5.0035948Modulating the transport property of flexible La0.67Ca0.33MnO 3thin film by mechanical bending Cite as: Appl. Phys. Lett. 118, 052404 (2021); doi: 10.1063/5.0024516 Submitted: 10 August 2020 .Accepted: 15 January 2021 . Published Online: 1 February 2021 Wentao Hua,1LuLu,2Lvkang Shen,2Jing Jin,2 HeWang,2Ming Liu,2 Chunrui Ma,1,a) and Chun-Lin Jia2,3 AFFILIATIONS 1State Key Laboratory for Mechanical Behaviour of Materials and School of Materials Science and Engineering, Xi’an Jiaotong University, Xi’an 710049, China 2School of Microelectronics, Xi’an Jiaotong University, Xi’an 710049, China 3Ernst Ruska Centre for Microscopy and Spectroscopy with Electrons, Forschungszentrum J €ulich, D-52425 J €ulich, Germany a)Author to whom correspondence should be addressed: chunrui.ma@mail.xjtu.edu.cn ABSTRACT Flexible epitaxial La 0.67Ca0.33MnO 3(LCMO) thin films are fabricated on an SrTiO 3buffered (001)-oriented fluorophlogopite substrate. The metal-to-insulator transition tends toward lower temperature when subjected to mechanical bending. Moreover, the transport behavior of the bent LCMO films in the insulating region follows the variable range hopping model and the resistivity increases with the reduction in thebending curvature radii because the applied strain aggravates the distortion of the LCMO crystal structure, decreases the hopping distance,and, hence, impedes the transport of charge carriers. The resistivity change induced by the mechanical bending can go up to 10 4% at 100 K and 105% at 10 K. Such a large resistivity change makes the flexible LCMO thin film promising as a mechanical-bending switch device at low temperature. Published under license by AIP Publishing. https://doi.org/10.1063/5.0024516 Thin films of mixed-valence perovskite manganites La 1/C0xAx MnO 3(A¼Sr, Ca, Ba) have attracted worldwide attention due to the abundant distinct physical phenomena, such as the colossal magneto-resistance effect, the transition from the paramagnetic to ferromag- netic state, the metal to insulator transition (MIT), phase separation, and so on. 1–4These ponderable physical mechanisms make it poten- tially applicable in the area of magnetic random-access memories,magnetoelectric devices, tunneling magnetoresistance devices, andswitching devices. 5–7Among these materials, La 0.67Ca0.33MnO 3 (LCMO) has become one of the mostly invested materials due to its complex pattern of spin, orbital, charge, and Jahn–Teller lattice distor- tions,8–10which induces a variety of exotic phenomena, such as a huge magnetoresistance effect. Moreover, as the representative of narrowbandwidth manganites with robust Jahn–Teller coupling, its electricaltransport is very sensitive to the external environment and exhibitsvarious novel physical phenomena. 11,12A small external perturbation can influence the Mn3þ/Mn4þexchange effect13and the Jahn–Teller distortion14and, thus, generate a large change in transport properties. So far, many researchers have devoted effort to investigate the effect ofthe external strain on its physical mechanism. One way was to takeadvantage of the lattice mismatch between the film and the substrateto build various interface strains. 15,16For example, Helali et al.fabricated LCMO on LaAlO 3,S r T i O 3,a n d( L a A l O 3)0.3–(SrAlTaO 6)0.7 to generate compressive strain, tensile strain, and negligible tensile strain, respectively. It is found that both tensile and compressive straincan increase the thermal activation energy and the variable range hop-ping (VRH) characteristic temperature in the paramagnetic state. 17 Another way to study the strain effect on transport behavior was tofabricate several samples with different thicknesses since the interfacestrain induced by the lattice mismatch between the film and the sub-strate is gradually released by formation of dislocations or defects withthe increasing film thickness. 18,19Wang et al. found that a dead layer about 6 nm thick was formed in LCMO epitaxial thin films and theconductivity of the films linearly increased with the film thickness due to the reduced effects of the dead layer. 20However, the properties of the films are affected not only by strain but also by the microstructureinduced by different growth modes and by crystalline quality on differ-ent substrates. Therefore, it is necessary to find a way to explore theinfluences of pure strain on the film properties. A flexible thin film not only is the basis of wearable/portable elec- tronic devices but also provides a test sample for modulating orexploring the novel physical properties by applying stretching strain ormechanical bending strain. Until now, two methods have been used tofabricate flexible epitaxial function oxide thin films. One is to transfer Appl. Phys. Lett. 118, 052404 (2021); doi: 10.1063/5.0024516 118, 052404-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplthe desired thin film from a rigid substrate to a flexible substrate through etching a buffer layer between the desired thin film and the rigid substrate.20Another is to directly fabricate the desired thin film on a flexible mica substrate.21Recently, Hong et al. investigated the effect of extreme tensile strain on La 0.7Ca0.3MnO 3membranes fabri- cated through transferring the epitaxial La 0.7Ca0.3MnO 3thin film onto the polyimide sheet and found that uniaxial and biaxial tensile strain benefits the formation of the antiferromagnetic insulating state and suppression of ferromagnetic metal.22However, few research studies contributed to investigation of the effects of mechanical bending strain on the transport properties of the LCMO thin films. Compared to the transferring method, the direct fabrication method can effectively avoid the contamination induced during the transferring process. In the present paper, the flexible epitaxial (111) LCMO thin films were fabricated on the SrTiO 3(STO)-buffered (001) fluorophlogopite single-crystalline substrates (F-Mica). The effects of mechanical bend-ing strain on the transport properties of the LCMO thin films were explored with and without the magnetic field. For preparation of the flexible epitaxial LCMO thin films, an epitaxial STO buffer layer was first deposited on the (001) F-Mica sub- strate at an oxygen pressure of 50 mTorr via a 248 nm KrF excimer pulsed laser deposition (PLD) system with the laser energy of 500 mJ. Then, the oxygen pressure and the laser energy were increased to 250mTorr and 650 mJ to grow the LCMO thin film on top of the STO buffer layer. The repetition rate of the laser was fixed at 5 Hz for the growth of both STO and LCMO. After deposition, the thin films were annealed at 1000 /C14Cf o r1 5m i na ta no x y g e np r e s s u r eo f2 0 0T o r ra n d then cooled down to room temperature at a rate of 25/C14C per min. The crystallinity and quality of the epitaxial LCMO thin films were charac- terized by high-resolution X-ray diffraction (HRXRD) using a PANalytical X’Pert MRD system. Reciprocal space mappings (RSMs) were conducted to determine the crystal structure by using a scanning line detector. The microstructure of the films was investigated by transmission electron microscopy (TEM). The transport properties were measured by the Van der Pauw method using a physical property measurement system (PPMS). The flexible film/F-Mica was peeled offmechanically from the substrate bulk with the assistance of the polyi- mide type. The bending states of the LCMO thin films were realized by attaching the films to an insulating double-faced adhesive tape on the copper molds with the bending curvature radii of 25 mm, 12.5 mm, and 7.5 mm. Figure 1(a) shows the XRD h–2hscanning pattern of the LCMO and STO thin film system on the F-Mica substrate. Only the (111) and(222) peaks of LCMO and STO and the {00 l} peaks of F-Mica can be seen, showing that the films grow with the {111} atomic planes parallel to the (001) plane of the F-Mica substrate. In order to investigate the in-plane orientation relationship between the film and the F-Mica sub- strate, uscans were performed around {002} reflections of STO and LCMO and {202} reflections of the F-Mica substrate as shown in Fig. 1(b) . It is clearly seen that the F-Mica peaks show threefold symmetry and the films exhibit a sixfold symmetry, which is indicative o ft h ei n - p l a n er e l a t i o n s h i po f[ 1 /C2210] LCMO //[1/C2210] STO//[010] F-Mica . According to the out-of-plane and in-plane orientation relationship between the films and the substrates, it can be derived that the fabri- cated films are epitaxial. This epitaxial relationship is also confirmed by the selected-area electron diffraction (SAED) pattern of the sample recorded along the [1 /C2210] zone axis of F-Mica, as shown in the inset ofFig. 1(c) . The thicknesses of the LCMO and STO layers are around 22 nm and 13 nm, respectively, which were measured from the cross- sectional TEM image [ Fig. 1(c) ]. From the symmetric RSMs taken around the (111) reflections of the multilayer and the (001) reflection of the F-Mica substrate in Fig. 1(d) , the (111) plane spacings of the LCMO and STO layers are measured to be 2.218 A ˚and 2.239 A ˚, respectively. In comparison with the corresponding (111) plane spacings (2.227 A ˚for LCMO and 2.255 A ˚for STO) of bulk materials,23 t h eL C M Oa n dS T Ot h i nfi l m su n d e r g oc o m p r e s s i v es t r a i n so f /C240.40% and /C240.71%, respectively, along the out-of-plane direction. From the asymmetric RSMs taken around the (213) reflections of the LCMO films and the (228) reflections of the F-Mica substrates, as shown in Fig. 1(e) , plane spacings of 1.884 A ˚and 1.887 A ˚are measured along the in-plane [1 /C2210] directions, respectively, for the LCMO and STO layers, which are larger than those of the LCMO (1.819 A ˚)a n d the STO (1.841 A ˚) bulk materials, leading to in-plane tensile strains of /C243.57% in the LCMO layer and /C242.50% in the STO layer. Figures 2(a) and2(b) (linear scale shown in Figs. S2–S5) show the data of resistivity measured in a range of temperatures from 10 to 350 K at 0 T in tensile and compressive mechanical strain. A normal MIT process occurs for the film in the flat state and the transition tem- perature is about 204 K, which is lower than the value of bulk material (265 K), probably due to the in-plane tensile strain of 3.57% induced by the lattice mismatch between the films and the substrate.24The transition temperature tends to lower temperature in the mechanicalbending state since the mechanical bending-induced tensile and com- pressive strains are likely to distort the lattices of the film and, hence, hinder the transport of charge carriers in the film, making LCMO reach the insulator state earlier when it is warmed from 10 to 350 K. This phenomenon consists of the Monte Carlo simulation reported by Sen and Dagotto that LCMO gradually reaches an insulating charge- ordered phase from a charge disordered metallic phase as strain FIG. 1. (a) XRD h–2hpattern of the LCMO thin film and STO buffer layers on the (001) F-Mica substrate. (b) The uscans taken around the {202} reflections of F- Mica and {002} reflections of LCMO/STO, respectively. (c) The TEM image of across-section of the sample and the SAED patterns taken from the areas covering the interfaces between the films and the substrate. (d) RSMs taken around the symmetric (222) of the LCMO thin film. (e) RSMs taken around asymmetric (213)reflections of the F-Mica substrate.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 052404 (2021); doi: 10.1063/5.0024516 118, 052404-2 Published under license by AIP Publishingincreases.25From Fig. 2(c) , it can be seen that the MIT temperature increases to 268 K in the flat state, indicating that the magnetic fieldenhances the double exchange and postpone it to enter the insulatorstate. However, it is unexpected that the MIT disappears in the testtemperature range under the mechanical bending strain as shown inFigs. 2(c) and2(d). The abnormal phenomena probably can contribute to the slowly decreasing resistivity with the increase in temperature. It is known that the magnetic field will provide the energy to assistcharge carrier to transport in the film and reduce the resistivity ofLCMO (Fig. S6), while the mechanical bending strain prevents thecharge carrier to transport and increase its resistivity. The competitionbetween energy from the magnetic field and mechanical bending strain merges MIT. In order to understand the transport mechanism under the effect of extra mechanical strain, the measured resistivity inthe high temperature region was simulated using the thermal activa-tion model (Fig. S7), 23small-polaron hopping (Fig. S8),26and VRH model.27It is found that the VRH model fits the data very well, as shown in Fig. 3 , implying that the transport mechanism in the LCMO films is the charge carrier hopping between localized electronic states. In the VRH model, qTðÞ¼q0eðT0=TÞ1=4,w h e r e T0represents thecharacteristic temperature and is proportional to the slope of a plot of ln(q)v s T/C01/4.27It is known that the average nearest-neighbor hopping distance R¼9 8paNEðÞkThi1=4and the localization length 1 a¼ð171UmV kT0Þ1=3;where Vis the unit-cell volume per Mn ion, kis the Boltzmann constant, and Um¼3JH=2 is the splitting energy between the spin-up and spin-down egbands. JHis the coefficient of Hund’s rule coupling, and Um/C252 eV in manganites. N(E) is the available den- sity of states, where the charge carrier can hop to. Based on the calcula- tion of Viret et al.,t h e N(E) value is around 9 /C21026m/C03eV/C01.26Here, we assume the volume of unit-cell conservation under the strain.Hence, the hopping distance can be obtained and is summarized inTable I with the fit value of T 0. It is found that the hopping distance decreases with the increase in mechanical bending strain, no matter tensile or compressive strain, except the compressive strain of 0.8%,which is slightly larger than the value of the compressive strain of0.4%. T 0exhibits an opposite trend to the hopping distance. It is known that T0is proportional to the hopping energy from one FIG. 2. Transport properties of the flexible LCMO thin film under extra mechanical tensile and compressive strain induced by convex and concave bending without themagnetic field (a) and (b) and with the 9 T magnetic field (c) and (d). FIG. 3. Logarithm of the resistivity vs T/C01/4plot of the LCMO thin film under extra tensile and compressive strain induced by various bending curvature radii abovethe transition temperature at 0 T (a) and (b) and 9 T (c) and (d). TABLE I. The VRH characteristic temperature and average nearest-neighbor hopping distance R of different tensile or compressive strains derived from the re sistivity of LCMO as a function of temperature at 0 T and 9 T. q¼q0/C2eðT0 TÞ1 4Tensile Compressive 0T 9T 0T 9T Strain T 0(107K) R (A ˚)T 0(107K) R (A ˚)T 0(107K) R (A ˚)T 0(107K) R (A ˚) 0 6.82 12.30 1.4 14.03 6.82 12.30 1.4 14.03 0.4% 7.23 12.24 1.76 13.77 9.33 11.98 3.3 13.070.8% 9.47 11.96 2.57 13.34 9.08 12.02 2.46 13.391.33% 10.63 11.84 3.63 12.97 12.91 11.65 4.28 12.79Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 052404 (2021); doi: 10.1063/5.0024516 118, 052404-3 Published under license by AIP Publishinglocalized state to another.28,29The reduction in the hopping distance and the increased hopping energy indicate that the applied strain aggravates the distortion of the LCMO crystal structure and impedes the transport of the charge carrier and enhance its insulatorbehavior. 22 Figure 4 exhibits the resistivity change ratio ðqbent/C0qflatÞ=qflat /C2100%, where qbentandqflatare the resistivity of the film in the bending and flat state, as a function of temperature from 10 to 200 Kwith and without the magnetic field. It is found that the resistivitychange ratio increases from /C2410 2%a t2 0 0 Kt o /C24104%a t1 0 0 Ka n d then it slowly increases to /C24105%a t5 0 Ka n da l m o s tr e m a i n s unchanged until 10 K. The change ratio is even higher with the appli-cation of a magnetic field. As shown in Fig. 2 , the huge resistivity change without/with the magnetic field can contribute to the different scenario of q(T) at low temperature. The LCMO thin film remains in a metallic state in the absence of extra mechanical strain, while theapplication of mechanical strain triggers an insulating state. As thecalculation by Sen and Dagotto is based on Monte Carlo simulation, there is a transition in the LCMO from a metallic ground state at small strain to an insulating state at large strain. 25That is to say, with the increased strain, the Jahn–Teller distortion drives the double-exchange-induced metallic state gradually to become a charge-ordered insulating state. Therefore, the suppressed ferromagnetic metallic phase and the activation of the charge-ordered insulating state underthe mechanical strain promote this huge resistivity change. If the flatstate with low resistivity is set as the “on” state, the bending state with large resistivity is set as the “off” state. The resistivity change shows the huge change from the metal state to the insulator state in the LCMOthin films through the mechanical bending. High sensitivity makeso u rd e s i g n e dL C M Ot h i nfi l m sp r o m i s i n gf o ra p p l i c a t i o ni nfl e x i b l e resistivity switches at low temperature. In conclusion, flexible epitaxial (111)-oriented LCMO thin films with STO buffer layers were fabricated on the F-Mica substrate using apulsed laser deposition system. Both extra tensile and compressive mechanical strains make the MIT peaks of LCMO become subtle, and the transition temperature turns to lower temperature. Although theapplied magnetic field provides energy to excite the charge carrier hopping and increases the MIT temperature, the energy is still not enough to overcome electron localization, and thus, the film exhibitsinsulator behavior at low temperature. The resistivity change by extra strain can go up to at least 10 5% relative to that of no extra strain at 50 K since Jahn–Teller distortion tends to be enhanced by the extramechanical tensile or compressive strain in the LCMO thin film. This huge resistivity change at low temperature suggests that the designed flexible LCMO thin film can be used as a bending-induced resistivityswitch at low temperature. See the supplementary material for the bending strain calculation, details of transport properties under mechanical tensile and compres- sive strain in the linear coordinate system, and the thermal activation and small-polaron hopping model data fitting. This research was supported by the Natural Science Foundation of China (Nos. 51702255 and 51390472), the National “973” projects of China (Nos. 2015CB654903 and 2015CB654603),Shaanxi Natural Science Foundation No. 2019JM-068, and the Fundamental Research Funds for the Central Universities. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1S. Jin, T. H. Tiefel, M. McCormack, R. A. Fastnacht, R. Ramesh, and L. H. Chen, Science 264, 413–415 (1994). 2R. von Helmolt, J. Wecker, B. Holzapfel, L. Schultz, and K. Samwer, Phys. Rev. Lett. 71, 2331–2334 (1993). 3M. A. Subramanian, B. H. Toby, A. P. Ramirez, W. J. Marshall, A. W. Sleight, and G. H. Kwei, Science 273, 81–84 (1996). 4A. S. McLeod, J. Zhang, M. Q. Gu, F. Jin, G. Zhang, K. W. Post, X. G. Zhao, A. J. Millis, W. B. Wu, J. M. Rondinelli et al. ,Nat. Mater. 19, 397–404 (2020). 5Y. Lu, X. W. Li, G. Q. Gong, X. Gang, G. Xiao et al. ,Phys. Rev. B 54, R8357 (1996). 6T. Sun, J. Jiang, Q. M. Chen, and X. Liu, Ceram. Int. 44, 9865–9874 (2018). 7M. Viret, J. Nassar, M. Drouer, J. Contour, C. Fermon, and A. Fert, J. Magn. Magn. Mater. 198-199 , 1–5 (1999). 8Z. Guo, D. Lan, L. L. Qu, K. X. Zhang, F. Jin, B. B. Chen, S. W. Jin, G. Y. 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A 108, 379–384 (2012). 18M. Bibes, L. I. Balcells, S. Valencia, S. Sena, B. Martinez, and J. Fontcuberta, J. Appl. Phys. 89(11), 6686–6688 (2001). FIG. 4. Resistivity change ratio between the LCMO thin films under tensile and compressive strain and no extra strain at 0 T (a) and (b) and 9 T (c) and (d).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 052404 (2021); doi: 10.1063/5.0024516 118, 052404-4 Published under license by AIP Publishing19S. Valencia, L. I. Balcells, B. Martinez, and J. Fontcuberta, J. Appl. Phys. 93(10), 8059–8061 (2003). 20H. Wang, L. Shen, T. Duan, C. Ma, C. Cao, C. Jiang, X. Lu, H. Sun, and M. Liu, ACS Appl. Mater. Interfaces 11, 22677–22683 (2019). 21J. Huang, H. Wang, X. Sun, X. Zhang, and H. Wang, ACS Appl. Mater. Interfaces 10, 42698–42705 (2018). 22S. S. Hong, M. Q. Gu, M. Verma, V. Harbola, B. Y. Wang, D. Lu, A. Vailionis, Y. Hikita, R. Pentcheva, J. M. Rondinelli et al. ,Science 368, 71–76 (2020). 23P. Mandal, B. Bandyopadhyay, and B. Ghosh, Phys. Rev. B 64, 180405 (2001).24S. Li, C. Wang, Q. Shen, and L. Zhang, Vacuum 164, 312–318 (2019). 25C. Sen and E. Dagotto, Phys. Rev. B 102, 035126 (2020). 26M. Viret, L. Ranno, and J. M. D. Coey, Phys. Rev. Lett. 75, 3910–3913 (1997). 27S. B. Li, C. B. Wang, D. Q. Zhou, H. X. Liu, L. Li, Q. Shen, and L. M. Zhang, Ceram. Int. 44, 550–555 (2018). 28P .K .S i w a c h ,H .K .S i n g h ,a n dO .N .S r i v a s t a v a , J. Phys. 18, 9783–9794 (2006). 29C. R. Ma, M. Liu, J. Liu, G. Collins, Y. M. Zhang, H. B. Wang, C. I. Chen, Y.Lin, J. He, J. C. Jiang et al. ,ACS Appl. Mater. Interfaces 6, 2540–2545 (2014).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 052404 (2021); doi: 10.1063/5.0024516 118, 052404-5 Published under license by AIP Publishing

5.0039420.pdf

J. Appl. Phys. 129, 095301 (2021); https://doi.org/10.1063/5.0039420 129, 095301 © 2021 Author(s).Breaking atomic-level ordering via biaxial strain in functional oxides: A DFT study Cite as: J. Appl. Phys. 129, 095301 (2021); https://doi.org/10.1063/5.0039420 Submitted: 02 December 2020 . Accepted: 10 February 2021 . Published Online: 02 March 2021 Kanishk Rawat , Dillon D. Fong , and Dilpuneet S. Aidhy ARTICLES YOU MAY BE INTERESTED IN Piezoelectric and electronic properties of hydrogenated penta-BCN: A computational study Journal of Applied Physics 129, 095101 (2021); https://doi.org/10.1063/5.0043450 Epitaxial (100), (110), and (111) BaTiO 3 films on SrTiO 3 substrates—A transmission electron microscopy study Journal of Applied Physics 129, 095304 (2021); https://doi.org/10.1063/5.0045011 Next generation ferroelectric materials for semiconductor process integration and their applications Journal of Applied Physics 129, 100901 (2021); https://doi.org/10.1063/5.0037617Breaking atomic-level ordering via biaxial strain in functional oxides: A DFT study Cite as: J. Appl. Phys. 129, 095301 (2021); doi: 10.1063/5.0039420 View Online Export Citation CrossMar k Submitted: 2 December 2020 · Accepted: 10 February 2021 · Published Online: 2 March 2021 Kanishk Rawat,1 Dillon D. Fong,2 and Dilpuneet S. Aidhy1,a) AFFILIATIONS 1Department of Mechanical Engineering, University of Wyoming, Laramie, Wyoming 82071, USA 2Materials Science Division, Argonne National Laboratory, Lemont, Illinois 60439, USA a)Author to whom correspondence should be addressed: daidhy@uwyo.edu ABSTRACT Oxygen vacancies are found to play a crucial role in inducing many functional properties at the heterointerfaces in complex oxides. Gaining better control over the properties requires an understanding of the atomic structure of oxygen vacancies at the heterointerfaces. In this paper, we elucidate the effects of the interfacial strain on the oxygen-vacancy ordering in fluorite δ-Bi2O3and perovskite LaNiO 2.5 using first-principles calculations. By applying biaxial strains, we find that the 〈110〉−〈111〉oxygen vacancy order in δ-Bi2O3is broken, resulting in a faster diffusion of oxygen ions. Similarly, the biaxial strain is used to leverage both ordered and disordered arrangements ofvacancies in LaNiO 2.5. Besides the vacancy order, we find that the biaxial strain can also be used to break the cation order in Gd 2Ti2O7, where Gd and Ti antisites can be created on the cation sublattice, which leads to enhanced radiation tolerance and higher oxygen diffusivity. These results indicate that the biaxial strain that is commonly present at heterointerfaces can be used to gain control over both ordered anddisordered arrangements of defects, potentially opening new opportunities to functionalize complex oxides. Published under license by AIP Publishing. https://doi.org/10.1063/5.0039420 I. INTRODUCTION Oxygen vacancies are critical for inducing various functional properties in ceramic oxides. One of the most common examples isthe oxygen ion transport in solid oxide fuel cells where oxygen atoms diffuse via the vacancy mechanism. 1–3The oxygen vacancies are specifically created in solid electrolytes such as zirconia andceria by adding trivalent dopants, to induce oxygen diffusion.Similarly, it has been found that the presence of oxygen vacanciescan lead to a change in the oxidation state of metal atoms, thereby creating charge carriers that can induce metal –insulator transition (MIT). This phenomenon has been observed in the reduction ofLaNiO 3to LaNiO 2.5due to oxygen vacancies forming Ni2+ions that promote MIT.4,5Oxygen vacancies are held responsible for the self-healing properties of pyrochlore oxides (e.g., Gd 2Ti2O7) that are potential candidates for storing nuclear waste.3,6–8Oxygen vacancies have also been considered one of the key reasons for theemergence of novel properties in thin-film heterointerfacialoxides. 9–11 Quite often, it has been observed that oxygen vacancies can arrange themselves to form ordered networks in a crystal lattice that limit material properties. For example, above 727 °C, δ-Bi2O3has the highest oxygen conductivity among various fluorite oxides used as solid electrolytes in solid oxide fuel cells. However, when doped with lanthanides to stabilize the fluorite phase at the desir-able temperature of 500 °C, the vacancies order in a 〈110〉−〈111〉 network, leading to a dramatic loss of ionic conductivity. 12–16 Similarly, oxygen vacancy ordering has also been found to affect electronic conductivity in LaNiO 3−x. In LaNiO 2.75, a higher elec- tronic conductivity and an MIT are observed in the presence of an ordered arrangement of oxygen vacancies as compared to lower conductivity and no observable MIT in the presence of a disor-dered arrangement of oxygen vacancies. 17Similarly, increasing the amount of oxygen vacancy disorders can decrease the electronic conductivity of paramagnetic SmNiO 3.18Besides oxygen vacancies, the ordering of atoms can also affect material properties. In partic- ular, pyrochlore oxides (with the chemical formula A 2B2O7)h a v e an ordered arrangement of A3+and B4+cations. The self-healing property of pyrochlores under irradiation is anchored in the abilityof A and B cations to easily switch their sites, i.e., forming antisites on the cation sublattice. 3,19Consequently, the oxygen vacancies that are locked due to A and B ordering are unlocked when the two cations switch sites, thereby leading to faster oxygen diffusion.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 095301 (2021); doi: 10.1063/5.0039420 129, 095301-1 Published under license by AIP Publishing.Thus, the ordering of both oxygen vacancies and cations can deter- mine the functional properties of ceramic oxides: breaking this order can be a powerful means of controlling functionality. Due to the vibrant interest shown by researchers in hetero- structures and the properties of materials at the interfaces of thinfilms, 20–23the role of interfacial strain in defect ordering is an attractive parameter to investigate. In recent years, researchers have leveraged interfacial strain to gain control over oxygen vacancies.For example, tensile strain in fluorite oxides is found to have loweroxygen vacancy migration barriers, affecting the kinetics of oxygendiffusion, thereby leading to faster oxygen conductivity. 3,24–26 Similarly, the thermodynamics of o xygen vacancies, i.e., the for- mation energy of oxygen vacancies, has also been correlated withinterfacial strain to manipulate the oxygen vacancy concentrationnear the interfaces. 9,27–32For instance, density functional theory (DFT) calculations have shown that strain can be used to increase the concentration of oxygen vacancies in CaMnO 327by lowering their formation energies. Similarly, DFT calculations have shownthat in SrCoO 3−δthin films, tensile strain results in lower forma- tion energies of equatorial oxygen vacancies (i.e., vacancies in theplane of biaxial strain) compared with apical vacancies (i.e., vacancies that are perpendicular to the plane of biaxial strain). 31 Thus, interfacial strain constitutes a common parameter through which the energetics of oxygen v acancies can be controlled. In this work, we show that interfacial strain can be used to break the ordering, of both oxygen vacancies and cations, in oxide materi- als. In particular, we show that tensile strain can (1) break the〈110〉−〈111〉oxygen vacancy order in fluorite δ-Bi 2O3, (2) allow for the creation of a metastable vacancy disorder in perovskiteLaNiO 2.5, (3) and break the A and B cation order in Gd 2Ti2O7 pyrochlore. This ability to break the order via strain opens a newpathway to control the energetics of oxygen vacancies andprevent the degradation of properties in heterointerfacial oxides.More importantly, a correlation between strain and atomic orderis developed in this work.II. CRYSTAL STRUCTURES A.δ-Bi 2O3 Figure 1(a) shows a 2 × 2 × 2 supercell of δ-Bi2O3, which is an oxygen deficient fluorite-based structure. The space group of δ-Bi2O3isF/C22m3 (No. 202),33and the lattice constant is 11.288 Å.33 In comparison with an ideal fluorite structure (such as in CeO 2) that has eight oxygen atoms in its unit cell, δ-Bi2O3has six oxygen atoms and two structural vacancies. The lack of two oxygen atoms is accounted for by the +3 valence state of Bi. Previous workshave shown that the vacancies form an ordered network in the〈110〉−〈111〉fashion, which requires a 2 × 2 × 2 supercell for a complete description of the structure. The 〈110〉−〈111〉ordered vacancy network has been observed from neutron-scattering exper- iments and is supported by DFT calculations. 33,34 B. LaNiO 2.5 The rare-earth nickelate LaNiO 3has a rhombohedral distorted perovskite-based structure. It has an R/C223cspace group (No. 167), a lattice constant of 5.250 Å, and α= 61.4°.35Oxygen deficient LaNiO 2.5exists in a monoclinic phase, with the c2/c space group (No. 15)35,36and with lattice constants a = 7.830 Å, b = 7.800 Å, c = 7.740 Å, and β= 93.7°.36However, LaNiO 2.5can be stabilized in a pseudocubic form when grown as a thin film. Recently, Tunget al. 4grew LaNiO 2.5on SrTiO 3and proposed a pseudocubic struc- ture of LaNiO 2.5(called structure A4as per its nomenclature), shown in Fig. 1(b) . Previous studies have described the structure of oxygen deficient LaNiO 2.5as similar to that of LaNiO 3but with Ni-O 4square planar coordination in the 〈110〉direction,37,38alter- nating with Ni-O 6octahedra. C. Gd 2Ti2O7 Gd2Ti2O7has a pyrochlore structure with space group Fd/C223m (No. 227).3Its structure can be described as a 2 × 2 × 2 supercell of FIG. 1. ( a )A2×2×2 δ-Bi2O3structure with 〈110〉−〈111〉ordered oxygen vacancies. (b) A 2 × 4 × 2 supercell of LaNiO 2.5showing structure A where the square planars are represented by the lighter gray shade, whereas the octahedra are represented by the darker gray shade. See text for a description of structure A. (c ) An ordered 1×1×1G d 2Ti2O7pyrochlore structure.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 095301 (2021); doi: 10.1063/5.0039420 129, 095301-2 Published under license by AIP Publishing.a fluorite structure. Due to +3 and +4 valencies of Gd and Ti, respectively, each 1/8th unit cell contains one structural oxygen vacancy. The cubic lattice parameter of Gd 2Ti2O7is 10.260Å.3 Figure 1(c) shows the ideal pyrochlore structure where Gd and Ti cations are ordered, i.e., they occupy 16cand 16dWyckoff sites, respectively. The oxygen atoms occupy 8band 48fsites, whereas the vacancies occupy 8asites. In the disordered structure, both cations are randomly distributed on the cation sublattice, andoxygen atoms and vacancies are randomly distributed on the anionsublattice. III. METHODOLOGY To investigate the effect of biaxial strain on these structures, spin-polarized DFT calculations are performed using the Viennaab initio Simulation Package (VASP). 39In all calculations, the Perdew –Burke –Ernzerhof (PBE) form of the generalized gradient approximation (GGA)40for the exchange –correlation functionals is used. Forδ-Bi2O3, a plane wave energy cutoff of 500 eV and a 3 × 3 × 3 Monkhorst –Pack41k-point sampling mesh are used for the 2 × 2 × 2 defect fluorite supercell containing 80 atoms. The total energy is converged with a 1 × 10−4eV tolerance. The 〈110〉−〈111〉δ-Bi2O3structure is derived from a previous work33and is found to have a lattice constant of 11.196Å. In this work, we also consider other vacancy-ordered structures, i.e.,vacancies ordered in 〈100〉,〈110〉, and 〈111〉directions. In these three structures, a 1 × 1 × 1 unit cell is used, because the ordering in these structures can be described by a 10-atom unit cell. In addi-tion to the vacancy-ordered structures, the disordered structures ofδ-Bi 2O3, i.e., containing the random positioning of oxygen vacan- cies, are also considered. For the three disordered structures, named D1, D2, and D3, 2 × 2 × 2 supercells containing 80 atoms are used. Although the disordering of the vacancies is done byhand, which may not represent complete randomness in thesupercell, our calculations reveal that as long as the strictly ordered structure is broken, the presence of some level of ran- domness destabilizes the ordered structure under biaxial strain.Consequently, the disordered structures become more stable,which leads to a change in the atomic-level properties. For all LaNiO 3and LaNiO 2.5calculations, a plane wave energy cutoff of 520 eV is used with a 4 × 2 × 4 k-point sampling mesh for a 2 × 4 × 2 pseudocubic supercell containing 80 atoms. The conver-gence criteria used is the same as that of δ-Bi 2O3,i.e., the total energy is converged with a tolerance of 0.0001 eV. LaNiO 3is modeled in a pseudocubic perovskite-based structure. Eight oxygen vacancies are introduced in the 80 atom system to obtain LaNiO 2.5. As mentioned previously, due to the ordering of oxygen vacanciesin LaNiO 2.5, there are alternating patterns of octahedral and square-planar coordinated Ni sites. While Tung et al.4presented two structures of LaNiO 2.5(i.e., structures A and B as per their nomenclature and renamed o1 and o2, respectively, in this work), we observed that other ordered structures could have lower energies.Therefore, we created ten other ordered structures of LaNiO 2.5,a s shown in Fig. S1 in the supplementary material section; these struc- tures are named o3, o4, o5, and so on. In addition, we consider five structures with disordered vacancies (named d1, d2, d3, and so on),as shown in Fig. S1 in the supplementary material section. The ener- gies of the oxygen-deficient cubic structures are evaluated at various biaxial strains. Under the assumption that the thin films are grownon cubic perovskite substrates such as SrTiO 3, the structures are relaxed at the +2% strain, while the cell volume and shape areconstrained. The total energies of the structures are compared to ascertain the stability of various vacancy-ordered structures under different strain conditions. For Gd 2Ti2O7, a plane wave energy cutoff of 500 eV is used with a 3 × 3 × 3 k-point sampling mesh for a 2 × 2 × 2 supercell containing 88 atoms. The same convergence criteria described above are used. The structure of Gd 2Ti2O7with a disordered arrangement of cations is taken from a previous work.3 When biaxial strain is applied in a given plane, the height of the unit cell in the out-of-plane axis changes according to thePoisson ’s ratio. We calculate the c-axis lattice constant by con- structing an energy vs volume curve by gradually varying the length of the supercell in the c-axis for a fixed x –y biaxial strain; the minimum in the curve is the obtained c-axis constant. Thesecalculations are performed for both x –y biaxial strains (i.e., −2% and +2%) for all δ-Bi 2O3,L a N i O 2.5,a n dG d 2Ti2O7ordered and disordered structures. An example of the energy vs volume curve for an o4 structure of LaNiO 2.5is shown in Fig. S2 in thesupplementary material section. It shows the energies of the structures as the c-axis length is varied. We find that when a +2% tensile strain is applied, the c axis relaxes from 3.832 Å to 3.680 Å, as shown in Fig. S2 in the supplementary material . In order to vali- date this method, we also compute the c-axis lattice constant forbulk LaNiO 3under the +2% biaxial strain and find it to be 3.800 Å, which is in agreement with that in the previous literature.4,42 Finally, we have performed molecular dynamics (MD) simulations to understand oxygen diffusion in δ-Bi2O3and Gd2Ti2O7. The simulations are performed using the Large-scale Atomic Molecular/Massively Parallel Simulator (LAMMPS) code.43 The interatomic potential and the MD simulation parameters for δ-Bi2O3such as time step, temperature, etc., are identical to previ- ous works.33,44The simulations were performed using a time step of 0.5 fs up to a total of 200 000 MD steps at 1000 K. Similarly,for Gd 2Ti2O7, the interatomic potential is also derived from previ- ous works.3,19The MD simulations are performed in a biaxially strained supercell, where the supercell is strained in the x and y directions, while the z axis is allowed to relax by fixing thevolume of the supercell. Mean square displacement (MSD) isused to measure oxygen diffusion given by the following equation: MSD¼ 1 NX (ri tþΔtþri t)2, (1) where ri tis the ion ’s position at time tand Nis the total number of diffusing ions. The diffusion constant Dis related to MSD by the relation MSD = 6 Dt.33 IV. RESULTS A.δ-BI2O3 Figure 2 shows a comparison of system energies among four ordered and three disordered structures of δ-Bi2O3. As mentionedJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 095301 (2021); doi: 10.1063/5.0039420 129, 095301-3 Published under license by AIP Publishing.above, in the ordered structures, the vacancies are ordered in the 〈100〉,〈110〉,〈111〉, and 〈110〉−〈111〉directions, whereas in the disordered structures, the vacancies are randomly distributed on the oxygen sublattice. In agreement with a previous work,33we find that the 〈110〉−〈111〉ordered structure is the lowest energy structure, as shown in Fig. 2 . The optimized lattice constant of 11.198 Å is also in good agreement with that in the previous DFT works.13,33 For oxygen vacancies ordered in the 〈100〉and 〈110〉direc- tions, the vacancies can lie in either the plane of the biaxial strainor perpendicular to it. Figure 3(a) shows the possible scenarios with unit cells of vacancies in the 〈100〉,〈110〉, and 〈111〉direc- tions and the plane of the biaxial strain indicated by the blue dotted lines. The directions of the applied strain corresponding tothe dotted line are shown for 〈100〉_xy, represented by the arrows. Since the 〈100〉vacancies lie in the x –y plane, if biaxial strain is applied in the x and y directions, the vacancies are in the plane of the strain, represented by 〈100〉_xy. If the biaxial strain is applied in the y –z plane, the 〈100〉vacancies are perpendicular to the plane of the biaxial strain and the orientation is represented by 〈100〉_yz. Similarly, in the structure with 〈110〉ordering, when biaxial strain is applied in the x and y directions ( 〈110〉_xy), the vacancies lie in the plane, and when the strain is applied in the y and z directions,the vacancies lie perpendicular to the plane of the strain ( 〈110〉_yz). In the 〈111〉ordered structure, the plane of the biaxial strain is inde- pendent of the orientation of vacancies; therefore, the system is strained only in the y –z plane, as shown in Fig. 3(a) . The energies for all ordered and disordered structures under different biaxial strains are compared in Fig. 3(b) . Under both 0% and −2% strains, the 〈110〉−〈111〉ordered structure has the lowest energy. However, under a +2% strain, the stability changes and the disordered structure, D1, has the lowest energy. Additionally, the disordered structures D2 and D3 have lower ener-gies than the 〈 110〉−〈111〉ordered structure at the +2% strain. It is interesting to note that while the stability of the structures changes from 〈110〉−〈111〉to disordered as the strain changes from−2% to +2%, the other three ordered structures, i.e., 〈100〉,〈110〉, and 〈111〉, remain higher in energy across all strain ranges. However, within these three ordered structures, their relative stabil- ity does change. While at 0% strain, the 〈100〉structure has the lowest energy, followed by 〈110〉and 〈111〉structures, under 2% strain, the 〈110〉_yz structure has the lowest energy followed by 〈110〉_xy, 〈111〉,〈100〉_xy, and 〈100〉_yz structures. A similar trend is observed when the samples are under tensile strain. These results indicate that (1) biaxial strain can be used to break theordered arrangement of vacancies to induce disorder in the anionsublattice and (2) different ordered arrangements can be stabilizedat different amounts of biaxial strains. The effect of change in the ordering of the vacancies on oxygen diffusivity is captured in MD simulations. Previously,Aidhy et al. 33showed that under 0% strain, oxygen atoms do not diffuse once the structure is locked in a 〈110〉−〈111〉ordered arrangement of vacancies. The MSD for the 〈110〉−〈111〉structure is reproduced in Fig. 4(a) ; it shows a plateau indicating that no oxygen diffusion takes place and the vacancies remain orderedthroughout the length of the simulation. Based on the DFT results that under tensile strain, the disor- dered structures could be relatively more stable than the ordered structures, it is hypothesized that once the 〈110〉−〈111〉ordered network of vacancies is broken by applying tensile strain, theoxygen vacancies will continue to diffuse. We test this hypothesisby performing MD simulations on 〈110〉−〈111〉ordered and two disordered (D1 and D2) structures, all under a 6% biaxial tensile strain. A higher tensile strain is chosen in order to allow apprecia-ble oxygen diffusion on a limited MD time scale. Figure 4(b) shows the MSD in the three structures. Instead of a plateau, continuousoxygen diffusion is observed in all three structures, indicating that the 〈110〉−〈111〉ordered network, which is otherwise stable under the 0% strain, is now broken, and the oxygen vacancies diffusethroughout the duration of the simulation. Identical MSD slopesare observed for the two disordered structures, indicating that allthree structures have disordered vacancies leading to high oxygen diffusion. This is indicative of the fact that biaxial strain is affecting the structure and the diffusivity of the oxygen atoms. It is to benoted that while the DFT results provide the stability of the struc-tures at 0 K, the MD simulations performed at elevatedtemperatures validate the prediction that the ordered network of vacancies is broken under biaxial strain [ Fig. 4(b) ]. An additional benefit of applying tensile strain is that it lowers the migration barriers of oxygen diffusion. Various previous studieshave shown that tensile strain reduces migration barriers, thereby increasing oxygen diffusivity. 24,45–50These calculations now show that tensile strain can also help unlock the ordered network,thereby further contributing to easier oxygen diffusion. B. LaNiO 2.5 While performing DFT calculations, Tung et al.4considered two structures of LaNiO 2.5distinguished by the ordering of oxygen vacancies. Their supercell consisted of LaNiO 2.5interfaced with SrTiO 3. Due to a lattice mismatch of +2% between the two materi- als, they applied a +2% biaxial strain on LaNiO 2.5. After relaxation, they found that structure A had a lower energy than structure B. Our calculations reproduce their observations. A comparison of FIG. 2. Comparison of the system stability among various ordered and disor- dered structures in δ-Bi2O3at 0% strain. The 〈110〉−〈111〉structure is the lowest energy structure, in agreement with that in previous work.33Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 095301 (2021); doi: 10.1063/5.0039420 129, 095301-4 Published under license by AIP Publishing.system energies of the two structures is shown in Fig. 5(a) . In this work, we relabel their structures A and B as o1 and o2, respectively. We have considered other ordered and disordered structures to understand the effect of biaxial strain on the ordering/disorder- ing of vacancies. It is hypothesized that varying the biaxial straincan lead to a change in the oxygen vacancy pattern, thereby leadingto a change in the stability of the structures. In particular, we haveconsidered twelve ordered and five disordered structures. These structures are schematically shown in Fig. S1 in the supplementary material section. Three biaxial strain conditions are considered, i.e., +2%, 0%, and −2%. The tensile strain (i.e., +2%) corresponds to the lattice constant of SrTiO 3applied to pseudocubic LaNiO 2.5. As discussed in Sec. III, the c-axis lattice constant is calculated corresponding to each strain condition for each of the seventeen structures. The relaxed c-axis lattice constants for all structures aregiven in Table I . Figure 5 shows a comparison of energies of 17 structures under three different strain conditions. While o1 has a lower energy than o2 under the +2% strain, as shown in Fig. 5(a) , their relative stability changes under the 0% strain, as shown in Fig. 5(b) . Their energy difference further increases at −2%, as shown in Fig. 5(c) , indicating that the o2 structure is expected to be more stable than o1 under compressive strain. This result indicates that the vacancy ordering changes with biaxial strain in LaNiO 2.5.While o1 is relatively more stable than o2 under a +2% strain, we find that it is not the lowest energy structure. Instead, we findthat o8 is the lowest energy structure, which has 7 meV/atom lower energy than o1. Furthermore, three other structures, i.e., o3, o4, and o11, have 6 meV/atom lower energies than o1. These threestructures have identical energies despite a different arrangementof oxygen vacancies, as shown in Fig. S1 in the supplementary material section. It is also observed that all of the disordered struc- tures, i.e., d1 –d5, have higher energies under the +2% strain. Thus, based on Fig. 5(a) , ordered vacancies can be expected in LaNiO 2.5 when grown on SrTiO 3. In contrast, under 0% strain, we find that the stability of the structures changes from ordered to disordered. Figure 5(b) shows that d3 is the lowest energy structure among all, followed by d4. The ordered structure, i.e., o8, is the third most stable structure.While the other three disordered structures are significantly higherin energy than ordered structures, Fig. 5(b) illustrates that disor- dered vacancies can be expected in LaNiO 2.5under 0% strain. The stability of disordered vacancies over ordered vacancies has been observed experimentally by Hirai et al.51in the SrFeO 2.5/DyScO 3 interface. Using high-angle annular dark-field scanning transmis- sion electron microscopy (HAADF-STEM), it was observed that the vacancies were completely disordered up to 5 nm thickness from the interface. As the strain relaxed at higher film thicknesses, FIG. 3. (a) 1 × 1 × 1 unit cells of 〈100〉,〈110〉, and 〈111〉vacancy-ordered structures showing planes of an applied biaxial strain with Bi atoms (purple), O atoms (red) and oxygen vacancies (white). The planes in which the biaxial strain is applied with respect to the vacancies are highlighted in the unit cells (b) Comp arison of system energies per atom of ordered and disordered structures at +2% biaxial strain, 0% strain, and −2% biaxial strain.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 095301 (2021); doi: 10.1063/5.0039420 129, 095301-5 Published under license by AIP Publishing.the authors showed the transition from a disordered to an ordered arrangement of vacancies. Thus, our DFT predictions qualitatively agree with the experimental observation, i.e., strain can affect theorder/disorder arrangement of vacancies. Finally, under a −2% compressive strain, we find that the o2 ordered structure is the most stable structure, sequentially followed by o7, o10, and d3, as shown in Fig. 5(c) . The change in stability from ordered to disordered and back to ordered vacancy structuresfrom +2% to 0% to −2%, respectively, indicates that the stability of the vacancy structures depends on the amount of strain applied,which can, in turn, be correlated to the chemical expansion of the oxygen vacancies. Previous works have shown that an oxygen vacancy generally occupies a larger volume than an oxygen atom,leading to chemical expansion in the material. 9,45–53Here, because the vacancies are ordered in different orientations, the biaxial straincan affect the chemical expansion and hence the relative stability of the structures. This could possibly be a reason for the stability of different structures under different amounts of strain, as showninFigs. 5(a) –5(c). C. Gd 2Ti2O7 Pyrochlore oxides have shown diverse electronic and atomic properties such as the MIT, topological insulators, fast oxygen-iontransport, and high radiation resistance for nuclear waste-storageapplications. 8,19,54However, gaining control over these properties in bulk pyrochlores has remained an open challenge due to the lack of simpler methods to stabilize the specific “disordered ”struc- ture in these materials needed to induce these properties. Based on the DFT results obtained in δ-Bi2O3and LaNiO 2.5, we explore the possibility of changing the cation ordering in the pyrochlore structure Gd 2Ti2O7. It was previously shown that cation disorder could be achieved by applying volumetric strain3where a disordered structure was found to be more stable than an orderedone under higher tensile strains. In this work, we focus on theeffect of biaxial strain on the relative stability of ordered and disor- dered structures. The disordered structure is taken from a previous work. 3The energy vs volume curve is plotted for the two struc- tures, as shown in Fig. 6(a) . For each amount of biaxial strain, the FIG. 4. Mean-square displacement of oxygen in the δ-Bi2O3structure at (a) 0% strain in the 〈110〉−〈111〉ordered structure and (b) 6% strain in ordered and disordered structures. FIG. 5. Comparison of system energies per atom among various ordered and disordered LaNiO 2.5structures at (a) +2%, (b) 0%, and (c) −2% biaxial strains.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 095301 (2021); doi: 10.1063/5.0039420 129, 095301-6 Published under license by AIP Publishing.c-axis lattice constant is uniquely determined, as discussed in Sec. III. We find that the stability changes from ordered to disor- dered at a volume of 1150 Å3. At the crossover point, the ordered structure is under a tensile strain of 6%, whereas the disorderedstructure is under a compressive strain of 4%. In the disordered structure, Gd and Ti atoms occupy antisites on the cation sublat-tice. As a result, Gd and Ti are not restricted to 16a and 16b Wyckoff sites. The oxygen atoms and vacancies are also not restricted to their corresponding 8band 48f, and 8aWyckoff sites, respectively. As a result, the oxygen vacancies are unlocked. Thesecalculations demonstrate that biaxial tensile strain can disorder thecation sublattice in Gd 2Ti2O7. The unlocking of the oxygen vacancies is observed in their dif- fusivity behavior. Figure 6(b) shows a comparison of oxygen MSD in the ordered and disordered structures of Gd 2Ti2O7. While significant oxygen diffusion is observed in the disordered structure, no diffusionis observed in the ordered structure. The lack of oxygen diffusion is due to higher oxygen migration barriers in the ordered structure compared with lower barriers in the disordered one. These diffusivityresults are consistent with those of previous works. 3,19 VI. DISCUSSION A.δ-Bi2O3 The breaking of the ordered arrangement of oxygen vacancies inδ-Bi2O3via biaxial strain opens an opportunity to achieve higher oxygen-ion conductivity. While δ-Bi2O3has the highest oxygen conductivity among known fluorite-based oxides, its conductivityis significantly lower at practically desirable temperatures of 500 oC because of the phase transformation from a delta phase to a mono-clinic one. While the addition of trivalent dopants can stabilize the delta phase at lower temperatures, the conductivity still decreases due to the decrease in the overall cation polarizability of dopants.Note that one of the main reasons for the high conductivity ofδ-Bi 2O3is the presence of 6s2lone pair electrons in Bi, which lends high polarizability.1,2,50,55,56The addition of low polarizable triva- lent dopants leads to a decrease in the overall cation polarizability, and hence, oxygen conductivity. Since the addition of trivalentdoping is necessary to stabilize the material at relevant tempera-tures, the resulting loss in conductivity due to ordering of thevacancies is undesirable. The observations in this paper show that tensile interfacial strain can be used to break vacancy ordering. The disordered structures exhibit lower energies at 2% tensile strain, sig-nifying thermodynamic stability over the ordered structure. This issupported by the MSD results, which show that while there is limited diffusion at 0% strain, significant diffusion is observed in both ordered and disordered systems under tensile strain. Thus, theTABLE I. Relaxed c-axis lattice constants (in Å) for all ordered and disordered LaNiO 2.5structures under +2% (x = y = 7.850 Å), 0%, and −2% (x = y = 7.500 Å) biaxial strains. Strain o1 o2 o3 o4 o5 o6 o7 o8 o9 o10 o11 o12 +2% 3.650 3.830 3.680 3.680 3.810 3.810 3.840 3.670 3.840 3.840 3.660 3.610 0% 3.830 3.830 3.830 3.830 3.830 3.830 3.830 3.830 3.830 3.830 3.830 3.830−2% 3.750 3.920 3.800 3.770 3.910 3.910 3.910 3.760 3.960 3.950 3.760 3.890 Strain d1 d2 d3 d4 d5 +2% 3.770 3.800 3.860 3.820 3.8250% 3.830 3.830 3.830 3.830 3.830−2% 3.900 4.000 3.990 4.000 3.970 FIG. 6. (a) Comparison of system energy per atom between the ordered and the disordered structures as a function of volume from DFT calculations. (b)MSD of oxygen in ordered and disordered Gd 2Ti2O7from MD simulations.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 095301 (2021); doi: 10.1063/5.0039420 129, 095301-7 Published under license by AIP Publishing.breaking of the ordered structures opens a door to nullify the loss of conductivity due to the addition of dopants. B. LaNiO 2.5 Recently, Golalikhani et al.5showed that oxygen vacancies play a critical role in the MIT in ultrathin LaNiO 3films. Using x-ray absorption spectroscopy, they detected the presence ofoxygen vacancies in LaNiO 3thin films. The presence of oxygen vacancies was suggested to be the cause behind the insulating behavior of ultrathin LaNiO 3thin films grown on LaAlO 3sub- strates. Tung et al.4performed DFT calculations in a LaNiO 3thin film grown on SrTiO 3. They found that thin films are necessary to cause transformation from LaNiO 3to oxygen-deficient LaNiO 2.5. They proposed two vacancy ordered structures for LaNiO 2.5. In this work, we advance the understanding and propose that both ordered and disordered arrangement of vacancies can be achievedvia interfacial strain. Our results for LaNiO 2.5indicate that at 0% strain, the disordered structure is thermodynamically stable. However, when the system is strained under both compressive and tensile conditions, different ordered systems become thermody-namically stable. Although our results show relative stability amonga select group of possible orderings in the material, we propose thatinterfacial strain can be used as an external factor to alter vacancy ordering. This opens up pathways to grow LaNiO 2.5on various substrates such as LaAlO 3or SrTiO 3and control the ordering of oxygen vacancies. As a result, this understanding opens up a possi-bility to induce newer functionalities in LaNiO 2.5. Nord et al.57observed a change in the structure of La0.7Sr0.3MnO 3(LSMO) thin films when grown on SrTiO 3. The LSMO thin films were subjected to a 1.2% tensile strain resultingfrom the smaller lattice constant of LSMO (3.876 Å) comparedwith SrTiO 3(3.925 Å). With the combined results from DFT calcu- lations and experiments, they found the presence of a brownmiller- ite phase with disordered oxygen vacancies existing 3 nm into the thin film from the interface. These examples support our predictionthat strain can be potentially used to achieve an order/disorderarrangement of oxygen vacancies near the interface. C. Gd 2Ti2O7 Previous work by Aidhy et al.3showed that the cation order- ing of Gd and Ti atoms could be disrupted by the application of volumetric strain. In this work, we show that biaxial strain can alsobreak cation ordering. This observation opens up a possibility tocreate cation antisites in pyrochlore thin films. Evidence of cationdisorder has been recently presented in Ho 2Ti2O7pyrochlore thin films grown on yttria stabilized zirconia.58Due to the lattice parameter mismatch between the two materials, a 2% strain in thethin film was observed. A large number of antisites near the inter-face, i.e., Ti sites on Ho sites and vice versa, were also observed.Similarly, Yang et al. 54reported the presence of Bi Irantisite defects in Bi 2Ir2O7thin films grown on YSZ. The disordering of cations due to tensile strain observed in our work is consistent with theserecent results. Thus, imparting tensile strain pyrochlore as thinfilms carries an opportunity to affect functional properties by low- ering the cation antisite energies and unlocking oxygen vacancies for fast ion diffusion.VII. CONCLUSION We have used biaxial strain to disrupt oxygen vacancy order- ing in fluorite-based oxide δ-Bi 2O3and rare earth nickelate LaNiO 2.5, and cation ordering in pyrochlore Gd 2Ti2O7. With a combination of DFT and MD simulations, we show that the ordered arrangement of oxygen vacancies in δ-Bi2O3can be broken via interfacial strain, thereby leading to faster oxygen diffusion. InLaNiO 2.5, we show that by varying biaxial strain, both ordered and disordered arrangement of vacancies could be achieved. Finally, we show that the ordered arrangement of cations in a pyrochlore struc- ture could also be broken by biaxial strain. Collectively, theseresults indicate that biaxial strain present at the interfaces could beused as a controlling knob to affect defect energetics and poten-tially unravel interesting functional properties in oxide thin films. SUPPLEMENTARY MATERIAL See the supplementary material for the ordered and disordered LaNiO 2.5structures with the position of oxygen vacancies along with the example of energy-volume curve used to determine relaxed c-axis constants. ACKNOWLEDGMENTS K.R. and D.S.A. acknowledge support by the National Science Foundation under Grant No. 1929112. They also acknowledge thesupport of computational resources from the Advanced Research Computing Center (ARCC) at the University of Wyoming. The work by D. D. Fong was supported by the U.S. Department ofEnergy (DOE), Office of Science, Basic Energy Sciences (BES),Materials Sciences and Engineering Division. The authors declareno competing financial interests. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1J. Jiang, D. G. Lim, K. Ramadoss, and S. Ramanathan, “Ionic conduction and unipolar resistance switching in δ-phase Bi 2O3thin films, ”Solid-State Electron. 146,1 3–20 (2018). 2N. M. Sammes, G. A. Tompsett, H. Näfe, and F. Aldinger, “Bismuth based oxide electrolytes —Structure and ionic conductivity, ”J. Eur. Ceram. Soc. 19, 1801 –1826 (1999). 3D. S. Aidhy, R. Sachan, E. Zarkadoula, O. Pakarinen, M. F. Chisholm, Y. Zhang, and W. J. Weber, “Fast ion conductivity in strained defect-fluorite structure created by ion tracks in Gd 2Ti2O7,”Sci. Rep. 5(1), 16297 (2015). 4I. C. Tung, G. Luo, J. H. Lee, S. H. Chang, J. Moyer, H. Hong, M. J. 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J. Appl. Phys. 129, 013103 (2021); https://doi.org/10.1063/5.0031063 129, 013103 © 2021 Author(s).Polarization effects in laser-induced plasma lasers based on elements from the 13th group Cite as: J. Appl. Phys. 129, 013103 (2021); https://doi.org/10.1063/5.0031063 Submitted: 28 September 2020 . Accepted: 16 December 2020 . Published Online: 06 January 2021 L. Nagli , E. Stambulchik , M. Gaft , and Y. Raichlin ARTICLES YOU MAY BE INTERESTED IN Resistor–capacitor modeling of the cell membrane: A multiphysics analysis Journal of Applied Physics 129, 011101 (2021); https://doi.org/10.1063/5.0033608 Target heating in femtosecond laser–plasma interactions: Quantitative analysis of experimental data Physics of Plasmas 28, 023101 (2021); https://doi.org/10.1063/5.0035356 Three-dimensional hybrid optoacoustic imaging of the laser-induced plasma and deposited energy density under optical breakdown in water Applied Physics Letters 118, 011109 (2021); https://doi.org/10.1063/5.0032513Polarization effects in laser-induced plasma lasers based on elements from the 13th group Cite as: J. Appl. Phys. 129, 013103 (2021); doi: 10.1063/5.0031063 View Online Export Citation CrossMar k Submitted: 28 September 2020 · Accepted: 16 December 2020 · Published Online: 6 January 2021 L. Nagli,1,a) E. Stambulchik,2 M. Gaft,1and Y. Raichlin1 AFFILIATIONS 1Department of Physics, Ariel University, Ariel 40700, Israel 2Faculty of Physics, Weizmann Institute of Science, Rehovot 7610001, Israel a)Author to whom correspondence should be addressed: levna@ariel.ac.il ABSTRACT We propose a model explaining polarization effects in laser-induced plasma lasers (LIPLs) of the 13th group elements, pumped by a linearly polarized laser beam. The model is based on considering optical transitions between magnetic sublevels involved in the pumping –generation cycle. The model reproduces experimentally observed LIPL polarization features under the np2P1/2, 3/2→n0s2S1/2pumping. On the other hand, polarization-resolved collisional-radiative modeling appears to be required for a quantitative explanation of the LIPL polarizationwhen the np 2P1/2, 3/2→n0d2D1/2pumping is used. Published under license by AIP Publishing. https://doi.org/10.1063/5.0031063 I. INTRODUCTION The lasing effect was demonstrated in laser-induced plasmas (LIPs) of elements from the 13th and 14th groups as well as Ca, Ti,Zr, Fe, Cu, Ni, and V plasmas. 1–3The unique generation mecha- nism of LIP lasers (LIPLs) allows one to study a variety of LIP properties and fundamental laser plasma physics problems. LIPLs based on elements from the 13th group having small spin –orbit splitting of the ground term (Al, Ga, In) under resonant pumping from the ground-term levels ( np2P1/2and np2P3/2), lase according to the three-level scheme (transitions ns2S1/2→n0p2P1/2 orns2S1/2→n0p2P3/2.1–6Thallium (Tl), the heaviest element of this group, possesses a large spin –orbit ground-term splitting (about 1 eV) and generates differently from other 13th group ele- ments.2The generation scheme for Al LIPL is presented in Fig. 1 . The scheme does not include infrared direct generation from5s 2S1/2to combined 3 s24p2P1/2;3/2and 3 s23d2D3/2;5/2 levels.6The inverse population of the upper lasing levels is created through acascade of collisional and radiative processes from higher excited states (the curved dashed lines). 4,5It was also shown that pumping from np2P1/2leads to unpolarized radiation while pumping from np2P3/2leads to a strongly polarized generation with a degree of polarization (DOP) and the polarization direction dependent onthe pumping transitions. 7For example, in Al LIPL ( Fig. 1 ), pumping 3 p2P3/2→5s2S1/2(266.039 nm) leads to generation at 396.15 nm with DOP ≈1 and the polarization vector Egparallel tothe pumping light polarization Ep,while pumping to 3 d2D3/2;5/2 (309.28 nm) leads to the same generation wavelength but with Eg⊥Epand DOP ≈−0.7.7A similar effect of polarization conser- vation in collisional processes was found in the generation of theoptically pumped alkali atoms lasers. 8–11The Tl LIPL generation is not polarized at all.2 The effects of LIPL polarization were studied in the absence of an external magnetic field and were also manifested in polarizedstimulated emission (SE) in the absence of an optical resonator. 1–3 A previous study7proposed that the observed polarization effects were caused by self-generated electric and magnetic fields, which break the magnetic sub-level degeneracy.11–15Here, we present a different model, based on fewer assumptions, explainingthe polarization that considers magnetic sublevels min the pumping –generation cycle. II. EXPERIMENTAL SETUP The experimental setup is similar to that described else- where 1,2,7and is schematically shown in Fig. 2 . In brief, a transverse pumping scheme is used. 5 × 20 × 2 mm3metallic aluminum (Al), indium (In), gallium (Ga), and thallium (Tl) as TlBr crystal (KRS-5)) samples wereplaced inside flat –flat optical resonator. The Al coated mirrors are used. The rear mirror has a 99.9% reflection, and the output mirror has about 10% reflection. The resonator was optimized forJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 013103 (2021); doi: 10.1063/5.0031063 129, 013103-1 Published under license by AIP Publishing.maximal lasing output. Nd:YAG 1.06 μm laser (LQ215-D, (pulse duration about 7 ns, pulse energy from 40 to 90 mJ/pulse)produces a plasma strip of about 5 mm in length and 0.5 mm diameter (irradiance from 1.5 to 3 J/cm 2) on a sample surface through a cylindrical lens. After a delay of 5 –10μs,(depending on the sample used), the plasma plume is pumped by an optical para-metric oscillator (OPO) (RADIANT 355 LD-UV, 5 ns duration, the spectral linewidth is about 5 cm −1, irradiance on plasma plume of about 0.1 J/cm2at resonant wavelengths of the plasma species. The plasma electron temperature Teand electron density nereach about 5000 K and 1016cm−3, respectively, at this time.2,3Unlike the previous experiments, the plasma radiation is transmitted to theShamrock SR 750-A spectrometer (grating 2400 l/mm), equipped with a fast ICCD camera (Andor DH320-18U-03) through a custom fiber bundle (round (SMA) to the ferrule (sleet) with a 500 μmc o r e and 1 m length). It was checked that the fiber also works as a polari-zation scrambler to eliminate spectrometer-induced polarizationeffects. The spectral and time resolution is about 0.01 nm and 1.5 ns, respectively. The line polarization is measured using the Thorlabs GL-10 calcite polarizer, placed before the optical fiber entrance. TheDOP is defined as DOP = (I II−I⊥)/(III+I⊥), where IIIandI⊥are the generation line intensities with Egparallel and perpendicular to Ep, respectively. A λ/2 plate alters the direction of the pumping polariza- tion vector. The geometry of the polarization measurements is shown in Fig. 3 . There, kpandkgare the pumping, and LIPL beams wave vectors, respectively, EpandEgare the pumping and generation polarization vectors. The yellow cylinder is the plasma plume aboveas a m p l es u r f a c e . III. RESULTS AND DISCUSSION As mentioned above, the upper lasing levels ’population inver- sion is formed by collisional and radiative processes from thepumped levels. 4,5For Al LIPL, e.g., these levels may be 5 s2S,3d2D, or 4 s2S(seeFig. 1 ). On the other hand, the plasma density is too low for efficient mixing of the ground-term2P1/2and2P3/2levels (see also Ref. 15). Therefore, it is evident that a resonant pumping from the2P1/2level will leave2P3/2strongly populated, and vice versa. Accordingly, the lasing can occur only for transitions to the same level from which one pumps as indeed, it has been experi- mentally observed.1–7 The following discussion is for Al LIPL; however, the same considerations apply to the other 13th group elements as well.Tl needs some additional considerations and will be discussed in Sec. III C . It was found that any resonant pumping from the ground level 3 p 2P1/2leads to generation at 394.40 nm, which is not polar- ized (DOP = 0), while pumping from the 3 p2P3/2level leads to gen- eration at 396.15 nm, the DOP of which depends on the pumping transition and its polarization direction.7Figure 4 presents exam- ples of the Al LIPL generation line spectra, measured underpumping at 266.04 nm (a) and 309.28 nm (b). In both cases, blacklines are the generation with E gparallel to the Ep,and red lines are the generation with Egnormal to the Ep. Table I lists measured DOP of Al LIPL for parallel (DOP II) and normal (DOP ⊥) direction of the pumping light polarization vector Ep, relatively to the lasing direction kg. Line notations are taken from Refs. 16and17. The polarization effects can be explained, considering magnetic sub-levels and assuming they practically do not mix due to collisionswith the plasma electrons during the pumping -generation cycle. From now on, we assume the quantization axis to lie along with the polarization of the pumping laser E p(Fig. 3 ). Figure 5 shows Al energy levels, including the magnetic sub-levels (it must be emphasized that there is no actual Zeeman splitting; themagnetic sub-levels are shown separately for the sake of clarity.Theπandσpolarizations are due to transitions with ΔM= 0 and ΔM= ± 1, respectively. Here, Mis the projection of the total angular momentum J. FIG. 2. Experimental setup. FIG. 1. A simplified scheme of the Al LIPL pumped at 265.30 nm, 266.04 nm, 308.20 nm, and 309.28 nm. The navy blue and blue arrows indicate pumpingfrom the ground-term 3 p2P1/2and 3 p2P3/2levels, respectively (the spin –orbit splitting 0.014 eV). The black dashed arrows designate collisional transitions, leading to a population of the 4 s2S1/2level, from which a lasing occurs to the 3p2P1/2or 3p2P3/2levels (the dashed navy blue and blue arrows, respectively).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 013103 (2021); doi: 10.1063/5.0031063 129, 013103-2 Published under license by AIP Publishing.A. Pumping polarization vector Epnormal to the lasing direction kg Let us start with pumping from2P1/2[seeFig. 5(a) ], which is the simplest case to consider [experiments geometry is shown in[Fig 3(a) ]. Indeed, the entire 2P1/2will be depleted, through the 2P1/2;−1/2→2S1/2;−1/2or2P1/2;1/2→2D*/2;1/2(for brevity, not shown in the figure), and similarly for the negative M’s:2P1/2;−1/2→2S1/2,−1/2,etc. (ΔM=0 transitions). Hence,2P1/2, +1/2, and2P1/2,−1/2are equally depleted, i.e., there is no preferred directionality in the system, and the lasing may occur with any polarization (DOP = 0). In other words, there should be no polarization. It is exactly what is observed in the experiments (see Table I ). We now consider pumping from np2P3/2ton0s2S1/2with lasing from n00s2S1/2tonp2P3/2[Fig. 5(b) ]. In this case,2P3/2;±1/2 FIG. 3. The polarization measurements ’geometry with the pumping beam polarization vector Epnormal (a) and parallel (b) to lasing direction vector kg. The yellow cylinder is the plasma plume. FIG. 4. Example of the Al LIPL generation line: (a) pumping 266.04 nm and (b) pumping 309.28 nm. Red curves Egare normal to Ep, and black curves Egare parallel to Ep.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 013103 (2021); doi: 10.1063/5.0031063 129, 013103-3 Published under license by AIP Publishing.become depleted through the np2P3/2;±1/2→n0s2S1/2, ±1/2 pumping, while np2P3/2, ±3/2remain populated. In principle, there could be lasing with the linear polarization (2S1/2, ±1/2→3p2P3/2;±1/2 ΔM=0) and the circular ones (2S1/2, ±1/2→2P3/2,∓1/2,ΔM=±1). However, each circularly polarized photon would be absorbed through the2P3/2, ±3/2→2S1/2, ±1/2processes. Thus, only the linearly polarized (ΔM= 0) photons survive and lase. The polarization degree is indeed close to unity in the corresponding experiments (see Table I ). Finally, let us discuss the scheme with pumping2P3/2to 2D3/2,5/2 (hereafter2D∗/2) with the lasing occurring from2S1/2 to→2P3/2; it is the most complicated case. At first glance, it appears that there should be no polarization at all since all2P3/2projections should be depleted through 2P3/2, ±1/2→2D∗/2,±1/2and2P3/2, ±3/2→2D∗/2,±3/2.However, we need to consider the “next order ”corrections, namely, excitations and de-excitations due to the plasma electrons (which, besides, might be preferably oscillating in one direction due to the laserfield present, further breaking the symmetry). 18This effect is well known in x-ray lasers.19–21A detailed polarization-resolvedcollisional-radiative (CR) model is likely required to explain the observations in this case. CR model should account for multiple levels of the neutral and at least singly ionized charge state of the atom, with all levelsresolved to their magnetic sub-levels in order to account for polari- zation effects to be inferred. The atomic processes accounted for must include electron impact excitation and de-excitation, electronimpact ionization and recombination, radiative recombination, andradiative decay (including stimulated one in the presence of thepumping laser field and later, the LIPL radiation). Also, collisions with the heavy particles (neutral atoms and ions) are likely very important in relaxation of the atomic state alignment by mixingmagnetic sub-levels belonging to the same level. 22Furthermore, the motion of the free plasma electrons in the laser field ’s presence acquires a directionality, which in turn introduces alignment effects in the atomic level populations on its own. Building such a compre- hensive computational model is a formidable task, which is beyondthe scope of the present study. There are no existing general-purpose CR codes with all the features listed above implemented toTABLE I. Polarization of the Al I LIPL for the pumping polarization vector Epnormal (DOP ⊥) or parallel (DOP II) to the lasing vector kg.2D*/2means closely spaced2D3/2and 2D5/2levels. Polarization Pumping transition, wavelength (nm) Lasing transition, wavelength (nm) DOP ⊥ DOP II 3p2P31/2→6s2S1/2237.84 4 s2S1/2→3p2P3/2396.15 0.8 No generation 3p2P1/2→4d2D*/2256.8 4 s2S1/2→3p2P1/2394.4 0 0.0 3p2P1/2→4d2D*/2257.51 4 s2S1/2→3p2P1/2396.15 −0.6 0.0 3p2P1/2→5s2S1/2265.25 4 s2S1/2→3p2P1/2394.40 0.0 0.0 3p2P1/2→3d2D*/2308.21 4 s2S1/2→3p2P1/2394.40 0.0 0.0 3p2P3/2→5s2S1/2266.04 4 s2S1/2→3p2P3/2396.15 1.0 No generation 3p2P3/2→3d2D*/2309.28 4 s2S1/2→3p2P3/2396.15 −0.7 0.0 FIG. 5. Energy schemes for pumping from2P1/2(a) and2P3/2(b), including the magnetic sublevels. The red lines are the ( π-polarized) pumping, and the blue solid and dashed arrows are the πandσ±circularly polarized generation, respectively. The blue dashed curves are the collisional processes.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 013103 (2021); doi: 10.1063/5.0031063 129, 013103-4 Published under license by AIP Publishing.the best of our knowledge. We hope the experimental results pre- sented in this study will serve as an incentive to implement such a model, which will certainly be important for analyzing radiation ’s polarization properties in the course of laser –matter interactions in a broad context. B. Pumping polarization parallel to the lasing direction It has been found that in this geometry [see Fig. 3(b) ], similar to the normal pumping discussed above, pumping from the 3 p2P1/2 level leads to an unpolarized generation at 394.40 nm. However, when pumping 266.04 nm (3 p2P3/2→5s2S1/2), the lasing entirely disappears, while pumping at 309.28 nm (3 p2P3/2→3d2D*/2,) the generated light is unpolarized. In the case of pumping from2P1/2, the same as in Sec. III A , considerations apply, with no polarization expected. For pumping from2P3/2to2S1/2and lasing from2S1/2to 2P3/2, we again repeat the same arguments as in Sec. III A . However, the quantization axis is now the same as the laser beam propagation direction. Since a photon may have only M = ± 1 projection in its direction, it immediately follows that thereshould be no lasing at all. In fact, for the very same reason, it canbe concluded that for any combination of the pumping and lasing transitions, only ΔM= ± 1 transition would emit photons propagating along the line of sight in this (pumping parallel tothe lasing direction) configuration. If everything is perfectlyaligned, there should be no difference between ΔM=+1 a n d ΔM=−1, and thus, the lasing should be unpolarized when allowed at all. Indeed, this is observed in our experiments. This concludes the discussion of Table I . C. Other species from the 13th group It must be emphasized that all considerations above are valid only for small spin –orbital splitting Δ LSof the ground nplevels (n= 3, 4, and 5 for Al, Ga, and In, respectively), such that ΔLS«Te under our experimental conditions. For these elements, the ground- term2P1/2;3/2levels are about equally populated, which prevents theσ±generation in n0s2S1/2→np2P3/2transitions under linearlypolarized pumping. Table II lists measured DOP ⊥of In LIPL for the pumping light polarization vector Epnormal to kg. We note that the lasing polarization properties are retained even when pumped to higher2S1/2or2D*/2states (see Tables I and II). Apparently, during the longer times required for forming the inver- sion population through a cascade of collisional-radiative processes in this case, the ground-term levels remain aligned. FIG. 6. A simplified scheme of the lasing levels of Tl LIPL pumped at 258 nm. The solid arrow is the pumping transition starting from the ground 6 s23p2P1/2 level. The curved dashed arrows are collisional transitions. The dashed blue arrows are the generation to the 3 p2P1/2and 3 p2P3/2ground levels.TABLE II. Polarization of In LIPL when pumping polarization vector Epis normal (DOP ⊥) to the lasing direction vector kgdirection. Pumping Lasing Transition Wavelength (nm) Transition Wavelength (nm) Polarization DOP ⊥ 5p2P1/2→8s2S1/2 246.01 6 s2S1/2→5p2P1/2 410.17 0.0 5p2P1/2→6d2D3/2 256.01 6 s2S1/2→5p2P1/2 410.17 0.0 5p2P1/2→7s2S1/2 275.39 6 s2S1/2→5p2P1/2 410.17 0.0 5p2P1/2→5d2D3/2 303.93 6 s2S1/2→5p2P1/2 410.17 0.0 5p2P3/2→8s2S1/2 260.17 6 s2S1/2→5p2P3/2 451.3 1.0 5p2P3/2→6d2D3/2 271.39 6 s2S1/2→5p2P3/2 451.3 −0.9 5p2P3/2→6d2D5/2 271.03 6 s2S1/2→5p2P3/2 451.3 0.75 5p2P3/2→7s2S1/2 293.26 6 s2S1/2→5p2P3/2 451.3 1.0 5p2P3/2→5d2D3/2 325.86 6 s2S1/2→5p2P3/2 451.3 −0.85 5p2P3/2→5d2D5/2 325.61 6 s2S1/2→5p2P3/2 451.3 −0.65Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 013103 (2021); doi: 10.1063/5.0031063 129, 013103-5 Published under license by AIP Publishing.In the case of In LIPL, we were able to pump 5 p2P3/2→6d 2D3/2, and2D5/2separately .It is found that in In LIPL under the 5 p 2P3/2→6d2D*/2pumping, the DOP sign changes from ( −) to (+). It needs additional theoretical considerations on the base of the CRmodel. However, in the Tl atom case, the ground configuration splitting is significant ( Δ LS=0.98 eV, see Fig. 6 ), and, therefore, the 6 p2P3/2states are populated very weakly. Consequently, pumping 6 p2P1/2→8s2S1/2(258.00 nm) leads to direct generation 8s2S1/2→6p2P1/2(323.97 nm), and 7 s2S1/2→6p2P3/2(535.05 nm); this is a four-level lasing scheme. By repeating arguments in Sec. III A , all magnetic sublevels of 6 p2P3/2are equally depopu- lated, and the reabsorption of the σ±photons, contrary to the Al LIPL, is absent. Therefore, according to the proposed model,both generations should not have been polarized, and it is exper-imentally confirmed. IV. CONCLUSIONS We studied polarization properties of the 13th group (Al, Ga, In, and Tl) LIPLs, pumped by resonant, linearly polarized light pulses in the absence of the external magnetic field. Measurementswere done in the transverse pumping geometry with the pumpingbeam polarization E pnormal or parallel to generation direction kg. For atoms with the spin –orbit splitting ΔLS«Te(Al, Ga, In) genera- tion occurs according to the three-level generation scheme. The polarization effects were explained, considering magnetic sub-levelsand assuming that they practically do not mix due to collisionswith the plasma electrons during the pumping –generation cycle. The absence of the polarization in the generation n 0s2S1/2,±1/2→np 2P1/2,±1/2under the pumping np2P1/2,±1/2→n00s2S1/2,±1/2is explained by equal depletion of the magnetic sublevels of np2P1/2 by the linearly polarized pumping light and equally probable transitions to these sublevels, giving πand σ±generation and as a result nonpolarized generation. Accordingly, pumping np2P3/2, ±1/2 ±3/2→n0s2S1/2, ±1/2/leaves the 2 P3/2, ±3/2states fully populated, preventing the σ±transitions (2S1/2, ±1/2→2P3/2,∓1/2, ΔM=ΔM=± 1) due to the self-absorption. As a result, only π polarization transitions (2S1/2, ±1/2→3p2P3/2, ±1/2ΔM=0) lase. The absence of polarization in the Tl LIPL generation, predicted by these considerations, confirms the proposed model in the Al,Ga, and In LIPL. Polarization properties of the 13th groupelements pumped to the 2D*/2states need additional theoretical consideration, likely requiring a polarization-resolved collisional- radiative model. DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material. REFERENCES 1L. Nagli and M. Gaft, Opt. Commun. 354, 330 (2015). 2L. Nagli1, M. Gaft, I. Gornushkin, and R. Glaus, Opt. Commun. 378, 41 (2016). 3R. Glaus, I. Gornushkin1, and L. Nagli, Appl. Opt. 56, 3699 (2017). 4I. Gornushkin, R. Glaus, and L. Nagli, Appl. Opt. 56, 695 (2017). 5I. B. Gornushkin and A. Y. Kazakov, J. Appl. Phys. 121, 213303 (2017). 6L. Nagli1, M. Gaft, Y. Raichlin, and I. Gornushkin, Opt. Commun. 415, 127 (2018). 7L. Nagli1 and Y. Raichlin, Opt. Commun. 447, 51 (2019). 8A. Mironov and J. Eden, in Frontiers in Optics/Laser Science 2016, paper JTh2A.12. 9A. E. Mironov and J. G. Eden, Opt. Express 25, 29676 (2017). 10A. E. Mironov, J. D. Hewitt, and J. G. Eden, Phys. Rev. Lett. 118, 113201 (2017). 11H. M. Milchberg and J. C. Weisheit, Phys. Rev. A 26, 1023 (1982). 12A. K. Sharma and R. K. Thareja, J. Appl. Phys. 98, 033304 (2005). 13A. K. Sharma and R. K. Thareja, Appl. Surf. Sci. 253, 3113 (2007). 14G. A. Wubetu, H. Fiedorowicz, J. T. Costello, and T. J. Kelly, Phys. Plasmas 24, 013105 (2017). 15N. Fortson and B. Meckel, Phys. Rev. Lett. 59, 1281 (1987). 16See http://physics.nist.gov/PhysRefData/ASD/lines_form.html for NIST, Atomic Spectra Database Lines. 17P. Smith, C. Heise, J. Esmond, and R. Kurucz, see http://www.pmp. uni-hannover.de/cgi-bin/ssi/test/kurucz/sekur.html . 18Y. Zel ’dovich and Y. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic (Phenomena, Dover Publications, Mineola, NY, 2002), pp. 382 –421. 19B. Rus, C. L. S. Lewis, G. F. Cairns, P. Dhez, P. Jaegle, M. H. Key, D. Neely, A. G. MacPhee, S. A. Ramsden, C. G. Smith, and A. Sureau, Phys. Rev. A 51, 2316 –2327 (1995). 20U. Fano and J. H. Macek, Rev. Modern Phys. 45, 553 (1973). 21K. Janulewicz, C. Kim, H. Stie, T. Kawachi, M. Nishikino, and N. Hasegawa, Proc. SPIE 9589 , 958901 (2015). 22S. Kazantsev and J. Hénoux, Polarization Spectroscopy of Ionized Gases (Springer-Science + Business Media, B.V, 1995).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 013103 (2021); doi: 10.1063/5.0031063 129, 013103-6 Published under license by AIP Publishing.

5.0031587.pdf

J. Appl. Phys. 129, 035701 (2021); https://doi.org/10.1063/5.0031587 129, 035701 © 2021 Author(s).Synthesis and characterization of amorphous Fe2.75Dy-oxide thin films demonstrating room-temperature semiconductor, magnetism, and optical transparency Cite as: J. Appl. Phys. 129, 035701 (2021); https://doi.org/10.1063/5.0031587 Submitted: 01 October 2020 . Accepted: 29 December 2020 . Published Online: 19 January 2021 Krishna Prasad Koirala , Aniruddha Deb , Sara Bey , Tatiana Allen , Ritesh Sachan , Venkatanarayana Prasad Sandireddy , Chenze Liu , Gerd Duscher , James Penner-Hahn , and Ramki Kalyanaraman ARTICLES YOU MAY BE INTERESTED IN Phase change materials in photonic devices Journal of Applied Physics 129, 030902 (2021); https://doi.org/10.1063/5.0027868 Observation of asymmetric explosive density evolution in the deflagration-to-detonation transition for porous explosives Journal of Applied Physics 129, 035901 (2021); https://doi.org/10.1063/5.0031340 Single-photon emission computed tomography in the scattering medium with the property of “scattering straight back” Journal of Applied Physics 129, 035101 (2021); https://doi.org/10.1063/5.0026165Synthesis and characterization of amorphous Fe2.75Dy-oxide thin films demonstrating room-temperature semiconductor, magnetism,and optical transparency Cite as: J. Appl. Phys. 129, 035701 (2021); doi: 10.1063/5.0031587 View Online Export Citation CrossMar k Submitted: 1 October 2020 · Accepted: 29 December 2020 · Published Online: 19 January 2021 Krishna Prasad Koirala,1,a) Aniruddha Deb,2Sara Bey,3 Tatiana Allen,3 Ritesh Sachan,4 Venkatanarayana Prasad Sandireddy,5Chenze Liu,6Gerd Duscher,6,7James Penner-Hahn,2 and Ramki Kalyanaraman5,6,b) AFFILIATIONS 1Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA 2Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, USA 3Department of Chemistry and Physics, University of Tennessee, Chattanooga, Tennessee 37403, USA 4Department of Mechanical Engineering, Oklahoma State University, Stillwater, Oklahoma 74078, USA 5Department of Chemical and Biomolecular Engineering, University of Tennessee, Knoxville, Tennessee 37996, USA 6Department of Materials Science and Engineering, University of Tennessee, Knoxville, Tennessee 37996, USA 7Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA a)Electronic mail: kkoiral2@vols.utk.edu b)Author to whom correspondence should be addressed: ramki@utk.edu ABSTRACT Recently, amorphous/disordered oxide thin films made from Fe and lanthanides like Dy and Tb have been reported to have a rich set of magnetic, optical, and electronic properties, as well as room-temperature magneto-electric coupling with multiferroics [A. Malasi et al. , Sci. Rep. 5, 18157 (2015); H. Taz et al. ,S c i .R e p . 6, 27869 (2016); and H. Taz et al. , Sci. Rep. 10,1–10 (2020)]. Here, we report the synthesis and detailed characterization of Fe 2:75Dy-oxide thin films prepared on various substrates using electron beam co-evaporation. The structure, chemistry, electric, magnetic, and optical properties were studied for the as-prepared and annealed (373 K, in air, 1 h)films of thickness 40 nm. High resolution transmission electron microscopy and electron diffraction study showed that the films were amorphous in both the as-prepared and annealed states. The electron energy-loss spectroscopy studies quantified that metal oxygen stoichiometry changed from Fe 2:75Dy-O 1:5to Fe 2:75Dy-O 1:7upon annealing. Synchrotron-based x-ray absorption spectroscopy investiga- tion confirmed that the as-prepared films were highly disordered with predominantly metallic Fe and Dy states that became slightlyoxidized with annealing in air. The as-prepared amorphous films demonstrated significantly high value of ordinary ( /difference10 cm 2/V s) and anomalous ( /difference102cm2/V s) Hall mobility and high electrical conductivity of /difference103S/cm at room temperature. The cryogenic magnetic property measurement shows two-step magnetization below 200 K, suggesting exchange-spring magnetic interaction. The nature of the field cooled and zero-field cooled curves suggested a spin-glass like transition between 78 K and 80 K, with a characteristic broad peak.The Tauc plot analysis from optical transmission spectra confirms the existence of an optical bandgap of /difference2:42 eV that increased slightly to/difference2:48 eV upon annealing. This rich set of transport, optical, and magnetic properties in these thin films is very exciting and points to potential applicability in low-cost multifunctional devices requiring a combination of transparent, semiconducting, and magnetic responses, such as in spintronics. Published under license by AIP Publishing. https://doi.org/10.1063/5.0031587Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 035701 (2021); doi: 10.1063/5.0031587 129, 035701-1 Published under license by AIP Publishing.I. INTRODUCTION Iron –lanthanide thin films, which are sensitive to oxidation, have been investigated due to their potential technological applica- tions in all optical switching and magneto-optical and thermo- magnetic recording devices.4–7Additionally, studies have also been focused on iron-based magnetic oxides due to the high temperature (above room temperature) Curie point.8,9However, the major focus of most studies aimed at utilizing the electronic and magnetic prop- erties of such materials has been toward optimizing synthesis and chemistry of the crystalline forms of the material. Recently, emergent properties were reported for amorphous FeTbDy-oxide (FTDO) thin film systems since they showed prom- ising rich characteristics of high carrier conductivity ( /difference5/C2104S/ m), very high amorphous-state Hall mobility ( /difference30 cm2/V s), good optical transparency ( .50%), and ferromagnetism at room temper- ature.1These unique set of properties and the relatively unknown role of the vastly available binary and ternary composition space for Fe and the lanthanides make this an interesting system to explore further for a variety of applications, including transparent electronic oxide devices which are currently dominated by other systems in the market like ITO, FTO, and ZTO.10,11The simultane- ous presence of room temperature semiconducting and ferromag- netic properties as well as the recent demonstration of stable coupling with BiFeO 3resulting in exciting magneto-electric coupling at room temperature provide added motivation to investigate the Fe–lanthanide oxide material system, especially as a promising candi- date for room temperature transparent spintronic applications.3 The rich properties shown by this oxide system are quite likely dependent on various factors, including the growth conditions, composition, and the oxidation state of the material.2Interestingly, the high conductivity and mobility in rare-earth iron-based oxide film is proposed to be dominated by oxygen mediated coupling of partially filled 3d orbitals of transition metals and 4f or 5d orbitals of lanthanides, which are considered highly localized and insensi- tive to angular variation in metal –oxygen –metal bonds.12,13If such an amorphous thin film material could be found with a useful combination of properties that are comparable to their crystalline forms, they could be of great value due to the ability to synthesize them at lower thermal budgets as well as eliminating the need to optimize processes to remove or reduce crystalline defects like grain boundaries. In this work, we investigated the iron –lanthanides system by trying to address the question of whether the ternary metal compo- sition is critical to the observed properties discussed above. We have chosen a binary metallic system made from Fe and Dy and investigated the properties of the disordered/amorphous oxide thin films synthesized by co-evaporation technique. The Fe-composition was chosen to maintain a general overall iron to lanthanide ratio close to the FTDO composition of 64% Fe to 36% lanthanides investigated by Malasi et al.1We utilized multiple characterization techniques to investigate and understand the structure, transport behavior, magnetism, and optical properties of Fe 2:75Dy-oxide thin films at room temperature as well as low temperatures. X-ray absorption spectroscopy (XAS) and transmission electron micros- copy investigations showed that the as-prepared films are amor- phous. XAS confirmed the increased oxidation of the as-preparedthin film on annealing in air. The as-prepared as well as annealed films showed both ordinary Hall mobility and anomalous Hall mobility at room temperature. The low-temperature resistivitymeasurement did not show hysteresis on heating and cooling,thereby confirming that there is no irreversible structural changebelow room temperature. At temperature below 200 K, the two-step magnetization was observed, suggesting spring-exchange type magnetic behavior. The cryogenic magnetic investigations showedspin-glass-like transition at 79.3 K and 78.1 K with broad peaks foras-prepared and annealed samples, respectively. The films exhibitedan optical bandgap in the visible range. II. METHODS A. Materials synthesis Fe 2:75Dy-oxide thin films of thicknesses 40 +0:5 nm were synthesized by co-depositing the metallic elements by electronbeam evaporation (Quad EV evaporator) under a background oxygen pressure of 3 /C210 /C08Torr. The films were deposited on Kapton, Quartz (Qz), and thermally grown SiO 2on Si substrates at room temperature. Prior to deposition, the Kapton foil was ultra-sonically cleaned in acetone for 30 min, followed by rinsing inde-ionized (DI) water for 30 min. Qz and SiO 2/Si were cleaned in acetone, ethanol, and DI water for 30 min each. Iron (99.995% purity) and dysprosium (99.99% purity) were obtained from AlfaAesar and MSE Supplies, respectively. The deposition rate of thefilms was typically 0.01 nm per second. Prior to deposition, inorder to achieve the Fe:Dy ratio of 2.75, the deposition rate was periodically calibrated in different powers and flux currents by atomic force microscopy (AFM). Following deposition, the sampleswere stored under ambient temperature and pressure conditions. B. Materials characterization X-ray absorption spectroscopy investigations were performed at the Sector-20-ID beamline of the Advanced Photon Source at Argonne National Laboratory. The sector 20 undulator is equipped with a Si (111) double crystal monochromator that is nitrogencooled and a Rh coated harmonic rejection mirror. The incidentintensity was monitored by a nitrogen filled ion chamber, while theenergy was calibrated using a Dy or Fe foil inserted downstream of the sample. The x-ray absorption spectra were measured in the fluorescence mode with a multi-element solid-state Ge detector.The extracted x-ray absorption near edge structure (XANES) datawere analyzed using the MBACK program, while for the extendedx-ray absorption fine structure (EXAFS) data were analyzed using the EXAFSPAK software package. 14,15The raw data were first imported in EXAFSPAK in order to perform the EXAFS analysis;FEFF9 was used to calculate the amplitude and phase parameters. 16 First of all, a pre-edge background, followed by a cubic splineEXAFS background were subtracted. The normalization of the EXAFS signals was achieved using an appropriate Victoreen func- tion to model the absorption decrease above the edge. 17 High resolution transmission electron microscopy (HRTEM) images, electron diffraction patterns, and electron energy-loss spec- troscopy (EELS) spectra were taken utilizing a Zeiss Libra 200MC at an acceleration voltage of 200 kV equipped with spectrumJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 035701 (2021); doi: 10.1063/5.0031587 129, 035701-2 Published under license by AIP Publishing.processing tools. EELS spectra were recorded in the scanning TEM mode and analyzed along the cross section of the Fe 2:75Dy-oxide films to investigate the homogeneity and atomicity of the thin filmsby the method used by Tian et al. 18A brief introduction to the methodology used to quantify the atomicity is presented in thesupplementary material . This method was previously verified to calculate the areal atomic density of the material system within 1% of accuracy. 18Cross-sectional TEM samples were prepared by the focused ion beam (FIB) method. For each film, a cross section ofabout 2 μm in length and /difference80 nm in thickness was milled out using Gallium as the ion source. Final polishing of the sample was done at a beam current of 20 pA at 10 kV in cross-beam ZEISS AURGIA. Prior to milling, a protective layer of carbon (0.5 μm thick) was deposited on the surface of the FeDy-oxide film. In order to study the transport properties of the samples, cryo- genic and room temperature resistivities were measured with the physical property measurement system (PPMS) from quantum design using the standard AC mode in a helium cooling cryostat.The van der Pauw configuration was utilized to measure the resis-tivity values of the films. In order to measure the Hall effect on thefilm system, the 4-probe transverse Hall resistivity ( ρ xy) was mea- sured at a constant current of 10 μA. High purity silver epoxy from SPI Supplies and gold wires of 0.25 mm diameter were utilized forboth contact electrodes and wiring. The Hall magnetoresistivitywas studied up to an applied field of 5 T, at room temperature. The magnetic properties of the thin films were investigated using the Superconducting Quantum Interference Device (SQUID)from Quantum Design. Investigations were performed to study thelow temperature magnetization effect, as well as field dependentmagnetization behavior of the system. Measurement was done using a plastic drinking straw as the sample holder. The linear optical transmission of the samples was measured by using the HR2000+ES spectrometer from Ocean Optics in thetransmission mode. III. RESULTS A. Structural properties The XAS measurements were performed on the Fe 2:75Dy-oxide thin films deposited on Kapton at both the Fe K-edge and the Dy L 3-edge. XANES spectra for the thin films for both edges are shown in Figs. 1(a) and 1(b). Fe-XANES shows a shift of the edge to a higher energy for the annealed film than theas-prepared film. This indicated that the annealed film is relativelyFe-oxidized than the as-prepared film. A change in the Dy-XANES[Fig. 1(b) ] for the samples was also observed. The intensity of 2p –5d transition at /difference7995 eV in the Dy-XANES spectra of the as-prepared sample was lower than the annealed samples, indicating a slightincrease in the oxidation state of the metal on annealing. The most striking observation from the EXAFS is how weak the signals are in comparison with those seen for ordered metal oxides. The nearest neighbor scattering is relatively weak, as seen both from the amplitude of the oscillations and also from the largevalues of the fitted Debye –Waller factors. There is almost no outer shell scattering as seen both from the nearly sinusoidal behavior of the EXAFS and also from the fourier transforms. From fits of the EXAFS data (as shown in Table I ), there is clear evidence for bothFe–O scattering at 1.97 Å and Fe –Fe scattering at 2.47 Å, albeit with unusually large Debye –Waller factors. 19–21The former is con- sistent with γ-Fe2O3as suggested by Okudera et al., while the latter Fe–Fe scattering is consistent with the short Fe –metal bond dis- tance in metallic iron ( /difference2:47 Å).22–24There was also evidence for longer distance Fe –Fe scattering at /difference2:6 Å. This is similar to the Fe–Fe distance found in the Fe 2Dy alloy and significantly shorter than the Fe –Fe distance of /difference2:9 Å found in Fe-oxides. These data establish qualitatively that the film gets converted from moreFe/less Fe 2O3to less Fe/more Fe 2O3on annealing and suggest the presence of at least a portion of the Fe as an Fe 2Dy alloy.22The dysprosium EXAFS is dominated by a single frequency, attributable to an average Dy –O distance of 2.28 Å, in good agreement with the average Dy –O nearest neighbor distance of 2.29 Å seen in Dy 2O3.25 Although it was possible to fit these data using two Dy –O shells at /difference2:34 Å, the short k range and relatively high noise level of these data do not support such an analysis. The Fe 2:75Dy-oxide thin films were also studied by transmis- sion electron microscopy. The HRTEM image and correspondingselected area electron diffraction (SAED) pattern of the as-preparedFe 2:75Dy-oxide films are shown in Figs. 2(a) and2(c). The HRTEM image of the as-prepared film showed that the film does not possess any crystalline phases or metal clustering. This was furtherconfirmed by the SAED pattern as shown in Fig. 2(c) . The ring electron diffraction pattern contained only a single broad diffused ring centered at 2.42 Å, generally observed for disordered/amor- phous atomic systems, which shows the absence of both long-rangeand short-range ordering. The intensity analysis of the diffusedpeak showed the value of full width at half maxima (FWHM) to be1.04 Å. This diffused broad ring can be interpreted to be a result of various Fe –O, Fe –Fe, Fe –Dy, and Dy –O bonds as also indicated by the XAS result as shown in Table I .F u r t h e r m o r e , Figs. 2(b) and2(d) show the cross-sectional HRTEM and SAED image for the annealedfilm. As evident from Fig. 2(b) , the HRTEM of the annealed film did not show the presence of any crystal line grains and clustering similar to the as-prepared film. However, the SAED pattern showed evidence of two diffused peaks, rather than a single broad peak, with one at2.99 Å and the other at 2.01 Å. The presence of two diffuse peakscould be an initiation of short-range atomic ordering due to structuralrelaxation through annealing. The EELS measurement along the cross section of the samples was used to quantify the distribution of iron, dysprosium, as well asoxygen on the films. The boxes shown by the yellow dashed linesinFigs. 2(a) and 2(b) indicate the regions from where the EELS spectrum mapping was done for the as-prepared and the annealed samples, respectively. The quantification and distribution of Fe, Dy,and O atoms for the as-prepared and the annealed films across thecross section of the films are shown alongside. As evident fromFigs. 2(a) and 2(b), the distribution of oxygen atoms is larger toward the surface region in both the as-prepared and annealed films. A plot comparing oxygen atoms distribution (in %) in theas-prepared and annealed films is presented in the supplementary material . The quantification of the EELS plot shows that the Fe to Dy ratio remains constant throughout the film, while relatively more oxygen to metals ratio on the surface side, quantitatively, by /difference8% in the as-prepared films and /difference14% in the annealed film. This can be attributed to the oxidizing nature of iron andJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 035701 (2021); doi: 10.1063/5.0031587 129, 035701-3 Published under license by AIP Publishing.dysprosium atoms at ambient temperature and pressure. As the films were stored in the ambient condition after deposition, thesurface Fe and Dy atoms, which are exposed directly to the atmo- sphere are likely to get oxidized. Moreover, parallel to the film surface, we did not see any change in the metal –oxygenstoichiometry. EELS maps with slightly varying Fe L 3, O K, Dy M 5 intensities are likely due to slight variations in the thickness of the sample. The overall quantification of the system showed that metals –oxygen stoichiometry of the annealed film was Fe2:75Dy-O 1:7compared to Fe 2:75Dy-O 1:5. FIG. 1. (a) Fe-XANES, (b) Dy-XANES, (c) Fe-EXAFS, and (d) Dy-EXAFS of the as-prepared and annealed films.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 035701 (2021); doi: 10.1063/5.0031587 129, 035701-4 Published under license by AIP Publishing.From EELS data taken along the cross section of the sample, the average white line ratio for Fe L- edges and the M 5/M4ratio for Dy M-edges were calculated and are shown in Fig. 3(a) and3(b). The white line ratio for Fe in Fe 2:75Dy-oxide as-prepared film was calculated to be 3.05 (close to pure Fe, L 32¼2:99).26This signifies that more of the Dy atoms are oxidized than Fe atoms, which is consistent with the fact that Dy atoms show more affinity to the oxygen atoms compared to Fe atoms. B. Transport properties The transport properties of the Fe 2:75Dy-oxide films on SiO 2/Si substrates were investigated in the temperature range of 3K–320 K for both as-prepared and annealed films. The electrical conductivity was measured while cooling the sample from a tem- perature of 320 K to 3 K and again heating the sample back to 320 K. The observed electrical conductivity of the as-prepared filmat 320 K was 8 :25/C210 3S/cm, which decreased to 4 :33/C2103S/cm as temperature was decreased to 3 K, as shown in Fig. 4(a) .A similar trend was also observed for the annealed sample, but with a slightly lower conductivity of 6 :71/C2103S/cm that decreased to 3:91/C2103S/cm as the temperature was lowered from 320 K to 3 K, respectively. Moreover, the room temperature conductivity wasobserved to be 7 :40/C210 3S/cm and 6 :20/C2103S/cm for the as-prepared and annealed films, respectively. Furthermore, no hys- teresis was observed in the heating and cooling cycles for theas-prepared or the annealed samples as is evident from Fig. 4(a) . The as-prepared film ’s behavior is consistent with semiconducting like behavior, with ordinary Hall mobility of 10.8 cm 2/V s and an anomalous Hall mobility of 156.5 cm2/V s. Compared to the as-prepared case, the annealed sample showed a slight decrease inboth ordinary (8.6 cm 2/V s) and anomalous Hall mobility 132.2 cm2/V s. Figure 4(b) shows the plot for transverse resistivity (magnetic field is perpendicular to the plane) with respect to the magnetic field strength of the antisymmetrized data. The transverse Hall resistivity showed a nonlinearity in magnetic field strength(B). This behavior is generally expected for a system with ferromag-netic interaction where resistivity is also dependent on magnetiza- tion of the system. The details of Hall mobilities calculations are presented in the supplementary material .C. Magnetic properties The magnetic properties of as-prepared and annealed Fe 2:75Dy-oxide films were investigated by measuring the magnetic field and temperature dependent magnetization at different tempera- tures between 5 K and 300 K. The magnetic hysteresis loops for dif- ferent temperatures (300 K, 200 K, 100 K, 50 K, and 5 K) arepresented in the supplementary material . At low temperatures (100 K, 50 K, and 5 K), the magnetic hysteresis loops exhibited akink under low applied fields, for both as-prepared and annealed films as evident from Fig. 5(a) measured for 50 K. The presence of a kink near applied low field value confirms that the magnetization atlow temperature is two steps as evident from Fig. 5(b) . For the high temperatures (300 K and 200 K), the hysteresis loop did not showany kinks. A plot of coercivity field (H c) and remanent magnetiza- tion (M R) with temperature are shown in Fig. 5(c) . The increased value of H cand M Rwith a decrease in temperature indicates the fer- romagnetic type interaction, which is generally contributed byanisotropy and/or magnetoelastic energy. The coercive field for theas-prepared sample tended to be larger than the annealed sample, while the remanence field was comparable for both samples. The zero-field cooled (ZFC) and field cooled (FC) magnetization data(M) was measured as a function of temperature (T) under an exter-nal applied field of 0.5 T from 3 K to 300 K. The ZFC and FC mag-netization was observed to gradually increase until the bifurcation of the curve, which started at 85 K, as shown in Fig. 5(d) . The maxima of zero field cooled curve for the as-prepared film appeared at79.3 K, while the annealed sample had a peak at 78.1 K, which repre-sents the blocking temperature of the thin films, with both peaks being extremely broad. In addition, the spin freezing temperatures which are associated with collective freezing of uncompensatedmoments, were observed at 7.8 K and 6.2 K for as-prepared andannealed films, respectively, as indicated in the low-temperaturemagnetization measurement shown in the supplementary material . D. Optical properties The linear optical transmission behavior of thin films was studied at room temperature. The transmission spectra were mea- sured as a function of wavelength for films deposited on Quartzsubstrates. The optical transparency of the films was measured tobe/difference20% and /difference23% for as-prepared and annealed films as shown inFig. 6(a) . Tauc plot analysis was used to estimate the indirect optical bandgap, as shown in Fig. 6(b) . The indirect optical bandgap was found to be /difference2:42 eV for the as-prepared film while the annealed film showed an increase in bandgap, which was esti-mated to be /difference2:48 eV within an uncertainty of 1.2%. The slight increase in the value of optical bandgap with annealing the sample up to 373 K may be due to structural relaxation on the system pro- ducing stronger bonding. IV. ANALYSIS AND DISCUSSION From the XAS, PDF measurement, and electron microscopy and spectroscopy study, it is evident that the vacuum co-depositedFe 2:75Dy-oxide films are disordered and become oxidized on annealing at ambient pressure with more of the surface Fe and Dy atoms being oxidized than in the inner side of the films. Unlike theTABLE I. Summary of the EXAFS analysis for the XAS spectra for both the Fe K-edge and Dy L 3-edge for Dy 2.75Fe–oxide thin films of the as-prepared and annealed states. Here, N is the coordination number, R is scatter distance (Å), σ2 (Å2) is the Debye –Waller factor, and F is the goodness of fit. Sample Path N R σ2×1 0−3F As-prepared Fe –O 0.7 1.96 6.4 24 Fe–Fe 1.8 2.45 7.2 Fe–Fe 1.1 2.61 8.1 Dy–O 3.0 2.28 8.7 90 Annealed Fe –O 1.1 1.97 6.6 21 Fe–Fe 1.8 2.47 7.5 Fe–Fe 1.0 2.62 7.8 Dy–O 3.0 2.27 7.5 69Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 035701 (2021); doi: 10.1063/5.0031587 129, 035701-5 Published under license by AIP Publishing.FIG. 2. (a) As-prepared film: cross-sectional HRTEM image of Fe 2:75Dy-oxide showing a plot with relative atomic percentage along the depth of the sample. The yellow box is the region from where the EELS spectrum map was taken; EELS spectrum mapping for Dy M 5, O K, and Fe L 3-edges, the intensity bar shows the index for electron counts; the light blue line in the yellow box separate the region along the depth of the sample with two distinct metal –oxygen stoichiometry. (b) Annealed film: cross- sectional HRTEM image of Fe 2:75Dy-oxide showing a plot with relative atomic percentage along the depth of the sample. The lower yellow box is the region from where the EELS spectrum map was taken; EELS spectrum mapping for Dy M 5, O K, and Fe L 3-edges, the intensity bar shows the index for electron counts; the light blue line in the yellow box separate the region along the depth of the sample with two distinct metal –oxygen stoichiometry; (c)selected area electron ring diffraction pattern of the as-prepared film, which shows the presence of a diffuse broad ring at 2.42 Å; (d) selected area electron ring diffraction pattern of the as-prepared fi lm, which shows the presence of two diffuse rings one at 2.99 Åand the other at 2.01 Å, suggesting the onset of short-range atomic ordering upon annealing.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 035701 (2021); doi: 10.1063/5.0031587 129, 035701-6 Published under license by AIP Publishing.previous investigation on FTDO thin films,1where the x-ray photo- electron spectroscopy (XPS) measurements showed the presence of significant amounts of Fe, Dy, and Tb cationic states on the thinfilm (.75%), here from XAS and EELS studies, we see that thin Fe 2:75Dy-oxide films are dominated by the metallic character of Fe, Fe2Dy with a possibility of some of Dy in the Dy 2O3form. One possible cause for this difference in observation can arise from the fact that, for XAS, we probe the bulk of the sample, while XPS isbeing more sensitive to the surface composition. This is further evi-denced from the EELS quantification we performed along the crosssection of Fe 2:75Dy-oxide films where the surface region of the films contained more oxygen atoms compared to the inner region.The low-temperature transport study showed the conductivity of the Fe 2:75Dy-oxide films increased with temperature indicating semiconducting nature of the thin films. In order to understand thesemiconducting behavior of these thin films under low temperatures,we fit the experimental data to the well-known variable rangehopping mechanisms. The variable range hopping Mott behavior 27 and Efros –Shklovskii28behavior are expressed, respectively, as σ(T)¼σMγexp (/C0TM=T)γ, (1) σ(T)¼σEγexp (/C0TE=T)γ, (2) FIG. 3. The energy loss peaks of (a) Fe L-edges; (b) O K-edge; (c) Dy M-edges measured for the as-prepared and annealed films. FIG. 4. (a) The low-temperature conductivity measurement of Fe 2:75Dy-oxide on the as-prepared and annealed films; (b) the transverse resistivity vs magnetic field strength shown for antisymmetrized data at 300 K.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 035701 (2021); doi: 10.1063/5.0031587 129, 035701-7 Published under license by AIP Publishing.where σMγandσEγare the Mott conductivity parameter and Efros – Shklovskii conductivity parameter, respectively. TMis the Mott char- acteristic temperature whose value depends on the localizationlength and density of state in the Fermi level, while T Eis the Efros – Shklovskii characteristics temperature whose value depends on thelocalization length and dielectric constant , [ γ¼1=(1þd); d being dimension of localized states] is related to the dimensionality of the localized state at valance band. For materials showing Mott behavior,theγvalue is close to 0.25, which refers to three-dimensional con- ductance, while for the material showing Efros –Shklovskii behavior, FIG. 5. (a) Measured magnetic hysteresis curve at 50 K showing hysteresis loop with the presence of kinks indicating the presence of two step magnetization on the both as-prepared and annealed samples; (b) the measured dM/dH vs applied field for the as-prepared sample at 50 K confirming the magnetization process is t wo steps; (c) plot of coercivity and remanence of hysteresis measured at different temperatures for both annealed and as-prepared samples; (d) observed FC –ZFC of the as-prepared and annealed thin films measured under an applied field of 0.5 T . The maxima of ZFC were observed at 79.3 K and 78.1 K for as-prepared and annealed thin fil ms, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 035701 (2021); doi: 10.1063/5.0031587 129, 035701-8 Published under license by AIP Publishing.γis close to 0.5 for all dimensions. However, the observed values were reported to be between 0.15 and 0.75, for an amorphous/ disorder system.29To find the characteristic temperatures (Mott or Efros-Shklovskii) of the Fe 2:75Dy-oxide thin films, the above Eqs. (1) and(2)can be rewritten as σ(T)¼σγexp (/C0Tγ=T)γ, (3) ln (W)¼/C0 ln (γTγ)þγ:ln (T), (4) where W¼[/C0dlnσ(T)=dln (T)]. The plot of ln ( W)v sl n ( T)i s shown in Fig. 7 . The ln ( W)v sl n ( T) only fit to the straight line at the low-temperature region from 3 K to 85 K with values of γto be 0.74 and 0.66 for as-prepared and annealed thin films, respectively.This confirms that at low temperatures ( ,85 K), the conductivity on films is more like governed by the Efros –Shklovskii type behavior. The characteristic temperatures (Efros –Shklovskii) were calculated to be 9.4 K and 9.1 K for as-prepared and annealed films, respectively.At the temperatures greater than 85 K, γvalue was greater than 0.75. This suggests that the high temperature ( .85 K) behavior is not gov- erned by the variable range hopping mechanism. To understand the high temperature behavior, thermal activation model was appliedusing σ(T)¼σ 0exp(/C0Ea=KT), (5) lnσ(T)¼lnσ0/C0Ea=KT, (6) where σ0,Ea,a n d Kare the pre-exponential value, activation energy, and Boltzmann ’s constant and analyzed by plotting ln{ σ(T)} vs 1/ T. The activation energy was found to be temperature dependent,which is plotted in the inset of Fig. 7(a) . It is evident that thermal activation energy is slightly higher for the as-prepared film than theannealed film. Furthermore, the negative slope of the Hall resistivity curve as shown in Fig. 4(b) confirms that the system confirms an n-type semiconducting behavior. In addition, the negative Hallmagneto-resistivity of 3.6% and 1.9% at saturating fields of 5 T wasobserved for the as-prepared and annealed samples, respectively.This is surprising in the fact that although magnetization of the films increases (with annealing), the magneto-resistivity is decreas- ing. This can be interpreted as annealing induced slight oxidation ofthe system leads to the decrease in the resistance, which may over-weigh the magnetization effect. The magnetization with respect to temperature contained the rich information about competitive magnetic behavior. The nature of the magnetic interaction between Fe (3d) and Dy(4f) atoms/ionsis generally considered weak (shielded by 5s and 5p orbitals) andnot well-understood. However, it is evident from Fig. 7(b) that Fe 2:75Dy-oxide films obey the Curie –Weiss behavior [1=χ¼C=(T/C0Tcw)] above the temperature of 150 K for the as-prepared and 125 K for annealed films with the Curie –Weiss temperature ( Tcw)o f/difference67:5 K for both films. Following calcula- tions, we obtained the effective magnetic moment to be 5.04 μB and 3.13 μB for the as-prepared and annealed films, respectively. The positive value of Curie –Weiss temperature establishes the fer- romagnetic interaction of Dy and Fe moments. Despite ferromag-netic correlation, we observed the two step magnetization processfor the temperature 100 K, 50 K, and 5 K. This kind of hysteresis loop is observed for the exchange-spring magnetic material due to the presence of hard and soft ferromagnetic phases. 30In fact, the soft magnetic phase nucleates the magnetic reversal at the lowerfield, while the hard magnetic phase reversal occurs at the highmagnetic field. It is expected in Fe 2:75Dy-oxide films due to the dif- ferent values of Dy and Fe moments and the structurally disordered nature of the films where different magnetic phases can co-exists. Itis evident from Fig. 5(c) that the coercivity value of the annealed sample is slightly smaller than the as-prepared sample below 300 K. FIG. 6. (a) The optical transmission (in %) of as-prepared and annealed films; (b) the Tauc plot for analysis of the optical bandgap. The optical bandgaps (ind irect) are /difference2:42 eV ,/difference2:48 eV for as-prepared and annealed films, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 035701 (2021); doi: 10.1063/5.0031587 129, 035701-9 Published under license by AIP Publishing.Annealing at lower temperature could initially release internal strain of the films, which results in the reduction of the coercivity. V. CONCLUSION In summary, we investigated the disordered rare-earth iron oxide Fe 2:75Dy-oxide in the as-prepared and annealed (373 K, air) thin film systems by x-ray spectroscopy and electron microscopy. Our results show that the as-prepared films are disordered withlack of both short-range and long-range ordering, and the oxida-tion and onset of the atomic ordering of the films can be observedwith annealing. In ternary FTDO thin film studied previously, the mechanism of semiconducting behavior was not well documented. 1 In Fe 2:75Dy-oxide films, we saw semiconducting behavior with vari- able range hopping mechanism (Efros –Shklovskii type) at low tem- peratures ( ,85 K) and thermally activated behavior at high temperatures ( .85 K) with anomalous Hall behavior at room tem- perature. Also, in FTDO, the cryogenic magnetic behavior was not investigated. Fe 2:75Dy-oxide films are rich in magnetic properties: we saw the Curie –Weiss behavior (ferromagnetic interaction in the paramagnetic regime) at room temperature, while FTDO wasreported to be ferromagnetic. 1,2The low-temperature magnetiza- tion (below 200 K) is in the form of two steps under low field, which indicates the exchange-spring type magnetic behavior due tothe presence of two magnetic phases (soft and hard) owing to thestructural disorder. The cryogenic magnetic behavior also shows that the system possesses spin-glass-like transition at 78 K –80 K also called blocking temperature. The Fe 2:75Dy-oxide thin filmsoffer the marginal optical transparency of 20% –25% and bandgap at the visible region, while FTDO was reported to be .50% trans- parent but with the higher value of indirect optical bandgap (/difference2:8 eV).1,2The overall studies indicate that Tb could be critical to maintain the transparency in the amorphous Fe –lanthanides oxide system, while Dy plays a more important role in maintainingand enriching the electrical and magnetic properties. Structurally disordered systems showing rich properties as these rare-earth ferrite oxide thin films can be a very effective pathway to realize thecost-effective production and generation of transparent spintronics,semiconductor devices, and magnetic sensors, which are integral parts of our future smart devices. SUPPLEMENTARY MATERIAL See the supplementary material for several details about PDF, EELS, Hall effect, magnetic hysteresis loop, and surface roughnessmeasurements. ACKNOWLEDGMENTS The authors acknowledge NSF Grant No. ECCS1607874. T.A. acknowledges support from the Oak Ridge National Laboratory user Grant No. CNMS2019-239, whereas R.S. acknowledgessupport from faculty startup funding at Oklahoma State University.G.D. acknowledges support from the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. This research used FIG. 7. (a) ln (W)v sl n ( T) plot of Fe 2:75Dy-oxide thin films. The blue and green straight line represents the linear fit line for the as-prepared and annealed films, respec- tively. At ,85 K, the conductivity follows the Efros –Shklovskii-like hopping behavior. The inset shows the thermal activation energy plotted against the temperature. (b) Variation of ZFC 1/ χwith T and corresponding Curie-Weiss fits (blue and green dotted lines) for the as-prepared and annealed films.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 035701 (2021); doi: 10.1063/5.0031587 129, 035701-10 Published under license by AIP Publishing.resources of the Advanced Photon Source Argonne National Laboratory (lines 20-ID and 11-ID-B), a U.S. Department of Energy (DOE) Office of Science User Facilities operated ContractNo. DE-AC02-06CH11357. J.P.-H. and A.D. would like to thankDr. Olaf Borkiewicz for his contributions and Kevin Beyer for hisassistance with PDF data collection at 11-ID-B. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1A. Malasi, H. Taz, A. Farah, M. Patel, B. Lawrie, R. Pooser, A. Baddorf, G. Duscher, and R. Kalyanaraman, “Novel iron-based ternary amorphous oxide semiconductor with very high transparency, electronic conductivity, and mobil- ity,”Sci. Rep. 5, 18157 (2015). 2H. Taz, T. Sakthivel, N. Yamoah, C. Carr, D. Kumar, S. Seal, and R. 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5.0033263.pdf

Chem. Phys. Rev. 1, 011305 (2020); https://doi.org/10.1063/5.0033263 1, 011305 © 2020 Author(s).Charge and energy transfer in the context of colloidal nanocrystals Cite as: Chem. Phys. Rev. 1, 011305 (2020); https://doi.org/10.1063/5.0033263 Submitted: 14 October 2020 . Accepted: 04 December 2020 . Published Online: 23 December 2020 Shawn Irgen-Gioro , Muwen Yang , Suyog Padgaonkar , Woo Je Chang , Zhengyi Zhang , Benjamin Nagasing , Yishu Jiang , and Emily A. Weiss COLLECTIONS This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN Emergence of new materials for exploiting highly efficient carrier multiplication in photovoltaics Chemical Physics Reviews 1, 011302 (2020); https://doi.org/10.1063/5.0025748 Ab initio methods for L-edge x-ray absorption spectroscopy Chemical Physics Reviews 1, 011304 (2020); https://doi.org/10.1063/5.0029725 Rational design of spinel oxides as bifunctional oxygen electrocatalysts for rechargeable Zn- air batteries Chemical Physics Reviews 1, 011303 (2020); https://doi.org/10.1063/5.0017398Charge and energy transfer in the context of colloidal nanocrystals Cite as: Chem. Phys. Rev. 1, 011305 (2020); doi: 10.1063/5.0033263 Submitted: 14 October 2020 .Accepted: 4 December 2020 . Published Online: 23 December 2020 Shawn Irgen-Gioro, Muwen Yang, Suyog Padgaonkar, Woo Je Chang, Zhengyi Zhang, Benjamin Nagasing, Yishu Jiang, and Emily A. Weissa) AFFILIATIONS Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3113, USA a)Author to whom correspondence should be addressed: e-weiss@northwestern.edu ABSTRACT As the number of hybrid systems comprising quantum-confined semiconductor nanocrystals and molecules continues to grow, so does the need to accurately describe interfacial energy and charge transfer in these systems. The earliest work often successfully captured at least qual-itative trends in the rates of these processes using well-known results from F €orster, Dexter, and Marcus theories, but recent studies have showcased how unique properties of nanocrystals drive interfacial energy transfer (EnT) and charge transfer (CT) to diverge from familiar trends. This review first describes how the nanocrystal-ligand system fits, at least superficially, into conventional models for EnT and CT, and then gradually introduces individual properties of nanocrystals that complicate our understanding of EnT and CT mechanisms. Thereview then explores instances in which features of nanocrystals that seem detrimental, such as trap states that introduce non-radiativerecombination pathways and strong spin–orbit coupling, can be controlled or used synergistically to produce a wider range of functionalitythan available in all-molecular donor-acceptor systems. Published under license by AIP Publishing. https://doi.org/10.1063/5.0033263 NOMENCLATURE QD Quantum Dot NPL Nanoplatelet BE Bright Exciton DE Dark Exciton EnT Energy Transfer TTEnT Triplet-Triplet Energy Transfer CT Charge Transfer MEG Multi-Exciton Generation I. INTRODUCTION In this review, we examine the progress in developing a meaning- ful framework and formalism for describing the dynamics of interfa-cial energy transfer (EnT) and charge transfer (CT) between colloidalsemiconductor nanocrystals—including zero-dimensional quantum dots (QDs) and quasi-2D nanoplatelets (NPLs)—and molecules adsorbed to their surfaces. In recent years, research in the nanocrystalfield has extended beyond probes of intrinsic electronic and opticalproperties into engineering of hybrid inorganic-organic colloidal sys-tems, and use of these hybrid systems in energy harvesting, sensing, quantum information, and photocatalysis. The language typically usedto describe the static and dynamic interactions across the inorganic- organic interface of QD-molecule systems is adopted from well- established theory in molecular systems. Swapping molecular donor oracceptor wavefunctions with effective wavefunctions for QDs, these traditional models have had success in predicting EnT/CT rates in many cases. Features that differentiate quantum-confined semicon-ductors from most molecules—such as the ability to accommodate multi-excitons and charged excitons (trions), and the large spin–orbit coupling—conspire to produce mechanisms absent in the all-molecular picture and requires us to consider EnT and CT pathways that involve more than the orbitals on the initial donor site and the final acceptorsite. Instead of trying to minimize this complexity and forcing QD behavior to conform to established molecular models, one can leverage it to enable function not possible with all-molecular architectures. This review aims to highlight ways that QDs serve as donors and acceptors in EnT and CT processes, often through processes that bypass the pathway mediated by “direct coupling” between the orbitals of the band edge exciton of the QD and the frontier molecular orbitals.We build upon previous reviews and comprehensive textbooks written on EnT and CT in semiconductor and molecular systems 1–4as well as research on QD electronic structure,5–8surface ligands,9–13and exci- ton dynamics14–17to focus on cases in which EnT and CT proceed in Chem. Phys. Rev. 1, 011305 (2020); doi: 10.1063/5.0033263 1, 011305-1 Published under license by AIP PublishingChemical Physics Reviews REVIEW scitation.org/journal/cprspite of relatively weak direct electronic coupling between donor and acceptor wavefunctions. We also consider phenomena such as Auger mediated electron transfer, trap mediated energy transfer, and surface mediated photocatalysis as examples of how experiment can push the- ory to evolve. The review is written in three parts, Scheme 1.S e c t i o n IIsets the foundation for the formalism used throughout the review and estab- lishes how coupling between donor and acceptor states is typically described. Section IIIintroduces features of QDs that distinguish them from molecules and explains how these features can mediate and influ- ence EnT and CT. Section IVdescribes how the features described in Sec.IIIinteract with each other and can be utilized synergistically. Overall, rather than wrangling the QD-ligand system into trends pre- dicted by the direct coupling scheme introduced in Sec. II,am o r e holistic theoretical understanding of their electron dynamics combined with recent advances in synthetic control will, we believe, allow for the creation of QD-molecule complexes with greater function than that of the sum of its parts. II. FORMALISM OF ENERGY AND CHARGE TRANSFER, AND AN OUTLINE OF RELEVANT QD ELECTRONIC STRUCTURE In modeling any dynamic process, one of the principal strategies is to treat only a fragment of the system in detail. This strategy under- lies, for example, the separation of the electronic and nuclear Hamiltonians in the Born-Oppenheimer approximation, reduction of the number of electronic states considered in describing energy trans- fer, or description of electron transfer from the perspective of one elec-tron subject to an effective potential. When decomposing a system into fragments, a careful dissection of the system may yield a mostly orthonormal basis set such that the electronic wavefunction describing each fragment does not overlap u mjun/C10/C11 ¼dmn. The single electron energies Emcorresponding to these wavefunctions umare sometimes called “site” energies and wavefunctions, respectively. In descriptions of electron and energy transfer, this partitioning results in “donor” and “acceptor” sites, which, in the QD-ligand hybrid system, typically are assigned to the nanocrystal core and organic ligand, respectively. We shall see later that ligands have the potential to strongly mix withQD surface states, complicating this description, but for the photoin- duced charge/energy transfer described directly below, we assume a weak electronic coupling between QD excitons as the “donor” sites and ligand molecular orbitals as “acceptor” sites. If we further limit our discussion to cases in which Fermi’s golden rule, wnm¼2p /C22hqEmðÞ Vnmjj2, applies (in addition to weak elec- tronic coupling, long time of interest s/C29½ 2p/C22hðÞ =ðEn/C0EmÞ/C138, second-order perturbation theory), in principal any energy or charge transfer rate can be calculated with knowledge of the density of states qand the matrix element Vnmjj ¼Ð WnHWmds/C12/C12/C12/C12). Marcus,18,19 Dexter,20and F €orster21theory are versions of Fermi’s golden rule22 adapted for calculating electron and energy transfer rates, respectively. Here we use EnT to demonstrate how the Vmatrix is applied. Other examples of how this formalism can be applied to other forms of exci- ton dynamics can be found in these useful works.23,24To calculate Vnmjj between two equivalent two-level systems, the wavefunctions for representing possible electron configurations can be expressed as Eq.(1)25and shown schematically in Fig. 1 , 1;3w1¼dad0ajj6d0adajj ðÞ =ffiffi ffi 2p ;1;3w4¼dada0 jj6da0dajj ðÞ =ffiffi ffi 2p ; 1;3w2¼daa0ajj6a0adajj/C0/C1 =ffiffi ffi 2p ;1;3w3¼add0djj6d0dadjj ðÞ =ffiffi ffi 2p ; (1) where j/C1 /C1 /C1j denote Slater determinants, superscripts denote the spin multiplicity, d/arepresent the orbitals of the donor/acceptor, d0/a0are corresponding excited orbitals, and the short bar over an orbital indicates an opposite spin. For the case of EnT, when the donor and acceptor are different species, the initial (R ¼reagent state) and final (P¼product state) wavefunctions can be expressed as a linear combi- nation of w1/C24w4. The electronic coupling term in the Hamiltonian can be written as26 HRP¼dd0aajj Hjjjddaa0j/C10/C11 ¼2dd0jaa0/C0/C1 |fflfflfflfflfflffl{zfflfflfflfflfflffl} 1/C02d0a0jad/C0/C1 |fflfflfflfflfflffl{zfflfflfflfflfflffl} 2þ/C1/C1/C1 ;(2) where term 1 is the Coulombic interaction between the charge transi- tion densities of the donor ( dd0) and the acceptor ( aa0), as in F €orster Part 1 QD* Molecule *0U Vbond Vπ–π1UPart 2 Part 3 +– 0L 1L+– 2L+– SCHEME 1. Images representing the three parts of this review. Section II(Part 1) introduces the electronic structure of excitonic states of QDs. Section III(Part 2) describes how the contributions of semiconductor character to QDs influence their electron dynamics and interactions with molecules. Section IV(Part 3) illustrates how certain elec- tronic, chemical, and structural features of QDs combine to produce unique photophysical phenomena.Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 1, 011305 (2020); doi: 10.1063/5.0033263 1, 011305-2 Published under license by AIP Publishingtheory. Within the dipole-dipole approximation, the electrodynamic interaction can be calculated using the transition dipole moments ofthe donor and the acceptor. The second term is the “Dexter” term thatoriginates from exchange interactions between donor and acceptorsites and accounts for the indistinguishability of the electrons in many-electron wave functions. The first term decays as R /C03, and the second term scales with orbital overlap between the donor and accep-tor sites as e /C0aR,w h e r e Ris the distance between the centroids of the donor and acceptor localized wavefunctions. The Dexter term there-fore is more steeply attenuated with respect to the donor-acceptor sep-aration than is Term 1. Dipole-forbidden or spin-forbidden energy transfer processes, including triplet-triplet energy transfer (TTEnT), can be formulated as a special case of the general energy transfer pre-sented in Eq. (2), with the key distinction that the exchange integral dominates 20because the low charge transition density ( aa0)o ft h e molecular acceptor’s triplet state minimizes coupling of transitionmoments to any of the exciton fine states of the QD. 27 The direct exchange coupling mechanism, which is proposed to mediate TTEnT from CdSe QDs to adsorbed acene ligands, does not straightforwardly account for indirect mechanisms involving interme- diate states, such as a charge-transfer pathway observed for QD-to-molecule TTEnT in other systems. 28,29Even for systems where no spectroscopic signatures from radical ion intermediates are observed,charge transfer states may still play a role in TTEnT. Lessons fromelectron transfer and singlet fission in all-organic systems teach us that even if “real” charge separated intermediates are never populated, “virtual” charge transfer states need to be considered as an additionalsuperexchange coupling pathway. 30–32 In principle, the electronic coupling matrix can be calculated, but in practice, the results depend drastically on the choice of donor andacceptor wavefunctions. Applied to calculating EnT and CT rates ofQD-ligand systems, states localized on the ligand are typicallydescribed using molecular orbitals, but descriptions of the QD exciton wavefunctions range from effective mass models 5,8to more accurate but computationally expensive tight-binding models33–36and atomis- tic approaches.37–41 The popularity of the effective mass model stems both from its intuitive explanation of the effect of QD size on its optical spectra andits ease of modification to include interactions such as spin–orbitcoupling, crystal field splitting, electron-hole exchange, and electro-n–phonon coupling. 5,42–44The effective mass model describes a carrier’s motion under a periodic lattice potential and quantum con- finement. The wavefunction is written as a product of the Bloch wave-function, representing the contribution from crystal structure, and anenvelope wave function, accounting for the confinement. The envelopefunction is expressed as a product of the spherical Bessel j L(then-th root/n;L) and spherical harmonic YL,Mfunctions, which introduce aseries of discrete energy bands represented by the three quantum numbers n,LandM. We can further simply the problem using the “single band approximation,” which assumes weak interactions among QD bands. The result is energies of individual electron/hole levels of En;L¼/C22h2/2 n;L 2mR2,w h e r e Ris the radius of the spherical QD and m¼meor mh. The single band effective mass model yields a bandgap energy (Eg) that is modified from the bandgap of the bulk semiconductor (Eg;0)a s Eg¼Eg;0þ/C22h2p2 2me;hR2; (3) where me;h¼memh meþmhis the reduced electron-hole mass. For QDs with small size, the confinement energy is much larger than the electron- hole binding energy. Consequently, the Coulomb electron-hole inter-action can be treated as a perturbation to the single band effective model, adding a correction term to the bandgap energy D eh¼/C01:765e2 e1R. T h es i n g l eb a n de f f e c t i v em a s sm o d e ls e r v e sa sa ne x c e l l e n t starting point, but it fails to explain certain experimental observationsthat require a more exact treatment of interband coupling, suchas temperature-dependent absorption and emission of QDs.Furthermore, as energy levels become more closely packed, for exam-ple in the valence band of CdSe QDs (where m h¼2.3me), interband coupling becomes too strong to neglect, and the multiband effective mass model is a better description. In the multiband model, the totalmomentum quantum number ( F) is used to describe QD energy lev- els:F¼JþL,w h e r e Jis the angular momentum of the envelope func- tion, and Lis the angular momentum of the Bloch function. The hole states are represented by a notation such as 1 S 3/2,w h e r et h el o w e ro f J andLis the S,P,D… state, and Fis written in the subscript. For CdSe, the lowest energy valence band is a P-type orbital ( L¼1), which is split by spin–orbit coupling ( DSO/C24400 meV) into a twofold degener- ateJ¼1/2 band and a fourfold degenerate J¼3/2 band. A band edge exciton of a CdSe QD comprises an electron in the 1Sconduction band and a hole in the 1 S3/2valence band, leading to a 2x4 ¼eightfold degenerate set of states. As shown in Fig. 2 ,t h i s degeneracy is lifted by a hierarchical series of terms (in decreasingorder of interaction strength): crystal field splitting, electron-holeexchange, and electron-phonon coupling. Specifically, the anisotropic crystal field splits the 1 S 3/2hole states into light ( jJmj¼1/2) and heavy (jJmj¼3/2) holes. The magnitude of the crystal field splitting depends on the composition, crystal structure, and shape anisotropy of thenanocrystal. Electron-hole exchange further lifts the degeneracy fortheJ mstates. Excitons formed by combining the S ¼61/2 electrons and Jm¼63/2 hole states have total angular momentum projections jNmj ¼1 and 2. Since the orbital angular momentum of a single photon is 1, optical transitions of the jNmj¼2 exciton are dipole forbidden; this exciton is therefore generally known as a dark exciton (DE), and thejN mj¼1 exciton is known as a bright exciton (BE). Since the electron and hole quantum numbers need to be of the same sign (thus havingparallel spins) in order to combine to jN mj¼2, the DE has triplet-like character, while the anti-parallel spins of the BE give it singlet-likecharacter. Due to strong spin–orbit coupling in QD, spin is howevernot a “good” quantum number and transitions between exciton statesamount to an exciton fine structure relaxation (EFSR) problem inD* Ad¢ a¢ a d y1 y2 y3 y4 D+ A–D+ A–DA* FIG. 1. Wavefunctions of electron configurations of 2-level system. D ¼donor, A ¼acceptor.Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 1, 011305 (2020); doi: 10.1063/5.0033263 1, 011305-3 Published under license by AIP Publishingwhich the total angular momentum quantum number Nis considered. In calculating EFSR rates, the coupling matrix is composed of two terms.45The first is the exciton–phonon coupling term, which describes the nonadiabatic coupling of the electronic states when the geometry is displaced from the equilibrium. This term is responsible for the nonradiative relaxation with no change in spin multiplicity. The second is the spin–orbit coupling term that leads to intersystem crossing. Even the multiband effective mass model has its limitations, mostly because it fails to capture the atomistic features of the QD. For example, dangling bonds can introduce mid-gap trap states, and sur- face ligands modify the electronic structure of the core.46Atomistic approaches, such as the pseudopotential method,39tight-binding model,47orab initio approaches, have been successful in incorporating some of this complexity into the QD’s electronic structure.48Of course, if these effects do not greatly modify EnT/CT rates, then the effective mass model is sufficient to describe the QD portion of the donor-acceptor system. As described in Secs. IIIandIVof this review, recent experimental observations have required theorists to move beyond this simplistic picture. III. HOW QD FEATURES AFFECT ENERGY/CHARGE TRANSFER IN QD-MOLECULE SYSTEMS In Sec. II, we sketched out some formalism for the simplest cases of EnT and CT, which occur through direct coupling of QD exciton states and ligand molecular orbitals. There are numerous examples where this description is sufficient to capture the dynamics of EnT/ CT. For example, Kodaimati et al. demonstrated that at distances greater than /C244 nm, EnT rates between QDs can be accurately predicted by F €orster theory.49Complications arise when multiple coupling pathways are active, which is often the case for EnT/CT to ligands on the QD surface. Although the potential for a variety of EnT/CT processes expands the capabilities of QDs, competing kinetic pathways and involvement of virtual or real intermediate states muddles predictions of net EnT/CT rates. Additional considerations specific to nanomaterials such as dimensionality and the intermolecu- lar structure of self-assembled ligands may require explicit treatment. In this section, we explore examples of how unique properties of QDs affect EnT/CT with a special emphasis on how an understanding of numerous interacting pathways is necessary to understand EnT/CT in QD-ligand systems.A. Exciton fine structure dynamics Many studies that leverage QDs as energy/charge donors occur at room temperature and as a result do not consider the influence ofintra-band transitions among thermalized exciton states, especiallyamong states of different spin multiplicity, on EnT/CT dynamics. This simplification is valid for most all-molecular cases since the rate of intersystem crossing is so slow that it does not compete with EnT/CTbetween donor and acceptor. In contrast, a QD’s fast spin-flip ratesalong with the non-uniform coupling of excitonic sublevels to acceptorstates may influence ultrafast CT/EnT dynamics since transitions within the exciton manifold may become a rate limiting step. If we start by only considering the jN mj¼1 and 2 exciton states (BE/DE) and a molecular acceptor state, it is clear that the electronic coupling[Eq.(2)] of the BE/DE with the molecule will differ due to the differing transition dipole moments of the BE/DE and differences in exchange integrals, so that direct EnT/CT will preferentially proceed through one of these states. In order to understand which cases of EnT/CT require explicit consideration of the QD’s spin-flip dynamics, the timescales of thesedynamics must be established. In the most studied QD systems, spher- ical CdSe QDs, temperature- and magnetic field-dependent measure- ments of the radiative lifetime constitute some of the earliest evidenceof multiple emitting states of QDs and provide a straightforwardmethod to extract BE/DE transition rates. 50For small ( <2 nm) QDs with a single zero phonon line emission at 2 K, the measured radiative lifetime is /C2470, and 20 ns at 140 K. This observation is rationalized in a three-state kinetic model, shown in Fig. 3 .A tl o wt e m p e r a t u r e s ,t h e BE does not emit before the state is depopulated by transition to theDE, so all emission is from the DE with a long radiative lifetime. 51For larger QDs, spin-flip rates have been reported to be an order of magni-tude slower, allowing for low temperature emission from the BE before the spin-flip can occur. 52The temperature dependence of the spin-flip rate has been modeled by Lounis et al. using the same three-state model as a product of the zero temperature spin-flip rate, c0, and the Bose-Einstein phonon number, NB¼1=½expðDE=kBTÞ/C01/C138,a tt e m - perature TwhereDEis the energy splitting between the BE and DE,1s (e)jm = 1s3 (h) 221+ jm =23+Nm = 0LNm = 0U N = 1 N = 2 Nm = 1L+Nm = 1U+ Nm = 2L+ FIG. 2. Lifting of degeneracy of exciton states through crystal field splitting, electron-hole exchange to form excitons with different total angular momentum N¼1, 2 (bright and dark excitons, respectively). 104 (a)A GF GAGFgthg0 (c) (b)Intensity (a. u.)103 102 101 0 40 80 120 Time (ns) FIG. 3. Emission decay dynamics from CdSe/ZnS QDs with various temperatures. The dark exciton-like decay was observed from a single QD with 16 K (b), com- pared to a single QD emission decay with 140 K (c). (a) is the ensemble emissiondecay spectra at 16 K. Reproduced with permission from Labeau et al. , Phys. Rev. Lett. 90, 257404 (2003). Copyright 2003 American Physical Society.Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 1, 011305 (2020); doi: 10.1063/5.0033263 1, 011305-4 Published under license by AIP Publishingwhich can range from 1 to 20 meV depending on QD size.53Since c0 is reported to be between 1 and 10 ns/C01,a n d kBT¼300K ðÞ /C2525 meV, room temperature spin-flip lifetimes range from /C241 to 500 ps, well within the range to affect EnT/CT dynamics. At room temper-ature, cross-polarized transient grating experiments have demon-strated that this ps-timescale spin-flip is attributed to hole flips, whileelectron spin-flips occur on a nanosecond timescale. 54These disparate timescales have enabled control of the spin polarization of radical pairs produced upon photoinduced charge transfer in QD-moleculesystems. For instance, Wu et al. generated the same charge separated species by excitation of the QD followed by a hole transfer to an aliza-rin surface ligand, or excitation of the ligand followed by an electrontransfer to the QD. 55The electron/hole transfer was slower than the hole spin-flip but faster than the electron spin-flip, so the radical pairgenerated by electron transfer was an overall singlet and the radicalpair generated by hole transfer was an overall triplet. 56 B. Triplet-triplet energy transfer (TTEnT) The electron-hole exchange interaction creates well-separated singlet and triplet states in molecules and correspondingly lifts thedegeneracy of bright and dark exciton states in QD. Althoughspin–orbit coupling and exciton–phonon coupling result in all excitonstates having finite singlet and triplet character, the net result is thatdark excitons nonetheless still have mainly triplet character. 33Direct TTEnT occurs through the Dexter exchange mechanism, and the spinangular momentum is conserved in the transfer process, so it is not surprising that QDs, with rapidly populated dark excitons, are effective triplet donors/acceptors for a variety of applications, including har-vesting of triplets generated from singlet fission and photochemicalupconversion. 28,57–59QDs also offer a significant advantage as photosensitizers/photocatalysts for triplet excited state organic trans-formations due to tunable exciton energies, relatively narrow line-widths allowing for high selectivity, and negligible singlet-tripletsplitting, which yields the highest possible triplet sensitization energyamong visible-light-driven sensitizers (up to 3.26 eV). 28,60 The presence of both singlet-like and triplet-like sublevels of QD excitons (all within k BT) limits the yield of TTEnT and complicates mechanistic interpretation, especially in cases where the excitons may also serve as precursors for redox reactions and F €orster-type singlet- singlet EnT, and/or serve as acceptors for back EnT/CT (frommolecule to QD). 61Sometimes these pathways work cooperatively. For instance, in QD-BODIPY (4,4-difluoro-4-bora-3a,4a-diaza-s-inda-cene)-pairs, FRET (F €orster Resonance Energy Transfer) and CT path- ways outcompete TTEnT, but the FRET pathway results in some yieldof the 3[BODIPY]/C3state as shown in Fig. 4 . The FRET sensitized 1[BODIPY]/C3transfers an electron back to the QD, forms a charge separated state in which a spin-flip event occurs, and spin-conserving charge recombination results in3/C3[BODIPY]/C3.62The myriad of pathways for generation of molecular triplets from QD donor statescomplicates predictive theory, but also presents opportunities forcreative design of systems for triplet-triggered photocatalysis, photonupconversion, and even quantum information. C. Dimensionality and anisotropy Advances in colloidal synthesis have expanded the scope of colloidal nanocrystals to include 1D nanorods/wires and 2Dnanoplatelets (NPLs) in addition to 0D QDs. The changes in nano- crystal shape introduce new knobs to tune EnT/CT between nanocrys- tals and molecules via modification of both the density of states andcoupling strength in Fermi’s golden rule. Since the density of electron/hole states changes from discrete states in 0D quantum dots to the sawtooth-like quasi-continuum in1D nanorods, to the step-like quasi-continuum in 2D NPLs, modifica-tions to both the overall rate and the distance dependence of EnT/CT can be observed. As mentioned in Sec. II, EnT between two transition dipoles results in a matrix coupling element that is proportional toR /C03, so that the overall rate of transfer goes as R/C06,b u ts y s t e m sw i t ha quasi-continuous density of states modify that dependence. Treating 2D materials as an assembly of incoherent point-like dipoles, these extensions of FRET unlock the possibility to control EnT distance bytuning nanocrystal shape. 63For example, the interaction between a 0D donor such as pyrene or a CdSe QD and a 2D acceptor like graphene(or 1D carbon nanotubes) yields a distance dependence of F €orster- type EnT rate of R /C04(or R/C05).64–66When 2D materials are utilized as both the donor and the acceptor, the EnT rate scales as R/C02.S u c hs h a l - low distance dependencies have led to intriguingly long F €orster radii: 33 nm in the case of EnT from CdSe/CdS NPLs to a monolayer ofMoS 2.67The control over density of states similarly allow for modifica- tion of the overall EnT/CT rates at a fixed distance.68Brumberg et al. measured electron transfer rates across the interfaces of nanocrystalfilms (0D QDs and 2D NPLs) and found that the fastest electrontransfer rate observed was at the NPL-NPL junction, due to its large orbital overlap and large density of acceptor states. 69Conversely, the CT rate from an NPL donor and a molecular acceptor was smallerthan that for a QD donor and molecular acceptor, because theincreased delocalization of the exciton within the large lateral area ofthe NPL decreased the percentage of the wavefunction that overlapped with the acceptor [ Fig. 5(a) ]. 70 Anisotropic confinement in nanocrystals further affects EnT and CT dynamics through the spatial orientation of the transition dipole, tuning the matrix coupling element in Fermi’s golden rule. Forinstance, films of CdSe NPLs have vastly different photoluminescence(PL) lifetimes when aligned face-to-face vs edge-to-edge, due to theadditional orbital overlap and the coupling of in-plane transition 2.12k2 k6 k0k1 k5k4k3E (eV) QD – 1BODIPY *QD* –BODIPY 500 nm exQD– –BODIPY + QD– –3BODIPY *1.89 1.52 0 Ground State QD–BODIPY FIG. 4. The efficient molecular triplet states generated through FRET, back electron transfer, and charge recombination, circumventing direct TTEnT between QD andligand. Reproduced with permission from Jin et al. , J. Chem. Phys. 152, 214702 (2020). Copyright 2020 AIP Publishing.Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 1, 011305 (2020); doi: 10.1063/5.0033263 1, 011305-5 Published under license by AIP Publishingmoments induced by face-to-face stacking [ Fig. 5(b) ].71In studies where NPLs are energy acceptors from 0D QDs, NPLs that are stackedface-to-face have a 50% enhancement in FRET yield compared toedge-to-edge stacked NPLs. 72The spatial orientation of the excitonic wavefunction within a nanocrystal also influences its electronic cou- pling to ligands. In photocatalytic Pt-CdSe nanorods, the rate of elec-tron transfer to Pt (the catalyst) on the nanorod tip is a factor of 20slower than to Pt randomly adsorbed along the length of the nano-rod. 73Given the number of properties in 1D and 2D materials that supersede or deviate from those observed in 0D QDs, it is likely thatcloser examination and modeling of these low-dimensional materials and their mixed-dimensional systems will unlock even more emergentbehavior for energy harvesting, photocatalytic, and other light-drivenapplications. 74 D. “Hot” exciton relaxation When attempting to truncate the series of states to consider in modeling an EnT/CT process, it is often assumed that internal relaxa- tion within a donor manifold occurs before EnT/CT occurs. In molec-ular systems, “hot” carriers quickly thermalize through vibrationalrelaxation and/or internal conversion to the lowest energy transitionso that only the lowest energy transition is considered. In quantum- confined systems such as QDs, carrier cooling via this mechanism is thought to be restricted due to the quantization of exciton energy lev-els with separations greater than the available phonon modes, so thatphonon-mediated relaxation requires multiphonon processes, a lowprobability event. This phenomenon, which is in many cases negatedor obscured by surface state-mediated relaxation, is referred to as the “phonon bottleneck.” 75This slowdown in phonon-mediated relaxa- tion opens the door for other kinetic processes, namely Auger recom-bination and multiple exciton generation, which similarly relax the hotexcitons on an ultrafast timescale but introduce additional consider-ations. In Auger recombination, the excess energy of electrons is trans-ferred to the hole, which cools more quickly due to a larger effective mass and smaller energy spacing. This process is generally treated with Fermi’s golden rule utilizing the Kane model, due to the sensitivity ofthe Auger relaxation rate to the overlap of electron and hole wavefunc-tions. 76In multiple exciton generation (MEG), predicted by Nozik in 2001 and subsequently observed by Schaller and Klimov in 2004 inPbSe QDs, 77,78the excess energy of a hot carrier generates a second exciton. This phenomenon has since been observed by others in a vari- ety of quantum-confined structures. However, the theoretical treat-ment of multiple exciton generation is still under debate, with bothcoherent and incoherent models being proposed. 79,80 Regardless of mechanism, it is known that multi-excitons can be generated on an ultrafast timescale from hot excitons ( /C24100 fs) in QDs, and their lifetimes are tens to hundreds of ps depending on thematerial. In certain cases, EnT/CT is competitive with Auger mediated bi-exciton relaxation so that multi-excitonic states can act as energy and charge donors. 81In the context of QD solar cells, designing fast EnT processes to extract multiple carriers from a single high energyexcitation could significantly reduce thermal losses and increase powerconversion efficiency. Several recent reviews have covered progress made in understanding these processes fundamentally and in the con- text of solar cells, so we will not discuss them further here. 82–84 Auger recombination also has a fundamental role to play in CT dynamics in QD-molecular adsorbate systems. In 2014, Lian, Prezhdo,and coworkers measured interfacial electron transfer rates from CdX(X¼S, Se, Te) QDs to three different adsorbed molecular electron acceptors. 85Variation of QD size and choice of acceptor molecule enabled tuning of the driving force for electron transfer between /C240–1.3 eV. The CT rate increased and then plateaued with increased driving force, even in the Marcus inverted regime, where the CT rateshould decrease with increased driving force. Ab initio calculations using a Cd 33Se33QD cluster86helped explain the phenomenon using the Auger-assisted electron transfer model, which assumes that elec-tron transfer is coupled to the generation of a “hot” hole through the 1000 100 00 100 1 01 04 5.2 nm 10.5 nm10 20 200 300 400 time (ns) k 2k 2face-down assembly face-down assemblyedge-up assembly edge-up assemblydrop casted 5000.1 0.01101000 100 10 1tCT µ S2Decay Time (ps) PL (count) KCT (109/s) 0400 300 200 100 0 50020 10 0 0 500 10001000 Surface Area (nm2)S (nm2)(a) (b) FIG. 5. (a) Inverse dependence of electron transfer rate to NPL surface area. Reproduced with permission from Diroll et al. , J. Am. Chem. Soc. 138, 11109 (2016). Copyright 2016 American Chemical Society. (b) PL lifetime of NPL isdependent on its orientation. Reproduced with permission from Goa et al. , Nano Lett. 17, 3837 (2017). Copyright 2017 American Chemical Society.Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 1, 011305 (2020); doi: 10.1063/5.0033263 1, 011305-6 Published under license by AIP Publishinglarge electron-hole Coulomb interaction in QDs. The “hot” acceptor states reduce the effective driving force in the highly exergonic CT pro- cess to eliminate the inverted region, and the remaining energy can be dissipated within the QD. A summary of the model is shown in Fig. 6(a).O l s h a n s k y et al. attempted to validate the Auger-assisted CT model by measuring the temperature dependence of hole transfer for a series of ligands to observe the predicted Arrhenius behavior and the activation-less regime at low temperatures. What they found however was that the temperature dependence observed in their Arrhenius plots, shown in Fig. 6(b) , was consistent regardless of driving force, implying a process independent of the ligand was involved. They pro- posed that a surface trap state may act as an intermediate, further com- plicating the picture.87,88While the exact mechanism is not clear, what is shown from these studies is that higher lying excited states play a larger role in EnT/CT in QD systems than in all-molecular systems. E. Carrier localization Surface or lattice-localized89trap states, which are often treated only as non-radiative pathways from the band edge exciton states, resulting in decreased PL quantum yield and emission intermittency (“blinking”), can be leveraged as EnT/CT intermediate states. Traps are generated from local structural defects, such as dangling bonds, vacan- cies, or anti-site defects in a bulk material, so that they are hard to cap- ture in an effective mass model apart from introducing them after the fact phenomenologically.90Trap states are most impactful when their energies are within the bandgap, since electrons or holes trapped within the conduction or valence band have the potential to quickly escape. Trap states localize charges spatially, which has an effect both on the dynamics of the charge being trapped, and on the conjugate charge through modification of electron-hole overlap. This spatial localization can allow for traps to act as intermediate states that effectively increase the wavefunction overlap between the QD and the molecular acceptor. Wuet al. designed a CdSe nanorod that trapped 99% of all holes with at i m ec o n s t a n to f <1p s( Fig. 7 ).91The trapping process out-competed all other recombination pathways, and the trap state had a long lifetime of 360 ns, so it served as a stable intermediate for eventualtransfer of the hole out of the QD to a phenothiazine acceptor withnear unity yield. Dopant sensitization is another useful process thathas been proposed to benefit from localization of excitonic carriers inclose proximity to the dopant atom. 92,93A weakened exciton electron- hole interaction can also lead to easier charge separation. Ye et al. showed that both electron and hole transfer kinetics may improvewith increasing concentration of surface hole trap sites. 94Ah i g h e rs u r - face trap concentration was correlated with better adsorption of coca- talysts leading to faster hole transfer, while favorable charge separationincreased electron transfer rates by a factor of 6.5. 1. The influence of ligands. a. Physical/structural effects of ligands. The arrangement of ligands o nt h eQ Ds u r f a c ec a nb et u n e dt oa d j u s tt h eo b s e r v e dC Tr a t ef r o mQDs to reactants. For instance, in a QD-photocatalyzed photo-redox C-C coupling reaction between phenylpyrrolidine and vinyl-sulfone, 95 Zhang et al. found that substituting the native oleate ligand of CdS QDs with octylphosphonate increased the initial rate of the reaction by a fac-tor of 2.3, due to disordering of the QD ligand shell. This disorderingmade the ligand shell more permeable to the phenylpyrrolidine substrate,accelerating the rate-limiting hole-transfer step. Ligands can also be usedto introduce interactions between donor and acceptor that increase theirelectronic coupling. For instance, the catalytic activity of a combinationof carboxylate-terminated CuInS 2QDs and positively charged Fe por- phyrin co-catalysts for reduction of CO 2to CO is a factor of 11 higher than that of the same, uncharged sensitizer and co-catalyst.96,97The authors modified the ionic strength of the catalytic reaction mixture(through addition of salt), and thereby correlated the catalytic activity tothe size of the electrostatically bound QD-porphyrin aggregates. b. Exciton delocalization by ligands. In contrast with our initial assumption that donor (QD) and acceptor (ligand) states can be neatly QD*(1Se,1Sh)–A QD–A QD*(1Se,1Sh)–A QD*(1Se,1Sh)–AQD*(1Se,1Sh)–AQD*–A QD*–A QD–AQD*(1Sh)–A– QD*(Eh,i)–A–QD* (1Sh)–A– QD* (Eh,i)–A– Eh,iEh,jQD* (1Sh)–A– QD* (Eh,j)–A–1P 1P 1S 1S 1S 1SET 1P 1P 1S 1S 1SAETkET kAETkR kRE G GG' 40 60 80 100 120 140λEReaction coordinate Rate (ns–1) 1/kbT (eV–1)10–1 10–2 10–3 Reaction coordinate Energy EnergyI) Conventional ET II) Auger-assisted ET(AI) (AII)(BI) (BII) (CII)(CI)(a) (b) FIG. 6. (a) Model for conventional (I) and Auger-assisted (II) ET from QDs to adsorbed acceptor molecules. “Hot” states effectively eliminate the Marcus inv erted region. Reproduced with permission from Zhu et al. , Nano Lett. 14, 1263 (2014). Copyright 2014 American Chemical Society. (b) Arrhenius plot of hole transfer rates of five different ligands. Solid lines rep- resent linear fits over the activated regime. Reproduced with permission from Olshansky et al. ,A C SN a n o 11, 8346 (2017). Copyright 2017 American Chemical Society.Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 1, 011305 (2020); doi: 10.1063/5.0033263 1, 011305-7 Published under license by AIP Publishingseparated, exciton delocalizing ligands (EDLs) directly modify the energies and spatial distributions of excitonic states in QDs.98–101 While some ligands modify exciton energies through electrostaticStark Shifts, EDLs extend the carrier density of QDs beyond their quantum-confined cores to coupled ligand states. The band edgeabsorbance peak of QD shifts to lower energy with no change in the physical size of the core, as a result of the hybridization between the frontier orbitals of EDLs and the valence or conduction band of QDs. Commonly used EDLs for QDs are conjugated S-containing ligands such as thiophenols and dithiocarbamates. EDLs can greatly accelerate the process of charge extraction from the QD excitonic state via extended conjugation of the electron or hole wavefunctions beyond the core of the QD. In 2016, Lian et al. found that the rate of hole transfer from a QD to bound hole acceptor mole- cule covalently linked to the QD was faster through an EDL ( s CS <300 fs) than through a carboxylate ( sCS/C241p s ) .102In 2017, Lee et al. observed that dithiocarbamate EDL capped-CdSe QDs participated in more efficient hole transfer to redox species in solution than MPA(mercaptopropionic acid)-capped QDs. 103In 2018, Azzaro et al.reported that treating CdSe QDs with phenyldithiocarbamate enhanced the exciton hopping rate among the QDs in a solid film to(1/200) fs /C01, 5 orders of magnitude larger than that in a film of oleic acid (OA)-capped QDs.104 One drawback for sulfur-based EDLs is that they quench the emission of many QDs as a result of their low oxidation potential and,therefore, limit QDs’ application in optical devices. As an alternative,Westmoreland et al. discovered the exciton delocalizing ability of N- heterocyclic carbenes (NHCs; see Fig. 8 ). 105They observed a batho- chromic shift of >100 meV of the bandgap of CdSe QDs upon treat- ment with MeNHC and attributed the effect to a “back-bonding”-type interaction of the chalcogenide orbitals of the QDs and the p-type orbitals of the NHC. The treatment of QDs with only MeNHCdecreased the photoluminescence quantum yield from 6% to /C244%. Functionalization of the NHC could lead to new QD-molecule donor-acceptor pairs with exceptional yields of photoinduced CT. IV. SYNERGISTIC EFFECTS Having covered how EnT/CT may generally be understood as mediated by direct coupling of exciton donor and ligand acceptor statesin Sec. IIand how individual features of CdSe nanocrystals complicate that simple picture in Sec. III, we now highlight how these seemingly divergent and competing features can be simultaneously utilized. Whileit is often ideal to modify individual aspects of QDs, the tuning of oneparameter often will have an effect on other QD properties. Here, we emphasize how a holistic understanding of the coupled parameter space m a ya l l o wf o rQ D st op e r f o r mf u n c t i o n sf a rb e y o n dw h a ti sp o s s i b l ei nmolecular systems, showcasing the full potential of QDs. Here we dis-cuss three phenomena that are intended to illustrate the combination ofsome of the individual phenomena we have discussed in this review intomore complex processes. These examples are only a peek at the possiblysynergistic phenomena in QDs and are chosen because their generalmechanisms have been investigated. There are certainly a much largerset of synergies that are not understood at this level, and many more emergent phenomena that have not been identified. A. Trap-mediated TTEnT Direct TTEnT from a QDs exciton to a molecule is typically sup- pressed by trap states, as they introduce additional non-radiative path- ways for the excitons to return to the ground state. Previous studies R CdSe CdSe WavelengthEmissionXX X XXXN N NNNX XXXN NNNN XXNNR R––– –––O O O OOOOOOO O O RR R FIG. 8. N-heterocyclic carbenes based exciton delocalizing ligands. 106 meV of bathochromic shift is observed due to the p-back bonding of the ligand. Reproduced with per- mission from Westmoreland et al. , J. Am. Chem. Soc. 142, 2690 (2020). Copyright 2020 American Chemical Society. (a) (b) Cds PTZ LUMO NH eV νs vac. S HOMO VBCB –4 –5 –6CR CR HTHT HTrHTr hν –+ FIG. 7. The trap mediated hole transfer process from CdS nanorods to PTZ (phe- nothiazine) molecules. The extraction of the trapped holes by PTZ molecules have99% efficiency because of the long lifetime of the trapped holes. Reproduced withpermission from Wu et al. , J. Am. Chem. Soc. 137, 10224 (2015). Copyright 2015 American Chemical Society.Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 1, 011305 (2020); doi: 10.1063/5.0033263 1, 011305-8 Published under license by AIP Publishingthat aim to optimize TTEnT through minimization of trap states or through passivation of surface traps indeed report enhanced TTEnTefficiencies. 106,107However, recently Jin et al. experimentally demon- strated that both band edge excitons and traps can be optimized toperform TTEnT in CdSe capped with 9-anthracene carboxylic acid(ACA), as shown in Fig. 9 . 108They found that a wide distribution of hole trap states are rapidly filled within 1 ps, outcompeting radiative decay. The decay of the traps’ spectral features was correlated with thegrowth of the ACA triplet’s features, so they proposed that the major-ity of TTEnT occurred from these trap states. Although TTEnT fromthe band edge excitons is a factor of 10 faster than from the trappedexcitons, the long lifetime of trapped excitons resulted in a >95% yield of TTEnT. Although this paper does not reveal a mechanism by whichthe traps were coupled to the ACA triplet state, it is reasonable that the trapped hole must have some triplet character to participate in Dexter EnT. B. Hot exciton processes modulated by dimensionality The promise of extracting “hot” (above-bandgap) carriers, makes nanocrystals particularly appealing for applications in energy harvest-ing and photoredox catalysis. In Sec. III, we discussed the conversion of hot excitons to multi-excitons that in turn served as EnT/CT donorswhen transfer rates outpace the decay of the multi-exciton states. 81 Similarly, in order to extract “hot” excitons, all relaxation pathways that expedite the relaxation of the carrier to the band edge become parasitic. In QDs and nanorods, phonon mediated hot carrier relaxa-tion occurs in hundreds of fs to a few ps. In NPLs, the higher densityof states promotes electron-phonon coupling such that phonon-mediated hot carrier relaxation speeds up to 60–70 fs. 109As a result, hot electrons have not been extracted from NPLs with molecularacceptors despite electron transfer on the hundreds of fs to a few pstimescale. 110This fast cooling rate is not insurmountable. In fact, trap- ping of hot electrons on the QD surface has been implicated in their photoluminescence blinking.111There remains however the opportu- nity to optimize the dimensions and the nature of the facets to speedup the CT rate or slow the carrier cooling rate. Strategies includetuning exciton-phonon coupling through the lateral size of the NPL,shown in Fig. 10 , and controlling CT through spatial orientation of molecular electron acceptors relative to NPL confinement dimension- ality. 112Okuhata et al. s h o w e dt h a tt h es a m em e t h y lv i o l o g e ne l e c t r o n acceptor had drastically different CT rates depending on which edge of a rectangular NPL the ligand resided.110Another possibility is to replace molecular acceptors with other 2D materials; 2D-2D hetero- junctions are known to exchange charge on the tens of fs timescale due to a large density of acceptor states.113 C. Ligand shells templated by nanocrystals While the orientation of ligands on QD surfaces has a direct influence on EnT/CT coupling due to changes in transition dipole ori- entation or spatial overlap of wavefunctions, additional interactions may exist between ligands and anisotropic nanocrystals. One example is how an ensemble of ligands’ dipoles may align to generate relatively large electrostatic fields, modifying the exciton energy. This behavior has been exploited to create energy level offsets in type II heterojunc- tions with the same size and type of QDs but different ligands [Fig. 11(a) ].114With NPLs, this effect is even larger, and the band edges have been predicted to shift by up to 5 eV [ Fig. 11(b) ].115This effect was attributed to the NPLs behaving as a parallel plate capacitor, where the electrostatic effect from the surface ligands on the large facet E (eV) 2.56 1.83 CdSe QD ACA Ground StateQD+–ACA– 3.48 eV QD––ACA+ 2.46 eV1ACA* 3ACA* hUBE Excition Trap ExcitonTrappingTET TET3.2 eV FIG. 9. Trap-mediated triplet energy transfer process. The long lifetime of the trapped charge carriers can overcome the slow kinetics for the triplet energy trans-fer process. Reproduced with permission from T. Jin and T. Lian, J. Chem. Phys.153, 074703 (2020). Copyright 2020 AIP Publishing. E (a) (b)p Shell S Shell ΔE EE LUMO HOMO [010] II y[100] II xpx = py = pz lx = ly = lz Se Shlx = ly > lz lx > ly > lzpx = py pe x ph xpzpz py pxSe –Sh pex – ph x FIG. 10. Modification of the excited state energy level driven by the unequal side length on the nanoplatelets. Tuning the s-p separation modifies coupling to longitu-dinal optical (LO) phonons. Reproduced with permission from Achtstein et al. , Phys. Rev. Lett. 116, 116802 (2016). Copyright 2016 American Physical Society.Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 1, 011305 (2020); doi: 10.1063/5.0033263 1, 011305-9 Published under license by AIP Publishingis exacerbated due to the anisotropic shape compared to isotropic QDs. Given that the surface area can be synthetically controlled, theband edge shift may be tunable through a combination of the synthetic reaction time and post-synthetic ligand exchange. Ligands can also be used to tune the degree of quantum confinement imposed by thethickness of the NPL. By changing the ligands in NPL films from the native oleate to a variety of halide, thiolate, and phosphonate ligands, the bandgap (along with an inter-subband transition) undergoes abathochromic shift due to delocalization from out-of-plane expansion of the potential well with some contributions from increases in biaxial strain. 116 A recent investigation by Jiang et al.117applied CdSe QDs as pho- tocatalysts and structural scaffolds for [2 þ2] photocycloadditions occurring from the triplet excited state of the molecular reagent. The authors achieved up to 98% diastereoselectivity for the previously minor syn-cyclobutane products of homo- and hetero-intermolecular[2þ2] photocycloadditions of 4-vinylbenzoic acid derivatives. Tethering of the substrates to the QD surface also resulted in up to 98% switchable regioselectivity for the products. This work is the ulti-mate demonstration of the simultaneous use of the QD as an opticalcoupling element, an energy donor, and a physical template for molec- ular self-assembly. The result is high-yielding, selective chemistry not possibly with molecular photocatalysts. V. CONCLUSION The field of electronic processes in nanocrystal-molecule systems is well-established yet fast-evolving. This review aims to demonstrate how QD-molecule systems have the potential to move beyond what is possible in all-molecular systems for light-powered generation of charge and energy conversion schemes. Section IIdescribed energy and charge transfer in these inorganic-organic hybrid systems with the theory developed for molecular systems, including well-known results from F €orster, Marcus, and Dexter. Section IIIshowed how an under- standing of EnT/CT in these systems requires additional consider- ations beyond the “direct coupling” of the initially prepared “donor” state and the final “acceptor” state. We explored topics such as how trap states act as kinetic intermediates and how ligands can affect exci- ton electronic structure in non-trivial ways. We also discussed how we might utilize, rather than minimize, unique features of QDs to access mechanisms of CT/EnT not available to molecules. Section IVlooked CDSe NPLs Band edges (eV) Pz/A (Debye/nm2)Cd–2 –4–6 –8 –10 –10 –5 0 5P z/AMoS2 CBM h=6.5ML HO SOAR h=3.5MLVBMSe h h(b) 3.5 Conduction BandEnergy (eV) ValenceBand4.0 4.5 5.0 5.5 6.0SH SH SH HSHS –F–S–CI–Br PbS QD–I SH SHSHC SHOH OSHNH2 H2N N(a) FIG. 11. Band edge of (a) QDs and (b) NPLs depending on ligand molecules. NPLs predicted to be much more tunable with ligand exchange than QDs. Reproduced with pe r- mission from Brown et al., ACS Nano 8, 5863 (2014). Copyright 2014 American Chemical Society. Reproduced with permission from Zhou et al. , Nano Lett. 19, 7124 (2019). Copyright 2019 American Chemical Society.Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 1, 011305 (2020); doi: 10.1063/5.0033263 1, 011305-10 Published under license by AIP Publishingfor ways to combine the features discussed in Sec. IIIsynergistically to enable new functions, including photochemical transformations of the molecules that adsorb to the nanocrystal surface. We hope that by dis- secting the QD-molecule system this way, we can inspire researchers to use quantitative models at all levels of theory to design the next gen- eration of hybrid materials with intentional connections among components. AUTHORS’ CONTRIBUTIONS All authors contributed equally to this manuscript. All authors reviewed the final manuscript. ACKNOWLEDGMENTS This work was enabled by funding from the Air Force Office of Scientific Research, Grant No. FA-9550–20-1–0364, the Department of Energy, Office of Science, Basic Energy Sciences, Grant No. DE-SC0020168, and the National Science Foundation through Northwestern’s Materials Research Science and Engineering Center (NU-MRSEC), Grant No. DMR-1720139. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1G. C. Schatz and M. A. Ratner, Quantum Mechanics in Chemistry (Dover Publications, Mineola, NY, 2002), p. xix. 2G. D. Scholes, Annu. Rev. Phys. Chem. 54, 57 (2003). 3N. A. Anderson and T. Lian, Annu. Rev. Phys. Chem. 56, 491 (2005). 4V. May and O. K €uhn, Charge and Energy Transfer Dynamics in Molecular Systems , 3rd rev. and enl. ed. (Wiley-VCH, Weinheim, 2011), p. xix. 5A. L. Efros and M. Rosen, Annu. Rev. Mater. Sci. 30, 475 (2000). 6A. M. Smith and S. Nie, Accounts Chem. 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5.0035739.pdf

Chem. Phys. Rev. 2, 011301 (2021); https://doi.org/10.1063/5.0035739 2, 011301 © 2020 Author(s).Structurally photo-active metal–organic frameworks: Incorporation methods, response tuning, and potential applications Cite as: Chem. Phys. Rev. 2, 011301 (2021); https://doi.org/10.1063/5.0035739 Submitted: 31 October 2020 . Accepted: 03 March 2021 . Published Online: 25 March 2021 Nicholas D. Shepherd , and Deanna M. D'Alessandro COLLECTIONS This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN The fluorescence quantum yield parameter in Förster resonance energy transfer (FRET)— Meaning, misperception, and molecular design Chemical Physics Reviews 2, 011302 (2021); https://doi.org/10.1063/5.0041132 Electronic structure modulation of MoS 2 by substitutional Se incorporation and interfacial MoO 3 hybridization: Implications of Fermi engineering for electrocatalytic hydrogen evolution and oxygen evolution Chemical Physics Reviews 2, 011401 (2021); https://doi.org/10.1063/5.0037749 Emergence of new materials for exploiting highly efficient carrier multiplication in photovoltaics Chemical Physics Reviews 1, 011302 (2020); https://doi.org/10.1063/5.0025748Structurally photo-active metal–organic frameworks: Incorporation methods, response tuning, and potential applications Cite as: Chem. Phys. Rev. 2, 011301 (2021); doi: 10.1063/5.0035739 Submitted: 31 October 2020 .Accepted: 3 March 2021 . Published Online: 25 March 2021 Nicholas D. Shepherd and Deanna M. D’Alessandroa) AFFILIATIONS School of Chemistry, The University of Sydney, Sydney, NSW 2006, Australia a)Author to whom all correspondence should be addressed: deanna.dalessandro@sydney.edu.au ABSTRACT Metal–organic frameworks (MOFs) are an important family of materials due to the properties that make them well suited to a range of applications. This includes structurally photo-active MOFs, which have properties that can be efficiently modulated through controlled lightirradiation, making them ideal due to the cost-effectiveness and noninvasive nature of this stimulus. The incorporation of structurally photo-active functional groups into MOFs has occurred through either guest inclusion, as pendant moieties, or as part of a ligand’s backbone. While initial studies into the incorporation of these groups focused on prominent photo-switches such as azobenzenes, the literature has expanded to include other classes described in the wider photo-switch literature, most notably spiropyrans (SPs). The incorporation of alter-native photo-switching classes has currently benefited the field through tuning the light responsive wavelength. Initial inquiries demonstratedsuitable function in gas sorption applications where irradiation could be exploited for inducing adsorption or desorption. Furthermore, thepotential applications explored in the literature have also recently expanded to include inquiries into other commercial functions, such as desalination [R. Ou et al. , Nat. Sustain. 3, 1052–1058 (2020)], photo-lithography [H. A. Schwartz et al. , Inorg. Chem. 56(21), 13100–13110 (2017)], and drug capture/release [X. Meng et al. , Sci. Adv. 2(8), 2–8 (2016)]. Published under license by AIP Publishing. https://doi.org/10.1063/5.0035739 TABLE OF CONTENTS I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 II. METHODS OF INCORPORATION . . . . . . . . . . . . . . . . . 2 A. Host-guest inclusion. . . . . . . . . . . . . . . . . . . . . . . . . . 2 B. Framework modification . . . . . . . . . . . . . . . . . . . . . . 3C. Thin films, SURMOFs, and composite materials . 4 III. PHOTO-SWITCHING BEHAVIOR. . . . . . . . . . . . . . . . . 5 A. Contemporary switching behavior . . . . . . . . . . . . . . 5 B. Visible light-responsive functional groups . . . . . . . 6C. Alternative switching behaviors . . . . . . . . . . . . . . . . 6 IV. POTENTIAL APPLICATIONS AND DEVELOPMENT OF FUNCTIONAL DEVICES. . . . . . 6 A. Gas sorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 B. Optical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8C. Magnetic and conduction properties . . . . . . . . . . . . 8D. Biochemical applications . . . . . . . . . . . . . . . . . . . . . . 9E. Other functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 V. SUMMARY AND OUTLOOK . . . . . . . . . . . . . . . . . . . . . . 10 AUTHORS’ CONTRIBUTIONS . . . . . . . . . . . . . . . . . . . . . . . . 11I. INTRODUCTION Metal–organic frameworks (MOFs) have garnered significant interest since the studies conducted by Hoskins and Robson 4,5as well as others.6These structures are defined by their assembly from metal/cluster-ligand coordination with multimodal ligands to yield extended, porous networks with characteristically high surface areas.7,8The inherent porosity and its associated host-guest chem- istry are typically crucial to potential applications, including electronics fabrication,9catalysis,10–12gas sorption,13–15separa- tions,16–21chemical sensing,22–27and drug delivery.28–30As origi- nally predicted from early inquiries,4,5component diversity has resulted in a vast library of published architectures. Furthermore, the robust nature of these materials makes them ideal platforms for additional synthetic experimentation. Such studies have demon- strated that alteration of assembly components31–34and/or frame- work post-synthetic modification (PSM)31,35–39can have a drastic impact on MOF properties, thereby allowing tuning for a specific function. Chem. Phys. Rev. 2, 011301 (2021); doi: 10.1063/5.0035739 2, 011301-1 Published under license by AIP PublishingChemical Physics Reviews REVIEW scitation.org/journal/cprTuning through synthetic modification has allowed incorpora- tion of molecular switching functionality into MOFs. These functionalgroups are appealing in developing materials where properties aremodulated through perturbation with an external stimulus. For thispurpose, various stimuli have been explored such as heat, 40,41pres- sure,41,42magnetic field,43redox potential,44–47and light.43,48–52The lattermost stimulus is advantageous for developing switching MOFsdue to the cost effectiveness and inherent abundance. 43 Interest in structurally responsive photo-switching compounds (a source for suitable guests/functional groups) stems from their potentialapplications, including chemical sensing, 53,54drug delivery,53,54sub- strate functionalization,54heterogenous catalysis,55molecular machines,56,57and data storage.58Photo-switching compounds that respond structurally are defined by their capacity to convert to a meta-stable state, via either p-bonding rearrangements and/or pericyclic rearrangements, 59upon stimulation with a select wavelength of light. The metastable chemical species typically exhibits variable propertiescompared to the original state. 60Popular structural photo-switch clas- ses include azobenzenes,53,60stilbenes,60SPs,60–62fulgides,60,63spi- rooxazines (SOs),60,64,65and diarylethenes.56,60,66–68Prominent classes with respect to MOF modification and their photo-switching pathwaysare summarized in Scheme 1. In addition to solution studies, photo- switching behavior has also been confirmed through single-crystal-to-single-crystal transformations for select discrete compounds. 66,69 Azobenzenes and diarylethenes were initially the most prominentphoto-switches in MOF modifications, although literature has recentlyexpanded to include SPs. 2,51,70 II. METHODS OF INCORPORATION Incorporating photo-switches into substrates is necessary for realizing potential applications, as previously noted by Feringa.71 Methods for photo-switch incorporation are broadly categorized intointrinsic and extrinsic functionalization, wherein photo-switchingfunctional groups are incorporated as either guest species or appendedmoieties, respectively. 51,72For extrinsic modifications, incorporation has involved a combination of pre-synthetic linker design and PSM.72 Further work has involved incorporating photo-active MOFs intofunctional devices, including films and surface-mounted MOFs(SURMOFs). 73,74Further device fabrication has included combining MOFs in complex material matrices and is an important step in pre-paring photo-switching frameworks for practical applications. A. Host-guest inclusion Diffusion has been an effective method for incorporating photo- switching guests into MOFs, as demonstrated by Lakmali andHettiarachchi in SP-modified MOF-74. 75Guest inclusion was con- ducted through refluxing the framework in a solution of dihydro- 10,30,30-trimethylspiro[2 H-1-benzopyran-2,20-(2H)-indoline] (SPH) to yield SPH@MOF-74.75Adjustment of the reflux time was noted to adjust the amount of SPH incorporated into the framework.75 Inclusion of photo-switching guests into MOF pore spaces has alsoinvolved layer-by-layer generation of the respective framework (typi-cally as a thin film) where encapsulation occurs during assembly. 76 Prominent examples include the encapsulation of azobenzene in HKUST-1 {[Cu 3(btc) 2], where btc3–¼benzene-1,3,5-tricarboxylate} conducted by Fu et al.77to yield azobenzene@HKUST-1 ( Fig. 1 ). This was achieved through a liquid-phase epitaxial pumping method to incorporate the photo-switch.77Further modification of the same MOF was also conducted with an ortho -fluoroazobenzene through dif- fusion of the photo-switch into the pore space.78 Additionally, photo-switching guests have been incorporated into MOFs through sublimation. This was initially conducted for azo- benzene incorporation into MOF-5 {[Zn 4O(bdc) 3], where bdc2-¼ter- ephthalate}, MIL-68(In) {[In(OH)(bdc)] /C11.0DMF /C1zH2O}, MIL-68(Ga) {[Ga(OH)(bdc)] /C10.9DMF /C1zH2O}, and MIL-53(Al) {[Al(OH)(bdc)]}.79 N N FFFF F F SS ON N OSSFFFFNN F F360—370 nm 260—400 nm 200—400 nm(a) (b) (c)Δ Δ Δ SCHEME 1. Photo-switches commonly incorporated in MOFs and their respective con- versions, classes illustrated are (a) azobenzenes, (b) diarylethenes, and (c) SPs.50,60 FIG. 1. Photo-switch encapsulation in azobenzene@HKUST-1 where the azoben- zene guest is indicated in green.77Gray spheres ¼carbon, red spheres ¼oxygen, and turquoise spheres ¼Cu(II). Hydrogens have been omitted for clarity.Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 2, 011301 (2021); doi: 10.1063/5.0035739 2, 011301-2 Published under license by AIP PublishingAll modified MOFs with the exception of MIL-53(Al) were capable of undergoing photo-switching consistent with azobenzene behavior.79 SP and SO species have also been incorporated into the same MOFsusing gas phase infiltration. 2,80In addition to facilitating insertion of the photo-switch, the authors noted that the elevated temperature allowed for effective solvent removal from the MOFs.2While this allowed for enhanced guest uptake, it also facilitated effective evalua-tion of host-guest relations and their impact on optical properties. 2 The exception in the study was MIL-53(Al), where the SP and SO coated the framework instead of diffusing into the pore space although the same photo-switching behavior was still observed.2,80Similar load- ing techniques with a range of fluorinated azobenzenes into MIL-53(Al) overcame this hurdle by incorporating the photo-switches intothe pore space. 81In the same study, MOF-5, MIL-68(In), and MIL- 68(Ga) were modified with the same photo-switches.81This work demonstrated that the use of photo-switches of various classes as guestspecies with their behavior intact is feasible. Aside from gas phaseincorporation, Walton et al. 82demonstrated high temperature photo- switching guest inclusion in the molten phase. The diarylethene species 1,2-bis(2,5-dimethyl-thien-3-yl)-perfluorocyclopentene (DTE) was sealed in a vessel with DMOF-1.82Heating to 130/C14Cw a ss u f fi - cient to allow DTE to enter the MOF’s pores, yielding DMOF-1@DTE.82. A study where JUC-120 was modified with SP compounds involved microwave-assisted crystallization inclusion. 83The most recent example of extrinsic SP incorporation into a framework was conducted through preparation of SSP@ZIF-8 {SSP@[Zn(2mim) 2], where 2mim/C0¼2-methyl-1 H-imidazole and SSP ¼10,30,30-trimethyl- spiro[chromene-2,20-indoline]-6-sulfonate}.84SSP was entrapped between positively charged zinc hydroxide nanostrands, whichwere converted into ZIF-8 through addition of the 2mim /C0 linker.84In addition, extrinsic modification has also been exploited as a means of coating MOF materials. This was demon-strated through coating Mg-MOF-74 {[Mg 2(dhtp)(H 2O)2]/C18H2O, where dhtp ¼dihydroxyterephthalate} and MIL-53(Al) with methyl red dye.85B. Framework modification MOF scaffold modifications typically occur through pre- synthetic ligand functionalization or PSM. The former has been effec-tive for incorporating azobenzene moieties as pendant functionalgroups of ligands (a selection of prominent ligands is provided inFig. 2 ). The first study where a photo-switching linker was used was the preparation of CAU-5 {[Zn 2(NDC) 2(azBIPY)], where NDC2– ¼naphthalene-2,6-dicarboxylate and azBIPY ¼(E)-3-(phenyldia- zenyl)-4,40-bipyridine} ( Fig. 3 ).86Other MOFs where azobenzene link- ers have been incorporated includeAzoMOF {[Zr 6(OH) 4O4(L)6], where L2/C0¼20-phenyl-diazenyl-1,10:40,100-terphenyl-4,400-dicarboxy- late},87{[Zn 4(tbazip) 3(bpe) 2(OH) 2]/C1bpe, where tbazip ¼5-[(4- tert- butyl)phenylazo]isophthalate and bpe ¼1,2-di(4-pyridyl)ethene},88 and Azo-UiO-66 {[Zr 6(OH) 4O4(azbdc) 6], where azbdc2–¼(E)-2- (phenyldiazenyl)terephthalate}.89The azobenzene moieties in these MOFs were incorporated as pendant functional groups.86,87,89,90The exceptions include JUC-62 {[Cu 4(AzTC) 2], where AzTC4–¼(E)-5,50- (diazene-1,2-diyl)diisophthalate},91ECUT-50 {[In(AzDC) 2] [H2N(CH 3)2], where AzDC2–¼(E)-4,40-(diazene-1,2-yl)dibenzoate and H 2N(CH 3)2þ¼dimethylammonium},55and {[Zn(AzDC)(4,40- BPE) 0.5], where AzDC2–¼(E)-4,40-(diazene-1,2-yl)dibenzoate and 4,40-BPE ¼(E)-4,40-(ethene-1,2-diyl)dibenzoate},92where the incorpo- rated azo functional group formed part of the respective linker’s back-bone ( Fig. 2 ). Other azobenzene-based switching moieties include ortho -fluoroazobenzenes, which were incorporated as part of an alter- native linker in F-azo-UiO-66(Zr) {[Zr 6(OH) 4O4(F2AzBDC) 6], where F2AzoBDC2–¼(E)-2-[(2,6-difluorophenyl)diazinyl]terephthalate},93 F-azo-MIL-53(Al) [Al(OH)(F 2AzoBDC)],93and {[Cu 2(F2AzoBDC) 2 (dabco)], where dabco ¼1,4-diazabicyclo(2.2.2)octane}.94 A similar methodology was also applied to incorporate diarylethene-based ligands into framework architectures, wherephoto-switching structures were either pendant or backbone func-tional groups ( Fig. 2 ). 95–99Currently, pendant diarylethene linkers have yielded a number of photo-switching MOFs that include UBMOF-1 {[Zn 4O(TPDC) 3], where TPDC2–¼9,10-bis(2,5-dime- thylthiophen-3-yl)phenanthrene-2,7-dicarboxylate}95and UBMOF-3 OO O O O O OOOOO HO HO HO HOHOOHOH OH OH OH OH N NNNN NNNN N N S SN S SOHO F FFF F FFFF F F O SSO HOOHF S S NNO OHN NNN HO O(a) (g) (h) (i) (j)(b) (c) (d) (e) (f) FIG. 2. Examples of azobenzene and diarylethene ligands designed for intrinsic photo-switch incorporation into MOFs.Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 2, 011301 (2021); doi: 10.1063/5.0035739 2, 011301-3 Published under license by AIP Publishing{[Zn 3(bdc) 3(TPDPy)], where TPDPy ¼4,40-[9,10-bis(2,5-dimethylth- iophen-3-yl)phenanthrene-2,7-diyl]dipyridine}.96Other studies include preparing {[Zn 4(bdc) 4(L)2/C14DMF /C1H2O]n, where L ¼4,40- [(perfluorocyclopent-1-ene-1,2-diyl)bis(5-methylthiophene-4,2-diyl)]-dipyridine} 98with a diarylethene linker where the photo-switching functional group was situated along the ligand backbone ( Figs. 1 and 2).97–99The same diarylethene linker has been successfully exploited in generating a DMOF series {[Zn 2(L)(L2)], where L ¼4,40- [(perfluorocyclopent-1-ene-1,2-diyl)bis(5-methylthiophene-4,2-diyl)]dipyridine or 4,4-[(perfluorocyclopent-1-ene-1,2-diyl)bis(thiophene-4,2-diyl)]dipyridine and bpdc 2–¼[1,10-biphenyl]-4,40-dicarboxylate or 4,40-oxybisbenzoate}.97,99Structures observed in the DMOF series varied based on the co-ligand used during synthesis.99 Most recently, ligands designed by Shustova and co-workers100,101 have also been applied to generate ligands with SP pendant groups. Aninitial study involved developing [1 0,30,30-trimethyl-6-nitro-40, 70-di(pyridin-4-yl)spiro[chromene-2,20-indoline] (TNDS) and [10,1000, 30,30,3000,3000-hexamethyl-6,600-dinitro-40,4000-di(pyridine-4-yl)-70,7000-bis- piro[chromene-2,20-indoline] (HDDB) ligands.100,101Subsequent coor- dination between TNDS, HDDB pyridyl groups and Zn(II) ions in thepresence of the appropriate co-ligand yielded {[Zn 2(DBTD)(TNDS)], where DBTD ¼30,60-dibromo-40,50-bis(4-carboxylatophenyl)- [1,10:20,100-terphenyl]-4,400-dicarboxylate} and [Zn 2(DBTD)(HDDB)], respectively.100,101 Additional modification has been conducted through PSM. The most prominent example with respect to SP incorporation involves modification of MOF-808 {[Zr 6(OH) 4O4(btc) 2]6þ, where btc3–¼ben- zene-1,3,5-tricarboxylate}102,103[Scheme 2(a)].70This involved coordi- nation of the precursor, 2–(2,3,3-trimethyl-3 H-indol-1-ium-1-yl) acetate, to the oxo-Zr(IV) clusters.70The SP moiety was generated through condensation between modified MOF-808 and 2-hydroxy-5-nitrobenzaldehyde. 70SP-modification of MOF-808 is an effective example of PSM for incorporation of photo-switches based on the exploitation of a combination of covalent and coordination reactions. Covalent PSM was also conducted to produce azobenzene func- tional groups in Cr-MIL-101_amide {[Cr 3(l3-O)(OH)(H 2O)2(L)3]/C1 nH2O, where L ¼(E)-2-[4-(phenyldiazenyl)benzamido]terephthalate} and Cr-MIL-101_urea {[Cr 3(l3-O)(OH)(H 2O)2(L)3]/C1nH2O, where L¼(E)-2-(3-(4-(phenyldiazenyl)phenyl)ureido)terephthalate} through conversion of the secondary amine attached to the 2-aminoterephthalate ligand [Scheme 2(b)]. 90Nucleophilic substitution of the chloride in azobenzoylchloride allowed for preparation ofCr-MIL-101_amide, while using 4-(phenylazo)phenylisocyanate as a reaction substrate yielded Cr-MIL-101_urea.90The same methodology was further applied to generate Cr-MIL-101_azo.104An additional example of covalent PSM involved {[Cu 2(SP-bpdc) 2(dabco)], where SP-bpdc2–¼2–(1–(2–(30,30-dimethyl-6-nitrospiro[chromene-2, 20-indol]-10-yl)ethoxy)-2-oxoethyl)-1 H-1,2,3-triazol-4-yl)-[1,10-biphe- nyl]-4,40-dicarboxylate}.105In this instance, the SP moiety was incor- porated via an azide click reaction with ethenyl groups attached to the bpdc2–linker.105 C. Thin films, SURMOFs, and composite materials In order to develop functional devices, fabrication of photo- switching MOF materials as thin films and/or membranes has been frequently explored. Thin film fabrication has been conducted using similar techniques to those described in the general MOF litera- ture.106,107Examples include azobenzene@HKUST-1 films where liquid-phase epitaxy was used to produce the MOF with the guestincluded. 77This simple method, or a derivative thereof, has been employed in the generation of various azo-MOF thin films described in the literature.73,108–110 This methodology was further extended to the growth of two azo-MOFs as SURMOFs by Wang et al.73The frameworks used for the study were {[Cu 2(NDC) 2(azBIPY)], where NDC2–¼naphthalene- 2,6-dicarboxylate} and {[Cu 2(AzoBPDC) 2(BIPY)], where AzoBPDC2–¼(E)-2-(phenyldiazenyl)-[1,10-biphenyl]-4,40-dicarboxy- late and BIPY ¼4,40-bipyridine}, although only the latter exhibited the characteristic photo-switching associated with the azobenzene moi- ety.73This impediment on photo-switching was attributed to the lack of steric interference from the NDC2–linker,73illustrating the impor- tance of careful ligand design in this context. It should be noted that the [Cu 2(NDC) 2(azBIPY)] MOF thin films are the only instance where photo-switching was inhibited. Intact photo-switching behavior has commonly been observed in efforts to develop MOF thin films. In addition, photo-active MOFs have been incorporated into composite materials. This consists of mixed matrix membrane studies recently published by Ladewig and co-workers.89,91,111The initial study involved investigation of the photo-activity of the azobenzene moiety in modified JUC-62,91followed by the incorporation of this frame- work, PCN-250 [Fe 3(l3-O)(AzTC) 6] and azo-UiO-66 into matrimid polymer matrices.89,111The incorporation of photo-switching MOFs into composite materials is an important step in realizing their poten-tial applications, 71and hence, has become the motivation to numerous (a) (b) FIG. 3. ( a) Molecular structure for CAU-5 where the photo-switching ligand ishighlighted in green.86(b) The MOF exhibits twofold interpenetration indicated by blue and red nets.86Gray spher- es¼carbon, blue spheres ¼nitrogen, red spheres ¼oxygen, and magenta spheres ¼Zn(II). Hydrogens have been omitted for clarity.Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 2, 011301 (2021); doi: 10.1063/5.0035739 2, 011301-4 Published under license by AIP Publishinginquiries wherein various substrates have been modified, including pol- ymers,112–116gels,117–119and nanoparticles (NPs).113,120 Further work conducted with photo-switch encapsulation in MOFs has involved controlled incorporation of an azobenzene- modified polymer guest. Uemura et al.121absorbed [poly-(ethylene glycol)]-tethered azobenzene (PEG-AB) into [Zn 2(terephthalate) 2(triethylenediamine)].121This was conducted with polymer in its thermodynamic state while irradiation allowed for par- tial release (29%) from the pore space.121Effective adsorption was attributed to the initially linear polymer strands being a more appro- priate fit for the MOF pore dimensions.121This was conducted over multiple adsorption/desorption cycles without deterioration of the sys- tem.121The controlled encapsulation is interesting as it presents a material that can be dissociated on command. III. PHOTO-SWITCHING BEHAVIOR Preserving photo-switching behavior following incorporation into MOFs is crucial to developing structurally photo-active architec- tures. In addition, it is also necessary to tune the photo-switching behavior observed in modified MOFs to optimize fatigue resistance. The expansion of exploited photo-switch classes has also allowed for incorporating moieties that are responsive to other environmentalstimuli, such as electrochemical potential,61,62,122,123pH,61,62and metal ions.61,62,124–131 A. Contemporary switching behavior To date, the photo-switches used have typically been UV- responsive with relaxation falling in line with a combination of P- and/or T-type behaviors. P-type photo-switching requires an alterna- tive light source (visible wavelengths for these classes).60Various azo- benzene and diarylethene-modified MOFs have exhibited this type ofphoto-switching behavior. Azo-MOF examples include {[Cu 2 (AzoBPDC) 2(AzoBiPyB)], where AzoBiPyB ¼(E)-4,40-[2-(phenyldia- zenyl)-1,4-phenylene]dipyridine}132and {[Zn(Im)(aIm)], where Im/C0 ¼imidazolate and aIm/C0¼2-phenylazoimidazolate}133where follow- ing irradiation with UV light, reversion could be triggered by irradia- tion at /C24455 and /C24525 nm, respectively. Similar behavior has been observed in most diarylethene-modified MOFs.95–98SSP@ZIF-8,84 [Zn 2(DBTD)(TNDS)],100,101and [Zn 2(DBTD)(HDDB)]100also exhib- ited P-type switching behavior where the SP opened to the merocya- nine (MC) state in response to UV, while ceasing irradiation allowed reversion. Other SP-modified MOFs have exhibited T-type switching behavior where heating has been required to regenerate the initial N NNO2 OHO O OO O O O OO O O RRRR H NN N R(b)(a) R RROO NNRR RRO NNO O O OH NH NOO O Where R = Cr-MIL-101-NH2CI NHNH2NO CI NNNO2O O SCHEME 2. PSM of MOF-808 and Cr-MIL-101-NH 2.70,90(a) In MOF-808, a SP precursor was coordination to the oxo-Zr(IV) secondary building unit before condensation with 2-hydroxy-5-nitrobenzaldehyde yielded the photo-switch.70(b) For Cr-MIL-101-NH 2, the addition of two azo reagents yielded two photo-active MOFs.90Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 2, 011301 (2021); doi: 10.1063/5.0035739 2, 011301-5 Published under license by AIP Publishingground state. This includes numerous MOFs with SP guests2and modified MOF-808.70DMOF {[Zn 2(L)(L2)], where L ¼4,40-[(per- fluorocyclopent-1-ene-1,2-diyl)bis(5-methylthiophene-4,2-diyl)]dipyr- idine and bpdc2–¼[1,10-biphenyl]-4,40-dicarboxylate} was an exception where the photo-switching ligands responded to both light and heating.97Similar behavior was observed in SO-modified MOFs where either heating or visible light could be used as stimuli for regen-erating the initial isomer. 80[Cu 2(SP-bpdc) 2(dabco)] also reverted back to its SP state through either heating or visible light.105 Select studies have indicated that the solid-state photo-switching behavior was consistent with solution studies as evidenced by similaractivation energies for isomerism in azo-MOFs. 134Other inquiries have demonstrated improved switching rate following incorporation into MOF lattices, including stilbene compounds (photo-switches thatare similar in structure and behavior to azobenzenes) incorporation into {[(ZnI 2)3(1)2]/C1x(C 6H5NO 2)]n,w h e r e1 ¼tris(4-pyridyl)tria- zine}.135The electron poor nature of the triazine ligand was expected to have a catalytic effect on stilbene photo-switching, resulting in a faster conversion as a guest in the lattice compared to solution stud- ies.135The exception was a dinitro-substituted guest which did not undergo photo-switching following inclusion into the pore space.135 Despite the observed enhancement of photo-switching rate in stilbenes,135photo-switch incorporation into a MOF lattice has also an adverse effect on conversion. Adverse effects include steric hindrance,which can impede structural rearrangements as observed in [Cu 2(NDC) 2(azBIPY)], where the NDC2–ligand impeded photo- switching of the azobenzene by blocking rotation of the azo group.73It should be noted that photo-switching may still proceed as observed for CAU-5,86DMOF-2,99DMOF-3,99and [Zn 4(bdc) 4(L)2/C14DMF /C1H2O]n,98despite twofold interpenetration. Photo-switch density in pore space has also provided steric limitations to switching behavior as observed in MIL-53(Al) with ortho -fluoroa- zobenzene derived guests.81Contrary to the typical photo-switching observed for the guest species, the modified MIL-53(Al) MOFs were inactive when irradiated due to the density from guest packing.81 In addition to steric restraints, the lattice environment has previ- ously stabilized the respective photo-switching moiety’s metastable state, thereby preventing full reversion to the corresponding initial ground configuration.95,133UBMOF-195and [Zn(Im)(aIm)]133exem- plify this, where visible light irradiation was insufficient for full rever- sion of the photo-switching ligands back to their uncyclised states. Another impediment to ideal photo-switching was demonstrated by Walton et al.136in UBMOF-2 {[Zn 4O(PhTPDC) 3], PhTPDC2–¼9, 10-bis(2-methyl-3-phenyl-thiophen-3-yl)-phenanthrene-2,7-dicarbox-ylate}, where the diarylethene linker was expected to convert to photo- inactive atropisomers following irradiation. The effects were evident in the poor fatigue resistance observed in the MOF over repeated photo-switching cycles. 136 B. Visible light-responsive functional groups Most photo-active MOFs in the literature include moieties that require UV irradiation for photo-switching. To circumvent the prob-lems associated with higher energy wavelengths (photochemical degra- dation), 137visible light-responsive photo-switches have been developed with prominent classes, including ortho -fluoroazoben- zenes,138hemithioindigoes,139and donor-acceptor Stenhouseadducts.140–143The first class is typically converted by green light while reversion requires blue light exposure (Scheme 3). Ortho -fluoroazobenzenes have recently been incorporated into MOFs for visible light-modulated behavior. This includes the use of the photo-switching linker in Scheme 3as an alternative ligand in UiO-66 and MIL-53(Al) syntheses to yield F-azo-UiO-66(Zr) and F- azo-MIL-53(Al), respectively.93[Cu 2(F2AzoBDC) 2(dabco)] thin films also exhibited the standard photo-switching behavior where irradia- tion with 530 nm allowed conversion to the cisstate, while reversion resulted from exposure to 400 nm.94In addition to these intrinsically modified MOFs, various frameworks have been developed with fluori-nated azobenzenes incorporated into their pore space. 78,81These mod- ified MOFs exhibited the same visible light–responsive photo- switching behavior commonly associated with discrete fluorinated azobenzenes,78,81with the exception of modified MIL-53(Al) where guest packing was too dense.81These MOFs represent an important step in developing visible light-responsive framework materials. In addition, SSP@ZIF-8 architectures were noted to be visible light–res- ponsive (to convert from the MC to the SP state) while dark condi- tions were adequate to generate the MC state.144 C. Alternative switching behaviors In addition to the use of visible light stimulated photo-switching moieties, select classes also present an opportunity for exploiting other useful stimuli. Diarylethene and SP moieties are potentially responsive to other environmental perturbations as both classes have undergone their characteristic photo-switching behavior in response to an applied electrochemical potential.61,62,122,123,145–148Furthermore, SPs and related photo-switches have also responded to pH,61,62metal ions,61,62 and pressure.61,62Alternative switching behaviors have recently been demonstrated in SP-modified Zn-MOF-74.75The incorporated SP guest exhibited thermochromic properties where heating up to 80/C14C facilitated conversion to the relevant MC configuration (reversed through UV irradiation).75Inquiries into these alternative switching stimuli are necessary for two reasons. The first of these is that it ena- bles exploration of other potential applications beyond those where photo-modulation are key. Also, these alternatives may interfere with photo-modulation of properties, presenting a barrier that would need to be overcome in select functions. IV. POTENTIAL APPLICATIONS AND DEVELOPMENT OF FUNCTIONAL DEVICES Initial application studies focused on exploiting photo- modulated uptake of target guests (CO 2in particular).78,87,91,93Recent studies have expanded to include other potential guests.78,108,149Other behaviors where photo-modulation has been demonstrated include O OF > 500 nm 405 nmO OO O O ONNF F NN F SCHEME 3. Photo-switching behavior for an ortho -fluoroazobenzene ligand where initial conversion occurs under green light and reverses with blue wavelengths.93Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 2, 011301 (2021); doi: 10.1063/5.0035739 2, 011301-6 Published under license by AIP Publishingoptical properties150and conductivity.84,94,105,151Proposed functions are partially dictated by the type of photo-switching behavior exhibited b yt h er e s p e c t i v eM O F s( i . e . ,P -v sT - t y p e ) . A. Gas sorption Numerous photo-switching MOF studies have focused on modu- lated gas sorption behavior, with CO 2being a favored target due to two factors. The first is the significance of CO 2as a greenhouse gas,13,36,152and second, the associated quadrupole would allow for effi- cient control based on photo-modulated intermolecular interactions with photo-switching moieties.78For gas sorption, both electrostatic interactions and steric factors have proven to be important in photo- switching in structurally responsive MOFs. The former has been dem- onstrated with azo-MOFs where the dipole increased from 0 to 3 Ddue to the UV-stimulated generation of the cisconfiguration. 78,87The increased dipole resulted in a significant improvement in CO 2Qst following UV irradiation forAzoMOF (from 29 to 38 kJ mol/C01).87 Despite improved CO 2binding affinity, UV irradiation also resulted in decreased uptake, which was attributed by Huang et al.87to the con- fined space of the polar adsorbent and adsorbate groups ( Fig. 4 ). The reduction in CO 2uptake in spite of the enhanced dipole in the cisstate w a sa l s on o t e df o rJ U C - 6 2 ,91F-azo-UiO-66(Zr),93and F-azo-MIL- 53(Al).93 DMOF contrasted with this behavior, where photo-switching of the diarylethene ligands resulted in an increase in CO 2uptake (up to four times adsorption).97The effect of an enhanced dipole was further exemplified through the incorporation of an SP species into MOF-808. Following modification, the photo-active MOF-808 architecture exhibited enhanced CO 2adsorption upon in situ conversion to theMC configuration.70As with several previously discussed studies, this was attributed to more favorable interactions with the MC state.70 More favorable interactions with CO 2were expected based on the zwitterionic nature of the MC configuration, which would have a more pronounced effect compared to the cisdiastereomer for azobenzenes. Reversion stimuli for photo-switching functional groups are also significant in dictating whether dynamic photo-modulation can be conducted. This was evidenced by contrasting uptake properties in PCN-123.153The azobenzene-modified MOF exhibited the typical reduction in CO 2uptake ( /C2453.9%).153Unlike other frameworks where dynamic switching was observed, ceasing UV irradiation for PCN-123did not allow a return to the original CO 2uptake.153Instead the T-type switching behavior required heating up to 60/C14Cf o r trans uptake to be observed in a subsequent measurement.153 P-type systems have an advantage in this respect, as their uptake behavior can be easily modulated during measurements, therefore demonstrating dynamic function. This was demonstrated by Wanget al. 132where gas mixtures were passed through a [Cu 2(AzoBPDC) 2 (AzoBiPyB)] SURMOF to determine separation factors for H 2/CO 2 and H 2/N2mixtures. For the former mixture, the observed separation factor was converted from 3 to 8 through UV-triggered switching of the azobenzene functional groups.132A st h ea z o b e n z e n el i g a n du n d e r - went P-Type photo-switching, the authors were able to cycle through the states in a single measurement. Similar results were obtained with the MOF for H 2/N2mixtures.132An additional benefit of this behavior that has not been demonstrated with isothermal data were that con-trolled irradiation intervals could be applied for more careful modula- tion of trans :cisratios (maximum conversion ratio was 37%:63%), allowing for stepwise modulated H 2selectivity.132 The study conducted by Wang et al.132demonstrates another sig- nificant property in these frameworks for application in gas phase sep- arations, that being selectivity. Selectivity for CO 2over N 2has been of interest due to the potential extraction of the former from flue gas streams (post-combustion sources).13CO 2selective photo-switching MOFs included the materials described by Ladewig and cow- orkers.89,111CO 2/N2selectivity based on ideal adsorbed solution the- ory (IAST) calculations for Azo-UiO-66 was approximately 37 and 100 at 273 and 298 K, respectively (considerably higher than the par- ent MOF).89An additional benefit was diminished N 2uptake in azo- UiO-66 compared to the parent MOF.89Similar behavior was observed in matrimid-MOF membranes (based on JUC-62 and PCN-250) where selectivity was approximately 60. 111 Studies have also indicated that steric hindrance can have an adverse effect on guest uptake during gas adsorption. DArE@PAF-1, while not a MOF, demonstrated that the photo-switch can instead provide hindrance.154Photo-switching was conducted under dynamic conditions although irradiation resulted in reduced CO 2uptake.154 This was attributed to the competition on the part of diarylethene guest molecules for the ideal binding sites in the framework.154Similar behavior has also been observed in select azo-MOFs where CO 2bind- ing to the carboxylate groups of the ligand and unsaturated metal sites was impeded by conversion to the cisconfiguration in modified PCN- 123 and IRMOF-100.155 Further interesting work involves exploiting the contrasting steric hindrance between ground and metastable states associated with aphoto-switch to useful effect. This was demonstrated with methyl red OO O O O ON NNVisible lightuv N O O FIG. 4. Photo-switching behavior of the azobenzene linker inAzoMOF where irradia- tion with a UV light source results in conversion to the cisconfiguration, resulting in reduced CO 2uptake despite an increased binding affinity. Reprinted with permis- sion from H. Huang, H. Sato, and T. Aida, J. Am. Chem. Soc. 139, 8784–8787 (2017). Copyright 2017 American Chemical Society.87Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 2, 011301 (2021); doi: 10.1063/5.0035739 2, 011301-7 Published under license by AIP Publishingcoated Mg-MOF-74 and MIL-53(Al).85These frameworks were ideal due to the high CO 2uptake and flexibility associated with Mg-MOF- 74 and MIL-53(Al), respectively.85Irradiation with visible light facili- tated CO 2adsorption in these architectures due to a gating effect on the part of the azobenzene species where visible light irradiation “opened” the pore apertures.85Contrasting behavior has also been observed for a series of SURMOFs {[Cu 2(AzoBPDC) 2(dabco)], [Cu 2(AzoTPDC) 2(dabco)], [Cu 2(AzoBPDC) 2(AzoBiPyB)], [Cu 2(bdc) 2(AzoBiPyB)], and [Cu 2(DMTPDC) 2(AzoBiPyB)], where [Cu 2(bpdc) 2(dabco)]}, where butanol uptake increased following con- version of the azobenzene moieties to their cisstate.74This was par- tially attributed to photo-switching density where, despite the stericimplications, greater azobenzene density resulted in a more effectiveincrease in butanol uptake. 74 Recent studies have also investigated adsorption of 1,4-butane- diol (in addition to 1-butanol) in photo-switching MOFs.78,108This includes an initial demonstration with [Cu 2(bdc) 2(AzoBipyB)].108In addition to undergoing the expected switching behavior, the increased dipole following switching increased the uptake for 1-butanol (5%) and 1,4-butanediol (20%).108Similar behavior was observed in HKUST-1 (with azobenzene guests) where an approximately8% increase in the uptake of 1,4-butanediol resulted from irradiation. 78 In both studies, variations in the uptake were attributed to an increasein the dipole upon switching. Applications in photo-modulated gas extractions were also demonstrated with unsaturated hydrocarbons in a diarylethene-modified MOF. 156UV irradiation reduced C 2H2selec- tivity over C 2H4from 47.1 to 3.0.156Alternatively, N 2physisorption could be modulated in [Zn 2(terephthalate) 2(triethylenediamine)] n, where conversion of the azobenzene guests resulted in improveduptake. 157These studies demonstrate that photo-switching behavior in MOFs can be applied for modulating gas sorption properties. Another study investigated adsorption/selectivity of CO 2,a c e t y - lene, and ethylene in ECUT-30.158The framework was noted to be selective for both gases over CO 2and for acetylene in preference to ethylene. In each of the three mixtures, selectivity (calculated by theauthors based on IAST) was noted to increase dramatically with UVirradiation. 158CO 2selectivity increased by approximately 50% in mix- tures with acetylene and ethylene.158For acetylene/ethylene mixtures, selectivity was noted to roughly double in favor of the former gas spe- cies following irradiation.158The initial inquiry into hydrocarbon adsorption behavior in photo-switching MOFs involved examiningCH 4uptake in Cr-MIL-101_amide and Cr-MIL-101_urea.90 Conversion of azobenzene moieties to the cisstate was noted to increase CH 4uptake, although partial photo-switching reversibility impeded function over multiple cycles.90Like the previously described studies regarding alcohol vapor adsorption, the saturated hydrocarbonstudy demonstrates the potential application in gas separations beyondCO 2mixtures. B. Optical properties In addition to gas phase properties, photo-switch incorporation into MOFs has allowed optical behavior modulation. {[Zn 2(ZnTCPP)(BPMTC) 0.85(DEF) 0.15], where H 4TCPP ¼tetrakis (4-carboxyphenyl)-porphyrin, BPMTC ¼bis(5-pyridyl-2-methyl-3- thienyl)cyclopentene, and DEF ¼diethylformamide} had diarylethene ligands that were incorporated as linkers between the 2D sheets.150 The photo-active diarylethene pillar ligands were capable ofmodulating fluorescent emission ( Fig. 5 ).150In the uncyclised state, the framework exhibited fluorescent emission at approximately 650 nm, which was quenched upon irradiation with a UV source.150 This was attributed to the photo-switch and was further supported bysimilar behavior being exhibited by the free diarylethene ligand. 150 The case study described here effectively illustrates the potential of photo-switches as a mechanism for efficient modulation of optical phenomena in MOF materials. C. Magnetic and conduction properties Photo-switching moieties have proven effective in modulating magnetic and conductive properties, commonly through synergistic effects with other components in the respective MOFs. Photo-modulated synergistic effects were demonstrated with variable mag- netic properties exhibited by endohedral metallofullerene guests, Sc 3C2@C 80and DySc 2N@C 80, incorporated intoAzoMOF.159For the former metallofullerene, spin relaxation was observed following azo- benzene switching.159Conversion to the cisform resulted in improved single-molecule magnet behavior from encapsulated DySc2N@C80.159 For both systems the enhanced polarity and aromatic interactions ofthe ligand’s ciss t a t ew e r ec a u s a t i v ef a c t o r si nv a r i a b l em a g n e t i cb e h a v - ior. 159The combined effect of the respective photo-switch and guest species in a MOF system were further demonstrated in [Cu 2(F2AzoBDC) 2(dabco)] where proton conduction could be photo- modulated.94The behavior in this case was altered through a synergis- tic effect between the pore space guest (either 1,4-butanediol or 1,2,3- triazole) and the state of the photo-switching moiety.94Green light irradiation of the ortho -fluoroazobenzene moiety to the cisdiastereo- mer, resulted in enhanced guest loading.94This in turn resulted in a NN S NS S NSUV visible light FIG. 5. Fluorescent behavior in [Zn 2(ZnTCPP)(BPMTC) 0.85(DEF) 0.15], where UV irradiation of the MOF quenched the 650 nm emission. Reprinted with permission from D. E. Williams, J. A. Rietman, J. M. Maier, R. Tan, A. B. Greytak, M. D. Smith,J. A. Krause, and N. B. Shustova, J. Am. Chem. Soc. 136, 11886–11889 (2014). Copyright 2014 American Chemical Society.150Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 2, 011301 (2021); doi: 10.1063/5.0035739 2, 011301-8 Published under license by AIP Publishinghigher guest density and, consequently, an increase in conductance of the framework in the cisisomer.94 Similar photo-modulated conduction properties were observed in a Cd(II) MOF series, ({[Cd(azbpy)(msuc)] /C12.5(H 2O)} n,w h e r e azbpy ¼4,4-azobispyryidine and msuc2/C0¼methylsuccinate ), ({[Cd(azbpy)(mglu)] /C15(H 2O)} n,w h e r em g l u2/C0¼methylglutarate ),a n d ({[Cd 1.5(azbpy) 2(glu)]/C1(NO 3)/C1MeOH}, where glu2/C0¼glutarate ).151 These frameworks exhibited semiconductor properties where the conductivity increased following conversion of the azobenzene groups to the cisconfiguration.151These MOFs contrasted with the photo-modulated behavior of [Cu 2(F2AzoBDC) 2(dabco)] as the con- ductive properties for the Cd(II) MOF series were intrinsic to theframeworks. 94,151For MOFs where their conductive properties are directly linked to the photo-switching moieties, the density of these functional groups are an important factor in their behavior.74 SP photo-switches have been more effective than azobenzenes in imparting photo-modulated conductivity due to the greater dipole dif- ference between the neutral (SP) and zwitterionic (MC) states. Thiswas demonstrated with SP@UiO-67 {SP@[Zr 6(OH) 4O4(bpdc) 6], SP¼10,30,30-trimethyl-6-nitrospiro[chromene-2,20-indoline]}.160UV irradiation to generate the MC state resulted in conductivity increasing by an approximate order of magnitude.160The authors noted that elec- trons acted as charge carriers in SP@UiO-67.160 Recent studies have further demonstrated the effectiveness of SP- MOFs in conduction applications. Recent materials are [Cu 2(SP- bpdc) 2(dabco)]105and SSP@ZIF-8 {SSP@[Zn(2mim) 2], where 2mim- ¼2-methyl-1 H-imidazole and SSP ¼10,30,30-trimethylspiro[chro- mene-2,20-indoline]-6-sulfonate}.84For [Cu 2(SP-bpdc) 2(dabco)], conversion of the SP moieties to their MC state allowed for increased proton conductivity through minimizing the mobility of aqueous and alcohol solvents.105Furthermore, the increase in conductivity was by approximately two orders of magnitude, therefore being more ideal for practical applications.105SSP-ZIF-8 exhibited enhanced proton conduction upon irradiating the MOF, where the zwitterionic nature of the MC state promoted transport through a combination of Grotthuss and vehicle mechanisms.84The photo-modulated behavior was further exploited as a control mechanism for a light source where the SP and MC configurations acted as “off” and “on” modes (Fig. 6 ).84The apparent effectiveness of SP-MOFs as proton conduc- tion materials demonstrates the necessity of exploring other organic photo-switching moieties. In addition to expanding accessible structures, the benefits also extend to other potential applications. Further experimentation with SSP@ZIF-8 (into membrane mate- rials from 8% to 10%) indicated that light-mediated conduction proper- ties also applied to Li(I).144Additionally, the MOF was able to separate Li(I) from other cations when the guests were in the MC state.144Li(I) ideal selectivity was calculated with respect to Na(I), K(I), and Mg(II) (with maximum values of 77, 112, and 4913, respectively).144 Ap r i o rs t u d yw i t h[ Z n 2(DBTD)(TNDS)] yielded similar results where the photo-modulated properties were employed for conductiv- ity applications.101[Zn 2(DBTD)(TNDS)] contrasted with SSP@ZIF- 8 as in the former, the SP moieties were incorporated intrinsically.84,101 Furthermore, in [Zn 2(DBTD)(TNDS)], conduction was not reliant on a guest species. Dolgopolova et al.101instead attributed the enhanced conductivity exhibited in the MC state to charge hopping between SP moieties (as facilitated by orbital delocalization and decreased spatialdistance).D. Biochemical applications Further promise for photo-switching MOFs stems from their potential application in biochemical functions. In addition to photo- modulated conductivity, which has been applied in mimicking biological systems, 3,161,162the contrasting properties before and after irradiation in photo-switching MOFs have been exploited for con- trolled species uptake and release.3,161A simple instance of this was conducted with propidium iodide (a luminescent dye), which wasincorporated as a guest into azo-IRMOF-74-III when its azobenzene moieties were in the cisconfiguration. 161Irradiation with visible light resulted in conversion of the azobenzene functional groups to theirtrans state, which resulted in the pore size decreasing from /C2410.3 A ˚to /C248.3 A˚. 161 Additional evidence for applications in controlled species uptake and release was provided by a more recent study with UiO-68-azo {[Zr 6(OH) 4O4(AzTPDC) 6], where AzTPDC2–¼(E)-20-(p-tolyldia- zenyl)-[1,10:40,100-terphenyl]-4,400-dicarboxylate}.3Rhodamine B was incorporated into the MOF as a guest.3The encapsulated rhodamine B was further trapped in the pore space through the addition ofb-cyclodextrin, a molecule that forms a stable complex with trans azo- benzene. 3Conversion of the azobenzene moiety to the cisisomer resulted in dissociation of the complex and release of b-cyclodextrin (Fig. 7 ).3This, in turn, allowed for release of the rhodamine B.3 The study also demonstrated that the photo-switching behavior may be effectively exploited through synergistic effects with othercomponents. Further exploitation of photo-switching MOFs in biochemical functions was demonstrated with a diarylethene-modified architec- ture. 162Similar to UiO-68-azo,3diarylethene and porphyrin-modified UiO-66 NPs were designed to act as a delivery system.162The diaryle- thene and porphyrin species acted as a switch and photosensitiser, respectively, and were incorporated as localized defects in the UiO-66 lattice.162The porphyrin allowed generation of1O2from3O2through a direct energy transfer process, which was quenched by UV ‘‘off’’ ‘‘on’’Dark Light NNO OSO3SO3 Dark Light FIG. 6. SSP was extrinsically incorporated into ZIF-8 membrane. The resulting photo-switching behavior exhibited by the SSP allowed for photo-modulated con-ductivity in membrane. The SP and MC states corresponded to “on” and “off”modes when used to control an LED. Adapted with permission from H. Q. Liang, Y. Guo, Y. Shi, X. Peng, B. Liang, and B. Chen, Angew. Chem. Int. Ed. 132, 1–7 (2020). Copyright 2020 Wiley. 84Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 2, 011301 (2021); doi: 10.1063/5.0035739 2, 011301-9 Published under license by AIP Publishingirradiation of the diarylethene switch.162The NPs were further incor- porated into B16 melanoma cells where the photo-modulated1O2 generation persisted.162Incorporation into a cellular system with photo-modulated function is significant given that it represents animportant step in realizing biochemical-based applications. The behavior of modified UiO-66 NPs observed by Park et al. 162is espe- cially important as it demonstrates the effectiveness of a substratesupport; notably the authors found that the discrete dyad was less efficient for conversion, in part because the MOF allowed for control of photo-switch and sensitizer molar ratios. E. Other functions Aside from commonly investigated applications, other functions have been described in the literature. Photo-modulated catalysis was demonstrated with ECUT-50, which was capable of increasing the rate for Knoevenagel condensation reactions. 55Aldehyde conversion occurred at <6% following 365 nm irradiation.55Prior to UV irradia- tion of the AzDC2–ligand, the authors reported 92.6% conversion.55 Similar behavior was also observed in the discrete ligand although effectiveness was diminished.55Catalytic behavior exhibited by theInCl 3precursor suggested a synergistic effect in ECUT-50.55Similar photo-modulated catalytic activity has been observed in discrete sys- tems with organic photo-switching units as exemplified by Ag@MS- Azo.163Here conversion of azobenzene to the cisconfiguration in Ag@MS-Azo enhanced catalytic activity for methylene blue reductionthrough disassociation with a-cyclodextrin. 163 In addition to the gas phase, photo-switching MOFs have also been applied in solution-based separations. Furlong and Katz149 achieved this with PSZ-1 {[Zn(L) 0.27(im)(NO 2-im) 0.70], where L ¼4,5- bis(2,5-dimethylthiophen-3-yl)imidazole-1-ide, im/C0¼imidazolate and NO 2-im¼nitroimidazolate}. UV irradiation of the diarylethene linker improved toluene (MePh), naphthalene, and pyrene aromatics retention when filtering a 1:1:1% solution through the MOF.149 Reversibility was demonstrated through visible light irradiation, wherePSZ-1 exhibited initial retention properties. 149Based on a lack in the variation in retention times, the authors proposed that the improved filtration behavior was based on electronic interactions.149The photo- modulated solution phase separation performance in PSZ-1 demon-strates a new potential application compared to the initial focus onCO 2capture. T h er e c e n t l yc o n d u c t e ds t u d yb yO u et al.1demonstrated that solution-based separations could be applied in a commercial context.Desalination potential was investigated in PSP-MIL-53 where incorpo-rated SP and poly(SP-acrylate) (PSP) guests allowed for photo- modulated NaCl adsorption from aqueous solution. 1Excitation of the incorporated SP to its respective MC state (under either UV light ordark conditions) resulted in adsorption of Na þand Cl/C0into PSP- MIL-53. Ion desorption occurred through reversion to the SP state(requiring visible light). 1Incorporation of both guests into PSP-MIL- 53 was necessary to minimize photodegradation, thereby ensuring function over multiple switching cycles. Practical demonstration ofmodulated desalination exemplified the future importance of structur-ally responsive photo-active MOFs as materials designed to alleviatesignificant social concerns such as access to drinking water. Photolithography was conducted with a series of SP-modified MOFs, MOF-5, MIL-68(In), MIL-68(Ga), and MIL-53(Al). 2Partial coverage was noted to allow for the generation of a pattern on theMOF surface due to coloration from SP conversion to the MC state. 79 Heating to revert the incorporated photo-switch to the SP stateresulted in loss of the generated pattern. 79Other substrates modified with photo-switching functional groups have already been subject tophoto-lithographic studies. 164Demonstration of this behavior in photo-active MOFs puts framework substrates on an equal footing with others exploited in the literature, including polymers,112–116 gels,117–119and NPs.113,120 V. SUMMARY AND OUTLOOK In summary, the recent developments in photo-switching MOFs represent significant strides toward practical application. This includesthe incorporation of these architectures in increasingly complex mate-rials, such as SURMOFs and composites with polymers. Additionally, attempts at tuning photo-modulated properties have been expanded to include the use of photo-switching moieties that either have zwitter-ionic states or are visible light–responsive. The latter provides a meansof enhancing the fatigue resistance in photo-switching MOFs whilethe exploitation of moieties with zwitterionic excited sates has allowed an expansion of potential applications. In particular, device fabrication OO N NNUVOO N O OO = Rhodamine B = β-cyclodextrinO FIG. 7. Abstract schematic of UV triggered rhodamine B release from UiO-68-azo where the cargo is initially held in place by b-cyclodextrin (through complex forma- tion with trans azo groups). UV irradiation results in conversion of the ligand’s azo- benzene groups to the cisconfiguration allowed dissociation of b-cyclodextrin and consequent release of rhodamine B. Adapted with permission from X. Meng, B.Gui, D. Yuan, M. Zeller, and C. Wang, Sci. Adv. 2, 2–8 (2016). Copyright 2016 American Association for the Advancement of Science.3Chemical Physics Reviews REVIEW scitation.org/journal/cpr Chem. Phys. Rev. 2, 011301 (2021); doi: 10.1063/5.0035739 2, 011301-10 Published under license by AIP Publishingfor electronics applications have advanced considerably with the devel- opment of SP-modified MOFs with photo-modulated conductivity. This is in addition to expansions in other areas, including photo- modulated gas extractions (where mixtures of interest have movedbeyond the purview of CO 2mixtures), and other functions (in areas such as biochemistry, catalysis, and magnetic materials). AUTHORS’ CONTRIBUTIONS All authors contributed equally to this manuscript. All authors reviewed the final manuscript. ACKNOWLEDGMENTS We gratefully acknowledge the support of the Australian Research Council (Grant No. DP180103874). The authors would like to thank the University of Sydney for resources and facilities necessary for continued research. 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5.0037937.pdf

