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5.0006183.pdf

J. Appl. Phys. 127, 193906 (2020); https://doi.org/10.1063/5.0006183 127, 193906 © 2020 Author(s).Low-energy switching of antiferromagnetic CuMnAs/GaP using sub-10 nanosecond current pulses Cite as: J. Appl. Phys. 127, 193906 (2020); https://doi.org/10.1063/5.0006183 Submitted: 28 February 2020 . Accepted: 06 May 2020 . Published Online: 20 May 2020 K. A. Omari , L. X. Barton , O. Amin , R. P. Campion , A. W. Rushforth , A. J. Kent , P. Wadley , and K. W. Edmonds COLLECTIONS Note: This paper is part of the special topic on Antiferromagnetic Spintronics. This paper was selected as an Editor’s Pick Low-energy switching of antiferromagnetic CuMnAs/GaP using sub-10 nanosecond current pulses Cite as: J. Appl. Phys. 127, 193906 (2020); doi: 10.1063/5.0006183 View Online Export Citation CrossMar k Submitted: 28 February 2020 · Accepted: 6 May 2020 · Published Online: 20 May 2020 K. A. Omari, L. X. Barton, O. Amin, R. P. Campion, A. W. Rushforth, A. J. Kent, P. Wadley, and K. W. Edmondsa) AFFILIATIONS School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom Note: This paper is part of the special topic on Antiferromagnetic Spintronics. a)Author to whom correspondence should be addressed: kevin.edmonds@nottingham.ac.uk ABSTRACT The recently discovered electrical-induced switching of antiferromagnetic (AF) materials that have spatial inversion asymmetry has enriched the field of spintronics immensely and opened the door for the concept of antiferromagnetic memory devices. CuMnAs is one promisingAF material that exhibits such electrical switching ability and has been studied to switch using electrical pulses of length millisecond downto picosecond but with little focus on the nanosecond regime. We demonstrate here the switching of CuMnAs/GaP using nanosecondpulses. Our results showed that in the nanosecond regime, low-energy switching and a high readout signal with highly reproducible behavior down to a single pulse can be achieved. Moreover, a comparison of the two switching methods of orthogonal switching and polarity switching was made on the same device, and it showed distinct behaviors that can be exploited selectively for different future memory/processing applications. Published under license by AIP Publishing. https://doi.org/10.1063/5.0006183 I. INTRODUCTION Writing of magnetic information using spin-polarized cur- rents, via current-induced spin transfer and spin –orbit torques, has been key to the development of magnetic random-access memory(MRAM) technologies. 1,2Recently, current-induced switching has been predicted and demonstrated in antiferromagnetic (AF) mate- rials, enriching the field of AF spintronics.3–19With the ability to manipulate AF domains electrically, memory-based spintronicapplications can be developed that utilize the terahertz dynamics 8 and multilevel response7of the AF order, while also producing no stray magnetic fields and being insensitive to external fields. Current-induced switching of AF materials relies on the generation of a spin –orbit torque with the same handedness on each magnetic sublattice. This can, in principle, be achieved in AFcrystals with sublattice inversion asymmetry, including CuMnAsand Mn 2Au,4–11,15,20as well as in AF/heavy-metal multilayer structures.12–14In CuMnAs, the switching of AF domains between metastable biaxial easy axes has been demonstrated with correlationshown between the directly imaged magnetic domain configurationand the electrical readout. 6The rotation of the AF momentsbetween two perpendicular orientations can be realized by applying current pulses along two orthogonal axes in a multi-contact device and can be electrically detected using anisotropic magnetoresistance(AMR), i.e., different electrical resistivities for current flowing parallel and perpendicular to the AF spin axis. 5Patterned CuMnAs devices have been shown as a proof-of-concept for a simple memory cellthat can be integrated with existing CMOS technology. 7 More recently, another form of switching in CuMnAs was demonstrated by reversing the polarity of a current pulse applied between a single pair of electrical contacts.9The resulting spin – orbit torque can cause motion of domain walls (DWs) separating the two orthogonal domain populations, resulting in measurablechanges of the AMR. This polarity-dependent switching methodoffers simpler device structures and also potentially lower switching currents compared with the orthogonal switching method. While electrical switching of CuMnAs/GaP has been previ- ously investigated using millisecond (ms) 5,9and picosecond (ps)7 pulses in detail, the nanosecond (ns) pulsing regime remains not yet fully studied. This regime is relevant and significant for a low-energy switching regime that can be easily integrated with standardJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 193906 (2020); doi: 10.1063/5.0006183 127, 193906-1 Published under license by AIP Publishing.microelectronics, as opposed to the relatively slow ms regime or the ps regime that would require either optical switching techniques or highly sophisticated electronics. In this paper, we investigate the switching behavior of a CuMnAs/GaP device in the nanosecond (ns) regime. Both switchingmethods of orthogonal switching that show a slow decay of signal and the polarity switching method that shows a highly non-volatile switching behavior are explored on the same device using our setup. II. DEVICE FABRICATION AND EXPERIMENTAL SETUP A 46 nm layer of CuMnAs was grown using molecular beam epitaxy (MBE) on a 2 00substrate of single crystal GaP (100).21,22 Prior to growing the CuMnA layer, the substrate was annealed at high temperature under P flux to remove the surface oxide layer. A 100 nm homo-epitaxial GaP buffer layer was grown to ensure asmooth surface for the CuMnAs. The hetero-epitaxial CuMnAslayer was then grown, followed by a 2.5 nm capping layer of Al toprevent surface oxidation. Following the growth process, 2 θ−ω x-ray diffraction scans were performed to confirm that the tetrago- nal structure was achieved with the c-axis out of plane and with the square ab plane of the tetragonal unit cell orientating at a planar45° offset to the 100 hi crystal axes of the cubic GaP unit cell. At 46 nm thickness, the CuMnAs layer is expected to have a biaxial magnetic anisotropy with in-plane easy axes separated by 90°, as opposed to the uniaxial anisotropy which is typically observed infilms thinner than ∼20 nm. 23,24The CuMnAs film was then patterned into eight-arm devices with a central junction of 3.3 × 3.3 μm2and a 2 μm arm width using photolithography and wet chemical etching. Optical micro-graphs of the device are shown in Fig. 1 . Multiple devices were fab- ricated on a single chip. The chip was mounted on a sample holder[Fig. 1(a) ], designed to allow nanosecond current pulses to be driven from the main pulse input terminal via any pair of arms of the device to the grounding output terminals (labeled on thebottom side of the holder with the green grounding line). Thecurrent path is selected either by manual toggle switches or by elec-tric relays controlled by an external Arduino controller module. By activating a toggle switch/relay at one of the four input paths and a toggle switch/relay at one of the grounding output ter-minals, the direction of propagation of the ns current pulse can beset. Two pulsing configurations were investigated in this work. Fororthogonal switching, the pulse is directed alternately along the [/C221/C2210] and [1 /C2210] axes of the CuMnAs film [dashed white arrows in Fig. 1(b) ]. For polarity pulsing, the pulse is directed alternately along the [1 0 0] and [ /C22100] directions [dashed blue arrows in Fig. 1(b) ]. In the latter case, an external invertor was connected to the pulse generator via a bridge connection to selectively invert the polarity of the signal before entering the waveguide. The amplitude and length of each pulse was set to be between 12 V and 14 V and 4 ns, respectively, using an Avtech pulse genera-tor. To generate a desired number of pulses, the pulse generator was set to be triggered externally by another function generator that can be automated using a computer software and can produce FIG. 1. (a) Photo of the waveguide sample holder showing the controllable toggle switches used to select the current pulse path. The inset above the image show s the shape of the 4 ns current pulse. 4 and 3 input pins on the left and right side of the holder, respectively, are used to connect probing DC current and voltag es. (b) Optical image of the eight-arm CuMnAs/GaP device. The upper image shows the full device with measurement schematic. The lower image shows a close-up of the cen tral junction, with the dashed arrows indicating the direction of pulsing for orthogonal (white arrows) and polarity (blue arrows). The scale bar is 4 μm.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 193906 (2020); doi: 10.1063/5.0006183 127, 193906-2 Published under license by AIP Publishing.a square signal with a sharp rising edge of 5 V. The two-wire resis- tance of the devices is typically around 475 Ω. This relatively high resistance with respect to the 50 Ωimpedance waveguide leads to some signal reflection as can be seen from the pulse waveform inthe inset of Fig. 1(a) . However, this signal distortion is within toler- able limits as the square shape of the pulse is fairly intact. In addition to the waveguide paths, the eight terminals of the device were connected to normal DC circuitry via additional input/output terminal pins. These can be seen on the right and left edge ofsample holder in Fig. 1(a) . The DC circuitry was primarily used for probing the transverse and longitudinal resistances of the device, R xy=Vxy/Iand Rxx=Vxx/I, following the application of current pulses. The transverse Vxyand longitudinal Vxxvoltages were contin- uously measured with a probing current Iof 500 μAa sd e p i c t e di n Fig. 1(b) . High impedance resistors and high-pass filters were added to decouple the high-frequency circuit and the DC circuit. All pulsing and measurements were performed in ambient conditions. III. RESULTS A. Orthogonal switching In studying the orthogonal switching in the ns regime, the dependence of the transverse resistance on the number of appliedpulses was investigated. Moreover, the switching robustness of the device was later tested to assess its reproducibility and stochasticity. To achieve the first objective, ten sets of measurements were per-formed, each consisting of a total of 30 switching attempts (15 pairsof switches), in which pulse trains were sent alternately across twoorthogonal axes with an interval of 1 min between each switching attempt. Between each set of measurements, the number of pulses was increased by an additional pulse such that the first set had asingle 4 ns pulse for each switching attempt, and the tenth set had atrain of 10 pulses. For sets 2 –10, the multiple pulses triggered at each switching attempt were separated by 1 μs. The amplitude of each pulse was set at 14 V producing a current of 29 mA, corresponding to a current density of 32 MA/cm 2in the arms of the device. Figure 2(a) shows the Rxytraces obtained for five pulses per switching attempt. After each pulse, the Rxyshows either an increase or a decrease depending on the polarity of the pulse, con- sistent with an expected AMR effect due to a transient rotation ofantiferromagnetic domains.6,24The Rxyrelaxes back to an equilib- rium value on a timescale of ≈20 s. Figure 2(b) shows the depen- dence of ΔRxy, defined as the difference between the first and last measured points after each pulse train, vs the number of pulses.The plot shows an asymmetry between the two orthogonal currentdirections and a non-monotonic dependence on the number ofpulses, which may indicate that both the switching and relaxation processes are thermally assisted. To further investigate the robustness and reproducibility of orthogonal switching in the ns regime, the same switching proto-col, with five pulses per switching attempt, was run for 100 switch-ing attempts. Figure 3 shows the resulting R xytrace indicating a consistent and highly reproducible switching as suggested by the histogram in the inset (further discussion later). The maximumΔR xywas calculated to be around 0.1 Ω, and the signal relaxes to its initial value in around 20 s. As indicated in Fig. 2(b) , any switching signal in this device was below the noise level for one and two pulses. However, consistent switching using a single pulse was s een in a duplicate device fabricated on the same CuMnAs layer in the same fabrication process. Figure 4 shows a total of 40 switching attempts (20 switching attempts on eachorthogonal axis) using a single 4 ns pulse of amplitude 12 V (J = 27 MA/cm 2). In this device, 20 switching attempts were per- formed along one axis with an interval of 1 min between eachattempt. Then, 20 switching attempts were performed on the secondaxis to show the reverse effect. Thi s process simulates a simple binary writing of a “leaky ”memory state with a single 4 ns electric pulse. A total energy of 1.2 nJ was needed for switching in the active area of2μm×2 μm × 46 nm of the CuMnAs device. This is orders of magni- tude less than energy obtained in the ms regime 7(where a 5 V USB, 50 ms pulse, and 46 mA were used to switch CuMnAs/GaP). The 1.2 nJ is also in the same order of magnitude of energy needed by fs-pulsing laser for ultra- fast optical switching.25T h ef a c tt h a tt h e energy required for the switching depends only weakly on the pulseduration from ns to fs indicates that current-induced heating plays arole in the switching of AF domains. 26 B. Polarity switching For the polarity switching geometry, first we investigated how changes in Rxycorrelate with the number of pulses, starting from a FIG. 2. Readout signal, Rxy, for orthogonal switching of the CuMnAs device. (a) 20 switching attempts using five pulses of current 29 mA and length of 4 ns. (b) Summary plot of the number of pulses vs ΔRxyfor orthogonal switching for pulsing along [ /C221/C2210] (red circles) and [1 /C2210] (blue circles). FIG. 3. Orthogonal switching using a train of five pulses of current 29 mA and length 4 ns for axis [1 /C2210] (blue) and [ /C221/C2210] (red). Inset: histogram showing the distribution of the readout signal ΔRxy.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 193906 (2020); doi: 10.1063/5.0006183 127, 193906-3 Published under license by AIP Publishing.single pulse to a train of 10 pulses, with each pulse separated by 1 μs.Figures 5(a) –5(c) show the Rxytraces for sets of measurements with 2, 3, and 4 pulses per switching attempt. Results showing ΔRxy vs the number of pulses for all the sets of measurements are sum- marized in Fig. 5(d) . As indicated in Figs. 5(a) –5(c),Rxyshows a step-like change between the high and low values depending on the polarity of the switching pulse. ΔRxyinitially increases with increasing number of pulses and saturates above four pulses. ΔRxyis defined here as the height of the step-like behavior which resembles the difference inR xybetween two consecutive switching events with opposite polar- ity. In order to compare ΔRxybetween polarity and orthogonal switching, a graph in Fig. 5(d) representing the full step height in orthogonal switching was plotted in the same graph (black circles).The step height in orthogonal switching is simply defined to be thechange in R xybetween the first point of switching of one arm and the first point of switching in the next switching event when the orthogonal arm is pulsed. Moreover, unlike in the case of orthogo-nal switching where the R xyrelaxes back between each pulse train, the results of polarity pulsing indicate no or very slow relaxation ofsignal at room temperature. Such an observation is consistent with previously reported results of polarity pulsing in the millisecond regime. 9To further investigate the retention time, the device was FIG. 4. Plot showing orthogonal switching in CuMnAs/GaP with a single pulse of current 25 mA and pulse width of 4 ns. T op: current pulses along the [ /C221/C2210] axis. Bottom: the [1 /C2210] axis. FIG. 5. Polarity switching readout Rxysignal for current along the [1 0 0] axis with two pulses (a), three pulses (b), and four pulses (c) of current 29 mA and a pulse width of 4 ns. (d) Summary plot of the number of pulses vs ΔRxy(Rxy_high–Rxy_low) for polarity switching (red) and orthogonal switching (black). (e) Plot showing retention of theRxysignal for several hours after polarity pulsing. (f ) Plot showing the measured R xxover the same time period.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 193906 (2020); doi: 10.1063/5.0006183 127, 193906-4 Published under license by AIP Publishing.switched to the high Rxyvalue and left for 15 h before measuring again. Rxydecreased only by 5% over 15 h [ Fig. 5(e) ]. This slight magnetic decay could be contributed by variations in the ambienttemperature as suggested by the increase in R xx[Fig. 5(f) ]. Plots in Fig. 5(d) clearly indicate a much higher ΔRxysignal obtained for polarity switching compared with orthogonal switching. This sug- gests that when orthogonal switching occurs, only a partial reorien- tation of domains occurs at the central region of the device.However, when polarity switching occurs it seems that domain wallmovement results in reorientation of a wider area of domainsleading to a higher readout signal. We also investigated the robustness, reproducibility, and sto- chasticity of the polarity switching of the device by continuouslyswitching it for 500 times while maintaining an interval time of 1min between each switching attempt. Figure 6(a) shows the R xy trace for the first 60 switching attempts across 60 min. The corresponding ΔRxyhistogram in Fig. 6(b) demonstrates the stochastic behavior of the readout voltage (see also Fig. S1 in thesupplementary material for the whole of the 500 switching events). It can also be seen in Fig. 6(a) that there is considerable back- ground drift. The origin of this drift is unknown but may be related to incomplete switching due to random pinning/depinning of domain walls or external factors such as changes in ambienttemperature, contact resistances, or current paths. The histogram plot in Fig. 6(b) obtained from the 100 switch- ing attempts indicates a clear difference in the statistical switching behavior between the orthogonal (purple) and the polarity (green)switching methods. The Gaussian curve fitting for the two histo-grams indicated a curve width, wc, at half the maximum of each histogram to be 0.01 Ωfor orthogonal and 0.06 Ωfor polarity switching. This indicates that a relatively more deterministic behav- ior is seen in the orthogonal switching compared with polarityswitching. As discussed above, this suggests that in orthogonalswitching the domain reorientation follows the same route in mostof the switching attempts, indicating that the switching energy barrier is more well-defined than in the case of polarity switching where a DW can face different pinning energy barriers in eachswitching attempt. Moreover, the different signal and stabilitybetween the two switching methods might also arise from themagnetocrystalline anisotropy for pulsing across different crystal orientations. 20 The relatively small signal obtained from orthogonal switching (compared with polarity switching) suggests that small domainregions are switching only. On the other hand, in the case of polar-ity switching, the wide distribution of the output signal indicates that domain motion in CuMnAs encounters pinning energy land- scapes where pinning seems to follow a thermally assisted switchingdistribution. IV. CONCLUSION In this paper, we demonstrated the ability of a CuMnAs device to switch consistently and reliably using only one to ten current pulses of length 4 ns. Switching using orthogonal current pulses along the [ /C221/C2210] and [1 /C2210] directions shows a slow decay of the readout signal, while polarity switching for current pulses alongthe [1 0 0] axis shows a highly non-volatile switching behavior.These results are promising for the development of a robust antifer- romagnetic MRAM. Further work is required to establish the role of magnetic anisotropy in determining the stability of the switchedstate. With the same medium showing both a “leaky ”memory state that can be tuned by temperature 10and a more stable switching behavior that can be controlled by polarity of the pulse,9both switching mechanisms can be utilized to perform memory and processing calculations simultaneously. SUPPLEMENTARY MATERIAL The supplementary material shows all the results for the polarity switching of Device I taken for 500 switches over 500 min. ACKNOWLEDGMENTS This work was supported by EU FET Open RIA under Grant No. 766566 and the UK Engineering and Physical SciencesResearch Council under Grant No. EP/P019749. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1A. D. Kent and D. C. Worledge, Nat. Nanotechnol. 10(3), 187 (2015). 2G. Prenat, K. Jabeur, P. Vanhauwaert, G. Di Pendina, F. Oboril, R. Bishnoi, M. Ebrahimi, N. Lamard, O. Boulle, and K. Garello, IEEE Trans. Multi-Scale Comput. Syst. 2(1), 49 –60 (2015). 3J. Shi, V. Lopez-Dominguez, F. Garesci, C. Wang, H. Almasi, M. Grayson, G. Finocchio, and P. Khalili Amiri, Nat. Electron. 3(2), 92 –98 (2020). 4J.Železný, H. Gao, K. Výborný, J. Zemen, J. Ma šek, A. Manchon, J. Wunderlich, J. Sinova, and T. Jungwirth, Phys. Rev. Lett. 113(15), 157201 (2014). 5P. Wadley, B. Howells, J. Železný, C. Andrews, V. Hills, R. P. Campion, V. Novák, K. Olejník, F. Maccherozzi, and S. Dhesi, Science 351(6273), 587 –590 (2016). 6M. Grzybowski, P. Wadley, K. Edmonds, R. Beardsley, V. Hills, R. Campion, B. Gallagher, J. S. Chauhan, V. Novak, and T. Jungwirth, Phys. Rev. Lett. 118(5), 057701 (2017). FIG. 6. (a)Rxysignal after applying 4 ns and four pulses indicating polarity switching for 60 attempts across 1 h. (b) Plot of histograms showing different values of ΔRxyfor orthogonal switching (purple) and polarity switching (green). Orange line is Gaussian fitting with wc(at half maximum) = 0.01 Ωfor orthogo- nal and 0.06 Ωfor polarity switching.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 193906 (2020); doi: 10.1063/5.0006183 127, 193906-5 Published under license by AIP Publishing.7K. Olejník, V. Schuler, X. Martí, V. Novák, Z. Ka špar, P. Wadley, R. P. Campion, K. W. Edmonds, B. L. Gallagher, and J. Garcés, Nat. Commun. 8(1), 1 –7 (2017). 8K. Olejník, T. Seifert, Z. Ka špar, V. Novák, P. Wadley, R. P. Campion, M. Baumgartner, P. Gambardella, P. N ěmec, and J. Wunderlich, Sci. Adv. 4(3), eaar3566 (2018). 9P. Wadley, S. Reimers, M. J. Grzybowski, C. Andrews, M. Wang, J. S. Chauhan, B. L. Gallagher, R. P. Campion, K. W. Edmonds, and S. S. Dhesi, Nat. Nanotechnol. 13(5), 362 –365 (2018). 10T. Matalla-Wagner, M.-F. Rath, D. Graulich, J.-M. Schmalhorst, G. Reiss, and M. Meinert, Phys. Rev. Appl. 12(6), 064003 (2019). 11S. Y. Bodnar, L. Šmejkal, I. Turek, T. Jungwirth, O. Gomonay, J. Sinova, A. Sapozhnik, H.-J. Elmers, M. Kläui, and M. Jourdan, Nat. Commun. 9(1), 1 –7 (2018). 12X. Chen, R. Zarzuela, J. Zhang, C. Song, X. Zhou, G. Shi, F. Li, H. Zhou, W. Jiang, and F. Pan, Phys. Rev. Lett. 120(20), 207204 (2018). 13P. Zhang, J. Finley, T. Safi, and L. Liu, Phys. Rev. Lett. 123(24), 247206 (2019). 14Y. Cheng, S. Yu, M. Zhu, J. Hwang, and F. Yang, Phys. Rev. Lett. 124(2), 027202 (2020). 15M. Meinert, D. Graulich, and T. Matalla-Wagner, Phys. Rev. Appl. 9(6), 064040 (2018). 16T. Moriyama, K. Oda, T. Ohkochi, M. Kimata, and T. Ono, Sci. Rep. 8(1), 1 –6 (2018).17L. Baldrati, O. Gomonay, A. Ross, M. Filianina, R. Lebrun, R. Ramos, C. Leveille, F. Fuhrmann, T. Forrest, and F. Maccherozzi, Phys. Rev. Lett. 123(17), 177201 (2019). 18M. Dunz, T. Matalla-Wagner, and M. Meinert, Phys. Rev. Res. 2(1), 013347 (2020). 19X. Zhou, X. Chen, J. Zhang, F. Li, G. Shi, Y. Sun, M. Saleem, Y. You, F. Pan, and C. Song, Phys. Rev. Appl. 11(5), 054030 (2019). 20X. Zhou, J. Zhang, F. Li, X. Chen, G. Shi, Y. Tan, Y. Gu, M. Saleem, H. Wu, and F. Pan, Phys. Rev. Appl. 9(5), 054028 (2018). 21P. Wadley, V. Novák, R. Campion, C. Rinaldi, X. Martí, H. Reichlová, J.Železný, J. Gazquez, M. Roldan, and M. Varela, Nat. Commun. 4(1), 1 –6 (2013). 22F. Krizek, Z. Ka špar, A. Vetushka, D. Kriegner, E. M. Fiordaliso, J. Michalicka, O. Man, J. Zubá č, M. Brajer, and V. A. Hills, Phys. Rev. Mater. 4(1), 014409 (2020). 23P. Wadley, V. Hills, M. Shahedkhah, K. Edmonds, R. Campion, V. Novák, B. Ouladdiaf, D. Khalyavin, S. Langridge, and V. Saidl, Sci. Rep. 5, 17079 (2015). 24M. Wang, C. Andrews, S. Reimers, O. Amin, P. Wadley, R. Campion, S. Poole, J. Felton, K. Edmonds, and B. Gallagher, Phys. Rev. B 101(9), 094429 (2020). 25Z. Ka špar, M. Surýnek, J. Zubá č, F. Krizek, V. Novák, R. P. Campion, M. S. Wörnle, P. Gambardella, X. Marti, and P. N ěmec, arXiv:1909.09071 (2019). 26A. V. Kimel and M. Li, Nat. Rev. Mater. 4(3), 189 –200 (2019).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 193906 (2020); doi: 10.1063/5.0006183 127, 193906-6 Published under license by AIP Publishing.

5.0004926.pdf

J. Chem. Phys. 152, 214106 (2020); https://doi.org/10.1063/5.0004926 152, 214106 © 2020 Author(s).Relativistic short-range exchange energy functionals beyond the local-density approximation Cite as: J. Chem. Phys. 152, 214106 (2020); https://doi.org/10.1063/5.0004926 Submitted: 15 February 2020 . Accepted: 15 May 2020 . Published Online: 03 June 2020 Julien Paquier , Emmanuel Giner , and Julien Toulouse ARTICLES YOU MAY BE INTERESTED IN Relativistic local hybrid functionals and their impact on 1s core orbital energies The Journal of Chemical Physics 152, 214103 (2020); https://doi.org/10.1063/5.0010400 The correlation factor approach: Combining density functional and wave function theory The Journal of Chemical Physics 152, 211101 (2020); https://doi.org/10.1063/5.0010333 A weight-dependent local correlation density-functional approximation for ensembles The Journal of Chemical Physics 152, 214101 (2020); https://doi.org/10.1063/5.0007388The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Relativistic short-range exchange energy functionals beyond the local-density approximation Cite as: J. Chem. Phys. 152, 214106 (2020); doi: 10.1063/5.0004926 Submitted: 15 February 2020 •Accepted: 15 May 2020 • Published Online: 3 June 2020 Julien Paquier,1,a) Emmanuel Giner,1 and Julien Toulouse1,2,b) AFFILIATIONS 1Laboratoire de Chimie Théorique (LCT), Sorbonne Université and CNRS, F-75005 Paris, France 2Institut Universitaire de France, F-75005 Paris, France a)julien.paquier@lct.jussieu.fr b)Author to whom correspondence should be addressed: toulouse@lct.jussieu.fr ABSTRACT We develop relativistic short-range exchange energy functionals for four-component relativistic range-separated density-functional theory using a Dirac–Coulomb Hamiltonian in the no-pair approximation. We show how to improve the short-range local-density approximation exchange functional for large range-separation parameters by using the on-top exchange pair density as a new variable. We also develop a relativistic short-range generalized-gradient approximation exchange functional that further increases the accuracy for small range-separation parameters. Tests on the helium, beryllium, neon, and argon isoelectronic series up to high nuclear charges show that the latter functional gives exchange energies with a maximal relative percentage error of 3%. The development of this exchange functional represents a step forward for the application of four-component relativistic range-separated density-functional theory to chemical compounds with heavy elements. Published under license by AIP Publishing. https://doi.org/10.1063/5.0004926 .,s I. INTRODUCTION Range-separated density-functional theory (RS-DFT) (see, e.g., Refs. 1 and 2) is an extension of Kohn–Sham density-functional theory (DFT)3that rigorously combines a wave-function method accounting for the long-range part of the electron–electron interac- tion with a complementary short-range density functional. RS-DFT has a faster basis convergence than standard wave-function theory (WFT)4and can improve over usual Kohn–Sham density-functional approximations (DFAs) for the description of strong-correlation effects (see, e.g., Refs. 5 and 6) or weak intermolecular interactions (see, e.g., Refs. 7 and 8). For the description of compounds with heavy elements, RS- DFT can be extended to a four-component relativistic frame- work.9–11In particular, in Refs. 9 and 10, second-order Møller– Plesset perturbation theory and coupled-cluster theory based on a no-pair Dirac–Coulomb Hamiltonian with long-range electron–electron interactions were combined with short-rangenon-relativistic exchange-correlation DFAs and applied to heavy rare-gas dimers. One limitation, at least in principle, in these works is the neglect of relativity in the short-range density functionals. It is thus desirable to develop appropriate short-range relativistic exchange-correlation DFAs for four-component RS-DFT in order to quantify the error due to the neglect of relativity in the short- range density functionals and possibly increase the accuracy of these approaches. As a first step toward this, in Ref. 11, some of the present authors developed a short-range relativistic local density-functional approximation (srRLDA) exchange functional based on calcula- tions on the relativistic homogeneous electron gas (RHEG) with the Coulomb and Coulomb–Breit electron–electron interactions in the no-pair approximation. In the present work, we test this srRLDA exchange functional on atomic systems, namely, the helium, beryllium, neon, and argon isoelectronic series up to high nuclear charges Z, using a four- component Dirac–Coulomb Hamiltonian in the no-pair approxi- mation. We reveal that, for these relativistic ions with large Z, the J. Chem. Phys. 152, 214106 (2020); doi: 10.1063/5.0004926 152, 214106-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp srRLDA exchange functional is quite inaccurate even for large val- ues of the range-separation parameter μ. We show how this func- tional can be improved by using the on-top exchange pair density as a new variable. Finally, we further improve the short-range rela- tivistic exchange functional by constructing a generalized-gradient approximation (GGA), achieving a 3% maximal relative energy error. This paper is organized as follows. In Sec. II, we lay out the formalism of RS-DFT for a four-component relativistic Dirac– Coulomb Hamiltonian in the no-pair approximation. In Sec. III, we give the computational details for our calculations. In Sec. IV, we test the srRLDA exchange functional and discuss its limitations. In Sec. V, we improve the srRLDA exchange functional by using the on-top exchange pair density. In Sec. VI, we construct and test short- range relativistic exchange GGAs. Finally, Sec. VII summarizes our conclusions. In Appendixes A and B, we derive the uniform coordi- nate scaling relation for the relativistic short-range exchange density functional and the expression of the on-top exchange pair density in a four-component no-pair relativistic framework. II. RELATIVISTIC RANGE-SEPARATED DENSITY-FUNCTIONAL THEORY Let us first discuss the choice of the relativistic quantum many- particle theory on which to base relativistic RS-DFT and the gen- eral strategy that we follow in this work. Clearly, since RS-DFT combines WFT and DFT, we need a relativistic framework, which is convenient for both of them. Relativistic Kohn–Sham DFT has been formulated based on quantum electrodynamics (QED),12–14 even though the no-pair approximation15,16is normally introduced at a later stage for practical calculations. As regards WFT, the best tractable relativistic framework is the recently developed effective QED Hamiltonian17–21incorporating all QED effects obtained with non-retarded two-particle interactions. In the present work, we will stick however to the most common choice of the four-component Dirac–Coulomb Hamiltonian in the no-pair approximation, which can easily be used for both WFT and DFT. This relativistic frame- work can be derived in several ways (see, e.g., Refs. 17, 18, and 20–22). In the effective QED approach, the Hamiltonian is written in second quantization with normal ordering with respect to the vac- uum state of empty positive-energy one-particle states and com- pletely filled negative-energy one-particle states while incorporating charge-conjugation symmetry. This Hamiltonian has a stable vac- uum state and is physically meaningful. In this approach, the no- pair approximation, corresponding to projecting this Hamiltonian onto the many-electron subspace generated by positive-energy one- particle states, is just a convenient approximation (but in principle not necessary), akin to the idea of restricting the orbitals entering the wave function to an active orbital subspace in the complete-active- space self-consistent-field method. The no-pair Dirac–Coulomb Hamiltonian is then obtained by further neglecting the effective QED one-particle potential corresponding to vacuum polarization and electron self-energy. By contrast, in the configuration-space approach, the Hamiltonian is written either in first quantization or, equivalently, in “naive” second quantization (i.e., without nor- mal ordering with respect to a stable vacuum state). The resultingHamiltonian has thus an unstable vacuum state, corresponding to empty positive-energy one-particle states and empty negative- energy one-particle states, and no bound states (the electronic states that should be normally bound being embedded in the continuum of excitations to positive-energy states and deexcitations to negative- energy states) and hence is per se unphysical. However, by projecting this Hamiltonian onto the many-electron subspace generated by the positive-energy one-particle states, we recover the same physically relevant no-pair Dirac–Coulomb Hamiltonian as the one obtained by starting with the effective QED approach. One drawback of the no-pair approximation is that the pro- jector onto the subspace of electronic states depends on the sep- aration between positive-energy and negative-energy one-particle states, and therefore, it depends on the potential used to generate these one-particle states. If the projector is applied to the Hamil- tonian, the whole resulting projected Hamiltonian is thus depen- dent on this potential. As mentioned in Ref. 14, this is problem- atic for formulating DFT since we cannot isolate, as normally done, a universal part of the Hamiltonian, and we thus cannot define universal density functionals. However, instead of thinking of the projector as being applied to the Hamiltonian, we can equivalently think of the projector as being applied to the considered many- electron state and optimize the projector simultaneously with the wave function. In this way, we can introduce universal density func- tionals, similarly to non-relativistic DFT, defined such that, for a given density, a constrained-search optimization of the projected wave function will determine alone the optimal projector without the need of pre-choosing a particular potential, at least for sys- tems for which positive-energy and negative-energy one-particle states can be unambiguously separated. Again, both the effective QED approach and the configuration-space approach can a priori be used for doing so. In the effective QED approach, the projec- tor would be optimized (by rotations between positive-energy and negative-energy one-particle states) using an energy minimization principle. In the configuration-space approach, the projector is opti- mized using a minmax principle.23–28In the present work, we fol- low the configuration-space approach and leave for future work the alternative formulation based on the effective QED approach. We thus start with the Dirac–Coulomb electronic Hamiltonian (see, e.g., Refs. 29 and 30) ˆH=ˆTD+ˆVne+ˆWee, (1) where the kinetic + rest mass Dirac operator ˆTD, the nucleus– electron interaction operator ˆVne, and the Coulomb electron– electron interaction operator ˆWeeare expressed using four- component creation and annihilation field operators ˆψ†(r)and ˆψ(r) without normal reordering with respect to a stable vacuum state. We thus write ˆTDas ˆTD=∫ˆψ†(r)[c(α⋅p)+βmc2]ˆψ(r)dr, (2) where p=−i∇ris the momentum operator, c= 137.036 a.u. is the speed of light, m= 1 a.u. is the electron mass, and αand βare the 4×4 Dirac matrices, α=(02 σ σ02)and β=(I202 02−I2), (3) J. Chem. Phys. 152, 214106 (2020); doi: 10.1063/5.0004926 152, 214106-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp where σ= (σx,σy,σz) is the three-dimensional vector of the 2 ×2 Pauli matrices and 02andI2are the 2×2 zero and identity matrices, respectively. Similarly, we write ˆVneand ˆWeeas ˆVne=∫vne(r)ˆn(r)dr, (4) where vne(r) is the nucleus-electron potential, and ˆWee=1 2∬wee(r12)ˆn2(r1,r2)dr1dr2, (5) where wee(r12) = 1/ r12is the Coulomb electron–electron potential, and ˆn(r)=ˆψ†(r)ˆψ(r)and ˆn2(r1,r2)=ˆψ†(r1)ˆψ†(r2)ˆψ(r2)ˆψ(r1) are the density and pair density operators, respectively. Introducing a set of orthonormal four-component-spinor orbitals {ψp(r)} that are eigenfunctions of a one-particle Dirac Hamiltonian with some potential, and assuming that this set of orbitals can be partitioned into a set of positive-energy orbitals and a set of negative-energy orbitals, {ψp(r)}={ψp(r)}εp>0∪{ψp(r)}εp<0, the no-pair15,16relativistic ground-state energy of an N-electron sys- tem can be defined using a minmax principle,23–28which we will formally write as E0=minmax Ψ+⟨Ψ+∣ˆTD+ˆVne+ˆWee∣Ψ+⟩. (6) In this equation, we search over normalized wave functions of the form ∣Ψ+⟩=ˆP+∣Ψ⟩, where ˆP+is the projector on the N- electron-state space generated by the set of positive-energy orbitals {ψp(r)}εp>0and |Ψ⟩is a general N-electron antisymmetric wave function, and the notation minmax Ψ+=minΨmax ˆP+=max ˆP+minΨ means a minimization with respect to Ψand a maximization with respect to ˆP+. This maximization must be done by rotations of the positive-energy orbitals {ψp(r)}εp>0with its complementary set of negative-energy orbitals {ψp(r)}εp<0. Here, we have assumed that the optimum of minmax is a saddle point in the wave-function parameter space [which can be calculated with a multiconfiguration self-consistent-field (MCSCF) algorithm28,31,32], so that the same energy is obtained whatever the order of min Ψand max ˆP+. Note that, in the non-relativistic limit ( c→∞), the energy gap between positive- and negative-energy orbitals of order 2 mc2goes to infinity and the maximization over ˆP+becomes useless, and thus, the min- max principle properly reduces to the non-relativistic minimization principle. Now, we attempt to formulate a relativistic DFT within this no- pair approximation. Following the spirit of the constrained-search formulation of non-relativistic DFT,33,34we propose to define the no-pair relativistic universal density functional as F[n]=minmax Ψ+→n⟨Ψ+∣ˆTD+ˆWee∣Ψ+⟩ =⟨Ψ+[n]∣ˆTD+ˆWee∣Ψ+[n]⟩, (7) where the minmax procedure is identical to that in Eq. (6) except for the additional constraint that Ψ+yields the density n, i.e., ⟨Ψ+∣ˆn(r)∣Ψ+⟩=n(r). In Eq. (7), Ψ+[n] is the optimal wave function for the density n. We will again assume that the optimum of minmax is a saddle point in the density-constrained wave-function parameter subspace. Of course, this functional is only defined for densities thatcome from a wave function of the form of Ψ+, which we will refer to asΨ+-representable densities. Note that, consistently by neglect- ing the Breit electron–electron interaction, we will only consider functionals of the density and not of the density current. The no- pair relativistic ground-state energy of Eq. (6) can be in principle obtained from F[n] as a stationary point with respect to variations overΨ+-representable densities, E0∈stat n{F[n]+∫vne(r)n(r)dr}, (8) where we have introduced the notation stat nto designate the set of stationary energies with respect to variations of n. Due to the min- max principle in Eqs. (6) and (7), we can only assume a stationary principle in Eq. (8), instead of the usual non-relativistic minimiza- tion principle over densities. This situation is in fact similar to the problem of formulating a pure-state time-independent variational extension of DFT for excited-state energies.35,36 We now define a no-pair relativistic long-range universal den- sity functional, similarly to Eq. (7), as Flr,μ[n]=minmax Ψ+→n⟨Ψ+∣ˆTD+ˆWlr,μ ee∣Ψ+⟩ =⟨Ψμ +[n]∣ˆTD+ˆWlr,μ ee∣Ψμ +[n]⟩, (9) with the long-range electron–electron interaction operator ˆWlr,μ ee=(1/2)∬wlr,μ ee(r12)ˆn2(r1,r2)dr1dr2, where wlr,μ ee(r12) =erf(μr12)/r12is the long-range electron–electron potential and μ is the range-separation parameter. In Eq. (9), Ψμ +[n]is the optimal wave function for the density nand range-separation parameter μ. We can thus decompose the density functional F[n] as F[n]=Flr,μ[n]+¯Esr,μ Hxc[n], (10) where ¯Esr,μ Hxc[n]defines the complementary relativistic short-range Hartree-exchange-correlation density functional. Plugging Eq. (10) into Eq. (8), we conclude that the no-pair relativistic ground-state energy of Eq. (6) corresponds to a stationary point of the following range-separated energy expression over Ψ+wave functions: E0∈stat Ψ+{⟨Ψ+∣ˆTD+ˆVne+ˆWlr,μ ee∣Ψ+⟩+¯Esr,μ Hxc[nΨ+]}, (11) where nΨ+is the density of Ψ+. For practical calculations, we will assume that the no-pair relativistic ground-state energy corresponds in fact to the minmax search over Ψ+, E0=minmax Ψ+{⟨Ψ+∣ˆTD+ˆVne+ˆWlr,μ ee∣Ψ+⟩+¯Esr,μ Hxc[nΨ+]}. (12) Even though we do not see any guarantee that this is always true, it seems a reasonable working assumption for practical calculations. In fact, it corresponds to what is done in practice in no-pair Kohn– Sham DFT calculations,37–46which corresponds to Eq. (12) in the special case of μ= 0, i.e., E0=minmax Φ+{⟨Φ+∣ˆTD+ˆVne∣Φ+⟩+EHxc[nΦ+]}, (13) where the wave function can be restricted to a single determinant Φ+and EHxc[n] is the relativistic Kohn–Sham Hartree-exchange- correlation density functional. Another special case of Eq. (12) is for μ→∞for which we correctly recover the wave-function theory of Eq. (6). J. Chem. Phys. 152, 214106 (2020); doi: 10.1063/5.0004926 152, 214106-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp As usual, we can decompose the complementary relativistic short-range Hartree-exchange-correlation density functional into separate components, ¯Esr,μ Hxc[n]=Esr,μ H[n]+Esr,μ x[n]+¯Esr,μ c[n]. (14) In this expression, Esr,μ H[n]is the short-range Hartree density func- tional (which has the same expression as in the non-relativistic case), Esr,μ H[n]=1 2∬wsr,μ ee(r12)n(r1)n(r2)dr1dr2, (15) where wsr,μ ee(r12)=wee(r12)−wlr,μ ee(r12)is the short-range electron–electron potential, and Esr,μ x[n]is the relativistic short- range exchange density functional, Esr,μ x[n]=⟨Φ+[n]∣ˆWsr,μ ee∣Φ+[n]⟩−Esr,μ H[n], (16) where Φ+[n]=Ψμ=0 +[n]is the relativistic Kohn–Sham single- determinant wave function and ˆWsr,μ ee=(1/2)∬wsr,μ ee(r12)ˆn2(r1,r2) dr1dr2is the short-range electron–electron interaction operator, and ¯Esr,μ c[n]is the complementary relativistic short-range corre- lation density functional. In Appendix A, we show that the rela- tivistic short-range exchange density functional Esr,μ x[n]satisfies a uniform coordinate scaling relation [Eq. (A6)], which represents an important constraint to impose in approximations. Even though the present formulation of relativistic RS-DFT seems reasonable for practical chemical applications, it obviously calls for a closer mathematical examination of its domain of valid- ity. In particular, it is clear that the minmax principle of the no-pair approximation in the configuration-space approach breaks down in the strong relativistic regime (i.e., for nuclear charges Z≳c). Of course, in the non-relativistic limit ( c→∞), relativistic RS-DFT properly reduces to non-relativistic RS-DFT. III. COMPUTATIONAL SETUP We consider the helium, beryllium, neon, and argon isoelec- tronic series, up to the uranium nuclear charge Z= 92. The electronic density n(r) naturally increases in the nucleus with Zand can be con- veniently measured with kFmax, i.e., the maximal value taken at the nucleus by the local Fermi wave vector kF(r)=(3π2n(r))1/3. The strength of the relativistic effects can be measured by comparing the local Fermi wave vector kF(r) to the speed of light c≃137.036 a.u. (with̵h=me= 1 a.u.): very little relativistic effects are expected in regions where kF(r)≪c, while strong relativistic effects are expected in regions where kF(r)≳c. To test the different functionals, we have first performed four-component Dirac–Hartree–Fock (DHF) calculations based on the relativistic Dirac–Coulomb Hamiltonian with the point-charge nucleus, using our own program implemented as a plugin of the software QUANTUM PACKAGE 2.0.47For the helium series, we use the dyall_1s2.3z basis set of Ref. 28 except for Yb68+and U90+for which the basis set was not available. For these systems, as well as for the beryllium, neon, and argon series, we construct uncon- tracted even-tempered Gaussian-type orbital basis sets,48following the primitive structure of the dyall-cvdz basis sets for He, Be, Ne, and Ar.49For each system and angular momentum, the exponentsof the large-component basis functions are taken as the geometric series ζν=ζ1qν−1, (17) whereζ1is chosen among the largest exponents from the dyall- cvdz basis set for the given element and angular momentum49,50and the parameter qis optimized by minimizing the DHF total energy. The small-component basis functions are generated from the unre- stricted kinetic-balance scheme.51The basis-set parameters are given in the supplementary material. Using the previously obtained DHF orbitals, we then estimate the short-range exact exchange energy Esr,μ x=1 2∬wsr,μ ee(r12)n2,x(r1,r2)dr1dr2, (18) where n2,x(r1,r2) is the exchange pair density given as n2,x(r1,r2)=−Tr[γ(r1,r2)γ(r2,r1)], (19) whereγ(r1,r2)=∑N i=1ψi(r1)ψ† i(r2)is the 4×4 one-electron density matrix written with the four-component-spinor occupied orbitals {ψi(r)}. This short-range DHF exchange energy is used as the ref- erence for testing the different exchange energy functionals, which are evaluated with the DHF density n(r) = Tr[γ(r,r)] (and the DHF exchange on-top pair density for some of them, see below) using an SG-2-type quadrature grid52with the radial grid of Ref. 53. IV. SHORT-RANGE EXCHANGE LOCAL-DENSITY APPROXIMATIONS The non-relativistic short-range local-density approximation (srLDA) for the exchange functional has the expression Esr,LDA,μ x[n]=∫n(r)ϵsr,HEG,μ x(n(r))dr, (20) where the non-relativistic short-range homogeneous electron gas (HEG) exchange energy per particle ϵsr,HEG,μ x(n)can be found in Refs. 1, 54, and 55. The relativistic generalization of this functional, referred to as the srRLDA, is Esr,RLDA,μ x[n]=∫n(r)ϵsr,RHEG,μ x(n(r))dr, (21) where the short-range RHEG exchange energy per particle ϵsr,RHEG,μ x(n)is given in Ref. 11 with arbitrary accuracy as system- atic Padé approximants with respect to the dimensionless variable ˜c=c/kF=c/(3π2n)1/3(we employ here the Padé approximant of order 6) with coefficients written as functions of the dimensionless range-separation parameter μ/kF. The dependence of ϵsr,RHEG,μ x(n) on the dimensionless parameters ˜candμ/kFis a consequence of the uniform coordinate scaling relation of Eq. (A6), which is valid of the RHEG. The relative percentage errors of the srLDA and srRLDA exchange functionals with respect to the short-range DHF exchange energy, i.e., 100 ×(Esr,DFA,μ x−Esr,μ x)/∣Esr,μ x∣, are plotted in Fig. 1 as a function of the dimensionless range-separation parameter μ/kFmax for three representative members of the neon isoelectronic series (Ne, Xe44+, and Rn76+). The relativistic effects go from very small for Ne to very large for Rn76+. J. Chem. Phys. 152, 214106 (2020); doi: 10.1063/5.0004926 152, 214106-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . Relative percentage error of the short-range exchange energy calcu- lated with the srLDA, srRLDA, srLDAot, and srRLDAot functionals for three representative members of the neon isoelectronic series (Ne, Xe44+, and Rn76+). Forμ= 0, the short-range interaction reduces to the full-range Coulomb interaction, and we observe that both the non-relativistic and relativistic LDA exchange functionals underestimate (in abso- lute value) the DHF exchange energy by 5%–10%. As previouslynoted,56the non-relativistic LDA exchange functional (evaluated with a relativistic density) fortuitously gives exchange energies with lower errors than the relativistic LDA exchange functional for sys- tems with significant relativistic effects (Xe44+and Rn76+). Whenμ increases, the srLDA and srRLDA exchange functionals show quite different behaviors for these relativistic systems. The relative error of the srLDA exchange energy changes sign with μand eventually goes to a negative constant for μ→∞, corresponding to an over- estimation in absolute value. By contrast, the relative error of the srRLDA exchange energy always remains positive and goes to a pos- itive constant for μ→∞, corresponding to an underestimation in absolute value. The more relativistic the system is, the larger this overestimation or underestimation is. While for Ne at large μboth the srLDA and srRLDA exchange functionals have almost vanish- ing relative errors, for Rn76+at largeμthe srLDA exchange energy is too negative by a little more than 5% and the srRLDA exchange energy is too positive by almost 20%. Clearly, both the srLDA and srRLDA exchange functionals are not accurate for relativistic systems. In non-relativistic theory, it is known that the srLDA exchange functional becomes exact for large μ,2which is one of the key advan- tages of RS-DFT. As apparent from Fig. 1, for relativistic systems, this nice property does not hold anymore for both the srLDA and srRLDA exchange functionals. This observation can be understood by using the distributional asymptotic expansion of the short-range interaction for large μ,2 wsr,μ ee(r12)=π μ2δ(r12)+O(1 μ3), (22) which directly leads to the asymptotic expansion of the short-range exact exchange energy Esr,μ x=π 2μ2∫n2,x(r,r)dr+O(1 μ4), (23) where n2,x(r,r) is the on-top exchange pair density. In the non- relativistic theory, considering the case of closed-shell systems for the sake of simplicity, the on-top exchange pair density is simply given in terms of the density as57 nNR 2,x(r,r)=−n(r)2 2, (24) and the srLDA exchange functional becomes indeed exact for largeμ, Esr,LDA,μ x[n]=π 2μ2∫nHEG,0 2,x(n(r))dr+O(1 μ4), (25) with the on-top exchange pair density of the non-relativistic HEG nHEG,0 2,x(n)=−n2 2. (26) In the relativistic theory, the on-top exchange pair density is no longer a simple function of the density, n2,x(r,r)=−Tr[γ(r,r)2], (27) which is not equal to −n(r)2/2, except in the special case of two elec- trons in a unique Kramers pair (see Appendix B). Therefore, for J. Chem. Phys. 152, 214106 (2020); doi: 10.1063/5.0004926 152, 214106-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp relativistic systems with more than two electrons, we see that the srLDA exchange functional is not exact for large μ[Eq. (25)]. The srRLDA exchange functional is also not exact for large μ. It takes the form Esr,RLDA,μ x[n]=π 2μ2∫nRHEG,0 2,x(n(r))dr+O(1 μ4), (28) with the on-top exchange pair density of the RHEG nRHEG,0 2,x(n)=−n2 4(1 +h(˜c)), (29) and the function11 h(˜c)=9 4[˜c2+˜c4−˜c4arcsinh(1 ˜c)(2√ 1 +˜c2−˜c2arcsinh(1 ˜c))]. (30) For an alternative but equivalent expression for nRHEG,0 2,x(n), see Eq. (A1) of Ref. 39. The srRLDA exchange functional is in fact not even exact at large μfor two electrons in a unique Kramers pair. In Sec. V, we show how to impose the large- μbehavior on the srLDA and srRLDA exchange functionals. V. SHORT-RANGE EXCHANGE LOCAL-DENSITY APPROXIMATIONS WITH ON-TOP EXCHANGE PAIR DENSITY In order to impose the correct large- μbehavior on the srLDA and srRLDA exchange functionals for relativistic systems, we need to introduce a new ingredient in these functionals, namely, the exact (relativistic) on-top exchange pair density n2,x(r,r), or equivalently the on-top exchange hole nx(r,r)=n2,x(r,r) n(r). (31) A simple way to use nx(r,r) to correct the srLDA exchange functional is to find, at each position r, the effective density neff(r) at which the on-top exchange hole of the HEG, nHEG,0 x(n) =nHEG,0 2,x(n)/n=−n/2, is equal to the on-top exchange hole of the inhomogeneous system considered, nx(r,r), i.e., nHEG,0 x(neff(r))=nx(r,r), (32) which simply gives neff(r) =−2nx(r,r). We then define the srLDA exchange functional with the on-top exchange pair density (srL- DAot) using this effective density as Esr,LDAot,μ x [n]=∫n(r)ϵsr,HEG,μ x(neff(r))dr. (33) This approximation could be considered either as an implicit func- tional of the density alone since nx(r,r) is an implicit functional of the density through the orbitals or as an explicit functional of both the density and the on-top exchange hole nx(r,r). This approxi- mation corresponds to changing the transferability criterion in the LDA: at a given point r, instead of taking the exchange energy perparticle of the HEG having the same density than the inhomoge- neous system at that point, we now take the exchange energy per particle of the HEG having the same on-top exchange hole than the inhomogeneous system at that point. Interestingly, this approxima- tion can be thought of as a particular application of the recently formalized connector theory.58,59 Similarly, we can correct the srRLDA exchange functional by finding, at each position r, the effective density nR eff(r) at which the on-top exchange hole of the RHEG, nRHEG,0 x(n) =nRHEG,0 2,x(n)/n=−(n/4)(1 +h(˜c)), is equal to the on-top exchange hole of the inhomogeneous system considered, nx(r,r), i.e., nRHEG,0 x(nR eff(r))=nx(r,r). (34) This equation is less trivial to solve than Eq. (32) since nRHEG,0 x(n) is a complicated nonlinear function of n(through ˜c). However, at each point r, a unique solution nR eff(r)exists since the function n↦nRHEG,0 x(n)is monotonically decreasing and spans the domain [−∞, 0] in which nx(r,r) necessarily belongs. In practice, we eas- ily find nR eff(r)by a numerical iterative method, and we use it to define the srRLDA exchange functional with the on-top exchange pair density (srRLDAot) as Esr,RLDAot, μ x [n]=∫n(r)ϵsr,RHEG,μ x(nR eff(r))dr. (35) Both the srLDAot and srRLDAot exchange functionals now fulfill the exact asymptotic expansion for large μ[Eq. (23)]. In fact, restor- ing the correct on-top value of the exchange hole could be beneficial for any value of μ, given the fact that the accuracy of non-relativistic Kohn–Sham exchange DFAs has been justified by the exactness of the underlying LDA on-top exchange hole (in addition to fulfilling the correct sum rule of the exchange hole).60Finally, we note that, in the non-relativistic limit ( c→∞), we have neff(r)=nR eff(r)=n(r) and all these short-range exchange functionals reduce to the non- relativistic srLDA exchange functional (i.e., srLDAot = srRLDAot = srRLDA = srLDA). The relative percentage errors of the srLDAot and srRLDAot exchange functionals for Ne, Xe44+, and Rn76+are reported in Fig. 1. The most prominent feature is of course the correct recovery of the large-μasymptotic behavior for both the srLDAot and srRL- DAot exchange functionals. It turns out the srLDAot and srRLDAot exchange functionals give very similar exchange energies for all val- ues ofμ. This comes from the fact that going from srLDA to srLDAot [Eq. (32)] tends to make the LDA exchange hole shallower and going from srRLDA to srRLDAot [Eq. (34)] tends to make the relativistic LDA exchange hole deeper, making finally for very close descrip- tions. The absolute relative percentage errors of the srLDAot and srRLDAot exchange functionals are always below 10%, and below about 2% for μ/kFmax≥0.5. VI. SHORT-RANGE EXCHANGE GENERALIZED-GRADIENT APPROXIMATIONS In order to improve over the short-range LDA exchange func- tionals at small values of the range-separation parameter μ, we now consider short-range GGA exchange functionals. We start with J. Chem. Phys. 152, 214106 (2020); doi: 10.1063/5.0004926 152, 214106-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the non-relativistic short-range extension of the Perdew–Burke– Ernzerhof (PBE)61theory of Refs. 62 and 63, referred to as srPBE, Esr,PBE,μ x[n]=∫n(r)ϵsr,HEG,μ x(n(r))[1 +fμ x(n(r),∇n(r))]dr, (36) with the function fμ x(n,∇n)=κ−κ 1 +b(˜μ)s2/κ, (37) where s= |∇n|/(2kFn) is the reduced density gradient and ˜μ =μ/(2kF)is a dimensionless range-separation parameter. In this expression, κ= 0.840 is a constant fixed by imposing the Lieb– Oxford bound (for μ= 0) and b(˜μ)=bPBE[bT(˜μ)/bT(0)]e−αx˜μ2, where bPBE= 0.219 51 is the second-order gradient-expansion coef- ficient of the standard PBE exchange functional, bT(˜μ)is a function coming from the second-order gradient-expansion approximation (GEA) of the short-range exchange energy and given in Refs. 64 and 65, andαx= 19.0 is a damping parameter optimized on the He atom. Forμ= 0, this srPBE exchange functional reduces to the standard PBE exchange functional,61and for large μ, it reduces to the srLDA exchange functional. A simple relativistic extension of this srPBE exchange func- tional can be obtained by replacing the srLDA part by the srRLDA one while using the same density-gradient correction fμ x(n,∇n), to which we will refer as srRLDA/PBE, Esr,RLDA/PBE, μ x [n]=∫n(r)ϵsr,RHEG,μ x(n(r)) ×[1 +fμ x(n(r),∇n(r))]dr, (38) which reduces to the srRLDA exchange functional for large μ. The srPBE and srRLDA/PBE exchange functionals have the same (incorrect) asymptotic expansions as the srLDA and srRLDA exchange functionals [Eqs. (25) and (28)], and we can thus use the same effective densities in Eqs. (32) and (34) to restore their large- μ behaviors, which define the srPBEot and srRLDA/PBEot exchange functionals, Esr,PBEot,μ x[n]=∫n(r)ϵsr,HEG,μ x(neff(r)) ×[1 +fμ x(neff(r),∇neff(r))]dr, (39) where∇neff(r) =−2∇nx(r,r), and Esr,RLDA/PBEot, μ x [n]=∫n(r)ϵsr,RHEG,μ x(nR eff(r)) ×[1 +fμ x(nR eff(r),∇nR eff(r))]dr, (40) where∇nR eff(r)=[dnRHEG,0 x(nR eff(r))/dnR eff]−1∇nx(r,r). In Fig. 2, we report the relative percentage errors of the srPBE, srRLDA/PBE, srPBEot, and srRLDA/PBEot exchange energies for Ne, Xe44+, and Rn76+. For Ne, where the relativistic effects are very small, all these functionals give almost the same exchange energy, as expected. For Xe44+and Rn76+, even though the srPBE and srRLDA/PBE exchange functionals are more accurate than the srLDA and srRLDA exchange functionals at μ= 0 (see Fig. 1), they eventually suffer from the same large inaccuracy as srLDA and FIG. 2 . Relative percentage error of the short-range exchange energy calculated with the srPBE, srRLDA/PBE, srPBEot, and srRLDA/PBEot functionals for three representative members of the neon isoelectronic series (Ne, Xe44+, and Rn76+). srRLDA as μincreases. This problem is solved by using the effec- tive densities from the on-top exchange pair density, the srPBEot and srRLDA/PBEot exchange functionals giving vanishing errors at largeμ. Similarly to what was observed for srLDAot and srRLDAot, J. Chem. Phys. 152, 214106 (2020); doi: 10.1063/5.0004926 152, 214106-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Relativistic correction factor ϕμ(n) to the density-gradient term [Eq. (42)] as a function of kFfor several values of μ. the srPBEot and srRLDA/PBEot functionals give very close exchange energies for all values of μ. Interestingly, we see that using the effec- tive densities also reduces the errors of srPBE and srRLDA/PBE at μ= 0, making srPBEot and srRLDA/PBEot quite accurate in thisfull-range limit. Thus, the srPBEot and srRLDA/PBEot exchange functionals are definitely an improvement over srLDAot and srRL- DAot. We observe a maximal absolute percentage error of about 3% for Rn76+forμ/kFmax≈0.2. In order to further reduce the errors, in particular for interme- diate values of μ, we now consider a relativistic correction to the density-gradient term in the srRLDA/PBEot exchange functional. We define a short-range relativistic PBE exchange functional using the on-top exchange pair density, referred to as srRPBEot, Esr,RPBEot, μ x [n]=∫n(r)ϵsr,RHEG,μ x(nR eff(r)) ×[1 +fμ x(nR eff(r),∇nR eff(r))ϕμ(nR eff(r))]dr, (41) where, in the spirit of the work of Engel et al. ,56we have introduced a multiplicative relativistic correction ϕμ(n) to the term fμ x(n,∇n)of the form ϕμ(n)=1 +a1(μ/c) ˜c2+a2(μ/c) ˜c4 1 +b1(μ/c) ˜c2+b2(μ/c) ˜c4. (42) Sinceϕμ(n) only depends on the dimensionless parameters ˜candμ/c, it does not change the uniform coordinate scaling of the functional, which still fulfills the scaling relation of Eq. (A6). FIG. 4 . Relative percentage error of the short-range exchange energy calculated with the srRPBEot functional for systems of helium, beryllium, neon, and argon isoelectronic series. J. Chem. Phys. 152, 214106 (2020); doi: 10.1063/5.0004926 152, 214106-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp After some tests, we chose to impose a1(μ/c) = b1(μ/c) to avoid overcorrections in low-density regions, which have very small relativistic effects. We started to determine the coefficients forμ= 0 by minimizing the mean squared relative percent- age error of the exchange energy with respect to the reference DHF exchange energy for seven systems of the neon isoelectronic series (Ne, Ar8+, Kr26+, Xe44+, Yb60+, Rn76+, U82+), giving a1(0) =b1(0) = 1.3824, a2(0) = 0.3753, and b2= 0.4096. The resulting relativistic correction factor ϕμ=0(n) can be seen in Fig. 3. It cor- rectly tends to 1 in the low-density ( kF→0) or non-relativistic (c→∞) limit and remains very close to 1 for kF≪c. In regions with very high densities, the relativistic correction factor ϕμ=0(n) induces a slight reduction of the effective density-gradient correction term in the functional, reducing a bit the relative error of the exchange energy for the heaviest systems. Forμ≠0, we have searched for coefficients in Eq. (42), which reduce the largest errors of the srRLDA/PBEot exchange energy observed at intermediate values of μ(see Fig. 2). We chose coeffi- cients depending on μ/cof the form a1(μ/c)=b1(μ/c)=a1(0)[1−erf(μ/c)], (43) a2(μ/c)=a2(0)[1−erf(μ/c)], (44) b2(μ/c)=b2(0)[1−βerf(μ/c)], (45) withβ=−4.235, which has been found by minimizing the mean squared relative percentage error of the short-range exchange energy for the same seven systems of the neon isoelectronic series and for four intermediate values of the range-separation parameter (μ/kFmax=0.05; 0.1; 0.2; 0.4). The resulting relativistic correction fac- torϕμ(n) is reported in Fig. 3. It still tends to 1 in the low-density limit but goes down to 0 when μ≫cin the high-density limit. The higher the value of μ, the faster it decreases as a function of kF. In Fig. 4, we report the relative percentage errors of the srRP- BEot exchange functional for systems of the helium, beryllium, neon, and argon series. For μ= 0, this functional achieves an error of at most about 1% for all systems, and it has the cor- rect large-μlimit. The maximum absolute percentage errors, which are found for intermediate values of μ, tend to grow with Zbut remain at most about 3% for the heavier systems. The srRPBEot exchange functional represents a significant improvement over the srPBEot and srRLDA/PBEot exchange functionals for the heavier systems. VII. CONCLUSIONS In this work, we have tested the srRLDA exchange functional developed in Ref. 11 on three systems of the neon isoelectronic series (Ne, Xe44+, and Rn76+) and compared it to the usual non-relativistic srLDA exchange functional. Both functionals are quite inaccurate for relativistic systems and do not have the correct asymptotic behavior for a large range-separation parameter μ. In order to fix this large-μbehavior, we have then defined the srLDAot and srRLDAot exchange functionals by introducing the exact on-top exchange pair density as a new variable. These functionals recover the correct asymptotic behavior for large μbut remain inaccurate for smallvalues ofμ. To improve the accuracy for small values of μ, we have then developed a relativistic short-range GGA exchange functional also using the on-top exchange pair density as an extension of the non-relativistic srPBE exchange functional. Tests on the systems of the isoelectronic series of He, Be, Ne, and Ar up to Z= 92 show that this srRPBEot exchange functional gives a maximal relative percent- age error of 3% for intermediate values of μand less than 1% relative error forμ= 0. Of course, in the non-relativistic limit ( c→∞), all the relativistic functionals introduced in this work properly reduce to their non-relativistic counterparts. Possible continuations of this work include further tests on atoms and molecules, extension to the Gaunt or Breit electron– electron interactions, development of the short-range relativis- tic correlation functionals, and use of a local range-separation parameter. SUPPLEMENTARY MATERIAL See the supplementary material for the parameters of the even- tempered basis sets constructed in this work. APPENDIX A: UNIFORM COORDINATE SCALING RELATION FOR THE RELATIVISTIC NO-PAIR SHORT-RANGE EXCHANGE DENSITY FUNCTIONAL Here, we generalize the uniform coordinate scaling relation of the non-relativistic exchange density functional66and of the non- relativistic short-range exchange density functional67to the case of the relativistic no-pair short-range exchange density functional of Eq. (16). Since the scaling relation involves scaling the speed of light c, we will explicitly indicate in this section the dependence on c. First, we introduce the non-interacting Dirac kinetic + rest mass energy density functional Tc s[n]defined by Eq. (9) in the special case of a vanishing range-separation parameter, μ= 0, Tc s[n]=minmax Φ+→n⟨Φ+∣ˆTc D∣Φ+⟩=⟨Φc +[n]∣ˆTc D∣Φc +[n]⟩, (A1) whereΦc +[n]is the relativistic Kohn–Sham single-determinant wave function. Let us now consider the scaled wave function Φc +,γ[n] defined by, for Nelectrons, Φc +,γ[n](r1,. . .,rN)=γ3N/2Φc +[n](γr1,. . .,γrN), (A2) whereγ>0 is a scaling factor. The wave function Φc +,γ[n]yields the scaled density nγ(r) =γ3n(γr) and is the minmax optimal wave function of ⟨Φ+∣ˆTcγ D∣Φ+⟩since it can be checked that ⟨Φc +,γ[n]∣ˆTcγ D∣Φc +,γ[n]⟩=γ2⟨Φc +[n]∣ˆTc D∣Φc +[n]⟩, (A3) and the right-hand side is minmax optimal by definition of Φc +[n]. Therefore, we conclude that Φc +,γ[n]=Φcγ +[nγ]. (A4) From the definition of the relativistic short-range exchange energy density functional Esr,μ,c x[n]=⟨Φc +[n]∣ˆWsr,μ ee∣Φc +[n]⟩−Esr,μ H[n], we then arrive at the scaling relation J. Chem. Phys. 152, 214106 (2020); doi: 10.1063/5.0004926 152, 214106-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Esr,μγ,cγ x[nγ]=γEsr,μ,c x[n], (A5) or, equivalently, Esr,μ,c x[nγ]=γEsr,μ/γ,c/γ x[n]. (A6) This scaling relation is an important constraint that is satisfied by our approximate density functionals. Besides, it shows that the low-density limit ( γ→0) corresponds to the non-relativistic limit (c→∞), while the high-density limit ( γ→∞) corresponds to the ultra-relativistic limit ( c→0). It also shows that, for a fixed value of the range-separation parameter μ, low-density regions explore the functional in the short-range limit ( μ→∞) and high-density regions explore the functional in the full-range limit ( μ= 0).APPENDIX B: ON-TOP EXCHANGE PAIR DENSITY IN A FOUR-COMPONENT RELATIVISTIC FRAMEWORK Using four-component-spinor orbitals ψi(r)=⎛ ⎜⎜⎜⎜ ⎝ψLα i(r) ψLβ i(r) ψSα i(r) ψSβ i(r)⎞ ⎟⎟⎟⎟ ⎠, (B1) the on-top value of the 4 ×4 one-electron density matrix has the expression γ(r,r)=N ∑ i=1ψi(r)ψ† i(r)=⎛ ⎜⎜⎜⎜ ⎝ψLα i(r)ψLα i(r)∗ψLα i(r)ψLβ i(r)∗ψLα i(r)ψSα i(r)∗ψLα i(r)ψSβ i(r)∗ ψLβ i(r)ψLα i(r)∗ψLβ i(r)ψLβ i(r)∗ψLβ i(r)ψSα i(r)∗ψLβ i(r)ψSβ i(r)∗ ψSα i(r)ψLα i(r)∗ψSα i(r)ψLβ i(r)∗ψSα i(r)ψSα i(r)∗ψSα i(r)ψSβ i(r)∗ ψSβ i(r)ψLα i(r)∗ψSβ i(r)ψLβ i(r)∗ψSβ i(r)ψSα i(r)∗ψSβ i(r)ψSβ i(r)∗⎞ ⎟⎟⎟⎟ ⎠, (B2) which leads to the density n(r)=Tr[γ(r,r)]=N ∑ i=1∣ψLα i(r)∣2+∣ψLβ i(r)∣2+∣ψSα i(r)∣2+∣ψSβ i(r)∣2. (B3) The on-top exchange pair density has the expression n2,x(r,r)=−Tr[γ(r,r)2]=−N ∑ i=1N ∑ j=1(∣ψLα i(r)∣2∣ψLα j(r)∣2+∣ψLβ i(r)∣2∣ψLβ j(r)∣2+ 2ψLα i(r)ψLβ i(r)∗ψLβ j(r)ψLα j(r)∗ +∣ψSα i(r)∣2∣ψSα j(r)∣2+∣ψSβ i(r)∣2∣ψSβ j(r)∣2+ 2ψSα i(r)ψSβ i(r)∗ψSβ j(r)ψSα j(r)∗+ 2ψLα i(r)ψSα i(r)∗ψSα j(r)ψLα j(r)∗ + 2ψLβ i(r)ψSβ i(r)∗ψSβ j(r)ψLβ j(r)∗+ 2ψLα i(r)ψSβ i(r)∗ψSβ j(r)ψLα j(r)∗+ 2ψLβ i(r)ψSα i(r)∗ψSα j(r)ψLβ j(r)∗). (B4) In the non-relativistic limit, each orbital has a definite spin state, i.e., ψi(r)=(ψLα i(r), 0, 0, 0)orψi(r)=(0,ψLβ i(r), 0, 0), and we recover the well-known expression of the on-top exchange pair density in terms of the spin densities nNR 2,x(r,r)=−N ∑ i=1N ∑ j=1(∣ψLα i(r)∣2∣ψLα j(r)∣2+∣ψLβ i(r)∣2∣ψLβ j(r)∣2) =−nα(r)2−nβ(r)2, (B5) or, for closed-shell systems, nNR 2,x(r,r)=−n(r)2/2. However, in the relativistic case, n2,x(r,r) can no longer be generally expressed explicitly with the density, as seen from the presence of terms mix- ing different spinor components in Eq. (B4). There are however two exceptions. The first exception is provided by one-electron sys- tems for which it is easy to check that n2,x(r,r) =−n(r)2, as in the non-relativistic case. The second exception is provided by sys- tems of two electrons in a unique Kramers pair, for which n2,x(r,r) =−n(r)2/2, as in the non-relativistic case. Indeed, for closed-shell systems, the one-electron density matrix can be decomposed into Kramers contributions,γ(r,r)=γ+(r,r)+γ−(r,r), (B6) whereγ+(r,r)=∑N/2 i=1ψi(r)ψ† i(r)andγ−(r,r)=∑N/2 i=1¯ψi(r)¯ψ† i(r), and ¯ψi(r)is the Kramers partner of ψi(r), ¯ψi(r)=⎛ ⎜⎜⎜⎜ ⎝−ψLβ i(r)∗ ψLα i(r)∗ −ψSβ i(r)∗ ψSα i(r)∗⎞ ⎟⎟⎟⎟ ⎠. (B7) In this case, the density can then be expressed as n(r) = 2Tr[γ+(r,r)] and the on-top exchange pair density as n2,x(r,r)=−2(Tr[γ+(r,r)2]+ Tr[γ+(r,r)γ−(r,r)]), (B8) where we have used Tr[ γ+(r,r)2] = Tr[γ−(r,r)2]. 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Phys. 152, 214106 (2020); doi: 10.1063/5.0004926 152, 214106-11 Published under license by AIP Publishing