J. Appl. Phys. 129, 010901 (2021); https://doi.org/10.1063/5.0037937 129, 010901 © 2021 Author(s).Spintronic terahertz emitter Cite as: J. Appl. Phys. 129, 010901 (2021); https://doi.org/10.1063/5.0037937 Submitted: 17 November 2020 . Accepted: 10 December 2020 . Published Online: 04 January 2021 Zheng Feng , Hongsong Qiu , Dacheng Wang , Caihong Zhang , Song Sun , Biaobing Jin , and Wei Tan ARTICLES YOU MAY BE INTERESTED IN Perspective: Ultrafast magnetism and THz spintronics Journal of Applied Physics 120, 140901 (2016); https://doi.org/10.1063/1.4958846 Ferrimagnetic insulators for spintronics: Beyond garnets Journal of Applied Physics 129, 020901 (2021); https://doi.org/10.1063/5.0033259 Pure spin current phenomena Applied Physics Letters 117, 190501 (2020); https://doi.org/10.1063/5.0032368Spintronic terahertz emitter Cite as: J. Appl. Phys. 129, 010901 (2021); doi: 10.1063/5.0037937 View Online Export Citation CrossMar k Submitted: 17 November 2020 · Accepted: 10 December 2020 · Published Online: 4 January 2021 Zheng Feng,1,2Hongsong Qiu,3Dacheng Wang,1,2Caihong Zhang,3 Song Sun,1,2Biaobing Jin,3 and Wei Tan1,2,a) AFFILIATIONS 1Microsystem & Terahertz Research Center, CAEP, Chengdu 610200, China 2Institute of Electronic Engineering, CAEP, Mianyang 621000, China 3Research Institute of Superconductor Electronics (RISE), School of Electronic Science and Engineering, Nanjing University, Nanjing 210093, China a)Author to whom correspondence should be addressed: weitan@csrc.ac.cn ABSTRACT While the technology of microwave and infrared sources is quite mature and has been widely used in our daily life for decades, sources that can work well across the terahertz (THz) range are still lagging behind, which is often referred to as the “THz gap. ”As one of the most pioneering THz setups, terahertz time-domain spectroscopy has been a vital tool to explore the properties of materials as well as their underlying physics. The mechanism is to use an ultrafast infrared pump pulse for exciting rapidly decaying currents inside either a nonlinear or a photoconducting medium, known as a THz emitter, which produces free-space coherent THz radiation. Most recently, a novel THzemitter emerges and rises, which is based on the spin-related effects in magnetic/nonmagnetic nanofilms and can cover the full range of theTHz band, named as spintronic THz emitter (STE). This perspective aims to elucidate the unique features and advantages of STE as well asits capability and potential to develop novel applications. We summarize the multidisciplinary efforts that have been made to improve the performance and function of STE, including but not limited to spintronics, optics, and electromagnetics. Distinct THz setups based on STE are reviewed, which may inspire various “real world ”applications in the near future. Published under license by AIP Publishing. https://doi.org/10.1063/5.0037937 I. INTRODUCTION The terahertz (THz) frequency region (0.1 –10 THz) lies in the electromagnetic spectrum between the microwave band and themid-infrared band. From a historical perspective, terahertz radia-tion is a relatively new description for far-infrared radiation, 1 which can be traced back to the 1980s.2–4Most recently, the THz band was announced to be open for experimental use for the next generation (6G) of wireless communication networks,5which is highly expected to offer data rate of more than 100 Gbps owing toits broader available bandwidth over the current microwave wirelesslinks. 6–9The large bandwidth (generally larger than 20 GHz) also offers the capability of high spatial resolution in radar and imaging systems, which supports the applications of defense and securityimaging. 10–12These “real world ”applications indicate that the THz technology would be seen in our daily life in the near future. Suchprogresses benefit from the rapid development of THz devices and components, especially continuous THz sources, such as Schottky-diode-based multipliers, 13–15transistor-based integrated circuits,16–18quantum cascade lasers,19–21and photomixers.9,22Additionally, various kinds of materials have unique absorption characteristics at the THz band, which originate from rotational and vibrational transitions in molecules. Such spectral fingerprinting fea-tures promote another branch of THz instruments, THz time-domain spectroscopy (THz-TDS), 2,4which provides a powerful tool for the fundamental study of condensed matter,23,24biomedical applications,25–27and non-destructive testing.28,29THz-TDS relies on THz emitters which generate temporal THz pulses when pumped byultrafast laser pulses commonly with durations between 10 and 120 fs. The setup is schematically shown in Fig. 1 . Typical solid THz emitters include photoconductive antennas utilizing photo-generatedelectrons and holes in semiconductors 30–32and nonlinear crystals for optical rectification,33–37such as ZnTe, GaP, LiNbO 3crystals, and organic crystals,38,39which are commercially available and used in hundreds of research laboratories worldwide. In addition, intenseTHz pulses can also be generated in gaseous 39–42and even liquid media43–46based on femtosecond-laser-induced plasma formation. One can see that four states of matter: solid, gas, plasma, and liquid,Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 010901 (2021); doi: 10.1063/5.0037937 129, 010901-1 Published under license by AIP Publishing.have already been used for THz wave generation by utilizing the charge degree of freedom of electron and its transient response. Fascinatingly, a novel THz emitter was recently proposed and demonstrated in 2013, which is based on the spin related effects inferromagnetic/nonmagnetic (FM/NM) nanofilms, and thus wasnamed as spintronic THz emitter (STE). The introduction of addi- tional spin degree of freedom presents unprecedented advantages, such as ultra-broad bandwidth (up to 30 THz) and flexible tunabil-ity via tailoring external magnetic fields. Up to date, while the THzemission efficiency has been enhanced to reach the same level ofmillimeter-thick ZnTe crystals, STEs are operating in several research laboratories. In addition, its flexible and low-cost fabrica- tion process as well as its excellent performance makes it promisingfor commercial setups and applications. In this brief perspective article, we first introduce the general principle of STE, while the extensive reviews on ultrafast magne- tism can refer to Walowski et al. 47We then review the progresses that have been made to improve the performance of STE and takeadvantage of its flexible tunability. Emerging physics and applica-tions are further discussed. Finally, we present our perspective onpotential pathways to further enhance the performance and demands for an extensive scope of THz applications. II. PRINCIPLE OF SPINTRONIC TERAHERTZ EMITTERS The core mechanism of STE is the conversion from the ultra- fast spin current to charge current, as suggested by its name. The spin current J sis the flow of the electron spin.48,49In a magnetized FM layer, the mobility of the majority spin-up electrons is higherthan that of the minority spin-down electrons, and hence, animpulsive J scan be generated in the FM layer with the pump of the ultrafast laser pulse. When it is injected into the neighboring NM layer, Jsis converted to the transversal charge current Jcdue to the spin–orbit coupling of the NM layer.50The sign of Jcdepends on the direction of the spins. The transient Jcwith a time scale ofsub-picosecond acts as an electric dipole, giving rise to the electro- magnetic radiation in the THz band into the free space. The most involved mechanism for the spin-charge conversion in STE is based on the inverse spin Hall effect (ISHE).50–55ISHE is the reciprocal effect of the spin Hall effect.56–58The spin-polarized electrons are deflected asymmetrically due to the relativistic spin–orbit coupling, such that a pure spin current through the material generates a transverse charge current. Heavy metals, e.g.,Pt 59and W,60have been reported to have large ISHE, and are there- fore commonly used in STE. A pioneering study by Kampfrathet al. reported the spintronic THz emission from the metallic ferro- magnetic structures Fe/Au and Fe/Ru. 50The electrons in the Fe layer are promoted by the femtosecond laser pulses from below theFermi energy to bands above it and begin to diffuse, as shown inFig. 2(a) . The diffusive current is spin-polarized because of the asymmetry of the spin-dependent density of the states 61,62in the Fe layer [see Fig. 2(b) ]. A net spin current is thus launched in the Fe layer and flows into the Au or Ru layer. The transportation of thespin-polarized hot electrons is described by the super-diffusionmodel. 63Because of the ISHE, this longitudinal spin current in the Au or Ru layer converts into a transient transverse charge current. Concurrent with the laser modulation of magnetization in the Fe layer, the spin angular momentum is also inevitably transferredinto the adjacent Au and Ru layer by the spin pumping effect, 64 which is independent of the moving carriers. According to the cal- culation, the spin current generated by spin pumping is two orders of magnitude smaller than that generated by diffusive hot elec-trons. 50By changing metallic cap layers with different carrier mobilities, one can largely tune the spatiotemporal shape of thespin transport, and thus can tailor the dynamics of the THz emis- sion. It was quite fascinating to find that the easy-to-fabricate and cost-efficient FM/NM heterostructure owns the capability to coverthe full THz band. 50 Note that, in addition to the FM/NM heterostructure, the bare FM films can also generate THz radiation.65–68The laser-induced FIG. 1. Schematic of THz-TDS. THz emitter plays the role of generating temporal THz pulses when pumped by ultrafast laser pulses.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 010901 (2021); doi: 10.1063/5.0037937 129, 010901-2 Published under license by AIP Publishing.ultrafast demagnetization process acts as a magnetic dipole for THz radiation, which was experimentally confirmed through the com- parison of the signals measured by THz emission and the time-resolved magneto-optical Kerr effect (TR-MOKE). 67 III. IMPROVEMENT OF PERFORMANCE In comparison with standard THz emitters, such as nonlinear optical crystals and photoconductive switches, STE shows a remarkable superiority in the ultra-broad bandwidth, which wasdemonstrated to cover the full range from 0.3 to 30 THz without agap, 51as shown in Fig. 3(a) . Hence, increased attention has been paid to this novel type of THz emitters. One of the key issues is to improve the THz emission amplitude as well as its efficiency to reach the same level or even exceed that of standard THz emitters. Considering the FM/NM heterostructure based on ISHE, the amplitude of the THz emission sensitively depends on the spinHall angle γ NMof the NM layer, absorptance Pabsof the laser power, and thicknesses d x(x = NM and FM) of the NM and FM layers, which has been expressed as51,69 ETHz/γNM/C1Pabs dFMþdNM/C1tanhdFM/C0d0 2λpol/C18/C19 /C1tanhdNM 2λNM/C18/C19 /C11 nairþnsubþZ0/C1(σFMdFMþσNMdNM)/C1e/C0(dFMþdNM)/sTHz, (1) with Z0being the impedance of vacuum, nair(nsub) being the index of refraction of air (the substrate), and σFM(σNM) being the electri- cal conductivity of the FM (NM) layer. The first term in Eq. (1)suggests that materials with larger spin-to-charge conversion efficiency can support higher THz emis-sion amplitude. The second term takes into account the laserabsorption P absin the metal layers. Only a fraction of the absorbed power contributes to the generated THz signal because only spin- polarized electrons near the FM/NM boundary will reach the NMlayer. Torosyan et al. estimated this fraction as 1/(d FM+dNM).69 The third term describes the relationship between spin polarization and the layer thickness dFM, with d0being the thickness of the “dead layer ”(non-magnetic regions at the interfaces in thin FM films) and λpolcharacterizing the saturation of spin polarization with layer thickness. The fourth term refers to the spin accumula- tion in the NM layer according to the Valet –Fert spin diffusion model,70,71where λNMdenotes the diffusion length of spin current in the NM layer. The fifth term accounts for the electromagneticradiation from a thin metallic layer, which is derived from theMaxwell equation via Green ’s function method. The last term accounts for losses of the THz wave in the metal layers. It can be described by a single exponential attenuation factor with an effec-tive inverse attenuation coefficient s THz. From Eq. (1), one can identify some general methods to improve the THz emission intensity: A. Selecting materials with larger spin Hall angle Various NM (heavy) materials, such as Pt, W, Ta, Ir et al. , have been investigated to maximize the THz output.51 Experimental results showed that the THz amplitude is almost pro- portional to spin Hall conductivity.51Among these, Pt offers the highest THz amplitude, and interestingly, the THz emission fromW shows the opposite sign related to its negative spin Hall angle[seeFig. 3(b) ]. B. Optimizing the thickness of the FM and NM layers Note that all terms in Eq. (1)except the first one are affected by the thickness of the FM and NM layers. The THz amplitude increases with the thicknesses of layers according to the third andfourth terms in Eq. (1), whereas it decreases according to the second, fifth, and last terms. Hence, there should exist an optimal layer thickness where all terms in Eq. (1)reach the best balance. In experiments, the THz emission is significantly enhanced at first by FIG. 2. Principle for STE. Scheme for (a) two dominate processes in typical FM/NM heterostructure: ultrafast laser pulse induced spin polarized current gen eration and spin-charge conversion based on ISHE, and (b) asymmetric spin-dependent density of the states in Fe. Reproduced with permission from Kampfrath et al. ,N a t . Nanotechnol. 8, 256 (2013). Copyright 2013 Springer Nature.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 010901 (2021); doi: 10.1063/5.0037937 129, 010901-3 Published under license by AIP Publishing.increasing the thickness of the NM and FM layer (as far as the FM layer exceeds the dead-layer thickness),51,69and soon hits the ceiling once the thickness reaches an optimized value. Beyond theoptimal thickness, the THz emission begins to taper off. The opti- mized thickness for both the NM and FM layers is a few nanome- ters in most of the literature [see Fig. 3(c) ]. C. Optimizing spin current transportation The pioneering FM/NM bilayer structure utilizes only the forward spin current. In 2016, Seifert et al. demonstrated a trilayer structure composed of W/Co 40Fe40B20/Pt, which offers a 40% higher THz amplitude benefiting from the opposite signs of thespin Hall angles of W and Pt and the constructively combined THzemissions contributed by the forward and backward spin cur- rents, 51as shown in the inset of Fig. 3(c) . On the other hand, Zhang et al. proposed a STE with the geometry of the exchange- coupled synthetic antiferromagnet (SAF) structure Pt(2 nm)/CoFeB(4 nm)/Ru(0.8 nm)/CoFeB(3 nm)/Pt(2 nm). 72When the magnetiza- tion directions of the two CoFeB layers are opposite, the THz emis- sions from the two Pt layers contribute constructively.In addition, the microstructural properties, including the crys- tallization, interfacial roughness, and interfacial intermixing alsoobviously influence the performance of STE. 73–75Sasaki et al. found that there is an enhancement of 1.5- to 6-times by annealing Ta(5 nm)/Co 20Fe60B20(tCFB)(tCFB= 4 and 10 nm) at 300 °C as com- pared to the as-grown ones.73The authors attribute the large enhancement to the increased mean free path λupof the hot spin-up electrons. The improved crystallization of Co 20Fe60B20after the annealing suppresses the disorder scattering and thus enlarges λupto a couple of nanometers ( λupis estimated to be 1 nm for amorphous Co 20Fe60B20). In addition, Li et al. demonstrated that the THz emission from a Co(10 nm)/Pt(3 nm) structure sensitivelydepends on the interface roughness. 74The roughness is controlled by modifying the Ar pressure during the sputtering. The perfor- mance deteriorates by a factor of 2 if the roughness increases from ∼0.1 nm to ∼0.3 nm. D. Improving laser pumping process The laser absorbance is rather limited in the bilayer and tri- layer structures, and a toy model showed that the laser absorptance cannot be larger than 50% in a metallic film with few nanometer- FIG. 3. Improving performance of STE via optimizing materials and structures. T erahertz amplitude as a function of (a) the NM material and (b) the thickness o f total struc- ture (including bilayers and trilayers). (c) Fourier spectra of THz signal covering the full range from 0.3 to 30 THz. Reproduced with permission from Seifert et al. ,N a t . Photonics 10, 483 (2016). Copyright 2016 Springer Nature.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 010901 (2021); doi: 10.1063/5.0037937 129, 010901-4 Published under license by AIP Publishing.thickness.54A stack of multiple spintronic THz devices is an enlightening strategy.52Taking a further step, Feng et al. proposed a metal-dielectric photonic crystal structure [SiO 2/Pt/Fe/W] n (n = 1,2,3, …), which is schematically shown in Fig. 4(a) . In such a structure, the interference between the multiple scattering wavessimultaneously suppresses the reflection and transmission of the laser and hence improves the absorption. 54The normalized THz amplitude from both experiments and model calculations vs thethickness of dielectric layer is also shown in Fig. 4(a) . A maximum of 1.7 times improvement was achieved compared to a singlePt/Fe/W emitter. Herapath et al. proposed to increase the laserabsorbance by integrating the W(2 nm)/CoFeB(1.8 nm)/Pt(2 nm) emitter with a dielectric lattice TiO 2(113 nm)/SiO 2(185 nm),76as shown in Fig. 4(b) . The thickness of the dielectric layers was opti- mized for a band from 900 to 1200 nm. The maximum THz fieldalmost doubles compared to the bare structure without dielectriclattice. E. Improving THz emission process While the above methods are targeted at the THz generation process, the integration of the STE with well-designed optical FIG. 4. Improving performance of STE via optimizing laser absorption and THz emission processes. (a) Metal-dielectric photonic crystal structure. (b) STE with dielectric cavity. (c) Antenna coupled STE. (d) Large-area STE (diameter of 7.5 cm). Panel (a) is reproduced with permission from Feng et al. , Adv. Opt. Mater. 6, 1800965 (2018). Copyright 2018 John Wiley & Sons. Panel (b) is reproduced with permission from Herapath et al. , Appl. Phys. Lett. 114, 041107 (2019). Copyright 2019 AIP Publishing LLC. Panel (c) is reproduced with permission from Nandi et al. , Appl. Phys. Lett. 115, 022405 (2019). Copyright 2019 AIP Publishing LLC. Panel (d) is reproduced with permission from Seifert et al. , Appl. Phys. Lett. 110, 252402 (2017). Copyright 2017 AIP Publishing LLC.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 010901 (2021); doi: 10.1063/5.0037937 129, 010901-5 Published under license by AIP Publishing.components could also improve the THz emission process. Torosyan et al. attached a hyper-hemispherical silicon lens to the substrate side of the Fe/Pt emitter with the film plane located at thefocal point of the lens. 69The strongly divergent THz beam was col- limated for an improved collection of the THz emission. Nandi et al. demonstrated the lumped-element, antenna- coupled STE pumped at the wavelength of 1550 nm with the laser spot diameter of nearly 10 μm.78As shown in Fig. 4(c) , the spin- tronic stack Pt(2 nm)/Co 40Fe40B20(1.8 nm)/W(2 nm) (grayish brown areas in the center) couples with the H-dipole (l a= 200 μm and gap size of w a=1 0 μm) and a slot-line (w a=2 5 μm) antenna. The peak-to-peak THz amplitude is enhanced by 2.42 times by using the 200 μm H dipole antenna. It was also surprising to find that the bilayer structure consist- ing of Co and light metal Al offers THz emission almost one thirdof that of the typical Co/Pt structure, 79although the spin Hall angle of Al is two orders of magnitude smaller than that of Pt. One of the reasons is that σAlis much smaller than σPtwhen the thick- ness is below 5 nm, thus having larger impendence for higher THzemission efficiency. F. Enhancing pumping energy Another direct way to significantly increase the THz emission amplitude is using large-area STE excited by high energy laser pulses,which was proposed by Seifert et al. , 77though it does not improve the efficiency. A trilayer structure W(1.8 nm)/Co 20Fe60B20(2 nm)/Pt (1.8 nm) was prepared on a fused silica substrate with a diameter of7.5 cm, as shown in Fig. 4(d) . The excitation by optical pump pulses of 5.5 mJ results in single-cycle THz pulses with a peak electric fieldof 300 kV/cm and pulse energy of 5 nJ. It is thus an easy-to-use yet powerful tool for THz nonlinear perturbation over the matter within the sub-picosecond time scale. IV. TUNABILITY OF SPINTRONIC TERAHERTZ EMITTERS One of the unique features of STE, which is distinct from the traditional THz emitters without spin process, is that the polariza-tion of the emitted THz pulses is independent of either the orienta-tion of the linearly polarized incident laser pulses or the crystalline anisotropy of the nano-films, but is always perpendicular to the external magnetic field. It provides an unprecedented degree offreedom to manipulate THz emission, including but not limited tothe manipulation of polarization, amplitude, and spectrum lineshape. A. Polarization rotation and conversion The most intuitive idea is to rotate the polarization of the gen- erated THz pulses by rotating the external magnetic field, whilekeeping the amplitude constant. A special case is that when the magnetic field reverses, the THz emission experiences a phase change of π. 52Taking a further step, Hibberd et al. demonstrated that when the STE is placed between two magnets of opposingpolarity, quadrupole-like polarization can be generated, 80as shown inFig. 5(a) . Kong et al. showed that in such a situation, the defects in the sample were supposed to result in a small anisotropy, thusinducing phase difference between orthogonal electric field.81 Consequently, elliptically polarized terahertz waves can be directly generated, whose chirality and azimuthal angle can be readilytuned [see Fig. 5(b) ]. Nevertheless, ellipticity can only vary within a small range which is limited by the phase difference. 81Then, Chen et al. presented a scheme composed of two spatially separated STEs with an orthogonal external magnetic field. By tuning the incident laser pump fluence and separation distance, almost equalamplitude and 90° phase difference were achieved, which give riseto the circular polarization. 82 An alternative method to generated arbitrarily polarized THz wave is to integrate STE with planar optical components, such as waveplates, liquid crystals, and metasurfaces. Qiu et al. demon- strated an integration of STE and large-birefringence liquid crystals,where the phase retardation can be continuously tuned over arange of [0, π/2] by applying a low voltage. 83By rotating the exter- nal magnetic field, the ellipticity of the THz emission can be varied from 0 to 1 continuously, as shown in Fig. 5(c) . However, the circu- lar polarization can only be realized at a single frequency. On theother hand, metasurfaces, which are ultrathin artificial structuresconsisting of planar subwavelength units with engineered electro- magnetic responses, offer more flexibilities in overcoming this limit. 84–87Co-designed with a STE, the metasurface composed of two layers of metallic wire gratings with twisted angles plays therole of a broadband quarter-wave plate. 88As a result, circular and elliptical terahertz waves with various chirality, azimuthal angle, and ellipticity can be converted from the original linear polariza-tion in a broad band. B. Amplitude and spectral line shape manipulation In principle, the THz amplitude generated in STE depends on the sample magnetization which determines the spin polarization. In practice, STE is commonly operating at an optimized conditionwhere the magnetization is completely saturated by an externalmagnetic field. Before saturation, the THz amplitude drops as the magnetic field decreases, 89–92which in turn offers a flexible route to tune the THz amplitude directly. Ferrimagnetic materials werealso employed (instead of ferromagnetic materials) for STE. Chenet al. investigated THz emission from Co 1−xGdx/Pt structures and showed that the amplitude can be tuned by tailoring the net spin polarization, which can be realized by changing the Gd fraction or temperature.89Schneider et al. demonstrated a STE based on Pt and terbium-iron (Tb xFe1−x) alloys.90The TbFe layer can change from an in-plane to an out-of-plane easy axis by varying Tbcontent, which changes the net in-plane magnetization and thus the THz amplitude. Based on a similar principle, a GdFe/Pt emitter was later demonstrated with a much higher THz amplitude. 91 Furthermore, Fix et al. demonstrated that a Pt/Gd 10Fe90/W/ Gd30Fe70/Pt emitter can be switched between a high- and a low- amplitude terahertz emitting state by tuning temperature.93For temperatures below a critical temperature, the Fe moments of the two Gd xFe100−xlayers are aligned antiparallel, leading to in-phase transient charge currents and thus high amplitude; for tempera-tures above such critical temperature, the Fe moments of the two ferrimagnetic layers are aligned parallel, leading to out-of-phase transient charge currents and thus low amplitude.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 010901 (2021); doi: 10.1063/5.0037937 129, 010901-6 Published under license by AIP Publishing.Meanwhile, inspired by the concept of metasurface whose electromagnetic responses depend on not only the material itself but also the artificial structures, researchers turned to make the STE itself patterned. Yang et al. presented that in a stripe-patternedFe/Pt sample, the THz amplitude can be modulated by nearly 40% when the angle between the stripe and the magnetic field is tuned from 90° to 0°. On top of that, the spectral line shape can be manipulated by a patterned STE as well,52as shown in Fig. 6(a) . FIG. 5. Polarization control of STE. Two magnets of opposing polarity induced (a) quadrupole-like polarization and (b) elliptically polarization. (c) Cir cular polarization gen- eration via integration of STE and large-birefringence liquid crystals. Panel (a) is reproduced with permission from Hibberd et al. , Appl. Phys. Lett. 114, 031101 (2019). Copyright 2019 AIP Publishing LLC. Panel (b) is reproduced with permission from Kong et al. , Adv. Opt. Mater. 7, 1900487 (2019). Copyright 2019 John Wiley & Sons. Panel (c) is reproduced with permission from Qiu et al. , Appl. Phys. Express 11(9), 092101 (2018). Copyright 2018 The Japan Society of Applied Physics.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 010901 (2021); doi: 10.1063/5.0037937 129, 010901-7 Published under license by AIP Publishing.It was attributed to the charge accumulation due to the discontinu- ity of metallic stripe, which generates a transient opposite electric field and consequently changes the initial transient charge current.Jinet al. demonstrated a nearly 90% amplitude modulation in a Pt/CoFeB/W stripe structure. 94 STE can also be modulated by external bias and laser pulses. Chen et al. demonstrated a hybrid emitter consisting of a FM/NM heterostructure deposited on a high-resistivity silicon substrate anda three-wire pattern with contact pads. The total THz emissionincludes two contributions: one from the FM/NM heterostructure,and the other from the biased semiconductor, 95as shown in Fig. 6(b) . The experimental results showed that in the frequency range of 0.1 –0.5 THz, the amplitude of THz emission can be mod- ulated by nearly two orders via external bias. Wang et al. studied the linear and nonlinear response of STE under illumination of apair of fs laser pulses, suggesting that both the amplitude and spec- tral line shape can be tuned by the additional laser pulses. 96Most recently, spin valve systems consisting of a magnetically soft free Felayer and a magnetically hard Fe fixed layer pinned by an antiferro-magnetic Ir 23Mn 77layer was demonstrated as switchable STEs.97 The principle is schematically shown in Fig. 6(c) . The relative mag- netization alignment of the two Fe layers can be varied by switch- ing the magnetization of the free Fe layer with external magneticfields in the range of a few mT. More than one order of magnitudeswitching was experimentally realized. 97 V. EMERGING PHYSICS In addition to the studies of STE consisting of FM/NM heter- ostructures based on ultrafast diffusive hot electron process and ISHE [as summarized in Fig. 7(a) ], novel spin related effects have also been explored for ultrafast spin current generation and spin-charge conversion, as they can be characterized by measuring THzemission. A. Emerging physics in ultrafast spin current generation The spin transport can also be generated along the tempera- ture gradient through an interface between a magnetic insulatorand an NM layer according to the spin Seebeck effect. 98,99This sce- nario was first implemented by Seifert et al. in the STE using the YIG/Pt structure [see Fig. 7(b) ].100The temperature excursion at the interface of YIG/Pt is about 50 K during laser excitation. Theresponse is quasi-instantaneous since thermalized electrons in thePt layer traverse the interface region within 4 fs and YIG spins reactwithout inertia. The increased electron temperature is then trans- ferred to phonons via electron –phonon scattering with a time cons- tant τ e-phffi370 fs. In previous works, ferromagnetic and ferrimagnetic materials are generally used for the injection of the spin currents in the STEs.Recently, Qiu et al. reported the optical generation of ultrafast spin current in the antiferromagnetic structure NiO(20 nm)/Pt(3 nm) at zero magnetic fields, 101as schematically shown in Fig. 7(c) . Antiferromagnets exhibit magnetic order with zero net magnetiza-tion and have a low magnetic susceptibility, so exceedingly high magnetic fields (of the order of 10 T) are indispensable in the con- ventional spintronics to break the degeneracy of the two magnoneigenmodes. 102,103In contrast, the perturbation of photons is more dramatic and the transient magnetization M(t) in the NiO layer can be induced through the magnetic difference frequency genera-tion process. The magnetization dynamics of the M(t) with a time scale of sub-picoseconds facilitates the generation of the ultrafastspin current. B. Emerging physics in ultrafast spin-charge conversion In analogy to ISHE, inverse Rashba –Edelstein effect (IREE) is the reciprocal effect of the Rashba –Edelstein effect and describes the generation of a charge current by a spin current in a two-dimensional electron gas in response to the Rashba spin –orbit cou- pling. 104,105ISHE describes the conversion when the spin current diffuses through the bulk material, whereas the IREE occurs at the Rashba interface. Efficient IREEs have been found at theAg/Bi, 106–108LaAlO 3/SrTiO 3,109and (MoS 2or WSe 2)/CoFeB110 systems. Two groups independently reported laser-induced broad- band THz radiation utilizing the interface IREE.107,108As shown in Fig. 7(d) , the femtosecond laser pulses are used to excite the heter- ostructure Fe(2 nm)/Ag(2 nm)/Bi(3 nm) or Fe(2 nm)/Bi(3 nm)/Ag(2 nm). The Fe film is magnetized by an in-plane magnetic fieldand hence facilitates the generation of spin currents, which subse- quently superdiffuse across the neighboring layer and arrive at the Rashba interface. Because of the IREE, this longitudinal spincurrent at the interface converts into a transient transverse chargecurrent. The reversed Rashba interface between Ag/Bi and Bi/Agstructures can give rise to the opposite polarities of the THz wave- forms. This observation reveals a novel mechanism for STEs based on interface states rather than bulk ones. In 2018, Wang et al. demonstrated ultrafast spin-injection and spin-to-charge conversion in topological-insulator/FM heterostruc-tures using THz emission spectroscopy [see Fig. 7(e) ], where the topological surface states play a crucial role. 111Compared with pure Bi2Se3and Co films, Bi 2Se3/Co heterostructure offers much larger amplitude of THz pulses. It was also found that the spin-to-chargeconversion efficiency is temperature independent in Bi 2Se3as expected from the nature of surface states. Cheng et al. observed an efficient spin current injection from Co to monolayer MoS 2via THz emission from Co/MoS 2structures.112Femtosecond laser pulses trigger far out-of-equilibrium carrier distribution in the Colayer. The minority-spin electrons tend to be quickly thermalized to a hot Fermi-Dirac distribution due to the spin asymmetry of the high-energy electron-electron scattering, while large amounts ofmajority-spin electrons persist for a longer time and diffuse to theCo/MoS 2interface, as schematically shown in Fig. 7(f) . The bandgap of MoS 2filters only the high-energy carriers where the population is almost fully spin-polarized, and the MoS 2layer acts as a converter of the spin to charge current, thereby facilitatingthe THz emission as a STE. VI. APPLICATIONS From the previous sections, one can find that the STE can provide not only high efficiency reaching the same level of 1-mm-thick ZnTe crystals in a wide range of laser pulse duration from 10 to 100 fs but also ultrabroadband spectra that is superiorJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 010901 (2021); doi: 10.1063/5.0037937 129, 010901-8 Published under license by AIP Publishing.FIG. 6. Amplitude and spectral lineshape manipulation of STE. (a) Stripe-patterned STE. (b) Three-wire patterned STE with contact pads on silicon substrat e. (c) Switchable STE based on spin valve design. Panel (a) is reproduced with permission from Yang et al. , Adv. Opt. Mater. 4, 1944 (2016). Copyright 2016 John Wiley & Sons. Panel (b) is reproduced with permission from Chen et al. , Adv. Opt. Mater. 7(4), 1801608 (2018). Copyright 2018 John Wiley & Sons. Panel (c) is reproduced with permission from Fix et al. , Appl. Phys. Lett. 117, 132407 (2020). Copyright 2020 AIP Publishing LLC.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 010901 (2021); doi: 10.1063/5.0037937 129, 010901-9 Published under license by AIP Publishing.than almost all conventional solid THz emitter.51,54Therefore, one of the most direct applications of STE is to replace ZnTe crystals inthe THz-TDS, which has already been realized in several researchlaboratories. Thanks to the flexible and low-cost fabrication process, larger-area (diameter of 7.5 cm) STE has been readily fabricated,which was reported to offer a peak electric field of 300 kV/cm. 77 The idea is quite straightforward: to increase the pump pulseenergy and the pump spot size simultaneously, while maintaining the power density and efficiency of STE almost the same as that of the small-size STE, and then to focus the THz pulse. Such amethod provides a new pathway to support strong-field THz appli-cations. For instance, THz pulses with a peak electric field of nearly 300 kV/cm have been employed to investigate the ultrafast quantum-paraelectric-to-ferroelectric phase transition in strontiumtitanate (SrTiO 3)113and magnon control in antiferromagnetic NiO.24The large-area STE has already met the requirements of such applications and possesses the potential to reach even strongerelectric fields. Being a nanofilm structure, STE exhibits significant advan- tages in the applications of THz imaging, as it can attach closely tothe object. A remarkable example is the application in THz near-field ghost imaging system, where the emitter is attached to theobject and excited by spatially encoded fs laser pulses. 114The setup is schematically shown in Fig. 8(a) . In such a situation, the sensing distance (the distance between the emission plane and the object)plays a key role in achieving high resolution. Conventional configu-ration uses millimeter-scale-thick nonlinear crystals for THz emis- sion. The patterned THz-wave is generated and propagates along the entire volume of the crystal, where the accompanied diffraction FIG. 7. Summary of current mechanisms for ultrafast spin current generation and spin-charge conversion. (a) Principle for typical STE. (b) Ultrafast spin S eebeck effect. (c) Ultrafast spin pumping. Inverse Rashba –Edelstein effect of (d) Bi/Ag interface, (e) topological insulator, and (f) MoS 2. Panel (a) is reproduced with permission from Seifert et al. , Nat. Photonics 10, 483 (2016). Copyright 2016 Springer Nature. Panel (b) is open access from Seifert et al. , Nat. Commun. 9, 2899 (2018). Panel (c) is reproduced with permission from Qiu et al. , Nat. Phys. (2020). Copyright 2020 Springer Nature. Panel (d) is reproduced with permission from Zhou et al. , Phys. Rev. Lett. 121, 086801 (2018). Copyright 2018 American Physical Society. Panel (e) is reproduced with permission from Wang et al. , Adv. Mater. 30, 1802356 (2018). Copyright 2018 John Wiley & Sons. Panel (f) is reproduced with permission from Cheng et al. , Nat. Phys. 15, 347 (2019). Copyright 2019 Springer Nature.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 010901 (2021); doi: 10.1063/5.0037937 129, 010901-10 Published under license by AIP Publishing.may influence the resolution. Instead, the propagation and diffrac- tion in STE are negligible due to the nanometer thickness. Based on this advantage, a resolution of 6.5 μm was reported in a ghost spintronic THz emitter-array microscope system.114Another example is the THz magneto-optic sensor/imager based on on-chipgeneration/detection scheme which is composed of a STE and an electro-optic detector. 115Since the magnetic field determines the THz polarization, the external magnetic field distribution in theproximity of the sensor surface can be recorded [see Fig. 8(b) ]. It provided the first on-chip magneto-optic imager for THz-TDS. Inaddition, Bai et al. reported an integration of an STE and asymmet- ric double-split ring resonator metasurface for label-free THz biosensing. 116 VII. CONCLUSION AND OUTLOOK In this perspective, we have briefly reviewed the recent pro- gresses in STE as well as its applications. Owning to the additionaladvantage of the spin freedom of electron besides that of the chargefreedom, STE presents unique properties, such as nanometer- thickness, ultra-broad bandwidth without phonon absorption, easy fabrication, and low cost. After comprehensive studies from variousperspectives (materials, spintronics, optics, and terahertz emission), the THz emission efficiency of STE has been significantly improved, which reaches the same level of millimeter-thick ZnTe crystals, andeven enters the scope of strong-field THz radiation. Meanwhile, STEoffers great flexibility in manipulating amplitude, bandwidth, andpolarizations of the emitted THz waves via various methods, includ- ing tailoring the external magnetic field, patterning the nanofilm, and integrating STE with metasurfaces. Based on the above progressesand advantages, STE has shown great application prospects in notonly conventional THz TDS and strong-field THz-pump system, butalso extraordinary scenarios such as ghost microscope system and on-chip magneto-optic imager. Looking to the future, it is clear that further improvement of the efficiency and strength of STE will have a strong impact onexpanding the scope of THz applications. For instance, (i) oneorder enhancement of efficiency can support STE to compete with photoconductive antennas, which are widely used in the commer- cial THz TDS but with high cost; (ii) higher efficiency and largerarea can support STE to reach the strong-field regime from 1 to10 MV/cm, which is expected to engineer new dynamic states of strongly correlated materials by modifying their intrinsic fields; (iii) higher signal to noise ratio can lead to higher resolution in the FIG. 8. (a) Ghost spintronic THz-emitter-array microscope. (b) THz magneto-optic sensor/imager. Panel (a) is open access from Chen et al. , Light: Sci. & Appl. 9, 99 (2020). Panel (b) is open access from Bulgarevich et al. , Sci. Rep. 10, 1158 (2020).Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 129, 010901 (2021); doi: 10.1063/5.0037937 129, 010901-11 Published under license by AIP Publishing.THz near-field imaging system [see Fig. 8(a) , for example]. Hence, ongoing effects need to be made to answer the question whether one order enhancement of efficiency is possible. The key pointsinclude but not limited to new (doped, alloy) materials with largerspin Hall angle, improved interface/heterostructure for better spintransport, stronger light –matter interactions for higher laser field localization and absorption, optimized substrate and antenna for higher THz wave emission efficiency, and new mechanisms forultrafast spin current generation and spin-charge conversion, etc.In addition, the unique properties of nano-film configuration, flexi-ble fabrication process, and magnetic-field tunability provide the potential for developing advanced applications, including but not limited to on-chip or conformal in situ sensing, detection, and imaging. A comprehensive and multidisciplinary study as well asimaginations is highly desired to promote both the fundamentalresearch and “real world ”applications of STE. Another important open question is whether the spintronic heterostructure can also operate as an ultrabroadband THz detector.It is well known that electro-optic crystals (such as ZnTe and InP),photoconductive antenna, and air plasma are widely used as THzdetectors, which are considered as the reverse versions of the genera- tion processes. Thus, it is quite intuitive to expect the possibility of spintronic THz detector (STD), in consideration of the excellent per-formance of STE, especially the extremely wide frequency range(more than 15 THz). How does it work? What are the key points for improving the detection sensitivity? How to reach the frequency range from 0.3 to 15 THz? If we are able to settle these questionsand realize high performance STD, greater value could be expectedand more fruitful applications would be inspired. AUTHORS ’CONTRIBUTIONS Z.F. and H.Q. contributed equally to this work. 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4.0000048.pdf

Struct. Dyn. 8, 014501 (2021); https://doi.org/10.1063/4.0000048 8, 014501 © 2021 Author(s).Carrier-specific dynamics in 2H-MoTe2 observed by femtosecond soft x-ray absorption spectroscopy using an x-ray free- electron laser Cite as: Struct. Dyn. 8, 014501 (2021); https://doi.org/10.1063/4.0000048 Submitted: 06 November 2020 . Accepted: 20 December 2020 . Published Online: 13 January 2021 Alexander Britz , Andrew R. Attar , Xiang Zhang , Hung-Tzu Chang , Clara Nyby , Aravind Krishnamoorthy , Sang Han Park , Soonnam Kwon , Minseok Kim , Dennis Nordlund , Sami Sainio , Tony F. Heinz , Stephen R. Leone , Aaron M. Lindenberg , Aiichiro Nakano , Pulickel Ajayan , Priya Vashishta , David Fritz , Ming-Fu Lin , and Uwe Bergmann ARTICLES YOU MAY BE INTERESTED IN Signatures of electronic and nuclear coherences in ultrafast molecular x-ray and electron diffraction Structural Dynamics 8, 014101 (2021); https://doi.org/10.1063/4.0000043 Shot noise limited soft x-ray absorption spectroscopy in solution at a SASE-FEL using a transmission grating beam splitter Structural Dynamics 8, 014303 (2021); https://doi.org/10.1063/4.0000049 Attosecond state-resolved carrier motion in quantum materials probed by soft x-ray XANES Applied Physics Reviews 8, 011408 (2021); https://doi.org/10.1063/5.0020649Carrier-specific dynamics in 2H-MoTe 2observed by femtosecond soft x-ray absorption spectroscopy using an x-ray free-electron laser Cite as: Struct. Dyn. 8, 014501 (2021); doi: 10.1063/4.0000048 Submitted: 6 November 2020 .Accepted: 20 December 2020 . Published Online: 13 January 2021 Alexander Britz,1,2 Andrew R. Attar,1,2,3 Xiang Zhang,4Hung-Tzu Chang,5 Clara Nyby,1,6 Aravind Krishnamoorthy,7Sang Han Park,8 Soonnam Kwon,8 Minseok Kim,8Dennis Nordlund,9Sami Sainio,9 Tony F. Heinz,1,3,10 Stephen R. Leone,5,11,12 Aaron M. Lindenberg,1,6,13Aiichiro Nakano,7 Pulickel Ajayan,4 Priya Vashishta,7David Fritz,2Ming-Fu Lin,2,a) and Uwe Bergmann1,a) AFFILIATIONS 1Stanford PULSE Institute, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA 2Linac Coherent Light Source, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA 3SUNCAT Center for Interface Science and Catalysis, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA 4Department of Materials Science and NanoEngineering, Rice University, Houston, Texas 77005, USA 5Department of Chemistry, University of California, Berkeley, California 94720, USA 6Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA 7Collaboratory for Advanced Computing and Simulations, University of Southern California, Los Angeles, California 90089, USA 8PAL-XFEL, Pohang Accelerator Laboratory, 80 Jigokro-127-beongil, Nam-gu, Pohang, Gyeongbuk 37673, South Korea 9Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA 10Department of Applied Physics, Stanford University, Stanford, California 95305, USA 11Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 12Department of Physics, University of California, Berkeley, California 94720, USA 13Department of Materials Science and Engineering, Stanford University, Stanford, California 94305, USA a)Authors to whom correspondence should be addressed :mfucb@slac.stanford.edu andbergmann@slac.stanford.edu ABSTRACT Femtosecond carrier dynamics in layered 2H-MoTe 2semiconductor crystals have been investigated using soft x-ray transient absorption spectroscopy at the x-ray free-electron laser (XFEL) of the Pohang Accelerator Laboratory. Following above-bandgap optical excitation of 2H-MoTe 2, the photoex- cited hole distribution is directly probed via short-lived transitions from the Te 3 d5/2core level (M 5-edge, 572–577 eV) to transiently unoccupied states in the valence band. The optically excited electrons are separately probed via the reduced absorption probability at the Te M 5-edge involving partially occupied states of the conduction band. A 400 6110 fs delay is observed between this transient electron signal near the conduction band minimum compared to higher-lying states within the conduction band, which we assign to hot electron relaxation. Additionally, the transient absorption sig nals below and above the Te M 5edge, assigned to photoexcited holes and electrons, respectively, are observed to decay concomitantly on a 1–2 ps timescale, which is interpreted as electron–hole recombination. The present work provides a benchmark for applications of XFELs for soft x-ray absorption stud ies of carrier-specific dynamics in semiconductors, and future opportunities enabled by this method are discussed. VC2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license ( http:// creativecommons.org/licenses/by/4.0/ ).https://doi.org/10.1063/4.0000048 INTRODUCTION The optically excited carrier relaxation and structural dynamics of semiconductors govern their optoelectronic properties andfunctionality in emerging device applications.1Femtosecond time- resolved x-ray spectroscopy and scattering are powerful techniques to track the electronic and structural dynamics of such materials in real time. With the recent development of ultrafast x-ray and extreme Struct. Dyn. 8, 014501 (2021); doi: 10.1063/4.0000048 8, 014501-1 VCAuthor(s) 2021Structural Dynamics ARTICLE scitation.org/journal/sdyultraviolet (XUV) sources using x-ray free electron lasers (XFELs) and high-harmonic generation (HHG), new opportunities are being explored for investigating charge-carrier dynamics in the condensed phase with carrier-, element-, and oxidation-state specificity.2–5The ability to probe the valence electronic structure via localized core levels using x-ray/XUV absorption spectroscopy (XAS) has been exploited to capture electron and hole carrier relaxation dynamics separately in bulk semiconductors6–12and to measure layer-specific dynamics within multi-component heterojunctions.13These early successes have been led primarily by applications of table-top HHG sources for XUV transient absorption and reflection spectroscopy.3 With the recent emergence of XFELs, which produce femtosec- ond x-ray pulses with /C246–8 orders of magnitude greater spectral brightness per pulse compared to HHG,14a new horizon is coming into view for ultrafast core-level spectroscopy of semiconductors. In addition to the potential for detecting weak signals using a high XFELphoton flux, the small spatial scale ( /C241–10’s lm) 15,16of the focused x-ray pulses, and the much longer penetration depths of higher-energy x-rays allow for smaller and thicker semiconductor samples, respec- tively, to be investigated. Furthermore, the small spin–orbit energy splittings of elemental core levels within the XUV range, which are often comparable to the bandgap of relevant semiconductors (1–3 eV), lead to overlapping spectral features that frequently complicate the interpretation of hole and electron dynamics in XUV spectros- copy.6–8,10,12,17Although deconvolution of the spin–orbit split transi- tions has been possible in some examples,6,10,17the overlapping spin–orbit features can be completely eliminated by using the higher- energy x-rays from XFELs at deeper core-level edges that are well- separated compared to the bandgap energies.18In the present work, we apply femtosecond optical-pump, x-ray-probe spectroscopy at the Te M5-edge (572–577 eV) using an XFEL to capture the carrier-specific dynamics of a prototypical layered semiconductor, 2H-MoTe 2.O u r study provides a benchmark for XFEL-based soft x-ray femtosecond transient absorption spectroscopy of semiconductor materials. MoTe 2is a transition metal dichalcogenide (TMDC) within the class of layered materials, like graphite, that are composed of two-dimensional sheets (monolayers) bound by weak van der Waals forces. In contrast to graphite and its corresponding monolayer form, gra- phene, which are both semimetallic, MoTe 2is stable in the bulk and as a monolayer in both a semimetallic phase (1T0) and semiconductor phase (2H).19,20The 2H-phase MoTe 2has a bandgap of /C240.9 eV in the bulk and 1.1 eV in the monolayer form,21which are similar to that of silicon (1.1 eV). Various applications based on both monolayer and multilayer MoTe 2have been investigated, including field-effect-tran- sistors,22,23photonic logic gates,24and phase-change devices.25The advancement of next-generation optoelectronics based on layered TMDC materials relies on a detailed understanding of the relaxation and transport of the carriers.24,26Therefore, recent studies using ultra- fast optical and THz absorption spectroscopies have examined carrier lifetimes in 2H-MoTe 2.27–29However, these measurements lack both carrier-specificity and sensitivity to hot electron and hole dynamicsprior to electron–hole recombination or trapping. Time-resolved XAS can distinguish electron and hole dynamics in semiconductors, including intraband carrier thermalization and cooling. 7,9,12To achieve this, an optical pump pulse first excites elec- trons into the conduction band (CB) and creates holes in the valence band (VB); a temporally delayed x-ray pulse then probes the changesin transitions of core-level electrons into the unoccupied states of the VB and CB. The effects of increased core !VB transitions due to the presence of photoexcited holes and decreased core !CB transitions due to excited electrons are collectively referred to as state-filling effects.6,10,17In a recent work, using XUV transient absorption spec- troscopy on 2H-MoTe 2, we captured the hot hole scattering dynamics within the VB, the subsequent electron–hole recombination, andcoherent lattice dynamics in 2H-MoTe 2induced by strong electro- n–phonon coupling.12The 1.5 eV spin–orbit splitting of the Te N 5and N4edges, which is comparable to the 2H-MoTe 2bandgap energy (0.9 eV), led to overlapping spectral features, particularly between the hole signal at the N 4edge and the electron state-filling signal at the N 5 edge. Unfortunately, this overlap precluded a detailed analysis of the electron thermalization and cooling dynamics. In the present work, we extend the method established by XUV transient absorption to the x-ray regime at the Te M 5edge of 2H-MoTe 2using an XFEL. We achieve a noise level of /C240.05% in measurements of the x-ray transmission, which surpasses the perfor-mance of XUV measurements on the same material and is crucial due to an order-of-magnitude smaller edge jump in absorption at the M 5edge compared to the N 5,4edge in the XUV. At the Te M 5edge, which is spectrally isolated by >10 eV from neighboring core-level edges, we show that the XAS closely maps the unoccupied density ofstates (DOS) of the CB in 2H-MoTe 2. In this time-resolved optical- pump, XAS-probe experiment, we identify a signature of hot electroncooling within the CB, which was not observed in the recent XUVstudy due to overlapping state-filling signals from the N 5and N 4 edges. In addition, we successfully capture the separate photoexcitedhole signal and trace the electron–hole recombination dynamics of2H-MoTe 2. EXPERIMENTAL METHODS AND CALCULATIONS Sample preparation and static XAS measurements using synchrotron radiation The static XAS of both the 1T0and 2H phases of MoTe 2are mea- sured in the present work, while pump–probe measurements arecarried out only on 2H-phase MoTe 2.S a m p l e so f2 H -a n d1 T0-MoTe 2 are synthesized via chemical vapor deposition (CVD) and directlydeposited on 100 nm thick Si 3N4substrates of 2 /C22m m2lateral size. This results in a homogenous thin-film polycrystalline sample for both phases. The structural phase of the samples is confirmed by their Raman spectra (see the supplementary material for more details).20 The static XAS measurements of the 2H and 1T0phases of MoTe 2 were initially carried out at Beamline 8–2 of the Stanford Synchrotron Radiation Lightsource (SSRL) synchrotron tuned to the Mo M 3and Te M 5,4edges. Further information about soft x-ray absorption spectroscopy at the SSRL can be found in the literature.30Several experimental approaches were considered for soft x-ray absorption measurements. In contrast to 3 dtransition metal L edges,31at o t a lo r partial fluorescence yield measurement of the Mo and Te M edges ischallenging because of their extremely low fluorescence yields of 0.3%and 0.1%, respectively. 32Thus, either a total electron yield (TEY) mea- surement or a transmission measurement would be more viable. Sincethe final goal is to incorporate these XAS measurements into pump–probe studies and the TEY measurement is more prone to artifactsinduced by the optical pump laser, 33w ef o c u si nt h i sw o r ko nt h e transmission measurement. To optimize the signal-to-noise ratio, theStructural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 014501 (2021); doi: 10.1063/4.0000048 8, 014501-2 VCAuthor(s) 2021MoTe 2layer thickness during sample synthesis is tailored to obtain approximately one absorption length of the x-rays at the M 5edge. The resulting 100 nm MoTe 2layer absorbs /C2420% of incident x-rays at energies just below the Te M 5,4absorption edges and /C2454% above, which results in a change in the transmission by the absorption edge jump of /C2434%.18However, we note that the change in transmission at the resonant M 5pre-edge—where the carrier dynamics are probed in the time-resolved measurements—is only 2%. Femtosecond time-resolved XAS using XFEL pulses The femtosecond time-resolved XAS measurements on 2H- phase MoTe 2were performed at the Soft X-ray Scattering and Spectroscopy (SSS) beamline of the Pohang Accelerator Laboratory (PAL)-XFEL. A simplified layout of the experiment is shown in Fig. 1 ; a detailed description of the soft x-ray beamline at PAL-XFEL can be found in the literature.34In brief, the x-rays are monochromatized (to a bandwidth of 0.1 eV) and focused onto the sample with a Kirkpatrik–Baez (KB) mirror system, resulting in an x-ray beam diam- eter of 110 lm/C280lm at the sample position, thus much smaller than the lateral extent of the thin-film MoTe 2sample. The mono- chromatized XFEL energy per pulse without further attenuation is measured as /C244lJ at 575 eV. In order to prevent x-ray-induced changes to the sample, including damage, the x-ray pulse energy is reduced by a series of Al filters (total thickness of 1.6 lm) before the sample, resulting in an average pulse energy incident on the sample of approximately 0.24 lJa tt h eT eM 5pre-edge. For an XFEL pulse dura- tion of /C2450 fs, this corresponds to a peak intensity of /C245/C21010W/cm2. The relative number of photons/pulse (I 0)o fe a c hi n c i d e n tx - r a yp u l s ei s measured using the ejected photoelectrons from a 150 nm thick Si 3N4 membrane detected with an MCP detector. The relative x-ray photons/pulse transmitted through the sample (I 1) is measured with a photodi- ode. Both detector signals are acquired with a 14-bit analog-to-digital converter (digitizer) and the integrated areas under the pulses are saved as I0and I 1.In the pump arm, the output of a commercial Ti:sapphire ampli- fier is frequency-doubled to generate 400 nm pulses with /C2450 fs pulse duration. The 400 nm (3.1 eV) pump excitation is chosen to producehot carriers in 2H-MoTe 2(bandgap of /C240.9 eV), with a large energy contrast between hot carriers and cooled, band-edge carriers. The 400 nm pump beam is focused by a lens and reflected by an annular mirror onto the sample at an angle of /C241/C14with respect to the x-rays. The laser spot size on the sample is 230 lm( f u l lw i d t ha th a l fm a x i - mum, FWHM), much larger than the x-ray spot size. For a typicalpump pulse energy of 34 lJ, the resulting excited carrier density is estimated to be 1.6 /C210 21carriers/cm3(see the supplementary material for details on the optical absorption at 400 nm). The x-ray transmission photodiode (I 1) is shielded from the transmitted and scattered 400 nm pump light with a free-standing 200 nm thick Alfilter. The temporal pump–probe overlap is measured via thex-ray-induced change of 400 nm reflectivity with a single-crystal YAGsample. This measurement was carried out at least twice per 12-h shiftand the drift in the centroid of the optical-x-ray temporal overlap wasconfirmed to be <100 fs on the timescale of a 6- to 12 h measurement. T h ep u m p – p r o b ei n s t r u m e n tr e s p o n s et i m ei sd e t e r m i n e dt ob e/C24200 fs, as described below. ELECTRONIC STRUCTURE CALCULATIONS Density functional theory (DFT) with the projector augmented wave (PAW) method 35implemented in the Vienna Ab initio Simulation Package (VASP)36,37was used to compute the density of states (DOS) for bulk 2H- and 1T0-phases of MoTe 2crystals without photoexcitation. Exchange and correlation effects were calculated using the Perdew–Burke–Ernzerhof (PBE) form of the Generalized Gradient Approximation (GGA). Wave functions were constructedusing a plane wave basis set with components up to kinetic energy of400 eV and the reciprocal space was sampled using a 3 /C23/C23 C-centered mesh with a 0.05 eV Gaussian smearing of orbital occu- pancies. DFT simulations of bulk 2H (1T 0)M o T e 2crystals were performed on 120-atom supercells measuring 17.58 A ˚/C212.19 A ˚ /C213.97 A ˚(17.48 A ˚/C212.67 A ˚/C215.43 A ˚) along the a-,b-, and FIG. 1. Schematic of the experimental setup at PAL-XFEL (not to scale). The incident x-ray intensity (I 0) of each pulse is measured using the ejected photoelectrons from an Si3N4membrane detected with an MCP detector. The transmitted x-ray intensity through the sample (I 1) is measured with a photodiode.Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 014501 (2021); doi: 10.1063/4.0000048 8, 014501-3 VCAuthor(s) 2021c-directions, respectively. Calculations were performed until each self- consistency cycle converged in energy to within 10/C07eV/atom and forces on ions are under 10/C04eV/A˚. The calculation of core-level absorption spectra at the Te M5edge was accomplished with DFT and Bethe–Salpeter equation (BSE) calculations using Quantum ESPRESSO and the Obtaining Core-level Excitations using Ab initio methods and the NIST BSE solver (OCEAN) software package.38–41The DFT-BSE calculation was conducted using norm-conserving scalar-relativistic PBE pseudopo- tentials with nonlinear core correction under GGA and a 6 /C26/C21k - point meshgrid.42–45In the calculation, the number of bands was set to 40 and the dielectric constant was set to 12.9. Convergence was achieved with an energy cutoff of 80 Ryd and a cutoff radius of 4 Bohr.A lifetime broadening of 0.1 eV was assumed in the BSE calculation. RESULTS AND DISCUSSION X-ray absorption spectroscopy of 2H- and 1T 0-phases of MoTe 2 The static XAS of the 2H- and 1T0-phases of MoTe 2thin-film samples at the Te M 5,4edges acquired at SSRL, are shown in Fig. 2 . The spectra cover the distinct M 5and M 4pre-edge features at 573 eV and 584 eV, respectively. The M 5and M 4pre-edge transitions corre- spond to the promotion of Te 3 d5/2and Te 3 d3/2core electrons, respectively, to the CB of semiconducting 2H-MoTe 2and to unoccu- pied states above the Fermi-level in 1T0-MoTe 2, as depicted schemati- cally in Fig. 2(b) . The broad, atomic absorption to the continuum hasits onset at higher energies, starting around the same energy as the M4-edge feature. This delayed onset is due to the centrifugal barrier.46 Note the dip in absorption at /C24568 eV in the 1T0-MoTe 2spectrum is caused by an imperfect I 0normalization during refill of the synchro- tron ring during that measurement. The inset of Fig. 2(a) shows in more detail the Te M 5pre-edge features of both structural phases of MoTe 2. In the semimetallic 1T0- MoTe 2,t h eT eM 5pre-edge onset is red-shifted compared to that of the semiconducting 2H-MoTe 2. This is consistent with the bandgap collapse in the 1T0phase compared to the 2H phase. To quantify this effect, we perform DFT calculations of the bulk DOS in both phases and extract the energy of the Fermi level, E F, in the case of bulk 1T0- MoTe 2and of the CB minimum, E CBM, in the case of bulk 2H-MoTe 2, relative to the vacuum. The corresponding DOS distributions are plotted in Fig. 2(b) and the energy difference between E CBM(2H) and EF(1T0) is 0.36 eV. In the inset of Fig. 2(a) , the vertical green dashed line denotes the energy difference between the calculated E F(1T0)a n d the measured core-level ionization potential of Te 3 d5/2(1T0), i.e., the energy of the transition represented by the purple vertical arrow on the left side of Fig. 2(b) . The black vertical dashed line marks the energy difference between the calculated E CBM(2H) and the measured core-level ionization potential of Te 3 d5/2(2H), i.e., the energy of the transition represented by the purple vertical arrow on the right side of Fig. 2(b) . Note that the Te 3 d5/2ionization potentials of the two phases are extracted directly from XPS measurements (see the supplementary material ,F i g .S 2 ) ,g i v i n gt h e0 . 1 5 e Vd i f f e r e n c en o t e di n Fig. 2(b) , FIG. 2. (a) Normalized Te M 5,4-edge x-ray absorption spectrum of 2H- and 1T0-phases of MoTe 2. Note the dip in absorption at /C24568 eV in the 1T0-MoTe 2spectrum is caused by an imperfect I 0normalization during refill of the synchrotron ring during that measurement. The inset shows an expanded plot of the Te M 5pre-edge feature. The green ver- tical dashed line denotes the energy of the Fermi level, E F,i n1 T0-MoTe 2relative to the Te 3d 5/2core level of the 1T0phase. The black vertical dashed line denotes the energy of the CB minimum, E CBM, in 2H-MoTe 2relative to the corresponding Te 3d 5/2core level of the 2H phase. (b) Calculated total DOS of bulk 2H- and 1T0-MoTe 2. The dashed hor- izontal lines mark the calculated energies of E F(1T0) and E CBM(2H) with an energy scale set relative to the Fermi level of 1T0-MoTe 2. The vertical purple arrows represent the onset of core-level transitions at the Te M 5edge of both phases, corresponding to the vertical dashed lines in (a). As the x-ray energy is increased above each onset, carriers are promoted to higher-energy states above E F(1T0) and E CBM(2H). Note the slight difference (0.15 eV) in the Te 3d 5/2core-level ionization potential in the 1T0-phase compared to the 2H-phase MoTe 2(see the supplementary material for more details), which gives a total difference of 0.51 eV in the expected XAS onsets of the two phases.Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 014501 (2021); doi: 10.1063/4.0000048 8, 014501-4 VCAuthor(s) 2021whereas the energy difference between E F(1T0) and E CBM(2H) of 0.36 eV comes from the DFT calculations. The overall expected red shift of the 1T0-MoTe 2M5pre-edge onset relative to that of 2H- MoTe 2is therefore 0.15 eV þ0.36 eV ¼0.51 eV, which is in good agreement with the experimental XAS. In both phases, although theM 5pre-edge transitions of Te(3 d5/2)!Te(5 p) character are dipole allowed, the absorption change (i.e., the edge jump) is weak at these resonances. As seen in Fig. 2(a) , the pre-edge features are /C2420 times weaker than the continuum resonance of the Te M 5,4edge. For a 100 nm thick sample, the x-ray transmission only changes by /C242% at the M 5pre-edge. The clear distinction between the 2H- and 1T0-phaseTe M 5-edge absorption illustrates the sensitivity of XAS to modifica- tions in the valence electronic structure. InFig. 3(a) , the DFT-calculated band structure of 2H-MoTe 2 along the C/C0K/C0M/C0Cpath is plotted. In Fig. 3(b) , the corresponding total CB DOS in 2H-MoTe 2is compared to the experimental absorp- tion spectrum of the Te M 5pre-edge, after applying a global energy shift to the CB DOS to match the onset of the experimental absorption spectrum. In addition, BSE calculations of the Te M 5pre-edge transi- tions are performed using the OCEAN software package.39,40The resulting OCEAN-simulated spectrum is plotted in Fig. 3(b) ,a f t e r applying a global shift to the energy scale to match the onset of the FIG. 3. (a) Band structure of 2H-MoTe 2along the C/C0K/C0M/C0Cpath. One representative VB !CB transition induced by the 400 nm pump pulse is shown as a blue arrow. (b) Experimental static XAS, CB DOS, and OCEAN-simulated spectra are plotted and horizontal dashed lines are drawn at representative energies withi n the CB, labeled CB 1, CB2, and CB 3. (c) A model of the probing scheme by x-ray transient absorption of 2H-MoTe 2. The VB and CB are shown schematically as filled and empty boxes and photoex- cited electrons and holes are drawn as filled or empty circles, respectively. Representative x-ray transitions from the Te 3 d5/2core level are shown with and without an X to rep- resent a decrease or increase in absorption relative to the static spectrum, respectively, due to state filling.Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 014501 (2021); doi: 10.1063/4.0000048 8, 014501-5 VCAuthor(s) 2021experimental absorption spectrum. The experimental M 5pre-edge s p e c t r u mh a st h r e em a j o rp e a k s ,l a b e l e dC B 1,C B 2,a n dC B 3for conve- nience, and these peak energies match the distribution in the CB DOS and the OCEAN-simulated spectrum. The agreement between the energy of the critical points in the CB DOS and the Te M 5pre-edge absorption indicates that the Te 3d 5/2core-hole is well screened and the M 5XAS can be considered as a map of the CB unoccupied DOS in the valence shell. Therefore, changes in occupation of the CB at a par- ticular energy within the band structure can be gauged by correspond- ing changes in the Te 3d 5/2!CB absorption via the state-filling description. As shown in the projected/partial DOS (PDOS) plotted in Fig. S4 of the supplementary material , the VB also contains significant Te character, allowing for the state-filling picture to be used to describe the Te 3 d5/2!VB hole signal observed following optical excitation as well. The contribution of Mo character in the DOS implies that the Mo M 3,2edge could also be used, in principle, to map changes in VB/ CB state occupation. However, as seen in Figs. S5 and S6 of the supple- mentary material ,t h eM oM 3,2edge is quite broad, leading to a mostly featureless pre-edge spectrum. The Mo M 3,2edge is also weak com- pared to the overlapping N K edge, which hinders the measurement in this region due to the use of Si 3N4as a substrate as well as a target for the I 0measurement. Capturing changes in the electronic structure with x-ray transient absorption In the femtosecond time-resolved experiments performed at the PAL-XFEL, we photoexcite 2H-MoTe 2with a sub-100 fs, 400 nm pulse and measure the time-delayed XAS of the excited sample. The probing scheme is schematically depicted in Fig. 3(c) .I nt h el e f t m o s t panel labeled t ¼0 fs, the above-bandgap pump excitation (h /C23400 nm ¼3.1 eV compared with the bandgap of /C240.9 eV) produces nonther- mal excited carriers with an energy distribution centered significantly above the CB minimum (electrons) and below the VB maximum (holes). The state-filling signal is expected to map this distribution: the reduced Te 3d 5/2!CB absorption due to the excited electrons will initially have a dominant contribution at higher energies in the CB. Over the course of 10’s–100’s of femtoseconds following initial excita- tion, the carriers are expected to thermalize and cool due to knowncarrier–carrier and carrier–phonon scattering processes, 1,12as depicted in the schematic of Fig. 3(c) , from the left panel to the middle panel. At these intermediate delay times following the thermalization/coolingprocess, the state-filling signal is expected to contribute closer to the Te 3 d 5/2!CB minimum in the case of the electrons. The 3.1 eV (400 nm) pump excitation is chosen to produce hot electrons with alarge energy difference between the initial hot electron signal (i.e., near Te 3 d 5/2!CB3) vs the cooled electron signal (Te 3 d5/2!CB1). Finally, electron–hole recombination will lead to the decay of the state-filling signal, as depicted in the rightmost panel of Fig. 3(c) .I n addition to the state-filling effects, lattice heating via carrier-phononscattering and nonradiative recombination can affect the core-level absorption edge. [Note that this is not shown in the schematic in Fig. 3(c) ] Lattice heating often manifests as a broadening/shifting of the core-level edge jump in absorption compared to room- temperature XAS, leading to a long-lived transient feature near the onset of the core !CB edge (see the supplementary material for more details). 6,7,10,17Femtosecond transient x-ray spectroscopy of 2H-MoTe 2 InFig. 4(a) ,t h eT eM 5- e d g ea b s o r b a n c eo f2 H - M o T e 2measured at PAL-XFEL without photoexcitation is plotted (ground state, blueline) and at a delay time of /C24400 fs after optical excitation (excited state, orange line). The excited-state spectrum is an average of twotransient spectra measured at nominal delay times of þ300 fs and þ500 fs (relative to the independent determination of time-zero described in the Experimental Methods and Calculations section).In the lower panel of Fig. 4(a) , the differential absorption spectrum at 400 fs delay, defined as DODðE;sÞ¼OD 400 fsðE;sÞ/C0OD staticðEÞ,i s plotted. This differential spectrum consists of three main features.First, an increased absorption at 572.5 eV, approximately 1 eV belowthe 3 d 5/2!CB onset. Second, a derivative feature centered at the 3d5/2!CB edge onset, with increased absorption just below the edge and decreased absorption just above the edge. Finally, it also shows ageneral decrease in the absorption at energies above the 3d 5/2!CB edge (between 574 eV and 576 eV). The observed changes in theabsorption amplitude are about /C247% of the M 5pre-edge feature (static 3d5/2!CB absorption). Thus, the expected optically induced tran- sient changes in the transmission are approximately 0.14% (i.e., 7% of the 2% change in transmission in the static pre-edge absorption). Due to the small absorption edge jump at the Te M 5pre-edge, the differ- ence spectrum in Fig. 4(a) required averaging over 10 h of data. The features observed in the difference spectrum at /C24400 fs delay time are interpreted in terms of the middle panel of the schematicshown in Fig. 3(c) . The increased absorption at 572.5 eV in the tran- sient spectrum, /C241e V b e l o w t h e 3 d 5/2!CB edge, is assigned to the photoexcited holes in the valence band. The derivative feature centered at 573.5 eV as well as the decrease in absorption above 574 eV arecaused by a combination of bandgap renormalization (red-shifting ofthe edge), broadening due to the presence of free carriers and lattice-heat effects, and state filling by the electrons in the CB. 6,10,17,47 InFigs. 4(b)–4(d) ,t h eDOD measured at three chosen energies indicated by arrows in Fig. 4(a) are plotted as a function of time delay. The three chosen energies are (i) 572.9 eV, which corresponds to the3d 5/2!VB hole signal close to the VB maximum, (ii) 573.9 eV, which corresponds to the 3 d5/2!CB1signal close to the bottom of the CB (near the K and M critical points), and (iii) 576 eV, which correspondsto the 3 d 5/2!CB3signal involving higher-energy states within the CB (near the Ccritical point). The features at 572.9 eV (3 d5/2!VB) and 576 eV (3 d5/2!CB3) both appear to rise within the pump excita- tion at time-zero, whereas the feature at 573.9 eV (3 d5/2!CB1) appears to be slightly delayed. The rise of the 576 eV signal is fit to aconvolution between a Gaussian function and a step function and theinstrument-response function (IRF) is determined to be /C24200 fs, defined by the full width at half maximum (FWHM) of the Gaussianfunction. The time traces of the signals at 572.9 eV and 573.9 eV areeach fit to a Gaussian broadened step function convolved with an exponential decay function. The onset of the 573.9 eV feature, defined by the center of the Gaussian broadened step function, t 0,i se x t r a c t e d as t 0¼400690 fs and this is compared to the fitted onset of the 576 eV signal at t 0¼0670 fs. The uncertainties in t 0are the standard errors of the extracted best-fit parameter. The total delay in the onsetof the 573.9 eV feature is therefore Dt¼4006110 fs with respect to the fitted onset of the 576 eV signal, with the corresponding uncer- tainty determined after error propagation.Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 014501 (2021); doi: 10.1063/4.0000048 8, 014501-6 VCAuthor(s) 2021The appearance of the 3 d5/2!VB and 3 d5/2!CB3signals at time-zero, within the IRF of the measurement, is consistent with a state-filling description where a distribution of carriers is producedimmediately with the pump excitation. The optical excitation opensnew transitions from the Te 3 d 5/2core level to the transient holes in the VB (3 d5/2!VB) and leads to a state-filling signal by the photoex- cited electrons reaching energies up to 2.2 eV above the CB minimum (3d5/2!CB3). The Dt¼40061 1 0f sd e l a yi nt h er i s eo ft h en e g a t i v e DOD feature near the band edge at 573.9 eV (3 d5/2!CB1), on the other hand, is attributed to the time required for the electrons to coolto the bottom of the CB by intra- and intervalley scattering through phonon emission, as depicted schematically in Fig. 3(c) (left to middle panels). The electron–phonon scattering leads to an electron distribu-tion closer to the CB minimum and to a delayed state-filling responseat these 3 d 5/2!CB1energies, which is expected on the few-hundred femtosecond timescale.1,12,48The measured electron cooling time of 4006110 fs assigned here for 2H-MoTe 2, following above-bandgapphotoexcitation, is comparable to the hole cooling time of 380 690 fs measured in the same material by XUV transient absorption.12The 4006110 fs delay measured in the present work between state-filling signals at CB 1compared to CB 3, which we attribute here to electron cooling, was not visible in the XUV measurements due to overlappingsignals of the Te 4d 5/2and 4d 3/2core levels. The 1.5 eV spin–orbit splitting of the Te N 5,4edges in the XUV prevents the distinction between CB 3and CB 1,e v e ni nt h es t a t i ca b s o r p t i o nm e a s u r e m e n t .W e note that the 400 nm excitation in the present work produces carrierswith considerably greater above-bandgap energy per photon com-pared to the broadband visible excitation (550–950 nm) used in the XUV experiment. After the initial rise of the transients observed in Figs. 4(b)–4(d) , the 3 d 5/2!CB3feature at 576 eV remains constant up to a delay time of 4 ps, whereas the time traces of the 3 d5/2!VB and 3 d5/2!CB1fea- tures are characterized by complete or partial decay on this timescale. The decay component of these two kinetic traces [ Figs. 4(b) and4(c)]i s FIG. 4. (a) Comparison between the Te M 5pre-edge absorption of 2H-MoTe 2in its ground state and /C24400 fs after 400 nm excitation (upper panel). The difference between ground- and excited-state spectra is shown in black circle and line (bottom panel). The error bars correspond to one standard deviation of the /C0log(I 1/I0) measurements of all XFEL pulses for each energy. This differential absorption trace is the average of /C2410 h of accumulated data. (b)–(d) DOD (solid colored circles) as a function of time delay measured at the energies indicated by matching colored arrows in (a). The time-delay step size is 266 fs in (b) and 133 fs in (c) and (d). The full lines are fits corresponding to Gaussian-broadened step functions to describe the onset of all three features and additionally convolved with an exponential decay term in (b) and (c ). The vertical dashed col- ored lines mark the time-delay center of the Gaussian-broadened step function for each feature. A typical delay scan collected at each monochromatiz ed x-ray photon energy setting is an average of /C241 h of accumulated data. Delay scans require shorter acquisition times compared to energy scans due to the 10 s wait time required to change monochromator energy setting, resulting in a significantly lower duty cycle for energy scans.Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 014501 (2021); doi: 10.1063/4.0000048 8, 014501-7 VCAuthor(s) 2021fit to a single exponential function ( ytðÞ¼y0þAexp/C0t=s/C0/C1 ,w h e r ey 0 and A are arbitrary constants) and the extracted time constant, s,i s found to be shole¼1.260.6 ps for the 3 d5/2!VB hole signal and selectron ¼1.260 . 8p sf o rt h e3 d5/2!CB1feature. These observations are consistent with electron–hole recombination and trapping, which occur on these timescales, i.e., leading to decay of the transient holes in the VB and of the photoexcited electrons in the CB. Although the carrier dynamics are expected to be dominated by an Auger recombination process,49the exponential decay fitting is chosen to compare to expo- nential time constants extracted from recent THz and visible pump– probe studies on similar defect-rich 2H-MoTe 2samples.27,28Our measured electron and hole lifetimes ( sholeandselectron ) are consistent, within the error bars, with the carrier-averaged lifetimes extracted from the THz and visible pump–probe studies and the hole lifetime measured by XUV transient absorption.12 In addition to the consideration of state-filling effects, carrier- phonon scattering leads to lattice heating, which can also affect the 3d5/2!CB absorption of 2H-MoTe 2, even in the absence of photoex- cited carriers.6,7,10,17The observation of a long-lived, negative plateau in the 3 d5/2!CBDO Ds i g n a l sa t5 7 3 . 9e Va n d5 7 6e Va f t e r /C241p si s associated with these lattice heating effects, which dominate after car- rier cooling and electron–hole recombination. To corroborate this assignment, the analogous observation of this effect was made at the Te N 5edge (Te 4 d5/2!CB) of 2H-MoTe 2using XUV transient absorption following both 800 nm and 400 nm photoexcitation (see thesupplementary material for details).12Similar long-lived, heat- induced effects are common in transient core-level absorption experi- ments of semiconductors.6,7,9 CONCLUSIONS AND OUTLOOK We measured ultrafast soft x-ray absorption spectra of polycrys- talline 2H-MoTe 2thin films following 400 nm excitation using an XFEL at PAL. The differential absorption spectrum at a delay time of 400 fs provides a direct observation of photoexcited holes in the VB, which are identified by short-lived Te 3 d5/2!VB absorption transi- tions appearing /C241e V b e l o w t h e 3 d5/2!CB absorption edge. The transient spectrum is also characterized by a negative DOD signal at the 3 d5/2!CB energies due to excited electrons in the CB, bandgap renormalization, and lattice-heating effects. The time traces of the Te 3d5/2!VB and Te 3 d5/2!CB1state-filling signals reveal hole and electron lifetimes of 1.2 60.6 ps and 1.2 60.8 ps, respectively, which is consistent with electron–hole recombination occurring on this time- scale. These carrier lifetimes are also consistent with THz and optical pump–probe measurements, which previously provided carrier- averaged lifetimes.27,28Furthermore, prior to electron–hole recombi- nation, we observe a 400 6110 fs delay of the 3 d5/2!CB1state- filling depletion involving states near the bottom of the CB compared to the 3 d5/2!CB3depletion involving higher-energy states within the CB. We interpret this finding as a direct signature of hot electron relaxation within the CB due to intra- and intervalley scattering by electron–phonon interactions. The observation of electron cooling dynamics in the present work provides additional and complementary information to the recent XUV transient absorption study of 2H- MoTe 2,12which did not distinguish these electron relaxation dynamics due to overlapping spectral features of the neighboring spin–orbit split N5and N 4edges.HHG-based XUV transient absorption spectroscopy has a num- ber of unique capabilities including: (1) demonstrated attosecond tem-poral resolution in the condensed phase, 50(2) simultaneous absorption at multiple elemental edges in a single measurement,12and (3) intrinsic pump–probe delay stability due to the single amplifiedlaser source for the pump and probe. The results presented here pro-vide a benchmark for femtosecond transient absorption spectroscopyof semiconductor materials using the advancing capabilities of XFELsthat operate at higher energies in the soft x-ray regime. Some of thedistinct, promising, yet complementary capabilities offered by XFELsinclude: (1) access to higher-energy core edges with large spin–orbitsplittings ( >10 eV), which was exploited in the present work to mini- mize the adverse overlap of adjacent core-level absorption, (2) muchhigher photon flux per pulse, allowing for increased sensitivity via pulse-to-pulse normalization, surface sensitivity via TEY measure- ments, 51and the possibility to detect weak transient signals due to the reduction of the shot noise level, (3) smaller focal spot sizes(/C241–10 lm), 15,16and longer absorption lengths at higher-energy x-rays, enabling future study of small semiconductor samples or indi-vidual domains of polycrystalline samples and thicker samples, respec-tively, and (4) greater tunability over the soft, tender, and hard x-rayenergies for accessing K, L, M, etc., edges of elements, which opens thepossibility of probing different orbital symmetries of the valence shelldue to dipole selection rules. High-repetition rate XFELs, such as the European XFEL and LCLS-II (currently under construction) will fur- ther advance the sensitivity of this approach. AUTHORS’ CONTRIBUTIONS A.B. and A.R.A. contributed equally to this work. SUPPLEMENTARY MATERIAL See the supplementary material for details on the sample prepara- tion and characterization, excited carrier density estimation, PAL-XFEL energy calibration, partial/projected density of states calcula-tions, static Mo M3-edge measurements, and XUV transient absorp- tion results with 400 nm pump-long lived signals. ACKNOWLEDGMENTS We thank Philippe Wernet, Kristjan Kunnus, Roberto Alonso- Mori, Dimosthenis Sokaras, and Das Pemmaraju for the fruitfuldiscussions on experimental and theoretical methods in soft x-rayabsorption spectroscopy. Furthermore, we thank Angel GarciaEsparza and John Vinson for their help with the OCEANcalculations. This work was supported by the Computational Materials Sciences Program funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-SC0014607 and a basic science research program funded by the Ministry ofEducation of Korea (Nos. NRF-2020R1A2C1007416 and2018R1D1A1B07046676). Computations were performed at theCenter for Advanced Research Computing of the University ofSouthern California and at the Argonne Leadership ComputingFacility under the DOE INCITE and Aurora Early Scienceprograms. Core-level absorption simulations were conducted at theMolecular Graphics and Computation Facility, UC Berkeley,College of Chemistry, funded by the National Institutes of Health(No. NIH S10OD023532). Further research at SLAC by A.A. andStructural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 8, 014501 (2021); doi: 10.1063/4.0000048 8, 014501-8 VCAuthor(s) 2021T.F.H. was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division, Catalysis Science Program under No. FWP 100435. H.-T.C. was recently supported by the W. M. Keck Foundation, Grant No. 046300. S.S. acknowledges fundingfrom the Instrumentarium Science Foundation and Walter Ahlstr €om Foundation. A.M.L. and C.N. acknowledge support from the U.S. Department of Energy, Basic Energy Sciences, MaterialsScience and Engineering Division under Contract No. DE- AC02–76SF00515. S.R.L. acknowledges support by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, under Contract No. DEAC02–05-CH11231, within the Physical Chemistry of InorganicNanostructures Program (No. KC3103) and the Chemical, Geosciences and Biosciences Division, within the Atomic, Molecular, and Optical Sciences Program. S.R.L. and H.-T.C. also acknowledge support by the Air Force Office of Scientific Research (Nos. FA9550–19-1–0314, FA9550–14-1–0154, and FA9550–15-1–0037), and the Army Research Office (No. W911NF-14–1-0383). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1J. Shah, Ultrafast Spectroscopy of Semiconductors and Semiconductor Nanostructures (Springer-Verlag Berlin Heidelberg, New York, 1999). 2K. Ramasesha, S. R. Leone, and D. M. 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5.0033592.pdf

Appl. Phys. Lett. 118, 013503 (2021); https://doi.org/10.1063/5.0033592 118, 013503 © 2021 Author(s).The origin of interlayer-induced significant enhancement of EQE in CzDBA-based OLEDs studied by magneto-electroluminescence Cite as: Appl. Phys. Lett. 118, 013503 (2021); https://doi.org/10.1063/5.0033592 Submitted: 18 October 2020 . Accepted: 16 December 2020 . Published Online: 05 January 2021 Xiantong Tang , Ruiheng Pan , Hongqiang Zhu , Xi Zhao , Linyao Tu , and Zuhong Xiong COLLECTIONS This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN Coherent optical interaction between plasmonic nanoparticles and small organic dye molecules in microcavities Applied Physics Letters 118, 013301 (2021); https://doi.org/10.1063/5.0027321 Emerging circularly polarized thermally activated delayed fluorescence materials and devices Applied Physics Letters 117, 130502 (2020); https://doi.org/10.1063/5.0021127 Heteroepitaxial growth of wide bandgap cuprous iodide films exhibiting clear free-exciton emission Applied Physics Letters 118, 012103 (2021); https://doi.org/10.1063/5.0036862The origin of interlayer-induced significant enhancement of EQE in CzDBA-based OLEDs studied by magneto-electroluminescence Cite as: Appl. Phys. Lett. 118, 013503 (2021); doi: 10.1063/5.0033592 Submitted: 18 October 2020 .Accepted: 16 December 2020 . Published Online: 5 January 2021 Xiantong Tang,1Ruiheng Pan,2Hongqiang Zhu,3XiZhao,1Linyao Tu,1and Zuhong Xiong1,a) AFFILIATIONS 1School of Physical Science and Technology, MOE Key Laboratory on Luminescence and Real-Time Analysis, Southwest University, Chongqing 400715, China 2Key Laboratory of Luminescence and Optical Information Ministry of Education, Institute of Optoelectronic Technology,School of Science, Beijing Jiaotong University, Beijing 100044, China 3Chongqing Key Laboratory of Photo-Electric Functional Materials, Chongqing Normal University, Chongqing 401331, China a)Author to whom correspondence should be addressed: zhxiong@swu.edu.cn ABSTRACT Over twelve-fold enhancement of external quantum efficiency (EQE) is observed in 9,10-bis(4-(9H-carbazol-9-yl)-2,6-dimethylphenyl)-9,10- diboraanthracene (CzDBA)-based organic light-emitting diodes (OLEDs) with an interlayer between the hole-transporting layer (HTL) and theemission layer, where the CzDBA emitter is a typically donor–acceptor–donor (D–A–D)-type thermally activated delayed fluorescence material. Analyses of the fingerprint magneto-electroluminescence traces indicate that the interlayer ensures the charge balance of the emission layer in devices, avoiding triplet-charge annihilation and contributing to the enhancement of EQE. Additionally, experimental results also show thatintersystem crossing (ISC) and reverse ISC (RISC) processes coexist in the device with an interlayer. Notably, ISC boosts with increasing biascurrents and working temperatures, respectively, exhibiting abnormal current and normal temperature dependences. This abnormal phenome-non is caused by the weakened RISC between charge-transfer states of CzDBA molecules at large bias currents. More interestingly, as bias cur- rents increase, ISC in the device without an interlayer first exhibits normal current dependences and then turns into an abnormal one, which may attribute to the competitive effects of exciplex at the HTL/CzDBA interface and excited states of CzDBA molecules. Our findings not onlyunravel the underlying mechanisms in D–A–D-type molecules but also provide ideas for designing highly efficient devices. Published by AIP Publishing. https://doi.org/10.1063/5.0033592 Over the past decade, aromatic compounds with thermally activated delayed fluorescence (TADF) characteristics have attracted considerable attention for their applications in high-efficiency organic light-emitting diodes (OLEDs). 1–5In these compounds, the reverse intersystem crossing (RISC) process from lowest triplet (T 1)t os i n g l e t (S1)s t a t e si sc r u c i a lf o rt h ee f fi c i e n tu t i l i z a t i o no fn o n - r a d i a t i v et r i p l e t excitons and can achieve approximately 100% internal quantum effi- ciency.1,2The mixing extent of T 1and S 1states can use the first-order perturbation theory that is expressed as k/HSOC=DEST,w h e r e k, DEST,a n d HSOCare the first-order mixing coefficient, the energy gap between S 1and T 1states, and the spin–orbit coupling value, respec- tively.3–5This relationship suggests that kis inversely proportional to DESTand efficient RISC requires small DESTvalues.5Therefore, many researchers are devoted to designing the donor–acceptor– donor (D–A–D)-type TADF materials with the intra-molecularcharge-transfer (CT) states to acquire the extremely small DESTvalues in recent years.6–8Compared to conventional donor- acceptor-type materials, D–A–D-type TADF molecules possess twisted bonding between the donor and acceptor units, causingsmaller DE STvalue, shorter TADF lifetime, and lower efficiency roll-off.6,7Among reported D–A–D-type TADF materials, 9,10- bis(4-(9H-carbazol-9-yl)-2,6-dimethylphenyl)-9,10-diboraanthracene (CzDBA) shows wide application prospects.6,8,9Nevertheless, in spite of the large amount of work devoted to designing and fabricating high- efficiency CzDBA-based OLEDs, the specific discussion about the microscopic mechanisms of spin-pair states in this D–A–D-type TADF material is still lacking. Herein, two CzDBA-based devices with/without an interlayer between the hole-transporting layer (HTL) and the emission layer were fabricated to investigate the underlying mechanisms Appl. Phys. Lett. 118, 013503 (2021); doi: 10.1063/5.0033592 118, 013503-1 Published by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplin D–A–D-type molecules. Using fingerprint magneto- electroluminescence (MEL) as a sensitive probing tool, we observedthe coexistence of intersystem crossing (ISC) and reverse ISC (RISC) in device with an interlayer. It is noteworthy that this ISC process boosts with increasing bias currents (i.e., abnormal current depen-dence) and working temperatures (i.e., normal temperature depen-dence). The reduced RISC process at high bias currents is responsible for this abnormal phenomenon. In contrast, weak ISC and the obvious triplet-charge annihilation (TQA) process occurred in the device with-out an interlayer. The former first decreases and then enhances withincreasing bias currents, which can ascribe to the comprehensive effects of exciplex at the HTL/CzDBA interface and excited states of CzDBA molecules by analyzing the electroluminescence (EL) spec-trum. The latter dominates in the whole current ranges and is detri- mental to the utilization of non-radiative triplet states. Accordingly, the absence of the TQA process in the device with an interlayercontributes to the utilization of triplet excitons by an efficient RISCprocess, making its EQE an order of magnitude higher than that of devices without an interlayer. Our work provides not only in-depth insight into the dynamics of spin-pair states in D–A–D-type OLEDsbut also ideas for designing high-efficiency spin optoelectronic devices. Two CzDBA-based devices had the following architectures: ITO/ poly(3,4-ethylenedioxythiophene):poly(styrenesulfonate) (PEDOT:PSS,40 nm)/N,N 0-bis(naphthalen-1-y)-N,N0-bis(phenyl)benzidine [NPB, (50/C0x) nm]/tris(4-carbazoyl-9-ylphenyl)amine (TCTA, xnm)/CzDBA (30 nm)/(1,3,5-triazine-2,4,6-triyl)tris(benzene-3,1-diyl)tris(diphe-nylphosphineoxide) (PO-T2T, 60 nm)/LiF (1 nm)/Al (120 nm), as shown in Fig. 1(a) . Here, the OLEDs with the interlayer thickness (x) of 10 and 0 nm correspond to Devices 1 and 2, respectively. In these two devices, NPB, CzDBA, and PO-T2T materials wereselected as the HTL, emission layer, and electron-transporting layer, respectively. The choice of TCTA as the interlayer was due to its high carrier mobility (3.0 /C210 /C04cm2/V s)10and moderate ioniza- tion energy (IE) of 5.7 eV,10causing that hole injection from NPB with the IE of 5.4 eV10to CzDBA with the IE of 5.9 eV,6which wasexpected to be efficient. Furthermore, one reference device based on classic tris(8-hydroxyquinoine)aluminum (Alq 3) material without the TADF characteristic had the architecture of ITO/PEDOT:PSS/ NPB (60 nm)/Alq 3(80 nm)/LiF/Al, which was fabricated for com- parison with Device 1. The similar fabrication and measurementmethods of these three devices were described in previous works. 11–13 Figure 1(b) shows the ultraviolet-visible absorption and photolu- minescence (PL) spectra of the pure CzDBA film and EL spectra of Devices 1 and 2. As can be seen, the PL spectrum of the CzDBA filmexhibits the emission band at 569 nm with a relatively narrow full width at half maximum (FWHM) of 87 nm, which can be explained by the rigid rod-like structure in which only the carbazole donor couldrotate freely. 6T h eE Ls p e c t r u mo fD e v i c e1h a st h es a m ee m i s s i o n peak and FWHM value as the PL spectrum of CzDBA. By contrast, the EL peak of Device 2 centers around 572 nm with an FWHM of111 nm, showing a slight red shift and obvious broadening. This result indicates the existence of other excited states besides CzDBA excitons in Device 2, which will be discussed and explained in detail later.Additionally, the current density–luminance–voltage characteristics ofDevices 1 and 2 are displayed in Fig. 1(c) .D e v i c e s1a n d2e x h i b i tl o w turn-on voltages of 2.9 and 3.1 V estimated at the brightness of 1 cd m /C02. Notably, a maximum luminance of 28 480 cd m/C02is acquired from Device 1 at the current density of 458 mA cm/C02,w h i c hi sn e a r l y an order of magnitude greater than that of Device 2 (3296 cd m/C02). Most importantly, although these two devices possess extremely simi-lar structures, the EQE maximum (7.40%) of Device 1 is approxi- mately twelve times higher than that (0.59%) of Device 2, as demonstrated in Fig. 1(d) . According to the literature studies, 14,15EQE of a fluorescent OLED can be expressed as EQE¼vS/C2gr/C2gPL/C2gout; (1) where vSis the exciton formation probability, gri st h ec h a r g er e c o m b i - nation efficiency, gPLis the PL quantum yield, and goutis the light out- coupling efficiency. Both Devices 1 and 2 selected single CzDBAmaterial as an emitter, leading to their EQEs with the same v S,gPL, andgoutvalues. Consequently, based on Eq. (1), we can conjecture that gr(i.e., charge balance in the emission layer) has a significant influence on the efficiency difference between Devices 1 and 2. Next, to further understand the origin of the significant enhancement of the EQE maximum in Device 1, we first perform the systematic measure-ments of Device 1 including the current and operating temperature dependences of MEL traces as follows. Figures 2(a) and 2(b) show the current-dependent MEL responses of Device 1 and the reference device operating at room tem- perature, respectively. Here, MEL is defined by the relationship ofMEL¼[EL(B)/C0EL(0)] /C2100%/EL(0), where EL( B) [EL(0)] is the EL intensity at field B[orB¼0]. Intriguingly, although the CzDBA and Alq 3emitters in these two devices have completely different char- acteristics, their MEL traces exhibit similar line-shapes. Specifically, their MELs rapidly rise in the low field range ( jBj<9 mT) followed by aw e a ki n c r e a s ea th i g h e r Band reach the saturation value at 300 mT. According to the literature studies,16–18the narrow FWHM ( /C249m T ) of these MEL behaviors is the most important feature for the hyperfine-induced ISC process between polaron pairs. Therefore, we can conclude that the ISC process exists in Device 1. In addition, wealso note that MEL amplitudes of these two devices obtained at FIG. 1. (a) Schematic diagram showing the energy level alignments of two CzDBA- based devices. (b) Absorption and PL spectra of CzDBA films with the thickness of100 nm and normalized EL spectra of Devices 1 and 2. (c) Current density(J)–luminance (L)–voltage (V) characteristics. (d) EQE-J plots.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 013503 (2021); doi: 10.1063/5.0033592 118, 013503-2 Published by AIP Publishing300 mT (MEL max) demonstrate opposite current dependences, as shown in Figs. 2(c) and2(d). Namely, the MEL maxvalue decreases with increasing bias currents in the reference device [ Fig. 2(d) ], indi- cating that the ISC process weakens at high currents (i.e., normal cur- rent dependence). On the contrary, the ISC process in Device 1 shows an abnormal current dependence [ Fig. 2(c) ], which reveals the pres- ence of other processes except for the ISC process and has never beenreported in the literature. In order to further explore the underlying microscopic mechanisms existing in MEL of Device 1, the measure- ments of its current-dependent magneto-conductance (MC) responseswere performed and the related measurement result is shown in Fig. S1(a) (see the supplementary material ). As can be clearly seen, MC behaviors at low fields are completely opposite to the ISC-determined MEL line shape [ Fig. 2(a) ] and can be ascribed to the role of the RISC process in Device 1. The detailed analyses for these MC responses areg i v e ni nN o t eS 1( s e et h e supplementary material ). These experimental results indicate that both the ISC and RISC process can coexist in Device 1. The underlying microscopic mechanisms between spin-pair states in Device 1 and the reference device are shown in Figs. 2(e) and 2(f), respectively. Under electrical excitation, the injected electron and hole carriers can form singlet (PP 1) and triplet polaron pairs (PP 3)a t the ratio of 1:3 in the emission layer. Then, the negligible spin exchange energy between PP 1and PP 3states could lead to the spin- mixing of PP states through the hyperfine-induced ISC process.19 Simultaneously, PP 1and PP 3can form CT-type S 1(CT 1)a n dT 1 (CT 3) states with the rate constants of kSandkT, respectively. In gen- eral, kTis larger than kS(i.e., kT>kS), resulting in the predominant ISC process (PP 1!PP3) from PP 1to PP 3,19,20as depicted in Note S2 (see the supplementary material ). However, the RISC process (CT 3!CT1)f r o mC T 3to CT 1dominates in the spin-flip process of CT states because of the longer lifetime of CT 3states than CT 1states. In fact, since the energies of CT 1and CT 3states in CzDBA molecules are very close,9ISC (CT 1!CT3) and RISC (CT 3!CT1)c a nc o e x i s tin devices. The ISC process is usually ignored because CT 1will quickly deactivate to the ground states and generate fluorescence. That is, ascompared to CT 3excitons, the lifetime of CT 1is very short. Additionally, the amount of CT 3e x c i t o n si sf a rl a r g e rt h a nt h a to fC T 1 excitons. Based on these analyses, it confirms that the ISC processbetween PP states (PP 1!PP3) and the RISC process between CT states (CT 3!CT1) coexist in Device 1, as shown in Fig. 2(e) .I nc o n - trast, the large energy gap between S 1and T 1in the reference device prevents T 1from directly converting to the S 1s t a t ev i at h eR I S Cp r o - cess. Therefore, the reference device only contains the ISC process(PP 1!PP3). As bias currents increase, the enhanced applied electric field can easily enlarge the distance between the electron and the holein PP 1states to dissociate into free charge carriers via the Onsager process,11,21,22weakening the ISC process. As for Device 1, MEL responses in Fig. 2(a) depend on the overlapping effect of positive ISC and negative RISC-determined line-shapes. Since the RISC processcan effectively be suppressed at high bias currents, 12,23the ISC process would boost with increasing bias current. Figure 3(a) shows the temperature ( T)-dependent MEL responses of Device 1. Apparently, MEL curves exhibit the ISC-dominated lineshape in the whole temperature range. The amplitude of MEL maxobvi- ously decreases with lowering temperature (i.e., the normal tempera-ture dependence), as shown in Fig. 3(b) . These experimental results can attribute to the comprehensive effects of decreased ISC and RISCprocesses at low temperatures due to their endothermic proper-ties. 12,16,17,21Furthermore, the inset of Fig. 3(b) plots MEL maxvs 1/T in a broad temperature range of 400 to 200 K. According to theArrhenius law, the relationship between MEL maxand Tcan be expressed as MEL max¼Aexp/C0Eact kBT/C18/C19 ; (2) where Ais the pre-exponential factor, Eactis the activation energy, and kBis Boltzmann’s constant.24–26The Eactvalue is estimated to be 40 meV, which is in agreement with the energy difference (39 meV)obtained by Cheng et al. from the theoretical calculation. 6 Room-temperature current-dependent MEL curves of Device 2 in low and high bias current ranges are shown in Figs. 4(a) and4(b), respectively. Compared to the single MEL line-shapes of Device 1[Fig. 2(a) ], the MEL curves of Device 2 [ Figs. 4(a) and4(b)] are obvi- ously composed of two parts including low and high field effects (i.e.,LFE and HFE). The LFE refers that MELs rapidly rise within the rangeofjBj<9 mT, while the HFE denotes that MELs slowly increase within the range of 9 mT <jBj<300 mT. In agreement with the pre- vious analyses for Device 1, here, the LFE in Device 2 still originatesfrom ISC-determined MEL curves. However, according to the litera-ture studies, 13,23the slowly increased HFE with the FWHM of /C24100 mT can be ascribed to the B-mediated TQA process. Very inter- estingly, current-dependent MC responses of Device 2 (see Fig. S3 inthesupplementary material ) have almost the same line shape as these MEL responses, further supporting the interpretation that ISC andTQA processes are assigned to dominate at the LFE and HFE, respec-tively. Notably, the RISC process is negligible in Device 2 becauseRISC-induced line-shapes cannot be observed in Figs. 4(a) and4(b) and the strong TQA process is detrimental to the utilization of non-radiative triplet states by the RISC process. To more clearly analyzethe significant impact of bias currents on these mechanisms, the MEL FIG. 2. Room-temperature current-dependent MEL responses of (a) Device 1 and (b) reference device. Their MEL values at 300 mT were extracted and are shown in(c) and (d), respectively. Schematic diagram showing the operating mechanisms of (e) Device 1 and (f) reference device.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 013503 (2021); doi: 10.1063/5.0033592 118, 013503-3 Published by AIP Publishingvalues at the LFE [i.e., MEL LFE¼MEL( B¼9m T )–M E L ( B¼0m T ) ] and HFE [i.e., MEL HFE¼MEL( B¼300 mT) – MEL( B¼9m T ) ] a r e summarized in Fig. 4(c) . Very interestingly, non-monotonic changes of MEL LFEand MEL HFEamplitudes suggest that both ISC and TQA processes first weaken and then enhance with increasing bias currentsin Device 2. Figure 4(d) shows the EL spectrum of Device 2, in which the peak is located at around 572 nm with the FWHM of 112 nm. Compared to the PL emission peak and FWHM of the CzDBA film [Fig. 1(b) ], the main /C24572 nm peak of Device 2 shows a small red shift and significant broadening. Using two Gaussian functions to fit wellthe EL spectrum of Device 2 [ Fig. 4(d) ], we can conclude that the EL spectrum factually contains two peaks, which are located at 568( 2 . 1 8e V )a n d6 3 7n m( 1 . 9 5e V )w i t hF W H Mv a l u e so f8 3a n d9 2n m , respectively. Undoubtedly, the former peak originates from the emis- sion of CT 1states in CzDBA molecules. Nevertheless, we consider that the latter peak must be generated by the emission of singlet exciplexformed at NPB/CzDBA interfaces. This inference can be proved bytwo factors. One is that the peak of 637 nm shows an obvious red-shift in comparison between the PL spectra from NPB (434 nm) and CzDBA (569 nm) [see the inset of Fig. 4(d) ]. The other is that the energy (1.95 eV) of this emission peak approximately equals the FIG. 3. (a) Temperature-dependent MEL responses of Device 1 at a bias current of 50 lA. (b) Temperature-dependent MEL maxextracted from Fig. 3(a) . The insets show the Arrhenius plot of MEL max. The linear fits (red line) can give the activation energy. FIG. 4. Current-dependent MEL responses of Device 2 in (a) low and (b) high bias current ranges. (c) MEL values at the LFE and HFE extracted from Figs. 4(a) and4(b) as a function of bias currents. (d) EL spectrum of Device 2 (navy solid line) and corresponding Gaussian fitting (olive and red dotted line). The inset shows the PL spectra of NPB and CzDBA films and the emission spectrum of the NPB/CzDBA exciplex. (e) Schematic showing the formation of excitons and exciplex in Device 2. (f) MEL co ntribution ratio as a function of bias currents.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 013503 (2021); doi: 10.1063/5.0033592 118, 013503-4 Published by AIP Publishingenergy difference (1.9 eV) between the highest occupied molecular orbit level ( /C05.4 eV) of NPB and the lowest unoccupied molecular orbit level ( /C03.5 eV) of CzDBA. Therefore, the NPB/CzDBA exciplex and CzDBA excited states can simultaneously exist in Device 2, as shown in Fig. 4(e) . Based on these detailed analyses, we conjecture that the competition effects between two kinds of spin-pair states (i.e.,NPB/CzDBA exciplex and CzDBA charge tranfer states) can lead to non-monotonic changes of the ISC and TQA processes. Very impor- tantly, g values (g þand g/C0) of positive and negative polarons whether in the CzDBA charge transfer states or at the NPB/CzDBA exciplex interface are theoretically different (i.e., Dg¼jgþ/C0g/C0j6¼0), but the Dg mechanism could be ruled out for interpreting the formation of h i g h - fi e l dM E L si nt h i sw o r k .R e l a t e dd i s c u s s i o ni ss h o w ni nN o t eS 3 (supplementary material ). In addition, the relative proportions of ISC and TQA processes in Device 2 can be reflected by the contribution ratio [ RLFE(HFE) ]o f their corresponding MEL values at the LFE and HFE, which can be expressed as RLFEðHFEÞ¼MEL LFEðHFEÞ MEL LFEþMEL HFE/C2100% ; (3) where the MEL LFEand MEL HFEvalues at different bias currents can be obtained from Fig. 4(c) . According to Eq. (3), the contribution ratios of each microscopic process under different bias currents are calculated and shown in Fig. 4(f) . Obviously, the contribution ratio of TQA is larger than that of ISC in the whole current range, indicating that the TQA dominates in Device 2. This is because the absence ofthe interlayer is not conducive to the hole injection in the emission layer, causing that a majority of electron carriers could interact with triplet states by the TQA process in Device 2. However, TQA is detri- mental to the utilization of non-radiative triplet exctions, leading to the low electroluminescence efficiency of Device 2. In summary, we have found that the EQE maximum value of Device 1 is about twelve times higher than that of Device 2. Via ana- lyzing the current-dependent MEL responses of these two CzDBA- based OLEDs with/without an interlayer, it can be concluded that the TQA process plays an important role in the efficiency difference of these two devices. Specifically, the presence of the interlayer makes the injection carriers more balanced in Device 1, leading to the absence of the TQA process and high EQE of the device. On the contrary, the nonexistence of the interlayer enables electrons to become majority carriers in Device 2, causing redundant electron carriers to interactwith triplet excitons via TQA. Furthermore, experimental results also show that the weakened RISC process at high bias current is responsi- ble for the abnormal current dependence of ISC in Device 1. However, ISC with the characteristic of normal and abnormal current dependen- ces in Device 2 can attribute to the competition effects of CzDBA exci- tons and NPB/CzDBA exciplex. This study reveals that using the interlayer to avoid the TQA process is quite important for obtaining larger EL efficiency in CzDBA-based devices. See the supplementary material for details on the analyses for current-dependence MC responses of Device 1 at room temperature,the explanation for the relationship between k SandkTin Device 1 and reference device, discussion for the effect of the Dg mechanism on the high field of MEL in Device 2, images of current-dependent MC ofDevices 1 and 2, and MC values at low- and high-field effects as a function of bias currents for Device 1. 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J. Appl. Phys. 129, 073104 (2021); https://doi.org/10.1063/5.0031514 129, 073104 © 2021 Author(s).Multiscale simulations of the electronic structure of III-nitride quantum wells with varied indium content: Connecting atomistic and continuum-based models Cite as: J. Appl. Phys. 129, 073104 (2021); https://doi.org/10.1063/5.0031514 Submitted: 01 October 2020 . Accepted: 27 January 2021 . Published Online: 18 February 2021 D. Chaudhuri , M. O’Donovan , T. Streckenbach , O. Marquardt , P. Farrell , S. K. Patra , T. Koprucki , and S. Schulz ARTICLES YOU MAY BE INTERESTED IN Role of strain and composition on the piezoelectric and dielectric response of Al xGa1−xN: Implications for power electronics device reliability Journal of Applied Physics 129, 075701 (2021); https://doi.org/10.1063/5.0033111 Machine learning for materials design and discovery Journal of Applied Physics 129, 070401 (2021); https://doi.org/10.1063/5.0043300 A first-principles understanding of point defects and impurities in GaN Journal of Applied Physics 129, 111101 (2021); https://doi.org/10.1063/5.0041506Multiscale simulations of the electronic structure of III-nitride quantum wells with varied indium content: Connecting atomisticand continuum-based models Cite as: J. Appl. Phys. 129, 073104 (2021); doi: 10.1063/5.0031514 View Online Export Citation CrossMar k Submitted: 1 October 2020 · Accepted: 27 January 2021 · Published Online: 18 February 2021 D. Chaudhuri,1,a) M. O ’Donovan,1,2 T. Streckenbach,3O. Marquardt,3P. Farrell,3 S. K. Patra,1 T. Koprucki,3 and S. Schulz1 AFFILIATIONS 1Tyndall National Institute, University College Cork, Cork T12 R5CP, Ireland 2Department of Physics, University College Cork, Cork T12 YN60, Ireland 3Weierstrass Institute (WIAS), Mohrenstr. 39, 10117 Berlin, Germany a)Author to whom correspondence should be addressed: debapriyachaudhuri.jobs@gmail.com ABSTRACT Carrier localization effects in III-N heterostructures are often studied in the frame of modified continuum-based models utilizing a single-band effective mass approximation. However, there exists no comparison between the results of a modified continuum model and atomistic calculations on the same underlying disordered energy landscape. We present a theoretical framework that establishes a connec- tion between atomistic tight-binding theory and continuum-based electronic structure models, here a single-band effective mass approxima-tion, and provide such a comparison for the electronic structure of (In,Ga)N quantum wells. In our approach, in principle, the effectivemasses are the only adjustable parameters since the confinement energy landscape is directly obtained from tight-binding theory. We find that the electronic structure calculated within effective mass approximation and the tight-binding model differ noticeably. However, at least in terms of energy eigenvalues, an improved agreement between the two methods can be achieved by adjusting the band offsets in the con-tinuum model, enabling, therefore, a recipe for constructing a modified continuum model that gives a reasonable approximation of thetight-binding energies. Carrier localization characteristics for energetically low lying, strongly localized states differ, however, signific antly from those obtained using the tight-binding model. For energetically higher lying, more delocalized states, good agreement may be achieved. Therefore, the atomistically motivated continuum-based single-band effective mass model established provides a good, computationally effi- cient alternative to fully atomistic investigations, at least at when targeting questions related to higher temperatures and carrier densities in(In,Ga)N systems. Published under license by AIP Publishing. https://doi.org/10.1063/5.0031514 I. INTRODUCTION In the past two decades, III-N-based semiconductors have attracted significant research interest given their potential for a variety of different applications. These applications include photo-voltaic cells and light-emitting diodes (LEDs). 1–3For instance, the active region of modern LEDs operating in the blue spectral regionis based on c-plane (In,Ga)N/GaN quantum wells (QWs). 2,4,5 Despite widespread application of these LED structures nowadays, to further improve their overall performance and efficiency, adetailed understanding of their fundamental properties is required, especially when moving into the ultraviolet or the green spectral region. While theoretical studies can provide guidance to achieve these goals, it is overall a very challenging task. Experimental inves- tigations give clear indications that the electronic and optical prop-erties of III-N materials and heterostructures are strongly affectedby carrier localization effects, originating from alloy fluctuations in III-N alloys. 6–9Thus, to achieve an accurate theoretical descriptionJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 073104 (2021); doi: 10.1063/5.0031514 129, 073104-1 Published under license by AIP Publishing.of these properties, it is important that the theoretical model accounts for localization effects.10–15As a consequence, a fully three-dimensional (3D) model is required, even when studyingIII-N quantum well (QW) structures. To target carrier localizationeffects in a such a 3D description, a variety of different theoreticalapproaches has been applied in the literature. These range from fully atomistic calculations 12,16to modified continuum-based models.10,11,17–20While atomistic modeling has been successfully applied to describe whole devices21,22as well as the influence of alloy fluctuations on the electronic properties of semiconductorheterostructures, 13,23their application generates a huge computa- tional effort, depending on the numbers of atoms involved. This applies, in particular, to the systematic evaluation of trends whenmodifying dimensions or chemical composition of a semiconductorheterostructure in large simulation cells. Especially in industryfocused device design activities, the huge computational effort of full atomistic device calculations is often not a viable route. Therefore, modified continuum-based approaches have found widespread application to account for alloy fluctuations and, thus,carrier localization effects in, for instance, (In,Ga)N/GaN hetero-structures. While the numerical burden in most cases is signifi- cantly reduced compared to atomistic approaches, this comes at a cost: the underlying atomistic structure is lost and the calculationsare carried out on a mesh where the information about the atomicspecies is replaced by an average alloy content. The determination of the local alloy content depends then on the chosen interpolation procedure and the same is true for the(local) material parameters. 18,23Overall, such an approach raises several questions, including how valid the use of bulk materialparameters in small spatial regions are or the fact that small scale alloy fluctuations are in general beyond the validity limits of continuum-based models. Nevertheless, especially single-bandeffective mass approximations have been often used in the literatureto study the electronic and optical properties of (In,Ga)N/GaNQWs with a strongly fluctuating energy landscape, constructed from bulk band parameters. In general, there exist different approaches in the literature for establishing such modifiedcontinuum-based models but there exists basically no analysis ofhow such an approach compares to the outcome of an atomisticmodel using the same underlying structure. Recent theoretical studies give indications that continuum-based models may under- estimate carrier localization effects in (In,Ga)N/GaN QWsystems, 24while, however, excellent agreement between atomistic and continuum-based modeling has been observed for nitride quantum dots of comparatively small dimensions.25 In this work, we fill this gap and establish a general theoretical framework that allows us (i) to connect an atomistic tight-binding(TB) model with modified continuum-based approaches (single-band effective mass, multi-band k/C1p) and (ii) to directly compare the results of these two approaches on the same input data set. The benefit of this approach is that we establish a modified continuum-based model that can be tailored and adjusted to provide a reason-able agreement with the atomistic model. This lends further trustfor the application of this framework in future studies, including, for instance, transport calculations of nitride-based heterostructures. More specifically, we have developed a method that allows us to extract an energy landscape from the atomistic TB model thataccounts for local strain and built-in potential fluctuations, which then serves directly as an input for continuum-based calculations. In doing so, the approach bypasses the complication of usinglocally averaged material parameters such as bulk band offsets orpiezoelectric coefficients since the continuum-based model isdirectly connected to the TB energy landscape which includes modifications in the band edges due to alloy fluctuations in the active region on a microscopic level. The continuum model,thus, operates on an atomistically derived energy landscape.Additionally, when connecting TB and single-band effective massapproximation (EMA), in principle, the only adjustable parameters left are the electron and hole effective masses. Furthermore, to transfer the atomistic energy landscape into the continuum-basedmodel, we use a finite element mesh (FEM) with as many nodesas lattice sites. Overall, our approach allows for multiscalemodeling 26–28of the electronic and optical properties of III-N het- erostructures in the picture of a modified continuum model with a benchmark loop to atomistic calculations. This, therefore, enablesus to adjust the model to design an “atomistically corrected ”con- tinuum model. In future studies, this may facilitate transportstudies by using drift-diffusion calculations to (i) account for alloy fluctuations and (ii) to allow for drastically reduced computational efforts when comparing to full atomistic device calculations. We show that even after calibrating the EMA against a virtual crystal approximation (VCA) TB model, the transition energies predicted by the EMA for the random alloy case significantly devi- ates from the TB results. This discrepancy is larger with increasingIn content, i.e., for longer wavelengths. However, we will show thatwhile preserving the average energetic separation between electronand hole states, a very good agreement between TB and EMA is achieved when the band offset in the (In,Ga)N region (QW region) is adjusted by a rigid shift that increases with increasing In content.This shows that the established framework can now be adjusted togive a good approximation of the TB results in terms of the ener-gies, which allows us to use it for future calculations. In addition to comparing energy eigenvalues, we have also analyzed carrier locali- zation effects predicted by the two above mentioned methods.To do so, we have calculated inverse participation ratios (IPRs) 7,29 for the first ten electron and hole states within TB and EMA. Our calculations show that in comparison to the TB model, the EMA significantly underestimates hole localization effects, especially for higher In content systems. For electrons, especially for lower Incontents, the situation is slightly different and a better agreementbetween TB and EMA is observed. However, this is only the case for the model that includes the rigid band offset shift. Nevertheless, we also find that the agreement between TB and continuum-basedmodel in terms of carrier localization effects improves for energeti-cally higher lying states. Thus, the developed and established modelshould provide an attractive approach to investigate (In,Ga)N/GaN QW systems at elevated temperatures and higher carrier densities where energetically high lying states become populated. The paper is organized as follows: in Sec. II, we introduce the theoretical framework that connects the atomistic and continuum-based models. The calibration of the EMA against the VCA TB model is presented in Sec. III A . Next, in Sec. III B , we compare the energy eigenvalues of the calibrated EMA with TB datafor (In,Ga)N/GaN QW systems with 5%, 10%, 15%, and 25%Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 073104 (2021); doi: 10.1063/5.0031514 129, 073104-2 Published under license by AIP Publishing.In, which exhibit random alloy fluctuations. The average normal- ized IPR values ( gIPR) for first ten electron and hole states in these systems are discussed and presented in Sec. III C . In Sec. IV,w e summarize our findings, while in the Appendix , details of our IPR value calculations are given. II. THEORETICAL FRAMEWORK FOR CONNECTING ATOMISTIC THEORY AND CONTINUUM-BASEDMODELS The aim of our study is to derive a modified continuum-based model that directly incorporates input from atomistic TB theory. Ingeneral, to study electronic and optical properties of a semiconduc-tor heterostructure, one is conventionally left with solvingSchrödinger ’s equation, ^Hψ¼(^Tþ^V)ψ¼Eψ, (1) where ^His the Hamiltonian of the system under consideration, and ^Tand ^Vare the kinetic and potential energy operators, respec- tively. The eigenenergy is denoted by E, and ψis the corresponding eigenstate. The aim of our framework is to extract a potential energy landscape ^Vfrom an atomistic TB model that can be used in a robust and computationally inexpensive continuum-based descrip-tion of Eq. (1). The procedure of extracting ^Vfrom a TB model is explained in Sec. II A. In Sec. II B, we outline how the obtained landscape is transferred to a FEM mesh and, thus, prepared for a continuum-based solver. A schematic illustration of the workflow isdisplayed in Fig. 1 . Here, we solve Schrödinger ’s equation within the framework of a single-band EMA, which has been implemented in the highly flexible plane wave based software package SPHInX, 30–33which is briefly explained in Sec. II C. We note that for the sake of a simplified discussion, and since it is a widely usedapproach in the literature to study the impact of alloy fluctuationson the electronic structure of (In,Ga)N QWs, we have limited ourself to a single-band EMA model. Of course, more sophisticated continuum models such as the eight-band k/C1papproach will facili- tate a more accurate description of the electronic properties ofheterostructures containing alloy fluctuations. For instance, it maybe of particular importance to take into account the nonparabolic behavior of the bands, which may become relevant in small structures with large band offset, e.g., small clusters with largeIn contents. However, the flexibility of SPHInX allows in principle to easily change the underlying Hamiltonian used in solving Schrödinger ’s equation so that the general approach presented here can be transferred to employ more sophisticated models.Additionally, given that single-band effective mass models are oftenapplied to study carrier localization effects in (In,Ga)N systems, any problem arising from the fact that a strongly fluctuating energy landscape presents in general a challenge for continuum-basedmodels, should be revealed by the analysis presented in this work. In addition to turning to multi-band k/C1pmodels and to avoid solving large scale eigenvalue problems, future studies may use the established framework to combine the TB energy landscape, mapped on a FEM mesh, as input for the recently introduced local-ization landscape theory 18,34to obtain an effective potential and the localized states on this landscape. All this can then serve,for example, as a starting point for transport calculations in future studies. A. Tight-binding model and local band edge calculations Atomistic theoretical studies have already shown that a single In–N–In chain, embedded in GaN, is sufficient to localize hole wave functions in an (In,Ga)N alloy. 7,35This indicates that in order to capture the localization effects in III-N systems accurately,the theoretical model ideally operates on an atomistic level. Whiledensity functional theory (DFT) provides such an atomistic and very accurate description, the computational demand of standard DFT approaches allows only to study systems of a few thousandatoms. Given that for QW or multi-QW systems not only the activeQW region but also the barrier material needs to be included, plussufficiently large in-plane dimensions, the relevant part of the system under consideration easily exceeds 10 000 atoms. Thus, to capture effects such as random alloy fluctuations on a microscopic level, we apply a nearest neighbor sp 3TB. The model is described in detail in previous works36,37and we only briefly summarize its main ingredients. TB parameters are obtained by fitting the TB band structures to III-N hybrid-functional DFT bandstructures as discussed in Refs. 38–40. As shown in the above refer- ences, the model has also been benchmarked for alloyed systems,by comparing, for example, the bandgap bowing of InGaN or InAlN systems with DFT and/or experimental data. In the case of an alloy, care must be taken when treating the TB matrix elements.Since for the cation sites (Ga, In) the nearest neighbors are alwaysnitrogen atoms, there is no ambiguity in assigning the TB on-siteand nearest neighbor matrix elements. This classification is more difficult for nitrogen atoms. In this case, the nearest neighbor envi- ronment is a combination of In and Ga atoms. Here, we apply thewidely used approach of using weighted averages for the on-siteenergies according to the number of In and Ga atoms. 41,42 Furthermore, the model accounts for local strain and polarization fields obtained from a valence force field (VFF)43model and a local polarization theory, respectively.38This model has been extensively tested and compared with experimental and DFT data for bothbulk and QW systems. 38,40As outlined above, the aim of our study is to establish a connection between the atomistic TB model and a continuum-based approach. The idea is to extract an energy FIG. 1. Schematic workflow from of our theoretical framework to connect an atomistic tight-binding model to a continuum-based Schrödinger solver (hereSPHInX). The connection between the atomistic and continuum-based grid is achieved by the finite element method, generating an atomistic finite element mesh that has as many nodes as atomic sites and which is here interpolated onan equidistant tensor-poduct mesh compatible with SPhinX.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 073104 (2021); doi: 10.1063/5.0031514 129, 073104-3 Published under license by AIP Publishing.landscape from TB that can be used as input for the continuum- based calculations. We stress that previous studies that establish and use modified single-band models for (In,Ga)N-based QWsdefine locally compositionally averaged material parameters such asband offsets or piezoelectric coefficients. There is obviously nogurantee that this represents a good and valid approximation. Our approach is different in the sense that we use a microscopic description of the energy landscape the carriers are “seeing ”in a disordered alloy. As we have already shown previously, this results,for example, in a bowing of valence and conduction bad edges dueto strain and built-in field fluctuations. Such bowing is usually not accounted for in modified continuum-based approximations. Thus, we go beyond the approximations made in “standard ”modified continuum models used in the literature in terms of obtaining amore refined description of the local energy landscape. To do so, our starting point is to derive a “local ”TB Hamiltonian, ^H local, that can be diagonalized at each lattice site. In a first step, a supercell of, for instance, an (In,Ga)N/GaN QW isgenerated, which may contain the relaxed atomic positions. Basedon this supercell, the corresponding TB Hamiltonian is generated.Diagonalizing this full TB Hamiltonian would give the single- particle states and energies. However, to obtain the local band edges and, thus, an energy landscape V(r), at each lattice site a local TB Hamiltonian, ^H local, which in the case of our nearest neighbor sp3TB model is a 8 /C28 matrix, is constructed from the full TB Hamiltonian. ^Hlocalnow describes the local environment of the atom at a given site and takes the form ^Hlocal¼E0H1/C04 int H1/C04y int E1/C04 ! : (2) Here, E0is a 4 /C24 matrix describing the on-site energies of s, px,py, and pzorbitals of the lattice site at which the energy land- scape will be calculated. The 4 /C24 matrix H1/C04 intdescribes the inter- actions (hopping matrix elements) between orbitals at the lattice site under consideration and the orbitals at its four nearest neigh- bors. Finally, the 4 /C24 matrix E1/C04contains the average on-site energies for sandp-orbitals of the nearest neighbors of the atom at which the local band edge is calculated. Given that these matrix ele-ments of ^H localare directly taken from the full TB Hamiltonian, the effects of (local) strain and built-in polarization fields are a priori included in the local band edges. Once this energy landscape isobtained, it is transferred to a regular wurtzite grid and passed toa continuum-based solver to obtain the electronic structure orperform transport calculations; it is not necessary to calculate strain and built-in fields in the continuum-based model. In this manner and as already stressed above, we circumvent the demandfor any averaging to find the “local ”In composition and then to calculate averages of elastic or piezoelectric constants to obtainthese fields. Again, any bowing of valence or conduction band edges seen in atomistic calculations of, e.g., III-N alloys, 38are directly encoded in the local TB band edges and should be trans-ferred to the continuum model. We note three important aspects ofthe procedure. First, given that we are using a nearest neighbor TB model, the interactions in the above local TB Hamiltonian are restricted to nearest neighbors to correctly reproduce the local bandedges of, for instance, an unstrained bulk system; the full TB Hamiltonian includes only interaction matrix elements between nearest neighbor anions and cations but not second-nearest neigh-bor cation –cation or anion –anion hopping matrix elements. If a second-nearest neighbor TB model is used, then interactions up tosecond-nearest neighbor would have to be included in the local Hamiltonian to obtain a correct description of even the unstrained bulk band edges. Second, the approach can be used for any straindependent TB Hamiltonian, even if the atoms are displaced fromthe ideal bulk positions, given that the local band edges are deter-mined from the matrix elements of the full TB Hamiltonian which depend on the relative position of the atoms (and the correspond- ing strain corrections). The only prerequisite is that local band edgeenergies are placed on a grid that is appropriate for the desiredcontinuum-based modeling. Finally, we note that there are differentways of calculating the local band edges. In the following, we have evaluated the local band edges at both anion and cation sites to achieve a higher resolution of the landscape. However, alterna-tive approaches could calculate the band edge energy only at eitherthe anion or cation sites. Future studies may now look at thesealternative schemes, while in the following we use the full anion cation structure. An example of the local conduction band edge (CBE) and valence band edge (VBE) calculated from the TB method via thelocal TB Hamiltonian for a simple VCA-type system in the absence of strain and built-in fields is shown in Fig. 2 (open circle, E CBE =VBE TB ) for a linescan along the wurtzite c-axis. Here, we use a 2.6 nm wide In0:15Ga0:85N QW; the cell size is approximately 10 /C29/C215 nm3. We note that the band edges obtained reveal a slightly softened QWinterface, which arises from the fact that at the interface between FIG. 2. Linescan of the potential energy profile in a (In,Ga)N/GaN quantum well with 15% In along the wurtzite c-axis. The system is treated within a virtual crystal approximation (VCA) without strain and built-in fields. The TB data is given by the black open circles, while the FEM mesh data using the TB data as input is shown by the red dashed line. (a) Conduction band edge (CBE); (b)Valence band edge (VBE).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 073104 (2021); doi: 10.1063/5.0031514 129, 073104-4 Published under license by AIP Publishing.GaN and (In,Ga)N N-atoms are exposed to varying numbers of Ga and virtual InGa atoms. Regarding the computational costs of this approach, we note that for determining the local band edges, the full TB Hamiltonianonly needs to be stored in the memory but does notneed to be diagonalized. For the local band edge calculations, only 8 /C28 matrices need to be diagonalized, which may even be distributed between different cores if needed in future studies. Finally, our VFFmodel is implemented in LAMMPS which is designed to run on alarge number of CPUs. 44We have recently relaxed (In,Ga)N/GaN QD systems with .1 000 000 atoms, using the same the VFF model applied here.45In the literature, VFF models underlying QD calculations have efficiently relaxed structures with .50 000 000 atoms.46Thus, overall when optimizing our approach further, large scale calculations with several million atoms will be within reach ofthis method in a numerically efficient manner. B. Connecting atomistic and continuum-based grid: Atomistic FEM mesh generation Having described the TB model and how the local band edges can be obtained from such a theory, we address in this section how this information is transferred to a finite element method (FEM) mesh, which can then be used for continuum-based calculations.Given that the TB energy landscape is known at each lattice site inthe TB supercell, we generate a so-called atomistic FEM mesh that has as many nodes as atoms in the system; the atomistic FEM mesh is generated using WIAS-pdelib and TetGen, 47see Figs. 3(a) and 3(b).Figure 4 depicts an example mesh for a TB model when applying a VCA. In this test system, the structure has126 780 atoms and the corresponding TetGen generated mesh has126 780 nodes and 891 188 tetrahedra. We note that the interface between the (In,Ga)N QW and GaN barrier region is again not sharp. As discussed already above, this is attributed to the atomisticeffect that the local environment of a N-atom at the well barrierinterface “sees”a varying number of Ga and in this case virtual InGa atoms, which now is also transferred into our atomistic FEM mesh and will also come into play when dealing with random alloys. The established atomistic FEM mesh can now be used to generate input for continuum-based models, including single- ormulti-band k/C1papproaches 25,30,33as well as a localization landscape theory description.18,48Using the WIAS-pdelib software again, the data from the atomistic FEM mesh are transferred to a3D equidistant uniform tensor-product point-set that is compatiblewith the plane waves based code SPHInX by interpolation, seeFig. 3(c) . Generally, we define a point p 0and ( n1,n2,n3) subdivi- sions in all the three directions ( x,y,z) to overlay n1/C2n2/C2n3 points on a part of the FEM mesh. Usually, we want not to cover the entire FEM mesh whereas the points outside will be ignored bySPHInX. To transfer the data, we generate a tensor mesh from then 1/C2n2/C2n3points with same number of points and then using linear mesh to mesh interpolation. The tensor mesh is necessary to use the point-neighborhood information for local efficient tensor-point to FEM-cell searching. On simpler terms, we overlay a subdo-main of the FEM mesh with points and linear interpolate the data from the FEM mesh to this points for SPHInX. The points are arranged in cuboid with axes ( x,y,z) with ( n 1,n2,n3) subdivisions. FIG. 3. Transformation of the tight-binding lattice to a FEM mesh and ultimately to a SPHInX compatible input point-set. We start with a point-set (a) defined by the atomistic lattice positions as given by tight-binding. Using T etGen a tetrahe-dral mesh (b) is generated, which has exactly the same number of nodes asatoms in (a). In doing so, the tight-binding input is exactly represented on the nodes of the atomistic FEM mesh. The data from the atomistic FEM mesh are then transferred to a 3D equidistant uniform tensor-product point-set (c) compat-ible with the plane waves based code SPHInX by interpolation.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 073104 (2021); doi: 10.1063/5.0031514 129, 073104-5 Published under license by AIP Publishing.To demonstrate that the underlying TB data is transferred success- fully into the FEM mesh and finally the mesh used for the continuum-based electron structure calculations, Fig. 2 shows (a) the CBE profile and (b) the VBE profile for a linescan along thec-axis of the 2.6 nm wide In 0:15Ga0:85N QW (already mentioned in Sec. II A). The TB band profile is given by the black solid line while the red dashed line is the mesh generated for the SPHInX calculations. As expected and required, the SPHInX compatiblemesh reproduces the TB landscape. Having established a connec-tion between the TB energy landscape and the mesh used in ourEMA calculations, we discuss how these data are now processed in the continuum picture within the SPHInX library. C. Continuum-based model The backbone of our continuum-based EMA calculations is the plane wave based software library SPHInX. 49,50This highly flex- ible package facilitates, in general, the three-dimensional calculationof strain and polarization fields as well as the electronic structure ofsemiconductor nanostructures. Here, arbitrary stiffness and piezo-electric tensors as well as multi-band k/C1pHamiltonians can be defined without any recoding. 32All these quantities can be defined in an input file in a human-readable meta-language. Given that the confining TB energy landscape contained is already known on a continuum-based grid, only the kinetic part ofthe Hamiltonian has to be provided. In what follows, we apply a single-band EMA for several reasons. First, it is a simple approach with just one adjustable parameter (the effective mass) that willallow a systematic method development study. Multi-band models,while we discuss and comment throughout the paper on their potential benefits, which would account for effects such as conduc- tion band valence band coupling or valence band mixing, arebeyond the scope of the present work. A multi-band study would increase number of free and adjustable parameters significantly (A i—valence band parameters; Kane parameters; etc.),51,52not to mention their composition dependence or the still large degree ofuncertainty in these parameters in the literature; 51all this would further complicate the comparison between continuum and atomis- tic results. Additionally, one needs to bear in mind that single-band effective mass models are widely applied in the literature whendescribing carrier localization effects in InGaN QWs. Thus, focus-ing on single-band effective mass models in comparison with anatomistic model allows us to flesh out potential problems with a one-band model in general. However, we also stress again that given the flexibility of the framework, follow-up studies can be easily extended to six- oreight-band k/C1pmodels, 25,53which may be targeted in future studies. We remind again that the TB energy landscape already contains (local) strain and built-in fields so that these quantities do not have to be calculated within the continuum model. However,the computational burden will increase significantly if the atomisticFEM mesh is used in combination with a multi-band k/C1pmodel in comparison to a single-band EMA. In fact for a multi-band k/C1p model, the computational burden may be similar to the TB model- ing on the active region of a full device structure. But, even the multi-band k/C1pmodel has a distinct advantage over the atomistic TB model, namely, that in such a continuum-based model the meshing in different spatial regions can be adjusted. This is in con- trast to the TB framework where one is bound to the atomistic res-olution. Thus, in the TB benchmarked continuum-based model,one may use the atomistic resolution in the active region but acoarser grained mesh in, for instance, the n- and p-doped regions of a device. We have already presented initial results for drift- diffusion calculations of an InGaN QW-based device. 54 In terms of the material parameters and the fact that we are using a single-band EMA, only the electron and hole effectivemasses have to be defined. In the following, we will use a constant effective mass throughout the whole simulation cell that will be adjusted by the average alloy content in the well. How these massesand their composition dependence are determined will be discussedin more detail below. Overall, we highlight again that the proposed framework is different from previous studies in the literature, given that we are directly transferring an TB deri ved energy landscape into the continuum-based solver. Thus, in our case and given that we areusing a single-band EMA, basically the only free input parame- ters in the continuum-based model are the effective masses. Furthermore, this approach now allows for direct comparison ofthe results of EMA and the TB model on the same alloy configu-ration (VCA and microscopic random alloy); such a comparisonwill be discussed in Secs. IIIandIV. III. RESULTS In this section, we present the findings on the electronic struc- ture of In xGa1/C0xN single QWs obtained within TB and continuum- based calculations. To study the impact of the In content on the results, values of 5%, 10%, 15%, and 25% In are considered in the following. Before turning to the random alloy analysis, we start FIG. 4. Atomistic finite element mesh using the tight-binding energy landscape for a c-plane (In,Ga)N/GaN quantum well with 15% In in the well in a virtual crystal approximation; the valence band edge profile is given in gray. Material domains and interface regions are indicated in the lower part of the figure.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 073104 (2021); doi: 10.1063/5.0031514 129, 073104-6 Published under license by AIP Publishing.with VCA calculations in Sec. III A . Given that the continuum- based calculations in this first, single-band approach contain only two free parameters, namely, the effective electron and hole masses,we use the VCA model system to analyze and calibrate the single-band EMA in general. The calibrated model is then used for ana-lyzing the impact of random alloy fluctuations on the electronic structure of (In,Ga)N QWs, and the results of these studies are pre- sented in Sec. III B. A. VCA comparison In this section, we present the outcome of our VCA studies. As already mentioned above, the aim is to calibrate the EMA against the TB data. While a very good agreement between TB and effective mass model is in general expected, a calibration step isessential for several reasons. First, without establishing very goodagreement between the EMA and TB model for a simple VCA case, it will not be clear if any potential differences between the two methods (in the random alloy case) stem entirely from the alloyfluctuations or are “pre-existing differences ”which may originate from the difference in the predicted/used effective masses inthe two models. The VCA comparison helps to eliminate such “pre-existing differences. ”Second, given that bulk effective masses are in general input parameters in any continuum-based model, weuse calculations in the absence of strain and built-in fields to poten-tially adjust the effective masses employed in our EMA model toreproduce the TB data. Performing such an analysis as a function of the In content xin the well allows us to establish a composition- dependent effective mass, which can then be used in calculationsaccounting for random alloy fluctuations. Since we are interested inestablishing the general framework, a position independent effective mass is applied, meaning that the effective mass in the well and in the barrier are identical. Here, several refinements are possible, e.g., having a position dependent effective mass, performing calculationsin the presence of strain and built-in field so that the effective masscontains corrections arising from these effects. However, the latter are usually not taken into account in standard approaches dealing with (In,Ga)N/GaN QWs in a continuum-based framework. It isimportant to note that above ansatz of a strain independenteffective mass is similar to previous works in the literature. 55,56 However, it differs from those studies by how the local band edges are treated. For instance, in the advanced continuum model of Ref.57, the effective masses in an EMA were also treated as strain independent. However, to achieve an excellent agreement betweenan EMA and an atomistic TB, nonlinear strain corrections wereincluded in the EMA. It is important to note that EMA and atom- istic TB model were treated independently in Ref. 57, which means that the strain effects in the local band edges are calculated sepa-rately in the continuum model and in the atomistic model. In ourframework, this is different since it is not necessary to calculate thestrain effects in the EMA model separately, they are build into the local band edges obtained from TB directly. This highlights again the benefit of our presented framework in comparison to previousliterature work. Therefore, our starting point for obtaining theeffective masses of In xGa1/C0xN as a function of x, is a linear, com- position weighted interpolation of the electron and hole masses for wurtzite InN and GaN via me,h InxGa(1/C0x)N¼xme,h InNþ(1/C0x)me,h GaN.Here, meare the electron and mhare the hole masses, taken from Ref.52, using equations in Ref. 58for determining the hole masses. For all calculations, we use as a test system a 2.6 nm wideIn xGa1/C0xN/GaN single QW. The simulation cell is approximately 10/C29/C215 nm3. The cell contains 126 780 atoms. The tensor- product mesh underlying all SPHInX-based EMA calculations uses a grid with a uniform step size of 0.2 nm, resulting in 50 /C244/C277 grid points. Figure 5(a) depicts the energies of the electron ground (Ee,α 0) and first two excited states (Ee,α 1,2) as a function of the In content x in the well obtained within EMA ( α¼EMA) and TB ( α¼TBM). AsFig. 5(a) shows, already when using the effective mass parame- ters from Ref. 52and the linear, composition weighted interpola- tion scheme for the effective mass we find a very good agreement FIG. 5. (a) Electron and (b) hole single-particle ground and first two excited states for a 2.6 nm wide In xGa1/C0xN/GaN quantum well in virtual crystal approxi- mation and in the absence of strain and built-in fields. The results are shown asa function of the In content x.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 073104 (2021); doi: 10.1063/5.0031514 129, 073104-7 Published under license by AIP Publishing.between EMA and the TB model. This is not only true for the ground state energies but also for the excited states. Turning to the hole energies Eh,α i(where i=0 denotes the ground state whereas i=1,2 are the first, second excited state), depicted in Fig. 5(b) , we find also a very good agreement between EMA (Eh,EMA i ) and TB (Eh,TBMi) results. We note that when neglect- ing spin –orbit coupling effects, the hole ground state is twofold degenerate in the TB model. However, given its single-band charac-ter, this effect is not captured in the EMA and would, therefore,require a multi-band model. Given the flexibility of our underlyingSPHInX framework such an extension on the continuum-based modeling can be implemented in a straightforward way. However, for the current work, we are mainly interested in the impact ofrandom alloy fluctuations on the electronic structure of (In,Ga)NQWs, for which also in the literature single-band approaches havebeen used, and we do not apply a two- or six-band model here. Given the good agreement between EMA and TB for electron and hole ground state energies also the ground state transitionenergies, ΔE α(x)¼Ee,α 0(x)/C0Eh,α 0(x), are in very good agreement over the full composition range considered. The calculated valuesdiffer by no more than 2 meV. Equipped with this calibrated EMA model, we present the results of calculations which account for random alloy fluctuations in Sec. III B. B. Random alloy case: Single-particle energies In this section, we compare the results from the calibrated EMA model with TB data for c-plane (In,Ga)N/GaN QWs in which random alloy fluctuations are considered in the well region. Again these calculations have been performed as a function of the In content in the well; we consider here the same In content rangeas in the VCA calculations, namely, 5%, 10%, 15%, and 25% anduse the (In,Ga)N/GaN QW structures studied in Ref. 9. For these systems, the TB model has shown to give good agreement with experimental data in terms of photoluminescence peak energies and full width at half maximum values. 35The simulation cell is approximately 10 /C29/C210 nm3and contains 819 20 atoms. For each In content, ten different random alloy configurations have been generated, allowing us to study the impact of the alloy micro- structure on the results. To avoid any preferential orientation orcorrelation of In atoms, we proceed as following. In the first step,we attribute to each cation site a random number. Then, in asecond step, the number of cation sites, n, that have to be occupied by In atoms on the grid to reflect the desired In content xis deter- mined. In the final step, the nlowest random numbers at the cation sites of the mesh are selected as In atoms while the remain-ing cations sites are Ga atoms. Using this procedure, in the follow-ing, we look at results averaged over the ten different microscopic configurations per In content. In the continuum-based calculations, we keep the grid spacing consistent with our calibrated VCAmodel, thus, the underlying SPHInX-based EMA calculations use atensor-product mesh with a uniform step size of 0.2 nm, resultingin 50 /C244/C250 grid points. 1. Electron single-particle energies Figure 6 shows the electron energies of the ground and first nine excited states for (a) 5%, (b) 10%, (c) 15%, and (d) 25% In. FIG. 6. Energies of the energetically lowest ten electron states in c-plane (In, Ga)N/GaN quantum wells with In contents of (a) 5%, (b) 10%, (c) 15%, and (d)25%. The results are averaged over ten different random alloy configurations. The data are shown for the TB model (solid black line), the single-band EMA without shift (solid blue line) and with shift (solid green line) of the band edges;more details are given in the main text. The ground states are marked as E e,0 TB for TB, Ee,0 EMA,NSfor EMA without shift, and Ee,0 EMA,Sfor EMA with shift.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 073104 (2021); doi: 10.1063/5.0031514 129, 073104-8 Published under license by AIP Publishing.The data are always averaged over the ten microscopic configura- tions considered. The TB data are given in black together with two sets of EMA results (green and blue), which will be explainedbelow. We first focus on the EMA results given in blue anddenoted by “No shift ”, abbreviated as NS in all the panels of Fig. 6 . This calculation corresponds to the situation where the TB energy landscape is directly used in the EMA calculations and the electron and later hole effective mass for the corresponding In content arechosen based on the VCA results discussed above. From Fig. 6 , several important aspects can be inferred. The most striking differ-ence between TB and EMA results (No Shift) is that the ground state energies E e,0 TBand Ee,0 EMA,NS differ significantly; this difference increases with increasing In content. However, while there is alarger difference between ground state energies, the energetic sepa-ration, ΔE GS, EX α ¼Ee,1 α/C0Ee,0 α, of the ground state Ee,0αand the first excited state between the two models differs, independent of the In content, by less then 4 meV. Note, in the expression ΔEGS, EX α , the superscript “GS”refers to the ground state, whereas the superscript “EX”refers to the excited states under consideration. Thus, while one may be tempted to increase the effective electron mass to obtaina better agreement between TB and EMA (No shift) ground state energies, such an increase in the mass will affect (reduce) the ener- getic separation between excited states in the EMA. Additionally,given that the electron energies in the case of the EMA are shifted tohigher energies when compared to the TB results, one may expect an earlier onset of carriers becoming more delocalized and, thus, may alter the description of carrier localization effects due to randomalloy fluctuations. All these (energetic separation of states; earlieronset of delocalization) are, however, important when studyingquantities such as the radiative recombination rate with increasing temperature or carrier density in c-plane (In,Ga)N/GaN systems, where the density of excited states plays an important role. 24,59Based on all this, and even though the EMA labeled “No Shift ”operates on the same energy landscape as the TB model, it gives energy eigenval-ues that on an absolute scale are very different from the TB model. The above seen deviation between TB and EMA exposes shortcomings of the single-band continuum model. The agreementbetween the continuum and the atomistic model may be improvedby moving to a multi-band band approach on the continuummodel side since aspects such as band nonparabolities would be captured. However, our aim in the current study is to establish (i) a general framework that allows us to bridge the gap betweencontinuum-based calculations and atomistic models and (ii) anEMA model that operates on the energy landscape obtained from the TB model with a minimum number of free and adjustable parameters while at the same time facilitating a good approxima-tion of the TB results. It has already been highlighted by Auf derMaur et al. 13that quantities such as the bandgap evolution in a VCA-type approximation may give a very different result as com- pared to an atomistic calculation that includes alloy fluctuations. In such a case, the bandgap bowing parameter may be adjusted(increased) to correct this. Here, we follow a similar approach toachieve a simple effective mass model that provides a good descrip-tion of the TB results and adjust the band offset in the QW region by a rigid, constant energy shift ΔE S CBO (conduction band) and ΔES VBO(valence band); all calculations have been repeated with the adjusted band offsets for electrons and holes. The results are showninFig. 6 in green and are labeled by “EMA S.”The applied ΔES CBO (conduction band) and ΔES VBO (valence band) shifts in the EMA model to the TB energy landscape in the QW region are summarizedinFig. 7 along with a quadratic fit of the form ΔE S,f α(x)¼axþbx2; the extracted coefficients aand bare also given in Fig. 7 ; bowing parameters for the band offsets are not unusual in III-N-based mate- rials as shown in the literature.38,40,60H e r e ,w eh a v eo b t a i n e d ΔES CBO from the average electron ground energy difference in the TB and the EMA NS(No Shift) models, cf. Fig. 6 . Applying this rigid shift to the band edges within the well and repeating the calculations resultsin a much better agreement between EMA S(green solid lines) and TB ground and excited state energies. For instance, in the 25% In content case, cf. Fig. 6(d) , the difference between the TB ground state energy, Ee,0 TB, and the ground state energy in EMA without applying the shift, Ee,0 EMA,NS , is 247 meV. Looking at the ground state energy when ΔES CBOis applied, Ee,0 EMA,S , we find a difference of only 69 meV with respect to the TB model. Also, the energetic separation between the ground and the first excited state is similar. For the TB model,we obtain ΔE GS,EX TB¼72 meV and for ΔEGS,EX EMA,S ¼68 meV. We note that especially at higher In contents ( .15% In), the deviations between the two models become larger when compared to the lower contents, even with the shift applied. But, as we will discuss below, on the energy scale of the transition energies, these deviations are ofsecondary importance. The additional benefit of applying ΔE S CBOis that the energy range over which the first ten electron states found is very similar between TB and EMA S. Thus, at least in terms of the energy eigen- values, the modified EMA with an energetic correction to the CBedge presents an attractive model to describe the electronic struc-ture of (In,Ga)N/GaN with random alloy fluctuations in the well to achieve a reasonable approximation of the atomistic data on average without increasing the numerical effort of the model. FIG. 7. Band offset correction for (a) conduction and (b) valence band edge as a function of the In content x. The data are fitted with the equation ΔES,f α(x)¼axþbx2. The obtained aand bvalues are given in (a) and (b) for the conduction [ ΔES,f CBO(x)] and valence bands [ ΔES,f VBO(x)], respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 073104 (2021); doi: 10.1063/5.0031514 129, 073104-9 Published under license by AIP Publishing.Additionally, having established the bowing parameters for the energy offset as a function of the alloy content allows us to apply the model in future studies with different In contents or largersystems without the need to perform a full TB calculation.As already discussed in Sec. II A, only the energy landscape needs to be extracted from the full TB Hamiltonian, which requires only storing of the full Hamiltonian but not diagonalizing it; only 8 /C28 sub-matrices are required to be diagonalized. 2. Hole single-particle energies Next, we turn our attention to the energies of the first ten hole states. The results from the three different models, discussed above for electrons, are shown in Fig. 8 for hole state energies in c-plane (In,Ga)N QWs with (a) 5%, (b) 10%, (c) 15%, and (d) 25% Incontent. The data displayed in the figure are again averaged overthe ten different microscopic configurations. The solid horizontalblack lines denote the TB results, while the blue and green lines give the results from the modified EMA models without and with applying a shift ΔE S VBOto the VBE in the well. Figure 8 reveals that without shifting the VBE in the well region, the EMA NS(blue) noticeable underestimates the ground state energy; in general, thisdifference increases with increasing In content [cf. Figs. 8(a) –8(d)]. Furthermore, without shifting the VBE in the EMA, the energetic separation between the ground and first excited state is, in general,smaller when compared to the TB results. For instance, in the10% In case, cf. Fig. 8(b) , in TB, this energetic separation is ΔE h,GS,EX TB ¼Eh,0 TB/C0Eh,1 TB/C2522 meV, while in the modified EMA NS one finds ΔEh,GS,EX EMA,NS ¼Eh,0 EMA,NS /C0Eh,1 EMA,NS /C259 meV. This could indicate that carrier localization effects, especially for states close to the VBE, are not well described in EMA NS. Thus, the wave func- tions calculated using EMA NSmay exhibit a more delocalized nature when compared to the TB wave functions. We will come back to this question further below when discussing the inverseparticipation ratio (IPR) values of the different states. All this again highlights the shortcomings of the single-band EMA which may be cured in part by applying a multi-band model. However, instead of targeting the problem with the computation-ally heavier multi-band model, we follow the procedure applied forthe electrons and construct a modified EMA, EMA S, which includes a shift of the VBE in the QW region. As one can infer from Fig. 8 , this model gives energies that are in reasonable agreement with the TB energies. The respective shifts are displayedinFig. 7(b) ; the data are fitted by ΔE S,f VB¼axþbx2, where and a andbare given in the figure. With these shifts applied, differences in ground state energies between TB and EMA Sare below 10 meV in the 5% [cf. Fig. 8(a) ], 10% [cf. Fig. 8(b) ], and 25% [cf. Fig. 8(d) ] In cases. Only for 15% we find a slightly larger difference betweenTB and EMA Sof approximately 19 meV [cf. Fig. 8(c) ]. However, this difference is significantly reduced compared to the 68 meVdifference between TB and EMA NSand further refinements can be made by adjusting the VBE shift further. However, to demonstrate the general strategy of our modified EMA, the achieved agreementbetween TB and EMA Sis sufficient for our purpose. But, we note also that while the agreement between the ground state energies is improved, the energetic separation between excited states may not be improved in general. Looking again at the 10% In case, Fig. 8(b) , FIG. 8. Energies of the first ten hole states closest to the valence band edge in (In,Ga)N/GaN c-plane quantum wells with In contents of (a) 5% , (b) 10%, (c) 15%, and (d) 25% In. The data are averaged over ten different random alloyconfigurations. The results are shown for the TB model (solid black line), thesingle-band EMA without (solid blue line) and with shift (solid green line) of the band edges; more details are given in the main text. The ground states are marked as E e,0 TBfor TB, Ee,0 EMA,NSfor EMA without shift, and Ee,0 EMA,Sfor EMA with shift.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 073104 (2021); doi: 10.1063/5.0031514 129, 073104-10 Published under license by AIP Publishing.in TB we find ΔEh,GS,EX TB /C2522 meV, while in the modified EMA S, the separation is ΔEh,GS,EX EMA,S ¼Eh,0 EMA,S /C0Eh,1 EMA,S /C2510 meV. Finally, we briefly discuss the average ground state transition energy, ΔE0 α¼Ee,0 α/C0Eh,0 α, which is displayed in Fig. 9 as a func- tion of the In content for the three different methods(α¼TB, EMA S,E M A NS). Overall, the graph shows the expected behavior that with increasing In content xthe transition energy shifts to lower energies, given the increase in built-in field and reduction in the bandgap of an (In,Ga)N alloy in general withincreasing In content. Also, as expected from our discussionabove on the electron and hole ground state energies, comparedto the TB transition energy (black squares), the EMA without the energetic shift to the band edges, EMA NS(blue circles), signifi- cantly overestimates the bandgap energy; this difference is morepronounced for higher In contents. On the other hand, the EMAmodel that includes the shift in the band edges, EMA S(green tri- angles), gives a very good description of the transition energy over the full composition range, inline with our analysis of the ground state energies above. Finally, we show the standard devia-tion ( σ α) of the distributions of the transition energies as a func- tion of the In content by color coded error bars in Fig. 9 . The error bar marked in black represents σTB, the green denote σEMA,S and the blue denote σEMA,NS .σαfor each method is calculated by usingthe following formula: σα¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP i(ΔE0 α,i/C0ΔE0 α)2 ns , (3) where ΔE0 α,iis the transition energy corresponding to each configu- ration for the different methods, ΔE0 αis the average ground state transition energy, and nis the total number of configurations (ten for each In content). In general, we find that the standard deviation ( σα) increases with the increase in In content. This is consistent for all the methods. For instance, σTBin the case for 5% and 15% In are 0.156 eV and 0.205 eV, respectively. Similarly,σ EMA,S for 5% and 15% In are 0.140 eV and 0.209 eV, respectively, whereas σEMA,NS for the same In content are 0.143 eV and 0.178 eV, respectively. Overall, the presented analysis exhibits shortcomings of the single-band EMA. However, instead of increasing the computationalload by moving to multi-band k/C1pmodels, we have proposed a simple modification of the EMA approach to achieve a good agree- ment with the atomistic TB results over the In composition range of 5%–25%. Furthermore, having established a composition-dependent band edge adjustment parameter for the EMA allows us now to usethis model in future calculations on the electronic and optical prop-erties or for transport studies of (In,Ga)N QWs, without having to perform a full TB calculation. Having discussed electron and hole energies, our analysis reveals that the developed modified continuum-based frameworkmay give a good approximation of the TB data. However, as already mentioned above, wave function localization effects may be different. This aspect is, for instance, important for the wave func-tion overlap, which impacts the (radiative) recombinationrates. 6,24,59In Sec. III C , we, therefore, focus our attention on wave function localization effects to further compare the outcome of modified EMA models with the TB data. C. Random alloy case: Inverse participation ratio (IPR) for electrons and holes In addition to comparing electron and hole energies, we also compare localization effects due to random alloy fluctuations. For the latter part, we employ the inverse participation ratio (IPR),29,61,62which provides a quantitative metric for this question. More details about the IPR value calculation, along with a detaileddiscussion why care must be taken when comparing IPR valuesobtained from atomistic and continuum-based models, are given in theAppendix . In this subsection, we present the normalized IPR values (gIPR) for the ground and excited states obtained from TB and EMA (with and without shift). The results are averaged over tendifferent alloy configurations for each In content. For normalizing the IPR values, we proceed as follows. The TB electron ground state IPR value of the 5% In system represents our reference point.Thus, all TB IPR values are normalized with respect to this IPRvalue. Furthermore, we assume that for the 5% In case, the aver- aged electron ground state of the EMA including the energy shift, EMA S, reflects the same carrier localization characteristics as the FIG. 9. Ground state transition energies in In xGa1/C0xN/GaN quantum wells as a function of the In content x. The black squares represent the tight-binding results ( ΔE0 TB), the green triangles give the results from the modified EMA with an energy shift applied ( ΔE0 EMA,S), while the blue triangles denote the EMA results without an energy shift of the valence band edge in the well ( ΔE0 EMA,NS). The data are averaged over ten different microscopic configurations per In content x. The standard deviation ( σ) of the distributions of the transition ener- gies for the three different methods are marked by the error bars colored inblack ( σ TB), green ( σEMA,S), and blue ( σEMA,NS).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 073104 (2021); doi: 10.1063/5.0031514 129, 073104-11 Published under license by AIP Publishing.averaged TB ground state for this In content. Thus also the IPR values of the EMA Sare normalized so that the average ground state IPR value of the 5% In system is also gIPR¼1. The obtained scaling factor to achieve this is applied to all EMA Sdata and also the results of the effective mass model without the band edge shift, EMA NS. If the localization features for the electron ground state are the same in EMA Sand EMA NS, EMA NSshould also give gIPR¼1 for the system with 5% In. A more detailed discussion of thisnormalization procedure is given in the Appendix . In doing so, normalized IPR values, gIPR, provide also a more intuitive represen- tation of the localization properties: if the gIPR value exceeds a value of 1 it is more strongly localized than the electron ground state at 5% In in the well; a value below 1 indicates that the statesare less localized when compared to the average electron groundstate at 5% In. The gIPR values for the first ten electron states are shown as a function of the state number in Fig. 10 for (a) 5%, (b) 10%, (c) 15%, and (d) 25% In. Before looking at the individual In contents, Fig. 10 clearly shows that with increasing In content the (normalized) groundstategIPR values predicted by all three methods increase. We attri- bute this to the fact that with increasing In content the piezoelectric field increases so that electron and hole wave functions localize at the opposing QW interfaces, in addition to localization effects dueto random alloy fluctuations. Turning now to the individual In contents and starting with the 5% In case, we find that the EMA model including the CBE shift (green triangles) both in terms of the magnitude of the gIPR values and its evolution with state number reflects well the TB data(black squares). Also the model without the CBE shift (bluesquares) gives a reasonable description of the average localization features of the TB model (black squares). However, with increasing In content, cf. Figs. 10(b) –10(d) deviations between TB and EMA models become more pronounced, especially for the energeticallylower lying states. For higher lying states (state number .5), espe- cially EMA S(including the CBE shift) describes these states very well; the EMA NS(no shift) provides also a reasonable description but always gives lower values. We attribute the latter to the fact thatthe first ten electron states obtained within EMA NS, as discussed in Sec.III B 1 , cover on an absolute scale also a very different energy range when compared to the TB energy values; this may also affect the results. However, overall our presented analysis shows that the use of a rigid CBE shift within the well not only improvesthe agreement in energy levels between TB and EMA but alsoimproves their average localization characteristics. Thus, the developed EMA should give on average a good approximation of the TB model. Having discussed the electron gIPR values above, we present these data now for the hole ground and excited states. We note thatwe are interested in studying the trends in localization characteristic by comparing the gIPR values for different In contents and between different models. Looking at quantities such as carrier localizationlengths for different In contents is beyond the scope of the presentstudy but has been recently analyzed in the literature. 35Figure 11 shows the average hole gIPR values for (a) 5%, (b) 10%, (c) 15%, and (d) 25% In in the well. The data are again averaged over the ten dif- ferent alloy configurations and are normalized with respect to theaverage electron ground state gIPR value for 5% In (see above). FIG. 10. Average normalized electron IPR values ( fIPR) for c-plane (In,Ga)N/ GaN quantum wells with varying In content: (a) 5% , b) 10%, (c) 15%, and (d) 25%. Black squares: tight-binding results; green triangles: single-band effective mass approximation in which the conduction band edge has been adjusted (seemain text for more details); and blue circles: single-band effective mass approxi-mation without the rigid conduction band edge shift. The data have been nor- malized to the average ground state electron IPR value of the 5% In case for tight-binding and shifted effective mass model, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 073104 (2021); doi: 10.1063/5.0031514 129, 073104-12 Published under license by AIP Publishing.Figure 11 reveals that the hole wave functions are far more strongly localized when compared to the electrons (see also figure insets and compare with Fig. 10 ). This finding is consistent with previous studies where charge densities of electron and hole wavefunctions have been inspected; 10,12,63this effect is also captured by modified EMA models. However, and independent of the In content, the gIPR values predicted by the EMA models are signif- icantly smaller when compared to the TB model, at least for thestates lying close to the VBE (energetically lowest lying hole states).For the ground states differences the EMA model (blue circles,green triangles) are smaller by a factor of order of 5 –10. For higher lying states, the differences are less pronounced and similar to the electrons, a good agreement between the three different methodsmay eventually be achieved. It should be noted that here higherlying states refer to the states that are located deeper in the valence“band ”and not the states that are significantly away from the Γ-point. Using the terminology of Ref. 7, by higher lying states we mean semi-localized states, where the impact of the alloy micro-structure on the wave function localization is reduced and theyapproach the charge density distribution of a “standard ”(no alloy fluctuations) QW. Several conclusions can now be drawn from this. First, using a modified continuum-based approach to analyze and explain lowtemperature experimental results may be difficult since in this casethe effects are dominated by states close to the VBE. These states may not be well captured in a modified single-band EMA. We note again that a better agreement may be expected when using a moresophisticated k/C1pmodel here. Second, while the modified EMA models describe the localization features of higher lying statesbetter, studying the evolution of radiative recombination or Auger effects as a function of the temperature may also be difficult since the density of localized states may not be well captured. However,when dealing with higher temperatures and/or high carrier densi-ties where now the physics are expected to be dominated by excitedelectron and hole states, the established modified continuum-based model should be in good agreement, in terms of energy eigenvalues and localization features, with the atomistic model. So using thismodel, e.g., in transport calculations at room temperature or beyond,our EMA Sshould provide a good starting point without the need to perform these calculations in a fully atomistic framework. IV. CONCLUSIONS In this work, we have established a multiscale approach that allows us to connect atomistic tight-binding models with modifiedcontinuum-based methods. More specifically, we have developed an approach that extracts an energy landscape from an atomistic tight-binding model, including local variations in strain andbuilt-in fields due to alloy fluctuations, on which single-bandeffective mass calculations have been performed to obtain theelectronic structure of c-plane (In,Ga)N/GaN quantum well struc- tures with different In contents. We stress that our developed ansatz goes beyond the widely used literature approach where theconnection between atomistic and continuum-based theory isbasically restricted to obtaining local alloy contents from an atomistic lattice that are used in continuum-based calculations. This local alloy information is often used to interpolate and FIG. 11. Average normalized hole IPR values ( fIPR) for c-plane (In,Ga)N/GaN quantum wells with varying In content: (a) 5%, (b) 10%, (c) 15%, and (d) 25%. Black squares: tight-binding results; green triangles: single-band effective mass approximation in which the valence band edge has been adjusted (see maintext for more details); and blue circles: single-band effective mass approximationwithout the rigid valence band edge shift. The data have been normalized to the average electron ground state electron IPR values of the 5% In case for the tight-binding and shifted effective mass model (see Fig. 10 ).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 073104 (2021); doi: 10.1063/5.0031514 129, 073104-13 Published under license by AIP Publishing.define basically bulk parameters, such as band offsets, on small local length scales, which in itself raises the question of validity of such an approach. With the model proposed here, we go beyondthe local bulk parameter averaging by generating an energy land-scape directly from tight-binding where the band edges are intrin-sically modified by the presence of In and Ga atoms in the structure on a microscopic level. Furthermore, our framework is general and can be used for any tight-binding ( sp 3,sp3s*,sp3d5s*) and any continuum-based model (si ngle- or multi-band). Finally, given that we have established an atomistic finite element mesh, itcan also be extended beyond electronic structure calculations to transport simulations in the frame of drift-diffusion models. Given the direct connection between atomistic tight-binding and single-band effective mass approximation in the sense that thecalculations are performed on the same energy landscape, single-particle states and energies can directly be compared. We find that even when using such an energy landscape, significant differences in the single-particle energies are observed. However, our data alsoshow that good agreement between a modified single-band effectivemass approximation and tight-binding can be achieved for the firstten electron and hole state energies after applying a rigid shift to the band edges. This provides now a simple recipe for future studies, given that we have also determined the composition depen-dence of the rigid energy shift for electrons and holes. Overall, thisallows us to use this further modified continuum model to achieve a good description of the single-particle energies in (In,Ga)N QWs without performing full TB calculations. Turning to carrier localization effects, here studied via the inverse participation ratio, we find that even when shifting the bandedges in the effective mass model, the continuum-based model underestimates the effects observed in the atomistic approach, espe- cially for higher In contents ( .15%); this effect is particularly pro- nounced for hole states. Thus, in situations where states near theconduction and valence band edge become important, for instance,at low temperatures or low carrier densities, the modified effective mass model may significantly underestimate the impact of carrier localization effects. This means also when studying quantities such asradiative or Auger recombination as a function of temperature orcarrier density, care must be taken when drawing conclusions from amodified continuum-based approach. However, in the case where energetically higher lying states become important, the here estab- lished continuum-based model can give a very good approximationof the atomistic results. We also expect that a better agreement canbe achieved by employing six- or eight-band k/C1pmodels rather than the single-band EMA used in the present work. Overall, the established model now presents an ideal starting point for furthercalculations on optical and transport properties of (In,Ga)N/GaNquantum well systems. ACKNOWLEDGMENTS The authors thank J. Fuhrmann (WIAS) for fruitful discussions. This work rece ived funding by the Deutsche Forschungsgemeinschaft (DFG) under Germany ’s Excellence Strategy EXC2046: MATH+, project AA2-5 (O.M.), Sustainable Energy Authority of Ireland and Science Foundation Ireland (Nos. 17/CDA/4789 and 12/RC/2276 P2).APPENDIX: INVERSE PARTICIPATION RATIOS In this appendix, we discuss (i) general aspects of the IPR value calculation in atomistic and continuum-based models;(ii) why care must be taken when comparing IPR values from the different approaches, and (iii) further remarks on the normalization procedure outlined in the main text of the paper. We start our dis-cussion with general comments on the calculation of IPR values. Turning to our atomistic TB model, in general, the IPR value of a given TB wave function ψ jcan be calculated as follows:7 Rj¼P i(P αjaiαj2)2 (P iP αjaiαj2)2: (A1) Here, the sum over iruns over the Nlattice sites/grid points in the simulation cell and aiαare the expansion coefficients for a given basis state/orbital αof the wave function ψjat the lattice site/grid point i; in the sp3TB model, αdenotes s,px,py, and pzorbitals. For the continuum-based EMA description, a similar approach canbe used and the sum becomes an integral, R j¼Ðjψj(x)j4d3x [Ðjψj(x)j2d3x]2: (A2) Please note, in the single-band EMA one is only left with one basis state, namely, an jSi-like basis state. In general and based on the above expressions, the larger the IPR for a given state, the stronger the wave function localization effect. Using the TB model as anexample, in the extreme case of a wave function localized to asingle site/grid point, the IPR value based on Eq. (1)isR j¼1. On the other hand, if the wave function is completely delocalized, thus distributed over the Nlattice sites/grid points of the simula- tion cell, the IPR of such a state is Rj¼1=N; in the continuum case Rj¼1=V, with Vbeing the volume of the system. In the main text, we have studied the localization effects of electron and hole wave functions both in TB and the modified continuum-based models by means of IPR values. Overall, and asmentioned above that care must be taken when comparing theatomistic and continuum-based data for several reasons. First, thenumber of grid points/lattice sites differs between the Sphinx mesh and the atomistic model. Furthermore, as already discussed above, in the continuum-based models, a complete delocalized state for agiven supercell would result in an IPR value of R EMA¼1=V,w h e r e Vis the volume of the simulation cell. In the TB model, a complete delocalized state would have an IPR value of RTB¼1=N. Therefore, when comparing the lattice and continuum cases, the quantity we are calculating is different; thus, a “normalization procedure ”needs to be established to connect NandV. Furthermore, it is important to note that the anion –cation structure is not resolved in the continuum-based models since they provide only an envelope func- tion. We have found in previous work7,64that electron wave func- tions are mainly localized on the cation planes with a smallerprobability density on the anion planes. The opposite is observed forthe hole wave functions. Again, even for a completely delocalizedstate, this would have to be considered when comparing IPR values between TB and continuum-based models in general.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 073104 (2021); doi: 10.1063/5.0031514 129, 073104-14 Published under license by AIP Publishing.To account for these intrinsic differences while still being able to compare trends in the IPR values between the different methods, we use the average electron ground state IPR value (averaged overthe ten different microscopic configurations) for the well with 5%In for calibration. We assume that on average the TB model andmodified EMA S, including CBE and VBE shifts, give very similar wave functions/localization effects for the electron ground state. In doing so, we can extract a “scaling factor ”for the modified continuum-based models to account for the fundamental differ-ences in these methods. The extracted scaling factor is then usedfor all other states and all other In contents since the volume of the supercells are kept approximately constant. Overall, the assumption that wave function localization aspects of the average electronground state for the 5% In are very similar should be reasonable.This assumption is motivated by (i) (local) built-in field and straineffects are directly transferred into the continuum-based model, (ii) that previous calculations have already shown that the electron wave functions are less strongly affected by alloy fluctuations andthat they reflect to a good first approximation an envelope functioncharacter, 7and (iii) that we have chosen a low In content system as a reference point for which carrier localization effects in the electron wave function should even be less important. We note also that we have inspected charge densities of electron ground state wave func-tions for different configurations in the 5% In case and found a goodagreement between those predicted by the continuum-based and the TB model. 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5.0036975.pdf

AIP Conference Proceedings 2319 , 040003 (2021); https://doi.org/10.1063/5.0036975 2319 , 040003 © 2021 Author(s).Observations of magnetic deformation of magnetars Cite as: AIP Conference Proceedings 2319 , 040003 (2021); https://doi.org/10.1063/5.0036975 Published Online: 05 February 2021 Kazuo Makishima ARTICLES YOU MAY BE INTERESTED IN Quantum gravitational wave function for the interior of a black hole and the generalized uncertainty principle AIP Conference Proceedings 2319 , 040001 (2021); https://doi.org/10.1063/5.0038344 Jeans instability of strongly coupled radiative plasma AIP Conference Proceedings 2319 , 040004 (2021); https://doi.org/10.1063/5.0036997 Vacuum polarization and persistence on black hole horizon AIP Conference Proceedings 2319 , 040006 (2021); https://doi.org/10.1063/5.0037424Observations ofMagnetic Deformation ofMagnetars KazuoMakishima a) 1)KavliIPMU,TheUniversity ofTokyo,5-1-5Kashiwa-no-ha, Kashiwa,Chiba,Japan277-8583 2)Department ofPhysics,TheUniversity ofTokyo,7-3-1Hongo,Bunkyo-ku, Tokyo,Japan113-0033 3)High-Energy Astrophysics Laboratory, Nishina Center, RIKEN, 2-1-1 Hirosawa, Wako, Japan 351-0198 a)Corresponding author: maxima@phys.s.u-tokyo.ac.jp Abstract. Through observations with the Suzaku andNuSTAR X-ray observatories, at least three magnetars have been found to show periodic phase modulations in their hard X-ray pulses, with a period Twhich is ∼104times longer than their pulse periods. Among multiple observation of each object, Twas kept constant, but the modulatio n amplitude varied. This effect can be interpreted as a manifestation of free precession of neutron stars that are axially deformed by ∼10−4in terms of the moment of inertia. Further interpreting this deformation as caused by internal magnetic stress, these neutron stars are inferred to harborintense toroidal magnetic fields reaching ∼10 16G. These observational results thus provide valuable information on the magnetars’ toroidal fields which are dif ficult to observe externally, and give clues to the X-ray emission mechanisms of magnetars. INTRODUCTION Neutron stars (NSs) are the densest of all the celestial objects, with the strongest surface gravity (except black holes). They must hence be highly spherical, particularly when spinning slowly with a period of P/greaterorsimilar0.3 sec so that the centrifugal force is negligible ( /lessorsimilar10−6) compared to the gravity. As a result, any aspheric deformation has not been detected so far from NSs. However, their strong magnetic field (MF), B, could deform them by a ratio of the magnetic energy∼(4πR3 NS/3)(B2/8π)to the self-gravitating energy ∼GM2 NS/RNS,w h e r e Gis the constant of gravity, while MNS∼1.4M⊙andRNS∼10 km are the typical NS mass and radius, respect ively. A detailed calculation predicts [1] ε≡ΔI/I3∼10−4×/parenleftBig B/1016G/parenrightBig2 (1) where εis a quantity called asphericity, Iis the moment of inertia, ΔI≡I1−I3is its deviation from spherical symmetry, with I3the component along the symmetry axis ˆ x3andI1that in the orthogonal directions. The deformation will be oblate ( ε<0) if the MF is poloidal, and prolate ( ε>0) if the MF is toroidal [1]. Isolated NSs are classi fied into three categories [2]; radio pulsars, millisecond pulsars, and magnetars [3], which are characterized by dipole MFs of Bd∼1012G,<1010G, and 1014−15G, respectively. Their values of Bdare estimated assuming that they spin down via emission of magnetic dipole radiation [2]. Magnetars [3], some 30 are known in the Galaxy and the Magellanic Clouds, are young NSs with the highest MFs, exhibitin g rotation periods of 2 −11 sec and rather large spin-down rates. They emit solely in X-rays, with enigmatic two-component spectra [4, 5] which consist of a thermal soft X-ray component (SXC) dominant in energies below ∼10 keV , and a non-thermal hard X-ray component (HXC; [6]) extending to /greaterorsimilar100 keV . The origin of these components is yet to be clari fied. Even considering magnetars, the values of Bdwould be too low to produce a detectable effect in εvia eq.(1). However, magnetars can harbor even stronger toroidal MFs up to Bt∼1016G, which are hidden inside the stars and invisible to us from outside. These toroidal MFs may be produced by strong differential rotation during the final stage of their formation [7]. The presence of such strong toroidal MFs inside magnetars is also suggested indirectly by some observations [8]. If so, magnetars would be deformed to ε∼10−4, which we might be able to detect. While studying magnetars using the Suzaku X-ray observatory [9], we unexpectedly encountered an intriguing effect [10, 11, 12]. In three magnetars, the HXC pulses have been found to suffer slow periodic phase modulations, whereas those in the SXC are strictly periodic. The phenomenon can be interpreted as a result of free precession of NSs that are axially deformed by ε∼10−4. Then, eq.(1) implies that they indeed harbor B∼1016G. OBSERVATIONS Using Suzaku , we have been conducting extensive X-ray studies of magnetars [4, 5]. Havi ng a broadband capability realized jointly by the Hard X-ray Detector (HXD; working in 10 to /greaterorsimilar100 keV [8]) and the X-ray Imaging Spectrom- eter (XIS; 0.5–10 keV), Suzaku is particularly matched to the two-component spectra of magnetars. The discovery mentioned above first came from the representative magnetar 4U 0142+61, followed by two objects. Proceedings of the 14th Asia-Pacific Physics Conference AIP Conf. Proc. 2319, 040003-1–040003-6; https://doi.org/10.1063/5.0036975 Published by AIP Publishing. 978-0-7354-4063-0/$30.00040003-1The representative Anomalous X-ray Pulsar, 4U 0142+61 Suzaku results in 2007 and 2009 With a relatively constant X-ray intensity and a very clear two-component spectrum [5, 6, 13], 4U 0142+61 is one of the X-ray brightest magnetars. It is also a typical example of so-called Anomalous X-ray Pulsars (AXPs), which constitute one of the two major subclasses of magnetars. The source was observed with Suzaku in 2007 [13] for a gross pointing of ∼200 ksec and a net exposure of ∼100 ksec. The blue curves in Fig. 1 show X-ray periodograms obtained on this occasion, using the XIS (panel a) and HXD (panel b). Thus, the SXC and HXC pulses were detected by the XIS and HXD instruments, respectively, at a consistent period of P=8.68878±0.00005 sec [13], which agrees with the spin-down trend from previous observations of this ma gnetar over past decades. In these periodograms, the pulse significance is evaluated with so-called Z2 mstatistics [10, 11, 12]; we fold the data at a trial period, Fourier transform the folded pro file, sum up the Fourier power up to a speci fied harmonic m, and normalize the summed power to the total number of photons. If the data contains no intrinsic periodicity, the value of Z2 mobtained in this way should obey a chi-square distribution with 2 mdegrees of freedom. We utilize m=4 for these data. The object was observed again with Suzaku in 2009 [10], under a very similar condition to that in 2007. The XIS again detected the SXC pulses at P=8.68891±0.00010 sec [10] (Fig.1a red). To our puzzle, however, the HXC pulses were not detected signi ficantly with the HXD (Fig.1b black). Theref ore, in the interval of 2 years, the HXC pulsation may have experienced some subtle changes, wit hout accompanied by noticeable intensity changes. Through careful inspections of the 2009 HXD data, we then noti ced that the HXC pulses become visible when we utilize limited portions of the overall data, and the pulse phase depends on the data portion to be used. We were hence led to assume that some unspeci fied mechanism modulates the HXC pulse phase periodically, so that the arrival time of individual pulse peak advances/delays by a small amount by Δt=Asin(2πt/T−φ) (2) where tis the time from the start of observation, Ais the modulation amplitude, Tis some long period, and φis the modulation phase at t=0. By scanning T,A,a n d φover some plausible ranges (e.g., 0 ≤A≤P/4), and changing P over a vicinity of the XIS-determined period, we repeat the pulse search trying to maximize Z2 4. FIGURE 1. Suzaku periodograms for 4U 0142+61 [10]. (a) The 1–10 keV XIS resultsin 2007/2009 (blue/red). (b) Those in 15–40keV with the HXD, in 2007/2009 (blue/lack). Red shows the “demodulated" 2009 result. FIGURE 2. Results of the demodulation analysis on the 15–40 keV HXD data of 4U 0142+61 in 2009 [10]. Panel (a) shows a color map of Z2 4as a function of φandA, with T=55 ksec fixed. Its projection onto the φandAaxes are given in panels (b) and (c), respectively. Panel (d) presents the maximum value of Z2 4 (black; left ordinate), and the associated value of A(green; right ordinate), as a function of T. The dashed brown line indicates Z2 4without demodulation. 040003-2Figure 2 summarizes the result of this “demodulation" analysis. As seen in panel (d), employing T=55±4k s e c increased the pulse signi ficance drastically to Z2 4=39.54, from Z2 4≈14 before demodulation, and against <Z2 4>=8.0 expected for random signals without periodicity. Panel (a) gives a color map of Z2 4with T=55 ksec fixed, and its projections are given in (b) and (c). Thus, φ≈70◦(though without physical meaning) and A≈0.7sec≈0.08Pare considered optimum. When the arrival times of the indi vidual HXD photons are corrected using these parameters and eq.(2), the HXC pulsation has been recovered in the periodogram as the red curve in Fig.1 (b). The effect of this demodulation is also clearly visible in Fig.3, which compares the folded 15–40 keV pulse pro files before and after the demodulation. Through a sort of Monte-Carlo simulation, but using the actual data themselves, we have con fined that t h ei n c r e a s ei n Z2 4has a>99% con fidence when fully taking into account “look elsewhere" effects. We went back to the 2007 HXD data [13], and con firmed that the HXC pulsation, which is detectable with the simple folding (Fig.2b blue), does not become more signi ficant via demodulation, using T=55 ksec or any other value of Tfrom 20 to 120 ksec. We conclude that the HXC pulses of 4U 0142+61 were subject to a signi ficant phase modulation in 2009, but the effect was insigni ficant ( A<0.9 sec though rather loose) in 2007. Furthermore, in both 2007 and 2009, the SXC pulses have been found to be precisely periodic, without any no ticeable phase variation. FIGURE 3. Pulse pro files (for 2 cycles) of 4U 0142+61 in 15–40 keV , observed in 2009 withthe HXD and folded at 8 .68899 sec. The black and red histograms show the results before andafter the demodulation, respectively [10]. FIGURE 4. The same as Fig.2(d), but using the 10– 70 keV NuSTAR data, and the harmonic number up to m=3 instead of m=4 [12]. Purple and green shows Z2 3(left ordinate) and A(right ordinate), respectively. Follow-up observations To recon firm the discovery from 4U 0142+61, we observed it again with Suzaku in 2013, under nearly the same conditions as in the previous two cases. Then, as summarized in Table 1, we have con firmed the HXC modulation effect at the same T[12], with a similar signi ficance to that in 2009; the best-estimate parameters (Table 1) gave Z2 4=38.5, against Z2 4=16.36 before the demodulation. As a result, the con fidence level of the discovery has increased to 99.99%, or the false detection probability decreased to <1×10−4, because the 2009 and 2013 data individually give a false probability of /lessorsimilar10−2.T h ev a l u eo f Asomewhat increased from 2009 to 2013 (Table 1). The three Suzaku observations hence suggest that Avaries with time, and it was probably very small in 2007. Again, the XIS data gave no hint of pulse-phase modulation over T=20−120 ksec, with a typical upper limit of A<0.3[ 1 2 ] . To remove an obvious concern that the HXC modulation could be some artifacts speci fict oSuzaku or the HXD onboard it, we analyzed an archival NuSTAR data set of 4U|0142+61, acquired about 8 months after the 2013 Suzaku observation. These NuSTAR data had already given a negative report on the 55 ksec phase modulation [14]. Then, our careful analysis of the 10–70 keV NuSTAR data using m=3 has given Fig.4. Thanks to the high signal-to-noise ratio realized by the hard X-ray focusing optics onboard NuSTAR , the pulse signi ficance is already extremely high, Z2 3≈465. Nevertheless, the signi ficance increased by the dem odulation correction with T≈55 ksec, and the chance probability of finding this increase at a value of Tthat is consistent with those from Suzaku is estimated as ∼2% [12] which is moderately small. Importantly, the optimum value of Ais very small (Table 1), and this fact is considered to have led to the previous negative detection [14]. In addition, the absence of this effect at /lessorsimilar10 keV was recon firmed. In a word, the NuSTAR data have signi ficantly reinforced [12] the Suzaku discovery. 040003-3TABLE I. Results of the HXC pulse-phase modulation studies. Object Observation P(msec)∗T(ksec) P/T(10−4)A/P(%) reference Suzaku 2007 8,688.78 – – <12 [10, 13] 4U 0142+61 Suzaku 2009 8,688.99 55 ±41 . 68 ±3[ 1 0 ] Suzaku 2013 8,689.14 54 ±31 . 6 1 4 ±4[ 1 2 ] NuSTAR 2014 8,689.17 54 .8±5.31 . 62 .0±0.9[ 1 2 ] 1E 1547−54 Suzaku 2009 2,072.135 36+5 −30.6 26 ±7[ 1 1 ] NuSTAR 2016 2,086.712 35 ±40 . 6 1 1 ±3i n p r e p . SGR 1900+14 Suzaku 2009 5,209.91 43 ±31 . 2 2 2 ±4i n p r e p . NuSTAR 2016 5,226.70 41 ±21 . 3 1 4 ±4i n p r e p . ∗: The pulse period, with typical uncertainties of ±(0.02−0.10)msec. 1E 1547−54 To search the data of other magnetars for similar phenomena, the fastest-rotating ( P=2.07 sec) magnetar 1E 1547 −54 with a rather transient nature was chosen. It was observed with Suzaku in 2009 January, during a prominent outburst [15], and yielded the highest HXD intensity among ∼10 magnetars observed with Suzaku . Analyzing the 15–40 keV HXD data in the same way as for 4U 0142+61, but employing m=2, we discovered that eq.(2) with T≈36 ksec and A≈0.25Pincreases the pulse signi ficance to Z2 2=45.60 [11], from Z2 2∼22 before the demodulation, and against<Z2 2>=4.0 expected for random signals. Even fully considering “look elsewhere" effects, this phenomenon is estimated to have a signi ficance at 99 .6%. The Suzaku XIS, an X-ray CCD instrument, was operated in this observation with a hight time resolution of 7.8 msec. The XIS data successfully reinforced the HXD discovery; the modulation amplitude decreased toward lower energies, from A∼0.25Pat 10 keV , to A∼0.12Pat 5 keV and /lessorsimilar0.07Pat<3k e V[ 1 1 ] ,r e flecting the decreasing contribution of the HXC to the X-ray flux. After the outburst terminated [5], 1E 1547 −54 was observed with NuSTAR , when the X-ray flux had decreased to ∼5% of that in the outburst. As given in Table 1, the NuSTAR data recon fir m e dt h ee s s e n c eo ft h e Suzaku findings, with Ahalved from the Suzaku epoch, although we do not know whether this is related to the outburst decay or not. Panels (a) and (b) of Fig.5 visualize the NuSTAR results, where the photons are “double-folded" into a map of pulse phase versus modulation phase. In the 4–7 keV map (panel b), the SXC pulses form a straight ridge running vertically, implying that the pulse phase is kept constant through the T=36 keV cycle. In contrast, the 8–25 keV pulses (panel a), representing the HXC, are subject to a clear wiggling through the 36 ksec cycle. FIGURE 5. Results of “double folding" analysis, where each photon is sorted into two-dimensional bins, according to its pulse phase (the arrival time modulo P, abscissa) and modulation phase (that modulo T, ordinate). The accumulated photon counts are color coded, where yellow is the highest and black the lowest. The values of PandTemployed in the folding are given in Table 1. (a) The 8–25 keV NuSTAR data of 1E 1547 −54. (b) The same as panel (a), but in the 4–7 keV band. (c) The 15–50 keV Suzaku HXD data of SGR 1900+14. (d) The same as panel (c) but using the 8–20 keV NuSTAR data. 040003-4SGR 1900+14 SGR 1900+14 represents another major subclass of magneta rs, called Soft Gamma Repeaters (SGRs), which are con- sidered to be younger and more active than the AXPs represented by 4U 0142+61. (The classi fication of 1E 1547 −54 is somewhat uncertain.) Including SGR 1900+14, three SGRs have so far exhibited historical “Giant Bursts", wherein explosive gamma-ray flux was so high that the Earth’s ionosphere was temporarily disturbed. Using two archival data sets of this object, one from Suzaku and the other from NuSTAR , we have discovered HXC pulse-phase modulation atT≈42 ksec (Table 1). As shown in panels (c) and (d) of Fig.5, the pulse peak indeed wiggles through the T=42 ksec cycle. Although the wiggling phase diff ers between the two data sets, this is because φrefers to the start of each observation. Thus, SGR 1900+14 has become a third magne tar that shows the HXC pulse-phase modulation. As a control study, we analyzed Suzaku data of the binary X-ray pulsar 4U 1626 −67. It has P=7.68 sec which is similar to those of magnetars, but is known to show no orbital Doppler effects. Then, unlike the magnetars, the X-ray pulses were found to be strictly periodic at any energy from 10 to 60 keV . DISCUSSION We have discovered a novel phenomenon of magnetars, i.e., the slow periodic pulse-phase modulation in their HXC. The effect was con firmed in the 3 objects, 4U 0142+61 (an AXP), 1E 1547 −54 (a transient magnetar), and SGR 1900+14 (an SGR), which altogether stand for different subc lasses of magnetars. Furthermore, their pulse-phase modulation effects are characterized by the following common properties. (i) The phase modulation is seen only in the HXC, and is absent in the SXC. (ii) The modulation period T, with P/T∼10−4, is constant for each object within errors. (iii) The modulation amplitude is typically A∼0.1P, but it varies considerably even in the same object. From these facts, we infer that we are observing some intrinsic property of these NSs that have ultra-strong MFs. Furthermore, we are c urrently analyzing Suzaku data of two more magnetars, 1RXS J170849 −4009 (an AXP) and SGR 0501+4516 (an SGR), and obtained a preliminary hint of the same effects in them. Therefore, the phenomenon is likely to be common to a majority of magnetars (whereas absent in the binary pulsar 4U 1626 −67). The stability of T(above item ii) suggests that the phase modulation is due to some celestial mechanics. However, the simplest interpretation, i.e., arrival-time delays due to binary motion, is immediately ruled out by the above items (i) and (iii). This agrees with a genera lly accepted view, that magnetars are isolated NSs without any binary companion [3]. We hence try to interpret the present discovery in terms of free precession , namely, the simplest and most basic behavior of an axisymmetric rigid body [16, 17] that is free from any external torque. Figure 6 (left) depicts a magnetar as an axisymmetric prolate object, where /vectorLis the angular momentum vector fixed to the inertial frame, and ˆ x3the symmetry axis, to be identi fied with the dipole magnetic axis which is fixed to the star. The ˆ x1and ˆx2axes are de fined in an appropriate way. Using these coordinates, let /vectorI=(I1,I2,I3)be the moment of inertia, with I1=I2/negationslash=I3andΔI≡I1−I3as in eq.(1). The system has two characteristic periods, given as Pfp≡2πI1/L,Prot≡2πI3/L (3) FIGURE 6. An illustration of a magnetar, as an ax ially symmetric prolate rigid body. ( Left)T h ec o n figuration of the angular momentum /vectorL(green), the symmetry axis ˆ x3(blue), the SXC emission region (cyan), and a possible HXC emission region (orange). (right ) The expected behavior of the SXC (blue) and HXC (red) pulses, using a flying and wobbling rugby ball as an analogy. 040003-5which are related as Pfp=(1+ε)ProtandProt=Pfp/(1+ε). Here, Protis the rotation period of the star around ˆx3, whereas Pfpis the precession period, with which ˆ x3rotates around /vectorLwhile keeping so-called wobbling angle α constant. Importantly, the pulse period which we observe is Pfprather than Prot, because the system symmetry around ˆx3would not allow us to observe Prot. The SXC is considered to be emitted from a region that is symmetric around ˆx3, as shown in cyan in Fig.6 (left). Then, the observed SXC intensity will indeed be precisely periodic at P=Pfp,a s represented by the blue curve in Fig.6 (right) which si mulates the visibility of the tip of a wobbling rugby ball. The HXC emission region, in contrast, is considered to violate the symmetry around ˆ x3, either in its position as the orange ellipses in Fig.6 (left) [2, 10], or in its beaming direction [11, 12]. Then, as represented by a red spot which is offset from the tip of the rugby ball (Fig.6 right), the HXC will come into our view slightly earlier or later than the SXC. As a result, the HXC pulses become subject to a slow phase modulation, like the red curve in Fig.6 (right). The period Tof this modulation, called slip period , is given as a beat between the two periods in eq.(3), as Tcos=/parenleftBig P−1 rot−P−1 fp/parenrightBig−1 cosα−1=Pfp εcosα. (4) In this interpretation, the r elative modulation amplitude A/Pbecomes a function of αand the degree of asymmetry around ˆ x3. Although α(not well constrained by the data) should be constant on time scales of years, the latter could vary, e.g., according to re-arrangement of the surf ace MFs, thus explaining th e observed variations in A. Combining eq.(4) with A/P∼10−4from (ii), and assuming cos α∼1, we estimate ε∼10−4. Then, from eq.(1), these magnetars are inferred to have B∼1016G. Since this is much higher than the observed Bd, the implied 1016G MF must be hidden inside the stars, in the form of toroidal MFs. Then, the deformation should be prolate ( ε>0) [1], which enables the precession to develop spontaneously w ithout external perturbation, because a prolate body would attain the minimum energy at α∼90◦when the energy is dissipated under a constant angular momentum. Invoking magnetic deformation and the consequent free pre cession, we have thus been able to explain the essence of our discovery; (i),(ii), and (iii). The results make the first discovery of axial deformation of NSs, and the first case of estimaing Btof magnetars which is otherwise dif ficult to know. This will provide important clues to the origin and evolution of the NS MFs, because the presence of Bt∼1016G must be explained. The discovery also implies that the SXC and HXC are emitted by distinct mechanisms, and the HXC emission geometry varies on time scales of years. These provide important information in clarifying the still unknown X-ray emission mechanisms of magnetars. CONCLUSION Through Suzaku andNuSTAR observations of three magnetars, we have detected periodical phase-modulation effects in their HXC pulsations. This novel discovery can be successf ully interpreted by presumi ng that these objects (and possibly many other magnetars too) harbor strong toroidal MFs of Bt∼1016G, and its magnetic pressure deforms the stars in prolate shapes by ΔI/I∼10−4. The results will provide a valuable probe to Bt, give some clues to the emission mechanisms of magnetars, and deepen our understanding of the NS MFs. REFERENCES 1. Ioka, K. 2001, Mon. Not. Roy. Astr. Soc. 327, 639 2. Makisima, K. 2016, Proc. Japan Academy, Ser, B. 92, 125 3. Mereghetti, S. 2008, Astro. Astrophys. Reviews 15, 225 4. Enoto, T., Nakazawa, K., Makishima, K., Rea, N., Hurley, K., & Shibata, S. 2010, Astrophys. J. Lett. 722, L162 5. Enoto. T. et al. 2017, Astrophys. J. Suppl. 231, Id. 8 6. Kuiper, L., Hermsen,W., den Hartog, P., &Collmar,W. 2006, Astrophys. J. 645, 556 7. Takiwaki, T., & Kotake, K. 2011, Astrophys. J. 743, Id. 30, 8. Rea, N., Vigan, D., Israel, G. L., Pons, J. A., & Torres, D. F. 2014, Astrophys. J. Lett. 781, L17 9. Takahashi, T., et al. 2007, Publ. Astr. Soc. Japan, 59, S35 10. Makisima, K., Enoto, T., Hiraga, J.S., Nakano, T., Nakazawa, K. , Sakurai, S., Sasano, M., & Murakami, H. 2014, Phys. Rev. Lett. 112, 171102 11. Makisima, K., Enoto, T., Murakami, H., Furuta, Y ., Nakano, T. , Sasano, M., & Nakazawa, K. 2016, Publ. Astr. Soc. Japan,68, Id. 12 12. Makisima, K., Murakami, H., Enoto, T., & Nak azawa, K. 2019, Publ. Astr. Soc. Japan 71, Id. 15 13. Enoto, T., Makishima, K., Nakazawa, K., Kokubun, M., Kawaharada, M., Kotoku, J., & Shibazaki, N. 2011, Publ. Astr. Soc. Japan 63, 387 14. Tendulkar, S. et al. 2015, Astrophys. J. 808, 32 15. Enoto, T., et al. 2010, Publ. Astr. Soc. Japan 62, 475 16. Landau, L. D., & Lifshitz, E. M. 1976, Mechanics, 3rd e d., vol. 1 (Oxford: Butterworth-Heinemann), ch. 4. 17. Butikov, E. 2006, Euro. J. Phys. 27, 1071 040003-6

5.0029727.pdf

J. Chem. Phys. 154, 044106 (2021); https://doi.org/10.1063/5.0029727 154, 044106 © 2021 Author(s).First principles theoretical spectroscopy of methylene blue: Between limitations of time-dependent density functional theory approximations and its realistic description in the solvent Cite as: J. Chem. Phys. 154, 044106 (2021); https://doi.org/10.1063/5.0029727 Submitted: 15 September 2020 . Accepted: 01 January 2021 . Published Online: 26 January 2021 Thiago B. de Queiroz , Erick R. de Figueroa , Maurício D. Coutinho-Neto , Cleiton D. Maciel , Enrico Tapavicza , Zohreh Hashemi , and Linn Leppert ARTICLES YOU MAY BE INTERESTED IN Reflections on electron transfer theory The Journal of Chemical Physics 153, 210401 (2020); https://doi.org/10.1063/5.0035434 Full-frequency GW without frequency The Journal of Chemical Physics 154, 041101 (2021); https://doi.org/10.1063/5.0035141 Relaxation dynamics through a conical intersection: Quantum and quantum–classical studies The Journal of Chemical Physics 154, 034104 (2021); https://doi.org/10.1063/5.0036726The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp First principles theoretical spectroscopy of methylene blue: Between limitations of time-dependent density functional theory approximations and its realistic description in the solvent Cite as: J. Chem. Phys. 154, 044106 (2021); doi: 10.1063/5.0029727 Submitted: 15 September 2020 •Accepted: 1 January 2021 • Published Online: 26 January 2021 Thiago B. de Queiroz,1,a) Erick R. de Figueroa,1Maurício D. Coutinho-Neto,1,b) Cleiton D. Maciel,2 Enrico Tapavicza,3,a) Zohreh Hashemi,4and Linn Leppert4,5,c) AFFILIATIONS 1Centro de Ciências Naturais e Humanas, Universidade Federal do ABC, Av. dos Estados 5001, 09510-580 Santo André-SP, Brazil 2Instituto Federal de Educação, Ciência e Tecnologia de São Paulo, Campus Itaquaquecetuba, Avenida Primeiro de Maio, 500, 08571-050 Itaquaquecetuba-SP, Brazil 3Department of Chemistry and Biochemistry, California State University, Long Beach, 1250 Bellflower Boulevard, Long Beach, California 90840, USA 4Institute of Physics, University of Bayreuth, Bayreuth 95440, Germany 5MESA+ Institute for Nanotechnology, University of Twente, 7500 AE Enschede, The Netherlands a)Authors to whom correspondence should be addressed: thiago.branquinho@ufabc.edu.br and enrico.tapavicza@csulb.edu b)mauricio.neto@ufabc.edu.br c)l.leppert@utwente.nl ABSTRACT Methylene blue [3,7-Bis(di-methylamino) phenothiazin-5-ium chloride] is a phenothiazine dye with applications as a sensitizer for photody- namic therapy, photoantimicrobials, and dye-sensitized solar cells. Time-dependent density functional theory (TDDFT), based on (semi)local and global hybrid exchange-correlation functionals, fails to correctly describe its spectral features due to known limitations for describing optical excitations of π-conjugated systems. Here, we use TDDFT with a non-empirical optimally tuned range-separated hybrid functional to explore the optical excitations of gas phase and solvated methylene blue. We compute solvated configurations using molecular dynam- ics and an iterative procedure to account for explicit solute polarization. We rationalize and validate that by extrapolating the optimized range separation parameter to an infinite amount of solvating molecules, the optical gap of methylene blue is well described. Moreover, this method allows us to resolve contributions from solvent–solute intermolecular interactions and dielectric screening. We validate our results by comparing them to first-principles calculations based on the GW+Bethe–Salpeter equation approach and experiment. Vibronic calculations using TDDFT and the generating function method account for the spectra’s subbands and bring the computed transition energies to within 0.15 eV of the experimental data. This methodology is expected to perform equivalently well for describing solvated spectra of π-conjugated systems. Published under license by AIP Publishing. https://doi.org/10.1063/5.0029727 .,s I. INTRODUCTION Phenothiazine dyes, such as methylene blue [MB, 3,7-Bis (di-methylamino) phenothiazin-5-ium chloride], are a technologicallyimportant class of π-conjugated heterocyclic molecules. They cover an extensive range of light assisted applications as sensitizers for photodynamic therapy,1,2photoantimicrobials,3and dye-sensitized solar cells.4These applications involve light absorption and J. Chem. Phys. 154, 044106 (2021); doi: 10.1063/5.0029727 154, 044106-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp electronic transitions in the molecule, which are strongly influ- enced by surrounding molecules and the medium in which these molecules are solvated.2,5It is therefore important to describe the electronic structure and dynamics of this class of molecules in realistic environments accurately.2 Density functional theory (DFT) and time-dependent DFT (TDDFT) have been extensively employed for such studies since they are, in principle, exact theories while computationally effi- cient.6–13Low-lying valence states of organic compounds of inter- mediate size are usually well described by exchange-correlation (xc) functionals within the local density approximation (LDA)7and generalized gradient approximation (GGA, also called semilocal approximation)14as well as by their linear combination with non- local Fock-like exact exchange (EXX), the global hybrids15,16(see, for instance, Ref. 17). However, these “standard TDDFT approxi- mations,” i.e., LDA, GGA, and global hybrid xc-functionals, yield significant deviations for π–π∗valence transitions of some sets of π- conjugated systems in comparison to experimental and theoretical reference data,18–24which is a statement that is also valid for MB25 and other related compounds.26,27The failure of standard TDDFT for intermediate or large π-conjugated systems has been the subject of an extensive number of studies.18,21–23,28,29 From the perspective of molecular topology, open chain π-conjugated systems can be categorized as either polyenes or polymethines.18Polyenes demonstrate π-density alternation in the bonds, but the structure is based on their σ-skeleton (alternating double and single bonds).18,29Polymethines, just as cyanine dyes, show similar alternating π-bonds but with the highest alternat- ing density in the atomic positions.18,29MB can be regarded as a carbocyanine related compound. Additionally, MB is expected to present even higher electron delocalization of the π-electrons in the ground state, in comparison to other phenothiazine dyes, due to the presence of two strongly electron-donating dimethyl-amino groups located diametrically alongside the molecule.5 For polymethines, the lowest allowed excitation is composed of a single electronic configuration, well characterized by a single- particle HOMO–LUMO transition.18,30Yet, standard TDDFT sub- stantially overestimates the transition energies.18,31,32These systems exhibit excited state charge densities that differ significantly from the ground state densities.18,29This error was later recast in terms of the very small magnitude of the exchange-correlation response kernel integrals.33,34This is a feature that is badly described by lin- ear response TDDFT since the xc kernel is calculated as the func- tional derivative of the xc potential at the ground state density (in principle, an exact linear response formulation if time propagated, but time-independent in the adiabatic approximation).35,36Similarly to the polyene case, pure GGAs perform slightly better than their corresponding hybrids.33 Besides the errors caused by the approximations described above (as compared to reference calculation methods), there are also errors arising from the incorrect and unrealistic description of the medium (as compared to experimental data).25In fact, electron-hole interactions can be significantly weakened by the effect of dielectric screening on the Coulomb potential.37For instance, the correct pre- diction of the S 0–S1transition energy in azobenzene derivatives in comparison to experimental data from calculations with the GW and Bethe–Salpeter equation (GW + BSE) requires the consideration of solvent polarization effects (dielectric screening).38Intermolecularinteractions also play a role. From the infrared absorption spectrum of MB in the gas phase in comparison to its hydrated crystalline state,39it was proposed that the functional group aside N+(CH 3)2 is H-bonding to a water molecule. Again, standard TDDFT approx- imations are ill-equipped for describing such effects since they can produce electron density delocalization errors and overesti- mate intermolecular interactions.33,40Finally, these approximations are not designed to appropriately describe the long-range dielectric screening,41–43scaling incorrectly at long distances [decaying faster than−(1/R), where Ris the electron–hole distance].44–46 More recently, (semi)local approximations have been com- bined with EXX using the error function [erf( ωr)] in the Coulomb operator. In these range-separated hybrid (RSH) functionals, a range separation parameter, ω, scales short- and long-range exchange terms such that (semi)local terms dominate at short range and EXX at long range [in this particular format, referred as the long-range corrected (LRC) functional].47–49This tempered mix of (semi)local and EXX takes advantage of the good performance of predomi- nantly (semi)local hybrid functionals for describing valence transi- tions15while ensuring the correct asymptotic of the potential ( −1/R), thereby improving preceding approximations in many respects.47–51 The range separation parameter can be empirically optimized47,52,53 or tuned from first principles, by choosing ωsuch that the eigenvalue of the Kohn–Sham frontier orbital located in the solute approaches the ionization energy.54,55The optimally tuned (OT) RSH function- als have been used with great success for the prediction of charge- transfer excitations for a variety of complex molecular systems,56–58 intermolecular interactions,59and dielectric screening.55,59In fact, the introduction of the non-local EXX term at long range in the RSH seems to be appropriate for introducing the correct field- counteracting behavior due to the environment. For instance, polar- izabilities and second hyperpolarizabilities are correctly predicted using this approach.60 In this article, we report first principles OT-RSH TDDFT calcu- lations for the low-lying excited states of MB in vacuum and water, aiming to describe the system realistically while gaining insight into the limitations of state-of-the-art TDDFT. We employed the RSH functionalωPBE (PBE exchange at short range)49optimally tuned for MB optimized in vacuum and solvated by water molecules. We circumvent problems associated with RSH tuning for solvated molecules by analyzing the orbital localization involved in the func- tional tuning. Furthermore, we rationalize and demonstrate that the spectroscopic features of MB represented in its aqueous medium are well characterized by the OT-RSH functional. We show that this is a consequence of the increased short-to-long GGA/EXX inter- change distance in solution ( ω−1), leading to higher weights of semilocal GGA exchange in the description of the MB molecule while treating the solvent by a balanced mix of semilocal and EXX at long range. Furthermore, we show that the zero-point vibra- tional energy shifts the spectrum to lower energies relative to ver- tical transitions and that vibronic contributions are responsible for a shoulder at higher energy (in comparison to the maximum absorp- tion). As reference data, we use a partially self-consistent GW + BSE approach with the optimally tuned ωPBE as the starting point (as benchmarked in Refs. 61 and 62) in addition to experimental data. Thus, we present an inexpensive TDDFT methodology to describe accurately the absorption spectrum of this important organic dye in an aqueous solution, which can be extended to describe J. Chem. Phys. 154, 044106 (2021); doi: 10.1063/5.0029727 154, 044106-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp excited states of other π-conjugated systems in a complex chemical environment. II. METHODS The details of the generation of the molecular configurations, molecular dynamics, calculation of vibrational spectrum, and cal- culation of excited states with TDDFT and GW + BSE and the data obtained are listed in the supplementary material. We also discuss our strategy for optimally tuning the RSH in explicit sol- vated systems, basis set convergence, and starting point issues in the GW + BSE calculations in the supplementary material. The geometries and vibronic spectrum were obtained from DFT/B3LYP/def2-SV(P) calculations (theory/xc-approximation/ basis set) followed by molecular dynamics simulations [isothermal– isobaric (NPT) ensemble with T = 298 K and P = 1 atm]. The ground and excited state vibrational spectra were computed using analytical and numerical second derivatives.63,64The structures for excited state calculations were obtained from MD frames and labeled according to their MD step after reaching thermal equilibrium and the number of solvating water molecules (in parenthesis), namely, MB-01(20), MB-13(20), MB-19(13), MB-27(20), MB-31(23), MB-38(26), and MB-46(21). These structures were generated with a solvation cutoff of 3.2 Å, where solvation cutoff is the maximum distance between an MB atom and an atom in the water molecule selected for the calculations. The structure MB-31 was taken as a representative structure and progressively solvated with additional cutoffs of 3.8 Å, 4.0 Å, 4.2 Å, 4.4 Å, 4.5 Å, 4.6 Å, 4.8 Å, 5.0 Å, 5.2 Å, and 5.4 Å, corresponding to 46, 58, 59, 67, 70, 71, 76, 84, 93, and 101 water molecules, respectively. The excited states were calculated from (i) TDDFT calculations with the functionals xPBE x+ (1−x) EXX + PBE c, 0≤x≤1, with the 6-31G(d,p) as basis set, (ii) TDDFT with theωPBE and 6-31G(d,p) as functional and basis set, and (iii) eigenvalue self-consistent G nWn+ BSE/ωPBE/6-311++(2d,2p) (method/starting point/basis set). III. RESULTS A. Ground state properties MB and solvent configurations were obtained using an itera- tive procedure to account for solute polarization described in detail in the supplementary material. We employ a protocol to induce the solvent perturbation on the solute by generating an average solvent electrostatic potential (ASEP) developed initially by Sánchez and co-workers.65,66We follow a sequential QM/MM iterative proce- dure similar to the one proposed by Coutinho et al.67,68to generate converged solvent configurations in equilibrium with induced solute polarization for fixed solute geometries. In their procedure, solvent effects are included using point changes whose configurations come from Metropolis Monte Carlo simulations while solute charges are computed using the CHELPG algorithm.69An iterative procedure is used, where at each iteration a new set of solute CHELPG charges is obtained at distinct sampled solvent configurations. They have applied their method for the calculation of condensed phase spectra of several systems with encouraging results.70–72Solvated MB displays a strong increase in dipole moment in the water when compared to vacuum, going from 2.55 D to 4.23 D as computed by our iterative procedure. A strong charge separation matches the increase of the dipole moment of the molecule, where the central nitrogen acquires a strongly negative charge compared to its vacuum value. CHELPG values increase from −0.60eto−0.91e. As expected, solute–solvent interactions are affected, resulting in a larger number of hydrogen bonds between the central nitrogen and water. These interactions are present in most configurations sam- pled from molecular dynamics and in the configurations selected for spectroscopic studies. B. Excited states of the isolated MB by exploring global hybrid functionals: Assignment of the failures and the OT-RSH proposal As we expect that standard TDDFT cannot describe the optical gap of MB well, we try to map possible error sources by calculat- ing the first excited state transitions (S 0–S1and S 0–T1), adding EXX stepwise in the hybrid functional. Figure 1 shows the S 1and T 1exci- tation energies calculated with TDDFT as a function of EXX/PBE x admixture for MB-31(0), as well as calculated from the G nWn+ BSE method [ ΔE(S 0–S1) = 2.25 eV and ΔE(S 0–T1) = 1.00 eV] and the experimental maximum absorption of MB in water (1.87 eV).73The experimental value taken in water does not represent the optical gap of the molecule in the gas phase, but solvent effects are not expected to exceed 0.1 eV–0.4 eV (see below). Thus, the optical gap of the molecule in the gas phase should be around 2.0 eV–2.3 eV, suggesting that G nWn+ BSE can be regarded as the reference data (also rationalized in Refs. 61 and 62). The S 0–S1transition energy approaches the G nWn+ BSE and the experimental values as the FIG. 1 . S0–S1and S 0–T1transition energies for MB-31(0) as a function of PBE exchange and exact Fock exchange admixture in the TDDFT approximation [Ex= xPBE x+ (1-x)EXX] and in comparison to G nWn+ BSE calculations and the experimental maximum absorption of MB in water. J. Chem. Phys. 154, 044106 (2021); doi: 10.1063/5.0029727 154, 044106-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp GGA portion increases, going from 2.87 eV when E x= EXX to 2.32 eV when E x= PBE x. Simultaneously, S 0–T1increases from the unrealistic value of ≈0.1 eV when E x= EXX to the reasonable value of≈1.2 eV when E x= PBE x. Yet, either TDDFT or GW + BSE esti- mates for the S 0–T1transition are considerably underestimated, in comparison with reference data (1.75 eV using CASPT2),74due to triplet instabilities.75–78 The superior performance of pure GGAs over HF indicates that charge transfer is not a dominant feature and that the tran- sition could be of double excitation character (when error can- cellations occur for GGAs).23However, the overestimation of the optical gap resembles the case of the cyanine dyes,18,33,34when the dynamical correlation is important, as well as the fact that the pure GGAs outperform their associated hybrids.33We could calibrate the global hybrid functional with high portions of GGA and pro- ceed further to study the influence of the solvent in the optical gap of MB. However, besides the lack of generality of such pro- cedure, from previous studies, we learn that the optimally tuned RSH can describe well intermolecular interactions59,79and dielectric screening,55differently from global hybrids.59Furthermore, short- contact intermolecular interactions with solvent molecules could lead to an increase in the charge-transfer character of the first excited state, another feature that is well described by RSH functionals in contrast to hybrid functionals,79especially if constructed with large portions of GGA. Figure 2 shows the electron–hole pair of the natural transi- tion orbitals (NTOs) of MB-31(0) for the S 0–S1transition from TDDFT/ωPBE calculations. The transition is almost entirely repre- sented by a HOMO–LUMO transition ( >90%) and described by the electron migration from the whole molecule to the central sul- fur and nitrogen atoms, an apparent valence excitation with little intramolecular charge transfer character. The TDDFT/ ωPBE cal- culations estimate the S 0–S1and S 0–T1transitions at 2.54 eV and FIG. 2 . NTO isosurface densities of the most significant electron–hole pair ( ⩾90%) of S 1for MB-31(0). Carbon atoms are represented in green, nitrogen in blue, sulfur in yellow, and hydrogen in white.1.22 eV, respectively. These transitions are off by ≈0.3 eV in com- parison with G nWn+ BSE, slightly worse than the pure PBE and desired accuracy (within ≈0.1 eV). We anticipate that the inferior performance of the OT-RSH with respect to the GGA will be attenuated or removed when inserting solvent effects. The optimal ωfor the solvated structures tends to be smaller since the frontier orbitals tend to be more delocalized.43Thus, the distance where the fraction of EXX is larger than the PBE exchange increases, such that a larger contribution to the energy of MB comes from the GGA exchange potential.43For instance, according to the optimal ωvalues, the GGA/EXX inter- change distance for the isolated molecule is at 1.3 Å and for the fully solvated system at 1.7 Å [taken from erf( ωoptr0) = 1/2, see ω optimization below]. Furthermore, the OT-RSH is indicated for explicitly solvated systems because GGAs tend to overestimate elec- tron density delocalization while EXX tends to underestimate it, such that the OT-RSH temper these limits showing small delocal- ization errors.33 C. Excited states from TDDFT/ ωPBE and the contributions from intermolecular interactions and dielectric screening Table I lists the TDDFT/ ωPBE transition energies of the first excited states with respective oscillator strengths for singlets of all configurations (note that triplet transitions are forbidden in our calculations due to the lack of spin–orbit coupling). Figure 3 illustrates the transitions in comparison with the experimental data. The excited states with strong oscillator strength ( ∼0.5) are always localized in MB, topologically similar to the transition of the isolated molecule, as noted by their electron–hole pair from the NTOs [see representative NTOs in Fig. 4-left for MB-31(101)]. The singlet transitions with low oscillator strength are of a mixed charge transfer character, described by the electron donation from a water molecule nearby the sulfur atom of MB [represented in Fig. 4-right for MB-31(101)]. For most of the configurations, this excitation is about 0.1 eV–0.4 eV distant from the excitation localized in MB, and its oscillator strength is limited to 10−2. Inter- estingly, for the configuration MB-31(23), this excitation is the low- est one and shows increased oscillator strength, 0.07, and it is much closer, in energy, to the one located in MB ( ΔE= 0.036 eV). This indicates that there might be a coupling between S 1and S 2, such that S 1increases in oscillator strength, pulling the absorption band to lower energies (similar to the coupling reported in Ref. 80). Note that by taking into consideration the structures generated with a solvent cutoff of 3.2 Å with 20–21 solvent molecules, the dispersion in the optical gap is about 0.3 eV, which is related to small molecular distortions and distinct solvent configurations (sim- ilarly to the configurational variances when collecting the exper- imental data, but here limited to a small number of sampling configurations). After looking at the optical gap dispersion of MB due to dis- tinct molecular arrangements and a variation of intermolecular interactions, we look at MB-31 as a representative configuration to investigate the influence of intermolecular interaction and dielectric screening in the first excited states. Thus, structures with various distance cutoffs with respect to solvent molecules were generated and the optimization parameter was calculated as a function of the J. Chem. Phys. 154, 044106 (2021); doi: 10.1063/5.0029727 154, 044106-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . First excited states and oscillator strengths (O.S.) for MB structures as calculated by the TDDFT/ ωPBE method. Note thatωEAis used for all calculations except for MB-31(101), in which ωext solvwas used instead, which is the optimal ω extrapolated to a completely solvated system. Molecule ωopt(×10−3a−1 0) S 1(eV) O.S. S1 S2(eV) O.S. S2 T1(eV) T 2(eV) MB-01(20) 182 1.6316 0.0008 2.3508 0.4767 1.0563 1.6293 MB-13(20) 182 2.3099 0.7991 2.4412 0.0181 0.9131 1.6807 MB-19(13) 191 2.3813 0.8285 2.5467 0.0014 0.9977 1.8134 MB-27(20) 182 2.5791 0.8002 2.6941 0.0056 1.2580 1.9027 MB-38(26) 179 2.2398 0.6958 2.3912 0.0009 0.8197 1.6489 MB-46(21) 179 1.9670 0.0055 2.3705 0.7052 1.0959 1.6642 MB-31(0) 203 2.5432 0.8694 2.7871 0.0063 1.2233 1.9838 MB-31(23) 179 2.1800 0.0701 2.2160 0.6484 0.7460 1.7490 MB-31(101) 150 2.1527 0.7135 2.3240 0.0079 0.8199 1.6509 number of solvent molecules. The optimization parameter could be represented by a mono-exponential decay as the number of solvent molecules ( n) increases, described as ω(n)=ω0exp(−n/δ)+ωext solv, whereω0is the optimal ωof the bare structure, δis a system depen- dent fitting parameter, and ωext solvis the optimal ωwhen extrapolat- ing the optimization to the fully solvated structure (“bulk” water).55 Interestingly, the exponential decay of the parameter ωas a function of the solvation shell number is not only observed for weakly inter- molecular interacting systems, as demonstrated in Ref. 55 but also for a system of polar molecules prone to intermolecular interaction. FIG. 3 . Excited state spectra for MB-31(0), in green; MB-01(20), MB-13(20), MB-27(20), and MB-46(21), in red; MB-31(23), in magenta; and MB-31(101), in blue, as calculated by the TDDFT/ ωPBE method, and experimental data, in black.Usingωext solvin TDDFT/ ωPBE calculations, one can account for the long-range dielectric screening of the media in the description of the excited states.55,81Figure 5 shows a good agreement between the approximate curve and the ΔSCF-calculated ω(n), which resulted in anωext solvvalue of 0.150 a−1 0. The comparison between the optical gap of MB as calculated from the G nWn+ BSE and TDDFT/ ωPBE methodologies is illus- trated in Fig. 6. The S 0–S1transition of MB-31(23) from the GnWn+ BSE calculation is at 2.05 eV, indicating a decrease in 0.2 eV due to intermolecular interactions and local screening. TDDFT/ωPBE predicts the S 0–S1transition of MB-31(23) at 0.36 eV below the transition for the isolated molecule. The OT-RSH is more influenced by the explicit solvation because the range separating parameter decays rapidly by including the first coordination shell, and the lowering of the parameter emulates long-range dielectric screening.55For instance, ωdecays from 0.203 to 0.179 a1 0by includ- ing the first coordination shell (88% of ω0), reaching 0.150 a1 0when the parameter is extrapolated to fully solvated structure (74% of ω0). FIG. 4 . NTO isosurface densities of the most significant electron–hole pair ( ⩾90%) of the S 1and S 2transitions for MB-31(101) as calculated by the TDDFT/ ωPBE method. Carbon atoms are represented in green, nitrogen in blue, sulfur in yellow, oxygen in red, and hydrogen in white. J. Chem. Phys. 154, 044106 (2021); doi: 10.1063/5.0029727 154, 044106-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . Optimalωas a function of the number of water solvent molecules surrounding MB-31 and exponential decay fits. Moving to the fully solvated structure, one notes an additional shift to the lower energy of about 0.03 eV, with all results indicating a modest contribution from long-range dielectric screening in the first excited singlet state. This is expected since MB is considered a dye with a small solvatochromic shift.5,82Note that the calcula- tion of MB-31(101) results in a S 0–S1with a deviation of 0.28 eV to higher energies in comparison to experimental data. Vibronic and zero point corrections shift this value further down, as described in Sec. III D. The performance of TDDFT for cyanine dyes has been previ- ously analyzed as follows:33An electronic transition with a strong HOMO–LUMO character in the adiabatic approximation is approx- imately given by the HOMO–LUMO gap ( εL−εH), added by the Coulomb electron–electron repulsion term and xc kernel integrals.33 For instance, for the two-level model HOMO–LUMO transition in the Tamm–Dancoff approximation,33 ETDDFT S1=(εL−εH)+ 2[LH∣r−1 12∣LH]+[LH∣fαα xc+fαβ xc∣LH], (1) where the second term on the right-hand side of the equation is a two-electron repulsion integral and the third term is the xc lin- ear response kernel ( fxc) integral ( Land Hrefer to the HOMO FIG. 6 . Diagram of the first excited state for MB-31(0) and MB-31(23) as cal- culated from the G nWn+ BSE and TDDFT/ ωPBE methods, in addition to the TDDFT/ωPBE calculation for MB-31(101), and experimental data.and LUMO Kohn–Sham orbitals, respectively, and αandβare spin labels). The HOMO–LUMO gap is the dominant positive term, the electron–electron Coulomb repulsion term is typically small but positive, and the xc kernel integrals are generally negative. The HOMO–LUMO gap for MB-31(0) as calculated from DFT/ωPBE is 4.5 eV (similar to the other configurations). Thus, the last term of the above equation, the xc kernel integrals, should be sufficiently large (and negative) in order to lower the optical gap in the direction of the reference data ( EGW+BSE S1 = 2.25 eV). The fact that the TDDFT/ ωPBE results are generally overestimating those of the GnWn+ BSE is similar to what has been observed for cyanine dyes, and has been attributed to too small contributions of the xc response kernel integrals.33,34 D. Vibronic contributions Another question regarding the absorption spectrum of MB is related to the shape of the envelope, in particular the cause of the line broadening and the shoulder between 625 nm and 606 nm (1.98 eV and 2.04 eV, or 16 000 cm−1and 16 500 cm−1, respec- tively) (Fig. 7). Within the generating function method,83,84the 0–0 transition occurs at ΔE0−0=ΔEadia−E0 ZPV+E1 ZPV, (2) where ΔEadia is the adiabatic excitation energy, and the last two terms are the zero-point vibrational energy (ZPV) in the ground FIG. 7 . Comparison between the calculated vibronic absorption spectrum in the gas phase and the experimentally measured spectrum in water.73Computed spec- tra have been downshifted by 0.5 eV (4046 cm−1) to align the main band of the broad spectrum with the experimental λmaxvalue. 00 0denotes the 0–0 transition. In the other peak assignments, the mode number according to Table S2 (supple- mentary material) is given with subscripts indicating its ground state vibrational quantum number and superscripts indicating its excited state vibrational quantum number of the transition; all other modes have both zero as initial and final quantum numbers. J. Chem. Phys. 154, 044106 (2021); doi: 10.1063/5.0029727 154, 044106-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp and excited states, respectively. The 0–0 transition (denoted 00 0in Fig. 7) serves as the reference point for the vibronic structure. Since the excited state ZPV energy (8.458 eV) is smaller than the ground state ZVP (8.532 eV), the 0–0 transition (2.351 eV) lies 0.16 eV below the TDDFT/B3LYP/def2-SVP adiabatic excitation energy of 2.424 eV. When applying a line broadening with a lifetime of 1000 au (red in Fig. 7), the peak maximum is shifted 0.019 eV to the blue, relative to the 0–0 transition using a lifetime of 100 000 au (blue in Fig. 7). In general, line broadening could be caused by conformational isomers85,86or by vibronic bands.83Our vibronic spectra calculations, including Duschinsky effects, show that both line broadening and the dominant shoulder are due to the vibronic effects. Furthermore, the line broadening is mainly caused by three vibrational modes, mode 11, mode 19, and mode 40, shown in Fig. 8 (Multimedia view)—(Table S2 lists all vibrational transitions in the supplementary material). By assigning modes 11 and 19 as the main causes for the broadening, our calculations are partially consistent with previous calculations of Franck–Condon factors by Dean and co-workers based on the displaced harmonic oscillator model; how- ever, mode 40 was not identified by their work.73In addition, several peaks with minor intensity ( <0.1 in Fig. 7) contribute to broadening and were not included in previous spectral simulations.73Regarding the dominant shoulder above 625 nm (16 000 cm−1), our calcula- tions show that it is mainly due to mode 68 and its combinations with modes 11 and 13 (Fig. 7). We also computed the vibronic spectrum using the ωPBE xc-functional [6-31G(d,p) basis set and ω=0.203 a−1 0] but within the displaced harmonic oscillator approximation because we could not determine the full Duschinsky matrix from the output. However, since the Duschinsky matrix for MB is close to unity, the approx- imation made is small. The ωPBE vibronic spectrum compares well to the TD-B3LYP spectrum (see the supplementary material). FIG. 8 . Dominant excited state vibrational normal modes and their fre- quencies calculated by TDDFT/B3LYP/def2-SVP. Multimedia views: https://doi.org/10.1063/5.0029727.1; https://doi.org/10.1063/5.0029727.2; https:// doi.org/10.1063/5.0029727.3; https://doi.org/10.1063/5.0029727.4As expected, the vibrational features are relatively insensitive to the xc-functional approximation. In addition, we computed the TD-B3LYP/TDA vibronic spectrum. Vibrational frequencies in the excited state are very similar in TD-B3LYP and TD-B3LYP/TDA, but some peak intensities are different in the two spectra due to dif- ferences in the Duschinsky matrix (see the supplementary material). This shows that the TDA affects the computation of the vibrational modes in the excited state. IV. CONCLUSIONS We present a strategy to predict the optical gap of MB in water using an optimally tuned range-separated hybrid functional for fully solvated structures within TDDFT. The addition of a realistic solvent environment, dielectric screening effects, and vibronic corrections are essential for bringing the results close to experiments. Solva- tion effects, which we take into account explicitly, are responsible for lowering the optical gap by about 0.2 eV–0.4 eV, resulting in an overestimation of the experimental data by 0.2 eV–0.3 eV depending on the structure used. Vibronic effects and zero-point corrections shift the adiabatic value down in energy by 0.16 eV, as estimated by TDDFT/B3LYP calculations. The overestimation then decreases to only∼0.15 eV with respect to experimental data. Vibronic cal- culations also show that in-plane mode 68, with a large compo- nent in ring CH bending, and mode 11 are responsible for the vibronic subband that is very characteristic of MB’s UV spectra in solution. The optical gap difference between G nWn+ BSE and TDDFT/ωPBE is reduced from 0.3 eV to 0.1 eV by going from the isolated MB to the solvated structure. The improvement in the performance of TDDFT/ ωPBE for the solvated structure occurs because the semilocal term of the functional takes over at larger distances, and dielectric screening is emulated by choosing the range separation parameter appropriately.55Since semilocal func- tionals are reported to treat better π-conjugated systems (although due to error cancellations),20,23,33we expect that such a strategy might be generalized, permitting to study these important systems at relatively low computational cost and with acceptable accuracy. The combination of the methodologies employed in this study is going to be relevant for the description of the absorption spectra of π-conjugated systems, especially in solution or in other chemical environments. SUPPLEMENTARY MATERIAL See the supplementary material for details about molecular configurations, vibronic spectra calculations and illustrative movies of the relevant vibrational modes, optimal tuning of the RSH func- tional and TDDFT calculations, GW + BSE calculations: start- ing point dependence and basis set convergence, and tables with data obtained in the optimal tuning of the RSH functional and GW + BSE calculations, and the vibronic modes. ACKNOWLEDGMENTS T.B.d.Q. acknowledges the support of CNPq (Universal Grant No. 404951/2016-3). Z.H. and L.L. acknowledge the support of the J. Chem. Phys. 154, 044106 (2021); doi: 10.1063/5.0029727 154, 044106-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Bavarian State Ministry of Science and the Arts through the Collab- orative Research Network Solar Technologies go Hybrid (SolTech), the Elite Network Bavaria (ENB), and computational resources pro- vided by the Bavarian Polymer Institute (BPI). M.D.C.-N. acknowl- edges the support of FAPESP (Grant No. 12/50680-5). This study was financed, in part, by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior–Brasil (CAPES), Finance Code 001. The authors are thankful to the University of Bayreuth-Theoretical Physics IV for computational time and Professor Stephan Kümmel for valuable discussions. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1A. Ormond and H. Freeman, Materials 6, 817 (2013). 2J. C. Dean, D. G. 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5.0036512.pdf

J. Chem. Phys. 153, 244102 (2020); https://doi.org/10.1063/5.0036512 153, 244102 © 2020 Author(s).Assessing the orbital-optimized unitary Ansatz for density cumulant theory Cite as: J. Chem. Phys. 153, 244102 (2020); https://doi.org/10.1063/5.0036512 Submitted: 06 November 2020 . Accepted: 01 December 2020 . Published Online: 22 December 2020 Jonathon P. Misiewicz , Justin M. Turney , Henry F. Schaefer , and Alexander Yu. Sokolov The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Assessing the orbital-optimized unitary Ansatz for density cumulant theory Cite as: J. Chem. Phys. 153, 244102 (2020); doi: 10.1063/5.0036512 Submitted: 6 November 2020 •Accepted: 1 December 2020 • Published Online: 22 December 2020 Jonathon P. Misiewicz,1 Justin M. Turney,1 Henry F. Schaefer III,1,a) and Alexander Yu. Sokolov2 AFFILIATIONS 1Center for Computational Quantum Chemistry, University of Georgia, Athens, Georgia 30602, USA 2Department of Chemistry and Biochemistry, The Ohio State University, Columbus, Ohio 43210, USA a)Author to whom correspondence should be addressed: ccq@uga.edu ABSTRACT The previously proposed Ansatz for density cumulant theory that combines orbital-optimization and a parameterization of the 2-electron reduced density matrix cumulant in terms of unitary coupled cluster amplitudes (OUDCT) is carefully examined. Formally, we elucidate the relationship between OUDCT and orbital-optimized unitary coupled cluster theory and show the existence of near-zero denominators in the stationarity conditions for both the exact and some approximate OUDCT methods. We implement methods of the OUDCT Ansatz restricted to double excitations for numerical study, up to the fifth commutator in the Baker–Campbell–Hausdorff expansion. We find that methods derived from the Ansatz beyond the previously known ODC-12 method tend to be less accurate for equilibrium properties and less reliable when attempting to describe H 2dissociation. New developments are needed to formulate more accurate density cumulant theory variants. Published under license by AIP Publishing. https://doi.org/10.1063/5.0036512 .,s I. INTRODUCTION ODC-12 is the most successful method to date of the density cumulant theory1–3(DCT) family of electronic structure methods. For a system of ooccupied orbitals and vvirtual orbitals, ODC-12 has the O(o2v4)scaling of coupled cluster with single and double excitations4,5(CCSD) but is consistently more accurate.3,6,7It has a simple, inexpensive analytic gradient theory.3It tolerates multiref- erence effects that leave CCSD qualitatively incorrect.8For these reasons, there has been interest in extending the success of ODC-12 both to achieve greater accuracy for weakly correlated molecules and to develop a method able to treat multiconfigurational molecules.8–10 To date, there has been only one published proposal of den- sity cumulant theory methods going beyond ODC-12 in accu- racy.9Reference 9 introduced a formally exact Ansatz for density cumulant theory and proposed that approximating it may yield the desired improvements to ODC-12. As a proof-of-concept, the authors implemented and benchmarked the ODC-13 method, which adds terms to ODC-12 and is derived from the aforementioned Ansatz . Unfortunately, the ODC-13 method did not improve on the success of the simpler ODC-12. The authors of Ref. 9 reported that ODC-13 was less accurate in the weakly correlated regime,as determined by comparison against experimental bond lengths and vibrational frequencies for diatomic molecules. The accuracy of the method across various correlation strengths was assessed by H2dissociation. For this system, ODC-13 was less accurate than ODC-12 past 0.9 Å and could not be converged beyond 1.3 Å. Refer- ence 9 observed that a particular exact relationship between two key intermediates of the theory, the 1-electron reduced density matrix (1RDM) and the 2-electron reduced density matrix (2RDM), was not satisfied in ODC-13. The authors suggested that violating that relationship might have caused the “unsatisfactory” performance of ODC-13. Additionally, the authors of Ref. 9 proposed but did not imple- ment an alternative scheme to approximate their Ansatz where that relationship is obeyed. This approximation consists of truncating a Baker–Campbell–Hausdorff (BCH) expansion to a finite number of commutators to obtain one part of the 2RDM while maintaining the aforementioned exact relation between the 1RDM and the 2RDM to determine the rest. As there have been no further studies of any post-ODC-12 methods of this Ansatz , it remains untested whether the failure of ODC-13 can be attributed to violations of this rela- tionship or whether the performance of ODC-13 indicates a more general complication in working with the DCT Ansatz advanced in Ref. 9. J. Chem. Phys. 153, 244102 (2020); doi: 10.1063/5.0036512 153, 244102-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp In this article, we study truncations of the orbital-optimized unitary coupled cluster Ansatz for DCT (OUDCT) proposed in Ref. 9. We begin in Sec. II with a thorough review of the equations of the OUDCT Ansatz , which are scattered across multiple papers.1–3,9 During this review, two new formal questions about the Ansatz arise, namely: 1. The residual equations for the OUDCT stationarity conditions contain terms with near-zero denominators, which all vanish in ODC-12. Do these vanish in the exact OUDCT theory? 2. A similar orbital-optimized variational unitary coupled clus- ter method can also be approximated by truncations of the BCH expansion to a finite number of commutators. What is the relationship between approximations of that method and OUDCT, truncated at the same degree? We then turn our attention to numerical studies of the perfor- mance of low-degree OUDCT truncations, with only double exci- tations. The truncation at degree two is the aforementioned ODC- 12 method and is our baseline for both accuracy and degree of truncation. After discussing our implementation of the methods in Sec. III, we perform numerical studies in Sec. IV. We investigate the following: 3. Do higher-degree truncations of the OUDCT Ansatz improve the accuracy for H 2? H 2is important both as a case where effects of triple and higher-rank cluster operators do not exist and as a model of variable correlation strength. 4. Do higher-degree truncations of the OUDCT Ansatz , restricted to doubles, improve the accuracy for systems with more than two electrons, where triples effects may be important? The results of our investigation lead us to conclude that OUDCT Ansatz truncations that include more commutators than ODC-12, up to five, improve upon ODC-13 for H 2dissociation but are still inferior to ODC-12 for moderate bond stretching. For equilibrium properties of weakly correlated molecules with more than two electrons, OUDCT approximations will not improve upon ODC-12 unless unitary cluster operators beyond doubles are accounted for. Furthermore, treating triple unitary cluster operators to four or more commutators in the BCH expansion will lead to singularities in the theory. II. THE ORBITAL-OPTIMIZED UNITARY DENSITY CUMULANT THEORY ANSATZ This section provides a self-contained exposition of DCT and the OUDCT Ansatz in particular, starting from an understanding of electron correlation at the level of Shavitt and Bartlett’s text4and a loose acquaintance with reduced density matrix (RDM) theory.11,12 Section II A derives the theoretical essentials of DCT, bypassing the intermediates κandτof Ref. 1. The degeneracies in the cumulant partial trace discussed in Sec. II A have not been discussed previ- ously. Sections II B and II C derive the Variational Unitary Cou- pled Cluster (VUCC) and Unitary DCT (UDCT) Ansätze . The latter should be compared with Ref. 9. Section II D discusses the addition of orbital optimization to the VUCC and UDCT Ansätze to produce OVUCC and OUDCT, focusing on the implications and advantages of doing so. Orbital optimization was added to DCT in Ref. 3. In thissection, we also discuss the possibility of singularities in the cumu- lant update equations, as these depend not only on the cumulant parameterization but also on the orbitals. Finally, Sec. II E formally analyzes the difference between UDCT and UCC truncated at the same degree. Throughout this section, we use primed indices to denote a quantity that must be computed in the basis of natural orbitals. A. Abstract density cumulant theory We begin by writing the Hamiltonian in the second-quantized form ˆH=hq pap q+1 4¯grs pqapq rs, (1) where hq pis the standard one-electron integral, ⟨ϕp∣ˆh∣ϕq⟩, and ¯grs pq is the antisymmetrized electron repulsion integral, ⟨pq||rs⟩. We use the notation introduced by Kutzelnigg in Ref. 13 for writing the vacuum-normal, particle-conserving second-quantized fermionic operators ( ap q=a† paqand apq rs=a† pa† qasar), and we also use the Einstein summation convention throughout this article. It follows from (1) that the energy expectation value of any normalized wavefunction Ψmay be written as E=hq pγp q+1 4¯grs pqγpq rs, (2) respectively defining the 1-electron RDM (1RDM) and 2-electron RDM (2RDM) with γp q=⟨Ψ∣ap q∣Ψ⟩ (3) and γpq rs=⟨Ψ∣apq rs∣Ψ⟩. (4) For exact wavefunctions, γpq rsis multiplicatively separable, not addi- tively separable, i.e., not size-consistent.14We may decompose it into size-consistent tensors with γpq rs=λpq rs+γp rγq s−γp sγq r. (5) It may be checked manually that if γpq rsis multiplicatively separable, λpq rsmust be zero.14These size-consistent tensors are called RDM cumulants and denoted with λ.14–21Because the 1RDM equals its cumulant, we shall usually refer to it with γ, as done in (5). When no superscripts or subscripts specify the rank of the tensor, λshall refer to the 2RDM cumulant and γshall refer to the 1RDM. Substituting (5) into (2) and using antisymmetry of ¯ggives E=(hq p+1 2¯gqs prγr s)γp q+1 4¯grs pqλpq rs. (6) This is an exact functional of γandλ. To find the exact ground-state energy, we want to minimize this functional over the set of γand λpossible, given the definitions of (3)–(5). A pair of γandλcon- sistent with those equations is said to be pure n-representable.22–27 However, the set of pure n-representable γandλhas a complicated structure, and there is no known parameterization that is necessary, sufficient, and computationally efficient. Accordingly, our strategy will be to take a parameterization that is necessary and sufficient, approximate it for computational efficiency, and vary the amplitudes until the derivative of the energy functional is zero. J. Chem. Phys. 153, 244102 (2020); doi: 10.1063/5.0036512 153, 244102-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp At first sight, we need to parameterize both γandλ. However, for a given λ, the set of γconsistent with it is strongly constrained. Whileγis not a function of λ(we will construct a counterexam- ple later in this subsection), the set of possible γis discrete for all but exceptional λ. This enables us to use implicit differentiation to treat (6) as a function of the λparameters alone for differentiation purposes, so we need only parameterize λ. We will begin by constraining the set of γthat are consistent with a given λ.1For an n-electron system, γpr qr=(n−1)γp q (7) and γp p=n. (8) [Equations (7) and (8) are easily proven by expanding Ψfrom (3) and (4) in terms of Slater determinants.] Inserting (5) into (7) yields, through straightforward algebra and an invocation of (8), dp q=(γ2−γ)p q, (9) where dp q=λpr qr. (10) The matrix dis quadratic in the matrix γ. The set of γconsistent with (9) for a given dmay be characterized as follows: If γis consistent with Eq. (9), then express Eq. (9) in an eigenbasis of γwhere eigen- vector vp′has eigenvalue γp′. The eigenvectors are called the natural spin-orbitals, and the eigenvalues are the natural spin-orbital occu- pation numbers. Then, the right-hand side of Eq. (9) is a diagonal matrix with entries γ2 p′−γp′. It follows that each eigenvector vp′is also an eigenvector of dwith eigenvalues Δp′=γ2 p′−γp′. This may be solved to yield γp′=1±√ 1 + 4Δp′ 2. (11) Choosing the + sign is consistent with γp′≥1 2, and choosing the − sign is consistent with γp′≤1 2. These choices are illustrated in Fig. 1. Therefore, for all γconsistent with Eq. (9), it is necessary that there exist some eigenbasis of dthat is also an eigenbasis of γwith eigen- values from (11). It is clear that the existence of such a deigenbasis is sufficient to satisfy (9) in the chosen eigenbasis and thus in any basis. Therefore, the set of γso constructed from dis precisely the set of solutions to (9). Equations (9) and (10) are merely necessary forγandλto be pure n-representable, not sufficient, but we shall not need sufficiency. Equivalently, solutions to (9) take the form γ=U⎡⎢⎢⎢⎣1+√1+4Δo 20 01−√1+4Δv 2⎤⎥⎥⎥⎦U−1, (12) where the matrix Uis some matrix of eigenvectors of d, Δ=U−1d U, (13) andΔoandΔvare the occupied and virtual blocks of Δ, the matrix of eigenvalues of d. A natural orbital taking the + solution of (11) FIG. 1 . The natural spin-orbital (NSO) occupation number γp′as a multi-valued function of the corresponding eigenvalue in the partial trace of the 2RDM cumu- lant, dp′. In general, a dp′eigenvalue is consistent with two possible occupation numbers, one suggesting an occupied orbital and the other suggesting a virtual orbital. is equivalent to it being a column of Uthat is multiplied against the (occupied) + block of (12) and analogously for the (virtual) −block. Depending on the eigenvalue structure of d, the set of possi- bleγfrom (12) may be either discrete or continuous. If there are no degeneracies in the dmatrix, then the eigenvectors of dare unam- biguous and so are the eigenvectors of γ. It remains only to choose whether a given natural spin-orbital occupation number should take the + sign (occupied-like) or the −sign (virtual-like) in (11). If the goal is to approximate some electronic state, these can be chosen by comparing the natural spin-orbitals to those of another approx- imation to the electronic state and choosing the signs to mimic the occupation numbers of the other approximation.28,29This other approximation may be a Hartree–Fock computation or a previous solution to (9). However, let us suppose that there is a degeneracy in the d matrix. There remains the discrete freedom in how many of an eigenspace’s eigenvectors take the + sign and how many take the −sign, but there is also a new continuous freedom in partition- ing the eigenspaces into + and −eigenvectors. A concrete exam- ple of this is a single determinant wavefunction, which has λ= 0. Then, (11) implies that the natural orbitals all have occupation num- ber 0 or 1, but the requirement that the wavefunction is a Slater determinant does not determine which orbitals are occupied and which are virtual. Instead, there is a continuous set of possible Slater determinants, as is well known from the continuous Hartree–Fock problem. This also shows that a pure n-representable λneed not uniquely determine the corresponding γ. This continuous freedom from degeneracies was not considered by previous research that derived (11).1,28,29 Let us use these results to write γas a (continuously) differen- tiable implicit function of λabout some neighborhood of a starting J. Chem. Phys. 153, 244102 (2020); doi: 10.1063/5.0036512 153, 244102-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp solution to (9). If we cast γin the basis of natural orbitals, we find that ∂ ∂dp′ q′γp′ q′=1 γp′+γq′−1. (14) This equation has singularities if γp′+γq′= 1, which are precisely the cases whered dγ(γ2−γ)fails to be invertible.d dγ(γ2−γ)is the matrix Ax in Theorem 9.28 of Ref. 30, so the hypotheses of the implicit function theorem are not satisfied in this case, and implicit differentiation fails. We may interpret these singularities via the discussion regard- ing solutions to (9). If p′=q′andγp′+γq′= 1, then orbital p′is half-occupied, and either choice of sign in (11) gives the same result. It is undetermined from (9) whether a change in dwill cause the occupation number to take the + sign and be slightly more occupied or to take the −sign and be slightly more virtual. Ifp′≠q′andγp′+γq′= 1, then p′and q′have a com- mon deigenvalue by (9), and we must decide how to split their degeneracy in the dmatrix into an occupied and a virtual orbital. Any slight perturbation of the dmatrix may break the degener- acy, causing unpredictable changes in how the orbital spaces break into an occupied and a virtual orbital. This costs differentiability because the eigenvectors are not even continuous with respect to changes in d. We may now use (14) in conjunction with (6) and (10) to min- imize the energy with respect to yet unspecified cumulant parame- ters. We can explicitly write the derivative of the energy with respect to the cumulant parameter tas ∂E ∂t=˜Fq p∂dp q ∂t+¯grs pq∂λpq rs ∂t, (15) defining ˜Fq′ p′=hq′ p′+¯gq′s′ p′r′γr′ s′ np′+nq′−1, (16) which must first be computed in the basis of natural spin-orbitals before being transformed back to the original orbital basis in which the cumulant was constructed.2It remains only to choose a cumu- lant parameterization. We note that the formula γ=κ+τused in the original DCT paper1has been entirely eliminated in this presentation. Previous work31also presented formulas without this decomposition but did not discuss them in detail. Contrary to the claims of Ref. 1, κisnota variable independent of λ, as the two strongly constrain each other. However, all previous DCT numerical studies2,3,6–9,32–35obeyed this constraint. The new constraint is discussed in the Appendix. B. Variational unitary coupled cluster Ansatz Assume that any wavefunction, Ψ, may be written as ∣Ψ⟩=exp(T−T†)∣Φ⟩ (17) for a reference determinant | Φ⟩where T=T1+T2+⋯ (18)and Tn=(1 n!)2 tij... ab...aab... ij.... (19) In other words, it is assumed that the unitary coupled cluster (UCC) Ansatz36–43is exact. The validity of this assumption has been studied by Evangelista, Chan, and Scuseria.44 The energy expectation value of this wavefunction is given by (2), where the RDM formulas (3) and (4) may be written as functions of the amplitudes t, γp q(t)=⟨Φ∣exp(T†−T)ap qexp(T−T†)∣Φ⟩ (20) and γpq rs(t)=⟨Φ∣exp(T†−T)apq rsexp(T−T†)∣Φ⟩. (21) By using the Baker–Campbell–Hausdorff expansion, (20) and (21) may be written as γp q(t)=∞ ∑ n=01 n!⟨Φ∣[⋅,T−T†]n(ap q)∣Φ⟩ (22) and γpq rs(t)=∞ ∑ n=01 n!⟨Φ∣[⋅,T−T†]n(apq rs)∣Φ⟩, (23) where the function [ ⋅,T](H) sends Hto [H,T]. The variational unitary coupled cluster (VUCC) Ansatz con- sists of approximating the functions (22) and (23), using those approximations in (2) to construct an approximate energy func- tion of the amplitudes t, and taking the energy as the variational minimum of that function. This is equivalent to the more usual definition, where the energy function is defined directly as E(t)=∞ ∑ n=01 n!⟨Φ∣[⋅,T−T†]n(H)∣Φ⟩, (24) but using RDM intermediates will facilitate comparison with DCT. C. Unitary density cumulant theory From (5), (22), and (23), we immediately have an exact function from the amplitudes ttoλ. Furthermore, this parameterizes only pure n-representable cumulants, and if the UCC Ansatz is exact, this parameterizes all pure n-representable cumulants. We can thus approximate the map from the tamplitudes to λand use density cumulant theory as developed in Sec. II A to approximate (3) and (4) and perform the variational unitary coupled cluster of Sec. II B. The only source of error is how we approximate the map from t amplitudes to the cumulant. Constructing the cumulant function by inserting (22) and (23) into (5) will lead to a large cancellation of terms. We can instead equate the connected terms on both sides of (23).9This is valid because that is the only way to divide the terms of (23) into pieces with the additive separability structure of (5). Every connected term must be assigned to the cumulant because it cannot arise as a prod- uct of disconnected pieces. No disconnected term can be assigned to the cumulant because by the linear independence of monomials in any variables (here the tamplitudes), the cumulant would not be J. Chem. Phys. 153, 244102 (2020); doi: 10.1063/5.0036512 153, 244102-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp zero as a polynomial in the amplitudes if its orbitals correspond to independent subsystems. Doing this yields the exact relation9 λpq rs(t)=∞ ∑ n=01 n!⟨Φ∣[⋅,T−T†]n(apq rs)∣Φ⟩C. (25) Let us make a few observations about these equations: 1. It is natural to approximate (25) by truncating its Taylor series expansion at some degree in the cluster operators T. These degrees in Tare what Ref. 9 meant by orders in perturbation theory. Although methods of the Ansatz can be analyzed in terms of the terms produced upon Møller–Plesset partitioning of the molecular Hamiltonian (MPPT),2,3and MPPT played a prominent role in the derivation of the cumulant approxima- tion of ODC-12,1,45MPPT is not necessary to formulate the Ansatz . 2. All DCT publications1–3,6–8,32–35excluding Ref. 9 parame- terized the cumulant in terms of parameters tij absatisfying λij ab=tij ab. This can be derived by approximating the UDCT cumulant parameterization (25) to two commutators and T=T2. The equation λij ab=tij abcan alternatively be interpreted as identifying the parameters as cumulant elements, hence why the parameters were written as λij abin many DCT publica- tions.1–3,7,32,33If the cluster operator includes T2andT4and is truncated after two commutators, or if it includes T2but is truncated after three commutators, approximations to (25) will no longer be consistent with λij ab=tij ab, and the amplitudes can no longer be identified with cumulant elements. For the exact Ansatz of Ref. 9, (25) shows that the parameters must be identified as unitary coupled cluster amplitudes. 3. As UDCT determines the energy by variationally minimiz- ing an approximation to (24), it can be regarded as a VUCC method. Specifically, UDCT constructs some number of cumu- lant diagrams and uses a power series of their partial trace to construct an infinite sum of non-cumulant (1RDM and prod- ucts of the 1RDM) terms. The mechanism of this summation is discussed in greater detail in Sec. II E. 4. One could approximate (25) differently for the purposes of constructing din (11) and of constructing the λcomponent of (23). This was done in the ODC-13 method of Ref. 9, which used a degree-four truncation and a degree-three truncation, respectively. Reference 9 attributed this truncation to differ- ent degrees for the poor performance of ODC-13. None of the other methods implemented in this work use this uneven truncation strategy. 5. Commutator truncations are not the only way to approx- imate (23) and the derived (25), although they are widely used.38,39,46–48For example, there is a recursive commutator approximation49–52where high-rank second-quantized oper- ators are projected out of commutators. If only RDMs at the converged amplitudes are necessary, there are also truncations of the inherently projective Bernoulli functional.53–56 D. Orbital-optimized unitary methods The tamplitudes appearing in (19)–(25) imply a division of orbitals into occupied and virtual spaces. While most electronicstructure methods relying on such a partition choose this divi- sion based on Hartree–Fock orbitals, it is possible to vary these orbitals over a computation. There are multiple possible criteria for what the converged orbitals of a computation are.1,57,58If we per- form VUCC or UDCT with the orbitals that minimize the energy, we call the resulting methods orbital-optimized variational unitary coupled cluster (OVUCC) and orbital-optimized unitary density cumulant theory (OUDCT),3respectively. The stationarity condi- tions are functions of the reduced density matrices;59therefore, DCT does not need to use (14) to compute the derivative of the energy with respect to orbital rotations. Orbital optimized methods are well-studied,3,59–67and the orbital-optimized unitary coupled clus- ter has recently received attention from quantum computing.68,69 The impact of orbital optimization in density cumulant theory, com- pared to an alternative orbital convergence criterion,1is studied numerically in Refs. 6 and 3. Because the orbitals are added as parameters, varying all unitary cluster amplitudes would lead to the dimension of the variational space being greater than the dimension of the total space of wave- functions, guaranteeing a redundancy. To remedy this, the T1are set to 0. We may qualitatively think of the T1amplitudes as correspond- ing to orbital rotations because a unitary cluster operator consisting only of T1amplitudes is simply an orbital rotation.70 Adding orbital optimization to the unitary transformation of (17) is a convenient choice for multiple reasons. First, because the exact unitary coupled cluster energy is a variational upper bound to the energy for any choice of cluster operators, the argument of Köhn and Olsen that orbital optimization costs reproducing the full configuration interaction limit does not apply.71Second, eliminating theT1amplitudes reduces the number of contractions that need to be considered in (25). Third, this means that in the gradient the- ory, it is not necessary to compute an orbital relaxation term.3,61 This both makes the analytic gradient theory simple and means there is no need to distinguish between the reduced density matri- ces delivered by the theory and “relaxed density matrices” including extra Lagrangian terms. Fourth, these operators do not need to be expanded in an infinite series in the manner of (25),3so we may completely avoid errors due to truncation of an infinite series with these parameters.9,34 The fifth reason is subtler and specific to UDCT. Because γis an implicit function of d, which is in turn a function of the amplitudes t,γis an implicit function of the amplitudes. When the denominator of (14) is not zero, the chain rule gives ∂ ∂tγp′ q′=1 γp′+γq′−1∂ ∂tdp′ q′, (26) where tis an arbitrary amplitude, and we are working in the basis of natural spin-orbitals of our current 1RDM. If orbital p′is occupied andq′is virtual, or vice versa, γp′+γq′−1≈0, and the denominator of (14) becomes very small, which may produce numerical issues. This calamity is avoidable. If∂ ∂tdp′ q′=0, then the right-hand side of (26) is zero, even if the denominator is very close to zero. For all previously studied DCT models, this is true in the occupied–virtual blocks for any choice of t, so we have ∂ ∂tγv o=∂ ∂tγo v=0. (27) J. Chem. Phys. 153, 244102 (2020); doi: 10.1063/5.0036512 153, 244102-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Let us call the orbitals used to define the amplitudes the reference orbitals . The origin of (27) is that the occupied–virtual and virtual–occupied elements of dare zero for any choice of t. This means occupied natural orbitals are linear combinations of occupied orbitals, and virtual natural orbitals are linear combinations of vir- tual orbitals. Combining these facts means that after moving to the current natural orbital basis for (26), do vanddv oremain identically zero even as the cumulant changes. Their derivative must there- fore vanish. This implies γv oandγo vvanish. Intuitively, optimizing the orbitals should account for the otherwise missing correlation in these blocks. Unfortunately, do vanddv obeing zero is not a general feature of the OUDCT Ansatz . To see this, let us borrow an idea from Ref. 1 and expand γin (9) byκ+τ, whereκis the 1RDM of Φ andτis the remainder. Using the Φ-normal operator ˜ap q,τmay be expressed as τp q(t)=∞ ∑ n=01 n!⟨Φ∣[⋅,T−T†]n(˜ap q)∣Φ⟩. (28) The occupied–virtual block of both sides of (9) is given by di a=τi pτp a, (29) and we must tell when this is nonzero. If only even rank operators are included in (18), then the total excitation rank of terms in (28) is odd no matter how many even rank operators are contracted against ˜ai a, so no complete contractions are possible. Consequently, τi ais zero by (28), and di ais zero by (29). However, if all operators are present in (18), then τi aterms exist and give rise to non-vanishing di athrough (29). There are di aterms of degree three in the amplitudes, but they vanish when T1= 0 due to their dependence on the ti aterm ofτi a. Terms without T1ampli- tudes, and thus nonzero even with optimized orbitals, first appear at degree four due to degree two terms of τi a. The term that is of degree three in T2and degree one in T3isdi a=(1 8tijk bcdtcd jk)(1 2tbe lmtlm ae) +(−1 2tbe lmtim be)(1 8tljk acdtcd jk). While inserting the series expansions of (28) into (29) can lead to cancellations, we do not observe cancella- tions in this example. This formula can also be derived directly, albeit tediously, from (10). We show this in the supplementary material. The implication is that continuing the OUDCT Ansatz will eventually lead to the terms with small denominators in (26) being multiplied by something that is not identically zero. These small denominators then must be computed, which is likely to lead to numerical problems. We have tested our results numerically by performing exact orbital-optimized unitary coupled cluster (OUCC) on the Be atom in the cc-pVDZ basis set. We find that the occupied–virtual block of the 1RDM, where orbital spaces are determined by the optimal reference, has a norm of 9.7 ×10−5. This is nonzero, to machine pre- cision. A commutator expansion of the 1RDM shows that the block is numerically zero to one commutator but has a norm of 9.5 ×10−5 after the second commutator. As discussed in Sec. II E, this is the commutator at which we expect the block to first become nonzero in OUCC, and this is consistent with di abeing nonzero at the degree four terms of the cumulant, (25).We close with a technical remark. In the special case that the occupied–virtual block of dis identically zero, we can always choose the natural orbitals such that each natural orbital is obtained by diagonalizing either the occupied block or the virtual block. If we make this choice, then the 1RDM derivative (14) simpli- fies into a series of equations for the occupied block and a series for the virtual block. This also leads to (27), the occupied–virtual block of the 1RDM being zero. This construction is identical with the one from (26) in the case that γp′+γq′≠1. However, if γp′+γq′= 1, (26) is not even defined, but constructing dby diagonalizing the blocks of γseparately remains valid. This block- diagonal procedure also gives a continuous γ, and if some infinites- imal change in dbreaks the degeneracy, this is the only choice ofγthat will be continuous in the direction of that infinitesimal change. E. Comparison of UDCT and UCC truncations Suppose that series (25) is evaluated to a certain number of commutators, and the 1RDM is generated from (11). Can we con- clude that this includes all the terms from evaluating (22) to the same number of commutators? For a general cluster operator, we cannot. Evaluating (25) to one commutator gives d= 0, which fails to generate the one com- mutator contribution to γi a,ti a. Evaluating (25) to two commuta- tors gives a block-diagonal d(by Sec. II D), which cannot gener- ate anyγi aterms, such as1 8tijk abctbc jk. Because the latter term does not contain T1amplitudes, determining dto degree ndoes not guar- antee thatγis determined to degree n, even for orbital-optimized methods. To explain this puzzling fact, recall that to construct γ, (12) requires the change-of-basis matrix Ugiven by the eigenvectors of d. We may expect that to determine γto degree nin the amplitudes, we require Uto degree n. Unfortunately, determining Uto degree nfrom (13) requires dto degree n+ 2 because dis nonzero only at degree two and greater. In general, determining dto degree n+ 2 suffices to determine γto degree n. In the supplementary material, we explicitly show that this is necessary forn= 1 and n= 2, if no restrictions are put on the cluster operator. If the cluster operator is restricted so that T1= 0 but no other restrictions are imposed, it is also necessary for n= 2. There is an important special case where determining γfrom d requires fewer commutators. If Uhas the same block-diagonal struc- ture as the central matrix of (12), then (12) simplifies into a single block-diagonal matrix, both blocks of which are power series in d.1 Then, if dis correct to degree n, the power series of γmust be as well. In this special case, which this paper will focus on, a degree n truncation of OUDCT includes all the degree nterms of OVUCC, plus terms of higher degree in the tamplitudes. It is even possible to identify which terms of higher degree are included in this case, which we shall consider later. III. COMPUTATIONAL IMPLEMENTATION To conduct the studies described in Sec. IV, we created (a) a Python program to perform OUDCT and OVUCC computations within a given commutator truncation, (b) a Python program to J. Chem. Phys. 153, 244102 (2020); doi: 10.1063/5.0036512 153, 244102-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp perform an exact orbital-optimized unitary coupled cluster (OUCC) computation, and (c) a Python program to compare the amplitudes of a truncated OUDCT or OVUCC computation with those deter- mined from a full OUCC computation. Because the equations for truncated unitary theories were already quite complicated,9we cre- ated a code generator to derive the necessary tensor contractions for the OUDCT and OVUCC computations. Below, we describe the code and the correctness checks we employed to ensure the accuracy of the results.72 A. Generation and implementation of truncated orbital-optimized unitary theories Our code generator first draws all possible fully closed diagrams of the normal ordered forms of (20) and (21). Diagrams that dif- fer only in the ordering of their operators are considered distinct. Subsequently, diagrams identical by time ordering are collected into a single expression. Diagrams of (21) are separated into discon- nected diagrams, which are not explicitly used in OUDCT but are explicitly used in OVUCC, and the connected diagrams, which are used in both theories. For OUDCT, the connected 2RDM diagrams are partial traced to obtain dof (10) as an explicit function of the amplitudes. The residual equations are obtained by differentiation of the energy expressions. No special code is generated for the orbital optimization, as those expressions are kept in terms of the reduced density matrices. From orbital and amplitude residuals, update steps to the orbital parameters and amplitudes were computed using a crude diagonal approximation to the exact Hessian giving denominators of signed “orbital energies” as in Møller–Plesset perturbation theory for OVUCC and unsigned diagonal elements of (16) for OUDCT, per Sec. 2.2 of Ref. 7. Direct inversion of the iterative subspace (DIIS)73is used to accelerate the convergence of the combined vec- tor of orbital amplitudes and tamplitudes. All tensor contractions use the opt-einsum package for efficiency,74and all integrals are obtained from a developer version of P SI4 1.4.75,76To ensure tight convergence, we required that the norms of the amplitude gradi- ent and orbital gradient were both under 1 ×10−12. To address convergence problems discussed in Sec. IV A, we enabled reading in amplitudes and overlap-corrected molecular orbital coefficients from previous computations. To confirm the accuracy of the generated equations, we per- formed various checks. To confirm the accuracy of our expres- sions for the reduced density matrices, we compared our degree- four expressions for (25) to those previously published.9We also implemented a code for projective unitary coupled clus- ter with Hartree–Fock orbitals and confirmed that our energies match the previously reported energies for the commutator trun- cations studied.37Although the conditions for determining ampli- tudes differ for projective and variational unitary coupled clus- ter, the function from amplitudes to energy is the same. Hence, by confirming the correctness of the functional for the projec- tive case, we have confirmed its correctness in the variational case. To confirm the accuracy of our expressions for the dmatri- ces, we performed OUDCT computations both with our explicit expression and by simply taking the partial trace of (10). In both cases, we observed the same energy. To confirm the accuracy of ourcomputed derivatives, we have computed the dipoles for all OVUCC and OUDCT truncations studied both by finite difference and ana- lytically, since both OVUCC and OUDCT automatically deliver relaxed density matrices suitable for property computations, by the Hellmann–Feynman theorem. If the orbitals or amplitudes do not variationally minimize the energy function, the Hellmann–Feynman theorem does not apply, and the two dipoles will differ. In all cases, we found that the two matched to 10 or more decimal places. Conse- quently, we have also been able to implement the analytic gradients of these theories. Excluding ODC-13, all OUDCT methods studied use the same parameterization of the cumulant for the intermediate d(10) as for reconstructing the RDM (5). As a consequence, the partial trace sat- isfies the equation γpr qr=γp q(γr r−1). We have numerically confirmed this for all OUDCT truncations. We note that OUDCT truncations do not necessarily satisfy (7) or (8), where nis the integer num- ber of electrons, but the OVUCC truncations do. This is because (7) and (8) are true as polynomials in the unitary coupled cluster amplitudes appearing through (20) and (21), so the partial trace of each degree in tmust be zero. OVUCC either includes all or none of the terms of a given degree, but OUDCT does not, as described in Sec. II E. B. Exact orbital-optimized unitary coupled cluster We implemented a scheme to obtain the exact OUCC orbitals and amplitudes via projective unitary coupled cluster. (When the cluster operator is not truncated, the variational and the projective unitary coupled cluster are equivalent. When only T1is removed, the same is true of their orbital-optimized variants.) Our algorithm consists of macroiterations and microiterations. In each macroiteration, we solve the projective UCC equations exactly through the microiterations. This gives us a wavefunction of form exp( T−T†)|Φ⟩. If the norm of T1is less than 1 ×10−8, we have converged to the exact amplitudes and orbitals, and the algo- rithm terminates. Otherwise, we make a guess to the exact orbitals as exp(T1−T† 1)∣Φ⟩and proceed to the next macroiteration. We solve the projective UCC equations in a given one-electron basis set following the prescription of Evangelista.37We construct the Hamiltonian, H, and T−T†in the basis of determinants. We then compute exp( T†−T)Hexp(T−T†)|Φ⟩, where Φis the ref- erence determinant, via the built-in matrix exponential and matrix multiplication operators of NumPy.77If exp( T−T†)|Φ⟩is an exact eigenstate, then exp(T†−T)Hexp(T−T†)∣Φ⟩=E∣Φ⟩, (30) so we select amplitudes such that the projection of exp( T† −T)Hexp(T−T†)|Φ⟩onto all excited determinants is zero. We take steps according to the formula ΔtI A=−rI A εI AϕI A, (31) where Iis equivalent to the occupation vector of the orbitals excited from, Ais equivalent to the occupation vector of the orbitals excited to,εis the sum of virtual orbital energies minus occupied orbital energies, and ϕI Ais the phase factor between the reference deter- minant and the relevant excited determinant. Convergence of the J. Chem. Phys. 153, 244102 (2020); doi: 10.1063/5.0036512 153, 244102-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp amplitudes within a given macroiteration is accelerated by DIIS.73 In all cases, we enforce convergence when the difference of two sides of (30) has norm less than 1 ×10−9. We find that by this point, the energy is converged to within 1 ×10−14Hartrees. As a final correctness check, the energy is then compared against the full configuration interaction energy from P SI4. C. Amplitude comparisons When comparing amplitudes from the exact OUCC and an approximate OUCC, the two amplitudes of the two methods are not in the same one-electron basis. To compare these quantities, after constructing them, we move all quantities to the basis of the approx- imate theory. We then find the difference of the two quantities and compute its Euclidean norm, the square root of the sum of squares of the elements. In the context of matrices, this has also been called the Frobenius norm. This metric is invariant to unitary choices of basis to perform the comparison in and couples how well the amplitudes match with how well the orbitals match. IV. RESULTS AND DISCUSSION In this section, we consider the OUDCT and OVUCC Ansätze truncated to T=T2, with between two to five commutators. We call these methods by names such as OUDCT nand OVUCC n, where ndenotes the number of commutators. OUDCT2 is also known as ODC-12,3and OVUCC2 is also known as OCEPA(0).78We shall also consider ODC-13,9which cannot be expressed as a single commutator truncation. A. H 2dissociation The dissociation of H 2has been previously used to model the performance of DCT over a range of electron correlation strengths.3,9,34OUDCT with only a T2operator will be exactly OVUCC with only a T2operator in the limit of an infinite com- mutator expansion. Because H 2is a two-electron system, if UCC is exact, OVUCC with only a T2operator will be exact as well. Due to this close relationship between the OVUCC and OUDCT Ansätze , we first compute the dissociation curve with the truncated OVUCC Ansatz for comparison. 1. Potential energy curves Previous experience with commutator truncations of projec- tive UCC with Hartree–Fock orbitals suggests that the stronger the correlation effects are, the less accurate the given UCC commutator truncation should be.37 OVUCC illustrates this trend as well as smooth convergence with respect to the number of commutators. The error in the energy curve for H 2is shown in Fig. 2(a). For every geometry, adding another commutator decreases the error of the energy compared to FCI. When going from OVUCC2 to OVUCC3, or OVUCC4 to OVUCC5, the decrease is roughly by a factor of five. When going from OVUCC3 to OVUCC4, the decrease is by a factor of 100 in the equilibrium region but diminishes to about a factor of two by around 2.5 Å. That odd and even rank truncations of VUCC will perform differently was theorized by Kutzelnigg.13We observe no conver- gence problems except for OVUCC2, also known as OCEPA(0).78 The poor performance of OCEPA(0) for H 2dissociation has been FIG. 2 . Dissociation curves of H 2from 0.6 Å to 2.5 Å computed with low-degree commutator truncations of the (a) OVUCC and (b) OUDCT Ansätze in the cc-pVDZ basis set. reported previously3and is unsurprising, as singularities are known to appear in CEPA(0) for bond dissociation.79 OUDCT displays markedly different behavior across two regimes, shown in Fig. 2(b). For near-equilibrium geometries, with the exception of OUDCT3, we observe improved accuracy as more commutators are added, as shown by the equilibrium geometry and harmonic vibrational frequency in Table I. However, as the bond stretches, the various OUDCT meth- ods behave dramatically differently. In agreement with previous studies,3OUDCT2, also known as ODC-12, has robust performance. In contrast to the other models, its error curve does not have an exponential shape. OUDCT3 can be converged, but the energy error increases sharply and is on the same order of magnitude as the energy errors of OVUCC2. OUDCT4 can only be converged with J. Chem. Phys. 153, 244102 (2020); doi: 10.1063/5.0036512 153, 244102-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Errors in the equilibrium bond length and harmonic vibrational frequencies of H 2, relative to FCI, for approximate OUDCT and OVUCC methods with T=T2, using the cc-pVDZ basis set. OUDCT2 OVUCC2 (ODC-12) OUDCT3 ODC-13 OUDCT4 OUDCT5 [OCEPA(0)] OVUCC3 OVUCC4 OVUCC5 re(pm) 0.03 −0.10 0.08 0.00 0.00 0.12 −0.03 0.00 0.00 ω(cm−1) −5 21 −20 −1 −1 −25 12 0 1 difficulty for internuclear distances greater than 1.6 Å. We were not able to converge the equations after 1.8 Å, even reading in the ampli- tudes and orbitals (after accounting for the change in overlap matrix) from previous computations. OUDCT5 is similar but with larger energy errors. ODC-13 also encounters significant error before diverging around 1.3 Å, which was attributed to large violations in Eq. (8) andγpq pq=n(n−1), which hold for pure n-representable RDMs.9 OUDCT4 and OUDCT5 follow these equations more closely than the OUDCT2 method by an order of magnitude. See the supplementary material for the errors in (8). While we observe that the partial trace failure of ODC-13 contributes to its poor per- formance, using more consistent approximations only delays the convergence problems. 2. Amplitude residuals It may seem puzzling that OUDCT2, also known as ODC-12, yields a relatively accurate H 2dissociation curve, but less severe truncations of the same Ansatz lead to more severe errors in dis- sociation curves. Intuition would suggest that OVUCC2 is already a good approximation to the exact OUDCT Ansatz , and better approximations to the Ansatz would give better energies. To identify the flaw in this intuition, consider the difference between the exact tamplitudes and the final tamplitudes of the approximate compu- tation as a fraction of the norm of the exact amplitudes, shown in Fig. 3(a) for OVUCC and in Fig. 3(b) for OUDCT. For all OVUCC and most OUDCT truncations, low error in the energies coexist with low error in the amplitudes, compared to the exact OUCC theory. For OVUCC, less severe commutator truncations decrease both errors across the entire curve, but for OUDCT, this decrease only occurs at weakly correlated geometries. For stretched geometries, OUDCT’s error is much worse. We must attribute this to OUDCT’s partial inclusion of terms of degree higher than the truncation level, as discussed in Sec. II E. At weakly corre- lated geometries, the small value of the amplitudes means these high degree terms are negligible, leading to good accuracy. Comparing Figs. 3(a) and 3(b), one would expect an energy error curve for OUDCT2 an order of magnitude worse than OVUCC3, let alone OVUCC4 or OVUCC5. This is not the case. In OUDCT2, an accurate H 2dissociation curve coexists with large errors in the amplitudes, indicating that the static correlation toler- ance of OUDCT2 does not result from well-approximating the exact OVUCC Ansatz , at least for H 2. This suggests that the effect of better approximating the exact OVUCC Ansatz will be difficult to predict, in agreement with Sec. IV A. FIG. 3 . The difference between the converged doubles amplitudes for (a) OVUCC and (b) OUDCT theories and the exact unitary T2amplitudes as a fraction of the norm of the exact OVUCC amplitudes for H 2computed with the cc-pVDZ basis set. J. Chem. Phys. 153, 244102 (2020); doi: 10.1063/5.0036512 153, 244102-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Alternative qualitative explanations for OUDCT2’s static cor- relation tolerance are beyond the scope of this work, but we intend to explore this in the future. B. Equilibrium properties of diatomics To assess the performance of OUDCT methods for equi- librium properties of diatomics in systems of more than two electrons, we have computed the equilibrium geometries and harmonic vibrational frequencies of eight diatomics using OUDCT and OVUCC methods with the cluster operator (18) truncated to T2 and the cc-pCVDZ basis set. To exclude non-Born–Oppenheimer effects and basis set convergence, we compare to high level ab initio results. Specifically, we compare to properties computed at the CCSDTQ(P)/cc-pCVDZ level with MRCC,80,81driven using P SI4.75 The necessary gradients were computed by finite difference of energies. Frequencies for these systems were computed using the DIATOMIC module of P SI4. For the OUDCT and OVUCC methods, we instead computed gradients analytically and fre- quencies by finite difference of gradients with a five-point stencil. The errors in the equilibrium geometries are shown in Fig. 4, and individual data points for equilibrium geometries and har- monic frequencies are given in Tables II and III, respectively. The ordering of methods in terms of accuracy is consistent across both sets of benchmark data. The best performance is displayed by OVUCC2, more commonly known as OCEPA(0).78The mean unsigned error in bond lengths is 0.50 pm, and the same in harmonic vibrational frequencies is 32 cm−1. The second best is OUDCT2 or ODC-12,3with respective mean unsigned errors of 0.79 pm and 54 cm−1. All higher order methods display slightly worse performance, on average. The degree three truncations do exceptionally poorly. The four and five commutator truncations have extremely similar performance. Mean signed geometry errors range from 0.88 pm to 0.91 pm, and mean unsigned harmonic frequency errors range from 65 cm−1to 68 cm−1. The ODC-13 method is also similar with 0.89 pm and 63 cm−1errors, respectively. We reach two conclusions from these data. First, the perfor- mance of ODC-13 for equilibrium properties of diatomic molecules is unrelated to point 4 of Sec. II C. The performance difference between the theory where the degree four connected terms of (23) are included (OUDCT4) and the theory where they are not (ODC-13) is statistically insignificant. Second, the similarity of these results upon increasing com- mutator truncations suggest that by four commutators, the equi- librium properties of these systems are well-converged to the exact result with respect to the number of commutators, and the differ- ence between OVUCC and OUDCT is negligible. This is supported by our findings that both energy and amplitudes are well-converged by four or five commutators for H 2near equilibrium in Sec. IV A. To outperform OUDCT2, OUDCT5 would need to lower Δabsby 12 pm and 13 cm−1for equilibrium bond lengths and harmonic frequencies, respectively. Accordingly, we expect that not even the exact OVUCC and OUDCT doubles theories (they are identical) can outperform OVUCC2 or OUDCT2. This strongly suggests that to improve beyond OVUCC2 and OUDCT2 within orbital-optimized FIG. 4 . The mean absolute error and standard deviation of the signed errors in the (a) geometries and (b) frequencies of diatomics, relative to CCSDTQ(P), for approximate OUDCT and OVUCC methods with T=T2, using the cc-pCVDZ basis set. unitary Ansätze , it will be necessary to consider cluster operators beyond doubles. To our knowledge, the only studies of unitary clus- ter operators beyond doubles are the recent work of Li and Evange- lista,82focused on their driven similarity renormalization group, and the non-iterative λ3correction considered within density cumulant theory.9 J. Chem. Phys. 153, 244102 (2020); doi: 10.1063/5.0036512 153, 244102-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II . Errors in the geometries of diatomics (pm), relative to CCSDTQ(P), for approximate OUDCT and OVUCC methods with T=T2, using the cc-pCVDZ basis set. Δabs denotes the mean absolute error, and Δstddenotes the standard deviation of signed errors. Molecule OUDCT2 (ODC-12) OUDCT3 ODC-13 OUDCT4 OUDCT5 OVUCC2 [OCEPA(0)] OVUCC3 OVUCC4 OVUCC5 N2 −0.51 −1.00 −0.60 −0.59 −0.61 −0.26 −0.90 −0.60 −0.62 CO −0.63 −0.92 −0.68 −0.66 −0.68 −0.50 −0.84 −0.66 −0.68 N2+−0.69 −1.36 −0.73 −0.76 −0.82 −0.14 −1.22 −0.78 −0.83 BO −0.84 −1.10 −0.88 −0.84 −0.86 −0.68 −1.01 −0.84 −0.86 CN −0.69 −1.45 −0.84 −0.87 −0.91 −0.23 −1.30 −0.88 −0.92 NF −0.65 −1.41 −0.95 −1.02 −1.02 −0.37 −1.29 −1.03 −1.03 NO −0.73 −1.32 −0.92 −0.89 −0.91 −0.47 −1.21 −0.90 −0.92 BeO −1.56 −1.81 −1.50 −1.39 −1.40 −1.31 −1.65 −1.39 −1.44 Δabs 0.79 1.30 0.89 0.88 0.91 −0.50 1.18 0.89 0.91 Δstd 0.33 0.29 0.27 0.25 0.25 0.37 0.26 0.25 0.25 TABLE III . Errors in the harmonic frequencies of diatomics (cm−1), relative to CCSDTQ(P), for approximate OUDCT and OVUCC methods with T=T2, using the cc-pCVDZ basis set. Δabsdenotes the mean absolute error, and Δstddenotes the standard deviation of signed errors. Molecule OUDCT2 (ODC-12) OUDCT3 ODC-13 OUDCT4 OUDCT5 OVUCC2 [OCEPA(0)] OVUCC3 OVUCC4 OVUCC5 N2 56 116 67 66 69 21 105 67 70 CO 67 91 72 68 70 55 85 68 70 N2+50 137 54 73 79 −38 124 75 81 BO 69 88 72 68 70 56 82 69 70 CN 46 106 56 59 63 −1 95 60 64 NF 20 66 40 47 47 3 61 48 47 NO 71 132 92 89 92 43 122 90 93 BeO 55 69 53 47 50 40 61 47 50 Δabs 54 101 63 65 68 32 92 66 68 Δstd 16 25 15 13 14 31 23 13 14 V. CONCLUSIONS In this research, we have studied the orbital-optimized uni- tary Ansatz for density cumulant theory (OUDCT) both formally and with numerical simulations of H 2dissociation and the equi- librium geometries and frequencies of diatomic molecules using low order truncations of the OUDCT Ansatz with a cluster oper- ator truncated to double excitations and de-excitations. We have also performed these simulations on analogous truncations of the closely related orbital-optimized variational unitary coupled cluster (OVUCC) Ansatz . We find the following: 1. The DCT Ansatz will encounter near-zero denominators in the gradient of the energy with respect to amplitudes if the occupied–virtual and virtual–occupied blocks of the 1-electron reduced density matrix (1RDM) are not identically zero. The OUDCT Ansatz does notpreserve this property once odd-rank cluster operators are added to the Ansatz . The terms that cause these problems will first appear at degree four in the Baker– Campbell–Hausdorff expansion of the density cumulant (25). IfT1is included in the cluster operator, they first appear at degree three. 2. The relationship between the OUDCT Ansatz and the OVUCC Ansatz is complicated by the presence of nonzerooccupied–virtual and virtual–occupied blocks of the 1RDM. If these blocks are identically zero, OUDCT truncated to n commutators is OVUCC truncated to ncommutators plus 1RDM and disconnected 2RDM terms of degree greater than nin the amplitudes. If these blocks are not identically zero, OUDCT truncated to ncommutators will have all 1RDM terms truncated to n−2 commutators but may miss terms at commutators n−1 and n. 3. Making less severe truncations of the OUDCT Ansatz does not uniformly improve the description of the H 2dissociation curve. While it is strongly improved near equilibrium, the degree four and five theories show worse performance and convergence problems not present for the simple two-commutator trunca- tion, ODC-12, away from equilibrium. The same is not true for OVUCC, where the same truncation procedure improves the entire curve. 4. Making less severe truncations of the OUDCT Ansatz with doubles does not improve the description of the equilib- rium properties of diatomics. Including the terms from three, four, and five commutators from OUDCT and OVUCC tends to cause a minor loss of accuracy compared to the two-commutator truncations, ODC-12 and OCEPA(0). Based on the rate of convergence with respect to commutator J. Chem. Phys. 153, 244102 (2020); doi: 10.1063/5.0036512 153, 244102-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp truncation, even the doubles-only OVUCC theory with no commutator truncation is likely inferior to ODC-12 and OCEPA(0). Let us remark on what these results mean for future develop- ments of DCT. If one decides to develop the theory via the OUDCT Ansatz , then to improve the description of molecules at equilibrium, the results of Sec. IV B advise against better approximating OVUCC doubles and in favor of including effects of higher-rank cluster oper- ators. Triples seem to be especially important in unitary theories, as in traditional coupled cluster theory,82and the non-iterative λ3 correction was seen to improve DCT results.9However, two special dangers then arise: 1. Triples approximations must avoid near-zero denominators in (16). We see three ways to control these singularities. First, the choice of energy-minimizing orbitals while neglecting T1 could be replaced in favor of the natural orbitals, where the occupied–virtual block of the 1RDM is identically zero. Sec- ond, choose the amplitudes non-variationally so that (16) and its singularities are irrelevant. Either of these options makes a parameter non-variational and results in a more complicated and expensive analytic gradient theory. Third, refuse to con- sider theories where the block-diagonal structure of the 1RDM is compromised during iterations. Such an approach cannot account for the nonzero terms in the occupied–virtual block of the 1RDM and cannot converge to the exact theory. Done per- turbatively, the success of the λ3correction9suggests that this route can still be quite accurate. For iterative approaches, the first terms in the BCH expansion that must be neglected are of degree four and have both doubles and triples. Neglecting this means that degree two terms in the 1RDM are also neglected, as shown in Sec. II E. Nonetheless, our unpublished numerical results indicate that including iterative triples to the degree two truncation of the cumulant (25) can still be quite accurate near equilibrium. 2. The cumulant parameterization determines the accuracy of the DCT theory, but the relationship between the degree of trun- cation of the OUDCT Ansatz and the accuracy of the resulting theory is not straightforward. Section IV A demonstrates that including more commutators in the OUDCT Ansatz can make the numerical results significantly less accurate. We expect the relationship to be even more complicated once triples amplitudes are included. However, if one wishes to improve static correlation tolerance, then the proper description of H 2is a prerequisite, and this can- not be accomplished by adding triples. Section IV A 2 makes clear that the static correlation tolerance of ODC-12 does not originate in the method well-approximating the OVUCC Ansatz . If one wishes to make a multireference generalization of ODC-12, there is not an obvious feature of ODC-12 that causes its static correlation tolerance and is therefore worth generalizing. Further theoretical developments in DCT are needed to provide a path to more and more accurate methods. We intend to investigate a new Ansatz more suitable for the description of static correla- tion and where the occupied–virtual blocks of the 1RDM can be guaranteed to be zero in a future publication.SUPPLEMENTARY MATERIAL The supplementary material includes the energies, cluster oper- ator norms, amplitude errors, and partial trace defects for H 2com- puted by various unitary methods; the optimized diatomic geome- tries by various unitary methods; an explicit demonstration of the role of the matrix Uin DCT’s construction of the 1RDM; and an explicit computation of the lowest order di aterms in the OUDCT Ansatz . ACKNOWLEDGMENTS We acknowledge support from the National Science Founda- tion, Grant No. CHE-1661604. We acknowledge helpful discussions with Professor Francesco Evangelista and Dr. Chenyang Li on iter- ative triples approximations in unitary theories, implementations of exact unitary coupled cluster theories, and the ability of truncated unitary coupled cluster to model static correlation. APPENDIX: ANALYSIS OF THE κAND τ DECOMPOSITION OF THE 1RDM IN DCT Many earlier DCT papers express the energy as a functional of an idempotent part of the 1RDM, κ, and the cumulant λ,1–3,6,7,32,33 withκsaid to be independent of λ.1,6,7,9,32Previously reported pure n-representability constraints on the arguments of this functional were incomplete. We first derive the constraints and then analyze previous DCT work in terms of the complete n-representability constraints within this formalism. Theκ,λformalism decomposes γasκ+τ.κis defined1,2to be the “best idempotent approximation” to γ.83That is,κfor an n-electron wavefunction is the 1RDM of a Slater determinant, and its occupied orbitals are the wavefunction’s nnatural spin-orbitals with the highest occupation numbers. κ,τ, and the cumulant partial trace dthen have a common eigenbasis of the natural spin-orbitals. τcan be determined by τp′=1±√ 1 + 4Δp′ 2−κp′, (A1) where pindexes the eigenvectors, and quantities in the natural spin- orbital basis are denoted with primed indices. κp′is 1 for an occupied natural spin-orbital and 0 otherwise, and Δp′refers to a deigen- value. Assuming that there are nnatural spin-orbitals with occupa- tion number ≥0.5 and all others have occupation number ≤0.5, (A1) simplifies to τi′=−1 +√1 + 4Δi′ 2, (A2) τa′=1−√1 + 4Δa′ 2. (A3) Here,τdepends on both κandλby (A1) and (10), respectively. In every case, κprescribes when to use the + solution of (11) and when to use the −solution.κalso prescribes how to resolve degen- erate eigenvectors in the dmatrix, if any, into occupied and virtual orbitals. J. Chem. Phys. 153, 244102 (2020); doi: 10.1063/5.0036512 153, 244102-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Using (6), the energy is then expressed as a functional of κ andλ,1 E(κ,λ)=hq p(κp q+τp q(κ,λ))+1 2¯grs pq(κp r+τp r(κ,λ)) ×(κq s+τq s(κ,λ))+1 4¯grs pqλpq rs. (A4) It is now necessary to consider what constraints must be placed onκandλ. Some are already known: 1.λmust be derived from some wavefunction, that is, be pure n-representable.1 2.κmust be derived from some wavefunction, that is, be an idempotent density matrix with trace n.1 However, the following have not been previously reported: 3. (A4) is only defined for pairs of κ,λderived from the same wavefunction. This is a stronger requirement than the above. For example, it was already known1thatκanddmust share an eigenbasis to satisfy this condition. This has been recognized as crucial in (A1) but has not been recognized as an additional constraint on the parameters of the energy functional. 4. Ifκanddhave a common eigenbasis, then contrary to previ- ous reports,1,6,7,9,32κandλare not independent. The set of λ allowed for a given κdepends on κ, and likewise, the set of κ allowed for a given λdepends on λ. Insisting that (A4) is variationally minimized with respect to vari- ations ofκunaccompanied by λvariations produces the DCT κ stationarity equation of Eq. (22) from Ref. 1. No DCT numerical studies varied κin this unphysical way.2,3,6–9,32–35In all cases, all variations of κwere coupled to vari- ations ofλso thatκanddcomputed from λhad a common eigen- basis, consistent with this constraint. To explain the matter, it is convenient to change the variables of (A4). First, define κp q(U)=(U†)p p′(κ′)p′ q′Uq′ q, (A5) whereκ′is an arbitrary but fixed idempotent density matrix with trace nandUis a unitary transformation. Then, varying κso that it remains an idempotent density matrix with trace nis equivalent to varying U, subject to it remaining unitary. κ′has always been chosen to be diagonal, so its orbitals do not mix occupied and vir- tual natural spin-orbitals. Choosing the primed indices to be natural spin-orbitals is consistent with this but not required. Second, define3 λpq rs(U,λ′)=(U†)p p′(U†)q q′(λ′)p′q′ r′s′Ur′ rUs′ s. (A6) Any wavefunction with cumulant λwill give cumulant λ′after the orbital rotation specified by the matrix Uand vice versa. It follows that for unitary U,λis pure n-representable if and only if λ′is. Using this, we define a new energy functional, E(U,λ′)=E(κ(U),λ(U,λ′)). (A7) These two functionals are related by a change of variables. While they have different functional dependence on their variables, theirphysical content is the same. However, E(U,λ′) follows previous DCT publications more closely.2,3,6–9,32–35They parameterized λ′as a finite polynomial in cumulant amplitudes and constructed λfrom (A6).2,3,6–9,32–35They allowed for variations of Uthat did not vary λ′, where variations of κthat did not vary λwere not allowed. This is true of both DC32and ODC3methods. The difference between the two is their Ustationarity condition. DC methods chose Uin E(U,λ′) to make the approximate E(κ,λ) stationarity with respect toκ, while ODC methods chose UinE(U,λ′) to make the approx- imate E(U,λ′) stationary with respect to U. Although DC meth- ods enforce stationary with respect to κvariations that violate n- representability, the rotations of Uused to satisfy that constraint preserve n-representability. Accordingly, the different orbital sta- tionarity condition does not directly affect the n-representability of DC methods. Let us consider E(U,λ′)’s constraints on its arguments. We now have the following: 1.λ′must be pure n-representable. 2.Umust be a unitary matrix. 3. (A7) is only defined for λ′that can be derived from a wavefunc- tion that also yields the κ′appearing in (A5). This implies that κ′andλ′have a common eigenbasis. In the typical case that κ′is chosen block-diagonal in the occupied and virtual blocks, this means the same must be true of d. 4. The set of admissible λ′is independent of U. It does depend on κ′, butκ′does not vary. All previous DCT numerical studies2,3,6–9,32–35parameterized λ′with block-diagonal d, although this was not mentioned as a necessary constraint for (A7). As discussed in Sec. II D, the general OUDCT Ansatz of Ref. 9 does not follow this constraint of block-diagonal d. 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5.0029339.pdf

J. Chem. Phys. 153, 234103 (2020); https://doi.org/10.1063/5.0029339 153, 234103 © 2020 Author(s).Index of multi-determinantal and multi- reference character in coupled-cluster theory Cite as: J. Chem. Phys. 153, 234103 (2020); https://doi.org/10.1063/5.0029339 Submitted: 11 September 2020 . Accepted: 04 November 2020 . Published Online: 15 December 2020 Rodney J. Bartlett , Young Choon Park , Nicholas P. Bauman , Ann Melnichuk , Duminda Ranasinghe , Moneesha Ravi , and Ajith Perera ARTICLES YOU MAY BE INTERESTED IN A route to improving RPA excitation energies through its connection to equation-of- motion coupled cluster theory The Journal of Chemical Physics 153, 234101 (2020); https://doi.org/10.1063/5.0023862 Electronic structure software The Journal of Chemical Physics 153, 070401 (2020); https://doi.org/10.1063/5.0023185 Coupled-cluster techniques for computational chemistry: The CFOUR program package The Journal of Chemical Physics 152, 214108 (2020); https://doi.org/10.1063/5.0004837The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Index of multi-determinantal and multi-reference character in coupled-cluster theory Cite as: J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 Submitted: 11 September 2020 •Accepted: 4 November 2020 • Published Online: 15 December 2020 Rodney J. Bartlett,1,a) Young Choon Park,1 Nicholas P. Bauman,1,2 Ann Melnichuk,1 Duminda Ranasinghe,3 Moneesha Ravi,1 and Ajith Perera1 AFFILIATIONS 1Quantum Theory Project, University of Florida, Gainesville, Florida 32611-8435, USA 2William R. Wiley Environmental Molecular Sciences Laboratory, Battelle, Pacific Northwest National Laboratory, K8-91, P.O. Box 999, Richland, Washington 99352, USA 3Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA a)Author to whom correspondence should be addressed: bartlett@qtp.ufl.edu ABSTRACT A full configuration interaction calculation (FCI) ultimately defines the innate molecular orbital description of a molecule. Its density matrix and the natural orbitals obtained from it quantify the difference between having N-dominantly occupied orbitals in a reference determinant for a wavefunction to describe N-correlated electrons and how many of those N-electrons are left to the remaining virtual orbitals. The latter provides a measure of the multi-determinantal character (MDC) required to be in a wavefunction. MDC is further split into a weak correlation part and a part that indicates stronger correlation often called multi-reference character (MRC). If several virtual orbitals have high occupation numbers, then one might argue that these additional orbitals should be allowed to have a larger role in the calculation, as in MR methods, such as MCSCF, MR-CI, or MR-coupled-cluster (MR-CC), to provide adequate approximations toward the FCI. However, there are problems with any of these MR methods that complicate the calculations compared to the uniformity and ease of application of single-reference CC calculations (SR-CC) and their operationally single-reference equation-of-motion (EOM-CC) extensions. As SR-CC theory is used in most of today’s “predictive” calculations, an assessment of the accuracy of SR-CC at some truncation of the cluster operator would help to quantify how large an issue MRC actually is in a calculation, and how it might be alleviated while retaining the convenient SR computational character of CC/EOM-CC. This paper defines indices that identify MRC situations and help assess how reliable a given calculation is. Published under license by AIP Publishing. https://doi.org/10.1063/5.0029339 .,s I. INTRODUCTION Single-reference coupled-cluster theory (SR-CC)1–3and its equation-of-motion coupled-cluster (EOM-CC) extensions4–6offer the best answers for the largest number of CC accessible prob- lems encountered in the electronic structure of molecules. They also are very attractive from the user’s viewpoint as their appli- cation requires no decisions except basis set and level of corre- lation. The former is now systematized with naturally extendable bases that permit extrapolation like the cc-pVXZ7–9and atomic nat- ural orbital (ANO) sets,10,11augmented by frozen natural orbitals (FNOs)12,13that can reduce the virtual space dimension by up to ∼40%. Even further improvement in the basis can be offered by F12 methods,14–17and localization schemes can lead to CC linearscaling.18–20Hence, given the basis, the next question is what level of excited determinants need to be included in describing the correlation: singles, doubles, triples, . . .leading to the hier- archy, CCSD, CCSD(T), CCSDT-3, CCSDT, CCSDTQ, and so forth, and their EOM extensions. This structure also permits con- venient calibration of CC methods to experiment as it is easy to run many calculations at a specified level—a theoretical model chemistry in Pople’s designation—and provide expected error bars for that level of calculation. These SR-CC and EOM-CC methods can be used with any single determinant starting point whether restricted Hartree–Fock (RHF), restricted open-shell (ROHF), unre- stricted (UHF), quasi-restricted (QRHF)21Kohn–Sham (KS), first natural determinant ( N), and Brueckner ( B), or even orbital optimized (OO–CC),22just to name the most useful ones. The J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp orbital insensitivity of CCSD and beyond makes the actual CC results for the state very similar for all choices, except when the reference function is qualitatively different, for example, as in RHF vs UHF in bond breaking. Thus, whatever benefit the choice of orbitals or single determinant can provide accrues to the SR-CC. The ease of using SR-CC methods is in notable contrast to multi-reference (MR) methods that require the identification of “active” orbitals to define the determinants that need to be treated at a different level than the other determinants. For example, the active orbitals required for bond breaking are likely to be very different from those required to get the best description of excited states. Even in bond breaking, those required for one dissociation limit (ABCD →AB + CD) will be quite different from those required for another path, (ABCD →ABC + D), and the union of such active orbitals is generally a hopeless proposition, severely complicating the determi- nation of reliable, relative energy differences. Hence, the choice of active orbitals is a new parameter in a calculation, so in addition to ensuring convergence with respect to the basis set and level of cor- relation, one must also ensure that the results are independent of a particular active orbital choice. Except for the choice of active orbitals, doing multi-reference configuration interaction (MR-CI) is conceptually straight-forward as using a MR space simply generates a new list of determinants (or spin-adapted configurations) to use in the CI, as long as all can be included at representative levels.23,24However, the rea- son that CC methods have effectively replaced CI methods is the exponential ansatz and its consequent size-extensivity,25–27which ensures that the correlation problem is usually better solved by CC at a lesser computational level than CI and that energy differ- ences on potential energy surfaces, between different states, or even between states with different numbers of electrons are better while being formally applicable to infinite systems such as crystals and polymers. MR-CC does not have the MR-CI formal issues such as size- inextensivity, but it still can suffer from active orbital choice and a lack of orbital invariance that plagues the theory.28Furthermore, the MR generalizations of CC can assume many different forms with different advantages and disadvantages (for a review, see the work of Lyakh et al.29), and it will be sometime before such MR- CC methods can be used with the same ease as SR-CC. Thus, an index that serves as a guide for the appropriateness of a given SR-CC calculation would be extremely valuable. Some MR problems occur when orbitals in a calculation become quasi-degenerate, call them I and A, which in the simplest case would correspond to the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO). Then, instead of a well-defined SR based upon one of the determinants, (core)I αIβ, the determinant (core)A αAβshould be highly weighted in the full CI wavefunction at some geometries and can be degen- erate at full separation like the σgandσuorbitals in H 2. When I and A are of the same symmetry, there will also be single excita- tions (core)I αAβ–(core)I βAα. This is a complete 2 ×2 MR space. The fact that the latter singlet coupled determinants can be removed at the cost of rotating the orbitals I and A30is one reason why the proposed index will emphasize the independence of orbital choice among other features. The MR solution to such a problem is to use the two determinants in a reference space, where both are treatedequivalently. The two-determinant CCSD (TD-CCSD)31,32does this for open-shell singlets and their triplet counterparts but exploits spin symmetry to simplify even this modest example of the Hilbert-space (State Universal) MR-CC.33The TD-CCSD approach was also pre- sented in its two-determinant Hilbert space form for generalized valence bond (GVB). Prior work in the direction of assessing multi-reference char- acter (MRC) falls into three categories: efforts based upon ana- lyzing the wavefunction, the energy, and quantities related to the density matrix like the von Neumann entropy.34In the first cat- egory, the largest amplitudes from T1and T2have always been listed in the ACES program system from its origin in 198235,36to help to identify such a situation and to offer a cautionary note on a calculation. T1tends to be associated primarily with the orbital choice as any single determinant can be related to any other via exp(T1)|Φ⟩=N|Φ⟩, where Nis a normalization factor. Hence, the primary focus is on T2since MR typically means at least two determinants being highly weighted that differ by a double exci- tation as in the above 2 ×2 problem and in a GVB37calculation among other situations. Examples include single bond breaking or low-spin, open-shell states. However, the demarcation line for too high an amplitude for a correlating determinant remains nebu- lous, and its value will change with the choice of orbitals. Genuine MRC should be fundamental, not simply a manifestation of orbital choice. Another wavefunction-based procedure is the use of the Shan- non index that has been proposed to have value in this context.38 Another is the so-called T1diagnostic.39–41The latter is based on HF-CCSD, rather than allowing for more general single determi- nant references. In the general context we consider, it will clearly fail to satisfy our requirements that the index should be insensitive to orbital rotation since a Brueckner rotation will force any index based solely on T1to vanish, yet the MR element in the calculation as judged by T2and other measures still persists.42 A second group of indices are based upon the energy and can be as simple as how important the (T) part of CCSD(T) is compared to CCSD or far more involved estimates that require high-order energy corrections,43such as the relative importance of quadruple and higher excitations. Since the objective is the full CI—and it will, in principle handle, any multi-reference issue—this would seem to be a valid but difficult measure to invoke. Mar- tin’s paper43should be consulted for a review of many other such indicators. What should be the properties of such an index? It should be (1) size-extensive to ensure that the index scales correctly with the number of electrons; (2) invariant to orbital rotations in the occupied and virtual space; (3) able to respond to changes in a molecule with geometry, as in bond breaking and transition states; (4) size-intensive for state differences, which should help to dis- tinguish between MR character within a cluster of units and the units themselves or between two different electronic states; (5) able to assist in separating required combinations of deter- minants for particular spin states as in low-spin states of molecules from more fundamental MR problems. J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Two other properties that are helpful are as follows: (6) insensitive to orbital transformations that mix the occupied and virtual spaces such as B,N, KS, QRHF, OO, and CAS- SCF; (7) quasi-independent of even the AO basis once a reasonable level like cc-pVTZ is reached. In brief, the problem either requires a true MR (as opposed to an MD) description or not. If that can be changed by rotating orbitals, then that amounts to a very low level of MR charac- ter that SR-CC methods can accommodate with different orbitals. For high-spin situations, SR-CC seldom has any numerical prob- lem, and even its spin-multiplicity is excellent,44,45but the spin- adaptation issue automatic in CI is very hard to manage in SR-CC for general open-shells,46like low-spin cases. This sometimes sug- gests the need for such a multi-determinantal reference. Future work should ask the question how important is having a spin- eigenfunction for a particular spin state that forces required com- binations of determinants compared to other more general MR aspects. One SR-CC approach to this problem is to “tailor” certain amplitudes as dictated by spin or spatial symmetry47,48so that at least that combination is a component of the correlated CC wavefunc- tion. This can significantly improve the computed spin multiplic- ity, e.g., to the degree that internally contracted MR-CC49provides such spin eigenfunctions unlike other MR-CC methods, one can ask how important is the spin-adaptation compared to the rest of the MRC in the wave function. If dominant, there is hope that alterna- tive spin algebra approaches for spin symmetry might be added to SR-CC. The above suggests that the best foundation for a meaningful index is the density matrix.50As the fundamental invariant in the system, the one-particle reduced density matrix is extensive, inten- sive as in EOM-CC excited states, invariant to rotations among occu- pied and virtual orbitals, and starts to become insensitive to rotations that even mix occupied and virtual orbitals due to having exp( T1) in the CC wavefunction. The latter pertains to the use of Brueck- ner, natural, Kohn–Sham, QRHF, CAS-SCF, or optimized orbitals (OO–CC), achieving invariance as the full CI is approached. Fur- thermore, the dimension of the one-particle reduced density matrix remains the same regardless of the excitation level of the cluster operator. Natural orbital occupation numbers identify orbitals unoccu- pied in a single reference determinant that have potentially large interaction with the electrons in the occupied orbitals in the ref- erence determinant.51These values define the extent of the multi- determinantal correlation problem, i.e., multi-determinantal char- acter (MDC), and will underlie the indices presented here. That information is taken from a CC density matrix. First, it will be used to define the external electron number (EEN), and the “variance” (¯V), as a measure of the electrons largely outside the first Nnatural spin-orbitals, to attempt to quantify the extent of the correlation. For open shells, the reference determinant will have different orbitals for different spins (DODS), providing an alpha and beta component to the density matrix. The EEN, which is size-extensive, and the vari- ance from a perfect single determinant of Nelectrons will warn the user of such a situation.From the occupation numbers of the natural orbitals, other indices are defined to offer a broad range of interpretive tools, particularly a multi-reference index (MRI) that is meant to distin- guish weak correlation (MDC) from stronger effects (MRC). With further development, it is hoped that the indices proposed here will reveal when a SR-CC calculation at a certain level should be adequate. That is, if CCSD shows too high a ¯Vor MRI for an open-shell singlet, will the corresponding index when applied to the TD-CCSD31,32or EOM-CCSD4–6wavefunction that correctly han- dles the two determinants show that the problem is removed? Besides obvious degeneracies as in this example, can the indices identify more general cases of “stronger” correlation? To address these questions, in the following, EEN, ¯V, and MRI will be presented and then illustrated by applications to several pro- totypical examples of what are usually called MR problems. The largest occupation number in unoccupied natural orbitals, NON is also presented. II. THEORY A. What level of theory is required to define MDC? In CC theory, the one- and two-particle response density matri- ces52,53for the corrections to a single determinant are defined in terms of TandΛ, the CC amplitudes, and the de-excitation or left-hand eigenvector of ¯H,e−THeT, γpq=⟨0∣(1 +Λ)e−T{p†q†}eT∣0⟩, (1) Γpqsr=⟨0∣(1 +Λ)e−T{p†qsr}eT∣0⟩. (2) The CC and Λamplitudes are, respectively, defined by ⟨Q∣¯H∣0⟩=0, (3) ⟨0∣(1 +Λ)|[¯H,ECC]|Q⟩=0, (4) where Qindicates the space of excited determinants orthogonal to the reference function. Note that the Λequations are linear and for- mally depend upon ECC, so unlike T, they can have disconnected terms in their amplitude equations. Once Λis determined, however, all terms in the computed density matrices in Eqs. (1) and (2) are properly linked and connected. The CC density matrices are not symmetric but give the same result when dotted into any symmetric operator. Nonetheless, for this work, they are symmetrized. Then, the symmetrized one-matrix, γ, is diagonalized to yield natural orbitals (NOs) and a set of occu- pation numbers, U†γU=n, which will provide one measure of the correlation problem. The tr γ=N, the number of electrons in the system. The distribution of electrons in the NOs can be indicative of how well the system is described by the first Nspin orbitals, the natural determinant, N. This determinant has maximum over- lap with the one-particle density matrix, so in that sense, it is the J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp “best” reference determinant for a SR-CC and density matrix analy- sis. The occupation numbers are doubly occupied for RHF exam- ples with occupation numbers up to 2, while for DODS cases as in UHF, the occupation numbers of the αorβblock have values that fall between 0 and 1. (Because of the non-Hermitian symme- try of CC theory, the response density matrix can sometimes have small negative occupation numbers, but that does not affect our analysis.) The critical quantity to compute is the variance relative to the density matrix for the Noccupied orbitals in the natural determinant, V=1 NTr(γocc−γ2 occ). (5) In the limit of perfect occupancy, the occupied density matrix is idempotent, γ2 occ=γocc, making the variance 0. Anything else gives a positive quantity for the variance. A little algebra shows that this quantity can be written in terms of the number of occupied and unoccupied orbital electrons, Noccor Nvirt, where Nocc=nocc ∑ ini, (6) Nvirt=nvrt ∑ ana, (7) N=Nocc+Nvirt=Nocc+EEN , (8) Tr(γ2 occ)=nocc ∑ in2 i. (9) Here, { ni} is the natural orbital occupation number for the i-th occu- pied n-spin orbital, while n awould be the occupation number for thea-th unoccupied orbital. Hence, ¯V=1 N(nocc ∑ ini−nocc ∑ in2 i). (10) The sum over virtuals, a, termed “external electron number” (EEN) will be used in the following. In terms of EEN, the variance can be obtained from the occupation numbers in the unoccupied orbitals, ¯V=1 N(nvrt ∑ ana−nvrt ∑ an2 a)=1 N(EEN−nvrt ∑ an2 a)=XEN−1 Nnvrt ∑ an2 a. (11) Both formulas give the same variance to numerical accuracy. Note XEN = EEN/N. The variance of Eq. (1) without the 1/ Nnormalization has been used in a difference context to describe the number of unpaired electrons by Takatsuka et al.54and Staroverov and Davidson.55 The question might arise about the quality of the computed nat- ural orbitals for defining the variance from a single determinant. It is well known that NO iterations were used by Bender and Davidson56 to maximize what could be obtained from a wavefunction limited to a fixed number of determinants in a CI calculation since the NOs mix occupied and unoccupied orbitals to try to obtain the most rapidly convergent representation for that CI. CCSD, on the other FIG. 1 . Natural orbital occupation numbers vs NO iterations for BN. hand, shows remarkable orbital insensitivity because of exp( T1) in the wavefunction and the fact that CC algorithms are never lim- ited to some number of determinants. These features pertain to a NO transformation, given the behavior of CCSD, shown in Figs. 1 and 2. The rapidity of the convergence of the NOs emphasizes their relevance for this study of MDC measure, in addition to the fact theNis the determinant that has maximum overlap to the corre- lated, reduced one-particle density matrix.50,51Of course, it should be understood that this analysis applies to any CC calculation with any reference determinant like HF as the NOs used are extracted from the computed CC response density, γ. Although intuitively appealing, EEN used by itself will clearly grow with the number of electrons. It is size-extensive as it is evalu- ated solely from summing linked diagrams, just as the CC energy is, so EEN for naphthalene (1.109 e) must be larger than that for ben- zene (0.686 e). However, XEN = EEN/N has the same value (0.016). For a cluster of M water molecules, the EEN value must be M times as large as it is for a single water molecule. Thus, there needs to be a suitable way to define a quantity that is inversely dependent on the number of electrons, N. Simply defining XEN = EEN/N alleviates part of the problem, except such a straight-forward “normalization” invalidates the expected property of additive separability that exten- sive methods must have. Whereas EEN(A) + EEN(B) = EEN(A + B), XEN(A) + XEN(B) does not equal XEN(A + B). The variance [Eq. (10)] naturally depends upon 1/N. Even though γoccin any CC calculation has to consist of linked diagrams, the extensive property of additive separability does not apply to ¯V because γocc2is not linked. FIG. 2 . Convergence of NOs with iterations for BN. J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp B. Computational details All results use the response density matrix throughout this paper and the cc-pVTZ basis set7–9with spherical basis func- tions, except when so-stated. All many-body perturbation theory (MBPT) and CC results are obtained with either the ACES II36or CFOUR57programs. For CAS-CC methods, CAS reference orbitals are obtained from the NWChem58program and fed into ACES II for the CC calculations. Full CI results with RHF reference functions are obtained using GAMESS.59Finally, all electrons are correlated in every example except when stated otherwise. C. Simple examples of density matrix occupation numbers To illustrate the role of natural orbital occupation numbers with simple but informative examples, consider the Be atom and the BH molecule. The Be atom has a well-known quasi-degeneracy between the 2s and 2p orbitals, which affects the occupation numbers as shown in Table I, causing the LUMO to have ∼0.06 occupation. This kind of value shows some quasi-degeneracy, but it is well-known that a SR-CCSD calculation is quite accurate, regardless. However, note that there is a notable difference in these values between CCSD and its MBPT perturbative approximations but little between CCSD and CCSD(T). The latter is to be expected here as Be mostly has a sepa- rated pair structure diminishing the effect of (T). To exploit the NO occupation numbers to define MDC, it is important that some easily computed approximate density matrices show many of the essen- tial features of the occupation number distribution near the full CI limit. The full CI for this example shows agreement with CCSD, CCSD(T), CCSDT-3, and ΛCCSD(T). The variance for Be is 0.0416, whereas the EEN is ∼0.06. EEN, itself, is particularly attractive as an extensive measure of “real” electrons that lie outside the perfect N determinant. Tables II and III show similar results for BH but at three geome- tries. At equilibrium, BH shows a similar behavior to Be. It manifests the quasi-degeneracy of the 3 σand 1 πorbitals, again giving an occu- pation number for the LUMO of ∼0.06. Here, the additional effect of (T) on top of CCSD is apparent but modest. Again, there is a notable difference between perturbation theory and CCSD, where the former’s inability to grow to the correct amplitudes causes a poor description of the correlated wavefunction. Unlike most stan- dard closed-shell systems where MBPT(2) provides ∼90% of the correlation energy, it is only 75% for BH. At equilibrium, the full CI occupation numbers show excellent agreement with the usual approximations. ¯Vis 0.0301.As shown in Table II, when the BH bond is stretched to 2.5 Å, the occupation of the 4 σgrows dramatically as the 4 σorbital begins to replace the nonbonding π(px/py) LUMO that occurred at equi- librium. This, of course, reflects the fact that the two determinants consisting of the HOMO 3 σand the LUMO 4 σwould be neces- sary to achieve correct separation into B + H. ¯Vhas now grown to 0.0641. This feature is even more dramatic as the bond is further stretched; ¯Vis now 0.1029. As the RHF based CC calculations are applied to larger bond distances, the discrepancies between the full configuration interaction (FCI) and the other CC approxi- mations grow. The problem with perturbative approaches is exac- erbated as the quasi-degeneracy of the problem increases. While CCSD and other infinite-order methods such as CCSDT or its iter- ative approximation, CCSDT-n, correctly approach the FCI limit, MBPT(2) and MBPT(4) cannot sufficiently grow. Because infinite- order calculations approach the desired occupancy of the natu- ral occupation numbers, which, in turn, suggests MDC, an index to capture its degree in a system should rely on infinite-order approaches. The discrepancies are particularly bad for CCSD(T) as one would expect from its incorrect dissociation for an RHF refer- ence. In fact, CCSD’s occupation numbers are substantially better at R = 6.0 Å. However, despite its perturbative element, the ΛCCSD(T) method is still relatively close to the FCI. This reflects using the Λvalues from its infinite-order solution to overcome much of the incorrect dissociation associated with the (T) correction.60–62 It is important to recognize from Table II that the last sev- eral columns refer to occupation numbers evaluated with different kinds of reference functions, B,N, QTP(00),63as an example of a KS reference, and CASSCF. Note that at the CCSD level, there is virtually no change in occupation numbers to at least three signif- icant digits. This orbital insensitivity of CCSD compared to CI is well-known, and it pays dividends in many applications of the the- ory, including that for occupation numbers and the indices that will be derived from them. Without such a condition, one would have indices that would vary with orbital choice, clearly a useless proposition. A final illustration for BH is offered by treating the occupation numbers using a UHF reference, shown in Table III. Such a refer- ence will dissociate correctly to B + H and will give rather different values for the occupation numbers. Note that now comparing the occupation numbers for the α-spin block from equilibrium and at 2.5 Å and 6.0 Å, there are only modest differences, indicating that in most respects, there is modest MDC left in the calculation. ¯Vis now only 0.0381 for the largest separation. TABLE I . The correlation energy (in hartree) and the NO occupation numbers of the three lowest energy orbitals of Be (RHF reference). Orbital MBPT(2) MBPT(4) CCSD CCSD(T) ΛCCSD(T) CCSDT-3 Full CI −0.033 562 −0.046 726 −0.050 685 −0.050 917 −0.050 896 −0.050 900 −0.050 936 1s (HOMO – 1) 1.999 1 1.999 0 1.999 1 1.999 1 1.999 1 1.999 1 1.999 1 2s (HOMO) 1.942 5 1.865 6 1.819 9 1.818 0 1.818 3 1.818 4 1.818 0 2p (LUMO)a0.016 4 0.029 9 0.058 6 0.041 9 0.041 8 0.059 2 0.059 3 aThe 2p is triply degenerate with identical occupation numbers. J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcpTABLE II . NO occupation numbers for the five lowest-energy orbitals of BH (RHF and other restricted references) at internuclear distances of 1.234 Å, 2.5Å, and 6 Å.a Reference orb.bHF B N DFT CAS Dist. Orb. MBPT(2) MBPT(4) CCSD CCSD(T) ΛCCSD(T) CCSDT-3 CCSDT CCSDTQ CCSDTQP FCI CCSD 1.234Å s(B) 1.9992 1.9991 1.9991 1.9991 1.9991 1.9991 1.9991 1.9991 1.9991 1.9991 1.9991 1.9991 1.9991 1.9991 σgc1.9735 1.9481 1.9582 1.9429 1.9432 1.9569 1.9552 1.9549 1.9549 1.9549 1.9583 1.9583 1.9582 1.9582 σgd1.9539 1.9093 1.8678 1.8700 1.8715 1.8593 1.8549 1.8539 1.8539 1.8539 1.8678 1.8678 1.8681 1.8681 px/py(B)e0.0167 0.0298 0.0592 0.0474 0.0467 0.0633 0.0650 0.0655 0.0655 0.0655 0.0592 0.0592 0.0590 0.0590 σuf0.0116 0.0098 0.0211 0.0128 0.0127 0.0221 0.0226 0.0227 0.0227 0.0227 0.0211 0.0211 0.0211 0.0211 2.500Å s(B) 1.9993 1.9992 1.9993 1.9992 1.9992 1.9993 1.9993 1.9993 1.9993 1.9993 1.9993 1.9993 1.9992 1.9992 s(B)c1.9521 1.8980 1.8879 1.8778 1.8794 1.8817 1.8757 1.8753 1.8753 1.8753 1.8888 1.8887 1.8885 1.8882 pz(B) + s(H)d1.9160 1.7905 1.7293 1.6778 1.6920 1.6967 1.6832 1.6829 1.6829 1.6829 1.7282 1.7281 1.7296 1.7305 px/py(B)e0.0163 0.0322 0.0501 0.0432 0.0427 0.0524 0.0536 0.0535 0.0535 0.0535 0.0493 0.0493 0.0497 0.0500 pz(B)−s(H)f0.0709 0.1700 0.2532 0.2744 0.2610 0.2880 0.2965 0.2970 0.2970 0.2970 0.2554 0.2555 0.2532 0.2518 6.000Å s(B) 1.9993 Unphysicalg1.9993 1.9992 1.9992 1.9993 1.9993 1.9993 1.9993 1.9992 Not 1.9993 1.9992 1.9992 s(B)c1.9454 1.8933 1.8604 1.8752 1.8868 1.8772 1.8754 1.8754 1.8754 converged 1.8936 1.8937 1.8938 pz(B) + s(H)d1.3739 1.0606 1.5415 1.0681 1.0181 1.0063 1.0043 1.0043 1.0043 1.0543 1.0573 1.0579 px/py(B)e0.0165 0.0458 0.0462 0.0428 0.0493 0.0513 0.0521 0.0521 0.0521 0.0453 0.0455 0.0451 pz(B)−s(H)f0.6135 0.9346 0.4439 0.8978 0.9821 0.9786 0.9801 0.9801 0.9801 0.9412 0.9380 0.9375 aThe response density matrices are used for all calculations. bReference orbitals used here are HF, B (Brueckner), N (Natural), DFT [QTP(00)], and CAS (CASSCF including HOMO and LUMO in the active space). For the B, N, DFT, and CAS references, the CCSD level of correlation is used to obtain NO occupation numbers. cThis orbital is 2 σat the internuclear distance 1.234 Å, but as the bond is stretched to 2.5 Åand 6Å, it simply becomes the 2s orbital on the boron atom. dThis orbital is 3 σat the internuclear distance 1.234 Åand separates to p z(B) + s(H). eThis corresponds to the degenerate nonbonding p xand p yorbitals on the B atom that are perpendicular to the internuclear axis. They have identical occupation numbers. fThis orbital is 4 σat the internuclear distance 1.234 Åand separates to p z(B)−s(H). gResults are not shown for MBPT(4) at 6 Åbecause the occupation numbers are unphysical. In the next table, UHF based MBPT(4) results are shown that overcome this problem. J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcpTABLE III . NO occupation numbers for the five lowest-energy orbitals of BH (UHF reference) at internuclear distances of 1.234 Å, 2.5Å, and 6 Å.a Dist. Orb. MBPT(2) MBPT(4) CCSD CCSD(T) ΛCCSD(T) CCSDT FCI 1.234Å 1σ, 1σ 0.9996, 0.9996 0.9996, 0.9996 0.9996, 0.9996 0.9995, 0.9995 0.9995, 0.9995 0.9996, 0.9996 0.9996, 0.9996 2σ, 2σa0.9868, 0.9868 0.9740, 0.9740 0.9791, 0.9791 0.9714, 0.9714 0.9716, 0.9716 0.9776, 0.9776 0.9774, 0.9774 3σ, 3σa0.9769, 0.9769 0.9547, 0.9547 0.9339, 0.9339 0.9350, 0.9350 0.9358, 0.9358 0.9274, 0.9274 0.9270, 0.9270 π[px/py(B)], π[px/py(B)]b0.0083, 0.0083 0.0149, 0.0149 0.0296, 0.0296 0.0237, 0.0237 0.0234, 0.0234 0.0325, 0.0325 0.0327, 0.0327 4σc0.0058, 0.0058 0.0049, 0.0049 0.0106, 0.0106 0.0052, 0.0052 0.0051, 0.0051 0.0113, 0.0113 0.0113, 0.0113 2.500Å s(B), s(B) 0.9997, 0.9997 0.9996, 0.9996 0.9997, 0.9996 0.9996, 0.9996 0.9996, 0.9996 0.9996, 0.9996 0.9996, 0.9996 s(H), s(B) 0.9975, 0.9816 0.9738, 0.9585 0.9416, 0.9472 0.9483, 0.9424 0.9374, 0.9430 0.9379, 0.9384 0.9377, 0.9377 s(B), p z(B) 0.9751, 0.9883 0.9660, 0.9762 0.8929, 0.8858 0.8276, 0.8513 0.8344, 0.8417 0.8429, 0.8424 0.8414, 0.8414 px/py(B), p x/py(B)b0.0083, 0.0073 0.0149, 0.0140 0.0248, 0.0240 0.0194, 0.0195 0.0192, 0.0193 0.0267, 0.0267 0.0267, 0.0267 pz(B), s(H) 0.0032, 0.0035 0.0055, 0.0052 0.1029, 0.1017 0.1376, 0.1236 0.1390, 0.1291 0.1477, 0.1470 0.1485, 0.1485 6.000Å s(B), s(B) 0.9997, 0.9997 0.9996, 0.9996 0.9996, 0.9996 0.9996, 0.9996 0.9996, 0.9996 0.9996, 0.9996 0.9996, 0.9996 s(H), s(B) 1.0000, 0.9822 1.0000, 0.9601 1.0000, 0.9512 1.0000, 0.9480 1.0000, 0.9484 1.0000, 0.9512 1.0000, 0.9639 s(B), p z(B) 0.9754, 0.9900 0.9463, 0.9808 0.9357, 0.9803 0.9322, 0.9790 0.9327, 0.9792 0.9357, 0.9803 0.8771, 0.9059 px/py(B), p x/py(B)b0.0082, 0.0072 0.0148, 0.0137 0.0251, 0.0234 0.0196, 0.0189 0.0195, 0.0188 0.0251, 0.0234 0.0564, 0.0532 pz(B), s(H) 0.0024, 0.0027 0.0038, 0.0042 0.0062, 0.0055 0.0038, 0.0044 0.0038, 0.0043 0.0062, 0.0055 0.0041, 0.0069 aThese orbitals are 2 σand 3 σat the internuclear distance 1.234 Åand separate to the pairs s(H) + s(B) and s(B) + p z(B). bThese orbitals correspond to the degenerate nonbonding p xand p yorbitals on the B atom that are perpendicular to the internuclear axis. They have identical occupation numbers, except at 6. Å. cThese orbitals are 4 σat the internuclear distance 1.234 Åand separate to p z(B) + s(H). J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Of course, it is always true that after the RHF to UHF instabil- ity occurs, UHF-CC calculations for a closed-shell molecule tend to show a wrong curvature in the spin-recoupling region due to spin- contamination from triplets and quintets, but slightly beyond that, no apparent difficulties in converging to the correct separated energy of B + H, but not the correct wavefunction. However, the density, the diagonal elements of the density matrix, is still correct at separation, and this property is reflected in the observed occupation numbers. D. Measure of MDC via the variance These simple examples demonstrate the role of the variance and EEN in assessing the accuracy of a CC calculation. The natural orbital analysis clearly defines EEN and MDC as it has a correspon- dence to the number of electrons that lie outside of the N-highest occupied natural orbitals, the natural determinant, N,while such an orbital approximation to the electronic structure still has qualitative meaning. This is the usual situation encountered, and the distinction from such a perfect N-electron determinant offers the first measure of MDC and eventually MRC. In the above RHF based CC results at 6 Å, only a FCI for a doubly occupied orbital enables separa- tion to B + H. However, size-extensive methods such as CC must give the correct energy subject to open-shell reference determinants, though not necessarily the correct path to separation. However, once one switches to UHF where correct separation is permitted, the n-particle reference determinant is now the UHF N. A compari- son of Tables II and III shows that the RHF FCI density matrix tends to largely occupy the last MO, now consisting of just the H atom, while the UHF shows the expected six spin–orbital occupancy. When counting the virtual occupancy, one has to be careful to rec- ognize that the 0.9375 occupancy of the orbital on the H atom in the RHF case is not part of the virtual space but a re-definition of the occupied space as the N-particle reference determinant has changed from shorter distances. At intermediate distances for closed-shell examples, one cannot expect that UHF based SR-CC calculations will fully overcome the MDC but typically do so once correct separation occurs to molecu- lar units that themselves do not suffer from MDC. This also suggests that near but even beyond the bifurcation point, one will do bet- ter for the spin multiplicity with the uncontaminated RHF reference CC result when still far enough from separation. Note that the exact energy result at any R is available from some SR-CC, where the ref- erence determinant is allowed to change along the path, whose best realization might be OO–CC,22,64since that might provide a bet- ter transition between the RHF and UHF reference-CC results than other choices. To use ¯Vto help calibrate MDC, one takes a selection of well- known single reference-based CC examples and then turns them into MR situations via bond stretching, rotation, or other consid- erations, among some other expected MR examples. The results are shown in Table IV, all subject to RHF reference functions except O 2. Table IV shows that when ¯Vvalues are less than ∼0.03, one would typically expect a SR example. The exception is H 4as a SR and MgO as a MR one. Several of the latter actually have smaller variances than the SR form of H 4but will be revealed as MR using other considerations. Table V shows the H 2matrix at equilibrium and stretched to a MR form at 5 Å. It shows the largest ¯Vamong the MRTABLE IV . Test set of 15 SR and 15 MR molecules.a SR MR ¯V XEN ¯V XEN He 0.008 0.008 Ne2+0.043 0.047 Ne 0.007 0.007 H 2 0.250 0.497 Ar 0.008 0.008 Li 2 0.084 0.164 H2 0.017 0.018 C 2 0.092 0.174 O2 0.013 0.013 N 2 0.072 0.103 N2 0.016 0.017 F 2 0.034 0.044 F2 0.012 0.012 HF 0.055 0.093 HF 0.010 0.010 BN 0.029 0.030 LiF 0.009 0.009 MgO 0.014 0.015 BeH 2 0.015 0.015 BeN−0.030 0.032 H2O 0.012 0.013 BC−0.036 0.038 H4 0.034 0.035 CN+0.035 0.038 BH 3 0.015 0.015 N 22+0.042 0.047 CH 4 0.015 0.015 H 2O 0.055 0.063 C2H4 0.017 0.017 C 2H4 0.030 0.033 aAn RHF reference is used for all systems except for O 2, where the UHF triplet was used for the ground state (3Σ− g). Geometries are shown in the supplementary material. examples partly because N is only 2. The variance here is based upon an RHF reference of N occupied orbitals, yet clearly, the full CI for H2has to have equal weights in its HOMO and LUMO at separation and that is impossible for an RHF function. Using a UHF reference, the appropriate reference UHF determinant shows that the two elec- trons are now distributed into four spin orbitals with ∼0.5ein each. Such a determinant can be viewed to be formed from σg+σuand σg−σu, each with an electron. In terms of the variance, this is now the reference single determinant. As seen from its density matrix, there is little occupancy of the “virtuals,” or as we will see, little MR character. For larger ¯Vand EEN values, its measured MDC effect might suggest an alternative approach. The latter include UHF-CC, EOM- CC, “tailoring” a SR-CC, ΛCCSD(T), or using a true MR-CC TABLE V . First five largest natural occupation numbers in the CCSD density matrix of H 2at the equilibrium bond length and 5 Åseparation. Spin NO(1) NO(2) NO(3) NO(4) NO(5) RHH= 0.7414 Å Alpha 0.982 19 0.010 02 0.002 99 0.002 10 0.002 10 Beta 0.982 19 0.010 02 0.002 99 0.002 10 0.002 10 RHH= 5.0Å Alpha 0.500 00 0.499 99 0.000 00 0.000 00 0.000 00 Beta 0.500 00 0.499 99 0.000 00 0.000 00 0.000 00 J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp method. Ideally, success for an alternative method like UHF-CC would be a reduction in ¯Vthat brings it below ∼0.03 (Fig. 3). The H 4SR example seems to be somewhat anomalous com- pared to the other SR examples and many others studied. However, the effect of 1/N in ¯Vand XEN raises the question that large num- bers of electrons will make N so large that their distinction toward any MDC or MRC feature might be lost. Note, simply adding triples and higher clusters cannot reduce the innate FCI ¯V. As better and better SR-CC approx- imations are made, the ¯Vextracted from the density matrix should be closer to the reference FCI value, when describ- ing the same electronic state. To help to assess the accuracies of the hierarchy of these approximations, one would like to know the percentage of the FCI occupation numbers accrued by the CC approximation, like σ=%(¯V(CC)−¯V(FCI))/¯V(FCI) ∼(EEN(CC)−EEN(FCI))/EEN(FCI). At low percentages, the approximation should be capable of fully accounting for the innate ¯V(FCI) value, while anything short of that presumably defines a fig- ure of merit that should be calibrated against the energy to offer an assessment of the quality of a calculation independently of the computed energy. The latter will require a reasonable estimate of the ¯V(FCI) value. As seen in the simple examples above, there is rapid convergence of a sequence of CC calculations to FCI occupa- tion numbers. Below, this issue is more fully addressed for the H 4 example.E. Illustration for H 4example An informative illustration of MRC is offered by the H4 model system, shown in Table VI, and extensively studied by Paldus and co-workers.65H4 serves as an excellent example where the degree of MRC can be controlled by geometry and occurs without any quasi-degeneracy among orbitals. At θ= 90○, the molecule becomes a square that causes a double excitation to grow to have the same weight as the reference function. At 180○, it is linear and SR. For this example, the hierarchy of methods, CCSD, CCSDT, and CCSDTQ, can be done, with the last equivalent to the FCI. In Table VI, the energies and ¯Vvalues are shown from CCSD, CCSDT, and CCSDTQ for H4. The expected behavior is observed. At the critical 90○point, the largest MRC measure arises with ¯Vbeing 0.140 compared to the SR value of 0.035. Note that there is very little change in ¯V between CCSDT and CCSDTQ results, where the latter is the FCI. There is a larger difference in the respective energies, with CCSDT often being below the FCI values in MR regions. In more SR regions, there is almost perfect agreement in energy. Energy error in CCSD is up to 10 mH in MR regions but less than 2 mH for SR cases. Further analysis of square H 4can be made as a function of the distance between the two H 2molecules, commonly known as the P4 model system.65This is shown in Table VII. Again, ¯Vis 0.140 in the FIG. 3 .¯V, XEN, and EEN of 15 SR (a) and 15 MR (b) test-sets. J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE VI . Correlation energies (in millihartree) and ¯Vvalues for the H4 model system at the CCSD, CCSDT, CCSDTQ, and ΛCCSD(T) levels of theory as a function of θ degrees. Correlation energy (millihartree) ¯V Θ SCF (a.u.) CCSD CCSDT CCSDTQ ΛCCSD(T) CCSD CCSDT CCSDTQ ΛCCSD(T) 90.0 −1.955 40 −130.36 −142.43 −140.68 −138.10 0.131 0.140 0.140 0.140 90.9 −1.964 51 −125.65 −134.80 −133.78 −131.54 0.117 0.139 0.136 0.133 91.8 −1.973 27 −122.12 −129.09 −128.53 −126.72 0.103 0.128 0.124 0.119 92.7 −1.981 68 −119.42 −124.87 −124.58 −123.14 0.091 0.113 0.110 0.105 93.6 −1.989 77 −117.29 −121.72 −121.57 −120.41 0.081 0.099 0.097 0.092 99.0 −2.031 83 −110.03 −112.31 −112.34 −111.81 0.053 0.058 0.058 0.056 103.5 −2.059 30 −106.76 −108.63 −108.67 −108.25 0.045 0.048 0.048 0.047 108.0 −2.080 99 −104.48 −106.17 −106.21 −105.84 0.041 0.044 0.044 0.043 117.0 −2.111 22 −101.53 −103.07 −103.09 −102.77 0.037 0.039 0.039 0.039 126.0 −2.129 61 −99.82 −101.29 −101.31 −101.00 0.036 0.037 0.037 0.037 135.0 −2.140 89 −98.80 −100.24 −100.26 −99.96 0.035 0.036 0.036 0.036 144.0 −2.147 87 −98.18 −99.61 −99.63 −99.34 0.034 0.036 0.036 0.036 162.0 −2.154 81 −97.56 −98.99 −99.01 −98.72 0.034 0.035 0.035 0.035 180.0 −2.156 64 −97.38 −98.80 −98.82 −98.53 0.034 0.035 0.035 0.035 MR region but 0.032 away from it. Once again, in the MR region, CCSDT falls below the FCI with CCSD as much as 10 mH above. A critical question to ask is the convergence of the energy with respect to the density matrix and the proposed MRC measures derived from it. For these two examples and BH, this is shown in Fig. 4. Many more such comparisons should be made to permit energy extrapolations. F. MRI index based on individual occupation numbers ¯Vmeasures MDC, and one would think large values would ultimately be indicative of MRC, but it appears that to further dis- tinguish between weak and strong correlation, one needs to exploit additional information about the role of the individual orbitalscontributing to MRC. Whereas ¯Vand EEN describe the total number of electrons that occupy the virtual orbital space beyond the N-orbitals of the first natural determinant, many MR problems are characterized by a few highly weighted occupation numbers, whose feature should also be assessed. Figure 5 illustrates the issue. All calculations are relative to RHF references, except O 2whose triplet state is described with a UHF. For SR situations, there are essentially no spin-orbital occupation num- bers lying between 0.1 and 0.5, while for the MR examples, one sees a quite different pattern, with a maximum occurring at 0.5 e. The latter occurs for H 2since subject to a doubly occupied σgHOMO, as the bond is stretched, the σuLUMO will become degenerate, so the two electrons will be split equally between the two orbitals, causing 0.5 e/spin orbital to be in the LUMO. Hence, relative to TABLE VII . Correlation energies (in millihartree) and ¯Vvalues for the P4 model system at the CCSD, CCSDT, CCSDTQ, and ΛCCSD(T) levels of theory as a function of distance between the two H 2fragments in the square configuration. Correlation energy (millihartree) ¯V DistanceaSCF(a.u.) CCSD CCSDT CCSDTQ ΛCCSD(T) CCSD CCSDT CCSDTQ ΛCCSD(T) 2.00 −1.955 40 −130.36 −142.43 −140.68 −138.11 0.131 0.140 0.140 0.140 2.10 −1.983 16 −118.91 −124.09 −123.83 −122.47 0.088 0.110 0.107 0.102 2.25 −2.018 74 −111.71 −114.13 −115.16 −113.58 0.057 0.064 0.064 0.062 2.50 −2.064 91 −106.15 −107.59 −107.64 −107.32 0.042 0.044 0.044 0.044 3.00 −2.122 75 −100.13 −100.97 −101.00 −100.82 0.036 0.036 0.037 0.036 4.00 −2.167 79 −93.89 −94.21 −94.22 −94.15 0.033 0.033 0.033 0.033 5.00 −2.178 87 −91.42 −91.53 −91.54 −91.51 0.032 0.032 0.032 0.032 6.00 −2.181 42 −90.59 −90.63 −90.63 −90.62 0.032 0.032 0.032 0.032 10.00 −2.182 15 −90.19 −90.19 −90.19 −90.19 0.032 0.032 0.032 0.032 aDistance between the two H 2fragments, in bohr. J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . Convergence of energy with the variance of the density matrix for dif- ferent CC approximations: (a) H 4(H4 model), (b) H 4(P4 model), and (c) BH. RHF, this indicates an MR problem. Later discussions will address UHF, but one can understand from the discussion of H 2above that now the reference single determinant Nis defined by the four spin- orbitals that will allow mixing between the original H 2HOMO and LUMO that will give a different count for highly occupied virtual orbitals. Nonetheless, the objective of an MRI is to assess the essen- tial role of such highly weighted occupation numbers beyond the ¯V measure. To accomplish this, one needs a way to properly weight the role of high occupation numbers that might dictate a MR problem, as well as for many repeated small values whose cumulative effect might suggest that. Making this assessment starts with the “proba- bility density function” (PDF) of occupation numbers for the chosen samples, shown in Fig. 6. Looking at Fig. 6, it is apparent that this density logarithmic PDF plot has much in common with a delta function, or highlysteeped Gaussian skewed to the right. On a normal scale, its delta function character is readily apparent since most orbitals have zero occupation. It should be apparent that what one expects from Figs. 5 and 6 is a very large number of near zero values. This separation allows the MRI to be relatively insensitive to the atomic basis set since the basis extent largely affects dynamical correlation. The func- tion is maximized at ni= 0.5 and retains large values for a broad range of occupation numbers above and below ni= 0.5 that should be indicative of a MR problem. Between these extrema, the function has meaningful non-zero values and allows distinguishing between cases when there are many modest values whose net result could signify a MR problem, even though there are no large individual contributions. As one would expect from the similarity to a delta function, the integral of the PDF the “Cumulative Distribution Function” (CDF), F(x), should be like a modified Heaviside distribution. This “custom” J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . Natural spin–orbital occupation numbers for (left) SR examples and (right) MR examples. distribution is shown in Fig. 7 after excluding occupation numbers <0.001 for visualization purposes and in Fig. 8. The CDF can be represented by the following equation: F(ni)=e(−λ)×(0.5−ni)q , (12) where niis the occupation number for each spin orbital, while rec- ognizing that the maximum occupation number that can occur for any spin-orbital is 0.5. The two fitting values in F(ni),λ, and q are 1.5×103and 6, respectively, determined from a “receiver operator characteristic”66plot that attempts to classify the samples into SR and MR while eliminating false positives and negatives. An alter- native procedure described in the supplementary material further establishes these parameter values. FIG. 6 . 15 SR and 15 MR examples of occupation numbers vs frequency of occurrence.It should be noted that the function F(x) [Eq. (12)] is defined between 0 and 1 and has maximum at 0.5 (see the supplementary material for details). But here the discussion is focused exclusively on the virtual orbitals. Modifying two parameters ( λand q) in Eq. (12) leads to the Heaviside step function. Summing up the values of F(ni) for each spin-orbital defines an index I, I=orb ∑ iF(ni). (13) While this index can be used to distinguish between SR and MR sit- uations, an index that is more easily interpreted can be defined by first taking the negative log(I) and then using the properties of the tanh function, MRI=−tanh(C+ log(I)). (14) FIG. 7 . Histogram of SR (blue) and MR (orange) onto the sloped Heaviside F(x). J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 8 . Plot of the distribution function F(x) [Eq. (12)] (Black) and the analogous form of the von Neumann entropy (orange). The von Neumann entropy distribution function is scaled so that the maximum is 1. This makes the multi-reference index (MRI) have the domain (−1, +1) with Ccalibrated such that the index is positive for a SR case and negative for a MR case. Figure 8 shows the CDF compared to the von Neumann entropy,34another potential density matrix-based metric whose dis- tribution is−∑niln(ni). It grows too rapidly as one moves away from ni= 0 or 1 and does not have the key structural features of the sloped Heaviside, as seen in Fig. 8. FIG. 9 . MRI vs −log(I) values for the test set of 15 MR (orange) and 15 SR (blue) species. Putting the above elements together, one obtains an MRI, and −log(I), to augment the ¯Vvalue in assessing MRC. These results are shown in Table VIII and in Fig. 9. ¯Vand MRI tend to agree for most examples, although the MR examples shown in bold tend to be ambiguous. Their ¯Vvalues are small, but the net effect of some occupation numbers tends to affect the proposed MRI. For example, consider BN, shown in Fig. 10. Note how the occupation numbers for the first few virtuals conspire into providing a MR MRI. This suggests that the other con- venient measure of MRC can be the size of the largest occupation TABLE VIII . Test set of 30 species used in the calibration of the MRI [Eq. (14)] along with their respective −log(I), ¯V, and MRI values.aThree molecules with boldface are further discussed in text. SR MR System –log( I) MRIb ¯V System –log( I) MRIb ¯V He 8.489 1.000 0.008 Ne2+−0.483 −0.998 0.043 Ne 7.943 0.999 0.007 H 2 −0.602 −1.000 0.250 Ar 7.115 0.996 0.008 Li 2 −0.602 −1.000 0.084 H2 7.727 0.999 0.017 C 2 −0.903 −1.000 0.092 O2 6.159 0.973 0.013 N 2 −0.618 −1.000 0.072 N2 5.862 0.951 0.016 F 2 −0.568 −1.000 0.034 F2 4.975 0.744 0.012 HF −0.602 −1.000 0.055 HF 7.516 0.998 0.010 BN 2.977 −0.777 0.029 LiF 7.434 0.998 0.009 MgO 2.809 −0.836 0.014 BeH 2 7.087 0.996 0.015 BeN–2.825 −0.831 0.030 H2O 7.220 0.997 0.012 BC–1.903 −0.971 0.036 H4 5.209 0.832 0.034 CN+1.548 −0.986 0.035 BH 3 7.077 0.996 0.015 N 22+0.640 −0.998 0.042 CH 4 6.979 0.995 0.015 H 2O 0.404 −0.999 0.055 C2H4 5.608 0.921 0.017 C 2H4 0.273 −0.999 0.030 aMolecular geometries and reference orbitals are listed in Table IV. bThe value of Cused was 4.016. J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-13 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 10 . 15 largest natural occupation numbers for BN. number in unoccupied natural orbitals, NON. This is the alternative to EEN that sums all such occupations, while NON focuses on only the largest individual value, as shown in Fig. 11. In Fig. 11, the reason that the three examples with small vari- ances, BeN−, BN, and MgO, show MR character is shown by the NON measure. See Appendix B for a summary of the use of the highest natural orbital occupation number, NON, for the 30 refer- ence examples. A value of 0.0583 discriminates between SR and MR 99.99% of the time, while 0.0337 is sufficient 99.73% of the time.G. Martin’s dataset of SR and MR molecules To offer an independent set of examples, consider Karton et al.’s predicted multireference molecules of their W4-11 test set, W4- 11-MR.67The W4-11 study used an energy criterion for determining the MRC based on the percent of the perturbative triples contri- bution to the total atomization energy, %TAE[(T)]. Compared to this, the density matrix-based index has several advantages. Con- sidering the rather dramatic difference between how close to the FCI’s natural orbital occupation numbers ΛCCSD(T) is compared to CCSD(T) and the known deficiencies of the latter when break- ing bonds subject to RHF references, the deficiency of %TAE[(T)] is not surprising. A better energy estimate would likely be given by the∼n7ΛCCSD(T) and CCSDT-3 methods. One of the most signif- icant disagreements found is for F 2O (see Table IX). The ¯Vvalue and the MRI considers F 2O to be single reference, whereas accord- ing to %TAE[(T)], it is highly multireference. This contradiction is also evident when other multireference indicators are consid- ered (see Table 1 of Refs. 68 and 69). Energy based criteria show more substantial deviation compared to non-energy based criteria. Furthermore, a quantity like %TAE[(T)] will depend on the basis set, whereas ¯Vand MRI are insensitive to basis. Note the relatively modest change from VTZ, to aVTZ, to aQTZ for −log(I) shown in Table IX with slightly more basis difference for a couple of the MRI values. FIG. 11 . NO occupation numbers for (a) SR and (b) MR examples. J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-14 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE IX . A comparison of −log(I), V, and MRI values for the W4-11-MR test set carried out using the cc-pVTZ (VTZ), aug-cc-pVTZ (aVTZ), and aug-cc-pVQZ (aVQZ) basis sets. −log(I) MRI ¯V System VTZ aVTZ aVQZ VTZ aVTZ aVQZ VTZ aVTZ aVQZ Be2 1.501 1.511 1.575 −0.987 −0.987 −0.985 0.042 0.042 0.042 B2 1.500 1.512 1.568 −0.987 −0.987 −0.985 0.033 0.033 0.033 BN 2.940 2.975 3.107 −0.792 −0.778 −0.721 0.029 0.029 0.028 C2 1.248 1.267 1.349 −0.992 −0.992 −0.990 0.037 0.037 0.037 OF 6.112 6.086 6.112 0.970 0.969 0.970 0.012 0.012 0.012 O3 3.687 3.712 3.791 −0.318 −0.295 −0.221 0.017 0.017 0.017 FO 2 4.203 4.272 4.345 0.185 0.251 0.318 0.014 0.015 0.015 F2O 5.430 5.415 5.454 0.888 0.885 0.893 0.013 0.013 0.013 FOOF 4.453 4.474 4.536 0.411 0.428 0.478 0.014 0.015 0.015 ClOO 5.527 5.500 5.563 0.907 0.902 0.913 0.012 0.012 0.012 Cl2O 5.696 5.665 5.692 0.933 0.929 0.932 0.010 0.011 0.011 OClO 5.773 5.727 5.754 0.942 0.937 0.940 0.011 0.012 0.012 c-HOOO 5.379 5.411 5.410 0.877 0.884 0.884 0.014 0.015 0.016 t-HOOO 5.056 5.245 5.121 0.778 0.842 0.802 0.015 0.015 0.015 S3 4.329 4.331 4.433 0.303 0.305 0.394 0.011 0.011 0.011 S4(C2v) 3.610 3.658 3.780 −0.385 −0.343 −0.232 0.011 0.011 0.011 O3and S 4stand out as barely MR problems via MRI with a small variance, much like the previous three ambiguous examples, BeN−, BN, and MgO. All have more electron occupation in the first few virtual orbitals than the corresponding SR examples, as shown in Fig. 11. The first two values for S 4are 0.070 11 and 0.035 96 and for O 3, 0.070 25 and 0.026 04. This makes the NON value >0.0583. Yet these are not as large as for the bond breaking examples. A SR-CCSDTQ calculation should certainly get both right with con- fidence, but ∼n7approximations like CCSDT-3(Q f) ought to work, too. Despite the results through aug-cc-pVQZ in Table IX, both referees questioned whether the computed indices would be sim- ilar for other types of basis set, with one arguing the superiority of the core corrected basis set (cc-pwCVTZ) for all electron cal- culations. The objective is the FCI in the basis, and the expec- tation is that one will get virtually the same indices from drop- ping the core or from all-electron calculations, as little of their numerical importance is associated with core electrons. To test a bit more, the two most ambiguous MR examples, O 3and S 4, are done with cc-pwCVTZ to compare. For O 3, the values are 3.738, −0.271, and 0.017 for −log(I), MRI, and ¯V, respectively, while S 4 gives 3.786, −0.226, and 0.012 for the same quantities. All are quite close to the results from aug-cc-pVQZ, further attesting to basis insensitivity. The other mandatory property of the proposed indices is an insensitivity to orbital choice already shown in Table II as a function of orbitals for N,B, RHF, KS, and CASSCF choices. The one excep- tion being broken symmetry UHF whose qualitative differences for bond breaking have already been discussed. Given the respective natural orbital occupations computed by any program, the routine available in the supplementary material will evaluate the various indices and the plots proposed.III. SOME NUMERICAL EXAMPLES A. Bond breaking of BH 3CO Consider the BH 3CO molecule (Fig. 12 and Table X). Unlike most molecules, it has the property that it is a closed shell that separates into closed shells (BH 3and CO), so a RHF reference is appropriate at any ground state (GS) geometry. This correct separation is reflected in the table of occupation numbers at equilibrium (1.5 Å), twice equilibrium, and at separa- tion. However, despite the fact that it has a prominent excited charge transfer state that separates into BH 3−and CO+, there seems to be no residual effect on ¯Vas all geometries show a value of 0.015 e−. The other two indices –log( I) and the binary MRI say the same thing. In fact, Fig. 12 makes it abundantly clear that all the indices are sim- ply flat as a function of R. This offers a kind of baseline for their values. The other critical observation is that at the separated limit, the extensivity of the EEN index is apparent as the EEN for BH 3CO at separation is equal to that for BH 3and CO. This is a useful fea- ture for a meaningful index, and since all results are obtained from (size-extensive) linked diagram expressions, it must follow. To the contrary, the ¯Vand MRI indices show an equally important “inten- sive” property. That is, the MRI index tries to measure “strong” correlation, while the orbital effects with occupations near 0 or 1 that primarily contribute to “dynamic” correlation are separated. This makes MRI and ¯Vto be nearly the same for the super-molecule and its component parts. Consider a cluster of repeated units, like non-interacting water molecules (Table XI). The “extensive” property guarantees that the EEN value will increase linearly with repeated units, while each unit is individually “intensive.” The distinction between the two is that an extensive property must be directly proportional to “size,” while J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-15 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 12 . Upper panel (a) shows BH 3CO PES and lower panel (b) shows EEN, −log(I), MRI, ¯V, and NON. TABLE X . Natural orbital occupation of BH 3CO at 1.5 Å, 2.5 Å, and 5.0 Å with RHF dissociation (first three rows). Next three rows indicate two fragments (BH 3and CO) and its sum (BH 3+ CO). Distance (Å) EEN −log(I) MRI ¯V NON 1.5 0.336 6.247 0.975 0.015 0.026 2.5 0.340 6.218 0.974 0.015 0.026 5.0 0.339 6.230 0.975 0.015 0.026 BH 3 0.126 7.007 0.995 0.015 0.010 CO 0.212 6.310 0.980 0.015 0.026 BH 3+ CO 0.339 6.231 0.976 0.015 0.026 the intensive unit needs to be independent of “size,” or alternatively, intensive properties correspond to differences between extensive units. “Size” can have many meanings, ranging from number of units to numbers of electrons. Hence, application of the procedure toTABLE XII . Behavior of −log (I), MRI, EEN, and V for LiF at three distances for two types of orbitals. The cc-pVTZ spherical basis is used with the CCSD wavefunction with an RHF and UHF reference. Dist. (Å) Reference −log(I) MRI EEN ¯V 1.564 RHF 7.434 0.998 0.110 0.009 3.128 RHF 7.291 0.997 0.116 0.010 5.500 UHF 7.730 0.999 0.078 0.006 FIG. 13 . Potential energy surface of LiF. Various indices at three points (black dots) are shown in Table XII. define an MRI for a “super-molecule” cluster of m water molecules demonstrates that to a very good approximation, the MRI is the same as it is for a single molecule. This behavior is shown in Table XI. The idea is that in a long, hypothetical molecule, if there is a region of MR character, the pro- posed indices should be able to capture that fact. This is tested by embedding a biradical into a cluster of water molecules. B. Lithium fluoride A molecule like LiF is a prototype for the very general issue of a curve crossing between two states of the same symmetry. At equilibrium, LiF corresponds to Li++ F−, while at separation in the gas phase, LiF →Li0+ F0. Does this make LiF a SR or MR prob- lem and at what geometry? Table XII reports the CCSD calculation performed on the LiF system using RHF and UHF orbitals. TABLE XI . Behavior of indices for two “extreme” situations. Model Example ¯V NON MRI m∗SR m∗(H2O) SR SR SR (m <∼1000) MR (m>∼1000) MR + m∗SR Biradical + m∗(Ne) SR (m >∼10) MR MR MR (m<∼10) J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-16 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Table XII reports the various MR indicators at the three dis- tances indicated in Fig. 13 for the PES. In this case, the charge- transfer state separates into closed shells. Thus, the excited state can be described by SR-CC, while the ground state separation is to open- shells. All indices show SR character at the points sampled, including the crossing point. In principle, one could have individual indicesfor the two crossing states, but the problem is hard to converge at the crossing point to obtain two separate solutions. The indices for the two states involved plus the broken sym- metry one indicate at the three geometries indicated by black dots that all are operationally single reference, with only the stretched geometry requiring a UHF reference. TABLE XIII . Multi-determinantal indices, ¯V, EEN,−log(I), MRI, and NON for six points. C2 N2 O2 F2 R′RCCSD UCCSD RCCSD UCCSD RCCSD UCCSD RCCSD UCCSD EEN −0.2 0.583 0.147 0.142 0.142 0.173 0.164 0.141 0.141 −0.1 0.526 0.216 0.176 0.175 0.204 0.191 0.167 0.167 0.0 0.487 0.315 0.231 0.231 0.264 0.235 0.219 0.217 0.1 0.526 0.298 0.331 0.331 0.383 0.287 0.331 0.255 0.2 0.664 0.407 0.555 0.321 0.687 0.245 0.535 0.153 0.4 1.504 0.208 2.050 0.148 1.954 0.146 0.884 0.142 ¯V −0.2 0.039 0.012 0.010 0.010 0.011 0.010 0.008 0.008 −0.1 0.038 0.018 0.012 0.012 0.013 0.012 0.009 0.009 0.0 0.037 0.025 0.016 0.016 0.016 0.014 0.012 0.012 0.1 0.040 0.024 0.023 0.023 0.023 0.017 0.017 0.014 0.2 0.049 0.032 0.036 0.022 0.037 0.015 0.025 0.008 0.4 0.083 0.017 0.095 0.010 0.067 0.009 0.032 0.008 −log(I) −0.2−0.343 6.932 7.254 7.254 6.667 6.815 7.524 7.524 −0.1 0.133 6.026 6.792 6.792 5.998 6.403 7.039 7.039 0.0 1.258 3.576 5.885 5.885 4.550 5.613 5.167 5.276 0.1 2.205 4.068 4.353 4.352 2.283 4.969 2.011 3.950 0.2 1.103 2.146 1.807 5.029 −0.106 4.899 −0.185 7.456 0.4−0.861 5.515 −1.044 6.874 −0.903 7.413 −0.601 7.562 MRI −0.2−1.000 0.994 0.997 0.997 0.990 0.993 0.998 0.998 −0.1−0.999 0.965 0.992 0.992 0.963 0.983 0.995 0.995 0.0−0.992 −0.414 0.953 0.953 0.488 0.921 0.818 0.851 0.1−0.948 0.052 0.324 0.324 −0.939 0.741 −0.964 −0.066 0.2−0.994 −0.954 −0.976 0.767 −0.999 0.708 −1.000 0.998 0.4−1.000 0.905 −1.000 0.993 −1.000 0.998 −1.000 0.998 NON −0.2 0.452 0.014 0.026 0.013 0.054 0.024 0.022 0.011 −0.1 0.350 0.024 0.039 0.020 0.074 0.031 0.043 0.021 0.0 0.246 0.047 0.061 0.031 0.111 0.041 0.090 0.044 0.1 0.200 0.057 0.102 0.051 0.191 0.046 0.203 0.063 0.2 0.210 0.095 0.195 0.038 0.394 0.047 0.412 0.012 0.4 0.825 0.022 0.871 0.009 0.985 0.008 0.769 0.007 J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-17 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 14 . PESs of diatomic molecules. The x axis (r′) is the distance with respect to the stable bond length (Re). The1Δgstate is used for O 2RHF-CCSD.C. Diatomic molecules Many of the main elements of MRC in quantum chemical calculations can be conveniently summarized by the behavior of homonuclear diatomic molecules as a function of internuclear dis- tance. To make the calculations as uniform as possible, the results for potential energy curves are represented in terms of the internu- clear distance variable, R′= (R−Re)/(R + Re) in Fig. 14. In the limit of complete separation (R = ∞), R′= 1, while at equilibrium, R′= 0. At R =−∞, R′=−1. A few points along this path can be identified as the critical ones for assessing different degrees of MRC (Fig. 15). As the molecule separates, those based upon the lowest energy UHF references maintain SR character, MRI = 1, except near the bifurca- tion from RHF to UHF, where one encounters spin contamination of the ground singlet state by higher-lying triplet and quintet states. This effect can be measured by ⟨S2⟩=¯S(¯S+ 1), from which the multiplicity is 2 ¯S+ 1=√ 1 + 4⟨S2⟩. Although a value close to this wrong multiplicity will persist all the way to separation, as seen by the potential energy curve (PEC), the energy curve and its associ- ated ¯Vand MRI are fine. In the event, one wants to also address FIG. 15 . MDC indices, EEN [(a) an d (b)], ¯V[(c) and (d)], −log(I) [(e) and (f)], and MRI [(g) and (h)]. Left panels obtained from RHF-CCSD, and right panels are from UHF-CCSD. J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-18 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the spin-contamination, insisting upon the RHF solution slightly beyond bifurcation helps, but only “symmetry tailoring” makes a significant improvement.47The latter is a kind of constrained SR- CC where a spin-constraint is imposed on the solution of the Tand Λamplitudes. Figure 14 shows the diatomic PECs for homonuclear diatomic molecules put on the same scale by virtue of the R′variable, subject to RHF and UHF references. At larger R,′the UHF-CCSD calcula- tions are lower in energy, tending to converge to the correct asymp- totic values. The dramatic difference between N 2’s RHF-CCSD and UHF-CCSD is notable. Figure 15 shows the differences in more detail, with some values of the indices shown in Table XIII. These values for the basic geometries are shown in Fig. 15. However, the figure shows larger R′values between 0.4 and 0.6. At these values, the EEN value tends to point out the actual number of electrons in the virtual orbitals near separations, with N 2being∼3, as one would expect, ∼2 for C 2, and∼1 for F 2. Yet once the broken symmetry UHF reference is used, the numbers remain modest. In the spin recoupling region from 0 to 0.2, however, the MRI abruptly changes from SR to MR. Its UHF equivalent, however, masks much of this issue, rapidly becoming SR at large R′. It is sometimes convenient to plot the natural occupation num- bers within the sloped Heaviside distribution function between 0 and 1.0 to demonstrate behavior as a function of geometry. Here, this is done for C 2using RHF and UHF references (Fig. 16). The plots show the very different behavior for the UHF reference CC results vs their RHF counterparts.Figure 16 demonstrates how the occupation numbers behave as a function of internuclear distance. The cluster of values near 0 and near 1, meaning completely unoccupied and fully occupied, have a role in introducing dynamic correlation, but little in assess- ing MRC. The latter depends upon those values falling in between. Because of the symmetry breaking introduced by the UHF refer- ence, there are few values compared to the RHF based calcula- tions, where we see four intermediate values. Note at negative values of R,′there is already an important distinction between RHF and UHF-CCSD. IV. EXAMPLES OF POTENTIAL MULTI-REFERENCE PROBLEMS A. Fe atom Far too many quantum chemists think that the presence of highly degenerate orbitals as occur in transition metal (TM) atoms demand that all calculations for such systems must use some MR method . This is wrong. The conventional wisdom is that five-fold degeneracy guarantees that any particle occupation of them, i.e., multiplets, will have to have low-lying, quasi-degenerate unoccupied orbitals that could cause MR character. Because ligands can quench some orbital and spin effects, an unambiguous test for MRC should occur for the atomic multiplets of the naked TM atoms. Yet, all of these multiplets (an exception is Ni) have been rather well described with SR-CC.70 FIG. 16 . Slopped Heaviside plots for the C2molecule at the given points. The x axis indicates the occupation number, and the y axis is the number of counts. Blue bars show the occurrences of occu- pation numbers contained in a bin. J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-19 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE XIV . Occupation values of neutral Fe atomic multiplets. The quintet states are calculated with a ROHF reference. CCSD is performed with the AHLRICHS-VTZ basis set. State Method −log(I) MRI ¯V NON 5F(d6s2) CCSD 5.031 0.768 0.009 0.017 CCSD(T) 4.283 0.261 0.010 0.021 5D(d7s1) CCSD 7.078 0.996 0.007 0.010 CCSD(T) 6.660 0.990 0.009 0.012 To illustrate, consider the two states of the Fe atom,5F(d6s2) and5D(d7s1) states. As Table XIV shows, both are easily SR. Even the seven-fold degeneracy in lanthanides and actinides seems to be largely under control in Peterson’s SR-CC calculations.71 B. BeH 2 An old pedagogical example of an attempt to treat an obvi- ously MR problem with SR CC theory occurs when the Be atom, with its s, p quasi-degeneracy, is inserted into an H 2molecule.72 The insertion requires breaking the H 2bond, which at separation means that the two determinants, | σg2| and | σu2|, would be degener- ate. These different effects are manifest into having two distinguish- able determinants for BeH 2, |a2|, Ref. a, and |b2|, Ref. b. These are composed of orbitals of different aandbsymmetry in the C 2vinser- tion path for the molecule, yet of the same total symmetry,1A1. A typical MR approach would naturally treat the two determinants as equally important components of the wavefunction, with compu- tation assigning different weights as a function of geometry. In the original smaller basis results,72the full valence CI at point E Ref. b had a weight of 0.724 and Ref. a had a weight of 0.560. However, SR-CC provides some alternative routes to the solution. First, either of the two closed-shell determinants can be used as the reference for a SR-CC calculation. This will provide two independently generated potential energy curves. Second, one can emulate the TD-CCSD result by building a QRHF reference determinant from taking the HOMO, a, and LUMO, b, of the priorSR-CC calculation and manually build two new orbitals (a + b) and (a−b) each occupied by one electron to define a symmetry bro- ken, open-shell QRHF reference or simply invoke UHF to enable all orbitals to break symmetry. For the sampling path defined in the original paper,72the com- puted energies in a larger cc-pVTZ basis are in Table XV, while Fig. 17 shows the PES. Note that the energies using reference Ref. a are correct for points A, B, C, and D, while using Ref. b, the CC results are correct for G, H, I, and J but are wrong at points E and F. Those two points are described better using UHF-based CC calculations. It is apparent from Fig. 17 that when the SR-CC is wrong, the result of the calculation can converge to another state. This is one of the perils of SR-CC in that the highly non-linear CC equations can have solutions that do not always correspond to the state of interest, here the ground state. Sometimes, the other solutions correspond to excited states, but sometimes, their solutions are simply fictious.73 This is one place where the application of the variational principle would have advantages. The only numerical tool one typically has in CC to get to the correct state is to exploit convergence controls. In a variational method like CI, one has more control to insist upon con- vergence to the ground state. However, somewhat like checking the stability of a HF solution, EOM-CC can be used to identify whether a ground state solution has negative excitation energies, indicating the existence of the lower solution. The indices for this PES are shown in Figs. 17(b)–17(e). When N is not a factor, i.e., side-by-side comparisons for the same number of electrons, the critical EEN value is ∼0.28, with those less being SR and larger values being indicative of MR character. Similarly, critical ¯Vvalues have SR examples <0.03, while the crit- ical NON value is 0.0583. Applying all these tools, for points B, C, and D, using Ref. a (blue curves in Fig. 17 and bold numbers in Table XV), all indicate SR with MR everywhere else. However, for Ref b (orange curve), despite the agreement with FCI at points G, H, I, and J (italic numbers in Table XV), the indices are slightly more ambiguous, as shown by MRI. The likely effect for this ambiguity is that the separated Be + H 2terminus of the pathway still has Be quasi- degeneracy, affecting the indices. Its NON value is 0.0298 for three p orbitals. TABLE XV . BeH 2PES (in a.u.) of two different configurations (1s orbital is dropped in the correlation) in a cc-pVTZ basis. The numbers in boldface and italic show the energies from the ground state configuration. RHF (configuration a) RHF (configuration b) BS-UHF Geometry CCSD CCSDT CCSDTQ CCSD CCSDT CCSDTQ CCSDTQ A −15.8476 −15.8483 −15.8484 B −15.8169 −15.8177 −15.8177 −15.3593 −15.3699 −15.3726 −15.8177 C −15.7508 −15.7521 −15.7522 −15.5055 −15.5183 −15.5065 −15.7522 D −15.6956 −15.6977 −15.6978 −15.5849 −15.5917 −15.5919 −15.6977 E −15.6662 −15.6711 −15.6709 −15.6502 −15.6226 −15.6235 −15.7122 F −15.6571 −15.6191 −15.6200 −15.6722 −15.6765 −15.6767 −15.7052 G −15.5637 −15.5690 −15.5693 −15.7328 −15.7345 −15.7346 −15.7346 H −15.5366 −15.5409 −15.5411 −15.7710 −15.7720 −15.7721 −15.7721 I −15.5152 −15.5157 −15.5158 −15.7893 −15.7895 −15.7895 −15.7895 J −15.5131 −15.5131 −15.5131 −15.7908 −15.7908 −15.7908 −15.7908 J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-20 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 17 . (a) PES, (b) EEN, (c) ¯V, (d) NON, and (e) MRI of BeH 2. C. Biradical phenylene dinitrene Several indices are introduced based upon occupation num- bers for different purposes. Ultimately, the objective is not just to call attention to MRC in calculations but also to help assess when a given result transcends any such potential error. The viewpoint is that for molecules, at least the “weak” correlation problem has been effectively solved to useful accuracy by the introduction ofcoupled-cluster and EOM-CC theory, but hidden MRC character can always be argued—justifiably or not—to compromise some con- clusions. Considering that the most definitive values for thermo- chemistry67,71,74,75have been obtained by performing complete basis set (CBS) converged calculations using SR-CC theory (albeit usually with composite versions), with no comparable or superior MR-CI results reported to potentially improve the answers, suggests that very few situations with large MRC were encountered. To justify alternative methods, one can always concoct special situations where some would hope there would be a problem. Some of those deserve scrutiny. For example, it was argued from the reduced density matrix community that the biradical, 1,4-phenylenedinitrene has enormous MRC and that CCSD(T) failed miserably for the singlet-triplet split- ting.76We studied this system (Table XVI), and it does show a large ¯Vand associated MRI and NON measure. However, applying the wide range of operationally SR-CC tools to the problem obtained the right answer for the singlet–triplet splitting using several different CC methods.77 The MRC is apparent, as all indices based on the RHF reference for the singlet ground state except maybe ¯Vsay so. The caution with ¯Vfor suitably large N values is that its useful distinction might be washed out, and this might apply here. Nonetheless, once a UHF is used for the singlet state, the MRC tends to be diminished but not removed. NON is particularly indicative of that. The only method that seems to be able to control the MR character is the doubly electron attached (DEA) similarity transformed EOM-CCSD (DEA- STEOM-CCSD). This method starts from a double cation reference for the ground state and then uses EOM to add the four determinants that can be made from the two quasi-degenerate orbitals at the EOM stage so that they can assume any value from the EOM diagonaliza- tion. In other words, it is multi-reference but achieves its capabilities within an operationally single reference CC framework.77As one would hope, DEA-EOM and its doubly ionized potential (DIP) EOM (DIP-EOM) analog add a new route to treating the most common 2 ×2 MR problems (Table XVII). For the triplet state, that is equally important here if one wants an accurate singlet–triplet separation, and it also has multi-reference character. Thus, this would appear to be a difficult problem to get right with SR-CC methods. At the RHF-CCSD and CCSD(T) levels, that is indeed the case, with the separations very far from the truth. However, once the UHF reference is used, the differences are rather dramatic. The experimental value for this problem is ±2 kcal/mol, and now the results fall within that range. Even, the ΛCCSD(T) relative to an RHF even achieves a value just within the error bars. The DEA-CCSD and DEA-STEOM-CCSD, respectively, give 0.03 and 0.34 when used for both the singlet and triplet states. In fact, this even offers the possibility of a “direct” calculation at one geome- try for such a splitting. The alternative DIP-EOM-CCSD based upon the double anion worked well, too, but it slightly prefers the triplet state as the ground state over the singlet one. D. H 8 Another case that initially caused some consternation is the dis- sociation of a chain of four hydrogen molecules, 4(H 2), exploding into eight infinitely separated H atoms.78Limacher et al . identified this as a prototypical example of strong correlation and used it to J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-21 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE XVI . Biradical, 1,4-phenylenedinitrene. Spin Reference Method Energy (a.u.) −log(I) MRI ¯V EEN NON Singlet RHF CCSD −339.374 65 −0.596 −1.000 0.023 1.541 0.353 CCSD(T) −339.506 87 −0.031 −0.999 0.022 1.345 0.162 ΛCCSD(T) −339.475 81 −0.602 −1.000 0.027 1.876 0.427 UHF CCSD −339.431 86 4.692 0.589 0.015 0.842 0.048 CCSD(T) −339.481 35 3.459 −0.506 0.018 0.999 0.068 ΛCCSD(T) −339.478 94 3.301 −0.614 0.018 0.999 0.071 DEA-STEOM-CCSD −339.448 35 4.545 0.485 0.014 0.792 0.051 Triplet UHF CCSD −339.433 08 4.366 0.337 0.015 0.857 0.057 CCSD(T) −339.480 61 3.119 −0.715 0.018 1.013 0.080 ΛCCSD(T) −339.478 05 3.275 −0.630 0.018 0.998 0.077 demonstrate that their geminal pair method describes the dissocia- tion correctly, while single reference CC methods fail. Limacher and co-workers concluded that both SR CCD and CCSD with the canon- ical RHF orbitals or the doubly occupied configuration interaction (DOCI) orbitals as reference orbitals “catastrophically” underesti- mate the dissociation energy around the inter-unit distance 4 Å where the bonds break. It is certainly true that RHF based CC cal- culations, where one insists upon doubly occupied orbitals that can- not possibly localize electrons onto the individual atoms as required by this study, have a problem with their dissociation, as already emphasized. However, as in many other examples, UHF-CC fixes this. A careful investigation of the dissociation reveals that there are no “catastrophic failures” of SR-CC describing the bond disso- ciation. Rather, one sees the expected behavior where RHF based CC calculations overestimate the dissociation energy and have some discontinuities, while UHF based CC is competitive with “pair” methods specifically designed to describe such bond breaking situations. The results for this system and index values are shown in Figs. 18 and 19. Here, the UHF based CC calculations still show a MR MRI, along with the other indices. However, one should not take that as indicative of an irresolvable problem for SR-CC. The TABLE XVII . S/T splitting from the energies in Table XVI (kcal/mol). S(RHF)/T(UHF) S(BS-UHF)/T(UHF)a CCSD −36.67 −0.76 CCSD(T) 16.48 0.47 ΛCCSD(T) −1.40 0.56 DEA-EOM-CCSD 0.03 DEA-STEOM-CCSD 0.34 DIP-EOM-CCSD −0.19 aThis indicates the energy difference is being measured from a UHF triplet calculation, vs a RHF-CC or UHF-CC singlet.potential energy surfaces here are quite reasonable using a UHF reference despite suffering from an index that is different from a standard SR example. The eventual resolution of how accurate the energy is vs the index values proposed here needs to be determined by many calculations. As emphasized, the indices defined here are meant to quantify an error in a CC calculation that some might attribute to an inadequate treatment of the perceived MR character, but when the result is close enough to the FCI, there is no further error regardless of the values of these indices. This is also illustrated by the above account of BeH 2where despite MRI arguing for a MR problem for the orange curve at points G, H, I, and J (see italic num- bers in Table XV), the numerical effect of CCSD and beyond is in excellent agreement with the FCI at those points. E. Open-shell singlets A partial fix for low-spin states of molecules is offered by using the symmetry “tailored” (TCCSD) method that fixes the amplitudes of cluster operators into Ψ0=ΣdjΦj79such that this “new” reference of the SR-CC wavefunction reflects the correct ratio of the princi- pal determinants that helps the overall spin symmetry of the SR-CC FIG. 18 . HF and CCSD energies vs R(H–H) of H 8→8H. J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-22 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 19 . NON (a), ¯V(b),−log(I) (c), and MRI (d) of H 8→8H. calculation. The most important of such examples are open-shell singlets as most excited states have this character and must be treated appropriately for spectral interpretation. Other situations such as low-spin doublets and the treatment of certain spatial states con- strained by using real functions in quantum chemical programs are discussed elsewhere.47 The TD-CCSD31,32was developed exclusively for open-shell singlets as the simplest illustration of the Hilbert Space MR-CC. Thetwo determinants correspond to |A αBβ|±|AβBα|. In addition, the EOM-CC for excited states always treats these two together for sin- glets and triplets. To the contrary, SR-CC implies that one of the two determinants is being used as the reference, with the other exiled to the orthogonal space, even though the correct solution should weight them equivalently and fulfill spin-symmetry. Putting the sec- ond of these two determinants into the Q space while using the RHF orbitals of a closed shell ground state to define the QRHF orbitals for the HOMO (A) and LUMO (B) means its initial second- order perturbation theory t-amplitude would have a zero denomina- tor, strongly pointing to divergence. However, despite this, infinite- order methods based on semi-canonical orbitals2can overcome such divergence with care and still provide reasonable results, but the question of how reasonable such SR-CC results are for this simplest case can be informative (see Table XVIII). Even when choosing one of the two degenerate determinants as a reference for a SR-CC calculation with a QRHF reference, the results are consistent with both the tailored and two-determinant MR calculation and the EOM that treats the two determinants equiv- alently. For the indices for the latter, it is important to recognize that the EOM density matrix, discussed in Sec. V, requires that the count of occupied orbitals be augmented to include the two degener- ate orbitals involved in the open-shell singlet. This matrix is shown in Table XVIII. That is, in the EOM density matrix, the four spin- orbitals with occupation 0.500 05; 0.495 94 define the N-electron Nreference determinant. Then, the occupation of the remaining virtuals goes into defining the indices. F. Transitions in bicyclo-butane For another example of MR character for a complicated tran- sition state, consider the conrotatory and disrotatory transitions in bicyclo-butane, initially studied by Piecuch80and Mazziotti81and shown in Table XIX. The details are published elsewhere.82The NON value for the ground state is only 0.0243, well below the 0.0583 cutoff for SR, but the RHF conrotatory threshold shows a value of 0.0598 and the disrotatory value is a large 0.1119, suggesting both components are MR, and this behavior persists when using larger basis sets. Note that the MRI index parallels the NON values, supporting the MR character (Table XIX). Once one introduces a UHF reference, the NON value for the DIS component is reduced to 0.0387, while the CON is SR at only 0.0216. This has the effect of reducing the DIS transition state (TS) value by∼12 kcal/mol, while the CON value is about the same for RHF and UHF at the CCSD level. Note that the NON and MRI indices for UHF-CCSD/TZ would say that this approximation has the most SR character for all three components. Is it possible that this feature could become a figure of merit for comparative results in future high-accuracy calculations? That is, one might argue that adding triples here, usually considered an improved calculation, could actually hurt the comparative barriers because the indices forΛCCSD(T) show MR character that might suggest a larger, or uncontrolled, error in the calculation. This kind of error analysis might become possible once such indices are well established. Here, adding triples with UHF- ΛCCSD(T) reduces the CON result by ∼3 kcal/mol in the modest DZ basis. Experiment for the CON TS is 41±2. J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-23 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE XVIII . Open-shell singlets of the H 2O molecule. HOMO and LUMO State Method References Energy −log(I) MRI ¯V EEN Alpha Beta B1 CCSD ROHF −76.050 03 6.987 19 0.994 76 0.013 01 0.132 53 0.976 00; 0.017 87 0.975 81; 0.017 68 CCSD UHF −76.050 52 7.081 74 0.995 66 0.012 75 0.129 80 0.977 60; 0.015 93 0.977 63; 0.016 19 TCCSD QRHF −76.040 44 7.338 25 0.997 40 0.010 69 0.108 49 0.981 80; 0.015 27 0.981 66; 0.012 00 TD-CCSD QRHF −76.040 34 7.320 66 0.997 31 0.010 89 0.110 60 0.981 50; 0.015 31 0.981 60; 0.011 80 EOM-CCSD RHF −76.039 26 8.007 12 0.999 32 0.003 36 0.033 68 0.500 05; 0.495 94 0.500 02; 0.495 97 A2 CCSD ROHF −75.971 45 7.316 95 0.997 29 0.011 57 0.117 46 0.982 43; 0.013 77 0.981 58; 0.012 87 CCSD UHF −75.971 98 7.317 04 0.997 29 0.011 57 0.117 44 0.982 74; 0.013 95 0.981 33; 0.012 66 TCCSD QRHF −75.966 89 7.391 14 0.997 66 0.010 50 0.106 51 0.983 02; 0.013 96 0.982 14; 0.010 88 TD-CCSD QRHF −75.966 65 7.372 15 0.997 57 0.010 68 0.108 38 0.982 15; 0.013 95 0.982 21; 0.011 09 EOM-CCSD RHF −75.965 09 8.006 81 0.999 32 0.003 36 0.033 64 0.500 17; 0.495 73 0.500 17; 0.495 73 A1 CCSD UHF/ROHF No convergence CCSD QRHF −75.962 79 6.080 02 0.968 28 0.015 60 0.159 83 0.964 90; 0.026 91 0.963 94; 0.032 42 TCCSD QRHF −75.948 68 7.135 30 0.996 10 0.011 50 0.117 01 0.979 41; 0.021 38 0.981 35; 0.013 88 TD-CCSD QRHF −75.948 84 7.167 42 0.996 34 0.011 55 0.117 42 0.981 50; 0.013 49 0.979 58; 0.019 04 EOM-CCSD RHF −75.948 07 7.964 23 0.999 26 0.004 01 0.040 3 0.538 88; 0.457 25 0.538 86; 0.457 27 V. ANALYSIS OF THE CONTRIBUTIONS OF T1,T2, AND T3AMPLITUDES TO THE DENSITY MATRIX To orient one to the density matrix, γpq, consider Fig. 20. These are all the contributions from T1,T2, and T3. It is straightforward to add those due to T4or higher with the unambiguous diagram gener- ation methods discussed in Refs. 1 and 3. The p,qvalues are virtual- occupied, D(v, o), virtual–virtual, D(v, v), and occupied–occupied, D(o, o). Diagrams 11 and 12 arising from the last two options rep- resent the major contributions. For D(v, v), one arises from singles (+λa itb i)and one from doubles, (+1/2λac ijtcb ij)(summation over the common non-target index is understood). D(o, o) replaces the above a,bformula by i,jand changes the sign. As the magnitude ulti- mately depends on Tr( γpq), its main contributors are the “diagonal” elements that arise from these expressions. In terms of orders of MBPT, both arise in second order in the perturbation, W, H0=∑ i≠jfij{i†j}+∑ a≠bfab{a†b}, (15) W=∑ iafa i{a†i+i†a}+1 4∑ pqrs⟨pq∣∣rs⟩{p†q†sr}. (16) Regarding off-diagonal elements, D(v, o), it is easy to see diagram- matically that the first-order correction for the most general non-HF case in the CCSD density matrix arises from T1andΛ1, (diagrams 1 and 2), though these will be second order in a HF reference case, along with the other second-order terms (diagrams 3, 11, and 12) and the third-order terms (diagrams 4, 5, and 6). The dominant diag- onal T2correlation effect occurs in second order in diagram 12 andT1in Diagram 11, while T1andT2are mixed to second order in dia- gram 3. T2also occurs in third-order in a,iin diagrams 5 and 6. The CC density matrix is only fully correct to third order once triples are added, (diagram 7) with T3evaluated at least at the CCSD(T) [or preferably, the ΛCCSD(T)] level. The dominant triples diagonal term, diagram 13, is fourth-order. Order distinction suggests that a better converged density matrix will correspond to a better energy, much like a variational energy is correct to second order, while its wavefunction is correct only to first, but since there is no rigorous variational condition in CC theory, one is left to argue about the comparative orders of perturbation theory for the energy and the density matrix. The outlier in the normal sequence, CCSD <CCSD(T) <ΛCCSD(T)<CCSDT-3 <CCSDT, ΛCCSD(T), has the advantage that it is only a non-iterative n7calculation and is correct to fourth- order for the energy, but because of its explicit dependence on the infinite-order solution of Λ, the approximation hides several impor- tant higher-order terms in it for the density matrix that do not occur in CCSD(T). These differences are apparent in Table II compared to the FCI values. So one would hope that ΛCCSD(T) and the itera- tive∼n7CCSDT-3 method, particularly with UHF references, would often provide good estimates of the FCI density matrices. The initial role of T4could also be accessed by the ∼n7non-iterative factorized approximation Q f.83This would take the energy to fifth-order and the density matrix to sixth-order. Continuing with perturbation theory, a caveat is that one should distinguish between HF references, RHF and UHF, and the non-HF ones that include ROHF.84The proper perturbation defini- tion differs in order, W, causing the T1term to now be second-order in the HF case because the Fock operator is fvrt occ=0. This makes the density matrix second-order as is the energy, with both initially aris- ing from T2(diagram 12), while the diagonal T1term (diagram 11) J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-24 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE XIX . Ground state (GS) and two transition states (CON and DIS) of bicyclo- butane. GS CON DIS ΔE (kcal/mol) RHF-CCSD/DZ 0 47.9 70.2 RHF-CCSD/TZ 0 49.7 73.9 UHF-CCSD/DZ 0 48.5 58.5 UHF-CCSD/TZ 0 50.3 61.1 RHF- ΛCCSD(T)/DZ 0 42.9 61.6 UHF- ΛCCSD(T)/DZ 0 47.1 60.5 EEN RHF-CCSD/DZ 0.4432 0.5298 0.6582 RHF-CCSD/TZ 0.4688 0.5450 0.6484 UHF-CCSD/DZ 0.4432 0.4825 0.4323 UHF-CCSD/TZ 0.4687 0.5013 0.4586 RHF- ΛCCSD(T)/DZ 0.4907 0.6296 0.8565 UHF- ΛCCSD(T)/DZ 0.4908 0.5969 0.4822 ¯V RHF-CCSD/DZ 0.0145 0.0171 0.0204 RHF-CCSD/TZ 0.0153 0.0176 0.0204 UHF-CCSD/DZ 0.0145 0.0157 0.0141 UHF-CCSD/TZ 0.0153 0.0163 0.0150 RHF- ΛCCSD(T)/DZ 0.0160 0.0200 0.0252 UHF- ΛCCSD(T)/DZ 0.0160 0.0191 0.0157 MRI RHF-CCSD/DZ 0.9748 −0.4651 −0.9955 RHF-CCSD/TZ 0.9700 −0.1793 −0.9871 UHF-CCSD/DZ 0.9749 0.7815 0.9784 UHF-CCSD/TZ 0.9700 0.8547 0.9737 RHF- ΛCCSD(T)/DZ 0.9601 −0.9272 −0.9995 UHF- ΛCCSD(T)/DZ 0.9601 −0.8160 0.9674 NON RHF-CCSD/DZ 0.0251 0.0667 0.1311 RHF-CCSD/TZ 0.0243 0.0598 0.1119 UHF-CCSD/DZ 0.0251 0.0435 0.0224 UHF-CCSD/TZ 0.0243 0.0387 0.0216 RHF- ΛCCSD(T)/DZ 0.0210 0.0877 0.1987 UHF- ΛCCSD(T)/DZ 0.0210 0.0773 0.0181is fourth order. For a Brueckner reference, T1= 0, so all major effects arise from T2. For the BH example with HF references at three geometries, the T2contributions are three-orders of magnitude larger than T1(see the supplementary material). Thus, the primary contributor to γpqisT2, whether a HF ref- erence or non-HF reference is used. This is the primary reason the so-called T1diagnostic40does not measure MRC, as was emphasized long ago.42The function of T1is to rotate orbitals, while T2intro- duces true correlation effects. To take a few examples, the most MR value in Wilson’s T1diagnostic analysis of few hundred molecules containing transition metals is NiSi,85with the very large value of 0.082 and maximum T1value of 0.148. Another measure used is the % of (T) for the atomization energy, also strongly indicating a MR example.85The indices defined here show 0.009 for ¯V, 0.038 for NON, and 0.760 for MRI, attesting to its SR character. For another example, consider the molecules BeO, MgO, and CaO. Yu and Truhlar studied various diatomic molecules contain- ing group 1 and 2 metal atoms.69They apply four different measures including the T1diagnostic (shown in the last column of Table XX that is MR is >0.02) and conclude these three molecules are MR by all four criteria. Our indices show that all RHF and UHF based exam- ples are SR except RHF-CCSD for MgO. It should be noted that the HOMO of HF orbitals is πfor BeO and CaO, whereas for MgO, it isσ. This is also shown in Martin’s work43(see the supplemen- tary material of Martin’s paper). This leads to different behaviors for the RHF-CCSD results of MgO. However, when the UHF is applied, MgO falls into the SR category as do most of the examples discussed in this paper. In the supplementary material, the numerical breakdown of γpqfor the BH example from Tables II and III at three differ- ent geometries and with RHF, UHF, and Brueckner orbitals is illustrated. In the case of EOM-CC, the density matrix assumes almost the same form as for the usual SR-CC ground state. However, it is given by⟨0|Lkexp(−T)p†qexp(T)Rk|0⟩for the k-th target state. So dia- grammatically, Λis replaced by LkandRkreplaces the bottom, linear Tvertex, that is, the linear in Tdiagrams 1, 2, 3, 7, 11, 12, and 13. The operator itself is now | ∼∼instead of |–. The wiggly line accounts for the density matrix operator being exp( −T)p†qexp(T) instead of justp†q, so the few contractions with the reference state, exp( T)|0⟩, are incorporated into the one-particle density matrix operator. For details, see Ref. 1. Since EOM-CC is not simply SR-CC, the concept of the single determinant reference for N-electrons is slightly modified. The main message is to understand that the underlying N-particle reference determinant now involves one more spin-orbital that is degener- ate with its complement, so the two ( ∼0.5) occupancies have to be FIG. 20 . CCSDT density matrix diagrams. J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-25 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE XX . MR Indices of MgO, BeO, and CaO. Indices are obtained from CCSD with cc-pVTZ and cc-pCVTZ basis sets. Mol. Dist. (Å) Reference Basis-set Eng. (a.u.) EEN V NON −LOG(I) MRI T1diag. BeO 1.3372aRHF cc-pVTZ −89.753 33 0.182 0.015 0.022 6.394 0.983 0.042b cc-pCVTZ −89.834 98 0.183 0.015 0.022 6.419 0.984 UHF cc-pVTZ −89.753 33 0.182 0.015 0.022 6.394 0.983 cc-pCVTZ −89.834 98 0.183 0.015 0.022 6.420 0.984 MgO 1.7679aRHF cc-pVTZ −274.714 20 0.294 0.014 0.081 2.809 −0.836 0.053b cc-pCVTZ −275.042 89 0.311 0.015 0.074 3.128 −0.710 UHF cc-pVTZ −274.719 60 0.223 0.011 0.052 4.921 0.718 cc-pCVTZ −275.048 58 0.243 0.012 0.047 5.232 0.838 CaO 1.821cRHF cc-pVTZ −752.061 11 0.281 0.010 0.024 6.325 0.980 0.037b cc-pCVTZ −752.290 74 0.301 0.011 0.023 6.314 0.980 UHF cc-pVTZ −752.061 11 0.281 0.010 0.024 6.326 0.980 cc-pCVTZ −752.288 62 0.294 0.010 0.023 6.387 0.983 aGeometry is taken from Ref. 43. bT1diagnostic are taken from Ref. 69, and the values are obtained from CCSD with aug-cc-pCVQZ basis set. cGeometries are taken from Ref. 86. added together to define the single determinant reference before counting the occupancy of the unoccupied virtuals. The same sit- uation will apply with other EOM models such as STEOM and DIP/DEA/EOM. VI. CONCLUDING REMARKS This paper defines several rigorous ways to identify MDC or MRC in molecular calculations. All aspire to measure the SR-CC limit of the full CI via its variance and related properties from the fiction of a perfect N orbital, natural determinant, N. The correlation problem built upon the latter reference best defines the limit of SR-CC theory. Adding further analysis of individual occupation numbers in virtual orbitals addresses how much of the measured MDC points to stronger correlation, like MRC, and how well various approximations in coupled-cluster theory accommo- date that ¯V,NON , and MRI. Critical values of the possible indices are defined, suggesting that the accuracy of a calculation might often be assessed if the indices for the calculation fall below those values. See Fig. 21 for an illustration of the convergence of the energy with respect to V, EEN, NON, MRI, and the level of CC applied for BH. Many more numerical studies of this sort should be done to ensure accurate extrapolations and “predictive” energies vs indices. Ultimately, the need for these indices for most practitioners is built on the assumption that SR-CC theory like CCSD(T) with a good basis is essentially right, provided the nebulous, ill-defined “MRC” is under control, hence the need for a measure of a potential error. But who is to say what a reliable index is? It is certainly not the T1 diagnostic, nor is it the % (T) in CCSD(T). However, by building on rigorous theory that embraces most of the invariance properties of the FCI, the four indicators proposed ought to introduce some rigor. Further work should do highly accurate SR-CC calculationsand MR-CC calculations to assess such indices and the real effects of MRC. The correlation between the computed energy for a prob- lem at a given truncation of the CC hierarchy, CCSD, CCSD(T), ΛCCSD(T), CCSDT-n, CCSDT, CCSDTQ-n, CCSDTQ, . . ., and the corresponding density matrix index provides a density matrix figure of merit that can be used to estimate the conver- gence of the energy, like multiplying the energy by the ratio, %(EEN(CC)−EEN(FCI))/EEN(FCI)∼%(¯V(CC)−¯V(FCI))/ ¯V(FCI), that should be indicative of the reliability of the computed energy, as shown in Fig. 21. To approximate the FCI occupation numbers, it appears that a UHF based CCSDT-3 is frequently ade- quate, but, of course, that does not mean its energy is the FCI one. However, one can hope to correlate the comparative errors in the energy in CCSD, CCSD(T), ΛCCSD(T), and higher approximations with the degree of convergence of the density matrix when several such numerical studies are made. However, there will be many cases where the indices might sug- gest a MR problem, but that does not mean the SR-CC calculation is specious. In this paper, there are several examples of SR-CC cal- culations for systems that according to the proposed indices retain unresolved MRC, but, in practice, CC’s rapid convergence to the FCI is frequently more than sufficient to offset the difficulty even at modest levels of cluster truncation like T and Q. The location of that crossing point to FCI warrants much further attention, but being able to quantify the degree of MR error along with other sources of error from basis sets to truncation of the cluster operator ought to help. Other common examples, including studies of the multiplets of transition metal atoms, demonstrate that despite high-orbital degeneracy, many SR-CC results provide the best results available, and the new indices can further attest to and quantify that. J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-26 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 21 . Differences with respect to FCI: energy (a), EEN (b), ¯V(c), and NON (d) of the BH molecule. The indices are calibrated by several obvious single and multi- reference examples. By defining all indices from the CC den- sity matrix, one is assured that the index properly accounts for invariance to orbital rotations among the occupied and unoccupied orbitals, will be size-extensive when evaluated from linked diagrams(EEN), and reflects the ¯V, MRI’s, and occupation number (NON) measure’s intensivity. All CC calculations benefit from the orbital insensitivity of CCSD and beyond in terms of mixing occupied and unoccupied orbitals. Consequently, these indices offer realistic assessments of MDC and MRC that are usually independent of the J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-27 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp choice of the single reference function in the CC calculation, whether RHF, UHF, ROHF, QRHF, OO, B,N, KS, or CAS-SCF. They are also relatively independent of the computational basis set, as shown in Table IX. The latter occurs because the critical larger occupation numbers are not particularly dependent on the basis set quality once obtained at a reasonable T or Q level. The basis set dependence of the energy and other properties primarily arises from the dynamic correlation, not any MRC. Consequently, the indices defined here offer a practical benefit for some of the usual computational issues encountered in calculations. Many numerical examples illustrate the role of the indices pro- posed. One rather universal observation is that DODS solutions based upon a UHF determinant in a CC calculation dramatically alleviates MRC, as shown in Fig. 22. Comparing this figure to the RHF Fig. 5 shows the drastic difference. For bond breaking issues of MR, this frequently requires a UHF that breaks spatial symme- try. This suggests that perhaps a two-step procedure would be use- ful: first use broken symmetry reference functions to describe com- plicated problems; and second, restore the symmetry—if deemed important—after accruing the numerical benefits, in analogy to work in nuclear physics termed “breaking and restoring symme- try.”87The UHF results here break spin and spatial symmetries, but there are several other symmetries that could be broken and potentially restored that have been considered by others.88 Although Fig. 22 shows virtually no indication of a MR prob- lem for any of the reference examples, this does not say there will never be one—as we have focused on mostly small, easily under- stood molecules—but unavoidable, genuine, MR effects do not hap- pen nearly as often as too many anticipate. So using SR- ΛCCSD(T) or SR-CCSDT-3 or some approximation with connected quadru- ples with converged basis sets for predictive thermochemistry will be fine for the majority of molecular studies, with UHF or ROHF for high-spin open shells and symmetry broken UHF for bond breaking. Transition states like in the bicyclo-butane example and singlet–triplet separations as in dinitrene are more demanding, showing MRC even for UHF based references, but operationally single reference CC results including EOM provide correctanswers. When problems are not accessible to various flavors of EE/IP/EA/DIP/DEA EOM-CC and STEOM,89further new devel- opments of SR-CC and EOM are warranted. Many of these, such as “tailored” CC,82,90adapted,91or “constrained” SR-CC by adding some desirable Lagrangian condition,92can often retain the ease of the SR-CC application with little numerical consequence. There is a need for a more proper definition of MRC than is commonly used. If the objective is to describe examples while pre- serving spin-symmetry , like in traditional MR-CI calculations, then there can appear to be many such “MR” problems involving near degeneracies, curve crossings, and other phenomena. However, once one allows “stable” broken symmetry UHF to be the “SR” in SR- CC, for most situations away from bifurcation, one now has a rou- tine SR-CC approach that will usually converge to the exact answer for the energy about as well as it does for closed shell or high- spin open-shell situations. Considering how “predictive” basis set limit quantum chemistry is used in applications today, this is a highly desirable alternative that does not require any selection or convergence of “active” orbitals, while providing size-extensive and intensive results that the MR-CI cannot. In the grander scheme of things, though spin symmetry is obviously extremely helpful for small molecules and particularly their excited states and associated spectral interpretation, when one considers the exorbitant numbers of electronic states that occur within a fraction of an eV for transi- tion metal excited states, an insistence upon pure spin symmetry per state would appear to be a dubious proposition. For some of the simpler, well-known MR situations, like biradicals,80,93,94and low-spin problems, the DIP/DEA/IP/EA/EE EOM-CC methods provide MR answers that also retain spin- symmetry. That is, these quasi-degenerate situations are described completely in a SR-CC, EOM-CC framework without using broken symmetry UHF solutions as in the DEA and DIP examples dis- cussed in Table XVI. With all these different approaches available, it appears that relatively few “true” MR problems occur in the elec- tronic structure of molecules that really demand MR-CC, as opposed tonucleons95orcrystals ,96where a description of phase transi- tions and superconductivity will likely require a genuine “strong” FIG. 22 . Natural orbital occupation number of SR (left) and MR (right) examples from BS-UHF-CCSD. J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-28 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp correlation theory of predictive accuracy. Nevertheless, the route toward such a theory should pass through SR-CC theory, and the indices proposed here will help to quantify any potential effect due to an unknown amount of residual MR character. SUPPLEMENTARY MATERIAL See the supplementary material for the geometries of 15 SR and 15 MR test-sets, the details of MRI including a simple python code, and CC density decompositions of the BH molecule. ACKNOWLEDGMENTS This work was supported by the U.S. Air Force Office of Scientific Research (Grant No. FA9550-14-10261) and the U.S. Army Research Office (Grant No. W911NF-16-1-0260). The calcu- lations were done at the University of Florida, HiPerGator high per- formance computer center with facilities partly funded by ARO. We appreciate Ernest Davidson informing us of related density matrix work by Takatsuka. APPENDIX A: ΛVST†IN THE ONE-BODY RESPONSE DENSITY MATRIX. Since the Λoperator is well approximated by T†for linear terms, another possible simplification in density matrix calculationsis to replace ΛbyT†. This avoids any need to compute both Λand Tin CC calculations, saving a factor of 2. Table XXI shows that this approximation is sufficiently reliable for the proposed index at equilibrium or near equilibrium. Thus, a working response density matrix expression for the correlation part of the wavefunction would beΔγpq=⟨0|(1 + T†)e−T{p†q}eT|0⟩and will be evaluated with CCSD since finite-order approximations are not adequate. One also consid- ers the CCSD(T) one-matrix to provide the density matrix to assess the initial effects of triples in calculations and CCSDT-3 to add an infinite-order T element in some cases. At significant bond sepa- ration, it can be important to use Λinstead of T†for the response density. In fact, just as ΛCCSD(T) replaces T†in CCSD(T) with Λoffers much improved results for demanding problems, it also provides a much better indicator of the correct result at the same non-iterative, ∼n7cost, as shown in Table XXI. The FCI values are in Table II. APPENDIX B: PLOTS OF OCCUPATION NUMBER CURVES AND NON MEASURE Some further illustrations help to fine-tune the NON measure of proposed occupation numbers. These are made using the inverse Gaussian Distribution (IGD) fit to SR, MR, and all examples, as shown in Fig. 23. Figure 23 shows three different fittings for the SR, MR, and all test-sets. The two parameters are chosen to minimize the root-mean- TABLE XXI . Comparison of BH natural orbital occupation numbers ΛvsT†for R(BH) = 1.234 (equilibrium) Å, 2.5 Å, and 6.0 Å. Boldface denotes the natural occupation numbers from occupied orbitals and the rests are from the first few virtual orbitals. 1.234 Å Λ 1.9991 1.9432 1.8715 0.0467 0.0467 0.0127 0.0127 0.0103 0.0090 T†1.9991 1.9429 1.8700 0.0474 0.0474 0.0128 0.0128 0.0103 0.0091 2.5 Å Λ 1.9992 1.8794 1.6920 0.2610 0.0427 0.0427 0.0171 0.0148 0.0108 T†1.9992 1.8778 1.6778 0.2744 0.0432 0.0432 0.0177 0.0152 0.0111 6.0 Å Λ 1.9992 1.8752 1.0681 0.8978 0.0428 0.0428 0.0267 0.0135 0.0091 T†1.9992 1.8604 1.5415 0.4439 0.0462 0.0462 0.0378 0.0178 0.0104 FIG. 23 . CDF plot for three test-sets, SR (blue), MR (red), and all examples (black). Left panel shows the normal CDF, and right panel shows 1-CDF in the log scale. J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-29 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE XXII . The optimized mean ( μ) and shape parameter ( λ) of the 30 molecule dataset. Normal opt. Weighted opt. μ λ μ λ SR 0.005 11 0.006 35 0.005 23 0.005 80 MR 0.006 47 0.003 45 0.037 94 0.002 02 All 0.005 63 0.004 62 0.020 32 0.001 09 FIG. 24 . CDF (top), PDF (middle), and PDF (0.01–0.05 region, bottom) from the optimized inverse Gaussian function are plotted.square deviation (RMSD) between the original data and that made from the new function, F(x; μ,λ). Blue lines optimize μ,λto min- imize the normal RMSD value, whereas the weighted optimization results in (red line) RMSD(weighted )=⌟roo⟪⟪op ⌟roo⟪mo⟨⌟roo⟪mo⟨⌟roo⟪⟨o⟪1 MM ∑ i(w(ni)(˜y(ni)−y(ni)))2. (B1) Mindicates the number of terms in the sample. niis the i-th natural occupation number. y(ni) is the empirical cumulative distribution function (ECDF) measure of the original data, and ˜y(ni)is the CDF value from the given function, F(x;μ,λ). The weight used here, w(ni), is the natural occupation number itself. Thus, it has small weights for the small occupation numbers and larger ones for those that should matter. If one sets w(ni) = 1, it becomes the normal RMSD. The two optimized parameters are listed in Table XXII, and the CDFs and PDFs out of the weighted optimization are plotted in Fig. 24. The last panel in the figure shows the region between 0.01 and 0.05. The decays of the PDF in the figure show how the distribution of SR, MR, and all examples differs. The CDF of the SR test-set indi- cates that the 99.73% (6 sigma) SR natural occupation numbers falls within a maximum occupation of 0.0337% and 99.99%. SR occupa- tions fall within 0.0583. It would appear to be safe to use a NON of 0.0583 as a criterion to discriminate between an SR and MR region. Table XXIII shows that a 0.0583 criterion includes all the maxi- mum natural occupation numbers (NON) in the SR test set. All MR maximum occupation numbers are larger than that criterion. TABLE XXIII . SR/MR measure based on the criteria (0.0337 and 0.0583). x = 0.0337 x = 0.0583 SR MR 99.73% (6 σ) 99.99% Mol. NON Mol. NON SR MR SR MR He 0.0037 H 2 0.4967 T T T T H2 0.0100 Li 2 0.4870 T T T T H4 0.0422 Ne2+0.1736 F T T T BeH 2 0.0098 HF 0.4354 T T T T BH 3 0.0087 H 2O 0.1505 T T T T Ne 0.0058 C 2 0.4877 T T T T HF 0.0126 BN 0.0770 T T T T H2O 0.0132 BeN−0.0798 T T T T CH 4 0.0108 BC−0.1034 T T T T LiF 0.0108 CN+0.1129 T T T T N2 0.0310 N 22+0.1479 T T T T C2H4 0.0323 N 2 0.4241 T T T T Ar 0.0092 C 2H4 0.1639 T T T T F2 0.0478 F 2 0.3212 F T T T O2 0.0229 MgO 0.0806 T T T T J. Chem. Phys. 153, 234103 (2020); doi: 10.1063/5.0029339 153, 234103-30 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material. REFERENCES 1I. Shavitt and R. J. Bartlett, Many-Body Methods in Chemistry and Physics (Cambridge University Press, Cambridge, 2009). 2R. J. Bartlett, in Modern Electronic Structure Theory, Part II , edited by D. R. 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5.0031196.pdf

AIP Advances 10, 120701 (2020); https://doi.org/10.1063/5.0031196 10, 120701 © 2020 Author(s).Recent advances in Re-based double perovskites: Synthesis, structural characterization, physical properties, advanced applications, and theoretical studies Cite as: AIP Advances 10, 120701 (2020); https://doi.org/10.1063/5.0031196 Submitted: 28 September 2020 . Accepted: 19 November 2020 . Published Online: 23 December 2020 Kai Leng , Qingkai Tang , Ying Wei , Li Yang , Yuting Xie , Zhiwei Wu , and Xinhua Zhu COLLECTIONS Paper published as part of the special topic on Chemical Physics , Energy , Fluids and Plasmas , Materials Science and Mathematical Physics ARTICLES YOU MAY BE INTERESTED IN Study on vorticity diffusion laws under oscillating flow and its influences on dissipation loss AIP Advances 10, 125001 (2020); https://doi.org/10.1063/5.0031633 Theoretical study of the chemical reaction mechanism and rate of SF n− + H2O (n = 3–6) under discharge AIP Advances 10, 095214 (2020); https://doi.org/10.1063/5.0018972 Magnetization switching depending on magnetic fields applied to ferromagnetic MnAs nanodisks selectively-grown on Si (111) substrates AIP Advances 10, 125003 (2020); https://doi.org/10.1063/5.0030841AIP Advances REVIEW scitation.org/journal/adv Recent advances in Re-based double perovskites: Synthesis, structural characterization, physical properties, advanced applications, and theoretical studies Cite as: AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 Submitted: 28 September 2020 •Accepted: 19 November 2020 • Published Online: 23 December 2020 Kai Leng,1Qingkai Tang,1Ying Wei,2Li Yang,3Yuting Xie,3Zhiwei Wu,1and Xinhua Zhu1,a) AFFILIATIONS 1National Laboratory of Solid State Microstructures, School of Physics, Nanjing University, Nanjing 210093, China 2National Laboratory of Solid State Microstructures, College of Engineering and Applied Science, Nanjing University, Nanjing 210093, China 3Kuang Yaming Honors School, Nanjing University, Nanjing 210093, China a)Author to whom correspondence should be addressed: xhzhu@nju.edu.cn ABSTRACT Recently, double perovskite (DP) oxides denoted A 2B′B′′O6(A being divalent or trivalent metals, B′and B′′being heterovalent transition met- als) have been attracting much attention owing to their wide range of electrical and magnetic properties. Among them, rhenium (Re)-based DP oxides such as A 2FeReO 6(A=Ba, Sr, Ca) are a particularly intriguing class due to their high magnetic Curie temperatures, metallic- like, half-metallic, or insulating behaviors, and large carrier spin polarizations. In addition, the Re-based DP compounds with heterovalent transition metals B′and B′′occupying B sites have a potential to exhibit rich electronic structures and complex magnetic structures owing to the strong interplays between strongly localized 3 delectrons and more delocalized 5 delectrons with strong spin–orbit coupling. Thus, the involved physics in the Re-based DP compounds is much richer than expected. Therefore, there are many issues related to the cou- plings among the charge, spin, and orbitals, which need to be addressed in the Re-based DP compounds. In the past decade, much effort has been made to synthesize Re-based DP compounds and to investigate their crystal structures, structural chemistry, and metal–insulator transitions via orbital ordering, cationic ordering, and electrical, magnetic, and magneto-transport properties, leading to rich literature in the experimental and theoretical investigations. This Review focuses on recent advances in Re-based DP oxides, which include their syn- thesis methods, physical and structural characterizations, and advanced applications of Re-based DP oxides. Theoretical investigations of the electronic and structural aspects of Re-based DP oxides are also summarized. Finally, future perspectives of Re-based DP oxides are also addressed. ©2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0031196 I. INTRODUCTION Perovskite oxides have been widely investigated during the past 70 years owing to their interesting multifunctional properties and versatile technical applications.1–3They possess a wide range of elec- trical properties such as insulating, semiconducting, metallic, and half-metallic behaviors, on the one hand, and ferromagnetic, ferri- magnetic, and antiferromagnetic behaviors, on the other hand.4–6Inaddition, they may demonstrate ferroelectric,7magnetic–dielectric,8 and multiferroic behaviors.9Thus, perovskite oxides have promis- ing applications in electronic/spintronic devices,10fuel cells,11solar cells,12and so on.13,14The above promising properties can be attributed to the high chemical and structural flexibilities of the per- ovskite structure. In an ideal cubic perovskite compound with the formula ABO 3, A-sited cations with a 12-fold coordination occupy the corners of a cubic unit cell and B-sited transition metal cations AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-1 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv sit at the center of the octahedron. Normally, the BO 6octahe- dra are required to distort and/or tilt in various possible direc- tions to accommodate the size mismatches generated by the sub- stituted cations at A and/or B sites. Therefore, almost all elements covered in the periodic table can be presented at the A or B site in the perovskite structure, opening up vast possibilities for the formation of perovskite compounds.15,16In recent years, with the research and development of quantum electronic/spintronic devices without dissipation, intensive research studies are focused on dou- ble perovskite (DP) oxides with the general formula A 2B′B′′O6 (A being divalent or trivalent metals, B′and B′′being 3d, 4d, or 5d transition metal cations) because of their intriguing mag- netic properties [e.g., ferromagnetic insulator (FMI) at high tem- peratures and spin polarized transport above room temperature]. DP oxides exhibit rich physical properties owing to a wide vari- ety of combinations of the metal ions at B′and B′′sites, which offers much tunable B′–O–B′′magnetic interactions via superex- change.10The first study of DP oxides was reported in the early 1950s, and it was widely carried out at the end of 1950s.17,18In the mid 1970s, a wide variety of DP oxides were reported. In 1998, Kobayashi et al.5reported a low-field room temperature magne- toresistance (MR) in an ordered Sr 2FeMoO 6DP compound; this finding stimulated the research studies on searching for new fer- romagnetic compounds with a DP crystallographic structure.19–22 Among the various DP oxides, rhenium (Re)-based DP oxides (e.g., A 2BReO 6, Sr 2(Fe 1−xCrx)ReO 6, Sr 2Fe1+xRe1−xO6) are a par- ticularly intriguing class owing to their high magnetic Curie tem- peratures, diverse electrical behaviors, huge magnetic anisotropy, and large carrier spin polarizations.23–25Moreover, the Re-based DP compounds containing 3 dand 5 dtransition metal ions at the B′and B′′sites have rich electronic structures and com- plex magnetic structures, which are ascribed to the strong inter- actions between strongly localized 3 delectrons and more delo- calized 5 delectrons with strong spin–orbit coupling. Thus, the involved physics in the Re-based DP compounds is much more complex than expected. Therefore, there are many issues related to the couplings among the charge, spin, and orbitals, which need to be addressed in the Re-based DP compounds.23–25In the past decade, many attempts have been made to study the structural chemistry, microstructure, and metal-to-insulator (MIT) transi- tion via orbital ordering, cationic ordering, and electrical, mag- netic, and magneto-transport properties of Re-based DP oxides, forming rich literature in the experimental and theoretical aspects. However, we note that only a few review articles of these com- pounds have been published. For example, Serrate et al. reviewed the FeMo-based and Re-based DP compounds with a focus on their magnetic properties and MIT behaviors,19whereas Karppinen and Yamauchi discussed their chemical aspects with emphasis on oxygen stoichiometry, cation ordering, and redox chemistry.26In this Review, a comprehensive review of the recent advances in Re- based DP oxides is provided, which includes their syntheses, phys- ical and structural characterizations, and advanced applications of Re-based DP oxides. Theoretical investigations of the electronic and structural aspects of Re-based DP oxides are also summa- rized. Finally, the summary and outlook of Re-based DP oxides are presented. It is expected that this Review will attract more interest in Re-based DP oxides and more researchers to enter this emerging field.II. SYNTHESIS METHODS OF Re-BASED DP OXIDES In the Re-based DP oxides of A 2MReO 6(A=alkaline earth metals such as Ba, Sr, Ca; M =transition metals such as Sc, Cr, Mn, Fe, Co, Ni, Zn), their rich substitutional properties are advan- tageous for fabricating DP oxide compounds in the form of bulk single crystals, polycrystalline ceramics, thin films, or nanocrystals. In this section, we will elaborate some methods used for synthesis of Re-based DP oxides. A. Re-based DP bulk oxides Re-based DP bulk oxides can be simply obtained by direct annealing the mixture of corresponding metal oxides at high tem- peratures, where the solid powders undergo both physical and chemical reactions to enable the thermal diffusion of ions or molecules. The solid-state reaction method is widely utilized to fab- ricate Re-based DP bulk oxides, which is environmentally friendly owing to without releasing toxic gases during the synthesized pro- cess. However, it has some disadvantages such as the final prod- ucts often with impurities and poor chemical homogeneity, non- distributed particle sizes, long reaction time required, and high annealing temperature (e.g., over 1000○C). These issues can be alle- viated by modifying the processing parameters of the solid-state reaction method (e.g., repeatedly grinding the starting materials; increasing annealing temperature and reaction time). For exam- ple, Kato et al.27synthesized the polycrystalline Re-based DP oxides such as Sr 2MReO 6and Ca 2MReO 6(M being transition metals) by solid-state reactions, where SrO, CaO, MO x, Re 2O7, and Re powders were used as the raw materials. The synthesized sam- ples were almost of pure phase despite a small amount of impu- rities (less than 2% in fraction) except Ca 2NiReO 6(about 8%). In another case of the Sr 2−xLaxFe1+x/2Re1−x/2O6(x=0.0–1.8) com- pounds, they were synthesized by the solid-state reaction method under an inert atmosphere, where La 2O3, SrCO 3, Fe 2O3, ReO 3, and Re were used as the starting materials. They were weighted based on the designed ReO 3/Re ratio to the theoretical chemical valence (5 +) of Re and calcined at 1100○C for 2 h in a stream of Ar gas.28To obtain a single-phase structure, the mixed pow- ders were repeatedly sintered at 1200○C for 3 h several times in the same atmosphere. To increase the density of bulk ceram- ics of Re-based DP oxides, Lim et al.29utilized the spark plasma sintering method to fabricate the bulk ceramic samples (e.g., Sr2FeReO 6and Sr 2CrReO 6). Spark plasma sintering technique is a high speed consolidation of the powders by using uniaxial pressure and pulsed electrical current under low atmospheric pressure.30,31 The remarkable features of the spark plasma sintering method make it suitable for fabricating Re-based DP ceramic oxides with high densification. B. Re-based DP thin films In recent years, Re-based DP thin films have attracted much attention owing to their high Tcvalue, large spin polarization, and half-metallicity, which have promising applications in spin elec- tronic devices. However, to date, only a few works reported on Re- based DP thin films, unlike their bulk counterpart. Recently, Sohn et al.25grew epitaxial Sr 2Fe1+xRe1−xO6(−0.2≤x≤0.2) thin films AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-2 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv on single-crystal (001) SrTiO 3substrates by pulsed laser deposi- tion (PLD). They demonstrated that the compositional ratio and cation ordering of the Sr 2Fe1+xRe1−xO6thin films were closely related to the oxygen pressure ( PO2). The thin films exhibited a ferromagnetic insulator (FMI) state with high- Tcferromagnetism and high saturation magnetization ( MS≈1.8μB/f.u.). In addi- tion, their room- Tsheet resistance was about three order larger as compared with that of metallic films. The stable FMI state was found in the cation-ordered Fe-rich films due to the formation of extra Fe3+–Fe3+and Fe3+–Re6+bonding states. In another study, highly ordered epitaxial Sr 2CrReO 6thin films were also deposited on single-crystal (001) SrTiO 3substrates with an epitaxial buffer layer of Sr 2CrNbO 6(30 nm–48 nm in thickness) by off-axis mag- netron sputtering.32The Sr 2CrReO 6films had 99% Cr/Re order- ing and high-quality crystallinity. Their electrical resistivity was strongly sensitive to the oxygen partial pressure. 18 000% mod- ulation in electrical resistivity was achieved at a temperature of 7 K (60% modulation at 300 K) as the oxygen partial pressure had 1% modulation during film growth. The results indicate that the electrical properties of Sr 2CrReO 6thin films are closely related to the content of oxygen vacancies. Orna et al.33also grew epitax- ial Sr 2CrReO 6films by PLD, which exhibited high crystallinity and large cationic ordering. Their MSvalue at 300 K was ∼1.0μB/f.u. and the electrical resistivity ( ρ) was 2.8 m Ωcm at 300 K, similar to those reported previously for the sputtered epitaxial Sr 2CrReO 6 films. C. Re-based DP powders Large magnetoresistances are reported in the polycrystalline Re-based DP oxides such as Sr 2FeReO 6and Sr 2CrReO 6, which resulted from the intergrain tunneling magnetoresistance (ITMR) effect controlled by the grain boundary magnetization.34It is also reported that the formation of nanosized particles can significantly enhance the ITMR effect in Re-based DP powders.35–37However, the synthesis of Re-based DP powders with high-purity and homogene- ity is proven to be challenging due to the strong refractive nature of the reduced rhenates (i.e., Re5+/Re6+). Recently, molten salt syn- thesis (MSS) is used to synthesize Re-based DP oxide nanoparti- cles (e.g., Sr 2FeReO 6, Ba 2FeReO 6, and Sr 2CrReO 6), and the eutectic mixture of NaCl–KCl salts acted as the reaction medium. Fuoco et al.37reported the influence of annealing temperature, holding time, cooling rates, and molten salt fluxes on the particle sizes, ordering degree at the B′/B′′site, and the physical properties of Sr 2FeReO 6, Ba 2FeReO 6, and Sr 2CrReO 6powders. They found that high purity and homogeneity were achieved in the Re-based DP oxide powders (e.g., Sr 2FeReO 6, Ba 2FeReO 6, and Sr 2CrReO 6) synthesized by the MSS method. The particle sizes of the Re- based DP powders were in the range from ∼50 nm to >1μm, which were dependent upon the different synthesized conditions and the compositions of DP oxides. Generally, small-sized parti- cles were synthesized under short reaction time, which had much large concentrations of grain boundaries, leading to strong temper- ature dependent resistivity. This new synthetic route allows one to deeply investigate the effects of the particle size and the ordered degree at the B′/B′′site on the magnetic properties of Re-based DP powders.III. DOPING EFFECTS IN Re-BASED DP OXIDES A. Hetero-valence dopants Spintronics is becoming a candidate technology based on the spin of the electron to carry and store information as 0 bits (spin up ↑) and 1 bit (spin down ↓). This new technology has several advan- tages over the current charge-based technologies.38–40To realize the full commercial spintronic devices, the essential components such as half-metallic spin injectors and magnetic semiconductors are highly required.41–43However, the main challenge to this new technology is the discovery of suitable candidates that have ferro- magnetic ordering and large carrier spin polarizations over room temperature. Re-based DP oxides such as Sr 2CrReO 6are considered the potential materials for spintronics owing to their high Tc value (∼635 K), half-metallicity, and large MS(=1.0μB/f.u.).44–48 Following the successful increase in the TCvalue in the Mo-based DP oxides by electron doping, similar research works were also carried out in the Re-based DP oxides.49However, the expected enhancement of TCwith lanthanide addition in the Sr 2CrReO 6 DP oxides was not realized. This means the magnetic interaction strength is decreased with lanthanide doping. The converse elec- tron doping scenarios observed in Sr 2CrReO 6and Sr 2FeMoO 6 may be ascribed to the presence of secondary phases in the final products of Sr 2-xLnxCrReO 6(Ln being La, Nd, or Sm). To confirm this assumption, Blasco et al.50fabricated two sets of polycrys- talline Sr 2−xLnxCrReO 6and Sr 2−xLnxCr1+x/2Re1−x/2O6samples (x =0.0–0.5; Ln =La, Nd, Sm) by the solid-state reaction method. They found that the synthesis of the Sr 2−xLnxCrReO 6(Ln=La, Nd, or Sm) single phase was impossible via conventional synthetic routes because of the formation of competitive Sr 2−xLnxCr1+x/2Re1−x/2O6 and Sr 11Re4O24phases. This was the reason why it is significantly hard to synthesize electron-doped DP oxides in the (Cr, Re)-based systems. Single-phase compounds were only obtained with the stoi- chiometric formula of Sr 2−xLnxCr1+x/2Re1−x/2O6, which crystallized in a fcc cubic cell with a shrinkage when the sizes of Ln ions were decreased. These samples exhibited spontaneous magnetization and large coercive field at room temperature, and their Curie temperature was weakly affected by Ln addition, whereas the MS value was decreased owing to the appearance of anti-site (AS) defects. The electronic states of both Cr3+and Re5+ions were not affected by replacing Sr by Ln, as confirmed by x-ray absorption spectroscopy (XAS) at the Cr Kedge and Re L1,2,3edges. Recently, the first-principles calculations demonstrate that the La-doped Ba2−xLaxFeReO 6(x=0.0, 0.5, and 1.0) compounds undergo a transition from half-metallic to insulating behavior as the La-doping content (x) is increased up to x =1.0.51The injections of electrons mainly go to Re t2gorbitals, whereas the Fe 3 dorbitals almost remain unchanged when increasing the La-doping content. There- fore, Fe exists as 3 +in doped and un-doped samples, while Re has different chemical states in Ba 2FeReO 6(5+), Ba 1.5La0.5FeReO 6 (4.5), and BaLaFeReO 6(4+) compounds. The remanent magnetization is decreased when increasing the La-doping content. B. Homo-valence dopants In the case of homo-valent dopants, polycrystalline Re-based DP Sr 2LnReO 6(Ln=Y, La, Nd, Sm–Gd, and Dy–Lu) samples are AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-3 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv also synthesized by the standard solid-state reaction, where the tran- sition metals such as Fe, Cr were used to replace lanthanide (such as trivalent Y, La, Nd, Sm–Gd, and Dy–Lu) at the B′site.52It is observed that the Ln3+and Re5+ions are orderly distributed at the B′ and B′′sites of the DP structure. The Sr 2LnReO 6compounds exhibit an antiferromagnetic behavior at 2.6 K–20 K. This can be ascribed to the magnetic interactions between Re5+ions. In the Sr 2YbReO 6 compound, the magnetic orderings of Yb3+and Re5+ions appear at 20 K, whereas, at around 90 K, Re5+ions are magnetically ordered in Sr2DyReO 6. By furthermore reducing the temperature, Dy moments are ferromagnetically ordered at 5 K. IV. STRUCTURAL CHARACTERIZATIONS OF Re-BASED DP OXIDES A. Introduction The structural characterizations of Re-based DP oxides include determinations of crystal structures, chemical compositions, and morphologies. There are several conventional methods for char- acterizing the crystal structures, which are based on the inactions between the incident x ray (or high-energy electron beam) and the investigated samples. X-ray diffraction (XRD) is a very com- mon method to determine the phase structures of all kinds of mat- ter ranging from fluids, to powders, and crystals. By analyzing the XRD data, the critical features such as crystal structures, crystal- lite sizes, lattice parameters ( a–c), lattice volume, and strain can be determined. Infrared and Raman spectra are two complemen- tary techniques, which provide the characteristic fundamental vibra- tions used for determination and identification of the crystal struc- tures at a molecular level. Scanning electron microscopy (SEM) and transmission electron microscopy (TEM) make use of a fine elec- tron beam to reveal the microstructures of the investigated samples, which work in electron back-scattering and transmission modes, respectively. Two kinds of high-resolution imaging modes named high-resolution TEM (HRTEM) and scanning transmission elec- tron microscopy (STEM) are often used to reveal microstructures (or structural defects) of Re-based DP oxides at an atomic level or even at a sub-Å level. Selected area electron diffraction (SAED) is widely utilized to determine the phase structure of the samples, whileconvergent beam electron diffraction (CBED) has a unique ability of determining the space group of crystals due to its great advan- tages of obtaining reliable experimental data from a single-domain area. The spectral analyzed techniques such as energy dispersive x- ray spectroscopy (EDS), electronic energy loss spectroscopy (EELS), and x-ray photoelectron spectroscopy (XPS) are used to deter- mine the chemical compositions and chemical bonding states of the Re-based DP oxide. In this section, we shortly review the recent advantages in the microstructural characterizations of Re-based DP oxides. B. Re-based DP bulk oxides To date, many Re-based DP bulk oxides have been synthesized by different methods, and their phase structures are first examined by XRD, from which the lattice parameters ( a–c), unit cell volume (V), and density of the sample in theory can be determined. As an example, XRD patterns of Ba 2FeReO 6and Ca 2FeReO 6DP com- pounds synthesized by the solid-state reaction are demonstrated in Fig. 1. As shown in Fig. 1(a), the XRD pattern of Ba 2FeReO 6can be well indexed based on a cubic cell ( Fm3m), and the lattice con- stant ( a) is determined to be about 8.054 Å. Ca 2FeReO 6[Fig. 1(b)] is found to crystallize in a monoclinic structure with the space group of P21/nand lattice parameters of a≈5.396 Å, b≈5.522 Å, c≈7.688 Å, andβ=90.4○.53It was also noticed that the impurity phase was not observed in the XRD patterns, implying the formation of a pure compound in each case. Figure 2 demonstrates the room tempera- ture XRD patterns of a series of Ca-doped Sr 2−xFeReO 6(x=0.0–2.0) samples.54It was observed that the samples with x =1.5 and 2.0 exhibited a monoclinic structural distortion and crystallized with the space group of P21/n, whereas the samples with x =0.5 and 1.0 had a pseudo-cubic structure. The lattice parameters ( a–c) determined from the XRD patterns are displayed in Fig. 3. In order to express the overall behavior of the investigated series, pseudo-cubic param- eters, aps(=a√ 2) and bps(=b√ 2), are used rather than the lattice parameters ( aandb) of the tetragonal and monoclinic samples. To describe the structural changes undergone in the series of Re-based A2MReO 6DP compounds (A being Sr, Ca; M being Mg, Sc, Cr, Mn, Fe, Co, Ni, Zn), the tolerance factor ( t) is introduced, which FIG. 1. X-ray powder diffraction patterns of double-perovskites of (a) Ba 2FeReO 6and (b) Ca 2FeReO 6. Reprinted with permission from Prellier et al. , J. Phys.: Condens. Matter 12, 965 (2000). Copyright 2000 IOP Publishing Ltd. AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-4 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv FIG. 2. XRD patterns of the Ca xSr2−xFeReO 6compounds with x =0.0–2.0 mea- sured at room temperature. From bottom to top, the value of x increases from 0.0 to 2.0 with an interval of 0.5. The inset shows the local XRD patterns (2 θ =22○–30○), showing the (111) diffraction peak and confirming the appearance of theP21/nstructure. Reprinted with permission from De Teresa et al. , Phys. Rev. B 69, 144401 (2004). Copyright 2004 American Physical Society. is written as t=(rA+rO)/√ 2[(rM+rRe)/2+rO], (1) where rA,rM,rRe, and rOrepresent the ionic radii of A, M, Re, and O ions, respectively. Figure 4 shows the tolerance factor dependent upon the cell volume, monoclinic structural distortion expressed by∣β-90○∣(β, the monoclinic angle), and averaged M-O-Re bond angles of the Re-based A 2MReO 6compounds.27It is noticed that the Sr-based compounds (e.g., Sr 2MReO 6) have larger cell vol- umes and M-O-Re angles (close to 180○) in comparison with the Ca-based compounds (e.g., Ca 2MReO 6). In addition, for each A- site ion, with the decreasing tolerance factor tin the A 2MReO 6 compounds, cell volumes continuously increased, whereas the M- O-Re angles monotonously decreased. For the compounds with t<0.98, their crystal structures are monoclinically distorted and the distortion degree of ∣β-90○∣increases with the reducing tvalue. During the growth of Re-based double perovskite oxides, AS defects are also formed in Sr 2FeReO 6(e.g., Fe occupies the nor- mal Re site, Fe Re, and Re occupies the normal Fe site, Re Fe) and Sr2CrReO 6(e.g., Cr occupies the normal Re, Cr Re, and Re occupies the normal Cr site, Re Cr), respectively, which are dependent upon the synthesis conditions (e.g., annealing temperature, holding time, and pressure) in the spark plasma sintering process. The formation of AS defects is harmful to magnetization. Therefore, it is critical to suppress the formation of AS defects during the growth process of the DP samples. In general, diffraction techniques are used to detect the AS defects (quantifying the amount of misplaced mag- netic ions) since some specific Bragg peaks such as (111), (113), and (331) diffraction peaks would emerge owing to the cationic ordering formed at the B′and B′′superlattices.55Recently, local FIG. 3. Lattice parameters ( a–c) of Ca xSr2−xFeReO 6compounds obtained from the room temperature XRD patterns plotted as the average radius (r A) of the A- site in A 2FeReO 6(open circle) and A 2FeMoO 6(solid triangle). To demonstrate the overall behaviors of the tetragonal and monoclinic samples ( x=0), pseudo- cubic parameters aps(=a√ 2) and bps=(b√ 2) are used to replace the lattice parameters aandb. The solid lines just show the trends. Reprinted with permission from Serrate et al. , J. Phys.: Condens. Matter 19, 023201 (2007). Copyright 2007 IOP Publishing Ltd. cationic ordering in the pristine and Re-excess Sr 2FeReO 6sam- ples was comparatively investigated by high-angle annular dark- field (HAADF)-STEM.56It is found that cationic ordering appears not only in the pristine sample but also in the Re-excess 15 mol. % Sr 2FeReO 6sample. The HAADF-STEM images obtained from the pristine Sr 2FeReO 6sample taken along [001] and [1 10] direc- tions are demonstrated in Figs. 5(a) and 5(b), respectively. It is noticed that only the Fe and Re columns (orange sphere) and Sr col- umn (green sphere) are resolved in the [001] projection, whereas, in the [1 10] projection, the Fe column (yellow sphere), Re column (blue sphere), and Sr column (green sphere) are clearly resolved. This confirms the Fe/Re ordering in the pristine Sr 2FeReO 6sam- ple. Figure 5(c) displays three kinds of intensity profiles of atomic columns, which are obtained from the pristine Sr 2FeReO 6sample (marked by a red line), the Re-excess Sr 2FeReO 6sample (marked by a blue line), and the simulated Sr 2FeReO 6image (marked by a green line). Thus, the Fe/Re cationic ordering at the B′and B′′sites in the pristine and Re-excess Sr 2FeReO 6samples is well confirmed. How- ever, in the pristine Sr 2FeReO 6sample, AS defects are also found, AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-5 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv FIG. 4. Tolerance factor dependent upon the cell volume, the monoclinic distor- tion expressed by ∣β-90○∣(β, monoclinic angle), and the averaged M–O–Re bond angles of the ordered A 2MReO 6DPs (A being Sr, Ca and M being Ca, Mg, Sc, Cr, Mn, Fe, Co, Ni, and Zn). Reprinted with permission from Kato et al. , Phys. Rev. B 69, 184412 (2004). Copyright 2004 American Physical Society. as revealed in Fig. 6(a). In column intensity profiles, the Re atomic columns with much higher intensities are clearly distinguishable, and between two adjacent Re columns, there are three Fe atomic columns with almost equal intensities, which are indicated by the red arrow. It is noticed that some Fe atomic columns exhibit much lower intensities as compared with their neighbors, which are indi- cated by the green or yellow arrow. The appearance of Fe atomic columns with unusual strong intensities indicates the formation of AS defects. Figures 6(b)–6(d) present a series of [110] projections of simulated HAADF-STEM images, which have the Fe–Re exchang- ing ratios at 10%, 15%, and 20%, respectively. It is observed that the intensity profiles of the Fe atomic columns marked by the green and red arrows indicate more AS defects, which match well with the simulated images with 15% and 20% Fe–Re exchanges. Besides the AS defects, the antiphase boundary (APB) was also observed in the Re-excess Sr 2FeReO 6samples. Figure 7(a) shows the HAADF-STEM image taken from an APB in the [1 10] direction projection, and Fig. 7(b) displays the intensity profile across the APB boundary. It is observed that the intensity profile in Fig. 7(b) exhibits quite sym- metry near the Sr atom, as demonstrated by a red arrow. However, across the APB, the intensities of the Re columns (column nos. 4–7) are found to be reduced gradually, whereas an inverse case appears for the Fe columns (indicated by blue arrows). This means the Fe/Re FIG. 5. HAADF-STEM images taken from the pristine Sr 2FeReO 6samples along (a) [001] and (b) [1 10] directions. Simulated HAADF-STEM images are shown as the insets. The Fe and Re columns (orange sphere) and Sr column (green sphere) are clearly resolved in the [001] projection, whereas, in the [1 10] projection, the Fe column (yellow sphere), Re column (blue sphere), and Sr column (green sphere) are clearly resolved. (c) Intensity profiles for the atomic chains of Re–Sr–Fe–Sr–Re along the [001] direction measured from the HAADF-STEM images taken from the pristine samples (red color) and the Re excess samples (blue color) in the [110] projection plane. The intensity profile of the simulated image is marked by green color. Reprinted with permission from Choi et al. , Microsc. Microanal. 19(S5), 25 (2013). Copyright 2013 Cambridge University Press. AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-6 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv FIG. 6. (a) HAADF-STEM image taken from the pristine Sr 2FeReO 6samples and their intensity profile of the atomic chains taken from the rectangle marked by red color. Simulated HAADF-STEM images of the Sr 2FeReO 6samples with Fe/Re disordering of (b) 10%, (c) 15%, and (d) 20% and the corresponding intensity profiles. Reprinted with permission from Choi et al. , Microsc. Microanal. 19(S5), 25 (2013). Copyright 2013 Cambridge University Press. FIG. 7. (a) HAADF-STEM image of the antiphase boundary (APB) observed in the Re-excess Sr 2FeReO 6samples and (b) intensity profile across the APB along the [001] direction. Reprinted with permission from Choi et al. , Microsc. Microanal. 19(S5), 25 (2013). Copyright 2013 Cambridge University Press.cationic disordering appears around the APB region due to the for- mation of strong antiferromagnetic coupling between the magnetic cationic pairs of Fe3+/Fe3+near the APB, leading to the reduced magnetic spin of Fe across the APB. Lim et al.29also reported that adding Re is an effective way to reduce the AS defects in Sr 2FeReO 6 (by 10.4% when increasing the amount of excess Re up to 15 mol. %), but it slightly increases the AS defects (by 0.9% in Sr 2CrReO 6 when increasing the amount of excess Re up to 10 mol. %).29Such a discrepancy can be ascribed to their different thermodynamic stabil- ities between Sr 2FeReO 6and Sr 2CrReO 6with the excess of Re. The microscopic structures of AS defects in Sr 2FeReO 6and Sr 2CrReO 6 are also examined by HAADF-STEM images. Figure 8(a) reveals the sequential atomic chains of Re–Sr–Fe–Sr–Re along the [001] direc- tion and Re–Fe–Re along the [110] direction since B-sited atoms (Re and Fe) are arranged along the ⟨110⟩direction. Figures 8(b)–8(e) show the HAADF-STEM images taken from Sr 2CrReO 6-Re excess 5 mol. %, Sr 2CrReO 6-Re excess 15 mol. %, Sr 2CrReO 6-no excess Re, and Sr 2CrReO 6-Re excess 10 mol. %. In the Sr 2CrReO 6-Re excess 5 mol. % sample [Fig. 8(b)], antiphase-boundary (APB)-like defects were observed, similar to that reported for Sr 2FeReO 656and Sr2FeMoO 6.57However, such defect structures were not observed in the sample of Sr 2CrReO 6with Re excess 15 mol. % [Fig. 8(c)], which was ascribed to the reduced sizes of AS defects with a larger amount of excess Re. In the pure Sr 2CrReO 6sample [Fig. 8(d)] and Re excess 10 mol. % Sr 2CrReO 6sample [Fig. 8(e)], the AS defects were not observed due to their wide distribution in the whole sam- ples, similar to the cases in other oxides.58,59The Fe/Re cation ordering at the B-sites (B′and B′′) of Sr 2FeReO 6was also exam- ined by using EDS element mapping formed by the Fe K αsignal at 6.405 keV and Re K αsignal at 8.652 keV, besides the nanod- iffraction patterns along ⟨011⟩pc.60Figure 9(a) is the HAADF-STEM AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-7 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv FIG. 8. (a) HAADF-STEM image obtained from Sr 2FeReO 6with excess 15 mol. % Re in the [1 10] projection, where Re (brown)–Sr (green)–Fe (gray)–Sr (green)–Re(brown) atomic chains along the [001] direction are well resolved from their Z contrast. The inset shows a schematic diagram of the atomic distributions in the projection plane of [1 10]. (b)–(e) HAADF-STEM images taken from Sr 2FeReO 6-5 mol. % excess Re, Sr 2FeReO 6-15 mol. % excess Re, Sr 2CrReO 6-0 mol. % excess Re, and Sr 2CrReO 6-10 mol. % excess Re samples, respectively. The inset of (b) shows an enlarged AS defect region indicated by the yellow rectangle, exhibiting an apparent difference from the well-ordered structure demonstrated in (a). Reprinted with permission from Lim et al. , Sci. Rep. 6, 19746 (2016). Copyright 2016 Nature Publishing Group. image taken from a perfect region of the Sr 2FeReO 6sample, and Figs. 9(b)–9(e) show the corresponding atomic-resolution elemental maps and nanodiffraction pattern. The compositional distributions of the Fe element (red) [Fig. 9(b)] and Re element (blue) [Fig. 9(b)]are stacked alternatively in the {111} planes, giving a composite color map, which matches well with the feature of the Z-contrast image shown in Fig. 9(a). The appearance of 1/2{ 111}pcsuper-diffraction spots marked by dotted circles in Fig. 9(e) also confirmed the FIG. 9. Structural and compositional characterizations of ordered Sr 2FeReO 6. (a) HAADF-STEM image viewed in the [1 10] projection; (b) and (c) atomic-resolution element maps of Fe (red) and Re (blue) elements, respectively. (d) Mixed Fe and Re maps, and (e) nanodiffraction pattern taken along ⟨011⟩pc. Reprinted with permission from Ho et al. , Ultramicroscopy 193, 137 (2018). Copyright 2018 Elsevier B.V. AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-8 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv alternative distributions of Fe and Re atoms in the {111} planes. The quantitative results of EDX spectra demonstrate the Fe/Re ratio of the selected region was 0.51:0.49, close to the statistical measure- ments. Therefore, the Sr 2FeReO 6phase can be further written as Sr2[Fe]B′[Re]B′′O6. The local electronic structures and geometries around the Cr and Re atoms in the double perovskite Sr 2−xLnxCr1+x/2Re1−x/2O6 doped with Ln ( =La, Nd, or Sm) were also investigated by x-ray absorption near edge structure (XANES) and extended x-ray absorp- tion fine structure (EXAFS) spectra.50Figure 10(a) presents the XANES spectra of Sr 2CrReO 6obtained at the Cr K-edge, where dif- ferent chemical valences of Cr ions are selected such as the Cr3+ ions in LaCrO 3and Cr 2O3, Cr6+ion in CrO 3, and metallic Cr. It is found that this edge is very sensitive to the local structures of the investigated samples. For example, in the CrO 3sample, a sharp peak appears in the pre-edge region at 5.993 keV, which resulted from the dipole-forbidden transition from 1 sto 3d. This indicates the tetrahedral coordination of Cr6+without an inversion center.61 As shown in Fig. 10(a), the positions of absorption edges are depen- dent upon the Cr oxidation states, which move to higher energy with increasing chemical valences of Cr ions. It is observed that the rising part of the Sr 2CrReO 6main edge located at 6.006 keV coincides with that of LaCrO 3due to both compounds having a similar perovskite structure; thus, the chemical valence of the Cr3+ ion is determined for Sr 2CrReO 6. The absorption spectra for the Sr2−xLnxCr1+x/2Re1−x/2O6samples (Ln =Nd or Sm) are identical tothose of Sr 2CrReO 6, and their edge positions are also presented [see the inset of Fig. 10(a)], where the parent compound data at 35 K and 295 K are also presented. It is found that the edge positions maintain almost a constant value, indicating the chemical states of Cr ions are stable despite replacing the Sr element by rare earths. This means Cr3+ions exist in all the investigated samples. To accu- rately understand the physical mechanisms of the pre-edge peaks of the Sr 2−xNdxCr1+x/2Re1−x/2O6samples, their pre-edge features are comparatively investigated with those of LaCrO 3, as shown in Fig. 10(b). The small peak denoted A was observed in all samples with nearly the same energy and intensity, which resulted from pure quadrupole transitions that occurred in related oxides, and was much sensitive to the occupied 3 dorbital.62In this way, the pres- ence of Cr3+in the double perovskite of Sr 2−xNdxCr1+x/2Re1−x/2O6 is confirmed. However, the other pre-edges of DP compounds have much difference from those of LaCrO 3. A strong peak (denoted the B peak) was observed in Sr 2CrReO 6, and its intensity was decreased with the increasing rare earth-doping concentration [see the inset of Fig. 10(b) for details]. The electrical conductivities of these samples were reported to be decreased as the Ln-doped concentration was increased.63This is consistent with the concomitant weak orbital overlapping between the Cr and oxygen atoms. Thus, the intensity of peak B reflects the covalent bond strength of the Cr–O bond. Recently, the sensitivity of the XANES spectra has been used to check the oxidation states of Cr and Re elements. The chemical shift of the CrKedge in the XANES spectra revealed the presence of Cr3+ions FIG. 10. (a) Normalized XANES spectra recorded at the Cr K-edge from metallic Cr (dotted-dashed line), Cr 2O3(dashed line), LaCrO 3(dotted line), Sr 2CrReO 6(thick line), and CrO 3(thin line), where the pre-edge features in the spectrum of Sr 2CrReO 6are denoted A and B, respectively. The data in the inset are the edge positions determined by the maximum of the first derivative of the normalized XANES spectra for the Sr 2−xNdxCr1+x/2Re1−x/2O6(filled squares) and Sr 2−xSm xCr1+x/2Re1−x/2O6(open squares) samples at 295 K. The edge position data for Sr 2CrReO 6at 35 K and 295 K are denoted by filled and open symbols, respectively. (b) Pre-edge structures in the normalized XANES spectra of Sr 2−xNdxCr1+x/2Re1−x/2O6(x≤0.5) and LaCrO 3after subtracting the background. The inset shows the area of peak B in Sr 2−xNdxCr1+x/2Re1−x/2O6 and Sr 2−xSm xCr1+x/2Re1−x/2O6series, which reflects the covalent bond strength of the Cr–O bond. Reprinted with permission from Blasco et al. , Phys. Rev. B 76, 144402 (2007). Copyright 2007 American Physical Society. AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-9 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv for all compounds [Fig. 11(a)], while the XANES spectra at the Re L1,2,3edges confirm the Re5+ions exist in all compounds [Fig. 11(b)]. This means the electronic states of both Cr3+and Re5+ions are not affected by Sr substitution by Ln elements. The Re–O and Cr–O bond lengths can be deduced from the EXAFS spectra, which FIG. 11. (a) Fourier transform of the k-weighted EXAFS spectra recorded at the Cr Kedge for the Sr 2−xSm xCrReO 6series at 35 K. Solid lines represent the best-fit simulations where the contributions between 1 Å and 4 Å are only considered in the structural analysis. (b) Fourier transform of the k-weighted EXAFS spectra taken at the Re L3edge for the Sr 2−xSm xCrReO 6series at 35 K. Solid lines represent the best-fit simulations where the contributions between 1.2 Å and 3.9 Å are only considered in the structural analysis. Reprinted with permission from Blasco et al. , Phys. Rev. B 76, 144402 (2007). Copyright 2007 American Physical Society.fit well with crystallography. These bond lengths do not change along with the Ln-doping concentration, which agrees with the as expected values for Re5+and Cr3+oxides. As a consequence, the spectroscopic study confirms that the chemical valences of Cr and Re elements are not changed before and after substituting Sr with Ln elements. C. Re-based DP thin films Re-based DP thin films have promising applications in the near-future spintronics due to their high Tcvalue, half-metallicity, and large spin polarization. In addition, during the growth pro- cess of thin films, the strain, chemical heterogeneity, and artificial structures can be well controlled to realize the novel or enhanced properties, which are much different from those observed in the form of bulk counterparts.64–68Recently, Sohn et al.25grew high- quality epitaxial Sr 2Fe1+xRe1-xO6(−0.2≤x≤0.2) thin films by PLD. Figure 12(a) demonstrates the XRD patterns of θ–2θscans for the films grown at 775○C but with different PO2, and Fig. 12(b) dis- plays the off-axis XRD θ–2θscans collected near the (111) diffraction peak of Sr 2Fe1+xRe1−xO6, where only the (111) peak is observed in the films grown with PO2≥15 mTorr. The appearance of the (111) diffraction peak confirms the Fe/Re ordering, which is confirmed by the HAADF-STEM image [Fig. 12(c)] taken from the film grown at 20 mTorr in the projection along the [1 10] direction. In Fig. 12(c), the atomic chains of Re–Sr–Fe–Sr–Re along the [001] direction are clearly distinguishable and the atomic chains of Re–Fe–Re along the [110] direction are also clearly observed, which are sandwiched between two adjacent Sr atomic layers. Such clear atomic ordering is highlighted by the yellow diamond in Fig. 12(c), which is well consistent with the atomic positions in the projected structure of Sr2FeReO 6along the [1 10] direction [Fig. 12(d)]. The Re and Fe atoms are alternatively distributed in the (111) planes, as marked by blue and red lines, respectively. In another work, Hauser et al.69grew (001)-oriented and (111)-oriented Sr 2CrReO 6epitaxial thin films by magnetron sputtering, which were characterized by XRD, as shown in Fig. 13. Figure 13(a) shows XRD θ–2θscans of (111)-oriented Sr2CrReO 6thin films grown on SrTiO 3single-crystal substrates, where only the {111} diffraction peaks from the film and substrate are observed, indicating the formation of the pure DP phase. In Fig. 13(b), Laue oscillations near the (004) diffraction peak of (001)- oriented Sr 2CrReO 6thin films are clearly observed, indicating a per- fect interface between the film and the substrate as well as a smooth film surface. The high crystalline quality of the Sr 2CrReO 6films is reflected by their rocking curves that are shown in Fig. 13(c). The full-width-at-half-maximum (FWHM) of the rocking curve for the Sr2CrReO 6(004) peak is only 0.0088○, which is smaller than 0.0093○ for the SrTiO 3(002) peak. For the Sr 2CrReO 6films grown under optimal conditions, both clear Laue oscillations and sharp rocking curves reveal the high quality of the Sr 2CrReO 6films.70This is also confirmed by HAADF-STEM images. Figure 14(a) displays a low- magnification HAADF-STEM image of the Sr 2CrReO 6film taken along the [1 10] direction, which exhibits flat surface and uniform thickness. The atomic structure of the Sr 2CrReO 6film is revealed by a high-magnification HAADF-STEM image, which demonstrates the atomic chains of Re–Sr–Fe–Sr–Re along the [001] direction and the atomic chains of Re–Fe–Re along the [110] direction, confirm- ing the clear Laue oscillations observed in Fig. 13(b). Pronounced AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-10 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv FIG. 12. (a) XRDθ–2θscans for the epitaxial Sr 2Fe1+xRe1−xO6films grown on SrTiO 3at 775○C and under different oxygen pressures. Diffraction peaks from the SrTiO 3substrate are denoted by asterisks. (b) Off-axis XRD θ–2θscans performed at ψ=54.7○ near the Sr 2Fe1+xRe1−xO6(111) peak, confirming the Fe/Re ordering. (c) HAADF-STEM image of the epitaxial Sr2Fe1+xRe1−xO6films grown under oxygen pressure of 20 mTorr. (d) Schematic diagram illustrating atomic distributions in the projected plane along the [1 10] direction. The alternating Re and Fe planes along the [111] direction are denoted by blue and red lines, respectively. The black arrow indicates the [111] direction. The yellow diamond outlined in (c) highlights the ordered distributions of Re ions. Reprinted with permission from Sohn et al. , Adv. Mater. 31, 1805389 (2019). Copyright 2019 WILEY-VCH Verlag GmbH & Co. KGaA. STEM image contrast is obviously observed in the inset of Fig. 14(b), where three distinct shadows are clearly resolved, which correspond to Sr, Cr, and Re atoms, respectively. This matches well with the crys- tal projection of the DP lattice in the [1 10] direction [Fig. 14(c)]. The direct observations on the Cr/Re ordering in the Sr 2CrReO 6 films reveal the high quality of the film. Similarly, epitaxial growth of Sr 2CrReO 6films on the SrTiO 3(001) substrate by PLD is also reported by Orna et al.33The epitaxial growth relationship between the Sr 2CrReO 6film and the SrTiO 3substrate is confirmed by high- resolution XRD patterns and STEM images, which is expressed as Sr2CrReO 6(001)[100] ∣∣SrTiO 3(001)[110]. The AS concentration in this epitaxial film was estimated to be 14%, and the Curie temper- ature ( TC) was determined to be 481 K, smaller than that reported for sputtered samples.71 D. Re-based DP oxide powders Recently, the half-metallic A 2FeReO 6(A=Sr and Ba) and Sr2CrReO 6powders were synthesized by the MSS method.37 Figure 15 demonstrates XRD patterns collected from the flux- synthesized and solid-state-synthesized A 2FeReO 6(A=Sr and Ba) and Sr 2CrReO 6powders. Rietveld refinements of these XRD data gave the crystal parameters such as unit cell parameters, atomic positions, fractional B′and B′′site occupancies, and order- ing degree. The Fe/Re ordering degree is reflected via the peak intensity of the (111) super-structural reflections in Sr 2FeReO 6andSr2CrReO 6, or the analogous (101) super-structural reflection in Ba2FeReO 6, and these super-structural reflections are labeled with asterisks in Fig. 15. Morphologies of the flux-prepared A 2FeReO 6 (A=Sr and Ba) and Sr 2CrReO 6powders under different flux- to-product molar ratios were examined by SEM, and the rep- resentative SEM images are shown in Figs. 16 and 17. Aggre- gated block-like particles were observed in the SEM images, and their sizes changed from about 50 nm to over 1.0 μm, which were dependent upon the particular DP oxide compositions and the flux-synthesized conditions. Generally, smaller particles could be synthesized under the larger flux-to-product molar ratio and faster cooling process, or shorter reaction time. Strong temper- ature dependence of magnetoresistivity was found in the flux- synthesized A 2FeReO 6(A=Sr and Ba) and Sr 2CrReO 6pow- ders, which are ascribed to the higher concentrations of grain boundaries present, as compared to the solid-state-synthesized samples. V. PHYSICAL PROPERTIES OF Re-BASED DP OXIDES A. Magnetic properties 1. Re-based DP bulk oxides The magnetic properties of Re-based DP bulk oxides are strongly related to the ordering of the B-site ions. Their net mag- netizations are decreased owing to the existence of B-site disorder. AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-11 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv FIG. 13. (a) XRDθ–2θscans of the (111)-oriented Sr 2CrReO 6film with a thickness of 1220 nm. The insets show details of the fitting near the Sr 2CrReO 6 (222) and (444) peaks, which overlap with the substrate peaks. The Cr/Re ordering parameter ξwas determined to be 0.99 by the Rietveld refinements (red or dark gray) of the XRD data of the (111)-oriented film. (b) XRD θ–2θ scans of the (001)-oriented Sr 2CrReO 6 film with a thickness of 190 nm. Laue oscillations near the Sr 2CrReO 6(004) peak are clearly observed. The inset shows the local off-axis θ–2θscans around the Sr 2CrReO 6(022) peak per- formed at a tilt angle ψof 45○in the (001)-oriented film. (c) Rocking curves of the Sr 2CrReO 6(004) peak (blue) and the substrate SrTiO 3(002) peak (black) for the (001)-oriented Sr 2CrReO 6(001) film with a thickness of 590 nm. Reprinted with permission from Hauser et al. , Phys. Rev. B 85, 161201 (2012). Copyright 2012 American Physical Society. The possible reasons can be classified as follows: (i) the B-site disor- der destroys the spin arrangements in the B′and B′′sublattices but has no influence on each magnetic moment at B′and B′′sites and/or (ii) each magnetic moment at B′and B′′sites decreased because of cation disorder, but the nature of the spin order at the B′and B′′sub- lattices is not affected. Since the magnetic properties of Re-based DP oxides have promising applications in spintronics, therefore, their fundamental magnetic properties have received much attention in the past decades, which have been investigated by the supercon- ducting quantum interference device (SQUID), a Quantum Design Physical Property Measurement System (PPMS) with a vibrating sample magnetometer (VSM). For example, under a static magnetic field increasing up to 30 T, high-field magnetizations of Re-based DP bulk oxides of A 2FeReO 6(A=Ca, Sr, BaSr) and Sr 2CrReO 6 were measured, which are shown in Fig. 18.47It is found that themagnetizations of BaSrFeReO 6, Sr 2FeReO 6, and Ca 2FeReO 6com- pounds measured at 4 K under 5 T are 2.98, 2.81, and 2.25 μB/f.u., respectively. At the maximum magnetic field of 30 T, the magneti- zation of BaSrFeReO 6[Fig. 18(a)] was measured to be 3.27 μB/f.u. This approached the saturated magnetization ( Ms) since this com- pound was nearly saturated under 30 T, as observed in the inset of this figure. To evaluate the Msvalue of this compound, the following expression is used:72,73 Ms(expt.)=Ms(1−2×AS), (2) where Ms(expt.) represents the saturated magnetization measured experimentally and Msrepresents the saturated magnetization with- out AS. In the present case, the AS value is 0.5% and Ms(expt.) is 3.27μB/f.u.; therefore, the Msvalue for the BaSrFeReO 6compound AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-12 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv FIG. 14. (a) and (b) HAADF-STEM images of a (001)-oriented Sr 2CrReO 6(SCRO) film with a thickness of 134 nm grown on the SrTiO 3(STO) substrate, viewed along the [1 10] direction at low-magnification and high-magnification, respectively. The inset in (b) highlights the atomic contrast of Sr, Cr, and Re atoms, matching well with (c) the atomic lattices in the projection plane along the [1 10] direction. Reprinted with permission from Hauser et al. , Phys. Rev. B 85, 161201 (2012). Copyright 2012 American Physical Society. is determined to be 3.30 μB/f.u. The M–H loop of the BaSrFeReO 6 compound was measured by a SQUID magnetometer up to 5 T, which is shown as an inset of Fig. 18(a) at the upper left corner. Mag- netic field dependence of the magnetization of the BaSrFeReO 6com- pound measured at 4 K and 100 K is shown as an inset of Fig. 18(a) at the lower right corner. The magnetization is expected to be increased when decreasing the temperature, which is confirmed by the slightly increased magnetization at 4 K as compared with that at 100 K, which allows one to ignore any kind of spurious paramagnetic con- tribution to the measurement at 4 K. The magnetization of the BaSrFeReO 6compound was measured to be 3.21 μB/f.u. at 100 K and under the maximum field of 30 T, which was still much higher than the assumed value of 3.0 μB/f.u. The magnetic data for the Sr 2FeReO 6 compound are displayed in Fig. 18(b), which are similar to the case of BaSrFeReO 6composition. The magnetizations of the Sr 2FeReO 6 compound under the maximum field of 30 T were measured to be 3.23μB/f.u. @4 K and 3.17 μB/f.u. @100 K, respectively, which exhibits a less saturated tendency than BaSrFeReO 6does. It is much more evident that the Ca 2FeReO 6sample does not achieve magnetic saturation under 30 T, as demonstrated in Fig. 18(c). Its magneti- zations under the maximum field of 30 T were about 3.12 μB/f.u. @ 4 K and 100 K. It is also noticed that the MSvalue @ 4 K is smaller than that @ 110 K across the magnetic field of 12 T–30 T, which can be ascribed to a structural transition undergone between the two monoclinic crystallographic structures in this compound below 120 K.74–76The above experimental results clearly demonstrate that the magnetic saturation of these materials can be only achieved by much high magnetic fields. In general, the higher magnetic field required for magnetic saturation and larger coercive fields are, the larger magnetic anisotropy is the compound possesses. Therefore, Re-based double perovskites exhibit a large coercive field and sat- uration field, which are comparable to those observed in perma- nent magnets, indicating a high magneto-crystalline anisotropy. In order to understand why the highest Curie temperature FIG. 15. XRD patterns and the corresponding Rietveld refinements for (a) Sr2FeReO 6powders synthesized at a 0.5:1 flux ratio for 12 h, (b) Ba 2FeReO 6 powders synthesized at a 1:1 flux ratio and radiatively cooled, and (c) Sr 2CrReO 6 powders synthesized at a 1:1 flux ratio. Experimental data are indicated by the circles, and the calculated patterns are denoted by the solid lines. Tick marks below the profiles indicate all the possible Bragg reflections, and the difference diffractogram is shown at the bottom. The main (111) or (101) superstructure reflec- tions are labeled with an asterisk for each phase. Reprinted with permission from Fuoco et al. , Chem. Mater. 23, 5409 (2011). Copyright 2011 American Chemical Society. AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-13 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv FIG. 16. SEM images taken from the Sr 2FeReO 6powders synthesized by a molten NaCl/KCl flux at a (a) 0.5:1 ratio for 3 h, (b) 0.5:1 ratio for 12 h, (c) 1:1 ratio for 3 h, and (d) 1:1 ratio for 12 h. Reprinted with permission from Fuoco et al. , Chem. Mater. 23, 5409 (2011). Copyright 2011 American Chemical Society. (TC≈635 K) is observed in the Sr 2CrReO 6compound, recently x-ray magnetic circular dichroism (XMCD), an element-specific magnetic measurement technique based on synchrotron radiation, has been utilized to directly measure each magnetic moment at the B′and B′′sublattices since XMCD technique has some advantages such as element-specific selectivity and the ability to extract the magnetic contributions from spin and orbital magnetic moments in complex magnetic materials using the magneto-optical sum rules.77,78As an example, the XMCD signal of Sr 2CrReO 6measured as a function of the applied magnetic field at the Re L2absorption edge and 10 K is displayed in Fig. 19,79from which the coercive field was determined to be 1.27 T. The intensity of the XMCD signal matches well with theM–H loop obtained by a SQUID at 5 K (see the inset of Fig. 19) and other measurements.27The MSvalue was measured to be 0.89μB/f.u., which is consistent with the theoretical value of 1.0μBcalculated based on a simple ionic model with Cr3+and Re5+ion antiferromagnetic coupling. In terms of the data of XMCD at the L2,3edges and sum rules, the magnetic moments of Re 5 d spin and orbital moments in the Sr 2CrReO 6compound were deter- mined to be −0.68μBand+0.25μB, respectively. The experimental results demonstrate that the Curie temperature of the A 2B′B′′O6 DPs scales with the magnetic moments of the non-magnetic 3 dor 5dtransitional metal ions at the B′′site. The physical pressure also has an impact on the magnetic prop- erties of Re-based DP bulk oxides such as A 2FeReO 6(A=Ca, Ba). For example, at room temperature, the Ba 2FeReO 6compound with a larger Ba cation at the A-site has a cubic crystal structure and exhibits both metallicity and magnetism, whereas the Ca 2FeReO 6 compound with a small Ca cation at the A-site has a mono- clinic structure and exhibits insulating behavior.80–82The higherTc(=540 K) and magnetic coercive field ( Hc=9 kOe) are observed in the Ca 2FeReO 6compound, indicating the co-existence of stronger exchange interactions and larger magnetic anisotropy in this com- pound. In Ca 2FeReO 6with a monoclinic structure, the rotations of FeO 6and ReO 6octahedra around band caxes are favorable, resulting in a smaller Fe–O–Re bonding angle of 156○.83,84Thus, the pdπhopping that controls the transport behavior in the un-rotated structure is decreased, resulting in the observed insulating behav- ior.85The distorted monoclinic structure with a lower symmetry makes the ReO 6octahedra be a tetragonal distortion and leads to an enhanced magnetic anisotropy as compared with the Ba compound. Escanhoela et al.86investigated the physical pressure dependence of the magnetic properties of ferrimagnetic A 2FeReO 6DP oxides with A=Ca or Ba by using XAS at the Re L2,3edge and pow- der diffraction technique. Figures 20(a) and 20(b) show the XMCD hysteresis loops of A 2FeReO 6(A=Ba, Ca) DP oxides measured at 10 K with different pressures. It was found that the Ba 2FeReO 6and Ca2FeReO 6compounds became magnetically harder when increas- ing the pressure. The coercive field of Ba 2FeReO 6increased from 0.2 T to 1.55 T as the pressure was changed from 1.5 GPa to 22 GPa, and that of Ca 2FeReO 6also increased about threefold under the pressure of 25 GPa. The pressure-dependent coercive field (Hc) is shown in Fig. 20(c) for both compounds. It is noticed that Ca2FeReO 6is under a pre-compressed state owing to the chemical pressure. Therefore, its coercive field change is smaller than that of the Ba 2FeReO 6compound for a given physical pressure. It was also found that the volume compression dramatically increased the mag- netic coercive field of these two polycrystalline samples at a rate of ΔHc/ΔV∼150 Oe/Å3–200 Oe/Å3. 2. Re-based DP thin films Hauser et al.69grew Sr 2CrReO 6films by the magnetron sput- tering method and measured their magnetic properties. Figure 21(a) shows the M–Hloop of Sr 2CrReO 6(001) films at 5 K, from which theMswas measured to be 1.29 μB/f.u., which is larger than the the- oretical value of 1.0 μB/f.u. obtained from ferrimagnetic ionic align- ments of Cr3+(3d3↑) and Re5+(5d2↓) ions. However, the present MSvalue is well consistent with the calculated value of 1.28 μB/f.u. that considers the contribution of spin–orbit coupling.87This means strong spin–orbit coupling should exist in Sr 2CrReO 6thin films. The coercive field ( HC) of the Sr 2CrReO 6(001) film at 5 K was measured to be 1.05 T, which means the Sr 2CrReO 6thin film is a hard ferrimagnet at low temperature. However, at 300 K, the HCis reduced to be 890 Oe, indicating that the Sr 2CrReO 6thin film is suit- able for magneto-electronic applications at room temperature. Orna et al.71also measured the magnetic properties of Sr 2CrReO 6thin films grown by the PLD method. Figure 21(b) shows the saturation magnetization ( Ms) dependence of the substrate temperature and the partial oxygen pressure. The Mshad the largest value ∼1.0μB/f.u. when the substrate temperature was between 700○C and 800○C and the partial oxygen pressure was 2.6 ×10−4Torr (open red squares), which is related to the AS fraction of the films. The AS concentra- tion in the thin films was estimated to be in the order of 14%. The inset illustrates the room temperature M–Hloops of the Sr 2CrReO 6 films grown at different temperatures. The coercive fields of all the Sr2CrReO 6films are found to be very small ( HC∼120 Oe) [see the inset of Fig. 21(b)]; however, the coercive field of bulk Sr 2CrReO 6 AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-14 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv FIG. 17. SEM images taken from the Ba2FeReO 6powders synthesized by a molten NaCl/KCl flux at (a) a 1:1 ratio for 6 h, (b) a 3:1 ratio for 6 h, and (c) a 3:1 ratio for 6 h and by quenching; Sr2CrReO 6using (d) a 1:1 ratio for 6 h, (e) a 1:1 ratio and slow cooling over 24 h, and (f) a 3:1 ratio for 6 h and quenching. Samples A, B, and D were radiatively cooled. Reprinted with per- mission from Fuoco et al. , Chem. Mater. 23, 5409 (2011). Copyright 2011 Ameri- can Chemical Society. is as high as 3.1 kOe at 300 K.73Under high partial oxygen pres- sure PO2, the Msvalue of Sr 2CrReO 6films can be larger than that of bulk Sr 2CrReO 6. This is ascribed to the CrO xoxide segregation at the film surface, and a similar case was reported in the Sr 2FeMoO 6 films.88 3. Re-based DP powders The magnetic properties of Sr 2FeReO 6, Ba 2FeReO 6, and Sr2CrReO 6powders synthesized by the molten salt method under different synthesized conditions were reported by Fuoco et al.37 Their M–Hloops are displayed in Fig. 22, which indicate the mag- netization saturation is not fully achieved for these compounds under 7 T. Previously, it was reported that these materials could only achieve the magnetic saturation under magnetic field as high as 30 T.47,89In Fig. 22(a), the magnetizations for the flux-prepared Sr2FeReO 6powders under different flux-to-product molar ratios and reaction time are measured to be in the range of 1.57– 2.17μBunder 7 T, which was lower than the theoretical value of 3.0μBobtained with the antiparallel spin alignments for the Fe3+and Re5+ions. Such smaller magnetizations are also bound up with the disordering degree at B′and B′′sites.54,90As for the flux- prepared Ba 2FeReO 6powders [Fig. 22(b)], they exhibit a lower Hc and larger Ms(from 2.10 to 2.40 μB) due to a higher ordering degree of the Fe/Re site, as compared to Sr 2FeReO 6. The spon- taneous magnetizations of the flux-prepared Sr 2CrReO 6powders are in the range of 0.74–0.80 μB, lower than the expected value of 1.0μBunder the antiparallel spin alignments of the Cr3+and Re5+ ions. It is found that the higher the ordering degree at the Cr/Re site is, the larger the values of MsandMrare in the flux-prepared Sr2CrReO 6powders. As a rule of thumb, the coercive field of the Sr2CrReO 6powders is much larger than that of either Sr 2FeReO 6or Ba2FeReO 6. B. Transport properties 1. Re-based DP bulk oxides Transport property measurements of the Re-based DP oxides were performed by the four-probe method in conjunction with AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-15 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv FIG. 18. (a)M–Hhysteresis loop of the BaSrFeReO 6compound measured at 4 K and the magnetic field up to 30 T. The bottom inset shows a zoom of the magnetization around 30 T measured at 4 K and 100 K. The line marks the expected value for the saturation magnetization without the contribution of orbital magnetization. The top inset shows a comparison between the measured magnetization by a SQUID up to 5 T and the data obtained in the high-field installation (red line). Similar results for (b) Sr 2FeReO 6, (c) Ca 2FeReO 6, and (d) Sr 2CrReO 6. Reprinted with permission from Teresa et al. , Appl. Phys. Lett. 90, 252514 (2007). Copyright 2007 AIP Publishing LLC. a Quantum Design PPMS system. With a constant applied mag- netic field, the temperature dependence of resistivity can be mea- sured. Prellier et al.53reported the electrical properties of ferri- magnetic A 2FeReO 6DP oxides with A =Ba and Ca. Figure 23(a) demonstrates the resistivity ( ρ) of the Ba 2FeReO 6ceramics vs the temperature under different magnetic fields (e.g., 0 kOe, 2 kOe, and 50 kOe). A metallic behavior was observed in the Ba 2FeReO 6 ceramics below 300 K. The inset of Fig. 23(a) presents the resis- tivity measured up to 385 K under no applied magnetic field, and the metallic behavior is still kept within this temperature region. On the contrary, Ca 2FeReO 6exhibited a semiconducting behavior across 5 K–300 K, as depicted in Fig. 23(b). The observed metal- lic behavior in the Ba 2FeReO 6compound is ascribed to the direct interaction between R e t2gand Re t2g, whereas the semiconduct- ing behavior observed in the Ca 2FeReO 6compound is attributed to the disrupted interaction between R e t2gand Re t2gby themonoclinic structural distortion (the degeneracy of t2gstates is lifted). To investigate the magnetoresistance (MR) effects in the Sr2FeReO 6and Ba 2FeReO 6compounds, the magnetic field depen- dent upon the MR ratios measured at different temperatures is demonstrated in Figs. 23(c) and 23(d), respectively. Here, the MR value is defined as MR(T,H)=[ρ(T,Hpeak)−ρ(T,H)]/ρ(T,H), (3) where Hpeak is the magnetic field corresponding to the maximal value of resistivity ( ρ). It should be noticed that both Sr 2FeReO 6 and Ba 2FeReO 6exhibit negative MR behavior, and their MR val- ues increase fast at low fields but slowly at high fields. This effect becomes much apparent at low temperatures, whereas it becomes less evident at room temperature (300 K); thus, the MR is much smaller. This is the typical characteristic of intergrain AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-16 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv FIG. 19. XMCD signal of the Sr 2CrReO 6ceramics measured as a function of the magnetic field recorded at the Re L2edge and 10 K. The inset shows the M–H loop of the same sample measured by a SQUID at 5 K. Reprinted with permission from Majewski et al. , Appl. Phys. Lett. 87, 202503 (2005). Copyright 2005 AIP Publishing LLC. MR.91It is also noticed that Sr 2FeReO 6has a significantly larger MR than Ba 2FeReO 6under comparable conditions, which can be ascribed to the metal–insulator boundaries in the metallic Ba-based compound.92The appearance of metallic or insulating behavior in the isoelectronic series of A 2B′B′′O6DP compounds depen- dent upon the A-site cation would enable them useful in magnetic devices. The transport properties of Re-based ordered DPs such as A2MReO 6, where A is Sr, Ca and M is Mg, Sc, Cr, Mn, Fe, Co, Ni, Zn, are also reported by Kato et al.27The temperature depen- dence of resistivities ( ρ) is shown in Fig. 24. It is noticed that the A 2MReO 6compounds with A =Sr exhibit higher conductiv- ity than the Ca-based ones. Insulating nature was observed in all the compounds except for the two compounds of Sr 2FeReO 6and Sr2CrReO 6, which are fallen into the classification of half-metals.91,93 It is observed in Fig. 24(a) that at room temperature, the resis- tivities of a series Sr 2MReO 6DPs (M being Mg, Sc, Mn, Ni, and Zn) are in the order of k Ω⋅cm, which has nothing to do with A- site ions. However, the Sr 2MReO 6DPs with M =Co, Cr, and Fe have relatively low resistivity. In the case of Ca 2FeReO 6compound, a temperature-driven metal–insulator transition appeared around 150 K, which was reflected by a turning point in the resistivity curve marked by a triangle in Fig. 24(b), similar to that reported previ- ously.74Figure 25 demonstrates the temperature dependent mag- netizations of A 2MReO 6(A being Sr, Ca; M being Cr, Mn, Fe, and Ni) DPs, which were measured under 1 T. The Msvalues of Sr2MnReO 6and Sr 2NiReO 6compounds were measured to be 2.0 and 1.0μB/f.u., respectively, which matched well with the expected values obtained from the antiferromagnetic coupling of Mn2+(or Ni2+) and Re6+ions. However, the measured magnetizations of Ca2MnReO 6and Ca 2NiReO 6at 1Twere much smaller than the cal- culated values from the antiferromagnetic coupling between Mn2+ (or Ni2+) and Re6+ions. This may be ascribed to the large coer- cive fields of these two compounds (e.g., 4.0 T for Ca 2MnReO 6and FIG. 20. XMCD hysteresis loops of the Re element measured under different pres- sures with the x-ray energy adjusted near the Re L2edge for (a) Ba 2FeReO 6 and (b) Ca 2FeReO 6samples. Loops are normalized to the saturation values. (c) Pressure-dependent coercive field Hcshown comparatively for both samples. Reprinted with permission from Escanhoela et al. , Phys. Rev. B 98, 054402 (2018). Copyright 2018 American Physical Society. 2.8 T for Ca 2NiReO 6, respectively) and the insulating nature as well as relatively low magnetic transition temperature ( Tc<150 K). On the other hand, Sr 2CrReO 6and Sr 2FeReO 6compounds exhibit metallic behavior and have high magnetic transition temperatures AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-17 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv FIG. 21. (a) In-plane M–H hysteresis loop measured at T=5 K for a 1220-nm-thick Sr 2CrReO 6(001) film. The measured saturation magnetization ( Ms) and coercive field (Hc) are 1.29μB/f.u. and 1.05 T at 5 K, respectively. The inset shows the MvsHcurve measured at T=300 K with Ms=1.14μB/f.u. and Hc=890 Oe. Reprinted with permission from Hauser et al. , Phys. Rev. B 85, 161201 (2012). Copyright 2012 American Physical Society. (b) Saturation magnetization ( Ms) vs substrate temperature ( Tsub) (with the oxygen pressure fixed at 2.6 ×10−4Torr, open red squares) and oxygen pressure PO2during deposition (with the substrate temperature fixed at 800○C, closed blue circles). The inset shows the room temperature M–Hloops of the (001)-oriented Sr 2CrReO 6films grown at different temperatures. Reprinted with permission from Orna et al. , J. Magn. Magn. Mater. 322, 1217 (2010). Copyright 2009 Elsevier B.V. (Tc=635 K and 400 K). This can be ascribed to that the exchange interactions between M ( =Cr, Fe) and Re spins are not disturbed and there is no gap around the Fermi level. The saturation magne- tizations of Sr 2CrReO 6and Sr 2FeReO 6compounds were measured to be 0.56 and 2.2 μB/f.u., respectively, lower than their theoreti- cal values of 1.0 and 3.0 μB/f.u. calculated based on the electronic configurations of Cr3+(or Fe3+) and Re5+ions within a simplest ionic model. Similarly, the Ca 2MReO 6(M being Cr, Mn, Fe, and Ni) DPs possess comparable saturation magnetization and higher magnetic TC, as compared with the corresponding Sr-based com- pounds. A metal–insulator transition is observed in the ordered A2MReO 6DPs (A being Sr, Ca; M being Cr, Mn, Fe, and Ni). This phenomenon can be ascribed to the formation of the half-metallic conduction band due to the hybridization between the down-spin t2g bands of Re and M ( =Cr, Fe) Cr3+around the Fermi level. This intu- itive interpretation is proven by theoretical electronic calculations of Sr2FeReO 6.91 In the cases of Ca 2FeReO 6and Ca 2CrReO 6compounds, their small structural distortion drives the one-electron bandwidth to be narrower than that in their Sr-based analogs, leading a Mott transi- tion in these compounds. For the Sr 2MnReO 6and Sr 2NiReO 6com- pounds, their highest occupied states are the up-spin egstates for Mn2+and Ni2+ions, respectively. To virtually excite the M3+–Re5+ state, the up-spin egelectrons of M2+ions are required to move toward the down-spin t2gstates of Re6+ions. Due to the lack of mixing states between t2gand egat the adjacent sites in a per- fect cubic perovskite, the above electronic transferring process is nearly prohibited in ordered DPs, which makes the Sr 2MnReO 6and Sr2NiReO 6compounds exhibit insulating behavior and used as Mott insulators. 2. Re-based DP thin films The transport properties of the Sr 2CrReO 6films were reported. Figure 26(a) shows the magnetization ratio of M/M(5 K) vstemperature, which is measured under 8 kOe. The inverse mag- netic susceptibility ( χ−1) vs temperature is demonstrated in the inset, showing two magnetic transition temperatures ( TC=508 K for the majority phase and TC=595 K for a secondary phase). In comparison, the TCfor bulk Sr 2CrReO 6is reported to be 620 K– 635 K24,91and 481 K for Sr 2CrReO 6films.33The observed higher TCmay be contributed from the local regions around a small num- ber of APBs, where there exists a stronger exchange interaction, enhancing the TCvalue. Figure 26(b) demonstrates the semi-log plots of the electrical resistivity of bulk Sr 2CrReO 6and thin films vs temperature. The room temperature bulk resistivity was measured to be 2.1 m Ωcm. With the decreasing temperature, the electrical resistivity increases, as reported by Kato et al.93In the Sr 2CrReO 6 film, the resistivity was increased more than two orders of mag- nitude (from 16.2 m Ωcm to 5.05 Ωcm) as the temperature was decreased from 300 K to 2 K. Such a behavior is the feature of a semiconductor (or insulator) with a gap at the Fermi level. The ln ρvs1000/ Tplot across the temperature range of 90 K–200 K is shown in the inset of Fig. 26(b), which exhibits nearly perfect lin- ear fitting. Such temperature dependence of resistivity is frequently observed in semiconductors owing to their thermal activation,94,95 which is given by ρ=ρ0exp(Ea/kBT), (4) whereρ0is the prefactor and Eais the activation energy. The Ea value was determined to be 9.4 meV from the linear fitting of the lnρvs 1000/ Tplot across the temperature range of 90 K–200 K. However, at temperatures below 90 K, the ln ρvs 1000/ Tplot does not exhibit a linear behavior; instead, a good linear fitting is found for lnρvs (1000/ T)0.25across a temperature range of 55 K–90 K, and for ln ρvs (1000/ T)0.20from 6 K to 25 K. This indicates a variable-range hopping model is appropriate in the whole temper- ature range (6 K–200 K), similar to those reported in the doped- semiconductors at low temperatures.96Sohn et al.25reported the transport properties of cation-ordered Sr 2Fe1+xRe1−xO6DP films. AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-18 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv FIG. 22. M–H loops of (a) Sr 2FeReO 6, (b) Ba 2FeReO 6, and (c) Sr 2CrReO 6 powders synthesized by the MSS method. The synthesized conditions in each curve are indicated by the flux-to-product molar ratio, reaction time, and cool- ing time if other than radiatively cooled. Reprinted with permission from Fuoco et al. , Chem. Mater. 23, 5409 (2011). Copyright 2011 American Chemical Society. Figure 27(a) displays the magnetization ( M)vs T curves mea- sured under 1000 Oe for the Sr 2Fe1+xRe1−xO6films with differ- ent stoichiometric ratios, and Fig. 27(b) shows the corresponding temperature-dependent sheet resistance RS. It was found that theferromagnetic ground state existed in the Fe-rich/ordered films, but it fully vanished in the Re-rich/disordered films. In addition, the metallic ground state was more vulnerable in the films with excess Fe ions than that with excess Re ions. The Fe-rich/ordered films (blue lines) underwent a magnetic transition around 400 K, but it completely disappeared in the Re-rich/disordered films (red lines). This was confirmed by the magnetization exhibiting no temperature dependence. The disappearance of long-range magnetic ordering in the Re-rich films was attributed to the cation disordering at the B- site, which substantially influenced the magnetic properties of DP oxides. In contrast to the magnetism, the Re-rich/disordered films exhibit the metallic ground state, as evident by the RS(T) curve in Fig. 27(b). However, the Fe-rich films behave like an insulator, and their room- Tresistivity is about three orders of magnitude larger than that of the stoichiometric or Re-rich samples. Therefore, in the present Sr 2Fe1+xRe1−xO6single-crystalline films, a metal–insulator transition is observed, which exhibits much more dramatic phenom- ena than that reported for polycrystalline Sr 2Fe1+xRe1−xO6ceram- ics.97The reason can be ascribed to the existence of many metal- lic grain boundaries in polycrystalline films but absence in the present films. Figure 27(c) summarizes the measured results for theMsand Rsvalues of the Sr 2Fe1+xRe1−xO6DP films; based on them, the Sr 2Fe1+xRe1−xO6single-crystalline films can be classified as paramagnetic metal (PMM, x <0), ferromagnetic metal (FMM, x≈0), and room-temperature ferromagnetic insulator (FMI, x >0) catalogs. 3. Re-based DP oxide powders Fuoco et al.37reported the transport properties of Sr 2FeReO 6, Ba2FeReO 6, and Sr 2CrReO 6powders synthesized by the MSS method, which were measured by polycrystalline pellets formed by pressing the powders but no annealing. Figure 28(a) shows their temperature dependent resistivity. A semiconducting-type behav- ior was observed in all the samples, which is consistent with previous reports. A stronger temperature dependence of resistiv- ity is observed in the flux-prepared Sr 2FeReO 6samples, which have resistivity about three times higher than that of the sam- ple synthesized by the solid-state reaction. The flux-prepared Ba2FeReO 6powders had a resistivity in the range of ∼0.07Ωcm to 0.1Ωcm across 300 K–550 K, and the Sr 2CrReO 6powders had a higher resistivity ( ∼13Ωcm to 3Ωcm). Figure 28(b) demonstrates the temperature dependent resistivity of Sr 2FeReO 6 powders synthesized by the solid-state method and flux-synthetic route, which was measured under magnetic fields of 0 T and 0.3 T across temperatures of 300 K–550 K. It was found that the flux-prepared Sr 2FeReO 6powders had larger resistivity, which also exhibited much evident magnetic field-dependence of resis- tivity. This phenomenon can be attributed to the high con- centrations of insulating grain boundaries in the flux-prepared Sr2FeReO 6powders. At room temperature, both Sr 2CrReO 6and Ba2FeReO 6powders exhibit a similarly large intergrain tunnel- ing magnetoresistance (ITMR) of up to ∼70% and∼65%, respec- tively, whose details are shown in Figs. 29(a) and 29(b), respec- tively. Such polycrystalline MR at low magnetic field is larger than that reported previously for Sr 2FeMoO 6(ITMR of ∼20% to 30% at 0.4 T) prepared from nanoscale particles of ∼29 nm to 45 nm.34 AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-19 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv FIG. 23. (a) Resistivity dependence of temperature for the Ba 2FeReO 6compound measured at different magnetic fields. The inset shows the resistivity mea- sured from 5 K to 385 K without magnetic field. (b) Resistivity dependence of temperature under zero-field for Ca 2FeReO 6. Reprinted with permission from Prel- lieret al. , J. Phys.: Condens. Matter 12, 965 (2000). Copyright 2000 IOP Publishing Ltd. (c) and (d) MR behaviors of Sr 2FeReO 6and Ba 2FeReO 6compounds measured at different temperatures. Reprinted with permission from Gopalakrishnan et al. , Phys. Rev. B 62, 9538 (2000). Copyright 2000 American Physical Society. C. Optical properties The optical behaviors of the Re-based DP oxides have promis- ing applications in various optical devices. It is reported that the optical properties of Re-based DP oxides are controlled by their inter-band transitions between the B-site (B′or B′′) transition metal cations and the oxygen anions, which reflect the electronic states’ energy distribution in valence/conduction bands.98–100Jeon et al.101 measured the reflectance spectra of Ba 2FeReO 6and Ca 2FeReO 6 polycrystalline samples at 300 K across the energy range of 5.0 meV– 6.0 eV, as shown in Fig. 30(a), from which the optical conduc- tivity spectra σ(ω) of the Ba 2FeReO 6and Ca 2FeReO 6samples can be obtained by the Kramers–Kronig (KK) transformation, which is shown in Fig. 30(b). Two spectral peaks are clearly observed in these two samples, which are marked by αandβfor Ba 2FeReO 6 and A, B for Ca 2FeReO 6. It is noticed that the intensities of the peaks (βand B) appearing on the higher energy side are muchlarger than those of the peaks ( αand A) located on the lower energy side. It is reported that the higher energy ( βand B) peaks resulted from the electric dipole allowed charge transitions from the O 2 pto the Re 5 dor Fe 3 dorbital states, whereas the lower energy (αand A) peaks are contributed from the d–delectronic tran- sition between the Re 5 dand/or Fe 3 dstates.102It is also noticed that the position of the B peak in Ca 2FeReO 6has a chemical shift (∼0.5 eV) toward higher energy as compared with that of Ba2FeReO 6, as marked by the arrow in Fig. 30(b). This indicates the projected density of states (DOS) of the unoccupied Re t2gbands in Ca 2FeReO 6is higher than that in Ba 2FeReO 6. The dc conduc- tivity data of Ba 2FeReO 6and Ca 2FeReO 6polycrystalline samples are also given in Fig. 30(b), as marked by black and red squares. At low-frequency, σ(ω) exhibits much differences between the two samples. However, as the photon energy was beyond 2.0 eV, the σ(ω) of Ca 2FeReO 6became much suppressed as compared with that of the metallic Ba 2FeReO 6, but theσ(ω) still had a finite Drude AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-20 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv FIG. 24. Temperature dependence of resistivity (ρ) of the ordered double perovskites (a) Sr 2MReO 6(M=Mg, Sc, Cr, Mn, Fe, Co, Ni, and Zn) and (b) Ca2MReO 6(M=Cr, Mn, Fe, Co, and Ni). Reprinted with permission from Kato et al. , Phys. Rev. B 69, 184412 (2004). Copyright 2004 American Physical Society. component at room temperature. To solve this issue, low temper- ature (10 K) conductivity spectra of Ca 2FeReO 6were measured in comparison with those at 300 K, as shown in Fig. 30(c). The low-frequency σ(ω) became more suppressed, the spectral peak was much narrow, and a gap opened at 10 K.103Such gap opening indicates the existence of an insulating ground state in the present Ca2FeReO 6sample. FIG. 25. Magnetization dependence of temperature under 1 Tfor Sr 2MReO 6and Ca2MReO 6compounds (M =Cr, Mn, Fe, and Ni). Reprinted with permission from Kato et al. , Phys. Rev. B 69, 184412 (2004). Copyright 2004 American Physical Society.Sohn et al.25measured the real part of optical conductivities σ1(ω) of the Sr 2Fe1+xRe1−xO6films by a spectroscopic ellipsome- ter across the energy of 1.2 eV–5 eV, as shown in Fig. 31(a). The inset shows the local density of states (DOS) for three bonding types (Fe3+–Re5+, Fe3+–Fe3+, Fe3+–Re6+) in Fe-rich films based on the density functional theoretical calculations.104–106For the origi- nal Re5+–Fe3+bonding of stoichiometric Sr 2FeReO 6, two types of optical transitions can be expected: (i) d–doptical transition from Re 5 dto Fe 3 dorbitals [indicated by A in the inset of Fig. 31(a)] and (ii) p–d optical transition from O 2 pto Re/Fe dorbitals [indi- cated by B in the inset of Fig. 31(a)]. As expected, two apparent spectral features are observed, one is a broad spectral band located below 2.5 eV (denoted A) and another one is a strong spectra peak located near 4.0 eV (denoted B) in the σ1(ω) of a stoichiometric film (green line, FMM film), which is consistent with the previous bulk data.101However, in the Fe-rich film (blue line, FMI film), a new spectral peak C appeared at the expense of weights of A and B peaks. Peak C could be originated from the optical transition from O 2 pto Fe 3 dorbitals in the Fe3+–Fe3+bonding, similar to the case of La3+Fe3+O3.107However, any absorption peak expected from the Re6+–Fe3+bonding, i.e., optical transition from O 2 pto Re/Fe dorbitals, is not observed in the present energy regime. This is attributed to that the O 2p states of the Fe3+–Re6+bonding are much far away from EFthan those of other two bonding states with strong O 2 p–Re 5 d1hybridization; thus, the corresponding optical transition requires higher energy than the present energy regime. Hauser et al.32measured the FTIR transmission spectra of a 200-nm thick Sr 2CrReO 6(001) film deposited on the SrTiO 3substrate. Its Tauc’s plot of ( αE)2vs E is shown in Fig. 31(b), where αis the absorption coefficient and Eis the incident photon energy.108A lin- ear fitting of the Tauc plot and its intercept of the extension of the linear part on the xaxis clearly indicate that the direct bandgap ( Eg) of the Sr 2CrReO 6film is 0.21 eV. These features observed below 0.17 eV likely resulted from phonon modes. The noise appearing around 0.29 eV and in the energy range of 0.44 eV–0.50 eV con- tributes to water vapor and CO 2present in the FTIR chamber. The present FTIR spectrum indicates that Sr 2CrReO 6is a semicon- ductor rather than a metal. Therefore, high temperature ferrimag- netism and semi-conductivity of the Sr 2CrReO 6thin films enable AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-21 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv FIG. 26. (a) Magnetization ( M) dependence of temperature for a 1220-nm-thick Sr 2CrReO 6(001) film measured at H=8 kOe. The inset in (a) shows the temperature dependent inverse magnetic susceptibility ( χ−1). (b) Semi-log plots of ρvsTof bulk Sr 2CrReO 6(black) and a 200-nm-thick Sr 2CrReO 6(001) film (blue) measured by the four-probe measured method. The inset in (b) shows a plot of ln ρvs 1000/ T, showing a good linear fitting with a linear correlation coefficient R equal to 0.999 99. The activation energy ( Ea) is deduced to be 9.4 meV. Reprinted with permission from Hauser et al. , Phys. Rev. B 85, 161201 (2012). Copyright 2012 American Physical Society. them to find promising applications in the fields of spin filters, mag- netic proximity switches, and energy efficient quantum electronic devices. VI. THEORETICAL STUDIES OF Re-BASED DP OXIDES In parallel with experimental investigations of Re-based DP oxides, theoretical calculations have also been carried out, allow- ing one to study electronic structures and structural aspects of Re- based DP oxides in theory. So far, several ab initio calculations such as the full potential linearized augmented plane-wave (FP- LAPW) method that is based on generalized gradient approximation (GGA),91,109–111local density approximation (LDA), LDA +U(the Hubbard Uterm), and LAD +U+SOC (spin–orbital coupling) calculations,101local spin density approximation (LSDA) and LSDA +Umethods,112and density functional theory (DFT) +Ucalcula- tions113are developed to investigate electronic, optical, mechanical, structural, and thermodynamic properties of Re-based DP oxides.As an example, the DOS of Sr 2FeReO 6calculated by the FP-LAPW method based on GGA is shown in Fig. 32.91It is found that at the Fermi level ( EF), the DOS exists only in the down-spin band, whereas there is no DOS in the up-spin band. In another word, there exists an energy gap ( ∼1 eV) between the occupied Fe egband and the unoccupied Re t2gband in the up-spin state [see the mid- dle part of Fig. 32]. The nature of half-metallicity means the carriers in half-metals are completely spin-polarized charge carriers in the ground state. In an ideal ferromagnetic half-metal, at the Fermi level between the two spin (up and down) bands, only one spin band is partially occupied, while the other does not have DOS across the Fermi level. Thus, the electrical conduction is controlled only by one spin direction of the carriers. In the half-metal Sr 2FeReO 6com- pound, theoretical calculations demonstrated that in the occupied up-spin band just below the EF, there mainly exist the Fe 3 delectrons with the localized spins on the Fe sites, whereas, in the down-spin band around EF, there mainly exist the hybridized Re 5 d t 2gand Fe 3d t2gstates. In addition, at the Fermi level, the partial DOS of Re (t2g) is 3–4 times larger than that of Fe (t 2g). This implies the FIG. 27. (a)M–Tcurves of the Sr 2Fe1+xRe1−xO6films measured under 1000 Oe, where the red, green, and blue lines indicate the M–Tcurves of Re-rich, stoi- chiometric, and Fe-rich films, respectively. (b) Semi-log plot of the sheet resistance RSas a function of temperature. Re-rich and stoichiometric films exhibit metallic behaviors, whereas the Fe-rich film exhibits an insulating behavior. (c) RS(300 K) and MSdependence of the x value in the Sr 2Fe1+xRe1−xO6films, from which the Sr2Fe1+xRe1−xO6single-crystalline films are classified as paramagnetic metal (PMM, x <0), ferromagnetic metal (FMM, x ≈0), and room-temperature ferromagnetic insulator (FMI, x >0) catalogs, respectively. Reprinted with permission from Sohn et al. , Adv. Mater. 31, 1805389 (2019). Copyright 2019 WILEY-VCH Verlag GmbH & Co. KGaA. AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-22 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv FIG. 28. (a) Resistivity dependence of temperature for Sr 2FeReO 6powders synthesized by the solid-state reaction and MSS method under a 1:1 flux-to-product molar ratio for 3 h, Ba 2FeReO 6synthesized at a 1:1 flux and radiatively cooled, and Sr 2CrReO 6synthesized at a 1:1 flux for 6 h and radiatively cooled. (b) Resistivity dependence of temperature for the Sr 2FeReO 6particles prepared by the solid-state reaction and flux-synthetic routes (1:1 M ratio, 3 h) under different magnetic fields. Reprinted with permission from Fuoco et al. , Chem. Mater. 23, 5409 (2011). Copyright 2011 American Chemical Society. electrical conduction of the Sr 2FeReO 6compound is dominated by the 5 dconduction electrons (Re5+=5d2). In order to understand the importance of the on-site Coulomb interaction energy U (4.0 eV– 5.0 eV), the crystal-field splitting energy, Δ(2.0 eV–3.0 eV), and the SOC (spin–orbit coupling, 0.3 eV–0.4 eV) in the electronic structures of the A 2FeReO 6DP oxides with A =Ca and Ba, the total energy and electronic band structures of the A 2FeReO 6DP oxides with A =Ca and Ba were calculated by LDA, LDA +U,and LAD +U+SOC methods.101Figures 33(a) and 33(b) show the calculated results for Ba 2FeReO 6by LDA and LDA +Umeth- ods, respectively. Different electronic structures are observed due to the different spin directions. It is found that a finite DOS exists at the Fermi energy in the down-spin bands (marked by red dashed lines), whereas, in the up-spin bands, a gap near the Fermi level is observed (marked by black solid lines). This indicates the charac- teristic electronic feature of a half-metal.5Similar results calculated FIG. 29. Temperature dependent resistivity measurements at 0 Tand 0.3 Tfor (a) Sr 2CrReO 6and (b) Ba 2FeReO 6particles prepared by a NaCl/KCl flux synthesis at 800○C for 6 h using a 1:1 flux:product molar ratio and radiatively cooled. Reprinted with permission from Fuoco et al. , Chem. Mater. 23, 5409 (2011). Copyright 2011 American Chemical Society. AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-23 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv FIG. 30. (a) Experimentally measured reflectance spectra of Ba 2FeReO 6(black line) and Ca 2FeReO 6(red line) at 300 K. (b) Optical conductivity spectra σ(ω) of Ba 2FeReO 6(black line) and Ca 2FeReO 6(red line), where the black and red squares represent the dc conductivities of Ba 2FeReO 6and Ca 2FeReO 6, respec- tively. (c) Optical conductivity spectra σ(ω) of Ca 2FeReO 6measured at 300 K and 10 K are denoted by red solid and blue dashed lines, respectively. Reprinted with permission from Jeon et al. , J. Phys.: Condens. Matter 22, 345602 (2010). Copyright 2010 IOP Publishing Ltd.by the LDA +Umethod were reported for Sr 2FeMoO 6.114,115How- ever, when the SOC becomes much strong in the Re-based DP oxides, the mixture of spin states and the half-metallicity can be neglected.116,117Figure 33(c) shows the calculated electronic struc- ture of Ba 2FeReO 6by using the LDA +U+SOC method. It is noticed that the SOC term leads to much changes in the electronic structure. In comparison with Fig. 33(b), great changes in the elec- tronic states are observed in Fig. 33(c), especially that two bands are nearly parallel to each other around the Fermi energy, as indi- cated by blue and magenta lines. Therefore, the SOC plays a crucial role in determining the electronic structures of Ba 2FeReO 6. Simi- larly, band structures of Ca 2FeReO 6with lattice distortion are also calculated by LDA, LDA +U, and LAD +U+SOC methods, which are shown in Figs. 34(a)–34(c).101In Fig. 34(b), the LDA +Ucalcu- lation predicts a semi-metallic ground state in Ca 2FeReO 6, in which the conduction band and valence band just get in touch with the Fermi level. Figure 34(c) shows the calculated electronic structure of Ca2FeReO 6by using the LDA +U+SOC method. However, no significant change is observed in the band dispersion as the SOC term is included. This means the Uterm is crucial, whereas the SOC term is less important in determining the electronic structure of Ca 2FeReO 6. To understand the role of lattice distortion in the electronic structure of Ca 2FeReO 6, theoretical calculations are also carried out by assuming that the bond angle of the Fe–O-Re bond in the Ca 2FeReO 6compound is equal to 180○rather than the experi- mental value of 153○. Figures 34(d)–34(f) show the calculated band structures of Ca 2FeReO 6by LDA, LDA +U, and LAD +U+SOC methods without considering lattice distortion. It is noticed that the band dispersions of Ca 2FeReO 6are almost the same as those of Ba2FeReO 6[see Figs. 33(a)–33(c)]. Such a similarity indicates that the lattice distortion has a great influence on the electronic structure of a real Ca 2FeReO 6compound. Therefore, in the real Ca 2FeReO 6 compound, the lattice distortion plays an important role in deter- mining the electronic structure. In contrast, there is very little lattice distortion in Ba 2FeReO 6, and its electronic structure is mainly con- trolled by electron correlation and SOC. Similar conclusions are also drawn in the A 2FeReO 6(A=Ca, Sr, and Ba) oxides based on the calculations by the LSDA and LSDA +Umethods.112To under- stand the correlation effects on the Re site, the partial DOSs of the Re 5dorbital in the A 2FeReO 6(A=Ca, Sr, and Ba) oxides are cal- culated based on fully relativistic Dirac approximation, as shown in Fig. 35. It is found that Ca 2FeReO 6has a much smaller Re band- width than the two compounds, A 2FeReO 6(A=Sr and Ba). There- fore, a strong electron correlation should exist in the Re 5 dbands in the Ca 2FeReO 6oxide, leading to much narrow bands near the Fermi level with pseudo-gaps just above it. These bands separate from almost disconnect states that are dominated by Fe contribu- tions above the Fermi level. A similar case happens for the Fe 3 d energy bands. It is found that the Fe t2gbandwidth is decreased by about 25% from Sr 2FeReO 6to Ca 2FeReO 6compounds. The Fe and Re dbandwidths are different among the A 2FeReO 6(A being Ca, Sr, and Ba) compounds, which can be ascribed to their dif- ferent crystal distortions. The John–Teller distortions are increased in the series of A 2FeReO 6(A=Ba, Sr, and Ca) oxides. The bond angle of the Fe–O–Re bond in the monoclinic Ca 2FeReO 6is about 156○,92which is much smaller than 180○in the cubic Ba 2FeReO 6and tetragonal Sr 2FeReO 6. Such low bond angle makes the Fe–Re bond overlapping decrease and the t2gbandwidths become narrowed. AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-24 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv FIG. 31. (a) Optical conductivity spectra of the Sr 2Fe1+xRe1−xO6films: Re-rich (x <0, PMM), stoichiometric (x ≈0, FMM), and Fe-rich (x >0, FMI) films. The inset schematically shows the local density of states (DOS) in the Fe-rich film with the Re5+–Fe3+, Fe3+–Fe3+, and Re6+–Fe3+bondings calculated by density functional theoretical calculations. Three expected optical transitions in the Re5+–Fe3+and Fe3+–Fe3+bondings are marked A, B, and C, respectively. Reprinted with permission from Sohn et al. , Adv. Mater. 31, 1805389 (2019). Copyright 2019 WILEY-VCH Verlag GmbH & Co. KGaA. (b) Tauc plot of ( αE)2vsEof a 200-nm-thick Sr 2CrReO 6 (001) film, which gives a clear bandgap Eg=0.21 eV. Reprinted with permission from Hauser et al. , Phys. Rev. B 85, 161201 (2012). Copyright 2012 American Physical Society. FIG. 32. The density of states of Sr 2FeReO 6and its elec- tronic structure are calculated by the full potential linearized augmented plane-wave method based on the generalized gradient approximation. The inset shows the schematic structure of ordered A 2B′B′′O6, where the transition metal atoms (B′and B′′) occupy the perovskite B site and form B′O6and B′′O6octahedra alternatively along the [111] direction. The Fermi level lies at the formed band exclu- sively by the Fe ( t2g↓)-O(2p)-Re ( t2g↓) sub-band. Reprinted with permission from Kobayashi et al. , Phys. Rev. B 59, 11159 (1999). Copyright 1999 American Physical Society. AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-25 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv FIG. 33. Theoretical band structures of the Ba 2FeReO 6compound calculated by using (a) LDA, (b) LDA +U, and (c) LAD +U+SOC methods. The band dis- persions of up-spin and down-spin are denoted by black solid and red dashed lines, respectively. The black solid line in (c) denotes the band dispersions formed by mixing up-spin and down-spin. EFindicates the Fermi energy. The two band dispersions near the Fermi level are marked by the blue and magenta lines, which are nearly parallel to each other. Reprinted with permission from Jeon et al. , J. Phys.: Condens. Matter 22, 345602 (2010). Copyright 2010 IOP Publishing Ltd. FIG. 34. Theoretical band structures of Ca 2FeReO 6with lattice distortion calculated by using (a) LDA, (b) LDA +U, and (c) LAD +U+SOC methods. The same band structures without lattice distortion are calculated by using (d) LDA, (e) LDA +U, and (f) LAD +U+SOC methods. The black solid and red dashed lines in (a), (b), (d), and (e) indicate the band dispersions of up-spin and down-spin, respectively. The black solid lines in (e) and (f) indicate the band dispersions of mixed up-spin and down-spin. EFindicates the Fermi energy. Reprinted with permission from Jeon et al. , J. Phys.: Condens. Matter 22, 345602 (2010). Copyright 2010 IOP Publishing Ltd. AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-26 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv FIG. 35. Re 5d partial DOSs in A 2FeReO 6oxides with A =Ba, Sr, and Ca, cal- culated by the LSDA relativistic Dirac approximation. Reprinted with permission from Antonov et al. , Phys. Rev. B 94, 035122 (2016). Copyright 2016 American Physical Society. In addition, the monoclinic structural distortion in the Ca 2FeReO 6 oxide lifts the degeneracy of the t2glevels on the Fe and Re sites. The above two factors result in more narrow energy bands for Fe and Ret2gin the Ca 2FeReO 6oxides, as compared with the A 2FeReO 6 (A=Sr and Ba) oxides. Thus, the Fe and Re t2gelectrons become much more localized and Ca 2FeReO 6undergoes a Mott transition. To understand the electronic, magnetic, and structural properties of the A 2FeReO 6(A being Ba, Sr, and Ca) oxides, x-ray absorp- tion spectra (XAS) and x-ray magnetic circular dichroism (XMCD) spectra are theoretically investigated at the Re, Fe, and Ba L2,3and Fe, Ca, and O Kedges by the LSDA +Umethod. Figure 36 shows the calculated XAS and XMCD spectra at the Re L2,3edges from the A 2FeReO 6(A=Ba, Sr, and Ca) oxides and their correspond- ing experimental spectra.118,119It is observed that the Re L3spectra (contributed from 2 p3/2→5d3/2,5/2 transition) have almost the same similarity and exhibit two peaks with the same edge energy posi- tion in A 2FeReO 6(A being Ba, Sr, and Ca) oxides. The first peakappeared at about 10 540.1 eV, and the stronger one appeared at 10 543.6 eV. The split energy is attributed to the crystal-field splitting ofdorbitals into t2gandegstates.120The Re L2XAS are composed of two peaks, and the low-energy peak exhibits relatively stronger intensity than the high-energy one. Theoretical calculations demon- strate an inverse peak relative intensity for the L3andL2XAS. The experimental dichroic L2lines of the A 2FeReO 6(A being Ba, Sr, and Ca) oxides exhibit an asymmetric negative peak with a shoul- der appearing on the higher energy side. Three peaks are found in the dichroic lines at the L3edge, which are two positive peaks with high energy and a negative one with low energy. This negative peak observed in Ba 2FeReO 6and Sr 2FeReO 6compounds embodies as a shoulder at low energy, but it exhibits a much large intensity in Ca2FeReO 6. The dichroism at the L2edge exhibits much stronger intensity than that at the L3edge in all three compounds. The XMCD spectra can be qualitatively interpreted by analyzing the correspond- ing selection rules, orbital character, and occupation numbers of individual 5 dorbitals.112The theoretical calculations match rela- tively well with the experimental XAS and XMCD spectra in the A2FeReO 6(A=Ba, Sr, and Ca) oxides in both the spectral shape and relative intensities. Indeed, theoretical calculated techniques now allow one to fundamentally understand the physical properties and structural aspects of Re-based DP oxides, and now, they become important and complementary methods to the experimental probes; in particular in some cases, the physical properties of Re-based DP oxides are only directly obtained by some theoretical calculations. In parallel with much advances in theoretical computational mod- els, the experimental scientists can grow Re-based DP oxide films by the laser-MBE method with the film growth controlled at the atomic scale and measure local physical properties at the nanoscale. It is expected, in the near future, the two trends have more chance to perform a lively theoretical–experimental dialog. VII. ADVANCED APPLICATIONS Owing to the diverse electrical behaviors and huge magnetic anisotropy, Re-based DP oxides have promising applications in oxide spintronics, nonvolatile data memory devices, and multifer- roic devices. For example, at room temperature, some A 2FeReO 6 oxides exhibit half-metallic ferromagnetism, which can be uti- lized as electrodes for fabricating the magnetic tunnel junctions (MTJs). In the MTJ structure, the tunneling current is dependent upon the relative orientation of the magnetization of two ferro- magnetic electrodes, and it is modified by applying a magnetic field.121,122Therefore, the MTJs constructed with Sr 2FeReO 6and Sr2FeMoO 6as ferromagnetic electrodes would exhibit high TMR ratios due to the larger magneto-crystalline anisotropy because of the Re ion. Furthermore, the Re-based and Mo-based DP oxides such as Sr 2FeReO 6and Sr 2FeMoO 6exhibit much different coer- cive fields; thus, in the MTJs, the Sr 2FeReO 6electrode with high coercive field may act as a pinned layer, while the Sr 2FeMoO 6 electrode with low coercive field acts as a free layer. The mag- netization of Sr 2FeMoO 6(free layer) can be rotated concerning the pinned layer (Sr 2FeReO 6) through a barrier in MTJs. As a consequence, it is possible to obtain different states of resistance, and each state can be functioned as a specific degree of spin, while, at the same time, the number of states can be applied in AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-27 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv FIG. 36. Experimental x-ray absorption spectroscopy (XAS) and x-ray magnetic circular dichroism (XMCD) spectra (open circles) recorded at the Re L2,3edges in the A2FeReO 6(A=Ba, Sr, and Ca) oxides as compared with theoretical calculations. The experimental dichroism taken at the Re L3edge and its theoretically calculated dichroism are enlarged by two times. Reprinted with permission from Antonov et al. , Phys. Rev. B 94, 035122 (2016). Copyright 2016 American Physical Society. AIP Advances 10, 120701 (2020); doi: 10.1063/5.0031196 10, 120701-28 © Author(s) 2020AIP Advances REVIEW scitation.org/journal/adv the storage or data processing system. Due to the two ferromag- netic electrodes (Sr 2FeReO 6and Sr 2FeMoO 6) having almost the same lattice parameters, epitaxial growth of such heterostructured MTJs can be easily achieved in different crystallographic direc- tions. This allows one to investigate electron tunneling as func- tions of not only the spin direction but also the orbital symmetry. It is expected that this direction will become an exciting topic in oxide spintronics. In an epitaxy bilayer of Ca 2FeMoO 6/Ca 2FeReO 6, as the insulating phase of Ca 2FeReO 6appears at low tempera- ture, the tunneling electrons across the insulator Ca 2FeMoO 6layer and the counter-electrode have strongly exchanged split sub-bands. Based on this phenomenon, spin-filtering devices can be devel- oped. Furthermore, such an epitaxial heterostructure provides eas- ier manipulating of the spin filtering than the other systems such as Al/EuS/Al or La 2/3Sr1/3MnO 3/NiFe 2O4reported previously.123,124 As the important components used for dissipationless electronic and spintronic devices, ferromagnetic insulators (FMIs) can filter the electronic charges to produce pure spin currents, manipulat- ing spins within nonmagnetic layers via the magnetic proximity effect,125–127creating the quantum anomalous Hall effect in con- junction with topological insulators.128,129For example, the cation- ordered DP Sr 2Fe1+xRe1−xO6thin films exhibit a FMI state and have high Curie temperature (400 K) and large Msvalue (1.8μB/f.u.). This enables them to be used in spin filters, magnetic proximity, and quantum anomalous Hall effects. Such new FMIs also find promising applications in spintronic, electronic, and energy efficient quantum devices. Another promising application of Re-based com- pounds is the development of multiferroic devices based on the mag- netoelastic coupling. The resistance modulations in the Re-based thin films can be induced by the strain, which is generated by a piezoelectric layer beneath the DP Re-based thin film. At present, the advanced applications of Re-based DP oxides are still at their embryonic stage, and many issues need to be resolved and some technical challenges lie ahead. As a consequence, there is a long journey before accomplishing the commercialization of Re-based DP oxides. VIII. SUMMARY AND OUTLOOK Here, we present a comprehensive overview of the recent advances in the Re-based DP oxides, focusing on their syntheses, structural characterizations, physical properties, advanced appli- cations, and theoretical studies on their electronic and structural aspects. The characteristic features of Re-based DP oxides such as metallic-like, half-metallic, or insulating behavior, high Curie tem- perature, and large carrier spin polarization enable them to find promising applications in oxide spintronic devices, multiferroic devices, and energy efficient quantum electronic devices. From the viewpoint of applications, the epitaxial growth of Re-based DP oxide thin films with high quality is highly required although the epitax- ial growth of Re-based DP oxide thin films has become an emerg- ing research field. It is expected that in the near future, remarkable advances will be achieved in the epitaxial growth of Re-based DP oxide thin films by the laser-MBE method with growth control at the atomic scale and enhanced properties. With the development of the aberration corrector (or Cs-corrector), the new generation HRTEM/STEM facility equipped with the Cs-corrector allows one to achieve a spatial resolution better than sub-Å and an energyresolution better than sub-eV, thereby making the structural charac- terizations of Re-based DP compounds at sub-Å available. Although the advanced applications of Re-based DP oxides are promising, they still remain in their early stage; a long journey lies ahead before the commercialization of Re-based DP oxides. However, the scientific and technical potentials of Re-based DP oxides are, for sure, great, and the future research in this field is very bright. It is hoped that this Review could attract renewed interest in the fundamental research of Re-based DP compounds, encouraging the scientific community to enter this impressive area. AUTHORS’ CONTRIBUTIONS K.L. collected the references and prepared the manuscript. Q.T., Y.W., L.Y., and Y.X. participated in sequence alignment. X.Z. designed the structure and modified the manuscript. All authors contributed to data interpretation and discussion and read and approved this manuscript. ACKNOWLEDGMENTS The authors are grateful for financial support from the National Natural Science Foundation of China (Grant Nos. 11674161 and 11174122), the Natural Science Foundation of Jiangsu Province (Grant No. BK20181250), and undergraduate teaching reform projects from Nanjing University (Grant Nos. X20191028402 and 202010284036X). The authors declare that they have no conflict of interest. DATA AVAILABILITY Data sharing is not applicable to this article as no new data were created or analyzed in this study. REFERENCES 1A. S. Bhalla, R. Y. Guo, and R. Roy, Mater. Res. Innovations 4(1), 3–26 (2000). 2A. R. Chakhmouradian and P. M. Woodward, Phys. Chem. Miner. 41(6), 387–391 (2014). 3J. P. Attfield, P. Lightfoot, and R. E. 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5.0040772.pdf