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AIP Conference Proceedings 2220 , 100005 (2020); https://doi.org/10.1063/5.0001764 2220 , 100005 © 2020 Author(s).X-ray photoelectron spectroscopy study of a layered tri-chalcogenide system LaTe3 Cite as: AIP Conference Proceedings 2220 , 100005 (2020); https://doi.org/10.1063/5.0001764 Published Online: 05 May 2020 Shuvam Sarkar , Vipin Kumar Singh , Pampa Sadhukhan , Arnab Pariari , Shubhankar Roy , Prabhat Mandal , and Sudipta Roy Barman ARTICLES YOU MAY BE INTERESTED IN Growth of Sn on Ni 2MnGa(100) AIP Conference Proceedings 2220 , 090013 (2020); https://doi.org/10.1063/5.0001788 Study of magnetoresistance in polycrystalline Fe intercalated TaS 2 AIP Conference Proceedings 2220 , 030001 (2020); https://doi.org/10.1063/5.0001191 Magneto-transport properties of proposed triply degenerate topological semimetal Pd 3Bi2S2 Applied Physics Letters 112, 162402 (2018); https://doi.org/10.1063/1.5024479X-Ray photoelectron spectroscopy study of a layered tri - chalcogenide system LaTe 3 Shuvam Sarkar1,a), Vipin Kumar Singh1, Pampa Sadhukhan1, Arnab Pariari2, Shubhankar Roy2, Prabhat Mandal2, Sudipta Roy Barman1,b) 1UGC -DAE Consortium for Scientific Research, University Campus, Khandwa Road, Indore -452001, Madhya Pradesh, India 2Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata -700064, West Bengal, India Corresponding author: b)barmansr@gmail.com a)shuvamsarkarhere@gmail.com Abstract. By employing x -ray photoelectron spectroscopy , we present a detailed study of the core level spectra of LaTe 3 in its charge d ensity wave phase at room temperature. The analysis of the Te 3d spectrum by curve fitting using least square error minimization reveal s that the Te atoms exist in two different charge states : the two Te atoms in the Te -Te layer has a valency of -0.5, whereas the Te atom in La -Te layer has a valency of -2. The La 3d spectrum shows three peaks for each spin orbit component , where the main peak and the satellite peaks appear due to different final states rel ated to charge transfer from d ifferent ligand Te states. INTRODUCTION Low dimensional metallic systems often undergo charge density wave (CDW) transition due to instabilities of the Fermi surface in presence of electron -phonon coupling or due to the nesting behavior of the Fermi surfac e[1]. Among many CDW systems [2,3,4,5], layered rare earth tri-chalcogenides such as tri-tellurides (R Te3) have drawn great interest of condensed matter scientists and their electronic and transport behavior have been extensive ly studied by angle resolved photoemission spectroscopy (ARPES), sca nning tunneling microscopy (STM) and transport measure ments [ 5,6,7]. RTe 3 crystallizes in a weakly orthorhombic layered quasi -two dimensional structure , which favors the formation of CDW instabilities due to its nested Fermi surface [1]. LaTe 3, a member of the RTe 3 family, shows a n incommensurate CDW ordering in the Te -Te 2D layer with an ordering vector of 2/7c* at room temperature (RT) along the c direction [6] and its CDW transition tem perature is reported to be above 450K [8]. ARPES study shows that the hole like bands are gapped near the ΓZ̅̅̅ high symmetry line due to the CDW formation along c* and a reconstruction of the Fermi surface occurs near the X̅ high symmetry point of the Brillouin zone (BZ) due to imperfect nesting in that region of the BZ and both electron - and hole -like band s cross the Fermi level [7]. The origin of such hole like carriers in RTe 3 family has been considered to arise due to the partial filling of Te 5p orb itals in an anionic configuration involving different charged states of Te atoms [7,9,10]. However, until date , no experimental evidence support ing this proposition has been reported . In the present study, we present high resolution XPS core level study of LaTe 3 to directly show that the Te atoms residing in different layers exhibit different charged states . EXPERIMENTAL Single crystals of LaTe 3 were grown via tellurium flux technique[ 6,11]. The XPS measurements have been performed on a freshly cleaved surface of LaTe 3 in ultra -high vacuum (UHV) at a base pressure of 1×10-10 mbar using R4000 electron analyzer from Scienta Omicron GmbH equipped with a monochromatic Al -Kα x-ray source in 3rd International Conference on Condensed Matter and Applied Physics (ICC-2019) AIP Conf. Proc. 2220, 100005-1–100005-4; https://doi.org/10.1063/5.0001764 Published by AIP Publishing. 978-0-7354-1976-6/$30.00100005-1transmission mode with 100 eV pass energy and 0.3 c urved slit. The in strumental energy resolution was about 0.3 eV at RT . The c ore level XPS spectra have been fitted using a least square error minimization procedure , as in our previous work [12], where ea ch component is represented by a Doniach -Sunjic (DS) line shape convoluted with a fixed Gaussian function representing the instrumental broadening. RESULTS AND DISCUSSI ON Figure 1(a) show s a wide range XPS s pectra measured on freshly cleaved LaTe 3 and the spin orbit components of La 3d, Te 3d and Te 3p core-level spectra are indicated. From the La 3d and Te 3d intensity ratio, considering the ir photoemission cros s sections, the sample is found to be stoichiometric . In Fi g. 1(b), the Te 3d5/2 spectrum , recorded with better statistics and smaller step size , has been shown . It exhibit s a main peak at 572.3 eV binding energy (BE) and a lower intensity shoulder at 571.6 eV, which was not visible in the wide range spectrum in Fig. 1(a) . This shoulder is not the Te 3d 3/2 spin orbit component , which appears at 582.5 eV binding energy [see Fig. 1(a)] , the Te 3d spin orbit splitting being 10.4 eV. So, we consider two peaks (Te a and Te b) with DS line -shape s to represent the two components of Te 3d 5/2 in our fitting routine. We obtain a fit ted curve ( black solid line in Fig. 1( b)) providing an excellent fit , as shown by the residual at the top of the panel. T he parameters obtained from the fitting are listed in Table I. It is interesting to note that the intensity ratio of the areas under Teb:Te a is 1.18:0.59, i.e. the intensity of Teb is twice that of Tea. Teb appear s at 0.7± 0.1 eV higher BE and its life time broadening is larger compared to Te a. FIGURE 1. (a) A wide range XPS spectrum of LaTe 3, wher e La 3d, Te 3p and Te 3d core-level peaks are indicated . (b) The experimental Te 3d 5/2 core-level spectrum (violet filled circles) along with the fitted curve and components obtained from the least square fitting. The residual is shown (black solid line) at the top of each panel . The yellow shaded region represents the Tougaard inelastic background. (c) The structure of LaTe 3, where the green filled circles represent La atom, and the violet and navy bl ue filled circles represent different types of Te atom s, as indicated . TABLE 1. The parameters obtained from the least square curve fitting of Te 3d 5/2 using two DS line shapes, where all the parameters such as DS asymmetry ( α), life time broadening ( ), binding energy (BE) , and intensity have been varied. In order to understand the origin of the two components of Te 3d, we need to examine the structure of LaTe 3. The basic layered structure is evident in the unmodulated phase , where the CDW modulation causes a small change in the Components α BE (eV) [±0.05 ] (eV) [±0.05 ] Intensity BE difference (eV) [ ±0.1 ] 0.06 3d5/2 Tea 571.55 0.25 0.59 0.7 3d5/2Teb 572.25 0.35 1.18 100005-2atom positions of Te -Te layer. LaTe 3 crystallizes in a quasi -2 dimensional layered orthorhombic structure (space group - cmcm ) in the unmodulated phase [Fig. 1( c)]. It consists of (i) two La-Te1 corrugated layers sandwiched between two sheets of (ii) Te 2–Te3 (Te-Te) 2D layers. The La-Te1 corrugated layers are bonded via van der Waals forces with the Te -Te layer . So while Te atoms in the Te -Te layer are bonded with each other and are in similar chemical environment, the Te1 atoms in the La -Te1 layer are strongly bonded with the La atoms . Thus, their chemical environments are quite different and t he origin of the two components ( Tea and Teb) could be related to th is. This can be justified as follows: i n the unit cell of LaTe 3, the number of Te 2+Te 3 atoms is twice of Te 1 atoms . Since intensity of Te b is double of Te a, this gives us a hint that the Te b component is related to the Te2+Te 3 (Te 2,3) atoms in the Te -Te layer . The Te a component is related to the Te1 atoms residing in the La-Te1 layer. We also note that Tea appear s at lower BE, indicat ing that it is more negatively charged than Teb. The outer shell configurations of La and Te are [Xe]5d16s2 and [Kr] 4d105s25p4. La attains its stable closed shell +3 valence configuration by donating three electrons to Te. However, the Te1 atom to which La is strongly bonded can accept only two electrons to attain closed shell configuration an d attain the valency of Te-2. The remaining extra electron is left to be accepted by the Te -Te layer , each of the Te 2 and Te 3 atoms notionally sharing half of this electron and attaining Te-0.5 valency . This also explains why the bonding between the La -Te1 and the Te -Te layers are weak. It implies partial filling o f the Te 5p states in the Te-Te layer , which is also responsible for the CDW. It may be noted that such partial filling of the Te 5p states has also been reported in angle resolved photoemissi on study of other member s of the RTe 3 family such as CeTe 3 and PrTe 3[7,10]. In Fig. 2, the La 3d core level spectrum shows an apparent double peak structure with the more intense peaks (M) at 853.4 and 836.6 eV and the lesser intensity satellite peaks (S) at relatively lower BE at 849.8 and 832.8 eV for the 3d3/2 and 3d 5/2 spin orbit components, respectively. We first consider two DS line shape components for each spin orbit component for the fitting, but unlike Te 3d, in this case the fitting is not g ood with the residuals showing systematic deviation in some regions, as shown by the black arrows in Fig. 2(a). So, we consider three peaks (M, S 1 and S 2) for each spin orbit component and find that the fitting is vastly improved, as shown by the residual that shows only random scatter [Fig. 2(b)]. Here, the χ2 value is reduced by about 60% compared to the fitting with two components [Fig. 2(a)]. From the fitting, we find that the three components for La 3d 5/2 appear at 836.6 (M), 834.8 (S1) and 832.2 (S 2) eV. The broad feature, centered at 843.5 eV, shown by orange line is related to the plasmon loss feature of La 3d 5/2. FIGURE 2: The XPS spectra of La 3d fitted with (a) two and (b) three components for each spin -orbit component. The residuals (black line) of the fits are shown at the top of each panel. The multi -peak feature in the La 3d spectrum generally arises due to the presence of different photoemission final states[13 -17]. In c ompound such as La2O3, where La is in 3+ state, Fuggle et al.[13] and Schneider et al.[14] have attributed the more intense higher BE component of the La 3d spectrum to a poorly screened 3d94f0 final state and the lower BE components to the well screened 3d94f1L final state (L marks the hole in the ligand i.e. oxygen atom), where charge transfer from the ligands to the La 4f occurs. A detailed model calculation based on the Anderson impurity model to determine the line shape of La 3d has been performe d for LaTe 2, which has almost similar layered 100005-3structure as LaTe 3, by Chung et al. [15]. They find that La is in 3+ valence state and the La 3d spectrum comprises of 3d94f1L and 3d94f0 final states, as in the case of La 2O3. However, in their calculation, only one type of ligand Te 5p states were considered. Following Re f. 15, we assign the higher BE main peak (M) to the 3d94f0 final state. The other two components S 1 and S 2 are related to 3d94f1L final state. These appear 2.6 eV apart possibly because there are two types of ligand Te atoms (Te 2,3 versus Te 1) in LaTe 3. The ligand 5p states related to fully filled Te-2 i.e. Te 1 are at lower energy compared to the partially filled 5p states related to Te-0.5 (Te 2,3). This is supported by angle resolved photoemission spectroscopy where the Te-2 5p bands are clearly separate d in energy from the Te-0.5 related 5p bands [18]. CONCLUSION In conclusion, our study of the Te 3d and La 3d XPS core level spectra of LaTe 3 establish es that the Te atoms exist in two different charge states. The Te atoms (Te 2,3) that make up the Te sh eet have partially filled Te 5p states . The components in the La 3d spectrum can also be explained on the basis of charge transfer from the different types of Te atoms (Te 1 and Te 2,3). Our results are consistent with formation of CDW in the Te sheets and existence of hole -like bands reported earlier in angle resolved photoemission study of RTe 3 family . REFERENCES 1. G. G rüner, Density Waves in Solids, Frontiers in Physics , Cambridge (1994 ). 2. J. A. Wilson, F. J. DiSalvo, and S. Mahajan, Adv. Phys. 24, 117 -201 (1975) . 3. S. V. Borisenko, A. A. Kordyuk, V. B. Zabolotnyy, D. S. Inosov, D. Evtushinsky , B. B üchner, A. N. Yaresko, A. Varykhalov, R. Fo llath, W. Eberhardt, L. Patthey, and H. Berger , Phys. Rev. Lett. 102, 166402 (2009). 4. S. W. D’Souza, A. Rai, J. Nayak, M. Maniraj, R. S. Dhaka, S. R. Barman, D. L. Schlagel, T. A. Lograsso, and A. Chakrabarti , Phys. Rev. B 85, 085123 (2012) . 5. N. Ru, J. -H. Chu, and I. R. Fisher, Phys. Rev. B 78, 012410 (2008) . 6. N. Ru , and I. R. Fisher, Phys. Rev. B 73, 033101 (2006) . 7. V. Brouet, W. L. Yang, X. J. Zhou, Z. Hussain, R. G. Moore, R. He, D. H. Lu, Z. X. Shen, J. Laverock, S. B. Dugdale, N. Ru , and I. R. Fisher, Phys. Rev. B 77, 235104 (2008) . 8. N. Ru, C. L. Condron, G. Y. Margulis, K. Y. Shin, J. Laverock, S. B. Dugdale, M. F . Toney, and I. R. Fisher , Phys. Rev. B 77, 035114 (2008) . 9. J. S. Kang, C. G. Olson, Y. S. Kwon, J. H. Shim, and B. I. Min, Phys. Rev. B 74, 085115 (2006) . 10. E. Lee, D. H. Kim, H. W. Kim, J. D. Denlinger, H. Kim, J. Kim, K. Kim, B. I. Min, B. H. Min, Y. S. Kw on, and J. -S. Kang, Sci. Rep. 6 (2016), 10.1038/srep30318 . 11. A. Pariari, S. Koley, S. Roy, R. Singha, M. S. Laad, A. Taraphder, and P. Mandal, arXiv:1901.08267 (2019) . 12. P. Sadhukhan, S. W. D′Souza, V. K. Singh, R. S. Dhaka, A. Gloskovskii, S. K. Dhar, P. Raychaudhu ri, A. Chainani, A. Chakrabarti, and S. R. Barman, Phys. Rev. B 99, 035102 (2019). 13. J. C. Fuggle, M. Ca mpagna, Z. Zolnierek, R. Lässer, and A. Platau , Phys. Rev. Lett. 45, 1597 -1600 (1980 ). 14. W.-D. Schneider, B. Delley, E. Wuilloud, J. -M. Imer, and Y. Baer, Phys. Rev. B 32, 6819 -6831 (1985) . 15. J. Chung, J. Park, J. G. Park, B. -H. Choi, S. J. Oh, E. J. Cho, H. D. Kim , and Y. S. Kwon, J. Kor. Phys. Soc. 38, 744 -749 (2001). 16. D. F. Mulli ca, C. K. C. Lok, H. O. Perkins, and V. Young, Phys. Rev. B 31, 4039 -4042 (1985). 17. M. F. Sunding, K. Hadidi, S. Diplas, O. M. Løvvik, T. E. Norby, and A. E. Gunnæs, J. Elec. Spectros . Relat . Phenom . 184, 399-409 (2011). 18. S. Sarkar, P. Sadhukhan, D. Curcio, M. Bianchi, A. Pariari, S. Roy, P. Manda l, P. Hofmann , and S. R. Barman (to be published). 100005-4

5.0008980.pdf

J. Appl. Phys. 127, 184305 (2020); https://doi.org/10.1063/5.0008980 127, 184305 © 2020 Author(s).Metal-functionalized 2D boron sulfide monolayer material enhancing hydrogen storage capacities Cite as: J. Appl. Phys. 127, 184305 (2020); https://doi.org/10.1063/5.0008980 Submitted: 30 March 2020 . Accepted: 25 April 2020 . Published Online: 12 May 2020 Pushkar Mishra , Deobrat Singh , Yogesh Sonvane , and Rajeev Ahuja ARTICLES YOU MAY BE INTERESTED IN Electromechanical properties of soft dissipative dielectric elastomer actuators influenced by electrode thickness and conductivity Journal of Applied Physics 127, 184902 (2020); https://doi.org/10.1063/5.0001580 Highly tunable thermal conductivity of C 3N under tensile strain: A first-principles study Journal of Applied Physics 127, 184304 (2020); https://doi.org/10.1063/5.0006775 Metal-ion subplantation: A game changer for controlling nanostructure and phase formation during film growth by physical vapor deposition Journal of Applied Physics 127, 180901 (2020); https://doi.org/10.1063/1.5141342Metal-functionalized 2D boron sulfide monolayer material enhancing hydrogen storage capacities Cite as: J. Appl. Phys. 127, 184305 (2020); doi: 10.1063/5.0008980 View Online Export Citation CrossMar k Submitted: 30 March 2020 · Accepted: 25 April 2020 · Published Online: 12 May 2020 Pushkar Mishra,1 Deobrat Singh,2,a) Yogesh Sonvane,1,a) and Rajeev Ahuja2,3,a) AFFILIATIONS 1Applied Material Lab, Department of Applied Physics, S. V. National Institute of Technology, Surat 395007, India 2Condensed Matter Theory Group, Department of Physics and Astronomy, Uppsala University, Box 516, 75120 Uppsala, Sweden 3Applied Materials Physics, Department of Materials and Engineering, Royal Institute of Technology (KTH), S-100 44 Stockholm, Sweden a)Authors to whom correspondence should be addressed: yas@phy.svnit.ac.in ;deobrat.singh@physics.uu.se ; and rajeev.ahuja@physics.uu.se ABSTRACT In the present work, we have systematically investigated the structural, electronic, vibrational, and H 2storage properties of a layered 2H boron sulfide (2H-BS) monolayer using spin-polarized density functional theory (DFT). The pristine BS monolayer shows semiconducting behavior with an indirect bandgap of 2.83 eV. Spin-polarized DFT with van der Waals correction suggests that the pristine BS monolayer has weak binding strength with H 2molecules, but the binding energy can be significantly improved by alkali metal functionalization. A system energy study indicates the strong bonding of alkali metals with the BS monolayer. The Bader charge analysis also concludes that aconsiderable charge is transferred from the metal to the BS monolayer surface, which was further confirmed by the iso-surface charge density profile. All functionalized alkali metals form cations that can bond multiple H 2molecules with sufficient binding energies, which are excellent for H 2storage applications. An ideal range of adsorption energy and practicable desorption temperature promises the ability of the alkali metal functionalized BS monolayer as an efficient material for hydrogen storage. © 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/10.1063/5.0008980 I. INTRODUCTION There are two major concerns across the world: one is the annihilation of non-renewable energy sources, and the other is climate change because of environmental pollution. Out of many available solutions, hydrogen (H 2) is outstanding due to having the highest energy density, high availability, and eco-friendliness, hence having the ability to become effective energy carriers.1–4However, some challenges such as storage, production, and transport limit the use of hydrogen as a clean renewable energy source. Out of all of these challenges, the toughest challenge is the too large scale storage of hydrogen within ambient conditions. The proposed target for H 2storage by the US Department of Energy (DOE) is specified with a system gravimetric capacity range of 4.5% –6.5%, system volumetric capacity of 30 g/l, operating ambient temperature of 233 K –333 K, min/max delivery temperature of 233 K –358 K, and max delivery from storage system of 12 bar for onboard vehicleapplications.5,6There are two main requirements for hydrogen storage: one is that the binding energy of hydrogen between chemi-sorption and physisorption should be within the 0.1 –0.2 eV/H 2 range,7and another requirement is the selection of a storage mate- rial of lighter elements. However, it is difficult to fulfill these two criteria at the same time, since hydrogen is bonded either too strongly with light elements, forming metal hydrides, or too weaklywith BN fullerenes and other heavier elements. 8 The emergence of 2D materials led to a new phase in materi- als science due to their exclusive physical and chemical proper- ties.9,10A number of recent studies have shown that 2D materials can store H 2in a huge amount.1,3,11–17Boron based 2D materials such as boron nitride (BN) and borophene were supposed to bepotential hydrogen storage materials because of their high surfaceto volume ratio, porous design, and particularly light weight. But pristine boron nitride and borophene have much less binding energy between H 2and the sheet; for boron nitride, it is 0.05 eV/H 2forJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 184305 (2020); doi: 10.1063/5.0008980 127, 184305-1 ©A u t h o r ( s )2 0 2 06H2molecules, and for borophene, it is 0.045 eV/H 2for adsorption of one H 2.18,19This difficulty has been solved by intermediate adsorption of light transition metals (TMs), alkali metals (AMs),alkaline metals (ALMs), and non-metals (NMs) between the sheetand hydrogen molecules that enhance the ultimate binding energyand efficiency of hydrogen storage. 18–25Here, we have chosen alkali metals (AMs) such as Li, Na, and K for intermediate adsorption, as there are a lot of significant factors for them and one of the mostimportant factors is that their binding energy ratio to correspond-ing cohesive energies is much higher than that of heavier elementssuch as Sc and Ti transition metals. This would minimize the chances of cluster formation. Another important factor is that they are light weighted (less atomic masses), which makes the storagematerial lighter and the weight % of hydrogen increases. The mech-anism of the interaction of hydrogen molecules with these metalions (Li +,N a+, and K+) justifies the advantage of taking Li, Na, and K for high hydrogen storage capacity. According to Niu et al.26 when metal cations like Li+,N a+, and K+interact with hydrogen molecules, charge polarization induces and the hydrogen moleculesbecome stretched and ultimately hydrogen coverage increases,which makes desorption also easier. According to Kubas ’s mecha- nism, 27transition metals such as Sc, Ti, Pd, and Pt interact with an H2molecule, which gives an electron to the vacant d orbital of a transition metal and the transition metal also transfers back anelectron to the H 2molecule. Hence, the bonding can be defined based on the hybridization of the d orbital of the metal with the s orbital of hydrogen. Also, the main difficulty of transition metalfunctionalization is that they tend to cluster on the material ’s monolayer. Apart from all these, there is another reason to avoidthe use of transition metals: they have high atomic mass, which reduces gravimetric density (weight %). Recently, a new layered monochalcogenide, boron sulfide (BS) has been investigated by density functional theory (DFT), whichcomprises two hexagonal layers of BS that is bonded with boron – boron atoms. The calculated phonon dispersion curve that has no imaginary frequency confirms the dynamic stability of the layered material. Its mechanical properties such as higher tensile strengthand stretchability make it strong and support novel applications.The ultimate tensile strength across the zigzag path was obtained as0.26, 28which is comparable to that of graphene (0.27)29and hexag- onal boron nitride (0.30).30 Encouraged by these fascinating features of the BS monolayer, we have investigated structural, electronic, and vibrational proper-ties and also explored the tendency to H 2adsorption with a pristine sheet. But the adsorption energy of hydrogen with a pristine sheet is very less; therefore, we functionalized the sheet with alkali metalssuch as Li, Na, and K and investigated the electronic properties,charge transfer mechanism of hydrogenation, and dehydrogenationprocess of the BS monolayer. II. COMPUTATIONAL METHODOLOGY We employed spin-polarized density functional theory (DFT) to investigate the structural and electronic properties and H 2 storage characteristics of the pristine and alkali metal furnished BS monolayer by using the Vienna ab initio Simulation Package (VASP)31,32software. Exchange correlation interactions wereapproached by generalized gradient approximation (GGA) with the PBE method defined by Perdew et al.33,34We utilized the van der Waals correction of the DFT-D2 method introduced by Grimme35 to escape the undervaluation of binding energy produced by the GGA method. The kinetic energy cutoff was taken as 500 eV in this calculation. The convergence criteria for force and energy werechosen as 0.001 eV/Å and 10 −6eV, respectively. For a sampling of the Brillouin zone, k-points were taken as 5 × 5 × 1 for both therelaxation and density of states (DOS) following the Monkhorst – Pack scheme. 36A vacuum of 15 Å was introduced to prevent the possible effects of the layers in a vertical direction. The bindingenergies of alkali metals (AMs), (E AM b), with the BS sheet were cal- culated by using the following equation: EAM b¼EAMþBS/C0EBS/C0EAM, (1) where E AMþBS,EAM, and E BSare the total energies of the BS sheet functionalized with alkali metals (AMs) Li, Na, and K, the totalenergy of AM elements, and the total energy of pristine BS sheet, respectively. The adsorption energy per H 2molecule is determined by the following equation: Ead¼EBSþAMþnH2/C0EBSþAM/C0nEH2 n, (2) where E BSþAMþnH2is the total energy of the hydrogen adsorbed metal functionalized BS sheet, E BSþAMis the total energy of the metal functionalized BS sheet, E H2is the energy of hydrogen mole- cules, and n indicates the number of hydrogen molecules. We havealso performed the Bader charge analysis for a better understandingTABLE I. Calculated lattice parameter, bond angle, and bond length of the BS monolayer. Lattice parameter (Å) Bond angle (deg) Bond length (Å) a = 3.041 /BSB = 102.426 B –B = 1.95 b = 3.041 /SBS = 102.426 B –S = 1.78 /SBS = 115.835 S –S = 3.43 FIG. 1. (a) T op and (b) side views of the pristine 2H-BS monolayer.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 184305 (2020); doi: 10.1063/5.0008980 127, 184305-2 ©A u t h o r ( s )2 0 2 0of the charge transfer mechanism between the BS sheet and alkali metals like Li, Na, and K. III. RESULTS AND DISCUSSION A. Structural, electronic, and vibrational properties of the pristine BS monolayer First, we examined the structural properties of the BS mono- layer to check the efficiency of the computational methodologyused in this work. The structural properties such as lattice parame-ter, bond angle, and bond length are described in Table I .W e found that our calculated results are in excellent agreement with an earlier study. 28,37The top and side views of BS are shown in Fig. 1 . To examine the electronic properties of the BS monolayer, we plotted the band structure and spin-polarized partial density ofstates (PDOS). Figure 2(a) presents the band structure and Fig. 2(b) shows the PDOS plot of the BS monolayer. We measured the bandgap using the PBE functional which is 2.83 eV, almost the same as previously reported. 28From the band structure, we observe that the valance band maxima (VBM) exist near the Γpoint and the conduction band minima occur near the M point. The band structure indicates that the BS monolayer is an indirect, wide bandgap semiconductor. From Fig. 2(b) , the PDOS plot also conveys the wide bandgap in the valance band in which the majorcontribution near the Fermi level originates from the S (3p) orbitals. To investigate the dynamic stability of the BS monolayer, we have calculated the phonon dispersion curve, which is shown in Fig. 3 . As it is noticeable from Fig. 3(a) , there is no imaginary fre- quency in phonon dispersion, indicating the dynamic stability ofthe BS monolayer. For further understanding of the phonon spec- trum, we have also calculated the vibrational density of states (VDOS), as shown in Fig. 3(b) .F r o m Fig. 3(b) , we illustrated that, in the acoustic branch, the S atom dominates over the boron atomand a highest peak at 1.53 states/THz was obtained around the11.22 THz frequency. However, in the optical branch due to the lighter mass, B dominantly contributes to the VDOS spectra. Two gaps are obtained in frequency, the first is around 12.9 THz – 17.7 Thz and the second is around 23.10 THz –27.60 THz. B. Hydrogenation of the pristine BS monolayer At first, we studied the sensory interaction of the H 2molecule with a pristine BS monolayer. In this analysis, we put the H 2mole- cule at nearly 1.5 Å above the monolayer. We have calculated the adsorption energy of the H 2molecules with the monolayer. The cal- culated value of adsorption is given in Table II , as illustrated in Fig. 4 , which shows that the interaction of the H 2molecule with the FIG. 2. (a) Band structure and (b) spin-polarized partial density of statesof boron (B-2p) and sulfur (S-3p) atoms in the pristine BS monolayer. FIG. 3. (a) Phonon dispersion curve and (b) vibrational density of states (VDOS) of the BS monolayer.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 184305 (2020); doi: 10.1063/5.0008980 127, 184305-3 ©A u t h o r ( s )2 0 2 0pristine BS monolayer is very poor. Therefore, we cannot use a pris- tine BS monolayer for further H 2storage applications in ambient conditions. For adsorption energy calculations, we used the formula Eads¼EBSþnH2/C0EBS/C0nEH2 n, (3) where E BSþnH2,EBS, and E H2denote the total energy of a complex system (hydrogenation of the BS monolayer), the energy of the BSmonolayer, and the energy of isolated H 2molecules. Here, n is the number of hydrogen molecules present on the surface of the BSmonolayer. C. Functionalization of BS with alkali metals (Li, Na, and K) To improve the adsorption energy of the H 2molecules, we first functionalized the monolayer with alkali metals (Li, Na, and K), andthen functionalized the BS sheet that would be utilized for hydrogenstorage. For achieving the lowest energy configuration, we placed asingle atom of alkali metals on all possible adsorption positions over the BS sheet including the top of the S atom, B atom, bridge of the B–S, and hallow site (four positions). Based on energy optimizationamong all four positions, the bridge of the B –Sp o s i t i o ni st h em o s t suitable position. Now, these optimized structures and the BS sheet functionalized with one alkali metal were used for hydrogen storage. All the optimized structures are shown in Fig. 5 . The calculated binding energies (E AM b) for single metal dopant adatoms Li, Na, and K above the bridge of the B –S position are −4.48 eV, −4.05 eV, −5.52 eV, respectively, which is represented in Table III . These binding energies (EAM b) are sufficiently high com- pared to their corresponding cohesive energies (E c)−1.63 eV (Li), −1.11eV (Na), and −0.93eV (K).38This makes it clear that the alkali metals Li, Na, and K are uniformly distributed over the BSsheet without being clustered. To investigate the electronic properties of the alkali metal functionalized BS monolayer, we sketched the spin-polarizedpartial density of states (PDOS). Figures 6(a) –6(c) represent the PDOS plot of Li@BS, Na@BS, and K@BS, respectively. As in pureBS, there is a wide bandgap of 2.83 eV; when we placed alkali metal atoms such as Li, Na, and K, this gap will be shifted from the con- duction band to the valance band. As shown in Fig. 6(a) in the case of Li, due to the low ionization energy, Li ’s 2s valance electron was donated to the BS sheet to fill the vacant conduction band, because the charge transfer ( ∼0.99 |e| from the Bader charge analysis) on the bottom of the conduction band moves toward the FermiTABLE II. Calculated adsorption energy and charge transfer of H 2molecule with the pristine BS sheet. SystemAdsorption energy per H2(eV)Charge transfer from sheet to per H 2molecule 1H2@BS −0.0393 0.00633 2H2@BS −0.0419 0.00562 3H2@BS −0.0448 0.00483 4H2@BS −0.0445 0.00454TABLE III. The binding energy (E b) of alkali metals to the BS sheet, Q acharge on the alkali metal atom before adsorption, and Q bcharge on the alkali metal atom after adsorption. ΔQ is the charge transfer to the BS sheet from the alkali metal atom. System EAM b(eV) Q a(a.u.) Q b(a.u.) ( ΔQ) (a.u.) Li@BS −4.48 1.000 000 0.007 631 0.9924 Na@BS −4.05 1.000 000 0.230 990 0.7690 K@BS −5.52 9.000 000 8.219 558 0.7804 FIG. 4. T op and side views of hydrogenation of a pure BS monolayer for (a) 1H 2, (b) 2H 2, (c) 3H 2, and (d) 4H 2.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 184305 (2020); doi: 10.1063/5.0008980 127, 184305-4 ©A u t h o r ( s )2 0 2 0FIG. 5. T op and side views of alkali metals (a) Li, (b) Na, and (c) K func- tionalized BS monolayer. FIG. 6. Projected density of states of the alkali metal functionalized BS sheet: (a) Li@BS, (b) Na@BS and (c) K@BS. FIG. 7. (a)–(c) T op views and (d) –(f) side views of isosurface charge density plots with an isovalue of 0.001 eV/Å3of the alkali metals (Li, Na, and K) functionalized BS sheets, respectively. Yellow and cyan show the accumulation and depletion of charges, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 184305 (2020); doi: 10.1063/5.0008980 127, 184305-5 ©A u t h o r ( s )2 0 2 0level (0.0 eV). The PDOS of Na and K with the BS sheet shows the same characteristic to that of Li and are shown in Figs. 6(b) and6 (c). These results revealed that the bonding between the alkali metal atom and the BS sheet is mainly ionic. This is also confirmedby the Bader charge analysis. 1. Bader charge analysis The charge transfer to the BS sheet from the alkali metals is determined by using the following equation: ΔQ¼Qa/C0Qb, (4) where ΔQ is a difference in the Bader charge and Q aand Q b ascribe the atomic Bader charges before and after adsorption of alkali metals, respectively. They showed that a considerable amountof charge shifted from the alkali metals to the BS sheet. We have plotted iso-surface charge density plots as shown in Fig. 7 . We observed that the charge donated by the alkali metals mostly accumulated on the nearest sulfur atoms of the sheet, and asmall part is gained by the boron atom. We also noticed that the Liatom transferred the maximum charge and the Na and K atoms transferred nearly equal charges that are also recommended by the Bader charge analysis.D. Hydrogenation and dehydrogenation of the functionalized BS monolayer In this segment, we will explain the adsorption and de-adsorption processes of H 2, with the alkali metals (Li, Na, and K) functionalizing the BS s heet. As can be observed from the Bader charge analysis and study of the partial density ofstates (PDOS), a large amount of charge transferred from thealkali metals to the BS sheet; as a result, a strong ionic bondformed between them, and the alkali metals were left in the cat- ionic states. Consequently, partly positive alkali metals (Li +, Na+,a n dK+) serve as a binding site for inserted H 2molecules. At first, we inserted a single H 2molecule in each alkali metal (Li, Na, and K) and enabled the system to achieve a ground stateconfiguration via full structure optimization. By attempting mul- tiple initial structures, it is observed that the vertical access of H 2molecules to the alkali metals is favorable. The inserted H 2 molecules become polarized when engaging cationic alkali metal atoms and bound with them via electrostatic and van der Wallsinteractions. To improve the storage capacity of H 2,w ei n s e r t e dm o r e H2molecules in the alkali metal functionalized BS sheet by a systematic approach as shown in Figs. 8 –10.I nt h es e c o n d step, we inserted two H 2molecules and enabled the systems again to go through full structure optimization. Similarly, we inserted three, four, five, and six H 2molecules in the alkali FIG. 8. (a)–(h) represent the hydroge- nation for 1H 2,2 H 2,3 H 2,4 H 2,5 H 2, 6H2, and 12H 2and dehydrogenation process of the K functionalized BS monolayer, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 184305 (2020); doi: 10.1063/5.0008980 127, 184305-6 ©A u t h o r ( s )2 0 2 0FIG. 9. (a)–(h) represent the hydroge- nation for 1H 2,2 H 2,3 H 2,4 H 2,5 H 2, 6H2, and 12H 2and dehydrogenation process of the Li functionalized BSmonolayer, respectively. FIG. 10. (a)–(h) represent the hydro- genation for 1H 2,2 H 2,3 H 2,4 H 2,5 H 2, 6H2, and 12H 2and dehydrogenation process of the Na functionalized BSmonolayer, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 184305 (2020); doi: 10.1063/5.0008980 127, 184305-7 ©A u t h o r ( s )2 0 2 0metal functionalized BS sheet and fully relaxed the structure. We have observed that a consistent change occurs in binding energy. Now, hydrogenation is done with 12H 2molecules, and when it had saturated, the value of the adsorption energyreached 0.106 eV for Li, 0.100 eV for Na, and 0.097 eV for Katoms [ Fig. 11(a) ]. After this, if we add more than 12H 2mole- cules, the adsorption energy becomes much smaller from theideal range of 0.2 eV/H 2–0.1 eV/H 27. The storage capacity of hydro- gen storage can be increased by both sides of the functionalization of alkali metal adsorption and further hydrogenation. Figures 8 –10 display the hydrogenation/dehydrogenation of all structuresstudied. We have also calculated the desorption temperature (T D) for all hydrogenated systems by using the von ’t Hoff equation as FIG. 11. (a) The variation of adsorption energy and (b) the corresponding desorption temperature as a function of H 2molecules for the metal functionalized by a single atom on the BS monolayer surface. FIG. 12. Spin-polarized density of states for 1H 2,2 H 2,3 H 2,4 H 2,5 H 2,6 H 2, and 12H 2molecules adsorbed on the Li functionalized BS monolayer. FIG. 13. Spin-polarized density of states for 1H 2,2 H 2,3 H 2,4 H 2,5 H 2,6 H 2, and 12H 2molecules adsorbed on the Na functionalized BS monolayer.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 184305 (2020); doi: 10.1063/5.0008980 127, 184305-8 ©A u t h o r ( s )2 0 2 0follows:39 TD¼Eads kbΔS R/C0ln p/C18/C19/C01 , (5) where K adsis the calculated adsorption energy (in Joules per H 2 molecule), k bis the Boltzmann constant, and the change in H 2 entropy from the gas to liquid phase is ΔS (75.44 J mol−1K−1). R is the universal gas constant (8.314 J mol−1K−1), and p is the equilibrium pressure, which is 1 atm. The values of T Dfor all systems are given in Table S1 in the supplementary material and also in Fig. 11(b) . Furthermore, we performed the Bader charge analysis for all considered systems by using the formula ΔQ=Q a−Qbwhere Q a and Q bare the Bader charges of hydrogen molecules before and after adsorption, respectively. We observed a considerable amountof charge transfer from the alkali metal functionalized BS to hydro- gen molecules as given in Table S1 in the supplementary material . Besides, we computed spin-dependent partial density of states to examine how the orbitals of the BS sheet atoms are attached tothe 1s orbital of the hydrogen molecules and alkali metals. Wesketched the PDOS plot for all alkali metal (Li, Na, and K) func- tionalized BS sheets as shown in Figs. 12 –14.F r o m Fig. 12 , in theconduction band near the Fermi level, when the number of hydro- gen molecules is less, the 1s sates of hydrogen strongly hybridized with the 2s state of the Li atom, as the number of hydrogen mole-cules increases the number of states/eV of hydrogen increases andhybridization moves toward higher energy and becomes weaker,which is verified by the decreasing order of adsorption energy, as shown in Table S1 in the supplementary material . In the valance band, a gap is observed from 0 eV to nearly 3.0 eV. A peak of the3s state of the S atom appears just before this bandgap. In the caseof Na, from Fig. 13 , we can see a similar trend, and in the case of 3H 2and 5H 2adsorption, a peak of the hybridized state of the 2p state of B and the 3p state of S is observed. In the case of the potassium (K) functionalized BS sheet, shown in Fig. 14 , for 12H 2in the conduction band, the asymmetric polarization of up and down states of the s-state of H 2is obtained and in all other cases, the trend is same as that of the Li and Na functionalized BS sheet. The outcome from these graphs reveals that, when the number of hydrogen molecules is less, the 1s state ofthe hydrogen molecules is strongly hybridized with alkali metals,2s, 3s, and 4s states, near the Fermi level. As the number of hydro-gen molecules increases, the number of states per eV also increases and now the hybridized states move away from the Fermi level. In the valance band near the Fermi level, a gap is observed fromnearly −2.5 eV to 0 eV in all PDOS graphs. V. CONCLUSIONS We have systematically investigated the structural, electronic, vibrational, and H 2storage properties of a BS material using spin- polarized density functional theory. Our DFT study with van derWaals correction suggests that the pristine BS monolayer has a weak binding with H 2molecules but the binding energy can be sig- nificantly improved by alkali metal functionalization. A systemenergy study indicates the strong bonding of the alkali metals andthe BS monolayer. The Bader charge analysis also concludes that aconsiderable charge is transferred from the metals to the BS sheet, which was further confirmed by charge density difference plots. All alkali metals form cations that can bond multiple hydrogen mole-cules with binging energies, which are excellent for H 2storage applications. The ideal range of adsorption energy and practicable desorption temperature promises the ability of the alkali metals to functionalize the BS monolayer as an efficient material for hydro-gen storage. SUPPLEMENTARY MATERIAL See the supplementary material for adsorption energies, desorption temperature, and the corresponding Bader charge analy-sis for nH 2molecule absorption on the functionalized BS monolayer. ACKNOWLEDGMENTS P.M. acknowledges SVNIT, Surat, for his institute research fel- lowship (No. FIR-DS17PH004). D.S. and R.A. acknowledge OlleEngkvists stiftelse, Carl Tryggers Stiftelse for Vetenskaplig Forskning (CTS), and Swedish Research Council (VR) for financial support. SNIC and HPC2N are acknowledged for providing FIG. 14. Spin-polarized density of states for 1H 2,2 H 2,3 H 2,4 H 2,5 H 2,6 H 2, and 12H 2molecules adsorbed on the K functionalized BS monolayer.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 184305 (2020); doi: 10.1063/5.0008980 127, 184305-9 ©A u t h o r ( s )2 0 2 0computing facilities. Y.S. acknowledges the Science and Engineering Research Board (SERB), India, for financial support (Grant No. EEQ/2016/000217). Computational facilities from theCenter for Development of Advanced Computing (C-DAC) Puneare also gratefully acknowledged. DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material . REFERENCES 1D. J. Durbin and C. Malardier-Jugroot, Int. J. Hydrogen Energy 38, 14595 (2013). 2C. B. Dutta and D. K. Das, Renew. Sustain. Energy Rev. 66, 825 (2016). 3N. A. A. Rusman and M. Dahari, Int. J. Hydrogen Energy 41, 12108 (2016). 4S. Dutta, J. Ind. Eng. Chem. 20, 1148 (2014). 5P. Chen, Science 285, 91 (1999). 6P. Mishra, D. Singh, Y. Sonvane, and R. Ahuja, Int. J. Hydrogen Energy 45, 12384 (2020). 7J. Zhou, Q. Wang, Q. Sun, P. Jena, and X. S. Chen, Proc. Natl. Acad. Sci. U.S.A. 107, 2801 (2010). 8F. E. Harris and H. S. Taylor, Physica 30, 105 (1964). 9A. J. Mannix, Z. Zhang, N. P. Guisinger, B. I. Yakobson, and M. C. Hersam, Nat. Nanotechnol. 13, 444 (2018). 10C. Zhai, J. Hu, M. Sun, and M. Zhu, Appl. Surf. Sci. 430, 578 (2018). 11P. Srepusharawoot, E. Swatsitang, V. Amornkitbamrung, U. Pinsook, and R. Ahuja, Int. J. Hydrogen Energy 38, 14276 (2013). 12T. Hussain, A. De Sarkar, and R. Ahuja, Int. J. Hydrogen Energy 39, 2560 (2014). 13P. Banerjee, B. Pathak, R. Ahuja, and G. P. Das, Int. J. Hydrogen Energy 41, 14437 (2016). 14P. Panigrahi, A. Kumar, A. Karton, R. Ahuja, and T. Hussain, Int. J. Hydrogen Energy 45, 3035 (2020). 15Y. Ye, C. C. Ahn, C. Witham, B. Fultz, J. Liu, A. G. Rinzler, D. Colbert, K. A. Smith, and R. E. Smalley, Appl. Phys. Lett. 74, 2307 (1999).16S. Satyapal, J. Petrovic, C. Read, G. Thomas, and G. Ordaz, Catal. Today 120, 246 (2007). 17D. Singh, S. K. Gupta, Y. Sonvane, and R. Ahuja, Int. J. Hydrogen Energy 42, 22942 (2017). 18W. 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5.0007128.pdf