Appl. Phys. Rev. 8, 011319 (2021); https://doi.org/10.1063/5.0040772 8, 011319 © 2021 Author(s).High-velocity micro-projectile impact testing Cite as: Appl. Phys. Rev. 8, 011319 (2021); https://doi.org/10.1063/5.0040772 Submitted: 15 December 2020 . Accepted: 02 February 2021 . Published Online: 18 March 2021 David Veysset , Jae-Hwang Lee , Mostafa Hassani , Steven E. Kooi , Edwin L. Thomas , and Keith A. Nelson COLLECTIONS This paper was selected as Featured This paper was selected as Scilight ARTICLES YOU MAY BE INTERESTED IN Selecting high velocity impact techniques Scilight 2021 , 121108 (2021); https://doi.org/10.1063/10.0003791 The photophysics of Ruddlesden-Popper perovskites: A tale of energy, charges, and spins Applied Physics Reviews 8, 011318 (2021); https://doi.org/10.1063/5.0031821 Attosecond state-resolved carrier motion in quantum materials probed by soft x-ray XANES Applied Physics Reviews 8, 011408 (2021); https://doi.org/10.1063/5.0020649High-velocity micro-projectile impact testing Cite as: Appl. Phys. Rev. 8, 011319 (2021); doi: 10.1063/5.0040772 Submitted: 15 December 2020 .Accepted: 2 February 2021 . Published Online: 18 March 2021 David Veysset,1,2,a) Jae-Hwang Lee,3 Mostafa Hassani,4 Steven E. Kooi,2 Edwin L. Thomas,5 and Keith A. Nelson6 AFFILIATIONS 1Hansen Experimental Physics Laboratory, Stanford University, Stanford, California 94305, USA 2Institute for Soldier Nanotechnologies, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA 3Department of Mechanical and Industrial Engineering, University of Massachusetts, Amherst, Massachusetts 01003, USA 4Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York 14850, USA 5Department of Materials Science & Engineering, Texas A&M University, 3003 TAMU, College Station, Texas 77842, USA 6Department of Chemistry, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139,USA a)Author to whom correspondence should be addressed: dveysset@stanford.edu ABSTRACT High-velocity microparticle impacts are relevant to many fields, from space exploration to additive manufacturing, and can be used to help understand the physical and chemical behaviors of materials under extreme dynamic conditions. Recent advances in experimentaltechniques for single microparticle impacts have allowed fundamental investigations of dynamical responses of wide-ranging samples,including soft materials, nano-composites, and metals, under strain rates up to 10 8s/C01. Here we review experimental methods for high- velocity impacts spanning 15 orders of magnitude in projectile mass and compare method performances. This review aims to present a com- prehensive overview of high-velocity microparticle impact techniques to provide a reference for researchers in different materials testingfields and facilitate experimental design in dynamic testing for a wide range of impactor sizes, geometries, and velocities. Next, we reviewrecent studies using the laser-induced particle impact test platform comprising target, projectile, and synergistic target-particle impact response, hence demonstrating the versatility of the method with applications in impact protection and additive manufacturing. We conclude by presenting the future perspectives in the field of high-velocity impact. Published under license by AIP Publishing. https://doi.org/10.1063/5.0040772 TABLE OF CONTENTS I. INTRODUCTION: FROM MACROSCALE TO MICROSCALE IMPACTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 II. MICRO-SCALE LAUNCHERS . . . . . . . . . . . . . . . . . . . . . . 4 A. Gas-gun systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 B. Sabot-less drag-acceleration systems . . . . . . . . . . . . 6 C. Laser-ablation systems . . . . . . . . . . . . . . . . . . . . . . . . 7 1. Laser-launched flyer plates . . . . . . . . . . . . . . . . . 72. Laser-induced particle impact test . . . . . . . . . . . 8 D. Electrostatic acceleration systems. . . . . . . . . . . . . . . 8 E. Other methods for projectile acceleration . . . . . . . . 10 F. Launcher performance summary . . . . . . . . . . . . . . . 11 III. HIGH-VELOCITY MICRO-IMPACT BEHAVIOR OF MATERIALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 A. Target response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1. Dynamic hardness of bulk materials. . . . . . . . . 112. High strain rate responses of nano-scale materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 B. Projectile response. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 C. Synergistic response . . . . . . . . . . . . . . . . . . . . . . . . . . 18 IV. PERSPECTIVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19SUPPLEMENTARY MATERIAL . . . . . . . . . . . . . . . . . . . . . . . 19AUTHORS’ CONTRIBUTIONS . . . . . . . . . . . . . . . . . . . . . . . . 19 I. INTRODUCTION: FROM MACROSCALE TO MICROSCALE IMPACTS To gain full comprehensive understanding of a material’s behav- ior, one must test its properties over a wide range of conditions. For engineering applications, materials and systems need to be tested under conditions that span the range expected during practical use, including environmental conditions (atmosphere, temperature, Appl. Phys. Rev. 8, 011319 (2021); doi: 10.1063/5.0040772 8, 011319-1 Published under license by AIP PublishingApplied Physics Reviews REVIEW scitation.org/journal/areradiation, external perturbation) and operational modes (deformation state and amplitude, cycle, strain rate, fatigue, and life expectancy). Often, a material’s properties and performance cannot be established under all realistic operational conditions but have to be inferred from standardized experiments under a limited range of conditions andthen extrapolated to more extreme conditions using dimensional anal- ysis and similarity laws. 1,2In the case of a material’s response to defor- mation, low rate testing can certainly hint at high-speed performance, but it has been shown repeatedly that unexpected material behavior can emerge at increasing strain rates and smaller size scales.3–6 Perhaps the first record of such an observation was by Galileo, who noted that a hammer blow leads to effects that would not be observed when the same enormous force is applied slowly under near static con- ditions (called quasi-static).7We intuitively understand that quasi- static testing of an object or a material does not take us far toward an understanding of dynamical behavior. In this review, we will describe experimental impact techniques that have allowed studies of materialbehavior under dynamic conditions (see Sec. II)f o l l o w e db ya n emphasis on single high-velocity microparticle investigations using the laser-induced particle impact test (LIPIT) (see Sec. III). Among dynamic testing methods, impact is perhaps the most ancient method and is still widely applied today under much morecontrolled conditions and expanded capabilities. Historically, the impact engineering field—only recently coined as such in the 1980s— has been driven by the fields of weapon design and tool crafting. 8 Impact weapons have evolved hand in hand with protective materialsand systems, which therefore necessitate field testing of both the weapon and the armor. Rudimentary weapons have been conceived following the simplistic view that the more massive and rapid theimpactor (the larger the translational kinetic energy, KE) the more the potential damage to the target and, likewise, the more massive the protection the better. Weapons are still conceived nowadays in this view, with so-called “kinetic energy” penetrators, consisting of a high-density material core (e.g., tungsten carbide or depleted ura-nium) for armor-piercing purposes. 9,10While the medieval wreck- ing ball is still largely used today, the industrial revolution saw the development of extensive KE impact tools for forging, mining, and construction/destruction applications, including drop forging11or pile driving.12In science, the fields of impact physics and shock physics thrived in the mid-twentieth century, not only motivated by a modern and more scientific approach to ballistics and shockbut also by more diverse subjects such as meteorite crater forma- tion or spacecraft protection. Most experimental tools for macro- scale high-velocity impacts still in use today were first introduced during this time period. The most popular and established impact methods for macro- scale testing are undoubtedly pendulum-based impact (Charpy, Izod), drop-weight impact, Taylor impact, ballistic impact, and plate impact, in order of increasing achievable strain rates. 13Pendulum-based Charpy and Izod tests, which differ by their specimen target geometry, and drop-weight tests follow a similar concept where a heavy non-deformable impactor is accelerated by gravity and strikes a target, which is often notched to initiate fracture of the specimen. The meth- ods allow studies and evaluations of impact resistance of materials and composites, mostly plastics and metals, and measurements of fracture toughness, ductility, strength, and energy dissipation 14–18for a wide variety of loading modes (tension, compression, shear, torsion stressstates). ASTM Standards have been written governing the details of experimental testing protocols and the interpretation of results from impacts with different impactor geometry and target materials and configurations (e.g., ASTM E23). Both pendulum-based and drop- weight instruments are widely used in research for material characteri-zation and in industry for material development and quality control. Specimen dimensions typically range from a few millimeters to centi- meters. However, impact velocities above a few meters per second are seldom achieved. Gas-based acceleration techniques can be categorized as a func- tion of projectile and target configuration, each allowing studies of dif- ferent aspects of impact and shock physics. The Taylor impact test (also referred to as the rod impact test) consists in firing, via a powder or gas driven gun, a mm- to cm-sized cylinder, having an aspect ratio L/D of >5, up to a few kilometers per second toward a massive, non- deformable (highly rigid) target. 19Because the projectile and not the target is the sample of interest, traditional Taylor testing is considered to be a reverse ballistic technique. Nonetheless, numerous studies also looked at symmetric impacts where both the projectile and the target are rod shaped and of interest.20,21Taylor impacts have mainly been used to establish materials’ constitutive models. A large range of strain rates can be achieved during testing, with very high strains at theprojectile-target interface and reduced strains at the rear-end of the projectile. 21–24Classical ballistic tests involve spherical, pellet, or bullet-shaped projectiles fired by guns. Upon exiting of the gun barrel, the projectile, usually accelerated in a sabot, detaches with launch velocities up to several kilometers per second, depending on the gas- gun and projectile caliber.25Detachment of the projectile from the sabot can be facilitated by rifled barrels that impart spin and associatedrotational KE. Studies are largely conducted to assess the ballistic resis- tance of materials and composites for protective purposes and for the fundamental understanding of material behavior under high-rate deformation and penetration. 26–31 Space exploration is constantly driving experimental and techni- cal development toward higher velocities for lm- to cm-sized objects for studies of protection against hypervelocity meteorites and debris, using not only gas-based technology but also electromagnetic systems or a combination of both.32–35Plate impact experiments have become the gold standard for testing in the field of shock physics. As in ballisticexperiments, plates are accelerated down a gun barrel in a sabot. 25 Traditionally, most plate impact experiments have been performed atnormal incidence for easier interpretation and simpler instrumenta- tion; however, increasingly complex configurations are being devel- oped for more complex sample loading with increased experimental control and more extensive diagnostic capabilities. 36–38Plate impact experiments are of particular interest due to the fact that upon high-velocity impact, shock waves are generated both in the impactor and the target under 1D-strain conditions, until the release of shock waves from the edges of the impactor disturbs the 1D state. Induced pres- sures can go above hundreds of GPa with associated strain rates above 10 7–108s/C01. Plate-impact experiments have been used to determine Hugoniot curves of materials,39–41measure dynamic spall strengths,42,43investigate high–pressure phase changes,44and study shock–induced chemistry.45Finally, Split-Hopkinson pressure bars are also widely used for shock studies as well as constitutive parameter determination, but compared to the methods mentioned above do not rely on direct impact on the target but rather on shock wave transitionApplied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011319 (2021); doi: 10.1063/5.0040772 8, 011319-2 Published under license by AIP Publishingthrough and reflection from an impacted rod and a confined target of interest.46 Implementing diagnostic techniques complementary to the launch method is key to fruitful and rich experimental investigations,which would otherwise be reduced to merely destroying specimens.Diagnostic methods can be distinguished between contact vs non-contact and in situ vsex situ . Non-contact methods are usually pre- ferred over contact ones to limit specimen response disturbance andinstrument failure, and to facilitate sample preparation. For instance,in the case of reverse ballistics as in rod impact experiments, where the sample of interest is the projectile, it is challenging to directly mount contact diagnostics on the projectile. 47 In situ diagnostics are more desirable than ex situ diagnostics as impact response is by nature dynamic, and post-impact characteriza- tions can hardly reveal the sequence of events. Most non-contact in situmethods rely on optical techniques such as high-speed photogra- phy (including thermal imaging) and interferometry,48or laser veloc- imetry.49These tools can offer high temporal resolution but suffer from poor spatial resolution and are limited to surface interrogationfor opaque specimens, although recent progress in femtosecond X-raymeasurements has been pushing forward experimental capabili-ties. 50,51A comprehensive review on diagnostic methods for macro- scale impacts was written by Field et al. and can be found in Ref. 13. With the rapid development of nano-technologies and the rich emerging micro-mechanisms and mechanical devices at smaller scales, the standard macro-scale projectile launchers are inappropriate because their projectiles are too large. Samples endowed with novelproperties through their nano-scale architectures are often producedin small quantities on the order of a picogram with a correspondingvolume of /C241u m 3(for a typical polymer density of 1 g/cm3). Considering these dimensions, macro-scale impact experiments arenearly impossible. However, the interesting mechanical responses ofthese materials often manifest at the microscale through size- dependent behavior and cooperative responses of nanoscale ele-ments. 52To probe such dynamic responses, impact experiments must be conducted at or below the micro-scale. A difference in size scalebetween the characteristic dimensions of the impactor and the (larger) target also facilitates interpretation. Micro-scale impacts also prove to be useful in a variety of engi- neering applications. First of all, hypervelocity micrometeorites and orbital micro-debris represent a threat to the integrity of spacecraftand to astronauts performing extravehicular activities, requiring novelmaterials design and protection. 53,54Second, the cold-spray additive manufacturing technique uses high-velocity metallic microparticles tobuild up coatings via impact bonding. 55Third, drug delivery methods using high-velocity drug-loaded microparticles can be used for needle- free epidermal immunization.56Finally, sand particles carried in pipe- lines or by the wind can erode inner walls of pipelines57or helicopter rotor blades.58All of these topics call for laboratory experimentation at the appropriate length scale. While the first experimental techniques in the late 1980s focused on microprojectiles for biological applications,59the emphasis on nanostructured materials and micro-manufacturing methods moti-vated the development of novel techniques to address fundamental materials science questions at the nanoscale. Here, we present a review of the techniques for high velocity (up to a few km/s) micro-projectileimpact testing, focusing on solid substrate-solid particle impact stud-ies.Figure 1 shows how complementary experimental techniques cover the field of impact physics to study and develop new applications(from sports and manufacturing to defense and space exploration) while providing insight into material deformation under extreme con- ditions. While gas-based techniques allow investigations related to bal-listics with associated strain rates of the order of 10 3–106s/C01,V a nd e Graaff accelerators on the other hand are relevant to hypervelocity 1100 s–1103 s–1 106 s–1109 s–1 1012 s–1 Impactor scalemn 1 mm 1 m 11 µmImpact Velocity (m/s)10100100010 000 Hypervelocity Crashworthiness Sports collisions Handheld device damageSpacecraft landing Wrecking ballBallistics Sand erosion Hail impactsDrug deliveryImpact weldingCold-spray Spacecraft entry Shot peeningMicrometeorite/ debris impacts MULTI-STAGE GAS GUNS LASER-INDUCED PARTICLE IMPACT TEST IMPACT NANOINDENTATIONPOWDER GUNSVAN DE GRAAFF ACCELERATORS FREE-FALL WEIGHTSGAS GUNSLASER-DRIVEN FLYERSPLASMA GUNS Ballis FIG. 1. Impact testing techniques (upper- case) and several applications (italic),along with lines of constant characteristicstrain rates. All-capital labels are associ- ated with shaded regions of the same color.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011319 (2021); doi: 10.1063/5.0040772 8, 011319-3 Published under license by AIP Publishingimpact events encountered in space with extreme strain rates above 1012s/C01. Micro- and nano-scale launching techniques, inspired by macroscale methods, can be grouped into three main categories: gas-based, laser-based, or electrostatic-based. In particular and as will beelaborated, laser-based methods present the advantage of high throughput and are capable of being operated in small facilities, even on standard laboratory optical tables. Through smaller projectiles andcomparable high velocities, these experimental platforms can surpass those on the macro-scale in terms of achievable strain rates, with reported rates up to 10 9s/C01, where distinct behaviors can be observed.60 In this review, we first describe the widely used methods for microparticle accelerations to high velocities. We then focus on the laser-induced particle impact test (LIPIT) platform for single-particleimpact investigations of materials’ behavior in the high-velocity regime. We give examples of micro-impact studies conducted with LIPIT focused on target, projectile, and synergistic target-particleimpact responses. We conclude with future perspectives in the field ofhigh-velocity microparticle impact physics. II. MICRO-SCALE LAUNCHERS A. Gas-gun systems Although gas-gun impact experiments were originally designed for macro-scale projectiles, they have, more recently, been adapted toallow microprojectile acceleration. This technique benefits from a longtradition and expertise in flyer-plate and ballistic impact experiments. Micro-projectile systems require minimum modifications to the gun, but attention must be paid to the flyer/sabot design. 61Similar to macro-scale ballistic experiments, these techniques consist in acceler- ating a sabot (carrying the micro-projectiles) through fast expansion of a compressed gas. In typical two-stage light-gas gun (LGG) systems,a piston is accelerated down a pump tube via fast burning of a powdercharge or an explosive detonation. The piston then compresses a light gas, which when sufficiently pressurized, causes rupture of a dia- phragm. The light gas expands into the launch tube, which has asmaller diameter than the pump tube, simultaneously accelerating a sabot carrying the micro-projectiles [ Fig. 2(a) ]. 62 Because the expansion velocity of a gas is inversely proportional to the square root of the gas molecular weight (i.e., the molecularmass), helium or hydrogen are commonly preferred.63The sabot and microprojectiles separate when the sabot is stopped toward the end of the barrel by a catcher plate62,64or a tapered section, releasing the now detached particles into vacuum at high velocities. Other sabot configu- rations rely on projectile-sabot separation outside the gun, either through aerodynamic-drag separation of a split sabot or via centrifugalforces acquired through sabot launch in a rifled tube. 64 In most experiments, multiple particles are accelerated in a single shot in so-called buckshot fashion.64,65On one hand, running experi- ments with multiple particles increases hit probability and can gener-ate extensive data acquisition for a single gun fire; however, this is at the cost of impact precision and control. The typical particle velocity spread is of the order of a few hundreds of meters per second, and theaiming angular precision a few degrees. On the other hand, single- particle systems are rarer because higher experimental precision is required to limit probability of particle deflection. These systems arenonetheless necessary for applications focused on single-particle impact events, such as micro-meteorite detection. 66For such purposes, a two-stage LGG was designed at the Japan Aerospace ExplorationAgency (JAXA), and spherical aluminum projectiles of 1.0–0.1 mm in d i a m e t e rc o u l db el a u n c h e da tu pt o7k m / s . 67 Higher velocities, further into the hypervelocity regime, have been achieved using a so-called three-stage gas gun at the SandiaNational Lab (USA). 68In this system, a sabot is accelerated in a two- stage LGG to impact a stationary flyer target [ Fig. 2(b) ]. The shock generated upon impact travels in the buffer and reaches the flyerthat holds the particles. Shock acceleration launches the particles via momentum transfer to yield hypervelocities relevant to micro- meteorite studies ( >10 km/s). 68–70This system has also been employed to fragment the flyer into micrometer-sized pieces, with reduced control on projectile size, velocity, and direction.68 Conversely, more compact systems involving compressed gas (with single stage guns) and sabot acceleration have been designed with lower velocity in the range of hundreds of meters per second, with the objective of needleless drug delivery with drug-loaded penetratingmicroparticles. 62 In sabot-based acceleration systems, the particle velocity is ulti- mately limited by the sabot velocity. As long as the particle mass is negligible compared to the sabot mass, sabot speeds are not affected by Expanding gas (i) (ii)Sabot Microparticles Catcher plate High-velocity particles(a) Gun barrelExpanding gas Sabot Flyer w/ particles Gun barrel High-velocity particles(i) (ii)(b) Buffer Graded-density impactorVent FIG. 2. Gas gun setups for millimeter- to micrometer-sized particle launch. (a) (i) Typical two-stage LGG: an expanding light gas gun propels a particle-car rying sabot. (ii) Upon sabot stoppage, particles separate and continue to travel at high velocities. (b) (i) Three-stage LGG: a sabot with a front graded-density impac tor, accelerated via light gas expansion, impacts a stationary target (buffer þflyer) holding microparticles. (ii) Shock propagation through the buffer to the back surface leads to detachment and accel- eration of the microparticles.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011319 (2021); doi: 10.1063/5.0040772 8, 011319-4 Published under license by AIP PublishingDistance from Nozzle Exit (mm)Velocity (m/s) 250 200 150 100 50 0 –50 0 5 10 15 20 25 Titanium particle cloud TPX cloudDirection of flight Direction of flight Direction of flight 10 mm 10 mmTitanium particles )b( )a( )d( )c(Distance from Nozzle Exit (mm)–5 0 5 10 15 20 25 –5 FIG. 3. Double-exposure raw image (a) and corresponding derived PIV map (b) of 99- lm polystyrene particles launched from a single-stage LGG. PIV velocity map yields the instantaneous particle velocity field at the exit nozzle.71Reprinted with permission from Mitchell et al. , Int. J. Impact Eng. 28, 581 (2003). Copyright 2003 Elsevier. (c–d) X-ray photographs showing 450- lm titanium particles, initially embedded in a polymethylpentene (TPX) matrix, launched using a three-stage gas gun.70(c) Photograph captured 100 mm from the barrel exit. (d) Photograph taken at a later time 440 mm from the barrel exit. The lead particle velocity was calculated to be 8.6 km/s. Rep rinted with permis- sion from Thornhill et al. , Int. J. Impact Eng. 33, 799 (2006). Copyright 2006 Elsevier. Gas/plasma flowPowder dispenser Drag acceleration (i) (ii) (b)(a)Powder DiaphragmsCompressed gas/plasma Gas/plasma flow Gas/plasma flowDrag acceleration FIG. 4. Drag-acceleration systems. (a) A compressed gas ruptures diaphragms holding microparticles, which are subse-quently accelerated by the gas flow. (b)Alternative method where particles are injected into the flow.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011319 (2021); doi: 10.1063/5.0040772 8, 011319-5 Published under license by AIP Publishingthe particle payload and particles conserve the sabot velocity after sep- aration. Consequently, the particle speed does not rely on the individ- ual particle mass but rather on the sabot acceleration capabilities of the gun.71However, it is noteworthy that for three-stage LGG systems, the particle velocity is mass-dependent since the acceleration relies on a momentum transfer mechanism,68which results in even higher parti- cle velocity spread when fragmented flyers are used. Outside of thegun, particle trajectories are affected by environmental conditions. For instance, most gas gun experiments dedicated to material behavior studies are typically operated under vacuum whereas drug-delivery- oriented instruments launch particles under atmospheric conditions. In the latter case, the air drag tends to slow down particles while increasing the velocity distribution. For most methods described above, particle velocities have been measured either through sabot velocity measurement using photosen- sors and a time-of-flight approach, 65,72as for macro-scale impacts or through high-speed photography with nanosecond exposure time. For instance, in the work by Mitchell et al. on microparticle-based drug delivery71and related works,73,74the particle velocity distribution was measured using particle image velocimetry (PIV). With two nanosecond-duration laser pulses (Nd-YAG) and a single CCD cam- era, stroboscopic images were taken at the exit nozzle [ Fig. 3(a) ]a n d particle velocity distribution could be estimated via image cross corre-lation [ Fig. 3(b) ]. Likewise, flash X-ray photography has been used with similar nanosecond time resolution with the added ability to pen- etrate optically opaque media 68[Figs. 3(c) and3(d)]. Besides particle- velocity measurements, these methods have limited capabilities for real-time in situ measurements of particle/target interactions and impact responses; studies instead mostly rely on postmortem examina- tions.65,74Indeed, the regime of high velocity micro-impact, necessi- tates both micro-scale spatial and nanosecond temporal resolution diagnostics, while requiring nanosecond timing and microscale aiming precision for the launch system. B. Sabot-less drag-acceleration systems To avoid sabot-related complications (fragmentation, diversion, collection), sabot-less methods have been developed to enable micro- particle acceleration. These systems are based on direct interaction between an expanding gas or a plasma jet with the particles, where the drag exerted by the gas or plasma accelerates particles to high veloci- ties. Particles can either be initially positioned in the course of the flow using stationary mounts or directly injected into the flow. For instance, Kendall et al. devised a hand-held contoured shock tube where par- ticles, instead of being mounted on a sabot, were directly enclosedbetween two diaphragms designed to burst under light-gas pressure, 73 as depicted in Fig. 4(a) . They were able to accelerate particles via drag forces, different materials, and diameters (e.g., gold 0.2–2.4 lm, poly- styrene 11–20.5 lm, glass 2.6 lm) to velocities ranging from 200 to 600 m/s. Motivated by drug delivery applications, particles were inten- tionally accelerated as a cloud (buckshot mode) to maximize the drug payload by shot (up to 2 mg), inevitably resulting in large velocity dis- t r i b u t i o n[ s i m i l a rt ow h a ti ss h o w ni n Fig. 3(b) ]. Other works, such as performed by Rinberg et al. ,75have also employed compressed gas, but particles were injected in the gas stream though a secondary chan- nel [Fig. 4(b) ]. Additionally, Rinberg et al. implemented a suction sys- tem to recover the helium gas at the exit of the nozzle and to limit gasinteraction with the target, which can be harmful when dealing with tissue. Drag acceleration is also at the center of the cold spray technique, which has been gaining interest in the coating industry since the 1980sand more recently in the additive manufacturing industry. 76Ah i g h - temperature compressed gas (typically helium, nitrogen, or air) is used as a propulsive gas to accelerate particles from a powder feedstock, most often toward a metallic target [as shown in Fig. 4(b) ]. The forma- tion of the deposited coating is via a solid-state process relying on par-ticle kinetic energy (and impact-induced plastic strain and adiabaticheating) rather than thermal energy of preheated particles. Because cold spray industrial systems accelerate simultaneously, in a continu- ous stream, a large number of particles [ Fig. 5(a) ] (with a range of par- ticle diameters and impact angles) in a hot (up to T /C241000 /C14C) gas, it has been difficult to answer fundamental physical questions with thosesystems: The individual particle velocity and temperature are not well controlled and are mostly deducted from gas dynamics or PIV mea- surements on many particles. A review detailing velocity measure-ments in cold spray has been recently written by Yin et al. 77Owing to their large number, particles can interact multiple times with otherparticles and the target. It has been therefore difficult to unravel the unit impact-bonding process using these systems. To overcome the complication of particle-particle interactions, wipe tests, where the tar-get is rapidly translated in the plane orthogonal to the spray jet, can be Distance from Nozzle Exit (mm)0 15 20 25 30 35 Distance from Center Line (mm)0 –5 –10 5 10 Direction of flight 0 100 200 300 400 500 600700Velocity (m/s) 5 mmPlasma flow(a) (b) FIG. 5. (a) PIV image showing velocity map converted from multi-frame imaging of aluminum particles exiting the helium nozzle of a cold spray system.86Reprinted with permission from Pattison et al. , Surf. Coatings Technol. 202, 1443 (2008). Copyright 2008 Elsevier. (b) Graphite particles (5–24 lm diameter) accelerated by a coaxial-gun-generated plasma flow. Particles are glowing due to plasma heating.Velocity (0.8 – 1.5 km/s) can be deduced from elongation of particle due to 12- ls camera exposure time.83Note that with such long-exposure imaging it is not possi- ble to distinguish single particles. A single streak could originate from two or more juxtaposed particles. Reprinted with permission from Ticos ¸et al. , Phys. Plasmas 15, 103701 (2008). Copyright 2008 AIP Publishing.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011319 (2021); doi: 10.1063/5.0040772 8, 011319-6 Published under license by AIP Publishingconducted, enabling postmortem observations and characterizations of single-particle deposits.78However, this technique does not circum- vent the lack of information about the history of the particle (particle velocity, impact angle, particle temperature). To solve these complica- tions, other methods for metallic-microparticle impact tests have been implemented (see Sec. II C 2 ). Following a similar concept, plasma-based systems have enabled purely gas dynamic acceleration of charge-neutral microparticles (essentially dust) through drag acceleration of a fast-expanding plasma jet with velocities up to a few kilometers per second. The attainable velocities are significantly higher than what is reachable by compressedgas systems 61and similar to multi-stage gas guns. Coaxial plasma accelerators (guns) were originally developed in the 1950s–1960s79 and have been proposed as plasma jet injectors for fusion applications to create, for instance, a spherically imploding plasma shell.80 Recently, plasma guns have been adapted to accelerate particles astracers to study plasma dynamics, where plasma properties could bededuced from particle trajectories. 81,82They have also been suggested as a promising tool for space propulsion or launch.83 Coaxial plasma guns consist of two concentric cylindrical electro- des. Before the shot, the coaxial gap is filled with a propellant gas (e.g.,deuterium), which becomes ionized when a switch connects a capaci- t o rb a n kt ot h ee l e c t r o d e s .T h i sg e n e r a t e sap l a s m at h a ti st h e na c c e l e r - ated by axial Lorentzian forces down the gun, where the highestvelocities are achieved near the center rod electrode. Particles can be injected into the plasma inside the bore of the gun, using for instance, a piezoelectric transducer to shake off particles from a dust container into the plasma 83or directly mounted, statically, in the gun barrel prior to plasma generation.84Even though relatively high velocities can be reached, the presence of a plasma, which can potentially impact the target and degrade the projectiles, significantly complicates the impact event [ Fig. 5(b) ]. Gas or plasma drag-entraining methods present the advantages of being able to accelerate large numbers of particles to relatively large velocities,85allowing material buildup in cold spray or statistical analy- sis for plasma studies.83However, they offer limited control of par- ticles’ trajectories, owing to gas dynamics instabilities and the presence and impact of hot gas or plasma on targets, which can both heat and chemically alter the target surface. The large uncertainties in impacttiming and aiming render these systems inadequate for single-impact observations and systematic investigations of unit impact events. Finally, contrary to sabot-based techniques, the maximum velocities reached through drag acceleration depend on both the mass and the cross-sectional area of the projectiles. C. Laser-ablation systems In essence, laser-ablation systems rely on the same physics and acceleration mechanisms (sabot acceleration/separation, drag accelera- tion, momentum transfer) as the gas and plasma guns described ear- lier. However, focused laser pulses can deliver large energies in much shorter times (down to the femtosecond or possibly less) with high peak power on areas that are much more localized (down to themicro-scale) than conventional methods. These characteristics make them a priori well suited for microscopic high-velocity launch, and moreover these approaches offer high controllability, reliability, andsafety.1. Laser-launched flyer plates When a laser pulse is focused on an absorbing material with suffi- cient peak power, the material is flash heated and can liquify, evapo- rate, sublimate, or be converted to a plasma. The sudden materialexpansion can serve, similar to gas gun or macro plate impact experi-ments, as a means to accelerate a flyer. Typically, in a laser-driven flyer(LDF) experiment, a nanosecond-duration laser pulse, e.g., from acommercially available Q-switched Nd:YAG laser with energies up to a few hundred megajoules, is focused on a metallic foil (typically a few hundred micrometers in diameter and up to a few tens of micrometersin thickness) glued (with, for example, a transparent epoxy) to a thick,rigid substrate that is transparent to the laser wavelength. 87–89Upon arrival of a high-fluence laser pulse, the metal absorbs the laser radia-tion, and a plasma is generated at the interface between the glue and the metallic film. The confined plasma rapidly expands and deforms the metallic foil, which subsequently tears off the rigid substrate and isdriven off as a flyer plate (see Fig. 6 ), with speeds up to a few kilo- meters per second. 90Ideal plate experiments require high control on the flyer profile, including its shape and the velocity. In other words,the flyer must remain flat as it travels from the launch pad (LP) to the target. Thus top-hat vs Gaussian focal spots have been preferred for uniform plasma generation and plate acceleration. 89Likewise, short LP-target distances in vacuum are desired to limit plate deformationduring propagation due to drag forces. 91In alternative design, metallic films have also been directly coated on optical fibers for their low cost,increased compacity, flexibility, and top-hat beam profile. 92 Earlier investigations, spear-headed by the Los Alamos National Laboratory (LANL)90and the Naval Research Laboratory (NRL)94in the U.S., were motivated by interest in impact explosive initiations orfast ignition, 90,95b u tr e c e n ti n t e r e s ti nL D Fh a sr e c e n t l yb e e ng r o w i n g more broadly in the shock physics community.91,96,97Indeed, LDF experiments are essentially similar to plate impact experiments with considerably shorter shock durations (usually of the order of tens of High-speed flyerGlueWindow (e.g., glass substrate)Laser pulse Metallic foil(i) Plasma generation (ii) )b( )a( Plasma Flyer Debris1 mmFoil tearing FIG. 6. (a) Schematic illustration of LDF launch pad (LP). (i) A laser pulse is focused through a transparent window to the back of a metallic foil, generating a plasma. (ii) The expanding/driving plasma accelerates the foil into free space. (b) High-speed photography of a flyer plate launched at 560 m/s. The bright plasmaand ablation debris are visible in the images. The interframe time is about 1.6 ls. The exposure (shutter) time is 1 ls, which results in blurring of the flyer.93 Reprinted with permission from Dean et al. , Appl. Opt. 56, B134 (2017). Copyright 2017 The Optical Society.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011319 (2021); doi: 10.1063/5.0040772 8, 011319-7 Published under license by AIP Publishingnanoseconds, depending on the plate thickness) but with generated shock pressures that can compete with macro-scale systems (up to/C24200 GPa 98), allowing studies of material behavior under dynamic (shock) loading. With a relatively simple LP assembly (down to three layers) and micro-scale localization of the laser damage, tens of shots can be per-formed with a single LP, increasing the number of experiments from a few a day in gas-gun systems to tens of experiments a day for LDF sys- tems. It should also be noted that these experiments can be conductedon a table-top apparatus, in contrast to much larger facilities required by gas-gun setups. On the other hand, inherent challenges remain, particularly regarding the flyer characteristics, including its dimen- sions, planarity, temperature (due to plasma heating), and integrity (extent of material loss through ablation). 96,99Experimental efforts have been dedicated to minimize the partial ablation/vaporization of the flyer caused by direct plasma heating and shock-induced spall- ation. Strategies include the addition of laser shielding layers (e.g., opa-que metallic layers) to avoid direct laser exposure and heat shielding layers (using low thermal diffusivity materials, e.g., epoxy). In such multilayer configurations, the flyer is accelerated by the shock gener-ated in the layers underneath it rather than by plasma expansion. 96,100 We note, however, that Curtis et al. have demonstrated that flyer abla- tion can be limited even in a simple configuration (as depicted inFig. 6 ) where the laser energy is mostly absorbed in the glass substrate rather than at the surface of the flyer. 89 Most diagnostic methods are inherited from macro-scale plate experiments, primarily VISAR or photon Doppler velocimeters (PDV)and high-speed photography. 89,93,101Increased timing control brought by all-optical systems, allowing high-precision synchronization, has also triggered the development of more elaborate diagnostic capabili-ties (e.g., in the Dlott group at the University of Illinois at Urbana- Champaign) in shock experiments, including for instance real-time emission spectroscopy 102and optical pyrometry.103 While most studies have been centered around the target’s shock state, a few works have been devoted to hypervelocity projectile pene- tration or impact resistance of materials using this technique almost exclusively directed toward protection from space debris.104–106 However, LDF systems have not been the experimental platforms of choice in general for hypervelocity investigations because flat projec- tiles are rarely encountered in hypervelocity environments. 2. Laser-induced particle impact test Several acceleration mechanisms to achieve fast-moving micro- particles had been demonstrated by techniques including laser ablation[LDF in Fig. 6(a) ] and laser-induced shock wave. 107However, the first introduction of a laser-driven microprojectile as a precisely defined high-strain-rate (HSR) microprobe was demonstrated by Lee et al. using the Laser-Induced Particle Impact Test (LIPIT).108As a quanti- tative HSR characterization method, the key progress advanced byLIPIT is the precise quantification of kinetic parameters of the well- defined microprojectile before, during ( in situ ), and after mechanical interactions with a specimen using ultrafast microscopy with hightemporal (nanosecond) and spatial (sub-micrometer) resolution. Although the first demonstration was done using an 800-nm, 250-ps duration ablation pulse from an amplified Ti:Sapphire laser system, 108 current LIPIT designs, with setups at MIT,109Rice University,110UMass Amherst,111University of Wisconsin-Madison,112and NIST,113use an ablation laser pulse from a Q-switched Nd:YAG sys- tem at either the fundamental (1064 nm) or the second harmonic(532 nm) wavelength. The launch pad (LP) assembly includes a glasssubstrate (typically 200 lm thick), an ablation layer (gold film), and a lightly crosslinked elastomer [polydimethyl siloxane (PDMS) or poly- urea, 20–80 lm thick]. When the laser pulse ablates the gold film, a plasma is generated and rapidly deforms the elastomer layer, which inturns ejects a particle to high velocity [see Fig. 7(a) ]. Separation is pro- vided by elastomer retractation by elastic forces as the elastomeric layer remains attached to the LP. With low particle-elastomer adhe- sion, the particle can freely travel toward a target. The elastomericlayer also serves as an effective thermal barrier between the hotablation-generated plasma and the projectile, so that the projectile ismaintained at its pre-launch temperature. The first version of the LIPIT initially developed by Lee et al. in 2010 at MIT 108i n s t e a dr e l i e do nd i r e c td r a ga c c e l e r a t i o n ,s i m i l a rt o sabot-less systems described in Sec. II B. A monolayer of silica spheres was deposited on a thin dye-doped polystyrene layer. Upon laser irra- diation and absorption by the dye, the polystyrene layer vaporized andmultiple particles were accelerated by the unconstrainted expandinggas/plasma [ Fig. 7(b) ]. PDMS launch pads were in use shortly after for much more precise single-particle launch and increased velocity con-trol. Aiming for higher velocities, Veysset et al. more recently designed a launch pad eliminating the elastomeric layer, which partially dissi- pates the plasma-induced acceleration forces. 114Higher velocities could be obtained for heat-resistant ceramic particles via direct plasmadrag acceleration. However, metallic particles (Al and Sn) melted dueto the direct contact with the hot plasma, which prevented accelerationto supersonic velocities. In the emergent LIPIT technique, the primary diagnostic is high speed photography either via single-detector multi- ple exposure with pico/femtosecond laser pulses [ Fig. 7(b) ] 108,110,115or using a high-speed multi-frame camera [ Fig. 7(c) ].109,116With LIPIT, particle velocities can be tuned by adjusting the laser energy. The high-est velocity reported is about 4 km/s with silica particles (3.7- lmd i a m - eter). 108Contrary to sabot-based gun schemes, the mass of the particle is not negligible compared to the mass of elastomeric layer in LIPIT, which is why maximum particle velocities depend on the particle mass(see Sec. II F). It should also be noted that with LIPIT, particles can also be accelerated to relatively low velocity down to less than 1 m/s(unpublished), given sufficiently low adhesion between the particle and the polymer. In contrast, because the LDF relies on foil ablation, the plasma must drive the foil sufficiently to detach the foil into a flyer,which results in a higher minimum velocity ( >0.1 km/s). 91 LIPIT has emerged as a fruitful tool for high-velocity micro- impact studies and has been applied to the study of a wide range ofmaterial behavior—for both projectile and target—in bulk polymers,gels, and metals as well as nano-composites, 2D layers, and fibers.Representative examples of LIPIT studies are presented in more detail in Sec. III. D. Electrostatic acceleration systems The use of an electrostatic acceleration method to launch micro- particles to hypervelocity was first described by Shelton et al. in 1960. 117The method was originally developed to study hypervelocity impacts of single microparticles and has been used extensively andprincipally in the field of micrometeorite impact studies 118–120andApplied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011319 (2021); doi: 10.1063/5.0040772 8, 011319-8 Published under license by AIP Publishingimpact-induced plasma physics.121Due to the high voltages required for both particle charging and acceleration, this acceleration technique must take place in vacuum, which not only provides a space-like envi- ronment but also frees the particle launching process from any con- current blast wave or propellant debris that must be accounted for in most other techniques. Electrostatic acceleration is achieved by firstcharging dust or microparticles and then passing the charged particles through an electric potential difference. The ultimate velocity ( v) that can be reached for a given particle is based on its charge-to-mass ratio ( m/q) and the magnitude of the potential difference ( U), where the kinetic energy of the particle 1 =2mv 2¼qU:117 Therefore, it is desirable to obtain as high of a charge-to-mass ratio as possible. The most efficient and effective route to reaching high charge states is to bring a microparticle into contact with an elec- trode held at high potential. The charging limit for most materialsis the threshold for ion evaporation (for positively charge particles) or electron field emission (for negatively charged particles). This also brings up a key limitation to this acceleration technique, namely that at least the surface of the particle needs to be conductive. Metallic particles are often used as well as conducting-polymer-coated 122or metal-coated insulating particles.123 There is a wide variety of electrostatic acceleration instruments reported in the literature over the years. They all include a dust orparticle source of single highly charged particles for acceleration and impact studies. Examples of dust/particle source designs can be found in Refs. 120and124. The charging of particles depends on the particle characteristic and dimensions (surface area), which results in different empirical mass-velocity relationships for different particle materials and dust sources (as illustrated in Fig. 9 , red solid lines). Often, many more charged particles are generated in these sources than make it into the acceleration region of the instruments. A series of pinhole apertures and electrostatic lenses are used to exclude any particles thatare not moving along the axis of the instrument. After a particle is extracted from a dust source, it is accelerated by passing through a large potential difference. In instruments that have demonstrated thehighest velocities, the voltage necessary is provided by a Van de Graaff generator, 124as illustrated in Fig. 8(a) . These are typically found in large shared-use facilities or national labs and require a large amountof laboratory space, extremely high voltages (up to 2–3 MV), and spe- cialized equipment. 35,125–127 The use of highly charged particles in the electrostatic accelera- tion technique allows various types of charged particle detectors to beused to measure charge and velocity and to extract particle mass. A series of image charge detectors are often employed to determine par- ticle properties after acceleration and before impact 120[Fig. 8(a) ]. An example of a pickup signal is shown in Fig. 8(b) , where the magnitude MicroparticleAblation filmGlass substrateLaser pulse High-speed particleElastomer(i) (ii)Plasma expansionPlasma generation Blastwave200 mm 0 ns 200 ns 400 ns 600 ns 800 ns 1000 ns 1200 ns 1400 ns100 mm 100 mmDt = 22 ns 3.8 km/s4.2 km/s(b) (a) (c) 825 m/sElastomerSlower out-of-focus particles FIG. 7. (a) Laser-induced particle impact test (LIPIT) schematic illustration adapted from Ref. 109(i) A laser pulse is focused on an ablation film that is quickly converted to a plasma. (ii) The plasma expands and pushes the elastomeric layer, which accelerates a selected particle to high velocity. Adapted from Veysset et al. , Sci. Rep. 6, 25577 (2016) under the terms of the Creative Commons Attribution 4.0 International License. (b) Double-exposure image showing acceleration of multiple p articles (3.7- lm diameter silica spheres) to a few kilometers per second. With a known time separation (22 ns) between the two illumination flashes (250-fs duration laser pulses ), the speed can be cal- culated by tracking particle displacement.108Reprinted with permission from Lee et al. , Nat. Comm. 3, 1164 (2012). Copyright 2012 Springer Nature. (c) Multi-frame image sequence showing an aluminum particle (12- lm diameter) as it impacts an aluminum substrate with a speed of 825 m/s, subsequently adhering to the target surface. Timing is shown at the top of each frame. The expanding and retracting polyurea elastomer is also visible in the images.116Reprinted with permission from Hassani-Gangaraj et al. , Scr. Mater. 145, 9 (2018). Copyright 2018 Elsevier.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011319 (2021); doi: 10.1063/5.0040772 8, 011319-9 Published under license by AIP Publishingof the signal reveals the particle charge q, and the delay between detec- tion times Dtgives the velocity. After detection and characterization of the particle parameters, electrostatic deflection systems can be employed to allow only particles of a selected charge and velocity, and therefore mass, to impact a chosen target [ Fig. 8(a) ]. Particles that do not meet the desired parameters can be deflected off the axis of theexperiment. This is a critical capability due to the fact that most ofthese instruments launch a stream of particles, and post-impact analy-sis would be difficult without particle discrimination and characteriza-tion. In principle, single-particle impact experiments can be conducted using Van de Graaff accelerators (VdGA). However, since particles are ejected from the dust generator at random times, in-flight detection offlying particles is necessary to trigger subsequent data acquisition, lim-iting real-time diagnostic capabilities. Single-particle discriminationfollowing in-flight detection requires a low rate of particle entry intothe acceleration column and hence limits the experimental rate to a few experiments a day. A higher impact rate with lower particle selec- tivity (wider selected mass and velocity distribution) has often beenpreferred. 120,124 Of all of the particle acceleration techniques described in this review, electrostatic acceleration provides the highest achievable veloc-ities, up to several tens of kilometers per second [see Fig. 8(c) ]a n d delivering impacts at characteristic strain rates of 10 12s/C01or higher.129 The technique is constrained by the requirement of using conductiveparticles and is typically limited to relatively small particle sizes of under a few micrometers in diameter. E. Other methods for projectile acceleration The set of methods described in the previous sections is certainly not exhaustive but includes the most widely adopted techniques for microparticle launch. Other methods have been developed motivated by the desire to reach even higher speeds to (1) replicate hypervelocity impacts, (2) develop a space launch platform of centimeter-scale objects, and (3) create higher shock pressure conditions by hyperveloc- ity impact. Because these methods aim for hypervelocities (beyond the high velocity studies presented here), we do not describe them in detail. Some of these methods, however, are worthy of notice. For instance, the railgun has long been seen as a promising technique for hypervelocity launch.130It involves accelerating a piston-like armature using Lorentzian forces, as for a plasma gun. Velocities up to 5–7 km/s for projectiles with masses of the order of one gram have been demon- strated, but this technique has not surpassed gas guns’ more consistent performance until now.131The inhibited shaped-charge launcher (SCL) and other explosive-based techniques have long been used to study hypervelocity impacts.132A SCL consists of the collapse of a shock-driven liner to create a micrometric jet (cylinder-like shape), with very limited control of shape, mass, velocity, and so forth. A brief (a) (b) Van de Graaff high-potential terminal Microparticle source(+q, m, v) Particle detection and selection units (PSU) Charge preamp. Electrostatic deflection Beam stop unselected part. To targetHV pulseTime (s)–6–5 –4 –3 –2–1 01 –7Charge q (10–14 C) 10–1510–1410–1310–1210–1110–9 10–10 4 2 681 0 2 0 3 0 50Dt (c) Iron particles Velocity (km/s)Mass (g)0246Det. 1 Det. 3 PSU Gate Det. 4 Min. charge detection limit FIG. 8. (a) Schematic illustration of a Van de Graaff (VdG) microparticle accelerator adapted from. Ref. 125. A large potential difference between the high-potential VdG termi- nal and a ground terminal accelerates charged particles toward a target. Hollow metal cylinders detect particle charge to allow velocity measuremen ts and mass estimations. Unwanted particles can be deflected farther down the beam path before reaching the target. Other deflection setups deflect only selected particles towa rd the target. Reprinted with permission from Keaton et al. , Int. J. Impact Eng. 10, 295 (1990). Copyright 1990 Elsevier. (b) Example of a single-particle signal generated upon passage through multiple charge detectors and PSU gate selection.120Reprinted with permission from Shu et al. , Rev. Sci. Instrum. 83, 581 (2003). Copyright 2003 AIP Publishing. (c) Mass-velocity dis- tribution obtained using a 2-MV VdG accelerator for iron particles.128Particles with low charge fall below the detector noise level and are not detected. Reprinted with permis- sion from Lee et al. , Int. J. Impact Eng. 44, 40 (2012). Copyright 2012 Elsevier.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011319 (2021); doi: 10.1063/5.0040772 8, 011319-10 Published under license by AIP Publishingreview, written by Schneider and Sch €afer, on these hypervelocity tech- niques can be found in Ref. 35. The Z-Machine, also using Lorentzian forces, has enabled the acceleration of micrometer-thick plates tohypervelocities above 10 km/s for equation of state measurements atpressures reaching /C241000 GPa. 133 F. Launcher performance summary A representative sample of maximum reported velocities as a function of projectile mass for different impact-testing systems overthe past 25 years is presented in Fig. 9 . Launch data are also catego- rized based on projectile shape and shot mode. Single-shot mode refers to cases where single-particle impact signatures can be recoveredwhereas buckshot/continuous mode represents cases where single-particle discrimination is not possible or reliable. We note here againthat sabot-based techniques such as single-stage and two-stage techni-ques accelerate particles almost independently of their mass, whereas for drag-based (plasma- and gas-drag accelerators, cold spray) and momentum-based techniques (3-stage LGG), lighter particles reachhigher speeds. Van de Graaff accelerators, LDF, and LIPIT methodsdemonstrate a stronger dependence on particle mass which is relatedto the maximum particle velocity through a nearly constant kinetic energy as shown in Fig. 9 . In the high-speed microprojectile regime (shown with a dashed square), LIPIT is uniquely positioned for single-projectile studies. III. HIGH-VELOCITY MICRO-IMPACT BEHAVIOR OF MATERIALS Here we describe and look into selected material behaviors that were revealed through high-velocity micro-impacts. We focus on thehigh-velocity regime and not on the hypervelocity regime ( >10 km/s) where the behavior is more extreme and where materials typically exhibit hydrodynamic behavior or are converted to plasma. Theintermediate regime of high velocity has recently gained interest with the active and rapid development of LIPIT platforms, which are uniquely positioned in this regime (see Fig. 9 ), allowing material study under previously under-explored deformation regimes. In Secs.III A–C , we distinguish three idealized impact configurations where deformations are (i) predominantly in the target (hard particle oncompliant target), (ii) predominantly in the particle (compliant parti-cle on hard substrate), and (iii) partitioned between particle and target,which have similar mechanical properties. A. Target response 1. Dynamic hardness of bulk materials Similar to what is routinely done in macro-scale ballistic impact experiments, hard/non-deformable projectiles can be aimed at 3Dquasi-semi-infinite targets. LIPIT was initially developed to study pen-etration resistance of a semi-infinite nano-layered material (see Sec.III A2 ) and soon adapted to bulk, isotropic materials. Following the premise of increased dissipation through dynamic glass transition ofelastomer by Bogoslovov et al. , 134several works have been dedicated to studying the impact and high-rate behavior of elastomers, namelypolyureas, polyurethane, and poly(urea-urethanes) (PUUs), under LIPIT to identify and leverage molecular attributes for energy dissipa- tion. For instance, Veysset et al. performed a series of experiments to visualize microparticles impacts on PUUs. 109,135High-velocity impacts resulted in particle penetration and subsequent rebound with no per-manent damage [see Fig. 10(a) ]. They demonstrated a correlation between the hard segment content of the PUUs and both the coeffi-cient of restitution (ratio of impact velocity and rebound velocity, indi-cator of energy dissipation) and the maximum penetration depth,related to material hardness. Further works by Hsieh et al.,W u et al., and Sun et al. aimed to elucidate the dynamic stiffening characteristics Max. Projectile Velocity (m/s) Projectile Mass ( g)10010–310–610–910–1210–1510–18102103104105 High-speed microprojectilesProj. shape: (Quasi-) Sphere CubeCylinderPlate Fragments Shot mode: Single Buckshot/ContinuousKE = 1 µJ KE = 1 mJ KE = 1 J KE = 1kJ KE = 1 nJ System: PDA LIPIT LGGGDAVdGA LDFCS3-Stage LGGZ-Machine SCL Railgun2-Stage LGG FIG. 9. (a) Launcher performance summary, including Van de Graaff accelerator (VdGA) and reported empirical mass-velocity laws for iron and conduction-po lymer-coated latex particles (red solid lines), plasma drag and gas drag accelerators (PDA, GDA), laser-induced particle impact test (LIPIT) and empirical mass- velocity law (violet solid line), cold-spray systems (CS), single- and multi-stage light-gas guns (LGG), laser-driven flyers (LDF), inhibited shaped-charge launchers (SCL) , railguns, and the Z-Machine. Systems are identifiable by symbol color. Symbol shapes represent projectile shapes. Open-symbols are for buckshot and quasi-continuous feed (or bu ckshot) systems. Solid symbols are for single-projectile systems. Lines of constant kinetic energy (KE) are plotted. See supplementary material for full data set and references.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011319 (2021); doi: 10.1063/5.0040772 8, 011319-11 Published under license by AIP Publishingof PUUs in view of segmental relaxation dynamics to unravel the role of hydrogen bonding in the dynamic glass transition and impact response.136–138In particular, the slower relaxation dynamics of the soft phase, undergoing a change from rubbery behavior at ambient conditions to a leather/glassy behavior under high rate deformation, was suggested to act as preferential molecular pathways for energy dis- sipation. Recently, temperature-controlled experiments by Sun et al. on polyurea revealed an increased energy dissipation (and related hardness) at the dynamic glass transition temperature [see Fig. 10(b) ].139The molecular relaxation at the origin of dissipation was identified as the a2relaxation corresponding to the soft segments near the hard-segment interfaces, as identified via broadband spectroscopy.This study therefore quantitatively confirmed the macro-scale experi- ments and thesis by Bogoslovov et al. 134Other soft materials beyond polymers, such as gels or biomaterials, have also been tested with LIPIT to determine their rate-dependent properties. Notably gelatin and synthetic gels, including poly(styrene-b-ethylene-co-butylene-b- ethylene) (SEBS) polymer-based gels with non-aqueous solvent, with varying concentrations were investigated.140,142Particle trajectories were observed in the transparent specimen [ Fig. 10(c) ]a n dp a r t i c l e trajectory models were evaluated. Recent experiments by Veysset et al. allowed the development of a refined model for particle impact in yield-stress fluids at intermediate Reynolds numbers [ Fig. 10(d) ].141In all gels tested, it was shown that, under micro-scale testing, theresistance to penetration, related to materials strength, significantly increased compared to macroscale experiments by orders of magni- tude, evidencing rate strengthening effects. Further insights into gel response can be achieved by numerical simulations where models are calibrated using particle trajectories and cavity dynamics.142 The dynamic impact hardness of metals has mainly been mea- sured over the past two decades through impact indentation using pendulum-based dynamic micro-indenters (PDMI).143PDMI has allowed fundamental studies on materials properties, despite experi- mental calibration complications and debate over hardness calcula-tions procedure. 140,144,145For example, Trelewicz and Schuh demonstrated the breakdown of the Hall-Petch strength scaling in nanocrystalline Ni–W alloys and subsequent emergence of an inverse Hall-Petch weakening regime.146However, in PDMI, the strain rate is limited to about 104s/C01because of relatively low impact velocities (up to a few millimeters per second).144In contrast, with higher achievable velocities, high-rate impact indentation studies can be conducted with LIPIT as demonstrated by Hassani et al.147In that study, hard/non- deformable alumina particles were used to impact pure copper and iron targets at velocities up to 800 m/s. Real-time imaging helped determine the plastic work leading to post-impact imprints on the sample surface [ Fig. 11(a) ], whose volumes were measured by confocal microscopy [ Fig. 11(b) ]. Using the conventional definition of hardness used in PDMI experiments, defined as the ratio of plastic work to 35-ns interframe time 50-ns interframe time 20 mm 50 mm (a) PUU target (b) Target Temperature (°C) Time (ns)% Energy Absorbed Penetration Depth ( mm)Vi (m/s) 0 100 150 5060 16 141210 864 2 0708090100 72 68 64 60Gel target(c) (d) 0 500 1000 1500 2000 2500630 m/s 440 m/s 215 m/s Modified Clift-Gauvin model FIG. 10. (a) Multi-frame impact sequence showing a silica particle (7.4- lm diameter) impacting a poly(urea urethane) target at 770 m/s and rebounding at reduced velocity.109 Reprinted with permission from Veysset et al. , Sci Rep. 6, 25577 (2016) under the terms of the Creative Commons Attribution 4.0 International License. (b) Percent impact energy absorption of polyurea target as a function of temperature, revealing glass-to-rubber transition around 115/C14C.139Reprinted with permission from Sun et al. , Appl. Phys. Lett. 117, 021905 (2020). Copyright 2020 AIP Publishing. (c) Multi-frame impact sequence showing a steel particle (13- lm diameter) penetrating a SEBS-based gel target at 630 m/s.141(c) Penetration trajectories in gel for three impact velocities and corresponding modeled trajectory curves. Reprinted with permission from Veyss etet al. , Exp. Mech. 60, 1179 (2020). Copyright 2020 Springer Nature.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011319 (2021); doi: 10.1063/5.0040772 8, 011319-12 Published under license by AIP Publishingindentation volume, hardness values could be estimated for character- istic strain rates of the order of 106–107s/C01[Fig. 11(c) ]. The material hardness was shown to significantly increase with strain rate above 103–104s/C01, which was attributed to a transition in the deformation mechanism from thermally activated to drag-dominated dislocation motion. 2. High strain rate responses of nano-scale materials Materials exhibiting nanoscale size effects or nanoscale phases are envisioned to be a breakthrough toward achieving high energy absorption for lightweight protective materials due to their mechanical anisotropy and high specific inter-phase interaction. As stated earlier, LIPIT was first devised and applied to detailed observations of poly- mer layered nanocomposites subjected to supersonic micro-projectile impacts.108The HSR responses and associated energy dissipation mechanisms of studied nano-scale materials have been categorizedinto two cases: primarily localized and primarily delocalized responses, as illustrated in Fig. 12 . Although the actual responses of the materials generally include both responses, this classification based on materials’ dominant deformation scale is useful to understanding energy dissipa- tion mechanisms depending on strain rates, as well as material proper- ties. Thus, the penetration energy ( Ep) can be expressed by EplugþElþEd,w h e r e Elis the localized energy dissipation at a vicin- ity of the direct impact area and Edrepresents delocalized energy dissi- pation beyond the direct impact area. The approximated kinetic energy transferred to a plug (as depicted in Fig. 12 ),Eplug, is supposed to equal qhApv2 plug=2, where q,h,a n d vplugare the specimen’s mass density, thickness, and velocity of a plug, respectively. The direct kinetic energy transfer to a plug of a specimen has been observed in penetration of free-standing thin-films of polystyrene,3,148polycarbon- ate,149multilayer graphene,110,150and graphene nanocomposites151 with hard, solid silica projectiles. A material’s physical responses become localized near the projectile when (i) the projectile speed is )b( )a( Debris Radially propagating cracksPrimarily localized deformation Primarily delocalized deformation Rounded cone Radial tensionTangential tensionFIG. 12. Schematic illustration of energy dissipation mechanisms during projectile perforation of materials primarily throughlocalized (a)3and delocalized response (b).110(a) Reprinted with permission from Hyon et al. , Mater. Today 21, 817 (2018). Copyright 2018 Elsevier. (b) Reprintedwith permission from Lee et al. ,S c i e n c e 346, 1092 (2014). Copyright 2014 The American Association for the Advancement of Science. –15 –10 –5 0 5 10 15(c) (a) (b) 10–41 10–2100102104106108 Strain Rate (s–1) Distance from Impact Center ( mm) Hardness (GPa)Height ( mm)10 Cu Fe LIPIT 0 –2 –42 CuFeCu Fe 5 mm FIG. 11. Dynamic impact hardness determination. (a) Post-impact craters and (b) surface after alumina-particle impacts on a copper substrate at 425 m/s and a n iron substrate at 657 m/s. (c) High-rate dynamic hardness as a function of deformation rate.147Reprinted with permission from Hassani et al. , Scr. Mater. 177, 198 (2020). Copyright 2020 Elsevier.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011319 (2021); doi: 10.1063/5.0040772 8, 011319-13 Published under license by AIP Publishingfast or comparable to the propagation speed of deformation ( /ffiffiffiffiffiffiffiffi E=qp orffiffiffiffiffiffiffiffiffi G=qp ) and (ii) the onset of material failure at the direct impact area (or projectile’s cross-sectional area Ap)i sf a s t e r . It was shown that for polymers having relatively slow propaga- tion speeds of deformation Epcan be effectively increased by enhanc- ingEl. A higher entanglement density and a higher molecular weight of polymers can delay the onset of brittle fracture at the direct impact area148and localized adiabatic heating can create a high rate visco- plastic flow around the perimeter. Thus, further plastic deformationaround the periphery of the projectile increases E l[Fig. 12(a) ].3,149In the case of elastic materials having high tensile strength and a low den-sity,E pcan be increased via enhancing Edand localized plastic defor- mation is relatively insignificant. For example, multilayer graphene (MLG), having an exceptionally fast propagation speed of deformation(tensile wave speed of 22 km/s), can rapidly transfer the projectile’skinetic energy to neighboring areas through the impact-induced conicdeformation [ Fig. 12(b) ]. 110Although this energy transfer is not energy dissipation through heat but rather energy delocalization, thetransferred energy has been suggested to eventually dissipate through various secondary processes such as cracking, folding, crease, andaerodynamic friction. Due to the high temporal (sub 100 ns) and spatial resolutions (sub 100 nm) required for the quantitative analysis of deformation inLIPIT studies of nano-scale materials, the real-time characterization ofdeformation has been challenging. However, because post-impact damage features at the vicinity of an impact or penetration region also provide essential information about characteristic deformation modesand energy dissipation mechanisms, SEM studies of post-LIPIT speci-mens are generally conducted. Figure 13 catalogs several representative damage features originating from relevant energy dissipation mecha- nisms. In a glassy polymer system such as polystyrene (PS), because E p is increased by higher fracture toughness and higher molecular weight, it was shown that PS can dissipate more energy via more extendedradial crazes [ Fig. 13(a) ]. In contrast, polycarbonate (PC) has signifi- cantly higher entanglement density than PS does, yielding a primary energy dissipation mechanism of PC, which can be seen from craze- (a) Molecular weight effect (polystyrene) (b) Entanglement effect (polystyrene, polycarbonate) (c) Orientation effect (ordered block copolymer) (d) Delamination effect (multi-layer graphene) (e) Phase percolation effect (graphene-oxide/silk)10 kg/mol 200 kg/mol PS PC m-projectile Granular10 mm 10 mm 2 mm 2 mm 10 mm 2 mm 500 nm 0 vol% 32 vol% 3 mm 3 mm 500 nm 500 nm1 mm m-projectile FIG. 13. Post-impact damage and deformation features of various materials. (a) Dense radial and tangential crazes appear around the penetration hole of a hig h molecular weight polystyrene (PS) freestanding membrane.148Reprinted with permission from Xie et al. , Macromolecules. 53, 1701 (2020). Copyright 2020 American Chemical Society. (b) Because of higher entanglement density, polycarbonate (PC) shows yield-dominant deformation compared to the craze-dominant features of PS.3Reprinted with permission from Hyon et al. , Mater. Today 21, 817 (2018). Copyright 2018 Elsevier. (c) Deformation features of block copolymer PS-b-PDMS lamellae when the impact direction is parallel (left) or perpendicular (right) to the plane of the lamellae.108Reprinted with permission from Lee et al. , Nat. Comm. 3, 1164 (2012). Copyright 2012 Springer Nature. (d) Higher energy dissipation of MLG is observed when delamination occurs in the penetration process.150Reprinted with permission from Xie and Lee., ACS Appl. Nano Mater. 3, 9 (2020). Copyright 2020 American Chemical Society. (e) When graphene-oxide flakes are added over the percolation threshold (32 vol%), smaller penetr ation damage and larger energy dissipation are possible.151Reprinted with permission from Xie et al. , Nano Lett. 18, 987 (2018). Copyright 2018 American Chemical Society.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011319 (2021); doi: 10.1063/5.0040772 8, 011319-14 Published under license by AIP Publishingless plastic deformation in Fig. 13(b) . For an anisotropic material with long-range ordering, the deformation responses are greatly affected by the angle of impact direction relative to the structural orientation. For example, a bulk lamellar nanocomposite consisting of glassy and rub-bery layers dissipates impact energy via strong reorientation eventswhen the impact direction is parallel to the layers, while extremelamellar compression is a dominant energy dissipation mechanism forthe perpendicular orientation [ Fig. 13(c) ]. Moreover, the nanocompo- site of the perpendicular orientation demonstrated 30% shorter pene- tration depth compared to the perpendicularly orientated case,indicating higher energy dissipation performance. Compared to thepolymers, high strength elastic nanomaterials such as MLG do notshow considerable plastic deformation features but a relatively smallnumber of straight radial cracks (typically fewer than six), propagating a long distance, multiple times the projectile’s diameter [ Fig. 13(d) ]. The long cracks and large penetration opening imply that extensiveconic deformation was made via impact energy delocalization [ Fig. 13(b) ]. Because the energy delocalization is achieved via prompt coni- cal deformation, the presence of the surrounding fluid (i.e., air)restricts the conic deformation and reduces E d. Thus, MLG under vac- uum can increase its energy delocalization performance to be 3 times higher than in air.150Despite the outstanding performance of MLG in enhancing Ed, due to its elastic and crystalline nature, MLG has the typical weakness of crystalline ceramics: low fracture toughness andhigh susceptibility to local defects. As a result, the onset of structuralfailure at the impact region is sensitive to local defects, and this local failure progresses to a global failure through rapidly propagating cracks. However, the weakness of MLG can be noticeably decreased byquasi-plastic deformation. As seen in Fig. 13(d) , when the initiation of cracks at the impact region is suppressed by quasi-plastic deformationthrough localized folding, interlayer sliding, and wrinkling of delami- nated layers of MLG, improved delocalization is possible. 150 Furthermore, one can envision the maximization of Epvia the combi- nation of local plasticity and global elasticity. Indeed, nanocompositesof plastic polymers and elastic nanomaterials are suggested to accom-plish the simultaneous enhancement of E landEd.151The nano-scale laminates comprised of alternating layers of silk fibroin and graphene oxide flakes exhibit enhanced hybridized dynamic responses when graphene oxide flakes reach their percolation threshold over whichimpact stresses can transmit through the continuous phase ofgraphene oxide flakes. Moreover, the fracture toughness at the impact region is also improved, as observed from a remarkably smaller mass loss [ Fig. 13(e) ]. Besides the membrane-like (2D) and the semi-infinite (3D) speci- mens, fiber-like (1D) specimens has also been characterized by LIPIT.Compared to 2D geometries, because precise measurement of instan-taneous positions of a projectile is possible via ultrafast stroboscopicimaging while interacting with a specimen, energy dissipation and force exchange can be quantified by first and second derivatives of the time-dependent positions ( Fig. 14 ). Using LIPIT, the collective dynam- ics of a CNT fiber (an ensemble of weakly interacting, aligned CNTs)were directly compared by Xie et al. to nylon, Kevlar, and aluminum monofilament fibers under the same supersonic impact conditions. 115 Although individual CNTs are an elastic strain-rate-insensitive mate- rial, strain-rate-induced strengthening arising from interactions between the individual carbon nanotubes was observed. Moreover, asthe CNT fiber surpassed the other three conventional fibers includingKevlar in specific energy absorption under the equal conditions, thedemonstrated performance of the CNT fiber at the microscopic scalecan also be realized in the scaled-up conditions. B. Projectile response In a reversed ballistic configuration, the dynamic yield strength of a projectile can be determined by analyzing its residual geometry after an impact onto a rigid flat target. This is the basis of the experi- ment conducted by Taylor 19using cylindrical projectiles during World War II. The Taylor impact test produces a non-uniform plastic defor-mation with the front part of the projectile crumpling upon impactand the rear part remaining undeformed. Considering the propagationof the elastic and the plastic waves into the projectile, Taylor explained the residual deformation and demonstrated that the shortening of the cylinder is proportional to the square of the impact velocity and theinverse of the dynamic yield strength. The original development wassubsequently extended by other researchers, 152–158making the Taylor impact test one of the main experimental approaches to determine theconstitutive behavior of materials at high strain rates. While classically conducted at macroscales, the Taylor impact test has inspired LIPIT researchers to study mechanics and materials phenomena withTaylor-like impact experiments at microscales. The decrease in the Time (ns)Force (mN))b( )a( (d)(e) (c) FIG. 14. In situ measurement of force exchange between a micro-projectile and a fiber specimen. Ultrafast stroboscopic micrographs of (a) pure aluminum, (b) nylon 6, 6, (c) Kevlar KM2, and (d) CNT fibers under microsphere impact around 500 m/s. (e) Instantaneous forces between the microspheres and the fibers are measured by tracking the positions of the projectiles.115Reprinted with permission from Xie et al. , Nano Lett. 19, 3519 (2019). Copyright 2019 American Chemical Society.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011319 (2021); doi: 10.1063/5.0040772 8, 011319-15 Published under license by AIP Publishingprojectile size from millimeters in the classic Taylor impact test to micrometers in LIPIT while maintaining the same level of impact velocities comes with the benefit of extending the achievable strain rates, form nominally /C24105in the former to /C24108s/C01in the latter. Figure 15(a) ,f r o mX i e et al. ,111shows the flattening of five ini- tially spherical polycrystalline Al 6061 microparticles as a result of impact onto a rigid sapphire target at various velocities. Post-impact characterization of the microstructures revealed localized deformation at the bottom of the particles along with insignificant plastic deforma- tion on the top. In the presence of such non-uniform plasticity, the analytical determination of material’s constitutive parameters requires significant simplifying assumptions. Finite element simulations, on the other hand, can be employed in conjunction with the experimentally measured profiles of the deformed particles for more precise and reli- able constitutive modeling. Xie et al.111and Chen et al.159used post- impact measurements of the flattening ratio of particles in a range ofimpact velocities, from 50 to 950 m/s to optimize a bilinear Johnson- Cook material model. It was found that, for a successful prediction of the deformed profile, the strain rate sensitivity parameter in theJohnson-Cook equation should be increased by an order of magnitude, from 2 /C210 /C03at low strain rates to 2.9 /C210/C02at strain rates higher than 600 s/C01. Wang et al.160also reported that the Johnson–Cook strain rate sensitivities measured by Klosky bar experiments lead to relativelyinaccurate deformation when applied to modeling LIPIT impacts. They used in situ observations of the deformation of commercially pure Ti particles impacting a rigid alumina substrate, such as shown in the snapshots of Fig. 15(b) , for constitutive modeling. Two material models namely, Johnson–Cook and Zerilli–Armstrong, were implemented in iterative finite element simulations combined with an optimization scheme to generate matching deformed profiles with the experiments [see Fig. 15(c) ]. Close to a fivefold increase in the Johnson–Cook strain rate sensi- tivity was found to be necessary to predict impact-deformation of Ti microparticles. The Zerilli–Armstrong model was found to havea better performance than the Johnson–Cook model, which was attributed to the micromechanical origin of the former compared to the empirical nature of the latter. Wang et al. 160also developed t = 0 ns(d) ((a) (b) 50 ns 100 ns 150 ns 200 ns 250 ns 300 ns 350 ns 400 ns 450 nsJC SimulationExperiments ZA Simulation X coordinate ( mm) Y coordinate ( μm)(c)180 m/s 530 m/s660 m/s290 m/s 420 m/s 50 mm FIG. 15. (a) Scanning electron microscope images of Al 6061 particles after impact onto sapphire showing the deformed shapes and flattening of the particles in a range of impact velocities. Red dashed lines indicate outlines of numerically simulated deformed particles.111Reprinted with permission from Xie et al. , Sci. Rep. 7, 5073 (2017) under the terms of the Creative Commons Attribution 4.0 International License. (b) Real-time observations of the impact-induced deformation of a 48- lm commercially pure titanium particle impacting a rigid alumina substrate at 190 65 m/s. The plot (c) next to the snapshots compares the experimentally measured deformed profile of the particle in (b) with the finite element simulations with optimized Johnson-Cook and Zerilli-Armstrong material models.160Reprinted with permission from Wang et al. , J. Appl. Mech. 87, 091007 (2020). Copyright 2020 ASME. (d) Cross-sectional bright field transmission electron microscope image of a silver microcube after a /C24400 m/s impact along the /C24[100] direction. Inset shows the impacted microcube from the top with a dashed line indicating the location of the cross section. Selected area diffra ction patterns confirm a variation from single-crystalline structure in the top region to the nanocrystalline structure at the bottom region.161Reprinted with permission from Thevamaran et al. , Science 354, 312 (2016). Copyright 2016 The American Association for the Advancement of Science.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011319 (2021); doi: 10.1063/5.0040772 8, 011319-16 Published under license by AIP Publishinga scheme to determine the exact strain rates regime experienced during LIPIT impacts by computing the incremental plastic work and the strain rate for each volume element at every time interval. For a Ti particle impact at 190 m/s, it was found the entire plasticwork is done at strain rates beyond 10 6s/C01. While the majority of the plastic deformation (almost 70%) occurs at the strain rate of /C24108, strain rates exceeding 109s/C01were also found in a smaller fraction (about 8.5%). The higher strain rate sensitivities found at the LIPIT-associated strain rates111,159,160compared to Klosky bar calibrations can be mech- anistically explained by the thermally activated motion of dislocations as carriers of plasticity.162The probability of dislocations to overcome barriers and obstacles in this mechanism follows an Arrhenius typerelation and decreases at higher strain rates. As a result, higher flowstresses are required to maintain plastic deformation, giving rise tostrain rate strengthening. What is more, the interactions of a moving dislocation with phonons and electrons at higher strain rates produce non-negligible drag forces, which in turn provide an additionalstrengthening mechanism. In addition to constitutive modeling, Taylor-like impact tests with LIPIT have been successfully used for studies of microstructural evolution and phase transformation under extreme conditions.Thevamaran et al. 161have used near-defect-free single crystals of Ag in near-perfect cubic geometry as projectiles and Si as rigid substrate. Figure 15(d) shows the cross-sectional transmission electron micros- copy analysis of the microstructure of an Ag micro-cube after animpact at /C24400 m/s along the [100] direction. A strong gradient in grain size can be observed along the height of the deformed micro- cube. The selective area diffraction patterns confirm a gradient micro- structure, from a single crystal on the top to nanocrystalline at thebottom. The grain sizes on the top and at the bottom were measuredto be/C24500 nm and /C2410 nm, respectively, demonstrating a LIPIT- induced gradient that is at least an order of magnitude steeper than the typical gradients produced by the more conventional surface mechanical grinding and surface mechanical attrition. The extremegrain refinement at the bottom of the projectile was attributed to ashock wave–induced severe plastic deformation that together with theadiabatic heat generation resulted in dynamic recrystallization. In another study, Thevamaran et al. 163found a martensitic phase transformation from a face-centered-cubic (fcc) structure to a hexago-nal-close-packed (hcp) structure during the impact of Ag microcubes at/C24400 m/s. The phase transformation was found to be orientation depen- dent. While impact along the [100] direction resulted in the martensitic phase, impact along the [110] direction did not trigger any phase Al on Ti 10 mm 10 μmZn on Zn 5 mm 10 mm Cu on Al100 mm 100 mm + 20 ns + 20 ns + 20 ns + 20 ns + 20 ns + 100 ns+ 100 ns+ 150 ns+ 150 ns + 10 s + 10 ns + 10 ns + 10 ns + 10 ns + 10 ns + 10 ns + 10 ns + 10 ns + 100 ns + 10 s(a) (c) (e)(b) Coefficient of Restitution0.000.050.100.15 0 200 400 600 800 1000 Impact Velocity (m/s)Al: 14 μm ± 2 μm VcrAl: 30 μm ± 7 μm Vcr 0 ns 50 ns 100 ns 150 ns 200 ns 250 ns 300 ns 350 nsW/Cu Cu/WTi/Cu Sn/CuCu/Al Cu/TiPenetration Splatting Al/Ni Al/Ti Al/CuNi/Ti Ni/Al Ni/Cu Ti/NiNi/Ni Ti/TiCu/Cu Sn/SnAl/Sn Al/AlAl/ZnZn/ZnAu/Au Ag/Ag R = 1/2 R = 2 R = 1 103 102 101 101102103 ρC02Yd(substrate) (MPa2)ρC02Yd(substrate) (MPa2)(d) FIG. 16. (a) In situ observation of the bonding moment in microparticle impact. Multi-frame sequences showing 45- lm Al particle impacts on Al substrate at 605 m/s (top) and 805 m/s (bottom), respectively, below and above critical velocity. The slower particle rebounds while the faster one bonds to the substrate. Materia l jetting is indicated with white arrows in the latter.116(b) Coefficient of restitution for Al particles of two different sizes as a function of impact velocity.116The drop of the coefficient of restitution to zero indi- cates the transition from rebound to bonding. Reprinted with permission from Hassani-Gangaraj et al. , Scr. Mater. 145, 9 (2018). Copyright 2018 Elsevier. (c) Exemplar deformed geometries of bonded particles in the three regimes of splatting, co-deformation, and penetration.179(d) Predictive map for impact-bonding regimes for similar and dissimilar materials with black data points indicating co-deformation, red data points penetration, and blue data points splatting. The diagonal l ine cutting the map into two cor- responds to an impact ratio of unity around which co-deformation is the operative mode. Dotted lines correspond to ratios of1=2and 2 where transitions from co-deformation to splatting and penetration occur, respectively.179Reprinted with permission from Hassani et al. , Acta Mater. 199, 480 (2020). Copyright 2020 Elsevier. (e) Multi-frame sequences showing a 10- lm tin particle impacting a tin substrate at 1067 m/s and the resultant splashing and material loss in the erosion regime.180Reprinted with permission from Hassani-Gangaraj et al. , Nat. Comm. 9, 5077 (2018) under the terms of the Creative Commons Attribution 4.0 International License.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011319 (2021); doi: 10.1063/5.0040772 8, 011319-17 Published under license by AIP Publishingtransformation. Molecular dynamic simulations showed that the impact-induced shock is the key driving force for generating avalanches of partial dislocations for the [100] impact, which, upon merging at the middle of the cube, initiate the phase transformation. The lack of abun-dant dislocation emissions and thus phase transformation in the [110]was attributed to a relatively low impact-induced hydrostatic stress. C. Synergistic response The synergistic deformation of metallic microparticles impacting onto metallic substrates can give rise to the conditions necessary for solid-state bonding. Successive impact-induced bonding at the micro- scale is now routinely exploited for deposition of coatings, structural repair, and additive manufacturing via cold spray process. The same conditions can be induced in a LIPIT experiment enabling fundamen- tal studies of impact-bonding that are otherwise challenging to con-duct using cold spray nozzles. Our intention in this section is not to overview the progress made in in the past decades in the field of cold spray, which have already been the subject of several books 164–168as well as succinct and thorough reviews.169–175Instead, our focus is to discuss how, in recent years, LIPIT-based experiments have helped extend the understanding of the mechanical and materials phenomena relevant to cold spray. A notable feature of the LIPIT is the ability to watch the interac- tion of a single particle and a substrate in situ and to develop under- standing therefrom. Hassani et al.116resolved the moment of impact bonding in real time with micro-scale and nanosecond-level spacio- temporal resolution. The snapshots in Fig. 16(a) demonstrate the tran- sition from rebound to bonding with increasing impact velocity for Alparticles impacting an Al substrate. The particle accelerated to a veloc-ity below the critical velocity for bonding rebounds despite a signifi- cant level of plastic flattening. The second particle accelerated to a velocity slightly above the critical bonding velocity adheres to the sub-strate with distinct unstable jetting and material ejection. These real-time observations confirm the critical role of jetting in impact-bonding and are the point of departure to follow-up discussions on the shock-induced nature of jetting and a proportionality between the critical velocity and the spall strength. 176–178The large interfacial strain from jetting displaces and fractures the surface oxide layer and createsfresh metallic surfaces while the impact-induced pressure brings parti-cle and substrate surfaces into intimate contact to form metallurgical bonding. The separation between the two regimes of rebound and bonding can be precisely identified when plotting the coefficient of restitution asa function of impact velocity. Figure 16(b) is an exemplar plot for Al particles impacting Al substrates. 116As impact velocity increases, more fraction of the kinetic energy of the incoming particles are dissipated by plasticity until a threshold where the coefficient of restitution drops tozero. Sun et al. attributed the deviation from the gradual decrease in the coefficient of restitution near the critical bonding velocity to the energydissipated by jetting. 185,186Critical velocities for 20 different combina- tions of particle and substrate materials were directly measured using the coefficient of restitution plots and reported in Ref. 179. Figure 16(b) also shows that particle size can affect the critical velocity for bonding.116Dowding et al.183conducted a focused study of the size effect on the critical velocity for Al and Ti particles impact- ing matched material substrates. It was found that as particle sizeincreases by a factor of /C244, Al and Ti critical velocities decrease by /C2425%. The power law scaling between the critical velocity and the par- ticle size measured via LIPIT experiments was comparable to those measured via cold spray nozzle experiments.184The mechanistic origin of the size effect is attributed to the fact that the higher kinetic energies carried by larger particles cause more local heating at the contact inter- face. Subsequently the local resistance of material to spallation decreases at higher temperatures. Dowding et al.184used the proposed proportionality between the critical velocity and the spallstrength 176,178to predict the size effect and reported a reasonable agreement between the theory and the experimental measurements. Jetting and material ejection were also observed during impact- bonding of gold particles on a gold substrate.185With gold not having a tendency to form a native oxide layer at the conditions of the experi- ment, this observation confirms that the spall fragments are ejected from the base metal. Nevertheless, the thickness of the native oxide layercan significantly affect the critical bonding velocity. Lienhard et al. 186 exposed Al powder particles to 300/C14Cf o ru pt o2 4 0m i ni nd r ya i r ,o rt o room-temperature with humidity levels as high as 50% for 4 d, and con- sequently conducted LIPIT experiments. No significant differences were found in terms of the critical bonding velocity, which was attributed to the similar characteristics of the passivation layer (thickness, uniformity, crystallinity, and composition) in different powder batches. An approxi- mately 14% increase in the critical bonding velocity, on the other hand, was measured for powder particles exposed to 95% relative humidity for 4 d. The latter exposure resulted in a 2–3 nm increase in the thickness of the passivation layer as well as changes in the chemistry and structure of that layer. The increase in the critical bonding velocity was theoretically justified by considering the proportionality between the critical velocityand the spall strength. 176,178 The geometry of particle and substrate interfaces is an important factor in determining the bond strength and can significantly vary from one pair of particle and substrate materials to another. Hassani et al.179conducted LIPIT experiments on nine dissimilar particle/sub- strate material pairs in addition to their previous matched particle/ substrate material impacts116,178,180and identified two limiting impact-bonding modes, namely, splatting and penetration. In both cases it is one material that accommodates a significant fraction of the impact-induced plasticity, i.e., the particle material in the splatting mode and the substrate material in the penetration mode. In between these two limits, lies a regime of co-deformation where a similar level of plastic deformation occurs in both particle and substrate materials upon impact-bonding. Shown in Fig. 16(c) are typical examples of the bonded particles in the three regimes of splatting, co-deformation, and penetration based upon which a spectrum of impact modes was pro- posed.179A dimensionless ratio—a function of density, bulk speed of sound, and dynamic yield strength of particle and substrate—was pro- posed to quantify the spectrum. Figure 16(d) shows a predictive map of impact-bonding modes based on the proposed ratio with the experi- mental observations also populated on the map. Co-deformation was found to occur for the material pairs with ratios around unity, whilesplatting and penetration are dominant for ratios smaller than 1=2and larger than 2, respectively. The timescale of deformation in metallic microparticle impact (usually on the order of 10–100 ns) is shorter than the time needed for the plasticity-induced heat to be conducted away from the interface. The highly adiabatic nature of the deformation can, as a result, lead toApplied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011319 (2021); doi: 10.1063/5.0040772 8, 011319-18 Published under license by AIP Publishinglocalized interfacial melting at high impact velocities, which in turn can interfere with solid-state bonding. In fact, the localized melting at substrate surfaces in Al particle impact on Zn and Sn was shown to hinder bonding.187The resolidification of the molten layer takes orders of magnitude longer than the time that the particle resides on the sur-face of the substrate. In other words, when localize melting occurs the particle is mostly in contact with a liquid interface whose low mechan- ical strength is easily overcome by a rapidly rebounding particle. A major fraction of the impact-induced plastic work dissipates as heat. If the heat is high enough to melt materials beyond the interfacialregions, then impact can cause material loss or erosion. The phenome- non, resolved by Hassani et al. , 180i ss h o w ni nr e a lt i m ei nt h es n a p - shots of Fig. 16(e) where a Sn particle impacts a Sn substrate at /C241 km/s. A splash with a cloud of ejecta is observed leading to a loss of material on the order of /C24100lm3. While the critical bonding velocity represents the transition from the rebound to the bonding regime, the threshold velocity to induce material loss, the so-callederosion velocity, marks a second transition from the bonding to theerosion regime. The critical and the erosion velocities together definethe lower and the upper bounds to the deposition window, i.e., the range of impact velocities that lead to successful material buildup in cold spray. Erosion maps developed based on the mechanistic originof material loss can be used to calculate the erosion velocity for givenparticle/substrate materials and a given particle size. 180 IV. PERSPECTIVES There are emerging opportunities in materials science that involve utilizing microscopic high-speed impact of a microparticlewith precisely defined collision parameters and geometries (projectile material, size, shape, temperature, impact velocity, impact angle), as a highly localized and quantifiable mechanical stimulus of high strainrates becomes available by LIPIT. While the momentum associatedwith the microscopic single impact is enough to produce HSR plastic deformation of a specimen, the resultant deformation volume ( /C241000 lm 3) is still small enough to be considered as a nondestructive charac- terization that can be combined with most high-resolution characteri-zation techniques. By virtue of this unique aspect of LIPIT, HSRelasto-plastic and visco-plastic behaviors of various materials, includ- ing metals, polymers, and nanomaterials, have been investigated, which would be difficult by other means. In contrast to the well-quantified kinetic parameters in LIPIT, the measurement of materialsstate parameters such as pressure, temperature, stress, and strain remain a challenge. Therefore, we foresee two major avenues for devel- opment of HSR studies using microparticle impacts. First, most techniques presented in Sec. IIrely on high-speed imaging for both launch characterization (particle size, speed, distribu-tion, etc.) and sometime impact response diagnostic (real-time imag- ing of penetration, target damage). While postmortem examinations inform greatly about the impact history, real-time measurements rep-resent the ultimate goal. Currently, laser-flyer techniques, compared toother methods, can leverage their optical nature to implement the most advanced synchronized optical diagnostics. These advanced diagnostics, such as optical pyrometry, real-time Raman spectroscopy,or femtosecond X-ray scattering measurements, 188have yet to be applied for LIPIT studies. Second, remaining challenges may be difficult to overcome solely with the advance of LIPIT, due to the strong locality of the transientfundamental quantities. The next major step in the understanding of the fundamental HSR materials properties is expected to be made through the synergistic combination of LIPIT and corresponding numerical modeling, where realistic HSR material parameters will beattained from the real-time LIPIT data, including projectile dynamicsand postmortem characterizations. For example, the ability to studyplastic deformation of projectiles both in situ and postmortem has provided LIPIT with unique potentials for Taylor-like impact experi-ments and constitutive models development and refinement at ultra-high strain rates. Further, as illustrated in Fig. 9 , no method exists for single micro- particle acceleration to velocities of the order of 10 km/s or higher.Reaching such velocities will open the door to more systematic studiesin aeronautics and space science. The LIPIT apparatus is not limitedby laser energies but rather by the resistance of launch pad to plasmaexpansion and by the resistance of the particle to acceleration.Through optimization of the launch pad design, using tougher poly- mers, for example, higher velocities might be reachable, hence posi- tioning LIPIT as a competitor in terms of velocity performance toplasma drag acceleration techniques. With higher velocities, the LIPITtechnique could find use in the study of materials and structures forhypersonic applications. New materials, such as nano-structuredrefractory alloys and ceramic composites, could be screened for perfor-mance in hypersonic environments and velocities. In addition, slightly larger samples could be impacted with particles launched to hyperve- locity by the LIPIT technique and atmospheric erosion and damagecould be characterized. The range of materials will continue to expand as novel nanoma- terials and architectures are designed and new physics are yet to beexplored under high rate dynamic conditions. For instance, LIPIT hasproven to be an effective platform for fundamental studies of the proc- essing science of cold spray. The strength of LIPIT in this context stems from the fact that it decouples the multivariable influence in acomplex deposition problem and enables studies of each variable at atime. While the effects of particle size, native oxide layer, alloying ele-ments, and mismatched material pairs on impact-bonding have beenisolated and studied through LIPIT, the role of several other parame-ters such as impact angle, particle shape, particle/substrate tempera- ture, and surface roughness are yet to be studied. The research efforts on this track have so far mainly focused on the impact behavior ofmetallic particles and substrates. With the rapid extension of kineticdeposition beyond metals and alloys, understanding impact and bond-ing in cases where particle and substrate belong to different classes ofmaterials with substantially different properties would be highly desir-able. Examples include metallic particles’ impact on polymers orceramic particles’ impact on metals. During material buildup, impacts of particles onto particles are by far more probable than impacts of particles onto the substrate. Therefore, studies of particle-particleinteractions also would be desirable. SUPPLEMENTARY MATERIAL The data points used to plot Fig. 9 are available as supplementary material online. AUTHORS’ CONTRIBUTIONS All authors contributed equally to this manuscript. All authors reviewed the final manuscript.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011319 (2021); doi: 10.1063/5.0040772 8, 011319-19 Published under license by AIP PublishingACKNOWLEDGMENTS D.V., S.E.K., and K.A.N. acknowledge funding support from the U.S. Army Research Office through the Institute for SoldierNanotechnologies, under Cooperative Agreement No. W911NF-18-2-0048. J.H.L. acknowledges funding support from the U.S.Department of Defense under Cooperative Agreement No.HQ0034-15-2-0007. M.H. acknowledges funding support from theU.S. Army Research Laboratory through a Cooperative ResearchAgreement No. W911NF-19-2-0329. The authors declare no conflict of interest. DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material . REFERENCES 1Y. T. Cheng and C. M. Cheng, “Scaling, dimensional analysis, and indentation measurements,” Mater. Sci. Eng. R Rep. 44(4–5), 91–149 (2004). 2Z. P. Ba /C20zant, “Scaling laws in mechanics of failure,” J. Eng. Mech. 119(9), 1828–1844 (1993). 3J. 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5.0029146.pdf

J. Chem. Phys. 153, 214107 (2020); https://doi.org/10.1063/5.0029146 153, 214107 © 2020 Author(s).Calculation of spin–orbit couplings using RASCI spinless one-particle density matrices: Theory and applications Cite as: J. Chem. Phys. 153, 214107 (2020); https://doi.org/10.1063/5.0029146 Submitted: 10 September 2020 . Accepted: 11 November 2020 . Published Online: 02 December 2020 Abel Carreras , Hanjie Jiang , Pavel Pokhilko , Anna I. Krylov , Paul M. Zimmerman , and David Casanova COLLECTIONS Paper published as part of the special topic on Up- and Down-Conversion in Molecules and MaterialsUAD2020 ARTICLES YOU MAY BE INTERESTED IN From orbitals to observables and back The Journal of Chemical Physics 153, 080901 (2020); https://doi.org/10.1063/5.0018597 Top reviewers for The Journal of Chemical Physics 2018–2019 The Journal of Chemical Physics 153, 100201 (2020); https://doi.org/10.1063/5.0026804 Electronic structure software The Journal of Chemical Physics 153, 070401 (2020); https://doi.org/10.1063/5.0023185The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Calculation of spin–orbit couplings using RASCI spinless one-particle density matrices: Theory and applications Cite as: J. Chem. Phys. 153, 214107 (2020); doi: 10.1063/5.0029146 Submitted: 10 September 2020 •Accepted: 11 November 2020 • Published Online: 2 December 2020 Abel Carreras,1 Hanjie Jiang,2 Pavel Pokhilko,2,3 Anna I. Krylov,3 Paul M. Zimmerman,2,a) and David Casanova1,a) AFFILIATIONS 1Donostia International Physics Center (DIPC), Manuel de Lardizabal Pasalekua 4, 20018 Donostia, Euskadi, Spain 2Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, USA 3Department of Chemistry, University of Southern California, Los Angeles, California 90089, USA Note: This paper is part of the JCP Special Topic on Up- and Down-Conversion in Molecules and Materials. a)Authors to whom correspondence should be addressed: paulzim@umich.edu and david.casanova@ehu.eus ABSTRACT This work presents the formalism and implementation for calculations of spin–orbit couplings (SOCs) using the Breit–Pauli Hamiltonian and non-relativistic wave functions described by the restricted active space configuration interaction (RASCI) method with general excitation operators of spin-conserving spin-flipping, ionizing, and electron-attaching types. The implementation is based on the application of the Wigner–Eckart theorem within the spin space, which enables the calculation of the entire SOC matrix based on the explicit calculation of just one transition between the two spin multiplets. Numeric results for a diverse set of atoms and molecules highlight the importance of a balanced treatment of correlation and adequate basis sets and illustrate the overall robust performance of RASCI SOCs. The new implementation is a useful addition to the methodological toolkit for studying spin-forbidden processes and molecular magnetism. Published under license by AIP Publishing. https://doi.org/10.1063/5.0029146 .,s I. INTRODUCTION The time-independent electronic Schrödinger equation pro- vides a central framework for the theoretical description and anal- ysis of a vast variety of molecular phenomena. Regarded as the first-principles foundation of quantum chemistry, the Schrödinger equation is itself an approximation that assumes instantaneous Coulomb interactions between charged particles. Hence, it is valid in the non-relativistic limit of quantum mechanics. Relativistic effects, neglected in non-relativistic quantum-chemical treatments, affect chemistry in profound ways.1–3Although more pronounced in molecules composed of heavier elements, relativistic effects4are also important in light molecules. In particular, spin–orbit cou- pling (SOC) facilitates many processes that are forbidden in the non-relativistic description. SOC mixes electronic states of differ- ent spins that do not interact in the non-relativistic picture. Thisinteraction manifests itself spectroscopically as spin–orbit split- ting of energy levels and intensity redistribution (intensity bor- rowing). For example, the electronic transition responsible for the purple color of iodine is formally spin-forbidden but becomes allowed due to strong spin–orbit interaction.3SOC also facilitates non-radiative and radiative transitions between different spin man- ifolds, leading to inter-system crossing (ISC)2and phosphores- cence.5These phenomena open spin-forbidden reaction and relax- ation pathways, which are relevant in astrochemistry, combustion chemistry, electronics, and catalysis.6–11In single-molecule mag- nets,12–14SOC introduces magnetic anisotropy and spin-reversal barriers, determining the relaxation time of magnetization, a cru- cially important property in practical applications of these magnetic materials. In molecules composed of light atoms, SOC can be described by using the so-called perturbative approach, i.e., by computing J. Chem. Phys. 153, 214107 (2020); doi: 10.1063/5.0029146 153, 214107-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp matrix elements of the Breit–Pauli Hamiltonian15,16using non- relativistic wave functions and diagonalizing the resulting matrix to obtain spin–orbit perturbed states. The set of non-relativistic wave functions form a spin–orbit diabatic representation; they can be described as zero-order states. The perturbed states, resulting from the diagonalization, form a spin–orbit adiabatic representa- tion. Effective algorithms17–19for the calculation of SOC matrix ele- ments often exploit the Wigner–Eckart theorem,20,21which enables the evaluation of the entire SOC matrix between two multiplets from just one transition, for example, between a singlet state and the M= 0 component of a triplet state. Recently, Pokhilko et al.19 reported an algorithm for evaluating SOCs by application of the Wigner–Eckart theorem to the reduced spinless density matrix gen- erated for one specific transition. Formulated in the spin-orbital form,22this approach is ansatz-agnostic and is applicable to tran- sitions between any types of open-shell states, as long as respective transition density matrices can be obtained. Pokhilko et al. applied this approach for computing SOCs using equation-of-motion coupled-cluster (EOM-CC) wave func- tions23within the Q-Chem electronic structure package.24Recently, this approach was extended25to compute SOCs between core- level states using EOM-CC augmented by core–valence separa- tion.26This algorithm has also been adopted by Meitei et al.27in a pilot RASCI (restricted active space configuration interaction) code within Psi4.28Here, we extend the implementation of Pokhilko et al.19to a production-level implementation of the RASCI family of methods29within Q-Chem, which enables the calculation of a large variety of open-shell and electronically excited states. In addi- tion to the calculation of SOCs, we also implemented the calculation of natural transition orbitals (NTOs) of the spinless transition den- sity matrix (1TDM) to provide insight into the mechanism of the spin–orbit interactions.30,31 Among the various flavors of RASCI methods, the spin-flip (SF) variant, RASCI-SF or simply RAS-SF, is of special interest. The SF approach32,33extends single-reference methods to strongly correlated states, traditionally treated by multi-reference methods. SF treatment uses a carefully chosen high-spin reference state, in which all strongly correlated electrons have the same spin, and then employs spin-flipping operators to access the desired low- spin manifold, providing a low-cost yet accurate description of the states of interest. Among various variants of SF methods,34–40 the RAS-SF approach41has particular utility. In RAS-SF, a small active space is defined to include the most strongly correlated elec- trons (i.e., the high-spin reference electrons and orbitals), which are treated at the complete active space (CAS) level of theory. A small subset of electronic configurations outside of this space is added to ensure balanced treatment of states with different char- acters, as described below. Since its introduction in 2009, RAS-SF methods have been employed to treat a variety of phenomena fea- turing a challenging electronic structure. Examples include dirad- icals,42,43polyradicals,44–46excitonic states in multichromophoric systems,47,48catalysis,49singlet fission processes,50–61and charge transfer.62,63 Following the EOM-CC philosophy,23the RAS-SF approach has been generalized to employ other (non-SF) types of excitation operators, such as excitation energy (EE), ionization potential (IP), and electron attachment (EA) operators, giving rise to the RASCI family of methods.29The present work further extends the scope ofapplications of the RASCI hierarchy of methods by enabling calcula- tions of SOCs using any type of RASCI wave function. We describe a general framework for the calculation of SOCs with RASCI wave functions and illustrate the utility of the new approach by application to selected atoms and molecules. This paper is organized as follows. We begin by briefly review- ing the RASCI formalism and describing the key steps of computing SOCs with the Breit–Pauli Hamiltonian. The essential computa- tional details are given in Sec. III. Section IV illustrates the perfor- mance of RASCI SOCs in doublet radicals, atoms, small diradicals, conjugated ketones and alcohols, and along the molecular torsion of tetramethyleneethane. Section V summarizes the main findings of this study. II. THEORY A. The RASCI method The RASCI ansatz41,64is defined by choosing a reference con- figuration ( Φ0, usually a Hartree–Fock determinant) and dividing the orbital space into three subspaces: RAS1, RAS2, and RAS3. The three RAS subspaces usually span the entire set of molecular orbitals, although the space can be reduced by freezing low-energy occupied and/or high-energy virtual orbitals. The RASCI wave function is produced by the action of an excitation operator that includes all possible electronic promotions within RAS2 ( ˆr0) and specified hole and particle excitations, i.e., configurations with vacancies ( holes ) and electrons ( particles ), in RAS1 and RAS3, respectively, ˆR=ˆr0+ˆrh+ˆrp+ˆrhp+ˆr2h+ˆr2p+ˆr2hp+⋯. (1) The configuration expansion within the RAS2 space is equivalent to full CI (FCI), whereas the truncated excitation set in RAS1 and RAS2 brings the essential electronic correlation while keeping the expan- sion compact (Fig. 1). The excitation operator connects the reference configuration Φ0with the target states { ΨI}, FIG. 1 . RASCI electronic configurations within the hole and particle approximation generated by the excitation operator ˆR=ˆr0+ˆrh+ˆrpacting on a reference Φ0. J. Chem. Phys. 153, 214107 (2020); doi: 10.1063/5.0029146 153, 214107-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp ∣ΨI⟩=ˆR∣Φ0⟩. (2) Several flavors of ˆRcan be used, i.e., excitation energy (EE), spin-flip (SF) operators performing single or multiple excitations of spin- α electrons to empty spin- βorbitals (or the other way around), as well as ionization potential (IP) and electron attachment (EA) opera- tors connecting the reference configuration with states with fewer and more electrons, respectively.29,65In particular, the action of a SF operator with ΔM<0 on a high-spin reference within the hole and particle approximation ( ˆR=ˆr0+ˆrh+ˆrp) has been successfully applied to a variety of strongly correlated systems, such as diradicals42,43and polyradicals,44–46multiexcitons in multichromophoric systems,47,48 and electronic states involved in singlet fission.50,51,53,54,56,57 The implementation of the RASCI ansatz within effective algo- rithms, such as analytic integral evaluation29,65and the resolution- of-identity (RI) approximation66resulted in effective computer codes capable of handling extended systems, such as carbon nan- otubes67and organic macrocycles.45,46,68 B. The Breit–Pauli spin–orbit Hamiltonian The Breit–Pauli spin–orbit Hamiltonian, originally introduced by Pauli,15,69is commonly employed in calculations of the spin– orbit interaction between the electronic states computed by non- relativistic quantum chemistry methods. In atomic units, the one- and two-electron spin–orbit terms of the Breit–Pauli Hamiltonian are HBP SO=1 2c2⎡⎢⎢⎢⎢⎣∑ ihSO(i)⋅s(i)+∑ i≠jhSOO(i,j)⋅(s(i)+ 2s(j))⎤⎥⎥⎥⎥⎦, (3) hSO(i)=∑ IZI r3 iI(riI×pi), (4) hSOO(i,j)=−1 r3 ij(rij×pi), (5) where cis the speed of light, riandpiare the coordinates and momentum of the ith electron, respectively, ZIis the atomic charge of the Ith nucleus, and rijandriIare the relative coordinates of electron iand electron jor nucleus I, respectively. The Breit–Pauli spin–orbit operator has one- and two-electron parts, which, in the second quantization form, can be written as follows:70 HSO=1 2c2⎡⎢⎢⎢⎢⎣∑ pqIpqa† paq+∑ pqrsJpqrsa† pa† qasar⎤⎥⎥⎥⎥⎦, (6) Ipq=⟨ϕp∣hSO(1)⋅s(1)∣ϕq⟩, (7) Jpqrs=⟨ϕp(1)ϕq(2)∣hSOO(1, 2)⋅(s(1)+ 2s(2))∣ϕr(1)ϕs(2)⟩, (8) where hSOandhSOOoperators are defined in Eqs. (4) and (5) and ϕp(i) denotes the pth spin-orbital with riandσispatial and spin coor- dinates, respectively. Details for computing one- and two-electron spin–orbit integrals ( IpqandJpqrs) can be found elsewhere.71Evaluation of the SOC between two electronic states requires the contraction of I pqandJpqrswith one- and two-particle transition density matrices. This expression can be reduced by invoking SOMF (spin–orbit mean-field) approximation,72 HSOMF=∑ pq[Ipq+∑ rsρrs(Jprqs−Jprsq−Jrpqs)]a† paq, (9) whereρrsis a density matrix of a vacuum determinant (see deriva- tion based on the separable part of the two-particle density matrix in Ref. 71), and with spin–orbit mean-field matrices defined as HSOMF pq=h1eSO pq+h2eSOMF pq , (10) h1eSO pq=Ipq, (11) h2eSOMF pq=∑ rsρrs(Jprqs−Jprsq−Jrpqs). (12) Expanding the contractions and integrating over spin variables in Eqs. (7) and (8), the mean-field Hamiltonian can be written as HSOMF=1 2∑ ¯pq[hSOMF L+,pqa† ¯paq+hSOMF z,pqa† paq+hSOMF z,¯p¯qa† ¯pa¯q+hSOMF L−,p¯qa† pa¯q], (13) where we assume restricted spin-orbitals, with the creation and annihilation of αandβspin-orbitals indicated with orbital indices without and with overbars, respectively. The spherical components L+andL−are defined as hSOMF L+=hSOMF x +ihSOMF y , (14) hSOMF L−=hSOMF x−ihSOMF y , (15) where hSOMF x ,hSOMF y , and hSOMF z denote the sum of one-electron and mean-field two-electron components.19If hSOMF z,pq=−hSOMF z,¯p¯q, (16) then Eq. (13) can be written in terms of triplet excitation operators as73 HSOMF=1 2∑ pq[hSOMF L+,pqT1,−1 pq+√ 2hSOMF z,pqT1,0 pq−hSOMF L−,p¯qT1,1 pq], (17) T1,−1 pq=a† ¯paq, (18) T1,0 pq=1√ 2(a† paq−a† ¯pa¯q), (19) T1,1 pq=−a† pa¯q. (20) J. Chem. Phys. 153, 214107 (2020); doi: 10.1063/5.0029146 153, 214107-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp In the case of restricted orbital spaces, condition (16) is satisfied by the one-electron terms, but for the two-electron mean-field part, it is ensured only for reference density matrices ρwith equal αand βoccupancies. The use of different spin-projected ρrsdensities in Eq. (8) results in hSOMF z,pq≠−hSOMF z,¯p¯q, introducing unphysical contribu- tions to the couplings. Only the densities from spin-pure states with the same number of αandβelectrons ( M= 0) guarantee the proper symmetry of the operator. 1. The Wigner–Eckart theorem in the space of tensor operators The evaluation of Eq. (17) can be implemented by the applica- tion of the Wigner–Eckart theorem to the T1,M′′ pq matrices ⟨ISM∣T1,M′′ ∣I′S′M′⟩=⟨S′M′; 1M′′∣SM⟩⟨IS∥T1,.∥I′S′⟩, (21) where I,S,Mindices indicate the Ith electronic state with spin Sand spin projection M,⟨S′M′; 1M′′|SM⟩is a Clebsh–Gordan coefficient, and⟨IS∥T1,.∥I′S′⟩is a spinless triplet transition density matrix, which we denote by u, following the nomenclature from Ref. 19, u≡⟨IS∥T1,.∥I′S′⟩=⟨ISM∣T1,M′′ ∣I′S′M′⟩/⟨S′M′; 1M′′∣SM⟩. (22) We construct ufollowing the prescription given by Pokhilko et al. ,19through the spin-conserving ααandββparts of the 1TDM (γ) between the states with the same spin projection, upq=1√ 2(γpq−γ¯p¯q)/⟨S′M; 10∣SM⟩, (23) γpq=⟨ISM∣a† paq∣I′S′M⟩. (24) The SOC between the two (spin-pure) electronic states is obtained as the contraction of the spinless umatrix with the appropriate one- and two-electron mean-field spin–orbit terms scaled by the associated Clebsh–Gordan coefficients, ⟨ISM∣HSOMF∣I′S′M′⟩ =1 2∑ pq[⟨S′M′; 1−1∣SM⟩hSOMF L+,pq+⟨S′M′; 10∣SM⟩√ 2hSOMF z,pq −⟨S′M′; 11∣SM⟩hSOMF L−,pq]upq. (25) Importantly, the spinless umatrix can be computed between two specific states from the interacting spin multiplets, | SM⟩and | S′M′⟩, and then used to obtain the entire SOC matrix for all pairs of the −S≤M≤Sand−S′≤M′≤S′states. The simultaneous calcu- lation of Nelectronic states is normally performed for the same spin projection ( M′=M) of states, as used in Eq. (23). For sys- tems with an even number of electrons, the M= 0 solutions are preferred, since they afford the computation of S≥0 states, i.e., spin singlets, triplets, and quintets. However, the Clebsh–Gordan coefficients ⟨S′M; 10| SM⟩vanish for S′=Swith M= 0, pre- cluding the computation of SOCs between same-spin states, e.g.,triplet–triplet or quintet–quintet. In these cases, a different spin projection needs to be used.74 2. Spin–orbit coupling constant The evaluation of key properties related to spin-forbidden pro- cesses, e.g., ISC rates, oscillator strengths, or magnetic anisotropies, require the calculation of SOCs for all multiplet components. While different Cartesian contributions and multiplet components depend on spatial orientation, the SOC constant (SOCC)75,76is rotationally invariant. In molecules, this constant can be obtained by taking the sum over all spin projections as follows: ∣SOCC∣2=∑ M,M′∣⟨SM∣HSO∣S′M′⟩∣2. (26) For atoms, one needs to sum over orbital angular momentum ( ML) and spin ( M) projections, ∣SOCC∣2=∑ ML,M′ L∑ M,M′∣⟨LM LSM∣HSO∣L′M′ LS′M′⟩∣2. (27) In atoms, the relationship between coupling constants and energy splittings can be obtained through the Landé interval rule,77 EJ−EJ−1=λJ, (28) whereλis a constant proportional to the one-electron spin–orbit coupling constant ζ(λ=±ζ/2S).78From the invariance of the (Frobenius) norm of the spin–orbit matrix with respect to unitary transformations and setting the non-relativistic energy to zero, i.e., ∑J(2J+ 1)EJ= 0, |SOCC|2in Eq. (27) can be expressed in terms of relative energies of Jstates ( EJ) as ∣SOCC∣2=∑ J(2J+ 1)E2 J. (29) Combining Eqs. (28) and (29) enables comparison between the cal- culated SOCC and the experimental level splittings in atoms. For the triplet (3P) and doublet (2P) atoms studied in Sec. IV B, ∣SOCC∣2=3ζ2. (30) For easy comparisons with previous studies,71,79the values for C, O, S, Si, F, and Cl presented in Table II correspond to ζ/2. III. COMPUTATIONAL DETAILS We implemented the computation of SOCs within the RASCI framework in a development version of the Q-Chem electronic structure package,24using general libraries developed for SOC cal- culations within EOM-CC.19The required integrals are evaluated using the King–Furlani algorithm,80as implemented by Epifanovsky et al .71Spin–orbit NTOs30are computed and analyzed using the libwfa library.81RASCI calculations were carried out with RAS1 J. Chem. Phys. 153, 214107 (2020); doi: 10.1063/5.0029146 153, 214107-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp and RAS3 subspaces including all occupied and virtual orbitals, respectively. Unless otherwise indicated, the core electrons were kept frozen. Details about molecular geometries are given in Sec. IV and in the supplementary material. SOC calculations were carried out with Dunning’s correlation-consistent basis sets:82–86cc-pVXZ (X = D, T, Q, 5) and aug-cc-pVXZ (X = D, T, Q), and the polarized core– valence bases: cc-pCVXZ (X = D, T, Q, 5). We note that, depending on molecular orientation, symme- try labels corresponding to the same orbital (or vibrational mode) may differ. Q-Chem’s standard molecular orientation is different from that of Mulliken.87For example, Q-Chem places the water molecule in the xz-plane instead of yz. Consequently, for C 2vsym- metry, b1andb2labels are flipped. More details can be found at http://iopenshell.usc.edu/resources/howto/symmetry. All reported symmetry labels follow the Q-Chem convention, which needs to be kept in mind when reading Secs. III B and IV E. IV. RESULTS A. Doublet radicals We begin by considering a simple example of doublet radi- cals OH, SH, and SeH with a doubly degenerate2Πground state derived by distributing three electrons in the πxandπyorbitals, as shown in Fig. 2. Spin–orbit interaction between the two degen- erate doublet states lifts the degeneracy and splits the interacting states, yielding the energy gap twice the magnitude of the interstate SOC. We computed SOCs using structures from Ref. 88 ( ROH = 0.9697 Å, RSH= 1.3409 Å, and RSeH= 1.5811 Å). To preserve orbital and state degeneracy, we describe these systems by the RAS- IP variant using the closed-shell Hartree–Fock reference configu- ration with an additional electron: [core](σs)2(σz)2(πx)2(πy)2. We carried out calculations including a variable treatment of electron correlation, by varying the size of the RAS2 orbital space. To inves- tigate the basis-set dependence of the results, we considered the cc-pVXZ and cc-pCVXZ series of bases (with X = D, T, Q, and 5). The computed SO splittings are shown in Table I, along with the corresponding experimental values. In agreement with the pre- vious study using multi-reference coupled-cluster method,88we observe that the double-zeta results are far from the basis-set limit. The computed SO splittings for OH obtained with cc-pVXZ and cc- pCVXZ (X = T, Q, and 5) agree well with the experimental value, with errors less than 2%. The splittings in SH and SeH, computed with the full correlation of the highest nsandnpelectrons ( n= 3 and n= 4 for SH and SeH), i.e., [7 e, 4o], show no differences between the cc-pVXZ and cc-pCVXZ values, but the errors relative to the experiment are larger (8%–10%). Correlation of the internal elec- trons affects the magnitude of SOCs in SH and SeH. Inclusion of the ( n−1) electrons, i.e., [15 e, 8o] in SH and [25 e, 13o] in SeH, increases the magnitude of the splitting and reduces the error rel- ative to the experiment. The improvement is more significant when using a polarized core–valence family of bases (cc-pCVXZ) because of the better description of the internal electrons. Correlating 2 s2p electrons in SeH ([33 e, 17o]) further improves the results, with errors of≤1% for all bases in the cc-pCVXZ series. FIG. 2 . Molecular orbital diagram illustrating the electronic configuration of the dou- bly degenerate2Πground states in doublet radicals. ndenotes the principal atomic quantum number and is equal to 2, 3, and 4 in OH, SH, and SeH, respectively. We note that the SOCs are rather sensitive to the treatment of electron correlation and that using unbalanced active spaces dete- riorates the results. Increasing the RAS2 space (or the active space in CASSCF) does not necessarily improve the accuracy, unless elec- tron correlation effects are properly handled, as shown by the results presented in Figs. S1–S3, where partial addition of atomic shells increases the errors with respect to the experimental values. Finally, we note that in all three molecules, full correlation of the 1 selectrons has a minor effect on the computed SOCs. The two-electron contribution to the SOC (and, consequently, to the splittings) is much smaller than the one-electron part and has an opposite sign76due to the charge signs of interacting particles (Fig. 3). The relative weight of the two-electron part in SeH is only 10%, but it is larger in lighter systems, i.e., 19% in SH and 35% in OH, illustrating, once again, the importance of two-electron SOC contributions in typical organic molecules. B. Spin–orbit splittings in atoms Table II shows the computed coupling constants for first- (car- bon, oxygen, and fluorine) and second-row (silicon, sulfur, and chlo- rine) atoms. For a balanced description of the3P multiplet, we used RAS- nIP with a high-spin quartet anion ( n= 1) reference for carbon and silicon atoms, and a singlet dianion ( n= 2) reference for oxygen and sulfur. To describe the2P states of fluorine and chlorine atoms, we used a singlet anion reference state (with fully occupied valence shell). Because triplet–triplet SOCs cannot be evaluated by the Wigner–Eckart theorem from the M= 0 components of the triplet states, we compute SOCs from the three high-spin ( M= 1)3P RAS- nIP solutions. Comparison of computed results with experimental splittings must be done with caution, since the spin–spin interaction can contribute to the energy splittings, especially for light atoms. The results are shown in Table II, together with pure SOC values derived from the experimental J= 0, 1, 2 state energy splittings by removing spin–spin interactions.79 The results show minor differences between the two considered basis sets (cc-pCVTZ and cc-pCVQZ). The RAS- nIP couplings for the C and O atoms are within 6%–8% of the experimental values. The accuracy for the second-row atoms is sensitive to the treatment of electron correlation. The SOCCs for the3P second-row atoms with the [3s3p] correlated space exhibit larger errors (17% and 14% for Si and S, respectively) than their first-row counterparts. Correlation of the 2 pelectrons increases the magnitude of the SOCC in silicon J. Chem. Phys. 153, 214107 (2020); doi: 10.1063/5.0029146 153, 214107-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Spin–orbit splittings (cm−1) for OH, SH, and SeH doublet radicals computed with RAS-IP with the cc-pVXZ and cc-pCVXZ (X = D, T, Q, and 5) basis sets.aCorrelated orbital space (RAS2) indicated in square brackets. The splittings are computed by considering SOCs between the two2Πstates. OH SH SeH Basis [7 e, 4o] [7 e, 4o] [15 e, 8o] [7 e, 4o] [25 e, 13o] [33 e, 17o] cc-pVDZ 133.64 345.53 355.46 1587.64 1608.33 1610.033 cc-pVTZ 139.75 348.40 369.03 1597.89 1647.63 1652.42 cc-pVQZ 140.99 344.80 364.91 1581.33 1638.98 1645.16 cc-pV5Z 141.70 345.29 384.22 1585.56 1683.89 1699.81 cc-pCVDZ 135.74 348.47 386.56 1609.39 1727.20 1745.46 cc-pCVTZ 140.66 347.25 385.29 1595.10 1710.75 1750.50 cc-pCVQZ 141.74 345.84 383.61 1587.37 1705.11 1755.22 cc-pCV5Z 141.60 345.16 382.65 1585.50 1703.57 1750.89 Experiment 139.2b377.0b1764.4c aFunctions with gand higher angular momentum omitted. bValues from Ref. 89. cValues from Ref. 90. by∼9 cm−1, reducing the errors to 4%–5%. The effect of correlat- ing 2 pelectrons is also large for sulfur; it increases the couplings by∼35 cm−1and reduces the error to 3%–4%. Correlating 2 selec- trons yields an additional 0.5 cm−1increase of the SOCC for Si and S atoms. Correlating 1 selectrons has practically no effect on the couplings, so that frozen-core approximation can be safely used in SOCCs calculations (Table S1). To quantify the effect of higher electronic states on the state energies, we computed SOCCs between the3P states with the low- lying1D and1S singlets in C, O, Si, and S (at the RAS-IP/cc-pCVTZ level). Table III compares the energy distribution of J= 0, 1, 2 FIG. 3 . Total (gray), one- (blue) and (mean-field) two-electron (orange) contribu- tions to the spin–orbit splittings (in cm−1) in OH, SH, and SeH computed with RAS-IP/cc-pCV5Z. The RAS2 space contains all occupied orbitals above the 1slevel: [7 e, 4o] (OH), [15 e, 8o] (SH), and [33 e, 17o] (SeH). Numbers over the total splitting bars indicate the fraction (in %) of total SOC vs one-electron contributions.obtained through the diagonalization of the SOC matrix of the nine- dimensional triplet space (RAS-IP[9s]) and including the couplings to1D and1S (RAS-IP[12s]). These second-order contributions to the energy splittings of the3P states are very small in all four atoms: less than 1%–2% of the RAS-IP[9s] values, in agreement with earlier multireference calculations.79 C. Singlet–triplet couplings in diradicals The SF excitation operator is particularly suitable for describing molecules with the diradical (or polyradical) character. Below, we TABLE II . Spin–orbit coupling constants expressed as ζ/2 (in cm−1) for first- and second-row atoms computed with RAS- nIP and with the cc-pCVTZ and cc-pCVQZ bases obtained from Eq. (30). Correlated orbital space (RAS2) indicated in square brackets. All electrons below the RAS2 space have been included in RAS1 (no frozen core approximation). Method Basis C(3P) O(3P) F(2P) [2s2p] cc-pCVTZ 14.86 71.11 134.19 [2s2p] cc-pCVQZ 15.06 71.24 135.92 Experimenta13.98 77.40 134.70 Method Basis Si(3P) S(3P) Cl(2P) [3s3p] cc-pCVTZ 61.34 222.04 271.71 [2p3s3p] cc-pCVTZ 70.47 188.63 296.94 [2s2p3s3p] cc-pCVTZ 70.91 189.17 297.78 [3s3p] cc-pCVQZ 61.10 221.04 271.51 [2p3s3p] cc-pCVQZ 69.94 185.90 296.70 [2s2p3s3p] cc-pCVQZ 70.36 186.43 297.55 Experimenta73.70 194.62 294.12 aPure spin–orbit values from Ref. 79. J. Chem. Phys. 153, 214107 (2020); doi: 10.1063/5.0029146 153, 214107-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE III . State energies (cm−1) obtained by diagonalizing the SOC matrix con- sidering3P states only (RAS-IP[9s]) and including1D and1S states (RAS-IP[12s]); RAS-IP/cc-pCVTZ. Atom J RAS-IP[9s] RAS-IP[12s] Expt.aExpt.b C 0 0.00 0.00 0.00 0.00 1 14.86 14.92 16.42 13.98 2 44.57 44.60 43.41 41.94 O 2 0.00 0.00 0.00 0.00 1 142.22 142.86 158.27 154.80 0 213.34 212.81 226.98 232.20 Si 0 0.00 0.00 0.00 0.00 1 70.91 71.39 77.12 73.70 2 212.72 213.15 223.16 221.10 S 2 0.00 0.00 0.00 0.00 1 378.35 385.32 396.06 389.24 0 567.52 563.45 573.64 583.86 aExperimental values from Ref. 91. bDerived from the Landé interval rule and using pure spin–orbit values from Ref. 79. investigate the performance of RAS-SF in the calculation of SOCs between the lowest singlet and triplet states of diatomic (BH and AlH) and triatomic (CH 2, NH 2+, SiH 2, and PH 2+) diradicals. 1. Spin–orbit couplings in BH and AlH The interest in BH and AlH molecules is motivated by their presence in stellar atmospheres.92–94More recently, small diatomics have been attracting considerable attention in the context of laser- cooling experiments and quantum information science.95BH and AlH have been investigated using different spectroscopies96–98and simulations.99These studies corroborated the diradical nature of their ground states. Hence, RAS-SF is expected to be a good approach for the description of their electronic properties, including the SOCCs for the 11Σ+to 13Πtransition. The calculations were carried out using the geometries from Ref. 99 ( RBH= 2.3289 bohrs and RAlH= 3.1103 bohrs). In RAS-SF and EOM-SF-CCSD calculations, we used a ROHF (13Π) triplet ref- erence. The RAS2 space included four electrons in five orbitals, i.e., the 2 s2porbitals of B (3 s3pof Al) and 1 sof H. Core orbitals were frozen in the RAS-SF and EOM-SF-CCSD calculations. Tables IV and V show the computed singlet–triplet energy gaps and SOCCs. The RAS-SF SOCCs agree well with the EOM-SF-CCSD values for BH, with a mean-absolute-percentage-error (MAPE) of 1.5% for the one-electron part and 2.7% for mean-field SOCCs. For AlH, the RAS-SF values are lower than EOM-SF-CCSD ones (both 1 eand mean-field SOCCs), with an MAPE of ∼3.0%. The effect of the basis set is similar for both methods, giving rise to differences of up to 10% (0.5 cm−1) for BH and 5% (1.1 cm−1) for AlH. Overall, these results indicate that RAS-SF provides accurate SOCCs for these diatomic diradicals with relatively weak basis-set dependence. This benchmark also suggests that accurate SOCCs can be computed at a low cost in a triple-zeta basis, provided that the frontier orbitals are included in the RAS2 space.TABLE IV . Vertical energy gaps (eV) and spin–orbit couplings (cm−1) between 11Σ+ and 13Πstates for the BH radical computed using RAS-SF and EOM-SF-CCSD with the cc-pVXZ and aug-cc-pVXZ (X = D, T, and Q) basis sets. The size of the active space (RAS2) is indicated in square brackets. RAS[4 e, 5o]-SF EOM-SF-CCSD Basis ΔE 1eSO SOMF ΔE 1eSO SOMF cc-pVDZ 1.116 8.33 3.47 1.291 8.09 3.51 cc-pVTZ 1.114 8.67 3.71 1.327 8.68 3.87 cc-pVQZ 1.083 8.81 3.82 1.336 8.85 4.00 aug-cc-pVDZ 1.058 8.36 3.48 1.309 8.06 3.49 aug-cc-pVTZ 1.076 8.69 3.72 1.330 8.66 3.86 aug-cc-pVQZ 1.078 8.84 3.83 1.336 8.85 4.00 TABLE V . Vertical energy gaps (eV) and spin–orbit couplings (cm−1) between 11Σ+ and 13Πstates for AlH computed using RAS-SF and EOM-SF-CCSD with the cc- pVXZ and aug-cc-pVXZ (X = D, T, and Q) basis sets. The size of the active space (RAS2) is indicated in square brackets. RAS[4 e, 5o]-SF EOM-SF-CCSD Basis ΔE 1eSO SOMF ΔE 1eSO SOMF cc-pVDZ 1.710 34.45 25.82 1.853 34.61 25.96 cc-pVTZ 1.726 33.39 24.99 1.895 35.03 26.30 cc-pVQZ 1.703 33.71 25.25 1.894 35.23 26.46 aug-cc-pVDZ 1.660 34.56 25.94 1.860 33.99 25.48 aug-cc-pVTZ 1.679 33.66 25.23 1.894 34.89 26.19 aug-cc-pVQZ 1.675 34.05 25.52 1.895 35.24 26.47 2. Spin–orbit couplings in CH 2, NH 2+, SiH 2, and PH 2+ Here, we analyze the SOCC between the lowest triplet (3B2) and singlet (1A1) states in triatomic XH 2(XH 2+) with X = C and Si (N and P) molecules. The calculations were carried out at the equilib- rium triplet-state (3B2) geometries;100–103optimized structures are given in Table S2 of the supplementary material. RAS-SF calcula- tions used the ROHF3B2reference and various RAS2 spaces. RAS1 and RAS3 include all occupied and all virtual orbitals, respectively. The results are collected in Table VI. The computed RAS-SF SOCCs agree well with the EOM-SF- CCSD results. The differences between the RAS-SF and EOM-SF- CCSD singlet–triplet SOCCs are on the order of 1 cm−1(error of 9%–14%) in CH 2and are slightly larger in NH 2+(15%–20% of error). Interestingly, the increase in the RAS2 size, which increases the amount of correlation included in RAS-SF, considerably reduces the differences between the RAS-SF and EOM-SF-CCSD values in SiH 2and PH 2+. The3B2/1A1couplings increase with the atomic number of the central atom. The ratio between the one- and two-electron parts grows linearly with Z(Fig. 4), with the absolute magnitude of the one-electron contribution being twice larger than the two-electron J. Chem. Phys. 153, 214107 (2020); doi: 10.1063/5.0029146 153, 214107-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE VI . Singlet–triplet SOCC (cm−1) between3B2and1A1states computed with RAS-SF and EOM-SF-CCSD with the cc-pVTZ basis set. Fully correlated electrons and orbitals (RAS2) are indicated in square brackets. Method CH 2 NH 2+SiH 2 PH 2+ RAS[2 e, 2o]-SF 9.73 15.27 50.28 110.31 RAS[4 e, 3o]-SF 9.87 15.50 51.65 112.82 RAS[6 e, 4o]-SF 9.36 14.51 51.28 109.46 RAS[6 e, 6o]-SF 9.54 14.96 53.55 113.96 RAS[12 e, 7o]-SF . . . . . . 58.56 119.38 RAS[14 e, 8o]-SF . . . . . . 58.73 119.66 EOM-SF-CCSDa10.86 18.26 56.74 119.97 aEOM-SF-CCSD values are from Ref. 19. part in CH 2and nearly five times in PH 2+. This behavior is expected, since one- and two-electron SO terms grow as Z4andZ3, respec- tively.15The slope of the linear fit could be used to extrapolate the two-electron contribution for larger Zvalues in the series. In general, the linear behavior in Fig. 4 justifies the use of effective Zin one- electron approximations to recover two-electron effects in heavy elements in which the one-electron term is dominant. As noted above, the use of different αandβdensities (ρ) in Eq. (12), such as for open-shell Hartree–Fock reference in RAS-SF, might violate L+/L−symmetry. The proper symmetry can be recov- ered by averaging between the two spin densities. Symmetrization ofα/βHartree–Fock densities has a minor impact on the com- puted singlet–triplet couplings in CH 2, SiH 2, NH 2+, and PH 2+, with changes with respect to (non-averaged) SOCs being around 0.01 cm−1or smaller (Table S3). D. Spin–orbit couplings between ππ∗and nπ∗states In this section, we investigate the performance of RASCI in describing the SOCs between singlet and triplet states of different FIG. 4 . Ratio of one- and two-electron contributions in the3B2/1A1SOCCs of CH 2, NH2+, SiH 2, and PH 2+diradicals vs the atomic number ( Z) of the central atom. SOCC values from the largest RAS2 space in Table VI. FIG. 5 . Molecules illustrating SOC between nπ∗andππ∗singlet and triplet states. Biacetyl ( BIA), butanone ( MEK ), and (2E)-2-buten-2-ol ( BOL ). character. We choose three organic compounds with nπ∗andππ∗ excited states (Fig. 5): a diketone, biacetyl, or butanedione ( BIA ), butanone, also known as methyl ethyl ketone ( MEK ), and one of its isomers (2E)-2-buten-2-ol ( BOL ). Conjugated ketones and alco- hols can exhibit strong SOCs even without the presence of heavy atoms. Carbonyl groups support low-lying nπ∗states, which can effectively couple to the ππ∗states with different spins, as explained by El-Sayed’s rules.5In particular, BIA shows phosphorescent emis- sion104due to the efficient ISC and radiative decay from the triplet manifold.105 Ground-state geometries of these molecules were optimized with B3LYP/cc-pVDZ and are given in the supplementary material. Vertical excitation energies and the SOCs were computed with the cc-pVTZ basis set for the TDDFT/TDA (Tamm–Dancoff approx- imation)106with the B3LYP functional, EOM-CCSD, and RASCI methods. The planar structure of the BIA ,MEK , and BOL backbone results in a clear separation between σ- andπ-type orbitals. The highest occupied molecular orbitals (HOMOs) of the ketone and diketone compounds have important contributions of the oxygen (n) lone pairs (Fig. 6) and π-type occupied orbitals are energeti- cally below them. This orbital ordering leads to low-energy n→π∗ excited states, with π→π∗states being higher in energy. The n- type orbital in BOL is lower in energy than the π-HOMO, which is mainly localized on the C = =C double bond. Consequently, the low- est singlet and triplet excitations in BOL are ofππ∗character. In all FIG. 6 . Occupied ( nandπ) and virtual ( π∗) frontier molecular orbitals of BIA,MEK , andBOL ; HF/cc-pVTZ. J. Chem. Phys. 153, 214107 (2020); doi: 10.1063/5.0029146 153, 214107-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp three molecules, the LUMO (lowest unoccupied molecular orbital) corresponds to a π-orbital (π∗). Table VII shows vertical excitation energies to the lowest nπ∗andππ∗states of BIA ,MEK , and BOL computed at the TDDFT/TDA (B3LYP), EOM-CCSD, and RASCI levels. The states of interest can be described as one-electron excitations from the ground state,57,81as indicated by the norms of the respective one- particle transition density matrix ( ∥γ∥), both for EOM-CCSD and for RASCI. Therefore, we regard EOM-CCSD as the reference method, as it has been shown to deliver accurate results for singly excited states.23,107–109We observe that B3LYP energies are system- atically red-shifted with respect to EOM-CCSD. On the other hand, the limited account of the dynamic correlation in RASCI results in the overestimation of excitation energies on the order of few tenths of an eV and up to ∼1 eV. Due to the limitation of the current Q-Chem implementa- tion,110the spin–orbit calculations with TDDFT/TDA (B3LYP) only include one-electron contributions of the Breit–Pauli Hamil- tonian. The RASCI and EOM-CCSD calculations include both one- electron and mean-field two-electron couplings. Overall, the cou- plings between the states of different character (1,3nπ∗/3,1ππ∗) are much larger than the couplings between the states with similar orbital occupations (1ππ∗/3ππ∗and1nπ∗/3nπ∗), in accordance with El-Sayed’s rules. Despite the limited account of dynamic correlation, the RASCI SOCCs agree rather well with the EOM-CCSD values (Table VIII). The strong nπ∗/ππ∗couplings in BIA ,MEK , and BOL molecules are of the order of several tens of cm−1, with relative errorsof RASCI with respect to the EOM-CCSD SOCCs of <8% for BIA andMEK . We observe larger differences for BOL , which can be related to larger differences between the EOM-CCSD and RASCI wave functions. TheBIA molecule has C2hsymmetry in which L+and L− belong to the Bgirreducible representation (irrep), and Lzis invari- ant under all the operations of the group ( Ag). Therefore, only the triplet states from the AgorBgirreps can couple to the ground-state singlet ( X1Ag) via the spin–orbit operator. The3nπ∗and3ππ∗states are antisymmetric with respect to inversion ( Auand Bu, respec- tively); hence, the SOC to the ground state is symmetry-forbidden. On the other hand, the spin–orbit coupling of3n′π∗(Bg) with X1Ag is strong. These trends are nicely reproduced by all three meth- ods, with one-electron SOCCs around 130 cm−1and the mean-field (total) couplings in the 80 cm−1–90 cm−1range. Similarly, non- vanishing couplings between the excited singlets and triplets are restricted to those for which the symmetry product of the two states belongs to the AgorBgirreps. In other words, only the states with the same parity, gerade (g) or ungerade (u), can interact through the spin–orbit Hamiltonian (this can also be seen from the pseu- dovector nature of ˆL). Therefore, non-zero couplings are obtained fornπ∗/nπ∗,n′π∗/n′π∗,ππ∗/ππ∗, and nπ∗/ππ∗pairs of states, in which the coupling strength is strongly dependent on the orbital occupations. Singlet–triplet couplings of the nπ∗/ππ∗type are rather strong, whereas all the others are much weaker ( <1 cm−1). As described by Pokhilko and Krylov,30the strength of the computed SOCs can be rationalized in terms of the NTO pairs of the TABLE VII . Vertical excitation energies ( ΔEin eV), norm of the transition dipole moment in parentheses (in a.u.), and ∥γ∥ values to the lowest nπ∗andππ∗singlet and triplet excited states of BIA,MEK , and BOL computed with B3LYP, EOM- CCSDa, and RASCI; cc-pVTZ basis set. The RAS2 space in RASCI calculations included the orbitals shown in Fig. 6. B3LYP EOM-CCSD RASCI State ΔE ΔE∥γ∥ ΔE ∥γ∥ BIA 3nπ∗2.15 2.68 0.939 2.98 0.974 3n′π∗3.90 4.35 0.941 4.85 0.967 3ππ∗5.00 5.36 0.958 6.43 0.979 1nπ∗2.71(0.027) 3.11(0.041) 0.932 3.51 (0.021) 0.968 1n′π∗4.50(0.000) 4.78(0.000) 0.932 5.28 (0.000) 0.953 1ππ∗7.76(0.899) 8.37(1.055) 0.927 9.15 (1.479) 0.950 MEK 3nπ∗3.87 4.18 0.944 4.83 0.982 3ππ∗5.97 6.21 0.963 6.90 0.994 1nπ∗4.47(0.013) 4.58(0.013) 0.937 5.34 (0.007) 0.976 1ππ∗8.14(0.409) 9.50(0.718) 0.935 10.50 (1.033) 0.971 BOL 3ππ∗3.97 4.14 0.962 4.20 0.997 3nπ∗8.28 9.14 0.938 10.30 0.976 1ππ∗6.70(1.469) 7.06(1.442) 0.944 8.07 (1.425) 0.980 1nπ∗8.70(0.050) 9.45(0.023) 0.934 10.72 (0.001) 0.971 aNorm of the one-particle transition density matrix averaged between A→BandB→A:∥γ∥2=∥γAB∥⋅∥γBA∥. J. Chem. Phys. 153, 214107 (2020); doi: 10.1063/5.0029146 153, 214107-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE VIII . Singlet–triplet SOCCs (cm−1) between ground state (GS), nπ∗andππ∗ singlet and triplet excited states computed with TDDFT/TDA (B3LYP), EOM-CCSD, and RASCI; cc-pVTZ basis set. B3LYP EOM-CCSD RASCI States 1 eSO 1 eSO SOMF 1 eSO SOMF BIA GS/3nπ∗0.00 0.00 0.00 0.00 0.00 GS/3n′π∗130.30 136.14 85.79 129.96 81.25 GS/3ππ∗0.00 0.00 0.00 0.00 0.00 1nπ∗/3nπ∗1.01 0.29 0.17 0.27 0.16 1nπ∗/3n′π∗0.00 0.00 0.00 0.00 0.00 1nπ∗/3ππ∗71.09 80.63 51.84 86.74 55.33 1n′π∗/3nπ∗0.00 0.00 0.00 0.00 0.00 1n′π∗/3n′π∗0.68 0.04 0.02 0.73 0.45 1n′π∗/3ππ∗0.00 0.00 0.00 0.00 0.00 1ππ∗/3nπ∗33.65 68.56 44.19 69.43 44.60 1ππ∗/3n′π∗0.00 0.00 0.00 0.00 0.00 1ππ∗/3ππ∗0.23 0.19 0.13 0.10 0.04 MEK GS/3nπ∗94.17 94.23 59.04 89.97 55.81 GS/3ππ∗0.51 0.44 0.08 0.42 0.04 1nπ∗/3nπ∗0.03 0.05 0.02 0.05 0.02 1nπ∗/3ππ∗67.83 78.80 50.38 73.10 46.57 1ππ∗/3nπ∗7.81 49.19 31.62 54.45 35.05 1ππ∗/3ππ∗2.28 0.49 0.31 0.08 0.05 BOL GS/3ππ∗0.67 0.59 0.01 0.57 0.00 GS/3nπ∗43.34 41.15 23.14 33.84 18.57 1ππ∗/3ππ∗0.09 0.08 0.02 0.01 0.03 1ππ∗/3nπ∗24.73 23.42 15.43 10.02 7.06 1nπ∗/3ππ∗20.52 16.62 11.07 3.84 3.44 1nπ∗/3nπ∗2.43 0.15 0.10 0.01 0.02 reduced spinless triplet transition density matrix ( u). The represen- tation of the leading singular values of u(Fig. 7) show how the large SOCs between low-lying states of BIA in Table VIII are obtained for those transitions with a change in orbital character, i.e., n-type hole toπ-type particle (or vice versa). The results for MEK are similar to those for BIA , but with only one low-lying singlet and one triplet of the n→π∗charac- ter. It is worth noting that while the RASCI and EOM-CCSD cou- plings are in very good quantitative agreement, the one-electron 1nπ∗/3ππ∗and1ππ∗/3nπ∗SOCCs obtained with B3LYP are consid- erably smaller, especially for the1ππ∗/3nπ∗pairs. This can be related to the mixing of electronic configurations in the B3LYP states, in contrast to the EOM-CCSD and RASCI states, which exhibit nearly pure nπ∗orππ∗characters. The largest differences between the RASCI and EOM-CCSD SOCs are obtained in BOL . RASCI cou- plings (1 eand SOMF) follow the same trends as the EOM-CCSD values, but they are systematically smaller, while the (one-electron) SOCs computed with B3LYP are close to EOM-CCSD. These results FIG. 7 . Spinless triplet NTOs, hole (left) and particle (right), with the largest singular values (ω) between GS/3n′π∗,1nπ∗/3ππ∗, and1ππ∗/3nπ∗pairs of states of BIA computed at the RASCI/cc-pVTZ level. Secondary NTO pairs can be found in Fig. S4. The isovalue of 0.020 was used. suggests that, in this case, the lack of dynamic correlation in RASCI negatively affects the accuracy of the computed SOCs. The lead- ing spinless triplet NTOs and their weights between singlet and triplet states of MEK andBOL can be found in Figs. S5 and S6, respectively. E. Molecular torsion in tetramethyleneethane Tetramethyleneethane (TME), which is one of the simplest non-Kekulé disjoint diradicals, has attracted interest as a potential organic magnet,111as a building block for electrically conductive polymers,112,113and has also been used to generate organoruthenium complexes with unique dianionic character.114Highly accurate elec- tronic structure calculations115–117established that the ground-state singlet and the lowest triplet in TME are very close in energy, with a singlet–triplet gap that varies with the molecular conformation due to the subtle changes in the interaction between the two radical elec- trons. These studies concluded that reliable calculations of singlet and triplet electronic states of TME require the inclusion of electron correlations between all six p-electrons. Here, we evaluate the profile of the SOCC between the nearly degenerate singlet and triplet states of TME along the torsional rotation about the central C–C bond with dihedral angles from 0 to 90○. We use geometries from Ref. 115, optimized at the CASSCF(6,6)/cc-pVTZ level for the ground-state singlet and the lowest triplet for different dihedral angles (supplementary material). J. Chem. Phys. 153, 214107 (2020); doi: 10.1063/5.0029146 153, 214107-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp RAS-SF calculations were carried out with the lowest ROHF septet as the reference, with six electrons in the six π-orbitals in RAS2 [Fig. 8(a)], and with the cc-pVTZ basis set. At the planar and fully twisted structures, TME has D 2hand D 2d symmetry, respectively. Only the three twofold symmetry rotations are preserved for any other torsional angle (D 2symmetry). For the sake of clarity, unless otherwise indicated, below we use the D 2labels (1A and3B1) to mark the two states of interest along the molecular torsion. Symmetry axes have been chosen by considering the D 2h structure on the xz-plane and the central bond aligned with the z axis. At the planar geometry (D 2hsymmetry), the two states have 1Agand3B1usymmetries, while Lz,Ly, and Lxoperators belong to the B1g,B2g, and B3girreps. At the D 2dsymmetry, the sin- glet and triplet have 1A 1and3B2symmetries, while the angular momentum operators belong to the A2andEirreps. Therefore, sym- metry selection rules forbid singlet–triplet spin–orbit interaction at the planar and orthogonal structures of TME, resulting in zero SOCCs. These symmetry restrictions for the singlet–triplet (1A/3B1) SOCs are lifted at the reduced D 2symmetry at intermediate dihe- dral angles ( Lz,LyandLx∈B1,B2, and B3). Figure 8(b) shows that the1A/3B1SOCC increases as soon as the molecule becomes bent.The computed SOCCs reach a maximum at ∼20○of∼0.35 cm−1 for the one-electron spin–orbit contribution and ∼0.20 cm−1for the total SOCCs computed for the singlet and triplet geometries. Beyond this point, the SOCCs drop to nearly zero at ∼45○. At this structure, the 7 aand 6 b1orbitals are nearly degenerate and the1A/3B1energy gap reaches the smallest value along the tor- sion coordinate.115At 45○twist, the triplet state is dominated by a single electronic configuration: (7a)1(6b1)1. In contrast, the singlet state,1A, is multiconfigurational, with two determinants with nearly equal contribution,115,117as indicated by the approxi- mately equal RAS-SF weights of the (7a)2(6b1)0and(7a)0(6b1)2 configurations (44% and 40%, respectively). Contributions to the 1A/3B1SOCC due to the interaction of (7a)2(6b1)0and(7a)0(6b1)2 with(7a)1(6b1)1cancel each other, similar to the SOCC cou- plings observed in C 2H4O intermediate.7Thus, the similar weight of the two configurations in1A at 45○results in nearly zero SOCCs.118The magnitude of the cancellation effect depends on the relative weights of the two most dominant NTO pairs, which exhibit similar values at 45○[ω1andω2in Fig. 8(c)]. The pro- file of the SOCCs in the 45○–90○torsion range is rather similar to the one in the 0○–45○region, but with lower SOCC maxima of ∼0.10 cm−1(∼0.22 cm−1) for the total mean-field (or one-electron) SOCs. FIG. 8 . (a)πorbitals of TME (isovalues of 0.05) obtained from the ROHF septet reference on the3B1optimized geometry with a 0○torsion angle. (b) One-electron and mean-field SOCCs along torsion angles calculated for the two optimized geometries. (c) Dominant NTO pairs of the spinless transition matrix of TME involved in the3B1↔ 1A transitions at different torsion angles, with isovalues of 0.05. The two largest singular values ( ω1andω2) between the NTO pairs at different torsion angles are presented in the table on the right. J. Chem. Phys. 153, 214107 (2020); doi: 10.1063/5.0029146 153, 214107-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Figure 8(c) shows the NTOs responsible for the1A1↔3B1SOCs at different torsion angles, indicating that the1A/3B1couplings arise mainly due to the 7 a→6b1orbital transition. V. CONCLUSIONS The present contribution expands the capabilities of the RASCI family of methods to the computation of SOCs using the Breit–Pauli Hamiltonian. SOCs are evaluated within the spin–orbit mean-field approximation, which considerably reduces the computational cost while introducing negligible errors relative to the full two-electron treatment. Our implementation is based on the application of the Wigner–Eckart theorem to triplet excitation operators and the gen- eration of a spinless triplet transition density matrix to obtain the entire SOC matrix for all spin projections. We tested the performance of RASCI SOCs in a variety of atomic and molecular systems. The results demonstrate that the RASCI method captures the key details of the electronic structure of correlated states, enabling accurate computations of SOCs within the limitations of the Breit–Pauli approximation. The level of cor- relation treatment in the RASCI ansatz affects the accuracy of the results. In particular, the correlation of inner electrons ( n−1 level) improves the computed values. On the other hand, core electrons can be safely disregarded in the computation of SOCs of valence states by using the frozen core approximation. Double-zeta basis sets are not sufficient, and triple-zeta or larger basis sets should be used for accurate results. The polarized core–valence family of bases (cc- pCVXZ) delivers a faster convergence with the size of the valence orbital space (X = D, T, Q, 5, . . .) than the cc-pVXZ family. The reported methodological developments broaden the scope of applications of the RASCI family of methods to include studies of spin-forbidden processes. Incorporating the SOC analysis into RAS- SF methods and utilizing the detailed wave function analysis of the spin–orbit perturbed RASCI states can provide deeper insight into the complicated strongly correlated systems. SUPPLEMENTARY MATERIAL See the supplementary material for the study of the basis- set dependence of SOCs in doublet radicals and atoms, optimized geometries for triatomic diradicals, spinless NTOs and optimized geometries of BIA ,MEK , and BOL molecules, and SOCCs and optimized geometries of TME. ACKNOWLEDGMENTS The authors are grateful for financial support from Eusko Jaurlaritza (Basque Government) through Project No. PIBA19- 0004 and the Spanish Government MINECO/FEDER (Project No. PID2019-109555GB-I00). The work in Los Angeles was supported by the Department of Energy (Grant No. DE-SC0018910). D.C. is thankful for financial support from IKERBASQUE (Basque Foun- dation for Science). We thank the anonymous reviewer for pointing out a mistake in the prefactor of the two-electron part of SOMF. The authors declare the following competing financial inter- est(s): A.I.K. is the president and a part-owner of Q-Chem, Inc.DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material. REFERENCES 1D. R. Yarkony, “Spin-forbidden chemistry within the Breit-Pauli approxima- tion,” Int. Rev. Phys. Chem. 11, 195–242 (1992). 2C. M. Marian, “Spin–orbit coupling and intersystem crossing in molecules,” Wiley Interdiscip. 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5.0031310.pdf

J. Chem. Phys. 153, 244118 (2020); https://doi.org/10.1063/5.0031310 153, 244118 © 2020 Author(s).A range-separated generalized Kohn–Sham method including a long-range nonlocal random phase approximation correlation potential Cite as: J. Chem. Phys. 153, 244118 (2020); https://doi.org/10.1063/5.0031310 Submitted: 29 September 2020 . Accepted: 29 November 2020 . Published Online: 29 December 2020 Daniel Graf , and Christian Ochsenfeld ARTICLES YOU MAY BE INTERESTED IN Toward chemical accuracy at low computational cost: Density-functional theory with σ- functionals for the correlation energy The Journal of Chemical Physics 154, 014104 (2021); https://doi.org/10.1063/5.0026849 Models and corrections: Range separation for electronic interaction—Lessons from density functional theory The Journal of Chemical Physics 153, 160901 (2020); https://doi.org/10.1063/5.0028060 Electronic structure software The Journal of Chemical Physics 153, 070401 (2020); https://doi.org/10.1063/5.0023185The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp A range-separated generalized Kohn–Sham method including a long-range nonlocal random phase approximation correlation potential Cite as: J. Chem. Phys. 153, 244118 (2020); doi: 10.1063/5.0031310 Submitted: 29 September 2020 •Accepted: 29 November 2020 • Published Online: 29 December 2020 Daniel Graf and Christian Ochsenfelda) AFFILIATIONS Chair of Theoretical Chemistry, Department of Chemistry, University of Munich (LMU), D-81377 Munich, Germany a)Author to whom correspondence should be addressed: christian.ochsenfeld@uni-muenchen.de ABSTRACT Based on our recently published range-separated random phase approximation (RPA) functional [Kreppel et al. , “Range-separated density- functional theory in combination with the random phase approximation: An accuracy benchmark,” J. Chem. Theory Comput. 16, 2985–2994 (2020)], we introduce self-consistent minimization with respect to the one-particle density matrix. In contrast to the range-separated RPA methods presented so far, the new method includes a long-range nonlocal RPA correlation potential in the orbital optimization process, making it a full-featured variational generalized Kohn–Sham (GKS) method. The new method not only improves upon all other tested RPA schemes including the standard post-GKS range-separated RPA for the investigated test cases covering general main group thermochemistry, kinetics, and noncovalent interactions but also significantly outperforms the popular G0W0method in estimating the ionization potentials and fundamental gaps considered in this work using the eigenvalue spectra obtained from the GKS Hamiltonian. ©2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0031310 .,s I. INTRODUCTION Density functional theory (DFT) is among the most popu- lar approaches for electronic structure calculations in the fields of solid state physics, computational chemistry, and materials science. However, standard density functionals show several shortcomings. For example, the unphysical Coulomb self-interaction is typically only incompletely corrected by approximate exchange–correlation (xc) functionals.1–10This spurious self-interaction results in a wrong asymptotic decay of the xc-potential,2,11,12which, in turn, results in a poor description of molecular properties such as ionization potentials.13Another well-known problem of standard density func- tionals is the missing description of long-range dispersion effects due to their nonlocal nature.14Furthermore, it should be men- tioned that since the pioneering work of Kohn and Sham15in 1965, hundreds of density functionals have been developed, which makes the selection of a suitable functional for a specific problem chal- lenging16and additionally limits its predictive power. Therefore,the development of a more broadly applicable method is highly desirable. Electronic structure methods based on the random phase approximation (RPA) have become increasingly popular in the last decade, providing a promising route toward a qualitative and quantitative improvement of standard density functional the- ory.17–48RPA methods contain an ab initio description of disper- sion interactions,49,50do not depend on any empirical parameters, and are applicable to vanishing electronic gap systems.22,51,52The random phase approximation belongs to the family of adiabatic- connection fluctuation-dissipation (ACFD) methods53,54that calcu- late the correlation energy using density–density response functions. The quantity to be approximated within these approaches is the frequency-dependent exchange–correlation kernel. The direct ran- dom phase approximation is the most simple approximation one can think of in this regard as it neglects the frequency-dependent exchange–correlation kernel entirely. It is hence often considered as the Hartree approximation for time-dependent density functional J. Chem. Phys. 153, 244118 (2020); doi: 10.1063/5.0031310 153, 244118-1 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp theory (TDDFT).24,55As a consequence, same-spin electron–hole pairs do not experience Pauli repulsion, making the direct RPA correlation hole too negative, which, in turn, results in an overcor- relation of electrons at short interelectronic distances.17,24,56–58Fur- thermore, RPA methods show slow convergence with respect to the size of the basis set19,59arising due to the explicit description of the correlation hole near the electron–electron cusp,60which requires a lot of one-electron basis functions with high angular momentum. As is common practice in the literature, we will, in the follow- ing, drop the “direct” term and use the terms direct RPA and RPA synonymously. Since DFT describes short-range electron–electron interaction well without the need for large basis sets, the idea of combining DFT and ab initio approaches arose some time ago.61–63The combina- tion of short-range DFT with long-range RPA methods is especially attractive in this regard. Not only do RPA methods describe long- range correlation exceptionally well,50but they are also perfectly compatible with the (nonlocal) exact exchange that corrects the spu- rious Coulomb self-interaction of standard DFT and leads to the cor- rect asymptotic −1/rdecay of the exchange-potential. This should be contrasted with standard global hybrids that decay as −ax/r12, where axis the fraction of exact exchange that leads to an underestimation of ionization potentials.64,65 In the last few years, a lot of work has gone into range-separated approaches based on the RPA by the Paris–Nancy group59,66–73and Scuseria and co-workers.74–79It was shown that the range-separated approaches show faster convergence with the basis set while at the same time improving the accuracy of intermolecular interactions, atomization energies, and barrier heights compared to their full- range versions. Very recently, our group contributed to this promis- ing field with a detailed benchmark of an efficient range-separated RPA method in the atomic/Cholesky orbital space.41It was shown that range-separated RPA performs more stably over the broad range of molecular chemistry included in the GMTKN55 dataset80 than standard full-range RPA, and the finding that range-separation leads to faster convergence with respect to the size of the basis set was confirmed. So far, range-separated RPA was performed almost exclu- sively in a post-generalized-Kohn–Sham (GKS) fashion where the orbitals are obtained by solving the GKS equations81including a short-range density functional combined with long-range (non- local) exact exchange. In these approaches, the long-range cor- relation part is completely omitted within the orbital optimiza- tion process. It was shown, however, that the reference orbitals and orbital energies have a strong impact on the performance of RPA.51,76,82–84Therefore, it might be advisable to include a long-range correlation potential compatible with the long-range exact exchange. He βelmann and Ángyán85recently presented a range-separated self-consistent RPA method based on the opti- mized effective potential (OEP) approach,82,86–102yielding a com- pletely local potential as required by a real Kohn–Sham (KS) method. In this work, we present a method that self-consistently min- imizes the total energy of our range-separated RPA functional41 with respect to the one-particle density matrix in the atomic orbital (AO) space. This leads to a variational generalized Kohn–Sham method that includes a nonlocal long-range xc-potential consisting of the exact exchange and the RPA correlation potential and henceaccounts for all parts of the potential in the orbital optimization process (full-featured). II. THEORY A. Range-separated atomic orbital random phase approximation In this section, we briefly review the theory underlying the range-separated atomic orbital (AO) resolution-of-the-identity (RI) RPA method presented recently by our group.41For a detailed description of range-separated RPA, the reader is referred to the paper of Toulouse et al.69 In the following, μ,ν,λ,σdenote atomic orbitals, i,jdenote occupied molecular orbitals (MOs), a,bdenote virtual MOs, i,j denote Cholesky orbitals, and M,Ndenote auxiliary resolution-of- the-identity functions. Nbasisdenotes the number of basis functions, Nauxdenotes the number of auxiliary RI functions, and Noccand Nvirtdenote the numbers of occupied and virtual molecular orbitals, respectively. Integrals are expressed in the Mulliken notation. Fur- thermore, Einstein’s sum convention103is used and the spin index is dropped for convenience. Our range-separated RPA method is based on the range- separated hybrid PBE (RSHPBE) functional of Goll et al. ,104which calculates the energy according to ERSHPBE=EH+EPBE,sr c +EPBE,sr x +EHF,lr x (1) with the Hartree energy EH, the short-range PBE-like exchange EPBE,sr x and correlation energy EPBE,sr c , and the long-range exact Hartree–Fock (HF) exchange energy EHF,lr x. The separation of the electron–electron interaction is achieved by partitioning the stan- dard electron–electron interaction operator vee= 1/r12into a short- range vsr eepart and a long-range vlr eepart using the error function and its complement61,62,105 vee=vsr ee+vlr ee=erfc(μr12) r12+erf(μr12) r12(2) with the range-separation parameter μand the interelectronic dis- tance r12. The RSHPBE functional, which is minimized in a standard gen- eralized Kohn–Sham scheme, does not contain long-range correla- tion and is thus in a second step corrected by a post-GKS long-range RPA correlation energy calculation using the RSHPBE orbitals and orbital energies. The long-range RI-RPA energy is given by ERPA,lr c =1 2π∫+∞ 0dωTr[ln(1−X0(iω)Vlr)+X0Vlr], (3) where Vlrrepresents the long-range electron–electron interaction in the auxiliary basis Vlr MN=(C−1)MP˜Vlr PQ(C−1)QN, (4) CMN=(M∣m12∣N), (5) ˜Vlr MN=(M∣vlr ee∣N), (6) J. Chem. Phys. 153, 244118 (2020); doi: 10.1063/5.0031310 153, 244118-2 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp and m12is the RI-metric. In the present method, the attenuated Coulomb metric m12=erfc(ωattr12) r12(7) withωatt=0.1a−1 0is used since it has been shown to constitute a good trade-off between accuracy and locality for fitting the full-range electron-electron interaction operator.48X0denotes the noninter- acting density–density response function in the zero-temperature case and is calculated in the imaginary time domain according to X0,MN(iτ)=G0,μν(−iτ)BM νλG0,λσ(iτ)BN σμ, (8) where G0(iτ) is the one-particle Green’s function in the imaginary time domain, G0(iτ)=Θ(−iτ)G0(iτ)+Θ(iτ)G0(iτ), (9) G0(iτ)=CμiCνiexp(−(εi−εF)τ), (10) G0(iτ)=−CμaCνaexp(−(εa−εF)τ) (11) with the occupied ( Cμi) and unoccupied ( Cμa) MO coefficients and the respective MO energies εiandεa.Θ(iτ) denotes the Heavi- side step function, and εFdenotes the Fermi level. The three-center integral matrix Bis given by BM μν=(μν∣m12∣M). (12) The response function of Eq. (8) is then transformed into the imaginary frequency domain by a contracted double-Laplace47or equivalently cosine106transform according to X0(iω)=∫+∞ −∞dτcos(ωτ)X0(iτ) (13) to perform the final frequency integration given in Eq. (3). B. Extending the RSHPBE functional by a long-range nonlocal random phase approximation correlation potential In Sec. II A, we reviewed our recently published range- separated AO-RI-RPA method. As mentioned earlier, this scheme lacks long-range correlation within the self-consistent optimiza- tion of the orbitals and orbital energies. In this section, we remedy this problem by extending the RSHPBE functional by a long-range nonlocal RPA correlation potential. The long-range RPA correlation potential is obtained by differ- entiation of the long-range RPA correlation energy with respect to the one-particle density-matrix P, VRPA,lr c =∂ERPA,lr c[P] ∂P. (14) To perform this differentiation, we first need to express the long- range RPA correlation energy in terms of the one-particle densitymatrix. This is easily achieved by expressing the one-particle Green’s functions according to G0(iτ)=Pexp(−τ(H−εFS)P), (15) G0(iτ)=−Pvirtexp(−τ(H−εFS)Pvirt) (16) with the two-center overlap matrix S, the unoccupied/virtual one- particle density matrix Pvirt, and the total Hamiltonian H. The total Hamiltonian of the present range-separated GKS method is given by H=h+J+VPBE,sr xc +VHF,lr x +VRPA,lr c =HRSHPBE+VRPA,lr c , (17) where hdenotes the core-Hamiltonian, Jdenotes the Coulomb potential, VPBE,sr xc denotes the short-range PBE-like exchange– correlation potential, VHF,lr x denotes the long-range exact (HF) exchange potential, and VRPA,lr c denotes the long-range nonlocal RPA correlation potential. As can be seen in Eqs. (15) and (16), the Green’s functions depend on the total Hamiltonian and hence on the long-range RPA correlation potential itself, making a straightforward differentia- tion impossible. This dilemma can be bypassed by a semicanoni- cal projection as described by Furche and co-workers.30We there- fore construct an intermediate Hamiltonian ˜Hby first calculating HRSHPBEusing the density obtained by diagonalizing the total Hamil- tonian and in a second step removing the occupied-virtual and virtual-occupied parts by projection according to ˜H=SPHRSHPBEPS+SPvirtHRSHPBEPvirtS (18) to ensure that both the total and the intermediate Hamiltonian yield the same density. Having two different Hamiltonians, Hand ˜H, of course means that there is an inconsistency in the GKS poten- tial defining the exchange–correlation energy of the functional and the derivative thereof resulting in two different GKS systems. With- out semicanonical projection, the two systems would in general differ in the potential and the respective density. Therefore, sem- icanonical projection is a straightforward approach to impose the weaker condition of requiring the two GKS systems to only yield the same density instead of being identical and hence a step toward full self-consistency. In order to keep the difference between the potentials of the two GKS systems as small as possible, we use the RSHPBE Hamiltonian as a basis of the intermediate Hamilto- nian, which differs from the total Hamiltonian only by the long- range RPA correlation potential. That omitting the RPA correla- tion potential in the construction of the intermediate Hamiltonian is a reasonable choice was already shown in previous work of our group.42 In the following, we will give the working equations for the cal- culation of the long-range RPA correlation potential in the atomic orbital space. The derivative of the long-range RPA correlation energy with respect to one element of the density matrix Pis given by VRPA,lr c,μν=Tr(∂ERPA,lr c ∂X0(iω)∂X0(iω) ∂G0(iτ)∂G0(iτ) ∂Pμν) (19) J. Chem. Phys. 153, 244118 (2020); doi: 10.1063/5.0031310 153, 244118-3 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp with the trace implying integration over imaginary frequency and imaginary time. The change in the long-range RPA correlation energy due to the variation of the one-particle Green’s function is described by the long-range correlation part of the RPA self- energy107denoted by Σlr c. Therefore, we can rewrite Eq. (19) as VRPA c,μν=Tr{∫+∞ −∞dτΣlr c(−iτ)∂G0(iτ) ∂Pμν}, (20) with Σlr c,μν(iτ)=−BM μλG0,λσ(iτ)Wlr,MN c(iτ)BN σν (21) andWlr cdenoting the long-range correlated screened Coulomb interaction given by Wlr c(iω)=Vlr[(1−X0(iω)Vlr)−1 −1], (22) Wlr c(iτ)=1 2π∫+∞ −∞dωcos(ωτ)Wlr c(iω). (23) The derivative of the one-particle Green’s function with respect to the ground state density matrix Pcan be split into three parts. The first part accounts for changes in the Green’s function while keeping the intermediate Hamiltonian fixed and is given by VRPA,lr c,1 =∫+∞ 0dτ(exp{+τ(˜H−εFS)P}Σlr c(iτ)+Y(−iτ)(˜H−εFS)) +∫+∞ 0dτ(exp{−τ(˜H−εFS)Pvirt}Σlr c(−iτ) +Y(iτ)(˜H−εFS)), (24) with Y(−iτ)=∞ ∑ k=1k−1 ∑ l=0τk k!((˜H−εFS)P)k−1−lΣlr c(iτ)P((˜H−εFS)P)l(25) Y(iτ)=∞ ∑ k=1k−1 ∑ l=0(−τ)k k!((˜H−εFS)Pvirt)k−1−l ×Σlr c(−iτ)Pvirt((˜H−εFS)Pvirt)l. (26) The second part includes the changes in the projection of the Hamiltonian HRSHPBEaccording to VRPA,lr c,2 =∫+∞ 0dτ(SPY(−iτ)SPHRSHPBE+HRSHPBEPSYT(−iτ)PS) +∫+∞ 0dτ(SPvirtY(iτ)SPvirtHRSHPBE +HRSHPBEPvirtSYT(iτ)PvirtS). (27)The third and last part arises from changes in the density entering HRSHPBEand is given by VRPA,lr c,3 =∫+∞ 0dτPμκYκγ(−iτ)Sγκ′Pκ′ν[(μν∣λσ)+(μν∣fxc∣λσ)] +∫+∞ 0dτPvirt μκYκγ(iτ)Sγκ′Pvirt κ′ν[(μν∣λσ)+(μν∣fxc∣λσ)], (28) with fxcdenoting the exchange–correlation kernel of HRSHPBE. The complete RPA correlation potential is then finally given by VRPA,lr c =VRPA,lr c,1 +VRPA,lr c,2 +VRPA,lr c,3 . (29) III. COMPUTATIONAL DETAILS The present range-separated self-consistent RPA method was implemented in the FermiONs++ program package developed in our group.108–110As mentioned above, we use the projec- tion of the RSHPBE104Hamiltonian ( HRSHPBE) of the standard GKS scheme in the post-SCF version as intermediate Hamil- tonian. This approach is referred to as range-separated self- consistent RPA (rsscRPA) in the following. In Sec. IV, we will compare the method with the range-separated PBE-like func- tional of Goll et al.104(RSHPBE); our standard post-GKS range- separated RPA presented in Ref. 41 and abbreviated as “RSHPBE + lrRPA;” the PBE functional;111,112ourω-CDGD RPA method (RPA) based on PBE orbitals;43and our self-consistent RPA42 (scRPA) using the projection of the HF-Hamiltonian as the inter- mediate Hamiltonian. All range-separated methods employed in this work use a fixed range-separation parameter of μ=0.5a−1 0. We used this value not only because it is a common and validated choice41,59,63,66–70,113–115 but also because it is physically reasonable. The average distance of valence electrons in molecular systems is around 1 a.u.–2 a.u.63 Since the inverse of the range-separation parameter approximately gives the distance where the range-separation is made, this would lead to values for μbetween 0.5 a−1 0and 1.0 a−1 0. There are also ways to determine the range-separation parameter non-empirically.116–119 In this context, it is interesting to note that Brémond et al.118 determined a value of 0.45 a−1 0for their RSX-PBE method, which is in good agreement with the empirically determined value of μ=0.5a−1 0. As atomic basis sets, the def2-TZVP, def2-TZVPP,120,121and aug-cc-pVQZ122,123basis sets are employed. For the resolution-of-the-identity that is used for 4-center integrals in the calculation of the RPA correlation energy/potential, the cor- responding auxiliary basis sets124–126are used with the atten- uated Coulomb metric127–129with an attenuation parameter ωatt=0.1a−1 0.48 For integrations along imaginary time and frequency as well as transformations between the two domains, we use optimized minimax grids with, in general, 15 quadrature points.43,106 J. Chem. Phys. 153, 244118 (2020); doi: 10.1063/5.0031310 153, 244118-4 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp IV. RESULTS AND DISCUSSION A. General main group thermochemistry, kinetics, and noncovalent interactions In this section, we assess the performance of the present range- separated self-consistent RPA method for general main group ther- mochemistry, kinetics, as well as noncovalent interactions and com- pare it to several other RPA schemes. To do so, we picked three test sets of each category in the large GMTKN55 database targeting basic properties and reaction energies for small systems (basic + small), reaction energies for large systems and isomerization reactions (iso. + large), reaction barrier heights (barriers), intermolecular noncova- lent interactions (intermol. NCIs), and intramolecular noncovalent interactions (intramol. NCIs), respectively. The def2-TZVP basis set was employed for all calculations in this section since larger basis sets are hardly usable for practical applications. A summary of the results is given in Tables I and II. The detailed results can be found in the supplementary material. We start our discussion with the results of the full-range meth- ods shown in Table I. As can be seen, standard RPA, with a total weighted mean absolute deviation according to weighting scheme 1 of Ref. 80 (WTMAD-1) of 4.10 kcal/mol, on average signifi- cantly improves upon the reference PBE calculations (WTMAD-1 of 9.58 kcal/mol). The largest improvement can be observed for noncovalent interactions, which is, of course, not surprising since it is well known that PBE lacks a correct description of disper- sion interactions. Self-consistently minimizing the total RPA energy with respect to the one-particle density-matrix (scRPA) decreases the WTMAD-1 value for the investigated subset of the GMTKN55 database further to 3.46 kcal/mol. Considering the results for dif- ferent categories shows that scRPA while on average improving upon post-KS RPA for the test cases included in the categories basic + small, iso. + large, and all NCIs has significant deficiencies in the calculation of barrier heights compared to standard post-KS RPA. An explanation for this observation might be that the inter- mediate Hamiltonian based on Hartree–Fock is not suitable for the description of the transition states. This assumption is supported by the fact that an evaluation of the barrier heights with the scRPA orbitals (RPA@scRPA) gives much better results. Note the differ- ence between the two approaches scRPA and RPA@scRPA: In the first approach, the response function and therefore also the correla- tion energy are evaluated with orbitals and their respective energies TABLE I . WTMAD-1 values for the full-range methods in kcal/mol for a subset of the GMTKN55 database grouped by categories. The def2-TZVP basis set was used for all calculations. A detailed list of the results can be found in the supplementary material. PBE RPA@PBE scRPA RPA@scRPA Basic + small 3.71 4.67 2.96 2.83 Iso. + large 9.70 2.64 1.77 4.03 Barriers 5.12 1.28 3.47 1.93 Intermol. NCIs 22.53 9.30 6.35 14.11 Intramol. NCIs 6.84 2.61 2.75 3.14 All NCIs 14.69 5.96 4.55 8.63 Total 9.58 4.10 3.46 5.21TABLE II . WTMAD-1 values of the range-separated methods in kcal/mol for a subset of the GMTKN55 database grouped by categories. The def2-TZVP basis set was used for all calculations. A detailed list of the results can be found in the supplementary material. RSHPBE rsRPA@ RSHPBE + lrRPA rsscRPA rsscRPA Basic + small 5.54 2.93 2.72 2.36 Iso. + large 6.71 3.10 3.10 3.06 Barriers 6.41 1.78 1.74 1.52 Intermol. NCIs 25.80 8.50 8.26 6.42 Intramol. NCIs 9.27 3.00 2.89 2.31 All NCIs 17.54 5.75 5.58 4.36 Total 10.75 3.86 3.74 3.13 stemming from the intermediate Hamiltonian, whereas in the sec- ond approach, the orbitals and orbital energies of a converged self- consistent RPA calculation are employed. Interestingly, the second approach gives much worse results in almost all other cases, yielding a total WTMAD-1 of 5.21 kcal/mol, which is even worse than RPA on PBE orbitals. Notably, the poor performance on intermolecular interactions stands out in this respect. In order to understand this observation, we plotted the signed errors of different approaches for the calculation of interaction energies of n-alkane dimers included in the ADIM6 test set in Fig. 1. To ease the following discussion, we make use of a notation for hybrid RPA methods similar to that employed in Ref. 37. The theoretical scheme that evaluates the Hartree–Fock energy with orbitals obtained from a self-consistent approach SC1 and adds the RPA correlation energy evaluated with orbitals/orbital energies obtained from a self-consistent approach SC2 is denoted as HF@ SC1 + cRPA@ SC2. The standard RPA@PBE approach, e.g., would in this notation be referred to as HF@PBE + cRPA@PBE. As can be seen in Fig. 1, standard RPA@PBE (Fig. 1, blue) performs very well on this specific test set, however, show- ing the typical underestimation of the dimer stability (underbind- ing). Ren et al.37found that this underbinding can be corrected by replacing HF@PBE with HF@HF, which amounts, to a large extent, to the single excitation correction.37This can also be observed for the ADIM6 test set (Fig. 1, brown), however, in this case, system- atically overcorrecting it. A similar behavior can be observed when replacing cRPA@PBE in the standard RPA with cRPA@scRPA (vio- let). Hence, it can be said that evaluating the HF energy with HF orbitals as well as evaluating the RPA correlation energy with scRPA orbitals/orbital energies increases the stability of the dimer com- pared to the monomers. When we now use the scheme HF@HF + cRPA@scRPA, a strong overcorrection can be observed, just as expected. In contrast, evaluating the RPA correlation energy with HF orbitals/orbital energies seems to decrease the stability of the dimer compared to the monomers, as can be seen by comparing RPA@HF, or in the other notation HF@HF + cRPA@HF (Fig. 1, red), with HF@HF + cRPA@PBE, demonstrating the significant dependence of the RPA functional on the reference potential. These findings can now be used to explain the behavior observed for the two self-consistent RPA approaches: RPA@scRPA is similar to the approach HF@HF + cRPA@scRPA and overcorrects in the HF as well as the cRPA part. The self-consistent RPA approach that uses J. Chem. Phys. 153, 244118 (2020); doi: 10.1063/5.0031310 153, 244118-5 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . Signed errors in kcal/mol for the ADIM6 test set using different methods. the orbitals and orbital energies of the intermediate Hamiltonian (based on the HF Hamiltonian) shown in orange, however, increases the stability of the dimer in the HF part and decreases it in the cRPA part. We now turn to the results of the range-separated methods presented in Table II. First of all, we note that, as expected, the addition of long-range RPA correlation to the RSHPBE energies sig- nificantly improves the performance in all cases and reduces the total WTMAD-1 from 10.75 kcal/mol for RSHPBE to 3.86 kcal/mol for RSHPBE + lrRPA. Note that the long-range RPA correlation energy in this case is evaluated using RSHPBE reference orbitals (see Sec. II A). Self-consistently minimizing the RSHPBE + lrRPA energy with respect to the one-particle density-matrix (rsscRPA) decreases the total WTMAD-1 to 3.74 kcal/mol although the effect on the single categories is sometimes small. Using the orbitals and orbital energies obtained by a converged rsscRPA calculation instead of the ones obtained by the intermediate Hamiltonian (rsRPA@rsscRPA) has a stronger impact and yields a total WTMAD-1 of 3.13 kcal/mol, which is the lowest value of all tested methods and can be seen as a first indication of the high quality of the underlying potential. At this point, we want to add some words considering the com- putational effort of the self-consistent RPA methods compared to their post-SCF counterparts: The time-determining step in our post- SCF RPA implementations used in this work is the calculation of the noninteracting response function that shows asymptotically lin- ear scaling43with the molecular size. This is to be contrasted to the asymptotically quadratic-scaling42calculation of the self-energy in the two self-consistent versions. In the case of the self-consistent versions, there is further an additional factor stemming from the number of SCF cycles needed until convergence is reached, which, in general, is on the same order as for standard PBE calculations. Note also that range-separation in both the self-consistent and the post- SCF variant does not have a significant impact on the computational efficiency. B. Ionization potentials and band gaps For a sound density functional, the negative of the highest occupied molecular orbital (HOMO) eigenvalue has to equal theionization potential in finite electron systems4,11,130and can hence be used as a test of the underlying exchange–correlation potential. Therefore, we compared the negatives of the HOMO eigenvalues obtained with several methods with experimental values for the GW27131test set. A summary of the results is given in Table III; detailed values can be found in the supplementary material. As mentioned above, the exchange–correlation functionals of standard density functional approximations such as PBE only partly correct for the unphysical Coulomb self-interaction, leading to a wrong asymptotic decay of the exchange–correlation potential. This spu- rious behavior is reflected in the poor quality of the HOMO energies as estimates of the ionization potentials with a MAD of 3.88 eV using the PBE functional. Within the RSHPBE functional, the asymp- totic region of the potential is described by the exact (HF) exchange potential that shows the correct asymptotic decay. This results in a significant improvement of the HOMO energies, yielding a MAD of 0.45 eV. Inclusion of the long-range RPA correlation poten- tial in the rsscRPA method improves the calculated HOMO ener- gies further and reduces the MAD to 0.26 eV. The new rsscRPA method also improves upon scRPA (MAD of 0.36 eV), demon- strating the benefits of range-separation in the electron–electron interaction. It should also be mentioned at this point that both self-consistent RPA methods significantly outperform the popular G0W0method (MAD of 0.59 eV) in approximating the ionization potentials considered in this work, which again gives rise to the TABLE III . Mean absolute deviations as well as maximum errors in eV of the calcu- lated ionization potentials compared to experimental values for the GW27131test set. The ionization potentials for RSHPBE, rsscRPA, PBE, and scRPA were calculated as negatives of the HOMO energies. All calculations were performed with the def2- TZVPP basis set. PBE, G0W0, and experimental values were taken from Ref. 131. Values for systems containing heavy elements such as Cs 2, Au 2, and Au 4were excluded due to technical reasons. RSHPBE rsscRPA PBE scRPA G0W0 MAD 0.45 0.26 3.88 0.36 0.59 MAX 1.04 0.86 6.73 0.84 1.26 J. Chem. Phys. 153, 244118 (2020); doi: 10.1063/5.0031310 153, 244118-6 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp assumption that the potentials obtained by these methods are of high quality. As a further test, gaps obtained as differences between the low- est unoccupied molecular orbital (LUMO) and the HOMO were calculated using the RSHPBE, rsscRPA, and the scRPA methods and compared to CCSD(T) total energy differences. The results are shown in Table IV. Again, both self-consistent RPA methods perform very well, significantly outperforming RSHPBE as well as G0W0. Also in this case, separation of the electron–electron inter- action shows to be beneficial, decreasing the MAD from 0.29 eV (scRPA) to 0.27 eV (rsscRPA). C. H 2dissociation curve It is well known that RPA captures static correlation to some extent and that this feature is connected to the spurious self- correlation inherent in RPA.10,57For example, RPA dissociates the H2molecule correctly in the sense that an infinitely stretched H 2 molecule has the same energy as two separate hydrogen atoms. How- ever, this comes at the cost of an unphysical bump in the dissociation curve. Furthermore, the description of the hydrogen atom itself is wrong, yielding a non-zero correlation energy. The one-electron self-correlation within RPA can, for example, be removed from the energy by the second-order screened exchange (SOSEX) approxima- tion.44,132,133Unfortunately, this leads to a larger static correlation error, underlining the connection between static correlation and self-correlation within RPA. We calculated the dissociation curve of the H 2molecule, which can be considered as the standard test for the description of static correlation of a functional, to investigate the performance of the self-consistent RPA methods discussed in the present work. The results are shown in Figs. 2 and 3. First of all, note the difference between the RPA@PBE (Fig. 2, right, blue) and the RPA-SOSEX@PBE (Fig. 2, right, black) curve. As can be seen, RPA-SOSEX describes the region around the mini- mum well but fails to capture static correlation effects at long bond lengths. In contrast, RPA shows a curve that is shifted to too nega- tive values on the absolute scale but shows a better agreement withTABLE IV . HOMO–LUMO gaps in eV and mean absolute deviations from refer- ence CCSD(T) values30(total energy differences). RSHPBE, rsscRPA, and scRPA calculations were performed with the def2-TZVPP basis set; CCSD(T) and G0W0cal- culations were performed with the aug-cc-pVTZ basis set. The CCSD(T) and G0W0 values were taken from Ref. 30. RSHPBE rsscRPA scRPA G0W0 CCSD(T) Li2 5.36 4.90 4.62 4.43 4.76 Na2 4.81 4.43 4.18 4.35 4.48 LiH 8.40 8.15 7.80 6.92 7.67 CH 3NO 2 11.97 11.03 10.84 9.82 11.41 MAD 0.55 0.27 0.29 0.70 ... the full configuration-interaction (FCI) curve on the scale relative to two separate hydrogen atoms (Fig. 3). As mentioned above, this behavior can be explained by the self-correlation within RPA, lead- ing to a too deep correlation hole and a wrong description of the hydrogen atom itself. Note also the unphysical bump in the dis- sociation curve obtained with RPA@PBE. Considering the curve obtained with RSHPBE + lrRPA (Fig. 2, left, blue) it becomes obvi- ous that the results are similar to those of RPA-SOSEX. The region around the minimum is described well; however, important cor- relation effects are missed in the long bond length regime. It is another appealing feature of range-separated RPA approaches that they are able to counteract the self-correlation problem within RPA, which is, of course, pronounced at short interelectronic distances. Therefore, range-separated RPA methods can be considered as cost- efficient alternatives to beyond-RPA methods including some kind of exchange. It should be mentioned, however, that range-separation is not able to remove the unphysical bump in the dissociation curve, although it is much less pronounced than in the full-range RPA. We now turn to the self-consistent RPA approaches dis- cussed in the present work. Self-consistently minimizing the total FIG. 2 . Dissociation curve of the H 2molecule calculated with different methods. All RPA calculations were performed with the aug-cc-pVQZ basis set. As reference serves a FCI/def2-TZVP dissociation curve. J. Chem. Phys. 153, 244118 (2020); doi: 10.1063/5.0031310 153, 244118-7 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Dissociation curve of the H 2molecule relative to two separate hydrogen atoms calculated with different methods. All RPA calculations were performed with the aug-cc-pVQZ basis set. As reference serves a FCI/def2-TZVP dissociation curve. range-separated RPA energy (Fig. 2, left, red) hardly changes any- thing compared to the post-GKS variant. If orbitals and orbital energies of a converged rsscRPA calculation are used to evalu- ate the total range-separated RPA energy (Fig. 2, left, brown), however, it becomes obvious that the small unphysical bump is removed. This behavior is even more pronounced in the full-range scRPA approach (Fig. 2, right, red and brown). Another interesting observation is that RPA@scRPA shows a very similar dissociation curve compared to RPA-SOSEX@PBE, which becomes especially apparent on the relative scale in Fig. 3. The shift to more nega- tive values on the absolute scale can again be explained by self- correlation, which is, of course, still present in the self-consistent version. As a final remark, we want to stress again the significant dependence of RPA on the reference or to be more specific on the orbital energies, which can be seen in the huge difference between RPA@PBE (Fig. 2, right, blue) and RPA@scRPA (Fig. 2, right, brown). V. CONCLUSION In this work, we presented the self-consistent minimization of our recently published range-separated RPA method (RSHPBE + lrRPA)41with respect to the one-particle density matrix in the atomic orbital space. The method extends the RSHPBE functional of Goll et al.104by a long-range nonlocal RPA correlation poten- tial in the orbital optimization process, making it a full-featured variational generalized Kohn–Sham method. The problem of impos- ing self-consistency on the long-range RPA correlation functional, which depends on the total Hamiltonian and hence on its own derivative, was bypassed by semicanonical projection30of the RSH- PBE Hamiltonian. The performance of the new method, termed rsscRPA, on general main group thermochemistry, kinetics, and noncovalent interactions was investigated using a subset of the large GMTKN55 database.80The overall performance of rsscRPA on this subset shows to be superior to that of the standard post-GKS vari- ant although the improvement is sometimes small. The method thatevaluates the RSHPBE + lrRPA functional using rsscRPA orbitals and orbital energies, termed rsRPA@rsscRPA, outperforms all other tested methods in this work including PBE, RSHPBE, RPA, RSH- PBE + lrRPA, and full-range self-consistent RPA, suggesting high quality of the orbitals, the orbital energies, and hence the underly- ing potential of rsscRPA. To further test the new method, ioniza- tion potentials and fundamental gaps calculated from the eigenvalue spectra of the GKS Hamiltonian were investigated. The method shows to give accurate results for the systems under investigation, significantly outperforming the popular G0W0method, which again implies high quality of the underlying potential. Finally, the behavior of the new method upon bond dissociation was investigated using the example of the H 2molecule. It showed that the performance is very similar to that of the post-GKS range-separated RPA. How- ever, the unphysical bump well known from the standard full-range RPA, which can also be observed in the range-separated variant, is removed. SUPPLEMENTARY MATERIAL Complete lists of the mean absolute deviations for a subset of the GMTKN55 database and the calculated ionization potentials of the GW27 test set are given in the supplementary material. ACKNOWLEDGMENTS The authors thank H. Laqua (LMU Munich) for helpful dis- cussions. Financial support was provided by the Excellence Clus- ter EXC2111-390814868, Munich Center for Quantum Science and Technology (MCQST) by the Deutsche Forschungsgemeinschaft (DFG). C.O. (Max Planck Fellow) acknowledges the financial sup- port of MPI-FKF, Stuttgart. DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material. J. Chem. 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J. Chem. Phys. 154, 064109 (2021); https://doi.org/10.1063/5.0041378 154, 064109 © 2021 Author(s).A phaseless auxiliary-field quantum Monte Carlo perspective on the uniform electron gas at finite temperatures: Issues, observations, and benchmark study Cite as: J. Chem. Phys. 154, 064109 (2021); https://doi.org/10.1063/5.0041378 Submitted: 22 December 2020 . Accepted: 17 January 2021 . Published Online: 12 February 2021 Joonho Lee , Miguel A. Morales , and Fionn D. 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Phys. 154, 064109 (2021); doi: 10.1063/5.0041378 Submitted: 22 December 2020 •Accepted: 17 January 2021 • Published Online: 12 February 2021 Joonho Lee,1,a) Miguel A. Morales,2and Fionn D. Malone2,a) AFFILIATIONS 1Department of Chemistry, Columbia University, New York, New York 10027, USA 2Quantum Simulations Group, Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore, California 94551, USA a)Authors to whom correspondence should be addressed: linusjoonho@gmail.com and malone14@llnl.gov ABSTRACT We investigate the viability of the phaseless finite-temperature auxiliary-field quantum Monte Carlo (ph-FT-AFQMC) method for ab ini- tiosystems using the uniform electron gas as a model. Through comparisons with exact results and FT coupled cluster theory, we find that ph-FT-AFQMC is sufficiently accurate at high to intermediate electronic densities. We show, both analytically and numerically, that the phaseless constraint at FT is fundamentally different from its zero-temperature counterpart (i.e., ph-ZT-AFQMC), and generally, one should not expect ph-FT-AFQMC to agree with ph-ZT-AFQMC in the low-temperature limit. With an efficient implementation, we are able to compare exchange-correlation energies to the existing results in the thermodynamic limit and find that the existing param- eterizations are highly accurate. In particular, we found that ph-FT-AFQMC exchange-correlation energies are in better agreement with a known parameterization than is restricted path-integral MC in the regime of Θ≤0.5 and rs≤2, which highlights the strength of ph-FT-AFQMC. Published under license by AIP Publishing. https://doi.org/10.1063/5.0041378 .,s I. INTRODUCTION Temperature-dependent properties of interacting fermions are fundamentally important in both experiments and theory. Typical phenomena at finite temperature (FT) include the Bardeen Cooper Schrieffer (BCS)-Bose-Einstein Condensation (BEC) crossover in the attractive 2D Fermi model,1the competition between stripe and superconducting orders in the 2D Hubbard model,2and plasmonic catalysis.3Furthermore, there is a growing interest in warm dense matter,4an extreme state of matter found in planetary interiors5 that can be created with high intensity lasers.6,7Understanding such phenomena at the theoretical level is challenging due to the del- icate interplay between electron–electron, electron–ion, quantum mechanical, and thermal effects, all of which can be equally impor- tant and often cannot be treated perturbatively. Thus, accurate com- putational approaches are required that are capable of capturing these effects.Density functional theory (DFT), as the workhorse of zero- temperature (ZT) electronic structure theory, is an ideal candidate: it is relatively cheap and accurate, it can be coupled with molecular dynamics to include ionic effects, and it can be rigorously formu- lated to incorporate thermal electronic effects.8Indeed, DFT has proven itself effective in simulating warm dense matter.9–12How- ever, questions remain regarding the accuracy of using approxi- mations made in thermal DFT, including the accuracy of the use of zero-temperature exchange-correlation functionals.13–16Comple- mentary approaches are therefore desired that can benchmark or supplement DFT results when necessary. Quantum Monte Carlo (QMC) methods are a promising class of such computational methods. They are, in principle, exact in a finite supercell and can explicitly include many-body and thermal effects in an unbiased way. Of the many flavors of QMC, real space path-integral Monte Carlo (PIMC) is perhaps the best known and well established.17PIMC, as a real space approach, works in the J. Chem. Phys. 154, 064109 (2021); doi: 10.1063/5.0041378 154, 064109-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp complete basis set limit, which is a considerable benefit at very high electronic temperatures. However, like all fermionic QMC meth- ods, it suffers from the sign problem that can only be overcome using the uncontrolled restricted path approximation,18which leads to the restricted PIMC (RPIMC) approach. This approach is similar in spirit to the fixed-node approximation in diffusion Monte Carlo (DMC),19but now a constraint is enforced using a trial thermal density matrix. The quality of this constraint is a priori unknown; however, results with free fermion nodes in the uniform electron gas (UEG) suggest that it is unreliable at high densities and at lower temperatures.20–22We note there have been promising devel- opments in extending the scope of PIMC to higher densities and lower temperatures through algorithmic developments23including the development of new algorithms such as the, in principle, exact real space permutation blocking PIMC24(PB-PIMC) and second quantized configuration PIMC25(CPIMC). In particular, PB-PIMC and CPIMC have been used as complementary approaches to simu- late the warm dense UEG above half the Fermi temperature, with the fermion sign problem preventing simulations below this. We note that restricted CPIMC may be a promising route to access lower temperature, given its results on small UEG supercells and the 33-electron supercell spin-polarized case.26Other interesting QMC approaches are the quantum chemistry inspired methods such as density matrix quantum Monte Carlo27–29(DMQMC) and Krylov- projected full configuration interaction quantum Monte Carlo.30 Like CPIMC, these approaches work in a second quantized space and work well at high densities; however, they often struggle to reach the complete basis set limit. All of these methods offer unbiased exact thermal expectation values, in principle; however, they all ulti- mately scale exponentially with the number of electrons, in general. See Ref. 31 for a review of the parameter regimes accessible to these new QMC methods. Recently, there has also been considerable interest in devel- oping finite-temperature deterministic quantum chemistry meth- ods that work in a finite basis set. These methods include sec- ond order perturbation theory,32–35coupled cluster theory,36–38and thermofield theory.39These are promising and offer a systematic approach for including electronic temperature effects, but they often struggle to reach the continuum limit [also known as the com- plete basis set (CBS) limit] due to the steep computational scaling with respect to the number of basis functions, M. For example, FT- CCSD scales like O(M6), which becomes prohibitively expensive to converge results to the CBS limit.37,38 Finite-temperature auxiliary-field QMC40,41(FT-AFQMC) is another promising QMC method. It works in the second quantized space and thereby suffers from the basis set incompleteness errors common to DMQMC and CPIMC. However, unlike DMQMC and CPIMC, it can be made to scale polynomially with system size at the cost of introducing an uncontrolled bias called the phaseless approximation.42,43Moreover, zero-temperature phaseless AFQMC (ph-ZT-AFQMC) has proven itself as one of the most accurate and scalable post Hartree–Fock methods.44–48In addition, the bias intro- duced by the phaseless approximation is typically much smaller than the fixed-node error in DMC,49which, in principle, should serve as a rough upper bound to the bias in RPIMC. Thus, it is important to assess the quality of ph-FT-AFQMC for realistic systems as to date, the applications have largely focused on model systems50,51or have not enforced the constraint,52–54which is not a practical approach asthe system size increases. Compared to FT-CCSD, ph-FT-AFQMC maintains favorable cubic scaling O(M3)for each statistical sam- ple,55,56which makes it better-suited for large-scale warm dense matter simulations. In this paper, we investigate the viability of ph-FT-AFQMC both in terms of its accuracy and its ability to reach the complete basis set limit. We stress that good performance for both of these metrics is critical if the method is to be practically useful for finite- temperature ab initio systems. To address these problems, we use the uniform electron gas (UEG) model as a test bed for ph-FT- AFQMC. To the best of our knowledge, our work is the first to apply ph-FT-AFQMC at finite temperature beyond lattice models with short-range interactions. The main goal of our work is to assess the accuracy of ph-FT-AFQMC when applied to the UEG model. Apart from being a foundational model in condensed matter physics,57–60the UEG offers a number of useful features from a com- putational point of view. First, the model can be tuned from weak to strong correlation as a function of the density parameter, rs, given that our previous work61has shown that ph-ZT-AFQMC is highly accurate for rs≤3–4 at zero temperature and we have an idea of the magnitude of errors we might expect. Second, basis set convergence can be easily investigated in a plane wave basis set by increasing the energy cutoff. Finally, the UEG at warm dense matter conditions has been the subject of intense study over the past decade.62In fact, many of the recent developments62in finite-temperature fermionic QMC methods were spurred on by a discrepancy between RPIMC and CPIMC20,21results for the warm dense UEG that has been incorpo- rated into finite-temperature exchange-correlation functionals.63–65 Because of this effort, there is a considerable amount of essentially exact data for energetic,66,67static,68–70and dynamic properties71,72 of the model in a wide range of densities and temperatures. Nonethe- less, despite this immense effort, there is a gap in accurate data below roughly half the Fermi temperature below, which no method could reach due to the Fermionic sign problem.73In this paper, we use ph-FT-AFQMC to partially fill this gap. This paper is organized as follows: In the first section, we out- line the ph-FT-AFQMC method, paying careful attention to how to implement it efficiently in a numerically stable fashion. Next, we carefully benchmark the method at various temperatures against exact results in small basis sets and investigate the low-temperature limit in comparison to the ph-ZT-AFQMC results. We also compare the ph-FT-AFQMC results to FT-CCSD and PIMC where possible. Finally, we compare our results against other QMC methods and available exchange-correlation parameterization in the thermody- namic limit and finish by outlining our perspective for the future of the method. II. THEORY We briefly summarize the phaseless approximation of AFQMC within the finite-temperature formalism (i.e., ph-FT-AFQMC). A. Finite-temperature AFQMC in the grand canonical ensemble 1. General formalisms The finite-temperature AFQMC algorithm aims to com- pute thermal expectation values based on the grand canonical J. Chem. Phys. 154, 064109 (2021); doi: 10.1063/5.0041378 154, 064109-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp partition function, Z=tr(e−β(ˆH−μˆN)), (1) where ˆNis a total number operator, μis a chemical potential, and the Hamiltonian involves one-body ( ˆH1) and two-body ( ˆH2) terms, ˆH=ˆH1+ˆH2. (2) The direct (deterministic) evaluation of trace in Eq. (1) scales expo- nentially since there are exponentially many states to consider. It is then natural to consider QMC algorithms to sample an instance of the terms in Z. A particular flavor of QMC that we focus on here is AFQMC, where the two-body propagator is expressed by the Hubbard– Stratonovich transformation,74 e−ΔτˆH2=∫dxp(x)e−√ Δτx⋅ˆv, (3) where ˆvis related to the two-body Hamiltonian as a sum of squared operators, ˆH2=−1 2Nα ∑ αˆv2 α, (4) where xis a vector of Nαauxiliary fields that are samples from the standard normal distribution, p(x). With the symmetric Trotter decomposition, the total propagator reads exp(−ΔτˆH)=∫dxp(x)ˆB(Δτ,x,μ), (5) where ˆBis defined as ˆB(Δτ,x,μ)=e−Δτ 2(ˆH1−μˆN)e−√ Δτx⋅ˆve−Δτ 2(ˆH1−μˆN). (6) For a given number of imaginary time slices n, we sample the auxiliary fields via MC, Z=∑ {x1,x2,...,xn}p(x1,x2,...,xn)tr(n ∏ i=1ˆB(Δτ,xi,μ)), (7) where p(x1,x2,...,xn) is the probability of sampling a specific path designated by auxiliary fields, x1,x2,...,xn, and the time step, Δτ =β/n, is determined by the temperature and the number of imagi- nary time slices. The evaluation of the trace in Eq. (7) is still difficult because it needs to consider all possible states in the Hilbert space despite the fact that every operator inside the trace is a one-body operator. One can show analytically that the trace can be written in terms of a determinant in the grand canonical ensemble,40,75,76 Z=∑ {x1,x2,...,xn}p(x1,x2,...,xn)det(I+n ∏ i=1B(Δτ,xi,μ)), (8) where Bis a matrix representation of ˆBin the single-particle basis. For a later use, we define the product of Bas A(nΔτ,{xk})=n ∏ i=1B(Δτ,xi,μ), (9)where we omit μin the argument of Afor simplicity. Note that for now, we defined Aonly at the end of the imaginary time propa- gation. Later, we will define Aalong the trajectory for an arbitrary imaginary time τ. We are mostly interested in computing expectation values based on the partition function in Eq. (8), not the partition function itself, ⟨ˆO⟩=tr(e−β(ˆH−μˆN)ˆO) Z. (10) The computation of expectation values can be easily achieved through an importance sampling procedure. To see this, we first write, ⟨ˆO⟩=1 Z∑ Xtr(ˆA(X)ˆO) =∑ Xtr(ˆA(X)ˆO) tr(ˆA(X))tr(ˆA(X)) Z, (11) where Xdenotes the set of auxiliary fields along each imaginary path. The field configurations Xare sampled by the MC algorithm where we write expectation values, ⟨ˆO⟩=∑iwiOL,i(Xi) ∑iwi, (12) where the local expectation value is defined as OL,i(Xi)=tr(ˆA(Xi)ˆO) tr(ˆA(Xi))(13) and walker weights wiare updated via the importance sampling procedure based on the distribution, tr (ˆA(X))/Z. In AFQMC, any expectation values are expressed as a function of the one-body Green’s function, Gij=⟨ˆciˆc† j⟩, (14) following the generalized Wick’s theorem.77Therefore, computing Gis sufficient to compute any expectation values. In terms of B(Δτ, x,μ), it can be shown that a sample of the one-body Green’s function (i.e., the local quantity) is75,76 G(nΔτ,{xk})=(I+A(nΔτ,{xk}))−1. (15) We note that a sample of the one-body reduced density matrix (1RDM), P, is obtained by P(nΔτ,{xk})=I−(G(nΔτ,{xk}))T. (16) Using Wick’s theorem, higher order reduced density matrices can be computed as products of the one-body Green’s function. 2. Phaseless approximation Since the value of determinants in Eq. (8) or equivalently tr(ˆA(X))for a given Xcan be either positive or negative (in our case complex),78the phase problem naturally arises in the finite- temperature algorithm. Therefore, it is important to impose the J. Chem. Phys. 154, 064109 (2021); doi: 10.1063/5.0041378 154, 064109-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp phaseless constraint43,51to remove the phase problem similar to the case of the ph-ZT-AFQMC algorithm.42This way, we can keep the overall scaling of the algorithm polynomial. We introduce a trial density matrix, ˆBT(Δτ,μT), and this is used to impose a phaseless constraint in the imaginary time evolution,43 which is defined as ˆBT(Δτ,μT)≡e−Δτ(ˆHT−μTˆN), (17) where ˆHTis some one-body operator and μTis the chemical poten- tial for the trial density matrix. Throughout the paper, we will assume that μTis tuned so that the thermal one-body density matrix from ˆBT(nΔτ,μT)has a trace of Nwith Nbeing the desired number of particles. We implement the imaginary time evolution of a path by the following algorithm. The central quantity in the constrained evolution is A. At an imaginary time τ=kΔτ, we define A(τ,{xi}k i=1)=(BT(Δτ,μT))n−k−1BT(Δτ,μT)×k ∏ i=1(B(Δτ,xi,μ)). (18) At each imaginary time step, we replace BTin the middle with B, that is, A(τ+Δτ,{xi}k+1 i=1)=(BT(Δτ,μT))n−k−1B(Δτ,xk+1,μ) ×k ∏ i=1(B(Δτ,xi,μ)). (19) In general, the product of Band/or BTrequires a special care for numerical stability, especially when simulating low tempera- ture. This can be achieved by the stabilization method described in Ref. 79, which we will describe further in Sec. II B. During this propagation, the importance function is deter- mined by the FT-AFQMC overlap ratio, Sk(τ,Δτ,{xi}k i=1)=det(I+A(τ+Δτ,{xi}k i=1)) det(I+A(τ,{xi}k−1 i=1)). (20) For a later use, we mention that we can equivalently write this ratio in terms of trace over all possible states in the grand canonical ensemble, Sk(τ,Δτ,{xi}k i=1)=tr(ˆA(τ+Δτ,{xi}k i=1)) tr(ˆA(τ,{xi}k−1 i=1)). (21) In practice, we employ the “optimal” force bias, which is a shift to the Gaussian distribution as well as the mean-field subtraction that enforces the normal ordering. The optimal force bias at τ=kΔτis ¯xk=−√ Δτ∑ α⎛ ⎝∑ pqPpq(τ)vα pq−¯vα 0⎞ ⎠, (22) and the mean-field subtraction is ¯vα 0=∑ pqvα pq(PT)pq. (23)We can then define the importance function (in the hybrid form80) as Ik(τ,Δτ,{xi}k i=1)=Sk(τ,Δτ,{xi−¯xi}k i=1)exk⋅¯xk−¯xk⋅¯xk 2. (24) The phaseless approximation ensures the reality and positivity of walkers using a modified importance function, Iph,k(τ,Δτ,{xi}k i=1)=∣Ik(τ,Δτ,{xi}k i=1)∣max(0, cosθk), (25) where θk=arg(Sk(τ,Δτ,{xi}k i=1)). (26) Using these, the n-th walker weight at τ=kΔτis updated via wn(τ+Δτ)=Iph,k(τ,Δτ,{xi}k i=1)×wn(τ). (27) This completes the description of the ph-AFQMC algorithm at finite temperature. 3. The T →0 limit Converging ph-FT-AFQMC to the ph-ZT-AFQMC limit as decreasing Tis highly desirable. This is due to the remarkable accu- racy of ph-ZT-AFQMC in a variety of systems benchmarked to date.44–48Reaching this zero-temperature limit would naturally sug- gest that the ph-FT-AFQMC algorithm is accurate at low tempera- ture. Unfortunately, reaching the zero-temperature limit is, in fact, difficult due to the differences in the phaseless constraint between two algorithms as we shall see below. We first define the zero-temperature overlap ratio at τ=kΔτ,42 ST=0 k(τ,Δτ,{xi}k i=1)=⟨ψT(N)∣ˆB(Δτ,xk, 0)∣ψ(τ,{xi}k−1 i=1,N)⟩ ⟨ψT(N)∣ψ(τ,{xi}k−1 i=1,N)⟩, (28) where Nis the number of particles and | ψT(N)⟩is the trial wave- function defined as ∣ψT(N)⟩=lim n→∞(ˆBT(Δτ, 0))n∣ψ0(N)⟩, (29) with |ψ0(N)⟩being some initial wavefunction and ∣ψ(τ,{xi}k−1 i=1,N)⟩=k−1 ∏ i=1ˆB(Δτ,xi, 0)∣ψT(N)⟩. (30) We note that we set μ=μT= 0 because the ph-ZT-AFQMC algo- rithm typically works in a fixed particle number space. In other words, the number of particles in | ψT(N)⟩and |ψ(N)⟩isNas speci- fied. Therefore, there is no need to have chemical potentials. This is then used to construct the phaseless importance function in Eq. (25), which defines the phaseless approximation at T= 0. The correspondence between the T→0 limit of ph-FT- AFQMC and ph-ZT-AFQMC can be understood by comparing J. Chem. Phys. 154, 064109 (2021); doi: 10.1063/5.0041378 154, 064109-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Eqs. (21) and (28). It is then ultimately equivalent to showing tr(ˆA(τ,{xi}k−1 i=1))=⟨ψT(N)∣ψ(τ,{xi}k−1 i=1,N)⟩where ˆAis defined through Eq. (19). In the limit of T→0, the number of time slices nbecomes∞. First, we consider the propagation in the middle of a path. Namely, we can assume 1 ≪k≪nin this case. One starts from lim T→0tr(ˆA(τ,{xi}k−1 i=1))=lim n→∞∑ M∑ α⟨ψα(M)∣(ˆBT(Δτ,μT))n−k+1 ×k−1 ∏ i=1ˆB(Δτ,xi,μ)∣ψα(M)⟩, (31) where the summation over Mis done over states in different particle number sectors and the summation over αcan be thought of as sum- ming over all possible states in the basis of | ψT(M)⟩. Assuming that μTandμare chosen so that it should pick out the N-particle sector, the action of (ˆBT(Δτ,μT))n−k+1to the left when n→∞(as well as 1 ≪k≪n) yields only one term out of the summation, that is, using Eq. (29), tr(ˆA(τ,{xi}k−1 i=1))→⟨ψT(N)∣k−1 ∏ i=1ˆB(Δτ,xi,μ)∣ψT(N)⟩ =⟨ψT(N)∣ψ(τ,{xi}k−1 i=1,N)⟩. (32) This then shows that the phaseless constraints are equivalent between ph-ZT-AFQMC and ph-FT-AFQMC algorithms when 1≪k≪n. Subtleties arise when 1 ≪k≪ndoes not hold. In the ph-FT- AFQMC algorithm, such a case always happens toward the com- pletion of a path. As an extreme case, let us consider the phaseless constraint at the last imaginary time step in the ph-FT-AFQMC algorithm. Namely, let us set k=n. With the same assumptions about μandμT, we have lim T→0tr(ˆA(τ,{xi}n−1 i=1))=lim n→∞∑ α⟨ψα(N)∣ˆBT(Δτ,μT) ×n−1 ∏ i=1ˆB(Δτ,xi,μ)∣ψα(N)⟩. (33) Not only does the summation over αnot easily truncates but also this limit no longer corresponds to the zero-temperature limit unlike in Eq. (32). This is indeed why one should not expect the ph-FT- AFQMC energy to approach the ph-ZT-AFQMC energy, in general. Because of these issues, even the importance function may need to be modified, while the same importance function was advocated in Ref. 52. Further investigation on this in the future will be interesting. Finally, there may be an additional complication when our assumptions about μandμTdo not hold exactly. In other words, it is possible to have fluctuation in the number of particles along an imaginary path. Such a fluctuation is not simply due to the fact that we are working in the grand canonical ensemble. Instead, it is due to the fact that the chemical potential used in ˆBT(i.e.,μT) is not necessarily the same as that of the many-body chemical potential used in ˆB. Therefore, on average, at a given imaginary time τ,Amay not yield the same number of particles as our target N. However, this is generally only secondary compared to the issue discussed in Eq. (33) (i.e., the issue of not projecting to α= 0) unless the under- lying system has a near-zero gap. When the system is metallic, thenumber of particle is very sensitive to both chemical potentials in ˆBT and ˆB. Therefore, along an imaginary propagation, the number par- ticle keeps changing even at very low temperature. Such a subtlety arises in other methods that work in the grand canonical ensemble such as low-order perturbation theory, which has been a subject of active research for some time.35Nevertheless, when the gap is not so small, in the limit of T→0, the number of particles changes very little as a function of μ. This makes the effect of particle fluctua- tion very small. In principle, one can work directly in the canonical ensemble that removes the need for the assumptions about μTand μ.54Nonetheless, the illustration of Eq. (33) still holds and some modification to the constraint is necessary in the context as well. These simple illustrations suggest that one may impose the phaseless constraint for T>0 based on a modified overlap ratio (at τ=kΔτ) (with the same assumptions about μTandμ), ˜Sk(τ,Δτ,{xi}k i=1)=tr((ˆBT(Δτ,μT))n∏k i=1ˆB(Δτ,xi,μ)) tr((ˆBT(Δτ,μT))n∏k−1 i=1ˆB(Δτ,xi,μ)), (34) where the trial density matrix is always multiplied up to the n-th power so that even in the last imaginary time step, one recovers the zero-temperature limit properly. However, the overall temperature in Eq. (34) is always lower than the physical temperature, which may break down in higher temperature regimes. Moreover, examining this constraint would be an interesting research topic in the future, but for the purpose of this work, we report numerical results based on the constraint in Eq. (20). B. Numerical stabilization It is well known that the standard determinant QMC (DQMC) algorithm suffers from numerical instabilities resulting from the repeated multiplication of the Bmatrices.81This issue can be over- come using the stratification method,79which we will now briefly describe. We first write an Amatrix via a column-pivoted QR decomposition (QRCP), A=QRΠ, (35) where Qis an orthogonal matrix, Ris an upper-triangular matrix, andΠis a permutation matrix. We then define A=QDT , (36) where D=diag(R), (37) T=D−1RΠ. (38) The QRCP decomposition is more expensive than matrix multipli- cations, so we perform the decomposition once in a while, and this frequency is controlled by a “stack” size parameter L. The number of stacks is then specified by nstack =β/L. The stack size Lis set such that one can perform the product of BL/Δτtimes without numerical instability. At an imaginary time τ, we are interested in computing J. Chem. Phys. 154, 064109 (2021); doi: 10.1063/5.0041378 154, 064109-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp A(τ)=nstack ∏ i=1Ai. (39) The product of stack { Ai} needs to be performed following the stratification algorithm: Compute the QRCP A1=Q1D1T1 for(do2≤i≤nstack) Compute Ci= (AiQi−1)Di−1 Compute the QRCP Ci=QiRiΠi SetDi= diag( Ri) Compute Ti=(D−1 iRi)(ΠiTi−1) end for In the end of this algorithm, we achieve A(τ)=QDT . (40) Finally, the Green’s function is also computed via a numerically stable form, G=(T−TQTDb+Ds)−T DbQT, (41) where Db(i,i)={1/∣D(i,i)∣if∣D(i,i)∣>1 1 otherwise(42) and Ds(i,i)={D(i,i) if∣D(i,i)∣≤1 sgn(D(i,i)) otherwise.(43) This can be derived from G−1=I+A (44) =I+QDT (45) =Q(Q−1T−1+D)T (46) =QD−1 b(DbQ−1T−1+Ds)T. (47) C. Exploiting the stack structure It is possible to reuse the QDT factorization of all stacks but one which is the central stack we are propagating. We write A(τ) as a product of two matrices (left and right blocks), A(τ)=AL(τ)AR(τ). (48) We note that the QDT factorizations of the left and right blocks are already given from the previous propagation. Namely, we already have AL=QLDLTL, (49) AR=QRDRTR. (50) Then, the numerically stable formation of A(τ) requires only two QRCP calls (as opposed to nstack calls described previously) and also far less matrix–matrix multiplications to do. The algorithm for propagating one time step is as follows:Compute TL=TLB−1 T Compute CCR= (BQ R)DR Compute the QRCP CCR=QCRRCRΠCR SetDCR= diag( RCR) Compute TCR=(D−1 CRRCR)(ΠCRTR) Compute CLCR=QLDLTLQCRDCR Compute the QRCP CLCR=QLCRRLCRΠLCR SetDLCR= diag( RLCR) Compute TLCR=(D−1 LCRRLCR)(ΠLCRTCR) We, therefore, achieve A=QLCRDLCRTLCR, (51) as in Eq. (40). The Green’s function can then be computed as before. D. Exploiting the low-rank structure He and co-workers82found that at low temperature, both ALand ARare low-rank, which can enable significant savings (O(M/N)). Such low-rank structures are reflected on DLandDR where with a certain threshold, we can approximate them as DL≈dL, (52) DR≈dR, (53) where dL/R(i,i)={DL/R(i,i)if∣DL/R(i,i)∣≥threshold 0 otherwise.(54) We denote the rank of dL/Rto be mL/Rand show how scaling reduc- tion can be achieved in terms of these ranks. From now on, we will only work in the reduced dimension provided by dL/R.dL/Ris an mL/R×mL/Rmatrix that is much smaller than the original M×M matrix. We write AL=qLdLtL, (55) AR=qRdRtR, (56) where qL/Ris a matrix of dimension M×mL/RandtL/Ris a matrix of dimension mL/R×M. To maximize cost saving, one needs to modify the stratification algorithm further. The most efficient prop- agation with stratification can be done as follows (assuming that we sampled B): Compute tL=tLB−1 T Compute cCR=BqRdR Compute cLCR=dLtLcCR Compute the QRCP cLCR=qLCRrLCRπLCR(qLCR ismL×mTandrLCRismT×mRwhere mT= min( mR,mL)) Compute qLCR=qLqLCR SetdLCR= diag( rLCR) Compute tLCR=(d−1 LCRrLCR)(πLCRtR) We then achieve A=qdt, (57) J. Chem. Phys. 154, 064109 (2021); doi: 10.1063/5.0041378 154, 064109-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp where qis an M×mTmatrix, dis an mT×mTdiagonal matrix, and tis an mT×Mmatrix. Using these reduced dimension matrices and the Woodbury identity, we can compute the Green’s function at a reduced cost, G=(I+qdt)−1(58) =I−q(d−1+tq)−1t, (59) where(d−1+tq)−1occurs in the dimension of mT×mTand other matrix multiplications are done at the cost of O(M2mT). Similarly, using the matrix determinant lemma, det(IM+qdt)=det(ImT+dtq). (60) The determinant evaluation occurs in the dimension of mT×mTas well. Furthermore, we note that the evaluation of ImT+tqdneeds to be done by the usual stabilization algorithm, ImT+dtq=(q−1t−1+d)tq (61) =(q−1t−1db+ds)d−1 btq. (62) From this, we evaluate the determinant as well as the Green’s function. For measurements, we use 1RDM, P, which is now expressed as P=tT(d−1+tq)−TqT. (63) This Pis also of low-rank, and this structure can be exploited to accelerate the local energy evaluation and other measurements. E. Uniform electron gas The uniform electron gas (UEG) model is usually defined and solved in the plane wave basis set. We will follow this convention in this work as well. The kinetic energy operator is defined as ˆT=∑ K∣K∣2 2a† KaK, (64) where Khere is a plane wave vector. The electron–electron interac- tion operator is (in a spin-orbital basis) ˆVee=1 2Ω∑ K≠0,K1,K24π ∣K∣2a† K1+Ka† K2−KaK2aK1, (65) where Ω is the volume of the unit cell. Finally, the Madelung energy EMshould be included to account for the self-interaction of the Ewald sum under periodic boundary conditions and21 EM=−2.837 297×(3 4π)1/3 N2/3r−1 s, (66) where Nis the number of electrons in the unit cell and rsis the Wigner-Seitz radius. We define the UEG Hamiltonian as a sum of these three terms,ˆH=ˆT+ˆVee+EM. (67) The two-body Hamiltonian ˆVeeneeds to be rewritten as a sum of squares to employ the AFQMC algorithm. It was shown in Ref. 83 that ˆVee=1 4∑ Q≠0[ˆA2(Q)+ˆB2(Q)], (68) where ˆA(Q)=√ 2π Ω∣Q∣2(ˆρ(Q)+ˆρ†(Q)) (69) and ˆB(Q)=i√ 2π Ω∣Q∣2(ˆρ(Q)−ˆρ†(Q)), (70) with a momentum transfer operator ˆρdefined as ˆρ(Q)=∑ σ∑ Ka† K+Q,σa† K,σΘ(Ecut−∣K+Q∣2 2), (71) where Θis the Heaviside step function and Ecutis the kinetic energy cutoff in the simulation. The HS operators ˆvare now ˆA(Q)and ˆB(Q). The local energy ϵn(τ) for the UEG then reads ϵn(τ)=EM+E1+E2, where the one-body energy, E1, is E1=∑ K∣K∣2 2PKK, (72) and the two-body energy, E2, is E2=1 2Ω∑ Q≠04π ∣Q∣2(ΓQ−ΛQ), (73) with the Coulomb two-body density matrix ΓQ, ΓQ=⎛ ⎝∑ K1PK1+Q,G1⎞ ⎠⎛ ⎝∑ K2PK2−Q,G2⎞ ⎠, (74) and the exchange two-body density matrix ΛQ, ΛQ=∑ K1K2PK1+Q,K2PK2−Q,K1. (75) The formation of ΓQcostsO(M2), whereas the formation of ΛQ takesO(M3)amount of work. Therefore, the evaluation of the exchange contribution is the bottleneck in the local energy evalua- tion. As noted in Ref. 84, the evaluation of the energy (and propaga- tion) can be accelerated using fast Fourier transforms, and we found this to be slower than an algorithm based on sparse linear algebra for system sizes considered in this work. Therefore, our implementation is exclusively based on sparsity. J. Chem. Phys. 154, 064109 (2021); doi: 10.1063/5.0041378 154, 064109-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Unlike the ph-ZT-AFQMC algorithm, finite-temperature esti- mators are all “pure” as opposed to “mixed,” which does not require any special treatments to measure expectation values of operators that do not commute with ˆH. For instance, the one-body and two- body energies can be straightforwardly read off from the local energy evaluation. III. COMPUTATIONAL DETAILS All simulations were performed with the open source PAUXY code.85HANDE was used to perform FCI calculations.86,87Q-Chem was used to crosscheck some of the mean-field finite-temperature calculations.88 We used 640 walkers along with the “comb” population control method.89,90ForΘ≤0.25, we found that the comb algorithm intro- duced significant population control biases, so we switched here to the pair-branch algorithm.91Multiple time steps were employed depending on rs, which are Δτ= 0.05 for rs≤1,Δτ= 0.025 for rs= 2, andΔτ= 0.005 for rs>2 all in reduced units. This choice was made to maintain the time step to be roughly the same throughout all rs values in the atomic unit. Given our time step error and population control bias study in the 54-electron super cell at zero temperature,61 we expect that the time step and population control biases will not affect the conclusions of our study. We typically averaged results over between 50 and 100 inde- pendent simulations, which typically were run with 160 CPUs for 10 hours. We used a free-electron trial density matrix and tuned the chemical potential μTsuch that ⟨ˆN⟩T=¯N. The interacting chemical potential was then tuned to also match ¯N. To determine the inter- acting chemical potential, we first use root bracketing based on a single step of the ph-FT-AFQMC algorithm (i.e., without consider- ing the error bars) until the chemical potential was determined to about an accuracy of 1%, which we call ˜μ. Following this, we per- form five poorly converged simulations (320 walkers averaged over 10 simulations) in the interval of [0.975 ˜μ, 1.025 ˜μ]. We next perform a weighted least squares fit to these five data points to determine the optimal chemical potential μ∗. Finally, we ran at this μ∗using a larger walker number and average over a large number of inde- pendent simulations to get the final results reported here. We find that the procedure works most of the time at the lower temperatures considered here, although some care needs to be taken at higher tem- peratures where the electron number varies more significantly with μ. The simulation input and output are available in Ref. 92. We will primarily focus on the regime of 0.5 ≤rs≤4 andΘ≤1, i.e., the warm dense regime, where Θ=T/TFandTFis the Fermi temperature of an unpolarized UEG. Box sizes were fixed via L=(4 3π¯N)1/3rsto enable comparison to results in the canonical ensemble. We note that the reduced unit system depends on the value of rsbecause the Fermi temperature depends on rsas kBTF=1 2(9π 4)2/31 r2s, (76) where kBis the Boltzmann constant. Due to this, for a given reduced temperature Θ, we observe the total path length [i.e., 1/( kBT)] in atomic unit to increase as O(r2 s), which makes the higher rsvalues computationally costly. A low-rank threshold of 10−6in Eq. (54) was used in every calculation.IV. RESULTS A. Assessing the phaseless constraint In ph-FT-AFQMC, the main source of error is the bias intro- duced by the phaseless constraints. The impact of this bias is heavily dependent on the quality of trial density matrices. Here, we will employ a simple non-interacting one-body trial density matrix based on the kinetic energy operator. Namely, (BT)K,K=e−Δτ(∣K∣2 2−μT), (77) which we call a free-electron (FE) trial density matrix with a trial chemical potential, μT. 1. A study of ¯N=2with M = 7 To assess the quality of ph-FT-AFQMC with the FE den- sity matrix, we can compare to exact full configuration interaction results (FT-FCI), which is possible in a very small finite basis set.93 We study the UEG model with only seven plane waves ( M= 7) and tune the chemical potential to reach an on-average two-electron system. The results of this comparison are plotted in Fig. 1 where we compare to a small system size ( ¯N=2) as a function of rsand Θ. At first look, the figure looks reasonable: ph-FT-AFQMC agrees with FT-FCI very well for low rs(high densities), which is consistent with results at zero temperature,61and the results disagree more as rs increases and as the temperature decreases. Since the T= 0 ph-ZT- AFQMC results are essentially exact for this system at all rsvalues considered here, these results unfortunately are a manifestation of the discrepancy between ph-FT-AFQMC and ph-ZT-AFQMC at the zero-temperature limit. For example, at rs= 3 and T= 0, the ph-ZT- AFQMC energy is −0.239 68(3) Eh/e, which is statistically identical to the FCI result of −0.239 68 Eh/e. This suggests that the phaseless constraint at finite temperature differs from that at zero temperature and that it is potentially considerably larger. In Sec. II A 3, we analytically showed that two factors (con- straint and chemical potential) can make the zero-temperature limit unreachable using the ph-FT-AFQMC algorithm. Incorporating other observations made in prior works,50we mention a total of four possible reasons for this disagreement: FIG. 1 . Comparison of ph-FT-AFQMC internal energies ( Eh) to exact (FCI) results as a function of temperature for ¯N=2for different values of rsandM= 7. J. Chem. Phys. 154, 064109 (2021); doi: 10.1063/5.0041378 154, 064109-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Imaginary time dependence of the average electron number and total energy in ph-FT-AFQMC for the rs= 3,N= 2, and M= 7 UEG model at β=20E−1 horΘ≈0.244, which is nearly the zero Tlimit in this small supercell. The three dif- ferent data sets correspond to three chemical potentials chosen around μ∗ FCI. In the lower panel, the horizontal line represents the FCI total energy at this density. We used 2048 walkers, the pair-branch population control algorithm, averaged over 30 independent simulations to obtain the error bars, and a time step of 0.05 E−1 h. 1. The issue of chemical potential mismatch : Since we work in dif- ferent ensembles (FT with the grand canonical ensemble and ZT with the canonical ensemble), it is important to tune μand μTproperly so that the number of electrons does not fluctuate so much along an imaginary time path as assumed throughout the discussion in Sec. II A 3. Based on Fig. 2, we suspect that this effect is quite minor since despite the fact that the aver- age particle number deviates from the desired value along the path, the effect of this deviation in other physical observables is negligible. 2. The fundamental differences between the two constraints : A sec- ond possibility is that the constraints are fundamentally dif- ferent even if μandμTare chosen properly. This is then the direct consequence of the illustration suggested by Eq. (33) in Sec. II A 3. Indeed, as can be seen in Fig. 3, we see that FIG. 3 . Imaginary time dependence of the average electron number and error in the total energy ( uph-FT-AFQMC (τ)−uFCI)) in ph-FT-AFQMC for the N= 2, M= 7 UEG model for rs= 1 (blue squares) and rs= 3 (orange circles). For rs= 1, we chose β=10E−1 h(Θ≈0.05), and for rs= 3, we chose β=40E−1 h(Θ≈0.122). The temperatures were chosen such that they corresponded to the zero-temperature limit. We used 2048 walkers, the pair-branch population control algorithm, averaged over 30 independent simulations to obtain the error bars, and a time step of 0.05 E−1 h.this appears to be the case. Toward the end of the path, we see that the ph-FT-AFQMC results begin to deviate substan- tially from the FCI result, with the magnitude of the deviation increasing with rs. This is consistent with the fact that larger rs requires longer imaginary time to project out the ground state. This longer projection time results in a larger portion of the path using a determinant ratio that does not resemble the zero- temperature overlap ratio. In the middle of the path, results are close to exact and the algorithm more closely resembles ph-ZT-AFQMC. 3. The difference between the trial density matrix and trial wave- function : A third possibility is that the FE trial density matrix is not appropriate to reproduce the correct zero-temperature trial wavefunction used in ph-AFQMC. This is an important concern, in general;50however, at the high densities and small system sizes considered here, we do not expect there to be any unrestricted Hartree–Fock solutions. Furthermore, for the UEG, RHF is equivalent to a free-electron trial in a closed shell system. Nevertheless, to verify this, we tested the thermal Hartree–Fock density matrix;94however, we see from Fig. 4 that this choice makes essentially no practical difference. 4. Time-reversal symmetry breaking : The final possibility is that as the phaseless constraint breaks imaginary time symmetry in the estimators50and we should instead average across time slices, as suggested in Ref. 50. Interestingly, we find that at rs= 3, this averaging procedure bias results to be above the exact value (see Fig. 5) and again does not resolve this dis- crepancy with the zero-temperature algorithm. This bias can also be understood in the context of Fig. 3, where the value of the energy along the imaginary time path is not symmet- ric with respect to τ. While the time-averaged estimators were found to be more accurate in the Hubbard model,50given our results, one should not expect this to be a general solution to this problem. Despite these concerns, it is also clear from Fig. 1 that ph-FT- AFQMC performs quite well with only small deviations seen from FIG. 4 . Comparison between the ph-FT-AFQMC internal energy per electron ( u) and the average particle number as a function of the chemical potential with a free- electron (FE) and thermal Hartree–Fock (THF) like trial density matrix. The vertical (horizontal) lines represent the FCI values for the chemical potential (energy and electron number). The system considered here is ¯N=2,M=7,rs=3with β=40E−1 h. J. Chem. Phys. 154, 064109 (2021); doi: 10.1063/5.0041378 154, 064109-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . Comparison between time slice averaged energies and the normal asym- metric estimator for the M= 7, ¯N=2, and rs= 3 benchmark system. Error bars, where not visible, are smaller than the markers. “No Average” refers to calculations where the global energy estimate is computed at τ=β, while “Average” averages the global estimate across different time slices, as suggested in Ref. 50. the exact result in the regimes we are interested in (i.e., Θ<1 and rs<3). 2. A study of ¯N=66with M = 57 We now study a larger UEG supercell considering ¯N=66 with M= 57. Obviously, such a parameter set-up would have a very large basis set incompleteness error and, therefore, should not be used to draw any physically meaningful results. Nevertheless, for this basis set size, FT-CCSD results are available for several Θand rsvalues37 to which we compare ph-FT-AFQMC results. We use the FE trial density matrix as before since we do not have any UHF solutions at zero temperature below rs= 3, as shown in the Appendix. The result of this comparison is shown in Fig. 6. We see that ph- FT-AFQMC agrees well with FT-CCSD for low rsin all temperatures FIG. 6 . Comparison between ph-FT-AFQMC and FT-CCSD exchange-correlation energies as a function of temperature for ¯N=66,M= 57. FT-CCSD energies are reproduced from Ref. 37. considered here. As we observed in the study of ¯N=2, the perfor- mance of ph-FT-AFQMC at low temperature may not be as good as the ph-ZT-AFQMC in the zero-temperature limit. Nonetheless, we found accurate results at low rsin the case of ¯N=2 and we observe consistently accurate results at a larger supercell ( ¯N=66) when compared against FT-CCSD. CCSD was shown to be accurate for lowrssuch as rs= 0.5 in the zero-temperature benchmark,95so this helps us build confidence on the performance of ph-FT-AFQMC at lowrs. However, the quality of FT-CCSD is expected to gradually degrade as rsincreases and Θdecreases, as indicated by its zero- temperature benchmark study.95Consequently, at rs= 4 FT-CCSD, exchange-correlation energies show erratic behavior changing the energy trend as a function of rscompletely. ph-FT-AFQMC, on the other hand, shows a smooth monotonic behavior as a function of rsin all temperatures. Given the superior performance of ph-ZT- AFQMC compared to CCSD in the UEG model,61this result is not surprising. FIG. 7 . Demonstration of low-rank com- pression efficiency (%) in the left, right, and total blocks for (a) rs= 0.5 and Θ= 0.5, (b) rs= 0.5 and Θ= 0.125, (c) rs= 4.0 and Θ= 0.5, and (d) rs= 4.0 andΘ= 0.125. All calculations are done withM= 1189, andμwas chosen so that the one-body trial density matrix satisfies ¯N=14. A threshold of 10−6was used in Eq. (54). J. Chem. Phys. 154, 064109 (2021); doi: 10.1063/5.0041378 154, 064109-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp B. Efficacy of the low-rank truncation In Fig. 7, we show the practical effectiveness of the low-rank truncation discussed in Sec. II D. We evaluate the compression per- centage based on the ratio between the rank of the left ( mL), right (mR), and total ( mT) and the number of basis functions ( M). Namely, ci=(1−mi M)×100%, (78) where i∈{L,R,T}. The set of parameters that we chose to look at this is¯N≃14,M= 1189, rs= 0.5, 4.0, and Θ= 0.5, 0.125. For the purpose of demonstration, tuning chemical potential is not important, so we setμ=μT, which ensures a correct average number of particles in the trial density matrix. Comparing Figs. 7(a) and 7(b), we see the effect of the tem- perature change at rs= 0.5. As expected, the lower the temperature, we observe the higher compression ratio. In fact, we are in the limit where M≫¯N, so the low-rank compression becomes very effective. As we move along an imaginary path, we see that the compres- sion efficiency decreases for the left (trial density matrix blocks) and increases for the right (sampled blocks). This is the feature of our propagation algorithm, which replaces ˆBTbyˆBeach time step. We observe nearly constant compression efficiency for the total block. We obtain about 80% compression in mTatΘ= 0.5 and nearly 100% compression in mTatΘ= 0.125. A similar conclusion can be drawn from Figs. 7(c) and 7(d). We see only small changes in the compression efficiency when changing from rs= 0.5 to rs= 4.0. While this technique is useful in speeding up the calculation, in general, one needs a small ¯N/Mratio and low temperature for a large saving. In our work, among larger calculations, the biggest saving was made in the case of ¯N=66 and M= 485 at Θ= 0.125, which will be presented below. For this particular example, we did not find the cost saving to be substantial due to its sizable ¯N/M, but the compression algorithm was still used for the numerical stability. C. Comparison to other approaches In Sec. IV A, we focused on assessing the accuracy of ph-FT- AFQMC on a very small supercell for a given finite basis set. The results from Sec. IV A suggest that ph-FT-AFQMC is quite accurate forrs≤3 and it is now important to assess the utility of this method in more realistic calculations specifically toward the CBS limit. FIG. 8 . Basis set convergence of the ph-AFQMC exchange-correlation ( uxc) energy at Θ= 0.5, ¯N=66. CPIMC results were converged to the basis limit.67. In Fig. 8, we investigate the magnitude of the basis set error as a function of rsatΘ= 0.5 where there exist previous exact CPIMC results as well as RPIMC data. We see that the ph-FT-AFQMC results are indistinguishable from the CPIMC results at rs= 0.5 when forM≥257 on the plotted scale. We also see that M= 485 is suffi- cient to converge results up to rs= 2 on this scale. As seen in previous studies, we find the RPIMC results to be too low; however, the mag- nitude of this bias seems to decrease with increasing rs. We can also ascribe the deviation of the FT-CCSD result from the PIMC data to the basis set size. We see that FT-CCSD and ph-FT-AFQMC agree well for M= 123 (the largest basis set considered in Ref. 38) up to rs = 1 with the FT-CCSD results deviating from the expected smooth trend with rsbeyond this. Unfortunately, we find obtaining ph-FT- AFQMC data beyond rs= 2 to be too expensive beyond M= 257 due to the smaller time steps required [equivalently the larger val- ues ofβrequired as rsincreases as in Eq. (76)]. Nevertheless, the results look promising and it is possible that reliable results could be obtained for even larger rsvalues if more computational resources were expended. With the confidence that the basis set error and phaseless error are under control in this parameter regime, we next look toward the FIG. 9 . Comparison between finite size corrected ph-FT-AFQMC (with M= 485), RPIMC,20and CPIMC67exchange-correlation energies uxcto the GDSMFB fit,65the KSDT fit,64and the corrected KSDT fit (corrKSDT).100Note that no twist averaging was performed. J. Chem. Phys. 154, 064109 (2021); doi: 10.1063/5.0041378 154, 064109-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp thermodynamic limit. To reach the thermodynamic limit, we use finite size corrections calculated from the finite-temperature ran- dom phase approximation.57,96The subject of finite size corrections is treated exhaustively elsewhere,62,66,97and we will not discuss them in any detail. In essence, we add a correction Δuxc(rs,Θ,N) to our QMC results where Δuxc(rs,Θ,N)=uRPA xc(rs,Θ,∞)−uRPA xc(rs,Θ,N), (79) where uRPA xccan be computed numerically through derivatives of the exchange-correlation free energy. Detailed equations and the code necessary for this are available in Refs. 97 and 98. We also verified the accuracy of these corrections across rsandΘ≥1 independently in our supplementary material using the existing CPIMC and PB- PIMC data.67 In Fig. 9, we compare our finite size corrected ph-FT-AFQMC results with M= 485 to the fit of Ref. 65 denoted here as GDSMFB and the RPIMC data of Ref. 20. For rs<2.0, we find an excellent agreement between our data and the GDSMFB fit, which is reassur- ing as the GDSMFB fit was generated using QMC data for the inter- action energy, not the internal energy, and also relied exclusively on finite-temperature PIMC data above Θ= 0.5, with a correction being applied to bridge the gap to the known zero-temperature result.58,99 We also compare to the KSDT fit from Karasiev et al. , which was originally fitted to the RPIMC data and subsequently reparameter- ized100(corrKSDT) to correct the zero-temperature limit and incor- porate the more accurate CPIMC and PB-PIMC data from Ref. 73. Similarly to previous work,65,100we find an excellent agreement between our QMC data and GDSMFB and corrKSDT, while we see some slight deviations at Θ= 0.25 from the KSDT fit. Note that we applied the same size corrections to the ph-FT-AFQMC and RPIMC data points and not the original size corrections from Ref. 20, which were subsequently found to be incorrect.66Thus, the RPIMC data do not generally agree with the parameterizations. Finally, we note that we observe a visible deviation of the ph-FT-AFQMC energy from the fits at Θ= 0.5 and rs= 3, which can be attributed to phaseless constraint bias and basis set incompleteness error. The agreement between ph-FT-AFQMC and the GDSMFB fit gives added confidence that this procedure was well founded and the fit of GDSMFB is highly accurate, particularly in the warm dense matter regime ( Θ≤0.5 and rs≤2). Similar to other studies,21,22,67we find that the RPIMC data are biased and generally too low; however, this bias becomes small past rs= 4. We emphasize that in this param- eter regime, previously no other methods than RPIMC could run and no reliable verification of the GDSMFB fit was available. This gives us confidence that the ph-FT-AFQMC method will be useful forab initio simulations of warm dense matters in the future. Finally, to assess the reliability of ph-FT-AFQMC for the calcu- lation of properties other than the total energy, we investigated the accuracy of the static structure factor S(K)=1 N⟨ˆρ(K)ˆρ(−K)⟩. (80) In Fig. 10, we compare our ph-FT-AFQMC results to the splined (exact) PB-PIMC and CPIMC results of Ref. 69, where we find an excellent agreement across rsatΘ= 1, which is the lowest temperature for which results exist. FIG. 10 . ph-FT-AFQMC static structure factor for the ¯N=34electron gas at Θ= 1 (markers) compared to the unbiased splined PIMC results (dashed lines) from Ref. 69. The K= 0 point has been omitted. V. CONCLUSIONS In this work, we have examined the accuracy of phaseless finite- temperature auxiliary-field quantum Monte Carlo (ph-FT-AFQMC) on the uniform electron gas (UEG) model over various Wigner-Seitz radii rsand temperatures Θ. The ultimate goal of this work was to access the regime where Θ≤0.5 and rs≤2.0. This is the regime where other commonly used QMC methods such as configuration path- integral MC (CPIMC) and permutation blocking PIMC (PB-PIMC) cannot be easily run. Furthermore, another popular flavor of QMC, restricted PIMC (RPIMC), typically exhibits a large bias for rs≤2.0, which is currently the only many-body calculations available in this regime. We summarize the findings and conclusions of our work as follows: 1. Difficulties in reaching the accuracy of the zero-temperature ph- AFQMC (ph-ZT-AFQMC) algorithm : We showed both analyt- ically and numerically that one should not expect the ph-FT- AFQMC energy to match the ph-ZT-AFQMC energy in the low-temperature limit even when the number of particles is tuned properly. 2. Utility of low-rank truncation : We demonstrated that the low- rank truncation method discussed in Ref. 82 is effective in the UEG Hamiltonian as well, especially when Θ<0.5. 3. Accuracy of ph-FT-AFQMC energies : We were able to use the ph-FT-AFQMC reliably to investigate Θ≤0.5 and rs≤2.0. Given the benchmark results on small basis sets that com- pare favorably to finite-temperature coupled cluster with sin- gles and doubles (FT-CCSD), ph-FT-AFQMC energies are expected to be accurate for a given basis set. Furthermore, we were able to run a large enough basis set so that the basis set incompleteness error is insignificant for the purpose of our study. Our work suggests that the bias in RPIMC is not neg- ligible for rs<2, and therefore, one must be cautious when using RPIMC for dense electron gas simulations as previously suggested by others.21,22,67 4. Validation of the GDSMFB fit65of the exchange-correlation energies at difficult parameter regimes : In the regime of rs≤2.0 andΘ≤0.5, no many-body methods have been able to verify J. Chem. Phys. 154, 064109 (2021); doi: 10.1063/5.0041378 154, 064109-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the GDSMFB fit whose parameterization relies on the unbi- ased PIMC data for Θ≥0.5. We found an excellent agreement between the GDSMFB fit and our ph-FT-AFQMC results. 5. Accuracy of ph-FT-AFQMC static structure factors : One bene- fit of working directly in the second quantized space is to be able to compute properties straightforwardly. As an example, we computed the static structure factor of a super cell of ¯N=34 and compared our results to the numerical exact PIMC results. We found a nearly perfect agreement between ph-FT-AFQMC and PIMC. Given our results, we are cautiously optimistic that ph-FT- AFQMC is a useful tool that is more scalable than other many-body methods based on the second quantization and can provide accurate results for the regimes where other methods either cannot run at all or cannot perform well. However, the remaining issues in ph-FT-AFQMC should not be ignored. Most notably, the inability to access the same zero- temperature limit as ph-ZT-AFQMC may become a more seri- ous issue in the future since ph-ZT-AFQMC has been shown to be accurate in many benchmark systems. In light of Eq. (33) and Ref. 54, it will be interesting to investigate different types of con- straints in the canonical ensemble, which can guarantee the same zero-temperature limit as ph-ZT-AFQMC. Furthermore, reaching the basis set limit for higher tempera- ture than Θ= 1.0 is excruciating, even though the imaginary time propagation is short. Since there is no more obvious low-rank struc- ture in the propagator, there seems not much that can be done to speed-up. Nevertheless, given the exceptional speed-up shown by the ph-ZT-AFQMC implementation for solids using graphics processing units (GPUs),101we expect that the analogous ph-FT- AFQMC implementation using GPUs will help greatly to ameliorate this situation. Looking to the future, one obvious extension will be to explore the dynamical structure factor that can be computed from analytically continued imaginary time displaced correlation func- tions in ph-FT-AFQMC.102–104These results would build upon recent promising advances in the analytic continuation of PIMC data72,105,106in the warm dense regime, where again we could poten- tially bridge the gap to lower temperatures. Another interesting avenue would be to explore the ability to compute free energy differ- ences much like is done in the interaction picture DMQMC.22While the extension of ph-FT-AFQMC to Fermi-Bose mixtures when the number of bosons is conserved was presented,51it will be interest- ing to see further development for the cases with a non-conserving boson number.107 SUPPLEMENTARY MATERIAL The supplementary material of this work is available, which contains the test of the RPA size correction used in this correction to the existing PIMC data. ACKNOWLEDGMENTS The work of J.L. was in part supported by the CCMS sum- mer internship at the Lawrence Livermore National Lab in 2018. J.L. FIG. 11 . Spin expectation value ( ⟨S2⟩) of UHF as a function of rsatΘ= 0 for the 66-electron supercell at M= 257 and M= 515. thanks Martin Head-Gordon and David Reichman for encourage- ment and support. We thank Hao Shi, Yuan-Yao He, and Shiwei Zhang for useful conversations on the FT-AFQMC algorithm. This work was performed under the auspices of the U.S. Department of Energy (DOE) by LLNL under Contract No. DE-AC52-07NA27344. The work of F.D.M and M.A.M. was supported by the U.S. DOE, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, as part of the Computational Materials Sci- ences Program and Center for Predictive Simulation of Functional Materials (CPSFM). Computing support for this work came from the LLNL Institutional Computing Grand Challenge program. APPENDIX: SYMMETRY BREAKING AT ZERO TEMPERATURE We studied the instability of RHF to UHF for the 66-electron supercell at zero temperature using the algorithm presented in Ref. 61. In Fig. 11, we present the spin expectation value ( ⟨S2⟩) of UHF as a function of rs. We found that there is no spin polarization occuring for rs<4.0 at zero temperature. As the focus of this study was the 66-electron supercell model for rs≤4.0, we did not consider UHF trial density matrices. 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5.0028664.pdf