J. Chem. Phys. 152, 214706 (2020); https://doi.org/10.1063/5.0007128 152, 214706 © 2020 Author(s).Grafting of iron on amorphous silica surfaces from ab initio calculations Cite as: J. Chem. Phys. 152, 214706 (2020); https://doi.org/10.1063/5.0007128 Submitted: 10 March 2020 . Accepted: 14 May 2020 . Published Online: 05 June 2020 Saber Gueddida , Michael Badawi , and Sébastien Lebègue ARTICLES YOU MAY BE INTERESTED IN CP2K: An electronic structure and molecular dynamics software package - Quickstep: Efficient and accurate electronic structure calculations The Journal of Chemical Physics 152, 194103 (2020); https://doi.org/10.1063/5.0007045 An effective Hamiltonian analysis of a Franck–Condon-like pattern in the IR spectra of phenol-alkylsilane dihydrogen-bonded clusters in the S 1 state The Journal of Chemical Physics 152, 194306 (2020); https://doi.org/10.1063/5.0005259 Dynamics of poly[n]catenane melts The Journal of Chemical Physics 152, 214901 (2020); https://doi.org/10.1063/5.0007573The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Grafting of iron on amorphous silica surfaces from ab initio calculations Cite as: J. Chem. Phys. 152, 214706 (2020); doi: 10.1063/5.0007128 Submitted: 10 March 2020 •Accepted: 14 May 2020 • Published Online: 5 June 2020 Saber Gueddida,a) Michael Badawi, and Sébastien Lebègue AFFILIATIONS Univ. Lorraine, LPCT, CNRS UMR7019, F-54506 Vandoeuvre-Les-Nancy, France a)Author to whom correspondence should be addressed: saber.gueddida@univ-lorraine.fr ABSTRACT Iron over silica catalytic systems have attracted considerable attention due to their activity and selectivity in different reactions, for instance, in the hydrodeoxygenation process. Here, the grafting mechanisms of iron under various forms (one atom, two atoms, or a cluster) on silica surfaces are studied using ab initio calculations. Various geometries with different locations of iron on the silica structure have been investigated, and it is found that a strong interaction between iron and the silanol groups takes place, mostly driven by the formation of Fe–O–Si bonds, and in few cases by nearby surface OH groups, creating Fe–OH–Si bonds. For the cluster, we show that the most favorable adsorption mode induces a strong effect on the silica surface accompanied with a large charge transfer, making it very stable and promising for a large panel of applications. Published under license by AIP Publishing. https://doi.org/10.1063/5.0007128 .,s I. INTRODUCTION Supported transition metal oxides are considered among the most important catalytic systems.1Most of these materials are often amorphous not only because of their low cost and high porosity but also because their performance is demonstrably superior to that of their crystalline counterparts.2Among the available supports, con- trolled mesopore structures such as MCM-41 and SBA-15 are very popular,2–10whereas crystalline silica polymorphs are almost never used, in part, because of their much lower specific area. A wide range of catalytically active transition metals such as Cr, Mo, W, Zn, and Nb have been grafted on various amorphous silica surfaces in order to improve their activity and selectivity,11–15which are important for various catalytic reactions, including the selective oxidations16and dehydrogenation17,18of alkanes and alkenes,17,19–23 as well as other selective oxidation reactions.24–27The grafting of iron on mesoporous silica is of peculiar importance as it covers a wide range of applications, including advanced oxidation processes,28 Fenton-like nanoreactors for enhanced cancer treatment,29,30new magnetic resonance imaging devices,31and energy production.32–35 In particular, it has been shown experimentally that Fe/silica cata- lysts are potentially active and selective for the hydrodeoxygenation (HDO) process.33–35The design of catalysts includes the choice of the support and the choice of active agents, such as iron nanoparticles, which can accelerate the chemical transformation and selectively obtain the desired products. One of the physical properties to be controlled is the size of the aggregates. A systematic study on the magnetic properties of isolated Fe nparticles as a function of particle sizes and structures was carried out in the past with standard density func- tional theory (DFT).36–41In addition, Fe naggregates with n ≤17 atoms have been studied computationally to reveal the origin of the observed magic numbers. Kim et al. demonstrated particularly high stability of clusters with 7, 13, and 15 atoms.42 However, little is known on the interaction between iron species and amorphous silica surfaces, which is necessary for opti- mizing the formulation of catalytic nanomaterials. This has moti- vated us to investigate the grafting mode and the electronic struc- ture of different iron species on various amorphous silica surfaces in order to gain a better microscopic understanding of the process involved. Our paper is organized as follows: in the first part, we briefly describe our calculation methods, and in the second part, we present our results for the supported Fe, 2Fe, and Fe clusters on amorphous hydroxylated silica surfaces. We first describe the structural proper- ties of different grafting geometries and then discuss their electronic J. Chem. Phys. 152, 214706 (2020); doi: 10.1063/5.0007128 152, 214706-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp and magnetic properties. Then, we present the Bader analysis of the charge density to determine the charge transfer between the Fe cluster and the silica surface. Finally, we offer our conclusions and perspectives. II. COMPUTATIONAL DETAILS Periodic spin polarized density functional theory (DFT) cal- culations were performed using the Vienna ab initio simulation package (VASP) code43by means of the projector augmented wave (PAW) method.44The exchange and correlation potential was cal- culated within the Perdew, Burke, and Ernzerhof45formulation of the generalized-gradient approximation (GGA-PBE). To correct the electron–electron interaction due to the iron localized dshell, we used the rotationally invariant PBE+U correction,46as implemented in the VASP code.47This approach has been successfully used to investigate a large variety of systems including oxides48,49and metal organic frameworks (MOFs).50The values of the parameters UandJ were set to 3 eV and 0.9 eV, respectively.51,52Van der Waals interac- tions were included by means of the DFT-D2 correction by Grimme et al. ,53as implemented in VASP.54 In this work, we investigated possible atomic models of iron species grafted on an amorphous hydroxylated silica surfaces. The amorphous silica-surfaces (SiO 2) considered in the present work were constructed by Comas-Vives55for different silanol densities. Here, we have considered only the surfaces with a silanol density of 3.3 and 4.6 OH/nm2(denoted hereafter SiO 2-4.6 and SiO 2-3.3).32 These silanol densities are typical of the experimental ones follow- ing different synthesis routes of mesoporous silicas.56–59Moreover, it has been shown recently that these surfaces are promising for the degradation of phenol under catalytic conditions.32Both SiO 2-4.6 and SiO 2-3.3 surfaces display all possible arrangements of silanol groups (nest-1, nest-2, geminal, vicinal, and isolated).32,57A single SiOH group is called an isolated silanol when the distance to the closer SiOH groups is more than ∼3.3 Å, which cannot be involved in mutual hydrogen bond, while the pair of silanols belonging to tetrahedra that share a common oxygen vertex is called vicinal. The geminal site is defined as two OH groups linked to the same surface silicon atom to give Si(OH) 2moiety. The nest-1 site is composed of a vicinal and an isolated silanol, while the nest-2 site is composed of a geminal silanol associated with one OH group (for more details, see Fig. S1 of the supplementary material).The SiO 2-4.6 surface was represented by a supercell containing 384 atoms, and for SiO 2-3.3, 375 atoms. These surfaces are typical for silica surfaces treated under vacuum conditions at 180–200○C for SiO 2-4.6 and at 200○C for SiO 3-3.3.55,60,61To model an iron cluster, we have chosen the Fe 13structure, which is the most sta- ble atomic arrangement among small Fe clusters, that adopts the D3dsymmetry.42The particular structural stability of the 13-atom cluster is observed also for other nanoparticles such as Ga 1362and Pt13.63In all cases, the periodically repeated slabs were separated by more than 20 Å of vacuum in the zdirection. The energy cutoff for the plane-wave expansion was set at 450 eV to ensure the con- vergence of our calculations. Due to the large size of the supercell, theΓpoint was used in the Brillouin-zone integration, and the total energy differences were converged within 10−6eV. For the atomic relaxation, the positions of the single iron atoms or all the atoms of the Fe 13cluster and the atoms of the first layer of the silica surface were relaxed by nullifying the forces on the atoms with a preci- sion of 0.03 eV/Å, while the other layers were kept fixed. A Bader analysis64,65of the charge density was performed to determine the charge transfer between the Fe 13cluster and the silica-surface. Our results were obtained with a 180 ×180×280 density grid with a charge transfer error estimated to be around ±0.01 e−. The analysis was performed using the reconstructed valence density in the PAW formulation.66 III. RESULTS AND DISCUSSION A. Grafting of a single iron atom Before finding the most stable structures of the silica supported iron cluster, we have investigated within the PBE+U+D2 approxi- mation the grafting of a single iron atom on the surface. The single iron atom is adsorbed on the silica-surface through its oxygen atoms by removing the corresponding hydrogen atoms to allow the forma- tion of Fe–O–Si bonds. Different geometrical configurations of the single iron atom on the surface were systematically investigated and can be divided into two groups with different adsorption positions: top (mono-grafting, hereafter named Fe-A and Fe-B) and bridge (di- grafting, hereafter named Fe-C and Fe-D), as shown in Fig. 1, for a silanol coverage equal to 4.6 OH/nm2. The Fe-A and Fe-C geome- tries are given in Fig. S2 of the supplementary material. In addition, since an iron ion can exist in low-spin (LS) or high-spin (HS) states, FIG. 1 . The left and the right panels show a single iron atom adsorbed on the amor- phous silica-surface SiO 2with a silanol density equal to 4.6 OH/nm2at top (Fe-B case) and bridge (Fe-D case) positions, respectively. J. Chem. Phys. 152, 214706 (2020); doi: 10.1063/5.0007128 152, 214706-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . PBE+U+D2 calculated HS and LS equilibrium distance d Fe–O(in Å) between the single iron atom and the substrate with a silanol density of 4.6 OH/nm2or 3.3 OH/nm2. ContactFe/SiO 2-4.6 Fe/SiO 2-3.3 oxygens Structure HS LS HS LS O1 Fe-A 1.79 1.79 1.82 1.78 Fe-B 1.81 1.77 1.85 1.80 O2 Fe-C 1.86–1.88 1.80–1.81 1.82–1.82 1.79–1.79 Fe-D 1.85–1.99 1.81–1.85 1.80–1.83 1.77–1.79 two sets of calculations were performed corresponding to each spin configuration. Our calculated Fe–O bond lengths are reported in Table I both for top and bridge grafting of a single iron atom (HS and LS states) on silica surfaces. The most stable top-grafted structure (Fe-B) on SiO 2-4.6 has a Fe–O bond length of 1.81 Å in the HS state and 1.77 Å in the LS state. These distances are slightly increased to 1.85 Å and 1.80 Å in the HS and the LS state, respectively, when the sin- gle iron atom is grafted onto SiO 2-3.3. However, for the bridging oxygen sites, the bond lengths of the most stable structure (Fe-D) of Fe/SiO 2-4.6 are in the range of 1.85–1.99 Å and 1.81–1.85 Å in the HS and the LS states, respectively, while for Fe/SiO 2-3.3, they are reduced to 1.80–1.83 Å for the HS state and 1.77–1.79 Å for the LS state. Therefore, the equilibrium distance d Fe–Ois slightly influenced by different parameters such as the grafting site and its location on the surface, the magnetic state of the iron atom, and the model used. In addition, the difference in the equilibrium dis- tance dFe–Ofor the HS and the LS states for both top and bridge adsorption positions are found to be 0.04 Å, while it is usually of about 0.2 Å in an octahedral environment.51This is due to the fact that the single iron atom is bonded only to one or two neighboring oxygen atoms, while in an octahedral environment, it is surrounded by six oxygen atoms. Our calculated geometrical parameters can be compared with the information delivered by structural observa- tions obtained from EXAFS, x-ray scattering, or nuclear magnetic resonance (NMR) spectroscopy. Our calculated grafting energies of a supported single iron atom on the amorphous silica-surfaces SiO 2-4.6 and SiO 2-3.3 for both HS and LS states are listed in Table II. The single iron atom is grafted by TABLE II . PBE+U+D2 calculated grafting energies (in eV) for an adsorbed single iron atom in its HS or LS state on amorphous silica surfaces SiO 2-4.6 and SiO 2-3.3 at top (structures: Fe-A and Fe-B) and bridge (structures: Fe-C and Fe-D) positions. ContactFe/SiO 2-4.6 Fe/SiO 2-3.3 oxygens Structure ΔEHS ΔELS Ion ΔEHS ΔELS Ion O1 Fe-A −1.67−1.04 Fe3+−1.66−1.01 Fe3+ Fe-B −1.68−1.02 Fe3+−1.93−1.71 Fe3+ O2 Fe-C −2.78−1.98 Fe2+−2.50−1.94 Fe2+ Fe-D −3.29−2.58 Fe2+−3.19−2.44 Fe2+dehydrogenation of surface silanols, following the reaction, Fe + SiO 2(H2)m→FeSiO 2(H2)m−n 2+n 2×H2, (1) where nis the number of the hydrogen atoms eliminated and varies between one and two for top and bridge positions, respectively, and mis the total number of H 2atoms of each silica-surface. The grafting energy ΔE is therefore given by ΔE=Esys+n 2EH2−Esurf−EFe, where Esysis the total energy of the [Fe/SiO 2(H2)m−n 2] system, E surfis the silica surface energy, and E Feand E H2are that of iron and H 2, respec- tively. According to the calculated energy values (negative values) for reaction (1) shown in Table II, the single iron atom is strongly bonded to the silica surface for both grafting modes, with the HS state being the most stable and with the bridge positions energeti- cally preferred. Overall, the most stable structure is predicted to be the Fe-D bridge site with grafting energies of −3.29 eV in the HS state and −2.58 eV in the LS state for the Fe/SiO 2-4.6 surface and for the Fe/SiO 2-3.3 surface, −3.19 eV and −2.44 eV for the HS and LS states, respectively. However, the most stable configuration in the top position is the Fe-B structure with energy values of −1.68 eV and−1.02 eV in the HS and LS states for Fe/SiO 2-4.6, and −1.93 eV and−1.71 eV for Fe/SiO 2-3.3. The spin-polarized density of states (DOSs) of the grafted single iron atom for both the HS and LS states in top (Fe-B) and bridge (Fe-D) positions are depicted in Figure 2. We found that the DOSs for both the HS and LS states where the sin- gle iron atom is in bridge position present similarities to that of an iron ion surrounded by six nitrogen atoms in an octahedral envi- ronment52(see Fig. S3 in the supplementary material). However, because here the single iron atom is bonded only to two neighbor- ing oxygen atoms, the occupied states of the t 2gand e gorbitals are strongly displaced toward the Fermi energy level. Indeed, the crystal- field splitting between the t 2gand e gstates is strongly reduced com- pared to an octahedral environment.52The same effect is observed also in the DOSs of both HS and LS states of Fe3+. We found a small positive magnetic moment induced on the oxygen atoms in direct contact with the iron atom of 0.14 μBin the HS state and 0.03 μBin the LS state for the top positions, while for the bridging sites, [0.06, 0.08] μBand [0.00, 0.00] μBfor the HS and LS states, respectively. The small magnetization that appears on oxygen is due to a direct ferromagnetic coupling between the silica surface and the iron atom. B. Grafting of two single iron atoms After finding the most stable configurations of an isolated Fe atom supported on a silica surface, two Fe atoms were studied as a function of their grafting sites and the distance between them. Since for the one atom case, the larger adsorption energies on each sur- faces ( −3.29 eV for SiO 2-4.6 and −3.19 eV SiO 2-3.3) are relatively close to each other, we have only considered the SiO 2-4.6 surface in the case of two Fe atoms. Several geometrical configurations were investigated as follows: the first iron atom is grafted at the most sta- ble top or bridge adsorption positions determined in the first part, and then, the second Fe atom is inserted at different sites and dis- tances from the first one. In the first two structures, the two iron atoms are grafted in top positions with different distances: d Fe–Feof 7.05 Å (structure 2Fe-A) and 2.47 Å (structure 2Fe-B). The second structure consists of the first iron atom in the bridge position and the J. Chem. Phys. 152, 214706 (2020); doi: 10.1063/5.0007128 152, 214706-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . PBE+U+D2 calculated spin-polarized density of states (DOSs) of the adsorbed single Fe atom for HS (solid red curve) and LS (dashed blue curve) states on SiO 2-4.6 at top (right) and bridge (left) positions. The Fermi level (E F) is the zero of energy. second one in the top position, with a distance separating them of 7.06 Å (structure 2Fe-C). The two single iron atoms of the last struc- ture (2Fe-D) are grafted in bridge positions with a Fe–Fe distance of 2.51 Å. Figure 3 shows the different adsorption configurations on SiO 2-4.6 at different sites and different distances. The corresponding Fe–O bond lengths are reported in Table III for all grafting structures of two single iron atoms on the silicasurface SiO 2-4.6: it is seen that when Fe is grafted in the top posi- tion, a small reduction of 0.02 Å is noticed from what is found for a single Fe on SiO 2. For the bridging site, the Fe–O bond lengths are in the range 1.88–2.15 Å, a variation of 0.15 Å is observed from those found for the adsorbed single iron atom. For the top–top adsorption structure, where the two iron atoms are separated by a relatively large distance of 7.05 Å, both HS and FIG. 3 . Two isolated iron atoms grafted onto a silica surface SiO 2-4.6 at dif- ferent sites and different distances: (2Fe-A) top–top positions with d Fe–Fe = 7.05 Å, (2Fe-B) top–top positions with d Fe–Fe= 2.47 Å, (2Fe-C) top– bridge positions with d Fe–Fe= 7.06 Å, and (2Fe-D) bridge–bridge positions with dFe–Fe= 2.51 Å. J. Chem. Phys. 152, 214706 (2020); doi: 10.1063/5.0007128 152, 214706-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE III . Computed bond lengths d Fe–Feand d Fe–O(in Å) within the PBE+U+D2 approximation for different grafting structures of the two single iron atoms on the silica surface SiO 2-4.6. Contact oxygens Structure d Fe–Fe dFe1−O dFe2−O O1–O 1 2Fe-A 7.05 1.82 1.81 O1–O 1 2Fe-B 2.47 1.79 1.79 O2–O 1 2Fe-C 7.06 1.81–1.82 1.78 O2–O 2 2Fe-D 2.51 1.85–2.05 1.81–1.94 LS calculations converged to the same state with a grafting energy of about −2.71 eV (see Table IV). The magnetic moment of the two iron atoms is found to be 5 μBfor each one, which corresponds to the HS magnetic state of the Fe3+ion. Therefore, increasing the number of Fe–O–Si links stabilizes the HS magnetic state. However, when the Fe–Fe distance is reduced to 2.47 Å, the magnetic moments are reduced to 4 μBfor the HS state and 0 μBfor the LS state, which cor- respond to the HS and LS magnetic states of Fe2+. For the case where one of the iron atoms is adsorbed in the top position and the sec- ond in the bridge one (structure: 2Fe-C), the magnetic moments are found to be 5 μBand 4 μB, respectively, which correspond to the HS magnetic state of Fe3+and Fe2+ions. The two HS and LS calculations are converged to the same state independently of the starting mag- netic order, with a grafting energy of −4.38 eV (structure: 2Fe-C). For the bridge–bridge adsorption structure (2Fe-D), both HS and LS calculations converged to the same state of Fe2+with a grafting energy of −4.66 eV. Overall, the structure 2Fe-D is the most stable grafting configuration for two iron atoms and corresponds to a Fe 2 dimer on the surface. The predicted electronic configurations and magnetic properties of iron species obtained from the calculations can be compared with experimental results such as the pair distri- bution function analysis (PDF), magnetic measurements, solid-state NMR spectroscopy, or TEM mapping. C. Grafting of an iron cluster Another attractive way to control the catalyst’s selectivity is the use of Fe nanoparticles.42Considering the interaction of the Fe 13 cluster with the silica-surface, several structural optimizations were started from different guesses with various adsorption structures [mono- (Fe 13-A), di- (Fe 13-B), tri- (Fe 13-C), and tetra-grafting (Fe 13- D)] on the silica surfaces SiO 2-3.3 and SiO 2-4.6. Notice that sincetetra-grafted species need the presence of four neighboring silanol sites, the probability to have four silanol groups close together in such a configuration is negligible for SiO 2-3.3 and therefore has not been considered here. Figure 4 shows the most stable configura- tions of the optimized structures of different grafting modes (Fe 13- A, Fe 13-B, Fe 13-C, and Fe 13-D) on the silica surface SiO 2-4.6. The interaction between the cluster and the silanol groups occurs by for- mation of Fe–O–Si bonds. However, in a few cases, nearby surface OH groups create Fe–OH–Si bonds. For the structure Fe 13-A, the optimized structure shows two Fe atoms in contact with one oxygen atom and a third one connected to an OH group. The Fe 13-B clus- ter is connected to the silica surfaces through two Fe–O–Si groups and one Fe–OH–Si. In the structure Fe 13-C, the optimized structure has four oxygens linked to four iron atoms, and therefore, 4 Fe–O– Si groups are established. The structure Fe 13-D shows that six Fe atoms are in contact with four oxygens and one connected to an OH group. 1. Equilibrium distances The calculated geometrical parameters of the free and adsorbed Fe13onto SiO 2-4.6 or SiO 2-3.3 systems (distances: d Fe–Fe, dFe–O, and dFe–OHin Å) for the most stable grafting structures are summarized in Table V. For the free cluster, the Fe–Fe bond lengths are in the range of 2.41–2.78 Å, and the averaged Fe–Fe distance is found to be 2.61 Å. According to the calculated equilibrium distances shown in Table V, the Fe–Fe bond lengths of the cluster are significantly influenced by the interaction with the surface: for the structures Fe13-A and Fe 13-B, the Fe–Fe bond lengths are in the range of 2.34–3.01 Å, and for the structure Fe 13-C and Fe 13-D, in the range of 2.29–3.23 Å. However, the averaged Fe–Fe distance of all grafted structures (2.59 Å) is barely influenced by the environment with only a small reduction of about 0.02 Å. These distances depend strongly on the number of Fe–O–Si and Fe–OH–Si groups: we notice an increase in some Fe–Fe bonds by 0.41–0.59 Å and a decrease in some others by 0.22–0.27 Å. The Fe–O bonds are in the range of 1.88–2.21 Å, while the Fe–OH bond lengths are in the range of 2.06–2.17 Å: these bond lengths are different from those found for an adsorbed single iron atom, an increment of 0.2 Å is noticed. 2. Grafting energies Table VI shows the grafting energies for the most stable graft- ing structures of the Fe 13cluster on the SiO 2-4.6 and SiO 2-3.3 silica TABLE IV . PBE+U+D2 calculated adsorption energies (in eV) of two isolated iron atoms grafted onto the silica surface SiO 2-4.6 for the most stable grafting structures. Initial State: HS Initial State: LS Contact oxygens Structure ΔE Ion(state) ΔE Ion(state) O1–O 1 2Fe-A −2.71 Fe3+(HS↑)–Fe3+(HS↑)−2.71 Fe3+(HS↓)–Fe3+(HS↓) 2Fe-B −4.46 Fe2+(HS↑)–Fe2+(HS↑)−2.89 Fe2+(LS)–Fe2+(LS) O2–O 1 2Fe-C −4.38 Fe2+(HS↑)–Fe3+(HS↑)−4.38 Fe2+(HS↓)–Fe3+(HS↑) O2–O 2 2Fe-D −4.66 Fe2+(HS↑)–Fe2+(HS↑)−4.66 Fe2+(HS↓)–Fe2+(HS↓) J. Chem. Phys. 152, 214706 (2020); doi: 10.1063/5.0007128 152, 214706-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . Fe 13icosahedral cluster grafted on an amorphous silica surface having a silanol density of 4.6 OH/nm2with the fol- lowing number of attached surface oxy- gen atoms: 2 (structure Fe 13-A), 3 (struc- ture Fe 13-B), 4 (structure Fe 13-C), and 5 (structure Fe 13-D). surfaces. The Fe 13cluster is grafted by dehydrogenation of surface silanols, following the reaction, Fe13+ SiO 2(H2)m→Fe13SiO 2(H2)m−n 2+n 2×H2, (2) where nis the number of hydrogens removed from the surface (between one and four) and mis the number of H 2of the silica- surface, 26 for SiO 2-4.6 and 23 for SiO 2-3.3. The adsorption energy ΔE is given by ΔE=Esys+n 2EH2−Esurf−Eclust, where Esysis the totalenergy of the whole system [Fe 13SiO 2(H2)m−n 2],EsurfandEclustare that of the free silica-surface and the free Fe 13cluster, respectively, andEH2is that of a dihydrogen molecule. According to the calcu- lated grafting energies for reaction (2) shown in Table VI, Fe 13is strongly bonded to the silica surfaces through their oxygen atoms. By comparing the energy values obtained for the three configurations of each grafting structure, we found that the grafting energies depend strongly on the location of the cluster on the surface. The grafting energies of the most stable grafting structures (Fe 13-A, Fe 13-B, Fe 13- C, and Fe 13-D) on SiO 2-4.6 are found to be −2.51 eV, −3.97 eV, TABLE V . PBE+U+D2 calculated equilibrium distance (in Å) d Fe–Fe, dFe–O, and d Fe–OHof the free and the grafted iron cluster on SiO 2-4.6 or SiO 2-3.3 for different grafting structures (Fe 13-A, Fe 13-B, Fe 13-C, and Fe 13-D). Fe13/SiO 2-4.6 Fe 13/SiO 2-3.3 Distance (Å) Free Fe 13 Fe13-A Fe 13-B Fe 13-C Fe 13-D Fe 13-A Fe 13-B Fe 13-C Max(d Fe–Fe) 2.78 3.01 2.95 3.14 3.23 3.04 3.44 2.87 Min(d Fe–Fe) 2.44 2.36 2.34 2.37 2.29 2.39 2.33 2.35 Avr d Fe–Fe 2.61 2.59 2.59 2.59 2.59 2.59 2.62 2.59 dFe–OH . . . 2.14 2.15 . . . 2.06 2.17 . . . . . . dFe–O . . . 2.08–2.06 1.86 1.86 1.88 2.01–2.01 1.84 1.90 . . . . . . 1.83 2.10 2.00–2.21 . . . 1.95 2.09 . . . . . . . . . 1.91 2.02–2.06 . . . 1.95–2.04 1.87 . . . . . . . . . 1.89 1.92 . . . 2.07 1.86 . . . . . . . . . . . . . . . . . . 1.86 . . . J. Chem. Phys. 152, 214706 (2020); doi: 10.1063/5.0007128 152, 214706-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE VI . PBE+U+D2 calculated grafting energies (in eV) for all possible grafting possibilities (structures: Fe 13-A, Fe 13-B, Fe 13-C, and Fe 13-D) on the silica surfaces. Initial and final contact oxygens Model Fe 13/SiO 2-4.6 Fe 13/SiO 2-3.3 O1→O2 Fe13-A1 −1.86 −1.90 Fe13-A2 −2.23 −2.08 Fe13-A3 −2.51 −2.52 O2→O3 Fe13-B1 −2.99 −2.57 Fe13-B2 −3.58 −4.13 Fe13-B3 −3.97 −9.43 O3→O4 Fe13-C1 −3.71 −3.77 Fe13-C2 −4.54 −4.64 Fe13-C3 −5.35 −5.07 O4→O5 Fe13-D −6.38 . . . −5.35 eV, and −6.38 eV, respectively, and on SiO 2-3.3,−2.52 eV, −9.43 eV, and −5.07 eV for the structures Fe 13-A, Fe 13-B, and Fe 13-C, respectively. These values suggest that the grafting structure Fe 13-B of the iron cluster on the SiO 2-3.3 silica surface is the most stable. This particular configuration will be discussed in more detail at the end of this section.3. Magnetic moments We have analyzed the magnetic properties of the free and the grafted cluster for all considered grafting sites. Figure 5 shows the computed magnetization density of a Fe 13cluster ferromagnetically coupled to the SiO 2-4.6 surface through their oxygen atoms at vari- ous adsorption sites. For all grafting structures, positive magnetiza- tion density is prominently seen around the iron atoms. There is a direct exchange between the interacted iron and oxygen atoms due to the hybridization between the Fe- dand O- porbitals. The oxygen atoms interacting with the iron cluster reveal a positive magnetiza- tion density (red isosurface), illustrating the ferromagnetic coupling between the cluster and the surface. However, for the Fe–OH–Si groups, the magnetic moment of the oxygen atoms is zero for all cases. For the most stable structure on SiO 2-4.6 (structure: Fe 13-D), the small magnetic moments induced on the contact oxygens (O 4) are found to be 0.04 μB, 0.02 μB, 0.02 μB, and 0.04 μB. In general, the magnetic moments induced by the iron atoms on the oxygens are in the range of 0.02–0.06 μB. The magnetic moments of the iron atoms for the free and adsorbed cluster on SiO 2-4.6 or SiO 2- 3.3 are listed in Table VII. The average iron magnetic moment for the free cluster is 3.06 μB, while for the adsorbed cluster on SiO 2- 4.6, the average iron magnetic moments are found to be 3.06 μB, 3.02μB, 2.99 μB, and 2.85 μBfor the grafted structures Fe 13-A, Fe 13B, Fe13-C, and Fe 13-D, respectively. For Fe 13/SiO 2-3.3, the average iron FIG. 5 . Isosurface plot of the magnetiza- tion density of Fe 13on SiO 2-4.6 for dif- ferent grafting structures. The color red represents spin up, and blue represents spin down. J. Chem. Phys. 152, 214706 (2020); doi: 10.1063/5.0007128 152, 214706-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE VII . PBE+U+D2 calculated magnetic moments (in μB) of iron atoms for the free and the adsorbed cluster iron on SiO 2-4.6 or SiO 2-3.3 amorphous silica surfaces for the most stable grafting structures (Fe 13-A, Fe 13-B, Fe 13-C, and Fe 13-D). Fe13/SiO 2-4.6 Fe 13/SiO 2-3.3 Atom n○Free Fe 13 Fe13-A Fe 13-B Fe 13-C Fe 13-D Fe 13-A Fe 13-B Fe 13-C 1 3.05 3.12 2.96 2.98 2.78 3.04 2.88 2.95 2 3.15 3.01 3.17 2.98 2.88 3.13 3.03 3.05 3 3.08 3.08 2.93 2.99 2.86 3.03 2.88 2.90 4 3.12 3.11 3.17 2.98 3.03 3.06 2.95 2.98 5 3.07 3.14 3.11 3.11 2.98 3.10 3.04 3.12 6 3.15 3.03 3.12 2.96 2.94 3.12 3.02 3.09 7 3.21 3.03 3.09 3.07 2.89 3.08 2.85 3.02 8 3.10 3.08 3.14 3.05 3.04 3.10 3.12 3.02 9 3.10 3.16 3.01 3.01 2.88 3.11 3.05 2.96 10 3.12 2.99 3.17 3.08 2.93 3.06 3.06 3.04 11 2.82 2.78 2.46 2.42 2.19 2.74 2.38 2.42 12 3.17 3.13 2.99 3.16 2.87 3.10 3.09 3.15 13 3.18 3.16 2.98 3.09 2.80 3.07 2.94 3.09 Tot 40.32 39.82 39.30 38.88 37.07 39.74 38.29 38.79 magnetic moments are found to 3.06 μB, 2.95 μB, and 2.98 μBfor the grafted structures Fe 13-A, Fe 13-B, and Fe 13-C, respectively. By com- paring the magnetic moment values of iron for the free cluster and their different adsorbed structures, the total iron magnetic moment is reduced by increasing the number of Fe–O–Si and Fe–OH–Si groups. This decrease is mainly due to the reduction of the mag- netic moment of the iron atoms in direct contact with the surface, due to their hybridization with the corresponding contact oxygens. In addition, we found that the magnetic moment of the center iron atom of the cluster is strongly influenced by the environment, which leads to an important reduction from 2.82 μBfor the free cluster to 2.19 μBfor the adsorbed one (Fe 13-D) on SiO 2-4.6. As a comple- mentary information, the partial and the total DOSs of the free and the adsorbed cluster on SiO 2-4.6 of the most stable configuration of each grafting structure are given in the supplementary material (see Fig. S4). 4. Charge transfer In order to determine the charge transfer between the Fe 13clus- ter and the silica surface, we have performed a Bader analysis of the charge density. The charge transfer to an active site can signif- icantly affect binding energies and therefore the catalytic activity. This is directly related to the fraction of charged sites, which evolves with the perimeter of the transition metal-support interface.67,68In Table VIII, we present the computed charge transfer from the sil- ica surface to the Fe 13cluster for the most stable geometries con- sidered. In all cases, the Fe 13cluster is negatively charged, and a large charge transfer from the silica surface SiO 2-4.6 to the cluster of about 0.82, 1.37, 2.07, and 2.82 electron for the structures Fe 13-A, Fe13-B, Fe 13-C, and Fe 13-D, respectively, is seen. For Fe 13/SiO 2-3.3, the charge transfer from the surface to the cluster is found to be 0.71, 2.64, and 2.09 electron for the structures Fe 13-A, Fe 13-B, and Fe 13- C, respectively. Our results show that the charge transfer dependson the number of oxygen linking to the Fe atoms and the number of Fe–OH groups. The Fe 13is strongly influenced by its interac- tion with the surface. The most affected iron atoms are those in direct contact with the oxygen atoms and more less for the closer ones. We found also that the central iron atom is positively charged, while the farthest iron atoms from the surface are unaffected. The charge transfer per atom of Fe 13is found in the range of 0.2–0.5 electron. TABLE VIII . Bader analysis of the charge density: computed charge transfer per atom (in e−) from both silica surfaces SiO 2-4.6 or SiO 2-3.3 to the Fe 13cluster for the most stable geometries considered (structures: Fe 13-A, Fe 13-B, Fe 13-C, and Fe13-D). Fe13/SiO 2-4.6 Fe 13/SiO 2-3.3 Atom n○Fe13-A Fe 13-B Fe 13-C Fe 13-D Fe 13-A Fe 13-B Fe 13-C 1 0.03 0.00 0.00 −0.01−0.03 0.03 −0.02 2 0.04 0.09 0.08 0.05 0.05 0.01 0.06 3 0.01 0.10 0.08 0.09 0.02 0.12 0.10 4 0.02 0.06 0.06 0.46 0.04 0.11 0.10 5 −0.04−0.02 0.40 0.38 −0.03 0.08 0.41 6 −0.01−0.00 0.40 0.37 0.01 0.34 0.47 7 −0.03−0.08−0.05 0.36 −0.07 0.09 0.03 8 0.26 0.48 0.32 0.49 0.27 0.50 0.31 9 0.34 0.39 0.28 0.24 0.31 0.44 0.28 10 0.02 0.31 0.49 0.45 0.01 0.42 0.48 11 −0.06−0.21−0.22−0.23−0.09−0.24−0.23 12 0.27 0.26 0.29 0.18 0.23 0.73 0.18 13 −0.03−0.01−0.06−0.01−0.01 0.01 −0.08 Tot 0.82 1.37 2.07 2.82 0.71 2.64 2.09 J. Chem. Phys. 152, 214706 (2020); doi: 10.1063/5.0007128 152, 214706-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 6 . The upper panel shows the Fe 13 cluster grafted on the amorphous sil- ica surface with a silanol density equal to 3.3 OH/nm2(a) before and (b) after relaxation of the system. (c) The lower panel shows PBE+U+D2 calculated total spin-polarized DOSs for Fe of the free (red solid curve) and adsorbed (blue dashed curve) cluster. (d) Isosurface plot of the magnetization density of Fe13/SiO 2-3.3. The color red represents spin up, and blue represents spin down. 5. The most stable grafting model A particular stable configuration (structure: Fe 13-B) is found on the SiO 2-3.3 silica surface that presents a grafting energy of about −9.43 eV. This is not only due to the number of Fe–O–Si groups established but also due to the deformation of the surface. Figures 6(a) and 6(b) show the atomic arrangement of the surface before and after the relaxation of the system. While initially the cluster is di-grafted on the surface (with two Fe–O–Si groups), the optimized structure shows the presence of six Fe–O–Si groups: the adsorption of the cluster has broken some Si–O–Si bonds and cre- ated new silanol sites with a strong effect on the silica surface, con- trary to the single iron atom or the other grafting structures of the cluster. This effect is observed on the total density of states depicted in Fig. 6(c). We found a displacement and broadening of all spin- polarized occupied and unoccupied peaks (spin-up toward higher energies and spin-down to lower ones). The total magnetic moment of the cluster (see Table VII) is also strongly reduced, due to the large number of Fe–O–Si groups, from 40.3 μBto 38.3 μB, and a significant charge transfer (of about 0.7 electron/atom, the largest seen here) is observed. IV. CONCLUSIONS In this work, we have shown that iron species (monomers, dimers, and Fe 13cluster) can be grafted on different amorphoussilica surfaces, which is relevant for certain applications in catalysis. Several configurations have been systematically investigated for each system (Fe/SiO 2, 2Fe/SiO 2, and Fe 13/SiO 2): we have obtained their equilibrium structures, the grafting energies, the DOSs, the magne- tization densities, and Bader analysis of the charge density to show the respective stabilities of different systems. We have shown that monomeric iron species in bridging positions are much more sta- ble than in top positions, independently of the silanol density of the surface. The two grafting modes of the single iron atom favor its HS magnetic states. We have also shown the coexistence of Fe2+(bridge) and Fe3+(top) on the silica surfaces, but Fe2+is the most stable ion. In addition, the electronic and the magnetic properties of two iron atoms depend strongly on their grafting sites and the distance between them. The grafting energy of 2Fe confirms the stability of the HS Fe2+on the silica surfaces. Then, we have also shown that the Fe 13clusters are strongly bonded to the silica surface. Their electronic and magnetic properties and grafting energies depend on the numbers of Fe–O–Si and Fe–OH–Si groups. We have found a large charge transfer from the surface to the cluster of about 3 elec- tron. The most stable grafting structure of the iron cluster shows a strong effect on the silica surface as we observed a reorganization of the silanol groups. The Fe/SiO 2models developed in this work can be used as starting points for further theoretical investigations of adsorption or catalytic process using amorphous silica, as will be shown in our future studies. J. Chem. Phys. 152, 214706 (2020); doi: 10.1063/5.0007128 152, 214706-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp SUPPLEMENTARY MATERIAL See the supplementary material for additional figures, includ- ing an explanatory scheme of different silanol sites existing on the silica surface, Fe-A and Fe-C geometries, DOSs of Fe–N6 for both HS and LS states, and the partial and the total DOSs of the free and the adsorbed iron cluster on the silica surface SiO 2-4.6 of the most stable configuration of each grafting structure. ACKNOWLEDGMENTS We would like to acknowledge fruitful discussion with Andreea Pasc, Nadia Canilho, and Younes Bouizi. This work was performed using the HPC Mesocenter “Explor” of the University of Lorraine. S.G. is grateful to the Lorraine University of Excellence for funding this work with the Lignin program. The authors acknowledge finan- cial support through the COMETE project (Conception in silico de Matériaux pour l’EnvironmenT et l’Energie) cofunded by the Euro- pean Union under the program “FEDER-FSE Lorraine et Massif des Voges 2014–2020.” There are no conflicts to declare. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1Z. Ma and F. Zaera, “Heterogeneous catalysis by metals,” in Encyclopedia of Inorganic and Bioinorganic Chemistry (American Cancer Society, 2014), pp. 1–16. 2B. R. Goldsmith, B. Peters, J. K. Johnson, B. C. Gates, and S. L. Scott, “Beyond ordered materials: Understanding catalytic sites on amorphous solids,” ACS Catal. 7, 7543–7557 (2017). 3M. Gierada, P. Michorczyk, F. Tielens, and J. Handzlik, “Reduction of chromia– silica catalysts: A molecular picture,” J. Catal. 340, 122–135 (2016). 4H. Guesmi and F. 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5.0012033.pdf

J. Chem. Phys. 152, 224304 (2020); https://doi.org/10.1063/5.0012033 152, 224304 © 2020 Author(s).Interaction of the HCO radical with molecular hydrogen: Ab initio potential energy surface and scattering calculations Cite as: J. Chem. Phys. 152, 224304 (2020); https://doi.org/10.1063/5.0012033 Submitted: 27 April 2020 . Accepted: 21 May 2020 . Published Online: 09 June 2020 Paul J. Dagdigian ARTICLES YOU MAY BE INTERESTED IN Coupled-cluster techniques for computational chemistry: The CFOUR program package The Journal of Chemical Physics 152, 214108 (2020); https://doi.org/10.1063/5.0004837 Observation of the elusive “oxygen-in” OCS dimer The Journal of Chemical Physics 152, 221102 (2020); https://doi.org/10.1063/5.0010716 The ORCA quantum chemistry program package The Journal of Chemical Physics 152, 224108 (2020); https://doi.org/10.1063/5.0004608The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Interaction of the HCO radical with molecular hydrogen: Ab initio potential energy surface and scattering calculations Cite as: J. Chem. Phys. 152, 224304 (2020); doi: 10.1063/5.0012033 Submitted: 27 April 2020 •Accepted: 21 May 2020 • Published Online: 9 June 2020 Paul J. Dagdigiana) AFFILIATIONS Department of Chemistry, The Johns Hopkins University, Baltimore, Maryland 21218-2685, USA a)Author to whom correspondence should be addressed: pjdagdigian@jhu.edu ABSTRACT The potential energy surface describing the interaction of the HCO radical with molecular hydrogen has been computed through explicitly correlated coupled cluster calculations including single, double, and (perturbative) triple excitations [RCCSD(T)-F12a], with the assump- tion of fixed molecular geometries. The computed points were fit to an analytical form suitable for time-independent quantum scattering calculations of rotationally inelastic cross sections and rate coefficients. Since the spin-rotation splittings in HCO are small, cross sections for fine-structure resolved transitions are computed with electron-spin free Tmatrix elements through the recoupling technique usu- ally employed to determine hyperfine-resolved cross sections. Both spin-free and fine-structure resolved state-to-state cross sections for rotationally inelastic transitions are presented and discussed. Published under license by AIP Publishing. https://doi.org/10.1063/5.0012033 .,s I. INTRODUCTION The formyl (HCO) radical has been known to be present in the interstellar medium (ISM) since the late 1970s.1It has been employed as a tracer of photodissociation dominated regions (PDR’s). PDR models have been employed to understand the chem- istry of vacuum ultraviolet (VUV) illuminated regions in our galaxy and in external galaxies.2–4This radical has been observed in other astronomical sources, including molecular clouds,1,5prestellar cores,6and the protostellar binary IRAS 16295-2422.7 The HCO radical is also of interest for its possible role in the formation of complex organic molecules in the ISM.8Our present understanding of the formation of this type of molecule in the ISM is regrettably limited. Two general production schemes have been proposed, namely, gas-phase chemistry initiated by evapora- tion of interstellar ices9,10and hydrogenation and/or radical–radical reactions on dust-grain surfaces.11,12 The investigation of molecular precursors is an important way to unravel how complex organic molecules are synthesized in the ISM. As illustrated in Refs. 13–16, the formyl radical has been pro- posed as the precursor for these larger molecules. However, the for- mation of HCO itself is not understood. Gas-phase chemistry6andreactions on the surface of dust grains12,13have been proposed as pathways for the production of HCO in the ISM. The application of a radiative transfer model, for example, as described by van der Tak et al. ,17must be employed with obser- vational data to derive quantitative estimates of the abundance of HCO. Such a model can also provide information on the density (usually mostly H 2) and kinetic temperature of the interstellar cloud. The millimeter and submillimeter spectroscopy of HCO has been reported, and transition frequencies in its rotational spectrum are precisely known.18 State-to-state rate coefficients for collision-induced rotational transitions in HCO are required for application of a radiative trans- fer model. To the author’s knowledge, these rate coefficients, either accurate or approximate rates, are not available in the literature for collisions of HCO with H 2or any other species. It should be noted that HCO and H 2can react through the endothermic process HCO + H 2→H2CO + H. However, the rate coefficient for this reac- tion has been estimated19to be 1.2×10−20cm3molecule−1s−1at 500 K. This reaction is entirely negligible under the conditions of the ISM. In this study, the calculation of the potential energy surface (PES) for the interaction of HCO( ˜X2A′) with H 2is reported. This J. Chem. Phys. 152, 224304 (2020); doi: 10.1063/5.0012033 152, 224304-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp surface is then employed to compute rate coefficients for HCO–H 2 rotational transitions, both without and with resolution of the fine- structure. This paper is organized in the following manner: Sec. II A presents the ab initio calculation of points on the HCO–H 2PES, while Sec. II B describes the fit of the calculated points to a form suit- able to use in quantum scattering calculations. Section II C describes various features of the PES, including the energy and geometry of the global minimum. Section III presents the formalism and methodology of the time-independent electron-spin-free quantum scattering calculations as well as the calculation of the fine-structure- resolved cross sections. Rate coefficients will be presented in a future publication. The computed cross sections are displayed and dis- cussed in Sec. IV. This paper concludes with a brief discussion in Sec. V. II. POTENTIAL ENERGY SURFACE A.Ab initio calculations The molecules were taken to be rigid in the calculation of the potential energy surface for the interaction of the ground electronic state of HCO( ˜X2A′), with H 2. The three-dimensional PES for HCO has been characterized through RHF-UCCSD(T) restricted Hartree- Fock coupled cluster calculations by Song et al.20They found the geometry at the global minimum to be described by the parame- tersrCH= 2.115 a0,rCO= 2.221 a0, and∠HCO = 124.6○. We employ this geometry of HCO in our calculation of the PES. This geometry is in good agreement with the experimental geometry determined by microwave spectroscopy21(rCH= 2.098 a0,rCO= 2.213 a0, and ∠HCO = 127.4○). The bond length of H 2was taken at the aver- age, rHH= 1.4487 a0, over the probability distribution of its v = 0 vibrational level.22 Five coordinates are required to describe the geometry of the HCO–H 2complex. We employ a modified version of the body- fixed coordinate system defined by Phillips et al.23and Valiron et al.24for the interaction of H 2O, and analogous molecules such as H 2S (see Ref. 25) that have C2vsymmetry, with H 2. Here, the origin of the body-fixed coordinate system is located at the cen- ter of mass of the HCO molecule. The orientation of the in-plane aandbinertial axes of HCO relative to the molecular framework of the molecule is illustrated in Fig. 1(a). The body-frame zand xaxes are chosen to lie along the aand binertial axes, respec- tively, of HCO, with the oxygen end of the molecule pointing in the−zdirection and the H atom having a negative xcoordinate. The HCO molecule, hence, lies in the xzplane of the body-fixed coordinate system. The ainertial axis was chosen as the body- fixed zaxis since this is the quantization axis for the rotational Hamiltonian. The coordinates defining the geometry of the HCO–H 2com- plex are illustrated in Fig. 1(b). The intermolecular separation (sepa- ration of the centers of mass) is denoted as R. The body-fixed angles (θ,ϕ) and (θ′,ϕ′) define the orientation of the collision direction and orientation of H 2relative to the HCO body-fixed coordinate system. The ranges of variation of these angles are 0○≤(θ,θ′)≤180○and 0○≤(ϕ,ϕ′)≤360○. The Jacobi vector Rlies along the zaxis when the H2molecule approaches the hydrogen end of HCO with θ= 0○and ϕ= 0○. The H 2molecular axis is parallel to the zaxis whenθ′= 0○ andϕ′= 0○. FIG. 1 . (a) Orientation of the in-plane inertial axes of HCO relative to the molec- ular geometry. “O” with an arrow indicates the origin of the body-fixed coordinate system. (b) Body-frame coordinate system defining the geometry of the HCO–H 2 complex. Explicitly correlated coupled cluster calculations with inclusion of single, double, and (perturbatively) triple excitations [RCCSD(T)- F12a]26,27were used to compute points on the HCO–H 2PES. The MOLPRO 2012 suite of computer codes28was employed to carry out these calculations. The aug-cc-pVTZ correlation-consistent basis set,29,30with aug-cc-VTZ/MP2FIT and cc-pVTZ/JKFIT as the den- sity fitting basis and resolution of the identity, respectively,31,32was used in the calculations. A counterpoise correction33was applied to correct for the basis set superposition error. Test calculations were also carried out with larger basis sets (aug-cc-pVQZ and aug-cc-pV5Z). In the initial calculations, the convergence with respect to the size of the basis set was examined. Figure 2 presents radial cuts computed with the aug-cc-pVXZ basis sets, where X = T, Q, 5,29 for an orientation close to that of the global minimum (determined through more extensive calculations discussed further below). It can be seen that the computed interaction is slightly more attrac- tive with the larger basis sets than with the aug-cc-AVTZ basis set; however, the differences in the computed binding energies for this orientation are less than 2 cm−1. The difference in the binding ener- gies computed with the aug-cc-AVQZ and aug-cc-AV5Z basis sets is 0.3 cm−1. For these radial cuts, the minimum in the interac- tion energy equals −205.7,−207.1, and−207.4 cm−1for calculations with the aug-cc-pVTZ, aug-cc-AVQZ, and aug-cc-AV5Z basis sets, respectively. Given the large number of points required to describe the PES, the aug-cc-AVTZ basis set was deemed sufficiently accurate for our purposes. Calculations of the full PES were carried out on a five- dimensional grid ( R,θ,ϕ,θ′,ϕ′) at 28 values of the intermolecular separation Rranging from 4 to 20 a0. The angles defining the orien- tation of the complex were chosen randomly.34The random number generator in MATLAB was employed, with the seed based on the cur- rent time. The calculations were carried out for 3500 random orien- tations (defined by values of θ,ϕ,θ′,ϕ′). Hence, the total number of points computed for the HCO–H 2PES was 98 000. J. Chem. Phys. 152, 224304 (2020); doi: 10.1063/5.0012033 152, 224304-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . The HCO–H 2interaction as a function of the intermolecular separation R for an orientation near to that of the global minimum [ θ= 95○,ϕ= 180○,θ′= 125○, ϕ′= 0○]. The inset shows an expanded plot around the minimum. B. Fitting the potential It is convenient for computing matrix elements of the inter- molecular interaction between scattering basis functions in time- independent quantum close-coupling scattering calculations to employ the bispherical harmonic expansion defined by Phillips et al.23for the expansion of the H 2O–H 2PES, with the normalization given by Valiron et al. ,24 V(R,θ,ϕ,θ′,ϕ′)=∑ l1m1l2lvl1m1l2l(R)¯tl1m1l2l(θ,ϕ,θ′,ϕ′), (1) where ¯tl1m1l2l(θ,ϕ,θ′,ϕ′)is a normalized spherical tensor, ¯tl1m1l2l(θ,ϕ,θ′,ϕ′)=αl1m1l2l(1 +δm10)−1∑ r1r2(l1l2l r1r2r) ×Yl2r2(θ′,ϕ′)Y(lr(θ.ϕ) ×[δm1r1+(−1)l1+m1+l2+lδ−m1r1]. (2) The normalization factor in Eq. (2) is24 αl1m1l2l=[2(1 +δm10)−1(2l1+ 1)−1]−1/2. (3) It should be noted that with our definition of the body-frame coor- dinate system (see Fig. 1), the quantization axis for the expansion of the HCO–H 2potential is the same as that for the rotational Hamiltonian. The indices l1,l2, and ldescribe the tensor ranks of the angu- lar dependence of the HCO orientation, the H 2orientation, and the collision vector orientation, respectively. Unlike the situation for the H2O–H 224PES, and also the H 2S–H 2PES,25there is no restriction on the value of m1other than | m1|≤l1. The symmetry with respect to reflection of H 2in the HCO plane is taken into account through the phased sum of ±m1in Eq. (2). The homonuclear symmetry ofH2requires that only even l2terms need to be included in the sum in Eq. (1). The allowed values of lare given by the vector addition of l1 and l2. The potential should be parity invariant, i.e., invariant to inversion of all coordinates through the origin.35This implies that the sum l1+l2+lmust be an even integer. However, in the case of the H 2O–H 2PES, this symmetry appears to be slightly broken (see Ref. 36), and some odd l1+l2+lterms were included in the angu- lar expansion.24It was not found necessary to include odd l1+l2+l terms in the angular expansion of the HCO–H 2interaction for an acceptable fit of the PES. The PES was found to be quite anisotropic, and for some ori- entations at values of Rnear to that of the equilibrium intermolec- ular separation, the potential is quite repulsive. Since highly repul- sive regions of the PES do not have to be accurately described for the scattering calculations,37we followed the procedure of Wormer et al.38and damped the interaction energy with a hyperbolic tangent function up to a maximum value Vmax. In this study, Vmaxwas set to 10 000 cm−1. For each value of the intermolecular separation R, the com- puted ab initio values of the interaction energy computed on the grid of 3500 random orientations were fit in a linear least-squares procedure to Eqs. (1) and (2) to determine the angular expansion coefficients vl1m1l2l(R). Fits were carried out with different numbers of expansion coefficients, determined by the maximum values of l1andl2in the sum in Eq. (1). An acceptable fit was obtained for lmax 1=12 and lmax 2=6 for all values of Rfor which the interaction energies were computed. In this case, the total number of angular terms in Eq. (1) is 1376. Some of the angular expansion coefficients were connected with a hyperbolic tangent switching function to the R−ndepen- dence expected for these terms. For example, the isotropic term (l1= 0, m1= 0, l2= 0, and l= 0) was connected to a R−6depen- dence at a long range. The terms involving multipole–multipole electrostatic interactions, for example, the dipole–quadrupole inter- action, were connected to the appropriate large- R R−ndependence. Other terms were damped to zero at large R. The expansion coeffi- cients vl1m1l2l(R)at arbitrary values of the intermolecular separation Rwere obtained by splining the values of the coefficients deter- mined on the grid of Rvalues for which ab initio energies were computed. The large number of expansion coefficients used to fit the PES will make the scattering calculations require an inordinate amount of time to run. In fact, such calculations are realistically not feasible with the current computational resources to map out the detailed energy dependence of the cross sections. As was done for the H 2S– H2PES,25we explored the removal of some angular coefficients to speed up the scattering calculations. Many of these angular coef- ficients were small and could be eliminated without a significant reduction in the quality of the fit. We employed the following procedure to reduce the number of coefficients. In the fit for R= 5.75 a0, an intermolecular sepa- ration near that for the minimum of the PES, coefficients whose magnitudes were less than a chosen threshold value were eliminated. It should be noted that this procedure does not bias the coeffi- cients since they are normalized.24The ab initio points were then refit with the set of ( l1m1l2l) terms corresponding to the remaining coefficients. This procedure yielded a slightly better fit than just J. Chem. Phys. 152, 224304 (2020); doi: 10.1063/5.0012033 152, 224304-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Fits of the HCO–H 2potential energy surface. lmax 1 lmax 2 Number of coefficients De(cm−1) Ab initioa206.9 12 6 1376 205.6 12 4 795 209.7 12b4 302 213.8 aaug-cc-AVTZ basis. bThreshold for retaining a coefficient equals 3 cm−1. See the text. using the reduced set of coefficients from the full fit: The rms devi- ation of the fit for R= 5.75 a0is 0.088 cm−1for the refitted, reduced set of coefficients, while it is 0.091 cm−1for the reduced set of coefficients from the full fit. Table I compares the number of required coefficients and the magnitude of the well depth for various fits of the PES. These well depths are compared with the ab initio well depth, which was obtained by calculations on a fine grid around the global minimum. We see in Table I that the fit with lmax 1=12 and lmax 2=6 gives a well depth very close to the ab initio value. The fits with smaller numbers of angular coefficients give slightly larger values of the well depth. In the scattering calculations presented in Sec. III, we employed the fit with lmax 1=12,lmax 2=4, and a threshold of 3 cm−1to reduce the number of angular coefficients to 302. As a more thorough check on the effect of fitting the ab initio points with a smaller number of angular coefficients, we compare in Fig. 3 contour plots of the PES near the geometry of the global minimum. Panel (a) of Fig. 3 presents a contour plot computed with the full set of 1376 coefficients, while the plot in panel (b) employed the reduced set of 302 coefficients. It can be seen that the differ- ences in these two plots are much smaller than the anisotropy of the potential. As a second comparison of the fit of the PES with the full and reduced set of angular coefficients, state–state inelastic cross sections computed as described in Sec. III with the two fits were compared. FIG. 4 . Cross sections for rotational excitation of the lowest HCO rotational level (000) in collision with ortho -H2(j= 1) at a collision energy of 100 cm−1, computed with the full set of 1376 angular coefficients. The identity of the final levels is as follows: 1, 1 01; 2, 1 11; 3, 1 10; 4, 2 02; 5, 2 12; 6, 2 11; 7, 3 03; 8, 3 13; 9, 3 12; 10, 4 04;11, 414; 12, 4 13; 13, 5 05; 14, 5 15; 15, 5 14; 16, 6 06; 17, 6 16; 18, 6 15; 19, 7 07. The lower panel shows the difference in the cross sections when computed with the full 1376 and reduced 302 set of angular coefficients. The top panel of Fig. 4 presents excitation cross sections for the low- est rotational level 0 00in collision with ortho -H2(j= 1) at a collision energy of 100 cm−1, computed with the full set of 1376 angular coef- ficients. The bottom panel of Fig. 4 displays the percent differences in these cross sections computed with the reduced 302-term set of angular coefficients vs the full set of 1376-term angular coefficients. FIG. 3 . Contour plot of the PES for an intermolecular separation R= 5.75 a0as a function of the polar angles θandθ′ for fixed values of the azimuthal angles ϕ= 180○andϕ′= 0○computed with (a) the full set of 1.376 and (b) a reduced set of 302 angular coefficients. J. Chem. Phys. 152, 224304 (2020); doi: 10.1063/5.0012033 152, 224304-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . Plot of the rms deviation of the fits and the absolute value of the angular average of the HCO–H 2PES as a function of the intermolecular separation R. We see that these differences are 2% or less, with the exception of the 1 01←000transition for which the difference is almost 4%. We deemed that these differences are acceptable in order to compute cross sections for transitions between the lower HCO rotational lev- els. It should be noted that a larger number of angular coefficients may be required if cross sections involving highly rotational levels are required. As a measure of the overall quality of the angular fits, Fig. 5 shows the root-mean-square deviation (rms) of the fit with 302 angular coefficients and the absolute value of the angular average of the PES as a function of R. The kink in the average value is due to a change in the sign of the average. The ratio of the rms deviation and the average is seen to be very small at large Rand increases as Rdecreases. The ratio of the rms deviation and the average is less than 0.1% for all values of R. The rms deviation was found to be virtually identical for fits of the full 1376 and reduced 302 angular coefficients. FIG. 7 . Geometry of the HCO–H 2complex at (a) the global minimum of the PES and (b) the secondary minimum. The complex is planar and lies in the body-frame xzplane. The origin is the location of the center of mass of the HCO molecule. C. Features of the potential energy surface In this section, we explore the PES by computing contour plots with various fixed values of the azimuthal angles. In all cases, the contour plots were computed using the full set of 1376 angular coefficients. The global minimum of the PES was found to have the follow- ing geometry: R= 5.865 a0,θ= 95.7○,ϕ= 180○,θ′= 127.8○, andϕ′ = 0○, with well depth De= 206.9 cm−1. There is a symmetry equiv- alent geometry, namely, R= 5.865 a0,θ= 95.7○,ϕ= 180○,θ′= 52.2○ (= 180○−127.8○), andϕ′= 180○, because of the homonuclear nature of the H 2molecule. The dependence of the HCO–H 2interaction energy around the global minimum as a function of the polar angles θandθ′withϕandϕ′set equal to 180○and 0○, respectively, is dis- played in Fig. 6(a). Figure 7(a) displays the geometry of the HCO–H 2 complex at the global minimum. The complex is planar in this case and lies in the body-frame xzplane. We can see from Fig. 6 that the PES is quite anisotropic, with the interaction energy at the equilib- rium intermolecular separation Re= 5.865 a0ranging from −207 to nearly 2000 cm−1. A secondary minimum was also found on the PES. This mini- mum has the following geometry: R= 6.310 a0,θ= 84.1○,ϕ= 180○, FIG. 6 . Interaction energy as a function of the polar angles θandθ′around (a) the global minimum ( R= 5.865 a0) and (b) the secondary minimum ( R = 6.310 a0). For both plots, the azimuthal anglesϕ= 180○andϕ′= 0○. The min- ima in each plot are indicated by the solid black circles. J. Chem. Phys. 152, 224304 (2020); doi: 10.1063/5.0012033 152, 224304-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 8 . Interaction energy as a function of Randθforθ′= 127.8○(the value for the global minimum), ϕ= 180○, andϕ′= 0○. θ′= 0○,ϕ′= 0○, with well depth 115.7 cm−1; with a symmetry equiv- alent geometry, R= 6.310 a0,θ= 84.1○,ϕ= 180○,θ′= 180○, and ϕ′= 180○. The dependence of the HCO–H 2interaction energy around the secondary minimum as a function of the polar angles θ andθ′is displayed in Fig. 6(b), while Fig. 7(b) displays the geometry of the HCO–H 2complex at the secondary minimum. As in the case of the global minimum, the complex is also planar in this case and lies in the body-frame xzplane. The saddle point between the two minima was found to have an energy of −19 cm−1. Hence, the bar- rier between the minima is 97 cm−1high with respect to the energy of the secondary minimum. For both minima, it can be seen from Fig. 7 that the hydrogen molecule is coplanar with the HCO radical and lies approximately perpendicular to the Jacobi vector. This orientation is consistent with the positive value of the quadrupole moment of H 2. It is also interesting to investigate the approach of the collision partners. Figure 8 displays the dependence of the interaction energy as a function of the intermolecular separation Rand the polar angle θfor fixed values of θ′,ϕ, andϕ′. We see that the collision partners can approach without a barrier for most values of θ, except for a very small barrier around θ∼45○. This plot also shows the strong anisotropy of the repulsive wall of the PES. III. SCATTERING CALCULATIONS Time-independent quantum scattering calculations were car- ried with the HIBRIDON suite of programs.39The theory of colli- sions of an asymmetric top molecule with a linear rotor has been derived by Phillips et al.23Time-independent quantum scattering calculations were carried out with a modification of the basis rou- tine recently employed25to treat H 2S–H 2collisions. The rotational constants for the ground vibrational level of HCO were taken from Ref. 18, and the H 2rotational constant was obtained from Ref. 40. HCO has doublet spin multiplicity, and each rotational level with n>0 is split into two fine-structure levels, labeled F1withtotal angular momentum j=n+ 1/2 and F2with total angular momentum j=n−1/2. In fact, each fine-structure level is further split by the hyperfine interaction due to the spin of the hydrogen nucleus.18The spin-rotation and hyperfine splittings are both much smaller than the rotational spacings.18In our scattering calculations, the fine-structure and hyperfine splittings were hence neglected, and the cross sections for collision-induced rotational transitions only were calculated from spin-free scattering calculations. Figure 9 displays the energies of the HCO low-lying rotational levels. This molecule is a near-prolate asymmetric top, and the prolate-limit projection quantum number kais an approximately good quan- tum number. The rotational angular momentum of H 2is denoted asj2. Full close-coupling calculations were carried out in order to compute spin-free state-to-state integral cross sections for tran- sitions between HCO rotational levels in collision with para -H2 (j2= 0) and ortho -H2(j2= 1). Scattering calculations were carried out for 390 energies up to 1250 cm−1for collisions with each of the H2nuclear spin modifications. The calculations were checked for convergence of the cross sections involving transitions between the 22 levels having energies ≤100.2 cm−1with respect to the size of the HCO rotational basis, spacing in the radial grid, and total number of partial waves included. Rotational levels of HCO up to n= 10, depending on the total energy, were included in the channel basis. The H 2rotational basis included j2= 1 only for ortho -H2. Partial waves up total angular momentum J= 182̵h, with all helicity values, were included in the calculations. The initial calculations for para -H2included both j2= 0 and 2 in the scattering basis for low collision energies; however, calculations at higher energies with this basis would take an inordinate amount of time. Hence, calculations on collisions of HCO with para -H2at higher energies included only j2= 0 in the scattering basis. We com- pare calculations of the cross sections for the 1 01←000and 2 02←000 transitions up to a collision energy of 150 cm−1with these two para - H2scattering basis sets in Fig. 10. It can be seen that the integral cross FIG. 9 . Low lying rotational levels of the HCO radical. The levels are grouped in columns according to the value of the prolate-limit body-frame projection quantum number ka. Each level is labeled by the rotational angular momentum nand the prolate- and oblate-limit projection quantum numbers kaand kc, respectively. The spin-rotation splitting is small and is not shown in this diagram. J. Chem. Phys. 152, 224304 (2020); doi: 10.1063/5.0012033 152, 224304-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 10 . Energy dependence of the spin-free cross sections for the (a) 1 01←000 and (b) 2 02←000transitions for collision energies up to 150 cm−1. Red: scattering basis includes j2= 0 and 2; blue, scattering basis includes j2= 0 only. sections obtained with these two scattering basis sets are very simi- lar in magnitude. The principal difference is that the energies of the scattering resonances are slightly shifted. In the case of C 2H–H 2col- lisions, the effect of inclusion of j2= 2 in the scattering basis was to increase the magnitude of the cross sections for the n= 0→n′= 1–4 transitions by ∼10%.48However, these cross sections are much smaller (1–7 Å2) than for HCO–H 2and could be more affected by inclusion of j2= 2 in the scattering basis. In view of this good agree- ment, the full set of calculations for collisions of HCO with para -H2 were carried out with only j2= 0 in the scattering basis. Cross sec- tions were computed up to a total energy of 1250 cm−1for collisions of both ortho - and para -H2. Since the spin-rotation splittings are much smaller than the rotational spacings, we have computed cross sections for fine- structure-resolved transitions through the recoupling method,41,42 which has been extensively employed to compute cross sections and rate coefficients for hyperfine-structure-resolved transitions.43Here, we obtain Tmatrix elements with inclusion of the electron spin from spin-free Tmatrix elements in the following way. The total angular momentum JTof the collision complex when the electron spinsof HCO is included equals J+s, where Jis the total angu- lar momentum of the collision pair without the electronspin. The Tmatrix elements describing the collision of the two molecules (with total angular momenta jandj2) including the electron spin can be calculated from the spin-free Tmatrix elements as follows:42 TJT j′n′k′ ak′ cj′ 2j′ Rl′,jnkakcj2jRl=∑ Jj′ 12j12(−1)jR+j′ R+l+l′+j2+j′ 2 ×([j][j′][j12][j′ 12][jR][j′ R])1/2(2J+ 1) ×{n j 2j12 l J j R}{n′j′ 2j′ 12 l′J j′ R} ×{jRn J s JTj}{j′ Rn′J s JTj′} ×TJ n′k′ ak′ cj′ 2j′ 12l′,nkakcj2j12l, (4) where [ x] = 2 x+ 1. The quantum numbers l,j12, and jRdenote the orbital angular momentum of the collision complex, the vector sum of n+j2, and the vector sum of j2+l, respectively. The ini- tial and final levels are indicated by unprimed and primed quantum numbers, respectively. FIG. 11 . Spin-free integral cross sections as a function of collision energy for the rotational excitation of HCO from its lowest rotational level, 0 00, to the next five rotational levels in collision with (a) para-H2(j2= 0) and (b) ortho -H2(j2= 1). The identities of the final levels are indicated in the legends. J. Chem. Phys. 152, 224304 (2020); doi: 10.1063/5.0012033 152, 224304-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Equation (4) was employed to compute Tmatrix elements involving collision-induced transitions between HCO fine-structure levels. The integral cross sections for transitions between HCO fine- structure levels induced by collisions with para -H2(j2= 0) and ortho - H2(j2= 1) were then calculated using the Tmatrix elements defined in Eq. (4) as follows: σj′n′k′ ak′ cj′ 2,jnkakcj2=π k2 i(2j+ 1)(2j2+ 1)∑ JT(2JT+ 1) ×∑ ll′jRj′ R∣TJT j′n′k′ ak′ cj′ 2j′ Rl′,jnkakcj2jR∣2. (5) Here, kidenotes the initial wavevector. IV. RESULTS Figure 11 presents the plots of the energy dependence of state- to-state integral cross sections for excitation of the lowest rotational level, 0 00, to the next five rotational levels (see Fig. 9) in collision with para -H2(j2= 0) and ortho -H2(j2= 1). These cross sections are quite large, reflecting the significant anisotropy of the PES (see Fig. 6). For most of the transitions, the cross sections rise sharply from their respective thresholds, reach a maximum, and then decrease slowly in magnitude at higher collision energies. There is a dense resonance structure due to quasibound levels built on the poten- tial well at low energies for collisions of HCO with para -H2(j2= 0),as has been seen in the energy-dependent cross sections for colli- sions of H 2with other molecules (see, for example, Refs. 44–48). The profiles of the resonances are not well recovered in our cal- culations since the energy grid spacing was 0.5 cm−1at low ener- gies. There are similar resonances in the energy-dependent cross sections for collisions of HCO with ortho -H2, but there are many more overlapping resonances with this H 2nuclear spin modifica- tion so that the energy dependence looks less structured. As a result, it is not possible to estimate the widths of the resonances in this case. The cross sections for collisions of HCO with ortho -H2(j2= 1) are much larger than those for collisions of HCO with para -H2(j2 = 0), as was also found for collisions of OH,44CN,45NH 3,46HCN,47 C2H,48and H 2S (Ref. 25) with H 2. In a molecule–H 2interaction, the full anisotropy of the PES is experienced in interactions of H2(j2>0) rotational levels, as compared to interactions with para - H2(j2= 0). In addition, the long-range interaction is stronger for ortho -H2(j2= 1) than for para -H2(j2= 0) since the former has a nonzero quadrupole moment, while the latter does not. It can be seen from Fig. 11 that the final state distributions are different for collisions of the HCO 0 00level with para -H2(j2= 0) and ortho -H2(j2= 1). The final level with the largest cross section (for energies>30 cm−1) is the 2 02level for the former collision partner, while the 1 10level has the largest cross section for the latter collision partner. FIG. 12 . Spin-free integral cross sections as a function of collision energy for the rotational excitation of HCO from (a) the 111level in collision with para-H2(j2= 0), (b) the 1 11level in collision with ortho - H2(j2= 0), (c) the 4 04level in collision with para-H2(j2= 0), and (d) the 4 04level in collision with ortho -H2(j2= 0). The identities of the final levels are indicated in the legends. J. Chem. Phys. 152, 224304 (2020); doi: 10.1063/5.0012033 152, 224304-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp It is also interesting to investigate the possible propensities in the state-to-state cross sections. Figure 12 displays sets of cross sec- tions as a function of the collision energy for the excitation of the HCO 1 11and 4 04levels in collision with para -H2(j2= 0) and ortho - H2(j2= 1). As seen in Fig. 11, the cross sections are much larger with the latter collision partner. For collisions of the 1 11level, dis- played in Figs. 12(a) and 12(b), the cross sections for the transition to the nearly isoenergetic 1 10level are large at low collision energies and decrease monotonically at higher energy. The transitions to the nearly isoenergetic 2 12and 2 11levels, with no change in the prolate projection quantum number ka, have cross sections very similar in magnitude and are nearly equal to the cross section for the 1 11→110 transition. All three of these transitions have Δka= 0. The cross section for the 1 11→404transition [see Figs. 12(a) and 12(b)], in which the projection quantum number kachanges, is seen to be much smaller than the cross sections for which Δka= 0. Hence, it appears that there is a propensity to conserve the HCO ka projection number in inelastic HCO–H 2collisions. Figures 12(c) and 12(d) display cross sections as a function of energy for excitation transitions out of the 4 04level in collision with para -H2(j2= 0) and ortho -H2(j2= 0), respectively. For both H 2colli- sion partners, the largest cross section is for the 4 04→505transition in which the kaprojection quantum number is conserved. Corre- spondingly, that the cross sections for transitions to the 1 11, 110, 212, and 2 11levels, for which Δka= +1, are much smaller than the cross section for the 4 04→505transition. For the 4 04initial level, we again see a propensity to conserve the kaquantum number. Another inter- esting feature of the cross sections plotted in Figs. 12(c) and 12(d)is that the cross sections for transitions having the same final ka quantum number, e.g., the 1 11/110and 2 ‘12/211pairs of the levels, are nearly identical. As discussed in Sec. III, fine-structure cross sections were com- puted by the recoupling method. Figure 13 compares the spin-free cross sections for the Δn= + 1, Δka= 0 transition from the 0 00, 101, 303, and 5 05initial levels and the corresponding fine-structure resolved cross sections for the j=n+ 1/2 initial level for each rotational level. We see that the spin-free cross sections for each rotational transition are larger than either of the corresponding fine- structure resolved cross sections. In fact, the sum of the cross sec- tions from a given fine-structure level to the j′=n′±1/2 final levels equals the spin-free cross section, as is true for any rotational transition. We see in Fig. 13 that for each transition, the fine-structure resolved cross section for the transition to the j′=n′+ 1/2 final level is larger than the cross section for the transition to the j′=n′−1/2 final level. As jincreases, the cross section for the j=n+ 1/2→j′ =n′+ 1/2 transition becomes increasingly larger than the cross sec- tion for the j=n+ 1/2→j′=n′−1/2 transition and approaches the magnitude of the corresponding spin-free cross section. The former transition has Δj=Δn, while Δj≠Δnapplies to the latter transition. The propensity for Δjto equal Δnis particularly strong for high rota- tional levels and applies to all fine-structure-resolved transitions in HCO–H 2collisions. A similar propensity rule was derived by Alexander49for fine- structure-resolved inelastic transitions in collisions of a diatomic molecule in a2Σ+electronic state with a structureless collision FIG. 13 . Integral spin-free and fine- structure-resolved cross sections as a function of collision energy for Δn= +1, Δka= 0 transitions from the (a) 0 00, (b) 1 01, (c) 3 03, and (d) 5 05initial lev- els in collision with ortho -H2(j2= 1). Dis- played in the figure are the spin-free cross sections for these transitions and the fine-structure resolved cross sections for transitions from the j=n+ 1/2 initial fine-structure levels. J. Chem. Phys. 152, 224304 (2020); doi: 10.1063/5.0012033 152, 224304-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp partner. The physical origin of this propensity rule is that in a molec- ular collision, the electron spin cannot be re-oriented since it is a spectator in the collision. In a fine-structure level of total angular momentum j=n+ 1/2, the rotational angular momentum nand the electron spin sare approximately parallel to each other, while for the fine-structure level j=n−1/2,nandsare roughly antiparallel. In a high rotational level, the angular momentum ncannot be colli- sionally re-oriented by a large angle so that the coupling of nands can be changed. By contrast, a small rotational angular momentum ncan be re-oriented by a large angle so that the coupling of nand scan be changed; this provides an explanation of why the cross sec- tions for the 1 01j= 0.5→202j′= 1.5 and j′= 2.5 transitions have similar magnitudes. V. DISCUSSION AND CONCLUSION In this paper, the calculation of the PES for the interaction of the HCO( ˜X2A′) radical with H 2is described, and the PES is pre- sented. The calculations employed an explicitly correlated coupled- cluster theory [RCCSD(T)-F12a] and assumed fixed geometries for the collision partners. The global minimum on the PES has a pla- nar geometry, and the dissociation energy Deof the HCO–H 2col- lision complex was found to be 206.9 cm−1, with an equilibrium intermolecular separation Re= 5.865 a0. Time-independent quantum scattering calculations were car- ried out with this PES in order to compute integral cross sections for transitions between HCO rotational levels induced by colli- sions by both nuclear spin modifications of the hydrogen molecule, para -H2(j2= 0) and ortho -H2(j2= 1), as a function of collision energy. As has been seen for many molecule–H 2collision sys- tems, the cross sections for a given transition in HCO were found to be much larger for collisions with ortho -H2(j2= 1) than with para -H2(j2= 0). The HCO( ˜X2A′) radical has a nonzero electron spin ( s= 1/2). Hence, each rotational level, with rotational angular momentum n, is split into two fine-structure levels with total angular momen- tum j=n+ 1/2 and j=n−1/2 (except for n= 0). Cross sec- tions between HCO fine-structure levels were computed from the electron-spin-free scattering calculations by a recoupling method that has been applied to hyperfine-resolved transitions in other molecules. These fine-structure-resolved cross sections display a propensity for Δj=Δntransitions, in analogy to the same propen- sity rule seen for collisions of linear molecules in2Σ+electronic states. A main motivation for this work is eventually to compute rate coefficients for transitions in HCO induced by collisions of the hydrogen molecule in order to provide collisional data for radia- tive transfer calculations on spectroscopic observations of HCO in the ISM. Since HCO rotational/fine-structure lines display a resolv- able hyperfine structure due to the spin of the hydrogen nucleus, it will be necessary to compute cross sections for hyperfine-resolved transitions in HCO from the scattering calculations presented in this work. SUPPLEMENTARY MATERIAL See the supplementary material for PES.tar , which contains a Fortran program for computing the HCO–H 2interaction energy for the entered values of R,θ,ϕ,θ′, andϕ′.ACKNOWLEDGMENTS The quantum chemistry calculations were performed on the Maryland Advanced Research Computing Cluster, which was funded by the State of Maryland and is jointly managed by Johns Hopkins University and the University of Maryland College Park. DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material. REFERENCES 1L. E. Snyder, J. M. Hollis, and B. L. Ulich, Astrophys. J. 208, L91 (1976). 2P. Schilke, G. Pineau des Forêts, C. M. Walmsley, and J. Martín-Pintado, Astron. Astrophys. 372, 291 (2001). 3S. García-Burillo, J. Martín-Pintado, A. Fuente, A. Usero, and R. Neri, Astrophys. J. 575, L55 (2002). 4M. Gerin, J. R. Goicoechea, J. Pety, and P. Hily-Blant, Astron. Astrophys. 494, 977 (2009). 5M. S. Schenewerk, L. E. Snyder, J. M. Hollis, P. R. Jewell, and L. M. Ziurys, Astrophys. J. 328, 785 (1988). 6A. Bacmann and A. Faure, Astron. Astrophys. 587, A130 (2016). 7V. M. Rivilla, M. T. Beltrán, A. Vasyunin, P. Caselli, S. Viti, F. Fontani, and R. Cesaroni, Mon. Not. R. Soc. Astron. 483, 806 (2019). 8E. Herbst and E. F. van Dishoeck, Annu. Rev. Astron. Astrophys. 47, 427 (2009). 9N. Balucani, C. Ceccarelli, and V. Taquet, Mon. Not. R. Soc. Astron. 449, L16 (2015). 10A. I. Vasyunin, P. Caselli, F. Dulieu, and I. Jiménez-Serra, Astrophys. J. 842, 33 (2017). 11R. T. Garrod and E. Herbst, Astron. Astrophys. 457, 927 (2006). 12R. T. Garrod, S. W. Weaver, and E. Herbst, Astrophys. J. 682, 283 (2006). 13N. Watanabe and A. 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Valiron, M. Wernli, A. Faure, L. Wiesenfeld, C. Rist, S. Kedžuch, and J. Noga, J. Chem. Phys. 129, 134306 (2008). 25P. J. Dagdigian, J. Chem. Phys. 152, 074307 (2020). 26T. B. Adler, G. Knizia, and H.-J. Werner, J. Chem. Phys. 127, 221106 (2007). 27G. Knizia, T. B. Adler, and H.-J. Werner, J. Chem. Phys. 130, 054104 (2009). 28H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, M. Schütz et al. , Molpro, version 2012.1, a package of ab initio programs, 2012, see http://www.molpro.net. 29T. H. Dunning, Jr., J. Chem. Phys. 90, 1007 (1989). 30D. E. Woon and T. H. Dunning, J. Chem. Phys. 98, 1358 (1993). J. Chem. Phys. 152, 224304 (2020); doi: 10.1063/5.0012033 152, 224304-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 31F. Weigend, Phys. Chem. Chem. Phys. 4, 4285 (2002). 32F. Weigend, A. Köhn, and C. Hättig, J. Chem. Phys. 116, 3175 (2002). 33F. B. van Duijneveldt, J. G. C. M. van Duijneveldt-van de Rijdt, and J. H. van Lenthe, Chem. Rev. 94, 1873 (1994). 34C. Rist and A. Faure, J. Math. Chem. 50, 588 (2012). 35C. Rist, M. H. Alexander, and P. Valiron, J. Chem. Phys. 98, 4662 (1993). 36S. Green, J. Chem. Phys. 103, 1035 (1995). 37M. Wernli, L. Wiesenfeld, A. Faure, and P. Valiron, Astron. Astrophys. 464, 1147 (2007). 38P. E. S. Wormer, J. A. Kłos, G. C. Groenenboom, and A. van der Avoird, J. Chem. Phys. 122, 244325 (2005). 39HIBRIDON is a package of programs for the time-independent quantum treat- ment of inelastic collisions and photodissociation written by M. H. Alexander, D. E. Manolopoulos, H.-J. Werner, B. Follmeg, P. J. Dagdigian, and others. More information and/or a copy of the code can be obtained from the website http://www2.chem.umd.edu/groups/alexander/hibridon.40K. P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules (Van Nostrand Reinhold, New York, 1979). 41M. H. Alexander and P. J. Dagdigian, J. Chem. 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5.0002892.pdf

J. Chem. Phys. 152, 174104 (2020); https://doi.org/10.1063/5.0002892 152, 174104 © 2020 Author(s).A basis-set error correction based on density-functional theory for strongly correlated molecular systems Cite as: J. Chem. Phys. 152, 174104 (2020); https://doi.org/10.1063/5.0002892 Submitted: 28 January 2020 . Accepted: 12 April 2020 . Published Online: 04 May 2020 Emmanuel Giner , Anthony Scemama , Pierre-François Loos , and Julien Toulouse COLLECTIONS This paper was selected as an Editor’s Pick The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp A basis-set error correction based on density-functional theory for strongly correlated molecular systems Cite as: J. Chem. Phys. 152, 174104 (2020); doi: 10.1063/5.0002892 Submitted: 28 January 2020 •Accepted: 12 April 2020 • Published Online: 4 May 2020 Emmanuel Giner,1,a) Anthony Scemama,2 Pierre-François Loos,2,b) and Julien Toulouse1,3,c) AFFILIATIONS 1Laboratoire de Chimie Théorique (UMR 7616), Sorbonne Université, CNRS, Paris, France 2Laboratoire de Chimie et Physique Quantiques (UMR 5626), Université de Toulouse, CNRS, UPS, Toulouse, France 3Institut Universitaire de France, Paris, France a)Author to whom correspondence should be addressed: emmanuel.giner@lct.jussieu.fr b)Electronic mail: loos@irsamc.ups-tlse.fr c)Electronic mail: toulouse@lct.jussieu.fr ABSTRACT We extend to strongly correlated molecular systems the recently introduced basis-set incompleteness correction based on density-functional theory (DFT) [E. Giner et al. , J. Chem. Phys. 149, 194301 (2018)]. This basis-set correction relies on a mapping between wave-function calculations in a finite basis set and range-separated DFT (RSDFT) through the definition of an effective non-divergent interaction corre- sponding to the electron–electron Coulomb interaction projected in the finite basis set. This enables the use of RSDFT-type complementary density functionals to recover the dominant part of the short-range correlation effects missing in this finite basis set. To study both weak and strong correlation regimes, we consider the potential energy curves of the H 10, N 2, O 2, and F 2molecules up to the dissociation limit, and we explore various approximations of complementary functionals fulfilling two key properties: spin-multiplet degeneracy (i.e., independence of the energy with respect to the spin projection Sz) and size consistency. Specifically, we investigate the dependence of the functional on differ- ent types of on-top pair densities and spin polarizations. The key result of this study is that the explicit dependence on the on-top pair density allows one to completely remove the dependence on any form of spin polarization without any significant loss of accuracy. Quantitatively, we show that the basis-set correction reaches chemical accuracy on atomization energies with triple- ζquality basis sets for most of the systems studied here. In addition, the present basis-set incompleteness correction provides smooth potential energy curves along the whole range of internuclear distances. Published under license by AIP Publishing. https://doi.org/10.1063/5.0002892 .,s I. INTRODUCTION The general goal of quantum chemistry is to provide reli- able theoretical tools to explore the rich area of chemistry. More specifically, developments in quantum chemistry primarily aim at accurately computing the electronic structure of molecular sys- tems. Despite intense developments, no definitive solution to this problem has been found. The theoretical challenge to tackle belongs to the quantum many-body problem due to the intrin- sic quantum nature of the electrons and the Coulomb repul- sion between them. This so-called electronic correlation problem corresponds to finding a solution to the Schrödinger equationfor an N-electron system, and two main roads have emerged to approximate this solution: wave-function theory (WFT)1and density-functional theory (DFT).2Although both WFT and DFT spring from the same Schrödinger equation, they rely on very dif- ferent formalisms as the former deals with the complicated N- electron wave function, whereas the latter focuses on the much simpler one-electron density. In its Kohn–Sham (KS) formula- tion,3the computational cost of DFT is very appealing since it is a simple mean-field procedure. Therefore, although contin- ued efforts have been made to reduce the computational cost of WFT, DFT still remains the workhorse of quantum computational chemistry. J. Chem. Phys. 152, 174104 (2020); doi: 10.1063/5.0002892 152, 174104-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The difficulty in obtaining a reliable theoretical description of a given chemical system can be roughly categorized by the strength of the electronic correlation. The so-called weakly correlated sys- tems, such as closed-shell organic molecules near their equilibrium geometry, are typically dominated by correlation effects that do not affect the qualitative mean-field picture of the system. These weak-correlation effects can be either short range (near the electron– electron coalescence points)4or long range (London dispersion interactions).5The theoretical description of weakly correlated sys- tems is one of the most concrete achievements of quantum chem- istry, and the main remaining challenge for these systems is to push the limit of the chemical system size that can be treated. The case of the so-called strongly correlated systems, which are ubiquitous in chemistry, is more problematic as they exhibit a much more complex electronic structure. For example, transition metal complexes, low- spin open-shell systems, and covalent bond breaking situations have all in common that they cannot be even qualitatively described by a single electronic configuration. It is now clear that the usual semilo- cal density-functional approximations of KS DFT fail to accurately describe these situations,6,7and WFT is king for the treatment of strongly correlated systems. In practice, WFT uses a finite one-electron basis set. The exact solution of the Schrödinger equation within this basis set is then provided by full configuration interaction (FCI) that consists in a linear-algebra eigenvalue problem with a dimension scaling expo- nentially with the system size. Due to this exponential growth of the FCI computational cost, introducing approximations is neces- sary, with at least two difficulties for strongly correlated systems: (i) the qualitative description of the wave function is determined by a primary set of electronic configurations (whose size can scale exponentially in many cases) among which near degeneracies and/or strong interactions appear in the Hamiltonian matrix and (ii) the quantitative description of the system requires also to account for weak correlation effects, which involve many other electronic con- figurations with typically much smaller weights in the wave function. Simultaneously addressing these two issues is a rather complicated task for a given approximate WFT method, especially, if one adds the requirement of satisfying formal properties, such as spin-multiplet degeneracy (i.e., independence of the energy with respect to the spin projection Sz) and size consistency. Beside the difficulties in accurately describing the molecular electronic structure within a given basis set, a crucial limitation of WFT methods is the slow convergence of the energy (and related properties) with respect to the size of the one-electron basis set. As initially shown by the seminal work of Hylleraas8and further developed by Kutzelnigg and co-workers,9–11the main convergence problem originates from the divergence of the electron–electron Coulomb interaction at the coalescence point, which induces a dis- continuity in the first derivative of the exact wave function (the so-called electron–electron cusp). Describing such a discontinu- ity with an incomplete one-electron basis set is impossible, and as a consequence, the convergence of the computed energies and properties is strongly affected. To alleviate this problem, extrapo- lation techniques have been developed either based on a partial- wave expansion analysis12,13or more recently based on perturbative arguments.14,15A more rigorous approach to tackle the basis- set convergence problem is provided by the so-called explicitly correlated F12 (or R12) methods4,16–20that introduce a geminalfunction depending explicitly on the interelectronic distance. This ensures a correct representation of the Coulomb correlation hole around the electron–electron coalescence point and leads to a much faster convergence of the energy than usual WFT methods. For instance, using the explicitly correlated version of the coupled clus- ter with singles, doubles, and perturbative triples [CCSD(T)] in a triple-ζbasis set is equivalent to using a quintuple- ζbasis set with the usual CCSD(T) method,21although a computational over- head is introduced by the auxiliary basis set needed to compute the three-electron integrals involved in F12 theory.22In addition to the computational cost, a possible drawback of F12 theory is its rather complex formalism, which requires non-trivial devel- opments for adapting it to a new method. For strongly corre- lated systems, several multi-reference methods have been extended to explicit correlation (see, for example, Refs. 23–27), including approaches based on the so-called universal F12 theory, which are potentially applicable to any electronic-structure computational methods.28–31 An alternative way to improve the convergence toward the complete-basis-set (CBS) limit is to treat the short-range correlation effects within DFT and to use WFT methods to deal only with the long-range and/or strong correlation effects. A rigorous approach achieving this mixing of DFT and WFT is range-separated DFT (RSDFT) (see Ref. 32 and the references therein), which relies on a decomposition of the electron–electron Coulomb interaction in terms of the interelectronic distance, thanks to a range-separation parameter μ. The advantage of this approach is at least twofold: (i) the DFT part deals primarily with the short-range part of the Coulomb interaction, and consequently, the usual semilocal density- functional approximations are more accurate than for standard KS DFT and (ii) the WFT part deals only with a smooth non- divergent interaction, and consequently, the wave function has no electron–electron cusp33and the basis-set convergence is much faster.34A number of approximate RSDFT schemes have been developed involving single-reference35–42and multi-reference43–48 WFT methods. Nevertheless, there are still some open issues in RSDFT, such as remaining fractional-charge and fractional-spin errors in the short-range density functionals49or the dependence of the quality of the results on the value of the range-separation parameterμ. Building on the development of RSDFT, a possible solution to the basis-set convergence problem has been recently proposed by some of the present authors,50in which RSDFT functionals are used to recover only the correlation effects outside a given basis set. The key point here is to realize that a wave function developed in an incomplete basis set is cuspless and could also originate from a Hamiltonian with a non-divergent long-range electron–electron interaction. Therefore, a mapping with RSDFT can be performed through the introduction of an effective non-divergent interaction representing the usual electron–electron Coulomb interaction pro- jected in an incomplete basis set. First, applications to weakly corre- lated molecular systems have been successfully carried out51together with extensions of this approach to the calculations of excitation energies52and ionization potentials.53The goal of the present work is to further develop this approach for the description of strongly correlated systems. This paper is organized as follows: In Sec. II, we recall the math- ematical framework of the basis-set correction and we present its J. Chem. Phys. 152, 174104 (2020); doi: 10.1063/5.0002892 152, 174104-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp extension for strongly correlated systems. In particular, our focus is primarily set on imposing two key formal properties that are highly desirable in the context of strong correlation: spin-multiplet degen- eracy and size consistency. To do this, we introduce (i) new func- tionals using different flavors of spin polarizations and on-top pair densities and (ii) an effective electron–electron interaction based on a multiconfigurational wave function. This generalizes the method used in previous works on weakly correlated systems50,51for which it was sufficient to use an effective interaction based on a single- determinant wave function and a functional depending only on the usual density, reduced density gradient, and spin polarization. Then, in Sec. III, we apply the method to the calculation of the poten- tial energy curves of the H 10, N 2, O 2, and F 2molecules up to the dissociation limit. Finally, we conclude in Sec. IV. II. THEORY As the theory behind the present basis-set correction has been exposed in detail in Ref. 50, we only briefly recall the main equa- tions and concepts needed for this study in Secs. II A–II C. More specifically, in Sec. II A, we recall the basic mathematical framework of the present theory by introducing the complementary functional to a basis set. Section II B introduces the effective non-divergent interaction in the basis set, which leads us to the definition of the effective local range-separation function in Sec. II C. Then, Sec. II D exposes the new approximate RSDFT-based complementary corre- lation functionals. The generic form of such functionals is exposed in Sec. II D 1, their properties in the context of the basis-set correction are discussed in Sec. II D 2, and the specific requirements for strong correlation are discussed in Sec. II E. Finally, the actual functionals used in this work are introduced in Sec. II F. A. Basic theory The exact ground-state energy E0of an N-electron system can, in principle, be obtained in DFT by a minimization over N-representable one-electron densities n(r), E0=min n{F[n]+∫drvne(r)n(r)}, (1) where vne(r) is the nuclei–electron potential and F[n] is the univer- sal Levy–Lieb density functional written with the constrained search formalism as54,55 F[n]=min Ψ→n⟨Ψ∣ˆT+ˆWee∣Ψ⟩, (2) where ˆTand ˆWeeare the kinetic and electron–electron Coulomb operators, and the notation Ψ→nmeans that the wave function Ψyields the density n. The minimizing density n0in Eq. (1) is the exact ground-state density. Nevertheless, in practical calculations, the accessible densities are necessarily restricted to the set of densi- ties “representable in a basis set B,” i.e., densities coming from wave functions expandable in the N-electron Hilbert space generated by the one-electron basis set B. In the following, we always consider only such representable-in- Bdensities. With this restriction, Eq. (1) then gives an upper bound EB 0of the exact ground-state energy. Since the density has a faster convergence with the size of the basis setthan the wave function, this restriction is a rather weak one, and we can consider that EB 0is an acceptable approximation to the exact ground-state energy, i.e., EB 0≈E0. In the present context, it is important to note that the wave functions Ψdefined in Eq. (2) are not restricted to a finite basis set, i.e., they should be expanded in a complete basis set. In Ref. 50, it was then proposed to decompose F[n] as, for a representable-in- B density n, F[n]=min ΨB→n⟨ΨB∣ˆT+ˆWee∣ΨB⟩+¯EB[n], (3) where ΨBare wave functions expandable in the N-electron Hilbert space generated by Band ¯EB[n]=min Ψ→n⟨Ψ∣ˆT+ˆWee∣Ψ⟩−min ΨB→n⟨ΨB∣ˆT+ˆWee∣ΨB⟩ (4) is the complementary density functional to the basis set B. Introduc- ing the decomposition in Eq. (3) back into Eq. (1) yields EB 0=min ΨB{⟨ΨB∣ˆT+ˆWee∣ΨB⟩+¯EB[nΨB]+∫drvne(r)nΨB(r)}, (5) where the minimization is only over wave functions ΨBrestricted to the basis set Band nΨB(r)refers to the density generated from ΨB. Therefore, thanks to Eq. (5), one can properly combine a WFT calculation in a finite basis set with a density functional (here- after referred to as complementary functional) accounting for the correlation effects that are not included in the basis set. As a simple non-self-consistent version of this approach, we can approximate the minimizing wave function ΨB 0in Eq. (5) by the ground-state FCI wave function ΨB FCIwithin B, and we then obtain the following approximation for the exact ground-state energy [see Eqs. (12)–(15) of Ref. 50]: E0≈EB 0≈EB FCI+¯EB[nB FCI], (6) where EB FCIand nB FCIare the ground-state FCI energy and density, respectively. As was originally shown in Ref. 50 and further empha- sized in Refs. 51 and 52, the main role of ¯EB[nB FCI]is to correct for the basis-set incompleteness error, a large part of which orig- inating from the lack of electron–electron cusp in the wave func- tion expanded in an incomplete basis set. The whole purpose of this work is to determine approximations for ¯EB[nB FCI], which are suitable for strongly correlated molecular systems. Two key require- ments for this purpose are (i) spin-multiplet degeneracy and (ii) size consistency. B. Effective interaction in a finite basis As originally shown by Kato,56the electron–electron cusp of the exact wave function originates from the divergence of the Coulomb interaction at the coalescence point. Therefore, a cuspless wave func- tionΨBcould also be obtained from a Hamiltonian with a non- divergent electron–electron interaction. In other words, the impact of the basis set incompleteness can be understood as the removal of the divergence of the usual electron–electron Coulomb interaction. As originally derived in Ref. 50 (see Sec. II D and Appendixes A and B), one can obtain an effective non-divergent electron– electron interaction, here referred to as WΨB(r1,r2), which repro- duces the expectation value of the electron–electron Coulomb inter- action operator over a given wave function ΨB. As we are interested J. Chem. Phys. 152, 174104 (2020); doi: 10.1063/5.0002892 152, 174104-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp in the behavior at the coalescence point, we focus on the opposite- spin part of the electron–electron interaction. More specifically, the effective electron–electron interaction associated with a given wave function ΨBis defined as WΨB(r1,r2)=⎧⎪⎪⎨⎪⎪⎩fΨB(r1,r2)/n2,ΨB(r1,r2)ifn2,ΨB(r1,r2)≠0 ∞, otherwise,(7) where n2,ΨB(r1,r2)=∑ pqrs∈Bϕp(r1)ϕq(r2)Γrs pqϕr(r1)ϕs(r2) (8) is the opposite-spin pair density associated with ΨB, and Γrs pq=2⟨ΨB∣ˆa† r↓ˆa† s↑ˆaq↑ˆap↓∣ΨB⟩is its associated tensor in a basis of spatial orthonormal orbitals { ϕp(r)}, fΨB(r1,r2)=∑ pqrstu ∈Bϕp(r1)ϕq(r2)Vrs pqΓtu rsϕt(r1)ϕu(r2) (9) and Vrs pq=⟨pq∣rs⟩are the usual two-electron Coulomb integrals. With such a definition, one can show that WΨB(r1,r2)satisfies 1 2∬ dr1dr2WΨB(r1,r2)n2,ΨB(r1,r2)=1 2∬ dr1dr2n2,ΨB(r1,r2) ∣r1−r2∣. (10) As shown in Ref. 50, the effective interaction WΨB(r1,r2)is necessar- ily finite at coalescence for an incomplete basis set and tends to the usual Coulomb interaction in the CBS limit for any choice of wave function ΨB, i.e., lim B→CBSWΨB(r1,r2)=1 ∣r1−r2∣,∀ΨB. (11) The condition in Eq. (11) is fundamental as it guarantees the correct behavior of the theory in the CBS limit. C. Local range-separation function 1. General definition The effective interaction within a finite basis, WΨB(r1,r2), is bounded and resembles the long-range interaction used in RSDFT, wlr ee(μ;r12)=erf(μr12) r12, (12) whereμis the range-separation parameter. As originally proposed in Ref. 50, we make the correspondence between these two interactions by using the local range-separation function μΨB(r)=√π 2WΨB(r,r) (13) such that the two interactions coincide at the electron–electron coalescence point for each r, wlr ee(μΨB(r); 0)=WΨB(r,r),∀r. (14) Because of the very definition of WΨB(r1,r2), one has the following property in the CBS limit [see Eq. (11)]: lim B→CBSμΨB(r)=∞,∀ΨB, (15) which is again fundamental to guarantee the correct behavior of the theory in the CBS limit.2. Frozen-core approximation As all WFT calculations in this work are performed within the frozen-core approximation, we use a “valence-only” (or no- core) version of the various quantities needed for the complemen- tary functional introduced in Ref. 51. We partition the basis set as B=C⋃V, where CandVare the sets of core and “valence” (i.e., non-core) orbitals, respectively, and define the valence-only local range-separation function as μval ΨB(r)=√π 2Wval ΨB(r,r), (16) where Wval ΨB(r1,r2)=⎧⎪⎪⎨⎪⎪⎩fval ΨB(r1,r2)/nval 2,ΨB(r1,r2)ifnval 2,ΨB(r1,r2)≠0 ∞ otherwise(17) is the valence-only effective interaction and fval ΨB(r1,r2)=∑ pq∈B∑ rstu∈Vϕp(r1)ϕq(r2)Vrs pqΓtu rsϕt(r1)ϕu(r2), (18) nval 2,ΨB(r1,r2)=∑ pqrs∈Vϕp(r1)ϕq(r2)Γrs pqϕr(r1)ϕs(r2). (19) One would note the restrictions of the sums to the set Vin Eqs. (18) and (19). It is also noteworthy that, with the present definition, Wval ΨB(r1,r2)still tends to the usual Coulomb interaction as B→CBS. For simplicity, we will drop the indication “val” in the notation for the rest of the paper. D. General form of the complementary functional 1. Generic approximate form As originally proposed and motivated in Ref. 50, we approx- imate the complementary functional ¯EB[n]by using the so- called correlation energy functional with multideterminant refer- ence (ECMD) introduced by Toulouse et al.57,58Following the recent work in Ref. 51, we propose to consider a Perdew–Burke–Ernzerhof (PBE)-like functional that uses the one-electron density n(r), the spin polarization ζ(r)=[n↑(r)−n↓(r)]/n(r)[where n↑(r) and n↓(r) are the spin-up and spin-down densities], the reduced density gradi- ents(r)=∇n(r)/n(r)4/3, and the on-top pair density n2(r)≡n2(r, r). In the present work, all these quantities are computed with the same wave function ΨBused to define μ(r)≡μΨB(r). Therefore, ¯EB[n]has the following generic form: ¯EB[n,ζ,n2,μ]=∫drn(r)¯εsr,PBE c,md(n(r),ζ(r),s(r),n2(r),μ(r)), (20) where ¯εsr,PBE c,md(n,ζ,s,n2,μ)=εPBE c(n,ζ,s) 1 +β(n,ζ,s,n2)μ3(21) is the correlation energy per particle with β(n,ζ,s,n2)=3 2√π(1−√ 2)εPBE c(n,ζ,s) n2/n, (22) J. Chem. Phys. 152, 174104 (2020); doi: 10.1063/5.0002892 152, 174104-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp whereεPBE c(n,ζ,s)is the usual PBE correlation energy per particle.59 Before introducing the different flavors of approximate functionals that we will use here (see Sec. II F), we would like to give some motivations for this choice of functional form. The form of ¯εsr,PBE c,md(n,ζ,s,n2,μ)in Eq. (21) has been originally proposed in Ref. 48 in the context of RSDFT. In the μ→0 limit, it reduces to the usual PBE correlation functional, i.e., lim μ→0¯εsr,PBE c,md(n,ζ,s,n2,μ)=εPBE c(n,ζ,s), (23) which is relevant in the weak-correlation (or high-density) limit. In the large-μlimit, it behaves as ¯εsr,PBE c,md(n,ζ,s,n2,μ)∼ μ→∞2√π(1−√ 2) 3μ3n2 n, (24) which is the exact large- μbehavior of the exact ECMD correlation energy.48,60Of course, for a specific system, the large- μbehavior will be exact only if one uses for n2theexact on-top pair den- sity of this system. This large- μlimit in Eq. (24) is relevant in the strong-correlation (or low-density) limit. In the context of RSDFT, some of the present authors have illustrated in Ref. 48 that the on- top pair density involved in Eq. (21) plays, indeed, a crucial role when reaching the strong-correlation regime. The importance of the on-top pair density in the strong-correlation regime has been also recently acknowledged by Gagliardi and co-workers61and Pernal and co-workers.62 Note also that ¯εsr,PBE c,md(n,ζ,s,n2,μ)vanishes when n2vanishes, i.e., lim n2→0¯εsr,PBE c,md(n,ζ,s,n2,μ)=0, (25) which is expected for systems with a vanishing on-top pair density. Finally, the function ¯εsr,PBE c,md(n,ζ,s,n2,μ)vanishes when μ→∞like all RSDFT short-range functionals, i.e., lim μ→∞¯εsr,PBE c,md(n,ζ,s,n2,μ)=0. (26) 2. Two limits where the complementary functional vanishes Within the definitions of Eqs. (13) and (20), any approximate complementary functional ¯EB[n,ζ,n2,μ]satisfies two important properties. First, thanks to the properties in Eqs. (15) and (26), ¯EB[n,ζ,n2,μ]vanishes in the CBS limit, independently of the type of wave function ΨBused to define the local range-separation function μ(r) in a given basis set B, lim B→CBS¯EB[n,ζ,n2,μ]=0,∀ΨB. (27) Second, ¯EB[n,ζ,n2,μ]correctly vanishes for systems with uni- formly vanishing on-top pair density, such as one-electron systems, and for the stretched H 2molecule, lim n2→0¯EB[n,ζ,n2,μ]=0. (28) This property is doubly guaranteed by (i) the choice of setting WΨB(r1,r2)=∞for a vanishing pair density [see Eq. (7)], which leads toμ(r)→∞and thus a vanishing ¯εsr,PBE c,md(n,ζ,s,n2,μ)[seeEq. (26)], and (ii) the fact that ¯εsr,PBE c,md(n,ζ,s,n2,μ)vanishes anyway when the on-top pair density vanishes [see Eq. (25)]. E. Requirements on the complementary functional for strong correlation An important requirement for any electronic-structure method is size consistency, i.e., the additivity of the energies of non- interacting fragments, which is mandatory to avoid any ambiguity in computing interaction energies. When two subsystems A and B dissociate in closed-shell systems, as in the case of weak intermolec- ular interactions, for instance, spin-restricted Hartree–Fock (RHF) is size-consistent. When the two subsystems dissociate in open-shell systems, such as in covalent bond breaking, it is well known that the RHF approach fails and an alternative is to use a complete-active- space self-consistent-field (CASSCF) wave function that, provided that the active space has been properly chosen, leads to additive energies. Another important requirement is spin-multiplet degeneracy, i.e., the independence of the energy with respect to the Szcompo- nent of a given spin state, which is also a property of any exact wave function. Such a property is also important in the context of covalent bond breaking, where the ground state of the supersystem A + B is generally of lower spin than the corresponding ground states of the fragments (A and B) that can have multiple Szcomponents. 1. Spin-multiplet degeneracy A sufficient condition to achieve spin-multiplet degeneracy is to eliminate all dependencies on Sz. In the case of the function ¯εsr,PBE c,md(n,ζ,s,n2,μ), this means removing the dependence on the spin polarization ζ(r) originating from the PBE correlation functional εPBE c(n,ζ,s)[see Eq. (21)]. To do so, it has been proposed to replace the dependence on the spin polarization by the dependence on the on-top pair density. Most often, it is done by introducing an effective spin polarization7,63–75(see, also, Refs. 76 and 77) ˜ζ(n,n2)=√ 1−2n2/n2 (29) expressed as a function of the density nand the on-top pair den- sityn2calculated from a given wave function. The advantage of this approach is that this effective spin polarization˜ζis indepen- dent from Szsince the on-top pair density is Sz-independent. Nev- ertheless, the use of˜ζin Eq. (29) presents some disadvantages since this expression was derived for a single-determinant wave function. Hence, it does not appear justified to use it for a mul- tideterminant wave function. More particularly, it may happen in the multideterminant case that 1–2 n2/n2<0, which results in a complex-valued effective spin polarization.64Therefore, following other authors,67,72,73we use the following definition: ˜ζ(n,n2)=⎧⎪⎪⎨⎪⎪⎩√ 1−2n2/n2ifn2≥2n2 0 otherwise.(30) An alternative way to eliminate the Szdependence is to simply setζ= 0, i.e., to resort to the spin-unpolarized functional. This low- ers the accuracy for open-shell systems at μ= 0, i.e., for the usual PBE correlation functional εPBE c(n,ζ,s). Nevertheless, we argue that J. Chem. Phys. 152, 174104 (2020); doi: 10.1063/5.0002892 152, 174104-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp for sufficiently large μ, it is a viable option. Indeed, the purpose of introducing the spin polarization in semilocal density-functional approximations is to mimic the exact on-top pair density,76but our functional ¯εsr,PBE c,md(n,ζ,s,n2,μ)already explicitly depends on the on- top pair density [see Eqs. (21) and (22)]. The dependencies on ζ andn2can thus be expected to be largely redundant. Consequently, we propose here to test the use of ¯εsr,PBE c,mdwith a zero spin polariza- tion. This ensures its Szindependence and, as will be numerically demonstrated, very weakly affects the complementary functional accuracy. 2. Size consistency Since ¯EB[n,ζ,n2,μ]is computed via a single integral over R3 [see Eq. (20)] that involves only local quantities [ n(r),ζ(r),s(r),n2(r), andμ(r)], in the case of non-overlapping fragments A + B, it can be written as the sum of two local contributions: one coming from the integration over the region of subsystem A and the other from the region of subsystem B. Therefore, a sufficient condition for size consistency is that these quantities locally coincide in the isolated fragments and in the supersystem A + B. Since these local quanti- ties are calculated from the wave function ΨB, a sufficient condition is that the wave function is multiplicatively separable in the limit of non-interacting fragments, i.e., ∣ΨB A+B⟩=∣ΨB A⟩⊗∣ΨB B⟩. We refer the interested reader to Appendix A for a detailed proof and discus- sion of the latter statement. In the case where the two subsystems A and B dissociate in closed-shell systems, a simple RHF wave func- tion ensures this property, but when one or several covalent bonds are broken, a properly chosen CASSCF wave function can be used to recover this property. The underlying active space must however be chosen in such a way that it leads to size-consistent energies in the limit of dissociated fragments. F. Actual approximations used for the complementary functional As the present work focuses on the strong-correlation regime, we propose here to investigate only approximate functionals that areSzindependent and size-consistent in the case of covalent bond breaking. Therefore, the wave functions ΨBused throughout this paper are CASSCF wave functions in order to ensure size consis- tency of all local quantities. The difference between the different flavors of functionals is only due to the types of spin polarization and on-top pair density used. Regarding the spin polarization that enters into the function εPBE c(n,ζ,s), two different types of Sz-independent formulations are considered: (i) the effective spin polarization˜ζdefined in Eq. (30) and calculated from the CASSCF wave function and (ii) a zero spin polarization. In the latter case, the functional is referred to as “SU,” which stands for “spin unpolarized.” Regarding the on-top pair density entering in Eq. (22), we use two different approximations. The first one is based on the uniform electron gas (UEG) and reads nUEG 2(n,ζ)≈n2(1−ζ2)g0(n), (31) where the pair-distribution function g0(n) is taken from Eq. (46) of Ref. 33. As the spin polarization appears in Eq. (31), we use theeffective spin polarization˜ζof Eq. (30) in order to ensure Szinde- pendence. Thus, nUEG 2will depend indirectly on the on-top pair den- sity of the CASSCF wave function through˜ζ. When using nUEG 2(r) ≡nUEG 2(n(r),˜ζ(r))in a functional, we will refer to it as “UEG.” The second approach to approximate the exact on-top pair density consists in using directly the on-top pair density of the CASSCF wave function. Following the work of some of the present authors,48,52we introduce the extrapolated on-top pair density ˚n2(n2,μ)=(1 +2√πμ)−1 n2, (32) which directly follows from the large- μextrapolation of the exact on-top pair density derived by Gori-Giorgi and Savin33in the context of RSDFT. Thus, the extrapolated on-top pair density ˚n2 depends on the local range-separation function μ. When using ˚n2(r)≡˚n2(n2(r),μ(r))in a functional, we will simply refer it to as “OT,” which stands for “on-top.” We then define three complementary functionals: (i) PBE-UEG that combines the effective spin polarization of Eq. (30) and the UEG on-top pair density defined in Eq. (31), ¯EB PBE-UEG=∫drn(r)¯εsr,PBE c,md(n(r),˜ζ(r),s(r),nUEG 2(r),μ(r)), (33) (ii) PBE-OT that combines the effective spin polarization of Eq. (30) and the on-top pair density of Eq. (32), ¯EB PBE-OT=∫drn(r)¯εsr,PBE c,md(n(r),˜ζ(r),s(r),˚n2(r),μ(r)), (34) (iii) SU-PBE-OT that combines a zero spin polarization and the on-top pair density of Eq. (32), ¯EB SU-PBE-OT=∫drn(r)¯εsr,PBE c,md(n(r), 0,s(r),˚n2(r),μ(r)). (35) The performance of each of these functionals is tested in the fol- lowing. Note that we did not define a spin-unpolarized version of the PBE-UEG functional because it would have been significantly infe- rior (in terms of performance) compared to the three other function- als. Indeed, because of the lack of knowledge on the spin polarization or on the accurate on-top pair density, such a functional would be inaccurate, in particular, for open-shell systems. This assumption has been numerically confirmed by calculations. III. RESULTS A. Computational details We present potential energy curves of small molecules up to the dissociation limit to investigate the performance of the basis-set correction in regimes of both weak and strong correlations. The con- sidered systems are the H 10linear chain with equally spaced atoms and the N 2, O 2, and F 2diatomics. The computation of the ground-state energy in Eq. (6) in a given basis set requires approximations to the FCI energy EB FCIand J. Chem. Phys. 152, 174104 (2020); doi: 10.1063/5.0002892 152, 174104-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp to the basis-set correction ¯EB[nB FCI]. For diatomics with the aug-cc- pVDZ and aug-cc-pVTZ basis sets,78energies are obtained using frozen-core selected-CI calculations (using the CIPSI algorithm), followed by the extrapolation scheme proposed by Holmes et al. (see Refs. 79–84 for more details). All these calculations are per- formed with the latest version of Q UANTUM PACKAGE84and will be labeled exFCI in the following. In the case of F 2, we also use the correlation energy extrapolated by intrinsic scaling (CEEIS)85as an estimate of the FCI correlation energy with the cc-pVXZ (X = D, T, and Q) basis sets.86The estimated exact potential energy curves are obtained from experimental data87for the N 2and O 2molecules and from CEEIS calculations in the case of F 2. For all geometries and basis sets, the error with respect to the exact FCI energies is esti- mated to be of the order of 0.5 mHa. For the three diatomics, we performed an additional exFCI calculation with the aug-cc-pVQZ basis set at the equilibrium geometry to obtain reliable estimates of the FCI/CBS dissociation energy. In the case of the H 10chain, the approximation to the FCI energies together with the estimated exact potential energy curves is obtained from the data of Ref. 88, where the authors performed MRCI+Q calculations with a minimal valence active space as reference (see below for the description of the active space). We note that even though we use near-FCI energies in this work, the DFT-based basis-set correction could also be applied to any approximation to FCI such as multireference perturba- tion theory, similar to what was done for weakly correlated sys- tems for which the basis-set correction was applied to CCSD(T) calculations.51 Regarding the complementary functional, we first perform full- valence CASSCF calculations with the GAMESS-US software89to obtain the wave function ΨB. Then, all density-related quantities involved in the functional [density n(r), effective spin polarization ˜ζ(r), reduced density gradient s(r), and on-top pair density n2(r)] together with the local range-separation function μ(r) are calculated with this full-valence CASSCF wave function. The CASSCF calcula- tions are performed with the following active spaces: (10e, 10o) for H10, (10e, 8o) for N 2, (12e, 8o) for O 2, and (14e, 8o) for F 2. We note that instead of using CASSCF wave functions for ΨB, one could of course use the same selected-CI wave functions used for calculating the energy, but the calculations of n2(r) andμ(r) would then be more costly. In addition, as the frozen-core approximation is used in all our selected-CI calculations, we use the corresponding valence-only complementary functionals (see Subsection II C 2). Therefore, all density-related quantities exclude any contribution from the 1s core orbitals, and the range-separation function follows the definition given in Eq. (16). It should be stressed that the computational cost of the basis-set correction (see Appendix B) is negligible compared to the cost of the selected-CI calculations. B. H 10chain The H 10chain with equally spaced atoms is a prototype of strongly correlated systems as it consists in the simultaneous break- ing of 10 interacting covalent σbonds. As it is a relatively small system, benchmark calculations at near-CBS values are available (see Ref. 88 for a detailed study of this system).We report in Fig. 1 the potential energy curves computed using the cc-pVXZ (X = D, T, and Q) basis sets for different lev- els of approximation, and the corresponding atomization energies are reported in Table I. As a general trend, the addition of the basis-set correction globally improves the quality of the potential energy curves, independently of the approximation level of ¯EB[n]. In addition, no erratic behavior is found when stretching the bonds, which shows that the present procedure (i.e., the determination of the range-separation function and the definition of the function- als) is robust when reaching the strong-correlation regime. In other words, smooth potential energy curves are obtained with the present basis-set correction. More quantitatively, the values of the atom- ization energies are within chemical accuracy (i.e., an error below 1.4 mHa) with the cc-pVTZ basis set when using the PBE-OT and SU-PBE-OT functionals, whereas such an accuracy is not yet reached at the standard MRCI+Q/cc-pVQZ level of theory. Analyzing more carefully the performance of the different types of approximate functionals, the results show that PBE-OT and SU- PBE-OT are very similar (the maximal difference on the atomization energy being 0.3 mHa) and that they give slightly more accurate results than PBE-UEG. These findings provide two important clues on the role of the different physical ingredients included in these functionals: (i) the explicit use of the on-top pair density originating from the CASSCF wave function [see Eq. (32)] is preferable over the use of the UEG on-top pair density [see Eq. (31)], which is somewhat understandable, and (ii) removing the dependence on any kind of spin polarization does not lead to a significant loss of accuracy, pro- viding that one employs a qualitatively correct on-top pair density. The latter point is crucial as it confirms that the spin polarization in density-functional approximations essentially plays the same role as the on-top pair density. This could have significant implications for the construction of more robust families of density-functional approximations within DFT. C. Dissociation of diatomics The N 2, O 2, and F 2molecules are complementary to the H 10 system for the present study. The level of strong correlation in these diatomics also increases while stretching the bonds, similar to the case of H 10, but in addition, these molecules exhibit more important and versatile types of weak correlations due to the larger number of electrons. Indeed, the short-range correlation effects are known to play a strong differential effect on the computation of the atom- ization energy at equilibrium, while the shape of the curve far from the equilibrium geometry is governed by dispersion interactions that are medium to long-range weak-correlation effects.5The dispersion interactions in H 10play a minor role on the potential energy curve due to the much smaller number of near-neighbor electron pairs compared to N 2, O 2, or F 2. In addition, O 2has a triplet ground state and is therefore a good candidate for checking the spin-polarization dependence of the various functionals proposed here. We report in Figs. 2–4 the potential energy curves of N 2, O 2, and F 2computed at various approximation levels using the aug- cc-pVDZ and aug-cc-pVTZ basis sets. The atomization energies for each level of theory with different basis sets are reported in Table I. Just as in H 10, the accuracy of the atomization energies is glob- ally improved by adding the basis-set correction, and it is remarkable that PBE-OT and SU-PBE-OT provide again very similar results. J. Chem. Phys. 152, 174104 (2020); doi: 10.1063/5.0002892 152, 174104-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . Potential energy curves of the H 10chain with equally spaced atoms calculated with MRCI+Q and basis-set corrected MRCI+Q using the cc-pVDZ (top), cc-pVTZ (middle), and cc-pVQZ (bottom) basis sets. The MRCI+Q energies and the estimated exact energies have been extracted from Ref. 88. The latter observation confirms that the dependence on the on-top pair density allows one to remove the dependence of any kind of spin polarization for a quite wide range of covalent bonds and also for an open-shell system such as O 2. More quantitatively, an errorbelow 1.0 mHa compared to the estimated exact valence-only atom- ization energy is found for N 2, O 2, and F 2with the aug-cc-pVTZ basis set using the SU-PBE-OT functional, whereas such a feat is far from being reached within the same basis set at the near-FCI level. J. Chem. Phys. 152, 174104 (2020); doi: 10.1063/5.0002892 152, 174104-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Atomization energies (in mHa) and associated errors (in square brackets) with respect to the estimated exact values computed at different levels of theory with various basis sets. System Basis set MRCI+Qa(MRCI+Q)+PBE-UEG (MRCI+Q)+PBE-OT (MRCI+Q)+SU-PBE-OT H10 cc-pVDZ 622.1 [43.3] 642.6[22.8] 649.2[16.2] 649.5[15.9] cc-pVTZ 655.2 [10.2] 661.9[3.5] 666.0[−0.6] 666.0[−0.6] cc-pVQZ 661.2 [4.2] 664.1[1.3] 666.4[−1.0] 666.5[−1.1] Estimated exact:a665.4 exFCI exFCI+PBE-UEG exFCI+PBE-OT exFCI+SU-PBE-OT N2 aug-cc-pVDZ 321.9 [40.8] 356.2[6.5] 355.5[7.2] 354.6[8.1] aug-cc-pVTZ 348.5 [14.2] 361.5[1.2] 363.5[−0.5] 363.2[−0.3] aug-cc-pVQZ 356.6 [6.1] 362.8[−0.1] 364.2[−1.5] 364.3[−1.6] Estimated exact:b362.7 exFCI exFCI+PBE-UEG exFCI+PBE-OT exFCI+SU-PBE-OT O2 aug-cc-pVDZ 171.4 [20.5] 187.6[4.3] 187.6[4.3] 187.1[4.8] aug-cc-pVTZ 184.5 [7.4] 190.3[1.6] 191.2[0.7] 191.0[0.9] aug-cc-pVQZ 188.3 [3.6] 190.3[1.6] 191.0[0.9] 190.9[1.0] Estimated exact:b191.9 exFCI exFCI+PBE-UEG exFCI+PBE-OT exFCI+SU-PBE-OT F2 aug-cc-pVDZ 49.6 [12.6] 54.8[7.4] 54.9[7.3] 54.8[7.4] aug-cc-pVTZ 59.3 [2.9] 61.2[1.0] 61.5[0.7] 61.5[0.7] aug-cc-pVQZ 60.1 [2.1] 61.0[1.2] 61.3[0.9] 61.3[0.9] CEEIScCEEISc+PBE-UEG CEEISc+PBE-OT CEEISc+SU-PBE-OT cc-pVDZ 43.7 [18.5] 51.0[11.2] 51.0[11.2] 50.7[11.5] cc-pVTZ 56.3 [5.9] 59.2[3.0] 59.6[2.6] 59.5[2.7] cc-pVQZ 59.9 [2.3] 61.3[0.9] 61.6[0.6] 61.6[0.6] Estimated exact:b62.2 aFrom Ref. 88. bFrom the CEEIS valence-only non-relativistic calculations of Ref. 90. cFrom the CEEIS valence-only non-relativistic calculations of Ref. 85. In the case of F 2, it is clear that the addition of diffuse functions in the double- and triple- ζbasis sets strongly improves the accuracy of the results, which could have been anticipated due to the strong breathing-orbital effect induced by the ionic valence-bond forms in this molecule.91It should also be noticed that when reaching the aug- cc-pVQZ basis set for N 2, the accuracy of the atomization energy slightly deteriorates for the PBE-OT and SU-PBE-OT functionals, but it remains, nevertheless, more accurate than the estimated FCI atomization energy and very close to chemical accuracy. The overestimation of the basis-set-corrected atomization energy observed for N 2in large basis sets reveals an unbalanced treatment between the molecule and the atom in favor of the molec- ular system. Since the integral over rof the on-top pair density n2(r) is proportional to the short-range correlation energy in the large- μ limit48,60[see Eq. (24)], the accuracy of a given approximation of the exact on-top pair density will have a direct influence on the accuracy of the related basis-set correction energy ¯EB. To quantify the qualityof different flavors of on-top pair densities for a given system and a given basis set B, we define the system-averaged CASSCF on-top pair density and extrapolated on-top pair density as ⟨n2,CASSCF⟩=∫drn2,CASSCF(r), (36a) ⟨˚n2,CASSCF⟩=∫dr˚n2,CASSCF(r), (36b) where ˚n2,CASSCF(r)=˚n2(n2,CASSCF(r),μCASSCF(r))[see Eq. (32)] andμCASSCF (r) is the local range-separation function calculated with the CASSCF wave function, and similarly, we define the system- averaged CIPSI on-top pair density and extrapolated on-top pair density as ⟨n2,CIPSI⟩=∫drn2,CIPSI(r), (37a) ⟨˚n2,CIPSI⟩=∫dr˚n2,CIPSI(r), (37b) J. Chem. Phys. 152, 174104 (2020); doi: 10.1063/5.0002892 152, 174104-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Potential energy curves of the N 2molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. The estimated exact energies are based on a fit of experimental data and obtained from Ref. 87. where ˚n2,CISPI(r)=˚n2(n2,CIPSI(r),μCIPSI(r))andμCIPSI (r) is the local range-separation function calculated with the CIPSI wave function. We also define the system-averaged range-separation parameters as ⟨μCASSCF⟩=1 N∫drnCASSCF(r)μCASSCF(r), (38a) ⟨μCIPSI⟩=1 N∫drnCIPSI(r)μCIPSI(r), (38b) where nCASSCF (r) and nCIPSI (r) are the CASSCF and CIPSI densities, respectively. All the CIPSI quantities have been calculated with the largest variational wave function computed in the CIPSI calculation with a given basis, which contains here at least 107Slater determi- nants. In particular, μCIPSI (r) has been calculated from Eqs. (16)–(19) with the opposite-spin two-body density matrix Γrs pqof the largest variational CIPSI wave function for a given basis. All quantities in Eqs. (36a)–(38a) were computed excluding all contributions from the 1s orbitals, i.e., they are “valence-only” quantities. We report in Table II these quantities for N 2and N for various basis sets. One notes that the system-averaged on-top pair densityat the CIPSI level ⟨n2,CIPSI⟩is systematically lower than its CASSCF analog⟨n2,CASSCF⟩, which is expected since short-range correlation, i.e., digging the correlation hole in a given basis set at the near FCI level, is missing from the valence CASSCF wave function. In addi- tion,⟨n2,CIPSI⟩decreases in a monotonous way as the size of the basis set increases, leading to roughly a 20% decrease from the aug-cc- pVDZ to the aug-cc-pVQZ basis sets, whereas ⟨n2,CASSCF⟩is almost constant with respect to the basis set. Regarding the extrapolated on-top pair densities, ⟨˚n2,CASSCF⟩and⟨˚n2,CIPSI⟩, it is interesting to note that they are substantially lower than their non-extrapolated counterparts, ⟨n2,CASSCF⟩and⟨n2,CIPSI⟩. Nevertheless, the behaviors of⟨˚n2,CASSCF⟩and⟨˚n2,CIPSI⟩are qualitatively different: ⟨˚n2,CASSCF⟩ clearly increases when enlarging the basis set, whereas ⟨˚n2,CIPSI⟩ remains almost constant. More precisely, in the case of N 2, the value of⟨˚n2,CASSCF⟩increases by about 30% from the aug-cc-pVDZ to the aug-cc-pVQZ basis sets, whereas the value of ⟨˚n2,CIPSI⟩only fluctu- ates within 5% for the same basis sets. The behavior of ⟨˚n2,CASSCF⟩ can be understood by noting that (i) the value of μCASSCF (r) globally increases when enlarging the basis set (as evidenced by ⟨μCASSCF⟩) and (ii) lim μ→∞˚n2(n2,μ)=n2[see Eq. (32)]. Therefore, in the CBS J. Chem. Phys. 152, 174104 (2020); doi: 10.1063/5.0002892 152, 174104-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Potential energy curves of the O 2molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. The estimated exact energies are based on a fit of experimental data and obtained from Ref. 87. limit,μCASSCF (r)→∞and one obtains lim B→CBS⟨˚n2,CASSCF⟩=lim B→CBS⟨n2,CASSCF⟩, (39) i.e.,⟨˚n2,CASSCF⟩must increase with the size of the basis set Bto even- tually converge to lim B→CBS⟨n2,CASSCF⟩, the latter limit being essen- tially reached with the present basis sets. On the other hand, the stability of ⟨˚n2,CIPSI⟩with respect to the basis set is quite remarkable and must come from the fact that (i) ⟨n2,CIPSI⟩is a good approxi- mation to the corresponding FCI value within the considered basis sets and (ii) the extrapolation formula in Eq. (32) together with the choice ofμCIPSI (r) is quantitatively correct. Therefore, we expect the calculated values of ⟨˚n2,CIPSI⟩to be nearly converged with respect to the basis set, and we will take the value of ⟨˚n2,CIPSI⟩in the aug-cc- pVQZ basis set as an estimate of the exact system-averaged on-top pair density. For the present work, it is important to keep in mind that⟨˚n2,CASSCF⟩directly determines the basis-set correction in the large-μlimit. More precisely, the correlation energy contribution associated with the basis-set correction is (in absolute value)an increasing function of ⟨˚n2,CASSCF⟩. Therefore, the error on ⟨˚n2,CASSCF⟩with respect to the estimated exact system-averaged on- top pair density provides an indication of the error made by the basis-set correction for a given system and basis set. With the aug- cc-pVQZ basis set, we have ⟨˚n2,CASSCF⟩−⟨˚n2,CIPSI⟩=0.240 for the N2molecule, while 2 (⟨˚n2,CASSCF⟩−⟨˚n2,CIPSI⟩)=0.190 for two iso- lated N atoms. We can then conclude that the overestimation of the system-averaged on-top pair density, and therefore of the basis-set correction, is more important for the N 2molecule at the equilib- rium distance than for the isolated N atoms. This probably explains the observed overestimation of the atomization energy. To confirm this statement, we computed the basis-set correction for both the N 2 molecule at the equilibrium distance and the isolated atoms using μCIPSI (r) and ˚n2,CIPSI(r)with the aug-cc-pVTZ and aug-cc-pVQZ basis sets. We obtained the following values for the atomization energies: 362.12 mH with aug-cc-pVTZ and 362.15 mH with aug-cc- pVQZ, which are indeed more accurate values than those obtained usingμCASSCF (r) and ˚n2,CASSCF(r). Finally, regarding now the performance of the basis-set correc- tion along the whole potential energy curves reported in Figs. 2–4, J. Chem. Phys. 152, 174104 (2020); doi: 10.1063/5.0002892 152, 174104-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . Potential energy curves of the F 2molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. The estimated exact energies are based on a fit of experimental data and obtained from Ref. 87. it is interesting to note that it fails to provide a noticeable improve- ment far from the equilibrium geometry. Acknowledging that the weak-correlation effects in these regions are dominated by dis- persion interactions that are long-range effects, the failure of the present approximations for the complementary functional can beunderstood easily. Indeed, the whole scheme designed here is based on the physics of correlation near the electron–electron coalescence point: the local range-separation function μ(r) is based on the value of the effective electron–electron interaction at coalescence and the ECMD functionals are suited for short-range correlation effects. TABLE II . System-averaged on-top pair density ⟨n2⟩, extrapolated on-top pair density ⟨˚n2⟩, and range-separation parameter ⟨μ⟩(all in atomic units) calculated with full-valence CASSCF and CIPSI wave functions (see the text for details) for N 2and N in the aug-cc-pVXZ basis sets (X = D, T, and Q). All quantities were computed within the frozen-core approximation, i.e., excluding all contributions from the 1s orbitals. System Basis set ⟨n2,CASSCF⟩ ⟨ ˚n2,CASSCF⟩ ⟨ n2,CIPSI⟩ ⟨ ˚n2,CIPSI⟩ ⟨μCASSCF⟩ ⟨μCIPSI⟩ N2 aug-cc-pVDZ 1.175 42 0.659 66 1.027 92 0.582 28 0.946 0.962 aug-cc-pVTZ 1.183 24 0.770 12 0.922 76 0.610 74 1.328 1.364 aug-cc-pVQZ 1.184 84 0.840 12 0.838 66 0.599 82 1.706 1.746 N aug-cc-pVDZ 0.344 64 0.196 22 0.254 84 0.146 86 0.910 0.922 aug-cc-pVTZ 0.346 04 0.226 30 0.223 44 0.148 28 1.263 1.299 aug-cc-pVQZ 0.346 14 0.246 66 0.212 24 0.151 64 1.601 1.653 J. Chem. Phys. 152, 174104 (2020); doi: 10.1063/5.0002892 152, 174104-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Therefore, the failure of the present basis-set correction to describe dispersion interactions is theoretically expected. We hope to report further on this in the near future. IV. CONCLUSION In the present paper, we have extended the recently proposed DFT-based basis-set correction to strongly correlated systems. We have applied the method to the H 10, N 2, O 2, and F 2molecules up to the dissociation limit at the near-FCI level in increasingly large basis sets and investigated how the basis-set correction affects the convergence toward the CBS limit of the potential energy curves of these molecular systems. The density-based basis-set correction relies on three aspects: (i) the definition of an effective non-divergent electron–electron interaction obtained from the expectation value over a wave func- tionΨBof the Coulomb electron–electron interaction projected into an incomplete basis set B, (ii) the fit of this effective inter- action with the long-range interaction used in RSDFT, and (iii) the use of a short-range, complementary functional borrowed from RSDFT. In the present paper, we investigated (i) and (iii) in the con- text of strong correlation and focused on potential energy curves and atomization energies. More precisely, we proposed a new scheme to design functionals fulfilling spin-multiplet degeneracy and size consistency. To fulfill such requirements, we proposed to use CASSCF wave functions leading to size-consistent energies, and we developed functionals using only Sz-independent density-like quantities. The development of new Sz-independent and size-consistent functionals has led us to investigate the role of two related quan- tities: the spin polarization and the on-top pair density. One important result of the present study is that by using func- tionals explicitly depending on the on-top pair density, one can eschew its spin-polarization dependence without loss of accu- racy. This avoids the commonly used effective spin polarization originally proposed in Ref. 64, which has the disadvantage of possibly becoming complex-valued in the multideterminant case. From a more fundamental aspect, this confirms that, in a DFT framework, the spin polarization mimics the role of the on-top pair density. Consequently, we believe that one could potentially develop new families of density-functional approximations, where the spin polarization is abandoned and replaced by the on-top pair density. Regarding the results of the present approach, the basis-set cor- rection systematically improves the near-FCI calculations in a given basis set. More quantitatively, it is shown that with only triple- ζqual- ity basis sets, chemically accurate atomization energies are obtained for all systems, whereas the uncorrected near-FCI results are far from this accuracy within the same basis set. In addition, it is shown that the basis-set correction gives substantial differential contribution to potential energy curves close to the equilibrium geometries, but at long internuclear dis- tances, it cannot recover the dispersion interaction energy missing because of the basis-set incompleteness. This behavior is actually expected as dispersion interactions are of long-range nature, and the present approach is designed to recover only short-range correlation effects.APPENDIX A: SIZE CONSISTENCY OF THE BASIS-SET CORRECTION 1. Sufficient condition for size consistency The basis-set correction is expressed as an integral in real space, ¯EB[n,ζ,n2,μ]=∫drn(r)¯εsr,PBE c,md(n(r),ζ(r),s(r),n2(r),μ(r)), (A1) where all the local quantities n(r),ζ(r),s(r),n2(r),μ(r)are obtained from the same wave function Ψ. In the limit of two non- overlapping and non-interacting dissociated fragments A + B, this integral can be rewritten as the sum of the integral over the region ΩAand the integral over the region Ω B, ¯EB A+B[n,ζ,n2,μ]=∫ΩAdrn(r)¯εsr,PBE c,md(n(r),ζ(r),s(r),n2(r),μ(r)) +∫ΩBdrn(r)¯εsr,PBE c,md(n(r),ζ(r),s(r),n2(r),μ(r)). (A2) Therefore, a sufficient condition to obtain size consistency is that all the local quantities n(r),ζ(r),s(r),n2(r),μ(r)areintensive , i.e., they locally coincide in the supersystem A + B and in each isolated fragment X = A or B. Hence, we must have, for r∈ΩX, nA+B(r)=nX(r), (A3a) ζA+B(r)=ζX(r), (A3b) sA+B(r)=sX(r), (A3c) n2,A+B(r)=n2,X(r), (A3d) μA+B(r)=μX(r), (A3e) where the left-hand-side quantities are for the supersystem and the right-hand-side quantities are for an isolated fragment. Such condi- tions can be difficult to fulfill in the presence of degeneracies since system X can be in a different mixed state (i.e., ensemble) in the supersystem A + B and in the isolated fragment.92Here, we will con- sider the simple case, where the wave function of the supersystem is multiplicatively separable, i.e., ∣ΨA+B⟩=∣ΨA⟩⊗∣ΨB⟩, (A4) where⊗is the antisymmetric tensor product. In this case, it is easy to show that Eqs. (A3a)–(A3c) are valid, as well known, and it remains to show that Eqs. (A3d) and (A3e) are also valid. Before showing this, we note that even though we do not explicitly consider the case of degeneracies, the lack of size consistency that could arise from spin-multiplet degeneracies can be avoided by the same strategy used for imposing the energy independence on Sz, i.e., by using the effective spin polarization˜ζ(n(r),n2(r))or a zero spin polarization ζ(r) = 0. Moreover, for the systems treated in this work, the lack of size consistency that could arise from spatial degeneracies (coming from atomic pstates) can also be avoided by selecting the same state in the supersystem and in the isolated fragment. For example, for the F2molecule, the CASSCF wave function dissociates into the atomic J. Chem. Phys. 152, 174104 (2020); doi: 10.1063/5.0002892 152, 174104-13 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp configuration p2 xp2 yp1 zfor each fragment, and we thus choose the same configuration for the calculation of the isolated atom. The same argument applies to the N 2and O 2molecules. For other systems, it may not be always possible to do so. 2. Intensivity of the on-top pair density and the local range-separation function The on-top pair density can be written in an orthonormal spatial orbital basis set { ϕp(r)} as n2(r)=∑ pqrs∈Bϕp(r)ϕq(r)Γrs pqϕr(r)ϕs(r), (A5) withΓrs pq=2⟨Ψ∣ˆa† r↓ˆa† s↑ˆaq↑ˆap↓∣Ψ⟩. As the summations run over all orbitals in the basis set B,n2(r) is invariant to orbital rotations and can thus be expressed in terms of localized orbitals. For two non- overlapping fragments A + B, the basis set can then be partitioned into orbitals localized on fragment A and orbitals localized on B, i.e., B=BA∪BB. Therefore, we see that the on-top pair density of the supersystem A + B is additively separable, n2,A+B(r)=n2,A(r)+n2,B(r), (A6) where n2,X(r) is the on-top pair density of fragment X, n2,X(r)=∑ pqrs∈BXϕp(r)ϕq(r)Γrs pqϕr(r)ϕs(r), (A7) in which the elements Γrs pqwith orbital indices restricted to fragment X are Γrs pq=2⟨ΨA+B∣ˆa† r↓ˆa† s↑ˆaq↑ˆap↓∣ΨA+B⟩=2⟨ΨX∣ˆa† r↓ˆa† s↑ˆaq↑ˆap↓∣ΨX⟩, owing to the multiplicative structure of the wave function [see Eq. (A4)]. This shows that the on-top pair density is a local intensive quantity. The local range-separation function is defined as, for n2(r)≠0, μ(r)=√π 2f(r,r) n2(r), (A8) where f(r,r)=∑ pqrstu ∈Bϕp(r)ϕq(r)Vrs pqΓtu rsϕt(r)ϕu(r). (A9) Again, f(r,r) is invariant to orbital rotations and can be expressed in terms of orbitals localized on fragments A and B. In the limit of infinitely separated fragments, the Coulomb interaction van- ishes between A and B, and therefore, any two-electron integral Vrs pq involving orbitals on both A and B vanishes. We thus see that the quantity f(r,r) of the supersystem A + B is additively separable, fA+B(r,r)=fA(r,r)+fB(r,r), (A10) with fX(r,r)=∑ pqrstu ∈BXϕp(r)ϕq(r)Vrs pqΓtu rsϕt(r)ϕu(r). (A11) So,f(r,r) is a local intensive quantity. As a consequence, the local range-separation function of the supersystem A + B is μA+B(r)=√π 2fA(r,r)+fB(r,r) n2,A(r)+n2,B(r), (A12) which implies μA+B(r)=μX(r)ifr∈ΩX, (A13)whereμX(r)=(√π/2)fX(r,r)/n2,X(r). The local range-separation function is thus a local intensive quantity. We can therefore conclude that if the wave function of the supersystem A + B is multiplicative separable, all local quanti- ties used in the basis-set correction functional are intensive, and therefore, the basis-set correction is size consistent. APPENDIX B: COMPUTATIONAL COST OF THE BASIS-SET CORRECTION FOR A CASSCF WAVE FUNCTION The computational cost of the basis-set correction is deter- mined by the calculation of the on-top pair density n2(r) and the local range-separation function μ(r) on the real-space grid. For a general multideterminant wave function, the computational cost is of order O(NgridN4 B), where Ngridis the number of grid points and NBis the number of basis functions.51For a CASSCF wave function, a significant reduction in the scaling of the computational cost can be achieved. 1. Computation of the on-top pair density For a CASSCF wave function Ψ, the occupied orbitals can be partitioned into a set of active orbitals Aand a set of inactive (doubly occupied) orbitals I. The CASSCF on-top pair density can then be written as n2(r)=n2,A(r)+nA(r)nI(r)+nI(r)2 2, (B1) where n2,A(r)=∑ pqrs∈Aϕp(r)ϕq(r)Γrs pqϕr(r)ϕs(r), (B2a) nA(r)=∑ pq∈Aϕp(r)ϕq(r)⟨Ψ∣ˆa† p↑ˆaq↑+ˆa† p↓ˆaq↓∣Ψ⟩, (B2b) nI(r)=2∑ p∈Iϕp(r)2(B2c) are the purely active part of the on-top pair density, the active part of the density, and the inactive part of the density, respectively. The leading computational cost is the evaluation of n2,A(r)on the grid, which, according to Eq. (B2a), scales as O(NgridN4 A), where NAis the number of active orbitals that is much smaller than the number of basis functions NB. 2. Computation of the local range-separation function In addition to the on-top pair density, the computation of μ(r) needs the computation of f(r,r) [see Eq. (A9)] at each grid point. It can be factorized as f(r,r)=∑ rs∈BVrs(r)Γrs(r), (B3) where Vrs(r)=∑ pq∈BVrs pqϕp(r)ϕq(r), (B4a) Γrs(r)=∑ pq∈BΓpq rsϕp(r)ϕq(r). (B4b) J. Chem. 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5.0002818.pdf