Appl. Phys. Lett. 118, 032404 (2021); https://doi.org/10.1063/5.0028664 118, 032404 © 2021 Author(s).Enhancement of YIG Pt spin conductance by local Joule annealing Cite as: Appl. Phys. Lett. 118, 032404 (2021); https://doi.org/10.1063/5.0028664 Submitted: 06 September 2020 . Accepted: 22 December 2020 . Published Online: 19 January 2021 R. Kohno , N. Thiery , K. An , P. Noel , L. Vila , V. V. Naletov , N. Beaulieu , J. Ben Youssef , G. de Loubens , and O. Klein ARTICLES YOU MAY BE INTERESTED IN Current switching of interface antiferromagnet in ferromagnet/antiferromagnet heterostructure Applied Physics Letters 118, 032402 (2021); https://doi.org/10.1063/5.0039074 Evidence of phonon pumping by magnonic spin currents Applied Physics Letters 118, 022409 (2021); https://doi.org/10.1063/5.0035690 Impact of the interplay of piezoelectric strain and current-induced heating on the field-like spin–orbit torque in perpendicularly magnetized Ta/Co 20Fe60B20/Ta/MgO film Applied Physics Letters 118, 032401 (2021); https://doi.org/10.1063/5.0035869Enhancement of YIG jPt spin conductance by local Joule annealing Cite as: Appl. Phys. Lett. 118, 032404 (2021); doi: 10.1063/5.0028664 Submitted: 6 September 2020 .Accepted: 22 December 2020 . Published Online: 19 January 2021 R.Kohno,1 N.Thiery,1K.An,1P.Noel,1 L.Vila,1 V. V. Naletov,1,2 N.Beaulieu,3,4J. Ben Youssef,4 G.de Loubens,3 and O. Klein1,a) AFFILIATIONS 1Universit /C19e Grenoble Alpes, CEA, CNRS, Spintec, 38054 Grenoble, France 2Institute of Physics, Kazan Federal University, Kazan 420008, Russian Federation 3SPEC, CEA-Saclay, CNRS, Universit /C19e Paris-Saclay, 91191 Gif-sur-Yvette, France 4LabSTICC, CNRS, Universit /C19e de Bretagne Occidentale, 29238 Brest, France a)Author to whom correspondence should be addressed: oklein@cea.fr ABSTRACT We report that Joule heating can be used to enhance the interfacial spin conductivity between a metal and an oxide. We observe that local annealing of the interface at about 550 K, when injecting large current densities ( >1012A=m2) into a pristine 7 nm thick Pt nanostrip evaporated on top of yttrium iron garnet (YIG), can improve the effective spin transmission up to a factor of 3. This result is of particularinterest when interfacing ultrathin garnet films to avoid strong chemical etching of the surface. The effect is confirmed by the following methods: spin Hall magnetoresistance, spin pumping, and non-local spin transport. We use it to study the influence of the YIG jPt coupling on the non-linear spin transport properties. We find that the crossover current from a linear to a non-linear spin transport regime isindependent of this coupling. Published under license by AIP Publishing. https://doi.org/10.1063/5.0028664 The transport of pure spin information through localized magnetic moments is at the heart of a new research topic calledinsulatronic (for insula -tor spin- tronic ). 1–4Interest stems here from the recognition that magnetic insulators are superior spin conductors compared to metals or semi-conductors. Among magnetic insulators, garnets, and, in particular, yttrium iron garnet (YIG), have the lowest known magnetic damping.5One can exploit here the spin Hall effect (SHE) to interconvert pure spin currents circulating inside the dielec- tric into charge currents, which can then be probed electrically. This is usually achieved by depositing a heavy metal electrode, advantageously Pt,6,7on top of the YIG surface. The Pt electrode allows us to probe the ensuing flow of spin escaping through the metal-oxide interface by spin pumping (SP),8spin Hall magnetoresistance (SMR),9–11spin Seebeck effect (SSE),12or spin–orbit torque (SOT) using non-local transport devices.1,13 The efficiency of the process is determined by the spin trans- parency of the interface and parameterized by the so-called spinmixing conductance, G "#. To optimize its strength, the YIG surface is usually treated by strong process such as Oþ/Arþplasma,14 annealing,15and piranha etching16,17prior to the metal deposition in order to achieve a good chemical and structural YIG jPtinterface. However, these treatments are performed on lm thick YIG samples and are difficult to implement once the thick-ness of the film is in the nanometer range as a modulation of sur- face roughness significantly disturbs its magnetic properties. An enhancement of SMR by global annealing at very high tempera- ture 18,19has been reported, yet this process is not necessarily com- patible at the device level. In addition to the cleaning of the YIGsurface, the use of sputtering technique to deposit the metal is known to lead to better G "#than the evaporation technique.20The latter, however, usually leads to lower resistivity, which should be favored when one wants to inject large current densities. Considering that evaporation is also advantageous to have better liftoff during nanofabrication, certainly a process allowing us to improve interfacial quality of evaporated Pt is desired. In this paper, we investigate the impact of local Joule heating at 550 K on the spin transport between thin YIG film and evaporated Pt. We use SMR and SP measurements to show a clear 3 times post-deposition enhancement of the interfacial spin transmission, which is i r r e v e r s i b l e .W eh a v ee x p l o i t e dt h i sf e a t u r et os t u d yt h ei n fl u e n c eo f the interfacial spin conductivity on non-linear spin transport proper- ties in lateral devices. We find that any enhancement of the coupling Appl. Phys. Lett. 118, 032404 (2021); doi: 10.1063/5.0028664 118, 032404-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplto the electrodes seems to play a negligible role in determining the non-linear characteristics of pure spin transport. We use a tYIG¼56 nm thick YIG film grown by liquid phase epitaxy on a 500 lmG d 3Ga5O12(GGG) substrate.9,21Ferromagnetic resonance experiments have shown a damping parameter ofa YIG¼2/C210/C04revealing an excellent crystal quality of the YIG film.22Two similar Pt nanostrips, respectively, Pt 1and Pt 2,a r e patterned by e-beam lithography to have a width of 300 nm andlength of 30 lm. A 7 nm thick Pt layer is then deposited by e-beam evaporation on the YIG film. The nanostrips are connected to Ti jAu (5 nm j50 nm) electrical contacts. The sample is mounted on a rota- tional stage and exposed to an in-plane magnetic field of l 0H0 ¼200 mT to fully magnetize the YIG film. All the magneto-transport experiments are performed at room temperature, T0¼300 K. The schematic of the sample geometry is shown in Fig. 1(a) . Transport measurements are performed by injecting 10 ms current pulses with a duty cycle of 10% into the Pt 1nanostrip via a 6221 Keithley current source which is synchronized to a 2182A Keithley nanovoltmeter. Wefirst investigate the configuration where the nanovoltmeter is con-nected to the same nanostrip to extract the magnetoresistive responsethrough V 1.23Figure 1(b) represents the evolution of the Pt 1resistance, RPt, as a function of the electrical current, I, showing a quadratic dependence due to Joule heating. The Pt electrode can also be used asa temperature sensor. The temperature rise of the Pt is simply inferredfrom the change of Pt resistance with T/C0T 0¼fPtðRPt/C0R0Þ=R0, where R0¼2:6kXis the Pt resistance at I¼0a n d fPt¼478 K is a thermal coefficient specific to our Pt nanostrip. In our structure, the local temperature reaches about 550 K when the current density isJ max¼1:2/C21012A/m2.The spin transparency of the interface can be evaluated from the SMR ratio defined as ðRu/C0RjjÞ=R0, which measures the relative change of the Pt resistance as a function of u, the azimuthal angle between H0(and thus the magnetization) and the x-axis. In this nota- tion, Rjji st h er e s i s t a n c ew h e n H0is applied parallel to the nanostrip direction ( jjcorresponds to u¼690/C14ory-axis). Figure 1(c) presents the SMR signal measured with a bias current of I¼100lA. The data in blue dots show the values obtained directly after the nanofabricationprocess. The angular dependence follows a cos 2ubehavior (see solid line fit), and the maximum SMR deviation is observed when H0is applied perpendicular to the nanostrip direction ( ?corresponds to u¼0/C14orx-axis) with a value of ðR?/C0RkÞ=R0¼/C02:9/C210/C05 extracted from the fit (blue line). From the theory of SMR,11,24the amplitude of the SMR ratio is expressed as R?/C0Rk R0¼/C0h2 SHE2ðksf2=tPtÞqg"#tanh2tPt 2ksf/C18/C19 1þ2ksfqg"#cothtPt ksf/C18/C19 ; (1) where q¼19:5lXcm is the electrical resistivity inside the Pt layer, with tPtits thickness, ksfits spin diffusion length, and hSHEits SHE angle. In our notation, g"#is an effective spin mixing conductance of the YIG jPt interface (see discussion below). Using SMR measurement, we then investigate the effect of 550 K local Joule annealing on the effective spin mixing conductance ofYIGjPt. The electrical annealing is provided by applying current den- sity pulses of J max¼1:2/C21012A/m2for about 60 min. Considering that the pulses are applied with a duty cycle of 10%, the cumulatedannealing time is only 6 min. The red dots show the SMR ratio mea-sured after annealing. The amplitude is increased to ðR ?/C0RkÞ=R0¼ /C08:9/C210/C05[red line fit in Fig.1(c) , which is 3 times larger than the value before annealing (blue line)]. This enhanced SMR is irreversible.Increasing the annealing time above an hour leads to negligible gain inthe SMR value. We also observe that the Pt 1resistance is changed by less than 1%, indicating no major structural changes in the Pt, whichsuggests that h SHEandksfremain the same throughout this treat- ment.25,26Knowing the product hSHEksfto be 0.18 nm,27we conclude that the spin mixing conductance is improved from g"#=ð2e2=hÞ ¼0:64/C21018m–2to 1.90 /C21018m–2.I tm e a n st h a t g"#/C28ðqksfÞ/C01 and thus the denominator of the right hand side of Eq. (1)is roughly 1. The enhanced value is comparable to the ones obtained after Arþ-ion milling process14and “piranha” etch,16,17which means that the observed increase in g"#is more a catching up of the deficit of spin conductance probably associated with the use of evaporation ratherthan an overall improvement of the result from what is obtained bysputtering. Next, we focus on spin pumping measurements using the same sample batch. The experiment is performed at a fixed fre-quency of 9.65 GHz in a X-band cavity while applying a static in-plane magnetic field perpendicularly to the Pt nanostrip ( x-axis). Conversely, the rf magnetic field h rfis applied along the Pt nano- strip ( y-axis). The rf power is fixed at 5 mW ( l0hrf¼2lT) to maximize the SP signal while minimizing non-linear effects suchas distorsion of the line shape (foldover effect). 28,29At the ferro- magnetic resonance condition, a flow of angular momentum fromthe YIG relaxes into the Pt. 30 FIG. 1. (a) Schematic of the YIG jPt structure with two nanostrips oriented along the y-direction. The first one (Pt 1) is connected to a current source, and it is used to measure the SMR ratio. The second electrode (Pt 2) is used to measure the non- local spin transport properties. (b) Resistance of the Pt 1nanostrip, RPt¼V1=I,a s a function of the injected current Iinside. (c) Angular dependence of the SMR ratio when an applied magnetic l0H0¼200 mT is rotated in the xyplane. The data show the result before (blue dots) and after (red dots) local Joule annealing at550 K. Solid lines are fit with cos 2u.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 032404 (2021); doi: 10.1063/5.0028664 118, 032404-2 Published under license by AIP PublishingThe normalized spin-pumping spectra ispare obtained by dividing the generated inverse spin Hall effect (ISHE) voltage by both h2 rfand the Pt resistance R0, i.e., isp¼VISHE=ðR0h2 rfÞ. T h es h a p eo ft h es p e c t r a ll i n ed o e sn o tf o l l o was i m p l e Lorentzian, which is attributed to inhomogeneous broadening. The main peak of the spectrum is identified as the Kittel mode (uniform precession), and we have measured its amplitude to estimate the effi- ciency of spin transmission through the interface. After characterizingt h ep r i s t i n eY I G jPt interface, we performed a local annealing by apply- ing pulse current density of J max¼1.2/C21012A/m2for various dura- tions (10, 30, and 60 min) following the same procedure as before. The effect of annealing on the spectra is shown in Fig. 2(a) . Similar to the SMR measurement, Joule annealing increases the spin pumping signal. From the amplitude of the main peak of the spectra and using the model in Ref. 30, one can estimate the enhancement of g"#from 0.60 /C21018m–2to 1.23 /C21018m–2, compatible with our previous SMR estimation. The interesting feature of this experiment lies in the measure- ment of the full width at half maximum (FWHM) of the main peak. As the amplitude of the peak becomes larger with the annealing time, FWHM remains constant over the whole annealing process [see Fig. 2(b) ]. The modulation of FWHM can be in general attributed to the extra damping induced by the coupling between YIG and Pt.31,32 The constant FWHM reveals that the linewidth is mostly controlledby the bare YIG film Aside from/In addition to the nanometric Ptnanostrip rather than the sole relaxation of precession dynamics at the YIGjPt interface. This suggests that the additional relaxation channel provided by the adjacent metallic nanostrip is a weak perturbation of the overall relaxation of the extended YIG thin film underneath. Since this finding differs from what is observed when the adjacent Pt covers the whole YIG film, 22we attribute the difference to finite size effects. For nanostructured Pt electrodes, the additional relaxation channel ofthe magnons in the YIG provided by the adjacent metal becomes weak when the lateral size of the Pt is smaller than the magnon wave- length. 33We emphasize that such scenario would then mostly concern the long wavelength magnons, such as those excited by the cavity.33 Finally we study the impact of local Joule annealing on the non- local spin transport properties. In these samples, the second Pt nano- strip (Pt 2), placed 2 lm away (edge to edge distance) from the firstnanostrip (Pt 1), is used as a detector of the spin current [see Fig. 1(a) ]. When a charge current is sent to the Pt 1nanostrip (injector), a spin accumulation is generated at the YIG jPt interface due to the SHE and its angular momentum is transferred from Pt to YIG.35The angular momentum is then carried in YIG by magnons, which are then detected via the ISHE voltage at the Pt 2nanostrip (detector). Again, the non-local spin transport is probed by applying pulses of electri- cal current, I, in the injector while simultaneously monitoring the non-local voltage Vu;I/C17V2ðu;IÞon the Pt 2detector as a function of the angle u. Similar to the SMR measurement, we use the background-subtracted voltage dVu¼Vu/C0Vjjto extract the spin contribution,13with Vjjbeing the voltage background drop mea- sured when the externally applied magnetic field, H0,i so r i e n t e d parallel to the Pt stripe. We distinguish SSE from SOT by defining two quantities based on the yzmirror symmetry:13Ru;I;Du;I /C17ðdVu;I6dV/C22u;IÞ=2, where /C22u¼p/C0u. Previously we have reported13that in 18 nm thick YIG film, injecting electrical current above a crossover threshold of the order of Jc¼6.0/C21011A/m2is sufficient to excite low energy magnetostatic magnons (in the GHz range). This phenomenon is characterized by the emergence a non-linear spin conduction as a function of the applied current. In the following, we want to use this incident change of the interface transmission to investigate the influence of non- equilibrium spin accumulation at the YIG jPt interface on the non- linear properties. L e tu sfi r s tc o n s i d e ri n Fig. 3(a) the pristine state where neither interface of YIG jPt at the injector nor the detector has been treated by Joule annealing. In the top panel (a) of Fig. 3 , we display the non-local Ru¼0;IandDu¼0;I(the values of RandDwhen H0is applied along the x-axis) as a function of the applied current in the injector. It can be seen that the D0;Ishows a quadratic rise due to its thermal origin while theR0;Iseems to evolve quasi-linearly with current Ion the range explore. The non-linear properties are analyzed by plotting in the inset ofFig. 3(a) the current dependence of r0/C1I=R0;I,w h e r e r0¼ð@R0;I=@IÞjI¼0is the slope of a linear regression through the R0;Idata measured at I<0.5 mA (see gray dashed lines). We define Ic as the intercept with the 75% decrease.13,34We report on the main graph the estimate of Ic, which is here not precise due to the low spin conductivity of the device. Next, we perform Joule annealing of the injector (Pt 1) for 60 min with the same procedure as before. The impact on the non-local signals can be seen in the second panel (b). We observe that the R0;Iis now 3 times larger, indicating that the excited magnon density in YIG is enhanced due to the higher spin transmission at the injector. Remarkably, now we are able to distin- guish clearly on the figure at Jc¼7/C21011A/m2,t h ec r o s s o v e r threshold from a linear to a non-linear spin transport regime, which is the signature of the participation of low energy magnetostatic mag- nons to spin transport. Such magnons are in principle solely excited by SOT exerted at the interface between YIG and the injector. We also note that the D0;I-signal produced by thermally generated magnons remains unchanged. This is expected because the spin conversion ofSSE signal occurs only at the detector Pt 2nanostrip (which is not annealed for this moment) whereas the injector nanostrip only plays the role of a heater. The last step, shown in panel (c), both injector and detector are annealed. The D0;Iis enhanced by a factor of 3 due to the higher spin conversion of the probing interface. As can be seen there too is that FIG. 2. The (a) shows spin pumping spectra at different annealing steps by inject- ing a current density Jmax¼1:2/C21012A/m2into the Pt nanostrip for three differ- ent annealing times. The inset (b) shows the normalized spin pumping spectra ateach annealing time.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 032404 (2021); doi: 10.1063/5.0028664 118, 032404-3 Published under license by AIP PublishingtheR0;Iis now 9 times larger than the reference [panel (a)]. It follows the fact that injection and detection of magnons at each YIG jPt inter- face are now 3 times more effective, leading to a factor of 9 increase in the SOT signals. Nonetheless, the crossover current density Jc13is not affected by the annealing, and it occurs at the same value for both cases(b) and (c). It supports the observation in SP experiments, where local annealing has not induced additional broadening the linewidth (inset ofFig. 2 ). It also highlights the striking difference of out-of-equilib- rium behavior between closed (nano-pillar) and open (extended films) magnetic geometries. A possible explanation compatible with the observed behavior could be an improved wettability of the Pt on YIG after local Jouleannealing. 36The effect could be parameterized by introducing an addi- tional extrinsic transmission coefficient 0 <T61 to the spin transpar- ency of the interface, T¼ g"#=G"#, which represents the effective contact area ratio between the YIG and the Pt. The reduction of the contact surface results in a smaller effective spin conductance g"#of the nanopatterned Pt electrodes, the intrinsic spin mixing conductanceG "#of the YIG jPt interface not being affected. This is distinct from the concept of effective spin mixing conductance introduced in SPmeasurements of YIG films fully covered by Pt, which originates from spin backflow.37Since the enhancement of a factor of 3 is consistently observed in all the devices, we believe that the change of contact area must have a geometrical origin probably linked to the nanofabrication process. Because the temperature does not exceed 550 K during theheating pulses, it is unlikely that the chemical and structural quality of the YIG surface is affected by this treatment. 38,39Thus we expect the local Joule annealing to affect only the interface between Pt nanostrip and the YIG layer,40which could result in a larger coverage of Pt onto the YIG surface. This improves the number of spin transmission chan- nels at the interface which poses similar spin mixing conductance. Through this mechanism the emission, reflection or absorption of the spin current can be enhanced. In summary, we consistently observed enhancement of the spin- induced voltage at the YIG jPt interface after local Joule annealing of evaporated Pt nanostrips. This enhancement possibly occurred through a higher coverage of Pt on the YIG surface, increasing the number of spin transmission channels available. These results can also explain the large difference in G"#;hSHE,a n d ksfin works performed via different deposition and measurement methods.41–43Additionally, the spin pumping measurements and non-local magnon transport measurements showed that an enhancement of the spin transmission does not necessarily involve an increase or reduction of the crossover threshold current to excite magnetostatic magnons. It points out the importance of the bare YIG film away from the Pt nanostrip in the relaxation of pure spin current transport. This work was supported in part by Grant No.18-CE24-0021 from the ANR of France. V.V.N. acknowledges support from UGA through the invited Professor program and from the Russian Competitive Growth of KFU. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1L. J. Cornelissen, J. Liu, R. A. Duine, J. Ben Youssef, and B. J. van Wees, Nat. Phys. 11, 1022 (2015). 2S .T .B .G o e n n e n w e i n ,R .S c h l i t z ,M .P e r n p e i n t n e r ,K .G a n z h o r n ,M . Althammer, R. Gross, and H. Huebl, Appl. Phys. Lett. 107, 172405 (2015). 3Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H.Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh,Nature 464, 262 (2010). 4A. Brataas, B. van Wees, O. Klein, G. de Loubens, and M. Viret, Phys. 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Non-local spin signals R0;I¼Ru¼0;I(left) and D0;I¼Du¼0;I(right) as a function of injected current Ifor different annealing steps. The inset of the left panel displays the current variation of r0/C1I=R0;I, where r0¼@IR0;IjI¼0is the initial slope of the spin current increase (see gray dashed line). The intercept with the0.75 level is a landmark that defines our so-called crossover current, J c.13,34The (a) panel represents non-local spin signals measured directly after the nanofabrica- tion. The (b) panel shows the result when the injector is annealed. The (c) panelshows the result when both injector and detector nanostrip are annealed.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 032404 (2021); doi: 10.1063/5.0028664 118, 032404-4 Published under license by AIP Publishing12K-i Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh, Appl. Phys. Lett. 97, 172505 (2010). 13N. Thiery, A. Draveny, V. V. Naletov, L. Vila, J. P. Attan /C19e, C. 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Yang, Nano Lett. 20, 7257 (2020). 34In a single mode model, r0/C1I=R0;Ifollows a parabola, which intercepts the abscissa at the threshold current density for damping compensation, Jth, and where the 75% cross-over landmark corresponds to half the threshold value Jc¼Jth=2. 35S. S.-L. Zhang and S. Zhang, Phys. Rev. Lett. 109, 096603 (2012). 36For the sake of completeness, we have also annealed the whole sample in an oven at the same temperature during an hour. No significant changes of the interfacial spin conductance were observed. indicating the importance of per-forming local Joule annealing. 37M. Haertinger, C. H. Back, J. Lotze, M. Weiler, S. Gepr €ags, H. Huebl, S. T. B. Goennenwein, and G. Woltersdorf, Phys. Rev. B 92, 054437 (2015). 38Y.-H. Rao, H.-W. Zhang, Q.-H. Yang, D.-N. Zhang, L.-C. Jin, B. Ma, and Y.-J. Wu, Chin. Phys. B 27, 086701 (2018). 39In this case, the annealing temperature of 550 K must correspond to the melt- ing temperature of an organic residue on the surface preventing contact between the Pt and the YIG. 40E. T. Papaioannou, P. Fuhrmann, M. B. Jungfleisch, T. Br €acher, P. Pirro, V. Lauer, J. L €osch, and B. Hillebrands, Appl. Phys. Lett. 103, 162401 (2013). 41M. B. Jungfleisch, A. V. Chumak, V. I. Vasyuchka, A. A. Serga, B. Obry, H. Schultheiss, P. A. Beck, A. D. Karenowska, E. Saitoh, and B. Hillebrands, Appl. Phys. Lett. 99, 182512 (2011). 42N. Vlietstra, J. Shan, V. Castel, J. Ben Youssef, G. E. W. Bauer, and B. J. van Wees, Appl. Phys. Lett. 103, 032401 (2013). 43L. Liu, R. A. Buhrman, and D. C. Ralph, “Review and analysis of measurements of the spin hall effect in platinum,” arXiv:1111.3702 (2012).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 032404 (2021); doi: 10.1063/5.0028664 118, 032404-5 Published under license by AIP Publishing

5.0038047.pdf

J. Chem. Phys. 154, 044302 (2021); https://doi.org/10.1063/5.0038047 154, 044302 © 2021 Author(s).Significant bonding rearrangements triggered by Mg4 clusters Cite as: J. Chem. Phys. 154, 044302 (2021); https://doi.org/10.1063/5.0038047 Submitted: 19 November 2020 . Accepted: 23 December 2020 . Published Online: 25 January 2021 Eva Vos , Inés Corral , M. Merced Montero-Campillo , and Otilia Mó COLLECTIONS Paper published as part of the special topic on Special Collection in Honor of Women in Chemical Physics and Physical Chemistry ARTICLES YOU MAY BE INTERESTED IN Computational density-functional approaches on finite-size and guest-lattice effects in CO2@sII clathrate hydrate The Journal of Chemical Physics 154, 044301 (2021); https://doi.org/10.1063/5.0039323 Full-frequency GW without frequency The Journal of Chemical Physics 154, 041101 (2021); https://doi.org/10.1063/5.0035141 Excitonic and charge transfer interactions in tetracene stacked and T-shaped dimers The Journal of Chemical Physics 154, 044306 (2021); https://doi.org/10.1063/5.0033272The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Significant bonding rearrangements triggered by Mg 4clusters Cite as: J. Chem. Phys. 154, 044302 (2021); doi: 10.1063/5.0038047 Submitted: 19 November 2020 •Accepted: 23 December 2020 • Published Online: 25 January 2021 Eva Vos, Inés Corral, M. Merced Montero-Campillo, and Otilia Móa) AFFILIATIONS Departamento de Química (Módulo 13, Facultad de Ciencias) and Institute of Advanced Chemical Sciences (IadChem), Universidad Autónoma de Madrid, Campus de Excelencia UAM-CSIC, Cantoblanco, 28049 Madrid, Spain Note: This paper is part of the JCP Special Collection in Honor of Women in Chemical Physics and Physical Chemistry. a)Author to whom correspondence should be addressed: otilia.mo@uam.es ABSTRACT The structure, stability, and bonding of the complexes formed by the interaction of Mg 4clusters and first row Lewis bases, namely, ammonia, water, and hydrogen fluoride, have been investigated through the use of high-level G4 single-reference and CASPT2 multireference for- malisms. The adducts formed reflect the high electrophilicity of the Mg 4cluster through electron density holes in the neighborhood of each metallic center. After the adduct formation, the metallic bonding of the Mg 4moiety is not significantly altered so that the hydrogen shifts from the Lewis base toward the Mg atoms lead to new local minima with enhanced stability. For the particular case of ammonia and water, the global minima obtained when all the hydrogens of the Lewis base are shifted to the Mg 4moiety have in common a very stable scaffold with a N or an O center covalently tetracoordinated to the four Mg atoms, so the initial bonding arrangements of both reactants have com- pletely disappeared. The reactivity features exhibited by these Mg 4clusters suggest that nanostructures of this metal might have an interesting catalytic behavior. Published under license by AIP Publishing. https://doi.org/10.1063/5.0038047 .,s INTRODUCTION Nanoscience is one of the areas that has undergone a spectacu- lar development in the last two decades, as a direct result of the novel and unexpected properties exhibited by nanoparticles, with a typi- cal size in the nm scale.1,2What makes nanoparticles so interesting is that their behavior and reactivity differ significantly from those of the bulk. In fact, the so-called nano-effects are observed when reach- ing a critical size.3Perhaps, one of the most paradigmatic examples to illustrate huge differences associated with the property change in the size scale is gold. Starting already with physical properties such as the melting point, whereas the melting point of bulk gold is 1064○C, the melting point of gold nanoparticles of about 1.5 nm is less than half of that value (500○C).4Even the typical yellow color of this metal changes to blue for nanoparticles of around 50 nm or smaller.1 The effects are also dramatic as far as the electron density distri- bution is concerned, as the nanoparticles do not have conduction bands and exhibit a rather peculiar reactivity. Indeed, whereas gold is usually considered an inert metal, gold nanoparticles are very reac- tive and behave as good catalysts5–8due to the presence of σ-holesassociated with regions of depleted electron density,9rendering them good electron acceptors. These features moved us to investigate the behavior of clus- ters involving alkaline-earth metals, because these elements, starting from Be and Mg, are characterized by being electron-deficient ele- ments able to form the so-called alkaline-earth (beryllium or mag- nesium) bonds.10,11The formation of alkaline-earth bonds results in a significant electron density redistribution of the Lewis base inter- acting with the Be or the Mg derivative.12,13As a consequence, the formation of alkaline-earth bonds modulates the strength of other non-covalent interactions, in which the Lewis base participates.14–18 They also contribute to create σ-holes,19to spontaneously produce radicals,20or to behave as extremely efficient electron21and anion sponges.22 In this study, we decided to focus our attention to Mg rather than Be clusters, due to the high toxicity of Be. Previous theoretical studies on small Mg nclusters found Mg 4and Mg 10to be magic clus- ters, due to the completion of the valence shells with 8 and 20 elec- trons, respectively, in agreement with the predictions of the jellium model of metal clusters.23Later studies using molecular dynamics J. Chem. Phys. 154, 044302 (2021); doi: 10.1063/5.0038047 154, 044302-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp confirmed the magic nature of Mg 4and Mg 10clusters.24By explor- ing the interaction of Mg 4clusters with conventional Lewis bases, namely, ammonia, water, and hydrogen fluoride, through the use of high-level ab initio calculations, we expect to have a first estimate of the catalytic potentiality of larger clusters, since it has been shown that bare Mg n(n<80) clusters can be efficiently obtained by the pickup of atoms into superfluid helium droplets.25 COMPUTATIONAL DETAILS It is well established that electron correlation effects are cru- cial to describe Mg 4,26whereas SCF calculations predict an almost unbound system.27For the sake of high accuracy, we decided, ini- tially, to use high-level G4 ab initio calculations to study the struc- ture and stability of the complexes formed upon the interaction of Mg 4clusters and the three Lewis bases indicated above. The G4 for- malism is a composite method in which electron correlation effects are accounted for by using Moller–Plesset perturbation theory up to fourth-order and CCSD(T) coupled cluster theory, with a final cor- rection for the Hartree–Fock limit evaluated using an extrapolation procedure and quadruple–zeta and quintuple–zeta basis sets.28The standard G4 theory uses B3LYP/6-31G(2df,p) optimized geometries, but since in our case we are dealing with weak interactions, where dispersion effects may be important as well as the use of diffuse func- tions, we decided to use a MP2/aug-cc-pVTZ formalism, instead of the standard B3LYP/6-31G(2df,p) one, to obtain the geometries and thermochemical corrections in our G4-type calculations. In fact, correlation effects were found to be crucial already to describe the bonding between Lewis bases and the single Mg atom, showing a considerable charge polarization of the electron density on the metal.29 Since, as mentioned above, the description of Mg nand Be nclus- ters represents a challenge from the theoretical viewpoint; due to the crucial role of electron correlation effects,26,30we have also explored whether a single-reference formalism would be appropriate. Quite unexpectedly, we found that single-reference procedures present for Mg 4and Mg 4–LB (LB = Lewis base) complexes a RHF-UHF insta- bility, to the best of our knowledge, not reported before in the lit- erature. Therefore, we have resorted to a multireference approach to accurately describe these systems. In particular, all the Mg 4– LB clusters were optimized with the complete active space second order perturbation (SS-CASPT2)31,32method combined with the cc- pVTZ basis set,33,34as implemented in Bagel software.35Different active spaces, including from 8 up to 12 orbitals, were employed for the construction of the complete active space self-consistent field (CASSCF) reference wavefunctions, depending on the Lewis base considered. The active spaces of the three families of adducts include 8 out of the 16 orbitals built from the linear combination of the 3s and 3p orbitals of the Mg atoms (the most correlated occupied and unoccupied orbitals). Additionally, the active spaces for Mg 4– H2O and Mg 4–HF complexes, respectively, include 2 and 1 pair ofσand σ∗orbitals sitting on the Lewis base. Final energies, in the case of Mg 4–NH 3clusters, were recomputed augmenting the active space in six extra orbitals, corresponding to the σand σ∗ orbitals of NH 3. The active space composition for the initial a1,w1, andhf1 adducts is collected in Figs. S1–S8 of the supplementary material. To reduce the computational cost of these calculations, we have excluded the 1s orbitals of all the heavy atoms from thecorrelation treatment, and we have employed the density fitting approach. The required auxiliary basis set for the density fitting calculation was generated using the PySCF code.36 The multireference character of the localized minima was quantitatively evaluated with the multireference diagnostic, M, from Tishchenko et al. , defined in this particular case (closed shell sys- tems), in terms of natural orbital occupation numbers of the most correlated occupied and unoccupied natural orbitals.37 To analyze the bonding characteristics of the complexes under investigation, we have initially used the molecular electrostatic potential (MEP), which corresponds to the (attractive or repulsive) potential that a unit positive charge experiences when approaching the cluster. This potential is defined as V(r)=∑AZA ∣RA−r∣−∫ρ(r′)dr ∣r′−r∣, (1) where Z AandRAare the charge and position of nucleus A and ρ(r) is the electron density, which usually provides reliable information on the electrophilicity or nucleophilicity of a given site of the sys- tem. The MEP is also a good procedure to detect the existence of σ-holes. A second useful approach is the quantum theory of atoms in molecules (QTAIM),38,39able to locate the critical points of the elec- tron density of any chemical system. The values of the electron den- sity at the so-called bond critical points (BCPs) are a good measure of the strength of the linkage and also provide information about its covalent character, through the values and sign of the Laplacian and the energy density. This information is nicely complemented by the one obtained by using other two alternative formalisms, namely, the Non-covalent Interaction (NCI) index formalism40,41and the elec- tron localization function (ELF).42The NCI is a method based on the use of the reduced density gradient that allows us to find regions in the real space where these interactions do actually take place, dis- tinguishing qualitatively and quantitatively between strong or weak attractive and repulsive interactions. The ELF divides the molecu- lar space in different kinds of basins that can be associated with core electrons, lone pairs (monosynaptic), and bonding regions (disynap- tic or polysynaptic basins). For all these techniques, AIM, NCI, and ELF, the one-particle reduced density matrix was obtained from the CASSCF wavefunctions. A more detailed description of these three procedures has been included in the supplementary material. RESULTS AND DISCUSSION Mg4cluster As previously reported in the literature,23,24,26,43our MP2/aug- cc-pVTZ geometry optimizations indicate that Mg 4clusters exhibit a tetrahedral structure, with Mg–Mg bond distances of 3.047 Å. The G4 calculated binding energy on this structure is 38.6 kJ/mol per atom, which is significantly lower than the value obtained using density-functional molecular dynamic methods (51.1 kJ/mol per atom),23but in rather good agreement—only slightly higher—than previous Born–Oppenheimer local-spin-density molecular dynam- ics reported values (34.7 kJ/mol and 38.6 kJ/mol per atom).24 As far as the bonding features are concerned, the NCI 3D plot in Fig. 1(a) shows that the Mg 4tetrahedral cluster exhibits a struc- ture, in which the four Mg atoms share their valence electron pairs leading to a typical metallic bonding arrangement. Consistently, the J. Chem. Phys. 154, 044302 (2021); doi: 10.1063/5.0038047 154, 044302-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . Bonding characteristics for Mg 4. (a) NCI 3D plot; (b) ELF(=0.75) plot; (c) AIM molecular graph; (d) molecular electrostatic potential on the 0.001 a.u. isodensity surface. In the ELF plots, the numbers correspond to the electron populations of the different monosynaptic (red lobes) or trisynaptic basins (green lobes). In the AIM molecular graph, the green, red, and blue dots indicate bond, ring, and cage critical points, respectively. The electron densities at these points are in a.u. In the MEP, blue areas indicate the most positive sites, i.e., electrophilic areas for a preferential nucleophilic attack. corresponding NCI 2D plot in the attractive region (see Fig. S9 in the supplementary material) presents a single peak at low density values. This metallic character is also seen in the ELF plot [Fig. 1(b)] showing different trisynaptic basins (green) at the edges of the tetra- hedron, whereas the population at the corners is located in monosy- naptic (red) basins. A value of the ELF equal to 0.75 was chosen to visualize the polysynaptic basins between Mg atoms, whereas at 0.85, only the monosynaptic Mg ones can be located. The molecular graph obtained with the AIM method [Fig. 1(c)] also corroborates this metallic character because the electron densities at the bond, ring, and cage critical points are very close, indicating a rather com- plete electron delocalization. Bond paths are slightly curved, simi- larly to what was observed in strained systems such as cyclopropane. Very interestingly, however, the molecular electrostatic potential of Mg 4[see Fig. 1(d)] shows at the corners of the tetrahedron’s well defined areas of positive potential (holes), associated with preferen- tial sites for nucleophilic interactions. Similar situations have been reported for other metallic clusters such as gold,9but likely due to the electron-deficient character of Mg, the values reported here for these Mg 4positive potentials are more than twice larger. Mg4–L (L = NH 3, H2O, HF) complexes As expected from the characteristics of the molecular elec- trostatic potential around the Mg 4cluster described in theComputational Details section, its interaction with the Lewis bases included in our study implies the nucleophilic attack of the base on one of the electrophilic Mg atoms. This is illustrated in Fig. 2 for the particular case of ammonia, but similar results are obtained when dealing with water or hydrogen fluoride, as illustrated in Fig. S10 of the supplementary material. A comparison of the NCI 3D in Figs. 1(a) and 2(a) shows that the interaction with ammonia does not perturb significantly the metallic bond observed for the isolated Mg 4cluster. A new and strong interaction (blue disc) appears between N and Mg. Some- thing similar is observed when looking at the ELF representation [Fig. 2(b)] though a new disynaptic basin between one of the Mg atoms and the N of ammonia appears, as a consequence of the nucleophilic–electrophilic interaction between the N-lone pair and the hole at the Mg atom. It can also be observed that this interac- tion slightly enhances the volume and population around the Mg atoms, so all basins are now polysynaptic (green). Consistently, the molecular graph [Fig. 2(c)] shows the formation of a new Mg–N bond, but the remaining Mg–Mg bonds in the Mg 4moiety are not very much altered with respect to those in the isolated cluster [recall Fig. 1(c)]. As it could be expected from the nature of the Mg 4–LB (LB = NH 3, H 2O, HF) interaction, the binding energy (BE), defined as the energy needed to dissociate each complex into two non- interacting systems, decreases remarkably for the corresponding FIG. 2 . Bonding characteristics for the Mg4–NH 3adduct showing: (a) the NCI 3D, (b) the ELF(=0.75) plot, and (c) the AIM molecular graph. Same conventions as in Fig. 1. J. Chem. Phys. 154, 044302 (2021); doi: 10.1063/5.0038047 154, 044302-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp adducts a1,w1, and hf1(see Fig. 3) from the best electron donor of the three bases, ammonia (61 kJ/mol), to the poorest electron donor, hydrogen fluoride (19 kJ/mol). However, the fact that the electron density distribution around the Mg 4moiety is not substantially altered in the adducts permits to guess that most of its bonding capacity is still almost intact. To investigate whether this is, indeed, the case, we have explored the existence and stability of new isomers that could be obtained by hydrogen shifts from the Lewis base toward the Mg 4moiety. The most stable minima found after successive hydrogen shifts are also shown in Fig. 3 ( a2-a6,w2-w5 , and hf2-hf3 ). Other less stable con- formations than those shown in Fig. 3 are presented in Fig. S11 of the supplementary material. In Fig. 3, we report both the high-level single-reference and multireference BEs for the adducts a1,w1, andhf1 and the rela- tive energies of the remaining complexes relative to them. It can be seen that both sets of values are similar, but more importantly, there are linearly correlated, as shown in Fig. S12 of the supplemen- tary material. However, taking into account that single-reference wavefunctions are affected, as mentioned above, by RHF/UHF insta- bilities, for the discussion that follows, we will use the multiref- erence values. This is supported by the analysis of the CASSCF wavefunctions of all the species investigated here, which, except for the Lewis bases, NH 3, H 2O, and HF, show from a moderate (0.05 <M<0.1: Mg 4,a1,w1,hf1) to a strong multireference character (M>0.1 rest of the minima); see Tables S1–S4 in the supplementary material. The values of the multireference diagnostic are consistentwith the weight of the close shell configuration, which is for all the calculated complexes below 0.85, with the second most impor- tant configuration corresponding to excitations among 3p orbitals sitting on the Mg 4moiety. Remarkably, the largest multiconfigura- tional character is found for the minima either showing three center H–Mg–H bonds or with a H bridging two non-directly bonded Mg atoms. Looking at the values in Fig. 3, it is very interesting to note that, whereas the stability of the adducts ( a1,w1, and hf1) follows, as pre- viously mentioned, the trend NH 3>H2O>HF, the stability of the most stable minima associated with a single hydrogen shift ( a2,w2, andhf2) follows the opposite trend, NH 3<H2O<HF. This can be understood by looking in detail at the bonding changes induced by the hydrogen shift. Let us take the complex with ammonia as a suitable example to do this analysis. In the process of going from the Mg 4–NH 3adduct a1to the Mg 4H–NH 2complex a2, one of the N–H bonds is replaced by a Mg–H bond, leaving the N atom with the capacity to bind to a second Mg atom. Indeed, in the new a2structure, once the new covalent Mg–H bond has been formed, the nitrogen atom of the remaining NH 2group is able to covalently bind two Mg atoms. Moreover, the ELF of this com- plex shows the existence of a disynaptic Mg–H basin [yellow lobe in Fig. 4(b)] and two disynaptic N–Mg basins (green lobes), each of them occupied by an electron pair. Consistently, the molecular graph of the a2complex [Fig. 4(c)] shows that the electron densities at the Mg–N BCPs are 53% higher (0.046 a.u. and 0.048 a.u.) than the value in the a1adduct [0.030 a.u., Fig. 2(c)]. In addition, coherently, FIG. 3 . G4 (blue) and CASPT2 (red) relative energies (kJ mol−1) of different isomers for the complexes between Mg 4and ammonia, water, and hydrogen fluoride with respect to the corresponding adducts, a1,w1,andhf1. For a1,w1,andhf1,the corresponding binding energies (kJ mol−1) are given within parenthesis. J. Chem. Phys. 154, 044302 (2021); doi: 10.1063/5.0038047 154, 044302-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . Bonding characteristics for the a2MgH–NH 2complex showing: (a) the NCI 3D plot, (b) the ELF(=0.85) plot, and (c) the AIM molecular graph. Same conventions as in Fig. 1. both Mg–N bonds in a2are 0.16 Å shorter that the Mg–N bond in a1. In summary, the hydrogen shift going from Mg 4NH 3to the most stable Mg 4H–NH 2complex increases not only the number of Mg–N bonds but also the strength of these linkages. It should be mentioned that similar bridged structures were reported in a previous study of (MgF 2)n(n = 2–3) by Francisco et al.44 The changes observed for the complexes with water and hydro- gen fluoride are qualitatively similar, but quantitatively stronger. Indeed, as illustrated in Fig. S13 of the supplementary material, the increase in the electron density of the Mg–O and Mg–F bonds on going from the adducts w1andhf1to the new isomers w2andhf2 (Fig. 3) is greater (85% and 91%) than that calculated for the N- containing analog. This is again reflected in the larger shortening undergone by the Mg–O and Mg–F bonds (0.20 Å for the former and 0.24 Å for the latter) than the one found for the Mg–N bonds. The conformers a3,w3, and hf3 only differ in the position of the shifted hydrogen, now bridging between two Mg atoms, and their relative stabilities are not much different from those of the isomers a2,w2, and hf2. The behaviors of the nitrogen containing complexes upon a second hydrogen shift differ significantly from those of the oxygen- containing ones, whereas for the latter, a second hydrogen shift to go from structure w3tow4implies a significant stabilization of the system (77 kJ mol−1at the CASPT2 level); this is not the case on going from structure a3to structure a4, where the process involves a decrease (48 kJ mol−1at the CASPT2 level) in the stability of the complex. These two complexes actually differ in their symmetry, ascomplex a4belongs to the C 2vsymmetry point group, whereas the w4complex is a D 2hstructure. In fact, the N and O basic sites appear in both cases bonded to the four Mg atoms. However, whereas the O atom lies in the same plane as the four Mg atoms, the N atom does not. As N is less electronegative, its volume is larger and cannot be accommodated at the center of the square defined by the four Mg atoms. These differences are evident when looking at the different ELF basin volumes around N and O (Fig. 5), which show that both nitrogen and oxygen in a4andw4present a ring-polysynaptic basin with a population ( ∼7.86 e) of practically four pairs. In addition, the N atom in a4is pentacoordinated through a covalent bond to one H atom, but in this case, the basic site is 0.78 Å above the center of the ring. The obvious consequence is that the bonding is nec- essarily much weaker. This description is corroborated by the fact that a third hydrogen shift connecting a4with a6leads to a signif- icant stabilization of the system (44 kJ mol−1at the CASPT2 level). As shown by the a6ELF plot in Fig. 5, the attachment of the third hydrogen atom to one of the Mg bonds breaks the symmetry of the system. Although N is still tetracoordinated to four Mg atoms, this time they lie on the same plane, rendering the system much more stable. As a final remark, it should be emphasized that structures a4–a6 andw4–w5present N and O atoms in a very different chemical envi- ronment with respect to the reactants, i.e., with respect to the iso- lated cluster plus the corresponding Lewis bases. Indeed, the afore- mentioned isomers are formed from the corresponding adducts after a single, double, and triple hydrogen shifts, respectively, leading to FIG. 5 . ELF(=0.85) plots for complexes a4,w4,anda6. Same conventions as in Fig. 1. J. Chem. Phys. 154, 044302 (2021); doi: 10.1063/5.0038047 154, 044302-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp much more ionic structures (see the QTAIM charges in Table S5), in which the nature of the original Mg 4cluster is completely lost. CONCLUSIONS The interaction of Mg 4clusters with conventional Lewis bases yields stable complexes reflecting the electrophilicity of the former, which, as other metal clusters, presents electro-deficient regions revealed by the MEP in the neighborhood of the metallic centers. The formation of these adducts does not significantly alter the metal- lic bonding of the Mg 4cluster so that successive hydrogen-shift processes from the Lewis base toward the Mg 4moiety result in the formation of new very stable isomers, in which two to four Mg atoms are covalently bonded to the basic site (N, O, and F) of the Lewis base. The global minima are those for which the basic site has the largest possible coordination number with the Mg atoms and exhibit binding energies of hundreds of kJ/mol at the CASPT2 level. In other words, the initial bonding arrangement of both the Mg 4cluster and the Lewis base has completely disappeared in the most stable struc- tures, and their stability is closely related to hypervalent N, O, and F atoms. SUPPLEMENTARY MATERIAL See the supplementary material for M diagnostic for the sys- tems investigated (Table S1–S4), selected active space used (Figs. S1– S8) NCI-2D plots for some selected systems (Fig. S9), NCI 3D, ELF, and AIM molecular graphs for Mg 4–H 2O and Mg 4–HF adducts (Fig. S10), additional local minima (Fig. S11), linear correlation between the CASPT2 and G4 calculated binding energies (Fig. S12), molecular graphs for some selected systems (Fig. S13), QTAIM charges for w1,w2,w3,w4, andw5Mg 4–H 2O minima (Table S5), and additional computational details. DEDICATION This paper is dedicated to all the mothers who fight for a better education for their children. ACKNOWLEDGMENTS This work was supported by the projects PGC2018-094644-B- C21, PGC2018-094644-B-C22, and PID2019-110091GB-I00 (MICINN) of the Ministerio de Ciencia, Innovación y Universi- dades of Spain, the Ramón y Cajal program, and the Formación de Profesorado Universitario (FPU) contract from the Ministerio de Economía, Industria y Competitividad of Spain. We are grate- ful to Professor Elguero, Professor Alkorta, and Professor Yáñez for helpful comments. The authors also thank the CTI (CSIC), the Red Española de Supercomputación, the CESGA Supercomputing Cen- ter (Finisterrae), and the Centro de Computación Científica of the UAM (CCC-UAM) for their computational support. DATA AVAILABILITY The data that support the findings of this study are available within this article and its supplementary material.REFERENCES 1G. E. Schmid, Nanoparticles. 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5.0035218.pdf

J. Phys. Chem. Ref. Data 50, 013101 (2021); https://doi.org/10.1063/5.0035218 50, 013101 © 2021 Author(s).Recommended Cross Sections for Electron– Indium Scattering Cite as: J. Phys. Chem. Ref. Data 50, 013101 (2021); https://doi.org/10.1063/5.0035218 Submitted: 27 October 2020 . Accepted: 03 December 2020 . Published Online: 29 January 2021 K. R. Hamilton , O. Zatsarinny , K. Bartschat , M. S. Rabasović , D. Šević , B. P. Marinković , S. Dujko , J. Atić, D. V. Fursa , I. Bray , R. P. McEachran , F. Blanco , G. García , P. W. Stokes , R. D. White , D. B. Jones , L. Campbell , and M. J. Brunger COLLECTIONS This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN Solution of Multiscale Partial Differential Equations Using Wavelets Computers in Physics 12, 548 (1998); https://doi.org/10.1063/1.168739 A method of flight path and chirp pattern reconstruction for multiple flying bats Acoustics Research Letters Online 6, 257 (2005); https://doi.org/10.1121/1.2046567 Chinese Abstracts Chinese Journal of Chemical Physics 33, i (2020); https:// doi.org/10.1063/1674-0068/33/06/cabsRecommended Cross Sections for Electron –Indium Scattering Cite as: J. Phys. Chem. Ref. Data 50,013101 (2021); doi: 10.1063/5.0035218 Submitted: 27 October 2020 Accepted: 3 December 2020 Published Online: 29 January 2021 K. R. Hamilton,1 O. Zatsarinny,1K. Bartschat,1M. S. Rabasovi´ c,2 D. ˇSevi´c,2 B. P. Marinkovi´ c,2 S. Dujko,2 J. Ati´ c,2D. V. Fursa,3 I. Bray,3 R. P. McEachran,4F. Blanco,5 G. Garc ´ıa,6 P. W. Stokes,7 R. D. White,7 D. B. Jones,8 L. Campbell,8 a n dM .J .B r u n g e r8,9,a) AFFILIATIONS 1Department of Physics and Astronomy, Drake University, Des Moines, Iowa 50311, USA 2Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia 3Curtin Insititute for Computation and Department of Physics and Astronomy, Perth 6102, WA, Austraila 4Laser Physics Centre, RSP, Australian National University, Canberra, ACT 0200, Australia 5Departamento de Estructura de La Materia, F ´ısica T´ ermica y Electr´ onica e IPARCOS, Universidad Complutense de Madrid, Avenida Complutense, E-28040 Madrid, Spain 6Instituto de F ´ısica Fundamental, CSIC, Serrano 113-bis, E-28006 Madrid, Spain 7College of Science and Engineering, James Cook University, Townsville, Queensland 4810, Australia 8College of Science and Engineering, Flinders University, GPO Box 2100, Adelaide, SA 5001, Australia 9Department of Actuarial Science and Applied Statistics, Faculty of Business and Management, UCSI University, Kuala Lumpur 56000, Malaysia a)Author to whom correspondence should be addressed: michael.brunger@ flinders.edu.au ABSTRACT We report, over an extended energy range, recommended angle-integrated cross sections for elastic scattering, discrete inelastic scattering proc esses, and the total ionization cross section for electron scattering from atomic indium. In addition, from those angle-integrated cross sections, a grand total cross section is subsequently derived. To constru ct those recommended cross-section databases, res ults from original B-spline R-matrix, relativ istic convergent close-coupling, and relativist ic optical-potential computations are al so presented here. Electron transport coef ficients are subsequently calculated, using our recommended database, for reduced electric fields ranging from 0.01 Td to 10 000 Td using a multiterm solution of Boltzmann ’s equation. To facilitate those simulations, a recommended elastic moment um transfer cross-section set is also constructed and presented here. Published by AIP Publishing on behalf of the National Institute of Standards and Technology. https://doi.org/10.1063/5.0035218 Key words: electron scattering cross sections; electron transport; recommended cross-section data; indium. CONTENTS 1. Introduction ............................ 2 2. Theory Details ........................... 3 2.1. OP model ........................... 3 2.2. SEP method ......................... 3 2.3. BEB approach . . . ..................... 3 3. Cross Section Assessment and Recommended Data .... 4 3.1. Elastic scattering . ..................... 4 3.2. Discrete inelastic cross sections ............. 7 3.3. Total ionization cross section .............. 1 2 3.4. TCS .............................. 1 74. Simulated Transport Coef ficients ............... 1 7 5. Conclusions ............................ 1 9 6. Supplementary Material ..................... 1 9 Acknowledgments . . ....................... 1 9 7. Data Availability . . . ....................... 1 9 8. References ............................. 1 9 List of Tables 1. Parameters used for the present BEB TICS calculation of atomic indium ........................... 4 J. Phys. Chem. Ref. Data 50,013101 (2021); doi: 10.1063/5.0035218 50,013101-1 Published by AIP Publishing on behalf of the National Institute of Standards and Technology.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jpr2. Recommended elastic ICS, MTCS, summed discrete inelastic (electronic-state) ICS, TICS, and grand total (TCS) cross sec- tions (310−16cm2) for electron scattering from indium . . 5 3. Angle-ICSs (10−16cm2) at 90 incident electron energies, selected to show any structure for electron –indium scatter- ing from the (5 s25p)2P1/2ground state ............ 8 4. Angle-ICSs (10−16cm2) for electron –indium scattering from the (5 s25p)2P1/2ground state ............... 1 0 5. Angle-ICSs (10−16cm2) for electron –indium scattering from the (5 s25p)2P3/2metastable state ............. 1 3 6. Angle-ICSs (10−16cm2) for electron –indium scattering from the 5p 3/2metastable state ................. 1 5 List of Figures 1. Angle-integrated elastic cross sections ( 310−16cm2) for electron scattering from In ................... 4 2. Summed discrete inelastic cross sections ( 310−16cm2) for electron-impact excitation of In ................ 73. Individual discrete inelastic cross sections ( 310−16cm2) for electron-impact excitation of In from the (5 s25p)2P1/2 ground state to the higher-lying excited states as denoted in legends (a) and (b) ....................... 7 4. Individual discrete inelastic cross sections ( 310−16cm2) for electron impact excitation of In from the (5 s25p)2P3/2meta- stable state to the higher-lying excited states as denoted in legends (a) and (b) . ....................... 1 2 5. TICSs (310−16cm2) for the process e−+I n→In++2e−17 6. Summary plot showing our recommended electron –In cross sections ( 310−16cm2) for elastic scattering, the sum over all discrete inelastic cross sections, the TICS, and the grand total cross section ................ 1 7 7. Calculated mean electron energies (above the thermal background) (a), rate coef ficients (b), drift velocities (c), and diffusion coef ficients (d) for electrons in In vapor at temperature T/equals1260 K (with thermal energy3 2kBT≈0.163 eV) over a range of reduced electric fields ................................. 1 8 1. Introduction In our recent experimental and theoretical study on the electron- impact excitation of the (5 s25p)2P1/2→(5s26s)2S1/2transition in indium (In),1we outlined a number of basic-science and applied rationales for why indium is a target of general interest. Of particularrelevance to this work, where we attempt to compile a complete cross- section database over a wide energy range, is the need to have such a complete database in order to conduct quantitative modeling in-vestigations for electron transport in indium under an applied electric field (e.g., Refs. 2and3) and for the collisional-radiative modeling of low-temperature plasmas where indium is one of the constituentspecies. 4Note that the importance of having such a comprehensive database available for these types of simulations is discussed in detail in Ref .5and indeed is one of the prime drivers behind the estab- lishment of the LXCat project.6Another important technological application of indium is its role as a tracer in two-line atomicfluorescence thermometry measurements. 7This approach employs two diode lasers with wavelengths of 410 nm and 451 nm, in order to excite the (5 s26s)2S1/2resonance state of indium atoms seeded into a flame. Owing to the typically greater oscillator strengths of atoms compared to molecules, strong fluorescence signals can be obtained at lower excitation energies. A particular plus of indium atoms is that itsspin –orbit coupling in the 5 pground state leads to an energy spacing of≈kT in standard combustion environments (2000 K –4000 K). 8 The only previous elastic angle-integrated cross-section (ICS) results available in the literature are due to Rabasovi´ cet al.9In that study, experimental ICSs were determined, from extrapolation and integration of their elastic differential cross sections, for incident electron energies ( E0) between 10 eV and 100 eV. Corresponding atomic optical-potential (OP) calculations, but now for E0/equals10 eV to 350 eV, were also reported.9As this previous study does not cover a comprehensive enough energy range for swarm or plasma simulation investigations and as further independent assessments of their results would be desirable, here, we report additional OP results and the results from a static-exchange plus polarization (SEP) theoretical approach, as well as the corresponding elastic cross sections from ourrelativistic B-spline R-matrix (DBSR) and relativistic convergent close-coupling (RCCC-75) computations from the work of Hamilton et al.1With these new theoretical results, we are con fident that a recommended elastic ICS database, for E0/equals0.001 eV –10 000 eV, can now be constructed. The situation is even worse for the case of excitation of the discrete inelastic states in indium. Aside from a set of nine angle-ICSs contained in the paper of ¨Og¨unet al. ,4five of which were for excitation from the (5 s25p)2P1/2ground state and a further four of which were for excitation from the close-lying (5 s25p)2P3/2metastable state, as well as our results for the single (5 s25p)2P1/2→(5s26s)2S1/2 excitation process discussed in Ref .1, we know of no other available results in the literature. The ICSs reported in Ref .4were calculated using the method of Gryzi´ nski,10,11which is not ab initio in its construction so that their data are unlikely to be accurate. As a consequence, we do not consider those results further here. On the other hand, we found excellent agreement between our DBSR, RCCC, and measured cross-section results for the (5 s25p)2P1/2→(5s26s)2S1/2 transition.1This gives us hope that these DBSR and RCCC com- putations can provide us with accurate and reliable data for a total of 42 discrete inelastic excitation processes as well as the summed ICS forthese discrete excitations (which can be compared with corre- sponding results from our atomic OP calculations), from which an extensive recommended cross-section database can be constructed. With regard to the total ionization cross section (TICS), how- ever, there has been a quite signi ficant body of earlier work already undertaken. This includes experimental results from the work of Vainshtein et al., 12Shimon et al.,13and Shul et al.,14as well as various types of calculations such as a semi-empirical result from the work of Lotz,15an empirical TICS from the work of Talukder et al. ,16a Deutsch –M¨ark method result from the work of Margreiter et al. ,17 and a binary encounter Bethe (BEB) formulation TICS from the work of Kim and Stone.18Note that in that latter study, plane-wave-Born (PWB) calculations for some of the more important autoionizing states were also undertaken, in order to present a more physical determination of the TICS. Furthermore, note that the measurement J. Phys. Chem. Ref. Data 50,013101 (2021); doi: 10.1063/5.0035218 50,013101-2 Published by AIP Publishing on behalf of the National Institute of Standards and Technology.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprof Shimon et al.13displays an unphysical shape, possessing a local minimum in the TICS at an energy where one might expect to find a maximum cross section. As a consequence, the results from the work of Shimon et al.13do not figure in our further deliberations. Un- fortunately, despite all this earlier work into indium ’s TICS, as we shall shortly see, the level of accord between those various mea- surements and calculations12–18is only marginal. Hence, in this paper, we also present TICS results from our RCCC-75, DBSR-214, and atomic OP calculations, as well as our own BEB calculation with a superior model chemistry over that used in Ref. 18to try and clarify matters prior to constructing a recommended TICS. The remainder of this paper is constructed as follows. In Sec. 2, we detail our theoretical approaches that were used to compute new cross sections for this investigation. Thereafter, in Sec. 3, a detailed comparison of all the available elastic ICS, discrete inelastic ICS, and TICS is provided with our recommended cross sections, which result from each of these comparisons, also being formed here. In Sec. 4,w e apply our recommended electron –indium database, to study the behavior of an electron swarm, under the in fluence of an applied external electric field, with various transport coef ficients2,3being derived. Finally, in Sec. 5, some conclusions from this investigation will be drawn. 2. Theory Details In Ref .1, an appropriate description of both our RCCC-75 and DBSR-214 calculations was given, to which we refer the interested reader. Note, however, that here we have extended our original RCCC-75 computations1to 7000 eV for both the elastic ICS and momentum transfer cross sections (MTCSs) and to 10 000 eV for the sum over all inelastic ICSs. In addition, as a part of this study, we re- ran our RCCC-75 calculations with a somewhat more sophisticated target description (dipole polarizability /equals53.128 a3 0vs 40.3 a3 0in Ref.1) than that employed in Ref. 1. However, at higher energies, the RCCC-75 elastic ICS and MTCS are basically identical irrespective of which target description is employed. Similarly, even though the summed discrete inelastic ICS is a little higher in magnitude, as a function of energy, with the new target description compared to that of Hamilton et al. ,1both sets of results have the same energy de- pendence at higher energies. As one of the main aims of this study is to determine a recommended database for e−+ In scattering, the above observations suggest that, in terms of possibly using the RCCC-75 results to effect a higher-energy extrapolation (see later), cross sec- tions from either target description are equally valid. Under those circumstances, we have continued to employ the RCCC-75 results from the work of Hamilton et al.1throughout this paper. Additional calculations using our atomic-optical approach, a SEP method, and our own application of the BEB procedure5are presented here, so a brief description of each of them is now given below. 2.1. OP model We have recently described our standard OP approach in our studies of the electron –beryllium,19electron –magnesium,20 electron –zinc,21and electron –bismuth22scattering systems. The generic details of this atomic OP method were given in those papers so that only the key points of this approach are summarized below. The projectile –atom interaction is described by a local complex potential given byV(r)/equalsVs(r)+Vex(r)+Vpol(r)+iVabs(r), (1) where the real part of the potential is comprised of the following three terms. Vsis the static term derived from a Hartree –Fock calculation23 of the atomic charge distribution. Vexis an exchange term that ac- counts for the indistinguishability of the incident and target electrons; it is given by the semi-classical energy dependent formula derived by Riley and Truhlar.24Finally, Vpolis a polarization potential for the long –range interactions that depend on the target dipole polariz- ability. For this study, the polarization potential of Ref .25, leading to results we denote as OP1, and that of Ref .26, leading to cross sections we denote as OP2, were both applied. The imaginary absorption potential accounts for the inelastic, both discrete and continuum, scattering events. It is based on the quasi-free model put forward by Staszewska et al.27but incorporates some improvements to the original formulation. These include allowing for the inclusion of screening effects, local velocity cor- rections, and the description of the electron indistinguishability,28 leading to a model that provides a realistic approximation forelectron –atom scattering over a broad energy range. 29 The present atomic optical model is non-relativistic in formu- lation and leads to angle-integrated elastic cross sections, the sum over all discrete inelastic angle-ICSs, and the TICS. Note that as indium is only a moderately heavy atom, the differences in the calculated scattering cross sections between a relativistic and non-relativistic treatment will not be signi ficant. As a consequence, the application of our non-relativistic OP approach is valid for this target. 2.2. SEP method Our SEP model includes both relativistic static and polarization potentials as well as the exchange interaction. The spin –orbit in- teraction in indium gives rise to two “ground-state ”levels, namely, (5s25p)2P1/2and (5 s25p)2P3/2with the j/equals3/2 state being ∼0.274 eV above the j/equals1/2 state. A linear combination of the wavefunctions corresponding to these two states was then employed in a 2-state Dirac –Fock multicon figuration calculation30,31to determine the ground-state con figuration of In within a frozen-core model. The static potential was then determined in the usual manner. The dipole polarization was determined using the relativistic non-perturbative polarized-orbital method for alkali and alkali-like atoms.26Finally, the exchange interaction was accounted for by anti- symmetrization of the total scattering wavefunction. This approach only yields elastic scattering cross sections, and since it does not include any inelastic processes, it is expected to become less reliable as the energy of the incident electron increases. 2.3. BEB approach Kim and Stone18calculated the TICS for indium using the BEB formalism.5,32The BEB approach is sensitive to the binding energies used in the calculation, so we have repeated the work of Kim and Stone18using an improved structural representation for indium based on the best available experimental data for the orbital binding en- ergies. Those values have been assembled from the available pho- toionization spectra33,34and information regarding the convergence of spectral lines to the ionization thresholds.35,36These values have been combined with orbital kinetic energies for atomic indium that were derived from a single point energy calculation of indium hydride J. Phys. Chem. Ref. Data 50,013101 (2021); doi: 10.1063/5.0035218 50,013101-3 Published by AIP Publishing on behalf of the National Institute of Standards and Technology.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jpr(InH, r/equals9.0 ˚A) in Gaussian 09,37with the model chemistry employing density functional theory (B3LYP)38and a double zeta valance polarized basis set.39The parameters used in the present BEB calculation are summarized in Table 1 . Within the BEB formalism,32the TICS is obtained by summing up the contributions from each populated orbital, with the ith orbital ’s contribution being given by Qi(ti)/equals4πa2 0Ni ti+ui+1 ηpqnR Bi/parenleftBigg/parenrightBigg2lnti 21−1 t2 i/parenleftBigg/parenrightBigg +1−1 ti−lnti ti+1/bracketleftBigg/bracketrightBigg .(2) In Eq. (2), the binding energy of the ionized orbital, Bi, is used to scale the incident electron-impact energy ( E0) and orbital kinetic energies (ui):ti/equalsE0 Biandui/equalsUi Bi, respectively. Niis the orbital occupation number, while Randa0are, respectively, the Rydberg constant and Bohr radius. A modi fication to the traditional BEB approach comes when dealing with heavier elements, where the scaled kinetic energy is corrected by the principal quantum number of the ionized atomic orbital ( ηpqn) if it is greater than 2. 3. Cross Section Assessment and Recommended Data 3.1. Elastic scattering InFig. 1 , we summarize the available experimental and theo- retical elastic angle-ICSs for electron –indium scattering, including original results from computations associated with this study. It is quite clear from this figure that between 10 eV and 90 eV, the ex- perimental data of Rabasovi´ cet al.9are, to within the cited error bars, in very good agreement with their optical-model SEPASo9compu- tation and our OP1, SEP, and DBSR-214 calculations. Agreement with our RCCC-75 calculation is also typically fair over this energy range. This level of accord, between experiment and theory, in that energy regime is by no means unique to indium, having also been observed by us in our recent study of elastic electron scattering from bismuth.22Similarly, but now for energies in the range 1 eV –10 eV, we find good levels of accord between our OP1, RCCC-75, and DBSR- 214 calculations. Below 1 eV, however, there is quite a signi ficant level of discrepancy between all the available theoretical results. While all the theories predict a signi ficant structure, which would arise due tothe temporary capture of the incident electron by the target, in the elastic ICS, the position (in the range ∼0.09 eV –0.2 eV) and mag- nitude ( ∼200310−16cm2–700310−16cm2) of that peak are seen to vary from one theory to another. Our non-relativistic OP1, OP2, andSEP calculations all show this structure arising in the ℓ/equals1 (i.e., p-wave) partial wave, suggesting that the origin of this feature is consistent with a p-wave shape resonance. For our relativistic RCCC and DBSR computations, the structure is in the J/equals2, parity /equals+1, partial wave of the total scattering system. As the (5 s 25p)2P1/2ground state has j/equals1/2, parity /equals−1, this leads to the projectile waves of either j/equals3/2, parity /equals−1,ℓ/equals1o rj /equals5/2, parity /equals−1,ℓ/equals3. Normally, we would expect it, consistent with our non-relativistic results, to be in ℓ/equals1, as ℓ/equals3 is too large for the centrifugal barrier to support a resonance. It is interesting to note that it is known40that the (5 s25p2)3 P0,1,2and (5 s25p2)1D2and1S0states of the negative indium ion are stable, with, for example, the3P0state having an electron af finity of 384 meV, the3P1state having an electron af finity of 460 meV, and the 3P2state having an electron af finity of 555 meV. Under these circumstances, a low-energy electron could simply bind to the indium atom to form In−, which may have consequences for electron swarm behavior2,3at low E/n0(E/equalsapplied external electric field and n0/equalsbackground gas density number). To quantitatively specify whether the structure we observe in Fig. 1 is a resonance or simply an artifact of our computational methods, we would need to do a sig- nificantly more accurate structure calculation for both In and In−andTABLE 1. Parameters used for the present BEB TICS calculation of atomic indium. See also supplementary material , Table S1 Orbital Bi(eV) Ui(eV) Ni ηpqn 5p 5.79 27.94 1 5 5s 11.16 53.70 2 5 4d5/2 24.4 282.35 6 4 4d3/2 25.7 282.35 4 4 4p3/2 86.0 441.12 4 4 4p1/2 95.4 441.12 2 4 4s 126.0 496.51 2 4 3d5/2 468.5 1735.41 6 3 3d3/2 476.4 1735.41 4 3 3p3/2 691.8 1844.65 4 3 3p1/2 731.3 1844.65 2 3 3s 854.4 1894.15 2 3 FIG. 1. Angle-integrated elastic cross sections ( 310−16cm2) for electron scattering from In. Results from the present relativistic SEP (purple dashed line); non-relativistic OP, (red dotted-dashed line) OP1 and (blue dashed line) OP2; andrelativistic RCCC-75 (blue dotted-dashed line) and DBSR-214 (black solid line)computations, as well as the experimental (black circles) and SEPASo theory (greendashed line) results from the work of Rabasovi´ cet al. , 9are plotted. J. Phys. Chem. Ref. Data 50,013101 (2021); doi: 10.1063/5.0035218 50,013101-4 Published by AIP Publishing on behalf of the National Institute of Standards and Technology.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprTABLE 2. Recommended elastic ICS, MTCS, summed discrete inelastic (electronic-state) ICS, TICS, and grand total (TCS) cross sections ( 310−16cm2) for electron scattering from indium. See also supplementary material , Table S2 E0(eV) Elastic ICS ( 310−16cm2) MTCS ( 310−16cm2)Σdiscrete inelastic ( 310−16cm2) TICS ( 310−16cm2) TCS ( 310−16cm2) 1.361310−325.6 25.4 25.6 4.080310−321.0 20.6 21.0 4.082310−321.0 20.6 21.0 6.800310−317.5 16.9 17.5 9.520310−314.9 14.2 14.9 0.012 25 12.9 12.2 12.9 0.017 69 10.4 9.69 10.4 0.023 13 9.40 8.69 9.40 0.028 57 9.89 9.07 9.89 0.034 01 12.1 11.1 12.1 0.039 46 16.8 15.6 16.8 0.044 90 25.5 23.9 25.5 0.050 34 41.1 39.2 41.10.055 78 69.5 67.3 69.5 0.061 23 123 120 123 0.063 95 165 162 165 0.069 39 300 298 300 0.080 27 703 703 703 0.085 72 696 697 696 0.096 60 432 432 432 0.104 8 304 302 304 0.110 2 251 249 251 0.121 1 188 183 188 0.142 9 131 124 131 0.164 6 106 98.4 106 0.210 9 84.2 74.0 84.2 0.273 5 71.3 59.7 0.00 71.3 0.276 2 70.7 59.1 2.25 72.9 0.278 9 70.0 58.2 3.46 73.5 0.287 1 68.2 55.8 5.61 73.9 0.319 7 63.3 49.6 10.7 74.0 0.401 4 59.8 44.0 18.5 78.3 0.496 6 61.9 43.9 20.0 81.9 0.761 9 66.3 44.2 16.7 83.0 1.075 68.9 44.3 13.5 82.4 1.673 70.6 42.7 10.1 80.6 2.626 69.0 36.2 7.53 76.6 2.721 69.0 35.8 7.04 76.0 2.762 67.0 34.1 7.82 74.8 3.018 65.8 33.0 7.23 73.1 3.020 64.1 33.0 7.73 71.8 3.023 62.1 32.9 9.48 71.5 3.034 66.6 32.3 7.64 74.2 3.184 68.0 32.1 6.77 74.8 3.186 71.3 31.8 6.97 78.3 3.189 63.0 31.5 8.97 72.03.192 64.5 31.2 7.66 72.1 3.211 65.6 30.8 7.20 72.8 3.765 60.8 25.8 7.99 68.8 3.769 62.0 25.6 7.62 69.6 3.791 61.2 25.2 8.22 69.4 3.804 58.3 23.9 10.1 68.3 J. Phys. Chem. Ref. Data 50,013101 (2021); doi: 10.1063/5.0035218 50,013101-5 Published by AIP Publishing on behalf of the National Institute of Standards and Technology.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprto perform our scattering calculations on a much finer energy grid. This is beyond the scope of this publication. Nonetheless, in spite of the above caveats, it is crucial in modeling/simulation applications to have a complete cross-section database.5As a consequence, given it is our most detailed relativistic ab initio computation, and in what follows, for energies below 1 eV, we recommend our DBSR-214 elastic ICS results. Note that at low energies, the dipole polarizability plays a very important role in the elastic scattering dynamics.41,42In our current work, the static dipole polarizability ( αd) of the In atom in the RCCC-75 model is 40.3 a3 0and in our DBSR-214 model is 61.3 a3 0, while the experimental value is ∼68.69 a3 0.43We believe that these differences between theory and experiment in αdexplain, at least in part, the different peak positions and magnitudes in the elastic ICS low-energy structures and further indicate that even more detailed structure descriptions are warranted at some stage.On the basis of the above discussion and the results shown in Fig. 1 , we form our recommended elastic ICS from our DBSR-214 result from 0.001 eV to 100 eV and from a suitably scaled, to maintain continuity (scaling factor /equals1.025), OP1 result from 100 eV to 10 000 eV. This recommended elastic ICS is listed in Table 2 f o ras e l e c t i o no f incident electron energies, and we e stimate the uncertainty on it to be ∼±2 0 %f o re n e r g i e sl e s st h a n3e Va n d ∼±15% for energies greater than 3 eV. Note that the sensitivity of our recommended cross sections to our choice of normalization energy was in vestigated both here and for all the later scattering processes we consi der. We found that our recommended cross sections, for E0⩾100 eV, were largely insensitive to our choice of normalization energy between 70 eV and 100 eV. This result gives us confidence in the robustness of our higher energy recommended cross sections. The elastic MTCS is also very important for electron transport simulations, with much of the discussion just given for the elastic ICS alsoTABLE 2. (Continued. ) E0(eV) Elastic ICS ( 310−16cm2) MTCS ( 310−16cm2)Σdiscrete inelastic ( 310−16cm2) TICS ( 310−16cm2) TCS ( 310−16cm2) 3.810 62.0 23.5 8.70 70.7 3.823 63.2 24.8 7.95 71.2 3.908 63.1 25.3 7.43 70.6 3.913 62.4 25.4 7.92 70.3 3.921 63.6 25.5 7.40 71.0 4.327 62.9 21.2 9.91 72.8 4.531 55.2 16.5 13.4 68.6 4.612 51.9 15.5 13.6 65.5 4.776 48.8 15.3 12.2 61.0 5.786 42.6 11.7 11.3 0.00 53.9 6.00 41.4 11.3 11.2 0.310 52.9 8.71 28.5 5.60 8.95 3.46 40.9 10.0 25.0 4.40 7.99 4.96 37.9 12.4 20.0 3.46 7.14 6.68 33.8 16.2 14.7 2.56 6.65 8.38 29.7 20.0 12.2 2.02 6.39 9.24 27.8 30.0 9.71 2.98 5.81 9.78 25.3 40.0 9.11 4.49 5.37 9.29 23.8 46.4 8.84 4.36 5.08 8.87 22.8 65.0 7.51 4.07 4.35 7.87 19.7 80.0 6.66 3.13 3.93 7.21 17.8 95.0 6.22 2.53 3.60 6.66 16.5 120 5.80 2.04 3.12 5.96 14.9 150 5.42 1.81 2.68 5.36 13.5 200 4.96 1.64 2.18 4.66 11.8 300 4.22 1.41 1.61 3.79 9.62 400 3.66 1.17 1.28 3.20 8.15 600 2.97 0.865 0.923 2.59 6.49 800 2.55 0.656 0.730 2.23 5.511 000 2.27 0.517 0.607 1.98 4.86 2 000 1.56 0.224 0.337 1.35 3.24 3 000 1.24 0.131 0.235 1.06 2.54 4 000 1.05 0.0877 0.179 0.887 2.12 5 000 0.924 0.0637 0.144 0.771 1.84 6 000 0.826 0.0489 0.120 0.682 1.63 8 000 0.693 0.0315 0.0888 0.561 1.34 10 000 0.605 0.0221 0.0701 0.482 1.16 J. Phys. Chem. Ref. Data 50,013101 (2021); doi: 10.1063/5.0035218 50,013101-6 Published by AIP Publishing on behalf of the National Institute of Standards and Technology.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprbeing applicable to it. As a consequ ence, we do not repeat that detail; rather, we simply note that our recommended MTCS is formed from our DBSR-214 calculation for 0.001 eV –100 eV and from our RCCC-75 result, again suitably scaled to ensure continuity (scaling factor /equals0.9564) for 100 eV –7000 eV. Finally, for 7 keV –10 keV, we use our scaled OP1 result (scaling factor /equals2.653) to complete our MTCS database. That recommended MTCS can also be found in Table 2 , with an uncertainty on it of ∼±20% for energies less than 3 eV and ∼±15% for energies greater than 3 eV. 3.2. Discrete inelastic cross sections We next consider the sum of all the discrete inelastic excited- state angle-ICSs, where only data from our OP1, OP2, RCCC-75, and DBSR-214 calculations are available. Those results are plotted in Fig. 2 , where several observations are immediately apparent. First, while our OP1 and OP2 results agree well with one another, both appear to predict an incorrect threshold for the opening of inelastic excitation, and as a consequence, both predict a maximum in the summed cross section to occur at too high an incident electron energy. This is not unexpected as the OP1 and OP2 models do not account for thefine-structure splitting of the ground state. So their lowest inelastic threshold is at around 3 eV for the (5 s26s)2S1/2state, while the actual value is ∼0.27 eV for the (5 s25p)2P3/2state. In addition, both the OP1 and OP2 results are generally in poor accord with our relativistic RCCC-75 and DBSR-214 results. In our recent work1on the electron- impact excitation of the (5 s25p)2P1/2→(5s26s)2S1/2transition, we found excellent agreement between our RCCC-75, DBSR-214, and measured angle-ICSs over their common energy range (to typically∼±10%). This level of accord between them is clearly not maintained inFig. 2 , a point that is in need of further interrogation, although we note that qualitatively the RCCC-75 and DBSR-214 results remain in fair agreement. The relatively marginal quantitative accord between our RCCC-75 and DBSR-214 computations, for the summed discrete angle-integrated inelastic cross sections in Fig. 2 , we believe is due to an inaccuracy of the quasi one-electron RCCC model for In, at least for some inelastic transitions that are important. One example of that is for the (5 s25p)2P1/2→(5s25d)2D3/2transition, where the optical oscillator strength in the RCCC target-state description has a value of 0.451, which is substantially higher than those from our DBSR cal- culations (0.341)1and the corresponding NIST value (0.36).35The effect of this carries through in the respective RCCC-75 and DBSR- 214 scattering results, where the magnitude of the RCCC (5 s25d)2D3/2 ICS would be anticipated to be greater than that for the DBSR-214 (5s25d)2D3/2ICS. This observation is entirely consistent, away from the in fluence of resonance-effects, with what we found in Fig. 2 .A sa consequence, for the summed discrete inelastic ICS, our recom- mended database is here formed from the DBSR-214 results fromFIG. 2. Summed discrete inelastic cross sections ( 310−16cm2) for electron-impact excitation of In. Present non-relativistic OP, (red dotted-dashed line) OP1 and (bluedashed line) OP2, and relativistic RCCC-75 (blue dotted-dashed line) and DBSR-214 (black line) computational results are plotted. FIG. 3. Individual discrete inelastic cross sections ( 310−16cm2) for electron-impact excitation of In from the (5 s25p)2P1/2ground state to the higher-lying excited states as denoted in legends (a) and (b). All results are from our DBSR-214 calculation, and they represent our recommended data for each of these processes. J. Phys. Chem. Ref. Data 50,013101 (2021); doi: 10.1063/5.0035218 50,013101-7 Published by AIP Publishing on behalf of the National Institute of Standards and Technology.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprTABLE 3. Angle-ICSs (10−16cm2) at 90 incident electron energies, selected to show any structure for electron –indium scattering from the (5 s25p)2P1/2ground state. See also supplementary material , Table S3a Cross section (10−16cm2) →(5s25p)2P3/2 →(5s26p)2P1/2 →(5s25d)2D3/2 →(5s24p)2P1/2 →(5s27s)2S1/2 →(5s27p)2P1/2 Energy (eV) →(5s26s)2S1/2 →(5s26p)2P3/2 →(5s25d)2D5/2 →(5s24p)2P3/2 →(5s24p)2P5/2 0.2743 0.00 0.2762 2.250.2789 3.460.3088 9.260.3524 14.50.3796 17.10.4476 19.90.5102 19.90.5714 19.20.6803 17.80.8027 16.20.8708 15.40.9796 14.3 1.116 13.2 1.265 12.11.483 10.91.673 10.11.823 9.572.041 8.932.259 8.402.653 7.443.022 8.25 0.003.025 8.01 1.373.033 7.01 0.7163.041 6.93 0.5063.061 6.86 0.3813.116 6.72 0.3643.186 6.47 0.4973.189 8.32 0.6573.770 5.82 1.743.804 8.25 1.803.815 6.59 1.60 3.908 6.07 1.37 3.913 6.01 1.913.921 5.88 1.523.945 5.96 1.70 0.003.946 5.93 1.71 0.03893.951 6.03 1.43 0.2493.973 6.06 1.36 0.2263.982 6.07 1.33 0.296 0.003.986 6.08 1.31 0.332 0.1824.014 6.10 1.27 0.314 0.2934.078 6.20 1.30 0.340 0.353 0.004.081 6.21 1.30 0.342 0.356 0.00832 0.00J. Phys. Chem. Ref. Data 50,013101 (2021); doi: 10.1063/5.0035218 50,013101-8 Published by AIP Publishing on behalf of the National Institute of Standards and Technology.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprTABLE 3. (Continued. ) Cross section (10−16cm2) →(5s25p)2P3/2 →(5s26p)2P1/2 →(5s25d)2D3/2 →(5s24p)2P1/2 →(5s27s)2S1/2 →(5s27p)2P1/2 Energy (eV) →(5s26s)2S1/2 →(5s26p)2P3/2 →(5s25d)2D5/2 →(5s24p)2P3/2 →(5s24p)2P5/2 4.082 6.21 1.30 0.343 0.356 0.0103 0.00348 4.095 6.21 1.24 0.385 0.365 0.0360 0.02424.150 6.31 1.25 0.444 0.396 0.0769 0.07134.337 7.47 1.31 0.476 0.407 0.151 0.213 0.004.340 7.50 1.31 0.477 0.408 0.153 0.215 0.05704.466 8.13 1.30 0.515 0.454 0.196 0.291 1.42 0.004.501 8.22 1.25 0.514 0.437 0.225 0.303 1.60 0.506 0.004.503 8.22 1.24 0.514 0.433 0.227 0.303 1.61 0.546 0.01494.531 8.06 1.25 0.512 0.425 0.243 0.288 1.63 0.937 0.08554.643 7.31 1.32 0.516 0.422 0.339 0.290 1.42 1.71 0.172 0.004.667 6.91 1.34 0.515 0.402 0.388 0.282 1.37 1.74 0.190 0.2164.748 5.53 1.31 0.564 0.555 0.403 0.496 1.27 1.66 0.205 0.841 4.776 5.27 1.27 0.550 0.442 0.311 0.320 1.24 1.64 0.191 0.971 4.818 4.92 1.33 0.475 0.421 0.401 0.359 1.24 1.64 0.217 1.11 0.004.830 4.83 1.33 0.474 0.444 0.402 0.376 1.24 1.64 0.197 1.14 0.1305.020 4.15 1.31 0.467 0.387 0.417 0.357 1.17 1.60 0.116 1.29 0.1305.075 4.04 1.31 0.490 0.421 0.451 0.376 1.15 1.58 0.148 1.26 0.09415.143 3.94 1.32 0.484 0.390 0.457 0.402 1.12 1.54 0.123 1.20 0.1245.238 3.84 1.35 0.468 0.389 0.522 0.427 1.08 1.50 0.111 1.14 0.09205.293 3.77 1.33 0.478 0.391 0.540 0.427 1.06 1.47 0.127 1.12 0.1115.347 3.72 1.33 0.500 0.427 0.568 0.457 1.04 1.45 0.120 1.09 0.09135.850 3.31 1.38 0.465 0.371 0.720 0.475 0.901 1.24 0.123 0.901 0.1006.94 2.82 1.40 0.445 0.373 1.02 0.479 0.692 0.936 0.113 0.674 0.1118.03 2.47 1.42 0.374 0.354 1.28 0.446 0.485 0.654 0.108 0.516 0.1138.71 1.93 1.43 0.370 0.342 1.40 0.400 0.411 0.545 0.0809 0.388 0.08199.66 1.69 1.46 0.353 0.322 1.49 0.379 0.309 0.382 0.0927 0.276 0.046110.7 1.50 1.49 0.333 0.313 1.55 0.322 0.230 0.274 0.0893 0.194 0.051312.4 1.20 1.49 0.315 0.298 1.62 0.280 0.167 0.208 0.0929 0.144 0.046014.8 0.962 1.51 0.320 0.279 1.73 0.232 0.140 0.181 0.0951 0.125 0.050417.6 0.785 1.53 0.319 0.258 1.83 0.196 0.115 0.156 0.101 0.103 0.0545 18.9 0.722 1.52 0.318 0.248 1.88 0.190 0.105 0.140 0.106 0.0949 0.0573 22.4 0.597 1.52 0.307 0.220 1.97 0.165 0.0718 0.0983 0.108 0.0650 0.058024.9 0.541 1.52 0.295 0.202 2.01 0.153 0.0588 0.0808 0.110 0.0522 0.056927.3 0.502 1.51 0.289 0.186 2.03 0.144 0.0491 0.0677 0.112 0.0427 0.056830.1 0.475 1.50 0.286 0.169 2.05 0.133 0.0402 0.0557 0.114 0.0342 0.057235.8 0.440 1.45 0.274 0.140 2.05 0.117 0.0258 0.0368 0.115 0.0209 0.057640.4 0.416 1.41 0.265 0.121 2.02 0.106 0.0179 0.0266 0.114 0.0135 0.057546.4 0.388 1.35 0.250 0.102 1.97 0.0936 0.0115 0.0184 0.111 0.00769 0.054750.2 0.370 1.31 0.238 0.0928 1.93 0.0874 0.00904 0.0153 0.108 0.00557 0.052757.8 0.337 1.23 0.215 0.0786 1.86 0.0775 0.00643 0.0120 0.102 0.00339 0.047665.0 0.309 1.17 0.195 0.0691 1.80 0.0702 0.00529 0.0105 0.0955 0.00250 0.043170.0 0.292 1.13 0.183 0.0642 1.76 0.0659 0.00477 0.00987 0.0912 0.00211 0.040380.0 0.261 1.05 0.164 0.0576 1.70 0.0589 0.00400 0.00889 0.0834 0.00158 0.035890.0 0.236 0.989 0.148 0.0552 1.63 0.0532 0.00341 0.00815 0.0765 0.00119 0.0324105 0.208 0.907 0.130 0.0519 1.54 0.0464 0.00274 0.00728 0.0682 7.83 310 −40.0286 115 0.192 0.860 0.119 0.0499 1.48 0.0428 0.00240 0.00682 0.0640 6.02 310−40.0267 J. Phys. Chem. Ref. Data 50,013101 (2021); doi: 10.1063/5.0035218 50,013101-9 Published by AIP Publishing on behalf of the National Institute of Standards and Technology.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprTABLE 4. Angle-ICSs (10−16cm2) for electron –indium scattering from the (5 s25p)2P1/2ground state. See also supplementary material , Table S3b Cross section (10−16cm2) →(5s27p)2P3/2 →(5s26d)2D5/2 →(5s24f)2F5/2 →(5s28p)2P1/2 →(5P27d)2D5/2 Energy (eV) →(5p26d)2D3/2 →(5s24f)2F7/2 →(5p28s)2S1/2 →(5s27d)2D3/2 →(5s28p)2P3/2 4.832 0.00 4.841 0.0343 0.004.848 0.0569 0.0207 0.004.857 0.0921 0.0528 0.03324.912 0.135 0.0709 0.08124.923 0.130 0.0767 0.0831 0.00 0.004.966 0.111 0.0984 0.0899 0.0212 0.02765.020 0.110 0.112 0.0967 0.0298 0.04615.038 0.114 0.109 0.0934 0.0345 0.0524 0.005.075 0.122 0.105 0.0867 0.0440 0.0653 0.02295.143 0.121 0.115 0.120 0.0308 0.0503 0.04485.184 0.130 0.110 0.109 0.0364 0.0545 0.04625.186 0.129 0.109 0.109 0.0365 0.0542 0.0464 0.005.187 0.128 0.109 0.108 0.0365 0.0540 0.0465 4.81 310 −40.00 5.190 0.127 0.108 0.108 0.0365 0.0534 0.0468 0.00180 0.00372 0.00 5.193 0.126 0.108 0.107 0.0366 0.0530 0.0470 0.00281 0.00656 0.00271 0.005.238 0.105 0.0968 0.0987 0.0373 0.0453 0.0511 0.0218 0.0600 0.0537 0.02445.293 0.114 0.110 0.108 0.0467 0.0508 0.0538 0.0313 0.0592 0.0644 0.03215.347 0.0984 0.103 0.0924 0.0463 0.0494 0.0448 0.0321 0.0635 0.0472 0.03215.401 0.120 0.125 0.110 0.0369 0.0528 0.0585 0.0381 0.0704 0.0633 0.03105.510 0.119 0.107 0.116 0.0425 0.0612 0.0354 0.0246 0.0442 0.0491 0.02295.551 0.0994 0.111 0.0974 0.0458 0.0651 0.0451 0.0231 0.0361 0.0427 0.02545.606 0.104 0.126 0.101 0.0416 0.0622 0.0559 0.0373 0.0369 0.0481 0.02855.660 0.0857 0.110 0.0990 0.0424 0.0560 0.0612 0.0277 0.0439 0.0468 0.03015.714 0.0910 0.0951 0.108 0.0507 0.0620 0.0506 0.0310 0.0566 0.0520 0.03405.850 0.105 0.136 0.115 0.0430 0.0836 0.0481 0.0409 0.0627 0.0589 0.04996.12 0.0950 0.136 0.125 0.0394 0.0617 0.0403 0.0299 0.0494 0.0547 0.04206.39 0.112 0.162 0.121 0.0494 0.0715 0.0453 0.0406 0.0650 0.0609 0.05506.94 0.122 0.185 0.121 0.0418 0.0647 0.0450 0.0381 0.0696 0.0589 0.04217.48 0.114 0.216 0.127 0.0545 0.0735 0.0357 0.0372 0.0732 0.0612 0.04318.03 0.116 0.255 0.120 0.0580 0.0830 0.0462 0.0429 0.0871 0.0604 0.04798.71 0.0975 0.276 0.138 0.0466 0.0776 0.0402 0.0351 0.0894 0.0540 0.03909.12 0.110 0.275 0.119 0.0457 0.0681 0.0253 0.0302 0.0943 0.0631 0.0451 9.66 0.0911 0.266 0.0920 0.0327 0.0712 0.0261 0.0158 0.0828 0.0381 0.0374 10.7 0.0802 0.260 0.0943 0.0306 0.0849 0.0242 0.0196 0.0835 0.0517 0.035011.8 0.0761 0.266 0.0885 0.0318 0.0898 0.0255 0.0139 0.0882 0.0483 0.030012.4 0.0739 0.256 0.0812 0.0301 0.0921 0.0240 0.0143 0.0848 0.0405 0.028813.7 0.0732 0.247 0.0690 0.0275 0.107 0.0244 0.0157 0.0713 0.0440 0.029614.8 0.0727 0.237 0.0590 0.0248 0.106 0.0255 0.0167 0.0687 0.0348 0.028915.6 0.0702 0.230 0.0539 0.0243 0.111 0.0261 0.0171 0.0619 0.0300 0.028516.2 0.0691 0.226 0.0496 0.0234 0.113 0.0265 0.0173 0.0592 0.0276 0.028817.6 0.0674 0.227 0.0427 0.0223 0.114 0.0269 0.0181 0.0548 0.0235 0.027418.9 0.0636 0.219 0.0395 0.0223 0.119 0.0288 0.0205 0.0492 0.0209 0.026221.1 0.0598 0.235 0.0354 0.0203 0.116 0.0301 0.0211 0.0494 0.0178 0.024522.4 0.0570 0.236 0.0335 0.0186 0.112 0.0300 0.0208 0.0483 0.0157 0.0235 J. Phys. Chem. Ref. Data 50,013101 (2021); doi: 10.1063/5.0035218 50,013101-10 Published by AIP Publishing on behalf of the National Institute of Standards and Technology.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprTABLE 4. (Continued. ) Cross section (10−16cm2) →(5s27p)2P3/2 →(5s26d)2D5/2 →(5s24f)2F5/2 →(5s28p)2P1/2 →(5P27d)2D5/2 Energy (eV) →(5p26d)2D3/2 →(5s24f)2F7/2 →(5p28s)2S1/2 →(5s27d)2D3/2 →(5s28p)2P3/2 23.8 0.0554 0.239 0.0315 0.0168 0.108 0.0302 0.0209 0.0465 0.0144 0.0228 24.9 0.0535 0.240 0.0309 0.0158 0.105 0.0307 0.0207 0.0457 0.0140 0.022126.3 0.0513 0.243 0.0301 0.0145 0.101 0.0313 0.0208 0.0447 0.0129 0.021227.3 0.0495 0.245 0.0293 0.0136 0.0975 0.0317 0.0209 0.0446 0.0125 0.020628.7 0.0476 0.246 0.0282 0.0126 0.0936 0.0322 0.0215 0.0448 0.0121 0.020230.1 0.0454 0.248 0.0271 0.0116 0.0897 0.0325 0.0213 0.0447 0.0113 0.019331.4 0.0434 0.249 0.0265 0.0108 0.0865 0.0329 0.0216 0.0450 0.0110 0.018432.8 0.0416 0.250 0.0257 0.0101 0.0833 0.0332 0.0216 0.0451 0.0106 0.017734.2 0.0399 0.251 0.0251 0.00943 0.0803 0.0334 0.0219 0.0452 0.0102 0.017035.8 0.0379 0.251 0.0243 0.00874 0.0768 0.0337 0.0222 0.0452 0.00979 0.016237.1 0.0363 0.251 0.0236 0.00823 0.0741 0.0337 0.0222 0.0451 0.00948 0.015638.8 0.0345 0.250 0.0228 0.00769 0.0711 0.0338 0.0225 0.0449 0.00913 0.014840.4 0.0328 0.248 0.0219 0.00720 0.0683 0.0337 0.0225 0.0445 0.00878 0.0141 42.0 0.0312 0.246 0.0212 0.00677 0.0657 0.0336 0.0224 0.0441 0.00848 0.0134 43.4 0.0300 0.245 0.0207 0.00644 0.0637 0.0336 0.0218 0.0436 0.00823 0.012945.0 0.0286 0.242 0.0201 0.00609 0.0615 0.0334 0.0218 0.0431 0.00796 0.012346.4 0.0275 0.240 0.0196 0.00584 0.0598 0.0332 0.0214 0.0426 0.00776 0.011848.0 0.0263 0.238 0.0190 0.00554 0.0578 0.0329 0.0212 0.0421 0.00751 0.011350.2 0.0248 0.235 0.0184 0.00519 0.0554 0.0325 0.0208 0.0415 0.00723 0.010752.1 0.0237 0.232 0.0178 0.00492 0.0535 0.0321 0.0203 0.0409 0.00700 0.010254.3 0.0225 0.229 0.0173 0.00464 0.0514 0.0316 0.0197 0.0402 0.00677 0.0096655.6 0.0218 0.227 0.0170 0.00449 0.0503 0.0313 0.0194 0.0398 0.00665 0.0093656.7 0.0213 0.225 0.0167 0.00437 0.0494 0.0311 0.0191 0.0395 0.00652 0.0091357.8 0.0208 0.224 0.0165 0.00426 0.0485 0.0308 0.0188 0.0391 0.00645 0.0089160.0 0.0198 0.220 0.0160 0.00404 0.0469 0.0303 0.0182 0.0385 0.00627 0.0085065.0 0.0180 0.213 0.0151 0.00363 0.0438 0.0290 0.0170 0.0371 0.00589 0.0077070.0 0.0165 0.206 0.0143 0.00330 0.0413 0.0278 0.0159 0.0357 0.00558 0.0070375.0 0.0152 0.199 0.0137 0.00302 0.0393 0.0266 0.0150 0.0343 0.00533 0.0064780.0 0.0142 0.192 0.0130 0.00279 0.0378 0.0255 0.0141 0.0329 0.00509 0.0060185.0 0.0133 0.186 0.0125 0.00259 0.0367 0.0244 0.0134 0.0316 0.00487 0.0056290.0 0.0126 0.180 0.0119 0.00242 0.0356 0.0234 0.0128 0.0305 0.00466 0.00529 95.0 0.0121 0.175 0.0114 0.00228 0.0345 0.0224 0.0122 0.0294 0.00447 0.00503 100 0.0117 0.170 0.0110 0.00216 0.0334 0.0215 0.0117 0.0283 0.00429 0.00482105 0.0115 0.165 0.0106 0.00206 0.0324 0.0207 0.0113 0.0273 0.00414 0.00466110 0.0114 0.161 0.0102 0.00197 0.0315 0.0200 0.0109 0.0264 0.00399 0.00455115 0.0116 0.158 0.00979 0.00189 0.0306 0.0193 0.0106 0.0257 0.00384 0.00449 J. Phys. Chem. Ref. Data 50,013101 (2021); doi: 10.1063/5.0035218 50,013101-11 Published by AIP Publishing on behalf of the National Institute of Standards and Technology.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprthreshold up to 100 eV and from 100 eV up to 10 000 eV we, as in Sec. 3.1, making use of a suitably scaled RCCC-75 cross section (scaling factor /equals0.7694) to facilitate the extrapolation to those higher energies and maintain continuity at 100 eV. We believe that the uncertainty estimate on this recommended ICS (again see Table 2 )i s∼±20%. Given our discussion immediately above, it is no surprise that for the individual discrete inelastic transitions, from both the ground- state (5 s25p)2P1/2level and the close-lying metastable (5 s25p)2P3/2 level, we have chosen to utilize our DBSR-214 calculations. Angle- ICSs for transitions from (5 s25p)2P1/2to a higher lying level i, ICS i, are plotted in Figs. 3(a) and3(b)and listed in the corresponding Tables 3 and4. A total of 21 discrete inelastic cross sections are presented here for the first time. Similarly, angle-ICSs from the (5 s25p)2P3/2state to a higher level iare plotted in Figs. 4(a) and 4(b) and listed in the corresponding Tables 5 and6. A further 21 discrete inelastic channels for excitation from the (5 s25p)2P3/2are also presented here for the first time. Near-threshold structures are observed in most of these inelastic ICS i(seeFigs. 3 and4). These structures are not pseudo-resonances;rather, they either originate from Feshbach resonances or are asso- ciated with the opening of higher-lying discrete electronic states (possibly Wigner cusps) as the incident electron energy is increased. Nonetheless, a more detailed study of these structures, beyond thescope of this paper, is required before any quantitative classi fications can be made. We believe the errors on these ICS i, on average, are ±15% for transitions originating from the ground (5 s25p)2P1/2state and ∼±20% for those transitions originating from the metastable (5 s25p) 2P3/2state. Note that the (5 s25p)2P3/2ICS is have been included here as we believe they will be needed in any quantitative kinetic-radiative study for a plasma in which indium is a constituent and may also be needed for our electron transport simulations in Sec. 4. While we do not explicitly show our extrapolations for each ICS i out to 10 000 eV, such extrapolations are simple enough to undertake.If we again pick our RCCC-75 summed inelastic ICS to perform the extrapolation, and again do the normalization at 100 eV, then for all E⩾100 eV, we find ICS i(E)/equalsICS i(100 eV ) ICS summed (100 eV )3ICS summed (E), (3) where all the values for the right-hand side of Eq. (3)can be obtained from Tables 2 –6as required. 3.3. Total ionization cross section InFig. 5 , we plot the available TICS for the scattering process e−+I n→In++2e−, including our present OP1, OP2, BEB, BEB+autoionization, RCCC-75, and DBSR-214 cross sections. It should be apparent that two experimental determinations, from Vainshtein et al.12and Shul et al.,14are available and that they disagree with one another (outside their reported uncertainties of ±18% and ±13%, respectively) in terms of their magnitudes. Vainshtein et al.12 determined the number density of their indium beam using the quartz crystal resonator method, which Lindsay and Mangan44noted can lead to problematic results. Shul et al. ,14however, employed a different approach that incorporates a fast neutral atom beam obtained by charge transfer of an energetic ion beam that is crossed by an ionizing electron beam. Unfortunately, as also noted by Lindsay and Mangan,44this approach does present formidable practical dif ficulties including being able to precisely ascertain the overlap between the electron beam and the fast neutral beam and with the possible presence of metastable species. Given the caveats in applying both those experimental pro- cedures, we ap r i o r i have no way of choosing between them. From a theoretical perspective, we find that our RCCC-75 TICS underestimates the magnitude of all the other TICS results. This can be understood by the fact that it currently does not incorporate many of the important autoionization channels that Kim and Stone18noted are crucial to consider in this case. The semi-empirical calculation of Lotz15and the present BEB results favor the measurement of Vainshtein et al.,12while the present BEB+PWB autoionization, the corresponding calculation from the work of Kim and Stone,18and the electron impact total single ionization (EITSI) results16favor the experiment of Shul et al.14Finally, in between (but outside their stated measurement uncertainties) the experimental results, we find in Fig. 5 the present OP1, OP2, and DBSR- 214 calculations and the Deutsch –M¨ark17computation. Note that, in principle, our OP1 and OP2 computations include those important autoionizing channels, while our DBSR-214 calculations incorporate most of them except for those that originate from the 4 dshell.FIG. 4. Individual discrete inelastic cross sections ( 310−16cm2) for electron impact excitation of In from the (5 s25p)2P3/2metastable state to the higher-lying excited states as denoted in legends (a) and (b). All results are from our DBSR-214 calculation, and they represent our recommended data for each of these processes. J. Phys. Chem. Ref. Data 50,013101 (2021); doi: 10.1063/5.0035218 50,013101-12 Published by AIP Publishing on behalf of the National Institute of Standards and Technology.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprTABLE 5. Angle-ICSs (10−16cm2) for electron –indium scattering from the (5 s25p)2P3/2metastable state. See also supplementary material , Table S4a Cross section (10−16cm2) →(5s25p)2P3/2 →(5s26p)2P1/2 →(5s25d)2D3/2 →(5s24p)2P1/2 →(5s27s)2S1/2 →(5s27p)2P1/2 Energy (eV) →(5s26s)2S1/2 →(5s26p)2P3/2 →(5s25d)2D5/2 →(5s24p)2P3/2 →(5s24p)2P5/2 1.005 68.7 1.127 70.21.290 71.91.399 72.81.562 73.81.767 74.61.903 75.02.079 75.22.365 75.22.474 73.22.748 72.6 0.002.753 69.3 2.712.757 71.8 1.672.761 72.5 1.14 2.767 72.8 0.873 2.787 73.1 0.5972.814 73.1 0.5282.912 74.4 0.6742.915 71.3 0.8173.154 71.0 1.793.522 61.2 2.363.636 70.8 1.753.655 69.8 2.043.660 69.2 2.543.670 68.8 2.41 0.003.674 69.1 1.92 0.2083.677 69.3 1.82 0.2953.685 69.4 1.79 0.3293.707 69.0 1.85 0.263 0.003.712 68.8 1.85 0.254 0.4633.804 69.1 1.77 0.362 0.818 0.003.807 69.1 1.77 0.368 0.838 0.0142 0.003.807 69.1 1.77 0.369 0.843 0.0176 0.00596 3.821 69.1 1.70 0.367 0.943 0.0451 0.0520 3.875 68.8 1.70 0.406 1.01 0.0978 0.1604.062 68.8 1.77 0.394 0.933 0.221 0.351 0.004.066 68.8 1.77 0.394 0.932 0.223 0.355 0.01434.093 68.7 1.78 0.397 0.953 0.236 0.377 0.09804.192 67.5 1.75 0.403 1.00 0.279 0.456 0.540 0.004.202 67.2 1.75 0.403 0.999 0.285 0.461 0.580 0.1064.227 66.5 1.69 0.391 0.989 0.287 0.499 0.643 0.311 0.004.229 66.5 1.68 0.389 0.987 0.287 0.503 0.648 0.334 0.02044.256 65.7 1.68 0.385 0.978 0.283 0.503 0.678 0.577 0.1074.369 61.9 1.76 0.370 0.935 0.330 0.551 0.675 1.39 0.197 0.004.392 60.4 1.80 0.344 0.838 0.337 0.583 0.670 1.49 0.208 0.424J. Phys. Chem. Ref. Data 50,013101 (2021); doi: 10.1063/5.0035218 50,013101-13 Published by AIP Publishing on behalf of the National Institute of Standards and Technology.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprTABLE 5. (Continued. ) Cross section (10−16cm2) →(5s25p)2P3/2 →(5s26p)2P1/2 →(5s25d)2D3/2 →(5s24p)2P1/2 →(5s27s)2S1/2 →(5s27p)2P1/2 Energy (eV) →(5s26s)2S1/2 →(5s26p)2P3/2 →(5s25d)2D5/2 →(5s24p)2P3/2 →(5s24p)2P5/2 4.420 58.3 1.85 0.437 1.05 0.353 0.609 0.660 1.51 0.232 0.835 4.474 56.9 1.74 0.389 0.946 0.359 0.540 0.635 1.54 0.246 1.494.501 56.5 1.72 0.404 1.01 0.350 0.574 0.611 1.52 0.163 1.704.544 55.9 1.77 0.409 0.934 0.384 0.684 0.590 1.53 0.238 1.98 0.004.556 55.6 1.76 0.428 0.934 0.397 0.698 0.581 1.52 0.244 2.04 0.1274.583 55.2 1.79 0.364 0.888 0.350 0.628 0.558 1.49 0.187 2.14 0.08204.692 53.8 1.80 0.329 0.867 0.366 0.664 0.491 1.38 0.148 2.35 0.07464.746 53.4 1.76 0.343 0.873 0.394 0.695 0.466 1.32 0.138 2.38 0.08724.801 53.0 1.74 0.318 0.846 0.394 0.726 0.445 1.27 0.188 2.36 0.07594.869 52.4 1.75 0.307 0.863 0.399 0.731 0.420 1.21 0.143 2.32 0.08745.018 51.3 1.74 0.307 0.819 0.430 0.839 0.377 1.11 0.168 2.19 0.07545.236 49.8 1.78 0.317 0.843 0.462 0.979 0.337 1.01 0.149 2.06 0.07615.386 48.8 1.78 0.304 0.826 0.481 1.06 0.318 0.956 0.153 1.96 0.0601 5.576 47.6 1.81 0.275 0.771 0.478 1.07 0.299 0.898 0.152 1.83 0.0744 5.848 45.9 1.84 0.272 0.774 0.486 1.19 0.275 0.827 0.142 1.71 0.06896.66 41.3 1.85 0.258 0.740 0.486 1.38 0.222 0.670 0.133 1.37 0.07927.21 38.3 1.87 0.253 0.751 0.524 1.66 0.187 0.547 0.120 1.20 0.07157.75 35.1 1.90 0.236 0.671 0.476 1.66 0.175 0.504 0.114 0.972 0.07368.84 30.7 1.91 0.216 0.583 0.429 1.76 0.103 0.335 0.108 0.679 0.06419.39 29.1 1.93 0.216 0.595 0.444 1.82 0.0827 0.274 0.104 0.538 0.058510.5 26.2 1.96 0.210 0.562 0.429 1.90 0.0593 0.187 0.109 0.364 0.052211.6 23.6 1.96 0.200 0.534 0.418 1.96 0.0484 0.151 0.109 0.309 0.045412.1 22.5 1.97 0.199 0.532 0.411 1.96 0.0457 0.138 0.112 0.286 0.046313.5 20.0 1.98 0.190 0.521 0.398 2.04 0.0414 0.125 0.112 0.257 0.044215.4 17.3 1.99 0.181 0.514 0.385 2.11 0.0380 0.112 0.115 0.239 0.041917.3 15.4 1.99 0.172 0.506 0.378 2.18 0.0329 0.0962 0.119 0.214 0.040418.6 14.3 1.98 0.164 0.496 0.378 2.23 0.0316 0.0897 0.122 0.199 0.038620.8 13.0 1.98 0.153 0.478 0.378 2.32 0.0257 0.0718 0.124 0.167 0.036522.2 12.5 1.97 0.146 0.466 0.374 2.35 0.0233 0.0630 0.125 0.150 0.034524.6 11.5 1.96 0.135 0.446 0.372 2.39 0.0195 0.0522 0.128 0.128 0.032627.1 10.9 1.94 0.125 0.428 0.367 2.41 0.0162 0.0432 0.130 0.109 0.0304 29.8 10.4 1.91 0.114 0.414 0.361 2.42 0.0133 0.0352 0.131 0.0922 0.0278 32.5 10.1 1.88 0.104 0.399 0.355 2.41 0.0109 0.0283 0.132 0.0774 0.025635.5 9.89 1.85 0.0949 0.384 0.347 2.39 0.00867 0.0221 0.132 0.0643 0.023438.5 9.71 1.80 0.0866 0.370 0.339 2.36 0.00694 0.0172 0.131 0.0538 0.021441.8 9.53 1.76 0.0788 0.355 0.329 2.32 0.00548 0.0131 0.129 0.0450 0.019444.8 9.34 1.71 0.0726 0.340 0.321 2.29 0.00452 0.0103 0.127 0.0390 0.017949.9 8.98 1.64 0.0638 0.316 0.306 2.22 0.00339 0.00712 0.122 0.0321 0.015657.5 8.34 1.53 0.0542 0.283 0.286 2.13 0.00256 0.00491 0.114 0.0269 0.013164.7 7.77 1.45 0.0477 0.257 0.269 2.05 0.00217 0.00398 0.107 0.0244 0.011469.7 7.40 1.39 0.0443 0.240 0.259 2.00 0.00199 0.00356 0.101 0.0232 0.010479.7 6.87 1.30 0.0393 0.216 0.241 1.92 0.00170 0.00297 0.0921 0.0215 0.0089989.7 6.51 1.22 0.0367 0.196 0.226 1.83 0.00148 0.00253 0.0841 0.0201 0.0079799.7 6.30 1.14 0.0355 0.181 0.213 1.75 0.00130 0.00220 0.0774 0.0188 0.00729115 6.14 1.05 0.0335 0.161 0.197 1.64 0.00109 0.00187 0.0696 0.0173 0.00685J. Phys. Chem. Ref. Data 50,013101 (2021); doi: 10.1063/5.0035218 50,013101-14 Published by AIP Publishing on behalf of the National Institute of Standards and Technology.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprTABLE 6. Angle-ICSs (10−16cm2) for electron –indium scattering from the 5p 3/2metastable state. See also supplementary material , Table S4b EnergyCross section (10−16cm2) →(5s27p)2P3/2 →(5s26d)2D5/2 →(5s24f)2F5/2 →(5s28p)2P1/2 →(5p27d)2D5/2 Energy (eV) →(5p26d)2D3/2 →(5s24f)2F7/2 →(5p28s)2S1/2 →(5s27d)2D3/2 →(5s28p)2P3/2 4.558 0.00 4.567 0.0635 0.004.573 0.105 0.0194 0.004.583 0.170 0.0496 0.04904.637 0.192 0.0800 0.1424.649 0.191 0.0836 0.148 0.00 0.004.692 0.187 0.0968 0.172 0.0314 0.01924.746 0.218 0.107 0.196 0.0616 0.03174.764 0.214 0.103 0.194 0.0671 0.0354 0.004.801 0.206 0.0938 0.189 0.0784 0.0430 0.02474.869 0.214 0.120 0.200 0.0588 0.0346 0.05244.909 0.211 0.112 0.207 0.0600 0.0381 0.05924.912 0.209 0.111 0.205 0.0598 0.0379 0.0589 0.004.913 0.208 0.111 0.204 0.0597 0.0377 0.0588 4.94 310 −40.00 4.916 0.205 0.111 0.202 0.0595 0.0374 0.0584 0.00185 0.00324 0.00 4.918 0.203 0.111 0.200 0.0593 0.0371 0.0582 0.00288 0.00571 0.00421 0.004.964 0.164 0.107 0.166 0.0556 0.0322 0.0531 0.0224 0.0522 0.0835 0.02905.018 0.204 0.101 0.184 0.0745 0.0432 0.0640 0.0194 0.0569 0.0944 0.05725.073 0.187 0.0935 0.167 0.0529 0.0350 0.0472 0.0193 0.0466 0.0917 0.05125.127 0.205 0.103 0.196 0.0729 0.0385 0.0585 0.0295 0.0555 0.109 0.04875.236 0.198 0.0994 0.175 0.0761 0.0472 0.0467 0.0191 0.0491 0.0837 0.03925.277 0.181 0.0998 0.177 0.0822 0.0496 0.0594 0.0194 0.0386 0.0702 0.04435.331 0.182 0.102 0.193 0.0842 0.0430 0.0736 0.0233 0.0424 0.0708 0.05995.386 0.160 0.104 0.186 0.0846 0.0454 0.0776 0.0220 0.0435 0.0688 0.05205.440 0.160 0.105 0.179 0.0863 0.0562 0.0674 0.0248 0.0449 0.0754 0.06055.576 0.189 0.115 0.223 0.105 0.0498 0.0592 0.0353 0.0507 0.0927 0.07835.848 0.181 0.116 0.237 0.0810 0.0440 0.0555 0.0296 0.0471 0.0965 0.06076.12 0.187 0.120 0.249 0.104 0.0578 0.0550 0.0324 0.0530 0.109 0.07986.66 0.193 0.129 0.276 0.0956 0.0474 0.0548 0.0295 0.0536 0.105 0.07117.21 0.183 0.128 0.311 0.111 0.0555 0.0444 0.0300 0.0554 0.116 0.06767.75 0.194 0.120 0.339 0.123 0.0619 0.0499 0.0308 0.0486 0.118 0.07468.43 0.155 0.155 0.369 0.117 0.0581 0.0370 0.0211 0.0597 0.132 0.06018.84 0.141 0.127 0.364 0.102 0.0449 0.0321 0.0281 0.0513 0.133 0.0613 9.39 0.109 0.110 0.335 0.104 0.0436 0.0273 0.0226 0.0458 0.111 0.0384 10.5 0.106 0.104 0.349 0.108 0.0403 0.0276 0.0221 0.0482 0.124 0.041211.6 0.0999 0.0991 0.343 0.116 0.0425 0.0283 0.0186 0.0467 0.122 0.034012.1 0.0981 0.0952 0.346 0.117 0.0403 0.0266 0.0177 0.0435 0.121 0.032613.5 0.0980 0.0847 0.330 0.126 0.0408 0.0278 0.0180 0.0406 0.107 0.034114.6 0.0991 0.0781 0.312 0.126 0.0398 0.0290 0.0174 0.0346 0.103 0.035115.4 0.0974 0.0733 0.307 0.130 0.0401 0.0290 0.0166 0.0318 0.0958 0.035115.9 0.0985 0.0695 0.298 0.132 0.0399 0.0298 0.0168 0.0291 0.0882 0.034317.3 0.0993 0.0651 0.303 0.132 0.0391 0.0302 0.0157 0.0264 0.0862 0.035618.6 0.0999 0.0598 0.291 0.136 0.0396 0.0319 0.0151 0.0223 0.0776 0.036620.8 0.0998 0.0581 0.305 0.132 0.0374 0.0330 0.0143 0.0206 0.0760 0.036922.2 0.0974 0.0567 0.306 0.127 0.0352 0.0335 0.0137 0.0193 0.0739 0.0363 J. Phys. Chem. Ref. Data 50,013101 (2021); doi: 10.1063/5.0035218 50,013101-15 Published by AIP Publishing on behalf of the National Institute of Standards and Technology.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprTABLE 6. (Continued. ) EnergyCross section (10−16cm2) →(5s27p)2P3/2 →(5s26d)2D5/2 →(5s24f)2F5/2 →(5s28p)2P1/2 →(5p27d)2D5/2 Energy (eV) →(5p26d)2D3/2 →(5s24f)2F7/2 →(5p28s)2S1/2 →(5s27d)2D3/2 →(5s28p)2P3/2 23.5 0.0961 0.0562 0.308 0.122 0.0332 0.0337 0.0134 0.0179 0.0728 0.0361 24.6 0.0944 0.0555 0.310 0.118 0.0316 0.0344 0.0130 0.0173 0.0714 0.035526.0 0.0927 0.0549 0.311 0.114 0.0299 0.0349 0.0126 0.0166 0.0704 0.035127.1 0.0916 0.0547 0.313 0.110 0.0285 0.0355 0.0122 0.0160 0.0701 0.034728.4 0.0908 0.0540 0.314 0.105 0.0270 0.0360 0.0119 0.0158 0.0700 0.034729.8 0.0889 0.0534 0.314 0.101 0.0255 0.0363 0.0114 0.0152 0.0695 0.034331.2 0.0873 0.0529 0.314 0.0970 0.0242 0.0367 0.0109 0.0150 0.0694 0.034032.5 0.0865 0.0526 0.314 0.0933 0.0231 0.0370 0.0105 0.0146 0.0694 0.033733.9 0.0858 0.0521 0.314 0.0898 0.0220 0.0372 0.0101 0.0143 0.0692 0.033635.5 0.0852 0.0516 0.312 0.0858 0.0208 0.0375 0.00966 0.0140 0.0688 0.033336.9 0.0842 0.0510 0.311 0.0828 0.0199 0.0375 0.00926 0.0137 0.0685 0.033038.5 0.0831 0.0501 0.309 0.0793 0.0189 0.0376 0.00884 0.0134 0.0679 0.032740.1 0.0818 0.0492 0.306 0.0762 0.0180 0.0375 0.00843 0.0131 0.0671 0.0323 41.8 0.0803 0.0483 0.303 0.0733 0.0171 0.0373 0.00805 0.0128 0.0664 0.0318 43.1 0.0787 0.0476 0.300 0.0711 0.0165 0.0372 0.00774 0.0125 0.0655 0.031244.8 0.0771 0.0468 0.297 0.0685 0.0158 0.0370 0.00740 0.0122 0.0647 0.030646.1 0.0755 0.0461 0.294 0.0666 0.0153 0.0367 0.00714 0.0120 0.0640 0.030047.8 0.0739 0.0453 0.291 0.0644 0.0147 0.0364 0.00683 0.0117 0.0632 0.029449.9 0.0719 0.0443 0.287 0.0618 0.0139 0.0359 0.00647 0.0114 0.0621 0.028651.8 0.0699 0.0434 0.283 0.0597 0.0134 0.0354 0.00617 0.0112 0.0611 0.027854.0 0.0674 0.0425 0.278 0.0575 0.0128 0.0348 0.00587 0.0109 0.0599 0.026955.4 0.0662 0.0419 0.275 0.0562 0.0124 0.0344 0.00569 0.0107 0.0593 0.026456.5 0.0651 0.0415 0.273 0.0552 0.0122 0.0341 0.00556 0.0106 0.0588 0.026057.5 0.0640 0.0410 0.271 0.0543 0.0119 0.0338 0.00543 0.0105 0.0582 0.025559.7 0.0620 0.0402 0.267 0.0526 0.0114 0.0331 0.00519 0.0102 0.0572 0.024764.7 0.0578 0.0383 0.257 0.0493 0.0105 0.0317 0.00471 0.00971 0.0548 0.023069.7 0.0538 0.0365 0.248 0.0467 0.00973 0.0302 0.00431 0.00923 0.0526 0.021574.7 0.0508 0.0349 0.238 0.0447 0.00910 0.0288 0.00398 0.00879 0.0504 0.020279.7 0.0479 0.0333 0.230 0.0434 0.00858 0.0275 0.00370 0.00837 0.0483 0.019184.7 0.0455 0.0319 0.222 0.0419 0.00816 0.0263 0.00346 0.00798 0.0463 0.018189.7 0.0434 0.0305 0.215 0.0405 0.00783 0.0251 0.00325 0.00762 0.0445 0.0172 94.7 0.0416 0.0293 0.208 0.0391 0.00757 0.0240 0.00308 0.00728 0.0428 0.0165 99.7 0.0400 0.0281 0.202 0.0379 0.00738 0.0230 0.00294 0.00696 0.0413 0.0158105 0.0387 0.0270 0.197 0.0367 0.00718 0.0221 0.00282 0.00668 0.0398 0.0153110 0.0377 0.0261 0.192 0.0356 0.00698 0.0213 0.00273 0.00642 0.0386 0.0149115 0.0368 0.0252 0.188 0.0346 0.00680 0.0205 0.00266 0.00617 0.0374 0.0145 J. Phys. Chem. Ref. Data 50,013101 (2021); doi: 10.1063/5.0035218 50,013101-16 Published by AIP Publishing on behalf of the National Institute of Standards and Technology.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprGiven the discussion above, it appe ars reasonable to assert that the measurements from the work of Vainshtein et al.12can constitute a lower bound on the true TICS, while those from the work of Shul et al.14 can be considered as an upper bound. In the latter case, this seems reasonable, as it is well known that PWB based calculations, without some appropriate scaling,5will overestimate the magnitude of the cross sections they calculate so that the BEB+PWB autoionization cross- section magnitude should also be too large. Under these circumstances, when coming to form a recommended TICS database, we will follow the approach of Itikawa (see e.g., Ref. 45), which essentially means that for energies from threshold to 200 eV, we take an average of the available experimental data, and then from 200 eV to 10 000 eV, we use a suitably scaled (scaling factor /equals1.014) OP1 result to effect the extrapolation to higher energies. Note that we have had to employ Itikawa ’sm e t h o d (successfully) in some of our recen t data compilations for electron scattering from some other atomic species.19,20,46The estimated un- certainty on this is ∼±22%, re flecting the error carried forward in taking the average of the TICS measurements.12,14Interestingly, this recom- mended TICS, to within our error jus t cited, is in quite good accord with the results from our OP1, OP2, and DB SR-214 calculations. The present recommended TICS can be found in Table 2 , and they are also plotted in Fig. 6 along with our recommended elastic ICS, MTCS, and sum over all discrete inelastic angle-ICSs.3.4. TCS The recommended TCS for E0/equals0.001 eV –10 000 eV is now simply formed by, at each incident electron energy, adding up the results for the recommended elastic ICS, the recommended sum over all discrete inelastic excitation ICS, and the recommended TICS, namely, summing up the results of columns 1, 3, and 4 of Table 2 . That recommended TCS can also now be found in column 5 of Table 2 ,a s well as being plotted in Fig. 6 . 4. Simulated Transport Coef ficients In what follows, we implement a well-benchmarked multi-term solution of Boltzmann ’se q u a t i o n2,47,48for the calculation of electron swarm transport coef ficients in gaseous In over a range of reduced electric fields E/n0, varying from 10−2Td to 104Td, where 1 Td /equals1 Townsend /equals10−21Vm2andn0is the neutral number density. The two- term approximation (TTA)47,48tends to break down at the higher E/n0 considered, although it remains accurate to within 20% for all transport coefficients, with the exception of some of the diffusion coef ficients. Under the TTA at high E/n0, these errors in diffusion can be as large as 51% for the flux transverse diffusion coef ficient, 69% for the bulk longitudinal diffusion coef ficient, and 70% for the flux longitudinal diffusion coef ficient. In our calculations, we assume isotropic scattering in the excitation and ionization processes, while we have included the anisotropic nature of elastic scattering through the use of the elastic MTCSs. We consider transport through an In vapor at temperature T/equals1260 K, which is in the vicinity of our previous crossed-beam experimental measurements.1,9As the corresponding thermal energy,FIG. 5. TICSs (310−16cm2) for the process e−+I n→In++2e−. Experimental data from the work of Vainshtein et al.12(blue diamonds) and Shul et al.14(red triangles) are plotted, along with earlier theoretical results from the work of Lotz15(green dotted-dashed line), an EITSI calculation16(blue dashed line), a Deutsch –M¨ark computation17(magenta dotted-dashed line), and a BEB+PWB calculation18(blue dotted line). Also plotted are our current BEB result (purple dashed line), OP1 (reddotted-dashed line) and OP2 (blue dashed line) results, RCCC-75 (blue dotted-dashed line) calculation, BEB+PWB (gray dashed line) calculation, and DBSR-214(black solid line) computation. See also the legend in the figure.FIG. 6. Summary plot showing our recommended electron –In cross sections (310−16cm2) for elastic scattering, the sum over all discrete inelastic cross sections, the TICS, and the grand total cross section. See also the legend in thefigure. J. Phys. Chem. Ref. Data 50,013101 (2021); doi: 10.1063/5.0035218 50,013101-17 Published by AIP Publishing on behalf of the National Institute of Standards and Technology.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jpr3 2kBT≈0.163 eV, is on the order of the energy of the first 5 p/parenleftbig/parenrightbig2P3/2 metastable state ( ∼0.274 eV), we consider it prudent to account for de- excitation/superelastic collisions in our Boltzmann equation solution. Indeed, by applying Maxwell –Boltzmann statistics to In vapor at the aforementioned temperature, we determine that 86% of In atoms are in the ground state, with the remaining 14% almost exclusively in the first 5p/parenleftbig/parenrightbig2P3/2metastable state. Note that we determine each de-excitation cross section from its corresponding excitation cross section by employing the principle of microscopic reversibility and detailed balancing.49We use our recommended elastic MTCS for elastic col- lisions with ground-state In atoms and obtain a separate elastic MTCS for In atoms in the first 5 p/parenleftbig/parenrightbig2P3/2metastable state by scaling our 5p/parenleftbig/parenrightbig2P3/2→5p/parenleftbig/parenrightbig2P3/2elastic ICS by the ratio of our recommended elastic MTCS to recommended elastic ICS. Similarly, while we use our recommended TICS for ionization of In atoms in the ground state, we shift the energy threshold of our recommended TICS down to ∼5.786 eV −0.274 eV /equals5.512 eV for ionization of In atoms already excited to the first 5 p/parenleftbig/parenrightbig2P3/2metastable state. The resulting calculated mean electron energies, rate coef ficients, drift velocities, and diffusion coefficients are presented in Fig. 7 .Figure 7(a) shows the difference between the mean electron energy ε ̄and the thermal energy of the In vapor,3 2kBT≈0.163 eV. In the low- field regime, near 10−2Td, this energy difference is very small, indicating that the electrons are in thermal equilibrium with the background In atoms. As E/n0increases,the mean electron energy decreases, reaching a minimum of ∼1m e V below the thermal background at ∼1.8 Td. We attribute this cooling to be primarily due to the 5 p/parenleftbig/parenrightbig2P1/2→5p/parenleftbig/parenrightbig2P3/2transition, resulting in a greater power output from the swarm due to excitations than input from the electric field, superelastic collisions, and elastic collisions in this regime. Eventually, as E/n0approaches ∼5.7 Td, the latter heating processes dominate enough to return the swarm to thermal equilibrium with the background. Then, as E/n0is increased further, the mean energy increases rapidly, slowing slightly in its ascent from ∼200 Td onward due to the signi ficant opening of the ionization channel. Figure 7(b) shows rate coef ficients for elastic momentum transfer, summed excitation, summed de-excitation, and ionization processes. The elastic momentum transfer rate coef ficient remains somewhat constant up to 1000 Td, before decreasing slightly at higher E/n0.T h e summed excitation and de-excitation rate coef ficients are identical close to thermal equilibrium, as is expected due to detailed balancing. These rate coef ficients begin to depart visibly from 10 Td onward, with an increase in excitation events and decrease in de-excitation events. Although it should be noted that this departure starts much earlier than this as it is the slight excess of excitation events at low E/n0that is responsible for the ∼1 meV cooling of electrons below the background. The ionization rate coef ficient is zero in the low- field regime, before becoming appreciable around roughly 200 Td. In the high- field regime, near 10 000 Td, ionization dominates with its rate coef ficient exceedingFIG. 7. Calculated mean electron energies (above the thermal background) (a), rate coef ficients (b), drift velocities (c), and diffusion coef ficients (d) for electrons in In vapor at temperature T/equals1260 K (with thermal energy3 2kBT≈0.163 eV) over a range of reduced electric fields. See also the legends for further details. J. Phys. Chem. Ref. Data 50,013101 (2021); doi: 10.1063/5.0035218 50,013101-18 Published by AIP Publishing on behalf of the National Institute of Standards and Technology.Journal of Physical and Chemical Reference DataARTICLE scitation.org/journal/jprthose for all other processes. Figure 7(c) shows the bulk and flux drift velocities of the swarm, both of which are observed to increase monotonically with E/n0, coinciding with one another up until the nonconservative effects of ionization manifest at around 200 Td. In thenonconservative regime from ∼200 Td onward, the bulk drift velocity exceeds the flux, suggesting that electrons are being preferentially created at the front of the swarm, shifting the center of mass in the direction of the applied field.Figure 7(d) shows bulk and flux diffusion coefficients in the directions longitudinal and transverse to the applied electric field. Of course, below ∼200 Td, the bulk and flux diffusion coefficients coincide. Below 0.1 Td, the transverse and longitudinal diffusion coef ficients are essentially equal due to the expected isotropy of the electron velocity distribution in this regime. Above 0.1 Td, both diffusion coef ficients begin to decrease slightly, reaching minima at roughly 10 Td and 20 Td for the longitudinal and transverse coef fi- cients, respectively. Past these minima, both diffusion coef ficients then proceed to rise monotonically with increasing E/n 0. In the noncon- servative regime above ∼200 Td, the bulk diffusion coef ficients exceed their flux counterparts, suggesting a preferential creation of electrons at the sides of the swarm, in addition to its front. 5. Conclusions We have compiled a complete angle-ICS database for electron –In scattering. As a part of that process, additional theoretical computations were undertaken, with these results also being reported here. While the need for having complete and accurate cross-section databases, for modeling a variety of electron-driven phenomena,50,51 is now well understood, recent work from the Madrid group52–54has reinforced that assessment. Interesting scattering results from this investigation include the very large shape resonances in the low-energy elastic ICS and MTCS, a series of near-threshold resonances in many of the discrete inelastic scattering channels we have considered, and the lack of consistent measurements for the experimental TICS in electron –In scattering. While there is no doubt that such experiments in In are dif ficult to undertake, further measurements of the TICS are clearly desirable. Finally, we have employed our recommended cross sections to study the behavior of a swarm of electrons, drifting through a background gas of In, under the in fluence of an applied electric field. This analysis was undertaken using a multi-term Boltzmann equation solution to determine the relevant transport coef ficients. Interesting results from this study included the need to allow for superelastic processes, the breakdown of the TTA in simulating the relevant transport coef ficients in some regions of E/n0, and that there was cooling in the mean electron energy of the swarm at ∼1.8 Td, which can be associated with the opening of the (5 p)2P3/2metastable channel. 6. Supplementary Material See the supplementary material for Excel tables of the present data. Acknowledgments The work of K.R.H., O.Z., and K.B. was supported by the United States National Science Foundation under Grant Nos. OAC-1834740 and PHY-1803844 and by the XSEDE supercomputer AllocationNo. PHY-090031. The (D)BSR calculations were carried out on Stampede2 at the Texas Advanced Computing Center. The work of D.V.F. and I.B. was supported by the Australian Research Council and resources provided by the Pawsey Supercomputing Centre withfunding from the Australian Government and the Government of Western Australia. F.B. and G.G. acknowledge partial financial support from the Spanish Ministry MICIU (Project Nos. FIS2016- 80440 and PID2019-104727-RB-C21) and CSIC (Project No. LINKA20085). This work was also financially supported, in part, by the Australian Research Council (Project No. DP180101655), the Ministry of Education, Science and Technological Development of the Republic of Serbia, and the Institute of Physics (Belgrade). 7. Data Availability The data that support the findings of this study are available from the corresponding author upon reasonable request. 8. References 1K. R. Hamilton, O. Zatsarinny, K. Bartschat, M. S. Rabasovi´ c, D. ˇSevi´c, B. P. Marinkovi´ c, S. Dujko, J. Ati´ c, D. V. Fursa, I. Bray, R. P. McEachran, F. Blanco, G. Garc´ıa, P. W. Stokes, R. D. 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5.0033536.pdf

J. Appl. Phys. 128, 244303 (2020); https://doi.org/10.1063/5.0033536 128, 244303 © 2020 Author(s).THz-range Faraday rotation in the Weyl semimetal candidate Co2TiGe Cite as: J. Appl. Phys. 128, 244303 (2020); https://doi.org/10.1063/5.0033536 Submitted: 16 October 2020 . Accepted: 03 December 2020 . Published Online: 28 December 2020 Rishi Bhandia , Bing Cheng , Tobias L. Brown-Heft , Shouvik Chatterjee , Christopher J. Palmstrøm , and N. P. Armitage COLLECTIONS This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN Issues in growing Heusler compounds in thin films for spintronic applications Journal of Applied Physics 128, 241102 (2020); https://doi.org/10.1063/5.0014241 Boris computational spintronics—High performance multi-mesh magnetic and spin transport modeling software Journal of Applied Physics 128, 243902 (2020); https://doi.org/10.1063/5.0024382 Ultrafast investigation and control of Dirac and Weyl semimetals Journal of Applied Physics 129, 070901 (2021); https://doi.org/10.1063/5.0035878THz-range Faraday rotation in the Weyl semimetal candidate Co 2TiGe Cite as: J. Appl. Phys. 128, 244303 (2020); doi: 10.1063/5.0033536 View Online Export Citation CrossMar k Submitted: 16 October 2020 · Accepted: 3 December 2020 · Published Online: 28 December 2020 Rishi Bhandia,1 Bing Cheng,1Tobias L. Brown-Heft,2Shouvik Chatterjee,3 Christopher J. Palmstrøm,2 and N. P. Armitage1,a) AFFILIATIONS 1The Institute of Quantum Matter, Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, Maryland 21218, USA 2Materials Department, University of California –Santa Barbara, Santa Barbara, California 93106, USA 3Department of Electrical & Computer Engineering, University of California –Santa Barbara, Santa Barbara, California 93106, USA a)Author to whom correspondence should be addressed: npa@jhu.edu ABSTRACT The Co 2family of ferromagnetic Heusler alloys has attracted interest due to their fully spin-polarized nature, making them ideal for applica- tions in spintronic devices. More recently, the existence of room temperature time-reversal-breaking Weyl nodes near the Fermi level was predicted and confirmed in these systems. As a result of the presence of these Weyl nodes, these systems possess a non-zero momentum space Berry curvature that can dramatically influence transport properties such as the anomalous Hall effect. One of these candidate com-pounds is Co 2TiGe. Recently, high-quality molecular beam epitaxy-grown thin films of Co 2TiGe have become available. In this work, we present a THz-range measurement of MBE-grown Co 2TiGe films. We measure the THz-range Faraday rotation, which can be understood as a measure of the anomalous Hall effect. We supplement this work with electronic band-structure calculations showing that the principal contribution to the anomalous Hall effect in this material stems from the Berry curvature of the material. Our work shows that this class ofHeusler materials shows promise for Weyl semimetal based spintronics. Published under license by AIP Publishing. https://doi.org/10.1063/5.0033536 I. INTRODUCTION Topological phases of matter have been of great current interest in condensed matter physics. One such manifestation of this non-trivial topological order is Weyl semimetals. These mate-rials arise when the conduction and valence bands of a materialtouch at a singular point near the Fermi energy in the Brillouin zone. 1This results in an effective Hamiltonian about this point where the energy has a linear dependence on crystal momentum.These band touchings are called Weyl nodes, and they act asmagnetic monopoles (sources and sinks) of an effective magneticfield, the Berry curvature, arising due to the momentum depen-dence of the Bloch functions in the Brillouin zone. 1,2A require- ment for the formation of Weyl nodes in a material is the breaking of inversion or time-reversal symmetry. While inversionbreaking Weyl semimetals such as TaAs have been discovered, 3,4 good examples of the theoretically simpler magnetic Weyl semi- metals have remained more elusive.The class of Heusler materials, a family of ternary intermetal- lics, has become a focus of research due to the wide variety of ele- ments that can be incorporated into them. Due to this flexibility,these materials have been found to exhibit a wide variety of phe-nomena ranging from half-metallicity 5to superconductivity6with potential applications in thermoelectrics7and spintronics.8They have also drawn interest due to their potential to host non-trivial topological phases because of their band structure ’s similarity to that of the III –V semiconductors.9This is due to the fact that the full Heusler with the chemical formula X 2YZ forms a zinc-blende lattice with two interpenetrating fcc sublattices as shown in the inset of Fig. 1 . Thus, Heuslers allow for a great range of properties and tunability while keeping many of the features of III –V semiconductors. Co2Heusler alloys such as Co 2TiGe are ferromagnetic with high Curie temperatures10and have drawn a great deal of interest due to their half metallic behavior, the property where the minority spin carrier experiences a bandgap, while the majority spin carrier isJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 244303 (2020); doi: 10.1063/5.0033536 128, 244303-1 Published under license by AIP Publishing.gapless.11–14Thus, these materials have the potential to be used as the basis for spintronic devices. First principles calculations haveshown that these Co 2alloys could potentially host Weyl fermions that would be accessible at room temperature.15,16More recently, a member of the family, Co 2MnGa, was confirmed to be a time-reversal-symmetry breaking Weyl semimetal through the exis-tence of “drum-head ”surface states. 17Furthermore, as these materi- als are magnetically soft, they also open opportunities to utilize an external magnetic field to manipulate the position of Weyl nodes. One potential manifestation of this Weyl physics is the anom- alous Hall effect (AHE).18The AHE occurs when a material exhib- its a Hall conductivity, σxy, that depends on its magnetization rather than an applied external magnetic field. At a microscopic level, the contributions to the AHE can be separated into threeparts: intrinsic, skew-scattering, and side-jump. 19The former con- tribution stems from the Berry curvature of the wave functions.Thus, this can be used to probe effects of the Weyl nodes that are theorized to exist within the band structure. However, this analysis is complicated by the fact that it is difficult to distinguish betweenthe intrinsic and side-jump contribution in DC experiments; there-fore, AC transport measurements can be an important tool to helpdistinguish between the different contributions to the AHE. In addition, an investigation of these materials ’THz-range frequency dynamics is important to understand these material ’s suitability for usage in next generation spintronic devices. Optical conductivitymay allow for a clearer analysis as by measuring the low-frequency conductivity of a material, it may be possible to distinguish between the intrinsic and side-jump contributions. 19II. EXPERIMENTAL DETAILS In this work, we present data from time-domain terahertz spectroscopy (TDTS) experiments conducted on high-quality,72.3 nm thick, thin films of Co 2TiGe grown by molecular beam epitaxy (MBE) on a substrate of MgO (001).20These experiments were conducted with a home-made time-domain terahertz spec-trometer with a 7 T magnet in closed cycle cryostat. 21By measuring the transmitted electric field through the sample and an appropri-ate reference substrate, we are able to calculate the complex trans- mission coefficient. From this, then the complex conductivity was calculated using an appropriate equation in the thin film limit. Inzero magnetic field, we use the standard expressionTωðÞ ¼ 1þn 1þnþZ0dσωðÞexpiω cn/C01 ðÞ ΔL/C2/C3 . Here, TωðÞis the complex transmission referenced to the substrate, nis the index of the refraction of the substrate, dis the thickness of the thin film, Z0is the impedance of the free space with a value of roughly 377 Ω, and ΔLis a small thickness difference between the sample substrate and the reference substrate. To examine the THz-range σxy(ω) of the material, we mea- sured the Faraday rotation of the THz light transmitted through our sample under a finite magnetic field. The magnetic field wasapplied parallel to the direction of propagation of light in theFaraday geometry. As the Faraday rotation is the angle that a line-arly polarized wave of light is rotated due to a magnetic field paral- lel to its direction of propagation, it is directly related to the transverse conductivity. In this geometry, utilizing a rotating wire-grid polarizer to modulate the polarization, we are then able tomeasure E xandEycomponents of the transmitted electric fields.21 By symmetry considerations of the material, we can constrain the Jones transmission matrix to have only two free parameters, Tyx andTxx, and calculate these matrix elements.22Then, to extract the Faraday rotation from the measured complex transmission, we usedthe expression θ F¼Re arctan Tyx=Txx/C0/C1 /C2/C3 . Figure 1 depicts the real and imaginary parts of the optical conductivity over the temperature range of 5 K –300 K. The overall magnitude of the conductivity increases with decreasing tempera-ture. The real part of the conductivity is flat and the imaginary partis small, consistent with the scattering rate for electrons well above the measured spectral region. The conductivity can then be fit with the Drude model for a free electron gas, σ(ω)¼ϵ 0/C0ω2 p iω/C0Γ/C0i(ϵ1/C01)ω"# , (1) where ωpis the plasma frequency, Γis the scattering rate, and ϵ1is the high frequency dielectric constant that accounts for the effects of higher band interband transitions on the low-frequency dielec-tric constant. The results of these fits are shown in Fig. 2 .A sw e can see, neither the plasma frequency nor the scattering rate showsa significant dependence on temperature. The plasma frequency shows about a 10% enhancement as the temperature decreases that is consistent with the overall increase in conductivity as the temper-ature decreases. The errors of the fit parameters are derived fromthe 95% confidence intervals derived from the fitting routine. InFig. 3 , we show the results of our polarimetry experiments. Figure 3(a) gives the Faraday rotation measured with the field FIG. 1. Complex conductivity of the Co 2TiGe thin film samples measured using TDTS. The real part of the conductivity (solid lines) is relatively flat in frequency and exhibits only a weak temperature dependence. The imaginary part of the conductivity (dashed lines) shows a similar temperature dependence. The insetfigure shows the Co 2TiGe Heusler crystal structure, with Co in red, Ti in green, and Ge in gray.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 244303 (2020); doi: 10.1063/5.0033536 128, 244303-2 Published under license by AIP Publishing.being swept either upward or downward. We see no sign of a hys- teresis as expected in a soft ferromagnetic thin film due to its in-plane easy axis consistent with DC Hall effect data shown inFig. 4 . This is consistent with an easy axis of these films lying in-plane, in the [110] direction as reported earlier. 20Starting from high fields, as the applied magnetic field decreases in magnitude, the magnetization relaxes to an in-plane direction, resulting in almost zero measured Faraday rotation due to the in-plane magne-tization. Figure 3(b) shows the Faraday rotation anti-symmetrized, where the up and down sweeps have been subtracted at oppositefields, i.e., θ ASHðÞ ¼ θupHðÞ /C0 θdown/C0HðÞ/C0/C1 =2, where θis the Faraday rotation and His the magnetic field. Figure 3(c) then shows the anti-symmetrized Faraday rotation averaged over the fre-quency range of 0.6 –1.2 THz, again showing the saturation that we saw earlier at fields above 5 kG. The Faraday rotation ’s saturation at low fields is consistent with the expectation that the AHE depends on the magnetization of the material. As the Faraday rotation can also be described by the expression, θ F/C25dσxyZ0 1þnsubþdσxxZ0, (2) we can see that in the zero frequency limit, it describes the Hall angle (if one neglects effects due to the substrate). To provide complementary information to our THz measure- ments, we used the open-source package Quantum Espresso23–25to calculate electronic band-structure calculations for Co 2TiGe under the density functional theory (DFT) framework. The projector-augmented wave method (PAW) was used with fully relativistic pseudopotentials, spin –orbit coupling, and the exchange functional was approximated using the generalized gradient approximation(GGA). To compute transport properties, we used the codeWannier90 26to project the calculated wavefunctions onto FIG. 2. The fit parameters a single Drude fit for the complex conductivity of the Co2TiGe thin film. The plasma frequency (a) and the scattering rate (b) both do not show a large dependence on temperature. FIG. 3. The Faraday rotation as measured using THz polarimetry. (a) The Faraday rotation averaged over the frequency range of 0.6 –1.2 THz as a function of the applied field. The saturation at high fields shows that the Faraday rotation is following the magnetization and thus is related to the anomalous Hall effect in the material. (b) The anti-symmetrized Faraday rotation where opposite fields have been subtracted. We see a clear saturation at fields above +40 kG. (c) Averaged anti-symmetrized Faraday rotation again averaged over the frequency range of 0.6 –1.2 THz.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 244303 (2020); doi: 10.1063/5.0033536 128, 244303-3 Published under license by AIP Publishing.maximally localized Wannier functions. From this, a 50 /C250/C250 adaptive grid was used to sample the Berry connection within theBrillouin zone. We then computed the anomalous Hall conductiv- ity numerically using Wannier90 with the Kubo –Greenwood formula as follows: σ AHC xy/C22hωðÞ ¼ie2 /C22hΩcNkX k,n,mfnk/C0fmk ðÞ εmk/C0εnk ðÞ εmk/C0εnk/C0/C22hωþiη ðÞ /C2Anm,x(k)Amn,y(k), (3)where Ωcis the cell volume, Ncis the number of kpoints used to sample the Brillouin zone, fnkis the Fermi –Dirac distribution func- tion for the band indexed by n,ηis a smearing parameter, and Amn(k)¼unkji∇kjumk hi is the matrix generalization of the Berry connection. For the DC limit as shown in Fig. 5(b) , we simply take the parameters ω¼0 and η¼0 in Eq. (3).I nFig. 5(a) , one can see a large complement of bands. Weyl nodes are located along the Γ/C0XandΓ/C0Kdirections. III. DISCUSSION The behavior of these Co 2TiGe thin films is consistent with the expected behavior for a magnetic metal. Through the measure-ment of the Faraday rotation, we understand the low-frequency response of the Hall effect. Above a low applied field, the Faraday rotation saturates, showing that σ xydepends on the magnetization. This suggests that the Faraday rotation is sensitive to the AHE andthus sensitive to the Berry curvature of the material. In the simplest case, for a pair of Weyl nodes near the Fermi level, the AHE contribution produced by a single pair of Weyl nodes due to the Berry curvature is given by σ xy¼e2K 2πh, (4) where Kis the wavevector between the Weyl nodes in the Brillouin zone. Using the Faraday rotation, we can estimate the Hall conduc-tivity using the expression provided previously and making the assumption that the magnetic field does not affect the magnitude of the longitudinal transport. We estimate σ xyas 83 :8Ω/C01cm/C01. For a single pair of Weyl nodes, this translates to a spacing of0:25 π a, where a, the lattice constant, is 5.830 Å. More generally, this can be seen as a measure of the total effective Berry dipole moment since Weyl nodes are monopoles of Berry curvature and the AHE contributions are additive. However, in Co 2TiGe, there are believed to be nine pairs of Weyl nodes15oriented such that some of their contributions partially cancel. In the upper half of the Brillouin FIG. 4. DC anomalous Hall effect data measured in the Hall bar geometry with the magnetic field in the [001] direction. FIG. 5. The results of the band-structure calculations presented in plots. (a) The first principles calculated band structure along symmetry directions wi th magnetization in the (001) direction and spin –orbit coupling turned on. Spin –orbit coupling was as implemented in Quantum Espresso. The zero level is referenced to the Fermi energy cal- culated using Quantum Espresso. (b) The calculated intrinsic DC anomalous Hall effect as a function of Fermi energy calculated using Wannier90. (c) T he optical intrinsic anomalous Hall conductivity calculated using Wannier90.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 244303 (2020); doi: 10.1063/5.0033536 128, 244303-4 Published under license by AIP Publishing.zone, there is predicted to be one Chern charge 2 Weyl node dis- placed along the kzdirection and two Chern charge 1 Weyl nodes with opposite relative charge in the kx–kzplane. Other Weyl nodes are related to these by the mirror symmetry in the kz¼0 plane and C4giving nine pairs in total. Using the Weyl node positions calculated via DFT,15we estimate a total Berry dipole moment of 1:12π a, which is of order of the observed value. Discrepancies with the actual value may arise from the fact that it comes from a differ-ence of large numbers and will have a proportionally larger uncer-tainty or the fact that some of the nodes (particularly the charge 2pair) may not give their full contribution to the AHE as they are far from E F. We can also compare this estimation of σxywith the measured DC values of ρxy. Here, using an average over a frequency range of 0.6–1.2 THz of the measured optical conductivity to approximate the zero frequency σxx, we obtain an estimated Hall resistivity of 3:15μΩcm. This gives a value very close to the measured Hall resis- tivity measured in the Hall bar geometry as depicted in Fig. 4 . Using the band-structure calculations, we can make a more detailed analysis of the contributions to the anomalous Hall effect.The expected Faraday rotation can be estimated using the measured optical conductivity and calculated intrinsic optical anomalous Hall conductivity in conjunction with Eq. (2). The results are plotted in Fig. 6 . The measured Faraday rotation shows a consistent offset with respect to the calculated Faraday rotation. We believe that the small mismatch is due to the contribution of skew-scattering and side-jump. However, it appears that the bulk of the contribution tothe AHE in this material stems from the intrinsic Berry curvature of this material. IV. CONCLUSION We have measured the THz-range optical conductivity of the Co 2Heusler Co 2TiGe using time-domain THz spectroscopy. These measurements were made possible by MBE-grown thin films ofthis system becoming available. We have also measured theTHz-range Faraday rotation of this material. Using DFT calcula-tions, we have demonstrated that the majority contribution to the Faraday rotation stems from the intrinsic Berry phase mechanism for the anomalous Hall effect of the material. ACKNOWLEDGMENTS We would like to thank T. McQueen for very helpful discus- sions regarding the DFT calculations. The work at JHU was sup-ported as part of the Institute for Quantum Matter, an EFRCfunded by the DOE BES (No. DE-SC0019331). The work at UCSB was supported by DOE BES (No. DE-SC0014388). We also acknowledge the use of shared facilities within the National ScienceFoundation (NSF) Materials Research and Science and EngineeringCenter (MRSEC) at the University of California, Santa Barbara,DMR 1720256, and the LeRoy Eyring Center for Solid State Science at Arizona State University. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1N. P. Armitage, E. J. Mele, and A. Vishwanath, “Weyl and Dirac semimetals in three-dimensional solids, ”Rev. Mod. Phys. 90, 015001 (2018). 2A. A. Burkov, “Weyl metals, ”Annu. Rev. Condens. Matter Phys. 9, 359 –378 (2017). 3S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.-M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, and M. Z. 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5.0040323.pdf

Rev. Sci. Instrum. 92, 034702 (2021); https://doi.org/10.1063/5.0040323 92, 034702 © 2021 Author(s).Operation analysis of the wideband high- power microwave sources based on the gyromagnetic nonlinear transmission lines Cite as: Rev. Sci. Instrum. 92, 034702 (2021); https://doi.org/10.1063/5.0040323 Submitted: 11 December 2020 . Accepted: 09 February 2021 . Published Online: 02 March 2021 Yancheng Cui , Jin Meng , Liyang Huang , Yuzhang Yuan , Haitao Wang , and Danni Zhu ARTICLES YOU MAY BE INTERESTED IN Picosecond high-voltage pulse measurements Review of Scientific Instruments 92, 034701 (2021); https://doi.org/10.1063/5.0028419 Identification of static nonlinearities by sinusoidal excitation with variable DC offsets Review of Scientific Instruments 92, 035103 (2021); https://doi.org/10.1063/5.0036696 A Bitter-type electromagnet for complex atomic trapping and manipulation Review of Scientific Instruments 92, 033201 (2021); https://doi.org/10.1063/5.0026812Review of Scientific InstrumentsARTICLE scitation.org/journal/rsi Operation analysis of the wideband high-power microwave sources based on the gyromagnetic nonlinear transmission lines Cite as: Rev. Sci. Instrum. 92, 034702 (2021); doi: 10.1063/5.0040323 Submitted: 11 December 2020 •Accepted: 9 February 2021 • Published Online: 2 March 2021 Yancheng Cui, Jin Meng, Liyang Huang, Yuzhang Yuan, Haitao Wang, and Danni Zhua) AFFILIATIONS National Key Laboratory of Science and Technology on Vessel Integrated Power System, Naval University of Engineering, Wuhan 430033, China a)Author to whom correspondence should be addressed: 360681625@qq.com ABSTRACT The wideband High-Power Microwave (HPM) sources, which combine the advantages of narrowband and ultrawideband sources, have drawn much attention. As a kind of wideband source, the gyromagnetic nonlinear transmission lines (GNLTLs) can directly modulate the incident pulses into radio frequency pulses without relying on the interaction between e-beam and microwaves. Due to the special working mechanism of gyromagnetic precession, the center frequency of the GNLTL can also be adjusted in a certain range. Based on classical magnetism and a simplified model of the GNLTL, this paper semi-quantitatively and theoretically analyzed the generation mechanism of HPM and illustrated the influences of the variations of parameters on the output microwaves. Then, a simple simulation based on 1-dimensional transmission line modeling method was carried out to study the performance of the GNLTL quantitatively, with the coupling of 1D telegraphist equations and the 3D Landau–Lifshitz–Gilbert equation. Simulation results preliminarily verified the conclusions derived from the theoretical analysis, and some working characteristics of the GNLTL were also obtained. This paper may help to understand the special working mechanism of the GNLTL and provide certain guidance for related simulations and experiments. Published under license by AIP Publishing. https://doi.org/10.1063/5.0040323 .,s I. INTRODUCTION High-Power Microwave (HPM) technologies have made great progress in the past few decades and still show great development potential with the rapid growth of application requirements in the military and industrial fields.1–3According to the traditional classi- fication of microwaves based on percentage bandwidth (pbw), the HPM can be analogously divided into narrowband (pbw <1%), wideband (1% <pbw<25%), and ultrawideband (pbw >25%).4 Obviously, due to the different performance in bandwidth, the nar- rowband and ultrawideband microwaves show the advantages of high energy and wide coverage on spectra, respectively. However, the narrowband microwaves are usually generated in special vac- uum tubes, and the electron beam interacting with microwaves is indispensable, which means the systems are usually complex. The ultrawideband microwaves, which get rid of the shackles of the electron beam, are usually generated directly by the on and off ofswitches, but the systems are usually huge and the frequency is not very high. Consequently, the wideband microwaves as a compromise have drawn attention gradually, and a number of wideband HPM sources have been built and studied in recent years. Due to the special advantages of nonlinear transmission lines (NLTLs) that do not require e-beams, which are more compact in structure and stable in operation, they have shown great potential in replacing traditional vacuum tubes. The gyromagnetic nonlinear transmission line (GNLTL) is a special kind of NLTL that has higher power capacity and is easy to be implemented compared to other types of NLTLs, such as the lines composed of nonlinear inductance or capacitance. In the first few years, the GNLTLs were mainly used as pulse sharpeners to get high-power pulses with sub-nanosecond or picosecond rising edges.5–7Eventually, it was discovered in exper- iment that the GNLTL can produce radio frequency oscillations, and then, a number of experiments have been carried out to study the generation of high-power microwaves. In the past few years, Rev. Sci. Instrum. 92, 034702 (2021); doi: 10.1063/5.0040323 92, 034702-1 Published under license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi researchers from Texas Tech University (TTU) have developed a systematic study on the GNLTL, including research on transmis- sion lines with different structures, the influence of parameters (such as temperature, excitation voltage, biasing field, and ferrite materi- als), and power synthesis.8–14The related work at TTU is instructive, and their research results provide a good reference for the design of GNLTLs. Russian scholars from the Institute of High Current Elec- tronics (IHCE) and the Institute of Electrophysics (IEP) have paid more attention to research with higher power levels (hundreds of MW to several GW), and they have achieved good results in power synthesis.15–20Teams from the UK, Ukraine, Brazil, and China have also carried out commendable investigations, which confirmed that the GNLTL performed well in the generation of wideband HPM.21–27 It is difficult to explain the principle of the GNLTL clearly in theory due to the complicated working mechanism. Most reports on the GNLTL focused on experimental research, and most of the theoretical explanations were based on the traditional applications of gyromagnetic precession. In fact, the application of using gyro- magnetic to generate HPM is different from the traditional situa- tions. The difference is that the angular magnetic field and the axial magnetic field in the former case are comparable, while the angu- lar magnetic field in the latter case is much smaller than the axial magnetic field. So, the traditional theory is not completely applicable here. In this paper, we try to qualitatively explain the physical mech- anism and working process of the HPM generation in the GNLTL, combined with the consideration of the aforementioned difference, and then analyzed the influence of the variations of some param- eters on the output microwaves. Furthermore, numerical simula- tions based on 1-dimensional Transmission Line Modeling (TLM) method were carried out to analyze the performance of the GNLTL quantitatively. II. THE HPM GENERATION BASED ON THE GNLTL A. Precession of the magnetic moments An atom composed of a nucleus and electrons is the smallest unit of an object. The electrons in atoms have two kinds of motion, namely, the orbital motion and the spin motion. According to the classical magnetism, the atomic magnetic moments are mainly gen- erated by the two motions of electrons.28If there is no external magnetic field, the atomic magnetic moments distribute randomly. The composition of all the moment vectors is the magnetization M. When an external magnetic field is applied to the magnet, all the moments will be aligned to the direction of the external field. For a uniaxial ferromagnetic crystal in equilibrium, the magnetization M is always parallel to the direction of the effective magnetic field Heff, which is mainly composed of the total external field, demagnetizing field, anisotropy field, exchange field, and other fields. For some rea- sons, if the direction of Mis changed and not parallel to Heff, then Mwill precess around Heffas a response. This is the so-called Lar- mor precession, and the process can be described by the following equation:28,29 dM dt=μ0γM×Heff, (1) whereμ0= 4π×10−7T m/A is the magnetic permeability of vacuum, γ=gqe 2me=−1.76×1011rad−1s−1T−1is the gyromagnetic ratio, gisLande factor (usually equals to 2), and qeandmeare the charge and mass of an electron, respectively. According to Eq. (1), Mwill precess perpetually around Heff, which we know is not the real case. Due to the damping mecha- nism, the energy of precession will decrease gradually and the angle between MandHeffwill also decrease, which means Mwill eventu- ally be aligned to the direction of the effective field Heff. To describe the damping mechanism, Landau and Lifshitz gave the following equation by adding a damping term M×(M×Heff) to Eq. (1):28,29 dM dt=μ0γM×Heff+αγμ 0 MsM×(M×Heff), (2) where Msis the saturation magnetization and αis the damping coef- ficient, which depends on the magnetic material. In fact, the expres- sion of the damping term is not unique. Gilbert put forward another equation using a M×dM dtterm,30 dM dt=μ0γM×Heff+α MsM×dM dt. (3) By simple transformation and simplification, Eq. (3) can be rewritten as dM dt=μ0γ 1 +α2M×Heff+αγμ 0 (1 +α2)MsM×(M×Heff). (4) It is easy to find that Eqs. (2) and (4) are equivalent when α is small. But when αis not small, the latter is closer to the real situation.30 B. Typical configuration of the GNLTL A simple 2D schematic diagram of the GNTL is shown in Fig. 1(a). In most cases, the GNLTLs are constructed as coaxial geometry consisting of an inner conductor, an outer conductor, and ferrites. The inner and outer conductors are usually made of brass or stainless steel. For high voltage application, a special material with high breakdown threshold is filled in the GNLTL to improve the insulation, such as SF 6, oil, and polymer materials. Ferrites act as a nonlinear and dispersive medium, and the material of ferrites can be MnZn, NiZn, or YIG. In addition, the ferrites are usually made into toroidal cores. Furthermore, solenoids or permanent magnets are necessary to provide the initial axial biasing magnetic field. The former are more widely adopted due to their simple structure and adjustable magnetic field. The latter maybe practical in some spe- cial occasions, but the arrangement of magnets needs to be designed specially. In the application of HPM generation, if an excitation pulse with a nanosecond or sub-nanosecond rising edge is fed into the GNLTL that has been pre-magnetized by the axial magnetic field, then one can observe a RF waveform at the output after the inci- dent signal has propagated through the GNLTL. The essence of this process is a dynamic motion of the magnetization vector under the action of the effective magnetic field (including external magnetic field, demagnetizing field, and anisotropy field). The external mag- netic field composed of the azimuthal field and the axial field is dominant. The former is induced by the incident pulse, and the lat- ter is induced by the solenoid. Distributions of the two important components of external field are shown in Figs. 1(b) and 1(c), respec- tively, and Fig. 1(d) is the composition of the two fields. The dynamic Rev. Sci. Instrum. 92, 034702 (2021); doi: 10.1063/5.0040323 92, 034702-2 Published under license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi FIG. 1 . (a) 2D schematic diagram of the GNLTL. (b) The azimuthal magnetic field induced by the incident pulse, displayed in a cross-sectional view. (c) The axial magnetic field induced by the solenoid, displayed in a quarter-sectional view. (d) The effective magnetic field in the GNLTL (including axial and azimuthal fields). process can be described as the following: (1) The solenoid is turned off, the external field is zero, there is no magnetization, and the mag- netic moments in ferrites distribute randomly. (2) Turning on the solenoid, then an axial field is generated inside the solenoid and the moments are aligned to the axial direction to form the magnetiza- tion vector M. If the current in the solenoid is large enough, the ferrites can be magnetized to saturation, which is considered true in the following analyses. (3) An excitation pulse is injected into the GNLTL and an azimuthal field that is perpendicular to the axial field is induced according to the right-hand rule. The azimuthal field will force the magnetization pre-aligned by the biasing field to deviate from the axial direction. (4) According to Eq. (1), once the direction ofMis changed, Mwill precess around Heffwith a certain frequency and oscillations that will be superimposed on the incident pulse. (5) Due to the presence of damping, the precession will gradually stop and the magnetization will return to the axial direction again, as described by Eq. (4).C. Working mechanism of the GNLTL In Sec. II B, the working process of the GNLTL is briefly intro- duced. In this part, we try to analyze the working mechanism of the GNLTL in detail. The precession of magnetization Mis placed in a Cartesian coordinate system shown in Fig. 2(a), where z,θ, and raxes represent the axial, azimuthal, and radial directions, respec- tively. The initial axial biasing magnetic field is supposed as Hz0, the azimuthal field is Hθ, the effective external field is the super- position of the two and the angle between Heff, and the zaxis is φ,φ=tan−1(Hθ Hz0). It should be pointed out that the azimuthal field does not change the magnitude of Mbut only its direction because the ferrites are saturated. The precession angle (the angle between M andHeff) isα, andα=φwhen damping is ignored. The precession ofMin the O−zθrcoordinate system is a 3D movement, and the components in different axes can be obtained by creating a new o′ −xyz′coordinate system, as shown in Fig. 2(b). The origin of the FIG. 2 . (a) The precession of Min the O−zθrcoordinate system. (b) The sit- uation in a new o′−xyz′coordinate system. Rev. Sci. Instrum. 92, 034702 (2021); doi: 10.1063/5.0040323 92, 034702-3 Published under license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi new coordinate system is the center of the circle drawn by the end point of Mduring the precession; the xaxis is parallel to the abline, which is the intersection line of the circle plane and the θ–O–zplane; theyaxis is perpendicular to the paper surface; and the z′axis is along the direction of Heff. The components of Mcan be calculated in three steps: first, cal- culating the components of Mon the x,y, and z′axes, denoted as Mx,My, and Mz′; then, calculating the components of Mx,My, and Mz′on the θ,r, and zaxes, denoted as Mx−θ,Mx−z,Mx−r,My−θ, My−z,My−r,Mz′−θ,Mz′−z, and Mz′−r; and finally, Mθ,Mz, and Mrcan be obtained by adding up all the components. According to Fig. 2, Mx=Mssinαcosβ, My=Mssinαsinβ, Mz′=Mscosα,(5) whereβis the angle between co′and the xaxis, co′is the projec- tion of Mon the circle plane, and β=ωt, whereωis the precession frequency. Because Mxis perpendicular to the raxis andα=φ, then Mx−θ=Mxcosφ=1 2Mssin(2φ)cos(ωt), Mx−z=Mxsinφ=Mssin2φcos(ωt), Mx−r=0,(6) whereφis the angle between xaxis and θaxis (it can be proved that the angle is equal to φ). Analogously, My−θ=0, My−z=0, My−r=My=Mssinφsin(ωt),(7) Mz′−θ=Mz′sinφ=1 2Mssin(2φ), Mz′−z=Mz′cosφ=Mscos2φ, Mz′−r=0.(8) The components of Mare Mθ=Mx−θ+My−θ+Mz′−θ =1 2Mssin(2φ)cos(ωt)+1 2Mssin(2φ), Mz=Mx−z+My−z+Mz′−z=Mssin2φcos(ωt)+Mscos2φ, Mr=Mx−r+My−r+Mz′−r=Mssinφsin(ωt).(9)For Eq. (9), the angle φis a fixed value defined by external fields, and both MθandMzcontain a constant term and a sine term, but Mr has only a sine term. According to the theory of demagnetization,28 there is no demagnetizing field on the axis of an infinite rod or in the circumferential direction of a closed-shape magnet. The demag- netizing fields on θandzaxes can be ignored but only on the raxis, Hr=−Mr. (10) Therefore, the magnetic flux in the raxis is constant, but the flux in the θandzaxes changes sinusoidally. Faraday’s law of elec- tromagnetic induction states that current will be induced when the magnetic flux enclosed by a closed loop changes, which means that there will be induced currents in the corresponding loops in the directions of θandzaxes. Figure 3(a) shows the magnetic flux distri- bution in the θaxis direction; the induced flux fills the space between the inner and outer conductors, but the primary flux is only inside the ferrites. It is assumed that Mθis uniform in the rdirection, and the flux change of a transmission line is given by Δϕθ=μ0ΔMθSf=μ0ΔMθΔz(rm−ri) (11) where Sfis the quarter section area of a ferrite bead with a length ofΔz. According to Lenz’s law, there must be a magnetic field with a reverse tendency to hinder the change of flux, and the field is provided by the induced current flowing in the inner and outer con- ductors. The induced flux Δϕθ−eis equal and opposite to Δϕθ, and the corresponding induced magnetic field is Hθ−e=Δϕθ−e μ0Sc=−ΔMθSf Sc=−ηΔMθ, (12) where Scis the quarter section area of a transmission line with a length of Δzandη=(rm−ri ro−ri)is defined as the filling factor of the GNLTL. Combining Eq. (9), the induced voltage is Vm=−dΔϕθ−e dt =−μ0Δz(rm−ri)dMθ dt =−ωμ0MsΔz(rm−ri)sin(2φ)sin(ωt). (13) Thus, the RF signal generated by the GNLTL is given by Eq. (13). Furthermore, there is also a sine variable in the zdirec- tion, which means induced field and current can also be generated in the corresponding loop, and the magnetic flux distribution in the zaxis direction is shown in Fig. 3(b). The induced eddy current is FIG. 3 . (a) Magnetic flux distribution in theθaxis direction. (b) Magnetic flux dis- tribution in the zaxis direction. “ ×” repre- sents the primary flux, and “ •” represents the induced flux. rois the inner radius of the outer conductor, rmis the outer radius of the ferrite, and riis the radius of the inner conductor. Rev. Sci. Instrum. 92, 034702 (2021); doi: 10.1063/5.0040323 92, 034702-4 Published under license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi azimuthal and the magnetic field is axial, which will affect the axial biasing field. According to the derivation of Eq. (12), the axial eddy field can be written as Hz−e=−ΔMzAf Ac=−kΔMz, (14) where k=Af Ac=r2 m−r2 i r2 o−r2 iis the area ratio of the magnetic material on the cross section of the GNLTL, 0 <k<1. From the semi-quantitative derivation above, it can be con- cluded that the HPM comes from the precession of the magneti- zation around the effective magnetic field. And the precession is a 3D process, which means that all components on the three axes will be affected by the precession and the components are coupled with each other: this is the essence of the tensor permeability. Due to the complexity of precession, the HPM generation is affected by many factors, including physical parameters of the GNLTL, excitation voltage, and induced field, which are also reflected in Eqs. (10)–(14). III. NUMERICAL SIMULATIONS OF THE GNLTL A. The coupling of telegraphist equations and LLG equation There are two processes in the generation of HPM in the GNLTL: one is the dynamic motion (precession) of Munder the action of Heff, which has been discussed above, and the other is the influence of the precession on the signals propagating in the transmission line. These two processes occur simultaneously and are coupled with each other. The signals propagating in the GNLTL are TEM-mode waves, and the classical telegraphist equations of transmission line can be written as di dz=−C0dv dt, dv dz=−dϕ dt,(15) whereϕis the magnetic flux per unit length, and it is assumed Mθ(r) is uniform in the rdirection; then, ϕ=L0i+rm ∫ riμ0Mθdr=L0i+μ0Mθ(rm−ri), (16) dϕ dt=L0di dt+μ0(rm−ri)dMθ dt. (17)Combining Eq. (4), we can get the following equations: ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩di dz=−C0dv dt dv dz=−L0di dt−μ0(rm−ri)dMθ dt dM dt=μ0γ 1 +α2M×Heff+αγμ 0 (1 +α2)MsM×(M×Heff).(18) The voltage and current of each node in the transmission line can be calculated by Eq. (18), and the induced flux is related to dMθ dtgiven by the Landau–Lifshitz–Gilbert (LLG) equation. More- over, the effective Heffcontains the axial biasing field, axial induced field, azimuthal induced field, excitation field, and radial demagne- tizing field. It is difficult to obtain an analytical solution of Eq. (18), so numerical simulations based on 1D Transmission Line Modeling (TLM) method were conducted. B. The 1D equivalent circuit model and the algorithm The transmission line modeling method is also called the trans- mission line matrix method, which is a conventional electromag- netic calculation method. The main principle of TLM is to discretize the transmission line into small segments, which are linked to the physical system by mathematical equations, and these segments are also related to the time step. According to Kirchhoff’s law and Mil- man’s theorem, voltage and current of the first node can be calcu- lated first, and then, the parameters of the subsequent nodes can be obtained, in turn, through the link relationships. Details of the TLM are given in Ref. 31. The GNLTL has a special transmission line structure, and the microwave signals propagating in the GNLTL can be calculated con- veniently by the TLM. A 1D equivalent circuit model of the GNLTL is shown in Fig. 4. The cascaded ladder work discretizes the GNLTL into nstages, and the length of each stage is Δz.Vsis the source andRsandRLare the resistance of the source and the load, respec- tively. R0,C0,L0, and G0are the resistance, capacitance, inductance, and conductance per unit length of the GNLTL, respectively. The interaction between the magnetization precession and the electric signal inside the GNLTL is reflected in the coupling between tele- graphist equations and the LLG equation, which is then dealt with the induced voltage sources Vm. Furthermore, the time step and the space step are constrained by Δt=Δz√L0C0, and Δtshould be small enough to make the modeling error negligible. FIG. 4 . The 1D equivalent circuit of the GNLTL. Rev. Sci. Instrum. 92, 034702 (2021); doi: 10.1063/5.0040323 92, 034702-5 Published under license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi FIG. 5 . A diagram of the algorithm. The algorithm to realize the model is a time stepping algo- rithm, and a diagram has been shown in Fig. 5. The algorithm first divides the input pulse into instantaneous samples that are d t(Δt) apart. Then, the GNLTL is divided into many segments, which are represented by the small rectangles. At each time step, the algo- rithm reads a single sample and assigns it to the corresponding segment. In addition, during the step, the equations (including the telegraphist equations, LLG equation, and link equations) discussed above are calculated. This process is repeated until the end of the input pulse. The algorithm takes a voltage reading at the end of the line at each time step, which is returned as the output of the GNLTL. Some other data such as the components of the magneti- zation can also be returned. There is a flowchart given in Fig. 6 to describe the calculation briefly. The core is to calculate the induced voltage based on the LLG equation and then add it to the telegraph equations. C. Simulation results and discussion A 30 cm-long GNLTL with SF 6insulated was constructed in software, and a trapezoidal excitation signal with 1 ns rising edge and 50 kV flattop was fed into the transmission line [the excitation FIG. 6 . A flowchart of the algorithm. pulse can be seen from Fig. 7(a)]. The parameters of the GNLTL and the simulation settings are detailed in Table I. Simulation results of the 30 cm GNLTL are shown in Fig. 7. It can be seen from Fig. 7(a) that oscillations were superimposed on the flattop after the input pulse has propagated through the GNLTL. Figure 7(b) shows the frequency spectrum of the output waveform. The center frequency is about 1.8 GHz, and the 3 dB bandwidth is about 23.6%. It was mentioned earlier that both Mθ andMzcontain a constant term and a sine term, but Mrhas only a sine term, which was verified in Fig. 7(c). In addition, the phases of MθandMzare complementary, which verified the energy coupling between azimuthal and axial magnetic fields. The curve in Fig. 7(d) is a filtered result, and the peak voltage of the microwave is about 25 kV. The influence of damping on the output microwaves was sim- ulated, and the results are shown in Fig. 8. It can be seen that the variation in damping does not have much effect on the center Rev. Sci. Instrum. 92, 034702 (2021); doi: 10.1063/5.0040323 92, 034702-6 Published under license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi FIG. 7 . Simulation results of the GNLTL. (a) The excitation pulse and the output waveform. (b) Frequency spectrum of the output waveform. (c) Components of the magnetization in φ,z, and rdirections. (d) 1 GHz high pass filtering on the output waveform. frequency, while the modulation depth shows a downward trend as the damping coefficient increases. In this paper, the modulation depth is defined as ( V1−V2)/(V1+V2), where V1andV2are the first and second peaks of the output oscillations, respectively. Then, the voltage conversion efficiency in the GNLTL can be described by the modulation depth. From the simulation results, we know that the ferrite with a low damping coefficient performs better on the efficiency of HPM generation. In fact, the damping coefficient αis proportional to the ferromagnetic resonance line width ΔH, which is a critical parameter for gyromagnetic ferrites and can be measured in a laboratory.28So, it can be concluded that a GNLTL loaded with a small line width ferrite material can generate microwave with highervoltage amplitude. This provides guidance for the selection of the magnetic material in experiments. Figure 9 shows the effect of saturation magnetization on the microwave output. It can be seen that the center frequency increases as the magnetic field increases, which is consistent with the con- clusions given in Refs. 18, 24, and 25. But on the other hand, the modulation depth shows a trend of first increasing and then decreasing, which means that the saturation magnetization has an optimal value. This trend of modulation depth is different from the conclusion that Vmis proportional to Msgiven in Eq. (13). The intrinsic mechanism has not been fully understood, and one guess is that the energy dissipation is related to damping. Further TABLE I . Parameters of the GNLTL and the simulation settings. Ferrite ε1μ1 Insulation ε2μ2 4πMs Hz0 l r o rm ri R0aG0aZ0 material (NiZn) (NiZn) material (SF 6) (SF 6) (Gs) (kA/m) α (cm) (mm) (mm) (mm) (Ω) (Ω) (Ω) NiZn 15 3 SF 6 1 1 3500 18 0.01 30 15 10 5 0 0 63.5 aThe GNLTL was assumed to be lossless to simplify the model. Rev. Sci. Instrum. 92, 034702 (2021); doi: 10.1063/5.0040323 92, 034702-7 Published under license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi FIG. 8 . The effect of damping on the output microwaves. (a) The domain waveforms. (b) The variations of center frequency and modulation depth. FIG. 9 . The effect of saturation magnetization on the output microwaves. (a) The domain waveforms. (b) The variations of center frequency and modulation depth. (The magnetization is usually expressed by 4 πMs, and the unit is Gs in Gaussian units.) FIG. 10 . The effect of initial axial biasing magnetic field on the output microwaves. (a) The domain waveforms. (b) The variations of center frequency and modulation depth. Rev. Sci. Instrum. 92, 034702 (2021); doi: 10.1063/5.0040323 92, 034702-8 Published under license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi TABLE II . Key parameters of the first reported GNLTL. Ferrite material Insulation material lr1(cm) ro−r1(mm) rm−r1(mm) ri−r1(mm) Hz0−r1(kA/m) NiZnaSF6 30 16 9 5 22 aεr1= 14,μr1= 4.8. studies especially experiments are still needed to give a detailed analysis. The effect of initial axial biasing magnetic field on the output microwaves is described in Fig. 10. The simulation results reveal that the center frequency decreases as the biasing field increases and, the trend is consistent with the conclusions given in Refs. 18, 24, and 25. The modulation depth increases initially, followed by a decrease trend. When the biasing field is 20 kA/m, which is very close to the azimuthal field (18 kA/m), the modulation depth is the largest. As a matter of fact, similar conclusion can also be obtained by Eq. (13). The conclusion is that the amplitude of the oscillation reaches the largest when the axial and azimuthal are equal. Specifi- cally, when Hθ=Hz0,φ=π/4 and then sin(2 φ) gets the maximum value (no other parameters are considered). This reveals two facts, one is that the center frequency can be adjusted easily by changing the axial biasing magnetic field. Although the center frequency can also be adjusted in other methods, the method referred above is the most convenient because the biasing field is provided by the solenoid whose current can be changed easily. The other fact is that the bias- ing field should be selected appropriately to get higher efficiency of microwave generation. Furthermore, the excitation voltage shouldbe compatible with the biasing field, and the insulation should also be guaranteed. IV. VALIDATION In order to verify the validity of the simulation research car- ried out above, two test simulations were developed based on the configuration parameters of the GNLTLs reported in Refs. 10 and 26. The two GNLTLs were constructed by two teams from differ- ent universities and are different in size, power level, and magnetic characteristics. Some key parameters of the first reported GNLTL are listed in Table II, and more detailed information can be found in Ref. 26. Figure 11(a) shows the excitation pulses including the reported experimental waveform and our fitted curve, and the peak voltage is about 70 kV. A GNLTL with the same structure and size was constructed in software, and then, the fitted excitation pulse was fed into the GNLTL. Comparison between the reproduced output waveform and the reported experimental result is shown in Fig. 11(b). It can be seen that the curve trend was calculated accurately by the simula- tion, and the first peaks of the two curves are in good agreement. FIG. 11 . First comparison between the reported experimental results and our reproduced results. (a) The input high voltage pulses. (b) The output waveforms. (4 πMs= 4800 Gs andα= 0.01). TABLE III . Key parameters of the second reported GNLTL. Ferrite material Insulation material lr2(cm) ro−r2(mm) rm−r2(mm) ri−r2(mm) Hz0−r2(kA/m) NiZnaSF6 90 3.9 3.175 1.5 15 aεr2= 15,μr2= 3. Rev. Sci. Instrum. 92, 034702 (2021); doi: 10.1063/5.0040323 92, 034702-9 Published under license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi FIG. 12 . The second comparison between the experimental results and the simulation results. (4 πMs= 3500 Gs and α= 0.01). At the same time, what cannot be ignored is that the oscillation fre- quency of the calculated waveform cannot be completely consistent with the experimental result. In the same way, the other test simulation was developed. The dimension of the second GNLTL is smaller than the first one, and the operating voltage is lower (10 kV–20 kV). The bias magnetic field and saturation magnetization are also smaller. Table III lists the key parameters of the second GNLTL, and more detailed informa- tion can be found in Ref. 10. The comparison of results is shown in Fig. 12, and the input and output waveforms were put together. It can be observed that the first few peaks of the two output wave- forms agree well. The latter part of the simulated output waveform can hold the oscillation at the top, while the experimental waveform decays quickly. Overall, the two comparisons validated that the proposed model can predict the wave trend and the amplitude of the out- put microwave, and there are also inconsistencies in some details. The sources of errors maybe multiple. On the one hand, it is diffi- cult to accurately detect the microwave signals with high-frequency and high power in experiments. On the other hand, the references did not give all the parameters of their GNLTLs, so that the simula- tions cannot fully reproduce the experimental results. In addition, it should be admitted that the simulations have inevitable errors due to the simplification of the physical model. However, it should be noted that due to the complex mechanism of GNLTLs, accurate simula- tion is relatively difficult and a common problem. The simulations discussed in this paper are basically credible, and the inaccuracy is acceptable to some extent, especially for the qualitative analyses and trend explorations. V. CONCLUSION In recent years, investigations of the wideband high-power microwave sources based on the GNLTL have made great progress due to the efforts of many researchers, and it is believed that the GNLTL has a great prospect in the application of compact HPMsources. A special working mechanism of the GNLTL brings special characteristics and performance but also makes the physical process more complicated. There is not much theoretical research but more experiments. More importantly, the HPM generation based on the GNLTL is different from traditional applications. This paper theo- retically analyzed the working mechanism of the GNLTL based on a semi-quantitative method. The process of microwave generation was explained clearly. Then, numerical simulations were developed in software. The simulation results verified the theoretical conclu- sions and can also guide the future experimental research. In par- ticular, test simulations were carried out to validate the proposed simulation model, and the not very big inaccuracy between the reported waveforms and the reproduced results tells that the model is practicable. Now, a physical GNLTL is under construction, and experimental research will be carried out soon. ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China under Grant No. 51907202. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1J. Benford, J. A. Swegle, and E. 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9.0000047.pdf

AIP Advances 11, 025211 (2021); https://doi.org/10.1063/9.0000047 11, 025211 © 2021 Author(s).Effect of surface modification treatment on top-pinned MTJ with perpendicular easy axis Cite as: AIP Advances 11, 025211 (2021); https://doi.org/10.1063/9.0000047 Submitted: 13 October 2020 . Accepted: 16 January 2021 . Published Online: 03 February 2021 H. Honjo , H. Naganuma , T. V. A. Nguyen , H. Inoue , M. Yasuhira , S. Ikeda , and T. Endoh COLLECTIONS Paper published as part of the special topic on 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials and 65th Annual Conference on Magnetism and Magnetic Materials ARTICLES YOU MAY BE INTERESTED IN Enhancement of magnetic coupling and magnetic anisotropy in MTJs with multiple CoFeB/ MgO interfaces for high thermal stability AIP Advances 11, 025231 (2021); https://doi.org/10.1063/9.0000048 Temperature dependence of the energy barrier in X/1X nm shape-anisotropy magnetic tunnel junctions Applied Physics Letters 118, 012409 (2021); https://doi.org/10.1063/5.0029031 W thickness dependence of spin Hall effect for (W/Hf)-multilayer electrode/CoFeB/MgO systems with flat and highly (100) oriented MgO layer AIP Advances 11, 025007 (2021); https://doi.org/10.1063/9.0000011AIP Advances ARTICLE scitation.org/journal/adv Effect of surface modification treatment on top-pinned MTJ with perpendicular easy axis Cite as: AIP Advances 11, 025211 (2021); doi: 10.1063/9.0000047 Presented: 6 November 2020 •Submitted: 13 October 2020 • Accepted: 16 January 2021 •Published Online: 3 February 2021 H. Honjo,1,a) H. Naganuma,1,2,3 T. V. A. Nguyen,1,2,3H. Inoue,1M. Yasuhira,1S. Ikeda,1,2,3,4and T. Endoh1,2,3,4,5 AFFILIATIONS 1Center for Innovative Integrated Electronic Systems, Tohoku University, Sendai 980-8572, Japan 2Center for Science and Innovation in Spintronics, Tohoku University, Sendai 980-8577, Japan 3Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan 4Research Institute of Electrical Communication, Tohoku University, Sendai 980-8577, Japan 5Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan Note: This paper was presented at the 65th Annual Conference on Magnetism and Magnetic Materials. a)Author to whom correspondence should be addressed: hr-honjo@cies.tohoku.ac.jp. Telephone: +81-22-796-3410, Fax:+81-22-796-3432 ABSTRACT We investigated the effects of surface modification treatment (SMT) on the perpendicular magnetic anisotropy (PMA) and the thermal tolerance of top-pinned magnetic tunnel junctions (MTJs) with a Co/Pt synthetic ferrimagnetic coupling reference layer. Applying an SMT to the bottom Pt layer increased the PMA of the overlying Co/Pt multilayer. X-ray diffraction spectrum analysis revealed that the SMT resulted in a higher crystallinity and smaller lattice spacing in the Co/Pt multilayer in the thinner bottom Pt layer, which may have increased the PMA in the Co/Pt multilayer. The tunnel magnetoresistance (TMR) ratio of the MTJ with SMT increased as the annealing temperature was increased up to 400○C; conversely, the TMR ratio of the MTJ without SMT decreased at an annealing temperature of 400○C. Evaluation of them-Hloops revealed that, after annealing at 400○C, the reference layers in the MTJs after SMT possessed better magnetic properties than those in the MTJs without an SMT; this is attributable to the higher PMA of the reference layers with SMT. EDX line analysis revealed that SMT suppressed Pt diffusion to the MgO barrier, resulting in a higher thermal tolerance and larger PMA of the reference layer. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/9.0000047 I. INTRODUCTION An embedded nonvolatile memory is an essential element for future electronics as it can significantly reduce the standby power, which is an important requirement in memory appli- cations. Spintronics-based memories are the most suitable for embedded nonvolatile memory applications owing to their low- voltage and high-speed operation capabilities.1–6CoFeB/MgO interface magnetic tunnel junctions (MTJs) with perpendicular mag- netic anisotropy (PMA) (i-pMTJs) have become the de facto stan- dard for modern spin-transfer torque magnetic random access memory (STT-MRAM) technology because they satisfy the require- ments of a low write current, high tunnel magnetoresistance (TMR) ratio, high thermal stability, and high scalability. Thus, CoFeB- MgO-based i-pMTJs have been intensively studied in the last decade.7–23Recently, spin–orbit torque MRAM (SOT-MRAM) hasalso been receiving attention as a promising candidate to replace high-speed SRAM in a high-speed memory system.24However, SOT-MRAM is still in its research and development phase because of its complex structure. STT-MRAM technology has a bottom-pinned structure, in which the pinned layer lies under the MgO tunnel- ing barrier, is widely used. Conversely, SOT-MRAM technology has an MTJ with a top-pinned structure, in which the pinned layer lies on the MgO tunneling barrier,24has been employed because a free layer must be positioned adjacent to the channel layer. The bottom-pinned structure exhibits high PMA and thermal tolerance due to the Co/Pt multilayer with high crystallinity formed on the thick buffer layer;25,26however, achieving high PMA is dif- ficult for the Co/Pt multilayer in a top-pinned structure because of the lack of a thick buffer layer below the Co/Pt multilayer. Increas- ing the Pt buffer layer (i.e., bottom Pt layer) thickness in bottom- pinned p-MTJs increases the PMA of the Co/Pt multilayer.12,25 AIP Advances 11, 025211 (2021); doi: 10.1063/9.0000047 11, 025211-1 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv However, in top-pinned p-MTJs, if the bottom Pt layer thickness exceeds 1.5 nm, the magnetic coupling between ferromagnetic lay- ers becomes insufficient.27Thus, the top-pinned structure exhibits a tradeoff between the PMA and magnetic coupling strength. Our previous study showed that a surface modification treatment (SMT) increases the perpendicular anisotropy and thermal tolerance of the Co/Pt layer, even for a relatively thin (3 nm) Pt buffer layer.28Con- sequently, in this study, to address the tradeoff between the PMA and magnetic coupling strength, we performed an SMT to develop a Co/Pt pinned layer with a high PMA for top-pinned p-MTJs. II. EXPERIMENTAL PROCEDURE We prepared four types of stacks (i.e., Stacks A–D). To eval- uate the PMA, Stacks A and B were prepared, which consisted of (in order from the substrate side) Ta(5)/CoFeB(1)/MgO/CoFeB(1) /Mo(0.3)/Co(0.5)/bottom Pt( t)/[Co(0.5)/Pt(0.3)] 4/Co(0.5)/capping layer, where the numbers in the parentheses represent the nom- inal thicknesses in nm. The thickness of the bottom Pt layer is denoted by t. Stacks C and D were the MTJs with a Co/Pt-based top- pinned synthetic ferrimagnetic coupling reference layer structure: Ta(5)/CoFeB(1)/MgO/CoFeB-based free layer/Mo(0.3)/Co(0.5)/Pt (t)/[Co(0.5)/Pt(0.3)] 4/Co(0.5)/Ru/[Co(0.5)/Pt(0.3)] 7/capping layer. For Stacks A and C, a 6-nm-thick bottom Pt layer was deposited on the bottom Co layer and exposed to Ar ion plasma (rf power of 50 W, dc-bias voltage to substrate of 400 V, and gas pressure of 0.8 Pa); then, the upper part of the stack was deposited without being exposed to the air. The post-SMT residual bottom Pt layer thick- ness ( t) was determined by Pt etching rate and processing time of SMT. Intermixing with Ta(5) in the bottom electrode caused the bottom CoFeB layer (under MgO of Stacks A and B) to become non- magnetic after annealing. All films were deposited onto a 300-mm ϕ thermally oxidized Si substrate by a DC/RF magnetron sputtering system at room temperature. After the deposition, Stacks A and B were annealed at 400○C for 1 h, and Stacks C and D were annealed at 300, 350, or 400○C for 1 h in vacuum. A vibrating sample mag- netometer was used to measure the m-Hcurves. The areal PMA energy density ( Kefft) was derived by calculating the areal differ- ence between the in-plane and out-of-plane m-Hcurves. The TMR ratio was determined using the current-in-plane tunneling method. The crystalline structures of the films were evaluated by implement- ing high-resolution X-ray diffraction (HRXRD) with a Cu-K αline (wavelength =0.1541 nm). III. RESULTS AND DISCUSSION Figure 1 shows Kefftas a function of tfor Stacks A and B after 400○C annealing. Evidently, Kefftfor Stack A with SMT was nearly constant for tranging from 0.5 to 2.5 nm. In contrast, Kefftfor the Stack B without SMT rapidly increased as tincreased up to 1.5 nm; it gradually increased for t≥1.5 nm. TheKefftvalues for Stack A with SMT were higher than those for Stack B without SMT for all the examined tvalues. To understand the physical mechanisms underlying these results, we employed HRXRD analysis to Stacks A and B. Figures 2(a) and 2(b) show out-of-plane X-ray diffraction spectrum for Stack A with SMT and for Stack B without SMT, respectively. In both cases, two strong diffraction peaks were observed; these peaks FIG. 1. Areal PMA energy density, Kefft, of the Co/Pt multilayer as a function of the bottom Pt layer thickness tfor Stack A with SMT and Stack B without SMT. were assigned to the Pt (111) and Co/Pt (111) diffraction planes. In Stack B without SMT, the intensities of both Co/Pt (111) and Pt (111) increased with increasing t. This indicates the presence of the Co/Pt multilayer with a heterogeneous structure that depends ont. Conversely, in Stack A with SMT, the intensities of Pt (111) and Co/Pt (111) are independent of t. We focused on the integrated intensity and lattice spacing of Co/Pt (111) because the structure of the Co/Pt multilayer is closely related to Kefft. The integrated intensity and lattice spacing of Co/Pt (111) results for the samples are illustrated in Figs. 2(c) and 2(d). The lattice spacing was calculated using the diffraction angles of Co/Pt (111) for Stacks A and B. The integrated intensity values for the Co/Pt (111) peaks for Stack A with SMT are larger than those for Stack B without SMT when t<1.0 nm (Fig. 2(c)). This indicates that the Co/Pt multilayer in Stack A with SMT has higher crystallinity even when t≤1.0 nm, resulting in higher Kefft. The lattice spacing of the Co/Pt (111) layer in Stack A with SMT is lower than that in Stack B when t>0.5 nm (Fig. 2 (d)). As has been previously reported, the magnetic anisotropy of the Co/Pt multilayer is correlated with the lattice spacing of the Co/Pt struc- ture;25thus, the X-ray diffraction structural analysis results suggest that the SMT-induced structural variation of the Co/Pt multilayer plays an important role in increasing PMA and robustness against annealing.28The influence of surface roughness on PMA may be small because SMT does not change the surface roughness of the Pt layer.28 Next, we evaluated the effects of SMT on the transport proper- ties as a function of annealing temperature to understand the robust- ness against annealing. Figures 3(a) and 3(b) show the annealing temperature dependence of the TMR ratio for Stack C with SMT and Stack D without SMT, respectively. For Stack C with SMT, the TMR ratio increased as the annealing temperature increased up to 400○C for tvarying from 1 nm to 2 nm. When t=0.5 nm, the TMR ratio degraded considerably after 400○C annealing. In contrast, for Stack D without SMT, TMR ratio degraded after 400○C annealing at all values of t. To elucidate the reasons for these phenomena, we evaluated the dependence of the magnetic properties on the annealing tem- perature. Figures 3(c) and 3(d) show the normalized magnetization AIP Advances 11, 025211 (2021); doi: 10.1063/9.0000047 11, 025211-2 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 2. Results of HRXRD analysis for (a) Stack A with SMT and (b) Stack B without SMT. Variations of (c) integrated intensity and (d) lattice constant of Co/Pt (111) for Stacks A and B with bottom Pt thickness. FIG. 3. Bottom Pt layer thickness dependence of TMR ratio for (a) Stack C with SMT and (b) Stack D without SMT annealed at various temperatures. m-Hloops for (c) Stack C with SMT and (d) Stack D without SMT annealed at 400○C. Inset arrows indicate the magnetization direction of each part of the magnetic layers. mversus the out-of-plane magnetic field Hloops for Stack C with SMT and Stack D without SMT, respectively; tranged from 0.5 nm to 2.5 nm at different annealing temperatures of 300, 350, and 400○C. For Stack C with SMT and tvalues ranging from 1.0 nm to 2.5 nm, a distinct magnetization plateau is observed. However, att=0.5 nm, the plateau region was not maintained, resulting in the reference layer’s instability against a perpendicular magnetic field. These conditions may have resulted in a degraded TMR ratiobecause a perfect antiparallel-magnetic configuration could not be realized between the free and reference layers. In the case of Stack D, the plateau region was not observed at any tafter 400○C annealing. This could be the reason for the degraded TMR ratio associated with the 400○C annealing treatment. To elucidate these behaviors, we performed cross-sectional TEM and energy dispersive X-ray (EDX) line analyses. Figures 4 (a) and 4(b) show TEM images of the stacks without and with SMT, AIP Advances 11, 025211 (2021); doi: 10.1063/9.0000047 11, 025211-3 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 4. (a) and (b) Cross-sectional TEM photographs; (c) and (d) cross-sectional depth profiles for Stack D without SMT and Stack C with SMT, respectively, after 400○C annealing. respectively. In Stack C with SMT, the Co/Pt multilayer had nearly perfect out-of-plane crystalline orientation, whereas in Stack D with- out SMT, the Co/Pt multilayer shows poor crystallinity. This results in a larger PMA of the Co/Pt multilayer in Stack C with SMT. Figures 4 (c) and 4(d) show depth profiles of Pt, Co, and Ru for the stacks. In the Stack D without SMT, a larger amount of Pt is observed in the CoFeB based reference layer, indicating Pt diffu- sion from the Co/Pt multilayer toward the MgO barrier. This could degrade perpendicular anisotropy of the reference layer. In contrast, a small amount of Pt in the CoFeB based reference layer is observed in Stack C with SMT, indicating a suppressed Pt diffusion, which leads to a higher PMA and TMR ratio after annealing at 400○C. The difference in the degree of Pt diffusion could be related to the crys- tallographic change in the Co/Pt multilayer owing to SMT on the bottom Pt layer. IV. CONCLUSIONS We investigated the effects of SMT on the thermal tolerance of top-pinned MTJs with a perpendicular easy axis. Applying SMT on the bottom Pt layer increased the PMA of the overlying Co/Pt layer. X-ray diffraction analysis revealed that the SMT increased the crystallinity of the Co/Pt layer when the MTJ had a relatively thin bottom Pt layer. The Pt (111) peak intensity for the MTJ without SMT increased with increasing Pt layer thickness, whereas that with SMT remained nearly constant, indicating SMT-induced homogeneity of the Co/Pt layer. These could result in higher PMA in the MTJ with SMT. The TMR ratio for the MTJ without SMT increased with increas- ing annealing temperature, whereas that for the MTJ without SMT degraded after 400○C annealing. SMT also improved the PMA of the reference layer. EDX line analysis revealed that SMT sup- pressed Pt diffusion to the MgO barrier, resulting in a higher thermal tolerance. In conclusion, SMT is an effective way to increase the PMA and thermal tolerance of a Co/Pt-based reference layer for top-pinned MTJs.ACKNOWLEDGMENTS This study was supported by a Center for Innovative Integrated Electronic Systems (CIES) Industrial Affiliation grant for STT- MRAM research, as funded by the CIES Consortium, JST-OPERA Program grant numbers JPMJOP1611 and CAO-SIP. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1H. Ohno, T. Endoh, T. Hanyu, N. Kasai, and S. Ikeda, in Proceedings of the 2010 IEDM, 2010, p. 9.4.1. 2T. Hanyu, T. Endoh, D. Suzuki, H. Koike, Y. Ma, N. Onizawa, M. Natsui, S. Ikeda, and H. Ohno, Proc. IEEE 104, 1844 (2016). 3T. Endoh, H. Koike, S. Ikeda, T. Hanyu, and H. Ohno, IEEE Trans. Emerg. Sel. Topics Circuits Syst. 6, 109 (2016). 4T. Endoh, H. Honjo, K. Nishioka, and S. Ikeda, in VLSI Technology Digest of Technical Papers, TMSF, 2019, p. 1. 5T. 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5.0031336.pdf

Appl. Phys. Lett. 118, 021901 (2021); https://doi.org/10.1063/5.0031336 118, 021901 © 2021 Author(s).Sublattice mixing in Cs2AgInCl6 for enhanced optical properties from first-principles Cite as: Appl. Phys. Lett. 118, 021901 (2021); https://doi.org/10.1063/5.0031336 Submitted: 29 September 2020 . Accepted: 29 December 2020 . Published Online: 15 January 2021 Manish Kumar , Manjari Jain , Arunima Singh , and Saswata Bhattacharya ARTICLES YOU MAY BE INTERESTED IN Low-cost, universal light-harvesting coating layer for thin film solar cells by employing micro- prism films Applied Physics Letters 118, 023301 (2021); https://doi.org/10.1063/5.0036223 Evidence of phonon pumping by magnonic spin currents Applied Physics Letters 118, 022409 (2021); https://doi.org/10.1063/5.0035690 Vertical p-type GaN Schottky barrier diodes with nearly ideal thermionic emission characteristics Applied Physics Letters 118, 022102 (2021); https://doi.org/10.1063/5.0036093Sublattice mixing in Cs 2AgInCl 6for enhanced optical properties from first-principles Cite as: Appl. Phys. Lett. 118, 021901 (2021); doi: 10.1063/5.0031336 Submitted: 29 September 2020 .Accepted: 29 December 2020 . Published Online: 15 January 2021 .Publisher error corrected: 19 January 2021 Manish Kumar,a) Manjari Jain, Arunima Singh, and Saswata Bhattacharyab) AFFILIATIONS Department of Physics, Indian Institute of Technology Delhi, New Delhi 110016, India a)Electronic mail: manish.kumar@physics.iitd.ac.in b)Author to whom correspondence should be addressed: saswata@physics.iitd.ac.in ABSTRACT Lead-free double perovskite materials (viz., Cs 2AgInCl 6) are being explored as stable and nontoxic alternatives of lead halide perovskites. In order to expand the optical response of Cs 2AgInCl 6in the visible region, we report here on the stability, electronic structure, and optical properties of Cs 2AgInCl 6by sublattice mixing of various elements. We have employed a hierarchical first-principles-based approach starting from density functional theory (DFT) with appropriate exchange-correlation functionals to beyond DFT methods under the framework ofmany body perturbation theory (viz., G 0W0@HSE06). We have started with 32 primary set of combinations of metals M(I), M(II), M(III), and halogen X at Ag/In and Cl sites, respectively, where the concentration of each set is varied to build a database of nearly 140 combina-tions. The most suitable mixed sublattices are identified to engineer the bandgap of Cs 2AgInCl 6to have its application in optoelectronic devi- ces under visible light. Published under license by AIP Publishing. https://doi.org/10.1063/5.0031336 Lead halide perovskites APbX 3(A¼CH 3NH 3þ;HCðNH 2Þ2þ; Csþ, and X ¼Cl/C0;Br/C0;I/C0) have created a huge sensation in the field of optoelectronics, particularly in photovoltaics owing to their suitable optical bandgap, long carrier diffusion length, high carrier mobility,and low manufacturing cost. 1–6Moreover, the bandgap is tunable with high defect tolerance.7,8These materials find applications in various optoelectronic devices, namely, solar cells,2,9,10light emitting diodes,11,12lasers,13,14and photodetectors.15,16In spite of their great potential in vast number of applications, there are two major chal- lenges: (i) instability against exposure to humidity, heat or light and(ii) toxicity of Pb. To tackle these issues, many works have been endeavored to find the alternative stable and environmentally sustain- able metal halide perovskites with fascinating optoelectronic properties akin to lead halide perovskites. 17–21 One of the approaches for removing Pb-toxicity is to replace Pb2þwith some other divalent metal. However, this replacement results in either indirect or large bandgap materials with degraded optoelectronic properties.22–24Substitutions of group 14 divalent cati- ons, viz., Sn2þand Ge2þ, have also been synthesized by researchers, but these are not stable at ambient conditions due to the easy oxidation to tetravalent Sn4þand Ge4þ,r e s p e c t i v e l y .25,26Another promising approach is to substitute a monovalent M(I) and a trivalent M(III) metal alternatively in place of two divalent Pb, which forms the doubleperovskite A 2M(I)M(III)X 6. Many high-throughput calculations have been performed on double perovskites for a variety of potential applications.27–29Recently, lead-free metal halide double perovskites have been synthesized, which are stable and environmentally benign,viz., Cs 2AgBiX 6(X¼Cl/C0;Br/C0;I/C0)a n dC s 2AgInCl 6.30–36Cs2AgBiX 6 perovskites possess indirect bandgap, which results in weaker absorp- tion and high non-radiative recombination loss.30–32In contrast, Cs2AgInCl 6has direct bandgap and long carrier lifetimes. However, its wide bandgap (3.3 eV) does not show an optical response in the visible region.34,35Alloying with suitable elements could be the best solution to reduce its bandgap and expand the spectral response in the visible light region. In recent studies, Cs 2AgInCl 6has been doped to tune its optical properties.36–40 In this letter, we have done the sublattice mixing by partial substi- tution of several metals M(I), M(II), M(III), and halogen X at Ag/In and Cl sites, respectively, to reduce the bandgap of Cs 2AgInCl 6, thereby, enhancing its optical properties. The charge neutrality condi- tion has been maintained by forming substitutional defects. We haveperformed hierarchical calculations using first-principles-based approaches, viz., density functional theory (DFT) with semi-local exchange-correlation (xc) functional (PBE 41), hybrid DFT with HSE0642,43and single shot GW44,45(G0W0) under the many body perturbation theory (MBPT). First, the structural stability analysis has Appl. Phys. Lett. 118, 021901 (2021); doi: 10.1063/5.0031336 118, 021901-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplbeen done by examining the Goldschmidt tolerance factor and octahe- dral factor. Since structural stability is not the sufficient condition to confirm the formation of perovskites, the decomposition energy46has been calculated, which reflects the thermodynamic stability of the materials. We have taken the difference between total energy of theconfigurations and their components (binary/ternary, in which they can decompose), which is opposite to what has been considered in Ref.46. Therefore, the configurations that have negative decomposi- tion energy are stable. Furthermore, to get better insights, we have investigated the reduction in bandgap via atom projected partial den-sity of states. Finally, by calculating the frequency dependent complex dielectric function, we have determined the optical properties of the materials that can be applied in the field of optoelectronics. The DFT calculations have been performed using the Vienna ab initio simulation package (VASP). 47The ion-electron interactions in all the elemental constituents are described using projector- augmented wave (PAW) potentials.48All the structures are optimized using generalized gradient approximation (PBE xc functional) until the forces are smaller than 0.001 eV/A ˚. Here, the PBE xc functional is used because the HSE06 xc functional is extremely slow for relaxing the structure. In the case of double perovskite Cs 2AgInCl 6,t h el a t t i c e constant is overestimated by 1.56% using PBE xc functional (and by1.22% using HSE06 xc functional) in comparison to experimental value obtained by Volonakis et al. 34Whereas, the PBEsol xc functional underestimates the lattice constant by 1%. The electronic self- consistency loop convergence is set to 0.01 meV, and the kinetic energy cutoff used is 500 eV for plane wave basis set convergence. A k-mesh of 4 /C24/C24 is used for Brillouin zone integration, which is generated using Monkhorst–Pack49scheme. Advanced hybrid xc func- tional HSE06 is used for the better estimation of bandgap as well as thermodynamic stability. Furthermore, we have checked the role of van der Waals (vdW) forces and configurational entropy, while ana- lyzing the stability of compounds. The latter has lesser effect on the stability. The consideration of vibrational energy contributes to seconddecimal place of the decomposition energy. It may change the number by very small amount, but neither changes the stability nor the hierar- chy of stability of the compounds. On the other hand, the van der Waals forces (two-body Tkatchenko–Scheffler 50) contribute to first decimal place of the decomposition energy. Most of the compounds’ stability has not been affected. However, in very few cases, it has minutely changed the stability of the compounds, which have decom-position energy value close to zero. In the case of determination of the optical properties, single shot GW (G 0W0) calculations have been per- formed on top of the orbitals obtained from HSE06 xc functional [G0W0@HSE06]. The polarizability calculations are performed on a grid of 50 frequency points. The number of bands is set to four times the number of occupied bands [for bandgap convergence, see Table S1 in the supplementary material ]. Moreover, the negligible effect of spin–orbit coupling (SOC) has been discussed. The double perovskite Cs 2AgInCl 6has a cubic structure with space group Fm/C223m. The corresponding sublattice is composed of alternate octahedra of InCl 6and AgCl 6,a ss h o w ni n Fig. 1(a) .O n partial substitution of different elements as shown in Fig. 1(b) (metals and/or halogens), the distortion is negligible (see Sec. II in the supplementary material ). Here, we have started with 32 primary set of combinations of metals M(I), M(II), M(III), and halogen X at Ag/In and Cl sites, respectively, where concentration of each set is varied tobuild a database of nearly 140 combinations. However, note that here, we have presented the results of 25% substitution for metals and 4% substitution for halogen atoms. This is because we have seen and thor- oughly checked that, with the increase in concentration of the externalelement, if the bandgap is increased (or decreased), the same trend isfollowed with further increase in concentration. Two such test casesare shown in Fig. S1. We have also reported this, to be the case in ourprevious experimental finding. 51Moreover, some combinations beyond 25% substitution are not considered in the following cases: (i)toxic elements [viz., Tl(III), Cd(II), and Pb(II)], (ii) elements that lead to instability on 25% substitution [viz., Co(II), Cu(II), Ni(II)], and (iii) elements that result in larger indirect bandgap (with respect to pristineCs 2AgInCl 6) on 25% substitution [viz., Ac(III), Ba(II), Ge(II), and Sn(II)]. The structural stability of all the configurations has been deter-mined by calculating two geometrical parameters, viz., theGoldschmidt tolerance factor ( t) and the octahedral factor ( l). For sin- gle perovskite ABX 3,t¼ðrAþrXÞ=ffiffiffi 2p ðrBþrXÞand l¼rB=rX, where rA;rB,a n d rXare the ionic radii of cation A, B, and anion X, respectively. In the case of double perovskites, rBis the average of the ionic radii at B sites. For stable perovskites, the ranges of tandlare 0:8/C20t/C201:0a n d l>0:41.52The Shannon ionic radii53have been considered to evaluate tandl. For the configurations we have consid- ered, tlies between 0.85 and 0.91, and lhas the value between 0.50 a n d0 . 5 9( s e eT a b l eS I Ii nt h e supplementary material ). Therefore, these probable structures are stable. In order to determine the thermodynamic stability, we have com- puted the decomposition energy ( DHD) using PBE and HSE06 xc func- tionals. We have substituted the external elements in Cs 8Ag4In4Cl24 supercell framework to model a solid solution.54In order to model the defected system, we have used an iterative procedure as shown inRefs. 5and55.T h eDH Dfor the decomposition of Cs 8Ag4In4Cl24into binary compounds is calculated as follows: DHDðCs8Ag4In4Cl24Þ¼EðCs8Ag4In4Cl24Þ/C08EðCsClÞ /C04EðAgClÞ/C04EðInCl 3Þ; (1) where E ðCs8Ag4In4Cl24Þ;EðCsClÞ;EðAgClÞ, and E ðInCl 3Þare the DFT energies of the respective compounds. The configurations havingnegative value of the DH Dare stable. The entropy of mixing is not FIG. 1. (a) Structure of Cs 2AgInCl 6, and (b) partial substitution with metals M(I), M(II), M(III), and with halogen X at Ag/In and Cl sites, respectively.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 021901 (2021); doi: 10.1063/5.0031336 118, 021901-2 Published under license by AIP Publishingconsidered here as it will not change the overall trend, i.e., the relative stability will remain same.56–58Figures 2(a) and 2(b) show the decomposition energy for the decomposition of Cs 2AgInCl 6and other mixed sublattices into binary compounds using PBE and HSE06 xcfunctionals, respectively (decomposition pathways are shown in thesupplementary material ). Only those elements, which lead to decre- ment in bandgap using PBE xc functional, are further considered withH S E 0 6x cf u n c t i o n a l . The quaternary compounds can be decomposed into ternary compounds. Therefore, we have also considered those pathways forthe materials [see decomposition pathways that are more probable (asper the smaller value of decomposition energy) in the supplementary material ], which have the favorable bandgap. The decomposition energy for the decomposition of Cs 2AgInCl 6a n do t h e rm i x e ds u b l a t t i - ces into ternary compounds is shown in Fig. 2(c) . For the decomposi- tion of Cs 8Ag4In4Cl24into ternary compounds, the DHDis determined as follows: DHDðCs8Ag4In4Cl24Þ¼EðCs8Ag4In4Cl24Þ/C02EðCsAgCl2Þ /C02EðCs3In2Cl9Þ/C02EðAgClÞ: (2) TheDHDhas the value of –2.48 eV/f.u. and –0.10 eV/f.u. for the decomposition of Cs 8Ag4In4Cl24into binary and ternary compounds, respectively. These negative values confirm that the perovskiteCs 8Ag4In4Cl24is stable. We have found that all the selected elements for sublattice mixing are stable with respect to the decomposition intobinary compounds (see Table S2 in the supplementary material ).However, for ternary decomposition pathway, Co(II), Ni(II), and Cu(II) are not stable [see Fig. 2(c) , where shaded region indicates the stable compounds, i.e., DH D<0]. This may be attributed to the smaller size of these cations that are unable to accommodate two octa- hedra with Cl 6, and the lowest octahedral factor of Ni(II) and Cu(II) ( s e eT a b l eS 2i nt h e supplementary material ). Also, Cu(I) and Ga(III) are less stable than pristine [see Fig. 2(c) ]. Moreover, we have noticed that Cu(I) is not stable at all (as positive value of DHD¼0.32 eV/f.u. (seeTable I ) when it has fully replaced the Ag, i.e., for 100% substitu- tion, which is in agreement with previous studies.46It is only stable for 25% substitution. Therefore, it is concluded that if the difference between sizes of the substitutional cations/anion and pristine’s cations/ anion is large, then that configuration would become unstable onincrement in concentration. A screening of various atoms for sublattice mixing has been done by calculating the bandgap first using generalized gradient approxima-tion (PBE) and, subsequently, with inclusion of SOC. The respective band gaps as obtained for pristine Cs 2AgInCl 6are 0.95 eV and 0.93 eV, implying insignificant SOC effect on its electronic properties.Also, as per existing literature, SOC has negligible effect in Ag/In- 28,59 and Au-60,61based double perovskite (see Fig. S3). Therefore, we have ignored the effect of SOC in our further calculations. However, the bandgap is highly underestimated by PBE xc functional due to the well-known self-interaction error. Therefore, we have further per-formed the calculations using hybrid xc functional HSE06 for those mixed sublattices, where, in comparison to pristine, the bandgap was reduced [see Figs. 2(a) and2(b), and Table S2 in the supplementary material ]. The calculated value of bandgap for Cs 2AgInCl 6is 2.31 eV using default exact Fock exchange of 25%, which is in good agreementwith previously reported theoretical value, but still underestimated in comparison to the experimental value (3.3 eV). 34We have also vali- dated that the bandgap becomes 3.19 eV on increasing the exact Fock exchange parameter to 40%. Despite the proximity of this value to that of experiments, it can be drastically changed for the systems havingdefects (substitution of different elements), and determining it accu- rately is not possible without the experimental inputs. In view of this, we have used the default 25% exact Fock exchange parameter for ourstudy, assuming this will give at least the correct trends. In the case of Cu(I) and Au(I) substitutional alloying at Ag site, the bandgap is reduced by /C240.8 eV, having a value of 1.51 eV and 1.54 eV, respec- tively. On the other hand, in the case of substitution of M(III) at In site, it does not have much effect on reduction in bandgap. OnlyCo(III) and Ir(III) substitutional alloying are able to reduce the FIG. 2. Decomposition energy ( DHD) for the decomposition of pristine and other configurations into binary compounds, and bandgap using the xc functionals (a)PBE and (b) HSE06. (c) Decomposition energy ( DH D) for decomposition into ter- nary compounds using HSE06 xc functional.TABLE I. Decomposition energy (for the decomposition into ternary compounds) of Cs2CuxAg1/C0xInCl 6. DHD(eV/f.u.) Compounds PBE HSE06 Cs2AgInCl 6 /C00.098 /C00.103 Cs2Cu0.25Ag0.75InCl 6 /C00.070 /C00.039 Cs2Cu0.50Ag0.50InCl 6 0.058 0.122 Cs2Cu0.75Ag0.25InCl 6 0.190 0.354 Cs2CuInCl 6 0.322 0.472Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 021901 (2021); doi: 10.1063/5.0031336 118, 021901-3 Published under license by AIP Publishingbandgap from 2.31 eV to 2.08 eV, whereas the rest are either increasing it or have no effect on the bandgap. In the case of M(II) substitutional alloying, one at Ag and other one at In site, only Zn(II) and Mn(II) are able to reduce the bandgap effectively, having a bandgap value of 1.87 eV and 1.77 eV, respectively. In the case of halogen substitution,viz., Br and I, it helps to reduce the bandgap to 2.10 eV and 1.85 eV, respectively (see Table S2 in the supplementary material ). In the afore- mentioned cases, while reducing the bandgap, the direct gap nature remains intact, except for Co(III) substitution. We have also calculated the bandgap using G 0W0@HSE06, which is overestimated and nearer to the experimental value. In the case of pristine Cs 2AgInCl 6,t h e bandgap is 3.77 eV. The comparison of bandgap computed using PBE, HSE06, and G 0W0@ H S E 0 6c a nb es e e nf r o mT a b l eS 3i nt h e supplementary material . The reduction in bandgap can be explained by observing the atom projected partial density of states (pDOS) (see Fig. 3 ). In the pris- tine Cs 2AgInCl 6, the Cl p-orbitals and Ag d-orbitals contribute to the valence band maximum (VBM) whereas the In and Ag s-orbitals con- tribute to the conduction band minimum (CBm) [see Fig. 3(a) ]. In the case of substitutional alloying of Cu(I) and Au(I), their d-orbitals are at higher energy level than the d-orbitals of Ag, thereby reducing the bandgap by elevating the VBM [see Figs. 3(c) and3(d)]. However, in the case of the M(III) substitution at In site, generally, the states lie inside the valence band (VB) or the conduction band (CB), and thus do not reduce the bandgap effectively. From Fig. 3(b) ,w ec a ns e et h a t the Ga(III) is reducing the bandgap by a negligible amount (as the states contributed by the Ga are lying inside VB and CB). Whereas, the Co(III) and Ir(III) substitutions at In site show a finite decrease in the bandgap. This is due to the Co d-orbitals and Ir d-orbitals contri-bution at CBm and VBM, respectively (see Fig. S4). In the case of M(II), there is a little contribution from d- and s-orbitals of M(II) at VBM and CBm, respectively, and therefore, reducing the bandgap by introducing the shallow states [see Figs. 3(e) and3(f)]. Moreover, the Mn states are asymmetric (with respect to spin states), which indicatesthat Zn(II) will be more stable than Mn(II). This can also be seen from the more negative value of DH Dfor Zn(II) in comparison to Mn(II). The bandgap reduction on substituting Br/I at Cl site is occurred by elevating the VBM, which is due to Br/I p-orbitals contribution at VBM [see Figs. 3(g) and3(h)]. The reduction in bandgap on mixing the halides is in line with the previous studies.31,34In all these cases, the defect levels are shallow, which is a desirable property for optoelec-tronic devices. Shallow defect states ensure that the recombination of photogenerated charge carriers is not prominent, and thus, the decre- ment in charge carrier mobility and diffusion will be insignificant. To obtain the optical properties, which are crucial for the perov- s k i t et ob eu s e di no p t o e l e c t r o n i cd e v i c e s ,w eh a v ec a l c u l a t e dt h ef r e -quency dependent complex dielectric function eðxÞ¼ReðeðxÞÞ þImðeðxÞÞusing G 0W0@HSE06 (the results obtained by HSE06 xc functional are shown in Fig. S5). Figures 4(a) and4(b)show the imagi- nary [Im( e)] and real [Re( e)] part of the dielectric function, respec- tively. The real static part (at x¼0) of the dielectric function is a direct measure of refractive index. Higher the refractive index, better will be the probability to absorb light. On alloying, the refractive index is increased (range: 1 :98/C02:15), and thus, the optical properties are enhanced. The static Re( e) is 2.05 for pristine Cs 2AgInCl 6, and the value has increased on alloying [see Fig. 4(b) and Fig. S6 and Table S4 in the supplementary material ]. The imaginary part reflects thetransitions from occupied to unoccupied bands. The absorption edge is red shifted, and hence, the visible light response has been achievedupon alloying [see Fig. 4(a) ]. The absorption spectra have also been obtained that corroborates with the red shift observed [see Fig. 4(c) ]. The absorption coefficient aðxÞis related to the dielectric function as follows: FIG. 3. Atom projected partial density of states (pDOS) using HSE06 xc functional of (a) pristine Cs 2AgInCl 6, (b) Cs 2AgGa 0.25In0.75Cl6, (c) Cs 2Cu0.25Ag0.75InCl 6, (d) Cs2Au0.25Ag0.75InCl 6, (e) Cs 2Zn0.50Ag0.75In0.75Cl6, (f) Cs 2Mn0.50Ag0.75In0.75Cl6, (g) Cs2AgInBr 0.04Cl5.96, and (h) Cs 2AgInI 0.04Cl5.96.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 021901 (2021); doi: 10.1063/5.0031336 118, 021901-4 Published under license by AIP PublishingaðxÞ¼ffiffiffi 2px cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ReðeðxÞÞ2þImðeðxÞÞ2q /C0ReðeðxÞÞ/C18/C19 1 2 :(3) This visible response is attributed to the reduction in bandgap, as shown in Fig. 4(d) . The other optical parameters such as refractive index, extinction coefficient, reflectivity, optical conductivity, and energy loss spectrum have also been obtained from the dielectric ten-sor (see Sec. XII in the supplementary material ). In conclusion, we have investigated the role of metals M(I), M(II), M(III), and halogen X in Cs 2AgInCl 6with mixed sublattices for inducing the visible light response by tuning its electronic properties, using state of the art DFT and beyond DFT methods. Many partiallysubstituted configurations help to tune the bandgap, thereby increasing the absorption. We have inferred that the sublattices with Cu(I) and Au(I) at the Ag site, Ir(III) at the In site, Zn(II) at the Ag and In sites simultaneously, Mn(II) at the Ag and In sites simultaneously, and Brand I substitutions at the Cl site have tuned the bandgap in the visible region. Hence, these can be considered as the most promising candi-dates for various optoelectronic devices, viz., tandem solar cells, LEDs, photodetectors, and photocatalysts. See the supplementary material for the details of decomposition pathways, bandgap, and various optical properties of pristineCs 2AgInCl 6a n do t h e rm i x e ds u b l a t t i c e s . M.K. acknowledges CSIR, India, for the senior research fellowship [Grant No. 09/086(1292)/2017-EMR-I]. M.J. acknowledges CSIR, India, for the senior research fellowship [Grant No. 09/086(1344)/2018-EMR-I]. A.S. acknowledges IIT Delhi for the financial support. S.B. acknowledges the financial support from SERB under core Research Grant (Grant No. CRG/2019/000647). Weacknowledge the High Performance Computing (HPC) facility atIIT Delhi for computational resources. FIG. 4. Spatially average (a) imaginary [Im ðeÞ] and (b) real [Re ðeÞ] part of the dielectric function, (c) absorption coefficient, and (d) bandgap obtained by G 0W0@HSE06 for the pristine Cs 2AgInCl 6and other mixed sublattices.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. 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5.0040497.pdf

Appl. Phys. Lett. 118, 073902 (2021); https://doi.org/10.1063/5.0040497 118, 073902 © 2021 Author(s).Smart power system of biocompatible and flexible micro-supercapacitor Cite as: Appl. Phys. Lett. 118, 073902 (2021); https://doi.org/10.1063/5.0040497 Submitted: 15 December 2020 . Accepted: 08 February 2021 . Published Online: 18 February 2021 Caifeng Chen , Hao Wen , Zhenkui Qu , Hao Wang , and Xiangyang Liu ARTICLES YOU MAY BE INTERESTED IN Free-standing and flexible graphene supercapacitors of high areal capacitance fabricated by laser holography reduction of graphene oxide Applied Physics Letters 118, 071601 (2021); https://doi.org/10.1063/5.0038508 Study of loss mechanisms on Sb 2(S1−xSex)3 solar cell with n–i–p structure: Toward an efficiency promotion Applied Physics Letters 118, 073903 (2021); https://doi.org/10.1063/5.0032867 Liquid–liquid phase transition in molten cerium during shock release Applied Physics Letters 118, 074102 (2021); https://doi.org/10.1063/5.0040437Smart power system of biocompatible and flexible micro-supercapacitor Cite as: Appl. Phys. Lett. 118, 073902 (2021); doi: 10.1063/5.0040497 Submitted: 15 December 2020 .Accepted: 8 February 2021 . Published Online: 18 February 2021 Caifeng Chen,1Hao Wen,1Zhenkui Qu,1Hao Wang,1,a) and Xiangyang Liu2,a) AFFILIATIONS 1Department of Physics, College of Physical Science and Technology, Research Institution for Biomimetics and Soft Matter, Fujian Provincial Key Laboratory for Soft Functional Materials Research, Xiamen University, Xiamen 361005, China 2Department of Physics, National University of Singapore, Singapore 117542, Singapore a)Authors to whom correspondence should be addressed: h_wang@xmu.edu.cn andphyliuxy@nus.edu.sg ABSTRACT Flexible micro-supercapacitor (MSC) is an ideal energy storage device for flexible and small-scale electronics, specifically some human health sensors, because of its flexibility, long working life, high power density, and high charge and discharge rate. In this work, a smart power sys-tem of MSC is developed. First, utilizing ink-jet printing and electrochemical deposition, flexible MSC is fabricated on the biocompatible sub-strate of a modified silk protein film, making the power system suitable for implantable devices. Second, aiming at the common drawbacks of small energy density and large voltage variation of MSC, a wireless charging component and a wireless inductor–capacitor (LC) voltage sen- sor are integrated with the MSC unit. Using pulse charging mode, charging and voltage detection can be performed at the same time. The LCvoltage sensor, using varactor diodes to realize voltage capacitance mapping, does not need extra ICs or consume any energy. Such a systemhas great application potential as the energy supply part of small devices implanted in the human body. Published under license by AIP Publishing. https://doi.org/10.1063/5.0040497 Nowadays wearable and flexible electronics, like skin electronics, 1 soft actuators,2health monitors,3and sensors,4are rapidly developed, causing a great demand for flexible energy storage devices. Flexiblemicro-supercapacitors (MSCs) can be integrated with micro-electronic devices to work as stand-alone power sources or as efficientenergy storage units complementing batteries and energy harvesters,leading to wider use of these devices in many fields. 5The energy den- sity of supercapacitors is lower than that of batteries, but they have ahigher power density, high charge and discharge rate, and long servicelife. Due to its miniaturization and integrability, MSC has great poten- tial in implantable devices. In order to overcome the low energy den- sity, some energy harvest parts are integrated to increase the workingperiod. 6However, the detection of voltage fluctuation of the superca- pacitor is usually out of consideration. To monitor the voltage of MSCcontinuously, conventionally it requires specified circuits and inte-grated chips to detect the voltage and needs wired connections or wi-fichips for data exchange. Such circuits have undoubtedly increased sys-tem complexity and power consumption, which is not suitable forapplication of small-scale implantable electronic devices. Therefore,designing convenient, battery-free voltage sensing components plays an important role in the power supply system of supercapacitor, which has a large fluctuation of voltage.In this regard, an inductor–capacitor (LC) wireless passive voltage sensor is developed to wirelessly monitor the voltage change of MSC. LCsensors are widely used to wirelessly transmit signals of pressure, 7tem- perature,8humidity,9biochemical,10and so on. Changes of capacitance, resistance, inductance, or coupling distance in the sensor are wirelessly and remotely detected by the readout coil, which makes them highly useful in sealed environments,11where physical access to the sensor is difficult, like medical sensing of intracranial pressure,12intraocular pres- sure,13–15intravascular blood pressure,16blood flow,17and so on. The other advantage of the LC sensors is that they do not require a power source for their operation, which alleviates the capacity shortage of MSC. And when the power is exhausted, one can still read the signal. For implantable devices, good flexibility and good biocompatibil- ity are two important factors for choosing the substrate materials ofsupercapacitors. Lots of soft materials are considered, such as paper, 18 textile,19polyethylene terephthalate,20polyimide,21and elastic materi- als.22Recently, silk films, as a natural structural polymer, have been successfully exploited in transistors, wireless sensor, and electronicskin. 23–26Silk fibroin (SF) has good biocompatibility, biodegradability, and unique mechanical properties.27Thus, it is a good attempt to fab- ricate silk-based micro-supercapacitor for flexible energy storage devi- ces, aiming for implantable electronic devices. Appl. Phys. Lett. 118, 073902 (2021); doi: 10.1063/5.0040497 118, 073902-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplHerein, an all-solid-state micro-supercapacitor is fabricated on the silk substrate. Inkjet-printing is adopted to pattern the electrode, while electrochemical deposition is used to functionalize the electrode. Then a smart power system is developed by integrating a LC voltage sensor anda charging coil with the fabricated MSC, in which we can detect the volt-age and charge it wirelessly, continuously and simultaneously. Pure silk films are naturally stiff and brittle in a dry state, easily resolve in solution, and cannot sustain relatively high temperature.And followed electrochemical process requires silk films to be toughwith better heat and chemical resistance. To enhance the toughness ofthe silk substrate, proper additive agent and treatment are adopted to make good chemical and heat resistances and better adhesion between conductive ink and the substrate. Polyurethane (PU), a biodegradablepolymer, which is widely used as a polymer scaffold in biological tissueengineering, shows low cytotoxicity both in vitro and in vivo . 28,29By combining polyurethane with silk fibroin, a flexible composite protein film (FCPF) is obtained. The preparation process of FCPF film is sche- matically illustrated in Fig. 1(a) .F i r s t , B. mori silkworm silk cocoons are cut into pieces and then dissolved in lithium bromide solution.Second, the dissolved solution is dialyzed to remove lithium bromideions to obtain pure regenerated silk aqueous solution. Third, PU solu- tion is mixed into SF solution. Finally, the mixture is cast on a flat frosted glass to fabricate micro-structured FCPF films. The digital photo of the FCPF film is shown in Fig. 1(b) .A n dt h e 3D confocal image of surface is shown in Fig. 1(c) . The surfaceroughness of the film is approximately 2.012 lm. The rough surface con- tributes to the enhanced adhesion between ink and the substrate. From Fig. 1(d) , the contact angle of Ag ink is 7.9 /C14. Small contact angle restrains coffee-ring phenomenon.30T h eh e a tr e s i s t a n c eo ft h eF C P Fi se v a l u a t e d by thermogravimetric analysis (TGA). As shown in Fig. 1(e) ,t h em a s s losses of the pure SF film and the FCPF film took place at 257.7/C14Ca n d 299.8/C14C, respectively, indicating that the thermal decomposition rate of the FCPF film is significantly lower than that of the pure SF film. Theproper reason is that silk fibroin molecules are uniformly dispersedbetween polyurethanes and interact with each other by hydrogen bond-ing to form a new cross-linked network structure, enhancing the thermal stability of the composite film. To prove that, Fourier transform infrared (FTIR) spectra are performed on pure SF films and FCPF films[Fig. 1(f) ]. For pure SF films, it shows characteristic peaks at 1648 cm /C01, indicating that the secondary structure is a-helices, while for the FCPF film it shows characteristic peaks at 1617 cm/C01, corresponding to the b- sheets structure. The secondary structure of silk fibroin consists of the major conformations including a-helix, b-sheet, and random coil, and b-sheet benefits silk’s water resistance and mechanical toughness. The FTIR spectra indicate that the incorporation of polyurethane is beneficialfor the conversion of silk fibroin from a-helices to b-sheets. Finally, the breaking stress and toughness of the two films are characterized by ten- sile test. As shown in Fig. 1(g) , the FCPF film shows better extensibility than the pure SF film, and such film is a good choice as a substrate forflexible electronics. FIG. 1. (a) Schematic illustration of the FCPF film preparation. (b)–(d) Photograph, 3D laser scanning microscope image and ink contact angle of the FCPF film. (e)–(g) Thermogravimetric Analysis (TGA), Fourier Transform Infrared (FTIR) spectra, and stress /C0strain plots of the FCPF film and the pure SF film.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 073902 (2021); doi: 10.1063/5.0040497 118, 073902-2 Published under license by AIP PublishingTo treat silk film as the substrate of micro-supercapacitors, proper fabrication method of electrode must be adopted. Diverse fabrication technologies, e.g., photolithography,31laser-scribing,21 mask-assisted patterning,32and inkjet-printing have been employed to fabricate planar MSCs. In this work, inkjet-printing technology is adopted because it allows for the precise control of the pattern geome- try with the merits of maskless, clean-room-free, low-cost, large-scale, feasible, and so on. The schematic illustration of the fabrication process is shown in Fig. 2(a) , and the photograph of the electrode is presented in the right. It contains eight pairs of interdigital “fingers,” and each owns a width of 350 lm, a length of about 8 mm, and a space (interval between two adjacent fingers) about 450 lm. The interdigitated finger electrode is patterned by inkjet-printing silver (Ag) nanoparticle ink onto the rough FCPF substrate. Then, Ni is electrochemically deposited on the Ag layer to enhance the conductivity. Next, a layer of MnO 2is electro- chemically deposited on the Ni/Ag layer to form a MnO 2@Ni/Ag hier- archical electrode. The morphologies of the three layers are presented inFigs. 2(b)–2(d) . The printed silver particles (around 20–50 nm) tightly stack together on the FCPF substrate. After the electrochemicaldeposition of Ni, it is shown that the Ni particles (around 60–120 nm) seamlessly cover the Ag conducting network. The conductivity of the electrode increases by 60 times after depositing Ni (Fig. S1). And resis- tance of Ni/Ag/FCPF only increases by less than 6% after being bent 1000 times, as shown in Fig. S2. After the electrochemical depositionof MnO 2, honeycomb-like nanostructure with a large specific surface area takes place. The crystal structure of the as-fabricated electrode is characterized by X-ray diffraction (XRD) [ Fig. 2(e) ]. The narrow peak at 2h¼20.8/C14corresponds to b-crystallites of silk.27The well- crystallized Ni is proved by the sharp peaks at diffraction angles of 44.507/C14,5 1 . 8 4 6/C14, and 76.37/C14, corresponding to (111), (200), and (220) planes of Ni (JCPDS No. 04–0850), respectively. Besides, five sharp peaks around 38.12/C14,4 4 . 2 8/C14,6 4 . 4 3/C14, 77.47/C14, and 81.536/C14correspond to (111), (200), (220), (311), and (222) planes of Ag (JCPDS No.04–0783), respectively. The x-ray photoelectron spectroscopy (XPS) analysis is conducted. As shown in Fig. 2(f) , two peaks located at 653.6 and 642.1 eV can be assigned to spin–orbit Mn 2 p 1/2and Mn 2 p3/2of Mn4þ, indicating the existence of MnO 2in the electrodes. To improve supercapacitive performance of the MnO 2@Ni/Ag electrode, 150 cyclic voltammetry (CV) scans are performed at a scan rate of 100 mV s/C01 between 0 and 1.0 V (vs Ag/AgCl) in the 1 M Na 2SO4aqueous electro- lyte, which is called Naþpreinsertion. In this process, MnO 2converts into Na xMnO 2, effectively improving supercapacitive perfor- mance.33,34As shown in Fig. 2(g) , the capacitance after the Naþprein- sertion (150th cycle) is significantly enlarged than that in the nonpreinserted state (first cycle). Subsequently, sodium sulfate/carboxymethyl cellulose (Na 2SO2/ CMC) gel is utilized as an electrolyte, drop-cast onto the interdigitated electrode, and encapsulated by medical tape to complete the planar MSC device. As shown in Fig. 3(a) , the MSC devices can operate at a FIG. 2. (a) Schematic illustration of the fabrication process of the MSC. Photograph of the electrode of MSC and its magnified optical image is shown in the righ t. (b)–(d) SEM images of Ag layer, Ni layer, and MnO 2layer. (e) X-ray diffraction (XRD) of the MnO 2@Ni/Ag/FCPF. (f) XPS fine spectrum of Mn 2 p. (g) CV curves before (first) and after (150th) Naþpreinsertion.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 073902 (2021); doi: 10.1063/5.0040497 118, 073902-3 Published under license by AIP Publishingwide potential window of 0–1 V with good capacitive behaviors and keep a similar shape. The areal capacitance calculated from galvano-static charging/discharging (GCD) curves at different current densities (0.1–1.2 mA cm /C02)[Fig. 3(b) ] is 11.64 mF cm/C02at 0.1 mA cm/C02and maintains 4.90 mF cm/C02when the current density is increased up to 1.2 mA cm/C02, which is four times higher than the symmetric MSCs on paper substrate reported previously.35And it exhibits good cycling sta- bility. As shown in Fig. 3(c) , 89.97% capacitance retains after 5000 cycles at a current density of 0.6 mA cm/C02, indicating the porous nanostructured MnO 2shows little expansion and shrinking on revers- ible charging/discharging reaction. The Nyquist chart in Fig. S5 con- sists of a typical semicircle in the high frequency region and a straight line at the low frequency. The intersection of the high-frequencyregion and the real axis represents the equivalent series resistance(ESR), which is 4.9 X. The large radius of the semicircle in the high- frequency region indicates a large charge transfer resistance, which may be caused by the long ion diffusion distance between theelectrodes. The mechanical test of the MSC is also performed. In Fig. 3(d) , the angle his defined as the bending angle. As shown in Fig. 3(e) ,t h e C Vc u r v e so ft h ed e v i c es h o wn e g l i g i b l ec h a n g ea tb e n d i n ga n g l e sfrom 0 /C14to 150/C14.M e a n w h i l e ,a ss e e ni n Fig. 3(f) , the device retains 92.8% capacitance after being bent 2000 times at 60/C14. One of the most appealing advantages of the inkjet-printer is the versatility of designing customized patterns; like that MSCs can be conveniently designed in series or parallel. As shown in Fig. 3(g) ,t h ec a p a c i t a n c eo ft w oM S C s devices in parallel is increased by twofold compared with a singleMSC. And the voltage window of two MSCs in series can extend to 2V [ Fig. 3(h) ]. Finally, the energy density and power density of the MSC are shown in the Ragone plot 32,35–44[Fig. 3(i) ] .T h ed e v i c ee x h i b - its maximum energy density of 1.62 lWh cm/C02at a power density of 0.05 mW cm/C02and maximum power density of 0.6 mW cm/C02at an energy density of 0.68 lWh cm/C02. Compared with other reports of planer interdigital micro-supercapacitors, this device shows compara-ble performance. Finally, a smart power system is developed by integrating a LC voltage sensor and a charging coil with the fabricated MSC, in whichwe can detect the voltage and charge it wirelessly, continuously, and simultaneously. The circuit diagram of the smart power system is shown in Fig. 4(a) , and the photograph is shown in Fig. 4(b) .U s i n g electromagnetic coupling, wireless energy transmission and wirelessvoltage sensing are realized. In the wireless charging part, a spiral FIG. 3. (a) CV curves of the device with different voltage windows at a scan rate of 50 mV s/C01. (b) GCD curves at different current densities (0.1–1.2 mA cm/C02). (c) Electrochemical stability at a current density of 0.6 mA cm/C02(the inset shows the GCD curve from the 3540th to the 3550th cycle). (d) Photograph of a device under bending. his the bending angle. (e) CV curves of MSC at different bending angles at a scan rate of 100 mV s/C01. (f) Mechanical stability at a bending angle of 60/C14at a scan rate of 100 mV s/C01. CV curves of two MSCs in parallel (g) and in series (h). (i) Ragone plot of the MnO 2MSC compared with other reports.32,35–44The area for areal capacitance cal- culation only includes the area of interdigital fingers.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 073902 (2021); doi: 10.1063/5.0040497 118, 073902-4 Published under license by AIP Publishinginductor is connected to a rectifier to convert alternating current (AC) to direct current (DC) and charge the MSC. In the LC voltage sensor part, a spiral inductor is connected to two voltage-dependent diodes(varactors), forming a resonant LC circuit. And the MSC is connectedaccording to the circuit diagram and provides a biasing voltage to thevaractors. The varactors are operated in reverse bias and isolatethe inductor coil to prevent shorting. Their capacitances are related tot h ea p p l i e dv o l t a g e( t h ev o l t a g eo fM S C )a n df u r t h e ra f f e c tt h er e s o -nant frequency of the LC circuit. Thus, the resonant frequency of LC voltage sensor is a function of the voltage of the MSC f 0VðÞ¼1 2pffiffiffiffiffiffiffiffiffiffiffiffiffi LC VðÞp : To wirelessly detect the resonant frequency of LC voltage sensor, an interrogator coil, which is connected to a vector network analyzer (VNA) through an SMA (SubMiniature version A) adaptor, is placed 0.5 cm above the inductor coil of the LC voltage sensor and is electro-magnetically coupled to it. The change of the sensor’s resonant fre-quency is remotely detected by measuring the input return loss (S 11)o f the interrogator coil. As shown in Fig. 4(c) , the resonant frequency shifts higher with the increase in voltage and seen from the sensitivity plot ofFig. 4(d) , it shows good linearity and the sensitivity is 3.5 MHz/V. Using pulse charging mode, charging and voltage detection can be performed at the same time. The charger sends out a pulse AC thatis captured by the charging coil and converted to a pulse DC which is applied to the MSC. The pulse DC is recorded by an oscilloscope andis shown in Fig. 4(e) . As seen, the duration of each pulse is 0.1 s with voltage of 2.9 V and the frequency is 1 Hz. When the pulse chargingstarts, the resonant frequency of the LC sensor is measured at thesame time by the VNA. As shown in Fig. 4(f) , before 0 s, the resonant frequency of the LC sensor stabilizes at 20.9 MHz. When the first pulse is applied to the MSC, the resonant frequency rapidly increases to 27 MHz, corresponding to 1.75 V which is deduced from the sensitiv-ity plot. After the pulse ends, the resonant frequency falls back to aminimal value which can be considered as the voltage of the MSC. Asshown by the red curve, the voltage of the MSC gradually increaseswith the number of pulses, approaching its maximum voltage around1.1 V. Exceeding 1.1 V, some other chemical reactions take place, pre-venting the voltage from further increasing, which is consistent with the GCD test of the MSC. In conclusion, a smart power system of biocompatible and flexi- ble MSC is developed. Modified silk film is used as the device sub- strate, which provides the overall biocompatibility and flexibility of thedevice. Inkjet printing and electrochemical deposition are used to pre-pare the MSC electrode materials. The areal capacitance of MSCdevice reaches 11.64 mF cm /C02at 0.1 mA cm/C02and retains 4.90 mF cm/C02when the current density is increased up to 1.2 mA cm/C02.A n d the device retains 92.8% capacitance of the initial value after being FIG. 4. (a) and (b) The schematic diagram and prototype of the smart power system. (c) Frequency plots of the LC voltage sensor. (d) Sensitivity plot of the LC vo ltage sensor. (e) Pulse DC recorded by an oscilloscope. (f) Resonant frequency change along with time. The right axis is the mapped voltage from sensitivity plot. Th e blue line marks the voltage value when pulse is on while the red line reflects the voltage of the MSC after the pulse ends.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 073902 (2021); doi: 10.1063/5.0040497 118, 073902-5 Published under license by AIP Publishingbent 2000 times at 60/C14. On the basis of the well-performed MSC, wire- less charging and LC voltage sensor are integrated to realize the design of smart power management. Using pulse charging mode, chargingand voltage detection can be performed at the same time. Most impor- tantly, the circuit requires few chips and does not consume energy. It has great application potential as the energy supply part of small device implanted in the human body. See the supplementary material for the conductivity test of the Ni/Ag electrode, cyclic voltammograms with different mass loadings of MnO 2, areal capacitance and volumetric capacitance of the device, Nyquist plot of electrochemical impedance spectroscopy of the device,formulas used to calculate the capacitance, energy density and power density, and detailed experimental process description. This work was supported by the National Nature Science Foundation of China (Grant No. 51907171). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1B. Xu, A. Akhtar, Y. Liu, H. Chen, W.-H. Yeo, S. I. Park, B. Boyce, H. Kim, J. Yu, H.-Y. Lai, S. Jung, Y. Zhou, J. Kim, S. Cho, Y. Huang, T. Bretl, and J. A.Rogers, Adv. Mater. 28, 4462 (2016). 2C. Yuan, D. J. Roach, C. K. Dunn, Q. Mu, X. Kuang, C. M. 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5.0036961.pdf

Appl. Phys. Rev. 8, 011302 (2021); https://doi.org/10.1063/5.0036961 8, 011302 © 2021 Author(s).Photothermal and photovoltaic properties of transparent thin films of porphyrin compounds for energy applications Cite as: Appl. Phys. Rev. 8, 011302 (2021); https://doi.org/10.1063/5.0036961 Submitted: 10 November 2020 . Accepted: 14 December 2020 . Published Online: 08 January 2021 Jou Lin , and Donglu Shi COLLECTIONS This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN Noise spectroscopy of molecular electronic junctions Applied Physics Reviews 8, 011303 (2021); https://doi.org/10.1063/5.0027602 Biological function following radical photo-polymerization of biomedical polymers and surrounding tissues: Design considerations and cellular risk factors Applied Physics Reviews 8, 011301 (2021); https://doi.org/10.1063/5.0015093 Hybrid printed three-dimensionally integrated micro-supercapacitors for compact on-chip application Applied Physics Reviews 8, 011401 (2021); https://doi.org/10.1063/5.0028210Photothermal and photovoltaic properties of transparent thin films of porphyrin compounds for energy applications Cite as: Appl. Phys. Rev. 8, 011302 (2021); doi: 10.1063/5.0036961 Submitted: 10 November 2020 .Accepted: 14 December 2020 . Published Online: 8 January 2021 JouLin and Donglu Shia) AFFILIATIONS The Materials Science and Engineering Program, Department of Mechanical and Materials Engineering, College of Engineering and Applied Science, University of Cincinnati, Cincinnati, Ohio 45221, USA a)Author to whom correspondence should be addressed :donglu.shi@uc.edu ABSTRACT To address the critical issues in solar energy, the current research has focused on developing advanced solar harvesting materials that are low cost, lightweight, and environmentally friendly. Among many organic photovoltaics (PVs), the porphyrin compounds exhibit unique structural features that are responsible for strong ultraviolet (UV) and near infrared absorptions and high average visible transmittance, making them ideal candidate s for solar-based energy applications. The porphyrin compounds have also been found to exhibit strong photothermal (PT) effects and recentlyapplied for optical thermal insulation of building skins. These structural and optical properties of the porphyrin compounds enable them to func-tion as a PT or a PV device upon sufficient solar harvesting. It is possible to develop a transparent porphyrin thin film with PT- and PV-dual-modality for converting sunlight to either electricity or thermal energy, which can be altered depending on energy consumption needs. A building skin can be engineered into an active device with the PT- and PV-dual modality for large-scale energy harvesting, saving, and generation. This review provides the current experimental results on the PT and PV properties of the porphyrin compounds such as chlorophyll and chlorophyllin.Their PT and PV mechanisms are discussed in correlations to their electronic structures. Also discussed are the synthesis routes, thin film deposi-tion, and potential energy applications of the porphyrin compounds. Published under license by AIP Publishing https://doi.org/10.1063/5.0036961 TABLE OF CONTENTS I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 II. PORPHYRIN COMPOUNDS AND THE DERIVATIVES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 A. Structural characteristics of porphyrin . . . . . . . . . . 4 B. The chemical characteristics and synthesis of porphyrins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 C. Electronic structures and absorptions of porphyrins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 III. PHOTOTHERMAL EFFECT . . . . . . . . . . . . . . . . . . . . . . . 5 A. The fundamental photothermal mechanism in porphyrins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 B. Photothermal thin film deposition . . . . . . . . . . . . . . 6C. The characteristics of the photothermal effect. . . . 7 1. Photothermal conversion efficiency ( g)...... 8 2. Specific absorption rate (SAR) . . . . . . . . . . . . . . 8 3. Specific photothermal coefficient (SPC) . . . . . . 84. U-factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10D. Biomedical applications of porphyrin compounds 10 IV. PHOTOVOLTAIC EFFECT OF PORPHYRIN COMPOUNDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 A. The fundamentals of photovoltaic effect. . . . . . . . . 11 B. The performance of a solar cell . . . . . . . . . . . . . . . . 11 1. Power conversion efficiency (PCE, g) ....... 1 1 2. Incident photon-to-current conversion efficiency (IPCE). . . . . . . . . . . . . . . . . . . . . . . . . . 12 3. Dye-sensitized solar cells (DSSCs). . . . . . . . . . . 12 4. Porphyrin-based photosensitizer . . . . . . . . . . . . 13 V. PHOTOTHERMAL AND PHOTOVOLTAIC DUAL MODALITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 VI. FUTURE PERSPECTIVES AND CONCLUSIONS . . . . 15 I. INTRODUCTION The pressing issues in energy sustainability call for novel approaches in generating inexhaustible energy without compromising Appl. Phys. Rev. 8, 011302 (2021); doi: 10.1063/5.0036961 8, 011302-1Applied Physics Reviews REVIEW scitation.org/journal/arethe energy sources in the future. Solar harvesting for energy sustain- ability has been an intense research area that focuses on various advanced material based technologies among which solar cells havebeen the major approach for converting photons directly to electricity by the photovoltaic (PV) effect. There are other photon energy conver- sion mechanisms such as the photothermal (PT) effect that has been extensively studied for biomedical therapeutics. 1–8In biomedical diag- nosis and therapeutics, the PT materials are typically made of photon- absorbing metallic conductors such as gold and graphene that exhibit strong photothermal effects capable of raising the local temperature tothe hyperthermia treatment range of /C2445 /C14C. These PT materials are often synthesized into small quantities in milligrams and dispersed in aqueous solutions for intravenous delivery. The excitation light source for photothermal treatment mainly employs near infrared (NIR) laser (typically 708 and 808 nm) for significant NIR absorption and tissuepenetration. The typical PT materials include inorganic materials (i.e., noble metals), 2,3carbon-based materials,4,5Fe3O4and its compo- sites,6,7organic materials (i.e., polymer-based nanomaterials),8,9and porphyrin compounds10–14. Only recently, the PT effect in the thin film form has been explored for energy applications with white light excitation.6,7,15–18 The residential and commercial building sectors account for about40% (or about 40 quadrillion British thermal units) of the total U.S.energy consumption. 19Thermal insulation has been conventionally achieved by various glazing technologies. Single-panes are, however, practically not viable due to rapid heat loss in the winter especially in northern America. A new concept of “optical thermal insulation” (OTI) was recently proposed and realized via PT film coated single- pane windows without any intervention medium.6,15–18Zhao et al. first reported the coating of a thin film containing Fe 3O4nanopar- ticles on glass for energy-efficient window applications.6As shown in Fig. 1 , if a spectral-selective PT thin film is applied on a window inner surface, the window surface temperature can increase from 25/C14Ct o>50/C14C upon solar irradiation. The window thermal trans- mittance depends on the temperature difference, DT, between the single-pane (without glazing) and the room interior. Due tophotothermal heating, the reduction in DT will effectively lower heat transfer through the building skin, characterized by a low U-factor (the U-factor is essentially the thermal transmittance) without dou- ble- or triple-glazing. OTI requires special optical spectra of the PT films, i.e., strong ultraviolet (UV) and NIR absorptions and high visible transmittance (AVT).19The strong absorptions in the UV and NIR regions ensure significant solar light harvesting for conversion to heat, while highAVT provides the visible transparency of the window. These particularspectral requirements of PT are also shared by transparent PV panels.Another common feature of PV and PT panels lies on a large two- dimensional (2D) surface area for sufficient solar light harvest. According to the National Renewable Energy Laboratory, supplyingall electricity needs of the U.S. with photovoltaic solar energy wouldrequire 1948 ft 2per person.20To effectively utilize surface areas, large- area building skins can be engineered into PT or PV panels for solar light harvesting, specifically those modern buildings with glass fac ¸ade (Fig. 2 ). This approach is particularly effective in the population- concentrated urban areas and megacities with high densities of build-ings. In this approach, the building skin is not only a “window”enframed within the building walls that compromise between lighting and heat loss, but also a PV or PT device capable of solar harvesting, energy conversion, and production of both heat and electricity simul-taneously or alternatively in dual modality modes. The key challenges in achieving successful dual-modality include particulate materials that have to be structurally and spectrally tailored in the PT and PV thin films on building skins. As is well-known, the basic requirement is efficient solar harvesting within certain spectra,typically in the three regions: ultraviolet (UV), visible, and infrared(IR), respectively, with corresponding wavelengths of 100–400 nm, 400–700 nm, and above 700 nm. According to the solar spectra, the infrared provides solar energy around 49.4% and visible light 42.3%,while UV radiation only contributes 8.3% at ground level. 21,22 Therefore, material design and structure development need to FIG. 1. Schematic working principle of photothermal film. FIG. 2. Optical photograph of a modern building with glass facade.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011302 (2021); doi: 10.1063/5.0036961 8, 011302-2consider harvesting most of the infrared radiation for efficient energy conversion and utilization. Transparent PT and PV thin films require high visible transmit- tance specifically for building skins. This unique characteristic necessi-tates spectral selective materials (and structures) that are stronglyabsorbent in the UV and NIR regions but highly transparent in thevisible band with a saddle-like (or U-shaped) spectrum, as schemati-cally represented in Fig. 3(a) . As can be seen in this figure, while gold nanoparticles exhibit strong absorption below 300 nm, there is anotherpeak at 520 nm, contributing to considerable visible light absorption.Much stronger absorption is observed in the UV range for grapheneand Fe 3O4nanoparticles, but it gradually decays to the minimum in the visible range up to near-infrared (NIR). All conducting materialswith high charge densities exhibit strong photothermal effects butlacking sufficient visible transmittance. Among most of the PT materi- als, chlorophyll shows a saddle-like spectrum with two peaks, respec- tively, at 415 nm (UV) and 664 nm (NIR), which is ideal for PT–PVdual modality building skin applications since it exhibits both the PTand PV effects [ Fig. 3(a) ].Chlorophyll belongs to the porphyrin compounds that are struc- turally characterized with a large ring molecule consisting of four pyr-roles, denoted as the porphyrin ring. The tetrapyrroles are smallerrings made of four carbons and one nitrogen. The porphyrin com-pounds typically have a metal ion at the center of the tetrapyrrolering. 23By alternating the chelated ion, different porphyrin compounds can be formed. Replacing the central metal ion is normally achieved by using strong acids as the reaction media.24Interestingly, some por- phyrins occur in nature such as chlorophyll, heme, and hemoglobin(HB). As is well-known, chlorophyll is an essential component inplants for photosynthesis. Its chemical structure is featured with amagnesium atom at the center of the tetrapyrrole ring. The saddle-likeabsorption [ Fig. 3(a) ] is particularly sought after for transparent solar harvesting devices. Similar to chlorophyll, most of the porphyrin com- pounds exhibit two main peaks in the absorption spectra, denoted asSoret bands (380–500 nm) and Q bands (500–750 nm). 25These optical characteristics tend to render the porphyrins with unique colors, mak-ing them important “dyes” in the paper and textile industries. 26The strong absorptions in these particular bands have also been utilized for FIG. 3. (a) Comparison of spectra of gold, graphene, and Fe 3O4; an ideal absorption spectrum (the dashed line) is schematically shown for highly transparent thin films, (b) absorption of Fe 3O4@Cu 2/C0xS showing its broad NIR absorptions (include the spectrum of Fe 3O4here for comparison), and (c) absorption spectra of porphyrins: chlorophyll, chlorophyllin, hemoglobin, phthalocyanine, and zinc-mesoporphyrin IX.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011302 (2021); doi: 10.1063/5.0036961 8, 011302-3photothermal applications in biomedicine, energy,11–16,18and photovoltaics.27–35,37 To enhance NIR absorption, Lin et al. modified Fe 3O4by addi- tion of Cu 2/C0xS and developed the Fe 3O4@Cu 2/C0xSt h i nfi l m s .17 Comparing with the Fe 3O4thin films, the Fe 3O4@Cu 2/C0xSt h i nfi l mi s spectrally characterized with a much stronger NIR absorbance [ Fig. 3(b)], contributing to an increased photothermal efficiency under sim- ulated solar irradiation. Due to the modified structure and NIR absorption, the photothermal effect is considerably enhanced inFe 3O4@Cu 2/C0xS.17For transparent PV and PT thin films, more refined absorptions near UV and NIR can be obtained in the porphyrin com- pound as shown in Fig. 3(c) . Both chlorophyll and chlorophyllin thin films have been developed with rather sharp peaks around 400 and 700 nm, giving distinctive optical characteristics of these materials fora variety of applications. 15,18Multilayer chlorophyll films were also deposited on glass substrates showing excellent photothermal property and transparency. The photothermal mechanism of porphyrins was identified based on the photon-activated molecular resonance.16 This review presents the most recent experimental results on structures and properties of the porphyrin compounds for promising energy applications. The fundamental photothermal mechanisms are discussed based on their photon–molecular interactions based on theRaman spectra. The photothermal properties of the porphyrin com- pounds are characterized and calculated including U-factor, specific absorption rate (SAR), specific photothermal coefficient (SPC), andphotothermal conversion efficiency. The current status of porphyrin- based solar cells is also discussed on the solar cell structure, I–V curve, and efficiency. The prospect of PT–PV dual modality film design will be introduced for future studies. II. PORPHYRIN COMPOUNDS AND THE DERIVATIVES A. Structural characteristics of porphyrin The porphyrin compounds are naturally occurring and charac- terized with the porphyrin ring structures 36that are typically found in chlorophyll37and heme of hemoglobin.38Due to the ring structure of the conjugated double bonds of porphyrins, porphyrin and its deriva- tives show unique absorption spectra, emission, charge transfer, and chelating properties. The chemical and physical mechanisms of por- phyrins are responsible for electron transfer, oxygen binding, photo-chemical, and photosynthesis. These unique structures and properties of the porphyrin compounds have been utilized in biomedical and energy applications including photodynamic/photothermal therapy(PTT), 39–41molecular electronic devices,42and solar energy conversion.27–35 Figure 3(c) shows the unique bands of porphyrins in the absorp- tion spectra. As shown in this figure, the Soret and Q bands are, respectively, in the 380–500 nm (blue) and the 500–750 nm ranges. The wavelength shifts and absorbance changes of porphyrins are dependent on the solvent, pH value, temperature, central metal ions, etc.23,43–46UV-vis spectra reflect the key structural features that are responsible for the chemical characteristics and physical properties of the porphyrin compounds. As the absorption peaks of the porphyrin compounds are within the main emission spectrum of solar radiation,the photons can be effectively harvested and converted to other forms of energy such as heat and electricity. For applications that require cer- tain transparency of the thin films, the absorptions are required to be isolated in the UV and NIR regions ( Fig. 3 ). For medical applicationssuch as photothermal therapy, strong NIR absorption is preferred for its deeper tissue penetration. 47 B. The chemical characteristics and synthesis of porphyrins The molecular structure of a porphyrin consists of a union of four pyrrole rings (one nitrogen and four carbon atoms) linked bymethine bridges ( ACH¼) to form a macrocycle ( Fig. 4 ). Porphyrins behave as the tetradentate ligands for chelating metals with four nitro-gen atoms holding the metal ions in the center. The central metal ionsact as Lewis acids for accepting electron pairs from dianionic porphy-rin ligands. The porphyrin structure shows two main linker positions,b-positions and meso-positions. There are eight b-positions located in the pyrrole group and four meso-positions located at the methinebridge. The natural porphyrins exhibit b-positions linked like chloro- phyll and heme, based on which the meso-porphyrins can be devel-oped as the functional artificial compounds. The porphyrin active sitesdepend on electronegativity, which can be controlled by selecting themetals coordinated to the central nitrogen atoms. Carrying divalent central metals can provide the electronegative porphyrin ligands to be substituted on the meso-carbon sites, resulting in stablemetalloporphyrin. 48–51 Tremendous synthetic procedures have been reported52–58based on the molecular structures of porphyrins. In principle, there aremany chemical strategies to synthesize and modify porphyrins, includ-ing different pyrrole, 52,53,57aldehyde,53,54,57dipyrromethene,55and tri- pyrrane.56The first study on porphyrin synthesis was carried out in 1935 by Rothemund52They reported porphyrin synthesis by mixing pyrrole and aldehyde, and heating them up in a sealed tube. This approach was later modified by using pyrrole and formaldehyde toform a porphyrin with meso-position substitutions. 53Further improvements were made by involving an acid catalyzed pyrrole andaldehyde condensation to react in open air. 54Consequently, Lindsey et al.57optimized this process by reacting pyrrole and benzaldehyde reversibly at room temperature. Lee et al. presented a one-flask synthe- sis of meso-porphyrins by using pyrrole and dipyrromethane. Boudifand Momenteau 56proposed a so-called “3 þ1” synthetic strategy of porphyrins. In this approach, the porphyrin macrocycle was obtainedby mixing acid-catalyzed condensation pyrrole with tripyrrane.Although either diformylpyrrole or tripyrrane must be symmetrical to FIG. 4. Molecular structure of porphyrin (M represent metal ions, such as Mg2þ, Cu2þ,F e2þ,Z n2þ, etc.).Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011302 (2021); doi: 10.1063/5.0036961 8, 011302-4remove isomers and recrystallization,58these porphyrins all exhibit two sub-equal intensity signals in the proton nuclear magnetic reso- nance (NMR) spectra. C. Electronic structures and absorptions of porphyrins According to H €u c k e l ’ sr u l e( t h ep l a n a rm o l e c u l ei sc o n s i d e r e d aromatic which has 4n þ2pelectrons), the porphyrin ring is a conju- gated molecule containing 18 pelectrons. Both Soret and Q bands attributing to the p–p/C3transitions can be explained by the Gouterman model (so-called four-orbital model). The Soret bands (or B bands) are given by a strong transition to the second excited state and the Q bands are associated with a weak transition to the first excited state. Gouterman25proposed the four-orbital model to explain the porphyr- ins absorption spectra based on the electron transitions between highest occupied molecular orbitals (HOMOs) and lowest unoccupied molecu-l a ro r b i t a l s( L U M O s ) ,a ss h o w ni n Fig. 5 . The relative transition energy is affected by the central metal ions and the substituents in the ring. There two symmetries are denoted as b 1(a2u) and b 2(a1u)i n HOMO and c 1(egx) and c 2(egy) in LUMO. Due to the configuration interaction of electronic transitions a 1u!egand a 2u!eg, the transi- tions between these orbitals provide two excited states. The mixing of orbital splits these two excited states in energy, resulting in two sym- metrical states (1Eu)(Fig. 6 ). The higher energy state with greater oscil- lator strength is responsible for the B-bands, while the lower energy state with weaker oscillator strength gives rise in the Q-bands. The absorption spectra are affected by the differences in the con- jugation and symmetry of a porphyrin.25,48,59The electronic structure of a metalloporphyrin can be changed by different central metal sub- stituent in the porphyrin ring. Shelnutt and Ortiz60reported that the a1uorbital energy would not be affected by metal substitution because it has nodes through the nitrogen atoms (pyrroles); therefore, it inca- pable of interacting with the metal orbitals. The a 2uorbital energy, on the other hand, is expected to be changed by metal substituents. Gouterman25reported that the a 2uorbital energies are influenced bythe p orbital of metals. The orbital energy rises with more electroposi- tivity, resulting in red-shift (Q bands) and intensified visible band. Theextent of delocalization of the electron a 2uorbital into the p zorbital of metal increases with electronegativity. The a 2ustabilization is also ascribed to the interaction with an empty metal p zorbital. The electronic structure of the metalloporphyrin can be modified by changing the substitutes at the b- or meso-positions of the ring.60,61 The a 1uorbital energy can be altered by changing the substituents at theb-positions since a 1uhas more b-carbons of pyrrole rings than a 2u. Comparatively, the a 2uorbitals are strongly influenced by metal and meso substituents because the symmetry allowed interacting with thep orbital of the metal and the considerable charge on the mesocarbons. 60Shelnutt and Ortiz reported that strong conjugation of the p-electron system of the vinyl group with the ring can improve the delocalization of the ring charge exceeding the a-carbons of the vinyl. This strong conjugation of the p-electron system further lowers the configuration-interaction (Cl) for protoporphyrin IX. Liao andScheiner 61found a strong electron effect in metalloporphyrin by replacing the meso phenyl group and b-pyrrolic hydrogen (H) with fluorine (F) atoms for its strong electron-withdrawing effect. RamKumar et al. synthesized A 3B type meso substituted porphyrin con- taining three thienyl groups and one phenyl group, and explained theorbital differences between thienyl ring and phenyl group by the den-sity functional theory (DFT). The p-orbital delocalization can beenhanced leading to red-shift in the emission spectrum due to the co- planar of the thienyl groups/phenyl groups with the dihedral angles. They also found that the a 2uand e gxenergy levels are associated with the redox phenomenon of the porphyrins. In particular, the lower a 2u energy results in lower oxidation potentials for the porphyrins, and thelower e gxenergy indicates higher reduction potential. Furthermore, the electronic structure of a metalloporphyrin can be altered by ligandcoordination to the axial positions on the metal. 61The ligands will enhance or inhibit the ability of metal to withdraw electrons from thep-system of porphyrins, so that the electron structure is modified via these axial ligands. The axial coordination to the high molecular sym-metry (square-planar, D 4h) is responsible for destabilization of dz2 orbital through interactions of r-bonding. III. PHOTOTHERMAL EFFECT A. The fundamental photothermal mechanism in porphyrins The photothermal effect is the physical manifestation of photon to thermal energy conversion through a material with an appreciable FIG. 5. Porphyrin molecular orbitals (MO) relevant to the Gouterman four-orbital model. Symmetry nodes are drawn in dash lines. FIG. 6. Orbitals and energy states diagrams showing possible transitions for por- phyrins. The actual relative energies depend on the substituents on the rings.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011302 (2021); doi: 10.1063/5.0036961 8, 011302-5absorption of light. Upon irradiation of various light sources, be it monochromatic or white light, the photothermal material can generate sufficient heat, thereby raising the local temperature within a short duration, provided that the material has considerable photothermalefficiency. The photothermal energy conversion for transition metals, especially those of nanoscale, is typically characterized by surface plas- mon resonance (SPR). The photothermal effect has been extensively studied for gold and graphene, and the energy conversion is attributed to the localized surface plasmon resonance (LSPR). 21,62,63Al o c a l i z e d plasmon is the result of the confinement of a surface plasmon in a nanoparticle smaller than the wavelength of the incident light. A nano-particle’s response to the oscillating electric field can be described by the so-called dipole approximation of Mie theory. 64The wavelength- dependent extinction cross section of a single particle defines the energy losses in the direction of propagation of the incident light due to both scattering and absorption.21,62,63,65–67However, Zhao et al. found a different photothermal mechanism for the porphyrin com- pounds that is associated with the porphyrin ring molecular structurebased on Raman data ( Fig. 7 ). 16 Figure 7 shows the Raman spectra of the chlorophyll, chlorophyl- lin, hemoglobin, and phthalocyanine samples. Raman peaks are typi- cally identified as molecular vibrations involving bonds of theporphyrins and can be significantly enhanced if the incident photonenergy is near the optical absorptions. Consistently, for 415 nm laser excitation, the Raman peaks are significantly enhanced for chlorophyll, chlorophyllin, and hemoglobin due to their absorption peaks near 400 nm, while those by the 514 nm laser are considerably reduced, indicating the strong associations between the photon absorptions and the porphyrin structures in these compounds. For phthalocyanine, the 633 nm laser results in much stronger Raman peaks corresponding to its absorptions in the NIR range. These results provide a physical base for the molecular oscillation-induced photothermal effects in the porphyrins. B. Photothermal thin film deposition Spin coating is an effective method for making uniform photo- thermal thin films of various kinds. The thickness of PT film can be well-controlled by rotational speed of spin processor and solution con- centration. Zhao et al. 15reported the multilayer chlorophyll PT films on a glass substrate via spin coating ( Fig. 8 ) and found that both the photothermal effect and the average visible transmittance (AVT) are interrelated via the chlorophyll concentration which plays a key role in the single-pane window application. They also investigated the struc- tures and properties of different porphyrin compounds such as chloro- phyllin, hemoglobin, and phthalocyanine, and developed their thin FIG. 7. Raman spectra of (a) chlorophyll, (b) chlorophyllin, (c) hemoglobin, and (d) phthalocyanine on aluminum foils, by 442, 514, and 633 nm lasers, respec tively.16 Reproduced with permission from Zhao et al. , J. Phys. Chem. C 124, 2 (2019). Copyright 2019 American Chemical Society.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011302 (2021); doi: 10.1063/5.0036961 8, 011302-6films by spin-coating.16As shown in Fig. 3(c) , the absorption of chlor- ophyllin (Chlin) is more pronounced than those of chlorophyll (Chl),hemoglobin (HB), and phthalocyanine (Phth) at the same concentra-tion. Therefore, stronger photothermal effect is observed in the chloro- phyllin thin films under the simulated solar light. They have alsocalculated various parameters of the porphyrin thin films based on the heating/cooling curves including U-factor, photothermal conversion efficiency, specific absorption rate, and specific photothermal coefficient. 16 C. The characteristics of the photothermal effect Figure 9 schematically depicts the heating/cooling curve of a pho- tothermal material under light irradiation. There are three stages inthe temperature changed ( DT) vs time plot: initial and rapid heating region (stage 1), steady state (stage 2), and cooling region (stage 3),respectively. Stage 1 indicates the heating of the sample due to the pho-tothermal effect is greater than heat loss, thereby a rapid increase intemperature. In stage 2, the photothermal heating is balanced by the heat loss through the sample surface, causing a temperature plateau in this region. In other words, the heat loss of the sample is equal to theheat gain by the photothermal effect. At the beginning of stage 3, thelight source is turned off, resulting in a Newtonian cooling behavior inthis region. The photothermal effect of the material is closely related to the optical absorption spectra and excitation photon energy. For laser excitation, more efficient photothermal effect can be generated if theincoming photon energy is near the high energy edge of the absorptionpeak. For white light irradiation, the photon harvesting takes a widerange of the absorption spectrum. Specifically, the B band and Q bandof chlorophyll are at approximately 415 and 664 nm, respectively [ Fig. 3(c)].Figure 10 shows the relatively temperature differences of chloro- phyll thin films and solution under 660 and 785 nm laser irradiationfor 10 min. As can be seen in this figure, the temperature increase ismuch rapid by the 660 nm laser than that by the 785 nm laser due tothe absorption peak near 660 nm. Therefore, to enhance both the pho-tothermal effect and transparency, the material must be spectral selec-tively tailored via structural design and compositional optimization to achieve strong absorptions in the non-visible regions such as NIR and high average visible transmittance (AVT). Figure 11(a) shows the relatively temperature differences of the multilayer chlorophyll thin films under white light irradiation.Consistently, thicker films (each layer is /C242lm) give greater DT gas FIG. 8. SEM cross-sectional image of the four-layer chlorophyll (Chl) film with poly- methyl methacrylate (PMMA) coated on glass substrate.15 FIG. 9. Schematic diagram showing three stages in the temperature vs time curve. FIG. 10. DTgvs time for (a) Chl in 1-butanol and (b) Chl-coated glass irradiated by 661 and 785 nm lasers.15Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011302 (2021); doi: 10.1063/5.0036961 8, 011302-7expected, but lower visible transmittance. Figure 11(b) shows a linear relationship between DTg,max vs visible transmittance for thin films of different layers. The same behaviors have been observed in other por-phyrin compounds of chlorophyllin, hemoglobin, and phthalocyanine samples as shown in Fig. 12 . The linear relationships between DT max and AVT for all thin film samples indicate that the photothermal behavior is highly correlated with the optical transmittance andabsorption. 1. Photothermal conversion efficiency ( g) The photothermal conversion efficiency, g,i sd e fi n e da st h er a t i o of the thermal energy generated by the sample to the incident photonenergy. The photothermal conversion efficiency for solar light wasdeveloped by Jin et al. 68and modified by Lin et al.17 g¼CgmgþCPT material mPT material þCpolymer mpolymer ðÞ DTmax IADt /C25CgmgDTmax IADt/C2100% ; (1)where g is glass, c is the specific heat capacity (J/g /C1/C14C), m is mass (g), DTmaxis the maximum change in temperature increase in the sample (/C14C), I is the incident light power density (W/cm2), A is the surface area of the sample, and Dt is the time required for a sample to reach the maximum temperature (s). The photothermal conversion efficien- cies have been determined for various PT materials and thin films.15,17 The biomedical applications of PT have been extensively studiedmainly for medical therapeutics. In a typical medical PT therapeutics, both the quantity of the materials ( /C24mg) and the total thermal energy required are significantly low for a highly localized tumor treat- ment. 1–8However, in energy applications, the gvalue is more critical as the total thermal energy generated is much greater, compared to that of the medical therapeutics, considering the large surface area of the thin films, such as the building skin. 2. Specific absorption rate (SAR) The specific absorption rate (SAR) of a photothermal material is a measure of the rate at which energy is absorbed by the material when exposed to an incident light and can be expressed as15,17 SAR¼CglassmglassþCPT material mPT material þCpolymer mpolymer ðÞ DTmax/C0CglassmglassþCpolymer mpolyme r ðÞ DTcontrol mPT material Dt; (2) where DTcontrol is the maximum change in temperature increase in the polymethyl methacrylate (PMMA) film without photothermal materi-als. This equation can be simplified as 15,17 SAR¼CglassmglassDTmax/C0DTcontrol ðÞ mPT material Dt: (3) SAR is an important parameter that characterizes the materials’ abili- ties in photon absorption. As is well-known, all materials and surfaces are capable of absorbing photon energy to certain degrees. However,in engineering applications, high SAR is required for rapid heat increase and pronounced photothermal effect for efficient energy con- version and heat generation. 3. Specific photothermal coefficient (SPC) Specific photothermal coefficient is an intensive material charac- teristic that varies from system to system. For energy conversion stud- ies, it is important to characterize a system with a material-specific FIG. 11. (a)DTgvs time for different layers of chlorophyll coating irradiated by simulated solar light. (b) Linear DTg, max vs average visible transmittance relationship of chloro- phyll coating.15Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011302 (2021); doi: 10.1063/5.0036961 8, 011302-8FIG. 12. Temperature difference DT vs time for (a) chlorophyll (Chl), (b) chlorophyllin (Chlin), (c) hemoglobin (HB), and (d) phthalocyanine (Phth) samples, with concentrations indicated; and DTmaxvs AVT for (e) chlorophyll, (f) chlorophyllin, (g) hemoglobin, and (h) phthalocyanine samples. All samples are illuminated by simulated solar light (0.1 W/ cm2).16Reproduced with permission from Zhao et al. , J. Phys. Chem. C 124, 2 (2019). Copyright 2019 American Chemical Society.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011302 (2021); doi: 10.1063/5.0036961 8, 011302-9parameter, regardless of its volume and concentration. In this way, the intrinsic properties of the materials can be well-characterized for pho- ton-to-heat conversion. Specific photothermal coefficient l(SPC) is defined as the thermal energy produced/converted.16SPC can be expressed as16 SPC¼_Q mass PT m aterial(4) where _Qis the heat produced via photothermal effect in unit of time (J/s) and mass PT material is the mass of photothermal material. SPC can be calculated based on the cooling curve in stage 3 as shown in Fig. 9 . The SPCs of some porphyrin compounds have been reported16to be 129 J/g /C1s for the chlorophyll film, 619 J/g /C1s for the chlorophyllin film, 180 J/g /C1s for the hemoglobin film, and 352 J/g /C1sf o r the phthalocyanine film. The higher SPC value indicates the greaterphoton-to-heat conversion ability. 4. U-factor The rate of heat loss by radiation, convection, and conduction through a medium is defined as U-factor. Since U-factor is essentiallythe thermal transmittance, the lower the U-factor, the better the ther- mal insulation. As a result of energy price increase, thermal insulation has become increasingly important for energy sustainability. According to the U.S. Advanced Research Projects Agency-Energy(ARPA-E), 69the energy and element consumption of buildings can be improved by replacing the double-pane window with a single-pane, specifically for the cold climate area in the U.S. The general require- ment of U-factor for window has been reported by NFRC70that the minimum requirement of U-factor is <1.7 W/m2K although the U- factor requirement varies for different climate zones.71Hence, the reduction of U-factor is a critical criterion in building skin designs. In accordance with ASTM C1199-14,72the general U-factor can be expressed as U¼1 1 hhþ1 hcþ1 UL; (5) where h hand h crepresent the interior and exterior heat coefficients, and U Lis the heat transfer coefficient of the windowpane. The U-factor calculation should be modified for vary materials and climate conditions.6,15–17TheU-factor equation can be expressed as17 U¼1 Tin/C0Tout Tin/C0Tg/C18/C19 /C21 1:46/C2Tin/C0Tg ðÞ L/C20/C210:25 þreT4 in/C0T4 g/C16/C17 Tin/C0Tg ðÞ2 43 5þ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:84/C2Tg/C0Tout ðÞ1 3/C16/C172 þ2:38/C2v0:89 ðÞ2r; (6) where Tinis the interior temperature, Toutis the exterior temperature, vis wind speed, Lis the height of window, ris Stefan–Boltzmann con- stant (5.67 /C210/C08W/m2/C1K4),eis the emissivity, and Tgis the inside surface temperature, respectively. Equation (6)c a nb ea p p l i e dt oe v a l u a t et h et h e r m a li n s u l a t i o no f a window with a temperature increase on the interior surface by pho-tothermal coating. Based on Eq. (6),L i n et al. 17estimated the U-factor of Fe 3O4and Fe 3O4@Cu 2/C0xS films with the following parameters: T in: 21.11/C14C; T out:/C017.78/C14C;v: 5.5 m/s; L:1 . 5 0m ,a n d e: 0.84. Assuming the original interior window temperature is 5/C14C, and T gis calculated to be 278.15 K þDTmax.T h e U-factors of Fe 3O4and Fe 3O4@Cu 2/C0xS films can be, respectively, reduced to 1.54 and 1.46 (W/m2K)17which are excellent values, all smaller than the minimum requirement of U-factor ( <1.7 W/m2K) reported by NFRC.70Furthermore, Zhao et al.15reported a concept of multilayer chlorophyll (Chl) films for energy-efficient window application. By applying the chlorophyllphotothermal coatings, they effectively lowered the U-factor of the single pane from 1.99 to 1.42 (W/m 2K). These results indicate a high possibility of single-panes even in the cold climate areas viathe solar-heated window surfaces based on the concept of optical thermal insulation without any interfering medium. The low U-factors obtained from these single-panes show promising poten- tials of the highly transparent, spectral selective thin films for opti- cal thermal insulation. 15–17D. Biomedical applications of porphyrin compounds The photothermal therapy (PTT) has been widely applied in bio- medical areas especially in cancer therapeutics. If a PTT agent can bedelivered to the tumor site via cell targeting, it results in effective can-cer cell killing via sheer heat generated locally. 73,74The cancer cells can be killed if the local temperature can reach /C2445/C14C. However, most of the PTT replies on strong photothermal materials such as gold andgraphene which pose significant toxicity and non-biodegradability.The bio-incompatibilities have raised considerable concerns in clinical applications. Porphyrin compounds are naturally occurring materials such as chlorophyll which is extracted from green vegetables. Not onlyare they environmentally green, but also non-toxic, making thesegreen materials ideal for medical applications. The PTT typicallyemploys the PT materials in the solution form which are delivered to the lesion sites via intravenous or direct injection. The photothermal heat is mainly generated by NIR band excitation using lasers for itsdeeper tissue penetration. The porphyrin compounds are known fortheir high packing density and photon absorption, 68making them excellent photosensitizers in PTT. They have also been used for mag- netic resonance imaging (MRI).10,75–78For instance, Mn-based por- phyrin is a good MRI and PTT agent because Mn3þions can generate sharp contrast in MRI while exhibiting the photothermal effects after three serial treatments.76Suet al.77reported the porphyrin andApplied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011302 (2021); doi: 10.1063/5.0036961 8, 011302-10graphene oxide (GO) composite as a good photothermal therapeutic agent. Compared to using GO or porphyrin separately, the combina-tion of GO and porphyrin has a broader light absorption region. Chuet al. 10extracted chlorophyll from plants and encapsulated it into plur- onic F68 (Plu) micelles for photothermal therapy and cancer imaging. Since chlorophyll can be extracted from most of the green vegetables, it is non-toxic and biocompatible; therefore, it is an attractive materialfor clinical applications. IV. PHOTOVOLTAIC EFFECT OF PORPHYRIN COMPOUNDS A. The fundamentals of photovoltaic effect In solar light harvesting and energy conversion, photovoltaics is best known for converting energy from the sunlight to electricity andsuccessfully applied for energy sustainability. However, the power con-version efficiency (PCE) of PVs varies widely from the silicon-based materials to polymers. 79According to the chart of best research-cell efficiency from NREL,79the best efficiency of organic cell achieved 17.5% and the best efficiency of a multijunction cell can attain up to47.1%. The most common polymer solar cells (PSC) are made ofP3HT and PCBM ([6,6]-phenyl-C61-butyric acid methyl ester) poly-mers with PCE about 13%. Although this is quite far from the 20% efficiency of the commercial solar panels, PSC has the advantages of light weight, low price, flexible for applications in biomedicine, remotemicrowatt sensors, wireless appliances, and visually enhanced finessefor architectural designs and building materials. 80–83 Structurally different from silicon-based inorganic PVs that involve the construction of a p–n junction, PSC typically utilizes con-ductive organic polymers for solar harvesting and charge transport toproduce electricity by the photovoltaic effect. Most of the organic solar cells (OSC) are based on the polymers with the pelectron system which is responsible for absorbing photons and transporting electronsby the conjugated pelectrons. Light absorption leads to excitation of p electrons from the highest occupied molecular orbital (HOMO) to thelowest unoccupied molecular orbital (LUMO) of the molecule. The light absorbance wavelength can be determined by the energy bandgap between the HOMO and the LUMO. 84,85 Charges are dissociated at the donor–acceptor interface by the dissociated driving force provided by the energy offset between theelectron donor and electron acceptor. The basic working principle in amolecular-based device can be summarized as follows ( Fig. 13 ): (1) exciton generation caused by light absorption; (2) exciton diffusiontoward a donor–acceptor interface; (3) exciton dissociation at the interface and charge carrier transport, and (4) charge carrier collection at the external electrodes. 86 B. The performance of a solar cell The performance of the cells is determined by the current–volt- a g ec u r v e so rI – Vc u r v e s .T h ec u r v ei so b t a i n e db yt h ec u r r e n ta n dvoltage of the cell measured through an external variable resistanceunder a standard intensity of incident light. The standard illumination for testing is 0.1 W/cm 2(AM 1.5), which is the sunlight intensity reaching earth surface through air mass at a solar zenith angle of42.8 /C14. The main parameters are provided by I–V curves, such as the power conversion efficiency (PCE), open-circuit voltage (V OC), short- circuit current (I SC), and fill factor (F.F.) ( Fig. 14 ).1. Power conversion efficiency (PCE, g) The power conversion efficiency is a parameter for representing the energy production of a solar cell. The efficiency can be calculated by the following equation: g¼Pmax Pin¼Isc/C2Voc/C2F:F: Pin/C2100% ; (7) where P maxis the maximum power per area (W/m2), P inis the light power per area (W/m2), ISCis the short circuit current (A), V OCis the open circuit voltage (V), and F.F. is fill factor (%). For ideal I–V curve, the maximum power per unit area can be expressed as Pmax¼Imax/C2Vmax; (8) where I maxand V maxare voltage and current where the generated p o w e ri sa tt h em a x i m u m . Short circuit current (I SC) is the highest current density obtained when there is no applied voltage. The short circuit current is affectedby the driving force for electron injection into the conduction band from the excited dyes. The open-circuit voltage (V OC) is the maximum FIG. 13. The principles of charge separation in organic solar cell.86 FIG. 14. Schematic of I–V curve.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011302 (2021); doi: 10.1063/5.0036961 8, 011302-11voltage obtained at no current flow. It is determined that the V OCwill be increased by having lower driving force or more positive redox potential.87It corresponds to the energy difference between the Fermi level of the semiconductor and the redox couple’s energy level. Fill fac- t o r( F . F . )i sd e fi n e da sar a t i oo ft h em a x i m u mp o w e rt ot h ep r o d u c to f its V OCand I SC, and it can be expressed as follows: F:F:¼Imax/C2Vmax Isc/C2Voc/C2100% : (9) 2. Incident photon-to-current conversion efficiency (IPCE) IPCE is an important parameter that presents the percentage of the absorbed light converted to current by a solar cell. IPCE indicates the spectral sensitivity of a solar cell by measuring intensity via a monochromatic light. IPCE can be expressed as IPCE %ðÞ¼1240/C2Jsc k/C2Pin/C2100: (10) Here, the J SCis the short-circuit photocurrent density. The per- formance of dye-sensitized solar cells (DSSCs) is dependent on thecharacteristics of the individual components above. Many investiga- tions focus on various materials parameters, such as substrate, photoa- node, 88–92dyes,31–35electrolyte,84,85and counter electrode.86–91 Research also focuses on the structure—properties relationships of theindividual components and interface effect on the performance of DSSCs. 3. Dye-sensitized solar cells (DSSCs) The engineering issues of DSSCs such as stability of dye and leak- age of electrolyte have been experimentally addressed for a variety of applications. A DSSC consists of photoanode, dyes, electrolyte, and counter electrode [ Fig. 15(a) ]. The working principle of DSSC can be described as follows [ Fig. 15(b) ]: (i) when the sunlight hits the device, the electrons in the dye molecules will be excited from ground state(HOMO) into excited state (LUMO); (ii) the excited dye molecules are oxidized (loss of electrons) and electrons are injected into the conduc- tion band of the semiconductor; (iii) the oxidized dye molecules are regenerated by electron donation from the redox mediator (I/C0), and two I/C0are oxidized to iodine (I 3/C0), and (iv) the I 3/C0diffuses toward the counter electrode and then it is reduced to I/C0by receiving electrons from counter electrode (cathode). Although the photon-to-electricity efficiency of DSSC is quite low, it has many advantages, includingnon-toxic, environmentally friendly, cost-efficient, simple manufactur- ing processes, etc. The substrate for a DSSC is a transparent conductive oxide (TCO) glass, made of indium tin oxide (ITO) or fluorine-doped tin dioxide (FTO). Metal oxide semiconductors are used for making pho- toanode of DSSC, namely, TiO 2,88ZnO,89SnO 2,90and Nb 2O591for transporting the electrons, which is also called the electron transport layer. Two main semiconductors commonly used for DSSC are TiO 2 and ZnO, and the bandgap of both materials is 3.2 eV.92Electrolytes are an essential mediator between the photoanode and counter elec-trode in DSSC. Electrolyte regenerates the dye sensitizer from oxidized state back to ground (steady) state by gaining electrons from redox mediator. 84The lost electrons are gained from the electrolyte, an oper- ation known as redox reaction. For example, the oxidized dyes can be regenerated by the iodide species I/C0.I/C0loses electrons to dyes, result- ing in formation of I 3/C0. The triiodide can be returned by gaining elec- trons from counter electrode (cathode), and the reaction can be expressed as I/C0 3þ2e/C0¼3I/C0 The efficiency of DSSCs depends on the kinetics of electron transfer at the liquid junction of the sensitized semiconductor/electro- lyte interface. For effective devices, the electron injection rate should be faster than dye degradation rate.85However, there are some chal- lenges for using metal-salt based liquid as electrolyte, in particular, dif- ficulties in tandem architecture, poor sealing, and photodegradation, resulting in low lifetime and efficiency of a device. A sputtered plati- num (Pt) is commonly used as a counter electrode in DSSC. The main FIG. 15. (a) DSSC structure and (b) schematic working principle of DSSC.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011302 (2021); doi: 10.1063/5.0036961 8, 011302-12purpose of the counter electrode is to catalyze reduction of I 3/C0to I/C0in redox electrolyte after electron injection. Since Pt is a rare and expen- sive metal, Pt-free counter electrodes are required to reduce the cost of DSSCs. Some alternative materials have been developed, for instance, transition metal compounds, carbon-based materials,86,87conducting polymers,88,89and composite materials.90,91 4. Porphyrin-based photosensitizer Photosensitizers play a critical role in DSSC in producing the photo-induced electrons and injecting the electrons into the conduc- tion band of the photoanode. The ideal photosensitizer should meet some requirements, particularly, wide light absorption range, good redox potential, high stability, and suitable anchoring property. Several groups have developed various porphyrin compounds as photosensi- tizers [ Fig. 16(a) ].31–35,91–99 Many porphyrin compounds have been used for developing solar energy conversion systems due to strong absorption in the visible region.92–96Kay and Gratzel31reported the first “porphyrin” DSSC based on many porphyrin complexes such as chlorophyll, chlorophyl- lin, pheophorbide, Zn-mesoporphyrin IX, etc. Porphyrin DSSC with Cu-2- a-oxymesoisochlorin attained a PCE of 2.6% in 1993,31which was further increased to 12.3% by using Zn(II)-porphyrin dye (YD2- o-C8) in 2011.32In 2014, a breakthrough efficiency of 13% for porphy- rin based DSSC was obtained by utilizing Zn-porphyrin derivative (SM371).33The Zn(II)-porphyrin derivatives are used frequently for DSSC for its appreciable PCE.97,98In particular, Santos et al.98studied the differences between Zn-porphyrin and metal-free porphyrin, and found faster injection dynamics of Zn-porphyrin due to its more nega- tive excited state oxidation potential than that of the metal-free porphyrins. Bessho et al.99investigated the photovoltaic effect of the Zn-porphyrin (YD-2), D205 dye, and YD-2/D205 cosensitizer. Theyfound that the J scand PCE can be enhanced by using YD-2/D205 co- sensitizer for its improved light harvesting. Other porphyrin compounds and its derivatives have also been investigated as photosensitizers, such as Mg(II)-porphyrin (chlorophyll),31,93,95Cu(II)-porphyrin (chlorophyllin),31,100,101Cu(II)- porphyrin (CuPc),102,103Ru-porphyrin (RuPc),104Ga-porphyrin (GaTsPc),105Fe-porphyrin (FeTsPc),106Ti-porphyrin (TiPc),107and metal-free porphyrin.98,108In comparison to Zn-porphyrins, PCE and stability of DSSC using other porphyrin compounds are, however, quite low. For instance, the PCE of the chlorophyll-based DSSC was reported to be only 0.73% by Amao and Komori in 2003.109Taya et al. improved the efficiency up to 1.077% in 2015.110Later on, Hassan and co-worker reported a PCE of chlorophyll-based DSSC about 2.62% in 2016.111Some studies indicated that PCE of chlorophyll-based DSSC can be improved by applying multiple dye112,113or doping metal.114 Despite the low PCE, the major advantages of chlorophyll-based photosensitizers include its green nature, low-cost, and high light har- vesting ability. There have been new strategies to enhance the light harvesting ability of the photosensitizer by using (i) push–pull porphyrins,115,116(ii) porphyrin dimers,108,117and (iii) multiple dyes.97,118,119 Push–pull (donor–acceptor) porphyrins are made of an electron- donating group (D), p-bridge ( p), and an electron-withdrawing group (A). A p-bridge is the conjugation units connecting the donor and acceptor. The spectra and electron transfer can be determined by the electronic interaction between the D and A groups. The D- p-A group of photosensitizers determines the procedures in an operating DSSC, which are oxidized-dye regeneration and hot-electron injection. The donor group is considered the dye-regeneration reaction after electron injection, and the acceptor group is to deliver the electrons from pho- tosensitizer to the semiconductor (e.g., TiO 2) for initiating the electron-hole separation process. Lee et al.115proposed a push–pull Zn porphyrin with strong electron-donating substituent and electron- FIG. 16. PV performance of (a) porphyrin-sensitized solar cell.91Reproduced with permission from Wang and Kitao, Molecules 17, 4 (2019). Copyright 2019 Multidisciplinary Digital Publishing Institute (MDPI); (b) multiple dyes (in black solid and dash lines) DSSC.119Reproduced with permission from Ogura et al. , Appl. Phys. Lett. 94(7), 073308 (2009). Copyright 2009 AIP Publishing. All J–V curves characterized under AM 1.5G illumination (1000 W/m2).Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011302 (2021); doi: 10.1063/5.0036961 8, 011302-13withdrawing substituent at meso-positions. They reported that por- phyrin’s absorption bands with a phenylethynyl bridge are red-shifted and broadened due to extension of p-conjugation. The best PCE of Zn-porphyrin DSSC has so far reached 6%, which is competitive to the DSSC using the commercial dye (N3). Later, the performance of push–pull Zn porphyrin DSSC was improved by the same group,116 and 7% PCE has been achieved. A porphyrin dimer is to combine two porphyrin moieties by a chemical bond. Koehorst et al.108proposed a spectrally enhanced por- phyrin heterodimers DSSC. This research employed a covalentlybonded Zn porphyrin (ZnP)/free base porphyrin (H 2P) heterodimers onto non-porous TiO 2layers (H 2P-ZnP-TiO 2) on ITO (Indium Tin Oxide). They reported that the photocurrent is contributed by Zn por-phyrin dimer, resulting in faster energy transfer. In addition, Mozer et al. 117proposed Zn–Zn porphyrin dimer (ZnP-ZnP) DSSCs and demonstrated that each porphyrin array contributes to current genera-tion in a cell. They found that the Soret band of dimers was broadened and the Q-bands showed a higher molar extinction coefficient than those of monomers. These results show the advantage of the porphyrinheterodimers in light harvesting. Light harvesting of a DSSC device can also be enhanced by utiliz- ing multiple dyes. Otaka et al. 118proposed the multicolored DSSCs with the red, purple, blue, green, and black color dyes in DSSCs. Ogura et al.119investigated a multiple dye system by applying terpyri- dine complex (black dye) and indoline dye (D131) in DSSC. Theyreported that the multiple dye system performs better IPCE than those with only black dye and showed an impressive light-to-electricity con- version efficiency (11%). They contributed this improved efficiency tothe combination of multiple dyes that broaden the absorption spectra range for enhanced photon absorption [ Fig. 16(b) ]. V. PHOTOTHERMAL AND PHOTOVOLTAIC DUAL MODALITY Some of the semiconducting materials exhibit both PT and PV effects. If designed appropriately, PT and PV dual modality is highly possible, specifically in some of the porphyrin compounds. The con- cept of PT–PV modality is schematically illustrated in Fig. 17 .A s shown in this figure, the PT–PV film is featured with several key char-acteristics: (1) it is highly transparent but only absorbing UV and NIR for energy conversion, which is particularly useful in building skin applications; (2) upon absorbing solar light in the UV and NIRregions, energy conversion seasonably takes two paths: photon-to-heat conversion in the winter for lowering the thermal transmittance and photon-to-electricity in warmer seasons as a solar panel; (3) thePT–PV dual modality can be switched easily by an electronic control unit depending upon the seasonal changes and energy needs, and (4) the PT and PV thin films can be integrated into the large-scale build-ing skin surface for effective solar harvesting that may compensate the low PCEs of the organic PVs. In the wintertime, the photothermal effect is utilized to heat the single-pane in order to reduce heat lossthrough the building skin based on optical thermal insulation. In this way, thermal insulation can be achieved optically without intervention medium. On the other hand, the undesirable solar infrared in summercan be compensated by the same thin film coating on the building skin but in a different modality: photovoltaic. Absorption of large infrared irradiation not only reduces cooling energy but generates elec-tricity for other appliances.Ever since the discovery of the PV effect, a variety of material systems have been identified, synthesized, developed, and engi-neered into many different types of solar harvesting devices forenergy applications. 120These include Si-based solar cells, copper zinc tin sulfide (CZTS), perovskite solar cells (PSCs), dye- sensitized solar cells (DSSCs), etc. Among all DSSCs, the porphy-rin compounds have demonstrated the most attractive propertiesfor photosensitizers. 30Functionalization of macrocyclic dyes in the porphyrin compounds contributes to strong absorptions in the vis- ible light region.28,29The first application of porphyrin and its derivatives for DSSC was reported in 1993 by Kay and Gratzel.31 They reported 2.6% of the overall photo-conversion efficiency(PCE) and attributed the low PCE to the Ohmic losses at high cur- rent densities. The PCE of Zn-porphyrin derivatives was increased up to 12.3% in 2011, 32and further improved to 13% in 2014.33 Although steady increase in PCE was achieved from 2009 to2014, 34challenging issues are to be addressed in the molecular design, particularly on the aging stability and inefficient photon collection in the Q band region (520 and >700 nm).35Overall, por- phyrin compounds have already been shown to have great poten-tial in developing advanced DSSCs by applying multiple dyes to extend light absorbing range for higher efficiency. 35 In recent years, transparent PV and PT films have gained great attention for their high potentials in efficient solar harvesting andenergy applications. The transparent organic photovoltaic (TOPV) thin films have been extensively studied. 121–123Due to its high AVT, the so-called building integrated photovoltaic (BIPV) has been devel-oped for building skins taking the advantages of large surface areaswithout interfering with the color requirements. 83It has been reported that efficiency of TOPV is about 6% while most of the commercial PVs/C2411%. To compensate this short coming in efficiency, the large surface area of building skin will be highly viable, specifically withPT–PV modality (or multimodality). Some of the porphyrin com-pounds are ideal material systems for PV–PT dual modality design as they are highly spectral-selective. FIG. 17. Schematic of PT–PV modality based on transparent porphyrin compounds.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011302 (2021); doi: 10.1063/5.0036961 8, 011302-14As described above, the common characteristics of both TOPV and transparent organic photothermal (TOPT) thin films are the strong UV and NIR absorptions for photon-to-electricity/thermal con- versions. With these spectral selective features, the so-called building-integrated photovoltaics (BIPV) have been developed. 124–128Yoon et al.127developed the amorphous silicon BIPV window ( Fig. 18 ). They reported that the inclined and horizontal BIPV double windows’ interior surface temperatures are about 2/C14C higher than that of the normal windows in winter. Moreover, Alrashidi et al.128studied the thermal performance of CdTe BIPV windows and compared its sur-face temperature with the single pane. They found the U-factor ofCdTe BIPV is lower than the single glazing from indoor and outdoorexperiments. Based on these results, the concept of PT–PV modality can be potentially applied in developing a multifunctional building skin as an active device, rather than a passive thermal barrier and light-ing source. This concept will pave a new path to next generationdevice-based building skin with structures tailored to a variety ofrequirements in architectural and civic engineering in many areas,such as energy sustainability, environmental control, bio-medical assessment, public health, living comfort, utility efficiencies, and eco- nomic considerations. VI. FUTURE PERSPECTIVES AND CONCLUSIONS Sustainability is defined by the United Nations as “meeting the needs of the present without compromising the ability of future gener-ations to meet their own needs.” With advances in physics and materi-als science, it is possible to generate inexhaustible energy for meetingenergy consumption forever. Presently, the renewability of energy relies on efficient harvesting of natural energy from sunlight, wind, rain, tides, waves, and geothermal heat. 129Since solar energy is clean, sustainable, and plentiful, harvesting solar light has been a majorapproach in generating renewable energy for various applications suchas generation of electricity, solar distillation, solar green houses, andsolar heating of buildings. 130–133In all solar-related technologies, novelmaterials have played key roles in providing all required properties for solar harvesting, conversion, and energy generation. Currently, the research is extensively focusing on developing advanced materials with new properties that enable multifunctions in solar harvesting and energy generation. For significant solar harvesting, large-areapanels are typically required, making building skins potentially viable substrates for both PT and PV panels. Both building- integrated photovoltaics (BIPV) and building-integrated photo- thermal (BIPT) systems are highly possible with the PT- and PV-dual modality designs. Advanced thin film technologies willenable both BIPV and BIPT to perform synergistically at the satis- factory level for sufficient energy saving and generation, seasonably altered and optimized. Recently, a study on the incident light angle dependence of the photothermal effect on several PT films has shown significance of the fac ¸ade orientation and inclination angle for successful integration of PV and PT technologies. 18 Although the silicon-based PV cells have achieved high conver- sion efficiency, there have been cost and environmental concerns in the manufacturing of the solar cells. It has been estimated that the solar energy costs approximately 8–10 times more than primary fuels (/C24$0.38 per kilowatt hour for solar, /C24$0.03 per kilowatt hour for gas).134The single largest cost is the solar panels themselves. Production of silicon PV panels is not “clean” as the by-product silicon tetrachloride can be both occupational and environmentally hazard- ous.135–137Porphyrin compounds exhibit both the PT and PV effects with significant AVT in the thin film form. They are also abundant in nature, environmentally friendly, and can be readily synthesized in laboratory straightforwardly and economically, making them highly desirable for both PV and PT applications. The photothermal effects of nanoparticles have been previously investigated via laser excitations and mainly utilized for medical ther- apy.6,7,39–41,73–78Several nanoparticle systems are well-known for pro- nounced photothermal effects, such as gold and graphene. These nanoparticles, even in extremely small quantities of milligrams in solu- tions, all exhibit strong photothermal effects, effectively raising the solution temperature by greater than 30/C14C under laser excitations. However, most of the photothermal materials have not been experi- mentally investigated in the thin film forms for energy applications. Furthermore, few studies have been carried out on the photothermaleffects of the metallic materials under white light excitations. Characteristically, most of the metallic materials are non-transparent with strong absorptions in the visible region. Porphyrin and its deriva- tives meet the transparency requirement by exhibiting selective absorptions only in the ultraviolet (Soret bands) and visible (Q bands) regions. The absorption spectra of porphyrins can be further selec- tively enhanced by modifying the conjugation and symmetry, substitu-ents, substituted positions, and ligands. With these unique characteristics, new strategies have been proposed and experimentally carried out to enhance the light harvesting ability in DSSCs, such as push–pull porphyrins by utilizing porphyrin dimers, or applying mul- tiple dyes. Some of these studies have shown a broadened spectrumwith red-shifted via push–pull porphyrins, resulting in an amplified molar extinction coefficient. Therefore, combinations of specific por- phyrins can be optimized for enhanced solar harvesting and utilized for PT and PV dual modality designs. The current research outcomes on the porphyrin compounds indicate their tremendous potentials in energy applications. The unique properties of porphyrin compounds FIG. 18. Schematic diagram of building integrated photovoltaic (BIPV).Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 011302 (2021); doi: 10.1063/5.0036961 8, 011302-15can be further utilized via large surface area of building skins for solar energy harvesting, conversion, and generation for next generation energy free systems. ACKNOWLEDGMENTS We acknowledge the financial support from the National Science Foundation Grants CMMI-1635089 and CMMI-1953009. 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10.0003526.pdf

Low Temp. Phys. 47, 250 (2021); https://doi.org/10.1063/10.0003526 47, 250 © 2021 Author(s).Features of kinetic and regulatory processes in biosystems Cite as: Low Temp. Phys. 47, 250 (2021); https://doi.org/10.1063/10.0003526 Submitted: 25 January 2021 . Published Online: 25 March 2021 L. N. Christophorov , V. I. Teslenko , and E. G. Petrov ARTICLES YOU MAY BE INTERESTED IN Thermodynamic uncertainty relation to assess biological processes The Journal of Chemical Physics 154, 130901 (2021); https://doi.org/10.1063/5.0043671 Schrödinger's equation as a diffusion equation American Journal of Physics 89, 500 (2021); https://doi.org/10.1119/10.0002765 Biomolecules and their complexes with nanostructures Low Temperature Physics 47, 179 (2021); https://doi.org/10.1063/10.0003517Features of kinetic and regulatory processes in biosystems Cite as: Fiz. Nizk. Temp. 47,2 7 3 –291 (March 2021); doi: 10.1063/10.0003526 View Online Export Citation CrossMar k Submitted: 25 January 2021 L. N. Christophorov, V. I. Teslenko, and E. G. Petrova) AFFILIATIONS Bogolyubov Institute for Theoretical Physics of the National Academy of Sciences of Ukraine, Kiev 03680, Ukraine a)Author to whom correspondence should be addressed: epetrov@bitp.kiev.ua ABSTRACT A feature of biological systems is their high structural heterogeneity. This is manifested in the fact that the processes observed at the nano- scopic level are noticeably multistage in time. The paper expounds an approach that allows, basing on the methods of nonequilibrium statis-tical mechanics, to obtain kinetic equations that enable describing the evolution of slow processes occurring against the background of faster ones. Vibrational relaxation in electronic terms and stochastic deviations of the position of the electronic energy levels of the system from their stationary positions are considered the most important fast processes. As an example, it is shown how the kinetics of one- and two-electron transfer through protein chains, the oxygen-mediated transfer of a triplet excitation in the pigment-protein complex, the kinetics oftemperature-independent desensitization of pain receptors, as well as conformational regulation of enzymatic reactions, can be described. Published under license by AIP Publishing. https://doi.org/10.1063/10.0003526 1. INTRODUCTION Biological systems manifest themselves as molecular devices that, on a nanometer scale, perform various types of physicochemi-cal processes, such as enzymatic reactions, conversion of chemical, mechanical, thermal and light energy into each other, synthesis of substances, etc. The use of physical mechanisms that control these (and other) processes is one of the most important areas of nano-technology. The main problem is that, although the basic physical mechanisms governing transitions in condensed matter and small inorganic and organic structures are well understood, the samecannot be said for biosystems. The reason lies in the significant het-erogeneity of these systems, which in the same nanoscale volumecan contain membranes, globules, chains, ligands, and other frag- ments. Each of the fragments or a group of fragments, as a rule, is responsible for the performance of its specific functions, includingthe transfer of energy and charges, conformation transformations,redox reactions, synthesis, etc. Heterogeneity leads to a direct or indirect relationships between the structural and functional charac- teristics of a biosystem. As a result, the characteristic time course of a particular process depends significantly on the indicated relation-ships. Thus, the system has a hierarchy of characteristic times dueto various types of interactions. This hierarchy allows the use of themethod of coarse-grained description, which makes it possible to reduce the number of parameters for describing a specific non- equilibrium process. The coarse-grained description methodcorresponds to Bogolyubov ’s concept of a hierarchy of relaxation times in both classical and quantum systems. 1,2 This article shows how the physical processes in biosystems can be described using averaged (coarse-grained) kinetic anddynamic equations. The general approach is illustrated by examples concerning the transport of electrons and excitations, as well as the dynamics of transient conformational processes. 2. COARSE-GRAINED KINETIC EQUATIONS In statistical mechanics, the average value of the physical quantity Ois defined as /C22O(t)¼tr^Oρ(t), where ^Ois its operator, andρ(t) is the nonequilibrium density operator (matrix) of the system. 1–4The trace (tr) operation assumes summation over all sta- tionary states jaiof the system. The time evolution of the density operator obeys the equation @ρ(t) @t¼/C0iL(t)ρ(t): (1) Here,L(t)¼(1//C22h)[H(t),...] is the Liouville superoperator of the system. Note that in Eq. (1)operators are defined by their matrix elements, ρa0a(t)¼(a0jρ(t)jaiand Ha0a(t)¼a0jhH(t)jai,s o ρ(t)¼Σaa0ρa0a(t)ja0iahjandH(t)¼Σaa0Ha0a(t)ja0iahj. Summation is carried out over all quantum numbers acorresponding to the degrees of freedom of the system under consideration.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,000000 (2021); doi: 10.1063/10.0003526 47,000000-250 Published under license by AIP Publishing.As applied to biosystems, it should be noted that despite the complexity of its structural units, the hierarchy of characteristic times of development of each process, as well as its localization in afunctional structural unit, makes it possible to physically describethe temporal evolution of the process. For this, a model is consid-ered in which a functional unit of the biosystem is considered as a system with a finite number of “working ”degrees of freedom, and the environment is a macroscopic condensed medium with manydegrees of freedom. As a result, the processes occurring in thesystem practically do not change the state of the environment,while the environment can significantly affect the transitions in the system. 5–8Thus, the system under consideration is an open system in which the temporal behavior of its characteristics is determinedby both the internal dynamic processes and contact with the envi-ronment, Fig. 1 . This contact leads to an exchange of particles and energy, as well as to shifts of energy levels of the system, including stochastic shifts. The basic Eq. (1)is applied to the whole structure “system + environment ”, the Hamiltonian of which H(t) depends on what physical models are used for the system (functional unit of the bio-system) and the environment (surrounding structure or/and solution). 2.1. Markov approximation In non-magnetic structures, an energy exchange between the system and the environment in most cases occurs through vibration quanta (phonons). This allows us to use the harmonic approxima- tion and thus simulate the environment as a reservoir of phononswith a set of frequencies { ω λ}. The establishment of the equilibrium distribution between the vibrational modes occurs in a rather short timeτvib/difference10/C012s. Therefore, on the time scale Δt/C29τvib, the phonons are in thermodynamic equilibrium and, thus, the environment is consid-ered as the heat bath (B). As a result, the Hamiltonian of the vibra- tional energy of the environment can be represented in the standard form H B¼X λ/C22hωλ(bþ λbλþ1/2), (2) where bþ λbλis the operator of creation (annihilation) of a phonon of the λth mode. The interaction of the vibrational states of the environment with the vibrational states of theelectronic terms of the system leads to the fact that during the characteristic time τ m∼τvibthe population Pmv(t) of the vth vibrational level of the mth electronic term becomes quasi-equilibrium.9This means that the ratio between the partial occupancies, Pmv(t)/Pmv0(t)¼exp[/C0/C22hωm(v/C0v0)/kBT], is the time- independent quantity ( kBand Tare Boltzmann ’s constant and absolute temperature, respectively). This means that the population of the vth level Pmv(t)=WmvPm(t) depends only on the equilib- rium probability Wmv¼exp(/C0/C22hωmv/kBT)/X vexp(/C0/C22hωmv/kBT), and the integral occupancy of the mth term, Pm(t)¼P vPmv(t). The temporal evolution of the latter occurs on the time scale Δt/difference τtrthat strongly exceeds the τvib. Thus, if the condition τtr/C29τvib (3) is satisfied, then a coarse-grained description related to the time scale Δt/differenceτtrcan be used. This allows us to restrict ourselves to analyzing the temporal behavior of integral state occupancies Pm(t) only. In addition, in this case, harmonic vibrations of the system(intrasystem phonons) are in thermodynamic equilibrium and,therefore, can also be attributed to the phonon reservoir. Accordingly, it is assumed that the vibrational modes with frequen- ciesω mbelonging to the mth electronic term can be included in theλ-modes of the bath Hamiltonian (4). In other words, the number of the vibrational level v= 0,1, 2, …in the electronic term exactly corresponds to the number of phonons nλof the λth mode associated with the term. The equilibrium density matrix of the phonon bath has the form ρB¼e/C0HB/kBT/trBe/C0HB/kBT: (4) The symbol tr Bdenotes the summation over all vibration states nλ=0 , 1 , 2 , …, which determine the multimode phonon states Πλjnλiof the heat bath. The average number of phonons with energy E λ¼/C22hωλis given by the Bose-Einstein distribution FIG. 1. Scheme shows that the investigated small system is open to the influ- ence of various types of fields, including stochastic ones. The main dissipative and relaxation processes in the system occur due to the exchange of vibrationswith the phonon reservoir (heat bath).Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,000000 (2021); doi: 10.1063/10.0003526 47,000000-251 Published under license by AIP Publishing.function /C22nλ¼trB(bþ λbλρB)¼[e/C22hωλ/kBT/C01]/C01: (5) The total Hamiltonian of the “system + environment ”can be represented in the form H(t)¼H0þΔH(t)þHSB, (6) where the time-independent part, H0¼HSþHB, (7) contains the component Hs¼P mEmjmimhj with Embeing the energy of the mth electronic term of the system in an electron-conformation state jmi. The second component is the bath Hamiltonian (2), which contains the vibration modes of the environment and the system. The eigenvalues and eigenstates of theH0are, respectively, EmþP λ/C22hω(nλþ1/2) and jmijnλi, where jnλi¼(nλ!)/C01/2(bþ λ)nλj0iis the bath state with nλ¼0, 1, 2, ..., the number of phonons ( j0i) denotes the phononless state, that is, the phonon vacuum). The contribution ΔΗ(t) is the result of the action of regular or stochastic fields, and HSBis the Hamiltonian of the interaction between the electronic and nuclear degrees offreedoms. Nucleus displacements have two dynamic effects. The first of them is the polaron shift, which reduces the electronic energy E (0) m of a system (found in the adiabatic approximation at a certain equilibrium position of the nuclei) to Em¼E(0) m/C0X λ(g(λ) m)2/C22hωλ: (8) Here, g(λ) m¼χ(λ) m//C22hωλis the displacement parameter, which is pro- portional to the constant of “transverse ”coupling χ(λ) mbetween the system and the environment (such a coupling does not lead tointrasystem transitions but only shifts its energy levels). The seconddynamic effect is the formation of the phonon assisted intrasystemtransitions accompanied by the creation/annihilation of the phonons. In the systems where transitions occur between nonadia- batic terms, the transition operator can be represented as 10,11 HSB¼Htr¼X m,m0(=m)Vm0meσmm0jm0imj,h (9) where Vm0mis the electronic transition matrix element. As a rule, the Hamiltonian (9)is used to describe the energy and charge transport in structures containing spaced centers of localization ofcharges or excitons. The transitions between adiabatic terms areassociated with the nonadia-batic operator 12 HSB¼Hnonad¼X m,m0(=m)X λχ(λ) m0meσmm0(bλ/C0bþ λ)jm0imj,h (10) where χ(λ) mm0, is the electronic nonadia-batic parameter. Hamiltonian (10) is most often used to describe the intersystem crossing. Both Hamiltonians contain the operator σmm0¼P λg(λ) mm0(bλ/C0bþ λ)which depends on the difference g(λ) mm0¼g(λ) m/C0g(λ) m0in the displace- ment of nuclei in the electronic states jmiandjm0i. The most complete information on the behavior of the system in time is reflected in the probability of occupation of the mth state of the system (state occupancy), Pm(t)¼mhjρs(t)jmi, where ρs(t)¼trBρ(t) is the density matrix of the system, and ρ(t) is the density matrix of the “system + environment, ”which evolves in accordance with the Liouville Eq. (1). Assuming HSBas a perturba- tion, in accordance with the approach based on the nonequilibriumdensity matrix method, 3,4,13,14we obtain the following integro- differential master equation for the required state populations: Pm(t)¼X m0ðt 0dτ[Gmm0(τ)Pm(t/C0τ)/C0Gm0m(τ)Pm0(t/C0τ)]:(11) A time behavior of the kernel Gmm0(τ)¼2 /C22h2Re[e/C0iΩmm0τ(τ)Λmm0(τ)Fmm0(τ)] (12) is controlled by the stationary transition frequency Ωmm0¼ (1//C22h)(Em/C0Em0) as well as the factors Λmm0(τ) and Fmm0(τ). The form of Λmm0(τ) is specified by the type of transitions caused by the system-bath interaction HSB. For example, in the case of transitions between nonadiabatic terms we have Λmm0(τ)¼jVm0mj2Y λX1 lλ¼/C01Φ(λ,lλ) mm0eilλωλt, (13) where quantity Φ(λ,lλ) mm0¼e/C0D(λ) mm0Ijlλj(z(λ) mm0)/C22nλþ1 /C22nλ/C20/C21lλ 2 (14) reflects the contribution of the processes carried out with participa- tion of lλphonons. The modified Bessel function Ijlj(z) and Debye-Waller factor D(λ) mm0¼(g(λ) mm0)2(2/C22nλþ1) depend on the argu- ment z(λ) mm0¼2(g(λ) mm0)2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi/C22n(/C22nþ1)p, which is thermally controlled. It is important to note that for lλ> 0 the function Φ(λ,lλ) mm0does not dis- appear at low temperatures. In fact, if /C22nλ!0, then according to the asymptotic of Ijlj(z)/C25(z/2)jlj(1/jlj!), which is valid for the Bessel function at z/C281, we obtain (note lλ=0): Φ(λ,lλ) mm0/C25exp[/C0(g(λ) mm0)2](g(λ) mm0)2jlλj/jlλj! /C2[/C22nλþ1Θ(lλ)þ/C22nλΘ(/C0lλ)]: (15) Ifmandm ’are the molecular terms with the corresponding vibra- tional levels, then the relaxation processes responsible for the estab- lishment of the equilibrium distribution among the vibrational levels of the terms are concentrated in the factor Fmm0(τ)¼e/C0γτ vib, (16) where γvib¼(τ/C01 mþτ/C01 m0) is the characteristic rate of the vibrational relaxation within the terms mandm ’.9Asγvib/differenceτ/C01 vib, the quantityLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,000000 (2021); doi: 10.1063/10.0003526 47,000000-252 Published under license by AIP Publishing.Fmm0(τ) decreases exponentially on the time scale Δτ∼τvib. The characteristic time τvibis much shorter than the time scale Δt∼τtr, on which the temporal evolution of occupancies takes place.Therefore, in Eq. (11), we can extend the upper limit of integration to infinity and neglect the non-markovity settingP m0(t/C0τ)/C25Pm0(t). This reflects the Markov nature of the trans- port process, in which the temporal behavior of the integral popu- lations is governed by the balance kinetic equations _Pm(t)¼/C0X m0[Kmm0Pm(t)/C0Km0mPm0(t)]: (17) The corresponding transition rate constants have the form Kmm0¼/C0ð1 0dτGmm0(τ), (18) where the integrand is determined by the Eq. (12). If the transition occurs between the non-adiabatic electronic terms, then using Eqs. (13) and(14) gives Kmm0¼2π /C22hjVm0mj2(FC)mm0: (19) The effect of phonons on the transition efficiency is concen- trated in the Franck-Condon factor,14–16which reads (FC)mm0¼1 /C22hY λX1 lλ¼/C01Φ(λ,lλ) mm0δΩ mm0/C0X λlλωλ ! : (20) In(20), the form of the FC factor is represented in the most frequently used version when Lorentzian L(Ω)¼(1/π)[γ/(Ω2þ γ2)] is replaced by delta-function δ(Ω). This simplification is due to the fact that the summation over lλin the interval [ /C01,þ1]m a y be replaced by the integration over the same infinite interval. 2.2. Stochastic influence In addition to the dynamic effects, one should also take into account the stochastic effect of the environment on the system. The stochastic movements of the structural groups of the environmentcreate time-dependent random fields, 8,11,17–21which can be caused by fluctuations of the energy of separate environmental groups.22 These random fields affect the system in such a way that its energy levels Emexperience stochastic displacements ΔE(t). Consequently, the Hamiltonian HSof the system acquires stochastic addition ΔH(t)¼P mΔEm(t)jmimhj, which is the time-dependent contri- bution to the overall Hamiltonian (7). In the model under consid- eration, the fluctuations in the environment lead to a stochasticchange in the number of phonons of each of the λth mode. Therefore, the mean vibration energy of the bath, /C22E, and the corre- sponding mean square energy fluctuation, δE2¼(E/C0/C22E)2¼kBT2(@/C22E/@T),23read /C22E¼X λ/C22Eλ, /C22Eλ¼/C22hωλ(/C22nλþ1/2), : (21)and δE2¼X λδE2 λ,δE2 λ¼(/C22hωλ)2[/C22nλ(/C22nλþ1)], (22) respectively. If we take into account that during vibrations of the structural groups of the environment, a part of the vibrationalenergy is transferred to the system, then the position of energylevels of the system becomes stochastic, E m(t)=Em+ΔEm(t). As a result, instead of regular factor Fmm0(τ)¼/C0iÐτ 0dτ0ΔΩ mm0(τ0) appears, which is a functional of the stochastic value ΔΩ mm0(τ)¼[(ΔEm(τ)/C0ΔEm0(τ))]//C22h.As a result, the integral occu- pancies are also stochastic occupancies P(m,t). Now, to find the kinetic master equation for the observed occupancies Pm(t), the original Liouville Eq. (1)must be averaged over the realizations of the random variable ΔEmm0(τ). In biosystems, a situation is often realized when transient pro- cesses occur at characteristic times τtr, which significantly exceed not only the characteristic times τvibof the establishment of equilib- rium vibrations, but also the characteristic times τstoch of stochastic changes associated with the movement of structure units of the environment. In particular, stochastic shifts occur over a widerange of characteristic times τ stoch of the order of 10−8−10−10s.24,25In this case, a coarse-grained description of the kinetics becomes possible. As shown in Fig. 2 , the behavior of the population P(m,t) on the time-scale Δt/differenceτstoch is random, while on the time-scale Δt/differenceτtrits average value Pm(t) reflects smoothed evolution over time. In our case, the time τtris related to the transi- tion rates caused by system-bath interaction HSB, and, thus, the time Δt/differenceτstoch characterizes stochastic variations of the factor Fmm0(τ). Therefore, according to the averaging proce- dure8,11,17,21,22,26and by virtue of the condition τtr/C29τstoch ; (23) we again come to Eqs. (11)and(12),w h e r en o w Pm(t)¼P(m,t) hihi , Pm0(t/C0τ)¼P(m0,t/C0τ) hihi and FIG. 2. Random deviations of the population P(m,t) of the mth state of the system from its average value Pm(t)¼P(m,t) hihi is due to the action of stochastic fields (discrete dichotomous and trichotomous deviations are shown). If condition (23) is satisfied, a smoothed (coarse-grained) description can be carried out on the time scale Δt/differenceτtr.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,000000 (2021); doi: 10.1063/10.0003526 47,000000-253 Published under license by AIP Publishing.Fmm0(τ)¼/C0iðτ 0dτ0ΔΩ mm0(τ0)*+*+ (24) are the averaged quantities. The basic principles of averaging stochastic functionals27–31 indicates that to calculate the Fmm0,(τ), it is necessary to know the mean amplitudes σjofΩmm0(τ) realizations and the stationary probabilities Wjof these realizations. In the important special case of the dichotomous random process, the quantity Ωmm0(τ) has only two realizations, σ1andσ2, which are executed with probabilities w1¼ν2/(ν1þν2) and w2¼ν1/(ν1þν2), respectively. Then, fol- lowing the exact results,8,26we obtain Fmm0(τ)¼sþesτ/C0sesþτ sþ/C0se/C0iΩ(av)τ, (25) where Ω(av)¼w1σ1þw2σ2is the average value of the Ωmm0(τ). The main temporal behavior of the factor Fmm0(τ) is specified by the quantities s+¼/C0(1/2)[ vc+rcosf/C0i(ΔwΔσ+rsinf)], (26) where vc¼(ν1þν2)/2, Δσ¼σ1/C0σ2, Δw¼w1/C0w2, r¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (v2 c/C0Δσ2)2þ4(ΔwΔσ)2q , and tan(2 f)¼2ΔwΔσ/(ν2 c/C0Δσ2): A fundamentally important role of the stochastic shift of the energy levels of the system lies in the generation of the dampingfactors γ stoch= (1/2)( νc±rcosw), contained in the quantities s+ and s−. The presence of exponential decay on the time scale Δτ/differenceτstoch¼γ/C01 stochshows that if the kinetic process is described by Eq.(11) and develops on a time scale Δτ/differenceτtrsatisfying inequality (23), then we can set P(m0,t/C0τ)/C25Pm0(t) and, as in the case of fast vibrational relaxation, the upper limit of integration can beextended to infinity. Thus, as a result of averaging, we come againto the balance-like kinetic equations (17). In these equations, the average rates of the m!m 0transition in a system interacting with a stochastic environment have the form (19) where, however, the integrand Gmm0(τ), Eq. (12), contains the stochastically averaged factor (24). In the case of a dichotomous fluctuations, the latter is given by the Eq. (25). 2.3. Thermodynamic fluctuations Specific reasons for the formation of the amplitudes σjof the stochastic variable ΔΩ mm0(τ)¼ΔEmm0(τ)//C22hcan be different. If the deviation of the environment energy is explained by the deviationof vibrational quanta from their mean values /C22n λ, then it can be assumed that the environment is in thermodynamic equilibrium and the vibrations of its structural groups are close to harmonic.Therefore, the stochastic addition δE mto the energy Emcan be caused by the fluctuation shiftP λ/C22hωλδnλin the vibrational energy of the environment. Since energy is conserved in each act of energy exchange between the environment (phonon reservoir) and thesystem, the condition EmþδEmþX λ/C22hωλ(nλþδnλ)¼Em0þδEm0þX λ/C22hωλ(n0 λþδn0 λ) (27) is satisfied even in the presence of random deviations of δnλ¼nλ/C0/C22nλandδn0 λ¼n0 λ/C0n0 λ. The phonon assisted transition m!m0occurs at a fixed energy difference Em/C0Em0 ¼P λ/C22hωλ(n0 λ/C0nλ).Hence, according to the Eq. (27), realiza- tion of the random energy deviations ΔEmm0(τ) occurs via sto- chastic values δEmm0¼δEm/C0δEm0¼¼P λ/C22hωλ(δn0 λ/C0δnλ). For noninteracting phonons, the averaging gives δn0λ¼δnλ¼0a n d δn0 λ,δnλ≃δn2 λδn0 λ,nλ. Then, according to the Eqs. (21) and (22) we get δEmm0¼X λδEλ¼0, (δEmm0)2¼2X λδE2 λ:(28) The expressions (28) show that the average thermodynamic fluctuations depend on the number of oscillator modes λ involved in the contact with the system. If the mean stochastic life-time τ(λ) stochof realization of each quantity δEλ(and thus δE2 λ) is approximately the same, then with a large number of oscillators the stochastic influence of the environment on them!m 0transitions should manifest itself in the form of white noise, while with a finite number of oscillators the stochastic effect will be discrete. In biosystems, discrete fluctuations are of particular impor- tance, since they relate to well-defined structural units of thesystem. For example, the study of protein fluorescence has showedthat the stochastic behavior of the fluorescence intensity reflects the fluctuation dynamics of proteins, and this dynamics is rather well manifested in the form of a dichotomous process covering thenanosecond and microsecond regions. 32Therefore, we will consider the role of such dichotomous fluctuations assuming that among the possible oscillatory modes λ, the most effective is mode λ*o f frequency ω*, which leads to fluctuations of the value (Em/C0Em0)//C22h¼Ωmm0. Hence, according to the results presented in the Eq. (28),w eh a v e δEmm0¼δE*and(δEmm0)2¼2δE2 *. In classi- cal physics, the linear frequency ν=ω/2πcorresponds to the fre- quency of collision of a particle with the walls of the potential well. The collision results in an exchange of energy between the oscilla-tor and the system. In the quantum case, assuming that at each col-lision with the wall the quantum of vibrational energy is acquiredby a system or the oscillator receives a quantum of energy from the system (one-phonon approximation), we can relate ν *¼κ/C22ω*/2π ¼(κ/2π/C22h)/C22E*to the frequency of realization of the one-phonon fluc- tuations (the temperature-independent parameter κ< 1 character- izes the efficiency of energy exchange between the system and the phonon reservoir). Positive and negative mean amplitudes of this fluctuation are σ+¼+σ*where σ*¼[2δE2 *]1/2//C22h. Hence, the sim- plest case of thermodinamic fluctuations of the vibrational energy of the environment leads to dichotomous stochastic displacementsLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,000000 (2021); doi: 10.1063/10.0003526 47,000000-254 Published under license by AIP Publishing.in the transition frequency ΔΩ mm0of the dynamic system. The fre- quency ν* and the amplitude σ*of these symmetric displacements are determined as (see also Refs. 11and21) v*¼κ(ω*/4π)(2/C22nþ1), σ*¼ω*ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2/C22n*(1þ/C22n*)p :(29) To be consistent with a standard symmetric dichotomous process, the escape frequencies and the amplitudes of the random deviationsmust be ν 1=v2=vc=ν*andσ1¼/C0σ2¼σ*, respectively. Then, based on the exact results, Eqs. (25) and (26), we obtain Ωav¼ ΔΩ mm0(τ) hihi ¼ 0 and s1,2¼/C0(1/2)[ v*+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2 */C04σ2 *p ]. When the classical limit ( /C22n*/C291) is satisfied, then Fmm0(τ)¼e/C0γτ stochcosσ*τ, γstoch≃κ(kBT/4π/C22h):(30) At the low temperature limit ( /C22n*/C28(κ2/8π2)/C281) we get Fmm0(τ)¼e/C0γτ stoch, γstoch≃(8πω*/κ)e/C0/C22hω*/kBT:(31) Averaged kinetic equations (20) with the transition rates deter- mined by the Eqs. (19),(12)–(14), and (24) are used below as a basis for a coarse-grained description of kinetic processes on a time scale Δt/differenceτtr[for example. condition (23)]. 3. RESULTS AND DISCUSSION This section shows how the above approach to describing the time behavior of the populations of the states of an open system isused to analyze kinetic and dynamic processes in various types ofbiosystems. To do this, in each specific case, a model is used that defines the states involved in the transitions and the interactions responsible for the transitions. As follows from Eqs. (12) and(19), the features of the temporal evolution of the state occupancies ofthe system are determined by the factors Λ mm’and Fmm’, which reflect the interactions between the electronic and nuclear degrees of freedom. 3.1. Single-electron transfer through protein chains There are many different types of charge transfer processes responsible for redox reactions in biosystems. Among them, aspecial place is occupied by reactions of one- and two-electron transfer between spaced redox centers. In many cases, such reac- tions are mediated by bridging structures (B) that connect thedonor (D) and acceptor (A) centers. The role of D and A centers isattributed to the redox groups, while the D –A connection is often carried out through protein chains 33–35or DNA.36,37To under- stand the physical mechanism of the formation of electron transfer between spaced redox centers, in the late 70s, a donor-acceptormodel of electron transfer through a protein chain was proposed,where the peptide groups of the protein chain played the role of a bridge for electron transfer. 38–41The model made it possible to explain the exponential drop in donor-acceptor electron transferrate kDAwith an increase in the number of repeating bridge units N. This indicates the presence of a superexchange interaction between redox centers and, thus, explains the long-range electrontunneling through the protein structure. Further improvement ofthe superexchange model allowed one to find the conditions for itsapplicability for various molecular DBA systems 42–46and to show that in the limiting cases the modified superexchange model covers the limiting case of deep tunneling and mimics tunneling througha rectangular barrier, thus making it possible to determine theeffective mass of a tunneling electron, as well as the height andlength of the barrier. 39,47If the energy gap between the highest energies of the donor and the lowest energy of bridging groups allows the temperature activated transfer of an electron from thedonor to the bridge, the transfer process in the DBA systembecomes noticeably more complicated. This is reflected in themixing of the superexchange and hopping transfer mechanisms. 40,41,48–50In the case of nonadiabatic D-A electron transport, the kinetics is described by the system of equations (17) with the hopping rates (19). The electronic states of the DBA system jmiare characterized by the presence of a transferred elec- tron on their structure units, i.e., m=D,A,B1,B2,…,Bnwhere D;D/C0BA,A¼DBA/C0,Bm;DB/C0 mA(symbols B;B1B2...BN and Bm;B1B2...B/C0 m...BNdenote a bridge chain without an extra electron and with the transferred electron located on the mth bridge unit, respectively). The normalization condition under which the system of N+ 2 kinetic equations (17) is solved has the form PD(t)þPB(t)þPA(t)¼1: (32) Therefore, if the integral population of the bridging units by the transferred electron is insignificant, i.e., PB(t)¼XN m¼1Pm(t)/C281, (33) and, therefore, PD(t)þPA(t)≃1, (34) then an electron transfer in the DBA system looks like a oxidation- reduction reaction only between D and A centers. In this case, theNbridging states jB miwith energies Emplay a virtual role in elec- tron transfer. The condition (33) is fundamentally important when obtaining analytical expressions for the bridge mediated electron-transfer rates.40,49,50One can show41how the (N+ 1)-exponential kinetics reduces to one-exponential kinetics, describing the time evolution only for PD(t) and PA(t). Such a reduction corresponds to a coarse-grained description of the of D−A electron transfer, which is characterized by the forward and backward D −A rates, kDAand kAD, respectively ( Fig. 3 ). WithinLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,000000 (2021); doi: 10.1063/10.0003526 47,000000-255 Published under license by AIP Publishing.the framework of a coarse-grained description, we get PD(t)≃1 KET[kbþkfe/C0kETt], PA(t)≃kf kET[1/C0e/C0kETt],(35) where the overall transfer rate kET¼kfþkb, (36) determines the time-scale Δt/differenceτtr¼k/C01 ETof the electron-transfer process. The forward rate kf¼k(hop) fþk( sup ) f(37) (and the backward rate kb¼kfe/C0(EA/C0EB)/kBT) contains two contribu- tions. One of them, k(hop) f¼k/C01k2 k1þk2/C18/C191 1þξ(N/C01), (38) contains the parameter ξ¼k1k2 k(k1þk2), (39) which characterizes the decrease in the electron transfer along the sequential pathway D !B1!B2...BN!Acarried out using the hopping rates k1=k1D,k−1=kD1,k2=kAN, and k¼kmm+1,(m/C01, 2, ...,N)(Fig. 3 ). The form of these rates is given by the Eq.(19). The second contribution, k(sup) f¼k3¼k( sup ) 0e/C0ζ(N/C01)(40) reflects the direct D !A pathway associated with a one-step hopping of an electron between the D and A centers. The parameter ζ¼/C02InjVBj ΔEDΔEB/C18/C19 (41) characterizes the decrease in the contribution associated with the superexchange D –A coupling. In the Eq. (41),VB=V mm±1 is the electron coupling between the nearest bridge units, and ΔED(A)=EB −ED(A)is the energy gap between the bottoms of electronic terms related to the bridge unit and the D(A) center. Figure 3 shows a good correspondence of the theory with experimental results onelectron transfer from D(OsII) to A(RuIII) through a biopolymer composed of proline units. 3.2. Two-electron bridge-mediated transfer Two-electron transport (TET) is a basic physical process that is responsible for redox reactions catalyzed by metalloenzymes. Inmost cases, TET is accompanied by the capture or release of one or more protons. For example, a hydride (:H −) transport is accompa- nied by the correlated transport of two electrons and a proton. Oneof the fundamental problems of multielectron transport is the eluci-dation of the mechanisms of correlated delivery of electrons fromdonor to acceptor groups during sequential and concert electron transfer. 51Following the approach presented in Ref. 52, as well as the above results concerning a coarse-grained description ofone-electron transfer, we will show how both mechanisms can berealized in a nanomolecular complex, where the transfer of two electrons from a donor to an acceptor occurs with the participation of a bridge structure. The results of the theory are used to explainthe experimental data on the reduction of mycothione reductase(MycR) with the oxidizing agent nicotinamide adenine dinucleo-tide phosphate (NADP). A model is used where the donor and acceptor centers can have one or two extra electrons, while the bridge has only one. Thisis possible if the D and A centers have polar groups, and the bridg-ing units are in a hydrophobic environment (this situation is quitetypical for a biosystem). As a result, two-electron transfer can be realized using two repeating one-electron routes. During the first route, one of the two extra electrons occupying the D center istransferred to the A center along the sequential pathway D 2/C0BA! D/C0B/C0 1A!D/C0B/C0 2A!...!D/C0B/C0 NA!D/C0BA/C0with forward (kD1,α,kNI) and backward ( k1D,β,kAN) rates as well as a one-step (superexchange) pathway D2/C0BA!D/C0BA/C0with forward ( kDI) and backward ( kID) rates. The first route forms an intermediate charging structure I;D/C0BA/C0, which switch on the second one-electron route. The latter includes the sequential pathway D/C0BA/C0!DB/C0 1A/C0!DB/C0 2A/C0!...!DB/C0 NA/C0!DBA2/C0with forward ( kI1,α,kNA) and backward ( ku,β,kAN) rates as well as a FIG. 3. The dependence of the overall rate KETof electron transfer from OsII n-donor to RuIII-acceptor (adapted from Ref. 41) occurs according to a mixed mechanism, including direct tunneling (rates k 3and k −3) and sequential hop- pings with the participation of Nunits of the bridge B (proline chain). For the considered DBA system, the probability of finding a transported electron on the bridge is insignificant and therefore the process is characterized by directone-step jumps with rates kfand k b.Details in the text.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,000000 (2021); doi: 10.1063/10.0003526 47,000000-256 Published under license by AIP Publishing.one-step (superexchange) pathway D/C0BA/C0!DBA2/C0with forward (kIA) and backward ( kAI) rates. Schemes of both routes are shown inFig. 4(a) , the rates kmm,are determined by the Eq. (19), (more detail form of the rates is given in Ref. 52). When the bridge states act as an intermediate and thus the condition (33) is satisfied, then the D(D2/C0BA)!I(D/C0BA/C0) and I(D/C0BA/C0)!A(DBA2/C0) transitions related to the routs j= 1 and j= 2, respectively, are controlled by route rates [ Fig. 4(b) ] k(j) f¼k(f,seq) j0 1þξj1/C0rN/C01 1/C0rþk(f, sup ) j0 e/C0ζj(N/C01), k(j) b¼k(j) fe/C0ΔEj/kBT,(r¼β/α):(42) They include the contributions caused by the one-electron sequen- tial and superexchange mechanisms of an electron transfer.Corresponding dependence on the number of repeating bridge units is characterized by the quantities ξj¼k(j) 1D(k(j) NA/C0α(1/C0r)) α(k(j) NAþk(j) 1D)(43) and ζj¼/C02l njVBj (ΔE(j) DΔE(j) A)1/2"# , (44) where the factors k(f,seq) j0¼k(j) D1k(j) NA k(j) NAþk(j) 1D, k(j) D1(NA)¼2π /C22hjV(j) 1D(AN)j2FC1D(j)(A(j)N),(45) and k(f, sup ) j0¼2π /C22hjV(j) 1DV(j) ANj2 ΔE(j) DΔE(j) AFCD(j)A(j) (46) are respectively the hopping and super-exchange one-electron transfer rates for the bridge with a single bridging unit ( N= 1). In Eqs. (43)–(46) the gaps ΔE(1) D¼E(D/C0B/C0A)/C0E(D2/C0BA), ΔE(1) A¼E(D/C0B/C0A)/C0E(D/C0BA/C0) and ΔED(2)¼E(DB/C0A/C0)/C0E(D/C0BA/C0), ΔEA(2)¼E(DB/C0A/C0)/C0E(DBA2/C0) specify the energy distances between the electronic working states of the entire DBA system, The route rates determine the coarse-grained kinetics on the time scale Δt/differenceτDIA/differenceK/C01 1,K/C01 2. The overall TET rates K1,2¼1 2(c1þd1+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (c1/C0d1)2þ4c2d2)q , (47) where c1;k(2) fþk(2) bþkAD,c2;kDA/C0k(2) f, d1;k(1) fþk(1) bþkDA,d2;kAD/C0k(2) b,(48) determine the kinetic process covering the three states jji,(j¼D, I, A), whose integral occupancies are related by the FIG. 4. Scheme (adapted from Ref. 52) of the tunneling and sequential path- ways of two electrons from the donor to the acceptor through the bridge with the formation of an intermediate state I (only one of the two electrons was trans-ferred). If the probability of populating the bridge with transferred electrons issmall, then, as in the case of one-electron transfer, a coarse-grained description of the process can be made using the rates kfand k b. Now these rates contain contributions from the “concert ”(two-electron coherent) and sequential transport mechanisms.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,000000 (2021); doi: 10.1063/10.0003526 47,000000-257 Published under license by AIP Publishing.normalization condition PD(t)þPI(t)þPA(t)¼1: (49) The corresponding scheme of kinetic processes is shown in Fig. 4(b) . These processes reflect the two-exponential temporal evo- lution of the integral occupancies Pj(t)¼PjþC(1) je/C0K1tþC(2) je/C0K2t(50) to their stationary values PD¼(c1k(1) bþd2k(2) f)/(c1d1/C0c2d2), PA¼(c2k(1) bþd1k(2) f)/(c1d1/C0c2d2)(51) andPI=1−PD−PA. With a weak population of the intermediate state (i.e. at PI(t)/C281), which happens if k(1) f,k(2) b/C28k(2) f,k(1) b, the transfer of two electrons from the D center to the A center mani- fests itself as a one-exponential kinetic process corresponding to the scheme shown in Fig. 4(c) . The temporal behavior of the two-electron occupancies, PD(t)≃1 kTET[kbþkfe/C0kTETt], PA(t)≃kf kTET[1/C0e/C0kTETt],(52) reflects a coarse-grained description on the time-scale Δt/difference τTET¼k/C01 TETwhere kTET¼kfþkbis the overall rate characterizing the two-electron transfer between D and A centers [cf. Figure 4(c) ]. Since kf(b)¼k(step) f(b)þkDA(AD), then kTET¼k(step) TETþk(conc) TET (53) The component k(step) TET¼k(step) fþk(conc) b, k(step) f(b)¼k(1) f(b)k(2) f(b) k(1) bþk(2) f,(54) is originated by the stepwise mechanism where the partial overall rates are determined by Eq. (42). Physically, these rates are identical to the one-electron transfer rate (37). The component k(conc) TET¼kDAþkAD (55) of the overall rate is associated with the concerted mechanism of a two-electron transfer [cf. Figs. 4(a) and4(b)]. The forward (back- ward) coherent rate kDA(AD)¼k(0) DA(AD)e/C0ζTET(N/C01), (56) exhibits an exponential drop with the attenuation coefficientζΤΕΤ=ζ1+ζ2determined by Eq. (44). The preexponential factor k(0) DA(AD)is the two-electron super-exchange transfer rate through a bridge with a single unit. 3.3. Proton-assisted reduction of mycothione reductase The enzyme mycothione reductase (MycR ;E) demonstrates specific electron-transfer pathways during redox reactions.53,54We consider one of them, associated with two-electron reduction ofenzyme by the oxidant nicotinamide adenine dinucleotide phos-phate (NADP). Experimental data shows 55that this reduction is controlled by the protonation/deprotonation of the active site Cys 34−S−S−Cys 39of enzyme (E ;MycR). In the oxidized enzyme Eoxthe−S−S−group binds the amino acids Cys 34and Cys 39, while in twofold reduced enzymes (E redH)−and E2/C0 redthe binding between these amino acids is broken. This transforms the active site into the protonated and deprotonated structures [Cys 34−S−,C y s 39−SH] and [Cys 34−S−,C y s 39−S−], respectively. Transformation of the oxidized enzyme E oxrequires several electron and proton coupled steps. Here, a special role belongs to flavin adenine dinuleotide (FAD), which, having received electrons from NADP, then participates as an electron donor at the final stage of two-electron transfer. Themeasured rate of the formation of the reduced enzyme E rediskred- =KTET≈130 s−1. This indicates that the characteristic time of the two-electron transport under consideration, τTET∼130 s−1, is much greater than the characteristic times of vibrational relaxation ( τvib) and protonation/deprotonation process ( τpr/dp). Let us denote by P(m) s(a)(t) the occupancies of the mth electronic state in the presence of protonated ( s=pr) or deprotonated ( s=dp) group a. Due to the inequality τTET/C29τpr/dp,τvib (57) the temporal evolution of the occupancies P(m) s(a)(t) change on the time scale Δt∼τΤΕΤin the same way as their integral occupancy Pm(t): P(m) s(a)(t)¼f(m) s(a)Pm(t), Pm(t)¼P(m) pr(a)(t)þP(m) dp(a)(t):(58) The weight of the protonation (deprononation) fraction in the mth electronic state is determined by the functions f(m) pr(a)¼1 1þ10pH/C0pK(m) a, f(m) dp(a)¼1 1þ10pK(m) a/C0pH,(59) (we use notations pH = −log10[H+] and pK(m) a¼/C0 log10K(m) a, where [H+] is the concentration of protons in the environment, and K(m) ais a constant characterizing the binding of a proton to the agroup of the system in the mth state). In the example under con- sideration, the first step of two-electron transfer is the binding of NADP to FAD −Eoxand the formation of the charged fraction NADP+−FADH−−Eox, which is affected by the protonation/Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,000000 (2021); doi: 10.1063/10.0003526 47,000000-258 Published under license by AIP Publishing.deprotonation of the group with pK ≃9.1. This group is supposed to be Arg, so the weight of the formed fraction is proportional to f(ox) pr(Arg). Fixing only the main formed fractions of the complex NADP −FAD−E, we can assume that the two-electron reduction of E include the departure of one electron fromNADP +−FADH−−Eox(≡D) and the formation of intermediate fraction NADP+−FAD*−−EH(≡I). After this, the transfer of the second electron occurs with the formation of the final two-foldreduced deprotonated fraction NADP +−FAD−E2/C0 red[≡A(dp)]. This sequential pathway D →I→A is characterized by the rate k(step) TET, Eq.(54). The gap EI−ED(A)between the intermediate and donor (acceptor) states exceeds several kBT. This leads to the inequalities k(1) f/C28k(1) band k(2) f/C29k(2) b. If additionally k(2) f/C29k(1) b, then k(step) TET/C25k(1) f. This rate is proportional of the weight of the proton- ated Arg during the formation of a two-electron donor center inthe complex NADP +−FADH−−Eox. Therefore, k(step) TET/C25f(D) pr(Arg)kD!I: (60) The coherent pathway D →A is associated with a one-step two-electron transfer. The corresponding rate k(conc) TET/C25kDA, Eq.(55) is formed when the amino acid Arg is protonated with weight f(D) pr(Arg)while the amino acid Cys 39is deprotonated with weight f(A) dp(Cys 39). Thus, k(conc) TET/C25f(D) pr(Arg)f(A) dp(Cys 39)kD!A: (61) Rates kD→Iand kD→Aare pH-independent values that can be estimated using quantum-chemical methods. In the semiphenome-nological approach, they are considered as fitting parameters. Figure 5 exhibits a bell-shaped behavior of k red=KTETversus pH, which corresponds to the theoretical expression (61) showing a concert (synchronic) mechanism of two-electron reductionFigure 5 . The bell-shaped pH-dependence of the rate of transfer of two electrons from NADP to enzyme MycR indicates a concert (coherent) transfer mechanism (adapted from Ref. 52).3.4. Oxygen-mediated transfer of triplet excitations in the pigment-protein complex Pigment-protein complexes with carotenoid (Car) and chloro- phyll (Chl) molecules play a decisive role in photosynthesis, partici- pating in the conversion of light quanta energy into chemical energy. One of the functions of carotenoids in carotinoid-containing photosynthetic proteins is the quenching of triplet exci-tations of chlorophyll with use of the triplet-triplet energy transfer(TTET) reaction 3Chl* +1Car→1Chl +3Car*. This prevents the formation of highly reactive singlet oxygen1O* 2resulting from the excitation-transfer reaction3Chl* +3O2→1Chl +1O* 2, which is especially effective under aerobic conditions in the absence ofcarotinoids. 56In the presence of carotenoids, singlet oxygen is quenched in accordance with reaction1O* 2þ1Car!3O2þ3Car*. Direct triplet transfer between Chl and Car molecules can prevent singlet oxygen formation only at rather small, ∼(3–4) Å, distance between these molecules. If it is not the case, then theTTET must be mediated by the bridging unit/units. In certainmembrane-bound protein complexes, the role of the bridging unit is often played by the oxygen molecule. Here, we discuss a triplet- triplet transfer between the Chl aand Car ( β-carotene) in cyto- chrome b 6fcomplex. In this complex, a molecular oxygen enters a specific intraprotein channel connecting the Chl aand Car.57This provides a sufficiently fast triplet-triplet3Chla*→3Car* excitation transfer and, simultaneously, prevents the formation of the1O* 2.I t is important that both a rather fast quenching of the Chl aby the β-carotene and protection against the formation of singlet oxygen are carried out under conditions when β-carotene is about 14 Å from Chl a. A possible physical explanation is discussed below based on the model of four electronic states of the complex [Chl a, O2,β-Car]. In their ground electronic states, the Chl aandβ-Car molecules have zero spin, and in the lowest excited states theirspins are equal to 1. In contrast, the O 2molecule has spins 1 and 0 in its ground and excited states, respectively. We denote by ms=0 , ±1 the projections of a spin S= 1, and by l¼π* x,π* y—the atomic orbitals of the O 2molecule. In complex [Chl a,O2,β-Car], there is no direct coupling between the spaced molecules Chl a≡Dand β-Car≡A, which appear in TTET as a donor and an acceptor of a triplet excitation, respectively. Regarding the coupling of the O 2 mediator with D and A molecules, the magnitude of this coupling is too small to noticeably change the electronic spectra of D andA. Consequently, the O 2−D(A) interaction can be considered as a perturbation and the following expressions can be used for excited initial (I), bridging (B), excited final (F) and ground (G) electronic states of complex [Chl a,O2,β-Car]: jI(ms)i≃j3D*(/C0ms)ij3O2(ms)ij1Ai, jB(l)i≃j1Dij1O* 2(l)ij1Ai, jF(m0 s)i≃j1Dij3O2(m0 s)ij3A*(/C0m0 s)i jG(m0 s)i≃j1Dij3O2(m0 s)ij1Ai:(62) Since TETT is considered in the absence of the magnetic field, the degeneracy with respect to the spin projection is not removed, and FIG. 5. The bell-shaped pH-dependence of the rate of transfer of two electrons from NADP to enzyme MycR indicates a concert (coherent) transfer mechanism(adapted from Ref. 52).Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,000000 (2021); doi: 10.1063/10.0003526 47,000000-259 Published under license by AIP Publishing.thus the energies of the above states are independent of msandl: EI≃E(3D*)þE(3O2)þE(1A), EB≃E(1D)þE(1O* 2)þE(1A), EF≃E(1D)þE(3O2)þE(3A*), EG≃E(1D)þE(3O2)þE(1A):(63) Considering complex [Chl a,O2,β-Car] as an open system, we arrive at kinetic equations (17), where, due to the presence of degeneracy, the role of mis played by I(ms),B(l),F(m ’s) and G(m0 s). Taking into account the fact that degeneracy is conserved duringtriplet energy transfer, we can analyze this transfer using the inte-gral occupancies P J(t)¼X mS¼0,+PJ(mS)(t), (J¼I,F,G), PB(t)¼X l¼π*x,π*yPB(l)(t):(64) The temporal behavior of these occupancies is governed by kinetic equations (17), where the transfer rates are58 KBI(F)¼3KO2D(A);ri(f), KIF(FI)¼3KD(A)A(D);rif(fi), KI(F)B¼2KD(A)O2;r/C0i(/C0f):(65) Here, elementary backward and forward rates are determined by Eqs. (19) and(20) as well as the relations KD(A)O2¼KO2D(A)e-ΔED(A)/kBT, KAD¼KDAe-ΔE/kBT,(66) where ΔED(A)≃E B/C0EI(F)(cf.Fig. 6 ) and ΔE≃E I/C0EFare the gap controlling the TTET. The characteristic time of the TTET in the b6fcomplex is τTTET <8n s57while the characteristic excitation decay times τdecfor the3Chl* a→1Chla,3Car*→1Car, and1O* 2!3O2transitions are (100–400)μs, (1–5)μs and 20 μs, respectively. As τTTET/C28τdec, (67) there are two different characteristic time scales for the transfer of excitations. This allows us to represent the temporary behavior of the occupancies of excited states as PJ(t)≃P(tr) J(t)P(t): (68) Occupancies P(tr) J(t) evolve on the time scale Δt/differenceτTTET and are characterized by the transfer rates (65), (cf. the scheme in Fig. 6 ). If we replace r/C0i(/C0f)!k(1(2)) f,ri(f)!k(1(2)) band rif(fi)!kDA(AD), then matematically the expression for P(tr) J(t) becomes of the same form as the expression defined by the Eq.(50). The overall transfer rates K1andK2are given by Eq. (47)where now c1¼rfþr/C0fþrfi,c2¼rif/C0rf, d1¼riþr/C0iþrif,d2¼rfi/C0ri:(69) These quantities determine the two-exponential regime of TTET via the excitation transfer rates (65) and(66). Despite the fact that molecular oxygen in the b6fcomplex facilitates the transfer of triplet excitation from Chl atoβ-Car, nevertheless, the formation of reactive singlet oxygen1O* 2was not detected in this complex. Within the framework of the physical model under consideration, this means a negligible probability PB(t) of the formation of bridge state B(1Chl1O* 21Car) mediating the TTET [cf. Figure 6(b) ]. The structure of complex b6fis such that an oxygen molecule fixed near heme bnis noticeably closer to β-Car than to Chl a,Fig. 6(a) . Therefore, the emptying of the state jBidue to the process B→A occurs much faster than the filling of the jBidue to the process I →B. As a result, the fraction of toxic oxygen1O* 2is prac- tically not formed ( Fig. 7 ), and TTET looks like a jump of triplet excitation from Chl a(D) to β-Car(A), occurring at a rate KTTET≃K2¼KD!A(1þe/C0ΔE/kBT): (70) FIG. 6. A possible way of transferring two triplet excitations from Chl atoβ-Car through the mediator structure in the pigment-protein complex b6f(a) and scheme of the transfer of triplet excitation from the initial state I =3Chl*3O21Car to the final F =1Chl3O23Car* with the participation of the bridge state B=1Chl1O2*1Car (b). Degradation of excitation brings the complex to the ground state G =1Chl3O21Car (b).Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,000000 (2021); doi: 10.1063/10.0003526 47,000000-260 Published under license by AIP Publishing.Forward TTET rate KD!A¼r(seq) ifþrif (71) contains contributions r(seq) if¼r/C0irf/(riþrf) and rifassociated with sequential and coherent routes, respectively. Which route is pre- ferred depends largely on the sign of the gap ΔED¼EB/C0EI, which in accordance with the Eq. (63) corresponds to the difference ΔEO2(1Δg!3Σ/C0 g)/C0ΔECh1a(S0!T) between energies of singlet-triplet excitations for Chl aand O 2molecules. If ΔED.0, then the involving of the sequential route in the TTET requires temperature activation, whereas the coherent route can work evenat low temperature. Qualitative estimations show 58that the sign of the gap ΔEDcan be positive or negative dependently on the polarity of the groups surrounding the complex [Chl a,O2,β-Car]. Therefore, to clarify the pathways of transfer of triplet excitation from Chl a,t oβ-Car in the pigment-protein complex b6f, experi- mental data are needed to describe the kinetics of transfer on atime scale of 0.1 –1 ns. The possible kinetics of such transport is shown in Fig. 7 . The TTET is carried out by converting two triplet excitations (TT) into two singlet (SS) excitations (I →B transition) and two singlet excitations into two triplet ones (B →F transition), Fig. 6(b) . Since there is no direct contact between mediator O and pigments Chl aandβ-Car, the TT →SS and SS →TT transitions occurs due to superexchange two-electron coupling between the oxygen molecule and each pigment. Presumably, these couplingsare formed with the participation of specific mediators [sheme Band tryptophan, Fig. 6(a) ]. Estimates show 58that taking into account these mediators allows one to obtain fairly realistic kinetics characterizing the TTET process in the b6fcomplex on the timescale Δt/differenceτTTET¼K/C01 TTET, see also Fig. 7 : P(tr) D¼1 K1K2(rirfiþrfrfiþrir/C0f), P(tr) O2¼1 K1K2(r/C0irfiþr/C0ir/C0fþrifr/C0f), P(tr) A¼1 K1K2(ririfþrfrifþrfr/C0i):(72) These values can be considered as the initial occupancies for the slow process of degradation of the excited states of the complex. To find the corresponding characteristic decay time τdec, we substitute P(tr) Jinto the Eq. (68). If the inequality (67) is satisfied, then taking into account the normalization condition P(t)+PG(t) = 1 and the fact that PG(t)¼/C0P JkJPJ(t)¼/C0P JkJP(tr) JP(t) we obtain PG(t)¼1/C0e/C0kdect: (73) Thus, the characteristic decay time τdec=k/C01 decis expressed through the rate kdec¼P(tr) DkDþP(tr) O2kO2þP(tr) AkA, (74) which characterizes the decay of excitation in the complex [Chl a, 02,β-Car]. With a small quasi-stationary occupation of the initial I and bridging B states of the pigment-protein complex, the decay ismainly associated with the 3Car*→1Car transition in the β-Car pigment. In this case, kdec≃kA. 3.5. Unique temperature-independent transitions Temperature independent transitions are usually associated with quantum tunneling processes. As applied to biologicalsystems, the tunneling was recorded in the 60s when considering the electron transfer from a high-potential cytochrome to an oxi- dized dimer of chlorophyll. 59The explanation and description of this process was carried out within the framework of the theory ofdonor-acceptor electron transport 15(the basic formulae of the theory follow from the Eqs. (19) and (20) form,m ’= D, A). However, in biosystems, temperature-independent reactions not associated with the processes of photosynthesis were also revealed.Here we discuss the desentization of ATP P2X 3receptors,60which takes place in a physiologically important region. The P2X 3receptors belong to the family of ionotropic recep- tors widely evolved in the peripherical nervous system. These receptors bind ATP molecules to specific gates of the selectivetransmembrane ion pores. The gates can be in the open (op) orclosed (cl) conformations. It has been shown experimentally (cf.Fig. 8 ) that different ATP-induced selective ionic currents I(t) man- ifest themselves as the same (almost identical) two-stage decrease. The corresponding temporal behavior of the probability of receptordesensitization, P d(t)=1−I(t)/I0, looks like60 Pd(t)¼1/C0[Ae/C0t/τ1þ(1/C0A)e/C0t/τ2] (75) with parameters A≃0.97, τ1= 14.7 ms and τ2= 230 ms FIG. 7. Kinetics of the triplet transfer in the complex looks like a transfer between donor (Chl) and acceptor (Car), that is, between I and F states only.This is due to the insignificant probability of population of the bridge state, in which singlet oxygen is formed (adapted from Ref. 58).Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,000000 (2021); doi: 10.1063/10.0003526 47,000000-261 Published under license by AIP Publishing.characterizing the pre-exponential weight and the two temperature-independent decay constants, respectively. To under-stand the physics of such unique temperature-independent behav-ior, we will follow the approach that takes into account the role of thermodynamic fluctuations in the formation of transitions between states of an open system. The description can be done onthe basis of a kinetic approach using balance equations (17), where the transition rates are evaluated with Eq. (19). A model is used in which the channel can be in only two physiologically important states (open and closed). 11,60,61In this case, the simplest explanation of the two-exponential temporalbehavior of desensitization is achieved if two types of channels,j= 1, 2, are responsible for desensitization. The current through the N jchannels of the jth type, Ij(t)¼NjijP(j) op(t), is determined by the current ijthrough an open separate channel and the probability P(j) op(t) that the channel is open. Since we get I(t)/I(0)¼ AP(1) op(t)þ(1/C0A)P(2) op(t) where A=N1i1/(N1i1+N2i2). To deter- mine the temporal behavior of probabilities P(j) op(t), we assume that transitions between the states m(s) = op( s) and m ’(s’) = cl( s’), are quasi-isoenergetic, but the states are degenerate in the number ofconformational substates s¼1, 2, ...,μ(j) OPand s0¼1, 2, ...,μ(j) cl. Then, according to expression (19), the equation describing the temporal behavior of the occupation of sth substate of the state op reads _P(j) op(s)(t)¼/C0Xμ(j) cl s0¼1[K(j) op(s)cl(s0)P(j) op(s)(t)/C0K(j) cl(s0)op(s)P(j) cl(s0)(t)]: (76) Transitions between substates s(s’) belonging to the state open (closed) occur for the corresponding characteristic times τ(j) op(τ(j) cl), which are much less than the characteristic times τj∼(10–100) msof the transition process op →cl. Therefore, the substate occupancy is defined as Pj m(s)¼(1/μ(j) m)P(j) m(t) where P(j) m(t)¼Pμ(j) m s¼1P(j) m(s)is the integral occupancy. Using this relation, we arrive at the equation _P(j) op(t)¼/C0(μ(j) op)/C01KjP(j) op(t)þ(μ(j) cl)/C01KjP(j) cl(t) where Kj¼P ss0K(j) op(s)cl(s0) /C25P ss0K(j) cl(s0)op(s). Taking the normalization condition P(j) cl(t)þ P(j) op(t)¼1 into account leads to the solution P(j) op(t)¼ [μ(j) cl/(μ(j) opþμ(j) cl)]e/C0k(j) trtμ(j) op/(μ(j) opþμ(j) cl) describing a decrease in the integral occupancy with the overall rate k(j) tr¼[(μ(j) opþμ(j) cl)/μ(j) opμ(j) cl]Kj. It is seen that if the degeneracy of the closed state significantly exceeds the degeneracy of the open state, then P(j) op(t)≃e/C0k(j) trt. This is the result of nonrecurrent kinetics, that leads us to the expression (75) where τ1,2¼[k(1,2) tr]/C01and k(j) tr≃(μ(j) op)-1Kj. Calculation of the transition rate Kjis based on expressions (19) and(12). We formally put m= op( s),m’= cl(s’) and estimate the K(j) mm0using the one-phonon approximation, in which Λm0m(τ)¼P λjM(λ) m0mj2[(/C22nλþ1)eiωλτþ/C22n/C0iωλτ λ]: Here, M(λ) m0m¼χ(λ) mm0= is the matrix element of the transition between adi- abatic terms mand m’. The transition is accompanied by absorp- tion or emission of a phonon of frequency ωλ, Eq. (10).I fa one-phonon transition occurs between the nonadiabatic terms, then M(λ) mm0¼Vmm0g(λ) mm0, where Vmm0¼V(j) op(s),cl(s0)is the matrix element of transition between the substates of diabatic terms (cf.Equation (9)and inset in Fig. 8 ). If random deviations of the energy difference E m−Em’, are caused by the thermodynamic fluc- tuations of the vibration energyP λ/C22hωλ(nλþ1/2), then in the clas- sical and quantum limits the factor Fmm0(τ) is given by Eqs. (30) and (31), respectively. Substituting the above expressions for Λm0m(τ) and Fmm0(τ) into the integral of expression (19) we obtain Kj¼X ss0K(j) mm0¼1 /C22h2X ss0X λjM(λ) mm0j2 /C2X ξ¼1,2(/C22nλþ1)γλ γ2 λþ[Ωmm0/C0ωλþ(/C01)ζσλ]2( þ/C22nλγλ γ2 λþ[Ωmm0/C0ωλþ(/C01)ζσλ]2) , (77) where γλ=γstoch. If among the set of modes λthere are modes λ=λ*, the connection with which leads to the appearance of reso- nance at Ωmm0¼+ω*þ(/C01)ζσ*, then the main contribution to the rate Kjwill be made by the resonance terms like ( /C22n*þ1)/γ*and /C22n*/γ*. According to the expression (30),w eh a v e γ*∼kBT.A s /C22n*þ1/C25/C22n**≃kBT//C22hωλ, we see that ( /C22n*þ1)/γ*and /C22n*/γ*are inde- pendent of temperature. Therefore, the rates Kjand thus, the degra- dation times τj¼μ(j) opK/C01 jare also temperature-independent characteristics of the process of desensitization. 3.6. Conformational regulation of enzymatic reactions Enzymes are natural nanocatalysts. Nowadays it is generally recognized that the unique peculiarities of their functioning are FIG. 8. T emperature-independent two-exponential kinetics corresponding to the onset of desensitization of P2X 3receptor (adapted from Ref. 61). The direction of the desensitization reaction is due to the nonrecurrent kinetics, when thedegeneracy μ(j) clof the closed state of the channel is much higher than the degeneracy μ(j) opof its open state.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,000000 (2021); doi: 10.1063/10.0003526 47,000000-262 Published under license by AIP Publishing.attributed to not only their active center specificity but also struc- tural changeability of the whole macromolecule. In this subsection we briefly trace the role of this distinctive property of biomacromo-lecules in regulatory mechanisms of enzymatic reactions. In thereactions of the enzyme ( E) with the substrate ( S), the probabilities of the free [ P E(t)] and bound ( PESj) states of the enzyme change while maintaining the normalization condition X jPj(t)¼1(j¼E,ES1,ES2,...): (78) In enzymatic reactions, the total concentration of enzyme, [ Et], does not change. Therefore, the condition (78) is identical to equal- ity [Et]¼[E]þP j[ESj], where [E]and [ ESj] are the concentra- tions of the free enzyme and the complex “enzyme + substrate ”in thejth state, respectively. In traditional enzymology, concentrations are used instead of corresponding probabilities PE=[ESj]/[Et] and PESj¼[ESj]/[Et] to describe the reactions of an enzyme with a substrate. The basis of enzymology is represented by celebrated Michaelis –Menten ’s scheme. Introduced more than a century ago,62it still serves as a starting point of any experimental or theo- retical investigation in the field. According to this scheme, a reac-tion of converting substrate Sinto product Pvia formation of complex ESwith enzyme-catalyst Eis described within simple chemical kinetics based on the mass action law as EþSOk kbES/C0kc!EþP, d[ES]/dt¼/C0(kbþkc)[ES]þk[S][E],(79) where [ S] is the substrate concentration. From Eq. (79), the famous expression for the stationary reaction velocity v=d[P]/dt=kc[Et] [S]/([S] + KM) immediately follows (here KMis Michaelis ’constant, KM=( k b+kc)/k, and [ Et]=[E]+[ES] is the total enzyme concen- tration). For a long time, the hyperbolic dependence v([S]) served as a validity test of investigation of any enzyme, and deviationsfrom it were considered as artefacts. The situation began changingin the second part of the last century, when the problem of regula-tion/control of enzymatic reactions gradually became central. 63In the first place, it concerned revisions of the mentioned hyperbolic dependence in favor of rather trigger-like ones, with considerablevelocity differences in narrower substrate concentration intervals.For this, in the first models of cooperativity, apart from allostery(the presence of several binding sites in an oligomeric biomole- cule), different conformations of the reaction states have been nec- essarily introduced. 64 Later, it turned out that “cooperativity ”(in the sense of sigmoid dependences on [ S]) could be exhibited even by non- allosteric enzymes with an only binding site.65,66For a prolonged period, this elegant idea based in fact on protein structured memory was beyond the enzymology mainstream and provoked avivid interest only recently (mostly caused by experimental confir-mations of the effect in some enzymatic reactions of vital physio- logical importance 67), up to introducing a special term “allokairy ”.68Revisions and extensions of the Michaelis –Mentenscheme were revived since the beginning of the present century, especially because of implementation of the single-molecule (SM) methods into enzymology. The latter, however, have not broughtfundamental changes in theoretical approaches to describing theregulatory role of structural changeability, as either ensemble orSM reactions were (and still are) mainly considered within the standard chemical kinetics schemes, often reduced to sets of linear equations for concentrations or population probabilities of reactionstates, respectively. Below we describe the generic models andeffects within this traditional framework and then discuss anadvanced approach based on the concept of molecular self- organization. The typical phenomena of conformational regulation can be seen even in reactions of monomeric enzymes possessing anonly binding site. 3.6.1. Conformational regulation in discrete schemes The simplest and traditional way to take into account the enzyme structural complexity implies introduction of several con-formational (sub)states of a free enzyme and/or enzyme-substratecomplex, with conformational transitions between them. For such schemes (which in fact correspond to the splitting of the Michaelis –Menten scheme into several conformational channels of the reaction) they use the mentioned rate equations. For ensemblereactions, especially for calculating their steady-state velocities, even the stationary solutions are often informative enough. Not much more complex is the case of SM reactions. As it can be concludedfrom numerous works exploiting this approach, the characteristiceffects of conformational regulation show up even under splittinginto only two channels; their larger numbers lead to more cumber- some expressions only, adding no truly new mechanisms. The generic minimal schemes of regulation 69,70related to nontrivial dependence of the reaction velocity on substrate concentrtion orunbinding rates are pictured in Fig. 9 (it is sufficient to show the enzyme states only). Scheme in Fig. 9(a) was proposed in Ref. 71. It illustrates a striking effect of reaction acceleration with rate k bof “unproductive ”substrate unbinding growing in a certain initial interval (this possibility was pointed out quite recently72). Here Eq.(79) should be obviously replaced by the set d[ES1] dt¼/C0(kbþkc)[ES1]þk[S][E], d[ES2] dt¼/C0(kbþkc)[ES2]þk[S][E](80) which, together with the total enzyme concentration conservation condition [ Et]¼[E]þ[ES1]þ[ES2], leads to the following sta- tionary reaction velocity: v¼k[S][Et][kb(kCþkc)þ2kCkc] k[S](2kbþkCþkc)þ(kbþkc)(kbþkC): (81) If the catalytic rates are of similar magnitude, kc/C25kC,then v (kb) monotonically decreases, as it is customary for the Michaelis – Menten scheme. If, however, the channels are markedly different in their catalytic rates, kC/C29kc,then there exist an interval of v(kb) growing. The effect becomes possible at substrate concentrationsLow Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,000000 (2021); doi: 10.1063/10.0003526 47,000000-263 Published under license by AIP Publishing.satisfying the condition k[S]>kckC(kC+kc)/ (kC−kc)2. This unex- pected result can be nevertheless explained rather simply. Indeed,without the possibility of escape from the “deadlock ”state ES 2with too long waiting times of the catalytic stage (thereby without a chance of a new start and subsequent catalysis via state ES1),the enzyme would be captured in this less functional state. In fact,scheme in Fig. 9(a) represents a particular case of a more general effect of decrease of the mean first passage time in random walkswith resetting. 73,74 Scheme in Fig. 9(b) is a simplified version of Rabin ’s scheme of“kinetic cooperativity ”63,65and uncovers important physical reasons of this distinctive deviation from the classical hyperbolicdependence. Unlike to the previous scheme, here different confor-mations of the free enzyme with different binding rates, k>k’,a r e introduced. It is assumed that “more active ”state E 2can slowly relax to “less active ”state E1. The higher substrate concentration, the longer the residence time in more active state. Under the condition α<(kb+kc)[(k/k’)−1] this results in a sigmoid dependence v([S]) (“cooperativity ”).75Various extensions of this scheme are frequently used (see for example, Refs. 67,68, and 76). It should be noted, however, that in the case of small number of conformation channels the sigmoid behavior is barelyseen (as in Fig. 9(b) where it is slightly discernible in the inset only). This is common for linear schemes of discrete conforma- tions, classical schemes of cooperativity included —in the latter,the more macromolecule subunits, the more pronounced cooperativity. Finally, scheme in Fig. 9(c) admits, in addition to the effects described above, the possibility of self-inhibition (again, in the case of markedly different catalytic rates). Under certain condition on the system parameters (see Ref. 69for details), the role of the less active channel can prevail with [ S] growing, and the reaction veloc- ity decreases, see the plot in Fig. 9(c) . With these three effects, the non-standard deviations from classical Michaelis –Menten ’s theory, related to conformational reg- ulation, are seemingly exhausted. Generalizations towards morecomplex discrete schemes (for example, Ref. 76) lead to very bulky expressions while hardly producing new meaningful physicalinsight. Almost all recent works in the field concern reactions of single enzymes, since this experimental technique is becoming dominant.It allows one to observe enzyme functioning in a serial regimeof consecutive conversions of substrates into products, oneby one, and to obtain arrays of turnover times and their proba- bility density f(t). The reciprocal thi /C01of the mean turnover time thi¼Ð1 0tf(t)dt(in fact, mean first passage time74) plays the part of the reaction velocity while the higher moments tmhi reflect “dynamical disorder ”,etc. For theoretical derivations of f(t), the same discrete schemes are used with the only difference that the kinetic equations are formulated for probabilities PE(t),PES(t)o f FIG. 9. Reaction schemes and corresponding velocity plots. Plot (a): k[S] = 10. kC=kc= 3 (curve 1); k C= 10, kc= 1 (curve 2). Plot (b): α=1 , k=1 , k’= 0.1, kb= 10, kc= 100. Inset: the initial part of the plot for small [S ] (see the text). Plot (c): α= 10,β=1 , k=1 , k’= 10, b= 10, B=1 , kc=1 , kC= 10.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,000000 (2021); doi: 10.1063/10.0003526 47,000000-264 Published under license by AIP Publishing.the reaction system residence in corresponding states. It should be noted, however, that the problem becomes non-stationary (the stage of the enzyme return into the initial free state is not takeninto account). Correspondingly, the conservation condition (78) does not hold true any longer, and attempts to preserve it (forexample, Refs. 77and78) lead to unnecessary inconsistence only. It is easy to show that thi /C01calculated for classical Michaelis – Menten ’s scheme (79) in such a way is equal to k[S]/([S]+KM), that is thi/C01¼v/[Et]. This important relationship, connecting the ensemble and SM results, was called “single molecule Michaelis – Menten equation ”and verified experimentally.77,79Remarkably, it remains valid in the presence of conformational splitting of the reaction pathway as well.69Thus, being interested in the reaction velocity, it is sufficient to calculate it in the stationary cases ofensemble schemes, which is much simpler. For example, calculationofv/[E t] performed in Ref. 69for the ensemble version of scheme inFig. 9(c) has resulted in the expression identical to that obtained forthi/C01in a much more laborious way. 3.6.2. Conformational regulation as an example of molecular self-organization All the schemes discussed above imply essentially non- equilibrium flow conditions (the part of the flow intensity is playedby substrate concentration [ S]). This provokes one to speculate about possible synergetic mechanisms of conformational regula-tion. According to Haken, 80the necessary conditions of self- organization phenomena include, apart from the flow, a pro- nounced temporal hierarchy and nonlinearity of the system dynam-ics. These conditions are practically omnipresent in biomolecularprocesses, enzyme functioning included. The spectrum of biomole-cule structural movements is very broad and includes, in particular, those much slower ones than elementary acts and turnovers of the reaction. Due to such structural memory, cumulative structuralchanges caused by consecutive substrate arrivals become possible.In turn, these changes entail changes in reaction rates, and this feedback ensures nonlinearity in the system. As a result, stationary non-equilibrium regimes of the enzyme functioning emerge whichare self-consistent with the flow. Intensity of the latter plays a roleof a control parameter, with its changes causing bifurcation phe-nomena like bistability, that is, emergence/disappearance and coex- istence of functional regimes with markedly different reaction velocities, etc. Far from being exotic, these quite natural ideas form the basis of our concept of molecular self-organization 75which was applied to describing the conformational regulation effects in primary reac- tions of photosynthesis (see for example, Refs. 81and82and refer- ences therein). Application of this concept to enzyme functioningis given in Ref. 83. It is supposed that slow structural changes can be represented by a continuous generalized structural coordinate x which the reaction rates become dependent on. The motion along this coordinate is certainly classical and governed, apart from the influence of standard thermal white noise F Lof Langevin ’s type, by the switching between structural potentials VE(x),VES(x) in the corresponding enzyme states. In other words, the substrate arrivals/ departures are the source of a specific dichotomous noise Ft,84,85 with its space of states { fE,fES}¼{/C0dVE/dx,/C0dVES/dx} andcharacteristics (reaction rates) dependent itself on the structural variable. Thus, the stochastic equation for xreads _x¼FtþFL, while the master equation for the dichotomous noise is _ρE(tjx)¼/C0k(x)ρE(tjx)þ[kb(x)þkc(x)]ρES(tjx), ρE(tjx)þρES(tjx)¼1, where ρE,ρESare the probability densities of realization of forces fE, fES, respectively. Then it can be shown84,85that the Michaelis – Menten scheme (79) can be represented by the following evolution equations for probability densities PE(x, t),PES(x,t): @PE @t¼DE@ @xdVE dxþ@ @x/C18/C19 PE/C0k(x)PEþ[kb(x)þkc(x)]PES, @PES @t¼DES@ @xdVES dxþ@ @x/C18/C19 PES/C0k(x)PEþ[kb(x)þkc(x)]PES, (82) where DE,DESstand for the coefficients of diffusion generated by the thermal noise. Eq. (82) can be considerably simplified by apply- ing the adiabatic approximation, obviously valid due to the slow-ness of structural movements. Then one arrives at Fokker –Planck ’s equation for distribution P(x,t)=P E(x,t)+PES(x,t) with the fol- lowing derivative of effective nonequilibrium potential Veff(x): dVeff dx¼dVE dxþdVES dx/C0dVE dx/C18/C19[S] [S]þKM(x): (83) To derive Eq. (83), it was supposed that kis non-distributed, k=k1[S] (this usually holds in enzyme reaction studies) while Michaelis constant became x-dependent, KM(x)=[kb+kc]/kj.A remarkable property of Veff(x) is its dependence on flow intensity [S]. Thus, if potentials VE(x),VES(x) are of a one-well shape, then, with [ S] growing, the shape of Veff(x), gradually transforming from that of VE(x) to that of VES(x), may become two-well in some inter- val of [ S]. In Ref. 83this is shown for the case when the depen- dence of Michaelis ’constant on xcan be reduced to exponential one, KM(x)≈e−x(while the part of generalized structural variable x is played by the affinity that changes due to cumulative actions of substrates). The corresponding emergence of stationary distributionP st(x) bimodality means the appearance of bistability, that is, coex- istence of two enzyme functioning regimes with markedly differentreaction velocities. With the strength of substrate-conformation interaction [difference in the positions of minima of V E(x),VES(x)] exceeding some critical value, the dependence v([S]) acquires a pro- nounced sigmoid shape (see Fig. 10 ), imitating the cooperativity effect. Also, allowance for realistic dependences kc(x) leads to the possibility of the self-inhibition effect.83 The above-described picture of reaction regime regulation rep- resents in fact a fold-type catastrophe in full analogy with non-equilibrium phase transitions of the 1st kind (see Ref. 83for details). As distinct from the typical examples of self-organization occurring in macrosystems, here we have this phenomenon at the level of single molecules.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 47,000000 (2021); doi: 10.1063/10.0003526 47,000000-265 Published under license by AIP Publishing.Experimental confirmations of conformational regulation mechanisms may be rather tricky since the latter are of hidden nature and imply concomitant measurements at very different timescales. For ensemble reactions the corresponding proof is oftenreduced to a collection of indirect evidences (for example, Ref. 81). In this sense, the new possibilities provided by SM technique lookmuch more promising. To this end, computer simulations of sto- chastic “SM trajectories ”(that is, thousands of turnovers in a model reacting system with realistic parameters which admit bist-ability) were undertaken in Ref. 86. Their statistical processing has revealed rather eloquent qualitative threshold-like changes in themean turnover times and behavior of characteristic temporal corre- lation functions (that is, just in the primary observables in SM experiments) exactly in the bistability area. Detection of suchanomalies in scanning a single-enzyme reaction subject to substrateconcentration changes would be the most straightforward confir-mation of molecular self-organization phenomena at work. 4. CONCLUSION The main goal of the work is to show how to describe and analyze the processes of charge/excitation transport and conforma-tional transformations in biosystems under conditions when trans- port and conformational changes occur against the background of faster processes. Fast processes, for example, relaxation among thevibrational levels of electronic terms of biomolecules, stochasticdeviation of the energy levels, as well as sorption-desorption ofprotons, lead to the establishment of a quasi-equilibrium between the probabilities of populating substates and thus facilitate the for- mation of integral occupancies of molecular terms of the system.Integral occupancies change on a time scale that is much longerthan the characteristic times of fast processes occurring between substates. As a result, we obtain averaged kinetic equations that contain the characteristics, which can be estimated from theexperiment. This refers also to the equations describing the binding of ligands or substrates to enzymes. We used the nonequilibrium density matrix method to obtain averaged kinetic equations and indicated criteria for the applicabil-ity of these equations to the description of various types of reac-tions in biosystems. As an application of the method, we analyzed one-electron and two-electron donor-acceptor transfer between redox centers, as well as the transfer of triplet excitation betweenpigments in the pigment-protein complex. In both cases, theimportant role of the bridge structure in the formation of hoppingand tunneling (coherent) paths of electron transfer and excitation is shown. The corresponding transfer rates and their dependence on the number of bridging units were found, and for two-electrontransfer, the dependence of the rate of the redox reaction on theprotonation of amino acids in the enzyme was also estimated. Acoarse-grained (averaged) description of transitions between con- formationally degenerate states of ion channels made it possible to propose a possible mechanism of channel desensitization. Within the conventional framework of enzyme kinetics, the generic schemes of conformational regulation of enzymatic reac-tions are revisited. 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AIP Advances 11, 015230 (2021); https://doi.org/10.1063/9.0000167 11, 015230 © 2021 Author(s).Magnetic phases in superconducting, polycrystalline bulk FeSe samples Cite as: AIP Advances 11, 015230 (2021); https://doi.org/10.1063/9.0000167 Submitted: 29 October 2020 . Accepted: 07 December 2020 . Published Online: 12 January 2021 Quentin Nouailhetas , Anjela Koblischka-Veneva , Michael R. Koblischka , Pavan Kumar Naik S. , Florian Schäfer , H. Ogino , Christian Motz , Kévin Berger , Bruno Douine , Yassine Slimani , and Essia Hannachi COLLECTIONS Paper published as part of the special topic on 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials and 65th Annual Conference on Magnetism and Magnetic Materials ARTICLES YOU MAY BE INTERESTED IN Superconductivity at in tetragonal FeSe under high pressure Applied Physics Letters 93, 152505 (2008); https://doi.org/10.1063/1.3000616 Mössbauer studies on magnetism in FeSe AIP Advances 11, 015114 (2021); https://doi.org/10.1063/9.0000108 Magnetic field sweep rate influence on the critical current capabilities of a Fe(Se,Te) crystal Journal of Applied Physics 128, 073902 (2020); https://doi.org/10.1063/5.0010324AIP Advances ARTICLE scitation.org/journal/adv Magnetic phases in superconducting, polycrystalline bulk FeSe samples Cite as: AIP Advances 11, 015230 (2021); doi: 10.1063/9.0000167 Presented: 4 November 2020 •Submitted: 29 October 2020 • Accepted: 7 December 2020 •Published Online: 12 January 2021 Quentin Nouailhetas,1,2 Anjela Koblischka-Veneva,1,3 Michael R. Koblischka,1,3,a) Pavan Kumar Naik S.,4,5 Florian Schäfer,6 H. Ogino,3,4Christian Motz,6 Kévin Berger,2 Bruno Douine,2 Yassine Slimani,7 and Essia Hannachi8 AFFILIATIONS 1Institute of Experimental Physics, Saarland University, Campus C 6 3, 66123 Saarbrücken, Germany 2Groupe de Recherche en Energie Electrique de Nancy (GREEN), Université de Lorraine, 54506 Vandvre-lès-Nancy, France 3Shibaura Institute of Technology, 3-7-5 Toyosu, Koto-ku, Tokyo 135-8548, Japan 4Research Institute for Advanced Electronics and Photonics, National Institute of Advanced Industrial Science (AIST), 1-1-1 Central 2, Umezono, Tsukuba, Ibaraki 305-8568, Japan 5Department of Physics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku City, Tokyo 162-8601, Japan 6Experimentelle Methodik der Werkstoffwissenschaften, Saarland University, Campus D 3 4, 66123 Saarbrücken, Germany 7Department of Biophysics, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam 31441, Saudi Arabia 8Laboratory of Physics of Materials - Structures and Properties, Department of Physics, Faculty of Sciences of Bizerte, University of Carthage, 7021 Zarzouna, Tunisia Note: This paper was presented at the 65th Annual Conference on Magnetism and Magnetic Materials. a)Author to whom correspondence should be addressed: m.koblischka@gmail.com and miko@shibaura-it.ac.jp ABSTRACT The FeSe compound is the simplest high-temperature superconductor (HTSc) possible, and relatively cheap, not containing any rare-earth material. Although the transition temperature, Tc, is just below 10 K, the upper critical fields are comparable with other HTSc. Preparing FeSe using solid-state sintering yields samples exhibiting strong ferromagnetic hysteresis loops (MHLs), and the superconducting contribution is only visible after subtracting MHLs from above Tc. Due to the complicated phase diagram, the samples are a mixture of several phases, the superconducting β-FeSe, and the non-superconducting δ-FeSe and γ-FeSe. Furthermore, antiferromagnetic Fe 7Se8and ferromagnetic α-Fe may be contained, depending directly on the Se loss during the sintering process. Here, we show MHLs measured up to ±7 T and determine the magnetic characteristics, together with the amount of superconductivity determined from M(T)measurements. We also performed a thorough analysis of the microstructures in order to establish a relation between microstructure and the resulting sample properties. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/9.0000167 I. INTRODUCTION The member of the iron-based superconductor (IBS) family with the simplest composition is FeSe (“11”).1,2This material is not only interesting from the basic physics point of view,3–5but also for applications as the grain boundaries (GBs) may not be not obstacles as for the cuprate 123-HTSc,6and the main properties are like other high-temperature superconductors (HTSc), that is, high upper critical fields Hc2, high current densities jc, and not being prone to flux jumps. Most importantly, the material is freeof rare-earths and toxic elements.7,8Thus, it is interesting to use FeSe for metal-sheathed tapes,9,10but also for bulk applications like superconducting trapped field (TF) magnets.11 Now, many problems for the sample preparation appear due to the relatively complicated phase diagram of FeSe.12–15It was found that the composition Fe 1Se1−δis extremely sensitive to the correct stoichiometry.14,15All this makes the preparation of single crystals very difficult,16and the preparation of bulk, polycrys- talline samples required for TF magnets is even more complicated. This was demonstrated by Diko et al.17using optical polarization AIP Advances 11, 015230 (2021); doi: 10.1063/9.0000167 11, 015230-1 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv microscopy together with x-ray data. Furthermore, antiferromag- netic (AF) Fe 7Se8and ferromagnetic (FM) α-Fe particles may be contained, depending directly on the Se loss during the sintering process. Thus, the preparation route to achieve good samples for applications must consider all the phase relations, and further- more, recent results18have demonstrated that the critical currents obtained by the simple solid-state sintering are similar to that of the direct competitor, MgB 2, without additional flux pinning sites, but not yet sufficient for most bulk applications. So, it is strongly required to learn more about the microstruc- tures achieved in the sintering processes to devise a better suited preparation technique. In the present contribution, we employ opti- cal and electron microscopy to gain a better understanding how the samples prepared by solid-state sintering are composed. The other important issue is how the presence of the secondary magnetic phases affect the superconducting properties. II. EXPERIMENTAL PROCEDURE Polycrystalline FeSe samples were prepared by solid-state sintering, starting from commercially available metallic Fe (99.9% purity) and Se (99.5% purity) powders. The powders were mixed together by ball-milling under Ar atmosphere. The resulting powder mixture was then pressed into pellets (5 mm diameter, 2 mm thick, 0.3 g) using uniaxial pressure of 10 MPa. The pellet was wrapped in Ta foil19and sealed into an evacuated quartz tube. The samples were then heated up to 700○C with a rate of 50○C/h and held there for 1 h. Then, the temperature was reduced to 400○C in 30 min and maintained there for 40 h to enable the growth of the β-FeSe phase. Finally, the quartz tube with the sample was air-quenched to reach room temperature in about 5 min to avoid a low-temperature phase transformation.14To find the optimum conditions for our furnace system, the Se content δwas varied between 0.9 and 1.2. All products were checked using x-ray diffraction (XRD) in a powder diffractometer (Rigaku Ultima IV), using Cu-K αradiation. Quantitative analysis was performed with the software Match! for Rietveld refinements. Typical XRD data are given in the supplementary material. For microscopy, the sample surfaces were mechanically polished using silicon carbide papers and diamond paste with alcohol as lubricant20For magnetometry, small samples ( ∼1.5×1×0.7 mm3) were cut from the big pellets. For resistance measurements, small bars (5 ×1×1 mm3) were prepared, and the contacts were fixed with silver paint, ensuring about 3.5 mm distance between the voltage pads. The magnetic characterization measurements were performed using SQUID mag- netometry (Quantum Design MPMS3) and the magneto-resistance characteristics were recorded using an Oxford Instruments 10/12 T Teslatron system with λ-plate. Polarization images were taken using a Zeiss Microplan system. Electron backscatter diffraction (EBSD) was performed employing a Zeiss Sigma VP microscope operating at 20 kV with Oxford Instruments EBSD detector and TSL analysis. The working distance was chosen to be 17 mm, the stepsize was 160 nm. III. RESULTS AND DISCUSSION Figure 1(a) presents magnetization hysteresis loops (MHLs) of a FeSe (initial composition Fe:Se =1:1.05) sample, taken at 4.2 FIG. 1. (a) Magnetization hysteresis loops (MHLs) of a polycrystalline FeSe (initial composition Fe:Se =1:1.05) sample at 4.2 K ( ■) and 20 K ( ●). (b) Superconducting hysteresis loop after simple subtraction. K (■) and 20 K ( ●). The 20 K-loop is clearly above the super- conducting region and shows ferromagnetic behavior with a small hysteresis. The 4.2 K-loop presents a combination of a ferromag- netic with a superconducting signal. At high fields above 6 T, both loops are nearly identical, but around zero field, the superconducting contribution (zero-field peak) is obvious. In (b), the simple sub- traction of both loops is shown. The resulting MHL is asymmet- ric, which may stem from the polycrystalline character. However, the simple subtraction assumes that the two contributions are fully independent from each other. Measuring a large number of such MHLs demonstrated that this is not the case. This effect of the interaction of the magnetic and superconducting contributions will need to be investigated in more detail, determining the Meissner currents. Figure 2 shows a plot of the magnetization (in units of μB/formula unit) as function of δ, the Se content in the formula Fe1Se1−δ, following Williams et al.15In this way, it becomes possi- ble to establish a relation between the prevailing magnetism and the Se content in the sample. The β-FeSe phase is non-magnetic, Fe 7Se8 is antiferromagnetic,21and especially, eventually unreacted (due to Se loss) α-iron is ferromagnetic. Furthermore, we must note that the remaining high-temperature phase δ-FeSe (space group P6 3/mmc, AIP Advances 11, 015230 (2021); doi: 10.1063/9.0000167 11, 015230-2 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 2. Magnetization as function of the Se deficiency δ(T=150 K, up to 9 T applied field). The solid line is a guide to the eye. The symbols ( ■) are the data of Ref. 15, the open symbols are the present work. The dashed lines give the saturation magnetization of our samples together with the precursor composition. NiAs-type), which may be present in the samples, gives a back- ground magnetization of 0.012 μB/f.u.,22which cannot be avoided as the conversion from δ-FeSe to β-FeSe cannot be complete.17The minimum of the curve in Fig. 2 is found, however, at 0.027 μB/f.u., which implies that there is an additional FM contribution, which may be α-Fe. This was not commented in Ref. 15. In our case, the minimum magnetization was 0.04 μB/f.u., and the corresponding x-ray results let us conclude that this point is on the side δ>0. Thus, the stoichiometry of Se in the starting mixture is crucial: If there is too much Se present in the sample, AF-Fe 7Se8will form, and if the Se loss is too high, unreacted FM α-iron will be found in the sample. The XRD analysis obtained via Match! gives 74.2 vol.-% β-FeSe and 25.8 vol.-% δ-FeSe for the sample Fe:Se =1:1.2, and 85.2 vol.-% β-FeSe and 14.8 vol.-% δ-FeSe for the sample Fe:Se =1:1.2, with the other phases treated as minor contribution. As the resulting particles or grains are in the nanometer range, it is difficult for com- mon XRD to detect these contributions. Additionally, most of the peaks in the XRD diagram overlap each other, so it is not trivial to distinguish these phases. One possibility to obtain this informa- tion are the peaks at 2 Θ=44.73○(α-Fe) and 50○(Fe 7Se8), which may enable to judge if δis positive or negative. Thus, we decided to run EBSD to obtain a spatially resolved information about the phase distribution in our samples. Figure 3 presents a polarization image, revealing twin pat- terns in the FeSe grains. (b) gives a SEM image, showing the FeSe grains (platelet-like shape) and the remaining pores in the sample. The density of the present samples is about 65% of the theoretical value. Figure 4 shows the EBSD results on the polycrystalline FeSe sample (Fe:Se =1:1.05). Image (a) is a gray-scale image quality map of the sample section (82 ×62μm2) studied, which resembles an SEM image, but in EBSD condition with 70○incline. Image (b) presents the phase mapping: The light blue color indicates the superconduct- ingβ-FeSe phase, pink stands for α-iron and blue for Fe 7Se8. Here it is interesting to note that both magnetic phases appear together inbetween the β-FeSe grains. The overall amounts detected by FIG. 3. (a) Polarization image of a polycrystalline FeSe (Fe:Se =1:1.05) sample, (b) SEM image. FIG. 4. EBSD analysis results on the sample Fe:Se =1:1.05. (a) Image quality map, (b) phase map, (c) IPF map for all phases together, (d) IPF map for β-FeSe sep- arately, and (e) the inverse pole figures in [001]-direction for β-FeSe (left), α-Fe (middle) and Fe 7Se8(right). EBSD are 54.4% β-FeSe, 17.4% α-Fe and 10.5% Fe 7Se8. The remain- der are pores, and non-indexed points including γ- and δ-FeSe. Note that these are data within the given surface. Therefore, this composition may vary with position. The dominating phase is always β-FeSe, and thus, purely superconducting paths exist in the sample. Image (c) presents an inverse pole figure (IPF) map in [001]-direction (i.e., perpendicular to the sample surface), show- ing the crystallographic orientation of all phases together, and (d) AIP Advances 11, 015230 (2021); doi: 10.1063/9.0000167 11, 015230-3 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv gives the orientation of the β-FeSe grains separately. The mea- sured Kikuchi patterns were indexed as β-FeSe (JCPDS file 030533, tetragonal, space group P4/nmm [129], PbO-type), α-iron (bcc) and Fe 7Se8(triclinic, space group P3 121). The color codes for the grain orientations are given in the stereographic triangles below the maps. Finally, (e) shows the inverse pole figures in [001]-direction forβ-FeSe, α-Fe and Fe 7Se8. From these maps, we see that the β-FeSe grains are not randomly oriented, but shows some distinct maxima. Furthermore, there is a needle-like arrangement of the grains with several intergrowths and other phases embedded within them. The strong orientation deviation leads to many high-angle grain bound- aries, which are obstacles to the current flow even if the grain cou- pling in FeSe should be much better than in cuprate HTSc samples. According to Diko et al. ,17theδ-FeSe formed in the first heating step undergoes a martensitic transformation below 457○C, thus some longer dwell time below this temperature is required to allow the growth of β-FeSe. Also, due to this transformation, there are only some major directions of the β-FeSe as seen in the corresponding pole figure. Having obtained these microstructure results, it is clear that in polycrystalline, bulk FeSe samples, there will always be some fer- romagnetic contribution present due to remaining δ-FeSe, even if the extra phases α-Fe and Fe 7Se8can be avoided by properly select- ing the starting composition. Thus, the superconducting properties of such samples will be always affected by this magnetic contribu- tion. This is especially evident when extracting data for Hc2, where superconductivity vanishes completely. Such data can be obtained from magnetoresistance measurements with a criterion to define the first onset of superconductivty. Typically, a criterion of 99% RNis applied,25where RNis the normal state resistivity. R(T,Ha)- curves of our samples were already published,26so we focus here on the extracted data for Hc2(sample Fe:Se =1:1.05). Figure 5 shows FIG. 5. Hc2(T)-data (■) obtained from resistance measurements on a FeSe sam- ple (Fe:Se =1:1.05). The green (WHH) and red (with spin-paramagnetic effect) fits determine Hc2(0). The inset presents R(T)and d R/dT(6 T applied field). The arrow points to the onset of superconductivity at ∼12 K. The data points at T=0 K give the results of other works: Hc2(0)=12 T ( Ha∥c,△), 29 T ( Ha/⊙◇⊞c, ▲)23and 59 T (FIC analysis, ●).24and the resulting data measured up to 10 T applied field. The onset of superconductivity takes place at ∼12 K, whereas the main super- conducting transition is found at 8.5 K (inset showing the resistive transition at 6 T applied field and its temperature derivative). The available field range is clearly not sufficient, so we need an extrapo- lation of the data towards 0 K. Commonly, the Werthamer-Helfand- Hohenberg (WHH) approach is employed for this purpose.27As shown by Cao et al. , this approach using the parameters α=0,λso=0 leads to a clear overestimation of Hc2in FeSe, yielding about 40 T for Ha∥con a signle crystalline sample. A similar value of ∼57 T (●) was obtained from a fluctuation-induced conductivity study using our experimental data on polycrystalline FeSe.24When consider- ing the spin-paramagnetic effect according to Ref. 28, then the fit using the parameters α=2.9 and λso=0 gives Hc2=25 T (randomly oriented polycrystal), which compares well to the result of Cao et al. with 12 T ( H∥c,△) and 29 T ( H/⊙◇⊞c,▲). The Hc2(0)-values of FeSe deduced with the spin-paramagnetic contribution thus compare well with the data of the other IBS materials.23,29 These results for Hc2(0)clearly demonstrate that the super- conducting properties of FeSe are affected by the magnetism of Fe, and in the case of bulk, polycrystalline samples, the situation is even worse due the presence of AF or FM phases. Thus, the supercon- ducting FeSe grains in such samples will always experience the local magnetic field created by the secondary phase particles. IV. CONCLUSIONS To conclude, we discussed the various phases appearing in superconducting bulk, polycrystalline β-FeSe samples intended for trapped field applications. By adjusting the Se-content in the pre- cursor material to the given furnace setup, it is possible to minimize the magnetic contribution, but a fully non-magnetic sample can- not be obtained due to the presence of needle-like δ-FeSe, which is not converted to β-FeSe during the heat treatment. The resulting microstructure is thus very complicated including several phases. To obtain good quality, bulk FeSe samples, which are useful for trapped field applications, a treatment leading to a higher sample density is strongly required. SUPPLEMENTARY MATERIAL See supplementary material for the x-ray data of the FeSe samples. ACKNOWLEDGMENTS This work is part of the SUPERFOAM international project funded by ANR and DFG under the references ANR-17-CE05- 0030 and DFG-ANR Ko2323-10, respectively. HO and SPKN grate- fully acknowledge the support by JSPS KAKENHI, Grant Num- ber JP16H6439. SPKN also wishes to thank JSPS for the fellowship (Grant No. P19354). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. AIP Advances 11, 015230 (2021); doi: 10.1063/9.0000167 11, 015230-4 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv REFERENCES 1F.-C. Hsu, J.-Y. Luo, K.-W. Yeh, T.-K. Chen, T.-W. 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5.0039532.pdf

J. Chem. Phys. 154, 084102 (2021); https://doi.org/10.1063/5.0039532 154, 084102 © 2021 Author(s).Efficient enumeration of bosonic configurations with applications to the calculation of non-radiative rates Cite as: J. Chem. Phys. 154, 084102 (2021); https://doi.org/10.1063/5.0039532 Submitted: 07 December 2020 . Accepted: 22 January 2021 . Published Online: 22 February 2021 Robert A. Shaw , Anjay Manian , Igor Lyskov , and Salvy P. Russo COLLECTIONS This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN r2SCAN-3c: A “Swiss army knife” composite electronic-structure method The Journal of Chemical Physics 154, 064103 (2021); https://doi.org/10.1063/5.0040021 Configuration interaction trained by neural networks: Application to model polyaromatic hydrocarbons The Journal of Chemical Physics 154, 094117 (2021); https://doi.org/10.1063/5.0040785 Spin relaxation in radical pairs from the stochastic Schrödinger equation The Journal of Chemical Physics 154, 084121 (2021); https://doi.org/10.1063/5.0040519The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Efficient enumeration of bosonic configurations with applications to the calculation of non-radiative rates Cite as: J. Chem. Phys. 154, 084102 (2021); doi: 10.1063/5.0039532 Submitted: 7 December 2020 •Accepted: 22 January 2021 • Published Online: 22 February 2021 Robert A. Shaw, Anjay Manian, Igor Lyskov, and Salvy P. Russoa) AFFILIATIONS ARC Centre of Excellence in Exciton Science, School of Science, RMIT University, Melbourne, VIC 3000, Australia a)Author to whom correspondence should be addressed: salvy.russo@rmit.edu.au ABSTRACT This work presents algorithms for the efficient enumeration of configuration spaces following Boltzmann-like statistics, with example appli- cations to the calculation of non-radiative rates, and an open-source implementation. Configuration spaces are found in several areas of physics, particularly wherever there are energy levels that possess variable occupations. In bosonic systems, where there are no upper limits on the occupation of each level, enumeration of all possible configurations is an exceptionally hard problem. We look at the case where the levels need to be filled to satisfy an energy criterion, for example, a target excitation energy, which is a type of knapsack problem as found in combinatorics. We present analyses of the density of configuration spaces in arbitrary dimensions and how particular forms of kernel can be used to envelope the important regions. In this way, we arrive at three new algorithms for enumeration of such spaces that are several orders of magnitude more efficient than the naive brute force approach. Finally, we show how these can be applied to the particular case of internal conversion rates in a selection of molecules and discuss how a stochastic approach can, in principle, reduce the computational complexity to polynomial time. Published under license by AIP Publishing. https://doi.org/10.1063/5.0039532 .,s I. INTRODUCTION There are several problems in chemical physics where one needs to enumerate points in an occupation or configuration space subject to some criterion on those configurations. Important exam- ples arise in statistical thermodynamics, where the calculation of partial partition functions over some subset of microstates is nec- essary for the determination of thermodynamical constants.1Sim- ilar quantum-statistical principles find applications ranging from path-integral molecular dynamics2and configuration interaction3,4 methods for bosons, with enumeration of configurations being a key difficulty.5,6The focus of this work will be that a number of molecular electronic properties can be determined from the knowl- edge of the configuration of quanta in vibronic modes subject to some energy constraint on the configuration.7–10In combinatorial mathematics, the problem of selecting integer occupations to sat- isfy a total “weight” is known as the knapsack problem.11In the simplest version of this, one has a knapsack that can carry a fixedvolume of objects, and one has a selection of objects to fill it with, each of which has an inherent volume and value. The problem is to fill the knapsack so as to maximize the total value while not exceeding the total possible volume. In our physical equivalent in this paper, the “knapsack” is the occupation vector, with each mode possessing an energy (equivalent to volume) and Franck–Condon (FC) factor12(equivalent to the value). We then wish not only to find the configuration that gives the best total value but also to enu- merate all possible configurations that satisfy the volume (energy) requirement. The difficulty with this problem is that it is of non-polynomial complexity,13and as such, no polynomial-time algorithm is known that can guarantee the correct solution. That is not to say that solutions do not exist; an obvious route would be to simply enu- merate every possible configuration and evaluate its volume and value, selecting the best possible solution from these. This has the added advantage that it solves our extended problem of determin- ing all the configurations within the energy criterion. However, the J. Chem. Phys. 154, 084102 (2021); doi: 10.1063/5.0039532 154, 084102-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp number of configurations increases exponentially with the num- ber of modes, rapidly making use of such a brute-force algorithm intractable. Specifically, in the 0–1 knapsack problem, (where each object or mode is either included or not), the number of configura- tions follows 2M, where Mis the number of modes. The best known heuristic algorithm (full polynomial-time approximation scheme) to solve the 0–1 problem scales as O(ϵ−1M1/ϵ)to get a solution with value at worst (1 −ϵ) times the optimum.14This does not solve the wider enumeration problem, however, and, by necessity, approaches the factorial scaling as the threshold ϵis tightened. Note that the more general bosonic problem, where each mode can have any inte- ger occupation, can always be rewritten in terms of the 0–1 problem by allowing copies of the modes. While the algorithms we present here are more widely appli- cable, we will focus on the specific problem of determining non- radiative rates. Understanding molecular photophysical processes is an important and difficult problem. This is particularly true in the field of exciton science15,16and in the design of optical devices and light-harvesting materials.17–19Typically, the goal is to either maximize or minimize the photoluminescent quantum yield of such devices, determined as the ratio of the radiative decay rate to the sum of radiative and non-radiative rates. The former comprise flu- orescence and phosphorescence, while the latter are predominantly internal conversion (IC, spin preserving) and intersystem crossing (ISC, spin flipping). These non-radiative processes are facilitated largely through coupling of electronic and vibrational states and thus occur when the molecule is excited into a configuration where the vibronic quanta satisfy an energy criterion. The simplest example would be the excitation from the ground to first excited electronic singlet states. The vibrational manifolds of each state then allow for an energy window of possible configurations that can result in internal conversion occurring, and each such configuration can then be weighted by a probability of resulting in either IC or fluores- cence. As such, the theoretical determination of the rate is naturally formulated as an example of the knapsack problem. Previous work by Valiev et al.20–22has focused on the solution of this problem using the Plotnikov, Robinson, and Jortner (PRJ) formalism for non-radiative energy transfer.9,10In their original and more recent papers, they give a scheme for the determination of the various quantum chemical properties that need to be calculated in determining these rates but focus more on the application of the method than on the algorithmic development for the exploration of configuration space. In this work, we will first briefly recap the theory underlying the formalism before analyzing the density of the configuration space. The main focus will then be on devising efficient algorithms for the full characterization of this space, both within this specific case study and also, more generally, for other knapsack-type physical problems. Finally, we will demonstrate the effectiveness of these algorithms and provide a fully open-source implementation, which can be found in Ref. 23. II. THEORY A. Background For a transition between two electronic states, | i⟩(initial) and |f⟩(final), with energy difference Eif, the general non-radiative ratein the PRJ formalism is written as20 knr=4 ΓfE(n)=Eif ∑ n∣Vif(n)∣2, (1) where the sum is over vibronic configurations n= (n1,n2,...,nM), Γfis the relaxation width of state f, and Vifis a coupling poten- tial that depends on which rate is being calculated. In this, we have assumed that the relaxation width does not depend strongly on n and that Eif≪Γf. These conditions generally hold true for tem- peratures around and below room temperature,24typically matching experimental conditions. Each nhas an associated Franck–Condon factor, determined using the Huang–Rhys (HR) factors, yj, of the Mvibrational modes, FC(n)=M ∏ j=1⎛ ⎝e−yjynj j nj!⎞ ⎠. (2) This factor effectively determines the extent of the vibrational over- lap, and thus the strength of the contribution to the rate, of a con- figuration. From this we see that modes increase with increasing yj and decrease with increasing nj. That is, modes with large HR factors can in general have larger occupations and still give significant con- tributions, or conversely, those with small yjare more likely to have low occupations. This will be important later, as it suggests a way to assess the importance of a configuration. There are then two physical regimes in which the PRJ formal- ism can be applied: under the Franck–Condon approximation or the Herzberg–Teller approximation.21For internal conversion, which we will be focusing on here, these two formulations are written as follows: The formulas for intersystem crossing are similar and can be found in Ref. 22, VIC,FC if(n)=−M ∑ j=1⎛ ⎝∑ νqm−1 ν⟨i∣∂ ∂Rqν∣f⟩Bνqj⎞ ⎠ ×(ωj(nj−yj)2 2yj⋅FC(n))1/2 , (3) VIC,HT if(n)=−M ∑ j=1M ∑ j′=1⎛ ⎝∑ νq∑ ν′q′(mνmν′)−1⟨i∣∂2 ∂Rqν∂Rq′ν′∣f⟩ ×BνqjBν′q′j′⎞ ⎠((nj′+yj′)2 2ωj′yj′)1/2 ×(ωj(nj−yj)2 2yj⋅FC(n))1/2 , (4) whereωjis the energy of the jth mode and Rνqis the qth coordinate of theνth atom with mass mν. Additionally, these depend on nuclear gradients along the vibrational modes, Bνqj, and vibronic couplings between the initial and final electronic states. Writing them in this way demonstrates how the rates can easily be simplified into dot products or matrix multiplications of a part that depends exclu- sively on the coordinates and a part that depends on the choice of configuration, kIC,FC=4 ΓfE(n)=Eif ∑ n[aFC(R)⋅zFC(n,y)]2, (5) J. Chem. Phys. 154, 084102 (2021); doi: 10.1063/5.0039532 154, 084102-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp kIC,HT=4 ΓfE(n)=Eif ∑ n[zHT(n,y)⋅AHT(R)⋅zFC(n,y)]2, (6) where the z,a, and Atensors are defined implicitly in Eqs. (3) and (4). Most importantly, all the rates, including the ISC ones, are modulated through a dependence on FC( n). Moreover, they are formed as a sum over configurations satisfying a fixed energy crite- rion, leading to our connection to the knapsack problem. Writing it asE(n) =Eifis somewhat disingenuous, however. For a system with a finite number of modes of fixed energy, the probability of find- ing a configuration (where occupations are necessarily integers) is infinitesimally small; this is due to the set of integers being count- ably infinite, while the set of real numbers (i.e., possible values of Eif) is uncountably infinite. Physically, asserting an exact energy cri- terion would be nonsensical anyway—the excitation is occurring between two electronic states with vibrational manifolds. Even at very low temperatures, the excitation band will have non-zero width, implying that in reality, the energy criterion is ∣E(n)−Eif∣≤δ, (7) whereδis some energy window reflecting the thermal variation in acceptable excitation energies. To determine the rates, we thus need to enumerate configuration space within this energy window. B. Density of configuration space Each configuration vector, n, can be thought of as a point in an M-dimensional configuration space, where each axis is rescaled by the weight or volume, wj, assigned to that object. The total volume of the configuration is thus the sum wjnj. The fixed-volume crite- rion then describes an ( M−1)-dimensional plane, and the expanded knapsack problem becomes finding all the lattice points that lie on that plane. If we extend to the range in Eq. (7), this describes a vol- ume in configuration space bounded by two such planes, and we wish to find all the lattice points within that volume. Intuitively, as M increases, the number of possible lattice points in either instance will increase too, and this will increase further if we use a wider window. In this section, we will look more rigorously at how many signifi- cant configurations are there in such a system and how the density of configurations behaves asymptotically. This will allow us to assess how successful any approximate methodologies are at characterizing the space. To determine the behavior, we consider how the volume of the acceptable region, and the number of lattice points within that region, increases with both Mand the window δ. The simplest two- dimensional case is shown in Fig. 1, where the hyper-surface is sim- ply a line described by ω2n2=E±δ−ω1n1. The acceptable volume is then the difference between the larger triangle and the smaller trian- gle. If we were to expand to three dimensions, these would be octants of a tetrahedron and, more generally in Mdimensions, a pair of reg- ularM-simplexes. The volume of such a simplex with side length L isLM/M! By defining our energy scale as E+δ≡1, the larger simplex has a volume of 1/ M!, and thus, the acceptable region as a proportion of this total volume is vM(δ)/VM=1−(1−2δ)M M!⋅M!=1−(1−2δ)M. (8) FIG. 1 . A representation of the enumeration problem when M= 2. The configu- rations are the lattice points, shown as bold black dots, while the energy criteria, E=Eif±δ, are described by the blue dashed lines. The “acceptable” region is then shaded in light blue, showing the density of configurations. As Mincreases, the size of this volume will also increase. Now, we consider the total number of lattice points within the larger simplex. If we consider the 2D case once more, then the num- ber of lattice points in the square with side E+δis simply the product of 1 + n1,max and 1 + n2,max . Similarly, in higher dimen- sions, the hypercube contains the product of all such 1 + ni,max. These maximum values are easily computed as ni,max=⌊E+δ ωi⌋. (9) TheM-simplex is then 1/ M! of the total volume of the hypercube25 and so, on average, contains 1 M!M ∏ i=1(1 +ni,max)∼(E+δ)M M!∏iωi such points. Finally, combined with Eq. (8), the acceptable region must, on average, contain ˜NM(δ)∼EM M!∏iωi[1−(1−2δ)M]⋅(1 +δ E)M . (10) Forδ≪E, we can expand the two terms in brackets as Taylor series. Retaining only the term linear in δ, this yields ˜NM(δ)∼2EMδ (M−1)!∏iωi+O(δ2), showing that the number of lattice points asymptotically tends to zero as the window shrinks, as expected. As mentioned earlier, the probability of a lattice point lying exactly on the hypersurface E(n) =Eis vanishingly small. In the regime of large M, we instead expand the brackets as binomial series. Approximating each weight as some fraction, ω≈αE/M, we get ˜NM(δ)∼2(M α)MM ∑ k=1M ∑ l=0(−2)k−1M! (M−k)!(M−l)!k!l!δk+lE−l. J. Chem. Phys. 154, 084102 (2021); doi: 10.1063/5.0039532 154, 084102-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Given the assumption again that δ≪E, this is dominated by the first term in the double sum, i.e., k=l= 0, This gives the much simpler asymptotic expression, ˜NM∼2MM αMM!∼2(e α)M , (11) where we have used Stirling’s approximation. From this, we see that for a fixed value of δ, the number of lattice points explodes expo- nentially in higher dimensions. Any approximations, therefore, that rely on finding a single acceptable point in the allowable region will therefore give increasingly worse results as the number of dimen- sions increases. Efficient strategies for finding only the most impor- tant configurations, using knowledge of the “values” of each point, are thus necessary to make this problem tractable. C. Franck–Condon weightings Having determined what the density of the configuration space is, the next step is to characterize the importance of various regions in that space. This will be problem specific depending on how we have defined the values in the knapsack problem. In the case of non-radiative rates, we want to find the configurations that give the largest contributions to the rate. As these are mediated primarily through Franck–Condon factors in all cases, we wish to find all con- figurations that have a factor within some threshold. We can rewrite the product in Eq. (2) as a sum in the exponent to get FC(n)=exp⎧⎪⎪⎨⎪⎪⎩∑ j(njlnyj−yj)⎫⎪⎪⎬⎪⎪⎭⋅M ∏ j=1(nj!)−1. All the lattice points in a region can be found as the union of all the lattice points on the planes defined by fixed | n| that intersect the region, where | n| =∑jnjis the total occupation number for that configuration. If we estimate the individual njas their average values, | n|/M, and yjby the average value, ¯y, we can use Stirling’s approximation to get FC(n)∼(M 2π∣n∣)∣n∣+M/2 (2πe¯y)∣n∣exp(−M¯y). (12) We note that any weighting function (i.e., probability distribution) defining the value of the objects in the knapsack problem will have this asymptotic form of envelope if its kernel follows the functional form, ρ(n)∼p1(∣n∣)⋅exp(p2(∣n∣))⋅M ∏ j=1(nj!)−1, (13) where p1and p2are arbitrary polynomials with real coefficients. This encompasses many different probability distributions, includ- ing those generated by partition functions in various ensembles1or, in general, where Boltzmann-like statistics are present.26 The importance of a given value of | n| is then Eq. (13) weighted by the number of acceptable configurations with that | n|. As will be discussed in more detail later, the number of such configurations fol- lows a binomial distribution, from some | n|minto |n|max, which are determined by the particular choice of target energy. By approximat- ing this symmetric binomial distribution as a normal distribution, the proportion, p(|n|), of total acceptable configurations with fixed |n| is given by FIG. 2 . Normalized importance factor for the Franck–Condon kernel ( ¯y=1) as a function of the total occupation, | n|, for various sized dimensions, M. p(∣n∣)∼√ αE Mπδexp{−E Mαδ(α∣n∣−M)2}, whereαis as defined earlier. Using the result from Eq. (11), we therefore have that the weighted importance value, P(|n|), is given by P(∣n∣)=p(∣n∣)⋅NM⋅ρ(∣n∣)∼p(∣n∣)(∣n∣ M)M/2−∣n∣ exp(−kM), (14) where the constant kis dependent on the kernel; for the Franck– Condon factors, k=¯y. From this, we see that the importance factor follows a form of gamma distribution, with the center of the dis- tributing shifting higher with the increase in dimension, M, and with the decrease in k. Figure 2 shows examples of these distributions for various val- ues of Mand | n|, with yminset at 0.5. This analysis shows that the FC factor acts as an envelope on configuration space, greatly reduc- ing the number of configurations that need to be considered, as those configurations outside the main envelope will not contribute significantly to the rate. From Eq. (12), we see that the width of the distribution follow M/2, implying that the number of signifi- cant | n| increases linearly with the dimension. However, this is also mitigated by the exponential, which adds a factor of e−yfor every extra mode, which is necessarily less than unity, as y>0. There- fore, theoretically, the problem is not of factorial complexity, as first seemed. III. METHODS The general problem we are trying to solve is to find all con- figurations nthat satisfy the energy criterion in Eq. (7), which have values,ρ(n), greater than some given threshold. The only assump- tions that we make are that the occupations, nj, are necessarily integers and that the “energy” can be written as ∑jnjwjfor posi- tive weights, wj. We have shown some of the theoretical proper- ties of such a system in Sec. II C, and in the present section, we J. Chem. Phys. 154, 084102 (2021); doi: 10.1063/5.0039532 154, 084102-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp will present algorithms for finding the solution. Implementations of these algorithms can be found in the open-source Knapsack software package.23 A. Screened brute force approach In the simplest brute-force approach to solving the problem, we enumerate every possible configuration and test it against the energy criterion. In this manner, no significant configurations can be missed. It may at first seem that there are infinitely many such configurations, but we can place an upper bound on the occupa- tion of each mode as no njwjcan be greater than the maximum allowed energy, E+δ. As such, Eq. (9) gives upper bounds for each occupation. There are then two possible algorithmic approaches to enumer- ating these for systems of arbitrary dimensions. The first is through hashing, where we loop an index, i(n) from 1 to N=∏jnj,max, where this index corresponds to a configuration. The configuration can be reconstructed as i(n)=n1+M ∑ j=2nj⎛ ⎝j−1 ∏ k=1nk,max⎞ ⎠. This is computationally efficient because you are simply increasing a counter, without the need for constructing Mnested loops. However, we can greatly improve the efficiency of the screening if we consider that nj,max should change depending on the occupa- tions for all k<j. That is to say that if the threshold is T, then given the occupation n1,n2,max should be calculated with a threshold of T/ρ(n)1and so on. The efficiency of this will then be affected by ourchoice of ordering of the modes, but this can be optimized by order- ing them beforehand such that the modes that give the strongest contributions to ρare first. In the case of the FC factors, this equates to ordering by decreasing the magnitude of the Huang–Rhys factors. We cannot use the index hashing approach with this kind of adap- tive screening, though, because the indexing no longer follows the simple form given above. In fact, even determining the exact num- ber of configurations, N, is a problem with the same computational complexity as finding all the configurations themselves. Instead, we can use a recursive approach combined with pre- tabulation of the maximum values of njfor each threshold up to the minimum threshold, T. This screened brute-force algorithm is described in Algorithm 1. The main downfall of this approach is that compilers have hard limits on the level of recursion allowed, mean- ing that there is a fundamental limit on the maximum possible num- ber of modes, M. Additionally, recursion, especially at high depths of recursion, can be notably slower than the hashing approach outlined earlier. The efficiency of this approach therefore needs to come from heavy screening, effectively based on limiting total occupation. We can estimate the total number of configurations that will be enumer- ated in this manner in the following, for example, where ρ(n) is the Franck–Condon factor. Taking logarithms of Eq. (2) and rearranging, we see that the maximum nfor a given threshold T=10−t≡e−˜tcan be estimated from yj−˜t=˜nj,maxlnyj−ln(˜nj,max!)≈˜nj,max[lnyj−ln˜nj,max+ 1], where we have again used Stirling’s approximation. Rearrang- ing and assuming a general maximum nof around 20 such that ALGORITHM 1 . Screened brute-force algorithm for finding configurations, n, that satisfy the energy criterion [Eq. (7)] within a threshold Ton the value ρ(n). 1: For each integer 0 ≤t≤−log 10Tand mode j, tabulate N(j,t)=nj,maxsuch that ρ(0,...,nj,max,..., 0)≥10−t 2: For each possible value of nj, tabulate T(j,n)=tsuch thatρ(0,...,nj=n,..., 0)≥10−t 3: Call Iterate (j=1,n=(N(1,tmax), 0, 0,...) 4:procedure ITERATE (j,n) 5:ifj=Mthen 6: fori←0,njdo 7: Let n′=nbut with nj=i 8: Check E(n′) against criterion 9: end for 10: else 11: fori←0,njdo 12: Let n′=nbut with n′j=i 13: Set tto the maximum of 0 or tmax−∑ k<j+1T(k,n′ k) 14: Set n′ j+1=N(j+ 1,t) 15: Call I TERATE (j+ 1,n′) 16: end for 17: end if 18:end procedure J. Chem. Phys. 154, 084102 (2021); doi: 10.1063/5.0039532 154, 084102-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp ln˜n<3≈ln 20, we get ˜nj,max(˜t)<˜t−yj 2−lnyj. (15) Therefore, following Algorithm 1, the total number of configura- tions can be estimated as ˜N=˜n1,max(˜t) ∑ n1=0˜n2,max(˜t1) ∑ n2=0...˜nM−1,max(˜tM−2) ∑ nM−1=0˜nM,max(˜tM−1), where the conditional threshold ˜tjis defined as ˜t−log10FC (n1,n2,...,nj−1, 0,...). This is seemingly a very complex sum, and we leave the somewhat involved algebraic manipulations to the supplementary material. However, the result is fairly simple, ˜N∼¯nM−1−(1 + ln ¯y+ ln ¯n)¯n2+O(¯n), (16) where ¯nis the maximum possible njand ¯yis the minimum yj. While this is clearly greatly reduced from the ¯nMscaling of the unscreened method, it is still exponential and will become unfeasible for large M. B. Reduction to quasi-polynomial time As a result, we want to find a way to reduce the scaling to some- thing computationally feasible. The key to this is the enveloping noted earlier in Eq. (12). This suggests that we can select a range of |n| such that ignoring all configurations with a total occupation out- side this range can be ignored without affecting the calculated value. The number of configurations NM(|n|) with a fixed | n| is given by NM(∣n∣)=1 (M−1)!M−2 ∏ k=0(∣n∣+k). (17) That is, the leading term goes as | n|M−2/(M−1)! As we will show shortly, this is strictly still asymptotically exponential in M, but the mantissa is close enough to unity that, for a pragmatic range of M, the complexity appears to behave polynomially. To see this, we note that for a fixed value of | n|, the leading term goes to zero as Mgoes to infinity and has a maximum at approx- imately | n| =M−1. The question then becomes as follows: How does the modal value of | n| depend on M. Differentiating Eq. (14)with respect to | n| and writing x= |n|/M, the | n|mode is found at the solution of 2Eα δx+ lnx−1 2x=2E δ−1. (18) We show numerical solutions for this for varying values of α and 2 E/δin Fig. 3. Notably—and this is shown rigorously in the supplementary material—there is a limiting solution for 2 E/δ sufficiently large. That is, for an energy window less than or equal to 2% of the total energy, the modal | n| is given by | n| =M/α. Equation (17) then tells us that the number of configurations asymp- totically follows ( e/α)M, which is exponentially increasing for α<e and decreasing for α>e. The value of αis essentially a measure of how much each individual mode contributes to the total energy on average. The behavior will therefore depend heavily on the spectrum ofωvalues: spectra with large spacings will have larger values of α, leading to a reduced number of configurations, while very densely packed spectra will have small αand much greater numbers of con- figurations. We will see later that the value of αin the case of the Franck–Condon factors is usually very close to, but slightly less than, e, and as such, the scaling appears locally to be polynomial, even though it is, strictly speaking, exponential; as such, we refer to it as “quasi-polynomial.” The algorithm for following such a procedure is also easily adapted from Algorithm 1 and can make use of the same screen- ing, yielding similar O(∣n∣2ln∣n∣)time savings. Only two changes are needed. We add an argument | n| to the Iterate procedure, and in line 6, we replace i←0,njwith i←max(∣n∣, 0), min(∣n∣,nM). Then, in line 15 where the recursion happens, we pass the new argument | n|−i. All we require, then, is a method of estimating the minimum and maximum values of | n| required to give values within a given threshold. Alternatively, we can try to estimate these by finding the |n| for which ρ(n) is at a maximum and estimate the spread of the Gamma distribution. For the former, we note that Eq. (17) tells us that, agreeing with intuition, the number of configurations increases with | n|, and so, it is much more important to find a tight bound on the maximum | n| than for the minimum | n|. In this regard, we can trivially compute a lower bound via the minimum target energy as ∣n∣min=⌈E−δ ωmin⌉. (19) FIG. 3 . The left panel shows numeri- cal solutions of equation for x= |n|/Min Eq. (18) for various values of α(marked on the lines) as a function of the energy parameters. These all tend to an asymp- tote in the large- E/δlimit, and these are plotted as a function of αin the right panel with the curve x= 1/αoverlaid. J. Chem. Phys. 154, 084102 (2021); doi: 10.1063/5.0039532 154, 084102-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp For the upper bound, we must analyze ρ(n). For the specific case of the Franck–Condon factor, from Eq. (2), that for a given threshold, T= 10−t, we have that, approximately, 10−t<exp⎛ ⎝∣n∣lnymax−∑ jyj⎞ ⎠(M ∣n∣)M , where we have estimated the individual njas |n|/Mand used the fact that 1/ n!<1/n. Under the reasonable assumption that ymaxhas no dependence on M, this rearranges to give the estimate ∣n∣max=M⋅exp⎧⎪⎪⎨⎪⎪⎩⎛ ⎝tln 10 −∑ jyj⎞ ⎠/M⎫⎪⎪⎬⎪⎪⎭. (20) As expected from earlier, this increases with Mbut is also somewhat affected by the values of yj. Interestingly, if we write the sum in the exponential as ∑jyj=M¯y, where ¯yis the average value of yj, we can expand the exponential as a Taylor series to see that for large M, ∣n∣max∼M−k+O(1 M). That is, the maximum increases roughly linearly with M. This, along with the analysis of the spread of the distribution of ρearlier (which is also linear in M), means that if we can determine approximately the modal | n|, the scaling from Eq. (17) will be considerably reduced. To do this, we use the method of Lagrange multipliers. In their most recent paper on the topic,22Valiev and co-workers demon- strated that they were using a very similar method to estimate the total contribution from the Franck–Condon factors. As a result, we are able to replicate their approximations directly for the calculation of non-radiative rates, and we will discuss in Sec. IV how our analysis here demonstrates that it becomes an increasingly poor approxi- mation as Mincreases. The method of Lagrangian multipliers is well-known and we do not need to discuss it in any detail. We wish to find the maximum of the distribution, ρ(n), which we assume to have kernel of the form given in Eq. (13). This is subject to the energy constraint in Eq. (7). The Lagrangian is thus L(n)=lnρ(n)−λ⎛ ⎝∑ jnjωj−E⎞ ⎠, whereλis a Lagrange multiplier. Differentiating with respect to nj and rearranging gives njexp{−p′ 1(nj) p1(nj)−p′ 2(nj)}=exp(−λωj), (21) which can then be solved for njand summed to give the modal value of | n|. The multiplier can be found by standard optimization means by finding the value of λsuch that Eq. (21) satisfies the energy criterion. The underlying assumption of this approach is that we are allowing njto take non-integer values; otherwise, the Lagrangian is discontinuous and thus undifferentiable. The resulting calculated njwill be fractional. Therein lies the problem with using this as a method for actually evaluating ρ—the configuration generated is completely invalid, and one is effectively approximating a sum across a distribution with its value at the maximum. As we know from the analysis earlier, the density of configurations increasesexponentially with Mfor nonzero δ, and as such, this type of esti- mate will be very poor for large M. On the other hand, it will also likely be poor for small Mbecause the nearest valid configurations— of which there will be relatively few—will not achieve this maximum value, and thus, the maximum will be a considerable overestimate. However, for our purposes, it is an excellent manner for determin- ing the two nearest integer | n| to the mode of the distribution and the value of ρat that point. Combined with our estimate for the maximum, we can then interpolate the rate at which the distribution decays by using the functional form in Eq. (13), effectively making the number of fixed-| n| values constant. We will therefore refer to this as the fixed-| n| algorithm. C. Stochastic sampling of configurations Another way to arrive at the conclusion of Eq. (18) but for gen- eralρof the form in Eq. (13) is to consider the following. If the spread ofρfollows M/2 and | n|maxfollows M, as per the previous analy- sis, we crudely expect the most important | n| to be at around M/2. The fixed-| n| algorithm from Eq. (17) will then follow the quasi - polynomial scaling; it is still strictly asymptotically exponential, but the asymptotics will only apply for relatively large M. If we wish to go to even larger systems, however, we need some non-deterministic or heuristic method of characterizing configuration space. The nature of the problem suggests that we can very simply stochastically sample the space by randomly selecting configura- tions. That is, for each j, we randomly select some njbetween 0 and nj,maxand then test the overall configuration to see if it satisfies the energy criterion. Such a method would naturally scale linearly with Mfor a fixed number of samples, as there are Mrandom numbers generated per sample. However, the number of samples necessary for such a uniform prior would necessarily scale factorially with M, making it a largely pointless endeavor. If we can find a starting guess in the manifold of acceptable configurations to seed the sampling, we could restrict the space that needs to be explored by only allow- ing samples of njclose to the guess. The natural manner to do this is to set a maximum number of occupations to change and/or the maximum amount that | n| is allowed to change. If a configuration so generated is “acceptable,” it is added to the pool of guesses from which the next sample is generated. However, from Eq. (11), we know that even this much reduced space scales exponentially with M. We can follow the determinis- tic approach of the previous sections, applying the distribution ρas the prior to our sampling procedure. We attach a weight to allow a mode to increment or decrement by one based on the change in the “value,” as defined by ρ, p± j=ln{ρ(n:nj) ρ(n:nj±1)}. (22) We take logarithms under the assumption that ρfollows the kernel in Eq. (13) and, as such, is exponential. The probability of the mode being selected is then simply P(nj→nj±1)=p± j ∑jp± j. (23) Equation (23) is certainly not the only possible choice of prob- ability distribution to enforce, especially as it divides incrementing J. Chem. Phys. 154, 084102 (2021); doi: 10.1063/5.0039532 154, 084102-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp and decrementing a mode into two separate probabilities. We have chosen to do it this way for two main reasons. First, modes that are least favorable to increment are most favorable to decrement, and vice versa; thus, having two essentially inverse probability distribu- tions make sense. Second, it allows us to sample for a fixed change in |n|: we select the change Δ|n| and then sample for k+increments andk−decrements such that k+−k−=Δ|n|. In this way, we can use the information from the fixed-| n| algorithm to further improve our sampling. In particular, we can use the Lagrangian estimate [Eq. (21)] as our initial starting guess and then sample away from this such that Δ|n| is weighted roughly quadratically, with a hard cutoff when we reach | n|minor |n|max. The stochastic algorithm is described fully in Algorithm 2. We note that the k±modes to increment/decrement are selected inde- pendently such that a mode can increase or decrease by more than one in a single sample, and it is possible to select the same sample multiple times. Algorithmically, this creates a problem of unique- ness when calculating the final value. There are two possible ways to approach this. The memory-intensive method is to set the group Gup as a hash table such that a configuration n′only gets added toGif it is not already in the table. The fixed memory approach, on the other hand, is to allow n′to be added to Gmultiple times and then, when the memory limit is reached, sort Gand screen out duplicates. Furthermore, at this point, configurations with ρbelow some threshold can either be discarded or written to file, with only the most “important” configurations kept in Gfor the next round of sampling. The additional expense in this approach, however, is that a considerable amount of writing to and reading from file is neces- sary, as the final value cannot be computed until all samples have been performed. The effectiveness of Algorithm 2 relies entirely on the fortuitous enveloping of configuration space by ρ(n) and the increasing pool of guesses in G. This means, however, that starting with only a single guess drastically reduces the rate at which ergodicity is approached (if ergodicity is reached at all). To this end, we use the traditional 0–1 knapsack problem fully polynomial time approximation scheme to generate Gguesses with values of Eevenly spaced between Eif−δ andEif+δ. The algorithm for doing so is well-known and described elsewhere.14 Finally, we must consider what number of samples will theo- retically be required to achieve a sufficient characterization of theconfiguration space. This is not an easy question to answer. Cer- tainly, we would expect, from Sec. III B, that the relevant space will still expand exponentially even with the restrictions on | n|. However, we would not necessarily expect the number of configurations with significant values of ρto increase in such a manner. Precisely, we would expect the number of samples needed to follow instead the following estimate: ˜Ns<∫∣n∣+Δ ∣n∣−Δρ(n)⋅ndn, where Δis some measure of the spread of the distribution ρ, and we are approximating this as a continuous distribution of possible n, hence the inequality. For the simplest possible version of the kernel in Eq. (13), that is ρ(n)=exp⎛ ⎝∑ jcjnj⎞ ⎠⋅∏ j(nj!)−1<∏ jexp(cjnj) nj for some constants cj. This equates to ˜Ns<∏ j∫¯n 0exp(cjnj)dnj=∏ j{1 cj[exp(cj¯n)−1]}, where ¯nis some estimate of the average maximum value of each nj satisfying the range | n|±Δ. Using the results of the previous sections, we know that | n|mode±Δ∼M/2. This means that | n| can range up to (asymptotically) MC for some constant C, making ¯n≈C. Replacing thecjwith their average, ¯c, this becomes ˜Ns∼(exp(C¯c)−1 ¯c)M , (24) which is still exponential in M. However, ¯cis typically negative; in the case of the Franck–Condon factors, cj= lnyj, and the yjare generally less than unity. This means that, like with the fixed-| n| procedure, the mantissa is close to 1, so ˜Nsappears to be polynomial until M gets very large. D. Parallelization It is important to note that both Algorithms 1 and 2 and, by extension, also the fixed-| n| algorithm are inherently parallelizable. Here, we briefly describe the strategy for each. ALGORITHM 2 . Algorithm for the stochastic sampling of configuration space weighted with a prior distribution defined via Eq. (23). 1: Generate some number of starting guesses in set G. 2:for all Samples do 3 Randomly select a guess n∈G 4: Sample Δ|n| from(∣n∣−∣n∣mode)2, such that | n|∈[|n|min, |n|max]. 5: Randomly select k+up to some fixed limit, and set k−=k+−Δ|n| 6: Sample k+modes to increment, k−modes to decrement, weighted as per Eq. (23). 7: Set n′=ˆk−ˆk+n. 8: if|E(n′)−Eif|≤δthen 9: Add n′toG 10: end if 11:end for J. Chem. Phys. 154, 084102 (2021); doi: 10.1063/5.0039532 154, 084102-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The screened brute force algorithm is perhaps the most com- plex because the task of determining exactly how many configura- tions will be screened is as complex as the enumerating itself. Cer- tainly, we cannot simply divide the work based on the maximum nof the first mode because the screening will necessarily result in fewer configurations for larger values of n1. Instead, we would pro- pose ordering the modes in terms of their maximum possible nfrom largest to smallest; in the case where ρis the Franck–Condon factor, this is equivalent to in the order of decreasing yj. The simple prob- lem of finding all bounds on the first kmodes can then be solved, and the work can be divided up accordingly. For example, if we take just the first two modes, for each value of n1, we would determine the maximum n2and then use n1n2as a heuristic to assess how many configurations there will be in total. In this way, a roughly equiv- alent number of configurations can be assigned to each thread or process, and Algorithm 1 can be carried out on each “chunk” inde- pendently. For the fixed-| n| version, the load balancing is simpler. We can quickly determine from Eq. (17) the maximum number of configurations for each | n| from | n|minto |n|max. As each | n| can easily be performed in parallel, the task is then dividing the total number of configurations evenly between each available thread or process. For the stochastic algorithm, every sample can, in principle, be done independently. However, there is a pooling of the config- urations into Gto be considered and whether the algorithm is per- formed entirely in-core or with regular dumps to file. There are then two distinct possible approaches. In the first, each thread or process performs a completely independent sample run, with its own pool of guesses. The initial guess pool could be the same for each run or be divided between all the runs. The advantage of this is that there is only overhead at the end when all of the samples from all the threads need to be combined. This works particularly well when all valid configurations are being written to file, as the sorting and screening greatly speed up the eventual recombination. The disadvantage is the duplicated effort, as the independent runs will likely find many of the same configurations, despite the stochasticity. The alternative then is to let all the threads share a guess pool. For every fixed number of samples, the thread-specific configurations are broadcasted out to all the other threads, effectively pausing the sampling on all threads while the update occurs. This does not work so well with writing to the file, however, as the records from each thread would need to be combined at every broadcast event, creating a considerable extra amount of file-based work. IV. RESULTS To test the efficiency of the new algorithms and the validity of our asymptotic analyses, we generated a random ensemble of sys- tems to be used with the density described in Eq. (2). For a selection of values of M(system size) from 20 to 50, the energies, ωi, and weights, yi, of each mode were selected from a uniform distribution on [0.01, 0.41] and a Pareto distribution with location 10, respec- tively. These were chosen to resemble realistic systems, as it would be exceptionally difficult to systematically find and compute such values for an ensemble of real molecules. In particular, the Pareto distribution for the weights generally results in a few modes with large weights (greater than 0.5), while most are less than 0.1, as is found in real molecular systems. The target energy for each modelsystem was then chosen as the mean of the ωimultiplied by a factor drawn from a normal distribution with location 16 and unit scale. This was based on the empirical observation that for polyacenes con- sidered elsewhere,20,21the excitation energy is roughly 16 times the average mode energy. For each value of M, ten such model samples were drawn and calculated using each of the algorithms described above, with the exception that the true brute force algorithm was only performed for the lowest value of M, as it is prohibitively expensive for larger M. All the calculations were run using the open-source knapsack soft- ware, running on four cores with 2 GB of memory. We also include calculations of the internal conversion rate, according to Eq. (5). For the molecular systems, electronic ground state ( S0) geome- tries were optimized using the B3LYP27,28and the def2-TZVP basis set,29in the Turbomole software package.30,31The first singlet excited state ( S1) was determined using TD-B3LYP with the same basis. These geometries were then used to calculate vibrational con- stants in the SNF package,32with the same functionals and basis, yielding the B-matrix in Eq. (3). This normal mode analysis was combined with the forces from the earlier optimizations to gener- ate the non-adiabatic coupling matrix elements needed in the same equation.33Finally, the S0toS1electronic excitation energy was cal- culated using DFT/MRCI34–36with BHLYP/def2-TZVP.37The ref- erence space was iteratively generated with ten electrons across ten orbitals, with a maximum excitation level of 2; all electron configu- rations with coefficients larger than 10−3were included at each step. Probe runs were calculated by discarding configurations with energy less than the highest reference energy; starting with a barrier of 0.6 Ehand then 0.8, with the finalized wavefunction built using a barrier of 1.0 Eh. Molecular orbitals with energies larger than 2.0 Ehwere not used. The number of modes included in the rate calculation was reduced by two means. First, a number of modes are equivalent by point group symmetry, typically reducing the number of modes by a factor of 2 (the order of the point group). Second, any modes with a Huang–Rhys factor of less than 10−6were discarded. A. Model systems For every model system, the brute force (where applicable), hybrid brute force, and fixed-| n| algorithms all gave values of the total density ∑nρ(n) that agreed to within a thousandth of a percent. For the stochastic algorithm, we first need to ascertain the appropri- ate number of samples to achieve convergence to the correct result. This was done by systematically increasing the number of samples from 108for a single model for each value of Mand determining at what number the density plateaus. Figure 4 shows this analysis for the three smallest system sizes. From the figure, we see that the density does flatten out and approaches the correct value ascertained from the non-stochastic algorithms. However, the deviation of the value after “convergence” is typically on the order of 0.1%, and adding additional samples does not help. The reason for this is the sensitivity to the pool of starting guesses—if the total | n| for each starting guess is too large, the algorithm “walks” to higher and higher | n|, missing the most important configurations as would be located using the fixed- |n| algorithm. As such, when generating the starting guesses as per step 1 of Algorithm 2, we took the following approach. The energy window was divided into 20 equally spaced chunks. We then used J. Chem. Phys. 154, 084102 (2021); doi: 10.1063/5.0039532 154, 084102-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . Convergence of the value of the density, ρ, with the number of samples used in the stochastic algorithm for a selection of the model systems described in the main text. The extrapolated asymptotes for each system are shown as dashed lines. the fully polynomial time algorithm for solving the 0–1 knapsack problem to find configurations close to each energy division. From this best guess, we scan for the nearest acceptable configuration with total | n| less than or equal to the target modal value of | n|, as determined by the Lagrangian method. This scan can be achieved rapidly by energy ordering the modes and iteratively adding or subtracting occupations so as to not change the energy by more than a given tolerance while approaching the correct total | n|. In practice, we have seen that this can typically be found in no more than 20 iterations. Seeding the guess pool in this way led to the promising results of Fig. 4; in contrast, using fewer guesses or not adjusting the value of | n| appropriately led to considerable under- estimates of the value of the density, sometimes of several orders of magnitude. Having determined the appropriate number of samples to use in the stochastic algorithm, we can now look at the scaling of both the computational time and the number of significant configura- tions found using each algorithm. We show these in Fig. 5. Note that we expect the hybrid brute force to find the most configurations,with consistently more configurations screened out due to insignifi- cance in both the fixed-| n| and stochastic algorithms. The amount of computational time should naturally scale roughly linearly with the number of configurations; however, the stochastic algorithm has a considerable overhead due to the need to sort and write to file con- figurations whenever the allocated memory is full. This is due to the redundancy inherent in the algorithm. We see these trends clearly in Fig. 5. Importantly, both the hybrid (screening-only) and fixed-| n| deterministic algorithms appear to show strictly linear log–log scaling of number of valid configurations, which implies polynomial behavior with respect to increasing system size. This demonstrates that the density of signif- icant configurations (which also satisfy the energy criterion) is not, in fact, exponential. The time taken—which encompasses all config- urations checked, not just those that contribute significantly—is not linear for the hybrid algorithm, however, with a clearly discernible slight increase in the gradient with the increase in M, indicating exponential behavior. The fixed-| n| algorithm, on the other hand, appears to be linear over the range of Mconsidered, with a straight line fit (with a gradient of 14.4) explaining over 99% of the observed variance. This reflects the predicted quasi-polynomial behavior due to restricting the configuration space. The results for the stochastic algorithm are less clear. There is a much greater spread of timings within each set of model systems due to the number of samples being required to reach convergence dif- fering between them. On the other hand, the number of significant configurations stays fairly consistent for each value of M, suggesting that the variability is largely due to the ease of finding suitable start- ing guesses. Interestingly, the number of configurations is higher for the stochastic algorithm than all but the full brute force algo- rithm, which implies that our importance criterion is not strong enough, and a more efficient sampling could be achieved by tighten- ing this. However, this effect lessens with the increase in M, and most importantly, there is clearly a point at which the time taken for the stochastic algorithm will be considerably less than that for the other algorithms. In fact, this has already started to happen for the largest value of Mconsidered in Fig. 5. This is because the cost of each individual sample does not increase with the system size (beyond a small amount of additional overhead in the sorting steps), whereas the recursive nature of the other algorithms means they get far more memory-intensive and thus expensive as Mincreases. Finally, we note that all three new methods perform orders of magnitude bet- ter than the full brute force approach for the smallest system (105 FIG. 5 . Log–log plots of the scaling in computational wall time (left) and num- ber of significant configurations (right) for the ensemble of model systems described in the main text. Full brute force results (black circles) are given only for the lowest value of M, as based on the number of configurations that would need to be checked for the next biggest, such a calculation would take several millennia. J. Chem. Phys. 154, 084102 (2021); doi: 10.1063/5.0039532 154, 084102-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 6 . Comparisons of the modal value of |n| (left) and total density ρ(right) as predicted by the Lagrangian procedure vs that found by tabulating all configura- tions from the fixed-| n| algorithm. times faster in the case of the fixed-| n| algorithm), and this improve- ment will only increase as Mdoes, given the factorial scaling of the naive method. It is also worth looking at the Lagrangian approximation of Eq. (21) in more detail, as it affects the efficacy of both the fixed- |n| and stochastic algorithms. Figure 6 shows the predicted modal |n| and the estimated density ρas determined using this approach, compared to that found from the hybrid brute force algorithm, across all model systems. Generally speaking, the Lagrangian pre- diction of | n| is within 1 of the empirical mode and is thus a very effective prediction for the overall distribution of significant |n|. However, the approximation to the density is very poor; it is on average two orders of magnitude too large or too small, with the nature of the error depending on the exact distribu- tion of weights and energies. As such, it neither provides a con- venient bound nor good approximation to the density. This is important in the context of non-radiative rates as this method has been used previously to estimate rates,22which being directly pro- portional to this density will also likely be in error by orders of magnitude. B. Example rates As an important final test, we apply the new algorithms to the determination of internal conversion rates for three molecular systems: anthracene, tetracene, and indole. We have selected these because they are molecules that are known to have a viable internal conversion pathway38–41and have a feasible number of vibrational modes for the calculations. Our intention here is not to assess the validity of the physical assumptions underlying the formalism inher- ent in Eq. (5), as that is not the topic of this paper and has been considered elsewhere.20,42In fact, it is very difficult to validate specif- ically the calculated internal conversion rate for a single pathway, as there is very limited experimental data for molecules that we can treat computationally, and moreover, these experiments will typi- cally be in solvent. Instead, we are demonstrating that for realistic systems, where the relevant parameters have not been randomly esti- mated as per the model ensembles above, the new algorithms give consistent results. In Table I, we tabulate the results from these algorithmic com- parisons, along with a summary of the relevant parameters. Cru- cially, we see that the hybrid and fixed-| n| algorithms consistentlygive the same results, with values of | n|maxand | n|mode determined more by the target transition energy than by the dimension M. As it is this parameter that controls the computational cost of the fixed-| n| method, this is promising. However, the number of configurations with a Franck–Condon factor greater than the fixed threshold (here chosen as 10−12) does increase noticeably with M. This is a partic- ular problem for the stochastic method, where we see that in the case of indole, convergence has not quite been achieved even at 2.5×1010samples. However, the stochastic results are consistently much closer to the deterministic results than those found through the Lagrangian approximation. This is particularly severe in the case of Indole where the sheer number of significant configurations has resulted in the Lagrangian result being six orders of magnitude too small. Finally, we see that the fractional energy parameter, α, which controls the scaling as per Eq. (18), is close to but less than e. This effectively means that, as expected, the scaling is still strictly expo- nential, but for this size of systems, the cost is not yet prohibitive. TABLE I . Internal conversion rates as calculated using Eq. (5) for anthracene, tetracene, and indole for each of the algorithms described in this work. Additionally, we give the number of modes M, target energy E, and mode-weight parameter αfor each along with algorithm parameters. The energy window δin all cases was 40 meV. The brute force rate for anthracene was 2.120 ×105s−1. Anthracene Tetracene Indole M 12 22 29 ES0→S1(eV) 3.172 2.435 4.689 α 1.8 2.5 2.4 |n|min 11 8 10 |n|max 18 16 24 |n|mode 14 12 19 No. samples 1 ×1085×1092.50×1010 No. config. 2.89 ×1052.02×1083.07×109 kIC,FC (s−1) Hybrid 2.120 ×1055.276 ×1061.258 ×109 Fixed-| n| 2.120 ×1055.276 ×1061.258 ×109 Stochastic 2.094 ×1055.080 ×1069.716 ×108 Lagrangian 1.825 ×1051.142 ×1076.885 ×103 J. Chem. Phys. 154, 084102 (2021); doi: 10.1063/5.0039532 154, 084102-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp These results are therefore promising for the application of these new algorithms to much larger systems. V. CONCLUSIONS In this work, we have developed three novel methods for the enumeration of bosonic configurations to calculate a density ker- nel with generalized Boltzmannian statistics. The two deterministic algorithms, described by Algorithm 1, are based on heavy screening of the configurations and division of the simplicial volume into fixed occupation slices, respectively. The third stochastic approach, given in Algorithm 2, is based on an importance sampling generated from a pool of guesses, using heuristic solutions to the knapsack prob- lem as a starting point. Our focus in validation has been on applying these new methods to the problem of calculating non-radiative rate constants for molecules, but the algorithms have been designed to be agnostic to the specific choice of the physical problem. Our asymp- totic analysis and subsequent numerical investigations demonstrate that the seemingly factorial scaling of these combinatorial problems is unphysical. In particular, for an ensemble of model systems with the simplest Boltzmann kernel, the number of significant configu- rations, in fact, seems to scale as a high-order polynomial. This has important consequences for future algorithmic developments, as it suggests that by improving the choice of the importance sampling procedure, the problem can eventually be solved in polynomial time. The efficient sampling and achievement of ergodicity in the stochastic approach are not simple, however, and there is still clearly much that could be improved. In particular, the sheer volume of the space to be sampled is so staggeringly large that the choice of appro- priate seeding guesses is itself a computationally intensive challenge. This is alleviated somewhat by our observations of where the max- imum total occupation can be found, as reflected in the fixed-| n| algorithm, as this necessarily reduces the space to be searched. It is not sufficient, however, to limit ourselves only to the distribu- tional maximum, as demonstrated by the poor performance of the Lagrangian approximation in many cases. One possible approach would be to solve a simpler problem in the subspace of kmodes with the largest weights and then perturbatively expand into the full space from there. This would then allow for more efficient directing of the sampling procedure to the important regions of configuration space. The difficulty comes from defining where the cutoff for the size of the subspace should be and assessing how this affects the final results. Similarly, the relevant molecules in new materials design are typically much larger than the three molecular systems we have con- sidered here, with Mtypically being in the range of 50–100, after symmetry and insignificant Huang–Rhys factors have been taken into account. We did not include such large molecules because of the sheer amount of computational effort in obtaining the neces- sary parameters to a sufficient accuracy, although we hope to do this in a follow-up study. Our model tests do show scalability up to the lower end of this range of M, and we would anticipate from extrap- olating the scaling behaviors that it should be feasible to consider these larger systems. However, it is an open question whether the sampling will be sufficient for the highest values of M. It would also be worthwhile to compare how changing the choice of polynomials in the kernel definition [Eq. (13)] affects the scaling parameter, α, and whether this can be used to improve sampling efficiency. Theseare mathematically challenging questions that suggest interesting pathways for future developments. SUPPLEMENTARY MATERIAL See the supplementary material (CSV file) for the input param- eters of all the molecular and model systems along with XYZ coordi- nates for the molecular systems and (PDF) additional derivations of results. Open-source implementations of all the methods described here can be found in the Knapsack software in the following GitHub repository: https://www.github.com/robashaw/knapsack (Last accessed: 11 February 2021) ACKNOWLEDGMENTS This work was supported by the Australian Government through the Australian Research Council (ARC) under the Center of Excellence scheme (Project No. CE170100026). This work was also supported by computational resources provided by the Australian Government through the National Computational Infrastructure National Facility and the Pawsey Supercomputer Center. DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material. REFERENCES 1J. P. Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complex- ity, Oxford Master Series in Statistical, Computational, and Theoretical Physics No. 14 (Oxford University Press, Oxford, New York, 2006). 2B. Hirshberg, V. Rizzi, and M. 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5.0037913.pdf