J. Appl. Phys. 127, 175303 (2020); https://doi.org/10.1063/5.0002818 127, 175303 © 2020 Author(s).Elastic strain control of electronic structure, and magnetic properties of [Pr1−xCaxMnO3/ SrTiO3]15 superlattices Cite as: J. Appl. Phys. 127, 175303 (2020); https://doi.org/10.1063/5.0002818 Submitted: 28 January 2020 . Accepted: 19 April 2020 . Published Online: 07 May 2020 Ravi Kiran Dokala , Shaona Das , Deep Chandra Joshi , Sayandeep Ghosh , Zhuo Yan , Yajun Qi , Sujit Das , and Subhash Thota ARTICLES YOU MAY BE INTERESTED IN Magnetocaloric properties of Ni 2+xMn1−xGa with coupled magnetostructural phase transition Journal of Applied Physics 127, 173903 (2020); https://doi.org/10.1063/5.0003327 Decoupling of Gd–Cr magnetism and giant magnetocaloric effect in layered honeycomb tellurate GdCrTeO 6 Journal of Applied Physics 127, 173902 (2020); https://doi.org/10.1063/5.0006592 Strong optical coupling between semiconductor microdisk lasers: From whispering gallery modes to collective modes Journal of Applied Physics 127, 173105 (2020); https://doi.org/10.1063/5.0004273Elastic strain control of electronic structure, and magnetic properties of [Pr 1−xCaxMnO 3/SrTiO 3]15 superlattices Cite as: J. Appl. Phys. 127, 175303 (2020); doi: 10.1063/5.0002818 View Online Export Citation CrossMar k Submitted: 28 January 2020 · Accepted: 19 April 2020 · Published Online: 7 May 2020 · Corrected: 12 May 2020 Ravi Kiran Dokala,1 Shaona Das,1Deep Chandra Joshi,2 Sayandeep Ghosh,1 Zhuo Yan,3Yajun Qi,3 Sujit Das,4,a)and Subhash Thota1,b) AFFILIATIONS 1Department of Physics, Indian Institute of Technology Guwahati, Guwahati, Assam 781039, India 2Department of Materials Science and Engineering, Uppsala University, Box 534, SE-75121 Uppsala, Sweden 3School of Materials Science and Engineering, Hubei University, Wuhan, Hubei 430062, China 4Department of Materials Science and Engineering, University of California, Berkeley, California 94720, USA a)Electronic mail: sujitdas@berkeley.edu b)Author to whom correspondence should be addressed: subhasht@iitg.ac.in ABSTRACT We report the growth, electronic structure, and in-plane magnetic properties of pulsed laser deposition grown 2D superlattice structures [Pr0:7Ca0:3MnO 3/SrTiO 3]15and [Pr 0:5Ca0:5MnO 3/SrTiO 3]15on (001) oriented SrTiO 3and LaAlO 3single crystal substrates. The x-ray reflec- tivity measurements reveal well-defined interfaces between the manganite and titanate layers along with the existence of Kiessig fringes, pro- viding the evidence for the smooth periodic superlattice structure. The reciprocal space mapping provides signature of tetragonal distortionin all the superlattices. The electronic structure determined from the x-ray photoelectron spectroscopy reveals divalent Sr and Ca, tetravalentTi, and mixed valent Mn with a pronounce shift of binding energy peaks toward the higher energy side in the superlattices grown on (001) oriented LaAlO 3as compared to those grown on SrTiO 3. These superlattices exhibit highly anisotropic ferromagnetic character. We used the law of approach to saturation to determine the anisotropy field (H K) and cubic anisotropy constant ( K1) for all the investigated superlat- tices. This analysis yields the highest H K/difference9 kOe and K1/difference8/C2105erg/cc for the [Pr 0:7Ca0:3MnO 3/SrTiO 3]15superlattice system. Furthermore, significant enhancement of the overall magnetic moment and a decrease in T C(,100 K) was observed in the case of LaAlO 3 grown superlattice, which indicates a substantial role of residual elastic strain on the magnetic ordering. Our results indicate that the strain induced elongation of MnO 6octahedra leads to finite possibility of non-orthogonal overlapping of orbitals in the presence of large crystal field splitting of eglevels, which, in turn, causes suppression of the ferromagnetic double exchange interaction. Published under license by AIP Publishing. https://doi.org/10.1063/5.0002818 I. INTRODUCTION The rare earth manganites R1/C0xMxMnO 3(R= rare earth, M¼Ca, Sr, Ba, Pr) have drawn immense scientific attention from fundamental and application perspectives, such as colossal magne- toresistance (CMR), superconductivity, and high electronic spinpolarization. 1,2In particular, the unusual electronic properties and colossal electro-resistance (CER) of La 1/C0xCaxMnO 3(LCMO) and Pr1/C0xCaxMnO 3(PCMO) make them a potential candidate for the non-volatile resistive random access memories (ReRAMs).3On the other hand, it is well known that superlattices and heterostructureshave their exotic properties and novel functionalities that are dif- ferent from their bulk counterparts.4–6Such features are attrib- uted to lattice mismatch between the film and the substrate, andthe substrate-induced strain influences the structural, electronic, magnetic, and transport properties. 7–10Superlattices are also con- sidered as model systems to study various physical phenomena inmagnetic interlayers like interfaci al coupling, spin-polarized tun- neling in heterointerfaces, and transport properties. Such proper-ties are very important from the perspectives of technological applications in the field of magnetic sensors, magnetic recording,Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 175303 (2020); doi: 10.1063/5.0002818 127, 175303-1 Published under license by AIP Publishing.and memory devices and in emerging technologies like spintronic devices as well as in fundamental physics.11–13The exchange-bias effect and antiferromagnetic interlayer/interfacial coupling phe-nomena are the key observations in manganite superlatticesreported in recent past. 14–17The electric control of the ferroelec- tric polarization between the magnetic and ferroelectric interfaces allows the modulation of charge concentration, thus enhancing the magnetic moment and increasing the ferromagnetic to para-magnetic Curie temperature (T C).15Until date, many researchers reported on the suppression of magnitude of magnetization andT Cfor La 1/C0xSrxMnO 3/SrTiO 3or La 1/C0xCaxMnO 3/SrTiO 3super- lattices as the thickness of Mn-oxide layer decreases.16Millis et al. revealed that T Cis very sensitive to the biaxial strain present in the sample.18,19It has been realized that elastic strain is a vital parameter that governs the electronic state of complex oxides andcan change bond angles and bond lengths, consequently altering the magnetic exchange interactions significantly. 20–22 Moreover, the studies related to the tailoring of microscopic phase separation in the manganites family have attracted immenseinterest due to their unique physical properties. Usually, suchphase separation leads to microscopic inhomogeneity due to the coexistence of insulating antiferromagnetic charge-ordered phase (originating from the Coulombic interaction of trivalent Mn andtetravalent Mn ions) and metallic ferromagnetic phase (arisingdue to Mn 3þ/C0O/C0Mn4þdouble exchange interaction).23–26Among the family of manganites, the following systems are vital com- pounds that sets the charge ordering with sharp phase boundaries:Nd 1/C0xSrxMnO 3,P r 1/C0xCaxMnO 3,C a 1/C0xSmxMnO 3,G d 1/C0xCaxMnO 3, and Eu 1/C0xCaxMnO 3.23Among these compounds, Pr 1/C0xCaxMnO 3 exhibits unique compositional and magnetic field dependence of electronic properties such as the irreversible nature of metal –insula- tor phase transition in the H-T plane and the robust charge-orderingstate between the compositions 0 :3/C20x/C200:5. 21It is well known fact that numerous external probes have a significant role on themetallic and charge order phases. In most of the cases, external mag- netic fields can drive the metal-to-insulator transitions in which both the magnetic structure and electrical conductivity vary markedly. 21 Kiryukhin et al. reported that illumination of x rays can switch the insulating antiferromagnetic state to the metallic ferromagnetic statebelow 40 K in Pr 0:7Ca0:3MnO 3.27On the other hand, the bulk com- pound Pr 0:7Ca0:3MnO 3(and 0 /C20x/C200:3) shows orthorhombic symmetry with complex canted magnetic ordering. Usually, twomagnetic transitions appear in the samples having a canted spinstructure. The first transition is the canted antiferromagnetic state occurring at 110 K, and the second one is the insulating canted ferro- magnetic state. Previous reports suggest that canting is proportionalto the doping concentration; however, beyond a critical compositionx crit,t h eP r 1/C0xCaxMnO 3system exhibits ferromagnetic behavior, but for the compositions between x¼0:3a n d0 :5, the antiferromag- netic insulating phase exists in the charge-ordered insulating state.28 In the form of thin films, the insulating charge-ordered state in Pr0:5Ca0:5MnO 3is highly susceptible to the substrate-induced strain and thickness of the film. Prellier et al. reported that the lattice strain influences the charge-ordering considerably in Pr 0:5Ca0:5MnO 3thin films grown on LaAlO 3(compressive) and SrTiO 3(tensile) using the pulsed laser deposition (PLD) method. These authors also reportedthat both charge-ordering insulating and metallic state coexist up tomoderate magnetic fields ( /C2050 kOe) in 75 nm thick Pr 0:5Ca0:5MnO 3 films grown on SrTiO 3substrates. However, the critical melting filed drastically reduces to 70 kOe for the film thickness of 110 nm.29On the other hand, the charge-ordering temperature decreases with thedecrease in lattice strain in the case of La substituted Pr 1/C0xCaxMnO 3 films.30Motivated by the above studies, in the present work, we focus mainly on the Pr 1/C0xCaxMnO 3system along with the insulating SrTiO 3layer with special emphasis on their combined superlattice configuration for two different compositions x¼0:5a n d0 . 3g r o w n by PLD technique. We explore the changes occurring on themagnetic ordering and electrical properties of the superlattice [Pr 1/C0xCaxMnO 3/SrTiO 3]15on (001) oriented LaAlO 3and SrTiO 3 single crystal substrates. The main advantage of choosing different substrates is that one can tune the homogeneous strain state of thesesuperlattices and study the interface driven modulated crystal struc-ture, electronic states, magnetic property, and transport property. A systematic study of the role of the epitaxial strain on the physical properties of the proposed superlattice configuration is completelynew and has not been reported in the literature until now. Ourresults and analysis provide the evidence that the residual strainbetween the substrate and thin films plays a major role in deciding the global magnetic ordering of the highly anisotropic [Pr 1/C0xCaxMnO 3/SrTiO 3]15superlattices. II. FABRICATION AND CHARACTERIZATION DETAILS [Pr1/C0xCaxMnO 3/SrTiO 3]15superlattices (of two compositions x¼0:3 and 0.5) were simultaneously grown on (001)-oriented LaAlO 3and SrTiO 3substrates by PLD technique from the stoichio- metric ceramic targets of Pr 0:7Ca0:3MnO 3,P r 0:5Ca0:5MnO 3, and SrTiO 3. We employed the excimer (KrF) laser of wavelength 248 nm and an energy density of 2 J/cm2for the deposition of the thin films. Schematic diagram of the superlattice is shown in theinset of Fig. 1(a) . The substrate temperature was fixed at 670 /C14C, and oxygen partial pressure was maintained at 0.15 mbar. In total, 15 double layers of Pr 1/C0xCaxMnO 3and SrTiO 3have been depos- ited on both the substrates. The structural characterization of thesesuperlattices have been carried out by using a Phillips X ’Pert x-ray diffractometer with Cu-K αradiation. All the magnetization measure- ments were performed using a superc onducting quantum interference device (SQUID) based magnetometer (MPMS) from quantum design.To probe the electronic structure of the superlattices, we performedx-ray photoelectron spectroscopy (XPS) measurements using the instrument Thermo Fisher Scientific 250 Xi. The same instrument has been employed to study the chemical state of the elements andthe surface chemical composition of all the synthesized superlat-tice structure. The surface topology of the superlattices wereexamined by means of an Atomic Force Microscope (AFM) from Oxford (Model-Cypher). III. RESULTS AND DISCUSSIONS Figure 1(a) shows the x-ray reflectivity curves of the superlat- tice on (001)-oriented SrTiO 3and LaAlO 3. The reflectivity spec- trum exhibits clear superlattice peaks originating from the chemicalmodulation from layer to layer along with the Kiessig fringes from the total thickness of the sample. These fringes provide evidence for a smooth and sharp layer structure and well-defined interfacesJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 175303 (2020); doi: 10.1063/5.0002818 127, 175303-2 Published under license by AIP Publishing.between the Pr 1/C0xCaxMnO 3and SrTiO 3layers. The total thickness of the samples has been calculated from d¼λ=(2Δθ), where “λ”is t h ew a v e l e n g t ho ft h exr a y sa n d Δθ¼θi/C0θi/C01,θiis the angle of theithorder fringe. We have evaluated the individual layer thick- nesses by calibrating the single Pr 1/C0xCaxMnO 3and SrTiO 3films before the start of superlattice growth. The magnitude of total thickness ( d) in the superlattice structure is 112 nm, whereas the individual layer thickness of Pr 1/C0xCaxMnO 3(d1)a n dS r T i O 3(d2) being 1.8 nm and 5.7 nm for the layers grown on (001)-orientedSrTiO 3and LaAlO 3, respectively. Figure 1(b) represents the θ/C02θdiffraction patterns near the (002) reflection of the [Pr1/C0xCaxMnO 3/SrTiO 3]15superlattices (SLs). Several distinct satellite peaks [marked as SL(1), SL(2) ...SL(/C01), SL( /C02), ...] around the main peak (002) are clearly noticeable, which indi- cates a periodic structure of the superlattice architecture along with reasonably smooth interfaces. The position of the SL(0) peakis independent of super period ( Λ) in both LaAlO 3and SrTiO 3. The modulation period of the superlattices estimated using the expression Λ¼(ni/C0ni/C01)λ=[2(sin θi/C0sinθi/C01)], where niis the number corresponding to ithorder oscillation of the fringe and θi is the angle of that particular ithorder fringe. Accordingly, we estimated the magnitudes of Λand these values are Λ¼7:5n m and 7.6 nm for the SL on SrTiO 3and LaAlO 3, respectively, which are in well agreement with those values obtained from the directmeasurements of d 1andd2(Λ¼d1þd2). It should be noted that the SL(0) peak position is different for the different single crystalsubstrates, reflecting the fact that strain-induced change play a sig- nificant role over the average out-of-plane lattice parameter ( c). The calculated out-of-plane lattice parameters ( c) from SL(0) peak position are 3.906 Å and 3.88 Å for the superlattices grown on(001) LaAlO 3and SrTiO 3, respectively. The in-plane ( a) and out-of-plane ( c) lattice parameters of the superlattices are also cal- culated from the reciprocal space map (RSM) around the (103) reflection as shown in Figs. 1(c) and 1(d). The average lattice parameters of the superlattices grown on LaAlO 3/SrTiO 3are a¼3:88=3:905 Å and c¼3:906=3:88 Å, respectively. The c-axis lattice parameters measured using this procedure are in good agree- ment with the ones calculated from θ/C02θdiffraction patterns. From these measurements, it is confirmed that the superlatticesgrown on LaAlO 3assumes a compressive strain state, (i.e., c=a.1), whereas the superlattices grown on SrTiO 3exhibit a tensile strain state ( c=a,1). Considering the pseudocubic bulk lattice constants of Pr 1/C0xCaxMnO 3(3.848 Å), we deduced the in-plane tensile strain that is equivalent to 0.8/1.5% for Pr 1/C0xCaxMnO 3superlattices grown on LaAlO 3/SrTiO 3, respectively. We have also studied the growth mechanism and surface mor- phology using the atomic force microscopic(AFM) measurements. Figures 2 and 3, respectively, represent the two-dimensional (2D) and three-dimensional (3D) AFM surface morphology images of[Pr 0:7Ca0:3MnO 3/SrTiO 3]15superlattices grown on (001) oriented (a) SrTiO 3and (b) LaAlO 3substrates. The AFM images (c) and (d) represents the composition x¼0:5, [Pr 0:5Ca0:5MnO 3/SrTiO 3]15 superlattices grown on (001) oriented SrTiO 3and LaAlO 3 substrates, respectively. All the samples exhibit uniform and homogeneous grain growth; however, from these 3D images, onecan clearly notice different growth mechanism in the case of [Pr 0:7Ca0:3MnO 3/SrTiO 3]15superlattices grown on (001) SrTiO 3as compared to the 3D columnar grain gr owth structure of superlattices grown on (001) LaAlO 3substrate. Furthermore, [Pr 0:5Ca0:5MnO 3/ SrTiO 3]15superlattices exhibit smaller grain size in both the cases (SrTiO 3and LaAlO 3)a sc o m p a r e dt ot h e[ P r 0:7Ca0:3MnO 3/SrTiO 3]15 superlattices in which grains are connected by interconnecting channels, exhibiting an island growth mode. The root-mean-squaresurface roughness of all superlattices lie between 0.40 and 0.6 nm,indicating that a comparatively homogeneous and smooth surface of the bottom Pr 1/C0xCaxMnO 3layer is favoring the high-quality growth of the top layer. In order to understand the electronic structure and local environment, we performed XPS measurements. Figure 4 shows the x-ray photoelectron intensity vs binding energy of core-level spectra of Sr-3 d[4(a),4(f),a n d 4(k)], Ti-2 p[4(b),4(g),a n d 4(l)], Ca-2 p[4(c),4(h),a n d 4(m) ], Mn-2 p[4(d),4(i),a n d 4(n)], and O-1 s[4(e),4( j),a n d 4(o)] for the superlattices FIG. 1. The crystal structure of [Pr 1/C0xCaxMnO 3/SrTiO 3]15superlattices. (a) X-ray reflectivity measurements of [Pr 0:7Ca0:3MnO 3/SrTiO 3]15superlattices grown on LaAlO 3(orange curve) and SrTiO 3(blue curve). The image shown in the inset provides schematic diagram of the superlattice structure. (b) θ–2θ x-ray diffraction (XRD) scans around the (002) reflection of superlattice grown on LaAlO 3(orange curve) and SrTiO 3(blue curve). The reciprocal space mapping (RSM) of the superlattice around the (103) reflection on LaAlO 3(c) and SrTiO 3(d).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 175303 (2020); doi: 10.1063/5.0002818 127, 175303-3 Published under license by AIP Publishing.[Pr0:7Ca0:3MnO 3/SrTiO 3]15g r o w no nL a A l O 3[4(a) –4(e)]a n d SrTiO 3[4(f) –4( j)], whereas Figs. 4(k) –4(o) represent XPS spectra of the [Pr 0:5Ca0:5MnO 3/SrTiO 3]15superlattices grown on LaAlO 3. In all the superlattices, the core-level Sr-3 dXPS spectra show three main peaks centered at 131.7, 133.4, and 137.3 eV. Here, theexperimentally obtained raw data are shown as scattered symbols, whereas deconvoluted theoretical fits are shown as solid lines. The first two peaks (131.69 and 133.43 eV) are corresponding tothe doublets of 3 delectronic states of Sr with the binding energy separation of /difference1:74+0:02 eV. These results confirm the divalent oxidation state of Sr. The third peak, which is having very less intensity located at 137.49 eV, is corresponding to the satellite peak of Sr 2þ. The intensity of these satellite peaks is higher in the case of LaAlO 3grown films as compared to those of SrTiO 3with slight shift in the peak positions. Nevertheless, all the core-level spectra ofSr-3dfor [Pr 0:5Ca0:5MnO 3/SrTiO 3]15superlattices grown on LaAlO 3 shifts significantly toward higher binding energy side. The core-level Ti-2pspectra exhibit two sharp peaks centered at 462.8 eV and 457.13 eV associated with the doublets Ti-2 p1=2and Ti-2 p3=2,r e s p e c - tively. The spin –orbit splitting Δ(2p3=2/C02p1=2)/difference5:70+0:02 eV suggests the tetravalent oxidation state of Ti. Moreover, a broad cusp observed across 471.5 eV is the satellite peak associated with Ti4þ. No significant change in the peak positions (or in intensity) wasobserved as the Pr compositions changes as well as with the changein the substrate. On the other hand, the photoelectron spectrum of“Mn ”from 2 pcore level is resolved into five peaks, four main peaks located at 640.15 eV(P 1), 641.15 eV(P 2), 652.29 eV(P 3), and 654.46 eV(P 4) and one broad satellite peak at 644.7 eV(S 1). For the deconvolution, we have applied a mathematical fitting con-straint on the full-width-at-half-maximum (FWHM) of the peak profile, i.e., the FWHM should range within 0.2 eV and this con- straint is released at the final iteration. The main peaks are corre-sponding to the two doublets of Mn. The binding energyseparation between the “Mn ”doublets are Δ(P 1/C0P3)/difference11:25 eV andΔ(P2/C0P4)/difference10:6 6 e V ,p r o v i d i n gt h ee v i d e n c ef o rt h em i x e d oxidation state of “Mn ”(i.e., Mn3þand Mn2þ). Generally perov- skites contain oxygen-deficient surface (10% oxygen vacancies) ascompared to the bulk. 31Since XPS is a surface sensitive technique the signature of Mn2þcan be noticed in the Mn-2 pcore-level spectrum. Another possible explanation would be the interfacial redox reaction between the electrodes (sputtered/deposited) and PCMO. The interfacial redox reaction leads to the formation ofan oxide layer on the metal electrode and an oxygen-deficientlayer on the PCMO film. 32–34The calculated Mn3þ/Mn2þratio by using the integrated intensities of Mn-2 p3=2spectrum is /difference1:84. Except the main peak P 1, no significant change in the peak positions (or in intensity) was noticed as the Pr compositions changes. TheCa-2p core-level photoelectron spectrum is deconvoluted into adoublet, Ca-2 p 3=2and Ca-2 p1=2, centered at 346.4 eV and 350.1 eV, respectively. The spin –orbit splitting between Ca-2 p3=2and Ca-2 p1=2 is/difference3:7 eV, confirming the divalent oxidation state of Ca in all the superlattice cases. Here, no significant alteration in these spectrawere observed with the choice of substrate. Finally, the O-1 sspec- trum is deconvoluted into three Gaussian and Lorentzian peaks. All deconvoluted peaks have the FWHM of /difference1:26+0:20 eV. The first peak at lower binding energy (528.33 eV) is associated withthe metal –oxygen (M –O) bonding, and the second peak (530.39 eV) FIG. 2. The two-dimensional atomic force microscopic (AFM) surface morphol- ogy images of (a) [Pr 0:7Ca0:3MnO 3/SrTiO 3]15superlattices grown on (001) ori- ented (a) SrTiO 3and (b) LaAlO 3substrates. AFM images (c) and (d) represent the [Pr 0:5Ca0:5MnO 3/SrTiO 3]15superlattices grown on (001) oriented SrTiO 3 and LaAlO 3substrates, respectively. FIG. 3. The three-dimensional AFM surface morphology images of [Pr0:7Ca0:3MnO 3/SrTiO 3]15superlattices grown on (001) oriented (a) SrTiO 3 and (b) LaAlO 3substrates. AFM images (c) and (d) represent the [Pr0:5Ca0:5MnO 3/SrTiO 3]15superlattices grown on (001) oriented SrTiO 3and LaAlO 3substrates, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 175303 (2020); doi: 10.1063/5.0002818 127, 175303-4 Published under license by AIP Publishing.is associated with the surface oxygen. Whereas the third peak (531.73 eV) signifies the presence of some excess oxygen at thesurface of the samples. The intensity of M –Op e a kf o r [Pr 0:7Ca0:3MnO 3/SrTiO 3]15grown on SrTiO 3is relatively lower as compared to other samples, suggesting the domination of surface and excess oxygen in the sample. Figure 5 shows the thermal variation of the magnetization M(T) of Pr 0:7Ca0:3MnO 3and SrTiO 3superlattices and Pr 0:5Ca0:5MnO 3and SrTiO 3superlattices grown on two different substrates (001) oriented SrTiO 3and LaAlO 3substrates. These measurements are performed under both zero-field-cooled (ZFC) and field-cooled (FC) conditions.We used two different external magnetic fields H DC¼1k O e a n d 10 kOe applied along the [100] direction (in-plane configuration).A clear bifurcation was noticed in both the M ZFC(T) and M FC(T) curves below 150 K for [Pr 0:7Ca0:3MnO 3/SrTiO 3]15superlattices due to the either very high anisotropy or finite-size effects ofthe system. Also, the [Pr 0:7Ca0:3MnO 3/SrTiO 3]15superlattice fer- romagnetic behavior with T C/difference170 K with a cusp at 148 K in MZFC(T) which shifts to 123 K with increasing the external field from 1 kOe to 10 kOe. However, the M(T) curves of[Pr 0:7Ca0:3MnO 3/SrTiO 3]15superlattice grown on LaAlO 3(001) substrate display no definite transition. Nevertheless, in all thesamples, we noticed a low-temperature peak across 57 K due tothe liquid oxygen that is paramagnetic in nature. 35This observa- tion is consistent with the XPS studies discussed above, where excess surface oxygen was detected in all the superlattices. Theinset of Fig. 5(b) shows the magnified view of the magnetization curves below 45 K. These plots clearly show a low-temperaturemaximum in M ZFC(T) at 17.36 K [14 K from the d( χT)/dT vs T plots (not shown)], which disappears in the presence of high mag- netic fields. The origin of this peak is due to the loosely bound spinsin the system. Moreover, no specific transition was observed in M(T)curves and no irreversibilities in M FC(T) and M ZFC(T) curves in the case of low Pr content system [Pr 0:5Ca0:5MnO 3/SrTiO 3]15, instead these samples grown on either LaAlO 3or SrTiO 3substrates exhibit weak-ferromagnetic behavior. In all these cases, we noticed thatsamples grown on LaAlO 3(001) show enhanced magnetic moments and is much prominent at high magnetic fields but exhibit lower ordering temperatures [T C/difference72:6K a n d 1 6 9 . 4K , for [Pr 0:7Ca0:3MnO 3/SrTiO 3]15g r o w no n( 0 0 1 )L a A l O 3and FIG. 4. The x-ray photoelectron spectra (XPS) of Sr-3 d[(a), (f ), and (k)], Ti-2 p[(b), (g), and (l)], Ca-2 p[(c), (h), and (m)], Mn-2 p[(d), (i), and (n)], and O-1 s[(e), ( j), and (o)] of superlattices [Pr 0:7Ca0:3MnO 3/SrTiO 3]15grown on LaAlO 3[(a)–(e)] and SrTiO 3[(f )–( j)]. (k) –(o) represent XPS spectra of the [Pr 0:5Ca0:5MnO 3/SrTiO 3]15 superlattices grown on LaAlO 3.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 175303 (2020); doi: 10.1063/5.0002818 127, 175303-5 Published under license by AIP Publishing.[Pr0:7Ca0:3MnO 3/SrTiO 3]15g r o w no n( 0 0 1 )S r T i O 3, respectively]. The differential susceptibility [d( χT ) / d Tv sT ]a n a l y s i sy i e l d st o slightly lower ordering temperatures T C¼158 K and the Pr0:7Ca0:3MnO 3and SrTiO 3superlattices grown on SrTiO 3(001) in line with Fisher ’s relation (heat-capacity, C M/differenceAd (χT)/dT).36 In ferromagnetic manganite heterostructures, double exchange interaction is the strongest for cubic symmetry (mediated by 180/C14 Mn –O–Mn bond) unit cell,37and the tensile strain reduces the TCferromagnetic manganite films [ Fig. 6(a) ]. In Fig. 6(b) ,w e have shown the schematic diagram of the d-orbitals to understand the spin configurations in the supperlattices of AFM(FM) states,which occur due to the in-plane tensile strain (without strain). Inour work, Pr 0:7Ca0:3MnO 3suffers less tensile strain on the LaAlO 3substrate than SrTiO 3substrates ( Fig. 6 ), which reflects difference in the magnetic properties of the superlattices grownon LaAlO 3and SrTiO 3. It is noted that the reduction of T Cfrom the bulk value for Pr 1/C0xCaxMnO 3is also result of finite layer thickness of Pr 1/C0xCaxMnO 3.38Previous neutron-diffraction studies on bulk Pr 0:7Ca0:3MnO 3provides evidence for the three different phase transitions in contrast to the present 2D superlat-tices case: (i) the change in the lattice symmetry across 200 K,(ii) collinear antiferromagnetic transition across 150 K, and(iii) the spin reorientation (from collinear and canted antiferro- magnetic spin state) transition across 110 K. 39The antiferromag- netic transition observed in the present case is in line with theabove result except other two transitions. Figure 7 shows the magnetization vs field hysteresis loops recorded at 5 K under the ZFC condition along the in-plane direction of all the superlattices after subtracting the diamagneticcontribution of the substrates using extrapolation method. All the samples exhibit finite coercivity (H C) with small asymmetry in the loops (slight shift of M –H loop along the field direction and moves downward). Detailed analysis of these results lead to the fol- lowing results. The superlattices [Pr 0:7Ca0:3MnO 3/SrTiO 3]15grown on SrTiO 3(001) exhibit larger H C/difference8:848 kOe and remanence magnetization M R/difference0:12μB/Mn with negligible exchange-bias field of H EB/difference80 Oe. Whereas all the other superlattices exhibit very small H C(650 Oe) and M R(0.05 μB/Mn) for LaAlO 3(001) grown [Pr 0:7Ca0:3MnO 3/SrTiO 3]15superlattices. For the system [Pr0:5Ca0:5MnO 3/SrTiO 3]15grown on SrTiO 3(001), H C/difference800 Oe and M R/difference0:007μB/Mn, whereas the same system grown on LaAlO 3(001) exhibits further low H C/difference800 Oe but higher MR/difference0:01μB/Mn. Nevertheless, all the superlattices grown on LaAlO 3(001) substrates exhibit very higher saturation magnetization (MS)v a l u e s[ M S¼0:73μB/Mn and 0.35 μB/Mn for LaAlO 3(001) and SrTiO 3(001) grown [Pr 0:7Ca0:3MnO 3/SrTiO 3]15superlattices, respectively]. The bulk polycrystals of Pr 0:7Ca0:3MnO 3exhibit quite larger magnetic moment (M FC/difference4μB/Mn) at low temperatures for HDC/difference30 kOe as compared to the present superlattice system (0.3μB/Mn and 0.5 μB/Mn) for [Pr 0:7Ca0:3MnO 3/SrTiO 3]15grown on SrTiO 3(001) and LaAlO 3(001) substrates, respectively.40 Our results are consistent with the previous studies by Jiang et al. on the bulk Pr 1/C0xCaxMnO 3system who reported significant decrease in magnetization values for x¼0:5a sc o m p a r e dt o x¼0:3, which is related to the decrease of unit-cell volume and enhanced structuralsymmetry (with tolerance factor, t/difference1). 41However, in the case of Pr0:67Ca0:33MnO 3layers grown on the LaAlO 3substrate with thick- ness/difference1400 nm exhibit M S/difference300 emu/cc at 10 K,42which is close FIG. 5. T emperature dependence of magnetization M(T) measured under zero-field-cooled (ZFC) and field- cooled (FC) conditions of the superlatti-ces [Pr 0:7Ca0:3MnO 3/SrTiO 3]15grown on (001) oriented (a) SrTiO 3and (b) LaAlO 3. Whereas (c) and (d) show the M(T) curves of [Pr 0:5Ca0:5MnO 3/ SrTiO 3]15superlattices grown on (001) oriented SrTiO 3and LaAlO 3single crystal substrates, respectively. All the M(T) measurements are performed attwo different external magnetic fieldsH DC¼1 kOe and 10 kOe under warming condition. The insets repre- sent zoomed view of the low tempera-ture magnetization data.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 175303 (2020); doi: 10.1063/5.0002818 127, 175303-6 Published under license by AIP Publishing.to [Pr 0:7Ca0:3MnO 3/SrTiO 3]15superlattices grown on the LaAlO 3 (001) substrate. In order to determine the magnetic anisotropy present in these superlattices, we used the Law of Approach to Saturation(LAS) model 43–45and fitted the experimentally obtained virgin magnetization isotherm curve of the hysteresis loop for H .HC with Eq. (1)given below. In general, near the saturation magnetiza- tion (M S), the magnetic moment of the samples can be expressed as follows: M¼MS1/C0a H/C0b H2/C18/C19 þχH: (1) In the above equation, the term a/H is linked with the struc- tural defects, whereas the magnetocrystalline anisotropy of the material is defined by the b/H2term and the last term represents the paramagnetic behavior of the system. Solid lines in Fig. 8 repre- sents the best fit obtained using Eq. (1)to the experimental data points (scattered symbols). The corresponding fitting parametersevaluated using the above analysis are listed in Table I .W e observed higher values for [Pr 0:5Ca0:5MnO 3/SrTiO 3]15superlattices grown on (001) oriented SrTiO 3and LaAlO 3substrates, which sig- nifies the presence of large structural defects as compared to the[Pr 0:7Ca0:3MnO 3/SrTiO 3]15superlattices grown on the similar substrates. The cubic magnetic anisotropy constant K1has been calcu- lated using the following equation: b¼8 105K2 1 μ2 0M2 S, (2) FIG. 6. (a) Variation of magnetic ordering temperatures (T C, shown in solid symbols) and saturation magnetization (M S, shown in hollow symbols) plotted as a function of lattice parameter ( a) of the [Pr 0:7Ca0:3MnO 3/SrTiO 3]15(red color) and [Pr 0:5Ca0:5MnO 3/SrTiO 3]15(blue color) superlattices grown on (001) oriented LaAlO 3and SrTiO 3substrates, respectively. (b) Schematic diagram of d-orbitals corresponding to the FM configuration (without strain) and AFM arrangement (in-plane strain) of spins in the superlattices. FIG. 7. Magnetic hysteresis (M-H) loops recorded at temperature (T¼5 K) for the [Pr 0:7Ca0:3MnO 3/ SrTiO 3]15superlattices grown on (001) oriented (a) SrTiO 3and (b) LaAlO 3. M-H loops shown in (c) and (d) corre- spond to the [Pr 0:5Ca0:5MnO 3/ SrTiO 3]15superlattices grown on (001) oriented SrTiO 3and LaAlO 3, respec- tively. Insets clearly show the coercive field (H C) and remanence magnetiza- tion (M R) at low fields.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 175303 (2020); doi: 10.1063/5.0002818 127, 175303-7 Published under license by AIP Publishing.where μ0i st h ef r e es p a c ep e r m e a b i l i t y .F o r[ P r 0:7Ca0:3MnO 3/SrTiO 3]15 superlattices grown on SrTiO 3(001), the magnitude of K1is considerably larger ( /difference8/C2105erg/cc) than the other superlattices {1 :42/C2104erg/cc and 1 :77/C2104erg/cc for [Pr 0:5Ca0:5MnO 3/SrTiO 3]15grown on (001)-oriented SrTiO 3and LaAlO 3substrates, respectively}. This may arise due to the ferro- magnetic behavior of the [Pr 0:7Ca0:3MnO 3/SrTiO 3]15superlattices. On the other hand, the remaining superlattices exhibit weak-ferromagnetic behavior in composition with the above samples.The superlattices [Pr 0:7Ca0:3MnO 3/SrTiO 3]15g r o w no n( 0 0 1 )o r i e n t e d LaAlO 3exhibit higher K1value than [Pr 0:5Ca0:5MnO 3/SrTiO 3]15. Furthermore, we evaluated the anisotropy field H Kusing the relation: HK¼2K1=μ0MS. Among all the systems, [Pr 0:7Ca0:3MnO 3/SrTiO 3]15 superlattices exhibit very high values of H K(/difference8:84 KOe), whereas for the other systems, the magnitude of H Kis significantly low (between 301 Oe and 378 Oe). Such a drastic reduction in H Kgen- erally occurs due to the decrease of the magnetic exchange interac- tions.45Previous studies on the angle dependent magneto-optical Kerr magnetometry studies on La 0:67Sr0:33MnO 3thin films grownon the (001) oriented SrTiO 3substrate reveal large anisotropy features induced by the defects in the crystal structure rather than the magnetoelastic effects.46The magnitude of K1obtained in the present case is comparable to that of La 0:7Ca0:3MnO 3thin films (4:5/C2105erg/cc) grown on (001) oriented SrTiO 3and other 2D manganite/ruthanate systems ( /difference4/C2106erg/cc) reported using the ferromagnetic resonance studies.47,48Conversely, in the present case, the magnitude of M Sincreases in the case of [Pr0:5Ca0:5MnO 3/SrTiO 3]15superlattices grown on LaAlO 3(001) as compared to the SrTiO 3(001) grown [Pr 0:5Ca0:5MnO 3/SrTiO 3]15 systems. More elaborately, the strain might change the Mn –O–Mn bond distances and bond angles, which, in turn, causes alteration in the effective spin exchange energy.21,49Also, the strain-induced elon- gation of MnO 6octahedra leads to crystal field splitting of eglevels as a consequence finite possibility of non-orthogonal overlappingbetween these orbital may takes place, which can suppress the ferro- magnetic double exchange coupling. 50–52Nonetheless, the interface between the Pr 1/C0xCaxMnO 3and SrTiO 3layers plays a role beyond the strain effects. Furthermore, the interface will affect the magneticorder significantly at this low thickness of the manganite layers ( /difference5 unit cells). Also, the degree of interdiffusion will affect magnetic ordering and a suppression of magnetic order with increasing inter- diffusion is expected across the interfaces of the superlattices studiedin the present case. 21,53,54 IV. SUMMARY In summary, we have successfully grown superlattices of [Pr0:7Ca0:3MnO 3/SrTiO 3]15and [Pr 0:5Ca0:5MnO 3/SrTiO 3]15on (001) oriented LaAlO 3and SrTiO 3single crystal substrates by pulsed laser deposition technique. On the basis of reciprocal spacemapping analysis, we concluded that these superlattices exhibitweak tetragonal distortion. The x-ray photoelectron spectroscopyresults confirm the divalent electronic state of Sr and Ca, the tetravalent state of Ti, and mixed valency in Mn with a strong shift of binding energy separation toward the higher energyside in (001) LaAlO 3grown superlattices as compared to those layers grown on (001) SrTiO 3.T h et e m p e r a t u r ea n dm a g n e t i c field dependence of magnetization results demonstrate that [Pr0:7Ca0:3MnO 3/SrTiO 3]15/SrTiO 3superlattices exhibit a ferro- magnetic character below 170 K with high degrees of anisotropy.Using the M-H isotherms and the law of approach to saturation forferromagnets, we have estimated the cubic anisotropic constant ( K 1) and anisotropy field (H K) for all the investigated systems, where the superlattices [Pr 0:7Ca0:3MnO 3/SrTiO 3]15/SrTiO 3exhibit the highest HK/difference9k O ea n d K1/difference8/C2105erg/cc. However, upon changing the residual strain (which is induced by the substrate), a drastic decreasein the magnetic ordering temperatures and significant enhancement in the overall magnetic moments are noticed. In this study, we have also shown that one can alter the magnetic structure of the superlat-tices under tensile strain, which may find potential utility in thefields of magnetoelectronic devices. ACKNOWLEDGMENTS R.K.D. acknowledges the FIST program of the Department of Science and Technology, India, for partial support of this work (Grant Nos. SR/FST/PSII-020/2009 and SR/FST/PSII-037/2016). FIG. 8. The M-H isotherms (scattered symbols) and the corresponding theoreti- cal fits (Eq. (1)) based on the law of approach to saturation, LAS (solid line) recorded at 5 K for the superlattices [Pr 0:7Ca0:3MnO 3/SrTiO 3]15grown on (001) oriented (a) SrTiO 3and (b) LaAlO 3. (c) and (d) correspond to the [Pr0:5Ca0:5MnO 3/SrTiO 3]15superlattices grown on (001) oriented SrTiO 3and LaAlO 3, respectively. TABLE I. List of various parameters obtained from the magnetization measure- ments. Saturation magnetization (M S), magnetic anisotropy constant ( K1), and anisotropy field (H K). Superlattices/substrateMS(μB/ Mn)K1(×104 erg/cc)HK(×103 Oe) [Pr0.7Ca0.3MnO 3/SrTiO 3]15/SrTiO 3 0.35 80 9.0 [Pr0.7Ca0.3MnO 3/SrTiO 3]15/LaAlO 3 0.74 5.77 0.31 [Pr0.5Ca0.5MnO 3/SrTiO 3]15/SrTiO 3 0.14 1.42 0.38 [Pr0.5Ca0.5MnO 3/SrTiO 3]15/LaAlO 3 0.21 1.77 0.33Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 175303 (2020); doi: 10.1063/5.0002818 127, 175303-8 Published under license by AIP Publishing.R.K.D and S.D. acknowledge the Central Instrument Facility (CIF) of the Indian Institute of Technology Guwahati for partial support of this work. All the authors acknowledge the support from theUGC-DAE CSR, Indore, for providing the temperature-dependentmagnetic measurements using the SQUID magnetometer.In particular, we express our gratitude and sincere thanks to D r .R .J .C h o u d h a r yf o rh e l p i n gu si nm a g n e t i cm e a s u r e m e n t s . Z.Y and Y.Q. thank the support from the National NaturalScience Foundation of China (No. 11974104). S.D. acknowledgethe U.S. Department of Energy, Office of Science, Basic EnergySciences, Materials Sciences and Engineering Division. REFERENCES 1R. von Helmolt, J. Wecker, B. Holzapfel, L. Schultz, and K. Samwer, “Giant neg- ative magnetoresistance in perovskitelike La 2/3Ba1/3MnO xferromagnetic films, ” Phys. Rev. Lett. 71, 2331 (1993). 2S. Das, A. Herklotz, E. Jia Guo, and K. Dörr, “Static and reversible elastic strain effects on magnetic order of La 0:7Ca0:3MnO 3/SrTiO 3superlattices, ”J. Appl. Phys. 115, 143902 (2014). 3A. Asamitsu, Y. Tomioka, H. Kuwahara, and Y. 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B 81, 214429 (2010).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 175303 (2020); doi: 10.1063/5.0002818 127, 175303-10 Published under license by AIP Publishing.

5.0006610.pdf

AIP Advances 10, 075105 (2020); https://doi.org/10.1063/5.0006610 10, 075105 © 2020 Author(s).Fluctuations of the wall shear stress vector in a large-scale natural convection cell Cite as: AIP Advances 10, 075105 (2020); https://doi.org/10.1063/5.0006610 Submitted: 19 May 2020 . Accepted: 11 June 2020 . Published Online: 02 July 2020 R. du Puits , and C. Bruecker AIP Advances ARTICLE scitation.org/journal/adv Fluctuations of the wall shear stress vector in a large-scale natural convection cell Cite as: AIP Advances 10, 075105 (2020); doi: 10.1063/5.0006610 Submitted: 19 May 2020 •Accepted: 11 June 2020 • Published Online: 2 July 2020 R. du Puits1 and C. Bruecker2,a) AFFILIATIONS 1Institute of Thermodynamics and Fluid Mechanics, Technische Universitaet Ilmenau, 98684 Ilmenau, Germany 2School of Mathematics, Computer Science and Engineering, City, University of London, London EC1V 0HB, United Kingdom a)Author to whom correspondence should be addressed: christoph.bruecker@city.ac.uk ABSTRACT We report first experimental data of the wall shear stress in turbulent air flow in a large-scale Rayleigh–Bénard experiment. Using a novel, nature-inspired measurement concept [C. H. Bruecker and V. Mikulich, PLoS One 12, e0179253 (2017)], we measured the mean and fluc- tuating part of the two components of the wall shear stress vector at the heated bottom plate at a Rayleigh number Ra = 1.58 ×1010and a Prandtl number Pr = 0.7. The total sampling period of 1.5 h allowed us to capture the dynamics of the magnitude and the orientation of the vector over several orders of characteristic timescales of the large-scale circulation. We found the amplitude of short-term (turbulent) fluctuations to be following a highly skewed Weibull distribution, while the long-term fluctuations are dominated by the modulation effect of a quasi-regular angular precession of the outer flow around a constant mean, the timescale of which is coupled to the characteristic eddy turnover time of the global recirculation roll. Events of instantaneous negative streamwise wall shear occur when rapid twisting of the local flow happens. A mechanical model is used to explain the precession by tilting the spin moment of the large circulation roll and conserva- tion of angular momentum. A slow angular drift of the mean orientation is observed in a phase of considerable weakening of mean wind magnitude. ©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/10.1063/5.0006610 .,s I. INTRODUCTION Since Ludwig Prandtl’s pioneering work, we know that the local heat transport at a surface with a temperature differing from that of the surrounded fluid is linked to the local momentum transport across the fluid layer close to the surface.1Measurements of the local wall shear stress (WSS) may, therefore, contribute to a better under- standing of the convective heat transfer process. However, this kind of data reflecting the dynamics of the local heat/momentum trans- port is rare, and to our knowledge, the present work is the first one providing measurement data of the instantaneous two-dimensional (2D) vector of the local WSS in thermal convection. Following Prandtl’s idea, Ludwieg carried out a first analysis of the relationship between the heat and momentum transport in thermal convection. Unfortunately, he did not have the appropriate metrology, and he could obtain only the time-averaged WSS from measurements of the profile of the velocity parallel to the wall.2For large-scale air convection studies such as in the so-called “Barrel ofIlmenau” (BOI), the existing database is still limited to mean veloc- ity profile measurements from which only a single component of the mean WSS could be derived. Due to the lack of sufficiently sensitive sensors of the WSS, the current status quo in such data knowledge is therefore solely available from Direct Numerical Simulations (DNS). Such simulations provide the local WSS vector information in time but usually for a limited simulation period of only a few tens of min- utes. First simulation data, published by Scheel and Schumacher,3 show the existence of singularities in the wall shear stress vector field similar to those reported in Bruecker.4These singularities are considered as footprints of large eruptions of fluid parcels from the wall, which significantly affect the heat transport.5It is, therefore, the authors’ conclusion that the information on the magnitude and the angle of the WSS vector as well as the information on its tempo- ral behavior are crucial to understand the local momentum and heat transport processes at the wall. In order to measure the instantaneous WSS in low-speed air flows, Bruecker and Mikulich developed a novel sensor that was AIP Advances 10, 075105 (2020); doi: 10.1063/5.0006610 10, 075105-1 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv particularly designed to be used in large-scale convection air flows such as in the Barrel of Ilmenau.6As the authors of the paper report, the sensitivity and the dynamic response of the sensor, which is based on a nature-grown dandelion pappus, were sufficiently good to resolve the dynamics of the very small WSS in thermal convection in air. The present work reports the first application of this sensor in a large-scale convection experiment in the BOI. It addresses the hitherto unknown dynamics of the WSS by simultaneously measur- ing the magnitude and the direction of the WSS vector. The results display the behavior of the modulation of the local WSS by the outer main wind and give insight into the statistics and dynamics of the turbulent boundary layer. The paper is organized as follows: In Sec. II, we describe the essentials of the measurement technology as well as the convection experiment wherein the sensor has been applied. Section III contains the results of our measurements, and in Sec. IV, we summarize our discussion. II. EXPERIMENTAL SETUP AND MEASUREMENT TECHNIQUE A. The large-scale Rayleigh–Bénard experiment “Barrel of Ilmenau” The WSS measurements were carried out in the so-called “Bar- rel of Ilmenau (BOI)” a Rayleigh–Bénard (RB) experiment using air (Pr = 0.7) as working fluid (see Fig. 1) and with the sensor mounted at the center of the bottom plate. The BOI consists of a virtually adiabatic container of cylindrical shape with an inner diameter of D= 7.15 m. A heating plate at the lower side releases the heat to the air layer, and a cooling plate at the upper side removes it. Both plates are carefully leveled perpendicular to the vector of gravity with an uncertainty of less than 0.15○. The thickness of the air layer Hcan be varied continuously between 0.15 m <H<6.30 m by moving the cooling plate up and down. The temperature of both plates can be set to values of 20○C<Th<80○C (heating plate) and 10○C<Tc <30○C (cooling plate). Due to the specific design of both plates (for more details, see du Puits et al.7), the temperature at their surfacesis very uniform and the deviation does not exceed 1.5% of the total temperature drop ΔT=Th−Tcacross the air layer. The variation of the surface temperature over the time is even smaller and remains below ±0.02 K. The sidewall of the convec- tion cell is equipped with an active compensation heating system that efficiently prevents a heat exchange between the interior of the RB cell and the environment. Glass windows in the top plate allow the optical access to the interior of the test section for the illumi- nation and for taking recordings. For our measurements, we used a smaller inset of diameter D= 2.5 m and height H= 2.5 m that was placed within the large-size test section (see also Fig. 1). The tem- perature at the bottom heating plate was set to T h= 25○C and at the top cooling plate to T c= 15○C, thus providing a temperature dif- ference of ΔT= 10 K. The Rayleigh number Ra = ( βgΔTH3)/(νκ) under these conditions is Ra = 1.58 ×1010, with the thermal expan- sion coefficient β= 3.421 ×10−3K−1, the gravitational acceleration g= 9.81 ms−2, the kinematic viscosity ν= 1.532 ×10−5m2s−1at 20○, and the thermal diffusivity κ= 2.163 ×10−5m2s−1. The particular benefit of the inset configuration is the fact that the vertical tem- perature distribution inside and outside the inset equals, therefore, the sidewall can be considered as fully adiabatic. The characteris- tic timescale of the flow in the test section is the so-called free-fall time unit, defined as Tf=√ βgΔTH, which is about Tf= 2.7 s for the current configuration. Another timescale is the character- istic eddy turnover time Teof the large circulation cell (LSC) in the form of a single recirculation roll, which is calculated from the mean wind U= 0.15 ms−1and the circumference of the cell to about Te= 50 s. B. The wall shear stress sensor The sensor including its calibration in the BOI is described in detail in Bruecker and Mikulich.6It follows the principle of an indi- rect WSS measurement by calculating the near-wall velocity gradient from the wall-parallel velocity at a given (short) distance from the wall. It is based on the flow-induced deflection of an elastically- mounted cantilever beam (inverted pendulum) that is built at his head from a pappus of micro-hairs (nature-grown dandelion pap- pus) (Fig. 2). To maximize the sensitivity, the sensor’s head consists FIG. 1 . (a) Sketch of the large-scale Rayleigh–Bénard experiment “Barrel of Ilmenau” with the smaller inset of D= 2.5 m. The origin of a Cartesian coor- dinate system is fixed with the center of the bottom wall (the location of the wall shear stress sensor) in the x,yplane and the zaxis pointing normal to the wall toward the top plate. (b) Mean velocity profile in the boundary layer of the BOI7 at the center of the cell. The plot shows the magnitude of the velocity vector at the centerline in different planes z par- allel to the surface of the wall. Inserted is a true-scale sketch of the sensor with its head at z 0= 7 mm, illustrating that it is fully surrounded by the linear part of the velocity profile. AIP Advances 10, 075105 (2020); doi: 10.1063/5.0006610 10, 075105-2 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 2 . Principle of the measurement concept using a wall-mounted cantilever beam with a pappus head (left). Pic- tures of the pappus sensor fixated with the stem in the flexible membrane at the bottom plate of the BOI (right). of a pappus of slender hairs with a diameter of a few tens of microm- eters, acting as an antenna. The mechanical behavior of the sensor is described in Bruecker and Mikulich6as a forced system with second- order response in overdamped condition (overdamped harmonic oscillator). A calibration of the mechanical model can provide the two unknown variables of the solution to the response function, the constant gain Kand the cut-off frequency fc, the frequency at which the sensor can no longer follow the excitation. A detailed view of the sensor is shown in Fig. 2. The stem and head were taken from a nature-grown dandelion with a pappus of radially arranged slen- der hairs (mean length l= 7 mm, mean diameter d= 30 μm).6It has a stem height of z0= 7 mm and the overall radial diameter of the pappus is about Dp= 14 mm. The Reynolds number Reof the flow around the individual hairs—simplified as thin cylinders of diameter d—is of the order of Red≈2 for air speeds of 1 ms−1. Thus, the drag is dominated by viscous friction and it scales, therefore, approximately linear with the flow speed.8–10The elastic joint, at which the stem’s foot is bonded, is made from rubber silicone (Polydimethylsiloxane, PDMS; Youngs modulus E≈1.5 MPa) and acts as a linear-elastic tor- sional spring with uniform bending stiffness in the radial direction. When the stem with the pappus is exposed to an air flow parallel to the wall, the resulting torque tilts the stem around the joint, similar to an inverted pendulum. As the tilt is proportional to the torque, the latter can be measured indirectly by the end-to-end shift vector ⃗Q(t) of the tip relative to the wind-off reference. We capture the tilting motion of the pappus by imaging its orbital motion from top, which provides the projection of the tip’s end-to-end vector in the hori- zontal x,yplane at z=z0with⃗Qx,y(t)=[Qx(t),Qy(t)]. For small tilt angles and a sufficiently small sensor scale with z0≪δ, these quantities are directly proportional to the wall shear stress compo- nents τx,y(see also in Skupsch et al.11). In 3D flows, the wall shear stress is a vector ⃗τ(t)=[τx(t),τy(t)]with the streamwise and the spanwise component (assuming the mean flow parallel to the wall in x-direction), respectively. Both components are defined by the wall- normal velocity gradients ∂ux/∂z∣z=0 mmand∂uy/∂z∣z=0 mmat the wall (in the plane perpendicular to the wall-normal coordinate z). Using a Taylor expansion, the information of the velocity field in the x,yplane close to the wall at a distance z=z0is related to the wall shear stress as follows:12 τx,y=μux,y(z0) z0+ O(z0)2, (1) with ux,y(z0)≈KQ x,y. (2)The second order term in Eq. (1) can be neglected in the viscous-dominated near-wall region (viscous sublayer). Previous flow studies in the BOI using Laser Doppler Velocimetry show a typical profile of the mean velocity at the position of the sensor, measured by using Laser Doppler Velocimetry, see Fig. 1. The lin- ear part of the profile as indicated by the dashed line represents the viscous sublayer close to the wall. The picture additionally displays a true-scale sketch of the sensor, which illustrates that the sensor is at the edge of the linear regime. Measurements by Ampofo and Karayiannis13in a similar low-turbulence convection flow as stud- ied herein show that the viscous sublayer thickness is of the order of 10% of the outer boundary layer, similar as observed in the BOI. The constant gain Kin Eq. (2) was measured in situ using a wind- generating device placed inside the BOI under isothermal conditions (see Fig. 3). The air flow is generated with a planar nozzle that gener- ates a Blasius-type wall-jet at the location of the sensor 20 slot heights hsaway.6Different jet velocities up to v= 1.50 ms−1have been set and the deflection ⃗Qx,yof the sensor head was measured. The results of the calibration procedure show a proportional increase of ⃗Qx,ywith the velocity of the Blasius jet at z0. Recall- ing that a linear relationship is expected between air velocity and pappus drag, a linear regression is applied to the measurements for the interesting range of velocities <0.8 m/s, which provides the gain K= 1000 s−1with the standard error of 5%. Beyond a velocity of about 0.8 ms−1, the recordings show that the configuration of the hairs starts to change over time and the linear relationship is no longer valid. This critical value is never exceeded in the convective airflow in the BOI. A step-response test with the sensor further provides the dynamic response, given as the magnitude and phase of the transfer function, see Figs. 3(c) and 3(d). The curves match the response of a second-order critically damped mechanical oscillator from which one obtains the cut-off frequency fc, at which the sensor can no longer follow the signal (the response starts to roll-off at −40 dB per decade). This is at a frequency of 100 Hz, which alternatively means a response time of approximately about τ95= 10 ms in reverse. Since the typical timescale of the small- est near-wall fluctuations has been measured in the past with about 0.5 s,14the sensor works completely in the range of constant ampli- tude response (gain) and zero phase-shift in the measurement range of f<2 Hz, capable to map the full dynamics of the flow. C. Optical setup for sensor imaging The optical setup for the tip-deflection measurements is shown in Fig. 1. The pappus sensor at the bottom plate was illuminated AIP Advances 10, 075105 (2020); doi: 10.1063/5.0006610 10, 075105-3 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 3 . Sketch of the wall-jet apparatus (a) for static sensor calibration shown in (b). The dynamic response was measured with a step-response test with the magnitude and phase given in (c) and (d), respectively. The dashed line in (c) indicates the theoretical response of a second-order critically damped system fitted to the measured parameters. by a defocused Laser beam (Raypower 5000, 5 ∼W power at λ= 532 nm, Dantec Dynamics, Skovlunde, Denmark) expanded to illuminate a spot of 50 mm diameter at the floor. A CCD cam- era (mvBlueFOX3-1031, Matrix Vision, Oppenweiler, Germany) placed on top of the cooling plate acquires the deflection of the sensor head in the wall-parallel x,yplane with a resolution of 2048 ×1536 px2and a frame rate of 10 Hz. The camera is equipped with a long-distance microscope (model K2/SCTM, Infinity Photo-Optical, Goettingen, Germany) that provides a resolution of 185 px/mm. A total of 54 000 images were recorded in a single mea- surement campaign. The images are streamed via USB ∼3 to the hard disc of a desktop. This equates to a maximum of 1.5 ∼h of the observation time per experiment. To avoid any vibrations during the recordings, the facility was left alone after starting the record- ing and no external disturbance could enter the RB cell. In order to remove any vibration induced by leaving and re-entering the facil- ity, the first and the last 2–3 min were rejected before we analyzed the data. The tip displacement vector in the images is obtained using a 2D cross-correlation method similar to that in the particle image velocimetry technique,15where we compare the quadratic subsec- tion of the sensor image between wind-off and wind-on situation. The shift in the tip position relative to wind-off is determinedwith subpixel accuracy using a 3-point Gaussian fit of the correla- tion peak in x- and y-direction, which has an uncertainty of about 0.05 px. A reference marker on the floor is used to correct for poten- tial vibrations of the camera during the recordings. After multipli- cation of the shift with the lens magnification, the vector ⃗Qx,y(t) of the sensor head is recovered for each time-step in the image sequence. In order to make our data comparable with velocity-gradient data recently obtained from PIV measurements, we consider in the following the viscosity-divided WSS τx,y/μ(known as the wall-shear rate) with the two components: τx(t)/μ=KQ x(t)/z0, τy(t)/μ=KQ y(t)/z0,(3) and we define the direction and the magnitude of the WSS as follows: Φ(t)=atan(τy(t)/τx(t)), Ψ(t)=1 μ∥τ∥=1 μ√ τ2x(t)+τ2y(t).(4) We capture our data with a sampling frequency of 10 Hz. In order to remove outliers, the data were filtered in time with a AIP Advances 10, 075105 (2020); doi: 10.1063/5.0006610 10, 075105-4 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv fourth-order Butterworth low-pass filter designed with a −3 dB cut- off frequency at 2 Hz. We have selected this particular frequency, since the typical timescale of the smallest near-wall fluctuations has been measured in the past with about 0.5 s.14Our long-term record- ing of totally 54 000 samples covers more than 100 LSC turnover times Te, and ensures sufficient statistical evidence even for the long timescales. III. RESULTS AND DISCUSSION Preceding the discussion, it is worth to note that earlier studies in the barrel with a similar aspect ratio indicate the existence of only a single LSC roller that was observed also to perform angular oscilla- tions around a mean direction. The normal flow mode present in the BOI is where the mean orientation of the LSC is locked in one par- ticular direction. Because of the modulation effect, which the outer flow enforces on the signal on the floor, the WSS signals should also reveal the footprint of this wiggling motion. Figures 4(a) and 4(b) show the complete time trace of the direction Φ(t) and the magni- tudeΨ(t) of the WSS (viscosity-divided WSS τx,y/μ) over a period of 1.5 hours. Overlaid in color is the low-pass filtered data ̃ΦLSC and̃ΨLSC(fourth order Butterworth low-pass filter designed with a −3 dB cut-off frequency at 0.003 Hz), based on the notation used in Shi et al.16Therefore, turbulent events happening close to the wall are filtered out (higher frequency), while the footprint of fluc- tuations of the mean wind direction and magnitude of the LSC remain. Both the original data, the direction Φ(t) and magnitude Ψ(t) of the WSS vector, fluctuate over time at a high frequency. Mean- while, the low-pass filtered WSS vector is almost perfectly aligned with the x-axis in phase A ( t= 0 s–3000 s). Beginning at t= 3000 s, a phase of a very slow drift of the angle ˜ΦLSCin counterclockwisedirection is seen, see phase B ( t= 3000 s–5000 s). This angular drift indicates a slow precession of the mean axis of the LSC, meanwhile the oscillations at higher frequencies persisted. Such a slow preces- sion mode can replace the normal flow mode present in the BOI. Initially, at t= 1000 s in phase A, the mean WSS magnitude amounts to¯Ψ=40 s−1. It decreases then slowly over a period of 2000 s further down to ¯Ψ=30 s−1at the end of Phase A (=3000 s) and finally reaches, in a rather short period, a minimum of ¯Ψ=20 s−1at t= 4000 s in phase B [see Fig. 4(b)]. The final slow-down lasted only about 1000 s (366 units of Tf), which is when the angle ˜ΦLSCchanged byπ. Figure 5 illustrates the complex behavior of the flow in the x, yplane as a trace plot of the original and low-pass filtered WSS vec- tor. As discussed before, the mean direction of the LSC in phase A (the red part of the time-filtered signal in Fig. 4) was almost con- stant toward north (positive x-axis), while in phase B the plane of the LSC shows a nearly constant angular drift in the counterclockwise direction. When correlating the onset of the angular drift with the mag- nitude of the WSS, the data let us conclude that the mean axis of the LSC started to rotate at a time, when the magnitude of the main wind started to critically slow down. If we again follow the argument of the outer modulation effect, then the magnitude of the low-pass filtered WSS is proportional to the characteristic velocity of the LSC (mean wind). From that, we can estimate the kinetic Energy ¯Ekinof the LSC as proportional to the square of the magnitude of the WSS with ¯Ekin∼Ψ2. The results show, therefore, that the average kinetic energy of the mean wind in phase B is reduced to about 50% of the energy in phase A. Such a slow-down was also observed by du Puits et al.7We hypothesize herein that the slow-down of the kinetic energy of the mean wind may have triggered the angular precession. The timescale of this precession is rather long, as it takes about 20 characteristic eddy turnovers of the LSC while the orientation drifts only along an angular arc of π. FIG. 4 . (a) Plot of direction Φ(t) and (b) magnitude Ψ(t) ofτ/μover a period of 1.5 h. Overlaid in color is the profile of the time-filtered signal of the direction ̃Φ(t) and the magnitude ̃Ψ(t). Two different characteristic phases are coded in color (phase A in red, phase B in blue). AIP Advances 10, 075105 (2020); doi: 10.1063/5.0006610 10, 075105-5 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 5 . Time trace of the viscosity- divided WSS vector τ/μ(t) in the x,y plane, comparable to the trace Q(t) of the sensor head. (a) τ/μ(t) in phase A; (b) time-filtered signal ˜τ/μ(t)in phase A (full red line) and in the successive phase B (dashed blue line). We further analyze the temporal behavior of the magnitude Ψ and the angle Φby computing their autocorrelation functions, Cxx(Δt)=lim n→∞1 nn ∑ i=1xi(t)xi(t+Δt). (5) We plot short in Figs. 6(a) and 6(b) exemplary short sequences of the time traces of Ψ(t) andΦ(t) together with the autocorrelation functions Cxx(Ψ) and Cxx(Φ) calculated from the full data. While theoscillations of the magnitude Ψ(t) seem to be rather irregular [see Fig. 6(a)], the plot of Φ(t) reveals a low frequency oscillation around the mean with a frequency of about 0.02 Hz [see Fig. 6(b)]. This oscillatory variation of the orientation of the WSS angle Φ(t) over a range of more than ±25○is similar as already observed in Shi et al.16 The timescale related to this oscillation corresponds to the charac- teristic turnover time Te≈50 s of the LSC. Its quasi-periodic nature is highlighted in the plot of the autocorrelation function Cxx(Φ) [see Fig. 6(d)], which shows strong periodic correlation peaks at FIG. 6 . (a) and (b) Time series of magnitude Ψ(t) and direction Φ(t) of the wall shear stress vector τ/μfor phase A, 400 s <t<700 s. (c) and (d) Autocorrelation function of the magnitude Cxx(Ψ) and the direction Cxx(Φ) for phase A. AIP Advances 10, 075105 (2020); doi: 10.1063/5.0006610 10, 075105-6 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv multiples of the time-lag Te[Fig. 6(d)]. The dynamical system has obviously two attracting states in the orientation of the LSC overlaid with a certain fraction of noise/turbulence.17 Furthermore, the result shows a long-term modulation of the angular oscillations. The peak amplitude of the correlation slowly decreases for increasing time lag to about zero at a time lag of Δt= 450 s (corresponds to 9 Te) and then increases again to a local peak correlation value Cxx(Φ) = 0.2 at Δt= 700 s (14 Te). The observed slow-down of the mean wind in phase B could be a consequence of this low-frequency modulation. In the following, we exclusively focus on phase A when the mean direction of the wind is constant on a large timescale and aligned with the x-axis with a mean WSS magnitude of ∣τx∣/μ =35 s−1and a rms of τ′x,rms/μ=18.3 s−1. An interesting feature is the observation of a negative streamwise wall shear stress τxas seen in the traces in Fig. 5(a), when the line crosses the second or third quadrant in the left sub-figure. Such events are observed in turbulent RB convection for the first time, but they were observed recently in turbulent boundary layer flows.4 Figure 7 shows the angular probability density function (PDF) of the yaw angle of the WSS as a wind rose plate with a mean direc- tion toward north (x-axis). The angle Φof the rays relative to north represents the yaw angle, while the length indicates the probability over all samples recorded in phase A. The magnitude Ψis overlaid in color. The graph is similar to that used by Bruecker displaying the measurements of the statistics of the wall shear stress in turbulent boundary layer (tbl) flows.4The distribution shows a type of cone, in which mean angles between ±25○around the x-axis predominate. However, there are also, even rarely, events of τ, in which the yaw angle exceeds ±90○. Thus, these rare events can be associated with events of large spanwise τy, first argued in a zero-pressure-gradient tbl.4The probability and the yaw angle of the rare events in thermal convection are quite similar to those reported from tbl measure- ments. It, however, remains open if the origin of such events is the existence of quasi-streamwise vortices as argued in the case of tbl.The observation herein indicates a rapid temporal variation of the local direction of the fluctuating wall shear stress, representing a high angular velocity of the WSS vector τduring these events. Figure 7(c) shows the PDF of the streamwise WSS normalized with the rms. It demonstrates a non-symmetric distribution with the proof of certain probability of negative streamwise WSS events. The measured PDFs shown in Fig. 7(c) can be well described by the following generalized extreme value (GEV) distribution:18 P(x′;λ,k)=1 λ(1 +kx′)−(1/k+1)e−(1+kx′)−1/k , (6) where the variable x′= (x−m)/λwith the shape parameter k, the scale parameter λ, and the location parameter m. The fit provides a shape parameter of k=−0.1907 ( λ= 0.937, m= 1.5403). For k<0, the distribution is reduced to the reversed Weibull distribution and has zero probability density for x>−λ/(k+m). From the fitted values of x, we find the corresponding upper bound with x >6.4553. Note that the herein observed distribution is reverse to the typical Weibull-type distribution observed in tbl, see the reference curve in Fig. 7(c) from Ref. 19. This means negative streamwise WSS events can have higher magnitude in convection flows. With respect to the occurrence of extreme events, it is quite difficult to analyze the data using classical conditional averaging methods or a fixed threshold definition due to this particular modu- lation of the magnitude and the orientation of the mean flow. Here, we try to discriminate the amplitude of the fluctuating WSS sig- nals into separate timescales. We distinguish between the periodic transitive dynamics represented in the low frequency dynamics of the mean wind and the small-scale turbulent fluctuations. To this end, we apply envelope functions with different time windows on τxto determine the amplitudes of these fluctuations on the differ- ent timescales. The envelope is calculated from the Matlab toolbox and uses a sliding time-window that connects within the window the local peaks (upper envelope for local maxima and lower envelope for local minimum peaks) with a smoothed spline.20 FIG. 7 . (a) Polar plot of probability distri- bution of the wall-shear angle in phase A (mean wind flow is in x-direction). (b) Zoom-in the image of the data given in the left. Angular steps are in 5○. The color indicates the magnitude in the ranges given in the legend bar. (c) PDF of the streamwise wall shear stress nor- malized with its rms. The solid line repre- sents the generalized extreme value dis- tribution according to Eq. (6) and with the parameters given in the text. The dashed line shows the results from Örlü and Schlatter for a zero pressure-gradient turbulent boundary layer flow.19 AIP Advances 10, 075105 (2020); doi: 10.1063/5.0006610 10, 075105-7 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 8 . Peak-to-peak amplitude of the fluctuations of τxat different timescales. (a) Time trace of τx(black line) with envelopes of the small-scale (ss) and large-scale (ls) fluctuations (thin blue and red lines, respectively). (b) Probability density function (PDF) of the small-scale fluctuations (the solid line is a Weibull fit with a scale parameter λ= 0.673 and a shape parameter k= 1.223), (c) PDF of the large-scale fluctuations (the solid line is a Gaussian fit with a mean value of ¯τx=2.85 and a standard deviation of σ(τx) = 0.92). For the low frequency dynamics, we chose a window of 15 s, while using a shorter time window of 0.5 s for the small-scale turbu- lent structures. One typical example of such an enveloping curve is plotted along with the original signal in Fig. 8(a). In order to analyze the amplitude of the fluctuations, we compute the absolute differ- ence between the upper and the lower envelopes | τx,max−τx,min| and determine the probability density function (PDF) for both time windows [see Figs. 8(b) and 8(c)]. The PDF of the small-scale (ss) fluctuations using a time window of 0.5 s is shown in Fig. 8(b) and that for the large-scale (ls) fluctuations is shown in Fig. 8(c). The ss fluctuations of the streamwise wall shear stress follow a Weibull distribution according to the expression P(x;λ,k)=k λ(x k)k−1 e−(x/λ)k , (7) (with the scale parameter λ= 0.637 and the shape parameter k= 1.223), the ls fluctuations are clearly Gaussian distributed. The latter indicates a normal distribution of the amplitude of the angu- lar fluctuations of the orientation of the LSC, as this contributes to the cyclic variation of τx. In conclusion, extreme events are more likely, if large excursions occur simultaneously for both statistical distributions. IV. CONCLUSION We have presented and discussed the first measurements of the instantaneous wall shear stress in a large-scale Rayleigh–Bénard experiment at Rayleigh and Prandtl numbers Ra = 1.58 ×1010andPr = 0.7, respectively. Using a novel, nature-inspired pappus sen- sor, we measured the magnitude and the orientation of the local wall shear stress vector at the center of the heated bottom plate. The results of our 1.5 h measurement series demonstrate that this vec- tor undergoes strong fluctuations in its magnitude as well as in its orientation. Important to note is that the sensor signal at the wall represents the sum of both the fluctuations on small timescales due to the turbulent nature of the boundary layer, and, in addition, the dynamics of the LSC due to the modulation effect of the outer flow onto the near-wall region. Therefore, our measurements allow also drawing conclusions on the magnitude and orientation of the main wind in the LSC. On average over a period of 3000 s (phase A), the mean wind is almost perfectly aligned with the x-axis. However, we observe a clear quasi-periodic angular precession of the orientation of the LSC in the range 50○–60○around the mean, each half-cycle taking exactly the time of one eddy turnover time Te≈50 s. The strong periodicity is manifested by the plot of the autocorrelation function Cxx(Φ), which shows periodic peaks at multiples of the eddy turnover times with values larger than 0.2 even after more than 900 s [see Fig. 6(d)]. Such a strong periodicity in the angular oscilla- tions has not been observed so far and motivated us to illustrate the dynamics of the LSC in a simplified mechanical model for further discussion. A schematic mechanical model is illustrated in Fig. 9 to discuss the observed regular oscillations. We hypothesize that the plane of the LSC with fluid rotating around its axis is represented by a rotat- ing disc, whose axis is initially aligned horizontally with the y-axis. An initial disturbance in the form of asymmetric lateral down- and upwash at the sides of the LSC (A1-A2) causes a torque that tilts AIP Advances 10, 075105 (2020); doi: 10.1063/5.0006610 10, 075105-8 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 9 . Simplified mechanical model of a tumbling rotating disc (tumbling LSC) to illustrate the modulation effect on the quasi-periodic angular precession of the wall shear stress vector at the wall. the spin momentum of the LSC in the horizontal plane and leads to a self-enforcing of this asymmetry. As the vortex axis reorients away from the horizontal plane, it generates a torque around the z- axis because of the conservation of angular momentum, which leads to a precession of the LSC. The cycle is reversed when the front of the LSC—while precessing—reaches the region A2 and counteracts the upwash, while the back of the LSC reverses the downwash A1. Hence, the system starts a cyclic clockwise–counterclockwise preces- sion motion around the z-axis, which correlates with the observed regular angular oscillations of the orientation of the WSS vector. Note that the diagonal orientation of the LSC is not contradicting previous observations that the orientation of the mean flow at the same instant and location is different at the bottom plate compared to the top plate, supporting the idea of a tilted or twisted circulation roll (Funfschilling and Ahlers21and Xi and Xia22). The long-term recording also allowed us to detect a very slow drift of the mean orientation in a certain phase (phase B) overlaid with the regular precessions described above. The angular drift in the counterclockwise direction takes about 30 times the eddy turn- over time for a 270○turn. This slow mode is accompanied by a decrease of the kinetic energy of the mean wind (imposed by the LSC) by about 50%. In the past, du Puits et al. reported at similar conditions also a critical weakening of the mean wind for a period of 4 h.23However, the authors could not link their observation to a modification of the angular orientation of the global recirculation. A possible explanation for this slow mode precession based on the proposed mechanical model could be a slight imbalance of the tum- bling cycle, which then leads to a net mean angular momentum. A transitional flow phenomenon like the reported rotation of the plane of the global recirculation has already been observed in turbulent Rayleigh–Bénard convection (RBC) in the past, see Refs. 21 and 22.However, they found that this occurs only very rarely. Insofar, it was rather a lucky coincidence that we could observe such a transition in our 90 min long measurement. Another phenomenon we observed in our long-term record- ings is the occurrence of local backflow in the boundary layer, while the large-scale circulation in phase A remains on average almost per- fectly aligned with the x-axis. Such local backflow events have also been detected recently in turbulent boundary layer flow along a flat wall, but this is the first time that such events could be documented in a temperature-gradient driven flow. Local backflow is correlated herein with large angular velocities of the wall shear stress vector, which we understand to be an indication for the existence of coher- ent vortical structures with a large inclination of the axes against the wall (nearly wall-normal vortex funnels). These short-term fluctua- tions have amplitudes that follow a highly skewed Weibull distribu- tion, while the amplitudes of fluctuations on the longer timescales are better fitted by a symmetric Gaussian. In both distributions, the ends of the tails can reach amplitudes of 3–4 times the rms of the mean streamwise wall shear stress. Such a coincidence of large values in both distributions indicates the high probability of rare excursions of the near-wall flow in magnitude as well in yaw angle. ACKNOWLEDGMENTS The position of Professor Christoph Bruecker is co-funded as the BAE SYSTEMS Sir Richard Olver Chair and the Royal Academy of Engineering Chair (Grant No. RCSRF1617/4/11), which is grate- fully acknowledged. We wish to acknowledge the support from the European Union under Grant Agreement No. 312778 as well as the support from the German Research Foundation under Grant No. PU436/1-2 (the camera was sponsored under Grant No. BR 1491/30-1). Moreover, we thank Vladimir Mikulich, Sabine Abawi, and Vigimantas Mitschunas for their technical assistance to run the experiment. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1L. Prandtl, “Bericht ueber Untersuchungen zur ausgebildeten Turbulenz,” Z. Angew. Math. Mech. 5, 136 (1925). 2H. Ludwieg, “Bestimmung des verhaeltnisses der austauschkoeffizienten fuer waerme und impuls bei turbulenten grenzschichten,” Z. Flugwiss .4, 73 (1956). 3J. D. Scheel and J. Schumacher, “Local boundary layer scales in turbulent Rayleigh–Bénard convection,” J. Fluid Mech. 758, 344 (2014). 4C. Bruecker, “Evidence of rare backflow and skin-friction critical points in near- wall turbulence using micropillar imaging,” Phys. Fluids 27, 031705 (2015). 5V. Bandaru, A. Kolchinskaya, K. Padberg-Gehle, and J. Schumacher, “Role of critical points of the skin friction field in formation of plumes in thermal convection,” Phys. Rev. E 92, 043006 (2015). 6C. H. Bruecker and V. Mikulich, “Sensing of minute airflow motions near walls using pappus-type nature-inspired sensors,” PLoS One 12, e0179253 (2017). 7R. du Puits, C. Resagk, and A. Thess, “Structure of viscous boundary layers in turbulent Rayleigh–Bénard convection,” New J. Phys. 15, 013040 (2013). 8E. Liebe, Flow Phenomena in Nature: A Challenge to Engineering Design , Com- putational Mechanics (Wit Press, Southampton, 2006). 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Schlatter, “On the fluctuating wall-shear stress in zero pressure- gradient turbulent boundary layer flows,” Phys. Fluids 23, 021704 (2011). 20MATLAB, version 9.0.0.341360 (R2016a), The MathWorks, Inc., Natick, MA. 21D. Funfschilling and G. Ahlers, “Plume motion and large-scale circulation in a cylindrical Rayleigh–Bénard cell,” Phys. Rev. Lett. 92, 194502 (2004). 22H. D. Xi and K. Q. Xia, “Azimuthal motion, reorientation, cessation, and rever- sal of the large-scale circulation in turbulent thermal convection: A compara- tive study in aspect ratio one and one-half geometries,” Phys. Rev. E 78, 036326 (2008). 23R. du Puits, C. Resagk, and A. Thess, “Mean velocity profile in confined turbulent convection,” Phys. Rev. Lett. 99, 234504 (2007). AIP Advances 10, 075105 (2020); doi: 10.1063/5.0006610 10, 075105-10 © Author(s) 2020