J. Chem. Phys. 154, 074702 (2021); https://doi.org/10.1063/5.0037913 154, 074702 © 2021 Author(s).Theoretical study of surface segregation and ordering in Ni-based bimetallic surface alloys Cite as: J. Chem. Phys. 154, 074702 (2021); https://doi.org/10.1063/5.0037913 Submitted: 17 November 2020 . Accepted: 31 December 2020 . Published Online: 16 February 2021 Dong Luan , and Hong Jiang COLLECTIONS This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN Recent progress in simulating microscopic ion transport mechanisms at liquid–liquid interfaces The Journal of Chemical Physics 154, 080901 (2021); https://doi.org/10.1063/5.0039172 r2SCAN-3c: A “Swiss army knife” composite electronic-structure method The Journal of Chemical Physics 154, 064103 (2021); https://doi.org/10.1063/5.0040021 From the dipole of a crystallite to the polarization of a crystal The Journal of Chemical Physics 154, 050901 (2021); https://doi.org/10.1063/5.0040815The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Theoretical study of surface segregation and ordering in Ni-based bimetallic surface alloys Cite as: J. Chem. Phys. 154, 074702 (2021); doi: 10.1063/5.0037913 Submitted: 17 November 2020 •Accepted: 31 December 2020 • Published Online: 16 February 2021 Dong Luan and Hong Jianga) AFFILIATIONS Beijing National Laboratory for Molecular Sciences, College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, China a)Author to whom correspondence should be addressed: jianghchem@pku.edu.cn ABSTRACT Ni-based bimetallic materials are promising for a series of important heterogeneous catalytic reactions because of their low cost and potential high activity. In order to understand their catalytic performances in catalytic processes, it is important to know the struc- tural properties of these bimetallic surfaces, including, in particular, how the guest metal is distributed in the nickle host at finite temperature. By using the cluster expansion model built on density-functional theory calculations, combined with Monte Carlo sim- ulation, we study the segregation and ordering behaviors in several frequently studied Ni-based bimetallic catalysts NiX (X = Fe, Co, and Cu). We found that Ni tends to segregate to the top most layer of the surface in NiFe and NiCo, while Cu tends to segregate to the topmost layer of NiCu surfaces. NiCo and NiCu lose short-range order quickly as the temperature increases. Under low tempera- ture, NiFe forms an ordered Ni 3Fe structure, which, however, disappears above 550 K because of the order–disorder transition. These findings can provide important information for the understanding of the stability and activity of Ni-based bimetallic catalysts at high temperatures. Published under license by AIP Publishing. https://doi.org/10.1063/5.0037913 .,s I. INTRODUCTION Forming a bimetallic surface is one of the most widely used approaches to tune the catalytic activity and stability of transi- tion metal heterogeneous catalysts.1–8For example, a lot of efforts have been invested in the exploration of Ni-based bimetallic cat- alysts for the dry reforming of methane (DRM),9–11which con- verts CH 4and CO 2to synthesis gas with a favorable CO and H 2 ratio and therefore has attracted a lot of interest in recent years.12,13 Nickle is the most widely investigated catalyst for DRM, but it suf- fers from rapid deactivation as a result of surface carbon depo- sition. Many noble metals are found to show high DRM activ- ity and also exhibit stronger resistance to carbon deposition than Ni,14but they are economically not feasible for large scale indus- trial utilization. Great progress has been achieved in the experimen- tal preparation and characterization of Ni-based bimetallic catalysts with significantly improved catalytic activity and/or stability for theDRM reaction,9,10,15,16but a deep understanding of the underlying microscopic mechanism is still missing, partly due to the highly complicated nature of DRM catalysis and the difficulty in estab- lishing quantitative structure–property relationship under reaction conditions.9 In order to understand the origin of the activity and/or stabil- ity of bimetallic catalysts, it is crucial to obtain information on their structures under practically relevant reaction conditions besides understanding the intrinsic chemical properties of bimetallic sur- faces. As it is well known, the chemical composition and ordering structure of a bimetallic surface usually differ from those of the bulk, a phenomenon known as surface segregation, which has cru- cial effects on its catalytic activities.5A lot of factors can contribute to the surface segregation of bimetallic catalysts, including the com- position, temperature, and reactive environments (the presence of gas phase reactants). For alloy nanoparticles, the size, morphology, and the support can also play crucial roles.17There are abundant J. Chem. Phys. 154, 074702 (2021); doi: 10.1063/5.0037913 154, 074702-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp experimental findings indicating that the surface compositions of bimetallic catalysts can be significantly modified in different reac- tive conditions and/or on different supports (see Refs. 5, 18, and 19 and references therein for more detailed discussions), but a com- prehensive understanding of underlying microscopic mechanisms of these phenomena is still very challenging. A lot of efforts have been invested in modeling bimetallic catalysts under reaction con- ditions based on different theoretical approaches. For example, Wang et al. investigated the segregation of Pt on the surfaces of Pt–Ni nanoparticles based on the Monte Carlo (MC) simulation using the modified embedded atom model (MEAM).20Divi and Chatterjee proposed to use the distribution coefficients obtained from MC simulation with the embedded atom model potentials to characterize the distribution of metal species in alloy nanopar- ticles.21Hwang and co-workers investigated atomic arrangements in AuPd and AuPt alloyed surfaces using MC simulations with first-principles based lattice Hamiltonian.22,23Similar approaches have been also applied to investigate adsorbate induced surface segregation in a series of studies.24–29There have also been sev- eral high throughput theoretical studies of environment-dependent surface segregation, usually based on simplified model bimetallic surfaces.30,31 Bearing in mind the application in the DRM process, in this work, we consider several Ni-based bimetallic catalysts. Most of the previous theoretical studies of bimetallic catalysts for DRM are based on highly simplified model surfaces or nano-clusters.15,32–34 In this work, we take a more realistic strategy and investigate the segregation and ordering behaviors of Ni-based bimetallic surfaces at finite temperature based on the cluster expansion model repre- sentation of bimetallic surfaces.35The CE model is widely used in the theoretical study of alloy materials with configurational (sub- stitutional) disorder,36,37and it has been also increasingly used for surfaces and nano-structured materials23–25,38,39(see Ref. 35 for a recent comprehensive review). As clearly demonstrated in previous works mentioned above, the presence of key intermediate species such as oxygen or carbon adsorbates can play crucial roles in the surface segregation of bimetallic catalysts under working conditions. To simplify the computational models, here we focus on the temper- ature effects on surface segregation without taking into account the effects of adsorbates. This article is organized as follows. In Sec. II, we briefly present the theoretical methods used in this work, including the cluster expansion method to describe the configurational dependence of bimetallic surfaces and the Monte Carlo simulation technique to characterize ordering behaviors at finite temperature, and give some computational details. In Sec. III, we present the cluster expan- sion models of NiX (X = Fe, Co, Cu) surfaces and discuss the seg- regation and ordering properties of these surfaces as revealed by MC simulation. Section IV summarizes the main findings of this work. II. THEORETICAL METHODS AND COMPUTATIONAL DETAILS A. The cluster expansion method We give a brief overview on the cluster expansion (CE) method using generic binary alloy A xB1−xas an example, and moresystematic formulations for general multi-component and multi- sublattice cases can be found in Refs. 36, 37, and 40–42. Consider an alloy with Nsites that can be occupied by A or B. The occupation of each site ican be represented by a pseudo- spin variable σi, which takes the value of +1 ( −1) when the site is occupied by A (B). The overall occupation configuration of Nsites can be described by the vector σ= (σ1,σ2,. . .,σN). An n-body cluster αdenotes a collection of n-sites, ( i1,i2,. . .,in). Taking into account the symmetry of the lattice, the total energy of the N-site system can be expanded as E(σ)=N∑ αmαJαΠα(σ), (1) where αdenotes symmetrically distinct clusters, mαis the multiplic- ity of the αcluster per site arising from the spatial symmetry of the lattice under study,36,43Jαare effective cluster interactions (ECIs), andΠα(σ)denotes the cluster function defined as Πα(σ)=1 Nm α∑ (i1<i2<⋯<in)∈ασi1σi2⋯σin. (2) For the total energy and many other properties, the expansion above can be truncated to consider only a few short-range two-body and three-body (and sometimes four-body) clusters within a cer- tain spatial cutoff to achieve an accurate representation. In practice, ECIs are obtained by fitting the energies of tens of representative supercell configurations in their locally relaxed structures calculated by density-functional theory (DFT).36,44For given training data, we use the k-fold cross validation (CV) technique to obtain opti- mal CE models without suffering from over-fitting.45To be more specific, the whole dataset is partitioned into kgroups of roughly equal size, and each time one group is used as the testing set and the remaining data as the training set; the k-fold CV score is then calculated by S(k) CV=⌟roo⟪⟪op ⌟roo⟪mo⟨⌟roo⟪mo⟨⌟roo⟪mo⟨⌟roo⟪⟨o⟪1 kk ∑ j=11 nj∑ i∈Sj(Ei−ˆE(i,j))2, (3) where Eiis the calculated DFT energy of structure i,njis the number of structures in the jth group, ˆE(i,j)is the predicted value of structure iobtained from a least-squares fit to the energies of structures excluding the j-group. It should be noted that the ener- gies presented in this work are all normalized by the number of primitive unit cells in each supercell structure. This process is repeated 50 times, and the CE model that gives the minimal aver- age CV score is selected as the optimal one to be used for fur- ther study. When kis equal to the total number of structures ( Ns), S(k) CVreduces to the leave-one-out cross validation (LOOCV) score that is often used in characterizing the predictive capability of CE models.46 The ATAT package43,44is used to facilitate the building of the optimal CE model. In particular, we use the default algorithm imple- mented in ATAT43to generate the training set to build the CE mod- els in which (1) starting from the primitive cell, all possible supercells with increasing size are generated iteratively; (2) for each supercell, statistically important configurations are selected (see Ref. 43 for J. Chem. Phys. 154, 074702 (2021); doi: 10.1063/5.0037913 154, 074702-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp more details) to perform DFT calculations and enter the training set; and (3) the iteration stops until a certain level of accuracy measured by the LOOCV is reached. As a result, the training sets generated in this way are typically made of structures of different supercell sizes. B. Monte Carlo simulation and order parameters The CE model for the total energy can be used to calculate the thermodynamic properties of alloy systems by using Monte Carlo simulation techniques. In this work, we are mainly interested in the distribution of the second metal in the Ni-based bimetallic sur- face at finite temperature. The central quantity in the statistical thermodynamics of alloy systems is the cluster correlation function (CCF), defined as the ensemble average of the corresponding cluster function ¯Πα, ξα(T)=∑σΠα(σ)exp(−E(σ)/kBT) ∑σexp(−E(σ)/kBT). (4) The CCF of the one-body cluster corresponding to the site iis related to the concentration of the species A (+1) or B ( −1) on the sitei,41 cA/B i=1 2(1±ξi). (5) Two-body CCF ξijindicates the pairing tendency on the sites iand j. For a fully disordered (f.d.) alloy, ξijis equal to the product of the one-body clusters on the sites iand j,41ξ(f.d.) ij =ξiξj. The deviation of ξijfrom ξ(f.d.) ij indicates the existence of short-range order (SRO). Alternately, SRO can also be char- acterized by Cowley–Warren parameters,47defined for a binary alloy as48 si,j=ξij−ξiξj 1−ξiξj, (6) which can be directly related to the measured intensity in diffraction experiments.48 All MC simulations are conducted using the emc246module of the ATAT package.44The simulation temperature range chosen is [200 K, 1500 K]. The bimetallic surface is simulated by a three-layer supercell box with 10 000 atoms per layer with a fixed composition of Ni:X = 3:1. The MC convergence criterion is set such that the preci- sion of the statistically averaged energy is within 10−5eV (see Ref. 46 for more details). C. DFT calculations All DFT calculations are conducted by using the Vienna ab initio simulation package (VASP), which uses the projector aug- mented wave (PAW) approach to treat core-valence interactions and the plane-wave basis to solve DFT equations.49–51Electron interac- tions are treated at the generalized gradient approximation (GGA) level with the PBESol functional.52The cutoff energy for the plane wave basis expansion is set to 450 eV, and the Methfessel–Paxton53 smearing with a width of 0.2 eV is used for the electronic occupation. Considering that pure Ni is ferromagnetic, all the calculations arespin polarized. The dipole correction54,55is considered in all sur- face calculations. We considered three Ni-based (NiX, X = Fe, Co, and Cu) FCC(111) bimetallic surfaces using the repeated slab mod- eled with six atomic layers and a 15 Å vacuum region. The top three layers are allowed to fully relax and their sites can be occupied by Ni and X randomly, and the bottom three layers are fixed and all occupied by Ni. The corresponding lattice constant of the bulk Ni is 3.461 Å, which is obtained by the optimization of the bulk Ni using the PBESol functional. All the slab structures are optimized until the energy difference between two ionic steps is smaller than 10−4eV. For structural relaxation, we use a 13 ×13×1k-mesh corresponding to the primitive surface unit cell, which is scaled to keep the density of k-points in the reciprocal space approximately equal in actual supercell calculations. As mentioned above, the struc- tures in the training set can have different supercell sizes, and in this work, the largest supercells considered are four times of the primitive surface cell. To avoid possible inconsistency in numeri- cal accuracy due to using different k-mesh for different supercells, the energies of all configurations initially relaxed by a sparse k-mesh are re-calculated by using a fine k-mesh of 24 ×24×1 corre- sponding to the primitive surface unit cell with the fixed geometric structure. III. RESULTS AND DISCUSSION A. Validation of the CE models As shown in Table I, for each bimetallic surface, about 80 con- figurations are generated with the Ni composition falling in the range of [0, 1] in the top three layers. The optimal CE model is obtained by checking the convergence of the average 10-fold CV score as a function of the cluster diameter, first considering the two- body clusters and then three- and four-body clusters in turn. As shown in Fig. 1, in general, the CV score decreases first as a func- tion of the cluster diameter and then increases or remains essentially constant at a certain diameter, indicating that further adding larger clusters does not improve the predictive capability of the CE model. An interesting feature is that for NiCo (NiCu), even only considering one-body clusters can already lead to a CV score of about 3.5 (8.5) meV/cell, which implies that the formation energy of NiCo (NiCu) bimetallic surfaces is mainly determined by the concentration of Co (Cu) in different surface layers. In the NiFe bimetallic surface, the two-body clusters play a more important role, but their contribu- tions become negligible when their diameter goes beyond the second nearest neighbors. In all three cases, short-range three-body clusters can slightly reduce the CV score, and the contributions of four-body clusters can be neglected. TABLE I . The number of configurations ( N), the average 10-fold CV score ( ⟨SCV⟩) and its variance ( σCV), and the root mean squared deviation (RMSD) corresponding to the final optimal CE model for each bimetallic surface considered in this work. Surface N ⟨SCV⟩(meV) σCV(meV) RMSD (meV) NiFe 81 6.4 0.38 4.2 NiCo 81 2.2 0.19 1.2 NiCu 76 2.9 0.13 2.1 J. Chem. Phys. 154, 074702 (2021); doi: 10.1063/5.0037913 154, 074702-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . The convergence of the average 10-fold CV score with respect to the cluster diameter of NiFe, NiCo, and NiCu bimetallic surfaces. The convergence is checked by considering two-, three-, and four-body clusters in turn. As a further validation of the CE models, we also investigate the convergence of the average 10-fold CV score as a function of the training set size by the following procedure: a subset of Nsstruc- tures is extracted randomly from the full dataset to calculate 10-fold CV score, and for each Ns, this is repeated 50 times, from which the average CV score and its variance as a function of Nsis obtained. The plots collected in Fig. 2 clearly show that the CV score converges well with respect to the training set size. Figure 3 illustrates the two- and three-body clusters included in the final CE models. We have also used Cook’s distances56to detect the possible presence of influential configurations (outliers), and the results are presented in Fig. S1 of the supplementary material, which shows that there are very a few outliers in the training set, and the values of ECIs in the CE models change very little by excluding those outliers. The final optimal CE models for NiFe, NiCo, and NiCu have the average 10-fold CV scores of 6.4 meV, 2.2 meV, and 2.9 meV, respectively, with variances of less than 0.4 meV, indicating their high accuracy. The insets in Fig. 2 show the CE fitted formation energies against the DFT calculated ones, which show very good agreement between them. These results clearly indicate the accuracy of the CE method to describe the configuration dependence of the energy on these bimetallic surfaces. B. Surface segregation Using the CE models we have obtained, we first investigate the surface segregation behaviors of NiX bimetallic surfaces, which are crucial for their catalytic properties. Previous theoretical stud- ies of surface segregation are often based on simple model cal- culations,61–63i.e., by comparing the energetic preference for the second metal atom to stay on the top layer with respect to stay- ing in the bulk, often modeled by the third layer. Surface segre- gation in bimetallic surfaces has also been investigated by using atomistic simulation techniques based on semi-empirical force fields.64The CE-based Monte Carlo simulation can be used to pro- vide more complete information on surface segregation at finite temperature. FIG. 2 . The average 10-fold CV and its standard error (both in meV) as a function of the training data size. The insets show the comparison of the CE fitted values against the DFT data using the final CE model for each bimetallic surfaces. One can first obtain some hints about the surface segregation trends from ECIs corresponding to one-body clusters. In terms of the convention used in this work, σ(Ni) = +1 and σ(X) = −1, which indicates that if the ECI of a one-body cluster is positive, then the occupation of the guest metal on the corresponding site is energeti- cally favorable, and vice versa. Table II collects the ECIs of one-body clusters on different bimetallic surfaces. There are three layers that can be occupied by Ni and X randomly, each with one distinct site, leading to three one-body clusters in every system. In NiFe, the one-body ECI on the top and second layers is strongly negative and becomes positive in the third layer, indicating that Fe tends to migrate into the bulk of the NiFe bimetallic surface, with the top lay- ers mainly occupied by Ni. In NiCo, the one-body ECI is negative on the top layer, positive in the second layer, and negligibly positive in the third layer, which implies that Co tends to accumulate in the second layer, and the top layer is favorably occupied by Ni. The one- body ECIs in NiCu show very different features, with the one on the top layer strongly positive and those on the second and third layer negative, indicating a strong tendency of Cu to occupy the top layer. J. Chem. Phys. 154, 074702 (2021); doi: 10.1063/5.0037913 154, 074702-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Illustration of two- and three-body clusters considered in the CE models. The differences in the one-body ECIs of different bimetallic surfaces can be interpreted in terms of the differences in surface energies and effective sizes of the component metals. The metal with smaller sur- face energy or larger atomic radius tends to appear on the top layer. Table III shows the surface energies of Fe, Co, Ni, and Cu calculated in this work. As a confirmation of our results, the surface energies reported in previous studies are also collected. As we can see from those data, there are noticeable quantitative differences in the values of surface energies from different theoretical studies, probably due to the use of different density-functional approximation and different DFT implementations, but their relative order is same and consis- tent with the segregation trends, as revealed from the one-body ECI values. TABLE II . The ECIs (in units of meV) of one-body clusters corresponding to the first (L1), second (L2), and third (L3) layers of the bimetallic surfaces considered in this work. Surface L1 L2 L3 NiFe −108.6 −27.5 8.2 NiCo −51.4 32.7 2.7 NiCu 87.6 −2.9 −36.2TABLE III . The FCC(111) surface energies (in units of J/m2) of the transition metals considered in our work. In this work, we use the PBEsol functional. SKRIVER57used the local-density approximation (LDA) implemented in the linear muffin-tin orbitals (LMTO) method with the tight-binding and atomic sphere approximations. VITOS58 used the LDA in the full charge density LMTO method. LEE59used the PBE GGA functional.60 Metal This work SKRIVER57VITOS58LEE59 Fe 2.90 3.28 . . . . . . Cu 1.57 2.31 1.95 1.44 Ni 2.30 2.63 2.01 1.92 Co 2.48 3.23 . . . . . . From the one-body CCFs calculated by Monte Carlo simula- tion, we can obtain layer-specific concentrations of the guest metal in terms of Eq. (5), which can provide more quantitative informa- tion on the surface segregation at finite temperature. Figure 4 shows the calculated X concentration of different NiX bimetallic surfaces as a function of temperature. For NiFe, the concentration of Fe in each layer is close to the overall composition of 0.25 at temperatures below about 550 K but exhibits strong segregation behaviors at high temperature. This apparently counter-intuitive phenomenon can be explained by considering two competing factors in the NiFe system. On one hand, Ni and Fe can form an ordered intermetallic com- pound Ni 3Fe, which leads to nearly equal Fe concentrations in dif- ferent layers at low temperature; on the other hand, it is energetically favorable for Fe to stay in the bulk (i.e., the third layer) in terms of the ECIs of one-body clusters in Table II. At temperatures above 550 K, corresponding to an order–disorder phase transition, which will be further clarified in Sec. III C, the second factor becomes important, which explains the lower Fe concentration in the first layer. Experi- mentally, it is found that Ni and Fe can form an ordered intermetallic compound Ni 3Fe below 773 K.65The theoretical phase transition temperature of 550 K agrees qualitatively to the experimental one, and the difference between them can be possibly attributed to the fact that here we are considering the NiFe bimetallic surface with bulk Ni as the substrate. The NiCo surface exhibits rather different features. For temperatures below 200 K, the top layer is almost fully occupied by Ni, and Co mainly stays in the second and third layers. As the temperature increases, the presence of Co on the top layer FIG. 4 . The concentration of the doping metal in the first (L1), second (L2), and third (L3) layers of NiX bimetallic surfaces as a function of the temperature calculated from Monte Carlo simulations. J. Chem. Phys. 154, 074702 (2021); doi: 10.1063/5.0037913 154, 074702-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp becomes significant. In NiCu, Cu strongly segregates to the top layer and begins to enter the second layer only at the very high temper- ature of about 700 K, which is consistent with the well established experimental findings that Cu tends to be enriched on the outer sur- face of Cu–Ni alloys under ultrahigh vacuum condition (see, e.g., Ref. 66 and references therein). Our simulation results also agree well with the experimental findings by Koschel et al. ,67who found that when growing thin Cu layers on Ni(111), annealing to 800 K is required to observe significant diffusion of Ni atoms into the top Cu layers. C. Short-range order Besides the surface segregation, short-range order can also have significant influences on the properties of bimetallic surfaces. In particular, SRO is directly related to the ensemble effect, which refers to the peculiar catalytic properties of bimetallic catalysts aris- ing from special local chemical structures that are not available on mono-metallic surfaces.3,68 As for surface segregation, we can obtain some clues on short- range order from ECIs of two- and three-body clusters, which are illustrated in Fig. 5, and their configurations are illustrated in Fig. 3. The ECI of a two-body cluster indicates the pairing tendency of metal atoms at the corresponding sites. A negative two-body ECI indicates that occupying the two sites of the cluster by the same metal atom is energetically favored.41Overall, the ECIs of NiFe, NiCo, and NiCu exhibit very different features. In NiFe, the ECIs of most two- and three-body clusters are positive. In NiCo, while the ECIs of FIG. 5 . The effective cluster interactions (ECIs) of two- and three-body clusters.two-body clusters can be either positive or negative, those of three- body clusters are mostly positive. In NiCu, the ECIs of all nearest neighboring two-body clusters are negative, but those of the next nearest neighboring two-body clusters are positive or nearly zero, while most of the three-body clusters have negative ECIs. We can therefore expect that the three bimetallic surfaces have very different short-range ordering behaviors. Different short-range ordering features in NiX bimetallic sys- tems are clearly illustrated in Fig. 6, which shows the snapshots of the top layer of NiX bimetallic surfaces during Monte Carlo simulations atT= 400 K and T= 1000 K, respectively. At T= 400 K, the top layer of the NiFe bimetallic surface exhibits a highly ordered struc- ture that is consistent with that of the ordered Ni 3Fe inter-metallic compound. At T= 1000 K, the concentration of Fe on the top layer is significantly reduced and its distribution becomes much more ran- dom. On NiCo, at T= 400 K, Co appears on the top layer with a low concentration in a singly dispersed way, but at T= 1000 K, the Co concentration in the top layer increases significantly, and a certain clustering tendency can be clearly noticed. The top layer of NiCu exhibits similar features at T= 400 K and T= 1000 K, mainly composed of Cu, and Ni forms isolated islands. We further characterize short-range ordering behaviors of dif- ferent bimetallic surfaces by using the Warren–Cowley SRO param- eters corresponding to the nearest neighboring two-body clusters within the first layer, second layer, and between the first- and second-layer, denoted as s1,1,s2,2, and s1,2, respectively, as shown in Fig. 7. As pointed in Sec. II B, a large deviation of these parame- ters from zero indicates strong short-range order, which is clearly the case in NiFe and NiCu, but much less in NiCo. The SRO param- eters in NiFe exhibit an abrupt change at about T= 550 K, and a similar distinct change is also observed in the Fe concentrations in different layers, which indicates that NiFe bimetallic surface exhibits an order–disorder phase transition at this temperature, as we have mentioned when discussing the surface segregation in NiFe. In con- trast to NiFe, SRO parameters in NiCo and NiCu exhibit smooth variation as a function of temperature. From the perspective of heterogeneous catalysis, the types of local chemical environments available on the surfaces at finite temperature play important roles in determining the catalytic FIG. 6 . The snapshots of the top layer of bimetallic surfaces from the Monte Carlo simulations at 400 K (top) and 1000 K (bottom). J. Chem. Phys. 154, 074702 (2021); doi: 10.1063/5.0037913 154, 074702-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 7 . The Warren–Cowley SRO parameters as a function of temperature for the nearest neighboring two-body clusters within the first layer ( s1,1), within the second layer ( s2,2), and between the first and second layers ( s1,2) in NiFe, NiCo, and NiCu bimetallic surfaces. FIG. 8 . The probability of forming Ni 3, Ni 2X, NiX 2, and X 3trimers as a function of temperature in the top layer of different NiX (X = Fe, Co, and Cu) bimetallic surfaces. activity of bimetallic surfaces. A comprehensive investigation of this issue is highly complicated and should be undertaken in the con- text of specific catalytic reactions. As well established in the liter- ature, the catalytic activity of transition metal surfaces for many important heterogeneous catalytic reactions can be related to the adsorption energies of the carbon and oxygen atoms, which adsorb on the hollow sites of the FCC(111) surface. As a simplistic treat- ment, one can assume that the adsorption energies of C and O are mainly determined by the chemical compositions of atoms around the hollow site. In Fig. 8, we calculate the probability of forming dif- ferent trimers (Ni 3, Ni 2X, NiX 2, and X 3) on the top layer of NiX bimetallic surfaces as a function of temperature, which can be cal- culated from the CCF of the smallest intra-layer three-body cluster. In NiFe, Ni 3and Ni 2Fe are dominant. In NiCo, Ni 3is dominant at low temperatures, but with increasing temperature, Ni 2Co becomes also important. In NiCu, all four configurations are present, and their probabilities vary significantly as a function of temperature. We can therefore expect that the catalytic properties of NiCu are mostly affected by the temperature. IV. CONCLUDING REMARKS To summarize, in this work, we have built the cluster expan- sion models for three Ni-based bimetallic surfaces based on DFTcalculations and used them to investigate the surface segregation and short-range ordering of these surfaces at finite temperature. We found that for NiFe and NiCo, it is energetically favorable for the guest metal (Fe or Co) to stay in the second or even third layer at high temperature, while Cu tends to segregate to the topmost layer in NiCu. NiFe exhibits an order–disorder transition at about 550 K, although significant short-range order is still present above the tran- sition temperature. Our study indicates that the local chemical envi- ronments on bimetallic surfaces strongly depend on the composi- tion and vary significantly as a function of temperature, which can have significant implications for the catalytic properties of bimetallic surfaces. We close this paper by remarking on the limitations of the current work and possible extensions in the future. First, we consid- ered the structure of idealized closed packed bimetallic surfaces at finite temperature. Real catalysts in working conditions are usually nanoparticles with some supports, often with complicated compo- sitions and structures. The morphology of nanoparticles and the presence of the support will certainly have significant effects on the chemical ordering of metal components. Second, we have not con- sidered the effects of the presence of surface adsorbed species on short-range ordering. For the DRM reaction, there will be a large amount of adsorbed oxygen (O∗) and carbon (C∗) on the surface of the catalyst, which can have significant effects on the chemi- cal ordering structure of bimetallic surfaces. As mentioned in the Introduction, the importance of considering the reaction condi- tions including the presence of adsorbates for understanding the structural and catalytic properties of bimetallic surfaces have been well recognized, and there have been already a series of theoreti- cal studies on this aspect that shed light on the important effects of adsorbates on chemical ordering of bimetallic surfaces or nanoparti- cles.25,27–29,31,39We can therefore expect that such treatments are also necessary to achieve more realistic modeling of the DRM process on bimetallic surfaces. Third, in this work, we considered bimetal- lic surfaces by mixing Ni with another 3d transition metal. There is also a lot of research interest in exploring bimetallic catalysts that mix Ni with noble metals and hopefully improve both the catalytic activity and carbon resistance of Ni-based catalysts for reactions like DRM.34,69–72In principles, those systems can be treated in a simi- lar way as in this work. On the other hand, because noble metals have significantly larger effective radii and much more different elec- tronic properties with respect to Ni, we can expect that Ni-noble bimetallic surfaces can be much more complicated and require more sophisticated theoretical techniques for realistic modeling. Last but not least, we have considered the chemical ordering at a finite tem- perature only from the equilibrium statistical thermodynamics per- spective, and it is also important to know the dynamics of bimetal- lic surfaces, e.g., how quickly a particular bimetallic surface evolves between different configurations. All these issues are important to obtain a complete picture of the catalytic properties of bimetallic sur- faces under working conditions and are left as the subjects of future studies. SUPPLEMENTARY MATERIAL See the supplementary material for Cook’s distance analysis of the data in the training sets that are used to build the CE models, J. Chem. Phys. 154, 074702 (2021); doi: 10.1063/5.0037913 154, 074702-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp and the comparison of ECIs obtained by using all data and those by excluding the outliers detected from Cook’s distance analysis. ACKNOWLEDGMENTS This work was supported by the National Key Research and Development Program of China (Grant No. 2016YFB0701100) and High-performance Computing Platform of Peking University. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1C. T. 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5.0040438.pdf

J. Chem. Phys. 154, 054314 (2021); https://doi.org/10.1063/5.0040438 154, 054314 © 2021 Author(s).Ab initio investigation of the CO–N2 quantum scattering: The collisional perturbation of the pure rotational R(0) line in CO Cite as: J. Chem. Phys. 154, 054314 (2021); https://doi.org/10.1063/5.0040438 Submitted: 13 December 2020 . Accepted: 14 January 2021 . Published Online: 05 February 2021 Hubert Jóźwiak , Franck Thibault , Hubert Cybulski , and Piotr Wcisło ARTICLES YOU MAY BE INTERESTED IN Rate constants for the H+ + H2 reaction from 5 K to 3000 K with a statistical quantum method The Journal of Chemical Physics 154, 054310 (2021); https://doi.org/10.1063/5.0039629 -machine learning for potential energy surfaces: A PIP approach to bring a DFT-based PES to CCSD(T) level of theory The Journal of Chemical Physics 154, 051102 (2021); https://doi.org/10.1063/5.0038301 Energy-transfer quantum dynamics of HeH+ with He atoms: Rotationally inelastic cross sections and rate coefficients The Journal of Chemical Physics 154, 054311 (2021); https://doi.org/10.1063/5.0040018The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Ab initio investigation of the CO–N 2quantum scattering: The collisional perturbation of the pure rotational R(0) line in CO Cite as: J. Chem. Phys. 154, 054314 (2021); doi: 10.1063/5.0040438 Submitted: 13 December 2020 •Accepted: 14 January 2021 • Published Online: 5 February 2021 Hubert Jó´ zwiak,1,a) Franck Thibault,2 Hubert Cybulski,3 and Piotr Wcisło1,b) AFFILIATIONS 1Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University in Toru ´n, Grudziadzka 5, 87-100 Toru ´n, Poland 2Univ. Rennes, CNRS, IPR (Institut de Physique de Rennes)-UMR 6251, Rennes F-35000, France 3Institute of Physics, Kazimierz Wielki University, ul. Powsta ´nców Wielkopolskich 2, 85-090 Bydgoszcz, Poland a)Author to whom correspondence should be addressed: hubert.jozwiak@doktorant.umk.pl b)Electronic mail: piotr.wcislo@umk.pl ABSTRACT We report fully quantum calculations of the collisional perturbation of a molecular line for a system that is relevant for Earth’s atmosphere. We consider the N 2-perturbed pure rotational R(0) line in CO. The results agree well with the available experimental data. This work constitutes a significant step toward populating the spectroscopic databases with ab initio collisional line-shape parameters for atmosphere-relevant systems. The calculations were performed using three different recently reported potential energy surfaces (PESs). We conclude that all three PESs lead to practically the same values of the pressure broadening coefficients. Published under license by AIP Publishing. https://doi.org/10.1063/5.0040438 .,s I. INTRODUCTION Detailed knowledge about the interaction energy in molecular systems is crucial for understanding a variety of physical phenom- ena. An accurate potential energy surface (PES) is important for the calculations of bound states of molecular complexes,1understand- ing the dynamics of the interstellar medium,2proper interpreta- tion of collisionally induced spectra,3and determination of shapes of the optical molecular resonances.4Nitrogen molecule, as the main constituent of Earth’s atmosphere, is of particular importance for the spectroscopic community. Collisions with N 2can perturb the absorption lines of less abundant molecules in the atmosphere, leading to the pressure broadening (and shift) of the spectra, and constitute the primary broadening mechanism in the troposphere.5 Accurate values of pressure broadening and shift coefficients are essential for reducing atmospheric-spectra fit residuals,6which might affect values of the quantities retrieved from the fit, such as the volume mixing ratio (VMR)6,7of the absorbing compound. This is especially important in terms of remote sensing applications, assubpercent accuracy of the VMR is needed to reliably identify the sources and sinks of the greenhouse gases.8,9Carbon monoxide is a trace gas in Earth’s atmosphere, which has an indirect impact on the concentration of methane. Indeed, CO reacts with hydroxyl rad- icals ( ⋅OH) and reduces their abundance in the atmosphere, which, in turn, leads to higher concentration of the CH 4molecules.10,11 Carbon monoxide is also an important gaseous pollutant and a useful tracer of various anthropogenic activities, such as fossil fuel combustion.12–14 Apart from remote sensing measurements, accurate pressure broadening coefficients of the nitrogen-perturbed CO lines are of particular importance in the analysis of the atmospheres of var- ious objects in the outer solar system. The atmosphere of Titan, the largest moon of Saturn, is dominated by nitrogen (94.2%) and methane (5.65%).15,16Trace amounts of the CO molecule were dis- covered there by Lutz et al .,17who identified several P and R lines of the 3-0 band from ground-based measurements. To accurately determine the CO concentration in Titan’s atmosphere, several studies were pursued, where pure rotational transitions,18,19in J. Chem. Phys. 154, 054314 (2021); doi: 10.1063/5.0040438 154, 054314-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp particular the R(0),20–23R(1),23,24and R(2)23,25lines, as well as tran- sitions from the fundamental band, were investigated.26–28Cur- rently,29the most accurate values of the VMR of carbon monoxide come from ground-based measurements using the Submillimeter Array,25from the SPIRE instrument on the Herschel satellite,30and the Composite Infrared Spectrometer (CIRS) on board the Cassini probe19and are given by (5.1 ±0.4)×10−5, (4.0 ±0.5)×10−5, and (4.7 ±0.8)×10−5, respectively. In all these investigations, the experimental values31–34of the pressure broadening coefficients of the N 2-perturbed CO lines were necessary to properly interpret the measured spectra. Carbon monoxide was also detected in the nitrogen- dominated35atmosphere of Triton, the largest moon of Neptune.36 The ground-based spectroscopic observations of Triton in the 2.32 μm–2.37μm region, using the European Southern Observatory Very Large Telescope (ESO VLT), determined the abundance of CO and CH 4to be at a level of a few hundredths of a percent of that of nitrogen. Accurate information about the relative abundance of CO and CH 4is especially needed for the analysis of seasonal changes in Triton’s atmosphere37and could be a subject of interest for future orbital missions to Neptune.38The atmosphere of Pluto shares some similarities to those of Triton and Titan, as it consists mostly of nitrogen, with trace amounts of methane (0.25%) and carbon monoxide.39Searches for the CO features in the atmospheric spectra of Pluto were subjects of various investigations in the millimeter40–43 and near-IR ranges.44,45The analysis of the pure rotational R(2) line by Lellouch et al. indicated a CO mole fraction of 515 ±40 ppm for a surface pressure of 12 μbars.43 As a molecular complex, the CO–N 2system was a subject of thorough theoretical and experimental investigations through- out the past years. The first pure rotational spectra of the CO–N 2complex were observed in 199646,47in the IR region. These works were followed by studies of the microwave and millimeter transitions in the complex,48–52and several stud- ies devoted to the transitions in the IR region.53,54We recall also two very recent studies of this complex in the millimeter range,55,56which provided an accurate test of the recently reported PESs.55,57,58 The theoretical investigations on the CO–N 2complex were reviewed in more detail in a previous paper.57Here, we only recall the importance of the study conducted by Fišer and Polák,59who investigated multiple orientations of the complex using the coupled- cluster singles and doubles including connected triple corrections [CCSD(T)] and the Møller–Plesset (MP) perturbation theory up to fourth order. The first four-dimensional (4D) PES was reported by Karimi-Jafari et al. ,60who employed the MP4 method with a basis set obtained from Dunning’s aug-cc-pVQZ basis set. In this investigation, the g and f functions were removed from the orig- inal basis set and, additionally, a set of 3s3p2d1f midbond func- tions was used. Nonetheless, as it was stated in Ref. 57, due to the method used, reduction in the basis set, and too small num- ber of grid points chosen by the authors, this PES should not be considered reliable enough to study various physical phenomena. Recently, three highly accurate PESs were reported.55,57,58Liuet al.58 calculated a 5D PES using the CCSD(T)-F12 method and the aug- cc-pVQZ basis set, claiming the discrepancies between theoretical and experimental energies of the IR transitions of the CO–N 2com- plex to be smaller than 0.068 cm−1. Surin et al .55reported a 4DPES calculated using the standard CCSD(T) method and the aug- cc-pVQZ basis set, supplemented with 3s2p1d midbond functions. The accompanying experimental results enabled the authors to assign several newly detected transitions. However, some significant discrepancies between the calculated and experimental rotational states were observed. The third, more recent potential,57was calcu- lated using the CCSD(T) method with the aug-cc-pVQZ basis set, extended with the 3s3p2d1f1g midbond functions. This 4D PES was fit to an analytical expression,57which performs well for geometries where the interaction energy does not exceed 100 μEh. The agree- ment between the theoretical and experimental values of the energy levels is significantly better than that for all the aforementioned PESs. Collisional broadening of the CO lines by nitrogen molecules has been studied in detail both theoretically and experimentally. A majority of the experimental data refers to the lines in the fun- damental band of CO.33,34,61–89N2-perturbed CO lines were also measured for the first84,90–94and second overtones.65,95–97In the case of pure rotational transitions, the pressure broadening coeffi- cients of the N 2-perturbed lines were determined for the first five R lines of CO. The first measurements of the nitrogen-, oxygen-, and air-broadened widths of the R(0) line were reported by Connor and Radford98and by Colmont and Monnanteuil.31These results were refined by Nissen et al.99in a comparative study of foreign- gas pressure broadening of the R(0) line, using Fourier-transform and radio-acoustical detection spectrometers. There are also a few papers regarding foreign-gas broadening of the R(1),32R(2)100, and R(4)101,102CO lines, in which a temperature dependence of the N 2 broadening coefficients was determined. Puzzarini et al.103reported a thorough study of various line-shape models in the analysis of the N2- and O 2-broadened R(0-3) lines at 296 K. The pressure broad- ening coefficients of the N 2-perturbed CO lines from the S branch were also determined using coherent anti-Stokes Raman scattering (CARS) spectroscopy.104 It has been recently shown for a benchmark system of helium- perturbed H 2that a full ab initio description of the shapes of rovi- brational transitions leads to subpercent agreement with experimen- tally measured spectra.105This successful theoretical approach is based on a very accurate PES, determined from first principles,106,107 quantum-scattering calculations, and a state-of-the-art model of collision-perturbed shape of rovibrational resonance,108–111which includes the speed-dependence of line broadening and shift112and the influence of velocity changing collisions.113Helium-perturbed H2lines were subjects of several recent investigations. In Ref. 114, the influence of PESs’ quality on shapes of particular rovibrational lines was studied in detail. The most accurate PES for this system and line-shape parameters for the rovibrational Q lines were reported in Ref. 107. Shapes of optical resonances from the O and S branches were determined in Ref. 115. Importance of centrifugal distortion (the dependence of the radial coupling terms of the PES on the rota- tional quantum number) was studied in Ref. 116. Finally, the first comprehensive database of the line-shape parameters (broadening, shift, their speed-dependence, and the complex Dicke parameter), generated from ab initio calculations, was reported in Ref. 117. The same methodology was applied to the cases of helium-perturbed lines of HD118,119and D 2.120,121Recently, subpercent agreement with the experimental shapes of Ar-perturbed rovibrational lines in CO was achieved.122 J. Chem. Phys. 154, 054314 (2021); doi: 10.1063/5.0040438 154, 054314-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Here, we report the ab initio quantum-scattering calculations of the collision-perturbed shape of the molecular line for a system that is relevant for the terrestrial atmosphere. We investigate the width of the N 2-perturbed pure rotational R(0) line in CO. Scatter- ing calculations are performed using three PESs.55,57,58Our results are in good agreement with the available experimental data.31,98,99,103 The shape of the pure rotational R(0) line is dominated by the colli- sional broadening, which is two orders of magnitude larger than the pressure induced shift. Hence, in this paper, we do not consider the pressure shift coefficient. Due to the complex structure of the close- coupled (CC) equations and a large number of channels that must be taken into account, the equations are solved using the coupled states approximation (CSA).123 This paper is organized as follows: in Sec. II, we briefly describe the PESs used in the calculations. In Sec. III, we discuss the details of the scattering calculations and we analyze the obtained generalized spectroscopic cross sections. In Sec. IV, we describe the calculations of the pressure broadening coefficient, and in Sec. V, we compare our results with the experimental data. We discuss the accuracy of our calculations in Sec. VI, and in Sec. VII, we summarize our results. II. POTENTIAL ENERGY SURFACE The PES for a system consisting of two rigid diatomic molecules (A and B) depends on four variables: the intermolecular distance between the centers of mass of the molecules, R, the angles between each of the molecular axes and the intermolecular axis, θAandθB, and the dihedral angle, ϕ. Let us identify subscripts A and B with CO and N 2molecules, respectively. The definition of the angles is presented in Fig. 1. Contrary to the PESs of Surin et al .55and Cybulski et al .,57which are 4D, the PES of Liu et al .58addition- ally takes into account the stretching of the CO molecule. How- ever, for the purpose of these calculations (we investigate a pure rotational transition in the ground vibrational state of CO), we do not consider any changes in the intramolecular distance of the molecule. In order to reduce the potential to a form suitable for the scattering calculations, the 4D PES is expanded over bispherical harmonics: V(R,θA,θB,ϕ)=∑ lA,lB,lAlA,lB,l(R)IlA,lB,l(θA,θB,ϕ). (1) The radial terms of the potential are denoted as AlA,lB,l, and the bispherical harmonics, IlA,lB,l, are defined as FIG. 1 . Geometry of the CO–N 2system.IlA,lB,l(θA,θB,ϕ=ϕA−ϕB)=√ 2l+ 1 4π∑ m(lAm lB−m∣lAlBl0) ×YlA,m(θA,ϕA)YlB,−m(θB,ϕB), (2) where Yli,m(θi,ϕi)are spherical harmonics and ( lim l j−m|liljl0) are the Clebsch–Gordan coefficients. In the case of the CO–N 2 system, lAis a non-negative integer, lBis a non-negative integer that takes only even values, and lsatisfies the triangular condition |lA−lB|≤l≤lA+lB. Additionally, the sum of these three indices is an even integer. The radial terms of the potential, AlA,lB,l(R), are obtained by the integration of the product of the bispherical harmonics and the full PES, over the angles θA,θB, andϕ:124 AlA,lB,l(R)=8π2 2l+ 1∫2π 0dϕ∫π 0dθAsinθA∫π 0dθBsinθB ×V(R,θA,θB,ϕ)IlA,lB,l(θA,θB,ϕ). (3) In the cases of Liu’s58and Surin’s55PESs, we used their analyti- cal fits. In the case of Cybulski’s PES,57the reliable fit is limited to energies smaller than 100 μEh; hence, we interpolated the orig- inal 10 000 energy points (which covered the intermolecular dis- tances from 5.0 to 30.0 a 0) with the reproducing kernel Hilbert space method (RKHS).125The values of the smoothness parameter, n, and the parameter that determines the long-range behavior of the radial terms, m, were set to the same values as the ones used in the paper of Surin et al.55 Finally, all the three PESs were prepared in a form of 205 radial terms, up to the term with lA= 10, lB= 8, and l= 18. The grid of the intermolecular distances for which the radial terms were calculated consisted of 230 points, from 4.25 a 0to 50 a 0. Figure 2 presents a comparison between isotropic and (two) anisotropic contributions to the three PESs in the vicinity of the minimum of the isotropic component. FIG. 2 . Comparison between isotropic ( lA,lB,l= 0, 0, 0, solid lines) and anisotropic (lA,lB,l= 1, 0, 1, dashed lines and lA,lB,l= 2, 2, 4, dotted lines) radial terms of the three considered PESs.55,57,58The two curves corresponding to the isotropic terms of Surin’s and Liu’s PES, as well as the curves representing the anisotropic terms for all the three PESs, are overlapping. J. Chem. Phys. 154, 054314 (2021); doi: 10.1063/5.0040438 154, 054314-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp III. QUANTUM-SCATTERING CALCULATIONS The quantum-scattering theory of two rigid diatomic molecules in a1Σelectronic state is well-known and thoroughly discussed in the literature.124,126–129Here, we briefly summarize the issue and we connect the results of the scattering calculations with the generalized spectroscopic cross section. The total wavefunction of the system is expanded in a basis set discussed in Ref. 124 or in Ref. 129. The expansion separates the intermolecular distance and the angles describing the geom- etry of the two diatomics. A substitution of the expanded form of the wavefunction into the Schrödinger equation leads to a set of CC equations for the radial part of the total wavefunction [see Eq. (9) in Ref. 124]. The equations are solved numerically for a wide range of kinetic energies. Boundary conditions on the radial part of the total wavefunction connect the solutions of the coupled equations with the scattering S-matrix [see Eq. (11) of Ref. 128]. The elements of the scattering S-matrix enter the formulas for the generalized spectroscopic cross section, whose real and imaginary parts are called the pressure broadening and shift cross section, respectively.130,131 In contrast to simpler systems of molecule–atom107,115,120,132or molecule–molecule with large rotational constants,109for a system of two diatomic molecules typical for Earth’s atmosphere, the number of basis levels (necessary to converge numerical calculations) grows considerably with the relative kinetic energy of the colliding pair. At present, this effectively hinders the possibility to study the room- temperature collisions. The memory resources and the CPU time required to perform such calculations exceed capabilities of typical work stations. It has enforced us to use an approximate method of solving the CC equations. There are several well-known methods for simplifying the cou- pled equations.123,133–141In this work, we have made use of the widely used coupled states (CS) approximation.123The theory behind the formulas is well-known, and the review of the literature can be found in Chap. 9 of Ref. 142. In the body-fixed frame of reference, the relative angular momentum operator, l2, couples channels with different values of the projection of the j12angular momentum, Ω, on the inter- molecular axis, where j12is the result of coupling of the rotational angular momentum of the two diatomic molecules, j12=j1+j2. Within the most common version of the CS approximation, the off- diagonal matrix elements of the relative angular momentum oper- ator, l2, coupling different values of Ω, are neglected. This leads to the coupled equations that are diagonal with respect to this quan- tum number and immensely speeds up the calculations. Addition- ally, a new quantum number, ¯l, which approximates the diago- nal matrix elements of the relative angular momentum operator as¯l(¯l+ 1), is introduced. This quantum number is used instead of the total angular momentum, J. Finally, within the CS approxi- mation, the coupled equations are block-diagonal with respect to ¯l and Ω. The rest of the procedure is similar to the case without the approximations. The S-matrix elements [connected with the space-fixed S-matrix elements by a unitary transformation, see Eq. (44) of Ref. 143 and the discussion therein] are obtained, and the generalized spectroscopic cross section is calculated. Due to the unitary transformation between the S-matrix elementsobtained within the CS approximation and calculated exactly in the space-fixed frame of reference, the formula for the general- ized spectroscopic cross section is quite different [see Eq. (96) of Ref. 144], σq 0(ji,jf,j′ i,j′ f,j2,Ekin) =π k2⌟roo⟪⟪op ⌟roo⟪mo⟨⌟roo⟪mo⟨⌟roo⟪⟨o⟪2j′ i+ 1 2ji+ 1∑ ¯l,Ω,Ω′∑ j′ 2,j12,j′ 12,¯j12,¯j′ 12(−1)j12+j′ 12+¯j12+¯j′ 12 ×(2¯l+ 1)√ [j12][j′ 12][¯j12][¯j′ 12](j′ 12¯j′ 12 q Ω−Ω′Ω′−Ω) ×(j12¯j12 q Ω−Ω′Ω′−Ω){j′ 12¯j′ 12q j′ ij′ fj′ 2}{j12¯j12q jijfj2} ×[δj12j′ 12δ¯j12¯j′ 12δjij′ iδjfj′ fδj2j′ 2−⟨jij2j12∣SCS ¯lΩ(ETi)∣j′ ij′ 2j′ 12⟩ ×⟨jfj2¯j12∣SCS∗ ¯lΩ′(ETf)∣j′ fj′ 2¯j′ 12⟩], (4) where [ x] = 2 x+ 1, the quantities in parentheses are Wigner 3-j symbols, and the quantities in braces are Wigner 6-j symbols. If line-mixing effects are not considered, the cross sections are cal- culated with ji=j′ iand jf=j′ f, which denote the initial and final states, respectively, of the spectroscopic transition. For the dipole R line analyzed in this work, the tensor order of the radiation–matter interaction, q, is equal to 1. The results presented here were obtained by solving the cou- pled equations, within the CS approximation, using the MOLSCAT code.145The modified log-derivative algorithm of Manolopoulos146 was used, with propagation beginning deeply in the repulsive wall of the potential at R= 4.25 a0and ending at R= 100 a0. The radial coupling terms of the PES were extrapolated for R>50a0in a Cn/Rnform, where Cnandnwere obtained from the fit using the long-range part of the AlA,lB,l(R)terms. We took advantage of the fact that there are two spin isomers of the nitrogen molecule, which give rise to two different symmetries of the rotational wavefunction. ortho -N2, with total nuclear spin I= 0 or I= 2, exhibits a rotational structure, which involves only even jvalues, while para -N2, with I= 1, yields rotational levels, which correspond to odd jvalues. This allowed us to perform scattering calculations with these two species independently and, thus, to reduce the basis set. At least two asymp- totically closed levels [( j1;j2) states, the energy of which is larger than the total energy of the scattering system and are, thus, energetically inaccessible at large intermolecular distances] were kept throughout the calculations. The rotational energy levels of both molecules were taken from the HITRAN database.147The generalized spectroscopic cross sections were obtained using the newly developed FORTRAN code.148 It is difficult to obtain, for a given relative kinetic energy, con- verged values of the imaginary part of σ1 0. The pressure shift cross section is very sensitive to the range and step of the propagator and the number of asymptotically closed channels in the basis set (see Sec. 5 of Ref. 115 and Sec. 4 of Ref. 119). The susceptibility of Im( σ1 0) to these factors is especially pronounced in the molecule–molecule scattering systems and might lead to relatively large uncertainties of the resulting pressure shift coefficients. This is the case for, analyzed here, the N 2-perturbed pure rotational R(0) line in CO. However, the pressure shift is at least two orders of magnitude smaller than J. Chem. Phys. 154, 054314 (2021); doi: 10.1063/5.0040438 154, 054314-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the collisional broadening of this particular line, and it has not been detected in any experiment.31,98,99,103Thus, in this study, we focus on the real part of σ1 0and the resulting pressure broadening coefficient, γ[see Eq. (6)]. We begin the discussion with a comparison of the pressure broadening cross sections obtained using three different PESs. Figure 3 presents the dependence of σ1 0on kinetic energy for j2= 0. For this particular value of j2, we were able to calculate the cross sections for the widest range of relative kinetic energies. As it can be seen in Fig. 3, all the three PESs result in almost the same values of the pressure broadening cross sections (the largest rel- ative difference between the cross sections is at a level of 1.5%). The same situation occurs for higher values of j2, with rela- tive differences between the values of σ1 0obtained with differ- ent PESs being at the level of 1%. Due to very small differences between the pressure broadening cross sections, in the following analysis, we will focus on the results obtained with the PES of Cybulski et al.57 Figure 4 presents the kinetic energy dependence of the calcu- lated generalized spectroscopic cross sections for various rotational levels of the perturber, j2. The range of the kinetic energies, for which the calculations of σ1 0are feasible, strongly depends on j2. For exam- ple, for j2= 0, we were able to calculate the cross sections up to kinetic energy values of 400 cm−1; for j2= 7, up to 250 cm−1; and forj2= 12, up to 120 cm−1. We note that, at large kinetic energies, the dependencies are linear on the log–log plot. This type of relation has already been reported in the studies of the argon-perturbed CO lines,122,149,150as well as in the studies of the Ar-perturbed CO 2151 or H 2-perturbed N 2152isotropic Raman lines. Moreover, it is seen from Fig. 4 that, for a given relative kinetic energy, the pressure broadening cross sections become j2-independent for larger j2. A tentative explanation for such behavior of the cross sections is as follows. The PBXS for this system is essentially determined by the contribution from the inelastic state-to-state cross sections (fact that we have checked making use of the random phase approxima- tion151,153,154). This is even more evident for larger j2values since the N2molecule rotates faster and averages the long-range anisotropic components of the PES, resulting in a small elastic dephasing FIG. 3 . Comparison between pressure broadening cross sections for j2= 0 calculated using three different PESs.55,57,58 FIG. 4 . Pressure broadening cross sections as a function of relative kinetic energy for several values of j2. The cross sections were calculated using the PES of Cybulski et al.57 contribution. Therefore, large j2values mostly sample the same, short-range (repulsive) part of the PES. We, thus, observe small dif- ferences between the PBXS for large j2values. This phenomenon was already noticed for several molecule–molecule systems.152–154 For instance, in the study of the C 2H2–H 2system,153a very weak j2-dependence of the cross sections for kinetic energies larger than 500 cm−1was reported (see Fig. 3 in Ref. 153 and the discussion therein). In fact, during the calculations of the thermally averaged pressure broadening coefficient at temperatures larger than 500 K, the cross sections for j2>3 were taken as the mean value of σq 0 with j2= 1, 2, and 3. Similar observations about less pronounced j2-dependence of σq 0at large relative kinetic energies were reported in the studies of the N 2–H 2and N 2–N 2systems (see Fig. 1 of Ref. 152 and Fig. 3 of Ref. 154, respectively). The conclusion that σ1 0(Ekin)obeys the power law allows us to extrapolate it for large kinetic energies. We fit the ab initio values of σ1 0to the single power law, σ1 0(Ekin)=A(E0 Ekin)b , (5) where E0= 100 cm−1. The cutoff energies are set from 30 cm−1to 180 cm−1, depending on the value of j2. In the case of the extrapo- lation with respect to the rotational quantum numbers of the per- turber, we make use of another observation that, for large values ofj2, for a given kinetic energy, the σ1 0cross section converges to a constant value. Thus, we use the following procedure to determine σ1 0for the given j2andEkin: for j2<12,σ1 0(Ekin,j2)is obtained by fitting the expression given in Eq. (5) to the results of the quantum- scattering calculations. For j2values larger than 12, we assume that σ1 0(Ekin,j2)=σ1 0(Ekin,j2=12). Figures 5 and 6 demonstrate the extrapolation procedure for the given j2andEkin, respectively. In Fig. 5, we present the ab initio and fitted values of σ1 0forj2= 0 and j2= 7. As noticed earlier, the power-law fit works well at sufficiently large values of Ekin. In Fig. 6, we show the values of the cross sections as a function of the J. Chem. Phys. 154, 054314 (2021); doi: 10.1063/5.0040438 154, 054314-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . Kinetic energy dependence of σ1 0forj2= 0 and j2= 7. The solid and dashed curves correspond to the ab initio and fitted [see Eq. (5)] values, respectively. For reference, we put the Maxwell–Boltzmann distribution at 298.15 K (gray curve). rotational levels of the perturber, for three kinetic energies: Ekin= 100 cm−1, 200 cm−1, and 300 cm−1. We note, following the discussion of Fig. 4, that the j2-dependence of the cross sections becomes less pronounced as j2increases. For these three particular values of Ekin, the cross sections for j2>5 differ from each other by less than 0.6%. In Table I, we show the calculated parameters of the power-law fit for j2<13 for the three different PESs.55,57,58These data can be used to reproduce the shape of the R(0) pure rotational line with different line-shape models (not considered here). Again, we can see that the three analyzed PESs lead to almost the same general- ized spectroscopic cross sections and that the dependence of the coefficient of the fit on j2is negligible for j2>5. FIG. 6 .j2-dependence of σ1 0for various kinetic energies. The points and the black dashed curves correspond to the ab initio and the extrapolated values, respectively.TABLE I . Coefficients obtained from the fitted cross section values with the power-law expression, Eq. (5), using three analyzed PESs. Values in parentheses correspond to the standard deviation error of the parameters. j2 Cybulski et al.57Surin et al.55Liuet al.58 A(Å2) 0 117.5(1.7) 116.2(1.7) 115.7(2.0) 1 124.7(3.3) 124.3(3.3) 123.7(3.2) 2 127.3(2.4) 127.2(2.1) 127.2(2.2) 3 127.4(3.5) 126.9(4.2) 126.2(3.7) 4 125.4(0.8) 124.8(0.8) 124.3(0.8) 5 125.9(1.0) 125.2(0.7) 124.8(0.7) 6 125.2(0.5) 124.9(0.5) 124.4(0.5) 7 126.3(1.2) 125.9(1.2) 125.5(1.2) 8 126.8(1.2) 126.6(1.2) 125.9(1.4) 9 125.9(1.0) 125.4(1.1) 124.7(1.1) 10 125.1(1.0) 125.0(0.9) 124.4(1.3) 11 125.6(1.9) 125.3(2.2) 125.2(2.4) 12 125.7(2.1) 125.7(2.3) 125.1(3.4) b 0 0.19(2) 0.18(2) 0.18(2) 1 0.19(3) 0.19(3) 0.19(3) 2 0.19(2) 0.19(1) 0.19(2) 3 0.20(3) 0.20(4) 0.20(3) 4 0.22(1) 0.21(1) 0.22(1) 5 0.23(2) 0.23(1) 0.23(1) 6 0.23(1) 0.23(1) 0.23(1) 7 0.24(3) 0.24(3) 0.24(2) 8 0.24(2) 0.24(2) 0.24(3) 9 0.21(2) 0.21(2) 0.21(2) 10 0.21(2) 0.21(2) 0.21(2) 11 0.20(3) 0.21(3) 0.20(3) 12 0.19(4) 0.19(5) 0.19(7) IV. LINE-SHAPE PARAMETERS Generalized spectroscopic cross sections are used to calculate the pressure broadening coefficients at a given temperature T, using the following formula:109 γ(v)=1 2πc1 kBT∑ j2pj22 π¯vpvRe∫∞ 0dvrv2 re−v2+v2 r ¯v2p ×sinh(2vvr ¯v2p)σ1 0(ji,jf,j2,vr), (6) where vis the speed of the active molecule, vpdenotes the speed of the perturber with the most probable value ¯vp=√ 2kBT/mp, and their relative speed is vrwith the mean value ¯vr.mpandkBare the mass of the perturber and Boltzmann constant, respectively. pj2 denotes the population of the perturbing molecule in a rotational state j2(at a given temperature): pj2=1 Z(T)wj2(2j2+ 1)e−Ej2 kBT, (7) J. Chem. Phys. 154, 054314 (2021); doi: 10.1063/5.0040438 154, 054314-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp andZ(T) is the partition function Z(T)=∑ j2wj2(2j2+ 1)e−Ej2 kBT. (8) In order to cover 99% of the population of the perturber’s rotational states at room temperature, one should extend the calculations up to j2= 22. The weight wj2comes from the symmetry of the total wave- function of the nitrogen molecule and the degeneracy of the total nuclear spin, I. The nitrogen nucleus is a boson with the nuclear spin equal 1, which gives the resultant total nuclear spin of the N2 molecule I= 0, 1, or 2. For I= 0 and I= 2 ( ortho -N2), only even values of the rotational quantum number of N 2,j2, are possible. This corresponds to the value of wj2=6.para -N2has the total nuclear spin I= 1, with only odd values of j2allowed. In this case, wj2=3. Speed-averaged values of the pressure broadening coefficients (γ0) are obtained either by averaging Eq. (6) over the velocity dis- tribution of the active molecule or directly from the generalized spectroscopic cross sections:109 γ0=1 2πc1 kBT¯vr∑ j2pj2Re∫∞ 0dxxe−xσq 0(ji,jf,j2,x=Ekin/kBT). (9) In Fig. 7, we compare the speed-dependent, Eq. (6), and speed- averaged, Eq. (9), values of the pressure broadening coefficient at 298.15 K. We also plot the Maxwellian distribution, which enters Eq. (9) and determines the speed-averaged value of the broadening coefficient. To compare our calculations with the experimental data, the influence of the speed-dependence of the line broadening on the collision-perturbed profile should also be taken into account. The speed-dependence of the broadening is manifested as line narrow- ing, i.e., the effective width of the profile is smaller than γ0. To account for this effect, we simulated the shape of the line with the weighted sum of Lorentz profiles (WSLPs) and fitted the sim- ulated profile with a simple Lorentz profile. The fitted width of the Lorentz profile (denoted here as γ†) corresponds to the values FIG. 7 .Ab initio speed-dependence of the collisional broadening, γ(v) (black solid line), for the N 2-perturbed pure rotational R(0) line of the CO molecule at 298.15 K. The speed-averaged value, γ0, is presented as the black dashed line. For refer- ence, we put the Maxwellian distribution at this temperature (in arbitrary units), as the gray solid curve. The results were obtained using the PES of Cybulski et al.57that are obtained from the experiments.31,98,99,103The influence of the speed-dependence of the collisional shift is negligible for this system. Here, we recall that the line-shape parameters are related to the standard, pressure-dependent broadening coefficients by Γ=γp, where pdenotes the pressure. V. COMPARISON WITH THE EXPERIMENTAL DATA In this section, we compare the obtained pressure broaden- ing coefficients with the available experimental data.31,98,99,103Con- nor and Radford98reported pressure broadening coefficients for the N2-, O 2-, and air-broadened R(0) lines in CO. The measurements were performed for several temperatures in the range from 202 K to 324 K, and the results were fitted to the empirical power-law relation, γ(T)=C(T0 T)d , (10) where T0= 300 K is a specific reference temperature. The study con- ducted by Colmont and Monnanteuil31was performed in a slightly narrower temperature range, from 220 K to 293 K. The authors derived the values of the fitting coefficients from Eq. (10) for the cases of self-, N 2-, O 2-, and air-broadened CO R(0) line, setting T0= 293 K. Over ten years later, Nissen et al .99reported results of the joint study conducted by the Kiel and Nizhny Novogrod groups. In this investigation, collisional broadening of the R(0) line in CO by several buffer gases was studied by using two experimental techniques in the time and frequency domain. The frequency-domain data were obtained using a radio-acoustical detection (RAD) spectrometer for the temperature range from 230 K to 300 K, while the time-domain data were acquired FIG. 8 . Comparison between the effective collisional width ( γ†) of the pure rota- tional R(0) line of the N 2-perturbed CO molecule with the pressure broadening coefficient (γ0) and the available experimental data. The estimated uncertainty of γ†(T) is represented by the shaded area (see Sec. VI for details). The results presented here were obtained with the PES of Cybulski et al.57 J. Chem. Phys. 154, 054314 (2021); doi: 10.1063/5.0040438 154, 054314-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II . Comparison of the theoretical and experimental power-law coefficients [Eq. (10)]. Values in parentheses correspond to the standard deviation error of the parameters. PES C(MHz Torr−1) d Surin et al.553.44(8) 0.71(14) Cybulski et al.573.45(7) 0.71(14) Liuet al.583.43(8) 0.71(14) Surin et al.55a3.29(7) 0.71(14) Cybulski et al.57a3.30(7) 0.71(14) Liuet al.58a3.28(7) 0.71(14) Experiment (Ref. 31)b3.26(10) 0.86(12) Experiment (Ref. 98) 2.99 (+14,−10) 0.74(10) Experiment (Ref. 99, FTMMW) 3.13 (1) 0.74(2) aThe results obtained after the inclusion of the speed-dependence of the broadening (see black curve in Fig. 8). bTheCcoefficient is calculated from the Canddvalues reported for T0= 293 K. using a Fourier-transform spectrometer in the millimeter wave region (FTMMW) at one temperature T= 296 K. Puzzarini et al .103conducted a thorough study of self- and foreign-gas broadening of the R(0-3) lines in the CO (12CO and13CO) molecules. The authors analyzed the shapes of the measured rovi- brational lines using the Voigt, Galatry,155and Rautian156profiles at 296 K. Figure 8 presents the comparison between the theoretical and experimental collisional width of the R(0) line of CO perturbed by N 2at various temperatures. In the case of the theoretical val- ues, we present the speed-averaged pressure broadening coefficient, γ0, and the effective collisional width of the line, γ†, as described in Sec. IV. In order to quantitatively compare the theoretical tem- perature dependence of the collisional width, we chose to present the power-law dependence of γ(T) by using coefficients fitted with Eq. (10). The theoretical and experimental values of the Cand d coefficients are given in Table II. The agreement between the theoretical and most of the experi- mental values31,99,103is very good. An exception is a slight difference between the theoretical collisional width and the results reported by Connor and Radford.98The inclusion of the speed-dependence of collisional broadening through the WSLPs leads to significantly better agreement with the experimental data. VI. ACCURACY OF THE CALCULATIONS In this section, we discuss the sources of the uncertainty of the pressure broadening coefficients reported in this paper. First, we analyze the sources of the uncertainty of the calculated gen- eralized spectroscopic cross sections, originating from the coupled states approximation, the initial parameters of the chosen propaga- tor, and the number of the closed levels used in the calculations. In the next step, we discuss the possible error originating from extrapolation of the kinetic energy dependence of the cross sections, and finally, we discuss the influence of neglecting the resonances observed at low kinetic energies. The resulting total uncertainty of the pressure broadening coefficient corresponds to the shaded area in Fig. 8.The most important contribution to the uncertainty of the final results comes from the coupled states approximation used while solving the coupled equations. The accuracy of the generalized spec- troscopic cross sections calculated within this approximation has been thoroughly discussed in the literature.151,157–159For instance, Roche et al .157have reported that, for low jvalues of the active molecule and relatively small values of kinetic energies, the cross sec- tions calculated within the CS approximation are underestimated by about 15%. However, as was confirmed in other studies using this approximation, i.e., in the cases of the molecule–molecule systems, the CS approximation works better in the domain of large kinetic energies and for larger values of j2. We have made several tests to check how much the CC and CS generalized spectroscopic cross sections differ. Overall, the agree- ment between the approximate and exact cross sections is better than that previously reported in the literature. The largest rela- tive difference between the CC and CS cross sections in the ana- lyzed energy domain is about 5.8% for Ekin= 150 cm−1andj2= 2. Apart from j2= 0, where the CS cross sections are underestimated by about 4.5%, the approximate cross sections are slightly larger than the CC values. The largest j2value, for which we could make such a comparison, was j2= 7, where we calculated the cross sec- tions for Ekin= 65 cm−1. Indeed, the CS works better for larger values of j2, although we have not observed any sign of conver- gence of the approximate cross sections to the exact values of σ1 0. To estimate the error introduced by the CS calculations, we took the relative differences between the CC and CS cross sections from those tests and averaged them with the statistical weights com- ing from the population of the nitrogen molecule in the consid- ered temperature range. This resulted in an estimated error of 2.7% coming from the approximate method of solving the coupled equations. Another contribution to the uncertainty of the results origi- nates from the initial parameters of the propagator that is used to solve the coupled equations. Let us recall that we have used the hybrid log-derivative propagator of Manolopoulos,146which can be controlled by choosing the starting ( Rmin) and end point ( Rmax) of the propagation, as well as the step size. The choice of the start- ing point of the propagation is driven by two factors. On the one hand, the propagation process should start possibly deep in the clas- sically forbidden region, where the PES is strongly repulsive. On the other hand, the quantum chemical methods, such as CCSD(T), are less reliable there since in the short-range region convergence problems tend to occur. This, obviously, affects the accuracy of the radial terms of the PES that enter the coupled equations. There- fore, one needs to find a starting point, which is a compromise between these two restrictions. In this study, we set Rmin= 4.25 a 0, which corresponds to the shortest ab initio distance in the R-grid of the PES of Surin et al .55We performed additional tests to check if a decrease in the value of Rminto 4.0 a 0or 3.5 a 0has any sig- nificant influence on the values of the generalized spectroscopic cross sections. Since the calculated σ1 0has remained unchanged in those additional runs, we can state that this parameter is chosen properly. The propagation should end in the region where the radial terms of the PES vanish and boundary conditions can be applied, and we used Rmax = 100 a0in our calculations. We have also performed additional tests to check how increasing (up to 200 a0) or J. Chem. Phys. 154, 054314 (2021); doi: 10.1063/5.0040438 154, 054314-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp decreasing (down 75 and even 50 a0) this parameter influences the generalized spectroscopic cross sections. The tests confirmed that increasing this parameter does not produce any significant differ- ence inσ1 0and the decrease of this parameter to 50 a0affects the pressure broadening cross section by less than 0.01%. We conclude that the range of the propagation has been chosen properly and does not contribute to the final error of the collisional broadening reported here. A change in the step size of the propagator mainly influences the collisional shift of the spectral line shape, which we do not con- sider here. Usually, the step size should be inversely proportional to the square root of the relative kinetic energy. However, an appro- priate choice of the step size becomes significant for kinetic energies smaller than 40 cm−1, where resonant structures make obtaining the convergence of the cross sections very difficult. In the kinetic energy domain considered here, the error introduced by the step size of the propagator is insignificant. As mentioned in Sec. III, in all the calculations, we kept at least two asymptotically closed levels. In the considered relative kinetic energy domain, this should be sufficient, but we have performed tests to check how additional closed levels (up to 6) influence the val- ues of the pressure broadening cross section. The additional asymp- totically closed levels changed the pressure broadening cross sec- tion up to about 0.7% of its referential value (for j2values larger than 7). To summarize, the total uncertainty of the calculated pressure broadening cross sections is estimated at a level of 3.5%. Another important contribution to the uncertainty of the pres- sure broadening coefficient originates from the extrapolation of the kinetic energy dependence of σ1 0. Let us recall that, for each j2, the power-law function was fitted to the values of σ1 0. The cutoff energy varied from 30 cm−1to 180 cm−1, depending on the j2value. In order to estimate the possible error associated with the range of the ab initio points used in the fitting procedure, we calculated the pres- sure broadening coefficients using the power-law dependence of σ1 0(Ekin)fitted to the data points with Ekin= 15 cm−1. The largest rel- ative difference between those values and the results presented here is at a level of 1%. In the present study, we neglected the resonant structures observed in the kinetic energy dependence of the pressure broaden- ing cross sections. These structures occur for kinetic energies smaller than 15 cm−1, although the exact range depends on j2. An example forj2= 9 is presented in Fig. 9. The Maxwell–Boltzmann distribu- tion, shown in this plot, clearly justifies the neglect of the points below 15 cm−1in the calculations of the line-shape parameters [see Eq. (9)] at room temperature. Nonetheless, we have performed tests to quantify the influence of taking into account all the data points, including the resonant structures. We conclude that the results (at room temperature) obtained in this way differ by about 0.5% with respect to the values reported in Sec. V. We have estimated all possible contributions to the final error of the pressure broadening coefficient. The generalized spectro- scopic cross sections are calculated with ∼3.5% uncertainty, which originates from the coupled states approximation (2.7%) and the contribution from the number of the closed levels taken into account while solving the coupled equations (0.7%). The cutoffs for the kinetic energies used in the power-law fit introduce an estimated error of about 1%. Neglecting the resonant structures of the cross FIG. 9 . Resonant structures observed in the pressure broadening cross sections (red curve) for kinetic energies smaller than 15 cm−1. The black line corresponds to the power-law function fitted to the points for which Ekin>30 cm−1. As a ref- erence, we draw the Maxwell–Boltzmann distribution at 298.15 K (gray curve). These results, corresponding to j2= 9, were obtained using the PES of Cybulski et al.57 sections for kinetic energies below 10 cm−1could affect the final results by 0.5%. Hence, the total uncertainty of the final pressure broadening coefficients is at the 5% level. VII. CONCLUSIONS We have presented the first-ever theoretical calculations of the pressure broadening of the pure rotational R(0) line of the N 2-perturbed CO molecule. Starting from an ab initio PES, we performed quantum-scattering calculations. Because of the computational complexity of the coupled equations, we made use of the well-known coupled states approximation. This allowed us to obtain the generalized spectroscopic cross sections for a wide range of kinetic energies and for the values of the rotational quantum number of the perturbing molecule j2≤12. We calculated the speed- averaged and speed-dependent values of the pressure broadening coefficient, and we obtained good agreement with the experimental data. One of the goals of this paper was to determine whether any previously reported PESs55,57,58reproduce the shape of the R(0) line significantly better than the other considered PESs. Our results sug- gest that all three PESs lead to almost the same results and it is impossible, at this level of accuracy of the scattering calculations, to determine which one is the most accurate. This work constitutes a large step toward theoretical investiga- tion of other N 2-perturbed pure rotational CO lines. Such studies are of particular importance for the physics of the terrestrial and extraterrestrial atmospheres and for populating the spectroscopic databases, such as the HITRAN database,147with ab initio collisional line-shape parameters. SUPPLEMENTARY MATERIAL See the supplementary material associated with this article for the tabulated radial terms of the three PESs. J. Chem. Phys. 154, 054314 (2021); doi: 10.1063/5.0040438 154, 054314-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp ACKNOWLEDGMENTS H.J.’s contribution was supported by the National Science Cen- tre in Poland through Project No. 2018/31/B/ST2/00720. H.C.’s con- tribution was financed by the National Science Centre in Poland within the OPUS 8 Project No. 2014/15/B/ST4/04551. P.W.’s con- tribution was supported by the National Science Centre in Poland through Project No. 2019/35/B/ST2/01118. 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5.0030949.pdf

J. Chem. Phys. 154, 034101 (2021); https://doi.org/10.1063/5.0030949 154, 034101 © 2021 Author(s).Excited states in variational Monte Carlo using a penalty method Cite as: J. Chem. Phys. 154, 034101 (2021); https://doi.org/10.1063/5.0030949 Submitted: 25 September 2020 . Accepted: 28 December 2020 . Published Online: 15 January 2021 Shivesh Pathak , Brian Busemeyer , João N. B. Rodrigues , and Lucas K. Wagner COLLECTIONS Paper published as part of the special topic on Frontiers of Stochastic Electronic Structure Calculations ARTICLES YOU MAY BE INTERESTED IN On the role of symmetry in XDW-CASPT2 The Journal of Chemical Physics 154, 034102 (2021); https://doi.org/10.1063/5.0030944 Using quantum annealers to calculate ground state properties of molecules The Journal of Chemical Physics 154, 034105 (2021); https://doi.org/10.1063/5.0030397 When do short-range atomistic machine-learning models fall short? The Journal of Chemical Physics 154, 034111 (2021); https://doi.org/10.1063/5.0031215The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Excited states in variational Monte Carlo using a penalty method Cite as: J. Chem. Phys. 154, 034101 (2021); doi: 10.1063/5.0030949 Submitted: 25 September 2020 •Accepted: 28 December 2020 • Published Online: 15 January 2021 Shivesh Pathak,1 Brian Busemeyer,2 João N. B. Rodrigues,3 and Lucas K. Wagner1,a) AFFILIATIONS 1University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA 2Center for Computational Quantum Physics, Flatiron Institute, York, New York 10010, USA 3Universidade Federal do ABC - UFABC, Santo André, S ˜ao Paulo 09210-580, Brazil Note: This paper is part of the JCP Special Topic on Frontiers of Stochastic Electronic Structure Calculations. a)Author to whom correspondence should be addressed: lkwagner@illinois.edu ABSTRACT In this article, the authors present a technique using variational Monte Carlo to solve for excited states of electronic systems. This technique is based on enforcing orthogonality to lower energy states, which results in a simple variational principle for the excited states. Energy opti- mization is then used to solve for the excited states. This technique is applied to the well-characterized benzene molecule, in which ∼10 000 parameters are optimized for the first 12 excited states. Agreement within ∼0.2 eV is obtained with higher scaling coupled cluster methods; small disagreements with experiment are likely due to vibrational effects. Published under license by AIP Publishing. https://doi.org/10.1063/5.0030949 .,s I. INTRODUCTION The ground and first few excited states determine the behav- ior of most systems in materials, condensed matter, and chemistry. A hallmark of correlated electron physics is the presence of many disparate excited states near the ground states, which may differ in complex ways that depend on exactly how electronic correla- tion is treated, as has been shown in the Hubbard model.1Meth- ods to access these low energy states in strongly correlated sys- tems are fundamental to understanding them. There is, thus, a need for scalable methods that can treat strongly correlated systems and non-perturbatively access excited states, which would generate new insights into long-standing challenging systems such as the low energy behavior of high-Tc cuprates/pnictides and twisted bilayer graphene. Quantum Monte Carlo techniques such as variational and dif- fusion Monte Carlo (VMC, DMC)2are based on many-body, non- perturbative wave functions. They offer low scaling, typically of orderO(N3−4 e), where Neis the number of electrons, and can offer impressive accuracy for that low scaling.3However, these techniques are mostly developed for ground state calculations. By far, the most common technique to approximate excited states is to fix a singleSlater determinant or linear combination of Slater determinants to approximate the ground state, while using a Jastrow4factor and dif- fusion Monte Carlo to address some of the electron correlations. This is the technique outlined in Ref. 2, and similar techniques are used in auxiliary field quantum Monte Carlo.5This technique works surprisingly well for many materials.6–12However, this technique depends on the quality of the wave function that generated the fixed trial function; if correlation included by the Jastrow factor would change the optimal Slater part of the wave function, then the ansatz is suboptimal, as demonstrated clearly in Ref. 13. Recently, and not so recently, there have been extensions to the variational and diffusion Monte Carlo (VMC, DMC) meth- ods to address excited states systematically. These techniques seek to improve the accuracy of excited states over the simple method explained above by optimizing the antisymmetric part of the wave function. Bernu and Ceperley proposed an algorithm based on dif- fusion Monte Carlo,14which has not been applied to many prac- tical cases, since it has a sign problem that leads to exponential scaling in the system size. Blunt et al.15proposed an algorithm using full CI (configuration interaction), which is also exponen- tially scaling. On the low-scaling side, Filippi and co-workers16–18 implemented a method similar to the state averaged complete active J. Chem. Phys. 154, 034101 (2021); doi: 10.1063/5.0030949 154, 034101-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp space self-consistent field (CASSCF) in VMC and demonstrated the technique on impressively large wave function expansions. How- ever, state averaging is not optimal when the optimal ground and excited orbitals are very different,19as often occurs in strongly cor- related systems. Neuscamman and co-workers19–22have proposed a low-scaling method that instead uses alternate objective functions to optimize excited states. While this technique does not suffer from the state averaging problem, it so far has only been applied to very few excited states and can experience difficulties converging to the correct state.23Finally, Choo et al.24used an orthogonalization step after each optimization step to access excited states on a lattice, but the algorithm has not been demonstrated on first-principles models. In this article, we implement and demonstrate a simple penalty method based on orthogonalizing to lower energy states to com- pute excited states using variational Monte Carlo. Similar techniques are commonly used in density matrix renormalization group cal- culations25but, to our knowledge, have not been applied in the variational Monte Carlo context. This technique obtains excited states one by one by enforcing orthogonality to lower energy states and can optimize general wave function parameters, including orbital parameters, as shown here. The scaling of the technique is O(NexNM e)+cO(N2 exNM e), where Neis the number of electrons, Nex is the number of excited states computed, cis a small constant, and Mis dependent on the wave function, three for a Slater–Jastrow wave function. The orthogonalization-based technique allows for access to multiple excited states and does not require state averag- ing. The technique is implemented in the pyqmc package, available online.26The method is applied to benzene with an ∼10 000 param- eter wave function, showing high accuracy compared to experiment and coupled cluster calculations on 12 excited states. The state- specific orbital optimization made possible by the new technique results in improvements in excited state energies from fixed node DMC. II. PENALTY-BASED OPTIMIZATION USING VARIATIONAL MONTE CARLO The method solves for each energy eigenstate one at a time by following the procedure outlined here: 1. First, the stochastic reconfiguration method27is used to find the VMC approximation to the ground state, | Ψ0⟩. 2. The first excited state | Ψ1⟩is found by optimizing the objec- tive function [Eq. (7)] with ⃗S∗=[0]andλset larger than the expected E1−E0. The algorithm is not very sensitive to the value ofλ, so we typically use λof order 1 hartree, substantially higher than excitation energies in the systems considered here. 3. The second excited state is found by optimizing the objective function [Eq. (7)] with ⃗S∗=[0, 0]and anchor states | Ψ0⟩and |Ψ1⟩. 4. Further excited states are found in the same way by orthogo- nalizing to the ones found in the previous steps. A. Objective function As shown in Fig. 1, it is straightforward to show that if | Φ0⟩is the ground state of the Hamiltonian, then so long as λ>E1−E0, the function FIG. 1 . The lower bound of E[Ψ] as a function of overlap with the first two eigenstates. The vertices are the first three eigenstates. arg min Ψ(E[Ψ]+λN2 ΨN2 Φ0∣⟨Ψ∣Φ0⟩∣2) (1) is equal to the first excited state | Φ1⟩, where E[Ψ] is the expectation value of the energy and NΨ=1/√ ∣⟨Ψ∣Ψ⟩∣. For completeness, we show this here. Consider the objective functional O[Ψ]=E[Ψ]+N2 ΨN2 Φ0λi∣⟨Ψ∣Φ0⟩∣2. (2) Then, the functional derivative of each term is given by δN2 ΨN2 Φ0∣⟨Ψ∣Φ0⟩∣2 δΨ∗=N2 Ψ(N2 Φ0⟨Φ0∣Ψ⟩∣Φ0⟩−N2 ΨN2 Φ0∣⟨Ψ∣Φ0⟩∣2∣Ψ⟩) (3) and δE[Ψ] δΨ∗=N2 Ψ(H−E[Ψ])∣Ψ⟩. (4) We have boldfaced the unpaired kets for clarity. Combining the two and settingδO δΨ∗to zero, we obtain (H−E[Ψ]−λN2 ΨN2 Φ0∣⟨Ψ∣Φ0⟩∣2)∣Ψ⟩+λN2 Φ0⟨Φ0∣Ψ⟩∣Φ0⟩=0. (5) TheH−E[Ψ] term ensures that this equation can only be satisfied by an energy eigenstate | Φi⟩. For i≠0, |Ψ⟩= |Φi⟩is a solution because ⟨Φi|Φ0⟩= 0. |Φ0⟩is a solution since ⟨Φ0|Φ0⟩= 1, which allows the λ terms to cancel. The value of the functional O[Φi] =Ei+λδi0. There- fore, ifλ>E1−E0, the global minimum is at | Φ1⟩. Because of the structure demonstrated in Fig. 1, there are no local minima in the complete Hilbert space. Similarly, the Nth excited state is given by ∣ΦN⟩=arg min Ψ(E[Ψ]+N−1 ∑ iλiN2 ΨN2 Φ0∣⟨Ψ∣Φi⟩∣2) (6) as long asλi>Ei−E0. We write the algorithm in terms of anchor states |Ψi⟩, where i = 0,. . .,N−1. These states are fixed during the optimization, and J. Chem. Phys. 154, 034101 (2021); doi: 10.1063/5.0030949 154, 034101-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp only the parameters of a single wave function | Ψ⟩are optimized. While ideally the anchor states are energy eigenstates, in general, they will be best approximations to them. We also find it useful to consider the objective function O[Ψ]=E[Ψ]+∑ iλi∣⃗Si−⃗Si∗∣2, (7) where Si=⟨Ψ∣Ψi⟩√ ⟨Ψ∣Ψ⟩⟨Ψi∣Ψi⟩(8) and⃗S∗is a set of target overlaps. For example, to obtain the Nth excited state, one would use N−1 anchor states each set to the best approximation to the N−1 lowest energy eigenstates and set ⃗S∗equal to a zero vector of N−1 length. B. Computation of the objective function and its derivatives using variational Monte Carlo In this section, we will explain how to evaluate the quantities needed using standard variational Monte Carlo techniques.2In this implementation, we sample a different distribution for each anchor state, ρi(R)=∣Ψi(R)∣2+∣Ψ(R)∣2. (9) Then, the unnormalized overlap is estimated in Monte Carlo ⟨Ψj∣Ψk⟩i≃⟨Ψ∗ j(R)Ψk(R) ρi(R)⟩ R∼ρi, (10) where Ris the many-electron coordinate and R∼ρimeans that Ris sampled from the normalized distributionρi(R) ∫ρi(R). Here, subscript i indicates that the overlap was estimated using ρi. The relative normalization of the wave function | Ψ⟩is, thus, Ni=⟨Ψ∣Ψ⟩i, (11) which is evaluated the same way as Eq. (10). For a parameter p of |Ψ⟩, ∂pNi=2Re⟨∂pΨ∗(R)Ψ(R) ρi(R)⟩ R∼ρi. (12) The overlap is given by Si=⟨Ψ∣Ψi⟩i Ai, (13) where Ai=√ ⟨Ψ∣Ψ⟩i⟨Ψi∣Ψi⟩i. The derivative of the unnormalized overlap is ⟨∂pΨ∣Ψi⟩=⟨∂pΨ∗(R)Ψi(R) ρi(R)⟩ R∼ρi. (14) The derivative of the normalized overlap is computed using the above-mentioned components as follows:∂pSi=⟨∂pΨ∣Ψi⟩i Ai−1 2⟨Ψ∣Ψi⟩i Ai∂pNi Ni. (15) Thus, all derivatives can be computed using only the wave function parameter derivatives. We also regularize all derivatives using the stochastic reconfig- uration27step to compute Rpq=⟨∂pΨ∣∂qΨ⟩ (16) for parameter indices pandq. C. Practical details The algorithm is outlined in Table I. We provide comments on steps that have some nontrivial considerations. In step 2, the param- eters are initialized. We typically initialize the parameters using an approximate excited state, typically either from an orbital promotion in a single determinant or from a small quantum chemistry calcula- tion. We have checked that it is possible to optimize starting from the ground state, but such a strategy is unnecessarily expensive, in particular, because the objective function of Eq. (7) is a saddle point at the ground state. In step 3(a), all the quantities in Sec. II B are computed. A Monte Carlo sampling of ρiis done for each anchor wave func- tion. The algorithm, thus, scales mildly with the number of anchor wave functions. The energy and its derivatives are averaged among all samples, so the most costly component of the calculation does not increase much with the number of anchor wave functions. It is important for the normalization of the wave functions to be similar; otherwise, the density in Eq. (9) is unbalanced. One could adjust weights in Eq. (9), but we find it more convenient to normalize all wave functions. We ensure that all anchor states have the same normalization and use the first anchor state as a reference. Before performing an optimization move, we first check whether N0is too far from1 2. The threshold is typically about 0.1. If it is too far, the parameters are rescaled to normalize the wave function and VMC is repeated with the renormalized wave function parameters. This is performed in step 3(b) in Table I. In steps 3(e) and3(f), we regularize the gradients of the nor- malization and the objective function. It is necessary to regularize both prior to the projection that will come in the next step. TABLE I . Implementation of the penalty-based optimization for excited states. 1. Choose target overlaps ⃗S∗ 2. Initialize ⃗p0 3.fori in range(nsteps), do (a) Compute Ni,⃗S,∇pN0,∇p⃗S,E, and∇pE (b)ifabs(N0-0.5)>threshold, normalize ΨN (c) Objective function is O=E+⃗λ⋅(⃗S−⃗S∗)2 (d) Construct gradient ∇pO=∇pE+∇p(⃗λ⋅(⃗S−⃗S∗)2) (e)∇pN←R−1∇pN (f)∇pO←R−1∇pO (g)∇pO←∇pO−(∇pO)⋅(∇pN) ∣∇pN∣2∇pN (h)⃗p(τ)←⃗pi−1−τ∇pO (i)⃗pi←arg minp(O(⃗p(τ)),τ) J. Chem. Phys. 154, 034101 (2021); doi: 10.1063/5.0030949 154, 034101-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp To prevent moves that change the normalization, we project out the derivative of the normalization from the objective function. Otherwise, the moves diverge from the equal normalization mani- fold, and it becomes difficult to evaluate the overlaps accurately. This is performed in step 3(g) in Table I.28 We find that line minimization, performed in step 3(i) in Table I, improves the performance of the algorithm significantly. We use correlated sampling to compute the objective function for various values of τand fit to a quadratic. We also reject moves that change the relative normalization by more than 0.3. III. DEMONSTRATION OF EXCITED STATE OPTIMIZATION USING VMC To demonstrate the technique, we apply it to two cases: H 2at varying bond lengths to check for correctness vs an exact solution and the excited states of benzene to demonstrate that it is capable of optimizing about 10 000 parameters on a system with 30 electrons in the calculation. A. Application to H 2 For application to H 2, the trial wave function was a simple com- plete active space (CASCI) wave function with two electrons and two orbitals taken from the restricted Hartree–Fock function. We take the three lowest excited states from this calculation, labeled ΨCASCI,i , where iruns from 0 to 2. This wave function was modified using a two-body Jastrow factor as parameterized in previous work29to obtain ΨCASCI-J,0=ΨCASCI,0 eU0. (17) We then optimized the determinant, orbital, and Jastrow parameters using a modified version of stochastic reconfiguration implemented inpyqmc to obtain ΨCASCI-J,0 . We then considered two Jastrow-based approximations to the excited states. The first, which we denote “Fixed,” is commonly used in the literature. It is given by ΨFixed CASCI-J, i =ΨCASCI,i eU0. (18) In this fixed wave function, no parameters are optimized at all, that is, the Jastrow factor is the same as the ground state, and the deter- minant and orbital coefficients are kept fixed. The second, we denote “Optimized,” begins with Eq. (18) and uses the penalty method to optimize determinant, orbital, and Jastrow parameters while ensur- ing orthogonality to lower states. In the supplementary material, a Snakemake workflow30is provided that performs the calculations shown here in pyqmc and pyscf.31We use aλof two hartree for these calculations, which were converged. For reference values, we used full configuration interaction (Full CI) to compute exact energies in a finite basis. We found that at the triple ζlevel, the energies were fairly well converged. This is slightly earlier than most materials due to the fact that H 2is very simple. In Fig. 2, we demonstrate the targeting capability of this tech- nique. Each point is a wave function generated in the Dunning cc- pvdz32basis as described above, with S∗set to various points on the Bloch sphere connecting the first three excited states. As expected, FIG. 2 . Targeted overlaps for H 2in the space of the three lowest eigenstates. The yellow points are wave functions, using the penalty method to set S∗. The xandy coordinates are the measured overlaps after optimization, and the energy E[Ψ] is the expectation value of the wave function after optimization. the superpositions of low energy wave functions fall on a plane, as sketched in Fig. 1, a critical check that the calculation is creating the desired wave functions. Comparisons between the CASCI, QMC, and full CI results are presented in Fig. 3. With a rather compact wave function, the optimized CASCI-J wave function obtains close agreement with the exact calculation, while optimizing significantly from the starting fixed CASCI-J wave function. In the case of the first excited state, simply adding a Jastrow factor to an existing CASCI wave func- tion does not improve the energies at all; optimization is required to obtain accurate results. B. Application to benzene excited states As a demonstration on a larger system, we computed the full πspace spectrum of benzene using our excited state optimization method. This set of 13 states contains a rich physical structure, with two different spin channels, single and double electron excitations, and states with ionic bonding in contrast to the covalently bonded ground state.33Some of these states, such as the1E1ustate, have been considered strongly correlated by previous authors.33As such, these excitations are a standard benchmark set for excited state methods and have been used to validate and compare electronic structure techniques for some time.34,35 To represent the wave functions, we used a multi-Slater– Jastrow wave function parameterization of the form ∣Ψ(⃗α,⃗c,⃗β)⟩=eJ(⃗α)∑ ici∣Di(⃗β)⟩. (19) The Jastrow factor J(⃗α)is a two-body Jastrow29factor. The deter- minants | Di⟩in the multi-Slater expansion were selected from a minimal CASCI calculation over the six πelectrons and six π orbitals in benzene. The single particle orbitals used in the CASCI were computed using density functional theory (DFT) with the J. Chem. Phys. 154, 034101 (2021); doi: 10.1063/5.0030949 154, 034101-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Comparison of FCI eigenvalues with energies of wave functions optimized with orthogonal optimization to target the first three eigenstates for H 2. The cc-pvtz basis of Dunning32was used. B3LYP functional, BFD triple- ζbasis, and BFD pseudo-potential.36 The CASCI and DFT calculations were carried out using pyscf.31 The parameters in the wave function are as follows: 108 Jastrow parameters ⃗α, 400 determinant coefficients ⃗c, and 9288 orbital parameters ⃗β. We used the parameterization of Eq. (19) to compute the ben- zene spectra using three different methods, with increasing cost. The first method, denoted as fixed, is a standard QMC excited state technique, where the coefficients ⃗α,⃗care first optimized on the ground state CASCI root with frozen orbital coefficients ⃗β, then the optimized Jastrow factor is multiplied with higher energy CASCI roots, and finally, these trial wave functions are used in VMC to compute excited state energies. This method does not allow for the state-specific optimization of any of the parameters ⃗α,⃗c, or⃗β.To understand the effects of orbital optimization in this system, we consider two parameter sets using the penalty technique. In the first, we fix the orbital parameters ⃗βto the DFT ground state orbital coefficients but allow the other parameters ⃗α,⃗cto be optimized; we denote the wave functions as VMC ⃗α,⃗c. Finally, we optimize all coef- ficients in Eq. (19), denoting those wave functions as ⃗α,⃗c,⃗β. All QMC calculations were carried out in pyqmc .26 The results of our excited state computations at the VMC level are shown in Fig. 4(a). We see a consistent 0.2 eV decrease in total energy across all 12 excited states going from the fixed parameter QMC method to the ⃗α,⃗cmethod using our new opti- mization technique. We find that optimizing ⃗α,⃗c,⃗βyields up to a 0.5 eV reduction in total energy relative to the fixed technique across nearly all of the excited states, a gain of around 0.3 eV due to orbital optimization. However, the differences between the excited states and the ground state are very similar between the frozen FIG. 4 . Benzene excited states computed using (a) VMC and (b) time step extrap- olated DMC for the full π-space spectrum. Different colors refer to increased parameter sets. ⃗αare the Jastrow coefficients, ⃗care determinant coefficients, and ⃗βare orbital coefficients, as denoted in Eq. (19). J. Chem. Phys. 154, 034101 (2021); doi: 10.1063/5.0030949 154, 034101-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp and optimized orbital calculations, which demonstrates the extent to which standard excited state techniques depend heavily on error cancellation. We computed the time step extrapolated DMC energies for the computed excited state wave functions, which are shown in Fig. 4(b). The additional benefit of optimizing the orbitals in DMC is seen primarily for just three states, states 1, 6, and 11, with reduc- tions in the energy of 0.14(3), 0.18(3), and 0.22(3) eV, respectively. The latter two of these states are the1B1u,1E1ustates, which have strong ionic bonding character among the πorbitals, unlike the ground state, which has covalent bonding character.33This differ- ence in bonding character is captured by the orbital optimization, leading to larger reductions in total energy in these states, while the other states have no reduction in total energy. The fact that only some excited states benefit from orbital optimization in DMC means that the energy differences are affected; ultimately, they are improved. Before comparing to the experiment, we note that most the- oretical results report the vertical excitation energy, which omits nuclear relaxation and vibrational effects. The transition from the ground vibrational level of the ground state to the ground vibra- tional level of the excited state is called the 0–0 excitation energy (E00) and is not necessarily the maximum intensity. The vertical excitation energy can be related to the 0–0 excitation energy as follows: E00(0→j)=Ej(⃗X0)−E0(⃗X0)(vertical) +Ej(⃗Xj)−Ej(⃗X0)(adiabatic) +̵h 2∑ i(ωj,i−ω0,i)(ZPVE), (20) where Ejis the Born–Oppenheimer energy of state j,Xjare the set of nuclear coordinates at the minimum energy of state j, andωj,iis the frequency of vibrational mode ifor electronic state j. The firstline is the fixed nuclei vertical excitation energy, the quantity which is directly computed in our work and in the cited theoretical calcu- lations, where the total energy of the jth excited state is computed at the ground state equilibrium nuclear positions ⃗X0. The second line is the adiabatic correction, which accounts for shifting of the nuclear positions in the jth excited state to the configuration ⃗Xj. The last line is the zero-point vibrational energy (ZPVE) correction and accounts for changes in the vibrational degrees of freedom between the ground and excited states. The experimental vertical excitation is obtained by subtracting the adiabatic and ZPVE energies from the 0–0 excitation energies. In Table II, we compare the vertical excitation energy com- puted from the experiment, the fully optimized DMC results in this work, and literature results of the large-basis coupled cluster (CC3),35complete active space perturbation theory (CASPT2),33 and time dependent density functional theory using the PBE0 func- tional (TDDFT-PBE0).41We also report computed ZPVE and 0–0 corrections using TDDFT with a 6-31g basis and PBE functional for the singlet states; we encountered difficulties converging the triplet states in TDDFT. We believe the TDDFT estimates to be reasonable, given the agreement with computed and measured values in the lit- erature:−0.15 eV39and−0.17 eV42adiabatic and ZPVE corrections for the1B2ustate and−0.56 eV43and−0.17 eV44for the3B1ustate. For the available vertical excitations, both the DMC and CC3 results are within 0.2 eV of the experimental values, while the CASPT2 and TDDFT methods have larger differences. The corrections due to adi- abatic and ZPVE effects are large enough that this conclusion might be reversed without these effects included. In Table III, we list the mean and standard deviation of the difference between CC3 and other methods. CC3 has been found to be very accurate for electronic excitations, including correlated doubles-like excitations for benzene.34,45The consis- tency between accurate, explicitly correlated methods such as CC3 and our new QMC technique, which makes different approx- imations, is encouraging, as is the fact that the QMC results TABLE II . Comparison of theoretically computed excitation energies with experimental values. All values are in eV. Maximum indicates the transition of maximum intensity. The adiabatic and ZPVE corrections are estimated using TDDFT with the PBE functional and 6–31g basis. Spectroscopy34,37–40Corrections Vertical excitation State Maximum E00 Adiabatic ZPVE Expt. DMC ⃗α,⃗c,⃗β CC335CASPT233TDDFT-PBE041 1B2u 4.90 4.72 −0.19 −0.18 5.09 5.15(3) 5.08 4.7 5.39 1B1u 6.20 6.03 −0.19 −0.33 6.55 6.62(4) 6.54 6.1 6.05 1E1u 6.94 6.87 −0.24 −0.32 7.43 7.72(4) 7.13 7.06 7.21 1E1u 6.94 6.87 −0.24 −0.32 7.43 7.63(3) 7.13 7.06 7.21 1E2g 7.80(20) 7.81 −0.21 −0.45 8.47 8.38(3) 8.41 7.77 7.52 1E2g 7.80(20) 7.81 −0.21 −0.45 8.47 8.34(3) 8.41 7.77 7.52 3B1u 3.94 3.65 −0.55 −0.19 4.39 4.15(3) 4.15 3.94 3.82 3E1u 4.76 4.63 4.89(3) 4.86 4.5 4.7 3E1u 4.76 4.63 4.96(4) 4.86 4.5 4.7 3B2u 5.60 5.58 6.08(4) 5.88 5.44 5.05 3E2g 7.49(25) 7.49(25) 7.74(4) 7.51 7.03 7.18 3E2g 7.49(25) 7.49(25) 7.60(4) 7.51 7.03 7.18 J. Chem. Phys. 154, 034101 (2021); doi: 10.1063/5.0030949 154, 034101-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE III . Agreement between theoretical vertical excitation energies. Optimizing more parameters using the penalty method improves agreement between CC3 and DMC, as seen in the mean and standard deviation of the difference between the excitation energies. ΔEj=Ej(m)−Ej(CC3) Method Parameters Mean (eV) rms (eV) CASPT2 −0.38 0.42 TTDFT-PBE0 −0.33 0.50 VMC None 0.34 0.46 VMC ⃗α,⃗c 0.12 0.35 VMC ⃗α,⃗c,⃗β 0.19 0.35 DMC None 0.24 0.35 DMC ⃗α,⃗c 0.18 0.31 DMC ⃗α,⃗c,⃗β 0.15 0.24 approach the CC3 results as the number of optimized parameters is increased. The cost of each excited state calculation is roughly a factor of 2–3 higher than the ground state calculation, as shown in Fig. 5. As a point of reference, the ground state optimization took 1.25 h on a 40 core processor for the fixed orbital optimization and 23 h on a modern 40 core processor for the orbital optimized calculation. One should keep in mind that as is typical in Monte Carlo, this cost is highly dependent on the desired uncertainty in the result. We chose very converged parameters, which results in a relatively high com- putational cost. The total relative cost over the ground state for all 13 states is a factor of 25 for the ⃗α,⃗ccalculation and 30 for ⃗α,⃗c,⃗β. Importantly, the relative cost does not increase rapidly with the state since, as mentioned in the Introduction, the overlaps are not very expensive to evaluate. FIG. 5 . Relative cost Ti/T0of computing excited states in VMC for the two differ- ent parameterizations considered in the benzene molecule, where Tiis the CPU time required to perform the calculation of excited state i.T0for the ground state were 1.25 h and 23.00 h on a single 40 processor node for the ⃗α,⃗cand⃗α,⃗c,⃗β parameterizations, respectively.IV. CONCLUSION We presented a scalable algorithm to compute approximate excited states of many-body systems using variational Monte Carlo. Our method is somewhat less complicated to implement than the linear method of Filippi16–18since the derivatives of the local energy are not required. Furthermore, our technique is capable of optimiz- ing complex parameters such as orbital coefficients in a state-specific manner and likely would be able to optimize parameters from wave functions such as backflow46,47and neural network forms,48–51as well as pairing functions52since the complexity is not much higher than in normal wave function optimization. In comparison to the method of Neuscamman,19–22this technique does not require a tuned parameter ω. One positive aspect of this is that the penalty technique can access several distinct but degenerate excited states separately. Degenerate ground states would also be detected by the penalty technique. The penalty technique is capable of optimizing wave functions with any overlap with a reference state. This capability may be use- ful in some circumstances, particularly for strongly correlated sys- tems and magnetic systems, in which energy eigenstates may not be easily representable by simple wave functions, but non-orthogonal basis states can be used to make relevant experimental predictions.9 Such wave functions are also appropriate for use in density matrix downfolding.53 The study on the benzene molecule revealed a few interest- ing physical insights. First, orbital optimization improves the nodal surface for excited states, particularly those with significant ionic character as compared to the ground state. Second, the classical association of the vertical excitation with the maximum intensity in spectroscopy leads to errors of up to 0.4 eV in the excitation energies of benzene. A fully quantum treatment of the Franck– Condon principle, as shown here, brings the experimental estimates in closer alignment with coupled cluster and quantum Monte Carlo results. SUPPLEMENTARY MATERIAL See the supplementary material for all collected data presented in the figures of the graph in a comma separated value format and scripts and a workflow for the H 2calculation for reproducibility. ACKNOWLEDGMENTS We would like to thank David Ceperley for helpful comments regarding the theorem of MacDonald54and William Wheeler for a careful reading of the code and bugfixes. This material is partially based on the work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Computational Materials Sciences Program, under Award No. DE-SC-0020177. B.B. was supported by Flatiron Insti- tute. Flatiron Institute is a division of the Simons Foundation. L.K.W. was supported by the Simons Collaboration on the Many- Electron Problem. This research is part of the Blue Waters sustained- petascale computing project, which is supported by the National Science Foundation (Award Nos. OCI-0725070 and ACI-1238993), the State of Illinois, and as of December 2019, the National Geospatial-Intelligence Agency. Blue Waters is a joint effort of the J. Chem. Phys. 154, 034101 (2021); doi: 10.1063/5.0030949 154, 034101-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp University of Illinois at Urbana-Champaign and its National Center for Supercomputing Applications. APPENDIX: PROPERTIES OF E(|⟨Ψ|Φ0⟩|2), AND THE APPROXIMATE VERSION E(∣⟨Ψ∣˜Φ0⟩∣2) Here, we present a few simple properties of the energy func- tional for reference. 1.E(|⟨Ψ|Φi⟩|2)≥E0−(E1−E0)|⟨Ψ|Φi⟩|2. This is the bounding line. 2. For any complete linear subspace, the minimization will form a line, and there are variational upper bounds.54 3. Broken symmetry wave function parameterizations (such as unrestricted Slater–Jastrow wave functions) do not usually comprise a complete linear space, so they may not form a line. DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material. REFERENCES 1Simons Collaboration on the Many-Electron Problem, J. LeBlanc, A. E. Antipov, F. Becca, I. W. Bulik, G. K.-L. Chan, C.-M. Chung, Y. Deng, M. Ferrero, T. M. Henderson, C. A. Jiménez-Hoyos, E. Kozik, X.-W. Liu, A. J. Millis, N. Prokof’ev, M. Qin, G. E. Scuseria, H. Shi, B. Svistunov, L. F. Tocchio, I. Tupitsyn, S. R. White, S. 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5.0029799.pdf

J. Appl. Phys. 128, 210901 (2020); https://doi.org/10.1063/5.0029799 128, 210901 © 2020 Author(s).Functional antiferromagnets for potential applications on high-density storage and high frequency Cite as: J. Appl. Phys. 128, 210901 (2020); https://doi.org/10.1063/5.0029799 Submitted: 18 September 2020 . Accepted: 13 November 2020 . Published Online: 02 December 2020 Hua Bai , Xiaofeng Zhou , Yongjian Zhou , Xianzhe Chen , Yunfeng You , Feng Pan , and Cheng Song ARTICLES YOU MAY BE INTERESTED IN Antiferromagnetic spintronics Journal of Applied Physics 128, 070401 (2020); https://doi.org/10.1063/5.0023614 Spin current generation and detection in uniaxial antiferromagnetic insulators Applied Physics Letters 117, 100501 (2020); https://doi.org/10.1063/5.0022391 Magnetization dynamics of nanoscale magnetic materials: A perspective Journal of Applied Physics 128, 170901 (2020); https://doi.org/10.1063/5.0023993Functional antiferromagnets for potential applications on high-density storage and high frequency Cite as: J. Appl. Phys. 128, 210901 (2020); doi: 10.1063/5.0029799 View Online Export Citation CrossMar k Submitted: 18 September 2020 · Accepted: 13 November 2020 · Published Online: 2 December 2020 Hua Bai, Xiaofeng Zhou, Yongjian Zhou, Xianzhe Chen, Yunfeng You, Feng Pan, and Cheng Songa) AFFILIATIONS Key Laboratory of Advanced Materials (MOE), School of Mate rials Science and Engineering, Tsinghua University, Beijing 100084, China a)Author to whom correspondence should be addressed: songcheng@mail.tsinghua.edu.cn ABSTRACT Antiferromagnets have drawn increasing attention in the last decade, for their advantages such as no stray field and ultrafast spin dynamics, giving rise to potential applications on high-density storage and high frequency. We summarize the recent progress on the control of antifer-romagnetic moments by electrical methods, including both electric current and electric field, which are important steps for the integrationof antiferromagnets toward high-density data storage. Several methods for distinguishing antiferromagnetic moments switching and artifactsare mentioned here. Then, we focus on the explorations of antiferromagnetic spin pumping and ultrafast spin dynamics. Such investigations would pave the way for applications with high frequency. Besides, the magnon transport in antiferromagnets is briefly introduced, which might be a basis of the antiferromagnetic logic. We conclude with a discussion of challenges and future prospects in antiferromagneticspintronics, which would stimulate in-depth studies and advance practical applications. Published under license by AIP Publishing. https://doi.org/10.1063/5.0029799 I. INTRODUCTION Antiferromagnet (AFM) is a class of magnetic material with no net magnetization, because the ordered moments are antiparal- lel between the adjacent atomic sites. Assuming that an AFM has two equivalent magnetic sublattices denoted as M Aand MB, then two orthogonal vectors can be defined: the magnetizationM AF=MA+MBand the Néel order N=MA−MB. For a long time, the essential features of AFMs have been commonly ignoredbecause they are overshadowed by ferromagnets (FMs). This per- ception was changed in the 1990s, when antiferromagnetic alloy began to be used as an exchange-coupling layer to pin ferromag-netic electrodes in magnetic tunneling junctions. It should be notedthat the AFM here just plays the role of supporting material, withits spin structure remaining almost unchanged during the deviceoperation. Recently, antiferromagnetic spintronics attracts continuous attention because of the potential applications for high-density storage and high frequency, stemming from the vanishingly smallstray field and ultrafast spin dynamics. 1,2In the last decade, severalapproaches were established for controlling AFMs, including mag- netic, optical, electrical, and mechanical methods ( Fig. 1 ), which are summarized in Table I . AFM is commonly considered insensitive to magnetic pertur- bations, and only a very large magnetic field could manipulate itsmoments. The magnetic state of AFM under an external magneticfield can be approximately described by a simple Stoner –Wohlfarth model, which combines antiferromagnetic exchange energy, anisot- ropy energy, and Zeeman energy. When applying an external field parallel to the easy axis of AFM, the Néel order remains along theeasy axis with zero net moment until the external field is largeenough to overcome the anisotropy field. Then, the Néel orderrotates to the direction perpendicular to the external field. This phenomenon is known as spin-flop, and the critical magnetic field is called spin-flop field H SF. Spin-flop transition makes it possible to control AFMs with weak anisotropy (such as α-Fe2O3and Cr2O3) by the magnetic field, which can be detected by spin Seebeck effect (SSE), spin Hall magnetoresistance (SMR), and non- local transport.3–6Besides, the FM/AFM exchange spring configu- ration facilitates the usage of a smaller external field to manipulateJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210901 (2020); doi: 10.1063/5.0029799 128, 210901-1 Published under license by AIP Publishing.antiferromagnetic moments, where tunneling magnetoresistance (TMR) and tunneling anisotropic magnetoresistance (TAMR) are used to monitor the switching behavior.7–9 Optical control of AFM is another important dimension in antiferromagnetic spintronics, which is proposed to show theadvantages of antiferromagnetic ultrafast dynamics. 10In contrast to FMs, the dynamics of AFMs perform as inertia-like motion, giving rise to the possible manipulation by ultrashort stimulus, such as terahertz (THz) pulses.11Note that THz pulses have magnetic field and electric field components, both of which can be used tocontrol AFM. 12,13 Although AFMs can be controlled by the magnetic field and THz pulses, the weak compatibility with complementary metaloxide semiconductor (CMOS) is the main roadblock. Therefore, itis necessary to realize the electrical manipulation of AFM, which isgreatly compatible with CMOS and essential in antiferromagnetic spintronics. Section IIdiscusses the control of AFM by electrical methods in detail, including current-induced spin –orbit torques and electric field. Later in Sec. III, we mainly discuss the antiferro- magnetic spin pumping and high frequency, as well as a brief intro-duction of non-local spin transport. Finally, we present our views of the future challenges and prospects of antiferromagnetic spin- tronics in Sec. IV. II. CONTROL OF ANTIFERROMAGNETIC MOMENTS BY ELECTRICAL METHODS A. Spin –orbit torques in antiferromagnets Using the electrical method to manipulate AFMs is essen- tial in antiferromagnetic spi ntronics. The current-induced spin –orbit torque (SOT), once widely used to switch the FM moments, 14may also induce the switching of antiferromagnetic moments by two mechanisms: the field-like torque with theform of dM A,B/dt∼MA,B×p15and the damping-like torque with the form of dM A,B/dt∼MA,B×(MA,B×p),16,17where MA,B represents the magnetization of each magnetic sublattice and p represents the spin polarization. Figure 2(a) illustrates the field- like torque-induced switching (Néel spin –orbit torque switching) process: when a charge current goes through AFMs with a locallybroken inversion symmetry (e.g., CuMnAs and Mn 2Au), oppo- sitely polarized spins would be generated in the two magnetic sub- lattice sites ( pA=−pB), giving rise to opposite directional effective fields (effective field is proportional to p).18Above the critical current density, the Néel order tends to be switched from theunstable N?p(M A,B?pA,B) state to the stable N//p(MA,B//pA,B) state [ Fig. 2(a) ]. The corresponding switching behavior can be monitored electrically by anisotropic magnetoresistance (AMR) orplanar Hall effect (PHE). 15,19–21Further experiments image the differences of the antiferromagnetic domains before and after thewriting current by x-ray magnetic linear dichroism photoemission electron microscopy (XMLD-PEEM), providing a direct evidence of the current-induced switching of antiferromagnetic moments. 22,23 FIG. 1. Schematic of the manipulation of antiferromagnetic moments by differ- ent methods. TABLE I. A summary of manipulation of antiferromagnets, showing the manipulation methods, mechanisms, and the detection methods. In the detection methods c olumn, the abbreviations are as follows: spin Seebeck effect (SSE), spin magnetoresistance (SMR), non-local transport (NLT), tunneling anisotropic magneto resistance (TAMR), tunneling magnetoresistance (TMR), Faraday effect (FE), planar Hall effect (PHE), anisotropic magnetoresistance (AMR), x-ray magnetic linear dichroism ph otoemission electron micros- copy (XMLD-PEEM), magneto-optical Kerr effect (MOKE), electric field-induced longitudinal resistance change ( E–Rxxloop), and electric field-induced magnetization change (E–Mloop). Manipulation Mechanism Detection Reference Magnetic field Spin-flop SSE, SMR, NLT 3–6 Magnetic field Exchange coupling TAMR, TMR 7–9 Optical THz magnetic field FE 12 Optical THz electric field PHE 13 Electric current Field-like torque AMR, PHE, XMLD-PEEM 15,22,23 Electric current Damping-like torque SMR, XMLD-PEEM, SSE, MOKE 25,27–29 Electric field Magnetoelectric coupling XMLD-PEEM 46,47 Electric field Ionic liquid gate voltage TAMR, E-Rxxloop 48–50 Strain Magnetic phase transition E-Mloop, E-Rxxloop, AHE 51,52 Strain Magnetic anisotropy switching AMR, PHE 53,54Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210901 (2020); doi: 10.1063/5.0029799 128, 210901-2 Published under license by AIP Publishing.Besides, terahertz electrical writing speed has been realized in an antiferromagnetic CuMnAs, paving the way for high-speed antiferro- magnetic memory.13Note that current would induce multi-domain switching in AFMs and the corresponding multi-level memorybehavior could be used in neuromorphic computing. 24 This field-like torq ue-induced switching only appears in AFMs with special crystal structures (i.e., global centrosymmetry plus broken sublattice inversion symmetry). For AFMs without aspecial crystal symmetry, manipulation of the Néel order couldbe achieved by damping-like torques in antiferromagnet/heavy metal (HM) heterostructure configuration (e.g., NiO/Pt). 25 Concomitant switching process is illustrated in Fig. 2(b) :w h e n an electric current is applied to the HM layer, the spin currentproduced by the spin Hall effect would inject into the AFM layer.In this scenario, spin polarization directions are the same for the two magnetic sublattices, leading to unstaggered field-like effec- tive fields, thus Néel spin –orbit torque-induced switching is absent. However, a detailed study of Landau –Lifshitz –Gilbert dynamics showed that the damping-like torques could effectivelyswitch the Néel order of AFM. 25Note that most experiments of damping-like torque-induced switching focus on antiferromagnetic insulator (AFMI), for which SMR offers a unique electrical detec-tion. 25,26Furthermore, several imaging methods were reported to detect the switching, such as XMLD-PEEM, SSE microscopy, andmagneto-optical Kerr effect (MOKE) microscopy. 27–29For the two-easy-axes AFMI (e.g., NiO), damping-like torques facilitate the switching of the Néel order between two metastable states, giving riseto two-state resistances. 25While for α-Fe2O3with three-easy-axes, three metastable states are degenerate, suggesting that damping-like torques would bring about tri-state switching of the Néel order and tri-state resistances.30It should be noted that damping-like torquescould also facilitate the switching of AFMs with the locally broken inversion symmetry (such as Mn 2Au), in which case field-like torques and damping-like torques may compete with each other.31 Besides, damping-like torque-induced switching is also observedin AFM Fe 1/3NbS 2without the HM layer, owing to a net spin polarization originated from the broken global crystal inversion symmetry.32Interestingly, the critical switching current density is as low as the order of 104Ac m−2. Recently, several studies show that some artifacts exist in the experiments of electrical control of AFM. The possible origins are electromigration, thermal effects, and the destruction of the device.30,33–35Based on the direct imaging evidence as described above,27–29the authenticity of the antiferromagnetic switching is proved. Particularly, the MOKE microscope not only clearly char-acterizes the switching of antiferromagnetic moments [ Fig. 3(a) ] but also visualizes the origin of the “sawtooth-like ”signals. As shown in Figs. 3(b) and3(c), with increasing current density, the electrical readout performs as a larger “jump ”and the MOKE image shows a larger area switching. The consistency between elec-trical readout and MOKE imaging could be further obtained in Fig. 3(d) , which solidly supports that the most contribution of the “sawtooth-like ”signal comes from AFM moments switching. Therefore, how to subtract artifacts from electrical readout signals is of great significance. One way is using the large externalmagnetic field to control the antiferromagnetic moments due to the spin-flop effect, then only artifacts can still contribute to the electrical signals. 30,32As shown in Fig. 4 , the moment-switching- induced signals can be separated by the subtraction of readoutsignals with and without a large magnetic field. We can see that a magnetic field of 9 T (which may be still not enough for pinning the AFM moments of NiO by spin-flop, but the highest field of our FIG. 2. The switching process of the Néel order via current-induced (a) field-like torques (Néel spin –orbit torques) and (b) damping-like torques. Here, the red and blue thin arrows with little circles represent oppositely polarized spins, and the red and blue thick arrows represent magnetic sublattices of antiferro magnet. The insets describe the two kinds of magnetic structures, i.e., antiferromagnets with (a) and without (b) the broken sublattice inversion symmetry. The main difference s between these two mechanisms (field-like torque-induced switching and damping-like torque-induced switching) can be summarized below: (1) field-like torque-ind uced switching (Néel spin – orbit torque switching) only appear in AFMs with special crystal structures (such as CuMnAs14), spin accumulation is generated in the AFM layer; damping-like torque-induced switching could appear in AFMs without special crystal structures, spin current comes from the adjacent heavy metal layer and (2) fie ld-like torque-induced switching tends to align the Néel order perpendicular to the current j(N?j); damping-like torque-induced switching tends to align the Néel order parallel to the current j(N/ /j ). (c) Spin –orbit torque-induced switching in noncollinear antiferromagnet. The states of magnetic octupole before and after switching are depicted as purple arrows in the corresponding antiferromagnetic layer.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210901 (2020); doi: 10.1063/5.0029799 128, 210901-3 Published under license by AIP Publishing.system) in Fig. 4(a) can clearly suppress the switching signals when the current density of 500 μs width pulse is below 5 × 107Ac m−2, indicating that the dominant contribution is from the AFMmoments switching. Further increasing the magnitude of theapplied current, we note that the suppression is not complete, because some thermal contribution emerges. Then, it is natural to understand that a large current pulse with the width of 10 ms iseasy to introduce strong thermal artifacts. 34Figure 4(b) shows a more clear distinction between electrical readouts with and without a large magnetic field in the configuration of Pt/ α-Fe2O3, because a small field ( ∼1 T) is enough to control AFM moments of α-Fe2O3.Hence, in this configuration, almost all electrical readout signals are induced by the moments switching of α-Fe2O3. Temperature-dependent experiment is another way to distin- guish whether the electrical signal is caused by AFM momentsswitching or artifacts. Here, magnetically intercalated transitionmetal dichalcogenides are suitable candidates for these control experiments. Applying current pulses below and above the Néel temperature, then we would get different resistance change, and thedistinction represents electrical signals from antiferromagneticmoments switching. 32Besides, some recent studies suggest that thermomagnetoelastic effects may also induce the switching of AFM moments33,36,37and even play a dominant role in some cases. Therefore, methods for distinguishing these two mechanisms(damping-like torques or thermomagnetoelastic effects) needfurther investigations. Apart from collinear AFMs, current-induced spin –orbit torques could also manipulate the spin orders in noncollinear antiferromag- nets, such as Mn 3GaN and Mn 3Sn.38,39Due to the special triangular spin structure, the “Néel order ”concept is unsuitable to describe the magnetic states of noncollinear antiferromagnets, which could bereplaced by the magnetic octupole. 40Taking Mn 3Sn for an example, current-induced switching of magnetic octupole is displayed in Fig. 2(c) : when an electric current is applied, the spin current pro- duced by the heavy metal would inject to the adjacent Mn 3Sn layer. With the assistance of the external magnetic field parallel to the elec- tric current, spin torques would induce the switching of the polariza- tion axis of the magnetic octupole.39An interesting point of noncollinear AFM is the existence of the anomalous Hall effectarising from the nonvanishing Berry curvature, which offers a detec-tion way with large readout signals. 41–43The electrical manipulations of collinear and noncollinear antiferromagnets shed light on the development of ultrafast and high-density memories. B. Electric field control of antiferromagnets The electric field provides another electrical way with ultralow energy consumption to control antiferromagnetic moments.44,45 Initially, the electric field control of AFM emerged in multiferroic materials with coupled ferroelectric and antiferromagnetic orders(e.g., BiFeO 3).46,47As illustrated in Fig. 5(a) , the electric field would manipulate electric moments and antiferromagnetic moments are switched concomitantly. However, it is necessary to investigate uni- versal methods for controlling antiferromagnetic moments by anelectric field because multiferroic materials are pretty rare. Then, people turn attention to the metallic antiferromagnets. To overcome the screening effect of metallic materials, a novel way that using ionic liquid as a dielectric gate is established, which can exhibit a large electric field effect to AFMs. 48–50Figure 5(b) displays the corresponding device configuration as well as the motion ofcations and anions when a gate voltage is applied. With the redis-tribution of charges, an electric double layer (EDL) is formed, playing the role of modulating the charge carrier density in the antiferromagnetic layer. The change of the charge carrier densitymay subsequently affect the electronic structure and magneticmoments of the AFM. Recently, a novel method that using electric field-induced fer- roelastic strain to manipulate AFMs attracts intense attention, FIG. 3. (a) Domain images show the initial state of a sample and the domain structure after jp= 1.5 × 1012Am−2writing pulse in −45° and +45° directions. (b) Electrical signal after subtraction a linear contribution for the accessiblecurrent densities. (c) Domain structure for different current densities after appli-cation of 15 pulses in the +45° direction. (d) Correlation of electrical signal and switching fraction calculated from the imaging. [Adapted with permission from F . Schreiber et al ., Appl. Phys. Lett. 117, 082401 (2020). Copyright 2020 AIP Publishing LLC.]Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210901 (2020); doi: 10.1063/5.0029799 128, 210901-4 Published under license by AIP Publishing.where the antiferromagnetic film is grown on a single crystal ferro- electric substrate.51–54For AFMs with special magnetic phase tran- sition (e.g., FeRh), two magnetic phases usually have different lattice parameters, indicating a possibility to modulate the magnetic phase transition by ferroelastic strain [ Fig. 5(c) ]. As a result, resist- ance states could be controlled by an electric field.55For AFMs with appropriate magneto-elastic responses (e.g., Mn 2Au and PtMn), electric field-induced ferroelastic strain would modulate magnetic anisotropy energy, leading to the switching of the antifer- romagnetic easy axis, as shown in Fig. 5(d) . Moreover, electric field-controlled antiferromagnetic tunneling junction has been the-oretically studied, with more than 500% TMR being predicted byfirst-principles calculations. 56Above efforts on the electric field control of AFM would open a door to antiferromagnetic memory with high density and ultralow energy consumption. III. ANTIFERROMAGNETIC SPIN PUMPING AND SPIN TRANSPORT A. Subterahertz spin pumping from antiferromagnet AFMs have the advantage of intrinsic ultrahigh spin dynamics frequency ( ∼THz) over FM ( ∼GHz), due to a combinationbetween the magnetic anisotropy and the large exchange interac- tion.2For AFMs with easy axes, the external microwave magnetic field along the easy axis would excite the antiferromagnetic reso- nance (AFMR). Theoretical analysis shows that the uniform preces- sional frequencies are separated into two regimes.57,58(1) When an external magnetic field H<HSF, AFMR has twofold degenerate modes (i.e., left-handed mode and right-handed mode) as showninFig. 6(a) , leading to two distinct branches of resonance frequen- cies. The corresponding relation between resonance frequency ω res and external magnetic field His described as follows: ωres¼γμ0ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi2HAHEp+γμ0H, (1) where γis the gyromagnetic ratio, μ0is the permeability of the vacuum, HAis the effective anisotropic field, and HEis the effective exchange interaction field. It should be noted that the spin-flop field HSFis dependent on HAandHEandHSF¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi2HAHEp. (2) When H>HSF, the antiferromagnetic orders are pinned at the direction nearly perpendicular to Hwith a tiny tilting, giving rise to a small net magnetization along the direction of H. In this case, the double-degenerate AFMR mode turns into the single-state resonance mode, which is similar to the ferromagnetic resonance FIG. 4. Measurement schematics and results of current-induced transverse resistance changes in (a) Pt/NiO and (b)Pt/α-Fe 2O3configurations. The trans- verse resistance differences between red lines (without an external field) and blue lines (with a large external field) suggestthat current can indeed induce switchingof the Néel order. Here, 9 T and 1 T external field are used to pin the Néel order of NiO and α-Fe 2O3.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210901 (2020); doi: 10.1063/5.0029799 128, 210901-5 Published under license by AIP Publishing.FIG. 5. Electric field control of antiferro- magnets by four different mechanisms: (a) magnetoelectric coupling in multi-ferroics materials; (b) ionic liquid gatevoltage; (c) ferroelastic strain-induced magnetic phase transition; and (d) mag- netic anisotropy switching. In panel (b),electric double layer (EDL) is formed atthe interface between ionic liquid and the antiferromagnetic layer. FIG. 6. Spin pumping from antiferro- magnetic resonance. (a) Schematics of the left-handed AFMR (LH-AFMR)mode, right-handed AFMR (RH-AFMR)mode, and QFMR mode. Here, σ LH and σRHare the spin polarizations associated with RH and LH chirality.(b) Magnetic-resonance frequency as afunction of the magnetic field μ 0H. AFMR or QFMR can be excited at four intersection points: RH-AFMR mode at(H R,ω1); LH-AFMR mode at ( HL,ω2); QFMR mode at ( HQ1,ω1) and ( HQ2, ω2). (c) Sample structure and mea- surement mechanism of antiferromag-netic spin pumping experiments.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210901 (2020); doi: 10.1063/5.0029799 128, 210901-6 Published under license by AIP Publishing.(FMR) mode and thus is denoted as the quasi-ferromagnetic reso- nance (QFMR) mode [ Fig. 6(a) ]. In this regime, resonance frequen- cies at different external magnetic fields satisfy the followingrelation: ω res¼γμ0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H2/C02HAHEp : (2) The magnetic field dependence of resonance frequency is depicted in Fig. 6(b) . Two horizontal lines with fixed frequencies (ω1andω2) are, respectively, intersected with the resonance fre- quency curves twice, at which points AFMR or QFMR are excited.Note that the chirality of the excited AFMR mode is determined by the frequency of microwave field. For example, the right-handed AFMR mode (RH-AFMR) is excited by microwave field with afrequency of ω 1(larger than intrinsic frequency) in the case of Fig. 6(b) , while the left-handed AFMR mode (LH-AFMR) is pro- duced by the microwave field with a frequency of ω2(lower than intrinsic frequency). In analogy to the ferromagnetic spin pumping experiments, AFMR also generates spin current, which could bedetected by the inverse spin Hall effect (ISHE) in the adjacentheavy metal layer [ Fig. 6(c) ]. Antiferromagnetic spin pumping was reported recently in antiferromagnetic insulators Cr 2O3and MnF 2.59,60In the study of Cr2O3, microwave field with only one frequency ∼0.24 THz (larger than intrinsic frequency) is applied to excite the antiferromagneticresonance. Therefore, the detected ISHE voltage only originates from the RH-AFMR mode. Reversing the direction of the magnetic field, the signal also changes its sign. Another group excited theAFMR of MnF 2with different frequencies and handedness of cir- cularly polarized microwave fields. Both RH-AFMR and LH-AFMRmodes can be stimulated in this scenario. It should be noted that two magnon modes are excited selectively depending on the hand- edness of microwave field, giving rise to the opposite spin polariza-tion. Above all, the chirality of the AFMR mode and the resultantspin polarization direction depend on three parameters: the mag-netic field polarity, the microwave frequency, and handedness.Antiferromagnetic spin pumping experiments provide a controlla- ble way to produce pure spin current at terahertz frequencies. B. Non-local spin transport and magnon-mediated spin torques in antiferromagnets Transport of spins, which is usually accomplished by spin- polarized charge current or magnon current, is at the center of spintronics. Compared with spin transport by the spin-polarized current, the magnon-mediated transport has a series of advantagessuch as Joule-heat-free and long-distance transport. 61A typical configuration of magnon-mediated non-local spin transport is dis-played in Fig. 7(a) , where two heavy metal stripes, respectively, play the role of spin source and spin detection. The efficiency of spin transport is signified by the intensity of ISHE voltage. Initially,magnon-mediated non-local spin transport mainly focused on fer-rimagnetic insulators (e.g., Y 3Fe5O12) due to the low damping constant.62Regarding the remarkable advantages of AFMs, more attention is paid to using AFMI as the medium of non-local trans- port. First, the non-local long-distance spin superfluid transportwas reported in canted easy-axis AFMI Cr 2O3.63Although the experimental temperature is much lower than the room tempera- ture (below 20 K), the spins can propagate over 20 μm. Soon later, diffusion-dominated non-local spin transport was reported in theeasy-axis α-Fe 2O3single crystal at 200 K.6It is found that the ISHE intensity can be controlled by the external magnetic field, and thehighest efficiency appears when the injected spin polarization is collinear with the Néel order. The above two non-local spin transport experiments are both based on AFMI with easy-axis, where two sublattices precess withdifferent cone angles, resulting in a finite angular momentum andoffering opportunity for the transport of spins. In contrast, it is considered that two magnon modes of AFMI with easy-plane cannot carry spin due to the absence of angular momentum.However, very recently, the non-local spin transport in the easy-plane AFMI α-Fe 2O3film was realized at room temperature, where the spin angular momentum manifests as circular precession of FIG. 7. (a) Schematic of non-local spin transport in an antiferromagneticinsulator. Two heavy metal (HM)stripes are placed on the top of the AFMI layer. Red and blue arrows illus- trate oppositely oriented spins. (b)Schematic of magnon-mediated spintorque switching. Note that the double headed arrows at the AFMI layer repre- sent the magnon transport behavior,and the thick blue arrows at FM layerillustrate the switching of ferromagnets. Here, magnon current is introduced by HM or topological insulator (TI).Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210901 (2020); doi: 10.1063/5.0029799 128, 210901-7 Published under license by AIP Publishing.sublattice spin.64The circular precession becomes linear precession and loses angular momentum after the propagation at a character-istic distance, which is determined by the difference of wavenum- bers of two eigenmodes. Those results show great potential of non-local spin transport in AFMI, such as magnonic memory devices with high operationfrequency and low energy dissipation. A milestone of magnonicmemory is the realization of magnetization switching induced by the magnon-mediated spin torque, 65which eliminates the Ohmic loss because of the absence of charge flow. Figure 7(b) illustrates the corresponding experimental configuration, in which spins gen-erated from HM or topological insulator (TI) are delivered through the precession of AFMI moments to the final FM layer. In analogy to spin transfer torque, the delivered spins would also induce spintorques in the FM layer and cause magnetization switching. Apartfrom the low energy consumption, large magnon-mediated spintorque efficiency was also reported, 65suggesting the great potential of magnon-spintronics for storage and logic with ultralow energy consumption. However, the spin current here is still introduced byapplying charge current into HM or TI, which may bring aboutJoule heat. Therefore, the next step is to realize all-magnon medi-ated spintronics. Recently, thermal-induced magnon torque in FM/ AFMI/FM trilayers was theoretically proposed, 66,67which could further decrease the energy dissipation of magnonic spintronics. IV. SUMMARY AND OUTLOOK Antiferromagnetic spintronics is emerging and promising, in which using AFM as the core layer might realize high-densitystorage and high frequency. Electrical control facilitates the com- patibility of AFM with common microelectronics technologies. The antiferromagnetic spin pum ping experiments signify apossible way of generating pure spin current at ultrahigh fre- quency. The realization of non-local spin transport and magnon-mediated spin torque switching would greatly reduce the energy consumption of the device. Apart from what we have summar- ized, there are still some open qu estions and challenges in anti- ferromagnetic spintronics: (1) All electrical manipulation and detection of antiferromagnetic moments have been realized, 15while the readout AMR signals are fairly weak. One solution is modulating the electronic structure of AFM to obtain larger AMR, where the topological AFMs with the tunable band structure are suitable candi-dates. 68,69As shown in Fig. 8(a) , these materials have two metastable states, and the band structures of these two statesare totally different: one has a Dirac cone and another has an energy gap, indicating a possibly large resistance change and AMR ratio. (2) The establishment of an antiferromagnetic tunneling junction is a key step for high-density data storage. Note thatcurrent-induced SOT would cause 90° switching of the Néel order, 15,25which means two orthogonal channels (four termi- nals) are necessary for the reversible switching of AFMmoments and data writing. Taking the reading electrode intoaccount, a total of five terminals are needed for the writing andreading data in the antiferromagnetic tunneling junction, which would greatly limit the storage density. Thus, how to establish the antiferromagnetic tunneling junction with less ter-minals is an emergent question. Two possible three-terminalantiferromagnetic tunneling junction configurations are dis- played in Figs. 8(b) and8(c): the first one uses the reversible SOT switching of noncollinear antiferromagnet for data FIG. 8. Possible materials and device configurations for high-density storage.(a) Large anisotropic magnetoresis-tance in topological antiferromagnets with the tunable band structure. (b) Three-terminal antiferromagnetic tunnel-ing junction device with current-inducedreversible magnetization switching of non- collinear antiferromagnet (e.g., Mn 3Sn39) for data writing. (c) Three-terminal anti-ferromagnetic tunneling junction devicewith electric field-induced ferroelastic strain for data writing (take Mn 2Au53for an example).Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210901 (2020); doi: 10.1063/5.0029799 128, 210901-8 Published under license by AIP Publishing.writing, and the second one uses electric field-induced antifer- romagnetic magnetization switching for data writing. Thesignal readout of these two configurations is, respectively, real-ized by TMR and TAMR. (3) Since antiferromagnetic spin pumping was reported, the recip- rocal effect that manipulating antiferromagnetic dynamics by the injected spin current is the next pursuing issue. An inter-esting application of the spin-controlled AFM dynamics is theantiferromagnetic spin-torque oscillator formed by the AFM/HM bilayer structure, where spin current from HM may excite the precession of AFM. 70,71When the spin torque is equivalent to the damping, AFM would show a stable precession with afixed frequency, bringing about oscillatory voltage or resistancesignals. Note that the typical AFMs have large anisotropy,which may cause large critical current density for spin-torque oscillating, thus AFMs with weak anisotropy (such as Fe 2O3 and MnF 2) are potential candidates for exploring antiferromag- netic spin-torque oscillators. (4) High spin dynamics frequency also suggests high velocity of domain wall (DW) motion in AFM. Theoretical studies show that both spin waves and spin –orbit torques could drive anti- ferromagnetic DW motion, and the velocity is two orders ofmagnitude larger than the ferromagnetic counterpart. 72–76So far, electrical control of DW motion is only reported in non- collinear antiferromagnet Mn 3Sn, while the velocity is less than 10 m/s,77lower than the expectation. Thus, attempting more novel antiferromagnetic materials for realizing DW motionwith higher speed is the next step. The investigation of ultrafastantiferromagnetic DW motion would provide a different avenue for the racetrack memory with high density and high writing speed. (5) Skyrmion is another potential data carrier of racetrack memory, which is pretty stable due to the topological protec-tion. Current-induced skyrmion motion has been well estab- lished for the ferromagnetic system, in which the critical current density is much lower than that of DW motion.However, the skyrmion Hall effect caused by the Magnus forcehampers the easy realization of ferromagnetic skyrmion race- track memory. This could be eliminated in antiferromagnetic or compensated synthetic antiferromagnetic skyrmions, 78,79 which needs to be experimentally explored further. (6) Recently, thermal-induced magnon torque in FM/AFMI/FM trilayers was theoretically proposed,66,67where external thermal gradient may produce magnon current in the antiferromag- netic insulator or upstream ferromagnetic layer. However,when the coherent spin current generated by HM/TI isreplaced by an incoherent thermal source, whether magnon-mediated spin torque can still induce magnetization switching is an open question. The corresponding experiments are worthwhile to be carried out. ACKNOWLEDGMENTS C.S. acknowledges the support of the Beijing Innovation Center for Future Chips, Tsinghua University. This work is sup- ported by the National Natural Science Foundation of China (NNSFC) (Grant Nos. 51671110 and 51871130).DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCE 1T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nat. Nanotechnol. 11, 231 (2016). 2V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, Rev. Mod. Phys. 90, 015005 (2018). 3S. Seki, T. Ideue, M. Kubota, Y. Kozuka, R. Takagi, M. Nakamura, Y. Kaneko, M. Kawasaki, and Y. Tokura, Phys. Rev. Lett. 115, 266601 (2015). 4S. M. 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5.0035702.pdf

J. Appl. Phys. 129, 025302 (2021); https://doi.org/10.1063/5.0035702 129, 025302 © 2021 Author(s).Epitaxial stabilization of (111)-oriented frustrated quantum pyrochlore thin films Cite as: J. Appl. Phys. 129, 025302 (2021); https://doi.org/10.1063/5.0035702 Submitted: 30 October 2020 . Accepted: 15 December 2020 . Published Online: 08 January 2021 Fangdi Wen , Tsung-Chi Wu , Xiaoran Liu , Michael Terilli , Mikhail Kareev , and Jak Chakhalian COLLECTIONS Paper published as part of the special topic on Topological Materials and Devices This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN Role of substrate strain to tune energy bands–Seebeck relationship in semiconductor heterostructures Journal of Applied Physics 129, 025301 (2021); https://doi.org/10.1063/5.0031523 Strongly correlated and topological states in [111] grown transition metal oxide thin films and heterostructures APL Materials 8, 050904 (2020); https://doi.org/10.1063/5.0009092 The importance of singly charged oxygen vacancies for electrical conduction in monoclinic HfO2 Journal of Applied Physics 129, 025104 (2021); https://doi.org/10.1063/5.0036024Epitaxial stabilization of (111 )-oriented frustrated quantum pyrochlore thin films Cite as: J. Appl. Phys. 129, 025302 (2021); doi: 10.1063/5.0035702 View Online Export Citation CrossMar k Submitted: 30 October 2020 · Accepted: 15 December 2020 · Published Online: 8 January 2021 Fangdi Wen,a) Tsung-Chi Wu, Xiaoran Liu, Michael Terilli, Mikhail Kareev, and Jak Chakhalian AFFILIATIONS Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA Note: This paper is part of the Special Topic on Topological Materials and Devices. a)Author to whom correspondence should be addressed: fangdi.wen@rutgers.edu ABSTRACT Frustrated rare-earth pyrochlore titanates, Yb 2Ti2O7and Tb 2Ti2O7, have been proposed as promising candidates to realize quantum spin ice (QSI). Multiple exotic quantum phases, including Coulombic ferromagnet, quantum valence bond solid, and quadrupolar ordering, have been predicted to emerge in the QSI state upon the application of a (111)-oriented external magnetic field. Here, we report on the successful layer-by-layer growth of thin films of the frustrated quantum pyrochlores, R 2Ti2O7(R¼Er, Yb, and Tb), along the (111) direction. We confirm their high crystallinity and proper chemical composition by a combination of methods, including in situ RHEED, x-ray diffrac- tion, reciprocal space mapping, and x-ray photoelectron spectroscopy. The availability of large area (111)-oriented QSI structures with planar geometry offers a new complementary to the bulk platform to explore the strain and the magnetic field-dependent properties in the quasi-2D limit. Published under license by AIP Publishing. https://doi.org/10.1063/5.0035702 I. INTRODUCTION In the context of modern condensed matter, there has been a growing interest in searching for exotic states and excitations inhighly frustrated material systems. For example, the excitations ofmagnetic monopoles due to the fractionalized dipolar degrees of freedom can be found in classical spin ice (CSI) materials such as Ho 2Ti2O7and Dy 2Ti2O7, the rare-earth pyrochlore oxides.1–10 Another intriguing state of correlated matter is the quantum spin liquid (QSL), characterized by emergent fractionalized excitationswith long-range entanglement. 11–17 Recently, a new class of QSLs named quantum spin ice (QSI) has been theoretically proposed by including quantum fluctuations,which act to prevent the spins from reaching a symmetry-breakingordered phase of CSIs. 9,18–24QSI has drawn much attention since it supports a QSL ground state described in the framework of compact lattice gauge theory with exotic excitations, including magnetic monopoles, U(1) gauge photons, and spinons.19,23,25–30 Interestingly, even if a QSI phase is not ultimately realized, due to its highly frustrated nature, proximity to the QSI state is anticipatedto induce several unusual quantum phases. 18,21,23,26,31–36 Experimentally, many frustrated quantum pyrochlores, includ- ing R 2Ti2O7(R¼Tb and Yb), have been put forward as promisingcandidates for QSI.9,20,29,30,37–46With the localized nature of f-shell moments and the strong spin –orbit interaction, these f-electron systems exhibit highly directional exchange interactions akin tothose found in Kitaev magnets, resulting in a strongly frustrated behavior. 9,47–50From the experimental viewpoint, however, more often than not, the exact nature of the ground state and the low-energy excitations in these quantum pyrochlores remains largely unresolved. One reason is the lack of detailed knowledge on the role of disorder. 42,51–54To illustrate, recently, it was found that dis- orders can promote a new phase in the candidate QSI pyrochlore Pr2Zr2O7, which is disordered and yet exhibits short-range antiferro-quadrupolar correlations and mimics the QSL-like fea-tures in neutron scattering. 52,55Another challenge is the difficulty of fabricating high-quality single-crystalline samples with low dis- orders, which hinders the intrinsic QSI physics and unavoidablygives rise to inconsistency between experiments. 56–63To address these challenges, it is, thus, critical to develop new methods of material synthesis that are distinct from the conventional solid-statesynthesis route to unveil the true nature of the frustrated quantum pyrochlores. One compelling approach that has attracted significant recent interest is to grow the pyrochlore materials as thin films orientedJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 025302 (2021); doi: 10.1063/5.0035702 129, 025302-1 Published under license by AIP Publishing.along the (111) direction.64–67The resulting films contain alternat- ing kagome and triangular atomic planes of magnetically active rare-earth ions, which are naturally formed in such orientationand are known to support emergent magnetic states, includingQSL. 64–66,68Based on this motif, many exotic phenomena including quantum kagome valence bond solid, Coulombic ferromagnet, quadrupolar, and monopole super-solids were proposed to emerge in the QSI films under a magnetic field applied along the (111)direction. 69,70Surprisingly, to date, the synthesis of (111) pyro- chlore thin films remains very limited and often relies on access tothe commercially unavailable pyrochlore substrate Y 2Ti2O7.71,72 Besides, the epitaxial control of the QSI films can be vital for the potential applications in the subfield of quantum informationtechnology. Here, we report on the successful layer-by-layer growth of high-quality thin films of a series of frustrated quantum pyro- chlores, R 2Ti2O7(R¼Er, Tb, and Yb), on the 5 /C25m m2(111) yttria-stabilized ZrO 2(YSZ) substrate. We confirm the high crystal- linity of thin films by in situ reflected high energy electron diffrac- tion (RHEED), x-ray measurements including x-ray diffraction(XRD), and reciprocal space map (RSM) and validate that all the films exhibit correct chemical composition by x-ray photoelectron spectroscopy (XPS). Our work offers a complementary to the bulkroute to resolve the puzzles in physics of frustrated quantum pyro-chlores and contributes to the possible realization of the long- thought QSI state. II. RESULTS AND DISCUSSION Three frustrated quantum pyrochlores, R 2Ti2O7(R¼Er, Tb, and Yb), were epitaxially stabilized on the YSZ substrates usingpulsed laser deposition with in situ RHEED control at the identical growth condition. Specifically, before the growth, substrates were heated up to 750 /C14Ca t1 0/C14C/min under 120 mTorr oxygen pres- sure. During the growth, the presence of the layer-by-layer growthmode was determined by high-pressure RHEED. All the reported films were deposited at the 18 Hz repetition rate interrupted by 15-s intervals. A high-frequency deposition enlarges the supersatu-ration limit, lowers the nucleation barrier, and thus markedlyimproves the growth. 73After the growth, samples were kept at the growth condition for 10 min and then cooled down to room tem- perature with the ramp rate of 15/C14C/min. It is noteworthy that no annealing was required in this process. Surprisingly, as shown in Fig. 1(b) , unlike a conventional RHEED pattern expected for interrupted growth, two oscillations were found to appear for all samples albeit with a different number of pulses required to complete a cycle. This unexpected result can be readily explained by the fact that since the rare-earth sites andTi sites have the same sublattice structure (but with a smalldisplacement), a Kagome layer is composed of rare-earth ions thatare coplanar with a triangle layer of Ti 4þions, which can be seen inFig. 1(a) . Therefore, a complete structural unit requires two inequivalent bilayers (BL) composed of one rare-earth kagomeplane (K R) plus one triangle Ti ions plane (T T)o rB L ¼KR-T T and another bilayer composed of kagome Ti (K T) and triangle rare-earth (T R) planes, thus overall 1 u.c. = K R-TT-KT-TR:As illus- trated in Fig. 1(a) , these BLs are not precisely on the top of eachother but rather stacked with a small offset. Further thickness analysis revealed that between each interval, we indeed deposited two BLs of material. Furthermore, this structural consideration is ingood agreement with the observed doublet structure of RHEEDintensity in all (111)-oriented films. The ability to see the two-peakstructure in RHEED oscillations during high-frequency deposition provides strong evidence for the layer-by-layer growth. After each deposition, RHEED intensity recovers to almost the same intensity, with a minor decay corresponding to the naturaldecay of the LaB 6electron source. This result implies that the surface roughness remained roughly constant for each unit cell. The RHEED patterns of all samples, including the YSZ substrate, are shown in the left part of Figs. 1(c) –1(f). Because of our fixed electron gun geometry, we could not adjust the electron incidentangle large enough to observe the ( /C01,/C01) and (1, 1) crystal peaks of YSZ. However, soon after the initial deposition, two evident streaky film peaks emerge at the ( /C01,/C01) and (1, 1) positions. The horizontal peak position matches the substrate peak position, whilethe vertical peak of the film can be observed because its shapebecomes streaky. A clear half-order reflections can be seen after thedeposition, which is as expected since the pyrochlore lattice constant is twice of that of YSZ. When cooled down to 300 /C14C, the RHEED pattern remained unchanged and disappeared at the room tempera-ture due to the strong charging effect. After the growth, each sample was characterized in four aspects: the surface morphology, crystallinity, chemical composition, and crystal structure. Other than determining the surface morphologybyin situ RHEED, atomic force microscopy (AFM) is another com- monly used probe. As seen in the right half of Fig. 1(c) ,t h eY S Zs u b - strate has a root mean square surface roughness S qof around 58 pm over a 1 :25/C21:25μm2area. The S qof Er 2Ti2O7(d), Yb 2Ti2O7(e), and Tb 2Ti2O7(f) are all of the same order of 60 –80 pm. Remarkably, since each BL corresponds to 80 pm, the surface rough-ness of all three films is within 1 BL, which is another manifestationof the atomically flat surface due to the layer-by-layer growth. To assure that during the deposition, ions are stabilized in the proper charge state, a detailed XPS analysis has been carried out onboth R (Er, Yb, and Tb) and Ti core-states. During the XPS analy-sis, all the recorded data were calibrated to the carbon 1 speak at 284.5 eV to eliminate the possible charging effect. The XPS core level data of Ti 4þin Er 2Ti2O7,Y b 2Ti2O7, and Tb 2Ti2O7are shown inFig. 2(a) . For better comparison, the Shirley-type background was subtracted from all Ti XPS spectra. Ti 2 p3=2peak and Ti 2 p1=2 peak were fitted with a fixed area ratio of 2:1. The measured binding energy for Ti 2 p3=2peak is around 458.8 eV, which is in good agreement with the reference Ti4þsystems.77,78Although tita- nium can easily lower its valency, the spectra show no sign of Ti3þ (/difference455 eV) present in any of the films. On the other hand, being in the group of lanthanides, Er, Tb, and Yb almost exclusively form compounds with an oxidation state ofþ3. Unlike the relatively simple line shape of Ti4þXPS, the XPS spectra of R sites (Er, Yb, and Tb) are significantly more complex.To overcome this issue, we compare our results to the referenceoxide systems with well-defined charge states. First, as shown in Fig. 2(b) , though the peak positions of Er 3þvary in different mate- rial systems, the XPS spectrum of Er 2O3shows a clear correspon- dence to our data.79–81The observed fine structure of Er3þcan beJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 025302 (2021); doi: 10.1063/5.0035702 129, 025302-2 Published under license by AIP Publishing.attributed to the hybridization of Eu ions into Er —O, Er —Er, and Er—Ti bonds and surface contamination.81For Yb 2Ti2O7, the single 4 d5=2peak can be clearly seen at around 185 eV, which is in good agreement with previous XPS result of Yb3þin Yb 2O3 [see Fig. 2(c) ].75,82,83It is worthy to note that if there is Yb2þ present in the material, an extra peak will show at around 181 eV, which is not the case for our result. Finally, the spectrum of Tb3þ 3dis consistent with the XPS spectra reported for Tb 2O3.76,84At this point, we can conclude that all synthesized films show correctR 3þ(i.e., Er3þ,Y b3þ, and Tb3þ) and Ti4þoxidation states. Next, we discuss the structural properties of the films. In the bulk, the reported lattice constants for Er 2Ti2O7and Yb 2Ti2O7are10.075 Å and 10.032 Å, respectively.85,86Their lattice constants cor- respond to the spacing of 5.82 Å for Er 2Ti2O7and 5.79 Å for Yb2Ti2O7for each BL grown along the (111) direction. However, from the bulk result, it is known that the lattice constant ofTb 2Ti2O7is very sensitive to the growth condition; specifically, depending on the synthesis technique, the lattice constant can vary from 10.127 Å87to above 10.15 Å.56,57,88As clearly seen in Fig. 3(a) ,E r 2Ti2O7and Yb 2Ti2O7show film peaks at higher angles than substrate peaks. The calculated spacing along the (111) direc-tion is estimated to be 5.80 Å for Er 2Ti2O7and 5.78 Å for Yb2Ti2O7. The smaller out-of-plane lattice spacing indicates tensile strain in the film, which is expected as pyrochlores have the lattice FIG. 1. (a) Schematic picture of R or Ti site sublattice in R 2Ti2O7. The triangle and Kagome structure along the (111) direction was marked out. A side view shows one of the (111) crystal planes that is universal to both A and Ti sites —when A sites are forming a triangle structure, the Ti sites are at corresponding Kagome positions and vice versa. (b) RHEED oscillations during the high-frequency growth, where each interval exhibits the growth of one BL. Note, only the kagome layers are ma rked for clarity. (c)–(f) RHEED pictures of YSZ, Er 2Ti2O7,Y b 2Ti2O7, and Tb 2Ti2O7, respectively. Electron beam incident from the (1,1, /C02) crystal orientation of YSZ. The white inset curve is a horizontal intensity line cut of the corresponding figure. To the right of (c) –(f) are the AFM of YSZ, Er 2Ti2O7,Y b 2Ti2O7, and Tb 2Ti2O7surfaces. With the same scanning area of 1 :25/C21:25μm2, the corresponding root mean square roughness obtained by AFM is as follows: YSZ /C058 pm, Er 2Ti2O7/C063 pm, Yb 2Ti2O7/C085 pm, and Tb 2Ti2O7/C080 pm.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 025302 (2021); doi: 10.1063/5.0035702 129, 025302-3 Published under license by AIP Publishing.constants smaller than that of the YSZ substrate (5.12 Å). Also, because Yb 2Ti2O7has a more pronounced lattice mismatch with YSZ, it has a smaller out-of-plane spacing, which is confirmed byour XRD result. Notably, the (333) and (555) peaks are not observed in the Er 2Ti2O7and Yb 2Ti2O7films. If we define the peak ratio between the (111) peak and the (333) peak as r, under cubic symmetry Fd/C223m, rETO¼99:56 and rYTO¼75:17, implying that the intensity of the (333) peak is indeed very low compared to the (111) peak.Moreover, even for the bulk crystals of Er 2Ti2O7and Yb 2Ti2O7, the (333) peak is barely observed in neutron or x-ray scattering measurements.89–91On the other hand, our XRD measurements on Tb2Ti2O7have revealed several variations in the structural prop- erties, including the lattice constant and intensity of the (333)peak. Here, assuming Tb 2Ti2O7is under Fd /C223m space group, and aTTO¼10:15 Å, the estimated rTTO¼96. However, our XRD result indicates that the value of rexp TTO¼10:44, which is much smaller than expected for the Fd /C223m space group. In addition, the estimated out-of-plane lattice spacing from (111), (222), (333),and (444) film peaks consistently yields d 111¼6:03 Å, which is by 2.6% larger than abulk¼10:15 Å ( d111¼5:86 Å). Based on the reciprocal space mapping (RSM) [see Fig. 3(c) ], the film appearsto be fully strained in-plane; therefore, the experimental unit cell volume of Tb 2Ti2O7is 1095 Å3, which is 4.7% larger than that of the bulk unit cell (1045.7 Å3). Despite the observed deviation in the lattice constant of Tb 2Ti2O7, the XRD scans show no extra peaks other than those of the (111)-oriented crystal, confirmingthe absence of a secondary chemical phase or domain separation in the Tb 2Ti2O7films. As shown in Figs. 3(a) and 3(b), all films demonstrate clear Kiessig fringes near the film peaks, which result from the interfer-ence of the x-ray beams reflected on the film surface and the inter-face between the film and the substrate. These fringes indicate that our films have high thickness homogeneity and small surface/inter- face roughness constant with the AFM data. With angle-dependentKiessig fringes, the film thickness is estimated to be 16 nm forEr 2Ti2O7, 20 nm for Yb 2Ti2O7, and 30 nm for Tb 2Ti2O7. Furthermore, the RSM shown in Fig. 3(c) implies a perfect align- ment between the substrate (042) and film peaks (084) horizontally, signifying that all samples are fully strained. The RSM data alsocorroborate that the out-of-plane lattice constants of Er 2Ti2O7and Yb2Ti2O7are larger than the out-of-plane lattice constant of the YSZ substrate and smaller for Tb 2Ti2O7. Altogether, we observed a unique crystal structure in the Tb 2Ti2O7film that deviates from the FIG. 2. (a) XPS of Ti 2p on Er 2Ti2O7,Y b 2Ti2O7,a n dT b 2Ti2O7films. All the background has been subtracted for better comparison. (b) XPS of Er 4d in Er 2Ti2O7sample in comparison to the data adapted from the reported Er 2O3(Kaya and Yilmaz, Nucl. Instrum. Methods Phys. Res. B 418,7 4–79 (2018). Copyright 2018 Elsevier). (c) XPS of Yb 4d in Yb 2Ti2O7sample in comparison to the data adapted from the reported Yb 2O3(Wang et al. , Mater. Chem. Phys. 77,8 0 2–807 (2003). Copyright 2003 Elsevier). (d) XPS of Tb 3d in Tb 2Ti2O7sample in comparison to the data adapted from the reported Tb 2O3(Cartas et al ., J. Phys. Chem. C 118, 20916 –20926 (2014). Copyright 2014 American Chemical Society).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 025302 (2021); doi: 10.1063/5.0035702 129, 025302-4 Published under license by AIP Publishing.commonly observed in the bulk Fd /C223m cubic structure. The origin of the enlarged unit cell volume and how the change in the crystal structure may affect its physical properties remain an open question to be studied in the future. III. CONCLUSION In conclusion, we have epitaxially grown new high-quality frustrated quantum pyrochlores R 2Ti2O7(R¼Er, Tb, and Yb) on (111)-oriented YSZ substrates in the layer-by-layer mode. Allsystems exhibit correct chemical composition on both rare-earth sites and Ti site. Structurally, all films are fully strained with high homogeneity and very small surface roughness. Among the threesystems, Er 2Ti2O7and Yb 2Ti2O7demonstrate tensile lattice strain as expected in the assumption of tetragonal distortion, while Tb2Ti2O7shows an unexpectedly larger lattice parameter for the Fd/C223m space group. Our work not only lays a solid ground forhighly frustrated lattices grown from pyrochlore materials into thin film forms in a (111) direction, but also offers a novel materials ’ platform to explore the effects of strain and magnetic field in the quasi-2D limit with the unique potential for the possible realizationof QSI. ACKNOWLEDGMENTS This work was supported by the Gordon and Betty Moore Foundation EPiQS Initiative through Grant No. GBMF4534. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. FIG. 3. (a) XRD of Er 2Ti2O7,Y b 2Ti2O7, and Tb 2Ti2O7films near the film peak (222) and the substrate peak (111). (b) Full range XRD of Er 2Ti2O7,Y b 2Ti2O7, and Tb2Ti2O7. (c)–(e) are the reciprocal space maps of Er 2Ti2O7,Y b 2Ti2O7, and Tb 2Ti2O7around the substrate peak (0 ,4,2) and the film peak (0 ,8,4). 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5.0024802.pdf

Rev. Sci. Instrum. 92, 013701 (2021); https://doi.org/10.1063/5.0024802 92, 013701 © 2021 Author(s).100 MHz large bandwidth preamplifier and record-breaking 50 kHz scanning rate quantum point contact mode probe microscopy imaging with atomic resolution Cite as: Rev. Sci. Instrum. 92, 013701 (2021); https://doi.org/10.1063/5.0024802 Submitted: 13 August 2020 . Accepted: 11 December 2020 . Published Online: 07 January 2021 Quan Feng Li , Yang Wang , Fang Wang , Yubin Hou , and Qingyou Lu ARTICLES YOU MAY BE INTERESTED IN Achieving μeV tunneling resolution in an in-operando scanning tunneling microscopy, atomic force microscopy, and magnetotransport system for quantum materials research Review of Scientific Instruments 91, 071101 (2020); https://doi.org/10.1063/5.0005320 Fast low-noise transimpedance amplifier for scanning tunneling microscopy and beyond Review of Scientific Instruments 91, 074701 (2020); https://doi.org/10.1063/5.0011097 A low noise cryogen-free scanning tunneling microscope–superconducting magnet system with vacuum sample transfer Review of Scientific Instruments 92, 023906 (2021); https://doi.org/10.1063/5.0041037Review of Scientific InstrumentsARTICLE scitation.org/journal/rsi 100 MHz large bandwidth preamplifier and record-breaking 50 kHz scanning rate quantum point contact mode probe microscopy imaging with atomic resolution Cite as: Rev. Sci. Instrum. 92, 013701 (2021); doi: 10.1063/5.0024802 Submitted: 13 August 2020 •Accepted: 11 December 2020 • Published Online: 7 January 2021 Quan Feng Li,1,a) Yang Wang,1Fang Wang,2Yubin Hou,3 and Qingyou Lu3,4,a) AFFILIATIONS 1Henan Key Laboratory of Photovoltaic Materials, School of Physics, Henan Normal University, Xinxiang 453007, People’s Republic of China 2College of Electronic and Electrical Engineering, Henan Normal University, Xinxiang 453007, Henan, People’s Republic of China 3Anhui Province Key Laboratory of Condensed Matter Physics at Extreme Conditions, High Magnetic Field Laboratory, Chinese Academy of Sciences, Hefei 230031, Anhui, People’s Republic of China 4Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China a)Authors to whom correspondence should be addressed: lqfeng@mail.ustc.edu.cn and qxl@ustc.edu.cn ABSTRACT The high-bandwidth preamplifier is a vital component designed to increase the scanning speed of a high-speed scanning tunneling microscope (STM). However, the bandwidth is limited not only by the characteristic GΩ feedback resistor RFbut also by the characteristic unity-gain- stable operational amplifier (UGS-OPA) in the STM preamplifier. Here, we report that paralleling a resistor with the tunneling junction (PRTJ) can break both limitations. Then, the UGS-OPA can be replaced by a higher rate, higher antinoise ability, decompensated OPA. By doing so, a bandwidth of more than 100 MHz was achieved in the STM preamplifier with decompensated OPA657, and a higher bandwidth is possible. High-clarity atomic resolution STM images were obtained under about 10 MHz bandwidth and quantum point contact microscopy mode with a record-breaking line rate of 50 k lines/s and a record-breaking frame rate of 250 frames/s. Both the PRTJ method and the decompensated OPA will pave the way for higher scanning speeds and play a key role in the design of high-performance STMs. Published under license by AIP Publishing. https://doi.org/10.1063/5.0024802 .,s I. INTRODUCTION The scanning tunneling microscope (STM) with high-speed (HS) scanning capability is a crucial tool in surface science and technology.1–5It has revealed the mechanism of dynamic phenom- ena at the atomic scale: the growth mechanism of graphene,4for example. Inevitably, it will also be a tool for high-speed readout in atomic-scale data storage devices.3The femtosecond (fs) laser STM6and electronic pump-probe radio STM can only investi- gate dynamic processes with a spectrum, but their imaging speeds are also conventional.7Accordingly, many groups are devoting themselves to the development of high-speed imaging STMs.8–12However, progress in this direction is greatly restricted by not only the limited bandwidth fBbut also the bad signal-to-noise ratio (SNR) in STM preamplifiers.4,13–16 The bandwidth is limited by several factors simultaneously. First, fB≤(2πRFCF)−1, where RFis the feedback resistance and CF = (CAF+CSF); in the latter, CAFis from an artificially added feed- back capacitor and CSFis the unavoidable stray capacitance in the feedback loop wiring.17Specific methods to increase fBinclude con- necting many small resistance feedback resistors in series and multi- stage amplifying.8Second, fBis also limited by fB≤GBP/G, where GBP is the gain bandwidth product of the operational amplifier (OPA) and G is the noise gain of the amplifying circuit. This formula Rev. Sci. Instrum. 92, 013701 (2021); doi: 10.1063/5.0024802 92, 013701-1 Published under license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi indicates that an OPA with higher GBP is required in the HS-STM preamplifier. Third, the STM preamplifier is a trans-impedance amplifier (TIA), while the preamplifier of the alternating current (AC)-STM belongs to the voltage amplifier.18Then, the bandwidth theory of TIA, which requires fB≤(GBP/2πRFCS)1/2, has to be taken into account, where CS=CCM+CDIFF +Cjis the total source capac- itance,19,20CCMand CDIFF separately are the common mode and dif- ferential mode input capacitance, and Cj(≈0.2 pF) is the tip–sample junction capacitance.20None of these bandwidth limitations can be ignored. The authors have decreased the RFfrom∼GΩ to 10 MΩ without the help of CAF, and excellent STM images have been obtained,11,21but the bandwidth was only a little more than 10 kHz. The authors have also tried a 10 kΩ order RF, which is frequently used in point contact mode STM/SPSTMs with several G 0conduc- tance22–24orμA tunneling current22,25,26(∼12.9 kΩ tunneling resis- tance24), to further increase the bandwidth. However, self-sustained oscillation (SSO) happens when the OPA model is OPA627 and there is no CAF. Even more unfortunately, no phase compensa- tion method is known to delete the SSO without a bandwidth cost, no matter internal compensation or external compensation.27The method of internal compensation by emitter degeneration, which is achieved by connecting a resistor in the emitter of the differen- tial input stage of an OPA, will decrease the open-loop gain AOL that indicates the superiority of the OPA. Internal dominant pole compensation, which is achieved by setting a capacitor in the inter- mediate stage, will decrease the GBP. External compensation by one-capacitor or two-capacitor is only suitable for voltage amplifiers. Limited by the above restrictions, the final bandwidth of the first stage (or the total) STM preamplifier is no more than 1 MHz. The highest frame frequency achieved is no more than 246 Hz,9and the highest line frequency is no more than 26 kHz11with poor atomic resolution.11,28 In this paper, the unique electronics property of the STM preamplifiers is analyzed. It varies at the different stages of the STM measurement. Then, the idea of paralleling a resistor with the tun- neling junction (PRTJ) to delete the SSO is presented. This method results in the noise gain of the STM preamplifier increasing from unity to dozens. Counterintuitively, this increase is a good thing since it means that a high-rate, decompensated OPA with better antinoise ability can be used in the STM preamplifier. Finally, with the help of PRTJ and the decompensated OPA, up to 100 MHz ultrahigh bandwidth is achieved in the STM preamplifier, which is better than existing methods.8,15,17High-clarity, atomic-resolution quantum point contact microscopy (QPCM) imaging is achieved with record-breaking line frequency and frame frequency. QPCM is excellent in chemical sensitivity22and can be used to hear the audiofrequency atom manipulation “sound” and investigate the origin of this “sound” better.23 II. PRINCIPLES AND DESIGN A. Electronic properties of the STM preamplifier The tip apex of the probe is far away from the sample initially in each STM measurement. Then, the coarse approach (CA) pro- cess is required, which makes the tip–sample distance smaller and smaller until the tunneling current is measurable. During this stage, the equivalent resistance rjCA(t) of the tunneling junction is close to∞, and almost no tunneling current can be detected at this CA stage. This is a typical unstable state in electronics in which the noise gain G = (1 + RF/∞)∼1; this is called the unity-gain state. Unfor- tunately, the CA stage has seemed inevitable in STM measurements up to now. The process following the CA stage is the scanning tunneling (ST) stage. The equivalent resistance rjST(t) of the tunneling junction is usually much less than the resistance RF. Then, the noise gain of the STM preamplifier usually is dozens, which results in the STM preamplifier being in a normal TIA state. B. Unity gain stable amplifiers The unstable unity-gain state requires a special unity-gain- stable (UGS) OPA under current STM preamplifier technology. UGS-OPA is a performance-compromised version of an OPA, designed for barely sufficient stability by complete compensation by the decompensated OPA. The UGS-OPA and decompensated OPA mostly appear in pairs.27,30,31The decompensated OPA has almost all the advantages over the UGS-OPA except that the hash required a minimum noise gain G min. For example, higher GBP, higher slew rate (SR), and less settling time (ST) indicate the speed superiority of the decompensated OPA. The higher open-loop gain AOL, less total harmonic distortion and noise (THD + N), higher common-mode rejection ratios (CMRR), and the higher maximum output voltage uomax indicate the other superiority of the decompensated OPA. As an example, OPA627 (a UGS-OPA) and OPA637 (a decompensated OPA) are compared, as shown in Table I. The bottom line of the first line of the table is the test conditions. OPA637 is dozens of times better than OPA627 in multiple indicators. It is a huge loss that decompensated OPAs could not be used in STM preamplifiers before now, especially in HS-STMs. C. STM preamplifier designs SSO happens when both the amplitude condition and the phase condition are satisfied in the STM preamplifier at the same time, TABLE I . Advantages of decompensated OPA over UGS-OPA, taking OPA627 and OPA627 as examples. GBP (MHz) SR (V/ μs) ST ( μs) A OL(dB) THD + N (ppm) CMRR (dB) uomax (V) Type of OPA 25○C 25○C G = −10 20 kHz 20 kHz 20 kHz 7 MHz G min OPA627BP (UGS type) 16 16 1.2 23 50 100 3 1 OPA637BP (Decompensated) 80 99 0.6 34 10 120 6 5 Rev. Sci. Instrum. 92, 013701 (2021); doi: 10.1063/5.0024802 92, 013701-2 Published under license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi regardless of whether the OPA is UGS.27,32In this situation, the minimum noise gain requirement of the picky decompensated OPA inversely indicates that increasing the noise gain can make the STM preamplifier more stable.30 A resistor is connected in parallel with the tunneling junction to increase the noise gain of the STM preamplifier. The bias voltage UB can be set in multiple ways, one of which is shown in Fig. 1. To bet- ter characterize the tunneling junction to be measured, the resistor mentioned above has to have constant resistance, noted as R0. The noise gain will increase from 1 to (1 + RF/R0) at the CA stage because of the parallel resistor configuration, which we denote byrj(t)//R0. This means that UGS-OPAs are no longer required. Instead, decompensated OPAs can be used not only to increase the fBbut also to augment the antinoise ability in the STM preamplifier, as shown in Fig. 1. The output uO(t) of this rj(t)//R0STM preamplifier can be divided into two parts uOSTM (t) and UOR0in theory by the super- position theorem: uO(t) = uOSTM (t) +UOR0. Constant UOR0=UB ×(1 + RF/R0) is caused by the constant current IR0, which flows through the constant resistor R0, as shown in Fig. 1. This UOR0value is independent of what sample is scanned, so it can be deleted to increase the contrast of the sample signal. The uOSTM (t), which is caused by the tunneling current ij(t) that flows through the tunneling junction rj(t), is the same as the output of a traditional STM pream- plifier, as shown in Fig. 1. The bad effect of the thermal noise current in the constant resistance R0will be analyzed in Sec. II D. D.SNR analysis of the rj(t)//R0STM preamplifier The thermal noise current caused by the constant resistance R0 can be calculated by inR0= (4 kTf B/R0)1/2= 1.62×10−4/(R0RF)1/2, where k= 1.38×10−23J/K, T≈300 K, and fBmax = 1/2πRFCsF≈1.6× 1012/RFforCsF≈0.1 pF21. IfRF/R0=N, then inR0max = 1.62×10−4× N1/2/RF, where Nis a constant. The signal of the STM output men- tioned above can be ideally expressed as simple harmonic uOSTM (t) =UAsin2πft+UC, while the frequency f, amplitude UA, and offset UCmay vary when the STM tip apex is scanning from one atom to another atom. The only part that reflects the resolution of the STM is the AC component UAsin2πft. Then, the effective value of the tunneling current ij(t) can be expressed as ij(t) =UA/√ 2RF. The negative effect FIG. 1 . Schematic diagram of the rj(t)//R0STM preamplifier. rj(t) represents the equivalent resistance of the tunneling junction. UBis the bias voltage. The ±USare the positive and negative power supplies. A decompensated operational amplifier is used in the STM preamplifier. The gradually changing shadow between the tip and sample illustrates the trend of the tunneling probability.ofR0on the SNR can be calculated as SNR R0min = (UA/√ 2RF)/inR0max = 4.4×103×N−1/2UA. Because Nis usually about 10, the fact that 100 mV UAis moderate in most STM measurements will make the SNR rise to about 100. When operated under LHe temperature, the SNR will increase tenfolds. The higher CMRR, higher A OL, and lower THD + N of the decompensated OPA compared to the traditional UGS-OPA will further ensure the high SNR . III. RESULTS AND DISCUSSION A. Higher stability STM preamplifier The stability of our rj(t)//R0STM preamplifier was verified first with a traditional UGS-OPA. The results are measured with a Tek- tronix TDS 3014C oscilloscope. SSO happens at about 90 kHz when OPA627 and a 1 MΩ feedback resistance RFis used. The wave resem- bles a square wave and is caused by the over saturated output of a sine wave, which is chopped, as shown in Fig. 2. The amplitude is so high that the scale rises to 3 V, as shown on the left side. The peak-to- peak value is still as high as about 13 V after being chopped, which means that no signal can be amplified and output in the conventional STM preamplifier circuit. When a 100 kΩ R0is introduced, SSO is eliminated. Then, a voltage amplifier is formed at the CA stage with no measurable tun- neling current. The output is so small that the scale is only 0.1 V, as shown in Fig. 2 on the right-hand side. The straight horizontal line output is mainly caused by the offset of the rj(t)//R0STM pream- plifier. No special shock absorption and electromagnetic shielding are needed, except that the STM preamplifier circuit box is made of cast aluminum. The alternating current component has strength no more than 10 mV, which is the noise of the system. The phase compensation achieved by the single resistor R0is different from the existing compensation methods, both internally and externally.27It can also be used in other TIAs to increase their stability. B. Radio frequency bandwidth STM preamplifier Decompensated OPAs, such as OPA637 or OPA657, can be introduced into the STM preamplifier to achieve higher bandwidth and higher SNR for the first time, benefiting from the highly sta- blerj(t)//R0design.21The higher stability enables the RFvalue to FIG. 2 . Comparison of stability between the traditional and rj(t)//R0STM preampli- fier. Both waves are characterized using an oscilloscope. Rev. Sci. Instrum. 92, 013701 (2021); doi: 10.1063/5.0024802 92, 013701-3 Published under license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi decrease from 1 MΩ to 10 kΩ, which further increases the fB. The resistor R0in both circuits has resistance 1 kΩ. Power spectral density measurements are used to characterize the bandwidth, as shown in Figs. 3(a) and 3(b). The bandwidths are about 8.9 MHz and 166 MHz. The three peaks in Fig. 3(b) may be caused by the potential instability in the corresponding fre- quency. The ultrahigh 166 MHz fBfits well with both the theory of fB= (2πRFCsF)−1= 159 MHz and the theory of fB≤GBP/G≈145 MHz, where the GBP of OPA657 is 1.6 GHz. Both bandwidths are also analyzed by the Bode diagram of the rj(t)//R0STM preamplifier under the single-pole OPA model and theAOL≫1 assumption.19When R0is introduced, the total input impedance changes from XCStoXCS//R0. However, the expression of the transfer function of the STM preamplifier19is still intact: u/i =ω02RF/(s2+ sω0/Q +ω02) with intact ω02= 2πGBP/[ RF(CF+CS)], where s = i ω. Only the expression of the quality factor Qchanged slightly, by RF//R0, into Q≈[2π×GBP×RF×(CF+CS)]1/2/(1+RF/R0+2π×GBP×RF×CF). When OPA637 is used, Qandω0are about 0.76 rad/s and 57.4 rad/s, respectively; when OPA657 is used, Qandω0are about 1.12 rad/s and 428 rad/s, respectively. Substituting each Qandω0 into the transfer function, we get u/iOPA637 = 3.28×1019/(s2+ 7.52 ×107s+ 3.28×1015) and u/iOPA657 = 1.8×1019/(s2+ 2.3×108s + 1.8×1017). The magnitude parts of each transfer function indicate that the bandwidths are about 9.3 MHz and 100 MHz, as shown in Figs. 3(c) and 3(d). The bandwidth of 9.3 MHz is in good agreement with the data in Fig. 3(a). We note that 100 MHz is less than 166 MHz; the lower value may result from the crude assumption of the single-pole OPA model, as shown by the single pole in Fig. 3(d). The frequency 166 MHz fBis so high that it becomes a challenge for other parts of the HS-STM system, such as the scanner and data acquisition (DAQ) card.9The DAQ card in our HS-STM system is a FIG. 3 . Power spectral density measurements and Bode diagrams of rj(t)//R0STM preamplifiers. (a) and (b) are the PSD measurements. (c) and (d) are the Bode diagrams. (a) and (c) are the output measured and calculated for the STM pream- plifier based on OPA637. (b) and (d) are the output measured and calculated for the STM preamplifier based on OPA657.PXI-6124 whose sampling rate is 4 MS/s. The resonant frequency of the scanner is about 100 kHz. C. Quantum point contact mode HS-STM The performance of the rj(t)//R0STM preamplifier was also confirmed by our home-built HS-STM11We used parameters RF = 10 kΩ, R0= 1 kΩ, and CAF= 0 pF, and an OPA637 was used, resulting in a bandwidth of about 9 MHz, as mentioned above and as shown in Figs. 3(a) and 3(c). High-clarity atomic resolution images are obtained, which have almost the same quality as those obtained using the conventional STM preamplifier,33as shown in Figs. 4(a)– 4(e). The frequency 9 MHz had broken the bandwidth record for an atomic resolution HS-STM. The images, which were raw data of a highly oriented pyrolytic graphite (HOPG) sample, were obtained in air under room tempera- ture with a mechanically cut Pt/Ir tip. The bias voltage was −220 mV. The different brightness areas in the image may be caused by the obliqueness of the sample. One of the most noteworthy reflections of the performance of the STM system was the peak–peak voltage upp= 2UA. The compromise magnitude in the line cut profile is upp ≅140 mV, as shown in Fig. 4(b). The 70 mV UAindicates that the SNR R0min is 4.4×103×N−1/2UA= 97, which means that the noise caused by R0is not worth worrying about at all. All the images in Fig. 4 were measured under a record-breaking frame rate of 250 frame/s4,9and a record-breaking line rate of 50 k FIG. 4 . The rj(t)//R0HS-STM image (raw data) based on the decompensated OPA. The sample is HOPG. (a) One frame HS-STM atomic resolution image (under conditions: 50 kHz fast scanning frequency, 250 Hz slow scanning frequency, 7.5 Å×7.2 Å). (b) Profile along the line AB in (a). About 150 mV uppis scaled to indicate the SNR of the STM system. UB=−220 mV. The lateral distance along the horizontal-axis is 7.9 Å. (d) is the opposite frame in the same period with (c). The time difference ∆t between (c) and (d) is 2 ms, which is half of the period. (e) is the image in the next half period of (d). The sine wave and the dotted lines in the atomic resolution images show the corresponding direction of scanning in the slow axis. Rev. Sci. Instrum. 92, 013701 (2021); doi: 10.1063/5.0024802 92, 013701-4 Published under license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi line/s;11,34both benefited from the unrestrained high bandwidth of therj(t)//R0STM preamplifier. A continuous scanning on the HOPG surface was also per- formed with a record-breaking slow axis scanning frequency of 250 Hz, as shown in Figs. 4(c)–4(e). There is little difference between each image. The frame frequency is twice as the scanning frequency to 500 Hz when counting the frames that are going back and forth, as shown in Figs. 4(c) and 4(d). The 70 mV UAindicates that the effective value of the tunnel- ing current signal is UA/√ 2RF= 5μA. This signal value means that the equivalent conductance of the tunneling junction is about 0.3G 0, which indicates that this STM is operating under the quantum point contact state. QPCM can be applied to many fields of surface sci- ence. For example, the tribological characteristic, which is related to speed,35,36has been expressed in terms of current.37,38As for the open problem of application with a weak tunneling current, multi- stage amplification can be employed. This improvement is our next goal. IV. CONCLUSION The higher stability of the rj(t)//R0STM preamplifier was ana- lyzed and testified, showing that it can increase the bandwidth. This research shows that a superior-performance, decompensated OPA can be used to further increase the bandwidth and the SNR of the STM preamplifier. The recording-break line rate and frame rate atomic resolution HS-STM images indicate that more interesting and important effects will be revealed. The rj(t)//R0method and decompensated OPA can also be generalized to almost all other types of STM preamplifiers to increase the SNR. ACKNOWLEDGMENTS This work was supported by the National Natural Sci- ence Foundation of China (Grant Nos. 11304082, U1932216, and 51627901), the National Key R&D Program of China (Grant Nos. 2017YFA0402903 and 2016YFA0401003), the Scientific Research Equipment Developing Project of the Chinese Academy of Sci- ences (Grant No. YZ201628), the Maintenance and Renovation Project for CAS Major Scientific and Technological Infrastructure (Grant No. DSS-WXGZ-2019–0011), and the Science and Technol- ogy Department of Henan Province (Grant Nos. 182102210367 and 182102210370). DATA AVAILABILITY The data that support the findings of this study are openly available upon reasonable request. REFERENCES 1L. Gao et al. , “Constructing an array of anchored single-molecule rotors on gold surfaces,” Phys. Rev. Lett. 101, 197209 (2008). 2R. Ma et al. , “Atomic imaging of the edge structure and growth of a two- dimensional hexagonal ice,” Nature 577, 60–63 (2020). 3F. E. Kalff, M. P. Rebergen, E. 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5.0034047.pdf

J. Appl. Phys. 128, 224105 (2020); https://doi.org/10.1063/5.0034047 128, 224105 © 2020 Author(s).Nonvolatile tuning of the Rashba effect in the CuInP2S6/MoSSe/CuInP2S6 heterostructure Cite as: J. Appl. Phys. 128, 224105 (2020); https://doi.org/10.1063/5.0034047 Submitted: 21 October 2020 . Accepted: 24 November 2020 . Published Online: 11 December 2020 Hong-Fei Huang , Yao-Jun Dong , Yang Yao , Jia-Yong Zhang , Xiang Hao , Han Gu , and Yin-Zhong Wu ARTICLES YOU MAY BE INTERESTED IN High-speed ultraviolet photodetectors based on 2D layered CuInP 2S6 nanoflakes Applied Physics Letters 117, 131102 (2020); https://doi.org/10.1063/5.0022097 Electric field control of the semiconductor-metal transition in two dimensional CuInP 2S6/ germanene van der Waals heterostructure Applied Physics Letters 114, 223102 (2019); https://doi.org/10.1063/1.5100240 Thickness dependence of domain size in 2D ferroelectric CuInP 2S6 nanoflakes AIP Advances 9, 115211 (2019); https://doi.org/10.1063/1.5123366Nonvolatile tuning of the Rashba effect in the CuInP 2S6/MoSSe/CuInP 2S6heterostructure Cite as: J. Appl. Phys. 128, 224105 (2020); doi: 10.1063/5.0034047 View Online Export Citation CrossMar k Submitted: 21 October 2020 · Accepted: 24 November 2020 · Published Online: 11 December 2020 Hong-Fei Huang,1Yao-Jun Dong,2Yang Yao,1Jia-Yong Zhang,1 Xiang Hao,1Han Gu,2,a)and Yin-Zhong Wu1,b) AFFILIATIONS 1Jiangsu Key Laboratory of Micro and Nano Heat Fluid Flow Technology and Energy Application, School of Physical Science and Technology, Suzhou University of Science and Technology, Suzhou 215009, China 2Changshu Institute of Technology, Changshu 215500, China a)Electronic mail: guhan@cslg.edu.cn b)Author to whom correspondence should be addressed: yzwu@usts.edu.cn ABSTRACT The van der Waals sandwich heterostructure CuInP 2S6=MoSSe =CuInP 2S6(CIPS/MoSSe/CIPS) has first been employed as a prototype to tune the Rashba effect. By nonvolatile controlling of the orientation of the polarization of the top ferroelectric CIPS monolayer, it is con-firmed that the Rashba effect can be switched on or off at the top position of the valence band (VB) around the Γpoint. More significantly, we find that the Rashba coefficient increases by almost one order of magnitude for the “on”state as compared with the freestanding MoSSe monolayer. Based on the results of first-principle calculations, it is obtained that the enhancement of the Rashba effect results from the charge transfer from the top CIPS layer to the MoSSe layer or the bottom CIPS layer, and the lifting of the d-orbit band of the light Cu atom leads to the disappearance of Rashba spin splitting at the top of the VB around the Γpoint. Furthermore, the polarization orientation of the bottom CIPS layer can greatly alter the bandgap of the sandwich structure. We hope the above nonvolatile and large amplitudetuning of the Rashba effect should be useful in the design of spintronic nano-devices. Published under license by AIP Publishing. https://doi.org/10.1063/5.0034047 I. INTRODUCTION Due to the d-orbits of the metal atom in two-dimensional transition-metal dichalcogenides (TMDs), strong spin –orbit cou- pling (SOC) effects have been found in these types of materials. 1 With the consideration of the spin –orbit interaction, spin degener- acy at the valence band and conduction band will be removed, and it is well known that there exist two types of spin splitting in theJanus TMD monolayer, 2,3one is the valley spin splitting around theK(K0) point, and the other is the Rashba splitting around the Γ point,4,5and the Rashba spin splitting appears due to the lack of out-of-plane mirror symmetry in Janus MXY (M = Mo, W; X=Y¼S, Se, Te) monolayers, which can be described by the Bychkov –Rashba Hamiltonian,6 HR¼αR(j~Ej)~σ/C1(~kk/C2~ez), (1) where αRis the Rashba coefficient, ~kk¼(kx,ky, 0) denotes the momentum of the electron within the 2D system, and ~σis thePauli matrix. The Rashba coefficient αRcan be obtained from the formula αR¼2ER kR, where ERis the energy difference between the maximum energy near the Γpoint and the cross energy of spin-up and spin-down in the valence band at the Γpoint and kRis the corresponding momentum offset. The Rashba effect was initiallyobserved in semiconductor heterostructures 7,8and has received growing interest because of its great significance in spin Field-Effect-Transistors (FETs).9The new members of Janus MXY monolayers with the intrinsic structure inversion asymmetry mayenrich the family of the Rashba effect and possibly improve theperformance of spin FETs. The spin splitting in the MXY TMD monolayer has been intensively investigated, 10and it is found that, for M¼W, the value of αRis larger than that for M¼Mo, and αRgrows along the series XY=SSe, STe, and SeTe. The above behavior of the Rashbaeffect resulted from the fact that the SOC is stronger for heavieratoms. Once a specific Janus TMD is selected, how to tune and obtain a giant Rashba effect should be very important and interest- ing. In theory, it is found that the Rashba effect can be tuned byJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 224105 (2020); doi: 10.1063/5.0034047 128, 224105-1 Published under license by AIP Publishing.using an external field and/or strain engineering. The Rashba spin splitting will be enhanced by a positive electric field or under a compressive strain.4,5,11However, this type of manipulation is vola- tile, and the enhancement of the Rashba effect will disappear whenthe applied electric field or the strain is switched off. The use of theferroelectric substrate BaTiO 3to manipulate the Rashba spin splitting in the 6 pstates of the Bi monolayer is demonstrated by first-principles calculations.12Although only a relative change in the splitting of about 5% is obtained for a different direction ofthe polarization in the BaTiO 3substrate, it provides a nonvola- tile way to tune the Rashba effect since the polarization direction of ferroelectric films can be nonvolatilely switched by an exter- nal field. It has also been confirmed that the sandwichBaTiO 3=SrRuO 3=BaTiO 3structure can be designed to tune the resistance state of the SrRuO 3layer by switching the orientation configurations of the BaTiO 3layers.13Following this way, here, the 2D ferroelectric CIPS monolayer is adopted to construct a CIPS/MoSSe/CIPS van der Waals heterostructure. First, van derWaals heterostructures 14have been broadly applied to design nano- devices because they are free from the dangling bond and are nolonger limited by the lattice mismatch between adjacent 2D layered materials. Second, the room-temperature ferroelectricity in the ultrathin CIPS film, which consists of an S cage with octahedralsites by Cu, In, and P –P in a triangular pattern, 15has been experi- mentally verified.16,17The vertical polarization of the 2D CIPS film reaches 4 μC=cm2,18which is much larger than that in other 2D ferroelectric films, such as the α/C0In3Se3film.19Furthermore, the ferroelectric field-effect transistor based on MoS 2and CIPS 2D van der Waals heterostructures has been demonstrated,20which means that the preparation of the CIPS/MoSSe/CIPS heterostructure will be possible in the experiment. Based on our results of first- principles calculations, we find that the Rashba effect can be electri-cally tuned in the CIPS/MoSSe/CIPS sandwich structure, and theRashba coefficient will be increased by one order in magnitude forselected configuration of polarization directions of the CIPS mono- layer. The energy band, spin textures, and the Bader charges are presented and discussed for all possible directions of the polariza-tion of the ferroelectric CIPS monolayer. It is analyzed that theenhancement of the Rashba effect in the above sandwich hetero-structure resulted from the charge transfer between the ferroelectric CIPS monolayer and the Janus MoSSe layer besides the strain mod- ulation. We hope this nonvolatile tuning of the Rashba effect usingthe ferroelectric substrate as well as the scheme of the van derWaals sandwich structure may be provided as a way to improve the performance of the spintronic device in the future. II. STRUCTURES AND COMPUTATIONAL METHODS The lattice constant of the free CIPS monolayer is 6.12 Å from our first-principles calculation, and the lattice constant of theMoSSe monolayer is 3.22 Å. Therefore, we use a 2 /C22 supercell of MoSSe and two 1 /C21 supercell of CIPS monolayers to construct the sandwich structure. The top view and side view of theCIPS/MoSSe/CIPS sandwich structure for different polarizationconfigurations are shown in Fig. 1 . In order to see the top view figures clearly, a semitransparent plane is inserted between the top CIPS monolayer and the middle MoSSe Janus film. From the topview of the sandwich structure in Fig. 1 , the P –P bond sites above the center of the triangle of Mo atoms, Cu atoms locate upward theremaining Mo atoms in the 2 /C22 supercell of MoSSe, and the space group of the constructed heterostructure is P3. 21Based on the results of density functional theory (DFT) calculations, it isobtained that the above constructed heterostructure for a differentpolarization orientation is stable and processes the lowest energy.Other stacking structures with the P3 space group, such as rotating CIPS along the P –P bond (axis) by π 6, will always increase the total energy. Therefore, only the structure with a different polarizationconfiguration in Fig. 1 is taken into consideration in the following. During the DFT calculation, the direction of the polarization ofCIPS films is preassigned, and all the atoms within the heterostruc- ture are fully relaxed. Here, we use “UU,”“UD,”“DD,”and“DU” to define the four polarization configurations in Figs. 1(e) –1(h), respectively, where “U”means the up direction of the polarization for the CIPS monolayer, and “D”stands for the down direction of the polarization. Regardless of the initial polarization orientations of the top and the bottom CIPS monolayers, the polarization con-figuration will be in the UU state or the DD state after an externalelectric field is applied across the heterostructure. 22,23Here, for sys- tematicness and comparison, the electronic structure and spin texture for the UD and DU configurations are also presented and discussed. The density functional theory (DFT) calculations are per- formed by using the Vienna ab initio simulation package (VASP)with the projector augmented-wave (PAW) potentials. 26We adopt the Perdew –Burke –Ernzerhof exchange function27and the plane wave cutoff is set to be 500 eV. The valence electrons are 3 d104s1 for Cu, 5 s25p1for In, 3 s23p3for P, 3 s23p4for S, 4 s24p64d55s1for Mo, and 4 s24p4for Se in our calculations. The vdW correction with DFT-D3 method28is taken into account. The Brillouin zone integration of the DFT calculations is performed using a Gamma- centered 8 /C28/C21kpoint mesh for the CIPS/MoSSe/CIPS hetero- structure. During the structural relaxation, the criterion for energyconvergence is 10 /C06eV, and the forces on all relaxed atoms are less than 0.01 eV/Å. Moreover, the vacuum separation between repeated slabs is set to be 25 Å, which avoids interactions between adjacent FIG. 1. T op views [(a) –(d)] and side views [(e) –(h)] of the CIPS/MoSSe/CIPS sandwich heterostructure for all possible configurations of the polarization orien- tation of the CIPS monolayer (UU, UD, DD, and DU). A semitransparent plane is inserted into the top CIPS layer and the middle MoSSe layer to see the topview figures more clearly.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 224105 (2020); doi: 10.1063/5.0034047 128, 224105-2 Published under license by AIP Publishing.layers. Due to the crystal symmetry, band structure calculations of the heterostructure along the special lines connecting the following high-symmetry points, namely, Γ(0,0,0), M(0.5,0,0), K(2/3,1/3,0), andK0(1/3,2/3,0) in the first Brillouin zone, are performed. III. RESULTS AND DISCUSSIONS After the full relaxation of the van der Waals heterostructure without consideration of spin –orbit coupling (SOC), we find that the middle MoSSe monolayer is compressed in the abplane, and a type-II band alignment24with a reduced energy gap occurs. The optimized lattice constants and the bandgaps for different polariza- tion configurations are listed in Table I . The band structures without the consideration of SOC are plotted in Fig. 2 , where the bottom of the conduction band (CB) is contributed by the In atom from the bottom CIPS monolayer, the minimum of the CB appears at the Γpoint, and the maximum of the VB locates at the Kpoint, which is contributed by the Mo atom. Therefore, all the band structures of the sandwich in Fig. 2 belong to the type-II indirect band structure.24Compared with the freestanding MoSSe monolayer, the bandgap will be decreased by half when the polarization of the bottom CIPS layer points upward, and the gap becomes even narrower when the polarization of thebottom CIPS layer points downward. With the consideration of SOC, the band degeneracy at the K andΓpoints are removed, which can be seen from the atom- projected band structures in Fig. 3 . The valley spin splitting at the maximum of VB at the Kpoint are 157.7 meV, 158.3 meV, 156.3 meV, and 155.0 meV for the UU, UD, DD, and DU configura-tions, respectively, which are almost the same as the valley spin split-tingλ kv¼168 meV for the freestanding MoSSe monolayer.3Unlike the valley spin splitting at the Kpoint, the Rashba spin splitting at the top of VB around the Γpoint will have a gigantic variation in magnitude for the UU and UD configurations, i.e., under the case ofthe polarization of the top CIPS film pointing up. As the polarization of the top CIPS film points down in Figs. 3(c) and3(d), the band splitting at the top of VB around the Γpoint seems similar to that in Figs. 3(a) and3(b); however, this cannot be defined as the Rashba spin splitting. As is well known, there are two criteria for the typicalRashba effect within two-dimensional systems: the first criterion is that the spin-polarization in the abplane will be antiparallel for thesplitted left band (LB) and the splitted right band (RB), and the second criterion is that the out-of-plane spin-polarization compo-nent on the top of VB near the Γpoint approaches zero. One can see from the spin textures under the lower right side of Fig. 3 ,t h a t the two criteria are both satisfied for the UU and UD configura- tions in Figs. 3(a) and3(b). For the DD and DU configurations in Figs. 3(c) and 3(d), the in-plane spin-polarization components are remarkably deviated from the antiparallel direction, and the out-of-plane spin-polarization component Sz ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi S2xþS2yþS2zp near the Γpoint is 0.27 and 0.35, respectively. Therefore, the band splitting at the top of VB around the Γpoint for the DD and DU configu- rations is not the Rashba spin splitting. The reason is that the highest VB around the Γpoint for the cases of DD and DU con- figuration is mainly contributed by the light Cu atom, while thehighest VB around the Γpoint is contributed by the heavy Mo atom for the UU and UD configurations, and the Rashba effect usually occurs for the heavy atom. Under the action of an external electric field across the heterostructure, the polarization configu-ration will be switched between the UU state and the DD stateregardless of its initial configuration. To sum up, one can natu-rally draw a conclusion that the Rashba effect at the top of VB around the Γpoint can be switched on or off by the change of polarization orientation of the top and bottom CIPS film in theCIPS/MoSSe/CIPS van der Waals heterostructure. Besides thefunction of switching on/off of the Rashba effect, it should also beemphasized that, for the “on”state, the Rashba effect is greatly enhanced as compared with the freestanding MoSSe monolayer, and the strength of the Rashba effect can be described by theRashba coefficient α Γ R, which is calculated and given in Table II for the heterostructure with UU and UD configurations. It isfound, from Table II , that the Rashba coefficient increases nearly one order in magnitude. Furthermore, the Rashba coefficients are nearly symmetrical along the Γ–Ma n d Γ–Kdirections, which differs from that in the ZnO/MoSSe heterostructure where a largedifference is found between α Γ/C0M R andαΓ/C0K R.29 From Table I , one can see that the middle layer MoSSe in the optimized heterostructure is compressed compared with FIG. 2. Atom-projected band structures of the CIPS/MoSSe/CIPS heterostruc- ture without SOC for (a) UU configuration, (b) UD configuration, (c) DD configu-ration, and (d) DU configuration. TABLE I. Lattice constants and energy gap for the relaxed CIPS/MoSSe/CIPS sandwich structure for different configurations of the polarization orientation of the CIPS monolayer. Configuration a(Å) b(Å) Strain of MoSSe E gap(eV) UU 6.290 6.290 /C02:64% 0.66(I) UD 6.293 6.293 /C02:59% 0.28(I) DD 6.290 6.290 /C02:64% 0.32(I) DU 6.286 6.286 /C02:69% 0.69(I) MoSSe (2 × 2) 6.46 6.46 … 1.61(D) Freestanding monolayer 1.63(D)5 CIPS (1 × 1) 6.12 6.12 … 1.64(D) Freestanding monolayer 1.66(D)25Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 224105 (2020); doi: 10.1063/5.0034047 128, 224105-3 Published under license by AIP Publishing.the freestanding MoSSe monolayer, and the strain σequals to /C02:6%. We have also calculated the Rashba coefficients of the MoSSe single layer under strain σ¼/C02:6%, which is αΓ/C0M R¼341 meVA/C14,αΓ/C0K R¼336 meVA/C14.C o m p a r e dw i t h αΓ/C0M R ¼465:5m e V A/C14andαΓ/C0K R¼468:2m e V A/C14in the sandwich heter- ostructure with the UU polarization configuration, we can con- clude that there must have been other factors for the enhancement of the Rashba effect for the UU configuration in theheterostructure as well as the contribution from the compressstrain. In the following, we will briefly explain the mechanism ofthe enhancement of the Rashba effect in terms of the interlayer charge transfer within the heterostructure, and the interlayer charge transfer is evaluated based on the Bader charge analy-sis. 31,32Bader ’s theory of atoms in molecules is often useful forcharge analysis, and the charge enclosed within the Bader volume is a good approximation to the total electronic charge of an atom. To give the interlayer charge transfer within the heterostructure,each layer is isolated from the sandwich structure and leaves theother layer as vacuum. Then, only the electron relaxation is per-formed for each layer, and the Bader charges for all atoms are cal- culated for each isolated monolayer. By subtracting the Bader charge of the original monolayer within the heterostructure from that of thenew isolated monolayer, the charge transfer, which is caused by theinterlayer van der Waals interaction, can be obtained. The specificcharge transfers between different layers are listed in Table III ,w h e r e the positive value denotes the loss of electron and the negative value stands for the gain of the electron. From Table III ,o n ec a n obtain that, due to the van der Waals interaction and the electronreconstruction, the charge in the top CIPS layer is transferred intothe middle MoSSe layer or the bottom CIPS layer, which will TABLE II. Rashba coefficient for the MoSSe freestanding monolayer and the sand- wich heterostructure with UU and UD polarization configurations. MoSSeUU configurationUD configuration freestanding monolayerof CIPS/ MoSSe/CIPSof CIPS/ MoSSe/CIPS αΓ!M R(meVÅ) 57.1, 67,30775465.6 448.6 αΓ!K R(meVÅ) 67.9, 67,30775468.2 456.2TABLE III. The gain and loss of electron (interlayer charge transfer) between differ- ent layers of the sandwich CIPS/MoSSe/CIPS heterostructure. Layer UU configuration UD configuration Top CIPS 1 7.4 × 10−35.6 × 10−3 Middle MoSSe −5.4 × 10−36.1 × 10−3 Bottom CIPS 2 −2.0 × 10−3−11.7 × 10−3 FIG. 3. Atom-projected band structure of CIPS/MoSSe/CIPS with SOC for different configurations (a) UU, (b) UD, (c) DD, and (d) DU. The enlargement of Rashba s plitting around the Γpoint is re-plotted on the top right corner for all configurations, and the in-plane spin texture at selected energy is shown on the lower right corner f or all configurations.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 224105 (2020); doi: 10.1063/5.0034047 128, 224105-4 Published under license by AIP Publishing.bring about a bias field pointing downward. The generated bias field will strengthen the asymmetry of the Janus MoSSe along the caxis5and further enhance the Rashba effect of the MoSSe monolayer. Since the precise control over the stacking structure is not easy to achieve, the heterostructures are inevitably mis-stacked. To probe the influence of the mis-stacking effect on the electronic structure and spin texture of the CIPS/MoSSe/CIPS heterostructure,we assume that the top CIPS monolayer is slightly shifted along theaaxis, as shown in Fig. S1 (see supplementary material ). Then, only the electronic self-consistent steps are performed. As one can see, from Fig. S2 and Table S1 in the supplementary material , the in-plane spin texture and the Rashba coefficients are little influ-enced for the UU polarization configuration, and we also find theRashba effect is switched off for the DD configuration. Therefore,the function of nonvolatile tuning of the Rashba effect is robust for the sandwich structures that are slightly mis-stacked. To find whether this tuning effect can also occur in a simple bilayer system,the MoSSe/CIPS bilayer is constructed, as shown in Fig. S3 in thesupplementary material . For a given orientation of the polarization of the CIPS film, the band structure and Rashba spin splitting are obtained after a full relaxation with the consideration of SOC. From Fig. S4 in the supplementary material , one can see that the polar direction brings little influence on the Rashba spin splitting,and only a little difference of the Rashba coefficient is found between the up state and down state of the CIPS film (see Table S2 in the supplementary material ). Therefore, the electrical tuning of the Rashba effect in the MoSSe/CIPS bilayer system does not work. In summary, we have verified that the CIPS/MoSSe/CIPS van der Waals heterostructure with different polarization configurations possesses very interesting band structures. The Rashba effect and the bandgap can be nonvolatilely tuned by switching the polariza-tion orientation of the CIPS monolayer. The performance ofswitching on/off of the Rashba effect at the top of VB around the Γ point and the large amplitude tuning of the Rashba coefficient can only be found in the above trilayer system and cannot be achieved in the MoSSe/CIPS bilayer system. We hope the idea of the van derWaals sandwich structure and further the nonvolatile method forthe tuning of the Rashba effect will shed light on the design ofspintronic devices in the future. SUPPLEMENTARY MATERIAL See the supplementary material for the corresponding atomic structures, electronic structures, spin textures, and Rashba coeffi- cients of the mis-stacked CIPS/MoSSe/CIPS heterostructure andthe MoSSe/CIPS bilayer. ACKNOWLEDGMENTS Thanks are due to the useful discussions with Professor Sheng Ju, Dr. Wei Xun, and Dr. Ju Zhou. This work is supported by theNational Science Foundation of China (Grant No. 11274054) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 19KJA140001).DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1Z.-Y. Zhu, Y.-C. Cheng, and U. Schwingenschlogl, Phys. Rev. B 84, 153402 (2011). 2A. Manchon, H. C. Koo, J. 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9.0000005.pdf

AIP Advances 10, 125230 (2020); https://doi.org/10.1063/9.0000005 10, 125230 © 2020 Author(s).Thermoelectric refrigerator based on asymmetric surfaces of a magnetic topological insulator Cite as: AIP Advances 10, 125230 (2020); https://doi.org/10.1063/9.0000005 Submitted: 26 September 2020 . Accepted: 07 November 2020 . Published Online: 30 December 2020 Takahiro Chiba , and Takashi Komine COLLECTIONS Paper published as part of the special topic on 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials , 65th Annual Conference on Magnetism and Magnetic Materials and 65th Annual Conference on Magnetism and Magnetic Materials ARTICLES YOU MAY BE INTERESTED IN Ambipolar Seebeck power generator based on topological insulator surfaces Applied Physics Letters 115, 083107 (2019); https://doi.org/10.1063/1.5109948 Ionic thermoelectric materials for near ambient temperature energy harvesting Applied Physics Letters 118, 020501 (2021); https://doi.org/10.1063/5.0032119 An attempt to simulate laser-induced all-optical spin switching in a crystalline ferrimagnet AIP Advances 10, 125323 (2020); https://doi.org/10.1063/9.0000003AIP Advances ARTICLE scitation.org/journal/adv Thermoelectric refrigerator based on asymmetric surfaces of a magnetic topological insulator Cite as: AIP Advances 10, 125230 (2020); doi: 10.1063/9.0000005 Presented: 4 November 2020 •Submitted: 26 September 2020 • Accepted: 7 November 2020 •Published Online: 30 December 2020 Takahiro Chiba1,a) and Takashi Komine2 AFFILIATIONS 1National Institute of Technology, Fukushima College, 30 Nagao, Kamiarakawa, Taira, Iwaki, Fukushima 970-8034, Japan 2Graduate School of Science and Engineering, Ibaraki University, 4-12-1 Nakanarusawa, Hitachi, Ibaraki 316-8511, Japan Note: This paper was presented at the 65th Annual Conference on Magnetism and Magnetic Materials. a)Author to whom correspondence should be addressed: t.chiba@fukushima-nct.ac.jp ABSTRACT Thermoelectric (TE) refrigeration such as Peltier cooler enables a unique opportunity in electric energy to directly convert thermal energy. Here, we propose a TE module with both refrigeration and power generation modes by utilizing asymmetric surfaces of a magnetic topological insulator (quantum anomalous Hall insulator) with a periodic array of hollows filled with two different dielectrics. Based on the Boltzmann transport theory, we show that its efficiency, i.e., the dimensionless figure of merit ZTexceeds 1 in the low-temperature regime below 300 K. The proposed device could be utilized as a heat management device that requires precise temperature control in small-scale cooling. ©2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/9.0000005 I. INTRODUCTION Thermoelectric (TE) devices are used in a wide range of appli- cations related to solid-state based power generation and refriger- ation. In particular, the TE refrigeration such as Peltier cooler has drawn attention due to a CO 2–free cooling technology for auto- motive applications, computer processors, refrigeration of biologi- cal samples, and various heat management systems.1,2The primary advantages of a Peltier cooler compared to a traditional vapor- compression refrigerator are flexibility and compactness owing to the lack of moving parts, enabling applications for small-scale cool- ing. TE cooling technology is based on the Peltier effect in TE mate- rials in which an electric current drives heat flow and creates the temperature difference at the hot and cold ends of a system. The efficiency of TE energy conversions is evaluated by the dimensionless figure of merit ZT.2,3Over the past several years, many new materials have been investigated for their use as TE mate- rials with high ZT.4So far, tetradymite–type chalcogenides such as Bi2Te3have been well known as a good TE material with ZT≈1,5–10 but have also drawn much attention as three-dimensional topolog- ical insulators (3D TIs) in recent years.113D TI is an electronic bulk insulator but has a linear energy dispersion near a single band- touching (Dirac) point on the surface due to strong spin–orbitinteraction. Recently, an ideal two-dimensional (2D) Dirac surface state in 3D TIs with a highly insulating bulk has been observed in (Bi1−xSbx)2Te3(BST) and Bi 2−xSbxTe3−ySey(BSTS).12By focusing on the TI surface states, some potential systems and devices to real- ize high-performance thermoelectrics so far have been theoretically proposed.13–20 According to the previous studies,14,15,20one of the simplest approaches to achieve a high ZTis the introduction of an surface band gap on the TI surface.14,15A system with massive Dirac elec- trons on a gap-opened TI surface can be realized by hybridization of the top and bottom surfaces.21,22This mechanism is applied to 3D TIs with many holes in the bulk14or to a superlattice made from a 3D TI and an empty layer.23A recent experiment has observed a large Seebeck coefficient in a ultrathin film of BSTS owing to the surface gap-opening by the hybridization effect.24In contrast, since a surface band gap is also induced by a magnetic perturbation that breaks the time-reversal symmetry, the application of a magnetic field should be the simplest approach. However, magnetic fields of ∼10 T induce a very small subgap (of the order of several meV) in the surface of 3D TIs.25An alternative approach is magnetic dop- ing into a 3D TI26,27or making ferromagnet contact with magnetic proximity effect,28–31which can induce a large surface band gap of the order of 100 meV. It is known that ferromagnetism in the AIP Advances 10, 125230 (2020); doi: 10.1063/9.0000005 10, 125230-1 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv magnetically doped 3D TIs can be developed through the carrier- mediated Ruderman–Kittel–Kasuya–Yosida (RKKY) mechanism and/or the carrier-independent bulk Van Vleck mechanism.32,33 In particular, the gap-opened magnetic TI surface exhibits the quantum anomalous Hall effect, characterizing the topological nature of 2D massive Dirac electrons,36and thus would be expected as a new platform for studying magneto–thermoelectric properties. In this paper, we propose a TE module utilizing asymmetric surfaces of a magnetic TI (quantum anomalous Hall insulator) in which a periodic array of hollows filled with two different dielectrics is introduced. A pair of these two surfaces that are adjoined with each other acts as a Π-shaped p-njunction with ambipolar con- duction,34,35which can be regarded as a thermocouple consisting of two dissimilar TE materials.37Thus, a serial connection of the thermocouple operates as a TE module with both refrigeration and power generation modes. By using the Boltzmann transport the- ory at finite temperatures, we show that ZTexceeds 1 in the low- temperature regime below 300 K. The proposed device could be uti- lized as a heat management device that requires precise temperature management. II. DEVICE PROPOSAL Here, we designs a TE module utilizing asymmetric surfaces of a magnetic TI. In Fig. 1, we summarize the concept of the pro- posed device. Figure 1 (a) shows the TE module made of a film of magnetic TI (quantum anomalous Hall insulator36) in which a periodic array of hollows filled with two different dielectrics is introduced. Such dielectric-filled hollows give rise to gap-opened metallic surface states, as shown in Fig. 1 (c) by yellow lines. In this paper, we call a pair of the two hollows connected by a lead a “topological thermocouple,” and its structure is schematically illus- trated in Fig. 1 (b). A pair of these two surfaces that are adjoined with each other acts as a Π-shaped p-njunction with ambipolar conduction, which can be regarded as a thermocouple consisting of two dissimilar TE materials. It is worth noting that recent experi- ments demonstrated one surface with positive carriers and the oppo- site surface with negative carriers in a heterostructure based on a magnetically doped 3D TI.38The difference in carrier types origi- nates from the structure inversion asymmetry (SIA) between the two adjoined surfaces in Fig. 1 (b), which is induced by the band bending imposed by the dielectrics.37,39The effective Hamiltonian for a pair of adjoined surfaces is H∓(k)=∓̵hvF(σxky−σykx)+mσz∓USIAσ0, (1) where ∓indicates TI surfaces attached to dielectric 1 ( −) and 2 (+),USIAdenotes the SIA between the two adjoined surfaces, σ0 is the identity matrix, and mcorresponds to the surface band gap. For simplicity, we do not consider the particle–hole asymmetry in the surface bands and assume that the gap-opened surface states have symmetric energy dispersions: E± s(k)=∓s√ (̵hvFk)2+m2 ∓USIAin which s=±labels the upper/lower surface bands, which are schematically depicted in Fig. 1 (c). Thus, a serial connection of the topological thermocouple can operate as a TE module with both refrigeration and power generation modes. To fabricate the pro- posed device, we might utilize the nanoimprint lithography which FIG. 1. (a) Schematic illustration (top view) of the TE module made of a film of magnetic TI. A periodic array of small square hollows filled with two different dielectrics is introduced into the magnetic TI. Each hollow harbors gap-opened metallic surface states (yellow lines) and is connected in series by leads (black solid and dashed lines). Refrigeration mode is shown here. (b) Schematic geom- etry of the fundamental element (topological thermocouple) consisting of two connected hollows with different dielectrics (1 and 2), possessing the p- and n- types metallic surface states. dis the distance between the two adjoined surfaces. (c) Corresponding k-dependent surface band dispersions around the Γpoint are depicted by blue lines in which μdenotes the chemical potential at equilibrium and USIAdescribes the structure inversion asymmetry (SIA) between the adjoined two surfaces due to band bending induced by the dielectrics. enables us to create a mold for making convex hollows. If the thickness is about 10 μm, many submicron hollows can be made by the mold. After molding, the electrode pattern is formed by photolithography in the submicron-scale. AIP Advances 10, 125230 (2020); doi: 10.1063/9.0000005 10, 125230-2 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv III. THERMOELECTRIC PROPERTIES To model the TE properties of the proposed device, we assume the emergence of ionic defects in the bulk of the TI as well as on its surface, taking into account the effect of element substitution of the 3D TI for systematic control of the Fermi levels.19,20Accordingly, based on the Boltzmann transport theory, we treat ionic disorder as a randomly distributed Coulomb-type long-range potential: Vc(r) =(e2/ϵ)∑i1/∣r−Ri∣with impurity concentration ncand the effec- tive lattice dielectric constant ϵ.20Assuming an ensemble averaging over random uncorrelated impurities up to lowest order in the scat- tering potential Vc(r), we can obtain the transport relaxation time20 τ(E± s)=τ(0) c(E± s)[1+3m2 (E±s)2]−1 , (2) whereτ(0) c(E± s)=E± s/(π2̵hv2 Fnc)denotes the transport relaxation time for the gapless surface state. According to the linear response theory, charge ( jp c) and ther- mal ( jp Q) currents ( p=−for electron and p=+for hole) can be described by linear combinations of an electric field Eand a tem- perature gradient ∇T: (jp c jp Q)=σp(1 SpT ΠpκpT/σp)(E −∇T/T), (3) where the electrical sheet conductance σp=e2Lp 0(in units of S=Ω−1) with electron charge −e(e>0), the Seebeck coefficient Sp=Lp 1/(eLp 0T)(in units of V K−1), the Peltier coefficient Πp=SpT (in units of V), and the thermal sheet conductance κp=[Lp 0Lp 2 −(Lp 1)2]/(Lp 0T)(in units of W K−1). For the application of Eand∇T along the xdirection, the coefficients Lp n(n=1, 2, 3)are obtained by Lp n=∑ s∫dk (2π)2τ(E± s)(v± s)2 x(−∂f(0) ∂E±s)pn(μ−E± s)n, (4) v± s=∇kE± s/̵his the group velocity of carriers, f(0)the equilibrium Fermi-Dirac distribution, and μthe chemical potential measured from the Dirac point ( E± s=0) of the original gapless surface band. Due to the heat transport by phonons, we need to include the ther- mal conductivity of phonons κph(in units of W K−1m−1) in the definition of ZT.3In the proposed device, the surface band structures of two adjoined surfaces are assumed to be symmetric so that ZTis equivalent to that of the individual surfaces and becomes a maxim. By using Eq. (4), the figure of merit on the TI surfaces is therefore given by20 ZT=σpS2 pT κp+dκph=(Lp 1)2 Lp 0(Lp 2+dκphT)−(Lp 1)2, (5) where dis the distance between the two adjoined surfaces, taking the role of a factor related to the surface-to-bulk ratio. Figure 2 (a) shows the calculated Peltier coefficient ∣Πp∣as a function of Tfor different values of m. As seen, the Peltier coefficient increases with increasing both Tandm. In this plot, based on the experiment in Ref. 38, we assume a carrier density 5.0 ×1011cm−2, which corresponds to μ≈65 meV, and take vF=4.0×105m s−1 FIG. 2. (a) Peltier coefficient and (b) thermoelectric figure of merit arising from a screened Coulomb impurity as a function of Tfor different m. In this plot, we set μ=65 meV and nc=1010cm−2. The details of the calculations are given in the text. as reported in Ref. 41. To decrease the heat transport due to phonons, we assume a thin film of 3D TI of thickness d=10 nm. It is noting that the topological surface dominates transport in thin films of a 3D TI with d≤14 nm was reported in recent experi- ments.40Figure 2 (b) shows the calculated thermoelectric figure of merit ZTas a function of Tfor different values of m. In contrast to the Peltier coefficient, ZThas a peak in the temperature range from 200 to 300 K. This is understandable because when the sur- face band gap opens, the thermal currents driven by the Peltier effect and a thermal gradient partially cancel through the relation (3) for E=0:jQ=(Lp 2−σpΠ2 p)(−∇T/T), leading to the maximization of ZT. Since the proposed device enhances the ZTin small scales in terms of d, we suggest that our TE module could be combined with optoelectronic devices such as cooling laser diodes that require pre- cise temperature changes1as well as be utilized for refrigeration of biological samples that require sensitive temperature control at localized spots. IV. SUMMARY In summary, we have proposed a TE module with both refriger- ation and power generation modes by utilizing asymmetric surfaces of a magnetic topological insulator (quantum anomalous Hall insu- lator). A pair of these two surfaces that are adjoined with each other acts as a Π-shaped p-njunction with ambipolar conduction, which can be regarded as a thermocouple consisting of two dissimilar TE materials. Thus, a serial connection of the thermocouple operates as a TE module. By using the Boltzmann transport theory, we demon- strated that its efficiency, i.e., ZTexceeded 1 in the low-temperature regime below 300 K. The proposed device could be utilized as a heat AIP Advances 10, 125230 (2020); doi: 10.1063/9.0000005 10, 125230-3 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv management device that requires sensitive temperature changes in a wide variety of applications for small-scale cooling. ACKNOWLEDGMENTS The authors thank S. Takahashi, S. Y. MatsushitaK. Tanigaki, and Y. P. Chen for valuable discussions. This work was supported by Grants-in-Aid for Scientific Research (Grant No. 20K15163 and No. 20H02196) from the JSPS. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1F. J. DiSalvo, “Thermoelectric cooling and power generation,” Science 285, 703 (1999). 2T. M. Tritt, “Thermoelectric phenomena, materials, and applications,” Annu. Rev. Mater. Res. 41, 433 (2011). 3H. J. Goldsmid, Thermoelectric Refrigeration Plenum (New York, 1964). 4J. Urban, A. Menon, Z. Tian, A. Jain, and K. 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5.0035496.pdf