5.0004346.pdf

J. Appl. Phys. 127, 173104 (2020); https://doi.org/10.1063/5.0004346 127, 173104 © 2020 Author(s).Strong two-photon absorption in ErFeO3 thin films studied using femtosecond near- infrared Z-scan technique Cite as: J. Appl. Phys. 127, 173104 (2020); https://doi.org/10.1063/5.0004346 Submitted: 11 February 2020 . Accepted: 17 April 2020 . Published Online: 06 May 2020 Anshu Gaur , Mahamad Ahamad Mohiddon , and Venugopal Rao Soma ARTICLES YOU MAY BE INTERESTED IN Mid-IR photothermal beam deflection technique for fast measurement of thermal diffusivity and highly sensitive subsurface imaging Journal of Applied Physics 127, 173101 (2020); https://doi.org/10.1063/1.5144174 Nano-spectroscopic and nanoscopic imaging of single GaN nanowires in the sub-diffraction limit Journal of Applied Physics 127, 173103 (2020); https://doi.org/10.1063/1.5128999 Excitonic effects on photophysical processes of polymeric carbon nitride Journal of Applied Physics 127, 170903 (2020); https://doi.org/10.1063/5.0005825Strong two-photon absorption in ErFeO 3thin films studied using femtosecond near-infrared Z-scan technique Cite as: J. Appl. Phys. 127, 173104 (2020); doi: 10.1063/5.0004346 View Online Export Citation CrossMar k Submitted: 11 February 2020 · Accepted: 17 April 2020 · Published Online: 6 May 2020 Anshu Gaur,1,a) Mahamad Ahamad Mohiddon,2 and Venugopal Rao Soma3,a) AFFILIATIONS 1School of Physics, University of Hyderabad, Prof. C.R. Rao Road, Gachibowli, Hyderabad 500046, Telangana, India 2Department of Physics, SRM University, Delhi-NCR, Sonepat 131029, India 3Advanced Center of Research in High Energy Materials (ACRHEM), University of Hyderabad, Prof. C.R. Rao Road, Gachibowli, Hyderabad 500046, Telangana, India a)Authors to whom correspondence should be addressed: gauranshu20@gmail.com andsoma_venu@uohyd.ac.in ABSTRACT Ultrafast nonlinear optical (NLO) characterization of ErFeO 3thin films deposited by the solgel assisted spin coating technique is reported herein. In the present investigation, we have used femtosecond (fs) pulses for studying the nonlinear absorption and refraction properties ofErFeO 3thin films. Intensity dependent open and closed aperture Z-scan experiments were performed on ErFO 3films in the visible and near-infrared wavelengths of 600 nm, 800 nm, and 1200 nm. To explain the observed NLO results, phenomenological models of simultaneous multi-photon and excited sate absorption are developed for three-level model systems suitable for excitation wavelengths with(600 and 800 nm) and without (1200 nm) linear absorption, respectively. Optical limiting (OL) characteristic is shown to exist through thephenomenon of two-photon absorption in a certain intensity range at the three wavelengths and under the simultaneous saturation oflinear absorption at 600 nm and 800 nm. The upper limit of the intensity for OL applications is demonstrated by the saturation effect. The non-linear absorption results are correlated with the linear absorption at these wavelengths, which involves electronic transitions between Fe d- and O p-orbitals. Published under license by AIP Publishing. https://doi.org/10.1063/5.0004346 I. INTRODUCTION Orthoferrites (OFs) with a chemical formula of RFeO 3 (R represents rare earth elements) are a representative of a standard magnetic system with an extremely high domain wall velocity(10 6m/s)1,2and spin reorientation transition (SRT).3These charac- teristics are unique among ferromagnetic materials. Rather, free wall-motion expresses the potential of OFs for faster data manipu- lation, whereas SRT (switching of an easy axis of magnetizationbetween a axis to c axis back and forth with a change of tempera-ture) can increase the information storage capacity. The peculiarity of OF ’s magnetic structure is delivered through mixed valency iron (Fe 2+,F e3+) and single valency rare earth (R3+) element-based two magnetic subsystems. The magnetic ordering in RFeO 3occurs through R –R magnetic moment interaction at very low tempera- tures, whereas room temperature magnetism is brought by ironmoments. It is shown by De Jong4and Kimel et al.5that the spin dynamics of OF-magnets can be tuned by the inverse Faradayeffect by the interaction with ultrashort laser pulses. 6The switching time of the domains, as a result of such an interaction, is reduced to few picoseconds, which is considered as another step forwardin the direction of utilizing OFs for faster magnetic data manipula-tion. Later on, a number of studies were performed on thecharacterization of laser induced spin dynamics of ErFeO 3, TmFeO 3, and other orthoferrites.6,7RE-orthoferrites, especially TbFeO 3,Y F e O 3,and ErFeO 3are though subjected to ultrafast laser pulses (at low temperatures), yet no work has been reported toassess the nonlinear optical (NLO) properties of these materials tothe best of our knowledge. A similar perovskite, orthochromites YCrO 38,9is shown to have two-photon assisted excited state absorption with nanosecond laser pulses at 532 nm excitationwavelength ( λ exc).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 173104 (2020); doi: 10.1063/5.0004346 127, 173104-1 Published under license by AIP Publishing.Materials with well-defined and strong NLO characteristics have revolutionized the field of laser and optics and formed ground of many modern days ’technologies such as ultrafast optics, ultrashort lasers, photonics, and optoelectronics. Depending uponwhether materials ’absorption increases or decreases with the intensity, two kinds of NLO effects, respectively, are classified, (1) reverse saturable absorption (RSA) suited to optical limiting, pulse shaping applications 10–12and (2) saturable absorption materials useful for mode locking applications.13,14The present work deals with the intensity and wavelength dependent study of NLO charac-teristics of ErFeO 3films deposited on the quartz substrate under the irradiation of femtosecond laser pulses in the standard Z-scan technique. Phenomenological models of non-linear absorption(NLA) including the saturation suitable at these wavelengths aredeveloped and applied to estimate the NL absorption and refractionparameters. For example, open aperture Z-scan spectra of ErFeO 3 at 800 nm are the attribute of linear and NL (two-photon) absorp-tion, which at higher input power is shown to saturate. A widerscope of the material is searched by intensity and wavelengthdependent characterizations. II. MATERIALS AND METHODS ErFeO 3films with a thickness of ∼1μm were deposited by the solgel assisted spin coating technique. The ErFeO 3precursor sol was prepared using high purity nitrates, erbium nitrate pentahydrate[Er(NO 3)3⋅5H2O, metals basis], and iron (III) nitrate nonahydrate [Fe(NO 3)3⋅9H2O]. Acetone and 2-methoxy ethanol were used as solvents. Stoichiometrically weighed erbium nitrate pentahydrate andiron nitrate nonahydrate were dissolved in acetone and 2 methoxyethanol solvents, respectively, at 80 °C on a magnetic stirrer. Thesetwo dissolved solutions were mixed drop by drop with a constant stirring. The pH value of the solution during the mixing was main- tained between 2 and 3 by adding diethylamine. The resultingsolution was cooled down to room temperature and then kept forstirring for 24 h. This sol was used to prepare ErFeO 3thin films by the spin coating technique. ErFeO 3thin films were coated on a quartz substrate at 3000 rpm for 1 min. These coated films were sub-jected to heat treatment at 100 °C for 2 h in a hot air oven to removethe residual solvents. Then, the films were heat treated at 350 °C for1 h to remove the organic materials. Annealing was carried at 700 °C for the duration of 2 h. The thickness of the film was measured by a surface profilometer (Ambios Tech., USA, Model XP-1). Thestructural characterization of the deposited films was carried outusing an x-ray diffractrometer (Bruker Model: D8 Discover) andmicro-Raman spectroscopy (WiTec Germany, Model alpha 300). The x-ray diffractograph was recorded in the grazing incident geom- etry with Cu k αradiation, incident on the film sample at an inci- dence angle of 0.5°. The Raman spectra of the ErFeO 3thin film were recorded in air using Nd-YAG laser at 532 nm in the backscatteringgeometry. A 100× objective was used to focus the sample and a Peltier cooled detector was used to record the Raman spectra. Linear absorption properties of ErFeO 3thin films were investigated in the wavelength range of 200 –1100 nm using a double beam UV –Visible spectrophotometer (JASCO model V-570). Atomic force microscopy measurements were carried in a contact mode using microscopy in a special configuration (WiTec Germany, Model alpha 300).Nonlinear refraction (NLR) and absorption properties of ErFeO 3thin films are characterized in closed and open aperture Z-scan configurations, respectively. The Z-scan experiments at800 nm were performed with a Ti-Sapphire pulsed laser beam(pulse width: 50 fs, repetition rate: 1 kHz). The amplifiers wereseeded with ∼15 fs pulses from the oscillator (Coherent, Micra). The femtosecond pulses at wavelengths of 600 and 1200 nm were derived from an optical parametric amplifier (Light conversion,TOPAS) operating at 1 kHz and with the above mentionedTi-Sapphire laser at its input. In the experimental arrangement,3 mm diameter laser beam was focused to the 1 μm thin film sample (deposited on a quartz substrate) using a plano-convex lens of 15 cm focal length. The transmitted light was collected by a pho-todetector (Si photodiode, SM1PD2A, Thorlabs) connected with alock-in-amplifier set to detect a signal of 1 kHz. The sample wasplaced on a motorized sample stage that can move (0.25 mm/step, 0.5 mm/step) in the beam direction (i.e., Z direction). The auto- mated sample stage and the photodiode are interfaced togetherthrough LabVIEW programming on a computer. The schematicand other details of the Z-scan experimental arrangement are pre-sented elsewhere. 15Input intensity dependent NLO characterization was performed at the wavelengths of 600 nm (2.07 eV), 800 nm (1.55 eV), and 1200 nm (1.03 eV). III. THEORETICAL BACKGROUND The closed aperture (CA) Z-scan data which characterize the NLR properties of ErFeO 3were fitted with the standard relation for transmittance derived by Sheik-Bahae et al.16The relation is devel- oped by solving the Lambert Beer-type differential equations forphase shift Δ wand the transmitted intensity Ιfrom the sample, given as dΔw dz0¼Δn(I)k, (1) dI dz0¼α(I)I, (2) where z0is the sample penetration depth, Δnis an intensity depen- dent part of the refractive index, and αis the total absorption coef- ficient of the sample which, in general, contains linear and NL contribution. Equations (1)and(2)when solved for Δwunder the constraint of cubic nonlinearity and no NLA results to on-axisphase shift at the focus and Z-scan transmittance at the exit surfaceof the sample as Δ w0(t)¼kΔn0(t)Leff, (3) T¼1/C04z zRΔ;0 1þz zR/C18/C192 ! 9þz zR/C18/C192 ! , (4) where Δn0(t)¼n2I00(t) with I00(t) being the on-axis intensity at the focus, k¼2π/λis the wave vector, Leff¼(1/C0e/C0α0L)/α0isJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 173104 (2020); doi: 10.1063/5.0004346 127, 173104-2 Published under license by AIP Publishing.known as the samples ’effective length α0being the linear absorp- tion coefficient, Lthe sample ’s actual length. CA data are fitted using Eq. (4)treating Δw0as a fit parameter. The open aperture (OA) Z-scan data are fitted by solving Eq.(2)for sample thickness z ’= L with αcontaining terms of linear, multiphoton, and excited state absorptions ( Fig. 1 ), α(I)¼α0þβIþγI2þα1þα2, (5) where α0,β, and γare the linear, two- and three-photon absorption coefficients, α1and α2are excited state absorption (ESA) coeffi- cients corresponding to the transitions from S 1(band) to S 2(band) and from S 2(band) to S 3(band), respectively. Equation (2)is solved at different sample positions (Z-values) with I 00/[1 + (z/z R)2] as the input intensity, where z Ris the Rayleigh length (= πw02/λ). Equation (5)in its current form is valid, when all the processes occur without any saturation effect and are independent of eachother, which are the case only at lower intensities and large totalpopulation density; in general, it is not valid. General expressionsfor NLA coefficients are obtained by solving the rate equations for the population at different energy states. Various possible transi- tions applicable to ErFeO 3in the four-level model system are depicted in Fig. 1 . The rate equations at S 3,S2,S1, and S 0states are given as dN3 dt¼γI3 3(hν)2þσE23N2I hν/C0N3 τ3, (6) dN2 dt¼βI2 2hν/C0N2 τ2þσE12N1I hν/C0σE23N2I hνþN3 τ3, (7) dN1 dt¼σ1(N/C0N1/C0N2/C0N3)I hν/C0N1 τ1/C0σE12N1I hνþN2 τ2, (8)dN0 dt¼/C0σ1(N/C0N1/C0N2/C0N3)I hν/C0βI2 2hν/C0γI3 3(hν)2þN1 τ1, (9) where N0,N1,N2, and N3are the population densities (No. of mol- ecules/cm3) at the S 0(ground state), S 1,S2, and S 3bands respec- tively, Nis the undepleted population density of ErFeO 3,his Planks ’ constant, νis the frequency of the light, τ1,τ2,and τ3are the relaxa- tion times at the S 1,S2,a n dS 3bands, respectively, σ1,σE12,andσE23 are the linear, ESA cross sections for S 1to S 2and S 2to S 3transitions, respectively. These equations are solved for N 1,N 2,a n dN 3under the steady state condition (i.e., dNi/dt = 0 ,i=1 , 2 , a n d 3 ) . O b t a i n e d populations are substituted in Eqs. (10)–(13) and then in (5), β¼σ2PA(N/C0N1/C0N2/C0N3)/hν, (10) γ¼σ3PA(N/C0N1/C0N2/C0N3)/(hν)2, (11) α1¼σE12N1, (12) α2¼σE23N2: (13) Depending upon whether LA is significant or insignificant at the chosen wavelengths, Eq. (5)is simplified to the one describing three NL phenomena maximum at a time. For example, for thecase of 1200 nm where LA is negligibly small [ Fig. 2(d) ], the rate equation corresponding to 1PA or N 1is not included since S 1will remain almost unpopulated (i.e., N1= 0) over a long period of time. The final form of absorption coefficients for 600 –800 nm ( α0≠0) and 1200 nm ( α0= 0) cases are obtained as α600,800 (I)¼α00þβ0IþαE12I IS1þη12I IS2/C18/C192"# 1þI IS1/C18/C19 1þη12I ISE1/C18/C19 þI IS2/C18/C192 1þη12þη12I ISE1/C18/C19 , (14) α1200(I)¼β0Iþγ0I2þαE23I IS2/C18/C192 þη23I IS3/C18/C193"# 1þI IS2/C18/C192 1þη23I ISE2/C18/C19 þI IS3/C18/C193 1þη23þη23I ISE2/C18/C19 , (15) where IS2,IS3, and ISEare referred to as the saturation intensities for 2PA, 3PA, and ESA processes, respectively, whose forms aregiven as I S1=hν/σ1PAτ1,IS2=√2(hν)2/σ2PAτ2, and IS3=3√3(hν)2/ σ3PAτ3,ISE1=hν/σE12τ2,ISE2=hν/σE23τ3, and η12=τ1/τ2η23=τ2/τ3 with σsand τsbeing the absorption cross sections and lifetime of the excited states, respectively. From Eqs. (14) and(15), it is clear that the three processes, 2PA, 3PA, and ESAs, are coupled to eachother in contrast to the models reported in the literature. 17–19 Furthermore, in the case of negligible 3PA, negligible ESA which is certainly the case of short-lived pulse (e.g., fs) excitation compared FIG. 1. Four-energy level model for an ErFeO 3material system. The possible transitions during excitation by femtosecond laser pulses of 600 nm (left),800 nm (middle), and 1200 nm (right) wavelengths are shown by vertical arrows.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 173104 (2020); doi: 10.1063/5.0004346 127, 173104-3 Published under license by AIP Publishing.to relaxation time of ESs ( ≈ps or ns)9,20and αE12,αE23¼0 and ISE1,ISE2/C29Iabove expressions are reduced to simpler forms as α600,800 (I)¼α00þβ0I 1þI IS1þI IS2/C18/C192 (1þη12), (16) α1200(I)¼β0I 1þI IS2/C18/C192: (17)The form of αin Eq. (16) governing the saturation of 1PA and 2PA is different from those reported in the literature.17–19In the limiting cases when β0is negligibly small and Ι≪ΙS2, it is reduced toα00/(1þI/IS1) and to β0I/[1þ(I/IS2)2] when there is no LA at λexc(α00≈0) and Ι≪ΙS1. The latter is nothing but Eq. (17) for pure 2PA. The linear and 2PA saturation processes for a homoge-neously broadened system are reported to be governed by models containing denominator terms 19of [1 + I/I S] and [1 + ( I/IS)2] and for an inhomogeneously broadened system, [ √(1 + I/IS)]17and [√1+( I/IS)2],17,18respectively. These forms are derived for the systems involving a single process of either of 1PA or 2PA, etc. Adding the above terms to explain the complex Z-scan graphs or to include the concurrently occurring absorption processes15,17,21–24 FIG. 2. (a) X-ray diffractogram, (b) Raman spectrum, and (c) AFM image, and (d) absorption spectrum of solgel assisted spin coated ErFeO 3film.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 173104 (2020); doi: 10.1063/5.0004346 127, 173104-4 Published under license by AIP Publishing.is merely an oversimplification. It is because under the multiple absorption situations the laser beam of intensity, I, or the photons simultaneously will be available for all the processes. In a similarway, the total number of molecules, N, will be distributed over each process of absorptions. 20,25,26In a specific comparison, the absorp- tion model presented by Rangel-Rojo et al. appears in a close rela- tion, however is not exactly the same, with the one developed in this paper. Rangel-Rojo et al. have developed the model for simul- taneous 1PA and ESA processes in terms of absorption crosssection and not in terms of absorption coefficient. Also, the satura-tion intensity parameter ( I S=hν/σ1PAτ21) is defined only for one of the processes, i.e., 1PA. If in their equation σr(=σESA/σ1PA),τr (=τ32/τ21) and ISterms are replaced with their definitions, pro- vided in the literature, we will not reach to Eq. (14) modified with β0= 0 and IS2≫I. This difference occurred because of the way the population density has been considered, while writing the rate of population increase at the FES and SES. The absorption is propor- tional to the instantaneous population at the lower energy state. Inour view, if the instantaneous population at the GS, FES, and SESare assumed to be defined by N 0,N1, and N2, respectively, such that N=N 0+N1+N2is a constant, the 1PA coefficient or cross section should be proportional to N0=N−N1+N2and not N0−N1because such a change of population is already being con- sidered in its instantaneous-type definition of N0,N1, and N2. One has to check that what changes it may cause in the estimated absorption parameters. However, it is not the focus of the present work. Finally, Eq. (2)can be solved for intensity Iby treating β0 and IS2as fit parameters if η12are known by the experiment or the literature. IV. RESULTS AND DISCUSSION A. Synthesis and structural characterization The crystalline phase of the 1- μm thick ErFeO 3thin film was confirmed through x-ray diffraction data. The XRD data of the ErFeO 3film measured in the grazing incident geometry mode are shown in Fig 2(a) . The diffractograph in the figure contains a set of four XRD peaks impregnated on a high background. The fourvisible peaks at 23.14°, 32.99°, 47.27°, and 59.16° correspond to(110), (112), (220), and (204) planes of an orthorhombic structure belonging to a Pnma space group (JCPDS file No. 001-074-1480). Using the peaks position and (hkl)-indices, lattice parameters arecalculated to be a = 5.26 Å, b = 5.58 Å, and c = 7.59 Å, which arefound to be consistent with the reported values. 27The absence of any peak at ∼30° and ∼61° discards the presence of a hexagonal phase, which is another possible structure for rare-earth ferrites.28,29 The lattice strain developed during the deposition of the film is esti- mated by comparing the XRD peak position of the recorded patternand the standard powder diffraction data given in the JCPDS data-base using the relation, Lattice strain ¼ (dobser/C0dref) dref/C2100, where dobs and drefare the interplanar spacing of the ErFeO 3sample and the reference corresponding to maximum intensity peak, i.e., (112).The strain is estimated to be +0.62% and positive sign of it representsits tensile nature. Such a low level of strain in the deposited ErFeO 3 film is due to the amorphous nature of the substrate and relativelygood thickness of the film. From the width of a strong XRD(112)-peak, the average crystallite size is calculated using Debye – Scherrer ’s equation and is found to be 73 nm. The Raman spectra of ErFeO 3film annealed at 700°C for 2 h are presented in Fig. 2(b) . There are five strong Raman active modes observed in the range of 200 cm−1to 600 cm−1, with no Raman active modes below 200 cm−1. Before, we will be able to associate these modes with any of the specific vibration of ErFeO 3, it is required to review the structure and Raman spectra detailsavailable in the literature. A detailed study 30,31on the Raman spectra has been reported at low temperatures, wherein Ramanshift mostly was below 200 cm −1. These peaks are the results of Raman scattering from magnon excitations and occur by the fluc- tuations in spin magnetic moments of rare earth ions.30,31These scattering studies on RFeO 3follow from its characteristic magnetic phenomena, e.g., spin reorientation, exhibited as a result of competi-tion between thermal and anisotropy energies. 31On the other side, limited reports are available on the Raman scattering from phonon modes.32,33It is known that ErFeO 3in a weakly distorted perovskite structure belonging to the space group of D 2h16orPnma34contains four formula units per unit cell, with R (=Er) atom occupying at thecenter of distorted octahedra formed by the oxygen atoms. Due to this, the normal phonon modes decompose and, in phonon mode representation, these modes at t h ec e n t e ro ft h eB r i l l o u i nz o n ea r e given by 31Γ=7A1g+8A1u+7B1g+8B1u+5B2g+1 0B2u+5B3g+1 0B3u. Among these modes, 24 are Raman active vibrational modes, 28 are infrared vibrational modes, and 8 are inactive modes.32These 24 Raman active modes belong to the irreducible representation of 7A 1g (xx, yy, zz), 7B 1g(xy, yx), 5B 2g(zx, xz), and 5B 3g(yz, zy) given by35 ΓRaman =7A1g+5B1g+7B2g+5B3g.The Raman modes depicted in Fig. 2(b) recorded at room temperature are related to the rotation and stretching operation associated wi th the lighter oxygen atom in the FeO 6octahedra. The occurrence of modes in different ranges follows the fact that the lighter atoms ha ve a high frequency mode compared to the heavier atoms. Weber et al. have carried out density functional theory calculations of phonon mode s in rare-earth orthoferrites and assigned the vibrational pattern to each mode.33Similar reports were published in the literature about the Raman spectra of RE ferrites including ErFeO 3, which have assigned the vibrational mode of orthoferrites30–32These modes are tabulated in Table I along with the experimentally observed data reported in Fig. 2(b) .A l lt h ed a t a reported in the literature were recorded on single crystals and so the measurements of the polarized Raman scattering were well controlled.However, in the present investigation, RE ferrite films were grown onan amorphous quartz substrate. The Raman scattering measurements were also performed in a typical configuration without any polaroid setup. Furthermore, the difference in the Raman modes position fromthe reported ones is expected to be due to the strain that develops inthe film, which alters the bond strengths of the ErFeO 3unit cell. A shift in the mode position to the h igher wave number side basically indicates an increase in the force constant of the bond. From the Raman spectra investigation, we confirmed that the ErFeO 3films are in the orthorhombic structure, whi ch supports the observations found in x-ray diffraction measurements. The quality of the ErFeO 3thin film surface was investigated by recording the morphology of the film by an atomic force micro- scope (AFM). Figure 2(c) depicts the AFM micrograph of the ErFeO 3thin film annealed at 700°C for 2 h. A uniform grainJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 173104 (2020); doi: 10.1063/5.0004346 127, 173104-5 Published under license by AIP Publishing.distribution throughout the film surface was observed. Grains are densely packed without visible voids and porosity over the entire surface of the sample. A line across the micrograph is drawn. From the data of line profile, average grain distributed and roughness arecalculated and found to be 410 nm and 18 nm, respectively. For thefilm deposited by the solgel assisted spin coating technique, 18 nm roughness is considered as relatively good quality of the film. The structural identity was further confirmed by the absorp- tion spectra analysis of the spin coated ErFeO 3thin film. The two possible phases, hexagonal and orthorhombic, exhibit fairly differ-ent absorption characteristics. 28Figure 2(c) shows the absorption spectrum of the ErFeO 3film as a function of photon energy (1 eV – 5.5 eV) and wavelength (200 –1200 nm). The absorption tendency represented by the absorption coefficient is significantly higher inthe UV region compared to those in the Vis or IR region. Thespectrum is characterized by a strong absorption peak at 4.6 eV along with three small humps in <4 eV-region. The spectrum after de-convolution suggests that ErFeO 3was characterized by four absorption bands, peaked at 1.84, 2.76, 3.4, and 4.6 eV indicated inFig. 2(d) . Small humps in the spectrum, which matched well with the literature, 29are associated with the absorption bands of ortho- rhombic ErFeO 3. It is reported that the absorption peaks of ortho- rhombic ErFeO 3are blue shifted compared to that of the hexagonal counterpart associated with stronger crystal field splitting at aFe-O6 cage. 28In the case of hexagonal structure, the crystal con- tains a Fe-O5 geometrical entity as a building structure, where an orthorhombic crystal is formed through the Fe-O6 octahedral cage.The axial interaction of O p-orbitals with Fe d-orbitals in the caseof an octahedral arrangement leads to the stronger crystal fieldsplitting and thereby electronic levels occur at higher energies than that in the hexagonal counterpart. Rai et al. 28found similar obser- vations on YFeO 3. Absorption peaks of h-YbFeO 3in their work occurred at 2.3, 2.9, 4.2, and 5.8 eV, whereas that of o-YbFeO 3 appeared at higher energies, e.g., at 2.7 eV corresponding to 2.3 eVfor a h-structure. 28Absorption peaks of h-ErFeO 3are reported to occur at 2.23, 2.94, 3.77, and 5.08 eV.29By similar correlation of YFeO 3, the peaks at 2.76, 3.40, and 4.60 eV observed in the present work are associated with o-ErFeO 3. The higher energy absorption peak corresponding to 5.08 eV of the h-structure is possibly super-imposed on the absorption saturation band of a glass substrate which commenced in Fig. 2(c) at 5.09 eV. Furthermore, these spec- tral characteristics correspond to specific transitions between differ-ent electronic energy bands of Fe, O, and Er atoms. Thelower-energy peaks at 1.84 and 2.76 eV are due to the absorption transition between Fe d and d orbitals, whereas the one at 3.40 eV have contribution of charge transfer (CT) transition from thevalence band formed by the occupied 2p oxygen orbitals to the conduction band formed by unfilled 3d iron orbitals. 36,37The absorption band at a higher energy of 4.60 eV (and the one sub- merged into the saturation region) is purely due to CT transitionbetween oxygen 2p and iron 3d orbitals. In the context of NLoptical characterization, the absorption bands centered at 2.76 eV, 3.40 eV, and 4.60 eV suggest the possibility of 2PA at 600 nm, 2PA and 3PA at 800 nm and 1200 nm. B. Nonlinear optical properties The CA Z-scan data of ErFeO 3film recorded with a low input intensity of 0.04 GW/cm2is shown in Fig. 3(a) . The input power level was chosen to be very low to suppress the NLA effects. Thenormalized transmittance presented in the diagram is the ratio ofZ-dependent transmitted power to the linear transmitted power; the linear one is measured keeping the sample far away from the focus where the NLA is absent. A “valley followed by peak ”-type of CA response in the figure is the result of intensity-dependent NLR.The sharp increase in the transmittance close to the focus observedinFig. 3(a) is associated with the increase in the sample ”s refractive index with the increase of the light intensity. On moving the sample toward the focus, the laser beam bends toward the axis sothat now more photons (increased intensity) are able to passthrough the aperture window. Thus, the transmittance increases and the graph transforms from the valley to the peak type. The NLR parameters are evaluated by fitting the experimental datausing Eq. (4). 16Fitted data are shown in the same figure by solid lines. The value of on-axis phase shift, Δ;0, the fit parameter, is estimated to be 0.65. The nonlinear refractive index, n2calculated using the defining relation of Δ;0, mentioned above, is 6.75 × 10−13cm2/W, where the positive sign represents the focusing nature of the sample. This n2of ErFeO 3value is found comparable to that of the thin film-ferrite, i.e., BiFeO 3,reported in the litera- ture,38which under 780 nm femtosecond laser pulse excitation exhibited the NLR with n2= 1.46 × 10−13cm2/W. Furthermore, order and purity (whether is of thermal origin) of the nonlinearityare also confirmed from the transmittance, i.e., the peak-valley,profile. 16The separation of peak to valley, ( ΔZp/C0v¼6:00 mm) equals to 1 :7ZR¼5:98 mm confirms that the nonlinearity in the refraction is caused by the third order process, where the overall refractive index is given as n¼n0þn2I, with n0being the linear refractive index. It is wise to mention that a separation close to1:2Z Rcorresponds to a fifth order NL process, whereas ΔZp/C0v higher than 1 :7ZRsuggests that the process may predominantly have dominance of the thermal effect.16Again, ΔZp/C0vof 5.98 mmTABLE I. Different phonon modes of ErFeO 3. Symmetry Reported theoretical mode (cm−1)31Main atomic motion33Experimental mode (cm−1) A1g … [010] pcFeO 6rotation in phase 211 A1g 273 O(1) x –z plane 287 B1g 322 [010] pcFeO 6rotation out of phase 329 A1g 434 Fe –O(2) stretching, in phase 450 A1g,B1g 505 Fe –O(1) stretching 573Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 173104 (2020); doi: 10.1063/5.0004346 127, 173104-6 Published under license by AIP Publishing.conveys that the origin of change in the refractive index is evidently not thermal in nature. The OA Z-scan data of ErFeO 3thin film were recorded at 800 nm at four input intensities of 0.13 GW/cm2, 0.18 GW/cm2, 0.49 GW/cm2, and 0.66 GW/cm2. The normalized transmittance data as a function of sample position with respect to lens ’focal point (i.e., Z = 0) are shown in Figs. 3(b) and 3(c). ErFeO 3in Fig. 3(b) illustrates reverse saturable absorption (RSA) or valley- type behavior at lower input intensities of 0.13 and 0.18 GW/cm2 where the sample transmittance drops continuously with increasing laser intensity, i.e., when the sample moves toward the focus. Thisvalley-type response can be the result of multi-photon absorption (MPA) 39–42or ESA,42–44both of which reduce the number of photons significantly in the transmitted beam. At a higher inputintensity of I 03= 0.49 GW/cm2, this RSA behavior gets limited to the off-focus region. The Z-scan graph turns to the SA-type (peak-type) in the near focus region for the saturation of the absorption process and thus transmittance is observed increasing on moving toward the focus. The NL effect of MPA or ESA is expected to sat-urate the same way as LA does on lacking the molecules at theinitial state. The complete transmittance curve of Fig. 3(c) is collec- tively referred as SA-in-RSA or W-shape behavior, 15which consis- tently exists at a higher input intensity of 0.66 GW/cm2with a relatively wider spread over a Z-coordinate. Admitting the onephoton absorption at 800 nm, the experimental data of Figs. 3(b) and 3(c) were fitted with the model containing the linear and instantaneous 2PA coefficients given by Eq. (16). In the case of Z-scan with short-lived laser pulses (e.g., femtosecond), the FIG. 3. Z-scan graph of solgel assisted spin coated ErFeO 3film at 800 nm in (a) closed aperture configuration at an input intensity of 0.04 GW/cm2and in [(b) and (c)] open aperture at an input intensity of (b) I 01= 0.13 and I 02= 0.18 GW/cm2and (c) I 03= 0.49 and I 04= 0.66 GW/cm2. Scattered symbols represent the experimental data where as solid lines correspond to the fitted data.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 173104 (2020); doi: 10.1063/5.0004346 127, 173104-7 Published under license by AIP Publishing.transition from ES is almost instantaneous and the difference between ESA and 2PA is of least significance.20The molecules at 800 nm can be excited by 2PA involving transition between Fe dand d orbitals and CT transition between Fe d and O p-orbitalsand pictorially to the upper part of band-S 1and/or lower part of band-S 2ofFig. 1 [since the absorption peaks at 2.76 and 3.40 are overlapped, Fig. 2(d) ].36,37The data are fitted with the model expression given by Eq. (16) taking the lifetime of excited states, S 1 and S 2, such that the ratio, η12=τ1/τ2= 10. The fitted curves are shown by the solid lines in the same figure. The Z-scan data atpeak intensities of 0.13 GW/cm 2and 0.18 GW/cm2were initially fitted with the model comprising 1PA and 2PA without any satura- tion effects (by setting Is1,Is2≫I0,I00). The data did not fit satis- factorily to the experimental values. To fit better, one more process,the saturation of LA was included in the model [by taking the rela-tively finite value of I S1in Eq. (16)]. The close match of the data with such a generalized model suggests the simultaneous presence of two NLA phenomena. The saturation effects however are suchlower that the transmittance curve remains in the valley-shape anddoes not transform to the RSA-in-SA shape, which is commonlyknown as signature graphs of the 1PA saturation process. 15,21 Saturation of LA is also a NL phenomenon which occurs due to the bleaching of the molecules at the ground energy state. At highintensities, the molecules spend longer time at the ES than at theGS and thus creating its deficit at the GS. Higher the intensity, more the deficit and higher will be the saturation and thus the transmission. Therefore, a competing “decreased absorption ” increases the wideness of the RSA valley and reveals the ErFeO 3’s ability for smooth better control of the intensity in OL applications.Furthermore, these materials can be used in pulse-shaping applica- tions. At higher input intensities, the saturation of 2PA is consid- ered to fit near-focus saturation behavior. The estimated NLparameters β 0,Is1, and Is2for all the four input intensities are quoted in Table II . It is observed that the β0-value increases on increasing the input intensity from 0.13 GW/cm2to 0.66 GW/cm2. The change is for the involvement of more and more molecules in the 2PA with increasing intensity. The 2PA cross section is calcu-lated using Eq. (10) except the instantaneous population density at GS (i.e., N–N 1–N2) is replaced by the undepleted population density N. The value of Nis calculated from the XRD data analysis and observed to be 8.04 g/cm3. It is common practice to considerN, the undepleted concentration, for the calculation of cross section.23The 2PA saturation intensity, IS2, for the cases of 0.3 and 0.4 GW/cm2follow a reverse trend of β0values as expected from the definition of IS2;IS2is inversely proportional to σ2PA. Conversely, σ1PA (IS1) under the saturation effect increases (decreases) when the input intensity increases from 0.18 to 0.49 and then to 0.66 GW/cm2. As mentioned above, the absorption sat- uration is caused by the bleaching of the molecules in the GS. Themolecules excited to SES by 2PA get de-excited to the original statein a shorter time compared to the molecule which is excited to theFES by 1PA. With the increase in the input intensity, as more mol- ecules involve in the 2PA, 1PA might be facilitated by these de-excited molecules, thereby leading to an increase in σ 1PAand a decrease in IS1. A decrease in IS1(when defined in the same way21,23) on increasing the input power is reported for other com- positions also.21Overall, the transmittance curve and the NLO parameters of ErFeO 3atλexc= 800 nm suggest that 2PA and satura- tion of 1PA can occur simultaneously while preserving the RSAbehavior. ErFeO 3can still be used for OL applications at input intensities in the range of 0.13 –0.18 GW/cm2. On the other hand, with W-shape, the optical limiting (OL) application can be of selec- tive nature where a range of intensities will be blocked by the mate- rial while out of the range will be allowed. The OA Z-scan experiments were also performed at the wave- lengths of 600 nm and 1200 nm, whose recorded graphs are shown inFigs. 4(a) and 4(b), respectively. ErFeO 3film, similar to the 800 nm-case, exhibited RSA characteristics at 600 nm. This RSAcharacteristic sustains at the three inputs of I 01= 0.21 GW/cm2, I02= 0.43 GW/cm2, and I 03= 0.66 GW/cm2, in the systematically increasing degree with the input intensity associated with the increase in the NLA cross section.15Under the same experimental conditions (initial beam spot size and converging lens focallength), the transmittance at 600 nm was observed to drop fasterwith respect to sample position (i.e., intensity) compared to theone at 800 nm, thereby giving a narrower appearance to the overall RSA response; this narrowness to a certain degree is related to Rayleigh length, Z R, thereby laser intensity profile, I(Z) (¼I00/{1þ(Z/ZR)2}); Z Rchanges from 6.08 mm to 4.56 mm. The RSA or OL characteristic is primarily associated with 2PA causedby the CT transition from Fe d-orbital to O p-orbital (and to the level S 3inFig. 1 ). Absorption by two photons is also reflected in TABLE II. NLA parameters of solgel assisted spin coated ErFeO 3films estimated from the fs Z-scan experiments at excitation wavelengths of 600 nm, 800 nm, and 1200 nm. GM—Goppert –Mayer unit of two-photon absorption cross section, 1 GM = 10−50cm4s λ(nm) I 0(GW/cm2)I S1(W/cm2) β0(cm/W), σ2PA(GM) I S2(W/cm2) 600 0.21 8 × 10146.9 × 10−10, 1.28 … 0.43 8 × 10148.5 × 10−10, 1.58 … 0.66 3 × 10139.7 × 10−10, 1.80 … 800 0.13 4.00 × 10112.38 × 10−8,3 3 … 0.18 4.50 × 10112.50 × 10−8,3 5 … 0.49 5.51 × 10101.04 × 10−7, 145 … 0.66 1.95 × 10102.75 × 10−7, 383 9.35 × 1011 1200 3.47 … 5.80 × 10−13, 0.5 mGM … 5.26 … 1.98 × 10−12, 1.84 mGM 9 × 1011Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 173104 (2020); doi: 10.1063/5.0004346 127, 173104-8 Published under license by AIP Publishing.the linearity of log eof the on-axis transmittance at the focus vs log e of input intensity with a slope close to 1; higher slopes of 2, 3, 4, and so on corresponds to higher order multi-photon absorptionprocesses. 41The valley-type data similar to the 800 nm-case are fitted by the model expression given by Eq. (16) involving linear absorption [S 0→S1, refer to Fig. 2(b) ] and 2PA (S 0→S3) with η13=τ1/τ3= 100. The fitted graph is shown in the same as Fig. 4(a) with solid lines and the estimated NL parameters are listed inTable II . The NLA coefficient of ErFeO 3atλexc= 600 nm is of the order of 10−10cm/W, 1 –2 orders of magnitude lower than that obtained at 800 nm. The observed decrease in the absorption coeffi-cient suggests that it is not only the LA at λ exc/2 that governs the NLA or MPA (discussed again at a later stage). 2PA cross-section not following the trend of LA is reported by Olesiak-Banska et al. for Au-nanoparticles.23The 2PA coefficient systematicallyincreased from 6.9 × 10−10cm/W to 9.7 × 10−10cm/W for a change of the input intensity from 0.21 GW/cm2to 0.66 GW/cm2. The RSA curve at the highest input intensity is fitted with the 1PA satu-ration to define correctly the wideness of the valley. NLA characteristics at 1200 nm on the other hand could have recorded at relatively higher input intensities. The OA Z-scangraph is shown in Figs. 4(b) and 4(c) at two input intensities of 3.48 and 5.26 GW/cm 2. ErFeO 3atλexc= 1200 nm demonstrated RSA behavior at the 3.48 GW/cm2and W-type strong saturation at 5.26 GW/cm2. Significantly wider RSA characteristics, spread over (−25 mm, 25 mm), is due to the higher Z Rat 1200 nm. The experi- mental data were fitted with 2PA and negligible LA [Eq. (17)]. β0in this case is observed to be of the order of 10−13with the value of 7.3 × 10−13at 1.89 GW/cm2. Smaller absorption coefficient is associated with the low LA ability at λexc/2, i.e., 600 nm FIG. 4. Open aperture z-scan graph of solgel assisted spin coated ErFeO 3films at (a) 600 nm and (b) 1200 nm. Scattered symbols represent the experimental data, whereas solid lines correspond to fitted ones.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 173104 (2020); doi: 10.1063/5.0004346 127, 173104-9 Published under license by AIP Publishing.[Fig. 2(b) ]. At higher input intensity, the strong absorption satura- tion is characterized by an absorption coefficient of 1.98 × 10−12cm/W and saturation intensity of 9 × 1011W/cm2. Like the previous cases, though the absorption saturates, the NLA coeffi-cient and thus the absorption cross section increases with theintensity. We expect an error of ±10% in the NLO coefficients pre- sented in this work due to the input power fluctuations, error in the estimation of beam waist and other fitting errors, etc. We alsochecked the contribution from the substrate alone and it was foundto be negligible. Regarding the wavelength dependent collective picture of the 2PA characteristics of the materials, it is reported that thebands become highly non-parabolic at 2PA particularly when 2 hν is significantly greater than E g. In this case, the 2PA coefficient is given as45 β0¼4πe4 /C22hc2/C18/C19P n2E3 g ! fhν Eg/C18/C19 : All the terms hold standard definition, except P is a constant, ω and n are angular frequency and refractive index at the radiationfrequency and fdefines the band structure and is a function of hν/E g. For non-parabolic model-case fis given as f(α)¼(2α/C01)1:5 α34 4531:5(4α/C01)α1:5 2 α/C01 3α/C01/C18/C192 þ1 (3α/C01)2() þ2 1532/C18/C19 1:5 α(2αþ1)1:5"# , where α¼hν/Eg. This relation in the same paper was used to compare the β-values of semiconductors (having bandgap E g, which satisfies the condition Eg/2 <hν<Egat the excitation frequency) considering non-parabolic model of bands. Later, this relation was applied to a series of semiconductors46by Stryland to show that β(normalized over band function fand the constant P) at a specific wavelength varies as inverse of cube of Eg. In other work, βof ZnO was shown to change with Ni doping for change in the bandgap following the above relation.47These works are dedi- cated to the comparison of βof materials having closely related bandgaps and excited by a common source/wavelength. On theother side, Wei et al. in a wavelength dependent study showed that βof CH 3NH 3PbX 3(X: Cl, Br, I) varies from 2.5 × 10−4cm/MW to 272 cm/MW for a change of f-parameter (h ν/Eg) from 0.524 to 0.76 following the above equation.48In the present case, the measure- ments are performed at different wavelengths which excite theErFeO 3system to different allowed energy bands as shown in Fig. 1 . Different bands are characterized with different bandgaps and band structure parameters. Also, because of overlappingabsorption peaks in the absorption spectrum [ Fig. 2(d) ], it was dif- ficult to determine band edge (and bandgap) using a Tauc plot [agraph between ( αhν) 2and hν49]. It is therefore rather complicated to draw a conclusion of the wavelength dependent NLA coefficient involving different bands. However, it is expected that differentband-structure parameter can be one of the primary reasons forobserved changes in the 2PA coefficients at 600 ( hν/E g600= 0.45) and 800 ( hν/Eg800= 0.46 for 3.4 eV and 0.56 for 2.76 eV) excitation wavelengths where Eg’s are taken from the absorption peaks [ Fig. 2(d) ]. Here, we also present a comparison of 2PA coefficients of few oxide ceramics. A similar composition YCrO 39, in a powder form, dissolved in ethylene glycol exhibited 3PA at 532 nm, with γ-absorption coefficient of 6.3 × 10−24m3/W2. 3PA absorption of YCrO 3is derived from 2PA involving CT transition (similar to ErFeO 3of the present work, at relatively closer λexc= 600 nm). Prior to another photon absorption, the YCrO 3system is reportedto undergo a radiative decay transition to a metastable state which acted as the initial state for ESA. BaTiO 3powders dissolved in eth- ylene glycol exhibited grain size dependent 2PA at 532 nm with β= 11.0 × 10−9cm/W and 6.0 × 10−9cm/W for grain sizes of 16 nm and 26 nm, respectively.50As an example to the NLO study of thin films, Ba 0.5Sr0.5TiO 3(BST)51exhibited 2PA at λexc= 800 nm with β= 1.6 × 10−10cm/W at the peak intensity of 2.0 × 1011W/cm2. The absorption characteristics of BST changed to 3PA with an increase of film deposition temperature that essentially enhanced the crys- tallinity of the film. However, in the present case of ErFeO 3, with a relatively lower peak intensity of 1.03 × 1011–1.44 × 1011W/cm2, the absorption coefficient is two orders higher than that of the BST film. This change may be associated with the higher (theoretical) density of ErFeO 3film and higher LA coefficient at λexc/2.52 Furthermore, we compare the NLA characteristics of ErFeO 3with one of the widely reported strong NLO ceramic oxides, i.e., zincoxide. NLO property of ZnO has been studied in various forms: colloidal solution of bulk powder, thin film, and composite, at dif- ferent wavelengths of 532 nm, 633 nm, and 800 nm, etc. Minet al. 53has reported a Z-scan study of ZnO nanorods of 400 nm length and 100 nm diameter grown on an indium titanium oxide(ITO) substrate by electro-deposition. 2PA coefficient at λ exc= 800 nm was observed to be 2.3 × 10−7cm/W at 1.2 GW/cm2. Lee et al. also reported a Z-scan of 1 μm long ZnO nanonrods grown on an ITO substrate at 800 nm and βwas 1.0 × 10−6cm/W with no mention of input intensity/power.54Maung et al.55 recorded a Z-scan of ZnO films of thickness of 2 μm at 532 nm and βas 4.86 × 10−5cm/W at 0.265 GW/cm2. Irimpan et al.56found 2PA in ZnO films deposited on glass at 532 nm with βof 4.59 × 10−3cm/W at 0.22 GW/cm2. Higher β-value in this particu- lar case seems partly associated with the large thickness of thesample since the absorbance of the sample was >2 and more signif- icantly involved nanosecond excitation probably invoking excited state absorption (not pure two-photon absorption). Rana et al. 47 found βof Ni doped ZnO nanorods dispersed in ethanol to be 7.6 × 10−9cm/W and 1.12 × 10−7cm/W for the rod lengths of 100Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 173104 (2020); doi: 10.1063/5.0004346 127, 173104-10 Published under license by AIP Publishing.and 200 nm corresponding to the doping level of 0 and 10 mol. %, respectively. The measurements were performed at 532 nm with an input intensity in the range of 0.1 –0.3 GW/cm2. The β-increase was associated with the changes in the bandgap by Ni doping.Collectively, the β-value of ZnO at 532 nm is 2 –3 orders higher than that at 800 nm though the NLA coefficient does not appear to be significantly different. 47In comparison, βof ErFeO 3thin films is comparable to that of ZnO at 800 nm and slightly lower at600 nm. The effect of preferred orientation on the NLO properties can be important for films and confined structures. 53,54For example, 100 and 500 nm diameter nanorods of zinc oxide exhibit 2PA with coefficient, β, of 2.3 × 10−7and 5.9 × 10−7cm/W, respectively.50,53If a Z-scan is recorded on a sample having oriented grains, the trans-mittance at λ exccan be different from the one with differently or non-oriented grained samples. This change in the NLO will be brought up by the variation in the bandgap, thereby the absorption characteristics. In the present case, due to the polycrystalline natureand∼1μm thickness of the film (confirmed from the Powder dif- fraction in a conventional θ–2θconfiguration), the anisotropy derived by a two-dimensional film-structure is believed to be minimum or insignificantly small. However, one may also expect preferred orientation in ErFeO 3to result during a Z-scan from the magnetic field linked with the laser beam itself. The strength of themagnetic field can be really significant in high intensity laser pulses. 57To speculate about the anisotropic effect, we need to re-examine the used Z-scan experimental condition (laser polariza-tion state and incubation/existing time) and the magnetic propertyof ErFeO 3. Z-scan graphs have been recorded with a plane polar- ized laser beam having electric and magnetic field vectors predomi- nantly confined in the plane of the film. The domain orientation by this linked magnetic field, if occurs, therefore will be confined inthe plane of the film. Furthermore, the Z-scan experiment lastedfor approximately 10 min and similar time was given as a gap tothe next Z-scan run. Along with this, during the scan, the ferrite film sample experiences pulsed optical signal of 50 fs duration at the repetition rate of 1 kHz. Thus, during one second (10 min) oftime, the magnetic field was available only for ≈50 ps (3 ns) which is really short for any significant domain orientation. Furthermore,ErFeO 3at room temperature exhibits a narrow M –H hysteresis loop with small coercive field and remnant magnetization. The magnetization does not saturate but keeps on increasing with fieldat higher intensities due to its AFM characteristic. 58Under all above mentioned experimental conditions, we expect the preferen- tial domain orientation due to laser-linked magnetic field and thus caused anisotropy in the NLO properties to be low or negligible.Also, we had performed Z scan 2 –3 times at each input intensity. If in one run, the ErFeO 3sample underwent preferred orientation, the next run might have change in the transmittance. However, we did not observe significant changes in the transmittance. Also, we have recorded the Z-scan graphs ( Figs. 3 and 4) in the increasing order of input intensity so that low intensity experiments were notaffected by the magnetic history of the sample. However, it alsomight be possible that changes in the transmittance due to such anisotropic effect are small and of the same order that occurs by small fluctuations in power and therefore, cannot be distinguished.It is also reported in the literature 59that a Z-scan performed withdifferent polarizations state (linearly, circularly, and elliptically) resulted into only small changes in the transmittance. The scan was performed on a dye sample with Ar-laser beam.59 V. CONCLUSIONS In summary, the present work demonstrates the use of solgel assisted spin coated 1 μm thick ErFeO 3films as a reverse saturable absorber at the wavelengths of 600 nm, 800 nm, and 1200 nm.The structural properties of the deposited film were confirmed bythe x-ray diffraction and micro-Raman spectroscopic techniques. The NLO properties of ErFeO 3film sample were characterized by femtosecond Z-scan experiments at different intensities at threeexcitation wavelengths of 600 nm, 800 nm, and 1200 nm. Theoptical nonlinearity of ErFeO 3is mainly of the third-order caused by 2PA which is shown to be saturated at the higher intensities at both 800 nm and 1200 nm. A generalized theory is developed for three simultaneously occurring processes, i.e., 1PA, 2PA, and ESA(2PA, 3PA, and ESA2) along with their saturation. These equa-tions are shown to turn to simple models, e.g., hyperbolic for thesaturation of linear absorption in a homogeneously broadened system, in the limiting conditions. 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5.0011433.pdf

Appl. Phys. Lett. 116, 232406 (2020); https://doi.org/10.1063/5.0011433 116, 232406 © 2020 Author(s).Field-free switching of magnetic tunnel junctions driven by spin–orbit torques at sub- ns timescales Cite as: Appl. Phys. Lett. 116, 232406 (2020); https://doi.org/10.1063/5.0011433 Submitted: 21 April 2020 . Accepted: 02 June 2020 . Published Online: 11 June 2020 Viola Krizakova , Kevin Garello , Eva Grimaldi , Gouri Sankar Kar , and Pietro Gambardella ARTICLES YOU MAY BE INTERESTED IN Non-magnetic origin of spin Hall magnetoresistance-like signals in Pt films and epitaxial NiO/ Pt bilayers Applied Physics Letters 116, 022410 (2020); https://doi.org/10.1063/1.5134814 Current-induced spin–orbit torque efficiencies in W/Pt/Co/Pt heterostructures Applied Physics Letters 116, 072405 (2020); https://doi.org/10.1063/1.5133792 Magnetization switching induced by magnetic field and electric current in perpendicular TbIG/Pt bilayers Applied Physics Letters 116, 112401 (2020); https://doi.org/10.1063/1.5140530Field-free switching of magnetic tunnel junctions driven by spin–orbit torques at sub-ns timescales Cite as: Appl. Phys. Lett. 116, 232406 (2020); doi: 10.1063/5.0011433 Submitted: 21 April 2020 .Accepted: 2 June 2020 . Published Online: 11 June 2020 Viola Krizakova,1,a) Kevin Garello,2 EvaGrimaldi,1Gouri Sankar Kar,2and Pietro Gambardella1,a) AFFILIATIONS 1Department of Materials, ETH Zurich, 8093 Zurich, Switzerland 2imec, Kapeldreef 75, 3001 Leuven, Belgium a)Authors to whom correspondence should be addressed: viola.krizakova@mat.ethz.ch andpietro.gambardella@mat.ethz.ch ABSTRACT We report time-resolved measurements of magnetization switching by spin–orbit torques in the absence of an external magnetic field in perpendicularly magnetized magnetic tunnel junctions (MTJs). Field-free switching is enabled by the dipolar field of an in-plane magnetized layer integrated above the MTJ stack, the orientation of which determines the switching polarity. Real-time single-shot measurements provide direct evi - dence of magnetization reversal and switching distributions. Close to the critical switching voltage, we observe stochastic reversal events due to a finite incubation delay preceding the magnetization reversal. Upon increasing the pulse amplitude to twice the critical voltage, the reversal becom es quasi-deterministic, leading to reliable bipolar switching at sub-ns timescales in zero external field. We further investigate the switching proba bility as a function of dc bias of the MTJ and external magnetic field, providing insight into the parameters that determine the critical switching voltage. Published under license by AIP Publishing. https://doi.org/10.1063/5.0011433 Current-induced spin–orbit torques (SOTs) allow for manipulating the magnetization of diverse classes of magnetic materials and devices.1–7 Recent studies on ferromagnetic nanodots and magnetic tunnel junctions(MTJs) have shown that SOT-induced switching can overcome spintransfer torque (STT) switching in terms of speed, reliability, and endur- ance. 8–16Fast and reliable deterministic switching of MTJs is especially important for the development of non-volatile magnetic random access memories.17–19However, when switching a perpendicular magnetization by SOT, a static in-plane magnetic field is required to break the torquesymmetry, which, otherwise, does not discriminate between up and down magnetic states. 2As such a field is detrimental for memory applications, various approaches have been proposed to achieve field-free SOT switch-ing, including exchange coupling to an antiferromagnet, 20–22RKKY,23 Dzyaloshinskii–Moriya24coupling to a reference ferromagnet, tilted mag- netic anisotropy,25geometrical asymmetry,26–28and two-pulse schemes.29–31An alternative approach, compatible with back-end-of-line integration of MTJs on CMOS wafers, is based on embedding a ferromag-net in the hard mask that is used to pattern the SOT current line. 32In such devices, the rectangular shape of the magnetic hard mask (MHM) provides strong shape anisotropy along the current direction, thus gener-ating an in-plane field on the free layer without imposing additional over- heads on the processing and power requirements of the MTJs [ Fig. 1(a) ]. In this Letter, we report on real-time measurements of field-free magnetization switching in three-terminal MTJs including a MHM.Whereas the switching dynamics has been extensively studied in two-terminal MTJs operated by STT, 33–38there are only a few studies addressing the transient dynamics and real-time reversal speed of indi-vidual SOT-induced switching events in three-terminal MTJs. 16,39In particular, the SOT-induced dynamics, reversal speed, and critical voltage in MTJs with perpendicular magnetization have not beeninvestigated in the absence of an external magnetic field. Our devices are top-pinned MTJs with 108% tunneling magneto- resistance (TMR) patterned into a circular pillar with a diameter of80 nm. The pillars are grown on top of a 190 nm wide b-W SOT- current line with a resistivity of 160 lXcm and a resistance of 370 X. The MTJ is formed by free and reference layers made of CoFeB with thicknesses of 0.9 and 1 nm, respectively, separated by an MgO tunnelbarrier with a resistance-area product of 20 Xlm 2,a ss h o w ni n Fig. 1(a). The reference layer (yellow) is pinned to a synthetic antiferro- magnet (black, SAF). The reference layer and the SAF generate an out- of-plane dipolar field /C2513 mT, which favors the antiparallel (AP) over the parallel (P) state of the MTJ. The hard mask used to pattern theSOT line incorporates a 50 nm thick Co magnet, 32which provides an in-plane field ( l0HMHM/C2540 mT) parallel to the SOT line [ Figs. 1(a) and1(b)]. Due to its high aspect ratio of 130 /C2410 nm2,t h em a g n e t i - zation direction of the Co hard mask remains constant after saturation.Moreover, the magnetization of the hard mask is not influenced by puls-ing the current through either the SOT line or the MTJ pillar. Electrical measurements are performed using a setup that com- bines real-time and after-pulse readout of the MTJ resistance, as Appl. Phys. Lett. 116, 232406 (2020); doi: 10.1063/5.0011433 116, 232406-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apldepicted in Fig. 1(c) and reported in more detail in Ref. 16.A f t e r - p u l s e switching measurements consist of an initialization pulse that sets the free layer magnetization in the desired state, which is verified by a dc measurement of the MTJ resistance, and a switching pulse, also followed by a dc resistance measurement. In the time-resolved measurements, a driving voltage supplied by a pulse generator with a rise time of 0.15 ns is split into two pulses with a well-defined amplitude, which are simulta- neously fed to the input electrodes of the three-terminal MTJ device.The pulse applied to the bottom electrode ( V SOT) drives the SOT rever- sal. Altogether, the pulse applied to the top electrode and VSOTdeter- mine the potential difference across the MTJ pillar ( VMTJ). By adjusting the ratio between these pulse amplitudes, we control the instantaneous value of VMTJ, which allows for studying phenomena emerging from the bias during SOT-driven reversal, such as voltage control of magnetic anisotropy (VCMA) as well as STT switching.16In this work, we restrict ourselves to switching at zero, low, and strong bias ( VMTJ¼0,/C00.5, and /C01:65VSOT). The pulse transmitted through the device is amplified and acquired on a 20 GHz sampling oscilloscope, which allows for monitor- ing the MTJ resistance in real-time in a time window defined by the width of the driving pulse. To facilitate the analysis of different switching events, a reference trace is subtracted to each voltage trace recorded dur-ing a switching pulse. The reference trace is obtained by maintaining the MTJ in its initial state, either P or AP, by the application of an external magnetic field opposing H MHM. The resultant trace is then divided by the voltage difference between the P and AP states [ Fig. 1(d) ]i no r d e rt o obtain the normalized switching signal Vsw. Time-resolved studies of current-induced switching can be per- formed by either pump-probe measurements,11–13which yield the average magnetization dynamics, or single-shot measurements.16,33–39 Although average time-resolved measurements provide information on the reproducible dynamic behavior of the magnetization and afford a higher signal-to-noise ratio compared to single-shot measurements,stochastic dynamical processes can only be revealed in studies carried out on individual switching events. In this study, we perform both types of measurements to highlight different aspects of the reversal dynamics. In order to provide a measurable tunneling magnetoresis- tance reading in single-shot measurements, we apply a small voltagebias V MTJon the MTJ to allow for current flow through the pillar. Depending on the sign of VMTJrelative to VSOT,t h i sb i a sc a ne i t h e r assist or hinder the SOT switching. Two effects are induced by VMTJ, namely, the STT and VCMA. As the sign of the STT depends on the orientation of the reference layer and the VCMA does not, these twoeffects can be disentangled from each other. 16Here, we set the sign of VMTJsuch that STT always opposes SOT switching for the chosen ori- entation of the reference layer and HMHM. We, thus, focus primarily on SOT-induced switching, unlike previous work in which STT was used to promote switching.16,30Moreover, in our configuration, the VCMA balances the effect of the SAF dipolar field. Figures 2(a) and2(b)show representative time traces of individual P-AP and AP-P switching events obtained in zero external field for differ- ent pulse amplitudes of VSOTand VMTJ¼/C00:5VSOT, which corre- sponds to a current density jMTJ<30% of the critical STT switching current. The switching traces reveal that the reversal of the free layer starts after a finite incubation time ( t0) followed by a single jump of the resis- tance during a relatively short transition time ( Dt), after which the mag- netization remains quiescent in the final state until the pulse ends. The reversal dynamics is, thus, qualitatively similar to that observed in the presence of an external magnetic field.16Accordingly, we attribute t0to the time required to nucleate a reversed domain and Dtto the time to propagate a domain wall across the free layer.11,16Noise in the time traces noticeably increases upon increasing VSOTand with the time elapsed from the pulse onset, which we associate with the rise of the device tem- perature during the pulse. Each reversal trace is fit by a linear ramp, and the characteristic times t0andDt,d e fi n e di n Fig. 1(d) , are extracted from the line breakpoints. The distributions of t0andDtrepresenting statistics over 200 single-shot measurements are plotted in Figs. 2(c) and2(d).A t low pulse amplitude, typical values of t0significantly exceed Dt.H o w e v e r , both the center and the width of the t0distribution can be reduced by more than one order of magnitude by increasing VSOT. To investigate the characteristic times in more detail, we extract t h em e d i a na sw e l la st h el o w e ra n du p p e rq u a r t i l e so fe a c hd i s t r i b u -tion and plot them as a function of jV SOTj.Figure 2(e) shows that the median t0decreases to below 1 ns for both AP-P and P-AP switching when increasing VSOTfrom 400 to 560 mV. At the lowest pulse ampli- tudes, the median t0of the AP-P switching configuration is twice as long as that compared to P-AP switching. This asymmetry, which is attributed to the dipolar field of the SAF, gradually reduces upon increasing VSOT. Additionally, our measurements show that such an asymmetry can be strengthened or eliminated by tuning VMTJ.T o demonstrate the potential of SOT-driven switching at increased VMTJ, we report in the same plot t0obtained at VMTJ¼/C01:65 V SOT(bottom curve). In this case, the difference between both configurations is mini- mized by VCMA, which favors AP-P at the expense of P-AP switch-ing, whereas the median value and its dispersion are reduced by the bias-induced temperature rise in the device. Note that STT has the same hindering effect on both switching configurations, as it favors the orientation of the free layer opposite to the final state defined by V SOT. Moreover, despite the presence of a strong opposing STT when VMTJ¼/C01:65 V SOT, which corresponds to jMTJ<96% of the critical FIG. 1. (a) Schematic and (b) TEM cross-sectional view of a field-free switching MTJ device. The field HMHM produced by the MHM is represented by gray arrows. (c) Simplified schematic of the measurement circuit for after-pulse and real-timedetection of SOT and/or STT switching. (d) Voltage difference between P and APstates measured by the scope and averaged over 300 switching events (black line). Single-shot switching voltage trace (blue line) prior to normalization (see the text) and definition of the characteristic times t 0andDt. The positive direction of the bias voltage VMTJis indicated in (a) and (c).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 232406 (2020); doi: 10.1063/5.0011433 116, 232406-2 Published under license by AIP PublishingSTT switching current in the absence of SOT, we do not observe writ- ing errors within our dataset. In contrast to t0,Dthas a much weaker dependence on VSOT. Figure 2(f) shows that 1 =Dtincreases linearly with VSOT,w i t ham o d e r - ate slope of 1.5 ns/C01V/C01. Linear scaling with current is indeed expected for the speed of SOT-driven domain walls in the flow regime.40–43 Supposing that the reversal initiates with the nucleation of a domain wallat one edge of the free layer, as shown in previous work, 11,44our data imply an average domain-wall propagation speed of 100 m s/C01induced by an SOT current density of 1 :7/C21012Am/C02atVSOT¼0.48 V. To demonstrate the reliability of switching for repeated events, we performed time-resolved measurements averaged over 1000 switching trials, as shown in Fig. 3 . Averaging the acquired waveforms allows us to decrease the bias down to VMTJ¼0:1V SOT,w h i c hc o r r e - sponds to a current density jMTJthat is <15% of the STT switching threshold. The time traces compare AP-P switching for different val- ues of VSOTgiven in multiples of the critical switching voltage ( Vc), corresponding to 50% switching probability. Each trace comprises aninitial delay and a smooth transition part without noticeable interme- diate states. Shortening of the delay and transition part of the averaged time traces indicates that the switching process changes from stochas-tic to almost deterministic upon increasing V SOT. We further observe a striking reduction of the total switching time from 15 ns to less than 1n s a t VSOT/C211:7Vc, which can be interpreted as switching at the sub-ns timescale in the great majority of 1000 trials, with the confi- dence given by the signal-to-noise ratio. Our measurements also allow us to compare the critical switch- ing voltage obtained by after-pulse, single-shot, and averaged time-resolved measurements. This comparison is relevant to assess thereliability of different methods employed to measure Vc, particularly for the more common after-pulse resistance measurements, in which t0orDtcannot be accessed. We define the critical switching time ( tc) as the pulse width corresponding to 50% switching probability in after-pulse measurements and as t0þDt=2 and the time required to reach Vsw¼0.5, in single-shot and averaged time-resolved measure- ments, respectively. Likewise, we define Vcas the corresponding SOT pulse amplitude. Figure 4(a) shows that, irrespective of the measure- ment method, all values of Vcf a l lo nt h es a m ec u r v ea n ds c a l e inversely with the critical time tcfor pulses shorter than 4 ns. Such a scaling is expected for the intrinsic regime, in which conservation of angular momentum gives 1 =tc/ðVc/C0Vc0Þ.8,45,46Here, Vc0is the intrinsic critical voltage that reflects the minimum amount of angular momentum required to achieve switching in the absence of thermalFIG. 2. Representative single-shot time traces for different pulse amplitudes of VSOTacquired at zero external field for (a) P-AP and (b) AP-P switching at VMTJ¼/C0 0.5VSOT. The characteristic times t0andDtare extracted by fitting the data to a linear ramp (black lines). (c) Statistical distribution of t0andDtfor P-AP and (d) AP-P switching. (e) Dependence of t0and (f) 1 =DtonVSOTextracted from the distributions. Symbols denote the median values, and shaded areas represent the lower and upper quartiles of the distributions. The bottom curves in (e) show t0measured with VMTJ¼/C0 1:65VSOT. The dashed line in (f) is a linear fit to 1 =Dt. FIG. 3. Averaged time traces of AP-P switching for 1.5-ns (left) and 15-ns-long (right) pulses at different VSOT pulse amplitudes close to Vcacquired at VMTJ¼0:1VSOT and zero external field. P-AP switching traces (not shown) yield similar results.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 232406 (2020); doi: 10.1063/5.0011433 116, 232406-3 Published under license by AIP Publishingeffects. A linear fit of the data in Fig. 4(a) gives Vc0¼390 mV, which corresponds to an intrinsic critical current of 1 :4/C21012Am/C02.F o r pulses longer than 4 ns, deviations from the linear behavior are attrib-uted to the onset of thermally activated switching. 8,45,46 Figure 4(b) shows that in both switching configurations, Vcreduces considerably upon increasing VMTJ.T h er e d u c t i o no f Vcis the largest for the shorter pulses and for AP-P switching (bottom panel) compared to P-AP switching (top panel). These observations can be explained by the combined impact of VCMA and heat generated by the bias. VMTJ>0 facilitates switching by weakening the anisotropy of the free layer, whereas the bias current simultaneously increases the temperaturein the device, thus lowering the switching energy barrier regardless of its sign. Therefore, when V SOTis negative (AP-P switching), both VCMA and temperature lead to a reduction of Vc; however, when VSOTis posi- tive (P-AP switching), the VCMA opposes the temperature-induced decrease in the switching barrier, resulting in a smaller reduction of Vc. Consequently, Vc, in both switching configurations, equalizes for VMTJ/C25/C01:65VSOT. This result shows that VMTJcan be efficiently used to realize symmetric switching conditions [also see Fig. 2(e) ]. Finally, we address the functionality of our devices under an external field ( Hx) by measuring the after-pulse switching probability (Psw) for different pulse amplitude and field conditions. We start by considering SOT switching at VMTJ¼0. Each point in the switching phase diagram in Fig. 5 illustrates the statistical result of 50 trials for a fixed pulse width (0.5 ns). The black boundary defines Psw¼0.5, which divides the diagrams into under- (gray) and over-critical (blue and yellow) regions. Since Hxpolarizes the MHM, which has a coercivity of about 20 mT, HMHM is always antiparallel to Hx[Fig. 1(a) ]. As a conse- quence, the switching polarity depends on the sign of jHMHM/C0Hxj, i.e., of the total in-plane field acting on the free layer. The diagram also allows for evaluating the strength of HMHM f r o mt h ed i f f e r e n c eb e t w e e n two Hxvalues resulting in the same Vc. In this manner, we estimate thatl0HMHM/C2540 mT. The diagram shows that bipolar switching is possible in a wide range of external fields with the exception of narrow intervals, in which jHMHM/C0Hxjapproaches zero. A typical transitionofPswfrom 0.01 to 0.99 occurs upon increasing VSOTby less than 80 mV, in contrast to STT switching, for which a 150 mV increase inV MTJis required using the same device and 15 ns-long pulses. Before concluding, we discuss a few possibilities to improve the switching speed and the design of field-free SOT devices. As shown inRef. 16, increasing the magnitude of the in-plane field significantly reduces the switching time for a given V SOT.I ng e n e r a l , HMHM can be increased by (i) optimizing the aspect ratio and thickness of the mag-netic layer, (ii) replacing the Co layer by a material with higher satura-tion magnetization, such as CoFe, and (iii) bringing the MHM closerto the MTJ. The optimal strength of H MHM will ultimately depend on the critical current, switching rate, and thermal stability of the freelayer set by the target application. Static measurements show that theTMR is not affected by the MHM and that the overall device proper-ties are more influenced by the design of the MTJ pillar rather than bythe hard mask itself. 32Better compensation of the out-of-plane stray field produced by the SAF would lead to a more symmetric switchingbehavior between the P and AP configurations. This can be achieved,e.g., by changing the thickness of the reference layer or the thicknessand number of repetitions of the SAF multilayer. Alternatively, asshown here, V MTJcan be used to balance the SAF field. The MHM approach is also compatible with dense designs, as shown in Ref. 32. Micromagnetic simulations further show that surrounding magnetshave a stabilizing effect on the magnetization of the hard mask.MHMs down-scaled to a volume of 50 /C2100/C225 nm 3and a pitch of 100/C2150 nm2have more uniform magnetization patterns than MHMs with a volume of 110 /C2390/C250 nm3and a pitch of 260/C2540 nm2. Future studies might establish if the MHM approach based on a single mask per MTJ is compatible with sharing the sameSOT write line between multiple MTJs, as proposed in Ref. 15. In summary, we have demonstrated field-free switching of per- pendicularly magnetized MTJs by SOT in real time. Single-shot time-resolved measurements show that the stochastic incubation delay nearthe critical voltage threshold ( V SOT/C25Vc) is several nanoseconds long for both the AP-P and P-AP switching configurations, whereas theFIG. 4. (a)Vcobtained from after-pulse probability (squares), real-time single-shot measurements (diamonds), and averaged time-resolved measurements (circles). The gray line represents a fit to the data as a 1 =tcfunction for tc<4 ns. (b) Vcfor SOT-dominated switching at different values of VMTJobtained from after-pulse prob- ability measurements for P-AP (top panel) and AP-P (bottom panel) switching. FIG. 5. SOT switching probability as a function of VMTJandl0Hxfor 0.5 ns-long pulses at VMTJ¼0. The color of each point represents the after-pulse switching probability Psw(out of 50 trials). The magnetization of the MHM was initially set to be negative (positive) for Hx<0(>0), resulting in HMHM>0(<0) as indicated by the arrows above the diagram. The AP/P labels denote the final state.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 232406 (2020); doi: 10.1063/5.0011433 116, 232406-4 Published under license by AIP Publishingactual transition time is about 1 ns. Upon increasing VSOTorVMTJ,t h e switching distributions narrow down, leading to reduced latency andquasi-deterministic switching. Averaged time-resolved measurementsshow that the total switching time can be reduced to 0.7 ns by increas-ing V SOTup to 1 :9Vcwith negligible assistance of either STT or VCMA. At timescales shorter than 4 ns, the critical switching voltageis found to scale linearly with inverse of the switching time, as expectedin the intrinsic regime. Real-time measurements and after-pulseswitching statistics as a function of pulse length are found to provide aconsistent estimate of the critical switching time. Measurements of theswitching probability as a function of V MTJand external field indicate that further improvements of the switching dynamics and reductionofV ccan be obtained by VCMA, increasing the dipolar field of the MHM, and compensating the dipolar field due to the SAF. This research was supported by the Swiss National Science Foundation (Grant No. 200020-172775), the Swiss GovernmentExcellence Scholarship (ESKAS-Nr. 2018.0056), the ETH Zurich(Career Seed Grant No. SEED-14 16-2), and imec’s IndustrialAffiliation Program on MRAM devices. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1A. Manchon, J. /C20Zelezn /C19y, I. M. Miron, T. Jungwirth, J. Sinova, A. Thiaville, K. Garello, and P. Gambardella, Rev. Mod. Phys. 91, 035004 (2019). 2I. 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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5.0009931.pdf