Appl. Phys. Lett. 117, 232408 (2020); https://doi.org/10.1063/5.0035496 117, 232408 © 2020 Author(s).Spin–orbit torque driven four-state switching in splicing structure Cite as: Appl. Phys. Lett. 117, 232408 (2020); https://doi.org/10.1063/5.0035496 Submitted: 29 October 2020 . Accepted: 30 November 2020 . Published Online: 09 December 2020 Yuhang Song , Xiaotian Zhao , Wei Liu , Long Liu , Shangkun Li , and Zhidong Zhang COLLECTIONS Paper published as part of the special topic on Spin-Orbit Torque (SOT): Materials, Physics, and Devices SOT2021 ARTICLES YOU MAY BE INTERESTED IN Enhanced spin–orbit torque switching in perpendicular multilayers via interfacial oxygen tunability Applied Physics Letters 117, 232406 (2020); https://doi.org/10.1063/5.0024950 Spin wave propagation in a ferrimagnetic thin film with perpendicular magnetic anisotropy Applied Physics Letters 117, 232407 (2020); https://doi.org/10.1063/5.0024424 Tunable spin–orbit torque switching in antiferromagnetically coupled CoFeB/Ta/CoFeB Applied Physics Letters 117, 212403 (2020); https://doi.org/10.1063/5.0031415Spin–orbit torque driven four-state switching in splicing structure Cite as: Appl. Phys. Lett. 117, 232408 (2020); doi: 10.1063/5.0035496 Submitted: 29 October 2020 .Accepted: 30 November 2020 . Published Online: 9 December 2020 Yuhang Song,1,2 Xiaotian Zhao,1Wei Liu,1,a) Long Liu,1,2Shangkun Li,3and Zhidong Zhang1 AFFILIATIONS 1Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, China 2School of Materials Science and Engineering, University of Science and Technology of China, Shenyang 110016, China 3Key Laboratory of Nanodevices and Applications, Suzhou Institute of Nano-Tech and Nano-Bionics,Chinese Academy of Sciences, Suzhou 215123, China Note: This paper is part of the Special Topic on Spin-Orbit Torque (SOT): Materials, Physics and Devices. a)Author to whom correspondence should be addressed: wliu@imr.ac.cn ABSTRACT We prepared a splicing structure by using electron beam lithography, in which Pt and Ta were spliced together as the spin source upon perpendicularly magnetized Pt/Co/Pt heterostructures. It is found that Pt and Ta can modulate both spin–orbit torque and Dzyaloshinskii–Moriya interaction on the same magnetic layer, respectively. The four-state magnetization switching is achieved with the combinations of two spliced parts, which is observed by using a magneto-optical Kerr microscope. The initial nucleation position determinesthe chirality. The Pt side stabilizes a right-hand N /C19eel domain wall and the Ta side stabilizes a left-hand N /C19eel domain wall. Our study provides a method for further increasing the storage density and studying spin–orbit torque laterally. Published under license by AIP Publishing. https://doi.org/10.1063/5.0035496 Spin–orbit torque (SOT)-induced magnetization switching is a promising technology for magnetic random-access memory (MRAM) application, 1–3which has separating read and write paths and higher operation speed, compared with conventional spin-transfer torqueMARM. 4It is necessary to realize a perpendicular magnetic anisotropy (PMA) to achieve high bit densities.5So far, a lot of structures with PMA have been widely investigated and the most common structureis the heavy-metal/ferromagnet (HM/FM) system. The heavy metallayer with a large spin–orbit coupling (SOC) is used as the source of SOT, from which the resulting effective fields are strong enough to control the magnetization in adjacent ultra-thin ferromagneticlayers. SOT can originate from the bulk spin Hall effect (SHE) 6,7 and the interfacial Rashba effect8,9and can produce two orthogonal components: the Slonczewski-like torque (damping-like torque) m/C2m/C2rðÞ and the field-like torque m/C2r;10,11where mis the unit vector of magnetization and ris the spin polarization vector. HSLðHDLÞ/ð m/C2rÞandHFL/rare effective fields corresponding to damping-like torque and field-like torque, respectively.12–14The Slonczewski-like torque that originated predominantly from the SHEis responsible for magnetic switching and current-driven domain wall(DW) motion.More recently, some brilliant works demonstrate that some asymmetric structures can improve the application of SOT, such as the wedge structure 12,15–17and lateral structure of post processing.18,19 To explore the effect of further increasing the asymmetry on SOT, wepropose a splicing structure, in which two kinds of heavy metals arespliced together as the spin source to control the same ferromagneticlayer (FM layer). Heavy metal Pt and Ta are the most popular candi- dates for studying the SOT mechanisms benefiting from the strong spin–orbit coupling. 20Moreover, the opposite signs of spin Hall angles (positive hSHfor Pt2,21and negative hSHfor Ta6) are suitable to study SOT in the splicing structure. In this work, a splicing structure with Pt and Ta was fabricated by electron beam lithography (EBL) upon the magnetic layer Pt/Co/Pt. Under a fixed in-plane magnetic field applied along the current ori-entation, Pt and Ta dominate SOT-induced switching, respectively. By applying different intensities of current pulses to the device, the mag- netic states can be switched between four states. As an amount of dataincreases rapidly, storing more information in smaller areas isexpected. Multi-resistance states driven by SOT show potential appli-cations in future multilevel memories and neuromorphic computingdevices, 22–28which can store more than two states in one memory bit. Appl. Phys. Lett. 117, 232408 (2020); doi: 10.1063/5.0035496 117, 232408-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplOur demonstrations are helpful for the development of low-power consumption and to increase the data storage density. The splicing structure with Pt and Ta was fabricated upon the multilayer Pt/Co/Pt as shown in Fig. 1(a) . Stacks of Ta (1)/Pt (1.5)/Co (0.8)/Pt (0.6) (thicknesses given in nm) as an independent FM layer29 were deposited on a thermally oxidized Si (001) substrate by dc mag-netron sputtering under a base vacuum lower than 4.0 /C210 /C07Torr. The Pt/Co/Pt structure was chosen as a magnetic layer because ofmany advantages as follows. First, the Pt/Co/Pt multilayer has a largePMA. Second, the spin currents can be canceled by a thin buffer Ptlayer and capping Pt layer, moreover, with the aid of a Ta seedlayer. 29,30Therefore, the SOT efficiency and the current-induced switching direction are dependent on the thicker HMs sputtering sub-sequently. 31Third, in Pt/Co/Pt, there are two approximately equal Pt/ Co interfaces, resulting in a negligible Rashba field.18,32,33The capping Pt plays an important role in preventing oxidization of the Co layer34 and keeps the PMA away from the diffusion of the Ta pad.35,36 After the FM layer grew, Ta (5)//Pt (5) were fabricated on it. At the first EBL, a rectangle (800 lm in length and 400 lmi nw i d t h )w a s designed, and the 5 nm thick Pt pad was deposited by dc magnetronsputtering. After the lift-off process, another same size rectangle wasdesigned to splice with the previous rectangle along one side. Then,5 nm Ta and 0.5 nm Pt were grown. The ultrathin Pt was added toprotect Ta from oxidation. Next, one 20- lm-wide Hall bar wasprepared at the juncture by EBL and Ar ion etching techniques. as shown in Fig. 1(b) . The optical micrograph of the fabricated Hall structure before depositing the electrode is shown in Fig. 1(c) ,i nw h i c h the Ta pad and Pt pad are 10 lm wide, respectively. Finally, a Ti (5)/ Au (50) electrode was deposited on the Hall bar. For electrical transport measurements, currents were injected by a Keithley 6221 current source, and anomalous Hall voltages were detected by nanovoltmeter Model 2182A. The current-driven switching loops were achieved by a pulse current with a duration of1 ms and an interval time of 1 s. The magnetic states were investigatedusing the anomalous Hall effect (AHE). A magneto-optical Kerr effect(MOKE) microscope was used to observe the current-induced magne- tization switching process. Figure 1(d) shows the AHE resistance ( R H) loop with a vertical magnetic field ( Hz). A small current (0.1 mA) was used to avoid the effect of joule heat. Our stacks exhibit a significant PMA. Two switch-ing steps are obtained, corresponding to different coercivity for two parts of the FM layer. The steps appear due to the magnetization reversal of the Pt side. The different critical currents of the Ta pad andPt pad are expected. Current pulse-induced magnetization switching with fixed in- plane fields ( H x) along the x-axis has been carried out. Current pulses of a width of 1 ms and an interval time of 1 s with varying amplitudes were applied to the device in the x-axis. We sweep the current flow FIG. 1. (a) Sketch of the splicing structure. (b) Optical micrographic image of the fabrication procedure, where the left pad is Ta and the right pad is Pt. The H all bar is located at the intermediate boundary. (c) Optical micrographic image of a Hall bar device before depositing the electrode. (d) Anomalous Hall resistance RHas a function of the out-of- plane field Hz.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 232408 (2020); doi: 10.1063/5.0035496 117, 232408-2 Published under license by AIP Publishingfrom 38 mA to/C038 mA and then back to 38 mA with Hx¼6500 Oe. Magneto-optical Kerr microscopy was used to acquire magnetic images of the device. A magnetic field Hz(þ500 Oe) along theþza x i s was applied to initialize the magnetic moments. After the magneticfield H zis removed, the images have been processed by differential processing. The initial gray color represents the spin up. When the magnetic state changes (along the /C0z axis), the grayscale value turns into white (Pt side) or black (Ta side) in this work. They also representspin down. Hereinafter, the magnetic state combinations (Pt and Ta) are marked by two arrows. The magnetic state of the Pt side is represented by the left red arrow and that of the Ta side is represented by the right blue arrow. Figure 2(a) shows the Kerr images of magnetic states ( ",#),(#,#), (#,"), and (",") and an anti-clockwise loop in positive H x (þ500 Oe) and Fig. 2(b) shows the Kerr images of magnetic states (#,"), (","), (",#), and (#,#) and a clockwise loop in negative Hx (/C0500 Oe). From the switching loop, the four different RHvalues are f o u n d .T h em a g n e t i cs t a t e s( ","), (#,"), (",#), and (#,#)c o r r e s p o n dt o 0.235X, 0.465 X, 1.42X,a n d1 . 6 4 X, respectively. When large current pulses (larger than 28 mA) are applied, the magnetic states of Pt and Ta sides are antiparallel to each other due to the opposite hSHwith Pt and Ta, and the RHvalues are intermediate values. When small cur- rent pulses (from 18 mA to 23 mA) are injected, only the Pt pad canreverse the domain and the magnetic states of Pt and Ta sides are par- allel to each other. This suggests that the value of current is larger than the critical current of Pt pad but smaller than that of the Ta pad, which FIG. 2. SOT-induced magnetic switching with in-plane magnetic fields. (a) The anti-clockwise switching loop and Kerr images in Hx¼þ 500 Oe. (b) The clockwise switching loop and Kerr images in Hx¼/C0 500 Oe. The magnetic states of the Pt side and Ta side are marked by red and blue arrows, respectively. (c) and (d) Operation of the SOT- driven four-magnetic state in Hx¼6500 Oe.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 232408 (2020); doi: 10.1063/5.0035496 117, 232408-3 Published under license by AIP Publishingcan be explained by the smaller coercivity of the Pt side and the effect of current shunting. The resistivity of Ta is larger than that of Pt,37 which results in a large current shunting through Pt. Note that spinsfrom Ta must pass through an ultra-capping Pt, and this may cause adecrease in SOT efficiency. The operation procedures to lead to anyone of the four magnetic states in positive H x(þ500 Oe) are shown in Fig. 2(c) .T h e(",#)a n d (#,") states are achieved by injecting a large positive current pulse (þ30 mA) and a large negative current pulse ( /C030 mA), respectively. The (",") and (#,#) states need combined currents ( /C030 mA and þ21 mA) and (þ30 mA and/C021 mA), respectively. The current den- sity of 30 mA is about 9.37 /C2106Ac m/C02and that of 21 mA is about 6.56/C2106Ac m/C02. After injecting the same form of pulses into the device, the magnetization switching under /C0500 Oe is opposite to that underþ500 Oe, as shown in Fig. 2(d) . The magnetic switching direction can be explained by a SHE þDzyaloshinskii–Moriya interaction (DMI) scenario.21,38–41When a current is injected into a heavy metal, the electrons with spin-polarization ralong the y-axis accumulate on the FM layer and gener- a t eaS O Te f f e c t i v efi e l d .T h ee f f e c t i v efi e l dc o r r e s p o n d i n gt oSlonczewski-like torque is given by H SL¼HSLðm/C2rÞ,w h e r e mis the unit vector of magnetization in the domain wall (DW) and ris the spin polarization vector. The magnitude of the effective field isexpressed as H SL¼/C22hhSHjJej=ð2jejMstFMÞ. With a large in-plane mag- netic field ( Hx) to overcome the effective DMI field ( HDMI), the DW moment in the N /C19eel-type walls will realign parallel to Hx. Therefore, the effective field is along the same direction for both up-down and down-up walls and facilitates domain expansion or contraction,depending on the polarities of J eandHx.Jerepresents electron flow. For the Pt pad, when the positive current ( þ30 mA) is injected into the heavy metal along the þx axis (electron flow along the /C0x axis), the electrons with spin-polarization ralong theþyaxis accumu- late on the HM/FM interface and mis parallel to Hxalong theþx axis. According to the formula HSL¼HSLðm/C2rÞ, the effective field toward theþz axis favors the spin up magnetic state ( "). The domain of spin up expands and domain of spin down shrinks as illustrated in Fig. 3(a) . Therefore, the magnetic state of the Pt side turns into spin up. Ta has an opposite sign of hSHto Pt, the electrons withspin-polarization ralong the/C0yaxis accumulate on the HM/FM interface, and the effective field is along the /C0z axis. Thus, the domain of spin down expands and domain of spin up shrinks as illustrated in Fig. 3(b) . The magnetic state of the Ta side turns into spin down ( #). Therefore, the magnetic state ( ",#) is obtained by injecting þ30 mA pulse under a positive Hx(þ500 Oe). Other cases can also be explained by this physical model. The direction of DW motion depends on the sign of HSLand DMI constants. The interfacial DMI is defined as EDMI¼/C0Dij/C1Si/C2SjðÞ ; where Dijis the vector of DMI constants and SiandSjare magnetic spin moments located on neighboring atoms. Under a determinate in- plane field, the spins of DW are along Hx. Therefore, the initial nucle- a t i o np o s i t i o nd e t e r m i n e st h ec h i r a l i t y .F o rt h es i t u a t i o ni nap o s i t i v e Hx (þ5 0 0O e ) ,t h ef o u rs w i t c h i n gp r o c e s s e sm a r k e da sA ,B ,C ,a n dDi nt h e switching loop are shown in Fig. 4(c) . The Kerr images and the schematic diagram of DW nucleation and propagation are shown in Figs. 4(a) and4(b), respectively. When a series of negative small pulses are applied in the /C0xa x i s (the electron flow is along the þx axis) and only the magnetic state of the Pt side is reversed in the A process. The reversed domain of spindown (#) nucleates at the left edge on Pt side and propagates to domain of spin up ( "). This suggests that the spin down state ( #)i s p r e f e r r e da n dt h ee f f e c t i v efi e l d( H SL)i sa l o n gt h e/C0za x i s .T h es i g no f rin the Pt pad is along the /C0ya x i sa ss h o w ni n Fig. 4(b) .A c c o r d i n g to the formula HSL¼HSLm/C2rðÞ , the direction of mis figured out to align theþxa x i s ,w h i c hi sp a r a l l e lt o Hxand consistent with the actual situation. The magnetization configuration of the Pt side is # " .T h i s is a typical N /C19eel DW of right-handed chirality, which corresponds to a positive DMI constant. When the negative current pulses are large enough to reverse the magnetic state of the Ta side as shown in process B, the reversed domain of spin up ( ") nucleates at the left edge of the Hall bar on the Ta side and propagates to the domain of spin down ( #), which means that the spin down state ( ") is preferred and the effective field ( HSL)i s along theþz axis. The sign of ris along theþya x i si nt h eT ap a da n d the direction of mis figured out to align along the þxa x i s ,w h i c hi s parallel to Hx. The magnetization configuration of the Ta side is " # , which is a typical N /C19eel DW of left-handed chirality corresponding to a FIG. 3. Schematic illustration of the DW texture for "#and#"N/C19eel DWs and spin transport of the spin Hall effect (SHE) with a positive current and in-plane magnetic field along the x-axis. (a) for the Pt side (b) for the Ta side. The domain wall textures are marked by blue arrows. The effective fields from Slonczewski-like t orque ( HSL) on the Pt side and Ta side are marked by yellow and brown arrows, respectively. The gray arrows reflect the direction of DW motion. The symbols /C12and/C10represent the direction of spin-polarization r.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 232408 (2020); doi: 10.1063/5.0035496 117, 232408-4 Published under license by AIP Publishingnegative DMI constant. It is found that the chirality of the Ta side is opposite to that of the Pt side in our splicing structure, which is inagreement with the results reported by Yu et al. 42They reported that DMI can be tuned by different capping materials and thicknesses andthe top FM/Ta and top FM/Pt interfaces show opposite signs of DMIconstants. In our work, FM/Ta and top FM/Pt interfaces can modifyDMI in the same FM layer. Although DMI is an interface effect, thecapping Pt on the FM layer (Pt/Co/Pt) is ultra-thin about 0.6 nm thin-ner than 1 nm, and the signs of DMI constants are mainly affected bythe upper Hall metal. 30 When the current pulses are positive (the electron flow is along the/C0x axis), the reversed domain nucleates at the other (right) edge of the Hall bar. The directions of rare reversed, and the directions of m are always parallel to Hx. The magnetization configurations of the Pt side and Ta side are # " (right-handed) and " # (left-handed), which are in accordance with processes A and B. A similar process isobtained in a negative H x(/C0500 Oe) and mare always parallel to Hx along the/C0x axis. The magnetization configurations of Pt and Ta sides are"!# (right-handed) and #!" (left-handed). It is concluded that the two pads in the splicing structure can modify DMI, respectively.In summary, we have achieved a four-state current-induced switching in a Pt//Ta splicing structure with an in-plane field, whichare marked as (","), (#,"), (",#), and (#,#). As observed by using the magneto-optical Kerr microscope, the reversed domain always nucle-ates at the initial edge of current flow and moves along the currentflow. The initial nucleation position determines the chirality. The Ptside stabilizes a right-hand N /C19eel wall and the Ta side stabilizes a left- hand N /C19eel wall in the same FM layer. Our results demonstrate that the spliced Pt and Ta can be regarded as two sources to determine theSOT and DMI, respectively. The possibility of the field-free switchingneeds to be studied more in our device. As a prospect, more spin Hallmetals or ferromagnets and antiferromagnets can be fabricated in thesplicing structure. Our study not only contributes to the increase inthe storage density but also proposes a lateral structure to investigate SOT. This work was supported by the State Key Project of Research and Development of China (No. 2017YFA0206302), and the National Nature Science Foundation of China under Project Nos. 51590883, 52031014, 51771198, and 51801212. FIG. 4. (a) SOT-induced DW nucleation and propagation in four processes. (b) Four parts of DW propagation and illustrations. Red arrows represent spin polar ization vector r, blue arrows represent the unit vector mof magnetization in the DW, and gray arrows represent the moving direction of DWs. The symbols /C12and/C10represent magnetic states. (c) Switching loop in Hx¼þ 500 Oe with states marked by A, B, C, and D.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 232408 (2020); doi: 10.1063/5.0035496 117, 232408-5 Published under license by AIP PublishingDATA AVAILABILITY The data that support the findings of this study are available within the article. REFERENCES 1I .M .M i r o n ,K .G a r e l l o ,G .G a u d i n ,P .J .Z e r m a t t e n ,M .V .C o s t a c h e ,S .A u f f r e t ,S . B a n d i e r a ,B .R o d m a c q ,A .S c h u h l ,a n dP .G a m b a r d e l l a , Nature 476, 189 (2011). 2L. Q. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 109, 096602 (2012). 3J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi, S. Fukami, T. Suzuki, S. Mitani, and H. Ohno, Nat. Mater. 12, 240 (2013). 4J. C. Rojas-S /C19anchez, P. Laczkowski, J. Sampaio, S. Collin, K. Bouzehouane, N. Reyren, H. Jaffre `s, A. Mougin, and J. M. George, Appl. Phys. Lett. 108, 082406 (2016). 5S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey, B. D. Terris, and E. E.Fullerton, Nat. Mater. 5, 210 (2006). 6L. Q. Liu, C. F. Pai, Y. Li, H. W. 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5.0022948.pdf

J. Appl. Phys. 128, 210902 (2020); https://doi.org/10.1063/5.0022948 128, 210902 © 2020 Author(s).Topological materials by molecular beam epitaxy Cite as: J. Appl. Phys. 128, 210902 (2020); https://doi.org/10.1063/5.0022948 Submitted: 28 July 2020 . Accepted: 09 November 2020 . Published Online: 03 December 2020 Matthew Brahlek , Jason Lapano , and Joon Sue Lee COLLECTIONS Paper published as part of the special topic on 2D Quantum Materials: Magnetism and Superconductivity 2DQMMS2020 ARTICLES YOU MAY BE INTERESTED IN III–V lasers selectively grown on (001) silicon Journal of Applied Physics 128, 200901 (2020); https://doi.org/10.1063/5.0029804 A practical guide for crystal growth of van der Waals layered materials Journal of Applied Physics 128, 051101 (2020); https://doi.org/10.1063/5.0015971 N-polar GaN/AlN resonant tunneling diodes Applied Physics Letters 117, 143501 (2020); https://doi.org/10.1063/5.0022143Topological materials by molecular beam epitaxy Cite as: J. Appl. Phys. 128, 210902 (2020); doi: 10.1063/5.0022948 View Online Export Citation CrossMar k Submitted: 28 July 2020 · Accepted: 9 November 2020 · Published Online: 3 December 2020 Matthew Brahlek,1,a) Jason Lapano,1 and Joon Sue Lee2,a) AFFILIATIONS 1Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA 2Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA Note: This paper is part of the Special Topic on 2D Quantum Materials: Magnetism and Superconductivity a)Authors to whom correspondence should be addressed: brahlekm@ornl.gov andjslee@utk.edu ABSTRACT Topology appears across condensed matter physics to describe a wide array of phenomena which could alter, augment, or fundamentally change the functionality of many technologies. Linking the basic science of topological materials to applications requires producing high-quality thin films. This will enable combining dissimilar materials while utilizing dimensionality, symmetry, and strain to create or controlthe electronic phase, as well as platforms to fabricate novel devices. Yet, one of the longstanding challenges in the field remains understand-ing and controlling the basic material properties of epitaxial thin films. The aim of this Perspective article is to discuss how understanding the fundamental properties of topological materials grown by molecular beam epitaxy (MBE) is key to deepening the knowledge of the basic physics, while developing a new generation of topological devices. A focus will be on the MBE growth of intrinsic materials, creation,and control of superconducting and magnetic topological phases. Addressing these questions in the coming decade will undoubtedlyuncover many surprises as new materials are discovered and their growth as high-quality thin films is refined. Published under license by AIP Publishing. https://doi.org/10.1063/5.0022948 I. INTRODUCTION During the past 40 years, topology has permeated condensed matter physics and has since risen to become one of the majortenets of the field. Topology is a very general concept used to clas- sify objects based on their broad properties, which is quantified by a single (or multiple) number(s) called the topological invariant. Inmathematics, this captures the similarities among soccer balls andoranges, which are distinguished from objects with holes such ascoffee cups and doughnuts. In the context of condensed matter physics, however, the attributes of geometrical objects, such as sur- faces, curvatures, etc., are analogously replaced by aspects of theelectronic wavefunction, which can be used to define a wide varietyof topological invariants. This has enabled understanding and pre-dicting a wide variety of electronic, magnetic, and superconducting phases. Historically, one of the first and most celebrated applica- tions of topology in condensed matter physics was the observationof quantized Hall resistance with vanishing longitudinal resistanceat ultraclean heterointerfaces. 1This has become known as the quantized Hall effect, which can be distinguished by the topological invariant called the Chern number or the Thouless –Kohmoto – Nightingale –den Nijs (TKNN) invariant.2The topological descrip- tion gave a full quantum mechanical explanation of the chiral edgestate responsible for the quantized transport: the bandgap created by the two-dimensional (2D) quantized cyclotron orbits gives rise to the non-zero Chern number and is spanned by gapless states that are bound to the one-dimensional (1D) edge. Such metallicstates localized on the boundary are a general feature of topologicalmaterials. 3,4After this initial success, important advances of topol- ogy quietly evolved until the early 2000s, when the quantumspin-Hall insulator was predicted in graphene 5and later in HgTe/ CdTe quantum wells.6Quantum spin-Hall insulators, which exhibit two spin-polarized, counter-circulating quantum Hall edge states,are described by a different topological invariant in the Z 2class.7 After experimental confirmation of quantized transport emanating from the 1D edge states,8the Z 2classification scheme was general- ized to three-dimensions9and gained the name topological insula- tors (TIs).10Experimental confirmation came rapidly through the observation of novel surface states within the bulk bandgap ofBi 1−xSbxusing angle-resolved photo-emission spectroscopy (ARPES) to map the surface band structure.11There are a variety of well-known electronic states that exist on surfaces of bulk mate-rials, 12,13which occur with a Z2topological number that is trivially zero. The topological surface states (TSSs), in contrast, that emerge on the surface of a so-called strong TI exhibit an odd number ofJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210902 (2020); doi: 10.1063/5.0022948 128, 210902-1 Published under license by AIP Publishing.linearly dispersing Dirac states, which are singly degenerate with the direction of the spin locked to the direction of the momen- tum.3,4This demonstrated that a large expanse of novel electronic states existed in materials both new, as well as those long studied inother contexts. This started off a near exponential rise in theoreticaland experimental studies of topological materials. Since the genesis of this field, many distinct states have emerged with a zoo of confirmed and candidate topological materi-als. 14Including TIs mentioned above, select examples that are at the forefront of research today include topological crystalline insu-lators, which possess spin –orbit induced band inversion; but unlike TIs, this crossing is protected by the mirror symmetry of the crystal lattice and exhibits an even number of 2D Dirac cones. 15,16Nodal systems are a very rich and diverse class of materials, where there isa three-dimensional (3D) band crossing that is protected by a sym-metry, either crystalline or time-reversal. 17These crossings, for example, can either occur at points or along lines of the Brillouin zone. When the crossing occurs at a point, this gives rise to Diracand Weyl semimetals, with the latter exhibiting novel Fermi arcsurface states. 18The crossing can also occur along a line, which are referred to as nodal line semimetals. Superconductors with well- formed bulk gaps also can exhibit topological structures with gapless states, which are manifestations of Majorana fermions aselectronic quasiparticle excitations. 19Higher order topological states are 3D materials, where the topological states emerge on the lower dimensional edges called hinge states.20Furthermore, topol- ogy extends outside the realm of atomic-scale materials to systemswhere the periodicity ranges from the micrometer-scale to even themillimeter-scale in the areas of topological photonics 21and pho- nonics,22respectively. The insight given by topological phases spans the fundamental physics of condensed matter but also pushes into the realm of high energy physics including, forexample, Majorana fermions, Weyl fermions, and axionicphenomena. As with nearly all discoveries in condensed matter physics, a slew of possible applications emerge. Topological materials are no exception. The novel character of the topological states can beapplied across a range of electronic, magnetic, optical and photonic,and phononic devices. The potentially useful applications of topo-logical materials include efficient spin generators and spin-charge converters in spintronic devices, 23basic elements to build qubits for fault-tolerant quantum computing,24,25energy-efficient micro- electronic components,26and novel information storage media based on topological spin-textures such as skyrmions.27Among these, applications in spintronics and quantum computing have gained much attention and have motivated studies of magnetismand superconductivity in topological material-based systems. Addressing both fundamental questions as well as pursuing practical applications necessitates creating high-quality materials as thin films. Advanced synthesis techniques, particularly, molecular beam epitaxy (MBE), can easily control growth at the submono-layer level with chemistries ranging across the periodic table. MBEis used to create systems where materials with dissimilar propertiesare joined at an atomically sharp interface. Furthermore, this level of precision and accuracy during synthesis opens many additional routes to control and manipulate topological systems, for instance,tuning dimensionality through thickness scaling or modifyingsymmetry at interfaces. Key examples are as follows: At an interface with a superconductor, Cooper pairs can leak across the interface, which represents a novel route to realize topological superconduc-tivity through combining a superconductor with a topologicalmaterial or a semiconductor that has strong spin –orbit coupling. Furthermore, when the thickness of a material is reduced below the scale of the wavelength of the electrons, the effective dimensionality is reduced, and the band structure is changed. This, as discussedabove, enabled the first realization of a 2D TI state in a HgTe/CdTequantum wells since band inversion in that system is driven byfinite thickness. Also, in the TI Bi 2Se3, finite thickness effects arise due to the wavefunctions on the top and bottom surfaces hybridiz- ing when the thickness of the films is than five monolayers orless, 28which is amplified by mixing In on Bi.29Additionally, 3D Dirac nodal metals have a Dirac point that is fourfold degenerate.When a symmetry, either time-reversal or spatial inversion, is broken, the single Dirac point splits into two Weyl points with twofold symmetry. 10,17,18,30This can be achieved in bulk crystals which lack either symmetry, but thin film systems can be con-structed to controllably break these as well. For example, interfacesgenerally break inversion symmetry, and carefully choosing materi- als with strong charge transfer can enhance such an effect. Combining topological materials as heterostructures with magneticmaterials can break time-reversal symmetry. Furthermore, consid-ering the rich array of magnetic phases and their domain struc- tures, this enables spatially tuning topological band structures in non-trivial ways. Finally, the goal of new electronic motifs based onthese exotic phases necessitates wafer-scale (lateral scale > cm) thinfilms to facilitate the micro- and nano-scale fabrication necessaryfor creating functional electronic devices. MBE is the gold-standard for the growth of epitaxial materi- als. This is due to the fact that MBE enables a high-level ofcontrol over the growth process, which, as discussed here, is a keyto understanding many aspects of topological materials. As shownin the schematic of a typical MBE growth system in Fig. 1(a) ,t h i s level of control is rooted in how the constituent elements are delivered to the growing surface of the thin film. Each element isindividually heated to a temperature where evaporation/sublima-tion takes place, which creates a beam of atoms (or molecules).These beams converge onto a substrate crystal with an atomically flat surface. There, the elements adsorb, undergo diffusion on the surface, and finally chemically bond. The temperature of the sub-strate can be tuned to an optimum value which, in a simplisticview, maximizes surface diffusion to confine nucleation and growth at the proper atomic sites. Furthermore, MBE growth takes place under ultrahigh vacuum (UHV) <1 × 10 −9Torr (for comparison, atmosphere is 760 Torr), where the mean free path isorders-of-magnitude larger than the chamber. This serves severalpurposes. First, the atomic beams suffer no scattering in route to the substrate, which enables highly uniform growth while mini- mizing both thermal leakage among the cells, and elementalcross-contamination. 31Under UHV conditions, the incorporation of contaminants from the background gas is minimized. As atypical example, at 1 × 10 −9Torr, it will take well over an hour to adsorb a monolayer of contamination (for growth of ultraclean materials vacuum conditions can be as low as 10−11–10−12 Torr32). Finally, systems are typically equipped with in situJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210902 (2020); doi: 10.1063/5.0022948 128, 210902-2 Published under license by AIP Publishing.diagnostics such as reflection high-energy electron diffraction (RHEED), which probes both the crystal quality as well as the morphology of the surface. This enables rapid feedback to opti-mize growth conditions. To grow the best quality topological materials requires precise characterization of the surfaces both from a structural perspective but also from an electronic point of view. This has spurred the development of powerful platforms for growing materials and char-acterizing all the surface properties in a single system withoutbreaking vacuum (in fact, such systems have a long history in semi- conductors and other areas, but topological materials have brought them to the forefront of research). An example is shown in theschematic in Figs. 1(b) –1(d), which is modeled after Oak Ridge National Laboratory ’s systems for growth and characterization of topological and correlated matter based on the designs ofScientaOmicron. Such a system includes a load lock that serves tobring samples from atmosphere to UHV without compromisingthe vacuum conditions. This is attached to an intermediate chamber, where samples are distributed to the various growth or characterization stations; this also serves as a buffer between thehigh pressure of the load lock and the UHV conditions of thevarious stations. Multiple chambers are not only for growing topo- logical materials, but also for growing metals, intermetallics, or novel complex oxide materials to serve as functional layers to be FIG. 1. (a) MBE schematic modeled after the ScientaOmicron EVO-50 (for clarity the nitrogen cryopanel is not shown). (b) –(d) Gaining a full understanding of topological materials requires a myriad of structural, morphological, electronic, and magnetic probes, as highlighted. This necessitates integrated instrum ents to grow atomically precise materials and heterostructures (b), transfer in situ to scanning probe (scanning tunneling microscope, STM), spectroscopic tools (angle-resolved photo-emission spectroscopy, ARPES and x-ray photo-emission spectroscopy, XPS), and transport systems (c), as well as many ex situ tools for characterization and device processing (d). Reproduced with permission from Bansal et al., Phys. Rev. Lett. 109, 116804 (2012). Copyright 2012 The American Physical Society; Lee et al. , Phys. Rev. B 92, 155312 (2015). Copyright 2015 The American Physical Society; Brahlek et al. , Phys. Rev. B 93, 125416 (2016). Copyright 2016 The American Physical Society; Brahlek et al. , Phys. Rev. Mater. 3, 093401 (2019). Copyright 2019 The American Physical Society; Dai et al. , Nano Res. 8(2015). Copyright 2015 Springer Nature; Lapano et al. , APL Mater. 8, 091113 (2020); licensed under a Creative Commons Attribution (CC BY) license; Liu et al. , Phys. Rev. B 96, 235412 (2017). Copyright 2017 The American Physical Society; Koirala et al. , Nano Lett. 15, 8245 (2015). Copyright 2015 The American Chemical Society; Lee et al. , Phys. Rev. Mater. 3, 084606 (2019). Copyright 2019 The American Physical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210902 (2020); doi: 10.1063/5.0022948 128, 210902-3 Published under license by AIP Publishing.interfaced with topological materials, for example, superconductors or magnets. Furthermore, there is a station dedicated to surface preparation for heating, sputtering, or simple depositions. In termsof characterization, ARPES is an indispensable tool for probing thesurface band structure. 33Typical systems can be equipped with He-lamp sources (primary line is He-I at 21.2 eV), laser sources (polarization control with energies ranging across 6 –11 eV), or even stationed at a synchrotron. Scanning tunneling microscopy(STM) enables direct real space probing of surface morphology,atomically resolved crystal structure, as well as directenergy-resolved imaging of the surface wavefunction that enables full maps of the quasiparticle interference spectrum, which has been a valuable probe for understanding the novel scatteringprocess of TSS in TIs 34as well as novel effects in 3D Dirac metals.35Furthermore, chemical information using x-ray photo- emission spectroscopy (XPS) is indispensable to elucidate how valence states are modified at heterointerfaces. Since many topolog- ical materials can rapidly degrade or change in air, in situ transport has recently been very critical at separating properties endowedduring growth from atmosphere effects, 36–38as well as probing highly unstable materials such as Na 3Bi39,40and the monolayer superconductor FeSe.41,42However, ex situ characterization systems are still critical, and scientists have performed profound work onstand-alone MBE systems. For example, state-of-the-art STMinstruments may be too sensitive to be practical when attached to such systems. Moreover, ARPES measurements performed on syn- chrotron beamlines with full control over energy, polarization, andspot size, are necessary to understand complexities of the matrix-element effects associated with the photoemission process. 43In such situations, in situ capping layers have proven very effective,44 as well as instruments called vacuum suitcases that maintain UHV conditions for weeks during transportation. Finally, transfer out ofthe MBE system is critical for a myriad of other probes such ashigh-magnetic-field magneto-transport, magnetometry, diffraction(x-ray, neutron, etc.), as well as device fabrication. It has been over a decade since the discovery of TIs, which reinvigorated the field of topological materials. In this time frame,many exciting things were revealed and problems have beensolved. The goal of this Perspective article is to frame future prob- lems in the field of topological materials that concern MBE growth . As such, we discuss the role that MBE has played in many key dis- coveries, while highlighting the scientific aspects and challengesthat led to these successes. This is a key to emphasizing openproblems that will expand scientific understanding as well as push toward applications of topological materials. This work is orga- nized as follows. Section IIhighlights how a detailed understand- ing of the synthesis is required to realize intrinsic properties oftopological materials. Since, by far, the tetradymite TIs are thebest studied, these materials are the key examples, and it is emphasized that most known and candidate topological materials have not been studied by MBE. Section IIIdetails current chal- lenges and opportunities faced in the field of realizing topologicalsuperconductors and the push toward Majorana fermions for usein quantum computing. Section IVconcerns the direction of inte- grating magnetism and topology, which enables control over time-reversal symmetry and is key for many possible applicationsin topological spintronics.II. SYNTHESIS OF TOPOLOGICAL MATERIALS BY MOLECULAR BEAM EPITAXY A. Toward intrinsic topological materials: Understanding and controlling defects The key to understanding and ultimately utilizing topological phenomena in real materials is isolating the unusual topological states such that they dominate the electronic properties. This fun-damentally requires precise control of the materials, which comesdown to positioning the Fermi level at a particular location of theband structure, with examples schematically shown in Fig. 2(a) .I n TIs and topological crystalline insulators, the novel character emerges only when the Fermi level is located near the 2D Diracpoint; similarly, in 3D Dirac and Weyl semimetals, the Fermi levelshould be close to the 3D Dirac or Weyl point for the electronicproperties to not be obscured by trivial states. Furthermore, access- ing the topological superconducting phases that host Majorana fer- mions requires positioning the Fermi level of the topologicalsuperconductors sufficiently close to the middle of the bulk energygap, as discussed in Sec. III. Finally, as discussed in Sec. IV,t o observe the quantum anomalous Hall effect requires controllably breaking time-reversal symmetry which opens a gap at the 2D Dirac point and results in the emergence of a 1D edge mode; yet ifthe Fermi level is far away from this location, quantization of thetransport properties cannot be observed. To achieve this level of control, considerable effort needs to be expended to optimize the growth conditions for each individual material, which ultimatelycomes down to understanding and minimizing defects. Such pro-gress is often slow and painstaking, but when mastered, historyshows that the results can be revolutionary. The band structure of a material fundamentally determines its topological class. 3,4In most topological materials, the bands that give rise to the non-trivial topological invariant are not necessarilyclose to the Fermi energy. This ultimately makes them impracticalfor device applications since the topological states are located far (>eV) from the Fermi level, making it impossible for those states to dominate the low-energy electronic properties. However, for asubset of these materials, the topological states are located within,or near to, the bandgap formed by the valence band and conduc-tion band. Key examples are the tetradymite TIs Bi 2Se3,B i 2Te3, and Sb2Te3,52topological crystalline insulators SnTe,16and 3D Dirac semimetals such as Na 3Bi and Cd 3As2,18many of which are semi- conductors with narrow bandgaps. Since all these materials arecharge balanced, the Fermi level of an ideal material, E F,ideal , should be intrinsically located within the bulk bandgap. Yet, defects in the form of vacancies, interstitial atoms, or impurities incorporatedduring the growth often dope the materials, thus shifting the Fermilevel from this ideal location toward the conduction or valencebands, as shown in Fig. 2(a) asE F,nonideal . This creates parallel con- duction paths effectively shorting out the electronic character of the topological states. For most topological materials, this has beenknown prior to the initial studies and has been found to be thetypical situation. Therefore, the challenge in this field is first tounderstand and address the defects incorporated during the synthesis. The formation of defects is not unique to topological materi- als, in that every material forms an array of defect states. ControlJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210902 (2020); doi: 10.1063/5.0022948 128, 210902-4 Published under license by AIP Publishing.and mitigation of defects have been major drivers toward under- standing basic phenomena in condensed matter physics. To dem- onstrate how charged defects lead to the occupation of the bulkconduction or valence bands, consider an isolated charge embed-ded in an insulator. This local charge is screened by the surround-ing electronic states. This is described by a hydrogen-like potential that is rescaled by the dielectric constant of the material, which has an effective Bohr radius given by a=ϵ(m e/m*)aB, where ϵis the dielectric constant, meis the electron mass, m* is the effective mass in the material, and aB≈0.5 Å is the Bohr radius in free space. The transition to a metallic phase (occupation of the conduction band or valence band) occurs when the density of such defects, N3D,i ssufficiently large that these local charge states overlap. This occurs at the so-called Mott criterion when ND≳(0.26/ a)3with NDbeing the critical density.53For semiconductors like Si, the critical density is of the order of 1017–1018cm−3.53In stark contrast, the critical densities for TIs are two orders of magnitude lower in the range of10 14–1015cm−3.52,54This results in the Fermi levels being well away from the Dirac point in nearly all samples. The highly sensitive nature of topological materials to defects is fundamentally intertwined with the properties that make themtopological. 55First, for band inversion, a necessary condition for most topological materials, the anion and cation electronegativity need to be similar,56which implies that the top of the valence band FIG. 2. (a) Schematics of topological band structures for select materials. The lower horizontal dashed line indicates the Fermi level positioned at the ide al location inter- secting only the topological states, and the upper dashed line indicates the Fermi level positioned at the non-ideal location, where it intersects th e bulk states due to charge doping by various defects. (b) In MBE growth, a major source of defects comes at interfaces, as highlighted by the yellow dashed lines in the scan ning transmis- sion electron microscopy image of the TI Bi 2Se3and the substrate Al 2O3. (c) and (d) Schematic of the virtual substrate method developed to dramatically reduce the density of interfacial defects (c), which can be seen in the scanning transmission electron microscopy image of the highly ordered interface among Bi 2Se3and the virtual substrate. (d). Reproduced with permission from Koirala et al. , Nano Lett. 15, 8245 (2015). Copyright 2015 The American Chemical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210902 (2020); doi: 10.1063/5.0022948 128, 210902-5 Published under license by AIP Publishing.is in close proximity to the bottom of the conduction band; this creates a situation where spin –orbit coupling, if sufficiently strong, can invert the bands. However, this is also the same criteria thatmakes the formation energy of defects relatively low, meaning thatsuch materials will readily accommodate defects. To highlight thechallenges of growing and controlling topological materials by MBE we examine the tetradymite TIs Bi 2Se3,B i 2Te3, and Sb 2Te3, which are the most widely studied topological materials and theformation of defects is the most well understood from both thefields of TIs as well as thermoelectrics. 52,56(The growth of II-VI HgTe/CdTe and narrow gap III-V topological systems have long been studied in the context of semiconductor devices, see Ref. 57.) The most common defects in these materials are vacancies on theanion site, for example, Se vacancies in Bi 2Se3, which are electron donors, and antisite defects where the cation (anion) can occupythe anion (cation) site which are, in most cases, acceptors (donors). 58,59Since both cations and anions can adopt different formal valence states and have similar electronegativity, predictingthe defect chemistry is not straightforward. For example, dependingon the details of the synthesis, Bi 2Te3readily forms all types of defects, which enable it to exhibit either n-type or p-type character. In contrast, Sb 2Te3is solely p-type due to the predominance of antisite defects. In Bi 2Se3, the large size mismatch among Bi and Se reduces antisite formation and favors Se vacancies giving a ubiqui-tous n-type character. Finally, the heavy atoms in these materials yield a high polarizability, and, thus, high dielectric constants: For example, Bi 2Se3ϵ≈113, Bi 2Te3ϵ≈290, and Sb 2Te3ϵ≈168.52 Through the Mott criterion, the high dielectric constant effectively lowers ND. Together, these attributes create a scenario which raises the defect density, while lowering the critical density ND. For bulk materials, this has been overcome by careful mixing schemes among the anion and cation sites along with resonant doping. Forexample, careful synthesis of both Sn-doped (Bi 1−xSbx)2(Te 1−ySey)3 and (Bi 1−xSbx)2Te2S, and Bi 2Se3under high Ar pressure have exhibited bulk carrier freeze-out at low temperatures and resistiv- ities in the range of 10 –100Ωcm, which is consistent with the Fermi level being inside the bulk bandgap.60,61 In general, bulk crystals are grown under quite different condi- tions compared to thin films, which manifests as the formation ofdifferent types of defects with different concentrations (see Ref. 62 for additional information concerning bulk synthesis). Compared to bulk crystals of the tetradymites which are typically synthesizedat temperatures in the range of 700 –800 °C, thin films are grown at much lower temperatures, typically in the range of 200 –300 °C. Due to the reduced kinetics, this can limit the formation of certain defects. A key example to illustrate this is the formation of the highenergy Se vacancies in bulk crystals of Bi 2Se3that form at the 3A locations (middle of the quintuple layer unit). In bulk crystals, thehigher growth temperatures enable the formation of these defects, and the quick cooling process tends to trap them. For thin films, in contrast, the low temperature growth prevents their formation andonly allows the formation of the lower energy Se vacancies withtypically reduced concentrations. 63 In contrast to bulk crystals, however, interfaces are a major source of defects for films. In particular, these occur in the form of chemical reactions, interdiffusion, and dislocations related to theprocesses of heteroepitaxy. This highlights the challenge ofunderstanding the structural and chemical relations between the film and substrate, and since many of the topological materials are layered van der Waals structures, these properties are particularlycomplex. In so-called van der Waals epitaxy, 64,65the lack of chemi- cal bonds orthogonal to the growth direction reduces the ability forthe individual layers to chemically adhere in-plane, which, in turn, reduces the effect of epitaxial strain. This generally results in strain- relaxation, where the film relaxes to the bulk lattice parameters,which typically occurs in the first one or two monolayers. This pre-vents utilizing the epitaxial mismatch between the substrate andthe film to distort the bonding environment, and, thereby, modify- ing electronic properties. There are advantages, however, for van der Waals epitaxy. The reduced sensitivity to lattice mismatchenables films to be grown on substrates with either large strains oreven amorphous surfaces, as well as other novel post-growth pro-cessing such as mechanical exfoliation of wafer-scale films 66or post-growth crystallization.67A major challenge arises, however, during the formation of the first monolayer: Growth occurs fromrandom nucleation points from which single monolayer islandsgrow laterally. These islands then merge to form the first mono-layer. Since the film and substrate have different lattice parameters, the boundaries where the mergers occur form defects, such as rota- tion and tilting between adjacent grains and secondary nanostruc-tures. 68,69Furthermore, for van der Waals epitaxy, understanding interfacial bonding and how to best mitigate dangling bonds prior to deposition is critical.65Key examples of this are the formation of Si–Se as an initial step in growing Bi 2Se3on Si (111)70,71and SiO 2,72and Ga –Se at the interface among Bi 2Se3on GaAs.73,74 These aspects, combined with any interfacial chemical reaction related possibly to unpassivated bonds, lead to a high number of defects confined to the interface, as can be seen in Fig. 2(b) ; subse- quent layers which are nearly commensurate with the first mono-layer typically have far fewer defects. Transport measurements forBi 2Se3films, for example, show that these interfacial defects are electron donors.45,75Through band-bending effects, these addi- tional electrons occupy both the TSSs at the substrate/film interface as well as moving the Fermi energy well into the bulk conductionband. 45,54,76The knowledge gained regarding the key signatures of these defects and how they are mitigated is important for futurestudies across topological materials, where it is critical to control the Fermi energy and place it sufficiently close to the Dirac point. Interfacial defect control has been achieved by choosing sub- strates with either a specific lattice match or chemical compatibility.In particular, substrates with a matching hexagonal surface struc- ture that have been used are α-Al 2O3,45,77Si (111),71,75,78InP (111),79–81SrTiO 3(111),82,83GaAs (111),84,85graphene-terminated SiC(0001),86CdS (0001),87as well as Si with amorphous SiO 2ter- mination.88For Bi 2Se3epitaxial strain on these substrates ranges from 0.25% mismatch on InP, to values as large as 15% on α-Al2O3. Despite this, the electrical properties are not found to vary significantly, in that minimum values of the sheet carrier den-sities are typically found to be in the range of 1 –3×1 0 13cm−2and mobilities in the range of 1000 –2000 cm2V−1s−1; based on the separation of the Dirac point from the conduction band minimum in Bi 2Se3, a sheet carrier density below 1 × 1013cm−2is necessary, but not sufficient, for the Fermi level to be below the conductionband minimum. 54,76Furthermore, with cracked Se (or Te), similarJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210902 (2020); doi: 10.1063/5.0022948 128, 210902-6 Published under license by AIP Publishing.results are found.89–91Furthermore, thickness independence of the sheet carrier density highlights that the source of defects are mainly the interface. In fact, this can be further seen by comparing epitax-ial growth of Bi 2Se3on substrates with the same in-plane symmetry to films grown on Si with amorphous SiO 2. Despite the lack of in-plane crystal structure, the large crystalline anisotropy of layered Bi2Se3enables it to grow with grains that are all oriented with their caxes perpendicular to the substrate surface, yet with random in-plane orientations. Surprisingly, transport properties are nomi-nally the same as epitaxial films grown on other substrates withsheet carrier densities around 1 × 10 13cm−2and mobilities around 2000 cm2V−1s−1.88This again highlights the high density of defects imparted at the interfaces. We next discuss two key strat-egies that have proven successful for Fermi-level control to realizeintrinsic topological properties. Understanding the limitation that the majority of defects are confined to the interface motivated the development of an MBE-grown virtual substrate that is both epitaxially as well aschemically matched to the layered tetradymite TIs. 49The key to isolating transport through the TI was the material In 2Se3, which has a polymorph that is nearly identical to the structure of the tet- radymites, and is a trivial band insulator with a bandgap of about 1.4 eV.92,93There were, however, several key challenges, which necessitated a careful understanding of the materials scienceinvolved in the growth, which is detailed schematically in Fig. 2(c) . The first was that the In 2Se3system has many polymorphs; stabiliz- ing the correct α-phase is challenging, since growth directly on Al2O3nucleates multiple phases of In 2Se3. Stabilizing the correct phase required first growing a thin layer of Bi 2Se3, which then favors the nucleation of α-In2Se3. The problem is that when a sub- sequent layer of Bi 2Se3is grown on top of this Bi 2Se3/In2Se3bilayer, the bottom Bi 2Se3will create an electrical short. Interestingly, the high volatility of Bi 2Se3relative to In 2Se3combined with the large bulk diffusion coefficient of layered materials enable a novel solu-tion: Bi 2Se3in the Bi 2Se3/In2Se3bilayer can be sublimated out from underneath In 2Se3using a high temperature anneal step following the growth of In 2Se3. This leaves a very high-quality In 2Se3virtual substrate for the subsequent growth of Bi 2Se3. However, a secon- dary challenge emerged: The high bulk diffusion caused smallamounts of In 2Se3to diffuse into Bi 2Se3. This is problematic since low concentrations of In in Bi 2Se3can drive it into a topologically trivial state.94This was subsequently solved by slightly lowering the growth temperature and growing a second layer composed of(Bi 0.5In0.5)2Se3, which together lowered the In diffusion well-below the critical value, where the topological phase transition occurs. Growing Bi 2Se3on this virtual substrate lowered the sheet carrier density by an order of magnitude to the range of 1012cm−2and raised the mobility of the TSS to the highest reported value, 16 000cm 2V−1s−1. The carrier density was sufficiently low, and combined with the high mobility enabled observing the quantum Hall effect in the high-magnetic field limit, as well as novel quantized Faradayand Kerr rotation angles, 95which is a direct signature of axion elec- trodynamics.96,97Recently, the virtual substrate was improved by moving to In 2Se3/(In 0.35Sb0.65)2Te3with a subsequently grown Ti-doped Sb 2Te3. This lowered the defect density to the range of 1011cm−2, but the mobility was lower at around 3000 cm2V−1s−1; this ultra-low carrier density enabled reaching the extremequantum limit, where a novel insulating phase emerged when the external magnetic field drove a splitting of the zeroth Landau level associated with the TSSs.98 As a means of electrically controlling defects, compensation doping schemes have also seen success, yet have faced key chal-lenges relative to how dopants are incorporated into the layered tet- radymite structure. One of the first schemes successfully used, which has long been understood and utilized in the context of ther-moelectrics, was balancing acceptor and donor defects throughalloying. 52As discussed above, based on the ability to readily form both antisite defects as well as Te vacancies Bi 2Te3can be made either n-type or p-type depending on the growth conditions. Sb2Te3is only p-type due to the propensity to form antisite defects. Therefore, creating the mixture (Bi 1−xSbx)2Te3is a direct route to controllably tune between n-type and p-type. This saw initialsuccess in lowering the carrier density relative to pure materials down to a value of 2 × 10 12cm−2forx≈0.95, yet the mobility remained relatively low initially at around 500 cm2V−1s−1.99Key improvements were achieved in transport by gate-control of theFermi level, 100which enabled the observation of the quantum Hall effect.101,102As detailed in Sec. IV, this ability to carefully balance defect types was a critical driver of the success in observing the quantum anomalous Hall effect in MBE-grown (Bi 1−xSbx)2Te3 films with magnetic doping. B. Emerging directions for molecular beam epitaxy growth of topological materials To close this section, we wish to point to open questions and emerging directions. As the field of topological materials hasexpanded, so has the number of candidate or confirmed systems.As such, a key direction is to diversify the topological materials grown as high-quality thin films. Beyond the tetradymites, a few materials have been thoroughly explored via MBE growth. Primematerial classes where significant efforts have taken place are therocksalt class of topological crystalline insulators, particularly SnTe, 103–109and the 3D class of Dirac and Weyl semimetals, partic- ularly Cd 3As2110–114and Na 3Bi.115–118Interestingly, the only MBE-grown topological material that was not studied prior to con-firmation of its topological states was Na 3Bi; the other materials have extensive histories prior to the genesis of the field of topologi- cal materials. This highlights the challenge of developing new mate- rials by MBE and shows that researchers typically only attemptgrowth of materials that are reasonably likely to succeed. In comparison to bulk crystal synthesis, where a scientist can pursue multiple growths in parallel across many different chemis- tries during a timeframe of a few weeks, 62growth and refinement of a particular class of MBE-grown materials takes place acrossmany months to years. This is largely limited by the throughput ofthe MBE process as well as elemental cross-contamination, whereeach chamber is dedicated to one specific chemistry, e.g., a single chamber must be dedicated to, for example, either arsenic, phos- phorous, or selenium/tellurium. Coupling this with the facts that(1) MBE chambers are only opened several times or less per yearfor maintenance or to exchange or recharge elements and that (2) the cost of each chamber ranges from hundreds of thousands to millions of dollars, naturally makes this a methodical area ofJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210902 (2020); doi: 10.1063/5.0022948 128, 210902-7 Published under license by AIP Publishing.research. As such, choices regarding material selection and experi- mental design need to be made carefully, which highlights the benefit of close collaboration with researchers performing bulk syn-thesis as well as theorists. To clarify the thought process regardingwhat might make a good candidate, we list some key questions anMBE researcher asks themselves before pursuing new materials: 1.Safety: The first question should always be are the elements haz- ardous? Adding toxic elements to an MBE system means that the researcher will have to come into close contact with themduring any maintenance for the foreseeable life of the MBEchamber, hence toxic elements —Hg, Tl, As, Pb, Os, Be, etc., anything radioactive, anything pyrophoric —Na, K, Rb, white-P, etc., should be avoided if possible. Elements such as Cl, F, and S are also corrosive and may damage pumps and seals within theMBE system, which adds an additional layer of complexity.Here, we stress that a researcher must take the proper precau- tions when considering adding toxic elements to their MBE systems. Carefully read safety data sheets on individual elementsand possibly reacted compounds; as discussed by May et al., 62 Hawley ’s Condensed Chemical Dictionary is a great resource for understanding hazards.119Consider that the source material may form flakes, dust, or possibly vapor, and carefully plan how to mitigate exposure. Finally, when considering using a toxicelement seek advice from colleagues in the MBE communitywho have experience dealing with such elements, as well aspeople in related fields and especially the safety staff at your institutions. 2. Has it been synthesized as a bulk crystal or as a thin film, and is there a thermodynamic phase diagram? These together hedgethat attempts at synthesis will be favorable, while simultaneouslyproviding important information about possible growth condi- tions, possible impurity phases, and defect structures. As dis- cussed above, knowledge of the bulk phase diagram andsynthesis guided the MBE growth of the tetradymites as well asmotivating the charge balancing of (Bi 1−xSbx)2Te3. 3. What elements are involved, is there a chamber compatible with that particular chemistry (in a sense rhetorical here), and arethose elements amenable to MBE? Regarding the latter, thequestion comes down to how the atoms can be delivered to thesubstrate in a controllable way that imparts a low kinetic energy. For the use in an MBE process, the elements should have suffi- ciently high vapor pressure, 10 −5–10−4Torr, below a tempera- ture of about 2000 °C. Accessible elements are most alkalinemetals, alkaline earths, most of the d-block excluding the 4dZr-Ru, and 5d Hf-Pt, which are refractory and require an e-beam evaporator or some other mechanism, 120rare earths, and most p-block elements other than some of the halides. 4. Is there a commercially available substrate where a low-energy surface is a good match to the surface symmetry and latticeparameter of the desired material? Is it possible to achieve atom- ically clean surfaces in situ , and is it available with high-quality atomically flat surfaces, all while being cost effective? 5. What is the volatility of each element as well as their com- pounds? The fewer the elements the easier, since the deposition rate of each element has to be carefully calibrated and slight deviations can result in incorrect stoichiometries. In specificsituations for some binaries (for example, the tetradymites fit this, but also the archetype-MBE material GaAs) where oneelement is highly volatile (Se or Te), at a given growth tempera-ture, the growth can be entirely controlled by the flux of theother atoms (Bi or Sb), where the excess unbonded Se or Te simply evaporates away resulting in an adsorption controlled growth window. This is the best-case scenario, and many mate-rials do not follow this, with common examples being theternary perovskite oxides, e.g., SrTiO 3, where the Sr to Ti ratio has to be manually controlled, with a typical error being >1%. Furthermore, as shown in the x-ray diffraction 2 θ–θscans in Fig. 3 , attempts at MBE growth of Co 3Sn2Se2at 400 °C from the ternary chalcogenide family of magnetic Weyl phases121,122 showed that highly volatile SnSe 2forms, then evaporates, result- ing in Co 7Se8at typical vapor pressures of Se; this highlights a specific unforeseen difficulty that can arise when exploring new topological materials by MBE. 6. Is the resulting material air stable? Since thin films both have a large surface to volume ratio and the fact that the novel states of topological materials are often on the surfaces, chemical reac- tions with the air can be extraordinarily detrimental. Whereasmuch understanding can be gained using in situ probes, as highlighted in Fig. 1 , most high-magnetic-field magnetometers and transport systems, x-ray diffraction systems, and clean room processes, for example, are ex situ . Much effort has to be expended discovering and understanding air-stability problemsas well as searching for and discerning the effects of cappinglayers, as has been demonstrated for the tetradymites, which arefairly air stable compared to, for example, Na 3Bi. Based on these guidelines, there are many important areas in science and technology that can be addressed by undertaking and refining the synthesis of new topological materials by MBE. Thetetradymite TIs are by far the most well explored family of topolog-ical materials, yet there are still many open questions, particularly regarding heterostructures with superconductors and magnets, and the ternary compounds such as MnBi 2Te4. These are discussed in the following sections. Of particular interest are the nodal semime-tals for which there is very little studied outside of the excellentwork done on thin films of 3D Dirac semimetals Na 3Bi and Cd3As2. For example, many intermetallic compounds, particularly the Heuslers are widely unexplored,123,124as well as square-net compounds,125and work is just emerging regarding candidate materials and understanding the challenges related to the growth.Furthermore, higher-order topological phases 20represent an excit- ing area where the precise control of surface morphology and island formation available in MBE can be employed to realize novelstates that emerge at 1D and 0D boundaries of materials. Beyondthese specific examples, novel large scale numerical efforts havepresented a near endless number of compounds available to explore, 126many of which are great candidates for MBE growth. III. TOPOLOGICAL SUPERCONDUCTORS BY MBE A. Introduction to topological superconductivity Interplay of topology and superconductivity results in an emerging class of quantum matter aptly called topologicalJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210902 (2020); doi: 10.1063/5.0022948 128, 210902-8 Published under license by AIP Publishing.superconductivity. Topological superconducting phases relate the topologically non-trivial structure of the gapless boundary modes(Majorana modes) to the gapped bulk with a non-zero topologicalinvariant. 3,4,19One of the most intriguing features is the emergence of Majorana fermions —particles that are their own antiparticles — which emerge as quasiparticles in these condensed mattersystems. 127,128Majorana fermions come in different forms. In con- trast to most known particles and quasiparticles, Majorana fermionsexhibit novel non-Abelian exchange statistics. When two indistin- guishable particles are exchanged, fermions and Bosons accumulate a phase of −1 and 1, respectively, with their quantum states unchanged, and anyons which belong to the Abelian group, accumu-late a non-integer phase between −1 and 1. Upon exchange of multi- ple Majorana fermions, which are non-Abelian anyons, the quantum state evolves to a unique state that depends only on their exchange(or braiding) trajectories. Confirmation of the existence of Majoranafermions and proof of non-Abelian statistics have been vigorouslypursued over the past decade. Achieving these milestones is critical for potential applications in topological quantum computing since it directly enables an extremely long decoherence time, which has longbeen a key challenge in this field. 24,25 Topological superconductivity can be realized in a variety of MBE-grown materials systems. This can be either single layers of intrinsic topological superconductors or engineered heterointerfaces. The latter includes non-superconducting topological materials orsemiconductors, both with strong spin –orbit coupling, which are interfaced with s-wave superconductors. Synthesis of high-qualitytopological superconductors (intrinsic or engineered) as well as con- clusive experimental evidence of key features have both been major challenges. Topological superconducting phases naturally emerge inspinless p-wave superconductors with triplet pairing symmetry (p-wave pairing in 1D and p x±pypairing in 2D). As first proposed by Kitaev in a 1D tight-binding chain with p-wave pairing, discrete states at zero energy ( E= 0) are bound to the ends of the chainunder certain conditions,129as illustrated in Fig. 4(a) . Such states are Majorana zero modes and can be realized in 1D intrinsic topologicalsuperconductors as well as in 1D superconductor/TI 130and 1D superconductor/semiconductor hybrid systems.131Creating interfaces that are transparent to superconductivity is one of the key challenges, and, by improving the interface in hybrid systems, enhanced super-conducting and Majorana-zero-mode features have been observed.In 2D topological superconductors, gapless chiral Majorana edgemodes can propagate along the edges of 2D topological supercon- ductors and emerge on the boundaries of vortices with Majorana zero modes at their cores as shown in Fig. 4(b) . 127,132Theory pro- posed that propagating chiral Majorana edge modes can arise ininteger quantum Hall or quantum anomalous Hall insulators when coupled with an s-wave superconductor. 133,134However, experimen- tal evidence of the resulting half-quantized conductance plateaus insuch systems has been under debate. 135,136In the vortex cores of 2D topological superconductors, Majorana zero modes give rise to zero-bias conductance peaks in, for example, scanning tunneling spectro- scopy. 3D (time-reversal invariant) topological superconductors are the superconducting counterparts of 3D TIs. They host 2D gaplessMajorana fermion surface states (Majorana cones) residing withinthe superconducting gap [ Fig. 4(c) ]. 137The use of interconnected MBE-ARPES and MBE-STM systems is critical to reveal the signa- tures of 2D and 3D topological superconductivity. For example, gapless surface states and zero-bias conductance peaks have beenobserved within magnetic vortices on the surface of β-PdBi 2thin films grown by MBE.138,139In the remainder of this section, we will discuss intrinsic and engineered topological superconductors systems enabled by MBE and further challenges for advanced studies. B. Intrinsic topological superconductors by MBE Superconductivity can be induced in the tetradymite TIs by doping or intercalation, resulting in possible topological FIG. 3. X-ray diffraction result for the attempted growth of the candidate topological material Co 3Sn2Se2on Al 2O3(peaks indicated with *), which resulted in the formation of Co 7Se8(peaks indicated by open squares). As shown schematically on the right, this is due to the formation and subsequent sublimation of highly volatile SnS e2.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210902 (2020); doi: 10.1063/5.0022948 128, 210902-9 Published under license by AIP Publishing.superconductivity with strong spin –orbit coupling. This has been observed in bulk crystals of Bi 2Se3where dopant atoms were inter- calated between van der Waals layers: Cu xBi2Se3with Tc≈3.5 K,140 SrxBi2Se3with Tc≈3.0 K,141NbxBi2Se3with Tc≈3.4 K,142and TlxBi2Te3with Tc≈2.3 K.143Among these, Cu xBi2Se3has been grown by MBE and extensively studied. In bulk crystals, an interca- lated Cu atom is a strong electron donor, which raises the carrier density and thereby stabilizes the superconducting phase. In con-trast, when grown by MBE, Cu is an electron acceptor and stronglylowers the carrier density, which highlights the amphoteric behav-ior of Cu in Bi 2Se3.52When the films are thin, this, in fact, lowers the Fermi level sufficiently to isolate transport through the TSS, yet precludes superconductivity.76Cu-doped Bi 2Se3shows the com- plexity that emerges for defects and dopants in the extremely dif-ferent growth regimes of bulk crystals and MBE. This highlights critical challenges that need to be understood to integrate super- conductivity in known topological materials via doping. Sr 2RuO 4is one of the promising candidates for intrinsic topo- logical superconductivity. This material has the same crystal struc-ture as the layered copper oxide (cuprates) high-temperature superconductor (La,Sr) 2CuO 4. However, the superconducting tran- sition temperature ( Tc), near 1 K, in Sr 2RuO 4is nearly two orders of magnitude lower than that of the cuprates.144Furthermore, in contrast to the antiferromagnetic ordering in the cuprates, Sr 2RuO 4 has ferromagnetic ordering that led to the exciting theoretical pro- posal of spin-triplet (odd-parity) superconductivity when the spins align and break time-reversal symmetry.145There has been experi- mental evidence of possible chiral spin-triplet pairing, which is alsocalled chiral p-wave pairing. However, there are still unsolved issues and consensus on the nature of the chiral p-wave superconductivity in Sr 2RuO 4has not been reached.146–149Nevertheless, there are many interesting aspects and open questions regarding Sr 2RuO 4 which can be addressed from the perspective of MBE growth. Forexample, T cof bulk Sr 2RuO 4can be as high as 1.5 K and is highly sensitive to defects, stoichiometry, and strain; MBE has good control over these aspects which has led to the growth of purecrystalline thin films by oxide-MBE.150Yet, not all Sr 2RuO 4thin films exhibit superconductivity due to the highly sensitive nature to defects. High-quality films can be synthesized by MBE with anadsorption-controlled growth window, where the Sr to Ru ratio iscontrolled by the formation and desorption of excess volatile Ru-Omolecules. This results in either SrRuO 3or Sr 2RuO 4depending on the temperature and flux ratio. These adsorption-controlled MBE films have shown superconductivity151–153with enhanced Tcup to 1.8 K when strained precisely on NdGaO 3(110).153Furthermore, epitaxial strain in Sr 2RuO 4can manipulate the Fermi surface topol- ogy.154Finally, Sr 2RuO 4is proposed to be classified as a topological crystalline superconductor where symmetry-protected Majorana fermions reside.155Experimentally, half-quantum vortices that are supported by spin-triplet paring was observed in Sr 2RuO 4,156 which can host Majorana zero modes in the vortex cores. However, direct evidence of Majorana bound states is lacking, which high- lights that there are many open questions regarding this noveltopological material. Advances in material synthesis and fabrication methods have allowed rapid development of highly crystalline 2D superconduc- tors prepared by van der Waals epitaxy and mechanical exfolia- tion. 157Among the 2D superconductors, only a few turned out to host intrinsically non-trivial topological band structures, includingFe-based high- T csuperconductors, transition metal dichalcogenide PdTe 2, and β-Bi2Pd. The superconducting transition temperature of bulk FeSe is 8 K. Astonishingly, however, this can be significantly enhanced byreducing film thickness down to a monolayer on the SrTiO 3 surface41as well as by electron doping the FeSe layer.42Interfacial electron –phonon interactions between the FeSe electrons and SrTiO 3phonons are thought to be a key ingredient to stabilizing the high- Tcsuperconductivity.158Furthermore, the interfacial atomic structure has to be carefully engineered by MBE growththrough precise control of substrate treatment, stoichiometry, growth mode, and post-annealing —all of which strongly affect reproducibility. 41Although, Tcof a monolayer FeSe film on SrTiO 3 FIG. 4. Schematics of Majorana modes that appear in 1D, 2D, and 3D topological superconductors (TSC). (a) In a 1D p-wave superconducting chain consisting of N sites, two Majorana operators, indexed as Aand B, are paired on each site, when in a non-topological phase. When the chain is in a topological phase, Majorana opera- tors on neighboring sites are combined, leaving two unpaired Majorana operators at the ends of the 1D chain. (b) In 2D topological superconductors, ch iral Majorana edge modes appear along the edges as well as on the boundaries of vortices. Majorana zero modes emerge in the vortex cores. (c) 3D topological superconducto rs exhibit Majorana fermion states on the surface.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210902 (2020); doi: 10.1063/5.0022948 128, 210902-10 Published under license by AIP Publishing.is reported as high as 100 K,159it has been very controversial because it strongly depends on the quality of the interface between FeSe and SrTiO 3and on the specific probes used to characterize the superconductivity. Monolayer FeSe films are highly sensitive toair exposure so that a protective capping layer is required forex-situ transport. However, the capping layer affects the FeSe film properties with T creduced by 10 s of Kelvin in comparison to Tc measured by in situ surface-sensitive techniques such as STM and ARPES. The challenge of probing highly unstable interfacial super-conductivity by a variety of techniques highlights the importance ofmulti-connected UHV systems equipped with in situ transport.Theory proposed topologically non-trivial states in band structures of FeSe and Fe(Te,Se) thin films. 160,161Integration of high Tcsuper- conductivity and non-trivial topology makes these materials uniqueand attractive. Signatures of Majorana bound states localized in thevortex cores have been demonstrated in Fe(Te,Se) on SrTiO 3,b y using interconnected MBE and low-temperature STM systems as shown in Figs. 5(a) and5(b).162,163Due to the high transition tem- perature, Fe-based high- Tcsuperconductors may serve as material platforms for a practical topological quantum computer operatingnear liquid nitrogen temperature. Challenges still remain in clarify-ing mechanisms of the high T csuperconductivity in Fe-based FIG. 5. Zero-bias conductance map (15 × 15 nm2) and dI/ dVspectra by STM on Fe(T e,Se) (a) and (b) and on β-Bi2Pd (c) –(e). (a) Zero-bias conductance map of a vortex on an Fe(T e,Se) surface with a perpendicular mag- netic field of 0.5 T . (b) Overlapping dI/dVspectra with increasing distance away from the center. The peak ismaximum at the vortex core center and decreases radiallyoutward. (c) STM tomography (400 × 400 nm 2)o fa2 0 unit-cell thick β-Bi2Pd film. The bright spots and lines cor- respond to Bi adatoms and anti-phase domain boundar-ies, respectively. (d) Normalized zero-bias conductancemap (350 × 350 nm 2) at 0.1 T . (e) Normalized dI/dV spectra measured from the center of a vortex core (top) to 40 nm away from the center (bottom). Reproduced withpermission from Lv et al. , Sci. Bull. 62, 852 (2017). Copyright 2017 Elsevier; Wang et al. , Science 362, 333 (2018). Copyright 2018 The American Association for the Advancement of Science.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210902 (2020); doi: 10.1063/5.0022948 128, 210902-11 Published under license by AIP Publishing.superconductors and in designing routes to use the Majorana bound states for a qubit. The 2D material PdTe 2is classified as a type-II Dirac semime- tal.164,165Interplay between the non-trivial topological nature of the electronic bands and superconductivity makes PdTe 2a promising candidate for a 2D topological superconductor. In contrast to interface-enhanced superconductivity in monolayer FeSe/SrTiO 3, monolayer PdTe 2is a narrow bandgap semiconductor and is not superconducting, whereas multilayer PdTe 2is metallic and super- conducting with Tc≈1.6 K.166,167A careful study of the thickness- dependent electronic properties of PdTe 2requires precise layer-by-layer growth control by MBE and in situ characterization techniques such as STM and ARPES. Superconductivity in PdTe 2is found to be insensitive to the substrate, which reflects the van derWaals epitaxy. Thin films of PdTe 2have been epitaxially grown on various substrates such as SrTiO 3(001), graphene-SiC (0001), and Bi2Te3.166–168Experimental studies revealed both type I and type II superconductivity in PdTe 2.166,169However, experimental demon- stration of signatures of topological superconductivity in PdTe 2 remains an immediate challenge. The layered material β-Bi2Pd is another 2D superconductor with Tc≈5.4 K. β-Bi2Pd exhibits several TSSs that have been experimentally confirmed by spin-resolved ARPES measurementson bulk crystals, and theoretically categorized to belong to the Z 2 topological class, analogous to strong TIs such as Bi 2Se3.170Thin films of β-Bi2Pd can be grown by MBE with different growth modes at different growth temperatures. The sticking coefficientof Bi is small at a substrate temperature above 200 °C, at whichsynthesis of β-Bi 2Pd required Bi-rich conditions (Bi/Pd flux ratio > 3). MBE growth of β-Bi2Pd films on SrTiO 3(001) at tem- peratures between 300 and 350 °C proceeds in the Volmer –Weber growth mode, where epitaxial islands with their thickness downto a single unit cell form on the substrate with various lateral sizescontrolled by the Bi/Pd flux ratio. 139In contrast, the sticking coef- ficient of Bi becomes comparable to that of Pd at room tempera- ture, and β-Bi2Pd thin film growth with the Bi to Pd flux ratio tuned to 2:1 proceeds in layer-by-layer fashion.171In addition to the typical bulk superconducting features, zero-bias conductancepeaks, possibly from Majorana zero modes, were observed in vor-tices of a β-Bi 2Pd film by in situ scanning tunneling spectroscopy as shown in Figs. 5(c) –5(e).139Furthermore, unique features of the superconducting state in β-Bi2Pd have been observed: First, the TSS superconducting gap is anomalously larger than the bulksuperconducting gap, 138which is unlike most topological super- conductors where the bulk superconducting gap is larger than the topological superconducting gap. The large superconducting gapobserved in TSSs may result in more stable Majorana zero modesthat persist to higher temperatures. Second, possible transport evi-dence for time-reversal-invariant topological superconductivity and Majorana surface states on a 3D topological superconductor have been recently demonstrated in bulk crystals doped by K. 172 Finally, Little –Parks devices showed the magnetic flux was half- quantized, which further indicated the unconventional nature ofthe superconductivity. 173The study of superconductivity in pris- tine and doped β-Bi2Pd films may enable the realization of the trivial-to-topological quantum phase transition in superconduct-ing states.IV. ARTIFICIALLY ENGINEERED TOPOLOGICAL SUPERCONDUCTORS Topological superconducting states can be artificially engi- neered by proximity-induced superconductivity in topological materials. When a 3D TI is interfaced with an s-wave superconduc- tor, effective spinless p-wave superconductivity emerges in the 2DTSSs, which can exhibit Majorana bound states in artificial vorti- ces. 174For 2D TIs, spin-filtered edge states proximitized by an s-wave superconductor can become 1D topological superconduc-tors that resemble Kitaev ’s model for a 1D spinless p-wave super- conductor. 1291D nanowires of 3D TIs and topological crystalline insulators can host topological superconducting states when in contact with an s-wave superconductor.130These engineered topo- logical superconducting states based on TIs and topological crystal-line insulators can be extended to other topological materials as well. Superconductors can be ex situ processed by e-beam evapora- tion, sputtering, and focused ion beam induced deposition to build superconductor-topological material hybrid systems. Ex situ super- conductors are critical to devise such hybrid systems using topolog- ical nanostructures 175in the forms of flakes, plates, ribbons, belts, and wires, either by mechanical exfoliation from bulk crystals, or by direct synthesis by vapor-liquid-solid (VLS) or chemical vapor deposition (CVD). The VLS growth methods typically use a metal (Au) particle catalyst that becomes liquid and collect source mate- rial in vapor phase. Above the solubility limit, the supersaturatedsource material starts to form a crystalline solid in the layer-by-layer fashion at the liquid/substrate interface. In the CVD growth method, the source material in the vapor phase directlydeposits on the substrate to form nanostructures or a film. To achieve a clean interface between a superconductor and a topologi- cal material processed ex situ , proper native-oxide removal is required via ion milling or chemical wet etching before the super- conductor deposition. However, precise control over the native- oxide removal is very challenging, and the resulting interface typi- cally contains defects and disorder as a result of the damage to the surface of the topological material as well as from the amorphousor the multi-grained nature of the polycrystalline superconductor. The defective interface can cause unwanted subgap states and sup- press features of topological superconductivity. With the atomicprecision of MBE layer-by-layer growth, thin-film-based superconductor-topological material hybrid systems can be synthe- sized, which is advantageous for further device development.Transparent interfaces with minimal defects can be obtained either byin situ MBE growth of both the topological material then the superconductor or by MBE growth of topological materials on aproperly surface-treated/desorbed ex situ superconductor, such as Bi 2Se3on NbSe 2(Fig. 6 ).176,177In contrast to the case of deposition and lift-off of patterned superconducting electrodes by ex situ fabri- cation, it is challenging to pattern the superconductor layer in MBE-grown superconductor-topological material hybrid systems.For example, when patterning a superconductor on top of a topo- logical material, selective etching of the superconductor is hard to achieve, and the remaining bare surface of the topological materialis likely to be damaged by the etching process. One solution to form clean junctions is to add a stencil/shadow mask for in situJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210902 (2020); doi: 10.1063/5.0022948 128, 210902-12 Published under license by AIP Publishing.superconductor deposition. A recent study demonstrated full in situlithography of superconductor-TI hybrid devices via MBE by using two monolithically integrated hardmasks for selective-areagrowth (SAG) of (Bi 1−xSbx)2Te3TI thin films and stencil lithogra- phy of a superconductor.178 Superconducting features induced in the topological states have been experimentally demonstrated in various superconductor-topological material systems. Josephson effects in superconductor-TIhybrid Josephson junctions have shown induced superconducting features linked to the TSSs. 179–182Furthermore, signatures of Majorana bound states, such as the fractional Josephson effect char-acterized by 4 πperiodic current-phase relation and suppression of odd Shapiro steps mediated by the 2D TI HgTe, 183tetradymite 3D TIs,178,184,185and Dirac semimetals Bi 1−xSbx186and Ca 3As2187as well as zero-bias conductance peaks in vortices on Bi 2Te3grown on NbSe 2[Figs. 6(c) and6(d)],188have been reported. Similarly, topological superconducting states can be artificially engineered by proximity-induced superconductivity in semicon-ductors with strong spin –orbit coupling. The spin-degenerate para- bolic band structure of a 1D semiconductor splits its up-spin and down-spin bands due to spin –orbit coupling. When an external magnetic field is applied, the associated Zeeman splitting opens anenergy gap at E= 0. At small energies within the gap, the spin- textured band structure can be considered effectively spinless. Interfacing an s-wave superconductor induces effective spinlessp-wave superconductivity in the semiconductor, and Majorana zero modes emerge at the ends of the 1D superconductor-semiconductor wire. 131,189Rashba spin –orbit-coupled semiconduc- tors of InAs and InSb nanowires are a natural choice to realize the engineered 1D topological superconductor. These 1D nanowire- based superconductor-semiconductor systems have been exten-sively studied and have demonstrated zero-bias conductancepeaks as possible evidence of Majorana zero modes. 190,191MBE is viable to grow InAs and InSb nanowires by the VLS growth method. Among these, MBE-grown InAs nanowires have been widely used. However, MBE growth of InSb nanowires is chal-lenging due to the narrow growth window. In contrast, InSbnanowires grown by metal-organic vapor phase epitaxy (MOVPE)are more accessible and have been widely used for Majorana physics studies. As in most of the other topological material systems, advances in 1D topological superconductors have beenmade possible by improvements in materials. Development of anepitaxial superconductor (Al) by MBE on InAs and InSb nano-wires 192,193has resulted in a highly transparent interface, which exhibited features of induced superconductivity and topological superconductivity, as shown in Fig. 7(a) .194–196To achieve clean Al-InSb junctions, MOVPE InSb nanowires were ex situ transferred to a multi-chamber MBE system, cleaned by atomic-hydrogen in UHV, and coated by epitaxial Al in situ by MBE with shadowing by adjacent nanowires.193 FIG. 6. (a) STM tomography of Bi 2Se3QLs grown on NbSe 2. (b) Schematic showing the layer-by-layer growth mode of Bi 2Se3along the red dashed line in (a). (c) Normalized zero-bias conductance map of a vortex coreon 5 QL Bi 2Te3/NbSe 2under 0.1 T at 0.4 K. (d) A series ofdI/dVspectra measured at locations along the black dashed arrow in (c). The zero-bias conductance peak bifurcates away from the vortex core. Reproduced withpermission from Wang et al. , Science 336, 52 (2012). Copyright 2012 The American Association for the Advancement of Science; Xu et al. , Phys. Rev. Lett. 114, 017001 (2015). Copyright 2015 The American PhysicalSociety.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210902 (2020); doi: 10.1063/5.0022948 128, 210902-13 Published under license by AIP Publishing.Toward the application of quantum computing based on superconductor-semiconductor systems, it is crucial to develop scalable platforms that can construct networks of 1D channels in a controllable manner. A 2D electron gas (2DEG) made of semicon-ductor heterostructures grown by MBE can provide high-mobilitysemiconductor channels and can be used, when interfaced with an s-wave superconductor, as a scalable platform for topological qubits consisting of multiple Majorana zero-modes. InAs and InSbquantum wells and InAs/GaSb double quantum wells have beeninvestigated for this purpose. 197–205For a transparent superconduc- tor–2DEG interface, a 2DEG needs to be placed close to the surface so that the wavefunctions of the superconductor and 2DEG overlap. If the quantum well top barrier between the superconduc-tor and 2DEG is very thin (none or just a few monolayers), strongcoupling between the two can be achieved, and, in return, the elec-tron mobility of the 2DEG is reduced due to a significant contribu- tion of remote ionized impurity scattering at the interface. With thicker top barriers, the electron mobility of the 2DEG improves,but the coupling between the superconductor and 2DEG is simi-larly reduced. An optimum thickness for the top barrier, whichmaximizes the mobility of the 2DEG, while maintaining a strong coupling with a superconductor is found to be around 10 nm in epitaxial Al-InAs 2DEG systems. In addition to the remote ionizedimpurity scattering at the surface, there are other scattering mecha-nisms that contribute to the electrical properties of a 2DEG. In comparison to InAs 2DEGs grown on InP and GaAs substrates, InAs 2DEGs grown on GaSb have a smaller lattice mismatch, and,thus, significantly reduced misfit dislocations and the related scat- tering. Nearly lattice-matched barriers of Al 1−xGaxSb can be used for the InAs 2DEGs grown on GaSb substrates, and smoother surface morphology and higher electron mobility were observed.206 The material choice for the quantum well barrier and capping layerdetermines the conduction band profile, types of defects forming in the barrier and at the interfaces and surface, and Fermi level pinning at the surface. For example, a near-surface InAs 2DEGwith 10 nm-thick AlGaSb top barrier and InGaAs capping (3 nm),grown on the GaSb substrate, showed an electron mobility morethan an order of magnitude higher than that of InAs 2DEGs with a 10-nm-thick InGaAs barrier, grown on InP. 207By etching Al into a narrow wire geometry and applying a global top gate to depleteelectrons in the outer areas, a 1D superconductor –semiconductor quantum wire can be effectively formed. Tunneling spectroscopy atthe end of the quantum wire has revealed signatures of Majorana zero modes, 208as shown in Fig. 7(b) . With a motivation for advanced quantum devices and a scal- able quantum computer, SAG of in-plane 1D wires of InAs andInSb 50,209–214by MBE, MOVPE, and chemical beam epitaxy has been investigated with a subsequent in situ epitaxial superconduc- tor. Advantages of SAG are scalability and flexibility in designing nanowire networks as well as reduced post-fabrication stepsthrough pre-fabrication of nanowire patterns on SiO xor SiN x dielectrics before growth, which could minimize defects introduced during device processing. There are certain devices that can be cleanly prepared by SAG. For example, three-terminal FIG. 7. Superconductor –semiconductor Majorana devices and signatures of Majorana zero modes (zero-bias conductance peaks) in (a) 1D nanowire-, (b) 2DEG-, and (c) SAG wire-based systems with epitaxial Al. (a) In 1D Al-InSb nanowire, a normal-superconductor (NS) junction is formed at the end of Al-covered InSb for tunneling spe c- troscopy of the Majorana zero mode. dI/dVconductance as a function of bias voltage and in-plane magnetic field revealed zero-bias conductance peaks, as shown in line- cuts, above a certain magnetic field ( ∼0.25 T). (b) Effectively identical quantum wire geometry for tunneling spectroscopy can be formed by local Al etching and global top gating on Al-InAs 2DEG. The corresponding conductance plot shows zero-bias conductance peaks above ∼2 T . (c) Three-terminal device (two NS junctions and one Al lead) fabricated using Al-InAs SAG wire, where Al covers only one facet on the triangular InAs SAG wire. Correlated splitting of the zero-bias peaks is seen in both the local ( gLL) and non-local conductance ( gRL). Reproduced with permission from De Moor et al , New J. Phys. 20, 103049 (2018), licensed under a Creative Commons Attribution (CC BY) license; Nichele et al. , Phys. Rev. Lett. 119, 136803 (2017). Copyright 2017 The American Physical Society; Ménard et al. , Phys. Rev. Lett. 124, 036802 (2020). Copyright 2020 The American Physical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210902 (2020); doi: 10.1063/5.0022948 128, 210902-14 Published under license by AIP Publishing.superconductor-semiconductor hybrid devices consisting of two normal leads at the ends of a nanowire and one in situ supercon- ductor lead in the middle of the nanowire can be prepared to inves-tigate the correlation of end-to-end subgap states and the splittingof zero-bias conductance peaks, as shown in Fig. 7(c) . 215,216In comparison to InAs and InSb nanowires by the VLS method, SAG wires contain more sources of defects and likely have lower electron mobility. SAG wires are directly grown on substrates with a signifi-cant, in most cases, lattice mismatch, resulting in misfit dislocationson the substrate surface and stacking faults along the {111} planesin the InAs or InSb wires. 50In addition, pre-fabrication processing including surface cleaning on the substrate can damage the surface and introduce disorder when growing SAG wires. Incompleteetching of nanowire patterns on SiO xor SiN xmay leave residues that inhibit semiconductor growth, and, thus, cause voids in thinnanowires or formation of grain boundaries in thick nanowires. An immediate challenge in SAG is to reduce the defects and disorder for better electron transport properties. The pursuit of experimental confirmation of Majorana fermions and non-Abelian statistics will continue through using intrinsic andengineered topological superconductors. Demonstration of non-Abelian statistics based on multiple Majorana bound states is directly related to the realization of a topological qubit. Due to theproposed fault-tolerant nature of topological qubits, confirming theseexperimental milestones will represent a great step forward in the fields of condensed matter physics as well as quantum information science. Toward this goal, foundational theoretical and experimentalwork has begun: creation/manipulation of Majorana zero modes atthe ends of 1D topological superconductors as well as in vortex coresin 2D and 3D topological superconductors, fractional Josephson effects mediated by topological materials, and chiral Majorana edge modes in quantum anomalous Hall insulators proximitized by asuperconductor. Up to the present time, robust signatures ofMajorana zero modes in superconductor –semiconductor hybrid systems have been evidenced by tunneling conductance measure- ments and Coulomb blockade measurements. Based on these achievements, a topological qubit may be realized in the near futureby using semiconductor nanowires grown by the VLS and SAGmethods and semiconductor 2DEGs. To obtain the theoretically pro-posed benefits of the topological protection and extremely long deco- herence time in a qubit, key challenges remain regarding improving materials, such as reducing defects in the semiconductor platformsand creating superconducting islands with clean superconductor-semiconductor junctions. Once a topological qubit is achieved, over- coming scalability and reproducibility across multiple devices will be critical to the development of multi-qubit systems. However, as withother industrial scale devices enabled by MBE, 217–219platforms such as semiconductor SAG wires and 2DEGs will be at the forefront ofquantum devices over the coming decade. V. MAGNETIC TOPOLOGICAL PHENOMENA AND TOPOLOGICAL SPINTRONICS A. Merging magnetism and topology at the atomic scale Ferromagnetism, where all spins in a material coherently align along the same direction, fundamentally breaks time-reversalsymmetry. 220,221Since degeneracies among energy levels are always protected by a symmetry, breaking these is a means for controlling the band structure of a material. This naturally leads to mergingmagnetism with topological materials as a route to control the elec-tronic character of the bulk and the boundary states. This is mostclearly demonstrated for the quantum anomalous Hall effect. 222,223 For the case of TIs, the bulk conduction and valence bands are inverted by strong spin –orbit coupling, which obey time-reversal symmetry; the corresponding degeneracy at the 2D Dirac point onthe surfaces for a 3D TI and edges for a 2D TI are protected by thissymmetry, in that they must exist so long as this symmetry is intact. Introducing time-reversal symmetry breaking in the form of ferromagnetism aligned perpendicular to the surface can break thisdegeneracy in a topologically non-trivial way, as shown in Fig. 2(a) . Take the case of the 3D TI: the resulting energy gap in the surfacestate enables the system to be described by another topological invariant, the Chern number. This places these materials in the same topological class as the quantum Hall states, and thus exhibit-ing chiral 1D quantized edge states, yet at a zero external magneticfield. This is called the quantum anomalous Hall effect, which wasfirst envisioned in Haldane ’s graphene model with a magnetic field woven into the unit cell. 224It took around a quarter of a century for experimental verification in MBE-grown tetradymite films thatwere carefully charge balanced and doped by transition metals tointroduce magnetism. 225Beyond the quantum anomalous Hall effect, a rich array of novel phenomena and possible technological applications emerge when magnetism and topological materials aremerged. As highlighted here, understanding and utilizing MBEgrowth is the beginning step to achieve these goals. As discussed in Sec. II, understanding the MBE growth from the perspective of defect formation and how it translates into changes of the topological properties was the key enabler for reali-zations of the quantized Hall effect. It was first predicted theoreti-cally that Bi 2Se3,B i 2Te3, and Sb 2Te3would become ferromagnetic when doped by the cations Cr or Fe.226When made thin, this mag- netic ground state would open a gap at the Dirac point which would give rise to a non-zero Chern number, thus hosting thequantum anomalous Hall state. Experimentally, this requireddoping the materials to stabilize a magnetic phase, while not reduc-ing spin –orbit coupling such that the topological properties are not endangered, i.e., inadvertent reduction of spin –orbit coupling can drive the system into a topologically trivial phase. Furthermore, thisrequired simultaneous Fermi level control such that it can be tunedinto the magnetic exchange-gap at the Dirac point. Each of these aspects has proven a distinct challenge. First, Bi 2Se3has the largest bulk bandgap, and the Dirac point is well-separated from the bulk bands. However, it is stronglyn-type, and it was only recent that thin films could be made withsufficiently low carrier density to place the Fermi level close to the Dirac point. Furthermore, doping transition metals simultaneously failed to stabilize ferromagnetism, raising the Fermi level deep intothe bulk conduction band, while weakening spin –orbit coupling to a point where the materials become non-topological. 227These aspects were overcome by understanding subtleties related to the origins of both ferromagnetism and the primary sources of charged defects in Cr-doped Bi 2Se3.228Doping Cr into Bi 2Se3raises the Fermi energy and simultaneously the Hall resistivity showsJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210902 (2020); doi: 10.1063/5.0022948 128, 210902-15 Published under license by AIP Publishing.signatures of paramagnetism emerging due to the net moments of the Cr. However, ferromagnetism does not emerge, which is likely related to raising the Fermi energy and suppressing the spin –orbit coupling strength, and, thereby, reducing the net ferromagneticinteraction. These challenges were overcome by utilizing the virtualsubstrate method to lower interfacial defects (see Sec. II) combined with Ca-doping, and adapting an interfacial strategy where mono- layers of (Cr 0.5Bi0.5)2Se3were grown at the interfaces of Bi 2Se3.229 These remote layers utilized the long-range magnetic interaction toamplify the ferromagnetic coupling, while leveraging the shortrange nature of spin –orbit coupling to protect band-inversion in the Bi 2Se3layer. Combined with lowering the global Fermi level, this stabilized a ferromagnetic phase in Bi 2Se3. This strategy high- lights a multifaceted origin for magnetism in the tetradymites,which occurs when J eff2-χL−1χe−1> 0 with Jeffbeing the effective exchange coupling and χLandχebeing the magnetic susceptibil- ities of the local moments and free electrons, respectively. For topo- logical materials χeshould be large due to the strong spin –orbit coupling through the van Vleck mechanism.226,230However, inter- facially induced magnetism indicates that ferromagnetism emergesdue to a combination of enhancing J effthrough the long range Ruderman –Kittel –Kasuya –Yosida interaction of the conduction electrons as well as maintaining a large spin –orbit coupling by keeping the bulk of the film undoped. Finally, although thequantum anomalous Hall effect was not observed in these films, a novel aspect was observed for the first time. The sign of the anoma- lous Hall effect is reversed in Bi 2Se3relative to all other magnetic topological materials, which is an important aspect for engineeringthe Berry phase contribution to the Hall effect and ultimately thequantum anomalous Hall effect. 226,230 Given the challenges with Bi 2Se3, early work toward realizing the quantum anomalous Hall effect focused on magneticallydoping Bi 2Te3and Sb 2Te3. The challenges with making Bi 2Se3mag- netic are less obtrusive in the telluride system since spin –orbit cou- pling is stronger, and, therefore, more robust to the weakening associated with doping with transition metals necessary for magne- tism. This has resulted in observing ferromagnetism in the purecompounds when doped by V, Cr, or Mn. 231–235Furthermore, the intrinsic ambipolar defect chemistry in the tellurides, discussed inSec. II, gives a chemical route to control the Fermi level. MBE growth of Cr 0.22(BixSb1−x)1.78Te3showed that with magnetic doping, the carrier type can be tuned continuously from p-to-ntype by increasing Bi content ( x), which maintains the ferromag- netic state. 233Furthermore, as the charge compensation point was approached, the Hall resistivity increased and saturated near about 3kΩ, about 1 tenth of the resistance quanta h/e2≈25 kΩ. This pro- vided an early hint that accessing the quantum anomalous Halleffect was feasible. However, this was limited by the ability tocontrol the defect chemistry of these tetradymite films, but the carrier density was sufficiently low to enable fine tuning to be done with an external gate voltage. SrTiO 3is intrinsically close to a ferro- electric instability, thus possessing a huge dielectric constant at lowtemperatures, ϵ≈20 000 at around 4 K. 236,237Experimentally, this enables microfabrication to be skipped as the gate voltage can be applied directly through the SrTiO 3substrates, even with millimeter-scale thickness. This enables controlling the number ofcarriers into the range of 10 13cm−2,238which is well above thedensity of carriers in MBE-grown tetradymite films.83Switching to SrTiO 3(111) substrates and growing a 5 QL Cr 0.15(Bi0.1Sb0.9)1.85Te3 film enabled the first observation of the quantum anomalous Hall effect, where the Hall resistance becomes quantized in units of e2/h, and the longitudinal resistance drops to zero, as shown in Fig. 8 .225 Despite the Curie temperature of Cr 0.15(Bi0.1Sb0.9)1.85Te3being about 10 K, the quantized phase only appeared when the tempera- ture was reduced by two orders of magnitude to less than∼100 mK. At 30 mK, the Hall conductivity was quantized at about 0.987 e 2/hat a maximum magnetic field of 18 T. Further work on V-doping showed improvements to the quantization at around 0.9998 e2/h.239This points to a complexity that arises in the band structure.239For the (Bi 1−xSbx)2Te3system the Dirac point is near the top of the valence band, which has a local maximum away FIG. 8. First observation of the quantum anomalous Hall effect on an MBE-grown 5 QL Cr 0.15(Bi0.1Sb0.9)1.85Te3film grown on SrTiO 3(111) and mea- sured at 30 mK. When the back-gate voltage is tuned the Hall resistance is quantized at high magnetic field in unit of h/e2(a) and the resistance vanishes (b). Reproduced with permission from Chang et al. , Science 340, 167 (2013). Copyright 2013 The American Association for the Advancement of Science.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210902 (2020); doi: 10.1063/5.0022948 128, 210902-16 Published under license by AIP Publishing.from the Brillouin zone center. Therefore, when the Fermi level is positioned within the exchange gap, there should necessarily be parallel bulk conduction which shorts the quantized edge transport.Ironically, the high levels of disorder may rescue the system fromthis non-ideality, in that they may localize the parallel bulk states atvery low temperatures due to Anderson localization. Moving away from homogeneous single-layer materials to heterostructures have enabled raising the transition temperaturessubstantially. Mogi et al started from a single-layer 8 QLCr 0.10(Bi0.22Sb0.78)1.90Te3which exhibited near-quantization (80%) at 0.5 K and full quantization at 50 mK.229Like the Bi 2Se3dis- cussed above which this work inspired, heavy Cr-doped layers [Cr 0.46(Bi0.22Sb0.78)1.54Te3] at the top and bottom layers of (Bi0.22Sb0.78)2Te3substantially increased the onset of the quantum anomalous Hall phase to better than 90% quantization at 0.5 K.Extending this again to a pentalayer structure, consisting of a tri- layer with the additional layers of (Bi 0.22Sb0.78)2Te3below and on top of the heavily Cr-doped layers further increased the quantiza-tion to near 100% for temperatures near 2 K. The rationale for thissuccess of this layering scheme likely was the larger Cr-dopingachievable with the spatial confinement in the pentalayer geometry, which could enhance the exchange gap, reduce sample inhomoge- neity, or both. Whereas early hints of the quantum anomalous Hall effect were observed in exfoliated bulk crystals, 240there are many exam- ples of novel phenomena that highlight the importance and unique ability of MBE to engineer materials at the atomic level. First is theenhanced transition temperature using the pentalayer geometry.Beyond this, the atomic scale control of MBE can enable indepen-dently tuning the magnetism at the top and bottom surfaces. For example, in the trilayer Cr-(Bi 1−xSbx)2Te3/(Bi 1−xSbx)2Te3/ V-(Bi 1−xSbx)2Te3heterostructure, the Cr-doped layer acts as a soft- ferromagnet and the V-doped layer a hard ferromagnet. As such,when the magnetic field is swept in a loop, the longitudinal resist-ance and Hall resistance are expected to exhibit a structure with multiple coercive fields where the top and bottom layers switch independently. When the magnetizations are antiparallel, thesystem formally obeys time-reversal symmetry. In this regime, azero-resistance Hall plateau emerges, and the resistance exhibits atremendous increase. 241,242This behavior is indicative of an axion insulating state where the surface states are all gapped. In a separate system, when the magnetic layers are spatially symmetric,Cr-(Bi 1−xSbx)2Te3/(Bi 1−xSbx)2Te3/Cr-(Bi 1−xSbx)2Te3the Hall effect exhibits an additional feature. When the system is in the QAH state and the magnetic field is swept close to the coercive field an addi- tional hump-like feature emerges.243This can be associated with the topological Hall effect, which emerges due to chiral spin tex-tures that arise as the system changes its magnetization. Achievingthis in a topological heterostructure requires global breaking of time-reversal symmetry, careful control over the Fermi level such that its in the exchange gap, and significant Dzyaloshinskii –Moriya (DM) interaction resulting from broken inversion symmetry andstrong spin –orbit coupling. Although the system is formally inver- sion symmetric, the topological Hall effect only emerges at large bottom-gate voltages. This likely points to the non-zero DM emerg- ing when there is a net electric field that breaks inversion symme-try. These examples together highlight how the precise control ofsynthesis enabled by MBE can drive discovery as well as shed light on the fundamental origins of emergent phases of topological matter. Going forward, many questions regarding magnetic topolog- ical materials remain open. First, what is the maximum tempera-ture the quantum anomalous Hall effect can be observed? Can this be as high as room temperature? This latter question seems tantalizingly plausible given that the quantum Hall effect can beobserved in graphene as high as room temperature. 244This was made possible by the large room temperature mobility and apply-ing a sufficiently strong magnetic field (45 T) such that the cyclo- tron energy is of order the temperature. 244Analogously, the cyclotron energy should be replaced by the exchange gap energy,Δ, for the quantum anomalous Hall effect. As shown in Fig. 2(a) , the exchange gap opened within the 2D surface band is spannedby the 1D chiral edge mode. Therefore, so long as the temperature is sufficiently small relative to Δ, then the quantized edge conduc- tion should be measurable. Temperature bounds can arise forseveral reasons. The first is that, obviously, the temperature needsto be below the Curie temperature; for many materials that canbe integrated with topological materials as heterostructures the Curie temperature can be much higher than room temperature. The second condition that determines the upper limit is thethermal activation of carriers across the exchange gap in the 2Dtopological surface bands (see Ref. 55). Considering a mobility around 2000 cm 2V−1s−1yields a resistance of 10 × h/e2for a carrier concentration of 1 × 1010cm−2, which corresponds to a deviation from perfect quantization by about 10%, and gives anupper bound for how much parallel conduction is acceptable toobserve the quantum anomalous Hall effect. Quantitatively, this shows that an exchange gap of at least 0.3 eV is necessary for observing quantized transport near room temperature. Such alarge gap, corresponding to a Curie temperature well over 1000 K,likely precludes this. However, considering an upper temperaturenear 77 K (liquid nitrogen temperature) corresponds to much smaller gaps, of order of 0.05 eV, which are well within theoretical predictions of heterointerfaces among TIs and magnets, whichhighlights open directions of MBE in the field. Finally, the quantum anomalous Hall effect has recently been observed in intrinsic materials without any doping. The two exam- ples are MnBi 2Te4245,246and twisted bilayer graphene precisely aligned to hBN.247The former is an intrinsically antiferromagnetic TI, which orders with a layered A-type structure.248,249The quantum anomalous Hall effect emerged in atomically thin flakes when exposed to an applied magnetic field, which resulted in the time-reversal symmetry breaking necessary to gap the surface bandstructure. This is a critical advance beyond the pioneering studiesthat first revealed the quantum anomalous Hall effect. In the gra-phene system, quantized transport was observed due to orbital magnetism that emerges in carbon systems and can be accessed when two layers of graphene are twisted to a precise angle of 1.15°and aligned to hBN. These two systems show the powerful tunabil-ity of atomically precise heterostructures and represent two areas,where key advancements can be made in MBE growth. Although at this moment it is not clear how to achieve precise alignment of the individual layers necessary for the bilayer/hBN system, high qualityMnBi 2Te4films should be possible to synthesize by MBE.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210902 (2020); doi: 10.1063/5.0022948 128, 210902-17 Published under license by AIP Publishing.MnBi 2Te4is a derivative of the tetradymite structure with an addi- tional layer of Mn-Te inserted into the quintuple layer structure.56 This forms a septuple layer composed of Te –Bi–Te–Mn–Te–Bi–Te, as shown in Fig. 9(a) . MBE growth will enable many routes to precisely tune the magnetic ground state and interfacial properties.However, initial reports of the growth relied on alternatingdepositions of Bi 2Te3and MnTe layers, which subsequently inter- mixed to form the septuple layers.250,251Going beyond this initial work, codeposition approaches252,253have confirmed that an adsorption controlled growth window can be achieved forMnBi 2Te4as well as the next member of the series MnBi 4Te7,a s shown in Figs. 9(b) –9(c) with stoichiometry confirmed by Rutherford backscattering spectroscopy.254With increasing Bi:Mn flux ratio, these films showed the evolution of the topological bandstructure as well as the magnetic ground state, which are key data that will help clarify some of the remaining mysteries in this mate- rial. This accomplishment further suggests that higher membersMnBi 6Te10, or MnBi 8Te13255–257may also be possible to synthesize either through adsorption controlled growth or through alayer-by-layer approach. Fully understanding the synthesis of thismaterial and how to integrate it with other functional materials asheterostructures will open many new routes to realize the quantumanomalous Hall effect as well as other exotic phases.B. Toward topological spintronics The unusual spin-textures found in topological materials make them attractive candidates for spintronic device applications,where the spin degrees of freedom are used to perform memory and logic operations rather than just the charge dynamics. 258 Moving away from the paradigm of charge-based electronics can offer many advantages including greatly reduced energy dissipation,faster operation, as well as novel routes to data storage. However,the field faces key challenges regarding generation, manipulation, and detection of spin currents as well as performing switching and gate operations since spin currents are not conserved, unlike chargecurrents. The field of “topological spintronics ”has emerged to address many of these challenges based on the novel character ofthe spin-polarized TSSs. In particular, the intrinsic locking of the direction of the spin to the direction of the wave vector, and thus the direction of the current flow, can mitigate many of the currentchallenges. For one, spin current generators can be realized in topo-logical materials, where spin-polarized currents can be producedthrough the spin-momentum locking of the TSSs as well as via spin Hall effect in the strong-spin –orbit-coupled bulk states. Switching can be enabled by the intercoupling between ferromag-netic layers and the spin-polarized surfaces. Magnetizationswitching of ferromagnetic layers have been demonstrated via FIG. 9. (a) MnBi 2Te4septuple layer unit. (b) X-ray diffraction 2 θ–θscans for various Bi:Mn flux ratios that exhibit the first two members MnBi 2Te4and MnBi 4Te7, as highlighted by circles and triangles, respectively. (c) Film stoichiometry plotted in atomic percentage for the MnBi 2Te4and MnBi 4Te7phases (indicated by horizontal dashed lines) was confirmed by Rutherford backscattering spectroscopy. Reproduced with permission from Lapano et al., Phys. Rev. Materials 4,111201(R) (2020). Copyright 2020 The American Physical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210902 (2020); doi: 10.1063/5.0022948 128, 210902-18 Published under license by AIP Publishing.spin-transfer torque exerted by topological materials. Realizing many of these goals requires understanding the detailed physics of spin-polarized transport in topological materials as well as thestrong physical and chemical interactions at the heterointerfacesamong topological and magnetic materials, which are necessary fora topological spintronic device. The current-induced spin polarization in TSSs can be electri- cally detected using a potentiometric geometry with a ferromagnet(FM) voltage probe on a TI channel, 259as illustrated in Fig. 10(a) . In this configuration, the direction of the current flow in the TIdetermines the direction of the spin polarization on the TI surface, and an external magnetic field aligns the magnetization of the FM voltage probe parallel or anti-parallel to the direction of the TI spinpolarization. Measured voltage from the FM voltage probe shouldswitch its sign when the relative direction of the TI spin polariza-tion and the FM magnetization changes. The amplitude of the voltage change when it switches is proportional to the projection of the spin polarization onto the FM. Experiments have revealed step-like hysteretic spin signals associated with the direction of spinpolarization in the TI and the magnetization of the FM voltageprobe, as shown in Fig. 10(b) . 46,260–266A high spin polarization ratio in the TSSs, due to the suppression of backscattering, has been experimentally confirmed by potentiometric measurements.Moreover, larger spin signals were observed when the chemicalpotential is tuned into the bulk bandgap and toward the Dirac point, due to reduced conduction through trivial bulk bands [Fig. 10(c) ]. 46We note that the sign of the measured spin signal is inconsistent between research groups.267Theoretical models to interpret such current-induced spin polarization as well as theexperimental results need to be reconciled. Spin-charge conversion is an essential component for spin- tronic technologies as it facilitates interfacing with current charge-based electronics. Currently, heavy metals are the materials mostcommon in such devices. When a charge current flows throughmaterials such as Pt, Ta, or W, for example, spin-transfer torque by spin-charge conversion in the heavy metal causes the magnetiza- tion of an interfaced FM to precess or reverse its orientation. 268In most materials, this is found to require a large current density,which causes deleterious heating. However, increased efficiency ofspin-charge conversion can be achieved by basing such devices on TIs. Efficiency from experimental studies of both charge-to-spin conversion and spin-to-charge conversion has been studied usingTIs. Schematics of representative device for charge-to-spin conver-sion and spin-to-charge conversion are illustrated in Fig. 11 . The charge-current-induced spin-polarized electrons in TIs can exert a spin-transfer torque to an FM (or spin –orbit torque when the origin of the spin polarization is related to spin –orbit coupling, such as the spin Hall effect), which works as a means ofcharge-to-spin conversion [ Fig. 11(a) ]. The first demonstration of a TI-based (Bi 2Se3/Pt bilayer system) spin-transfer torque was mea- sured by ferromagnetic resonance [ Fig. 12(a) ]. This exhibited a record efficiency for this effect at room temperature.23The spin-charge-conversion efficiency can generally be quantified bythe spin Hall angle (or spin torque ratio) θ k;(2e//C22h)JS,k/J ¼(2e//C22h)σS,k/σ, where JS,kandσS,kare the parallel components of the spin current density and spin conductivity, respectively, absorbed by the FM, and Jandσare the charge current density and charge conductivity, respectively, in the spin current sourcematerial. The spin-transfer torque induced by the TI is strong enough to switch the magnetization of the adjacent FM metal layers with in-plane and perpendicular magnetic anisotropies atroom temperature. 269,270Current-induced spin-polarized electrons in TIs can exchange-couple to an adjacent FM layer. Such TI/FMbilayers can exhibit unidirectional magnetoresistance (UMR), R UMR,271which is dependent on the TI current direction. This can be seen in Fig. 11(b) where high and low resistance states depend on the relative direction of the TI spin polarization and the FMmagnetization. For the charge-to-spin conversion process FIG. 10. (a) Schematic of potentiometric geometry for electrical detection of current-induced spin polarization. (b) Spin-dependent voltage changes its s ign according to the direction of the DC bias current. Step-like jumps occur at magnetization switching of the FM probe. (c) The voltage change, step height in (b), is la rger when the Fermi level is near the Dirac point. Reproduced with permission from Lee et al. , Phys. Rev. B 92, 155312 (2015). Copyright 2015 The American Physical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210902 (2020); doi: 10.1063/5.0022948 128, 210902-19 Published under license by AIP Publishing.associated with UMR, the figure-of-merit is expressed as RUMR per current density ( j) per total resistance ( R):RUMR/j/R. In the Bi 2Se3/ CoFeB bilayer system, RUMR/j/Rwas found to be twice as large in comparison to the best reported Ta/Co bilayers.272 Spin-to-charge conversion can be achieved in TI-based spin pumping devices where precessional magnetization dynamics of anFM injects a spin current into a TI, as shown in Fig. 11(c) . The injected spin current can be converted into a charge current in the TI through the inverse Edelstein effect. This is facilitated by the 2Dsurface states in combination with the inverse spin Hall effect bythe bulk states. In Bi 2Se3/CoFeB spin pumping devices, the result- ing spin Hall angle at room temperature turned out to be greater than that of heavy metal-based spin pumping devices.273From experimental reports of spin-charge conversion, the measured spinHall angle based on TI/FM systems widely varies from 0.001to 2. 274Different mechanisms of spin-charge conversion with a single material may result in different values of the spin Hall angle. Materials quality and TI-FM interface quality can also play animportant role in determining the spin-charge conversion effi-ciency, as discussed next. Thin films of MBE-grown TIs are promising platforms for large-scale device fabrication. One of the major challenges in fabri- cating TI/FM bilayer devices is to make clean interfaces betweenthe materials. Oxides and the subsequent surface damage associatedwith their removal, and defects introduced at the interface duringfabrication significantly degrade the efficiency of spin-charge con- version. One solution to avoid the degradation is to grow in situ FM layers on TI films. TI/magnetic-TI bilayers of (Bi,Sb) 2Te3/ Cr-doped (Bi,Sb) 2Te3grown by MBE have been demonstrated to show spin –orbit torque with magnetization switching of the magnetic TI layers [ Fig. 12(b) ]275,276as well as large UMR.277 Although highly efficient spin-charge conversion was observed inTI/magnetic-TI bilayer devices, there are key limitations. The most prominent is the low Curie temperature of the ferromagnetic TI layer (below 40 K). This requires development of other in situ mag- netic materials for room-temperature applications. An alternativesolution to achieve a clean interface is in situ capping of TIs directly after MBE growth, using materials such as Se on Bi 2Se3. This preserves air-sensitive surfaces of TIs81and opens up the use of various magnetic materials prepared ex situ by other deposition techniques such as e-beam evaporation and sputtering. There aremultiple routes to remove the Se capping layer from Bi 2Se3leaving atomically clean surfaces.44The Se capping layer can be removed by heating the substrate at 200 –250 °C under Se flux with in situ monitoring such as RHEED in an MBE chamber; the recovery ofthe diffraction pattern of Bi 2Se3is a clear indicator of Se-desorption. The Se beam flux prevents further creation of Sevacancies and likely facilitates removal of Se-O. 44Although Se beam flux and in situ monitoring are not available in many cases, a gentle pre-sputtering of the Se-capping layer combined with carefulcontrol of substrate temperature and desorption time could recovera clean Bi 2Se3surface. Subsequent in vacuo deposition of the FM layer will likely be a key in obtaining a clean interface between Bi2Se3and the FM layer. There are a few issues in spin-charge conversion using Bi2Se3/FM-metal bilayer systems. One major issue is the coexis- tence of bulk conduction and surface conduction in Bi 2Se3thin films. The Fermi level in Bi 2Se3films is typically located slightly above the conduction band edge. Because Bi 2Se3intrinsically has a strong spin –orbit coupling, spin accumulation at the surface through the spin Hall effect in the bulk, which, when combinedwith the spin polarization by TSSs, is not negligible. In fact, it could positively enhance the overall spin Hall angle since the sign of the spin polarization is the same as that of the TSSs. In FIG. 11. Schematics of (a) spin-transfer torque, (b) UMR, and (c) spin pumping devices. Charge-to-spin conversion occurs in spin-transfer torque and UMR dev ices, whereas spin-to-charge conversion occurs in the spin pumping device.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210902 (2020); doi: 10.1063/5.0022948 128, 210902-20 Published under license by AIP Publishing.contrast, if the Fermi level is located further up in the bulk bands, Rashba spin-split states may play an additional role. Particularly,these states reduce the overall spin polarization signal since itsorientation is opposite relative to the TSSs. To avoid this negativecontribution, the Fermi level needs to be near or below the con- duction band edge, which can be achieved by minimizing defects at the interface, preferentially using TIs that are more tolerant tosuch defects, or by carefully choosing materials with optimumband alignment. The second major issue is the shunting of chargecurrent via the FM metal. This occurs since the conductivity of t h eF Mi so r d e r so fm a g n i t u d el o w e rt h a nB i 2Se3. Despite the effi- cient spin-charge conversion in Bi 2Se3, the overall current density through the Bi 2Se3/FM-metal bilayer structure is significantly higher than what is in the Bi 2Se3layer. To overcome the shunting issue, insulating (or highly resistive) FMs are desirable for Bi2Se3/FM bilayer systems.TI/FM-insulator bilayers with a clean interface can be pre- pared by MBE. FM insulators have been found in the forms ofoxides, nitrides, sulfides, and dilute magnetic semiconductors.However, there are key challenges related to the integration ofthese with topological materials. Typical growth temperatures of crystalline FM insulators are much higher than that of the Bi-chalcogenide TIs. In addition, lattice matching with substratesor epilayers is critical for the growth of FM insulators. Given theseconditions, MBE growth of TIs on FM insulators is more applica-ble, in most cases, than the growth of FM insulators on TIs. FM insulators for TI/FM bilayer structures prepared by MBE include yttrium iron garnet (YIG) Y 3Fe5O12, M-type hexagonal ferrite BaFe 12O19, the dilute magnetic semiconductor (Ga 1−xMn x)As, and the Heisenberg FM insulator EuS.278–282 For MBE growth of TIs on crystalline magnetic oxides such as YIG and BaFe 12O19films, which can be prepared by sputtering or FIG. 12. (a) Schematic (top) showing how an in-plane current in a TI generates spin accumulation on the surface, which in turn exerts torque on the adjacent FM la yer. In a ferromagnetic resonance measurement at room temperature (bottom), the symmetric component corresponds to in-plane torque, consistent with th e torque induced by TSSs. (b) In a Cr-doped TI bilayer heterostructure, magnetization switching through the spin –orbit torque was demonstrated at 1.9 K. Reproduced with permission from Mellnik et al. , Nature 511, 449 (2014). Copyright 2014 Springer Nature; Fan et al. , Nat. Mater. 13, 699 (2014). Copyright 2014 Springer Nature.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210902 (2020); doi: 10.1063/5.0022948 128, 210902-21 Published under license by AIP Publishing.pulsed laser deposition, TIs can be grown by either one-step or two-step growths. In the case of large lattice mismatches between two layers, two-step growth may produce smoother crystalline TIfilms. 71Spin pumping into Bi 2Se3from YIG revealed that the spin-to-charge conversion efficiency drops rapidly when the Bi 2Se3 thickness decreases below 6 QLs. The efficiency remains almost constant with Bi 2Se3thickness being greater than 6 QLs.278This result correlates with a decrease in the spin polarization with ahybridization gap in Bi 2Se3under 6 QLs283and implies that the TSSs play a dominant role in spin-to-charge conversion in Bi 2Se3/ YIG bilayer systems. Spin-to-charge conversion in this system was further investigated using Cr-(Bi 1−xSbx)2Te3on YIG, and the com- bination of composition control and electrostatic gating enabledtuning the surface and bulk contributions. 284Wang et al. found that the spin Hall conductivity did not change substantially whenthe Fermi level was tuned across the bulk bandgap. It was concluded that the spin-to-charge conversion originates from either the full spin –orbit coupled bulk states or the spin-momentum-locked TSSs. Strong interfacial exchange couplingrevealed induced magnetism in the TI layer. In the Bi 2Se3/ BaFe 12O19bilayer system, hysteretic anomalous Hall effect as well as magnetization switching of the BaFe 12O19layer by spin –orbit torque from Bi 2Se3was observed even at 300 K.280The spin –orbit torque switching efficiency is comparable with Pt/BaFe 12O19bilayer systems at 300 K, however, the efficiency becomes 30 times higher at 3 K most likely due to enhanced surface conduction at lower temperatures. TI/FM-insulator bilayer systems are an ideal platform to study the breaking of time-reversal symmetry and opening of an energygap at the Dirac point due to exchange coupling. This was shown using (Bi,Sb) 2(Te,Se) 3films, with compositions tuned to reduce bulk conduction, grown on highly resistive FM (Ga,Mn)As. Withthe magnetization oriented normal to the interfaces an exchangegap was opened, which was indicated by a modification of aquantum correction to the magnetoresistivity called the weak anti- localization effect. 281In 3D TIs, the strong spin-momentum locking causes a coherent cancellation of backscattering due to theaccumulation of phase factors of ± πfor loops of opposite orienta- tion, which effectively reduces the probability for resistive backscat-tering. The emergence of an energy gap at the Dirac point modifies the Berry phase to be π(1/C0Δ/(2E F)) with Δbeing the gap size and EFbeing the Fermi energy, and weak localization is expected to arise as the Fermi level approaches the energy gap.285In the (Bi,Sb) 2(Te,Se) 3/GaMnAs bilayer system, the crossover between weak antilocalization and weak localization was observed and attributed to the gap opening by breaking time-reversal symmetry.281 High spin-charge-conversion efficiency in the tetradymite TIs is promising for spintronic applications in computation, logic, andmemory devices. However, high efficiency alone may not guarantee that TI-based devices can immediately be embedded in the current technology and replace the metal-based spintronic devices. One ofthe challenges is the high resistivity of tetradymites, which couldcause additional power consumption, compared to the metal-basedspintronic devices. More experimental effort is required to under- stand and utilize exotic new topological materials in current spin- tronic device motifs, as well as in new approaches such astopological antiferromagnetic spintronics. 285For example, in thetopological material α-Sn, resonant spin pumping from Fe through Ag into MBE-grown α-Sn films induced lateral charge current due to the inverse Edelstein effect.286The resulting spin Hall angle was well above those of heavy metals and comparable with that fromspin-pumping in Bi 2Se3/FM-metal systems. Furthermore, the topo- logical Kondo insulator SmB 6shows TSSs with insulating bulk at low temperatures. Similarly, spin injection experiments on SmB 6 revealed spin-to-charge conversion by the inverse Edelstein effect linked to the surface states.287These studies highlight many func- tional aspects of the spin-polarized band structure across the familyof topological materials which can be unlocked by mastering MBE growth. VI. MOLECULAR BEAM EPITAXY OF TOPOLOGICAL MATERIALS OVER THE NEXT DECADE To conclude, the goal of this Perspective article is to highlight the strong interrelation of MBE growth and future advances in the field of topological matter. This is demonstrated by many historicalexamples of how MBE growth has led to serendipitous discoveriesof topological phases. These subsequently pushed the developmentof theory for both understanding the physics of materials and pre- dicting new phases of matter. Over the past decade and a half, this strong interrelation among theory and experiment has blossomed.This synergy has led to rapid progress in predicting topologicalphases in certain materials as well as baseline confirmation, partic-ularly related to novel band structures. As such, there are many open questions that can be addressed using thin films grown by MBE, and certainly many surprises awaiting discovery. This pre-dicts an exciting decade to come that is full of new discoverieswhich will push closer to practical applications of topologicalmaterials. Particularly, attempting and refining the growth of new mate- rials is an open area that is very broad in scope. This is historicallya well-proven route to new discoveries. Currently, only a handful ofknown and predicted topological materials have been synthesized by MBE, with the majority of the work focused on the tetradymite family including Bi 2Se3,B i 2Te3, and Sb 2Te3. These materials have been at the basis of most of the exciting discoveries, and certainlywill continue to play an important role. This is especially clear withthe recent exciting discoveries made in the ternary tetradymites series, particularly the intrinsic magnetic TI, MnBi 2Te4. Refinement of the growth of this material and higher order compounds,254as well as integration with strong magnetic materials will open thedoor to answering fundamental questions, particularly, what is themaximum temperature the quantum anomalous Hall effect can be observed. Beyond TIs, many new topological phases have emerged and represent a totally unexplored space for MBE growth. Thebroad class of nodal metals is the biggest example of these, withpreliminary work being confined to the 3D Dirac semimetals, butleaving very little work done on the Weyl semimetals and the myriad of other nodal metals. Therefore, implementing MBE ’s ability to control dimensionality, symmetry, and proximity-coupling across interfaces will no doubt lead to many interestingobservations. As the diverse character of the topological states gives many new routes to perform computation or data storage, topologicalJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 210902 (2020); doi: 10.1063/5.0022948 128, 210902-22 Published under license by AIP Publishing.devices represent an area of critical importance. Implementation of any such device, either a prototype or mass production, will ulti- mately require a high-quality thin film either as a single slab ofmaterial or as a heterostructure of multiple materials with dissimi-lar properties. Achieving this can lead to new technologies rangingfrom spintronics to quantum computation. Current barriers are the materials, which are fundamentally difficult to control since the materials-character that makes them topological often lowers thebarrier for the formation of defects while amplifying their deleteri-ous effects. 55Therefore, a new generation of materials is needed, which are robust during synthesis, air stable, and stable during nec- essary processing and integration with other materials. Furthermore, the typical scales for the bandgap (<0.3 eV), dielectric constants(>100), and effective masses ( ∼0.1m e) have been shown to be barri- ers to intrinsic properties dominated by the topological states, andrealizing room temperature operation of a topological device will require optimizing these key parameters. 55The timeframe to over- come such challenges can be given by considering other fields: Thefield of quantum computation has only recently advanced to thelevel of integrating multiple quantum devices into a single processorthat can realistically compete with a classical computer. 288,289This triumph took nearly 30 years290of dedicated effort to understand and master relatively simple materials such as Al/Al 2O3and Si. In contrast, the subfield of topological quantum computation has a rel-atively short history and has not found its silicon , let alone achieved the experimental milestone of a topological qubit. But rapid theoret- ical and experimental progress has been made to determine motifsthat can serve as the basis of a topological qubit as well as identify-ing several materials systems with promising properties. As such,the future will be filled with excitement as MBE growth is refined, the unusual topological states are better understood, and new applications and devices emerge. ACKNOWLEDGMENTS We would like to thank Roman Engel-Herbert, Seongshik Oh, Brian Sales for insightful comments, as well as Anjali Rathore forassistance with literature review. This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division (manuscriptpreparation and MBE growth) and by the Laboratory DirectedResearch and Development Program of Oak Ridge NationalLaboratory, managed by UT-Battelle, LLC, for the U.S. Department of Energy (structural characterization). 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5.0028908.pdf

J. Chem. Phys. 153, 244305 (2020); https://doi.org/10.1063/5.0028908 153, 244305 © 2020 Author(s).Accurate ground state potential of Cu2 up to the dissociation limit by perturbation assisted double-resonant four-wave mixing Cite as: J. Chem. Phys. 153, 244305 (2020); https://doi.org/10.1063/5.0028908 Submitted: 09 September 2020 . Accepted: 13 November 2020 . Published Online: 22 December 2020 P. Bornhauser , M. Beck , Q. Zhang , G. Knopp , R. Marquardt , C. Gourlaouen , and P. P. Radi ARTICLES YOU MAY BE INTERESTED IN Full-dimensional, ab initio potential energy surface for glycine with characterization of stationary points and zero-point energy calculations by means of diffusion Monte Carlo and semiclassical dynamics The Journal of Chemical Physics 153, 244301 (2020); https://doi.org/10.1063/5.0037175 A comparison of experimental and theoretical low energy positron scattering from furan The Journal of Chemical Physics 153, 244303 (2020); https://doi.org/10.1063/5.0027874 Directing excited state dynamics via chemical substitution: A systematic study of -donors and -acceptors at a carbon–carbon double bond The Journal of Chemical Physics 153, 244307 (2020); https://doi.org/10.1063/5.0031689The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Accurate ground state potential of Cu 2 up to the dissociation limit by perturbation assisted double-resonant four-wave mixing Cite as: J. Chem. Phys. 153, 244305 (2020); doi: 10.1063/5.0028908 Submitted: 9 September 2020 •Accepted: 13 November 2020 • Published Online: 22 December 2020 P. Bornhauser,1 M. Beck,1 Q. Zhang,1 G. Knopp,1 R. Marquardt,2 C. Gourlaouen,2 and P. P. Radi1,a) AFFILIATIONS 1Paul Scherrer Institute, Photon Science Department, CH-5232 Villigen, Switzerland 2Laboratoire de Chimie Quantique, Institut de Chimie, UMR 7177, Université de Strasbourg/CNRS, 4, Rue Blaise Pascal - CS90032, 67081 Strasbourg Cedex, France a)Author to whom correspondence should be addressed: peter.radi@psi.ch ABSTRACT Perturbation facilitated double-resonant four-wave mixing is applied to access high-lying vibrational levels of the X1Σ+ g(0+ g)ground state of Cu 2. Rotationally resolved transitions up to v′′= 102 are measured. The highest observed level is at 98% of the dissociation energy. The range and accuracy of previous measurements are significantly extended. By applying the near dissociation equation developed by Le Roy [R. J. Le Roy, J. Quant. Spectrosc. Radiat. Transfer 186, 197 (2017)], a dissociation energy of De= 16 270(7) hccm−1is determined, and an accurate potential energy function for the X1Σ+ g(0+ g)ground state is obtained. Molecular constants are determined from the measured transitions and by solving the radial Schrödinger equation using this function and are compared with results from earlier measurements. In addition, benchmark multi-reference configuration interaction computations are performed using the Douglas–Kroll–Hess Hamiltonian and the appropriate basis of augmented valence quadruple ζtype. Coupled-cluster single, double, and perturbative triple calculations were performed for comparison. ©2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0028908 .,s I. INTRODUCTION The bond energy of a molecule is a fundamental thermochem- ical quantity in chemistry: it tells us how much energy is released when a bond is made or, alternatively, how much energy is needed to break that bond. About one decade after the seminal paper of Gilbert Newton Lewis in 1916,1who suggested that a pair of elec- trons shared by two atoms is responsible for the formation of a chemical bond, quantum mechanics laid the ground for our under- standing of the creation and destruction of molecules,2–6which has been continuously refined since then. A recent comprehen- sive review by Frenking and Shaik contains the relevant advances up to our present understanding of bonding.7Bond energies can be determined experimentally via traditional calorimetry.8How- ever, the quantum mechanical understanding of the chemical bondhas allowed us ever since its beginning to assess bond energies spectroscopically, such as with the more recently developed stim- ulated emission pumping experiments9–17or zero electron kinetic energy (ZEKE) experiments.18–21 Quantum chemical models in combination with the rapid growth of computing power provide an alternative to experimental determination. For molecules composed of first and second row ele- ments, accurate bond dissociation energies are accessible using the coupled-cluster methods with single and double excitations, while treating triple excitations perturbatively [CCSD(T)22–24]. The high- est possible accuracy is achieved by considering correlation con- sistent25or geminal basis sets26in addition to essential corrections from spin–orbit interactions, relativistic effects, and quantum elec- tron dynamics. Coupled cluster theory combined with geminal basis sets can indeed be considered to be among the most accurate and J. Chem. Phys. 153, 244305 (2020); doi: 10.1063/5.0028908 153, 244305-1 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp powerful method to obtain accurate information on the chemi- cal structure and properties of medium sized molecules, when the ground state is essentially single reference, even when late-transition metal atoms are included.27 In a study on transition metal containing dimers, it is empha- sized, however, that coupled cluster computations are usually impractically expensive for most problems in transition metal chem- istry and suggest that the method yields similar but not better results than the much less expensive density functional theory (DFT).28 When compared with experimentally determined dissociation ener- gies, the mean unsigned deviations for the transition metal contain- ing dimers were ≈4 kcal/mol (1700 hccm−1) from that study ( his the Planck constant and cis the speed of light in vacuum). To warrant our quantitative understanding of the chemical bond, the poor accuracy reported in Ref. 28 must be improved, both theoretically and experimentally. Such an understanding will reflect our knowledge of the nature of the chemical bond, which depends on the computational approaches used and the experimental verifi- cation of the calculated observables. In this context and in addition to covalent and ionic bonding, a third class of electron pair bond, called charge-shift (CS) bonding, was introduced.29CS bonding is characterized within valence bond theory by the occurrence of large resonance energies associated with the mixing of covalent and ionic components of the bonding wavefunctions, which is expected to take place with compact electronegative or lone-pair rich species, and we refer the reader to the original literature for further explanations on the concept of CS bonding.29Recent computational investiga- tions on the nature of single bond transition-metal dimers30have shown that, for the isoelectronic coinage metal dimers, Cu 2, Ag 2 and Au 2, a significant contribution between 40% and 50% to the total bonding energy is attributed to CS bonding. This assertion can hardly be verified experimentally, as the aforementioned reso- nance energy is not observable. One can nevertheless address calcu- lated and measured values for equilibrium bond lengths and bond energies. Currently, perhaps the most reliable measurement for the dis- sociation energy for dicopper, one of the best studied transition metal dimers, dates back to 1986.31In that work, Rohlfing and Valentini applied UV excitation of Cu 2produced by laser vapor- ization in a molecular beam and observed in emission long pro- gressions to the ground state. Vibrational levels up to v≤72 were measured by dispersed fluorescence. The determined vibrational origins were fitted to the near-dissociation equation reported by Le Roy and Lam.32The resulting value for the dissociation energy, De= (16 760±200) hccm−1, at the vibrational quanta at the disso- ciation limit of vd= 128±5 is considerably the most reliable value ofDein the literature. We report in the following on an experimental investigation, which yields a significantly increased accuracy for the bond dis- sociation energy by applying a non-linear spectroscopic method and by taking into account results from a recent deperturbation study of high lying energy levels of the copper dimer.33In addi- tion to the determination of De, an accurate potential energy func- tion is obtained to assess the structural and dynamical properties of this important transition metal species and to provide a further stringent assessment for high level computations. Multi-reference configuration interaction (MRCI) calculations were reported in a previous study.34A variation of those calculations as well as newCCSD(T) calculations will be discussed in the present work. Results from the present calculations as well as from previous theoretical work will be critically reviewed in light of the new experimental results. Experimental investigations are based on the characterization of highly excited states in the spectral region between 37 400 and 38 050 cm−1, which have opened ways to access spectroscopically vibrational levels of the ground state close to the dissociation limit.33 Accurate molecular constants and the term symbols for the I and J states35have been determined. Furthermore, approximately 1000 rovibronic transitions in the vicinity of the J state of the two main isotopologues63Cu 2and65Cu63Cu were assigned. The deperturba- tion of strongly interacting rotational levels of the J ( v= 0–2) state revealed dark states that were preliminarily assigned to high vibra- tional levels of the G 0+ ustate. As discussed in our previous work,34 the G 0+ ustate emerges from the avoided crossing of the B1Σ+ u(0+ u) state and an ion-pair state. Close to the equilibrium bond length, the B state has a stronger ionic character, while the G state is strongly covalent. At larger distances, however, in the region of the avoided crossing, the latter becomes ionic. Transitions from neutral into ion- pair states are typically strong. Therefore, strong emission from the B1Σ+ u(0+ u) state to low vibronic levels of the ground state X1Σ+ g (0+ g) is observed. By implication, the shallow potential of the G 0+ u state should expose most ion-pair character at the outer wall at even larger internuclear distances beyond the avoided crossing. By the Franck–Condon principle, vertical transitions to low vibrational lev- els are not expected to occur for the G state. Alternatively, as will be shown in this work, significant transition strength to high vibra- tional levels in the ground state emerges and allows for spectroscopic detection of ground state ro-vibrational levels up to the asymptotic limit. In the theoretical section (Sec. II), we briefly recall the key elements used in the MRCI and CCSD(T) calculations. In the experimental section (Sec. III), we outline the two-color resonant four-wave-mixing method to perform Perturbation-Facilitated Optical–Optical Double-Resonance (PFOODR) spectroscopy. The nonlinear spectroscopic technique is favorably suited to perform background-free and highly sensitive stimulated emission pump- ing (SEP) type investigations in a molecular beam environment. PFOODR has been applied initially on alkali metal dimers to access triplet states from the electronic singlet ground state.36–38 Triplet↔singlet transitions are spin-forbidden, but both singlet and triplet states become accessible from the ground state by an intermediate singlet–triplet mixed state. Pulsed or continuous-wave lasers have been used to perform PFOODR spectroscopy by fluo- rescence excitation,39dispersed fluorescence,38,40ion detection,41or continuous-wave optical triple resonance spectroscopy.42Applying non-linear four-wave mixing to perform PFOODR has been demon- strated by some of us in an investigation of the dark triplet manifold of C 3that exhibits a singlet ground state43and was also used to characterize the first quintet–quintet band of C 2.44 II. THEORETICAL METHODS Technical details of the MRCI method used to obtain the theo- retical results presented in this report on the X1Σ+ gground state of Cu 2are given in Ref. 34. In essence, they are based on a restricted J. Chem. Phys. 153, 244305 (2020); doi: 10.1063/5.0028908 153, 244305-2 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp active space calculation, composed of 22 electrons in 18 orbitals (RAS–22,18) and using the Douglas–Kroll–Hess Hamiltonian45,46 and the appropriate basis of augmented valence quadruple ζtype (AVQZ-DK) as defined in the MOLPRO program package.47These methods led to unprecedented accuracy of the theoretical descrip- tion of the 0+ uand 1 umanifolds in excited Cu 2and a solid theoretical justification of the assignment of the A′state. We refer to Ref. 34 for further explanations on details of the calculations. Such an accuracy was achieved because the active space was optimized at the MC-SCF level upon inclusion of a fourth state with 1Σ+ gand a third state with1Σ+ usymmetry label. These are charge transfer states in the vicinity of the equilibrium structure (Ref. 34; Table I therein). We term this calculation “NpC.” As the set of active orbitals include the 4p subshell, when the Cu–Cu distance is elongated, the fourth1Σ+ gand third1Σ+ ustates change character and become the gerade and ungerade states of the “sp” asymptote, i.e., yielding the dissociation products Cu(3d104s) + Cu(3d104p). This asymptote is about 3.8 eV higher than the ground state asymptote leading to Cu(3d104s) + Cu(3d104s)48and 2.3 eV higher than the manifold of “sd” asymptotic states leading to Cu(3d104s) + Cu(3d94s2), to which the several 0+ uand 1 ustates belong to, which were calculated in Ref. 34. It turns out that the (22,18) active space is very likely not suf- ficient for a correct description of the “sp” asymptote. Higher lying orbitals would be needed to render this calculation more accurate. This would make the active space even larger and is beyond the scope of the present work. As a side effect, in the absence of higher orbitals, the quality of the ground state itself suffers when one attempts to optimize all states in the asymptotic region. To guarantee a suffi- ciently good description, in the present work, we decided to remove the fourth1Σ+ gand third1Σ+ ustates from the active space during orbital optimization at the MC-SCF level. We term such a calcu- lation “N.” The active space contains effectively only the ground state and all singly excited states leading to neutral “ss” and “sd” asymptotes (see Table I in Ref. 34). CCSD(T) calculations were performed on the basis of sin- gle reference Hartree–Fock functions for Cu 2at the equilibrium bond length of about 225 pm and the separated fragments using the Douglas–Kroll–Hess Hamiltonian and the AVQZ-DK basis set compatible with it, as implemented in the MOLPRO program package.47 III. EXPERIMENT The experiments are performed in a molecular beam apparatus designed for simultaneous linear and non-linear spectroscopic mea- surements of stable and transient species and have been described in detail previously.34,43,44,49,50Briefly, Cu 2is prepared in a home- built metal cluster source by laser-vaporization.50The second har- monic of a Nd:YAG laser (Continuum, NY81; ≈100 mJ/pulse) is focused through a 500 mm focal lens onto a copper disk target (99% purity). The target is translated and rotated by electric motors (Maxon) to control carefully the rate at which fresh surface is sam- pled. A pulsed valve (Series 9 general valve, Parker-Hannifin) is used to introduce helium (6.0, Messer Schweiz AG) carrier gas that is expanded with the copper plume in a near supersonic expan- sion through a 1 mm diameter nozzle into the vacuum. The heliumbacking pressure behind the pulsed valve is 50 bars. Cop- per dimers are probed with the two-color resonant four-wave mixing (TC-RFWM) technique ≈5 mm downstream from the nozzle. The optical setup has been described recently43and is only summarized here. TC-RFWM is performed by using two dye lasers (NarrowScan, Radiant Dyes), which are simultaneously pumped by a single Nd:YAG laser (QuantaRay Pro 270-10, Spectra-Physics). The bandwidth of the dye lasers is specified to ≈0.04 cm−1. A com- bination of optical components establishes a forward BOXCARS51 configuration.52Three parallel propagating laser beams pass along the three main diagonals of a parallelepiped and cross at a small angle of∼1○. Doppler broadening is minimized by arranging these beams orthogonally to the propagation direction of the molecular beam. The two laser beams of equal frequencies are referred to as PUMP beams and the third beam as PROBE beam. Phase matching conditions govern the direction of the SIGNAL beam that propa- gates roughly along the fourth (dark) diagonal of the parallelepiped. The SIGNAL beam is detected by a photomultiplier after remov- ing scattered light and unwanted fluorescence by spatial and spec- tral filters on its nearly 5 m path. Fluorescence signals are observed perpendicular to both the molecular beam and laser propagation through a 1 m spectrometer (SPEX) and detected by an additional photomultiplier tube. IV. EXPERIMENTAL RESULTS A. Dispersed fluorescence and four-wave mixing As mentioned in the introduction, high-lying vibrational lev- els of the G 0+ ustate gain transition strength by perturbation with the J 0+ ustate and are, therefore, accessible from the ground state of Cu 2.34Fig. 1 depicts a dispersed fluorescence (DLIF) spectrum upon excitation of the P(18) G(62)−X1Σ+ g(0+ g)(v′′=0)transi- tion of65Cu63Cu at 37 445.03 cm−1(lower trace, blue). The notation G(62)is used to refer to the level that is presumably the v′= 62 vibrational level of the G state. The assignments and characteriza- tion of the high lying levels of the G state are subject of a separate publication.33 The DLIF spectrum shown in Fig. 1 demonstrates the large Franck–Condon overlap between G(62)and high lying vibrational levels of the ground state X1Σ+ g(0+ g). Upon excitation, substantial emission up to v′′≈102 for J′′= 18 and 16 is observed. Above the asymptotic atom limit, broad diffuse features are present that origi- nate from bound-free transitions.53The Condon internal diffraction produces an interference pattern that extends substantially beyond the dissociation limit. A detailed report of the phenomenon in Cu 2 is subject of a forthcoming publication. Due to the small rotational constant ( B′′≈0.1 cm−1), the DLIF spectrum is not rotationally resolved. However, the long progres- sion with large emission to high v′′levels opens ways to perform rotationally resolved and isotopologue-specific four-wave mixing experiments to characterize precisely the ground state potential up to≈98% of the dissociation limit. Individual TC-RFWM spectra are shown in Fig. 1 (upper traces, the ordinate is shifted to ease visual comparison). For the applied SEP scheme, the PUMP laser is operated at a fixed-wavelength resonant with a particular rotational J. Chem. Phys. 153, 244305 (2020); doi: 10.1063/5.0028908 153, 244305-3 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . Dispersed fluorescence and TC-RFWM spectra to high lying vibrational levels in the ground state X1Σ+ g(0+ g)of65Cu63Cu. Fluorescence is excited by using the P(18) line of the G(62)−X(v′′= 0) transition. Significant emission is observed to levels up to v′′= 102 of the ground state (upper trace, blue). The inverted trace is a simulation taking into account the line positions and intensities determined in this work. The offset traces show separate SEP-type TC-RFWM scans of the PROBE laser. As for dispersed fluorescence, the PUMP lasers are tuned to the P(18) rotational line of the G(62)−X(v′′= 0) band. The inset is an expanded representation of the four-wave mixing PROBE scan displaying the P(18) and R(16) SEP lines from the excited J′= 17 level. Broad diffuse features beyond the asymptotic limit originate from bound-free transitions (see text for details). transition within the target species, while the PROBE laser is scanned over the spectral region of interest. The SEP scheme makes use of the same excited intermediate state, and therefore, the possible final states are governed by strict two-photon selection rules from the initial state.50,54 As for the DLIF spectrum in the figure, the P(18) G(62) −X1Σ+ g(0+ g)(v′′=0)transition is selected to populate J′= 17 of the upper state with the PUMP laser. Since the excited G(62)is an 0+ gstate in the appropriate Hund’s case (c) notation, only one R and one P branch transition to J′′= 16 and 18 of each ground state vibration is optically allowed. To exemplify a typical measurement to determine accurate molecular constants, the spectral region for transitions to v′′= 101 is reproduced at full rotational resolution in the inset of Fig. 1. The P(16) and R(18) lines are unambiguously assigned in a straightforward manner, and their spectral positions yield the ori- gin and the rotational constant for the X1Σ+ g(0+ g),v′′= 101 of the 65Cu63Cu isotopologue. In general, several low lying J′′-levels are measured for each vibration v′′by pumping selected rota- tional levels in the G(62)state. The molecular constants are then determined by performing non-linear least squares fits to the line positions using the PGOPHER55software from Colin Western. To characterize the ground state potential, 347 SEP type tran- sitions have been measured for the two main isotopologues of Cu 2 spanning the range of vibrational levels from v′′= 36 to 98 and v′′ = 36 to 102 for63Cu 2and65Cu63Cu, respectively. Levels with v′′ ≧83 are obtained by pumping selected J′of the G(62)intermedi- ate level. In some cases, the v′= 0 level of the J 0+ ustate is accessedas the common intermediate level. The level mixing in the J, v′= 0∼G(62)system provides sufficient ion-pair state character for spe- cific rotational levels such that they are equally suitable to access high lying vibrational states of X1Σ+ g(0+ g). Lower ground state vibrational levels ( v′′= 36–61) are obtained by exciting rotational levels of v′= 8, 20, and 28 of the G 0+ ustate. These levels have sufficient FC-overlap with the ground state X1Σ+ g(0+ g),v′′= 0 level and can be excited directly. In fact, Rohlfing and Valentini31have reported excitations of the G-states up to v′= 39 and vibrationally resolved fluorescence emission to levels as high as v′′= 72. In addition to the experimentally determined molecular con- stants in this work, high-resolution data from Ram et al.56are included for the analysis. The authors applied Fourier transform emission spectroscopy and obtained precise constants in the range ofv= 0–3 for both isotopologues. The complete line list is pro- vided in the supplementary material to this publication. The result- ing vibronic origins Gvand rotational constants Bvfrom the least- squares fit55for the two main isotopologues are listed in Tables IV and V. The root mean square (rms) value of the fit is 0.02 cm−1, which is less than the linewidths of the dye lasers. Statistical uncer- tainties are given in units of the last significant figure. No allowance is given for systematic errors in the calibration. The constants include the centrifugal distortion constants (up to Ov), which are obtained by the analysis of the effective Gvand Bvvalues, as described in Sec. IV B. Figure 2 displays a Birge–Sponer plot. The measured ΔGv′′+1 2 (=Gv′′+1−Gv′′)values vs v′′+1 2are shown for63Cu 2(red cir- cles) and65Cu63Cu (blue squares). The Gvvalues for the heavier J. Chem. Phys. 153, 244305 (2020); doi: 10.1063/5.0028908 153, 244305-4 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 .ΔGv′′+1 2vsv′′data are plotted for63Cu2(red circles) and65Cu63Cu (blue squares). The heavier isotope values are shifted by applying isotope rela- tions. The green line represents a Dunham extrapolation for v<60 assuming a Morse potential with the parameters ωeandωexe. To account for the nega- tive curvature for v>60, the Dunham parameters ωeyeandωezeare included for the extrapolation procedure (red line). Birge–Sponer extrapolation by using Eq. (1) of order 6 (light blue). The physically correct 1/ R6behavior at the asymp- totic limit is included for two NDE curves: NDE(LIF) is obtained from vibrationally resolved fluorescence measurements (Ref. 31) for v′′≤72. The black solid trace depicts the NDE approach to the rotationally resolved data up to v= 102 in this work. The inset shows an enlarged view of the dissociation region (see text for details). isotopologue are shifted in accord with the usual isotope relation- ships. Up to v≈60ΔGv′′+1 2is, in a first approximation, linear. In this region, the potential is described by a Morse potential with Dunham parameters57ωe= 266.30(5) cm−1andωexe= 0.9939(9) cm−1. A fit with these parameters gives the straight line [Morse (v<60), solid green line] in Fig. 2. The dissociation energy De for a Morse potential is hcω2 e/4ωexe=17 838(18)hccm−1at the vibrational level ωe/2ωexe≈134. It is obvious, that the extrapo- lation strongly overestimates the dissociation energy and fails to describe ΔGv′′+1/2values for higher ground state vibrations. A rel- atively large rms value of the residuals (1.3 cm−1) indicates that the linear approximation is inaccurate by neglecting the negative curva- ture of the higher lying vibrational levels. For an improved result, it is necessary to extend the Dunham parameters to ωeyeandωeze (DH, red line). The fit results are listed in Tables I and II. The fitparameters ωe,ωexe,ωeyeandωezeare in agreement with the results obtained from the vibrationally resolved fluorescence measurements from Rohlfing and Valentini.31The first anharmonicity parameter, ωexe, is close to the Fourier transform value from Ram,56while the second anharmonicity parameter ωeyeis higher. The parameters are expected to be more accurate by considering the larger range of vibrational levels. The parameter pertaining to the rotational con- stant, Be, and hence the equilibrium distance reis in accordance with the high resolution study of Ram et al. The vibration–rotation interaction parameter, αeis about 2% smaller than the reported value of Ram. Again, the difference is most likely due to the larger dataset measured in this work. The parameters listed in the last column [“theory (this work)”] were obtained by using the analyti- cal potential Vrdefined in Eq. (19) below. The agreement with the parameters obtained from the analysis of the experimental data is satisfactory. An estimation of the dissociation energy is obtained by a Birge– Sponer extrapolation to the expression ΔG(v+1 2)=∑ak(v+1 2)k ;k=0, 1,...,n. (1) Applying the equation with polynomials of the order n= 4, 5, and 6 and integrating under the fitted ΔG(v+1 2)curves yields an average dissociation limit, De, of 16 118(25) hccm−1. The average (v+1 2) intercept is evaluated to vD= 108.2(3). The Birge–Sponer fit of order 6 is shown in Fig. 2 (solid, light blue). However, it has been argued that the Birge–Sponer method is limited because the correct long- range behavior of the potential is not adequately described.32,58–62 In Sec. IV B, the dissociation energy is evaluated more accurately by taking into account the appropriate asymptotically dominant inverse-power contribution to the potential. B. Determination of the ground state potential function of Cu 2 To analyze the vibrational energies and rotational constants (Gvand Bv, respectively) listed in Tables IV and V, the proce- dures reported by Ji and co-workers63and Liu et al.64are adapted and briefly outlined in the following. The applied “near-dissociation expansion” (NDE) has been introduced by Le Roy (Ref. 65 and references therein) and contains the theoretically known dissocia- tion behavior. Compared to the conventional Dunham expansion,66 the NDE expressions are more reliable for an extrapolation to high vibrational levels beyond the observed data. The X1Σ+ g(0+ g)state dissociates to the Cu(2S) + Cu(2S) atomic limit. Thus, the vibrational spacing lying near dissociation is mainly governed by a long-range van der Waals dispersion (or induced- dipole–induced-dipole) potential of the form V(r)=De−C6/r6, (2) where Deis the dissociation energy and C6is the leading long-range potential coefficient. A reliable value estimate for the latter is avail- able from a recent ab initio study.67The dispersion interaction is computed at the coupled cluster level [CCSD(T)] and amounts to J. Chem. Phys. 153, 244305 (2020); doi: 10.1063/5.0028908 153, 244305-5 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Fit parameters of the X1Σ+ g(0+ g)state of63Cu2from a Dunham analysis of GvandBv,v<60 values listed in Tables IV and V and J≤50. Both isotopologues are fitted simultaneously. The values for the heavier isotopologue are shifted by using the usual isotope relationships. The rms uncertainty of the residuals is 0.03 cm−1, which is smaller than the specified laser bandwidth. All values are in cm−1except rein Å (see text for details). Parameter RohlfingaRambExp.(this work) Theory(this work) ωe 266.43(59) 266.4594(29) 266.487(14) 266.610 4(4) ωexe 1.035(30) 1.0350(16) 1.042 15(87) 0.978 05(3) ωeye×1031.70(58) 0.92(28) 1.806(19) −0.637 3(7) ωeze×105−1.78(37) ... −1.81(1) −0.229 1(5) Be ... 0.108 7808(24) 0.108 86(10) 0.108 796(3) αe×104... 6.2009(57) 6.061(24) 5.812(3) γe×107... 24.2(16) −8.2(13) −0.66(10) re ... 2.219 27(3) 2.218(1) 2.219 11(6) aReference 31. bReference 56. 1.154×106hccm−1Å6. An accuracy of this theoretical value was not established by the authors. However, as has been observed by Le Roy and Lam32and verified in this work, the results are very insensitive to this value ( vide infra ). The equation for the vibrational energies of isotopologue- αis given by G(α) v=De/hc−K(α) 0(v)×F(α) 0(v(α) D−v), (3) and for the rotational constant, B(α) v=(μ1/μα)K(α) 1(v)×F(α) 1(v(α) D−v) (4) in which v(α) Dis a non-integer effective vibrational index at dissocia- tion andμαis the reduced mass. For the C6-potential of Cu 2adapted in this work, K0andK1are given by the equation K(α) m(v)=(μ1/μα)3/2−mXm(v(α) D−v)3−2mcm−1, (5) TABLE II . Dissociation energies for Cu 2from experiment. Deare listed for an extrap- olation to the asymptotic limit using a Morse potential ( v<60) with parameters ωeandωexe, a Dunham expansion using parameters of Table I, a Birge–Sponer approach [average of orders 4,5, and 6; Eq. (1)], and the NDE procedure (see text for details). Method De/hccm−1vD RMS Morse ( v<60) 17 838(18) 134 1.3 Dunham ( v<60) 16 331 111.3 0.033 Birge–Sponer 16 118(25) 108.2(3) 0.12(2) NDE (LIF)a16 760(200) 128(5) ... NDE (this work) 16 270(7) 113.6(10) 0.07(2) aReference 31.where Xm=Xm×⎡⎢⎢⎢⎢⎣(μ u)6 (C6 hccm−1Å2)2⎤⎥⎥⎥⎥⎦−1/4 (6) and Xmare known numerical factors (7931.949 and 546.64 for m = 0 and 1, respectively).65 The rational polynomial with orders LandM, F(α) 0(v(α) D−v)=⎛ ⎝1 +∑L i=τp0 i(vD−v)i 1 +∑M j=τq0 j(vD−v)j⎞ ⎠S , (7) is applied for the vibrational energies. For the rotational constants, an exponential expansion F(α) 1(v(α) D−v)=exp[L ∑ i=1p1 i(vD−v)i] (8) is used. The expansion parameters p0 i,p1 i,q0 i, the effective vibra- tional index, vD, and the energy at the dissociation limit Deare fitted by taking into account the constants of both main isotopo- logues (Tables IV and V) simultaneously. The expansion parame- ters for the isotopologues are related by p(α) i=(μ1/μα)i/2p(1) iand q(α) j=(μ1/μα)j/2q(1) j, where63Cu 2is the reference isotopologue withα= 1.v(α) Dfor different isotopologues are related by68 (v(α) D+ 1/2)=√ μα/μ1(v(1) D+ 1/2). (9) The exponent power SisS= 1 for an “outer” expansion or S= 2n/(n−2) for an inner expansion n= 6, the power of the asymp- totically dominant leading inverse power term appropriate for the ground-state dissociation of Cu 2. Together with the order of the polynomials Land M, the chosen model determines how fast the transition to the limiting behavior takes place. In order to deter- mine the best model that reproduces the term energies to within the experimental accuracy, all possible combinations of the expansion J. Chem. Phys. 153, 244305 (2020); doi: 10.1063/5.0028908 153, 244305-6 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE III . Parameters defining the NDE functions for the vibrational Gvand rota- tional Bvconstants. The values are obtained by using a dispersion constant C6of 1.154×106hccm−1Å6. The dissociation energy Deis the average of four NDE models that reproduce the measurements within an accuracy of rms <0.1 cm−1(see text for details, dots stand for irrelevant parameter values). Parameter Gv(m= 0) Bv(m= 1) p1 1.98 ×10−1−1.53×10−2 p2 −4.38×10−4−3.82×10−4 p3 −7.47×10−62.18×10−5 p4 1.34 ×10−7−5.31×10−7 p5 −9.82×10−107.59×10−9 p6 3.76 ×10−12−6.41×10−11 p7 −6.07×10−152.95×10−13 p8 ... −5.67×10−16 q1 4.96 ×10−2... q2 4.53 ×10−3... vD 113.6(10) ... De/hccm−116 270(7) ...orders L and M for the inner and outer Padé approximants were tried. A python program has been written, which performed the least-squares fits to the Gvand Bvvalues in Tables IV and V by applying the lmfit software.69192 models for the rational polyno- mial orders L= 0, 1, ..., 7 and M= 0, 1, ..., 5 in Eq. (7) with a leading expansion coefficient of τ= 162were computed for inner and outer Padé expansions in order to obtain model-averaged esti- mates of DeandvD. Only four models reproduced the term energies of both isotopologues simultaneously to within an accuracy of rms <0.1 cm−1. The best fit by applying an inner Padé approximant with nine parameters (L = 7 and M = 2) is listed in Table III and repro- duces the data within the experimental accuracy (rms = 0.05 cm−1). The complete dataset of the model calculations is provided in the supplementary material to this report. The averaged dissociation energy of the four models is 16 270(7) hccm−1and the averaged limiting vibrational level at dissociation is vD= 113.6(10). For the rotational constants, Bv, an exponential NDE expansion with eight parameters is applied [Eq. (8)]. An excellent fit is obtained with an uncertainty for the residuals of rms = 2 ×10−4cm−1. The NDE coefficients are then used to compute a Rydberg– Klein–Rees (RKR) representation of the potential energy function TABLE IV . Molecular constants for63Cu2obtained by the present analysis. All values are in cm−1. v′′Gv Gv(RKR) ΔGv Bv Bv(RKR) ΔBv Dv×108Hv×1013Lv×1018 0 0.000 0(1) 0.036 −0.036 0.108 471 (2) 0.108 466 0.000 004 7.249 −0.151 −0.016 1 264.393 8(9) 264.430 −0.036 0.107 855 (2) 0.107 853 0.000 002 7.245 −0.162 −0.014 2 526.724(1) 526.765 −0.042 0.107 242 (2) 0.107 238 0.000 004 7.240 −0.168 −0.016 3 787.011(1) 787.051 −0.041 0.106 628 (2) 0.106 623 0.000 004 7.236 −0.169 −0.019 36 8 261.12(2) 8 261.151 −0.033 0.086 9 (1) 0.086 970 −0.000 086 7.534 −0.525 −0.212 37 8 454.20(1) 8 454.247 −0.047 0.086 39 (9) 0.086 363 0.000 023 7.562 −0.533 −0.247 38 8 645.31(2) 8 645.361 −0.054 0.085 8 (1) 0.085 755 0.000 008 7.592 −0.539 −0.284 39 8 834.44(2) 8 834.488 −0.048 0.085 2 (1) 0.085 145 0.000 053 7.626 −0.543 −0.321 44 9 750.13(1) 9 750.105 0.021 0.082 1 (1) 0.082 065 0.000 028 7.853 −0.556 −0.518 45 9 927.20(2) 9 927.173 0.030 0.081 4 (1) 0.081 443 −0.000 031 7.911 −0.559 −0.556 47 10 275.23(2) 10 275.183 0.047 0.080 1 (1) 0.080 191 −0.000 098 8.042 −0.568 −0.628 48 10 446.06(3) 10 446.106 −0.048 0.079 9 (2) 0.079 561 0.000 367 8.115 −0.575 −0.661 49 10 614.97(5) 10 614.960 0.008 0.079 1 (3) 0.078 928 0.000 162 8.193 −0.585 −0.692 51 10 946.46(3) 10 946.414 0.046 0.077 7 (2) 0.077 652 0.000 001 8.366 −0.615 −0.748 52 11 109.09(3) 11 108.990 0.098 0.076 8 (1) 0.077 009 −0.000 178 8.460 −0.635 −0.772 53 11 269.51(2) 11 269.447 0.061 0.076 3 (1) 0.076 362 −0.000 061 8.561 −0.661 −0.794 54 11 427.85(2) 11 427.771 0.076 0.075 6 (1) 0.075 711 −0.000 125 8.667 −0.691 −0.814 58 12 039.55(6) 12 039.433 0.120 0.072 5 (5) 0.073 057 −0.000 537 9.153 −0.878 −0.879 60 12 332.10(1) 12 332.048 0.049 0.071 72 (2) 0.071 695 0.000 029 9.437 −1.018 −0.914 61 12 475.08(2) 12 474.988 0.092 0.070 90 (5) 0.071 005 −0.000 103 9.589 −1.102 −0.936 83 14 999.9(2) 15 000.001 −0.085 0.052 9 (2) 0.052 915 −0.000 034 16.285 −7.674 −7.799 84 15 083.8(1) 15 083.815 −0.025 0.051 80 (10) 0.051 891 −0.000 086 16.881 −8.491 −9.068 87 15 317.2(2) 15 317.197 −0.027 0.048 6 (2) 0.048 656 −0.000 096 18.986 −11.702 −14.762 89 15 457.2(1) 15 457.312 −0.149 0.046 3 (1) 0.046 346 −0.000 002 20.724 −14.751 −21.167 97 15 886.6(2) 15 886.739 −0.104 0.035 5 (2) 0.035 498 0.000 047 32.602 −46.729 −136.574 98 15 925.2(2) 15 925.085 0.137 0.033 7 (2) 0.033 916 −0.000 184 35.046 −56.144 −185.031 J. Chem. Phys. 153, 244305 (2020); doi: 10.1063/5.0028908 153, 244305-7 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp up to the dissociation limit employing the software RKR170from Le Roy. Since the measured transitions contain levels with high rota- tional quantum numbers and large internuclear distances (see the line list in the supplementary material), higher-order centrifugal distortion constants are included in the analysis. Therefore, (i) the radial Schrödinger equation is solved for the evaluated potential by using the program LEVEL.71(ii) The computed centrifugal distor- tion constants (up to Ov) are held fixed in a subsequent least-squares fit procedure, as described in Sec. IV A, in order to obtain refined GvandBvvalues. (iii) The RKR potential and centrifugal distortion constants are re-evaluated by applying the optimized NDE param- eters. The procedures (i) to (iii) are repeated until conversion is achieved, which occurs after two iterations. The important advantage of the NDE approach is its capabil- ity to extrapolate Gvand Bvvalues beyond the measured values and construct a physically reliable RKR potential up to the dissocia- tion asymptote. The ΔGv+1 2curve vs v′′shown in Fig. 2 (black solidline) illustrates the effect of the inverse-power expansion in the long range. The point of inflection at v≈110 yields a higher number of vibrational levels below the dissociation. Note that the earlier appli- cation of the NDE equations based solely on vibrational term values up to v≤7231and lacking rotational characterization significantly overestimates the dissociation limit (violet solid line in Fig. 2). The RKR potential obtained with the converged GvandBvval- ues in Tables IV and V is depicted in Fig. 3 (NDE, red solid line). The turning points from the RKR inversion procedure are marked with red dots in the figure. In addition, rotationless vibrational wave functions are shown for a few vibrational levels ( v= 0, 20, 50, and 102). The wave function for the highest observed vibrational level atv= 102 shows clearly that the molecule remains predominantly close to the outer turning point. The probability of residing close to the inner wall is much smaller. Obviously, Franck–Condon factors for vertical transitions from low-lying vibrational levels are strongly impeded. The optical–optical double-resonance method applied TABLE V . Molecular constants for65Cu63Cu obtained by the present analysis. All values are in cm−1. v′′Gv Gv(RKR) ΔGv Bv Bv(RKR) ΔBv Dv×108Hv×1013Lv×1018 0 0.000 0(1) 0.046 −0.046 0.106 795 (2) 0.106 800 −0.000 005 7.028 −0.144 −0.015 1 262.357 3(9) 262.413 −0.055 0.106 199 (2) 0.106 200 −0.000 001 7.024 −0.155 −0.013 2 522.705(1) 522.753 −0.049 0.105 593 (3) 0.105 599 −0.000 007 7.019 −0.160 −0.015 3 781.030(1) 781.076 −0.046 0.104 993 (3) 0.104 999 −0.000 006 7.015 −0.161 −0.018 36 8 207.34(2) 8 207.392 −0.055 0.085 7 (1) 0.085 799 −0.000 128 7.297 −0.499 −0.190 37 8 399.50(2) 8 399.557 −0.053 0.085 2 (1) 0.085 207 0.000 007 7.323 −0.507 −0.223 38 8 589.71(2) 8 589.772 −0.064 0.084 7 (1) 0.084 614 0.000 079 7.351 −0.513 −0.256 39 8 777.97(2) 8 778.033 −0.060 0.084 18 (10) 0.084 018 0.000 159 7.383 −0.517 −0.291 42 9 331.07(3) 9 331.033 0.039 0.081 8 (2) 0.082 221 −0.000 459 7.498 −0.526 −0.401 44 9 689.85(2) 9 689.812 0.035 0.080 8 (2) 0.081 013 −0.000 189 7.594 −0.530 −0.474 45 9 866.23(2) 9 866.213 0.017 0.080 3 (1) 0.080 405 −0.000 098 7.649 −0.532 −0.510 48 10 383.35(2) 10 383.355 −0.001 0.078 72 (10) 0.078 569 0.000 147 7.840 −0.546 −0.610 49 10 551.71(2) 10 551.684 0.022 0.077 98 (9) 0.077 952 0.000 024 7.913 −0.554 −0.640 51 10 882.22(3) 10 882.196 0.025 0.076 7 (2) 0.076 708 0.000 016 8.075 −0.580 −0.694 52 11 044.38(3) 11 044.356 0.026 0.076 2 (2) 0.076 081 0.000 115 8.164 −0.598 −0.717 53 11 204.46(3) 11 204.436 0.023 0.075 5 (1) 0.075 451 0.000 036 8.258 −0.620 −0.738 54 11 362.45(3) 11 362.422 0.025 0.074 9 (1) 0.074 816 0.000 074 8.358 −0.647 −0.757 57 11 823.87(5) 11 823.670 0.200 0.072 3 (4) 0.072 886 −0.000 612 8.692 −0.761 −0.805 60 12 265.60(8) 12 265.518 0.078 0.071 2 (6) 0.070 908 0.000 327 9.082 −0.938 −0.850 61 12 408.58(8) 12 408.409 0.170 0.069 5 (6) 0.070 236 −0.000 763 9.225 −1.012 −0.869 83 14 945.40(9) 14 945.419 −0.021 0.052 9 (3) 0.052 739 0.000 152 15.439 −6.868 −6.696 84 15 030.47(2) 15 030.484 −0.010 0.051 85 (7) 0.051 754 0.000 098 15.984 −7.580 −7.748 85 15 112.57(7) 15 112.639 −0.069 0.051 0 (3) 0.050 745 0.000 233 16.573 −8.387 −9.013 86 15 191.9(1) 15 191.843 0.053 0.049 5 (4) 0.049 709 −0.000 160 17.210 −9.306 −10.541 87 15 268.14(4) 15 268.057 0.082 0.048 43 (9) 0.048 647 −0.000 219 17.902 −10.358 −12.405 88 15 341.25(6) 15 341.239 0.016 0.047 5 (2) 0.047 556 −0.000 090 18.654 −11.567 −14.699 89 15 411.35(8) 15 411.346 0.001 0.046 4 (3) 0.046 433 −0.000 049 19.475 −12.965 −17.543 95 15 765.2(1) 15 765.098 0.079 0.038 8 (4) 0.038 937 −0.000 115 26.472 −28.495 −61.662 96 15 812.51(8) 15 812.541 −0.029 0.037 7 (3) 0.037 538 0.000 164 28.140 −33.245 −79.287 99 15 934.6(1) 15 934.501 0.112 0.032 9 (4) 0.033 025 −0.000 152 34.574 −56.007 −187.234 100 15 968.3(1) 15 968.312 −0.037 0.031 6 (4) 0.031 405 0.000 243 37.365 −68.339 −260.671 101 15 998.7(1) 15 998.702 0.007 0.029 7 (4) 0.029 721 −0.000 053 40.603 −84.766 −373.519 102 16 025.59(8) 16 025.693 −0.101 0.027 9 (3) 0.027 971 −0.000 103 44.400 −107.231 −553.902 J. Chem. Phys. 153, 244305 (2020); doi: 10.1063/5.0028908 153, 244305-8 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Adiabatic potential energy functions for the ground state of Cu 2. The green line is a cubic spline interpolation of the energy points calculated ab initio (“N” calculations, see text); the black line is for the “NpC” calculations. The red line is the result from the RKR-evaluation of the raw experimental data. Turning points for the experimentally determined vibrational term values (Tables IV and V) are indicated by red dots. The largest evaluated outer turning point is r= 573 pm at v′′= 102. Dissociation occurs at De≈16 270 hccm−1obtained with the “near- dissociation equation” (NDE). Vibrational wave functions (from evaluation of the RKR potential for J= 0) are shown for v′′= 0, 20, 50, and 102 (see also Tables IV and V and text for further explanations). in this study circumvents this hindrance by taking advantage of gateway states of mixed J ∼G character, which enable optical pump- ing from initially low ( v= 0–2) to high ( v≥62) lying vibrational states. V.AB INITIO RESULTS A. Ground state potential energy function The black and green solid lines shown in Fig. 3 are spline interpolations Vspline of the X1Σ+ gground state energies calculated ab initio within the “NpC” and “N” scheme, respectively. The ref- erence energy is −3307.4167 Ehfor the “NpC” and −3307.4174 Eh for the “N” calculation, both occurring at r= 220 pm. The dissoci- ation energy in the X1Σ+ gground state is approximately De≈16 700 hccm−1for the “NpC” and 16 600 hccm−1for the “N” calculation. Close to equilibrium, the RKR and ab initio potential energy functions agree well on the scale of the graph. The agreement is also reasonably good asymptotically for the potential function obtained from the “N” calculation (green line), which leads to a dissociation energy that is about 300 hccm−1larger than the RKR result. From the “N” calculations, one obtains an interpolated equilibrium bond length of 222 pm, and the first three vibrational terms Gvfor the 63Cu 2isotopologue are 272 cm−1, 533 cm−1, and 788 cm−1forv= 1,2, and 3, respectively, while the RKR potential yields the same value for the equilibrium bond length and term values that are 8 cm−1, 6 cm−1, and 1 cm−1lower (see Table IV). The terms from the “NpC” calculations are 258 cm−1, 519 cm−1, and 779 cm−1forv= 1, 2, and 3, respectively. Qualitatively, the potential function from the “N” cal- culation hence agrees fairly well with the RKR result. We understand the poor agreement of the “NpC” and RKR potential functions in the intermediate region between 300 pm and 500 pm as stemming from the lack of accuracy of the former, which is induced by a too strong compromise in attempting to optimize higher lying states leading to the “sp” asymptote, as discussed above. A comparison with results from previous studies might be use- ful. In Ref. 72, the “experimental” RKR potential for the electronic ground state of K 2is compared with a theoretical potential func- tion obtained from a one-electron pseudo-potential treatment of the electronic structure. The result presented in Fig. 1 of that work indi- cates an apparent good agreement. However, the lines for the two potentials compared in that figure are scaled such that the asymp- totes are identical, leaving a difference of about 200 hccm−1at the equilibrium. When the potentials are shifted so that the equilibrium energies are equal, as in Fig. 3, the ab initio potential turns out to be flatter than the RKR potential. In Fig. 2 of Ref. 73, a potential function obtained from a very high level ab initio calculation for the ground state of F 2is com- pared with a RKR potential. The agreement is perfect. Closer analysis of Table X of Ref. 74 shows that the excellent agreement is related to inclusion of correlation energy obtained from an extrapolation to the full configuration interaction (FCI) and complete basis set limit that the authors of those works dubbed “correlation energy extrapolation by intrinsic scaling (CEEIS).” This additional correla- tion energy that, in essence, mimics FCI has an important variation of the order of 10 m Eh(10 millihartree, 1 m Eh≈219hccm−1) toward smaller interatomic distances, which effectively renders the theoretical potential energy function stiffer, leading to a good agree- ment with the “experimental” RKR potential similar to the potential function from the “N” calculations for Cu 2in the present work. Cur- rently, the implementation of CEEIS to the copper dimer case is not practicable, however. We note that, while in F 2, spin–orbit coupling also contributes to a variation on the order of m Ehalong the ground state poten- tial energy function (see Table IV in Ref. 73), this coupling is much smaller in the ground state of Cu 2from the present calculations, which is also confirmed in Ref. 75. B. Dissociation energy The dissociation energies are De≈16 700 hccm−1for the “NpC” and 16 600 hccm−1for the “N” MRCI-calculation. These values are about 400 hccm−1and 300 hccm−1larger, respectively, than the highly accurate value obtained from the measurements pre- sented in the present work and the corresponding value of the RKR- potential. Table VI presents other theoretical results for the copper dimer. The best theoretical result ( De≈16 299 hccm−1) seems to be from the CCSD(T) calculation of Ref. 75 in which a pseudo-potential approach was used in connection with extrapolation to the basis set limit. The best value obtained from the valence bond calculation in Ref. 30 is De≈14 235 hccm−1and deviates significantly from the present theoretical and experimental results. J. Chem. Phys. 153, 244305 (2020); doi: 10.1063/5.0028908 153, 244305-9 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE VI . Dissociation energies of Cu 2from theory. De/hccm−1Method Authors References 16 700 MRCI,’NpC’ This work 16 600 MRCI,’N’ This work 16 063 CCSD(T), AVQZ-DK This work 16 096 CCSD(T), aug-cc-pVQZ-PP Peterson, Puzzarini75 16 299 CCSD(T), pseudo-potential Peterson, Puzzarini75 14 235 Valence bond Radenkovi ´cet al.30 In addition to the MRCI treatment, all electron CCSD(T) calcu- lations were carried out using the same AVQZ-DK basis set, which is comparable to the aug-cc-pVQZ-PP basis of Ref. 75. The result- ing energies are −3307.4736 Ehfor Cu 2(X1Σ+ g) at 223 pm and −1653.7002 Ehfor Cu(4s), rendering De≈16 063 hccm−1. Note that the result from Ref. 75 using the equivalent basis is De≈16 096 hc cm−1. The pseudo-potential treatment is thus only slightly superior to the treatment adopted in the present work. The CCSD(T) calcu- lations are inappropriate to render the potential energy functions depicted in Fig. 3, however, because of the multi-configurational character of the wave function for r→∞, as discussed below. We might conclude that, albeit rather costly and involved, cal- culations at the MRCI or CCSD(T) level are necessary to warrant an accuracy of at least 300 hccm−1to 400 hccm−1for the calculated value of the dissociation energy (1 kcal mol−1). C. The nature of the Cu–Cu bond With these results in mind, we may now turn to the discus- sion of the nature of the bond in the copper dimer. The MRCI wave functions are of the general form Ψj≈∑ ICIjΦI(CSF), (10) where ΦI(CSF)are spin and symmetry adapted configuration state functions (CSFs). Table VII gives the main coefficients for two wave functions of1Σ+ gcharacter having the lowest energies. CSF 1 describes essentially a σg[4s]-bond, CSF 2 is the σu[4s] anti-bonding configuration state, CSF 3 describes an attenuatedσg[4s]-bond and an enhanced σg[3dz2]-bonding configuration state, and CSF 4 is the anti-bonding state corresponding to CSF 3. Close inspection of Table VII shows that the ground state (state 1) is essentially a single-configuration state with a doubly occupied σg[4s] orbital, a typical σ-bond case generated by two 4s frontier orbitals. Roughly, the Cu–Cu bond is to 90% a σg[4s]-bond and to 10% a σg[3dz2]-bond. The CSFs in Table VII are molecular orbital configuration state functions. They may be expressed in terms of valence bond covalent and ionic states as follows: Φ1(CSF)=Φ(ion)[4s]+Φ(cov)[4s]√ 2, (11) Φ2(CSF)=Φ(ion)[4s]−Φ(cov)[4s]√ 2, (12) Φ3(CSF)=Φ(cov)[3d4s]−Φ(ion)[3d4s]√ 2, (13) Φ4(CSF)=Φ(cov)[3d4s]+Φ(ion)[3d4s]√ 2, (14) where Φ(ion)[4s] and Φ(cov)[4s] are ionic and covalent valence bond configuration states, respectively, involving two 4s frontier orbitals. Φ(ion)[3d4s] and Φ(cov)[3d4s] are ionic and covalent valence bond configuration states, respectively, involving a pair of 4s and 3d z2 frontier orbitals. TABLE VII . Leading configuration states in the two lowest1Σ+ gstates of Cu 2with their expansion coefficients at equilibrium and the dissociation asymptote and relevant orbital occupation scheme. Equilibrium Asymptote CSF Occupation scheme CI1 CI2 CI1 CI2 Φ1(CSF)σg[3dz2]2σg[4s]2σu[3dz2]20.874 −0.068 0.665 −0.084 Φ2(CSF)σg[3dz2]2σu[3dz2]2σu[4s]2−0.096 0.054 −0.640 0.085 Φ3(CSF)σg[3dz2]2σg[4s]2σu[3dz2]1σu[4s]10.082 0.874 0.078 0.644 Φ4(CSF)σg[3dz2]1σg[4s]1σu[3dz2]2σu[4s]20.012 0.217 0.081 0.646 J. Chem. Phys. 153, 244305 (2020); doi: 10.1063/5.0028908 153, 244305-10 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Combining Eqs. (11)–(14) with the coefficients from Table VII, one obtains, at equilibrium, Ψ1≈0.686 Φ(cov)[4s]+ 0.066 Φ(cov)[3d4s] + 0.550 Φ(ion)[4s]+ 0.049 Φ(ion)[3d4s], (15) Ψ2≈−0.086 Φ(cov)[4s]+ 0.771 Φ(cov)[3d4s] −0.010 Φ(ion)[4s]+ 0.465 Φ(ion)[3d4s]. (16) At equilibrium, state 1 is hence a nearly 50% mixture of the covalent and ionic valence bond states formed by the two 4s frontier orbitals with normalized weights w(cov)≈0.6862+ 0.0662∝61% and w(ion)≈0.5502+ 0.0492∝39%. At the asymptote, combining with the coefficients from Table VII, one obtains Ψ1≈0.923 Φ(cov)[4s]+ 0.112 Φ(cov)[3d4s] + 0.018 Φ(ion)[4s]+ 0.002 Φ(ion)[3d4s], (17) Ψ2≈−0.120 Φ(cov)[4s]+ 0.912 Φ(cov)[3d4s] + 0.001 Φ(ion)[4s]+ 0.001 Φ(ion)[3d4s]. (18) Atr→∞, state 1 represents hence essentially the dissocia- tion of a bond formed by two 4s orbitals (“ss”-channel), while state 2 describes the dissociation of a bond formed by the 4s and 3d z2 orbitals (“sd”-channel). Not shown in Table VII is that, at the asymptote, the ground state results from an important mixing of the reference states, which describe the “ss” and “sd”-channels at the multi-configurational self-consistent-field level of calculation. In a single reference multi- configurational interaction calculation, this mixing would not appropriately be taken into account, and consequently, the energy at the asymptote would potentially be considerably higher than that calculated here, leading to a larger bond dissociation energy. D. Discussion The description of the electron pair bond as a superposition of covalent and ionic states was originally proposed by Pauling76within valence bond theory. When there is an overwhelming weight of one of the two forms in the total wave function, the bond may be termed either covalent orionic , depending on whether the covalent or ionic configuration state functions are preponderant. When the weights of these forms become similar, the bond is better described as a charge- shift bond , which is a typical situation for electron rich bonding partners such as in F 2.29Our results reported in Eqs. (15) and (16) indicate that the bonding in Cu 2should be considered to be of the charge-shift type, which is in agreement with results from valence bond calculations.30In the latter work, however, the weight of the covalent state (structure 1 in Table II of Ref. 30) is 71%, whereas that of the ionic states (structures 2 and 3) is only 26%. The weight of the covalent part in a valence bond calculation, where frontier orbitals rather than molecular orbitals are optimized, is somewhat larger than that obtained within a molecular orbital calculation, suchas in the present work, where molecular orbitals rather than fron- tier orbitals are optimized. The best value obtained in Ref. 30 for the binding energy in Cu 2is 14 235 hccm−1(40.7 kcal mol−1; Table I of Ref. 30). From Table VII, one concludes that the configuration state functions involving singly occupied 4s orbitals contribute to (non- normalized) ∼(0.8742+ 0.0962)∼77% of the ground state wave function, whereas those involving each singly occupied 4s and 3d z2 orbital contribute to less than 1%—the remaining 22% are dis- tributed among many excited configuration state functions not shown in the table. The small contribution from the singly occu- pied 4s and 3d z2orbitals is only one part of the “orbital splitting” inferred in Ref. 30. By this procedure, the occupation of two dif- ferent orbitals is meant, which are localized on the same center. It allows us to improve on seizing electron correlation. The other part stems from 4s and 4p zcorrelation, visible in the MOLPRO output, but not influential in the characterization of the wave function. Both the covalent and ionic portions of the ground state wave function of dicopper stem from the interaction between the 4s orbitals. One should also note that a similar analysis of the MRCI ground state of dihydrogen yields w(cov)≈58% and w(ion)≈42%. In Ref. 29, the covalent part in the total wave function of H 2isw(cov) ≈76%. Hence, the weight of the covalent part in the total wave func- tion of H 2turns out to be larger in a valence bond calculation than in a molecular orbital calculation, too, similar to the case of Cu 2. How- ever, while the bond in the latter is considered to be of charge-shift type,30that of the former is considered to be of covalent charac- ter.29From the analysis of the present molecular orbital calculations, both molecules would be considered to be of charge-shift type. One obvious consequence of this analysis is that one cannot rely on the relative weights of the covalent and ionic portions of the wave func- tion alone to conclude about the nature of the bond. Analysis of bond critical points is necessary in addition, as well as the parti- tioning of the total energy into pure covalent, ionic, and charge-shift contributions. Another consequence is that the charge-shift attribute of a bond depends on the calculation method, as much as does the analysis of the bond critical points and energy partitioning. These are not observable quantities, however, and so, the “disclosure of the true nature of the chemical bond,” as suggested in Ref. 29, might not be possible from experimentally available information. E. Alternative analytical potential energy function The RKR potential energy function discussed in Sec. IV B is one possibility to represent the ground state potential of Cu 2in terms of turning points at given energies. Such representations are very accurate but not easily available, in general, and deriving them for polyatomic molecules is practically not feasible. For polyatomic molecules, one often needs to resort to simplified, compact ana- lytical representations of potential energy surfaces in one or more dimensions. One dimensional analytical potential energy functions along a bonding coordinate, such as the Morse potential, are valuable ingredients of representations of potential energy hyper-surfaces77 and it is useful to test how accurately analytical representations can describe the interaction potential of molecules such as Cu 2, where the spectroscopic structure can be very well assessed experimentally up to the dissociation energy. J. Chem. Phys. 153, 244305 (2020); doi: 10.1063/5.0028908 153, 244305-11 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp In the case of Cu 2, the Morse potential was found to be insuf- ficiently flexible to describe simultaneously even the lowest spec- troscopic terms and the dissociation energy. Instead, the following analytical function, Vr(r), which describes a Morse-like potential, is more flexible: Vr(r)=Ve(1 +w(r)[y(r)[y(r)w(r)−2]]+e(r)), (19) where w(r)=exp(−B(r−re)3 r2), (20) y(r)=exp(−A(r−re)), (21) e(r)=Eexp⎛ ⎝−[R6 r]6⎞ ⎠. (22) The quantities Ve,re,A,B,E, and R6are adjustment parame- ters. They are varied in the definition domains Ve≥0,re≥0,A≥0, B≥0, 0≤E≤1, and R6≥0. The function Vr(r) was originally pro- posed in Ref. 78, where graphical representations are discussed for varied forms that this function can take, depending on the choice of parameter values. The term described by the function e(r), Eq. (22), captures the correct asymptotic Vr∼−C6/r6behavior related to the dispersion or induction interaction between two neutral fragments upon dissociation of a diatomic molecule, as discussed in Refs. 78 and 79. The adjustment to the ab initio energy points was carried out with a modified version of the Levenberg–Marquardt algo- rithm80in which additional non-linear constraints among adjustable parameters can be incorporated. These are the dissociation energy, De≈Ve(1 +E), (23) theC6coefficient, C6=VeE R 66, (24) and the Morse potential expression for the fundamental transition, ˜ν1≈ωe−2ωexe=A 2πc√ 2Ve μ−h A2 4π2cμ, (25) which is only an approximation for the actual value that can be obtained for it from the solution of the Schrödinger equation [in Eq. (25),μis the reduced mass of the dicopper molecule]. Optimal parameter values are collected in Table VIII. To obtain these values, the aforementioned non-linear conditions were enforced with De= 16 270 hccm−1,C6= 1 154 000 hccm−1Å6, and ˜ν1=264 cm−1. The function Vris depicted in Fig. 3. It overlaps almost per- fectly with the RKR potential in the vicinity of the equilibrium dis- tance as well as in the asymptotic region. The agreement is some- what poorer between 350 pm and 500 pm. Vryields accurate fitTABLE VIII . Parameter defining the Vrpotential energy function of Eq. (19) and their values. Ve/hccm−116 245 re/Åa2.220 A/Å−11.428 8 B/Å−10.497 17 E 0.001 52 R6/Å 6.0b a1 Å = 100 pm = 10−10m. bManually fixed value. parameters from a Dunham analysis, which are reported in the col- umn “theory (this work)” in Table I. The theoretical parameter val- ues deviate within 1% from the experimental ones with the exception ofωeye,ωeze, andγe, which could be related to the poor representa- tion of the potential by the analytical function in the intermediate range of bond distances. All in all, the function defined by Eq. (19) can be considered to yield a satisfactory overall description of the potential. VI. CONCLUSIONS In this work, 347 double-resonant transitions to excited vibra- tional levels in the X1Σ+ g(0+ g)state have been measured with rota- tional resolution. By accessing high-lying perturbed intermediate states, SEP-type transitions to vibrational levels of the ground state X1Σ+ g(0+ g)up to v′′= 102 were accessible. The substantially increased accuracy of the measured band positions and the extended vibrational energy range up to ≈98% of the dissociation energy allow for a precise determination of the potential function and the dissociation energy. The NDE analysis applied simultaneously for the63Cu 2and 65Cu63Cu isotopologues reproduces the measured level energies within the experimental accuracy and leads to the determination of a faithful Rydberg–Klein–Rees (RKR) representation of the potential function up to the dissociation. A compact analytical representation in the form of Eq. (19) is also proposed, which yields slightly less accurate ro-vibrational term values in the higher energy domain, while reproducing the experimental dissociation energy and thus being valuable for its simplicity and compactness. In parallel, high-level ab initio calculations at the internally contracted multi-reference configuration-interaction level of theory have been performed. Calculations based on an active space con- taining the ground and singly excited states that lead to the neu- tral “ss” and “sd” asymptotes yield a ground state potential curve in good agreement with the experiment. Attempts to account for the higher lying states leading to the “sp” limit without inclusion of higher lying orbitals, such as with the “NpC” calculations reported in Ref. 34, compromise the accuracy of the ground state potential in the asymptotic region. To obtain the missing correlation energy calcula- tions within larger configuration interaction spaces, approaching the complete basis set limit, such as with the CEEIS method,74would be required. Such calculations, however, are not currently feasible for Cu 2. We hope to be able to improve, in future work, the accuracy of J. Chem. Phys. 153, 244305 (2020); doi: 10.1063/5.0028908 153, 244305-12 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp both the ground and excited state calculations of dicopper within a more adaptive approach of the full configuration interaction space by a selective inclusion of 4p orbitals. The congruence of the theoretical and experimental results sheds some light on the nature of the Cu–Cu bond. We clearly showed that bond formation in Cu 2is governed by the interaction of two 4s frontier orbitals leading to a doubly occupied σ-orbital in the ground state at equilibrium. Close to equilibrium, the wave function thus obtained is an almost 1:1 mixture of covalent and ionic con- figuration state functions, which is quite typical for homonuclear bonds of essentially monovalent species. Configurations involving pairs of 4s and 3d orbitals play a role in the assessment of the corre- lation energy but are of minor importance with respect to the bond type, which in valence bond theory would be best termed a mixed covalent-ionic bond. The attribute “charge-shift” was given to the bond type in dicopper as a result of valence bond calculations.30It is compati- ble with the findings from the present calculations using molecular orbital theory. However, the bond in dihydrogen could be classi- fied in the same way within molecular orbital theory, if solely the weights are considered by which the ionic and covalent configura- tion state functions intervene in the multi-reference configuration interaction wave function of the ground state at the equilibrium dis- tance. Yet, dihydrogen is said to be covalent within valence bond theory.29Clearly, bond types are calculation method dependent attributes depending on the system. Observables must not depend on the calculation method, when calculations are sufficiently accu- rate. To the best of our knowledge, a direct experimental obser- vation of the type of a homonucleus chemical bond has not yet succeeded. SUPPLEMENTARY MATERIAL See the supplementary material for observed and calculated lines for the two main isotopologues of Cu 2in the range of vibra- tional levels from v′′= 36–98 and v′′= 36–102 for63Cu 2and 65Cu63Cu, respectively, a table that contains also the PUMP tran- sitions to the intermediate excited state for the SEP-type double resonance measurements, and a table that lists the expansion param- eters for the 192 model calculations, which were performed to determine NDE polynomials that reproduce the term energies for both isotopologues simultaneously to within an accuracy of rms <0.1 cm−1. 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5.0029951.pdf

J. Appl. Phys. 129, 035105 (2021); https://doi.org/10.1063/5.0029951 129, 035105 © 2021 Author(s).Impacts of atomic and magnetic configurations on the phase stability of Fe– Pd shape memory alloys: A first-principles study Cite as: J. Appl. Phys. 129, 035105 (2021); https://doi.org/10.1063/5.0029951 Submitted: 17 September 2020 . Accepted: 01 December 2020 . Published Online: 15 January 2021 Chun-Mei Li , Shun-Jie Yang , Yang Zhang , and Ren-Zhong Huang ARTICLES YOU MAY BE INTERESTED IN First-principles study on photoelectric and transport properties of CsXBr 3 (X = Ge, Sn) and blue phosphorus van der Waals heterojunctions Journal of Applied Physics 129, 035302 (2021); https://doi.org/10.1063/5.0036217 Interpreting Kelvin probe force microscopy on semiconductors by Fourier analysis Journal of Applied Physics 129, 034301 (2021); https://doi.org/10.1063/5.0024073 A numerical fitting routine for frequency-domain thermoreflectance measurements of nanoscale material systems having arbitrary geometries Journal of Applied Physics 129, 035103 (2021); https://doi.org/10.1063/5.0030168Impacts of atomic and magnetic configurations on the phase stability of Fe –Pd shape memory alloys: A first-principles study Cite as: J. Appl. Phys. 129, 035105 (2021); doi: 10.1063/5.0029951 View Online Export Citation CrossMar k Submitted: 17 September 2020 · Accepted: 1 December 2020 · Published Online: 15 January 2021 Chun-Mei Li,a) Shun-Jie Yang, Yang Zhang, and Ren-Zhong Huang AFFILIATIONS College of Physics Science and Technology, Shenyang Normal University, 253 Huanghe North Street, Shenyang 110034, China a)Author to whom correspondence should be addressed: cmli@synu.edu.cn ABSTRACT The effects of local atomic and magnetic configurations on the phase stability and elastic property of the face-centered cubic (fcc) and two body-centered tetragonal [face-centered tetragonal (fctI) and fctII, with 0 :9,c=a,1 and 0 :71,c=a,0:9, respectively, in the fct unit cell] phases of Fe 1/C0xPdx(0:28/C20x/C200:34) shape memory alloys are systematically investigated by using the first-principles exact muffin-tin orbital method in combination with the coherent potential approximation. It is shown that, considering four types of atomic configurations in a fcc unit cell, the two with one random sublattice are both preferable in each xbelow 300 K. When T¼300 K, the one with three random sublattices also changes to be stabilized for x/C200:30, whereas that with four random sublattices becomes stable in most of these alloys until T/C21600 K. Upon tetragonal distortions, in these fully disordered alloys, both the fctI and fctII phases are unstable. The fctI phase is found for 0 :29/C20x/C200:33, having only the configuration with one random sublattice on the same layer with the Pd site in the unit cell, whereas the fctII phase is obtained for x/C200:30, possessing all the configurations with one, two, and three random sublattices. These results representing the phase diagram of these alloys, their determined equilibrium lattice parameters, and elastic constants of the threephases at 0 K are in line with the experimental and theoretical data, and their estimated structural ( T M) and magnetic ( TC) transition tem- peratures are also close to the experimental data. Adding 4% magnetic disorder in Fe 0:70Pd0:30, the fctII structure is effectively prevented, whereas the thermoelastic martensitic transformation of fcc –fctI can still be retained at 0 K. Published under license by AIP Publishing. https://doi.org/10.1063/5.0029951 I. INTRODUCTION Ferromagnetic shape memory (FSM) alloys of Fe –Pd have attracted a lot of experimental and theoretical research. To date, magnetic field induced reversible strain of about 3% has been observed in these alloys.1Although it is lower than that (about 10%)2observed in Ni –Mn –Ga, Fe –Pd FSM alloys possess many other excellent performances, such as good biocompatibility, highresistance to corrosion, high ductility, and low brittleness. 3–5They have significant potential for eventual technological applicationsin actuators or sensors. 4,6Such Fe –Pd thin films have been the focus of research primarily for shape memory devices in bioapplications.7,8 The FSM effect in Fe –Pd alloys originates from the reversi- ble martensitic transformation (MT) from face-centered cubic (fcc) to body-centered tetragon al (bct), which was measured with0:9,c=a,19,10in its equivalent face-centered tetragonal (fct) unit cell, and is generally called “fct”(named “fctI”here) struc- ture since it is close to the fcc one. Similar to Ni –Mn –Ga FSM alloys, this structural phase transition is accompanied by a remarkable softening of the tetragonal shear elastic constant [C0¼(C11/C0C12)=2] and a divergent increase of the elastic anisot- ropy ( A¼C44=C0) of the cubic parent phase,11,12when Tdecreases toward the MT temperature ( TM). It has been confirmed that a low twin boundary energy together with a high elastic energy is the key prerequisite for the nanotwinning of the adaptive martensite in Fe0:70Pd0:30alloys.13However, it is unfortunate that the MT from fcc to fctI in Fe 1/C0xPdxalloys takes place only in a very narrow com- positional range (0 :29,x,0:32).10Moreover, further irreversible phase transformation upon cooling, which results in another bct (with 0 :71,c=a,0:9 in its equivalent fct unit cell, called “fctII ” here) or the body-centered cubic (bcc) phase, also tends to occur,9Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 035105 (2021); doi: 10.1063/5.0029951 129, 035105-1 Published under license by AIP Publishing.similar to the alloys with x/C200:29 where the fcc –fctII or fcc –bcc phase transition generally happens at low temperatures. Then, in the alloys with 0 :29,x,0:32, the stability of the fctI phase is restricted by the above two phase transitions, which deterioratestheir shape memory property. Discovering the parameters affect-ing their thermoelastic MT of fcc –fctI and then trying to find out ways to expand the relative stability of the fctI martensite are, thus, desirable. Based on the first-principles calculations, the crystal structure of Fe 1/C0xPdxalloys is found strongly dependent on their local atomic configurations. FctI-Fe 0:50Pd0:50was calculated in the ordered but not the disordered composition,14,15in line with the experimental measurements.16In the off-stoichiometric composi- tions (0 :29,x,0:32), upon tetragonal distortions,14their par- tially disordered L12structure (with the fcc unit cell where the face-centered positions occupied by one atom but the cubic corners occupied by another atom, such as Cu 3Au) was confirmed to transit to the fctII structure as well as the ordered L12/C0Fe0:75Pd0:25 alloy,17whereas their fully disordered structure was generally found to transit directly to the ground bcc structure.13,14,18Gruner et al. had also investigated the influence of local lattice distortions and found that the lattice relaxation could even change the ground state of the disordered Fe 65:7Pd34:3alloy from fcc to fctII.18Generally, in a disordered alloy, the local atomic configurations can be varied bymany factors, such as the quenching and annealing conditions, ambient temperatures, and so on, which can further change the long-range atomic disordering degree and then the stability of thesystem. Thus, although in bulk samples of these off-stoichiometricFe 1/C0xPdxalloys, the fctI phase has been confirmed experimentally with the disordered structure by Matsui and co-workers;3,19–21this phase is still hardly obtained theoretically in their ideally fully dis- ordered compositions.14,18,19The further investigations of more dif- ferent atomic configurations and their effects on the fcc –fctI–fctII phase transitions are, therefore, essential to design these alloys withdesirable shape memory property. Dependent on the site occupations, the local magnetic spins of atoms in a cubic structure can also change under tetragonallattice distortions. For example, in the L2 1phase of Ni –Mn –In FSM alloys,22the excess Mn atoms on the In site are ferromagnetic (FM) coupling with those on their own sites, whereas in the tetrag- onal martensite, they turn out to be anti-ferromagnetic (AFM) cou- pling with each other. In the disordered L12/C0Fe1/C0xPdxalloys,14the magnetic moments of Fe atoms on two different layers have beencalculated to be the same in the cubic structure but different in the tetragonal one. In these alloys, the local magnetic configurations of Fe atoms might be then also associated with their phase stability. In this paper, we will systematically investigate the effects of dif- ferent local atomic and magnetic configurations on the phase stabil-ity and elastic property of fcc-, fctI-, and fctII-Fe 1/C0xPdx (0:28/C20x/C200:34) FSM alloys and then try to explore the physical mechanisms driving their x-dependent MT. The rest of the paper is arranged as follows: in Sec. II, we describe the first-principles method we used and the calculation details; in Sec. III, the site pref- erences of the three phases are determined for all the studied alloys, their equilibrium properties and elastic constants are presented, both the composition and temperature dependence of their free energiesare explored, and the influences of the atomic and magneticdisorders on the fcc –fctI–fctII phase transitions are discussed. Finally, we summarize the main results of this work in Sec. IV. II. METHODS AND CALCULATION DETAILS Based on density functional theory (DFT), all the present total energy calculations are performed by using the first-principlesexact muffin-tin orbitals (EMTO) method. 23–25EMTO is an improved screened Koringa –Kohn –Rostoker method. Within its theory, the single-electron Kohn –Sham equations are solved by use of Green ’s function technique, the effective potential is described by the optimized overlapping muffin-tin potential,23,26and the total energy is calculated by means of the full-charge density tech- nique.23With these improvements, the EMTO method is suitable to describe accurately the total energy change with respect to aniso-tropic lattice distortions. Moreover, compared to the other first-principles plane wave methods, it can also conveniently incorporate the coherent potential approximation (CPA), 25,27,28which greatly facilitates the calculations of the systems with chemical and mag-netic disorders. The accuracy of the EMTO –CPA method for the equation of state and elastic properties of disordered alloys hasbeen demonstrated in a number of former works. 14,25,28–31It is, therefore, chosen to investigate the disordered Fe 1/C0xPdxbinary alloys here, although within this method, ionic relaxations, whichmight be beneficial in view of the different atomic radii of Fe andPd, are not considered. For the present application, the exchange-correlation potential is described with the generalized gradient approximation (GGA) by Perdew, Burke, and Ernzerhof (PBE). 32The EMTO basis sets include s,p,d, and fcomponents, and the scalar-relativistic and soft-core approximations are adopted. The valence electrons aretreated with Fe 3 d 64s2and Pd 4 d10. Both the muffin-tin potential sphere radii ( RFe mtand RPd mt) and the atomic radii ( RFe wsand RPd ws)o n the Fe and Pd sublattices are optimized by the usual setup, i.e., allof them are equal to the Wigner –Seitz radius of the Voronoi poly- hedron around each site ( R ws), and they also do not change upon any volume-conserving (i.e., Rwsis fixed) lattice deformation adopted here. The usual setup also facilitates us to compare theenergies of each off-stoichiometric Fe 1/C0xPdxalloy with the different atomic configurations. Otherwise, if an unusual setup ( RFe mt=Rws, RPd mt=Rws,RFe ws=Rws,o r RPd ws=Rws) is chosen, the Fe and Pd atoms on the different sites may have different muffin-tin radii and atomic radii, which would further vary the energy differencesbetween each alloy with the different atomic configurations.Green ’s function is calculated for 32 complex energy points distrib- uted exponentially on a semicircular contour. In the one-center expansion of the full-charge density, the number of orbitals is trun- cated at 8. The Brillouin zone is sampled by a 17 /C217/C217 uniform k-point mesh without using any smearing technique. In a fcc unit cell, there are four sublattices: S 1(0.5, 0, 0.5), S 2 (0, 0.5, 0.5), S 3(0.5, 0.5, 0), and S 4(0, 0, 0). The fcc –fct phase tran- sition along the Bain path33is then obtained by expanding (con- tracting) the fcc lattice along one of the cubic axes ( c) and contracting (expanding) along the two others ( a). Here, aand cin both the fctI and fctII structures are defined by means of the face- centered unit cell as well. Since in the real case (in the long-range atomic ordering), a disordered Fe 1/C0xPdxalloy can be theoreticallyJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 035105 (2021); doi: 10.1063/5.0029951 129, 035105-2 Published under license by AIP Publishing.assumed to possess a structure with several different local atomic configurations mixing in some degree which may vary according toits production process, we consider four types of local atomic con- figurations here, in order to investigate the site preference of the three phases as well as the effects of the different atomic configura-tions on the fcc –fctI–fctII phase transitions. In a fcc unit cell, they are defined as shown in Table I . Types I and II mean the occupa- tions with only one random sublattice (S 1and S 3, respectively), III possesses three random sublattices (S 1,S2, and S 3, i.e., L12struc- ture), and IV corresponds to the fully disordered configuration.Upon tetragonal distortions, the symmetry corresponding to theconfiguration of II remains unchanged, whereas those of I, III, andIV change to be Pmmm (group 47), P4/mmm (group 123), and I4/mmm (group 139), respectively. Taken fcc-Fe 0:70Pd0:30alloy with the configuration of IV as an example, it has four sublatticeswith the same concentration of Fe 0:70Pd0:30. When the magnetic spins of the Fe and Pd atoms are fully ordered, its each sublattice isdescribed with Fe " 0:70Pd" 0:30, whereas when they are partially disor- dered,34it changes to be (Fe" 0:70Pd" 0:30)1/C0y=2(Fe# 0:70Pd# 0:30)y=2, where y is called the magnetic disordering degree. The equilibrium lattice parameters ( aand c/a), bulk modulus (B), and magnetic moments of both the cubic and tetragonal struc- tures are determined by fitting the calculated total energies vs volume (nine data points) to a Morse function.35The single-crystal elastic constants can be derived by the second derivative of the elec-tronic total energy with respect to strains. In our work, the volume-conserving lattice deformations and corresponding mathematical formula, which have been described clearly in the previous paper, 36 are adopted. The corresponding polycrystalline elastic constants are estimated by means of the Hill average method.23 The site preference is determined by comparing the formation energies ( Ef) per atom of Fe 1/C0xPdxalloy with the different site occupation configurations, which is evaluated by Ef¼Eel/C0(1/C0x)EFe/C0xEPdþTSmix, (1) where Eel,EFe, and EPdare the electronic total energies per atom of the alloy, bcc-Fe, and fcc-Pd, respectively, andTS mix(¼1 4kBTP4 i¼1[xilnxiþ(1/C0xi)ln(1/C0xi)], with xibeing the composition at each of the four sublattices and kBbeing the Boltzmann constant) is the chemical mixing entropy per atom ofthe alloy. The electronic entropy ( TS el) term is neglected since at ambient condition, the electronic temperature effect is negligible. The free energy ( F) of the alloy can be then simply evaluated with F¼EfþFvib, where the phonon vibrational free energy, Fvib,i sexpressed by Fvib¼/C0 kBT[D(ΘD=T)/C03ln(1/C0e/C0ΘD=T)]þ9 8kBΘD, (2) withΘDand D(ΘD=T) corresponding to the Debye temperature and the Debye function,35,37,38respectively. The temperature dependence of the magnetization ( M) and susceptibility ( χ) is evaluated using the Uppsala Atomistic Spin Dynamics (UppASD) program.39–42Within this method, the itiner- ant electronic system is mapped to an effective classical Heisenberg model, where the 0-K interatomic exchange parameters ( Jij, with i and jbeing 1, 2, 3, and 4, representing the four sublattices of Fe1/C0xPdxalloys, respectively) include the interactions between the atoms within the tenth-nearest neighbors, calculated using the magnetic force theorem43implemented in the EMTO –CPA program.23The motion of the magnetic moment is described using the Landau –Lifshitz –Gilbert (LLG) equation.39,40In our previous works, we have successfully evaluated the curie temperatures ( TC) of the L21phase of Ni 2Mn 1/C0xFexGa,44Ni2/C0xCoxMn 1:60Sn0:40,45 and Co 2Cr(Ga 1/C0xSix)31FSM alloys with the UppASD method. In the present application, all the parameters are set as in Ref. 31. III. RESULTS AND DISCUSSIONS A. Crystal structure of Fe 0.75Pd0.25 In order to ensure the accuracy of our calculations, we first compare the evaluated a,c/a,B, and magnetic moments per atom of Fe and Pd ( μFeand μPd) and those ( μ)o f L12- and L10/C0Fe0:75Pd0:25alloys with their previous first-principles full- potential linearized augmented-plane wave (FLAPW) results and the available experimental data, as shown in Table II . In both the structures, our aand c/avalues evaluated with the chosen parame- ters are in good agreement with the previous FLAPW results,17,46 and in the L10structure, they are also very close to the experimen- tal data. Since both the lattice thermal expansion and the atomic disorder can expand the volume of Fe 1/C0xPdxalloys,47,48the presentTABLE I. Four types of atomic configurations in a fcc unit cell of the disordered Fe1−xPdx(0.28≤x≤0.34) alloys. Type S 1 S2 S3 S4 Symmetry Group I Fe+Pd Fe Fe Pd P4/mmm 123 II Fe Fe Fe+Pd Pd P4/mmm 123III Fe+Pd Fe+Pd Fe+Pd Pd Pm /C223m 221 IV Fe+Pd Fe+Pd Fe+Pd Fe+Pd Fm /C223m 225 TABLE II. Equilibrium lattice parameters ( a, in Å, and c/a), bulk modulus ( B,i n GPa), and magnetic moments per atom of Fe and Pd ( μFeandμPd,i nμB) and those ( μ,i n μB)o f L12- and L10−Fe0.75Pd0.25, in comparison with the first- principles full-potential linearized augmented-plane wave (FLAPW) results from Refs. 17and 46and the experimental data from Refs. 7and 49. Phase Method a c/a B μμ Fe μPd L12 PBE 3.747 156.0 2.10 2.70 0.30 PBE( spd) 3.771 153.0 2.14 2.76 0.29 PBE(frozen) 3.787 160.3 2.14 2.76 0.29 LDA 3.626 202.9 PBEsol 3.675 174.6 FLAPW463.740 155.9 2.16 2.76 0.36 Exp.493.818 L10 PBE 4.058 0.761 168.2 2.18 2.88/2.58 0.36 FLAPW174.060 0.78 Exp.74.045 0.77Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 035105 (2021); doi: 10.1063/5.0029951 129, 035105-3 Published under license by AIP Publishing.avalue of L12/C0Fe0:75Pd0:25is a little smaller than the measured data of the disordered cubic alloy.49 The frozen-core approximation and the relatively small basis set (spd) tend to increase aofL12/C0Fe0:75Pd0:25inTable II , whereas the LDA50and PBEsol51approximations decrease abut increase B of the cubic alloy. In Fig. 1(a) , the equilibrium volume of the L12 phase is also shown to get a little bigger when it is calculated with a larger or a smaller RPd mtand RPd ws. Moreover, the “abnormal ”setting (RPd mt=1:0RwsorRPd ws=1:0Rws) can increase greatly c/aof the L10 structure in Fig. 1(b) . The calculations with the normal settings (RPd mt¼1:0Rwsand RPd ws¼1:0Rws)i n Figs. 1(a) and1(b), neverthe- less, well represent the aandc/avalues of the two structures, and in Fig. 1(b) , the evaluated electronic total energy (about /C00:44 mRy) of theL10phase relative to the L12one is also in line with the reportedtheoretical data (around /C00:55 to/C00:37 mRy).17,18Our previous calculations using the normal setting together with a 13 /C213/C213 k-point mesh successfully reproduce the elastic constants of L10/C0Fe0:50Pd0:50alloys as well.14In the present work, a 17 /C217/C2 17k-point mesh is chosen for the tetragonal structure as well as the cubic one. We then assume that the present EMTO calculations are reasonable enough to ensure the accuracy of the results of the off- stoichiometric Fe 1/C0xPdxalloys below, although the method has the limitation in the relaxation of atomic positions.18,52 B. Site preference of off-stoichiometric Fe 1−xPdx 1. The parent phase Figure 2 compares Efof the parent phase of Fe 1/C0xPdx (0:28/C20x/C200:34) alloys with the four types of atomic occupations of I –IV (EI f,EII f,EIII f, and EIV f). In Fig. 2(a) ,a t0 K , EI fand EII fare always the same in each xand with the increase of the atomic dis- ordering, Efwith a negative value increases in each alloy. At finite temperature, the chemical mixing entropy term, TSmix, should further prefer the relative stabilities of the more disordered configu- rations. As shown in Fig. 2(b) , at 300 K (around or above TMof these alloys), the differences between EI f(EII f),EIII f, and EIV fdecrease in all these alloys. Above x¼0:30,EI f(EII f) is still the lowest in each x, whereas for x/C200:30,EIII fturns out to be comparable with FIG. 1. Electronic total energy ( Eel)o f L12/C0Fe0:75Pd0:25alloys change with respect to Wigner –Seitz radius ( rws) (a) as well as that of the alloy change against c=aupon tetragonal distortions (b), calculated with the different potential sphere radius ( RPd mt) and atomic radius ( RPd ws) on the Pd sublattice. FIG. 2. x-dependence of formation energies ( EI f,EII f,EIII f, and EIV f) of the parent phase of Fe 1/C0xPdx(0:28/C20x/C200:34) alloys with the four types of atomic configurations of I –IV at 0 K (a) and 300 K and 600 K (b), respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 035105 (2021); doi: 10.1063/5.0029951 129, 035105-4 Published under license by AIP Publishing.EI f(EII f) and when x¼0:28, the former ( /C01:98 mRy) is even a little smaller than the latter ( /C01:97 mRy). Around the room tem- perature, the fully disordered configuration of IV is still the mostunstable in all these alloys. Furthermore, when Tincreases to 600 K, E IV fgets close to both EI f(EII f) and EIII fin each x, and for x/C200:32,Efeven slightly decreases with the increase of the atomic disordering in each alloy. Then, in the parent phase of these alloys,the configuration of IV is supposed to be stabilized for T/C21600 K. With increasing x,E falways decreases at each temperature, which is independent on the site occupation. The parent phase then gets more and more stable with the Pd addition in Fe 1/C0xPdxalloys. As shown in Table III , it is clear that the present avalues of the parent phase of Fe 1/C0xPdxalloys with the configuration of I (II) are also much closer to the measured data around the room tem-peature. 5,7,53The difference of abetween each alloy with the con- figurations of I (II) and III is less than 0.001 Å, whereas the configuration of IV tends to expand the volume of these alloys incomparison with all the other arrangements. 2. The tetragonal phase Figure 3(a) compares the EI f,EII f,EIII f, and EIV fvalues of the parent phase of Fe 0:70Pd0:30alloys change with respect to c=aupon tetragonal distortions. Here, EI f(EII f) corresponding to c=a¼1i sa s reference at both 0 K and 300 K. At 0 K, the atomic disorder alsodisfavors the stability of each tetragonal structure. Contrary to the parent phase, E I fand EII fare generally different in each c=a=1, which can be attributed that there μFe,μPd, and then μcorrespond- ing to the configurations of I and II are not the same in Fig. 3(b) . Thus, only the sublattices of S 1and S 2are equivalent for the excess Pd atom in each partially disordered tetragonal structure. As shown inFig. 3(a) , the minima of EI f,EII f, and EIII fare all around c=a¼0:75/C00:85, meaning the fctII structure.7,10Nevertheless, similar to the previous first-principles calculations,13,14,18EIV fhas its minimum value around c=a¼0:71, corresponding to the bcc structure. At 0 K, the EI f,EII f, and EIII fvalues corresponding to the fctII structure are estimated about /C00:14 mRy, /C00:14 mRy, and /C00:10 mRy, respectively. When Tincreases to 300 K, the corre- sponding EIII fdecreases to about /C00:21 mRy with a merely /C00:07 mRy lower energy than the evaluated EI fand EII f (/C00:14 mRy) of the structure. Thus, around and below the room temperature, the three partially disordered configurations of I –IIIare supposed to be all stabilized in the fctII phase of Fe 0:70Pd0:30 alloys. Shown in Fig. 3(a) , in the curves of EI f/differencec=aand EII f/differencec=a there is also a local minimum around c=a¼1:02 and 0.97, respec- tively, meaning the intermediate tetragonal phase. Here, this phase apparently favors the two configurations with one random sublat- tice at 0 K. Considered the four types of configurations, the fullydisordered fcc structure has been nevertheless confirmed to be FIG. 3. 0-K and 300-K formation energies ( EI f,EII f,EIII f, and EIV f) (a) and total magnetic moments ( μI,μII,μIII, and μIV) (b) per atom of Fe 0:70Pd0:30alloys with the four types of configurations of I –IV change with respect to c=aupon tetrago- nal distortions. EI f(EII f) corresponding to c=a¼1 is as reference at both 0 K and 300 K.TABLE III. Equilibrium lattice parameter ( a, in Å), bulk modulus ( B, in GPa), and magnetic moments ( μ,i nμB) of the parent phase of Fe 1−xPdx(0.28≤x≤0.34) alloys with the configurations of I (II), III, and IV , respectively, in comparison with the available experimental data from Refs. 5,7, and 53. aB μ x I (III) IV Exp.5,7I (III) IV I (III/IV) Exp.53 0.28 3.7514 (6) 3.761 3.75 164.4 (9) 165.7 2.06 (5/7) 2.05 0.29 3.7553 (6) 3.765 3.75 164.5 (8) 165.9 2.04 (4/5)0.30 3.7593 (6) 3.768 3.756 164.1 (8) 166.1 2.02 (0/2) 2.02 0.31 3.7631 (6) 3.772 164.3 (8) 166.2 2.00 (0/1) 0.32 3.7670 (4) 3.775 164.7 (8) 166.4 1.99 (8/9) 1.980.33 3.7701 (8) 3.779 3.76 165.0 (0) 166.6 1.97 (7/7)0.34 3.7741 (9) 3.782 165.3 (1) 166.7 1.95 (5/5)Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 035105 (2021); doi: 10.1063/5.0029951 129, 035105-5 Published under license by AIP Publishing.stabilized till T/C21600 K, and it transits directly to the ground bcc structure upon tetragonal distortions. Above results are reasonable since experimentally, in Fe 0:71Pd0:29alloys, the number of both fctI and fctII variants have also been confirmed to be much less thanthat of the parent phase, 19and based on the first-principles calcula- tions, Stern et al. proposed the fctI phase might be more ordered than the parent phase in Fe 0:75Pd0:25alloys as well.54Next, we, therefore, concentrate on investigating the fcc –fctI–fctII phase tran- sitions of these Fe 1/C0xPdxalloys with the configurations of I –III. C. Martensitic transformation In order to investigate the effects of different local atomic con- figurations on the fcc –fctI–fctII phase transitions of Fe 1/C0xPdx (0:28/C20x/C200:34) alloys, their EI f,EII f, and EIII fvalues change with respect to c=aare calculated and compared in Fig. 4 , taking thosecorresponding to c=a¼1 as reference in each x. Here, it should be noted that in the real case, around and below the room tempera- ture, the parent phase still possesses the fcc symmetry (i.e., the con-figuration of III) in the long-range atomic ordering since there thelocal atomic configurations of I and II are confirmed to be both sta-bilized in the phase. As shown in the figure, for x¼0:28, the global minima of E I f,EII f, and EIII fall appear around c=a¼0.75 – 0.80, corresponding to the ground fctII structure. With increasingx, their minimum values increase. As a result, when x¼0:31, the obtained fctII structure turns out to possess a higher energy thanthe parent phase in Figs. 4(a) and4(b).I nFig. 4(c) , the fctII struc- ture does so when x¼0:32, whereas for x¼0:31, it still possesses a slightly lower energy ( /C00:01 mRy) than the fcc phase. The more disordered configuration of III seems to stabilize the fctII structurein the compositions with a little higher Pd content. Considered allthe configurations of I –III in the parent phase, upon tetragonal dis- tortions, the fctII structure is, thus, stabilized by the electronic energy for x/C200:31. Evaluated with the three types of configura- tions, their equilibrium properties of the fctII phase are similar. Forx¼0:28, the experimental a(4.087 Å) and c/a(0.746) values are available, 7being in good agreement with the present ones corre- sponding to the three types of site occupations (4.117 Å and 0.759, 4.104 Å and 0.767, and 4.114 Å and 0.761, respectively). As shown in the inset of Figs. 4(a) and 4(b), respectively, in the curves of EI f/differencec=aand EII f/differencec=aof all these alloys, there is also a local minimum around c=a¼1:02 and 0.97, respectively, which should correspond to the fctI phase. Calculated with theconfiguration of III, nevertheless, they become to be not obviousand even disappear for x/C210:30 in the inset of Fig. 4(c) . From the first-principles calculations, the c=aratio of the intermediate tetrag- onal structure was also found to be a little larger than 1 in the ordered Fe 0:75Pd0:25and disordered Fe 0:70Pd0:30alloys by Chepulskii and co-workers.13,55However, as shown in the inset of Fig. 4(a) , the present ΔEfvalue of the structure estimated with the configuration of I is no more than 0.02 mRy in the absolute value in each x, being smaller than those evaluated with the arrangement of II shown in Table IV . There, these calculated ΔEfvalues of fctI-Fe 1/C0xPdxalloys are also in line with the previous first- FIG. 4. EI f(a),EII f(b), and EIII f(c) of Fe 1/C0xPdx(0:28/C20x/C200:34) alloys change with respect to c=aupon tetragonal distortions. Here, Efcorresponding toc=a¼1 is as reference for each alloy with the three types of configurations.TABLE IV . Equilibrium lattice parameters ( a, in Å, and c/a), bulk modulus ( B,i n GPa), magnetic moments ( μ,i nμB), and relative formation energies ( ΔEf, in mRy) to the parent phase of fctI-Fe 1−xPdx(0.28≤x≤0.34) alloys with the configuration of II, in comparison with the available experimental data from Refs. 5and 7. x a c/a B μ ΔEf 0.28 3.824 0.965 164.2 2.06 −0.033 0.29 3.819 0.968 164.3 2.04 −0.038 Exp.73.84 0.95 0.30 3.814 0.969 164.3 2.03 −0.042 Exp.53.822 0.95 0.31 3.810 0.970 164.1 2.01 −0.047 0.32 3.805 0.972 164.1 1.99 −0.051 0.33 3.802 0.974 164.2 1.97 −0.053 Exp.73.81 0.97 0.34 3.797 0.976 164.4 1.95 −0.050Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 035105 (2021); doi: 10.1063/5.0029951 129, 035105-6 Published under license by AIP Publishing.principles result (around /C00:050 mRy56), and their aand c/a values are close to the experimental data.5,7The experimental fctI phase is then supposed to be stabilized by the configuration of II inthe present work. As shown in Table IV , with the Pd addition, μof the phase always decreases as well. In the whole composition rangeof 0 :28/C20x/C200:34,Bkeeps almost constant and ΔE fchange no more than 0.02 mRy. adecreases, whereas c/aincreases with x.A bigger 1 –c=aof fctI-Fe 1/C0xPdxalso corresponds to a higher experi- mental TM,57,58similar to Ni –Mn –Ga alloys.59,60 InFigs. 4(a) –4(c), respectively, it is also noted that when c/a decreases to 0.85, EI f,EII f, and EIII fof these alloys with 0 :28/C20x/C20 0:32 suddenly change to decrease or decrease steeply with further decreasing c/a. The discontinuity of the curves of Ef/differencec=aaround c=a¼0.85 could be mainly attributed to two reasons here. One is the fast increase of μof these alloys with decreasing c/aaround c=a¼0:85 [see in Fig. 3(b) ] further accelerates the decrease of Ef with decreasing c/athere. Another one is that the tetragonal struc- tures with 0 :71,c=a,0:85 are closer to the fctII structure, whereas those with 0 :85,c=a,1 are nearer to the fctI structure, i.e.,c=a¼0:85 is right in the middle of the c/avalues of the two tetragonal phases and there, a turning point then tends to appear in the curves of Ef/differencec=aof these alloys. D. Elastic properties 1. Elastic constants of the austenite and martensite Table V shows x-dependence of the elastic constants of the parent phase of Fe 1/C0xPdxalloys with the three types of configura- tions of I, II, and III, respectively. In each x, the calculated C11,C12, and C44with the different occupations change very small. With increasing x,C11and C44increase, whereas C12decreases. As a result, C0increases with xas well, mechanically favoring the parent phase getting more and more stable with the Pd addition. In each x, the C0value calculated with the configuration of I ( C0I) is a little larger than the correspondent of II ( C0II), and both of them are much smaller than the value ( C0III) evaluated with the configura- tion of III. Correspondingly, upon tetragonal lattice distortions,then the fctI phase may be not easily obtained in the alloys with the configuration of III [ Fig. 4(c) ], whereas it is more stable in the compositions with the configuration of II [ Fig. 4(b) ]. For x/C210:30, both C 0Iand C0IIare much closer to the data measured at about300 K,11,12confirming again that there the parent phase favors the configurations of I and II around the room temperature. For the fctI phase with the configuration of II shown in Table VI ,C11,C12,C33,C44, and C66all increase, whereas C13 decreases with increasing x. However, for the fctII phase shown in Table VII , evaluated with each type of configuration, C13,C44, and C66always increase with x, whereas C33decreases with the Pd addi- tion. C11andC12change oppositely with respect to x:C11increases, whereas C12decreases with increasing xin the alloys with the con- figuration of I; whereas in those with the arrangements of II andIII,C 11decreases, whereas C12increases with x. The present elastic constants of both the fctI and fctII phases are in line with the avail- able theoretical results19,61shown in the two tables. The stability criteria for cubic crystals requires that C11.jC12j,C11þ2C12.0, and C44.0, and for tetragonal crys- tals requires that C11.jC12j,C33.0,C44.0,C66.0, C11þC33/C02C13.0, and Cs(¼C11þC12þ2C33/C04C13).0. From our calculations, the elastic constants of both the parent and fctII phases satisfy all of the conditions. Nevertheless, when x=0.28, Csof the fctI phase is about /C010:5 GPa, meaning at 0 K, fctI-Fe 0:72Pd0:28is mechanically unstable. Experimentally, the fctI structure is indeed seldom observed in the compositions with x/C200:28. Furthermore, we calculate the shear modulus ( G), Young modulus ( E), and ΘDof the three phases of alloys by means of the Hill average method.23It is found that in each phase, all of them TABLE V . x-dependence of the elastic constants (in GPa) of the parent phase of Fe 1−xPdx(0.28≤x≤0.34) alloys with the configurations of I, II, and III, respectively, in com- parison with the available C0values measured at about 300 K from Refs. 11and 12. C11 C12 C44 C0 x I II III I II III I II III I II III Exp.11,12 0.28 166.7 166.1 170.1 161.5 161.7 161.7 95.4 90.1 96.7 2.6 2.2 4.2 0.29 169.4 168.8 172.8 160.8 161.2 160.5 97.7 90.7 97.9 4.3 3.8 6.1 0.30 171.8 171.3 175.4 160.3 160.5 159.4 99.7 91.0 98.4 5.7 5.4 8.0 6.00.31 173.9 173.6 177.7 159.5 159.7 158.4 101.8 91.5 99.3 7.2 6.9 9.70.32 176.3 176.1 179.8 158.9 159.0 157.4 103.9 92.2 100.4 8.7 8.5 11.2 9.50.33 178.6 178.4 182.3 158.2 158.3 156.4 106.1 92.9 100.7 10.2 10.0 13.0 0.34 180.9 180.8 184.2 157.5 157.6 155.5 108.3 93.6 101.7 11.7 11.6 14.3 9.0TABLE VI. x-dependence of the elastic constants (in GPa) of fctI-Fe 1−xPdx (0.28≤x≤0.34) alloys with the configuration of II, in comparison with the theoreti- cal results from Ref. 61. xC 11 C12 C13 C33 C44 C66 0.28 177.3 150.6 164.7 160.2 91.1 66.2 0.29 179.2 151.0 162.7 164.5 91.9 67.4 0.30 181.0 151.4 160.5 168.9 92.7 68.60.30 61183.4 131.6 159.2 165.4 99.3 65.2 0.31 182.6 151.6 158.1 173.1 93.3 69.8 0.32 184.5 151.8 155.9 177.4 94.0 71.0 0.33 186.1 151.9 154.7 180.3 94.8 72.20.34 188.6 152.7 151.9 186.3 95.7 73.4Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 035105 (2021); doi: 10.1063/5.0029951 129, 035105-7 Published under license by AIP Publishing.possess the same trends of change against x, meaning the alloying effect on their stability is similar in the three phases of alloys. InFig. 5 , their Θ Dvalues are especially shown as a function of x. With the different configurations, the calculated trends of ΘD/differencexkeep almost unchanged in the three phases. With increasing x,ΘDof the parent and fctI phases always increases whereas that of the fctIIphase decreases monotonically, in good agreement with the experi-mental results. 62,63Evaluated with the same configuration of II, the fctII structure possesses the biggest ΘDvalue in each alloy below x¼0:32, whereas for x/C210:32,ΘDof the fctI structure turns out to be the largest in each x. In the small range of 0 :29/C20x,0:32,ΘD of the parent and fctI phases is comparable, and the former is still a little smaller than the latter. From a mechanical point of view, forx,0:32, the fctII phase is then stabilized at 0 K, whereas for x/C210:32, the fctI phase changes to be the most stable at low temper- atures. In each x, the smaller Θ Dvalue of the parent phase could produce relatively lower Fvibof the system, and then energetically promote the stability of the phase with the increase of temperature.For Fe 0:69Pd0:31alloys with the configuration of III, the evaluated 0-KFvibof the fcc phase (2.03 mRy) is lower than that (2.19 mRy) of the fctII phase even about 0.16 mRy. Then, when both Eeland Fvib are taken into account, the fctII phase with the configuration of III should be energetically stabilized for x,0:31 as well. 2. Free energies of the austenite and martensite Figure 6 compares Fof the three phases of Fe 1/C0xPdxalloys with the configuration of II change with respect to xat 0 K, 200 K, and 400 K, respectively. At 0 K, the fctII phase possesses the small- estFforx/C200:30, whereas above x¼0:33,Fof the parent phase changes to be the smallest in Fig. 6(a) . The fctII phase then corre- sponds to the ground state of the alloys with x/C200:30, whereas for x.0:33, the alloys are stabilized by the parent phase even at 0 K. For 0 :29/C20x/C200:33, the fctI phase is shown slightly lower than the parent phase in energy at 0 K, and their energy difference alsobecomes smaller and smaller with the increase of x. In these alloys, the thermoelastic MT of fcc –fctI is thus prone to occur at low tem- peratures, and their T Mvalues should decrease with the Pd addi- tion, in line with the experimental results.10,58 As shown in Figs. 6(b) and6(c), with increasing T, the ener- gies of both the fctI and fctII phases relative to the parent phasegradually increase in each alloy. When T¼200 K in Fig. 6(b) , the fctI and fctII phases change to be stabilized merely for 0 :29,x/C20 0:31 and x/C200:29, respectively. When Tfurther increases to 400 KinFig. 6(c) , the parent phase even turns out to possess the lowest F in all these studied alloys. Then, they are all stabilized by the parentphase instead for T/C21400 K. Furthermore, we calculate the free energy differences between the fctI and parent phases ( ΔF fctI/C0/C0fcc) as well as the fctII and fctITABLE VII. x-dependence of the elastic constants (in GPa) of fctII-Fe 1−xPdx(0.28≤x≤0.32) alloys with the configurations of I, II, and III, respectively, in comparison with the previous theoretical results from Ref. 19. C11 C12 C13 C33 C44 C66 x I II III I II III I II III I II III I II III I II III 0.28 268.6 250.9 258.5 84.1 96.3 92.7 154.0 158.0 154.9 198.8 189.2 196.4 93.1 91.4 95.3 13.5 14.1 13.0 0.29 269.4 246.4 255.3 81.5 99.1 93.8 154.6 158.2 156.1 196.2 187.4 192.9 93.4 92.0 96.0 13.7 14.9 13.50.30 269.6 237.9 251.7 79.9 105.8 95.4 155.6 158.3 157.5 194.0 185.4 189.6 93.6 94.1 96.6 13.9 15.9 13.9 0.31 270.1 228.2 248.6 77.6 109.8 96.6 156.3 158.4 158.5 191.3 179.7 186.6 93.8 96.4 97.9 14.2 16.9 14.2 0.3125 19268 81.6 167 175 98.7 12.9 0.32 270.2 219.4 244.9 75.3 114.4 97.7 156.7 158.5 159.3 188.9 175.2 183.4 93.9 97.5 98.6 14.5 17.4 14.6 FIG. 5. x-dependence of the ΘDvalues of Fe 1/C0xPdx(0:28/C20x/C200:34) alloys with the parent (a), fctI (b), and fctII phases (c), respectively, evaluated with the configurations of I, II, and III for the parent and fctII phases and that of II for thefctI phase, in comparison with the available experimental data from Ref. 62.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 035105 (2021); doi: 10.1063/5.0029951 129, 035105-8 Published under license by AIP Publishing.phases ( ΔFfctII/C0fctI) of these alloys with 0 :29/C20x/C200:33 change with respect to T.A ss h o w ni n Fig. 7(a) ,w i t h Tincreasing from 0 to 400 K, ΔFfctI/C0fccgradually increases to be positive in each alloy and at each temperature, it always increases with x. As a result, the estimated TMvalues with ΔFfctI/C0fcc¼0 gradually decrease with the increase of x, in agreement with the available experimental data.58As shown in Fig. 7(b) ,f o r x¼0:29 and 0.30, the calculated ΔFfctII/C0fctIalso increases to be positive with Tincreasing from 0 to 400 K. The evalu- ated critical temperatures for ΔFfctII/C0fctI¼0 are about 208 K and 80 K, respectively, also in line with the reported phase diagram of thetwo alloys. 10Therefore, the vibrational free energy term is confirmed to play an important role on the relative stabilities between the three phases of Fe 1/C0xPdxalloys at finite temperature. E. Effect of magnetic disorder on the martensitic transformation In the present work, the temperature dependence of Mas well asχof Fe 0:70Pd0:30alloys with the configuration of II is especially calculated. In Fig. 8 , its obtained normalized magnetization ( M=M0,with Mand M0being the magnetization at any temperature and 0 K, respectively) and χare shown as a function of T.TCis then esti- mated about 710 K, in good agreement with the experimental data(723 K 5). For the alloy, the fcc –fctI and fctI –fctII phase transitions have been confirmed to occur at about 280 K and 80 K, respectively, where its evaluated M=M0inFig. 8 is about 0.92 and 0.98, FIG. 6. Free energies ( F) of the parent, fctI, and fctII phases of Fe 1/C0xPdx (0:28/C20x/C200:34) alloys with the configuration of II change with respect to xat 0 K (a), 200 K (b), and 400 K (c), respectively. FIG. 7. T emperature dependence of the free energy differences between the fctI and parent phases ( ΔFfctI/C0fcc) (a) and the fctII and fctI phases ( ΔFfctII/C0fctI) (b) of Fe 1/C0xPdx(0:29/C20x/C200:33) alloys with the configuration of II. FIG. 8. T emperature dependence of normalized magnetization ( M=M0, with M andM0being the magnetization at any temperature and 0 K, respectively) and magnetic susceptibility ( χ) of the parent phase of Fe 0:70Pd0:30alloys with the configuration of II.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 035105 (2021); doi: 10.1063/5.0029951 129, 035105-9 Published under license by AIP Publishing.respectively. Correspondingly, when the alloy occurs the two phase transitions upon cooling, the magnetic disordering degree of the system, y(/C251/C0M=M0), changes to be about 8% and 2%, respec- tively, which should further vary Fof the alloy and then give a non- negligible contribution to the above two phase transitions as well. In order to investigate the magnetic disordering effect on the phase stability of Fe 0:70Pd0:30alloys, we further calculate the trends ofEI f/differencec=a,EII f/differencec=a, and EIII f/differencec=aof the alloy change with respect to y. Here, we assume that at low temperatures, the curves ofM=M0vsTare similar in the parent phase with the three config- urations of I –III, and upon tetragonal distortions, they have almost no great change in each tetragonal structure. The magnetovolume effect64is also supposed to be much small, and then it is neglected. As shown in Fig. 9 , the EI f,EII f, and EIII fvalues are all calculated with the equilibrium volume of the corresponding parent phasewith y¼0, and their values corresponding to c=a¼1 are as the respective reference in each y. It is found that for c=a/C210:95, all of them have no great change with the increase of yin each structure. However, for c=a,0:95, they increase with yin each c=aandthere, they also increase faster and faster with the decrease of c/ain each y=0, in comparison with the case of y¼0. As a result, their values corresponding to the fctII structure increase with y. When y/C213%, both E I fand EII fof the tetragonal structure change to be positive [ Figs. 9(a) and 9(b), respectively], and the corresponding EIII fdoes so as well for y/C214% [ Fig. 9(c) ]. Then, above y¼4% (T/C25200 K) with each of the three partially disordered configura- tions, the fctII structure always possesses a little higher electronicenergy than the parent phase. As shown in Fig. 9(b) , the magnetic disordering also seems to slightly suppress the relative stability of the fctI phase (around c=a¼0:97) to the parent phase and then, disfavors the MT from fcc to fctI. However, this effect is not as great as it on the relativestability of the fctII phase in the figure. Even when y¼4%, E II fof the fctI phase still possesses a negative value, meaning in the corre-sponding partially magnetic disordered Fe 70Pd30alloy, the fctI phase is also lower than the parent phase in energy and there, the fcc–fctI phase transition can be still retained at 0 K. Therefore, it is concluded that a properly small amount of magnetic disorder inFe 1/C0xPdxalloys can effectively prevent the fctII phase but retain the thermoelastic MT of fcc –fctI at low temperatures. IV. SUMMARY Using the first-principles EMTO –CPA method, we have sys- tematically investigated the effects of local atomic and magnetic configurations on the phase stability and elastic property of fcc-,fctI-, and fctII-Fe 1/C0xPdx(0:28/C20x/C200:34) FSM alloys, and then explored the physical mechanisms driving their x-dependent MT. The main results are summarized as follows: (1) Considered the four types of typically ideal atomic configura- tions of I –IV in a fcc unit cell, the two with one random sub- lattice (I and II) are both preferable for each xbelow 300 K and when T¼300 K, and that with three random sublattices (III) also changes to be energetically comparable with them for x/C200:30; the fully disordered configuration of IV is stabilized in most of these alloys until T/C21600 K. Upon tetragonal dis- tortions, the fctI phase is obtained for 0 :29/C20x/C200:33, having only one random sublattice on the same layer with thePd site (II); the fctII phase is found for x/C200:30, where the configurations of I, II, and III are all stabilized. (2) The determined equilibrium properties and elastic constants of the fcc, fctI, and fctII phases are all in line with the availableexperimental and theoretical data. With increasing x, the tetra- gonality 1 /C0c=aof the fctI phase decreases, whereas C 0of the fcc phase increases. Then, it should be more and more difficult for the tetragonal lattice deformation to occur in the parentphase with the increase of x, corresponding to the experimental T Mdecreasing with the Pd addition. (3) The composition and temperature dependence of the free ener- gies of the fcc, fctI, and fctII phases evaluated with the configu- ration of II represent the phase diagram of these alloys, andtheir estimated T Mvalues are also in agreement with the avail- able experimental data. It confirms that the phonon vibrational free energy term plays an important role in the relative stabil- ities between the three phases at finite temperatures. FIG. 9. EI f(a),EII f(b), and EIII f(c) of Fe 0:70Pd0:30alloys change with respect to c=aupon tetragonal distortions, evaluated with y¼0%, 1%, 2%, 3%, and 4%, respectively. Here, Efforc=a¼1 is as reference in each case.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 035105 (2021); doi: 10.1063/5.0029951 129, 035105-10 Published under license by AIP Publishing.(4) The magnetic disorder suppresses the relative stabilities of both the fctI and fctII phases. In comparison, this effect on the fctIIphase is, relatively, much greater than that on the fctI phase. InFe 0:70Pd0:30alloys, by adding 4% magnetic disorder, the fctII structure is effectively prevented, whereas the thermoelastic MT of fcc –fctI can still be retained at 0 K. The estimated TC value of the alloy is also close to the measured data. ACKNOWLEDGMENTS The authors would like to acknowledge the financial support from the National Science Foundation of China (NSFC) underGrant Nos. 11674233 and 51301176, the China Postdoctoral Science Foundation under Grant Nos. 2013M530133 and 2014T70264, and Natural Science Foundation of Liaoning Province underGrant No. 2019-MS-287. The MoST of China under Grant No.2014CB644001 is also acknowl edged for financial support. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1T. Kakeshita, T. Fukuda, and T. Takeuchi, Mater. Sci. Eng. 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