J. Chem. Phys. 152, 234308 (2020); https://doi.org/10.1063/5.0009931 152, 234308 © 2020 Author(s).Reinvestigation of the Rydberg W1Π(ν = 1) level of 12C16O, 13C16O, and 12C18O through rotationally dependent photodissociation branching ratio measurements Cite as: J. Chem. Phys. 152, 234308 (2020); https://doi.org/10.1063/5.0009931 Submitted: 04 April 2020 . Accepted: 31 May 2020 . Published Online: 17 June 2020 Pan Jiang , Xiaoping Chi , Guodong Zhang , Tonghui Yin , Lichang Guan , Min Cheng , and Hong Gao ARTICLES YOU MAY BE INTERESTED IN Experimental and theoretical investigations of HeNeI 2 trimer The Journal of Chemical Physics 152, 234307 (2020); https://doi.org/10.1063/5.0008760 Ab initio potential energy surface and microwave spectrum of the NH 3–N2 van der Waals complex The Journal of Chemical Physics 152, 234304 (2020); https://doi.org/10.1063/5.0011557 Rotational state-changing collisions of C 2H− and C 2N− anions with He under interstellar and cold ion trap conditions: A computational comparison The Journal of Chemical Physics 152, 234303 (2020); https://doi.org/10.1063/5.0011585The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Reinvestigation of the Rydberg W1Π(ν= 1) level of12C16O,13C16O, and12C18O through rotationally dependent photodissociation branching ratio measurements Cite as: J. Chem. Phys. 152, 234308 (2020); doi: 10.1063/5.0009931 Submitted: 4 April 2020 •Accepted: 31 May 2020 • Published Online: 17 June 2020 Pan Jiang,1,2Xiaoping Chi,1,2Guodong Zhang,1,2Tonghui Yin,1,2Lichang Guan,1,2Min Cheng,1,2,a) and Hong Gao1,2,a) AFFILIATIONS 1Beijing National Laboratory for Molecular Sciences (BNLMS), Institute of Chemistry, Chinese Academy of Sciences, Beijing 100190, China 2University of Chinese Academy of Sciences, Beijing 100049, China a)Authors to whom correspondence should be addressed: chengmin@iccas.ac.cn and honggao2017@iccas.ac.cn ABSTRACT A recent high resolution photoabsorption study revealed that the Rydberg W1Π(ν= 1) level of carbon monoxide (CO) is perturbed by the valence E′′1Π(ν= 0) level, and the predissociation linewidth shows drastic variation at the crossing point due to the interference effect [Heays et al. , J. Chem. Phys. 141(14), 144311 (2014)]. Here, we reinvestigate the Rydberg W1Π(ν= 1) level for the three CO isotopologues,12C16O, 13C16O, and12C18O, by measuring the rotationally dependent photodissociation branching ratios. The C+ion photofragment spectra obtained here reproduce the recent high resolution photoabsorption spectra very well, including the presence of the valence E′′1Π(ν= 0) level. The photodissociation branching ratios into the spin-forbidden channel C(1D) + O(3P) show sudden increases at the crossing point between the W1Π(ν= 1) and E′′1Π(ν= 0) levels, which is in perfect accordance with the drastic variation of the linewidth observed in the recent spectroscopic study. Further analysis reveals that the partial predissociation rate into the lowest channel C(3P) + O(3P) shows a much more prominent decrease at the crossing point, which is caused by the interference effect between the W1Π(ν= 1) and E′′1Π(ν= 0) levels, than that into the spin-forbidden channel C(1D) + O(3P), and this is the reason of the sudden increase as observed in the photodissociation branching ratio measurements. We hope that the current experimental investigation will stimulate further theoretical studies, which could thoroughly address all the experimental observations in a quantitative way. Published under license by AIP Publishing. https://doi.org/10.1063/5.0009931 .,s I. INTRODUCTION Carbon monoxide (CO), the second most abundant molecular species in the universe, plays an important role in determining the properties of an interstellar medium. Specifically, its photoabsorp- tion and photodissociation (PD) in the vacuum ultraviolet (VUV) region is believed to be the main reason for carbon and oxygen isotope heterogeneities as observed in the solar system;1–4it is also the main source of the excited atomic species C(1D,1S) and O(1D, 1S), that have been observed to exist in cometary coma.5,6Besidesthe relevance in the fields of astrophysics and astrochemistry, CO has also been served as a prototype diatomic molecule for studying indirect predissociation dynamics.7–9Due to the above mentioned broad interest in many different research fields, the photoabsorp- tion and photodissociation of all CO isotopologues in the VUV region have attracted numerous experimental studies with vari- ous techniques, which measured the photoabsorption line positions, photoabsorption cross sections, predissociation linewidths, band oscillator strengths, and their isotope dependences.10–14Accom- panying these experimental efforts has been a great amount J. Chem. Phys. 152, 234308 (2020); doi: 10.1063/5.0009931 152, 234308-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp of theoretical work performed to understand the complicated spectroscopic features of CO and the predissociation dynam- ics that involves numerous Rydberg–Rydberg, Rydberg–valence, and spin–orbit interactions.15–20Despite these experimental and theoretical works, detailed quantitative understanding of the pre- dissociation process has only been achieved at several low-lying vibronic levels.19,20 Of all the excited Rydberg states of CO, the W1Πstate, which is a 3sσRydberg state with an A2ΠCO+excited core, plays a particular role in the predissociation process of CO. Due to its excited ionic core, the W1Πstate has a relatively larger equilib- rium bond length than that of the ground electronic state. This not only makes direct photoexcitation from the ground vibronic state populate relatively high vibrational levels in the W1Πstate21,22 but also makes this state interact strongly with the valence states of1Πand3Πtypes, thus playing an important role in the predis- sociation process of CO.8,16,18Recently, Eidelsberg and co-workers have used the high-resolution VUV Fourier transform spectrom- eter installed on the DESIRS beamline at SOLEIL to measure the accurate band oscillator strengths and predissociation linewidths of the W1Πstate for various CO isotopologues, and strong rota- tion, parity, and isotope dependences of these parameters have been observed and documented.23–25The photoabsorption of the W1Π (ν= 1) level was investigated specifically in detail, and an accidental rotational perturbation was identified for all the five CO isotopo- logues, namely,12C16O,12C17O,12C18O,13C16O, and13C18O.26This rotational perturbation was attributed by them to the interaction with theν= 0 level of a previously unobserved valence1Πstate, E′′1Π, which has been predicted to exist theoretically before.16,27 The rotational predissociation linewidths show drastic changes at the crossing point (or the accidental perturbation) between the W1Π (ν= 1) and E′′1Π(ν= 0) levels, which was attributed to an interfer- ence effect between the two interacting levels due to the fact that both of them predissociate through couplings with the repulsive states. A recent theoretical calculation based on solving a coupled vibrational Schrödinger equation using accurate potential–energy curves for all the states involved was performed by Lefebvre-Brion and Kalemos, and the rotational dependences of linewidths at the crossing point were very well reproduced.28Despite these state- of-the-art experimental and theoretical works, the explicit multi- channel pathways from the Frank–Condon region to each of the dissociation limits are still not clear. By combining the tunable VUV laser radiation source with a time slice velocity-map imaging (TS-VMI) setup, Jackson, Ng, Gao, and co-workers have systematically measured the photodissocia- tion branching ratios into the three lowest dissociation limits C(3P) + O(3P), C(1D) + O(3P), and C(3P) + O(1D) for12C16O,13C16O, and 12C18O.29–39These measurements are important not only for astro- chemical relevance4,40but also for unraveling the complicated pre- dissociation dynamics of CO.33,38Especially, branching ratio mea- surements can help distinguish the dissociation pathways into each individual channels; this is particularly useful for understanding the predissociation process of CO for which multi-dissociation path- ways or channels often coexist simultaneously.38The recent theo- retical calculation of the photodissociation branching ratios for the W1Π(ν= 1) level of12C16O reproduced the experimental values very well.31,41This is achieved by substantially modifying the coupling strength between the triplet k3Πand 43Πlevels, which adiabaticallycorrelates with the C(3P) + O(3P) and C(1D) + O(3P) dissociation limits, respectively.41However, the above studies were not up to high enough rotational levels to reach the crossing point between the W1Π(ν= 1) and E′′1Π(ν= 0) levels as identified in the recent high resolution spectroscopic study.26Similar interference effects as iden- tified between the W1Π(ν= 1) and E′′1Π(ν= 0) levels, which dras- tically change the predissociation linewidths at the crossing point, have also been observed before in23Na39K42and N 2.43To the best of our knowledge, how this interference effect can influence the relative quantum yields between different channels (if multiple channels are present simultaneously) has never been investigated before. Thus, it would be interesting to see how this interference effect could affect the branching ratios of the W1Π(ν= 1) level of CO at the crossing point with the E′′1Π(ν= 0) level, and this is the main content of the current study. II. EXPERIMENTAL METHODS The experimental apparatus, which consists of a tunable VUV laser radiation source and a TS-VMI setup, has been described in detail before.33,44Briefly, a supersonic molecular beam of different pure CO isotopologues (Yuan-Hua in China,12C16O>99.9%; Cam- bridge Isotope,13C16O,13C = 99%,18O<5%; Linde Gas North America LLC,13C16O,13C>99%,18O<5%; and Sigma-Aldrich, 12C18O>95%,18O>99%) is generated by a Parker general valve (Series 9), which has a nozzle diameter of 0.5 mm and operates at a stagnation pressure of 30 psi and a repetition rate of 10 Hz. After passing through two conical skimmers with diameters of 2 mm, the CO beam crosses with the VUV beam perpendicularly in the photodissociation and photoionization (PD/PI) region of the TS- VMI setup. After absorbing a sum-frequency VUV photon, the CO molecule is excited to the complex region of the W1Π(ν= 1) and E′′1Π(ν= 0) levels and then undergoes predissociation to form C atoms in the3P and1D states. Both C(3P) and C(1D) can be directly ionized by absorbing a second sum-frequency VUV photon in the same laser pulse. The C+ions are extracted and focused by the elec- trostatic VMI lens onto a conventional imaging detection system, which consists of a 50 mm MCP detector, a P47 phosphor screen, and a CCD camera. The tunable VUV laser radiation source is generated by the two-photon resonance-enhanced four-wave mixing scheme using pulsed Xe gas as the nonlinear medium. A 10 Hz Nd:YAG laser (Quanta-Ray, Pro-270-10E) is used to pump two dye lasers (Sirah, Cobra-Stretch) at the same time. The output from the first dye laser is frequency-doubled through a BBO crystal and fixed at 222.568 nm (ω1), which is resonant with the two-photon transition of Xe: (5p)5(2P1/2)6p2[1/2](J = 0) ←(5p)6 1S0. The output from the sec- ond dye laser ( ω2) is tuned in the wavelength range 676 nm–685 nm, which will have the sum-frequency VUV (2 ω1+ω2) cover the com- plex region of the W1Π(ν= 1) and E′′1Π(ν= 0) levels. The exact wavelength of ω2is monitored by using a wavemeter (HighFinesse WS-6). Both of the ω1andω2laser beams are focused into a T-shape channel housed in a vacuum chamber. The T-shape channel is con- nected to the nozzle of a second Series 9 Parker general valve, and the Xe gas is pulsed into the channel when the two laser beams arrive. The four-wave mixing process occurs in the T-shape channel and the VUV laser radiation is generated. There are not any dispersion J. Chem. Phys. 152, 234308 (2020); doi: 10.1063/5.0009931 152, 234308-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp setups between the T-shape channel and the PD/PI region, and thus, all wavelength components are present in the PD/PI region, includ- ingω1,ω2, difference-frequency VUV (2 ω1−ω2), sum-frequency VUV (2ω1+ω2), and the tripling VUV (3 ω1). In this study, we need a much hotter molecular beam than before31,36in order to reach into the crossing region between the W1Π(ν= 1) and E′′1Π(ν= 0) levels. To achieve this, we change the delay of the molecular beam so that the hot portion of the beam crosses with the VUV beam; we also try to “worsen” the working condition of the pulsed valve, for example, by using worn-out plas- tic tips instead of good ones. By doing all these efforts, we are able to collect C+ion photofragment spectra up to much higher rotational levels than before.31,36 III. EXPERIMENTAL RESULTS A. C+ion photofragment spectra The C+ion photofragment spectra (C atom photofragment excitation or PHOFEX spectra) are obtained by monitoring the intensities of the C+ion signals in the Time-of-Flight (TOF) mass spectra while scanning the VUV wavelength. The C+ion photofrag- ment spectra obtained in the current experiment in the complex region of the W1Π(ν= 1) and E′′1Π(ν= 0) levels for12C16O,13C16O, and12C18O are presented in Figs. 1(b), 2(b), and 3(b), respec- tively. The rotational level assignments as shown by the droplines are adopted from Ref. 26. Due to the very large differences of sig- nal intensities between the low and high rotational levels and also between the W1Π(ν= 1) and E′′1Π(ν= 0) levels, it is not possi- ble to measure the whole spectra in a single scan under common experimental settings. We collect the spectra in several piecewise energy ranges and then properly connect them to form the C+ion photofragment spectra, as shown in Figs. 1(b), 2(b), and 3(b). In each piecewise range, we optimize the voltage setting on the MCPdetector and the threshold setting of the multichannel scaler [P7888- 2(E), FAST ComTec GmbH] to make sure that the signal is strong enough while not saturating the multichannel scaler. Thus, the rel- ative intensities of different peaks, especially those between the W1Π(ν= 1) and E′′1Π(ν= 0) levels, are not accurately determined in Figs. 1(b), 2(b), and 3(b); only the peak positions and linewidths are relatively accurate. The C+ion photofragment spectra as shown in Figs. 1(b), 2(b), and 3(b) agree well with the recent high resolution photoabsorp- tion spectra by Heays et al.26A direct comparison between them are presented in Figs. S1, S2, and S3 in the supplementary material for12C16O,13C16O, and12C18O, respectively. Most of the absorp- tion structures as observed by Heays et al.26are very well reproduced in our C+ion photofragment spectra as shown by the vertical lines, including those responsible for the E′′1Π(ν= 0) level. Those sud- denly narrow rotational lines at the crossing points as identified by Heays et al.26are also well-resolved here as indicated by red arrows in Figs. 1(b), 2(b), and 3(b). Those peaks labeled with stars were not observed in the absorption spectra, and their images contain many weak ring structures, which do not match any of the pho- todissociation channels due to single sum-frequency VUV photon absorption. The origins of these peaks are not known at the moment, which might be caused by certain multiphoton processes, consid- ering that there are many different wavelengths of laser radiations present in the PD/PI region. The satisfactory degree of agreement between the C+ion photofragment spectra and the high-resolution photoabsorption spectra provides us a good chance to investigate how the branching ratios into the channel C(1D) + O(3P) depend on the rotational levels, especially at the crossing point between the W1Π(ν= 1) and E′′1Π(ν= 0) levels. B. Branching ratio measurements The TS-VMI images are collected at most of peak positions as identified in the C+ion photofragment spectra and then converted to FIG. 1 . (a) Photodissociation branching ratios into the channel C(1D) + O(3P) and (b) C+ion photofragment spectrum in the complex region of the W1Π(ν= 1) and E′′1Π(ν= 0) levels for12C16O. The error bars represent the standard devia- tion (1σ) of three independent measure- ments. The rotational assignments are adopted from Ref. 26. The red arrows show the crossing points between the W1Π(ν= 1) and E′′1Π(ν= 0) levels as identified in Ref. 26. The peaks labeled with stars might be due to certain multi- photon process (see text for details). J. Chem. Phys. 152, 234308 (2020); doi: 10.1063/5.0009931 152, 234308-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . (a) Photodissociation branching ratios into the channel C(1D) + O(3P) and (b) C+ion photofragment spectrum in the complex region of the W1Π(ν= 1) and E′′1Π(ν= 0) levels for13C16O. The error bars represent the standard devia- tion (1σ) of three independent measure- ments. The rotational assignments are adopted from Ref. 26. The red arrows show the crossing points between the W1Π(ν= 1) and E′′1Π(ν= 0) levels as identified in Ref. 26. The peak labeled with star might be due to certain multi- photon process (see text for details). the total kinetic energy release (TKER) spectra from which the two dissociation channels C(3P) + O(3P) and C(1D) + O(3P) can be dis- tinguished according to their different kinetic energy releases. The branching ratios of different photodissociation channels are deter- mined by integrating the areas of the corresponding peaks in the TKER spectra after normalizing them to the different photoioniza- tion cross sections of C(3P) and C(1D). In this study, photoioniza- tion cross sections of 16 Mb and 29 Mb are used for C(3P) and C(1D), respectively.45,46The uncertainties of these photoionization crosssections are fairly large, which are ∼30% for C(3P) and could be even larger for C(1D). The obtained branching ratio values for dissocia- tion into the channel C(1D) + O(3P) are listed in Tables S1, S2, and S3 in the supplementary material and plotted vs the VUV photoex- citation energy in Figs. 1(a), 2(a), and 3(a) for12C16O,13C16O, and 12C18O, respectively. The uncertainties are the standard deviations (1σ) of three independent measurements, which did not include the contribution from the photoionization cross sections as mentioned above. FIG. 3 . (a) Photodissociation branching ratios into the channel C(1D) + O(3P) and (b) C+ion photofragment spectrum in the complex region of the W1Π(ν= 1) and E′′1Π(ν= 0) levels for12C18O. The error bars represent the standard devia- tion (1σ) of three independent measure- ments. The rotational assignments are adopted from Ref. 26. The red arrows show the crossing points between the W1Π(ν= 1) and E′′1Π(ν= 0) levels as identified in Ref. 26. The peak labeled with star might be due to certain multi- photon process (see text for details). J. Chem. Phys. 152, 234308 (2020); doi: 10.1063/5.0009931 152, 234308-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp We previously reported the branching ratio values for the W1Π(ν= 1) state of12C16O and13C16O at relatively low rota- tional levels, and no obvious rotation and parity dependences were noticed.31,36This agrees well with the observations of the current study as shown in Figs. 1(a) and 2(a) in the low J′region (J′is the rotational quantum number of the upper level). This study extends the measurement to much higher J′levels. Even though individ- ual rotational levels cannot be resolved from each other due to the predissociation broadening effect, complicated, while on the other hand, interesting rotational dependences of the branching ratios can be observed, as shown in Figs. 1(a), 2(a), and 3(a). For12C16O, the branching ratios into the channel C(1D) + O(3P) generally increase slowly when going to higher J′levels and those of the W1Π(ν= 1) state are slightly larger than those of the E′′1Π(ν= 0) state, as shown in Fig. 1(a). For13C16O and12C18O, which are very similar to each other due to their similar reduced masses, the percentages into the channel C(1D) + O(3P) keep constant up to very high J′levels and only start to increase slowly at about J′= 9, as shown in Figs. 2(a) and 3(a). The branching ratio values for the W1Π(ν= 1) and E′′1Π(ν= 0) states are almost equal to each other in the cases of 13C16O and12C18O. Aside from the insensitivity of the branching ratios on the rotational quantum numbers in most parts of the absorption spec- tra, the most prominent features observed in the current study are the sudden increase in the branching ratio values for the channel C(1D) + O(3P) at the crossing points between the W1Π(ν= 1) and E′′1Π(ν= 0) levels, as shown in Figs. 1(a), 2(a), and 3(a). A direct comparison of the TKER spectra of the Q(6) and Q(7, 8) rotational lines for the W1Π(ν= 1) state of13C16O is presented in Fig. 4. The relative intensities of the peaks corresponding to the channels C(3P) + O(3P) and C(1D) + O(3P) are completely reversed when J′changes from 6 to 7 and 8, implying a sudden variation of the branching ratio values. For12C16O, the sudden branching ratio increase occurs at FIG. 4 . The total kinetic energy release (TKER) spectra at the rotational transi- tions of Q(6) (blue curve) and Q(7, 8) (red curve) for13C16O. The heights of the two peaks corresponding to the channels C(3P) + O(3P) and C(1D) + O(3P) are rescaled by setting the higher peak in each spectrum to 1.the band head of the R-branch [R(5)] and at the rotational transi- tions of P(7) and Q(5, 6) of the W1Π(ν= 1) band; the R(4) line of the E′′1Π(ν= 0) band also shows a slight increase compared with those of R(1, 2, 3), as shown in Fig. 1(a). For13C16O and12C18O, it occurs at R(7), the band head of the R-branch, and at P(8, 9) and Q(7, 8) of the W1Π(ν= 1) band, even though the increase at R(7) is not as much as those at P(8, 9) and Q(7, 8), probably due to the substantial overlapping with other transitions at the band head. All the above rotational transitions, where the branching ratio sud- denly increases, are in excellent coincidence with the crossing points between the W1Π(ν= 1) and E′′1Π(ν= 0) levels, as identified by Heays et al. ,26where the predissociation linewidths show drastic variations. IV. DISCUSSIONS In the high-resolution photoabsorption study of Heays et al.,24,26the predissociation linewidth is observed to have a sudden decrease at the crossing point, and they attributed this to the inter- ference effect between the interacting W1Π(ν= 1) and E′′1Π(ν= 0) levels, both of which predissociate through coupling with the repul- sive states. By considering the excellent coincidence between the current study and the study by Heays et al. ,26we tentatively attribute the sudden variations of the branching ratio values as observed in this study also to the interference effect between the W1Π(ν= 1) and E′′1Π(ν= 0) levels. As the branching ratio represents the rel- ative ratio between the partial predissociation rates (or linewidths) into the two dissociation limits C(3P) + O(3P) and C(1D) + O(3P), the sudden jump of the branching ratio value at the crossing point indicates that the interference effect between the W1Π(ν= 1) and E′′1Π(ν= 0) levels as observed by Heays et al.26should have affected the partial predissociation rates into the two limits differently. This level of partial predissociation information is only available when combining the branching ratio measurement with the high resolu- tion photoabsorption spectroscopic study, as demonstrated in our recent study.38 Due to the severe overlapping between rotational lines, it is not possible to do a quantitative analysis as we did before in the Rydberg 4p(2) and 5p(0) complexes’ region of13C16O,38while a quasi-quantitative analysis of the rotational dependences of the par- tial predissociation rates is still available. Here, we take the Q-branch of the W1Π(ν= 1) band of13C16O as an example, for which the rotational lines are relatively better separated from other lines, to illustrate how the interference effect affects the two dissociation limits differently (similar analyses on the Q-branches of12C16O and12C18O are presented in Figs. S4 and S5 of the supplementary material). The partial predissociation rates into the two dissociation channels C(3P) + O(3P) and C(1D) + O(3P) for the Q-branch of the W1Π(ν= 1) band of13C16O are calculated by kT(J′)×Bri(J′) [Eq. (1) in Ref. 38], where kT(J′) is the total predissociation rate for the upper rotational level J′;Bri(J′) is the photodissociation branching ratio of channel ifor the upper rotational level J′. The total predissociation rates are calculated using the fitted equation in Table 3 of Ref. 24, and the branching ratios are from the current measurement, as listed in Table S2 in the supplementary material. The rotational dependences of the partial predissociation rates thus calculated are plotted vs J′(J′+ 1) in Fig. 5, and the error bars have J. Chem. Phys. 152, 234308 (2020); doi: 10.1063/5.0009931 152, 234308-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . The rotational dependences of the partial predissociation rates (in unit of 1011s−1) into the dissociation channels C(3P) + O(3P) (red dots) and C(1D) + O(3P) (blue dots) for the Q-branch of the W1Π(ν= 1) band of13C16O, J′is the rotational quantum number of the upper level. inherited the experimental uncertainties from both the total pre- dissociation rate measurements in Ref. 24 and the branching ratio measurements in the current study. Similar plots of the partial pre- dissociation rates vs J′(J′+ 1) for the Q-branches of12C16O and 12C18O are shown in Figs. S4 and S5 in the supplementary material. As shown in Fig. 5, the overall shapes of the rotational depen- dence curves of the partial predissociation rates of both C(3P) + O(3P) and C(1D) + O(3P) are very similar to that of the total pre- dissociation linewidths as measured in the high resolution photoab- sorption studies.24,26They increase linearly as a function of J′(J′+ 1) and show sudden decreases at the crossing point between the W1Π (ν= 1) and E′′1Π(ν= 0) levels and then start to increase linearly as a function of J′(J′+ 1) again after the crossing point. Aside from the crossing point, the partial predissociation rates into the two chan- nels show similar linear dependences on J′(J′+ 1), and this explains why the branching ratios are almost independent of J′, as shown in Figs. 1(a), 2(a), and 3(a), even though the total predissociation rates are observed to be linearly dependent on J′(J′+ 1). At the crossing point, the interference effect causes much more drastic variations on the partial predissociation rate into the channel C(3P) + O(3P) than that into the channel C(1D) + O(3P), and this is the reason why the branching ratio values show sudden jumps at the crossing points, as shown in Figs. 1(a), 2(a), and 3(a). While13C16O and12C18O are similar to each other due to their similar reduced masses (see Figs. 5 and S5 in the supplementary material), an obvious isotope effect can be noticed at the crossing points when comparing12C16O with13C16O and12C18O. For12C16O, the partial predissociation rate into the channel C(1D) + O(3P) also shows a dramatic decrease at the crossing points, which are comparable to that into the channel C(3P) + O(3P) in magnitude (see Fig. S4), while for13C16O and 12C18O, the magnitudes of the decrease in the partial predissociation rates for C(1D) + O(3P) is much smaller than that for C(3P) + O(3P) (see Figs. 5 and S5, respectively). This explains why the increments of the branching ratio values at the crossing points for13C16O and12C18O are much larger than that of12C16O, as shown in Figs. 1(a), 2(a), and 3(a). Before the study by Heays et al. ,26the similar interference effect was first observed by Kasahara et al.42between the mutually inter- acting B1Πand b3Πstate of23Na39K, both of which predissociate through coupling with the dissociative continuum of the c3Σ+state, and the predissociation linewidth changes drastically around the maximum perturbation. Vieitez et al.43observed the interference effect between the o1Π(ν= 1) and b1Π(ν= 9) states of14N2. Com- pared with23Na39K and14N2,42,43where only one dissociation limit is available, the predissociation process of CO at the W1Π(ν= 1) level is much more complicated26,28,41for which both of the disso- ciation limits C(3P) + O(3P) and C(1D) + O(3P) are energetically available, and for each of them, more than one dissociation path- ways can be involved. This is also illustrated in Fig. 5, i.e., both the partial predissociation rates of C(3P) + O(3P) and C(1D) + O(3P) have rotational independent and also rotational dependent parts, which should originate from different dissociation pathways. From Fig. 5, we can conclude that the characteristics of dissociation into the channel C(3P) + O(3P) dominate the properties of the dissoci- ation process of the W1Π(ν= 1) state, and those into the channel C(1D) + O(3P) only play a minor role. The rotational independent part of the lowest channel could be due to interactions with the E′1Πand k3Πstates, which adiabatically correlate with the channel C(3P) + O(3P),28and the rotational dependent part, which is inde- pendent of the parity, might be through coupling with the D1Δstate, as suggested by Heays et al.26The detailed dissociation pathways into the spin-forbidden channel C(1D) + O(3P) have been much less investigated before, and the rotational independent part might be due to coupling with the 43Πstate, which adiabatically correlates with the channel C(1D) + O(3P).28 The coupled channel method with ab initio calculated poten- tial curves was used to correctly reproduce the linewidths of pre- dissociation of the W1Π(ν= 1) state of CO,28and the branching ratio values at low J′levels of the W1Π(ν= 1) state of12C16O have also been well reproduced by substantially adjusting the coupling strength between the k3Πand 43Πstates.41However, these calcu- lations only consider electronic states of1Πand3Πsymmetries, and the possible roles of electronic states of other symmetries were not included, for example, the D1Δstate, which was proposed to be the possible reason for the rotational dependent part of the predissocia- tion linewidths of the W1Π(ν= 1) state.28It would be interesting to see if the coupled channel method can reproduce the rotation and isotope dependent branching ratios, as shown in Figs. 1(a), 2(a), and 3(a), especially the sudden jumps at the crossing points. This will provide unprecedented detailed information on the predissociation dynamics of CO. V. SUMMARY In this study, we have employed a tunable VUV laser radia- tion source and a TS-VMI setup to measure the photodissociation branching ratios in the complex region of the W1Π(ν= 1) and E′′1Π(ν= 0) levels for12C16O,13C16O, and12C18O. With optimized experimental conditions, C+ion photofragment spectra similar to the recent high resolution photoabsorption spectra are achieved; the rotational dependences of the branching ratio value have been J. Chem. Phys. 152, 234308 (2020); doi: 10.1063/5.0009931 152, 234308-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp obtained up to the crossing region between the W1Π(ν= 1) and E′′1Π(ν= 0) levels as identified in the high resolution photoab- sorption study.26At the crossing points, the branching ratios into the spin-forbidden channel C(1D) + O(3P) show sudden jumps in otherwise smooth dependence functions of the rotational quantum number. We decompose the total predissociation rates into partial predissociation rates of individual channels of C(3P) + O(3P) and C(1D) + O(3P) at the Q-branch of the W1Π(ν= 1) band using the method we proposed before.38We find that both the partial predis- sociation rates of the two available channels show sudden decreases at the crossing points due to the interference effect between the W1Π(ν= 1) and E′′1Π(ν= 0) levels, while the decrease for the chan- nel C(3P) + O(3P) is much more prominent than that for the channel C(1D) + O(3P), and this is the reason why the branching ratio into the channel C(1D) + O(3P) shows sudden jumps at the crossing points. We hope that the current experimental study can stimulate more theoretical studies, which can fully and quantitatively address the multichannel characteristics of the interference effect between the W1Π(ν= 1) and E′′1Π(ν= 0) levels. This is important for under- standing the numerous accidental perturbations as observed in the predissociation process of CO. SUPPLEMENTARY MATERIAL See the supplementary material for Figs. S1–S5 and Tables S1–S3. ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Grant No. 21803072), the Program for Young Outstanding Scientists of Institute of Chemistry, Chinese Academy of Science (ICCAS), and Beijing National Laboratory for Molecular Sciences (BNLMS). We thank Professor Yang Pan (National Syn- chrotron Radiation Laboratory, University of Science and Technol- ogy of China) for instrumentation support. DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material. REFERENCES 1R. N. Clayton, Nature 415, 860 (2002). 2H. Yurimoto and K. Kuramoto, Science 305(5691), 1763 (2004). 3J. R. Lyons and E. D. Young, Nature 435, 317 (2005). 4J. R. Lyons, E. Gharib-Nezhad, and T. R. 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5.0008130.pdf

J. Chem. Phys. 152, 204701 (2020); https://doi.org/10.1063/5.0008130 152, 204701 © 2020 Author(s).First-principles study of the adsorption of 3d transition metals on BaO- and TiO2- terminated cubic-phase BaTiO3(001) surfaces Cite as: J. Chem. Phys. 152, 204701 (2020); https://doi.org/10.1063/5.0008130 Submitted: 18 March 2020 . Accepted: 03 May 2020 . Published Online: 22 May 2020 Rafael Costa-Amaral , and Yoshihiro Gohda The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp First-principles study of the adsorption of 3 d transition metals on BaO- and TiO 2-terminated cubic-phase BaTiO 3(001) surfaces Cite as: J. Chem. Phys. 152, 204701 (2020); doi: 10.1063/5.0008130 Submitted: 18 March 2020 •Accepted: 3 May 2020 • Published Online: 22 May 2020 Rafael Costa-Amarala) and Yoshihiro Gohdaa) AFFILIATIONS Department of Materials Science and Engineering, Tokyo Institute of Technology, Yokohama 226-8502, Japan a)Authors to whom correspondence should be addressed: amaral.r.aa@m.titech.ac.jp and gohda.y.ab@m.titech.ac.jp ABSTRACT The deposition of transition metals (TM) on barium titanate (BaTiO 3, BTO) surfaces is involved in the development of several BTO-based devices, such as diodes, catalysts, and multiferroics. Here, we employ density functional theory to investigate the adsorption of 3 dTM on both BaO- (type-I) and TiO 2-terminated (type-II) surfaces of cubic BaTiO 3(001) at low levels of surface coverage, which is important to comprehend the initial stages of the formation and growth of TM overlayers on BTO. The most stable adsorption site is identified for each adatom on both surfaces. Our discussion is based on analyses of structural distortions, Bader charge, electron density difference, magnetic moments, work function, density of states, and adsorption energies. For the type-I surface, most of the adatoms bind covalently on top of the surface oxygens, except for Sc, Ti, and V atoms, which adsorb preferentially on the bridge site, between O ions, to form two polar TM–O bonds. On the type-II surface, the TM are located at the fourfold hollow site, which allows the formation of four TM–O interactions that are predominantly ionic. Upon the adsorption, we noticed the formation of in-gap states originated mostly from the adatom. When electrons are transferred to the substrates, their conduction bands become partially occupied and metallic. We observed a decrease in the work function of the type-II surface that is fairly proportional to the charge gained, which suggests that the BTO work function can be manipulated by the controlled deposition of TM. Published under license by AIP Publishing. https://doi.org/10.1063/5.0008130 .,s I. INTRODUCTION Barium titanate (BaTiO 3, BTO) is a wide-gap semiconductor that has a perovskite structure (ABX 3) and presents four phases: above its Curie point (130○C), it crystallizes into the paraelec- tric cubic phase ( Pm3m); below this temperature, BTO changes to tetragonal ( P4mm), then to orthorhombic ( Amm 2) at 5○C, and to rhombohedral ( R3m) at about −90○C.1BTO is the most consistently studied perovskite material due to its wide use in several fields, such as nonlinear optics,2electronics,3data storage,4and electrochem- istry.5Because many of these high-tech applications are aligned toward thin-film geometries, where surface features are of growing importance, efforts were directed at studying BTO surfaces. Among the BTO surfaces, the (001) and (111) surfaces have been the most investigated experimentally.6However, the former is much more stable than the latter.7The BaTiO 3(001) surface has twopossible terminations, namely, the BaO-terminated surface (type-I) and the TiO 2-terminated surface (type-II), where both are nonpo- lar and present a comparable range of thermodynamic stability.8,9 To understand the well-known degradation of ferroelectric prop- erties of BTO in thin-film10and particulate11geometries, the sur- face properties of both tetragonal and cubic phases of BTO were investigated from first principles, where the results indicated that intrinsic surface effects should not be responsible for the decrease in ferroelectric polarization.8Electronic properties and chemical states of several BTO surfaces have been experimentally assessed using electron-energy-loss spectroscopy, Auger spectroscopy, and low-energy electron diffraction.12–14Aside from clean surfaces, some progress has also been made regarding the effect of surface defects on BTO, such as the formation of oxygen vacancies and their role in the origin of metallic states in BTO.15Additionally, it has been shown that the stability of BTO polar surfaces with different structures and J. Chem. Phys. 152, 204701 (2020); doi: 10.1063/5.0008130 152, 204701-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp compositions can be manipulated by varying the substrate ferroelec- tric polarization.16Studies about the use of strain effects to change the properties of BTO thin-films have also been reported.17 The wide range of technological applications that explores the physics of metal/BTO interfaces, such as in Schottky diodes18and multiferroics,19,20has motivated research on the interaction of tran- sition metals (TM) with BTO surfaces as well. Furthermore, the dec- oration of substrates with TM adatoms or nanoparticles has been shown as a viable route to tune the surface properties and extend the applicability of several materials.21–23Earlier studies have addressed, from first principles, the deposition of TM on the surfaces of per- ovskites, such as Ag,24Ta,18W,18Ir,18and Pt18on BaTiO 3(001) or Pd25and Pt26on SrTiO 3(001), where the authors identified the on-top O site as the preferred adsorption site on both AX and BX 2terminations. However, because (1 ×1) surface unit cells were employed, their calculations concern TM interacting in the mono- layer (ML) regime or in surface coverages of at least 0.5 Ml. Hence, many aspects of the TM interaction in low-coverage regimes, which are important to understand the initial stages of the formation and growth of TM overlayers on BTO and other perovskites, remain unexplored so far. In this study, we perform density functional theory (DFT) cal- culations on the deposition of all the first row TM adatoms on both type-I and type-II surfaces of BaTiO 3(001). Here, we restrict our- selves to the paraelectric phase and focus on the surface properties. By considering all the TM from Sc to Zn, we aim to assess diverse adsorption characteristics due to the wide variety of valence states provided by the gradual filling of the dorbitals along the period. Unlike previous studies,18,24–26we take into account all the high- symmetry adsorption sites for both surfaces, from which the most stable one is identified for each TM adatom. The structural dis- tortions, charge transfer, work function, density of states (DOS), magnetic moment, and binding energy are presented for the lowest- energy configurations. According to our results, not all the TM should adsorb on top of the oxygen adsorption site of the type-I surface, as we found that more electropositive TM such as Sc, Ti, and V can be stabilized by two TM–O bonds on the bridge site between O ions. On the type-II surface, there is a strong prefer- ence for the fourfold site, above the subsurface Ba ion, where the TM form four bonds with nearby oxygen atoms. As indicated by the charge transfer analysis and binding energies, the interaction on the TiO 2-terminated surface is predominantly ionic for all the adatoms and much stronger when compared to the adsorption on the type- I, which is driven mainly by covalent interactions for most TM and a large contribution of dispersion forces in the case of Zn. Here, we also show how the TM adsorption affects the geometries of both sur- faces and how their electronic properties can change and, in some cases, be tuned, by the deposition of TM. II. THEORETICAL APPROACH AND COMPUTATIONAL DETAILS A. Total energy calculations Our total energy calculations were based on spin-polarized DFT within the generalized gradient approximation27(GGA) pro- posed by Perdew–Burke–Ernzerhof28(PBE) for the exchange- correlation energy functional. Because conventional semilocaldensity functionals are unable to properly describe dispersion inter- actions, which should not be neglected even for ionic systems,29we employed the DFT-D3 scheme proposed by Grimme,30as imple- mented in the Vienna ab initio simulation package (VASP).31,32 The DFT-D3 correction is a pairwise force field, where the disper- sion coefficients take into account the variations in the local envi- ronment. The Kohn–Sham self-consistent equations were solved using the all-electron projected augmented wave33(PAW) method as implemented in VASP, version 5.4.4,34–36which describes the electron–ion interactions by PAW potentials provided within the code. A list with the employed PAW potentials can be found in the supplementary material. The equilibrium volume of the cubic BaTiO 3bulk phase was obtained using a plane wave cutoff energy of 830 eV and a 8×8×8k-point mesh to integrate the Brillouin Zone (BZ). For the total energy calculations for surfaces, we employed a cutoff energy of 466 eV, along with a 3 ×3×1k-point mesh, due to the increased size of the unit cell. The equilibrium geometries were obtained once the atomic forces were smaller than 0.025 eV/Å on every atom, whereas we adopted a total energy convergence criterion of 10−5eV. For the work function calculations, the energy convergence criterion was set to 10−6eV. Finding the lowest energy configuration in magnetic sys- tems is a non-trivial task because multiple magnetic solutions can be found. We overcame this problem by setting calculations with differ- ent starting values of magnetic moment on the adatom and choosing the solution with the lowest energy. The BaTiO 3(001) surfaces were modeled using the repeated slab geometry. We employed a (2√ 2×2√ 2)R45○unit cell, with a 7-layered slab separated by a 15 Å vacuum region, which allowed us to investigate the adsorption of TM at low surface coverages of 1 16ML and1 24ML for the type-I and type-II surfaces, respectively. The surface coverages are defined with respect to the number of sur- face atoms. This setup also reduces the interaction of the TM with its images to less than 2 meV while yielding fairly converged surface properties, as shown in Fig. S1. To avoid the formation of spurious dipole interactions along the nonperiodic directions, we preserved the inversion symmetry of the cells by adsorbing the transition met- als on both sides of the slab. To evaluate the adsorption energies, we adopted the energy of an isolated atom in its ground state given by the GGA, which was obtained using an orthorhombic box with 20×21×22 Å3. The computed electronic configurations are provided in Table I. B. Adsorption sites on BaTiO 3(001) In the unit cell of the cubic perovskite BTO, the Ba atom is located at the corners of the cube, the Ti atom sits at the body- center position, and the O atoms sit at the face-centered positions. In the [001] direction, the BTO slab is composed by the stacking of BaO and TiO 2planes, resulting in two possible surface terminations, both of which are predicted to be formed depending on whether the growth occurs in a Ba- or Ti-rich medium.8,9Hence, we inves- tigate the adsorption site preference of TM adatoms on the high- symmetry adsorption sites of BaO- (type-I) and TiO 2-terminated (type-II) surfaces of BaTiO 3(001), as illustrated in Fig. 1. For the BaO plane, we considered four adsorption sites, e.g., topBa I, topO I, bridgeBaO Iand bridgeOO I, and six adsorption sites for the TiO 2plane, J. Chem. Phys. 152, 204701 (2020); doi: 10.1063/5.0008130 152, 204701-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . The electron configuration for the ground state of the isolated TM computed from PBE, most stable adsorption site, adsorption energy, Ead(in eV), effective Bader charge of the TM, QTM Bader(ine), local magnetic moment of the adatom, mloc(inμB), and the change in work function due to TM adsorption, ΔΦ(in eV). A graphical representation of the data within is displayed in Fig. S6. Type-I Type-II TM Config. Site Ead QTM Bader mloc ΔΦ Site Ead QTM Bader mloc ΔΦ Sc [Ar]3 d14s2bridgeOO I−4.48 0.93 0.53 −0.51 hollowBa II−10.42 1.88 0.00 −2.22 Ti [Ar]3 d34s1bridgeOO I−5.04 0.82 0.00 −0.59 hollowBa II−10.80 1.66 0.90 −1.53 V [Ar]3 d44s1bridgeOO I−4.69 0.88 2.80 −0.60 hollowBa II −9.74 1.66 1.62 −1.76 Cr [Ar]3 d54s1topO I −4.48 −0.15 3.95 −0.11 hollowBa II −8.78 1.33 3.40 −1.38 Mn [Ar]3 d54s2topO I −4.45 −0.13 4.84 0.10 hollowBa II −8.27 1.49 3.60 −1.58 Fe [Ar]3 d74s1topO I −4.18 −0.19 3.77 0.04 hollowBa II −7.59 1.30 2.57 −1.36 Co [Ar]3 d84s1topO I −3.42 −0.24 2.73 −0.14 hollowBa II −6.50 0.99 1.10 −1.20 Ni [Ar]3 d94s1topO I −2.97 −0.24 0.00 −0.56 hollowBa II −5.85 0.90 0.00 −1.14 Cu [Ar]3 d104s1topO I −1.95 −0.22 0.81 −0.17 hollowBa II −3.80 0.96 0.52 −1.27 Zn [Ar]3 d104s2topO I −0.61 −0.11 0.00 −0.21 hollowBa II −2.37 1.19 0.00 −1.43 namely, topTi II, topO II, bridgeTiO II, bridgeTiBa II, bridgeBaO II, and hollowBa II. Initially, to map the energetic preference of the adsorption sites, the adatoms were constrained to relax their positions only perpendic- ularly to the surface. The remaining atoms, except those from the middle-layer which were kept frozen, were allowed to fully relax. Then, for the most stable adsorption site, we removed the constraint on the adatoms and allowed the configurations to further relax. To FIG. 1 . Top and side views of the BaO- (type-I) and TiO 2-terminated (type-II) sur- faces of BaTiO 3(001) employing a (2√ 2×2√ 2)R45○surface unit cell. The Ba, Ti, and O atoms are indicated in green, blue, and red, respectively. The orange circles mark the position of the adsorption sites.investigate the stability of the obtained configurations, we displaced the adatom by 0.1 Å on the surface plane and allowed the structure to relax once more, without any symmetry constraints except for the inversion one. Most of the systems relaxed to configurations that were virtually the same as the ones before the displacement, e.g., the energy differences between the configurations were below 0.01 eV per adatom. However, new local minima were obtained for Co on the type-I and Cr, Co, Ni, Cu, and Zn on the type-II surface, which had their stability verified by an additional displacement-relaxation procedure. For the lowest-energy configurations, we performed energetic, structural, and electronic analyses. III. RESULTS AND DISCUSSION A. Bulk and surface properties For the cubic phase of BTO, the PBE + D3 functional yields an equilibrium lattice parameter of a0= 4.02 Å, which is in better agreement with the experimental value, a0= 4.00 Å,37when com- pared to previous DFT-PBE calculations,38e.g., the relative error ofa0reduces from 1.0% to 0.5%. The obtained cohesive energy, Ecoh=−32.51 eV, is also consistent with the value of −32.70 eV from a previous study.38The Bader charge analysis39shows that Ba, Ti, and O atoms present effective charges of 1.57 e, 2.13 e, and−1.23 e, respectively, which differ from their respective formal charges 2, 4, and−2 because of the covalent contribution within the Ba–O and Ti–O bonds. In fact, if we define the ionicity as the ratio between the effective and formal charges, it is straight forward to see that the Ba–O bond is 78% ionic, while the Ti–O bond is about 53%. From the total and local density of states (LDOS) of bulk BTO, illustrated in Fig. S2, we notice that the energy band between −11.0 and−9.5 eV consists basically of Ba 5 pstates with a small contribu- tion of O 2 sand 2 pelectronic states. The valence band is mostly com- posed of O 2 pstates, whereas the conduction band carries mainly Ti 3dcharacter, which is consistent with previous reports.15The presence of bonding states in the lower portion of the valence band (between −4.50 eV and −2.0 eV) and corresponding antibonding J. Chem. Phys. 152, 204701 (2020); doi: 10.1063/5.0008130 152, 204701-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp states in the conduction band indicates a larger covalent contri- bution in the Ti–O interaction, which is in line with our Bader results. Upon the formation of the BaO- and TiO 2-terminated surfaces, the interlayer distance between the two topmost layers is reduced by 4.96 and 5.66%, respectively, which is in qualitative agreement with the relaxation pattern found in previous studies.15,40Here, we define the work function, Φ, asΦ=Vvac−εF, where Vvacis the Coulomb potential at the middle of the vacuum region and εFis the Fermi energy of the slab. The type-I surface presented a work function of 2.78 eV, while the value of 5.34 eV was found for the type-II surface. The substantial difference in Φbetween the two sur- faces can be explained by the formation of surface dipoles as follows: For the type-I surface, the topmost BaO and TiO 2layers present, in a (1×1) unit cell, net charges of 0.24 eand−0.37e, respectively, which forms a positive surface dipole that decreases the work function; the opposite is found for the TiO 2-terminated surface, where the sur- face and subsurface layers have net charges of −0.22 eand 0.36 e, thus presenting an outwardly pointing negative charge. A brief dis- cussion about the DOS of both BTO surfaces can be found in the supplementary material. B. Relative stability of the adsorption sites In Fig. 2 and Table S2, we present the relative total energies of the considered adsorption sites with respect to the most stable configuration for each TM on both type-I and type-II surfaces. For the type-I surface, most of the 3 dTM adsorb preferentially on topO I, followed by bridgeBaO I, bridgeOO I, and topBa I, which is the less sta- ble high-symmetry adsorption site. However, for the Sc, Ti, and V atoms, the bridgeOO Iis the most stable adsorption site. This occurs because the TM–O interaction has a more ionic nature for these ele- ments (as will be shown), causing a displacement of two adjacent FIG. 2 . Relative total energies with respect to the most stable configuration of the TM on the high-symmetry adsorption sites of both type-I and type-II surfaces. The adsorption sites are shown in Fig. 1.oxygen atoms from the BTO surface to stabilize the adatom. Thus, it is clear that the TM–O interaction determines the adsorption site preference. Previous LDA-FLAPW calculations indicate that the topO Isite of BaTiO 3(001) surface is also favored by monolayers of 5 d TM, such as Ta, W, Ir, and Pt.18 Concerning the type-II surface, the most stable adsorption site is above the Ba atom, namely, hollowBa II, as it allows the TM to be closer to the substrate and to form four TM–O interactions. In fact, the energy trends shown in Fig. 2 can be mostly explained in terms of the interplay between the number of TM–O bonds that are formed in each adsorption site and the break of the substrate packing caused by the displacement of the surface atoms (especially the O), which have positive and negative impacts on the stability of the systems, respectively. Exceptions are found for the TM at the extremities of the chemical period due to a too strong or a too weak interaction with the substrate. To understand the interaction of TM on barium titanate sur- faces and its effect over the substrate properties, we evaluate the structural, energetic, and electronic properties of the lowest-energy configuration for each TM/BaTiO 3system. The results are summa- rized and discussed in Secs. III C–III F. C. Structural parameters To evaluate the structure of the adsorbed systems, we calculate the shortest TM–O distances, dTM−O, and defined three parameters to quantify the geometric distortions caused by the TM-BTO inter- action, namely, the magnitude of the displacement vector for the O, Ti, and Ba atoms that are most affected by the adsorption, ΔO,ΔTi, andΔBa, respectively. In Fig. 3, we illustrate these geometric parame- ters and how they vary along the period. Refer to the supplementary material, Figs. S4 and S5, for the lowest-energy configurations of all the TM/BTO systems. As shown in Fig. 3, the TM adsorption on the topO Isite causes the oxygen atom right below the adatom to displace 0.15–0.43 Å toward it, whereas the binding on bridgeOO Iinduces the two oxygen atoms nearby the site to shift about 1.42–1.57 Å to stabilize the TM. As a consequence, the Ti atom underneath each displaced O atom moves 0.07–0.22 Å downwards due to the weakening of the corre- spondent O–Ti interaction. Minor shifts of 0.03–0.15 Å are found for the neighboring Ba atoms. Upon the adsorption on hollowBa IIof the type-II surface, the four O atoms around the site shift toward the TM adatom by amounts that can reach 0.37–0.44 Å. The surface Ti atoms move away by 0.16–0.21 Å, while the Ba ion right below displaces between 0.21 and 0.31 Å to inner regions to reduce the Coulombic repulsion with the TM. We notice that the values of ΔOandΔTifor the topO Isite in the type-I surface decrease monotonically from Sc to Zn, suggest- ing a weakening of the TM-BTO interaction along the TM period. The distortion parameters for the type-II exhibit a parabolic behav- ior that peaks around the middle of the period (Mn and Fe) and might suggest a larger covalent contribution on these systems, as we will point in the DOS section. The values of the dTM−Orange from 1.73 Å up to 2.06 Å for type-I, while for type-II, they vary from 1.85 Å to 2.02 Å. Except for the Zn case, the TM–O bond length is shorter in the BaO-terminated surface, which is expected because the adatom is sharing electrons with four O atoms on the hollowBa IIsite, leading J. Chem. Phys. 152, 204701 (2020); doi: 10.1063/5.0008130 152, 204701-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Geometric parameters used to characterize the adsorption of TM on BaTiO 3(001) surfaces, namely, the shortest distance between the TM and the O atoms, dTM−O, the magnitude of the displacement vector for the O, Ti, and Ba atoms, ΔO,ΔTi, andΔBa, respectively. to individually weaker TM–O interactions. We notice some corre- lation between the increase in TM–O distances and the decrease in distortion parameters for the late transition metals. However, there is no overall trend of dTM−Oalong the period, especially because the atomic radii of the early 3 dTM should be strongly affected by the larger ionic contribution of their interaction, as will be shown in Sec. III D. D. Charge transfer and work function To assess the ionicity of the TM-BTO interaction, we used the Bader charge analysis39to calculate the effective charges of theadatoms, which are found in Table I. Upon the TM adsorption on the type-I surface, most of the adatoms receive charge from the sub- strate with negative values of QTM Bader ranging from −0.11 to −0.24 e. The charge comes mostly from the O atoms of the surface, which lose about 0.01–0.04 eeach. Sc, Ti, and V transfer 0.93 e, 0.82 e, and 0.88eto the substrate, respectively, especially, to the Ti and O atoms of the subsurface, which receive around 0.03–0.08 eand 0.01–0.04 e, respectively. On the other hand, the adatoms lose between 0.90 eand 1.88 ewhen adsorbed on the hollowBa IIsite, thus indicating a com- paratively larger ionic contribution to the adsorption. On the type-II substrate, most of the charge stays delocalized among the Ti atoms around the adsorption site and positioned at the third topmost layer, where each atom receives 0.02–0.05 eand 0.05–0.08 e, respectively. The substantial difference of the direction of charge flow between the substrates can be explained by the larger (by 0.12 e) negative charge of the oxygen atoms on the type-I surface, which leads to an inferior oxidizing capacity when compared to the surface oxygens of the more open TiO 2-terminated surface. We also found that the charge transferred to the type-II substrate reduces gradually with the increase in electronegativity observed along the 3 dperiod (see Fig. 4). Similar behavior was observed for TM interacting with less ionic substrates such as the MoS 2monolayer.41As for the type-I surface, we observe a poor correlation due to the differences in the chemical environment of the bridgeOO Iand topO Isites, which leads to a much more ionic TM-BTO interaction for the former case. In Fig. 5, we present the contour of the electron density differ- ence,Δρ, for TM/BTO systems, which is defined as Δρ(r)=ρTM/BTO(r)−ρBTO(r)−ρTM(r), (1) FIG. 4 . Top panels: plot of the TM effective Bader charge, QTM Bader, vs the correspon- dent values of Pauling electronegativity. Bottom panels: plot of the work function change upon adsorption, ΔΦ, vsQTM Bader. The straight lines correspond to the linear fitting of the points, and the respective coefficients of determination are also given. J. Chem. Phys. 152, 204701 (2020); doi: 10.1063/5.0008130 152, 204701-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . The electron density difference of the TM adsorbed on the most favorable site on both BaO- (type-I) and TiO 2-terminated (type-II) surfaces of cubic BaTiO 3(001). The magenta and yellow colors indicate accumulation and depletion of charge density, respectively. An isosurface value of 0.004 e/Å3was adopted. where ρTM/BTO(r)represents the electron density of the adsorbed system, while ρBTO(r) and ρTM(r) are the electron density of the iso- lated substrate and adatom, respectively, with the atomic positions kept fixed as obtained in the TM/BTO configuration. Among the common features presented by the TM on topO I, we notice that the adsorption promotes mostly local changes in the charge density that include the increase in ρbetween the adatom and adsorbate, which is generally accompanied by the transference of charge density from states that resemble dxzanddyzorbitals to regions near the top half part of the adatom to reduce the electronic repulsion. As expected from the Bader results, the regions of charge transfer are compar- atively larger for the bridgeOO Icases. Likewise, the TM adsorption on the type-II surface promotes a significant accumulation of charge within the TM–O bond region and a large depletion of electron den- sity near the adatoms, which is also consistent with their cationiccharge and the ionic bond picture of the TM–O interaction. Interest- ingly, the adsorption promotes larger changes in ρon the Ti atoms of the inner layers instead of the ones closer to the hollowBa IIsite, which is in accordance with our Bader results for charge flow within the substrate. The calculation of a system’s work function provides informa- tion about the absolute electron energy-level with respect to the vac- uum, which plays an important role in heterostructured interfaces or active surfaces, employed in solar cells, oxide electronics, electrocat- alysts, diodes, and so on.42To understand how the TM adsorption affects the work function of BTO, we calculated the work function change, ΔΦ, as the difference of the work functions of the adsorbed system and clean surface, which is summarized in Table I. For most of the cases, the interaction of the adatom with BTO decreases the substrate work function, e.g., by 0.11–0.60 eV (type-I) and J. Chem. Phys. 152, 204701 (2020); doi: 10.1063/5.0008130 152, 204701-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 1.14–2.22 eV (type-II). The only exceptions occur on type-I for the Mn and Fe cases, where the work function is found to slightly increase by 0.10 eV and 0.04 eV, respectively. The significantly larger changes observed for the TiO 2-terminated surface seem to be corre- lated with the charge transferred to the substrate (Fig. 4), which is expected when the adatom–substrate interaction has a highly ionic contribution and promotes the formation of surface dipoles pointing toward the vacuum region.43However, such a general trend is not found for the type-I surface, especially among the topO Iresults, prob- ably due to the more covalent nature of the TM-BTO interaction, which could be easily disturbed by the presence of secondary surface dipoles and the electric polarization of the adatom semicore states.44 Hence, a possible tuning of BTO work function through deposition of TM seems to be much more consistent on the type-II surface. E. Density of states In Fig. 6, we present the spin-polarized local DOS (LDOS) for the two topmost layers of the lowest-energy TM/BTO systems. Theprojected spd DOS for the TM adatoms are displayed in Fig. S7. Upon the adsorption on the topO Isite, delocalized dstates from the adatom are formed within the energy range of the substrate valence band. In particular, for the Co and Ni cases, they span in energy lev- els as low as −6 eV. For the Sc, Ti, and V on the type-I surface, the BTO conduction band becomes partially occupied due to the elec- tron transfer from the adatom into the 3 dorbitals of the subsurface Ti atoms and, to some extent, due to the O and Ti displacements, as the stretch of M–O bonds can change significantly the charge bal- ance between the atoms.45Likewise, previous results46found that the interaction of the same atoms on MoS 2shifted the Fermi energy of the adsorbed systems to higher energies above the bottom of the sub- strate conduction band. As the adatoms 3 dorbitals are filled along the period, they shift to energy levels closer to the valence states of the substrate, which favors the hybridization between the TM 3 d orbitals and O 2 pstates, as indicated by the overlap of peaks between −2 eV and εF. Similar features were found in the interfaces between BTO and 5 dtransition metals.18In the case of Zn, the 3 dorbitals FIG. 6 . The local density of states (LDOS) for the TM adsorbed on the most favorable site on both BaO- (type-I) and TiO 2-terminated (type-II) surfaces of cubic BaTiO 3(001). The dashed line indicates the Fermi energy, which is located at 0.00 eV. J. Chem. Phys. 152, 204701 (2020); doi: 10.1063/5.0008130 152, 204701-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp lie below the BTO valence states, so the small peak around −1.2 eV is due to the hybridization between Zn 4 sand O 2 p, which leads to a weaker interaction with the substrate, as the adsorption energy results demonstrate. Because of the stronger ionic character, the TiO 2-terminated surface becomes metallic as some charge is transferred from the TM to the lower portion of the conduction band, which is consistent with the aforementioned increase in the electronic density around the Ti atoms (see Fig. 5). We also point out that the system contain- ing Cr should present a more pronounced conductive properties due to the larger up-spin DOS at the Fermi level. By comparing the DOS for the same adatom on different surfaces, we notice that the Sc, Ti, and V adsorption on the type-II surface presents a lesser degree of covalency. However, the hybridization features are enhanced for the remaining TM, e.g., the adatom states broaden and peaks that are absent in the clean surface DOS become more prominent, especially in the region from about 1–2 eV below εF. Furthermore, the changes of the substrate Ti valence band states along the period are more noticeable on the type-II due to the large distortions undergone by the surface (see Fig. 3). For both substrates, most of the systems present states located within the bandgap region of cubic BTO, com- monly referred to as in-gap states, which are mostly originated from the TM adatoms. As shown in Table I, most of the TM/BaTiO 3systems present a net magnetic moment, which comes primarily from the adatoms because the ground state of both BTO surfaces is non-magnetic. However, we observe that the degeneracy of the substrate spin states is broken when the adsorption presents more covalent features, such as in the Fe and Co cases. While it is expected that Zn/BTO systems present no magnetic moment due to the closed-shell configuration of the isolated Zn atom, interestingly, the adsorption of Sc on type-II and Ni on type-I and type-II surfaces yields DOS with no exchange- splitting, even though they present initial magnetic moments of 1 μB and 2 μB, respectively, in their free atom states. F. Adsorption energy of the TM on BaTiO 3(001) surfaces The interaction energy of an adsorbate on a substrate can be useful to evaluate the effectiveness of a surface in stabilizing heteroepitaxial metallic films. Thus, we computed the adsorption energy of the TM adatoms on BaTiO 3(001) surfaces, Ead, as follows: Ead=ETM/BaTiO 3 tot −EBaTiO 3 tot −ETM tot, (2) where ETM/BaTiO 3 tot is the total energy of the adsorbed system, while EBaTiO 3 tot andETM totrepresent the total energies of the clean (type-I or type-II) surface and the TM free atom, respectively. The results are summarized in Table I. On the type-I surface, the magnitude of the adsorption energies ranges from 4.45 eV up to 5.04 eV for the early 3 dTM. The values start to drop significantly past Mn, as electrons are being paired in thedvalence orbitals, until it reaches the lowest value, e.g., −0.61 eV for the Zn adatom. This low binding energy, along with our previ- ous findings such as the absence of a strong hybridization with the adsorbate states and the small values of the distortion parameters on the type-I surface (see Fig. 3), might suggest a larger contribution of dispersion forces to the interaction. For the remaining adatoms on type-I, the values of Ead, the effective charges, and the evidenceof TM 3 dand O 2 phybridization point to a covalent-bond picture with an increased ionic contribution for Sc, Ti, and V atoms. If we consider the same TM, the magnitude of the binding energies is comparatively larger on the TiO 2-terminated surface and, in general, decreases monotonically with the increase of the atomic number. The trends found on both BTO surfaces are completely different from the ones found for 3 dTM on graphene47and MoS 2(001)46in which the binding energies along the period presented a “V-shaped” curve, i.e., the binding energies increased on either side of Cr/Mn couple. The Ti adatom presents the largest binding energies on both type-I and type-II surfaces, namely, −5.04 eV and −10.80 eV, respec- tively, and its smaller atomic radius48(1.4 Å) when compared to Sc (1.6 Å) is seemingly responsible for the larger stability. In Fig. 7, we present the plots of the structural and elec- tronic properties against the values of Ead. Although one expects that shorter bond distances are associated with a stronger interac- tion, we could not find a general correlation between the TM–O bond distances and the binding energies, probably due to the inter- play between factors that change the interatomic distances, such as atomic radius, charge transfer, and hybridization. Some order seems to exist for the systems with weaker interaction, however. Concerning QTM Bader, for the type-II surface, we found an overall correlation between the effective charge of the adatoms and the respective adsorption energy for the TM with dopen-shell config- uration, which provides further evidence about the importance of Coulomb interactions for their adsorption. The very stable semi- closed (3 d104s1) and closed-shell (3 d104s2) configurations of Cu and Zn adatoms increase the energetic cost associated with their FIG. 7 . The top, middle, and bottom panels present, respectively, the plot of the shortest distance between the TM and the O atoms, dTM−O, effective Bader charge of the TM, QTM Bader, and change in work function, ΔΦ, against the adsorption energy, Ead. The blue lines are added to highlight trends among the atoms. J. Chem. Phys. 152, 204701 (2020); doi: 10.1063/5.0008130 152, 204701-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp oxidation, thus shifting them to less negative values of Ead. As a con- sequence of their ionicity, some degree of correlation between the interaction strength and the work function change is found on the type-II surface. Contrarily, the left panels of Fig. 7 indicate a lack of correlation for the type-I surface that corroborates with the idea of a hybridization-driven adsorption for most of the TM. Although our results concern the surfaces of paraelectric phase, BTO is a ferroelectric material at room temperature. Hence, it is important to infer the probable effects of the ferroelectricity on the interaction of TM on BTO. For polar surfaces of both BaTiO 3and PbTiO 3, the relative stability of several compositions and recon- structions were found to be sensitive to the presence and orienta- tion of the ferroelectric polarization.16However, the effect of the ferroelectric polarization should be much smaller on the non-polar surfaces of BTO because of the lesser electrostatic instability associ- ated with the surface charges. For the tetragonal BTO (001) surfaces, it was shown that the ferroelectric polarization normal to the sur- face is suppressed by the depolarization fields49and that the energy associated with the surface relaxation is many times larger than the bulk ferroelectric well depth.8Because the energetics and structural changes of the substrates due to the adsorption of TM are even larger than the clean surface effects, the trends observed for the surfaces of paraelectric BTO are likely to be found for the non-polar surfaces of tetragonal BTO as well. IV. CONCLUSIONS Through DFT calculation, we investigated the adsorption of 3dtransition metals on both BaO- (type-I) and TiO 2-terminated (type-II) surfaces of paraelectric BaTiO 3(001). We identified the most stable adsorption site for each adatom among all the possible high-symmetry sites of each surface and performed analysis of sur- face energetics, structural changes, charge transfer, work function, and density of states to understand the nature of their interaction and its effects on the substrate properties. Although previous stud- ies18,24–26predict that TM monolayers adsorb on top of the oxygen surface atoms, our results indicated that the preferred adsorption site changes at lower surface coverages, such as1 16ML and1 24ML, for the type-I and type-II surfaces, respectively. For instance, while most of the TM still prefer the on-top O site, the adsorption of more elec- tropositive TM such as Sc, Ti, and V on the type-I surface should occur at the bridge site between two O ions because they are sta- bilized by two significantly polar TM–O bonds. A covalent-bond picture for Cr up to Cu on the on-top O site is supported by changes in the substrate geometry, small values of charge transfer, and evi- dence of hybridization between the TM 3 dand O 2 pstates. On the other hand, the adsorption of Zn on type-I seems to be governed by dispersion forces. Larger differences were found on the type-II surface as all the adatoms presented a strong preference for the fourfold hollow site above the subsurface Ba ion, which allows the TM to stay closer to the substrate and form four TM–O interactions. The binding energies are significantly larger, where the magnitudes range from 2.37 eV (Zn) up to 10.80 eV (Ti), and correlate well with the elec- tronic charge transferred from the TM to the substrates. Hence, although they present enhanced hybridization features when com- pared to type-I, their interaction should be predominantly ionic, especially for low-valent early transition metals.The adsorption of most 3 dTM decreases the BTO work func- tion by 0.11–0.60 eV (type-I) and 1.14–2.22 eV (type-II) except for Mn and Fe on type-I, where a slight increase of Φis promoted. The considerable reductions found on the type-II surface correlate fairly well with the amount of the electronic charge transferred to the sub- strate, thus suggesting the deposition of TM adatoms as a possible route to tune the work function of the TiO 2-terminated surface. For the Sc-, Ti-, and V-decorated type-I and all TM-decorated type-II surfaces, our results predict a metallic behavior due to the transfer of electrons into the lower part of the conduction band. Although our calculations focused exclusively on surfaces of the cubic phase of BTO, our general conclusions are likely to be applicable to the (001) surfaces of the tetragonal phase as well because the adsorption energetics are many times larger than the ferroelectric well depth, reported to be around 0.03 eV.8Our results contribute to improv- ing the understanding on the initial stages of the formation and the growth of TM on BTO surfaces, which is crucial to enhance the tuning capabilities of devices based on TM/BTO interfaces. SUPPLEMENTARY MATERIAL See the supplementary material for additional results of ener- getic, geometric, and electronic properties. ACKNOWLEDGMENTS This work was financed, in part, by CREST-JST (Grant No. JPMJCR18J1), JSPS-KAKENHI (Grant No. 17K04978), and CSRN in Osaka University. The calculations were partly carried out by using HPC resources at the Institute for Solid State Physics (ISSP), The University of Tokyo. 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5.0004777.pdf

Appl. Phys. Lett. 116, 190502 (2020); https://doi.org/10.1063/5.0004777 116, 190502 © 2020 Author(s).Hybrid superconductor-semiconductor systems for quantum technology Cite as: Appl. Phys. Lett. 116, 190502 (2020); https://doi.org/10.1063/5.0004777 Submitted: 14 February 2020 . Accepted: 30 April 2020 . Published Online: 13 May 2020 M. Benito , and Guido Burkard COLLECTIONS Paper published as part of the special topic on Hybrid Quantum Devices Note: This paper is part of the Special Issue on Hybrid Quantum Devices. This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN Developing silicon carbide for quantum spintronics Applied Physics Letters 116, 190501 (2020); https://doi.org/10.1063/5.0004454 Cp2Mg-induced transition metal ion contamination and performance loss in MOCVD-grown blue emitting InGaN/GaN multiple quantum wells Applied Physics Letters 116, 192106 (2020); https://doi.org/10.1063/1.5142505 Breathing modes in few-layer MoTe 2 activated by h-BN encapsulation Applied Physics Letters 116, 191601 (2020); https://doi.org/10.1063/1.5128048Hybrid superconductor-semiconductor systems for quantum technology Cite as: Appl. Phys. Lett. 116, 190502 (2020); doi: 10.1063/5.0004777 Submitted: 14 February 2020 .Accepted: 30 April 2020 . Published Online: 13 May 2020 M.Benito and Guido Burkarda) AFFILIATIONS Department of Physics, University of Konstanz, D-78457 Konstanz, Germany Note: This paper is part of the Special Issue on Hybrid Quantum Devices. a)Author to whom correspondence should be addressed: guido.burkard@uni-konstanz.de ABSTRACT Superconducting quantum devices provide excellent connectivity and controllability, while semiconductor spin qubits stand out with their long-lasting quantum coherence, fast control, and potential for miniaturization and scaling. In the last few years, remarkable progress has been made in combining superconducting circuits and semiconducting devices into hybrid quantum systems that benefit from the physical properties of both constituents. Superconducting cavities can mediate quantum-coherent coupling over long distances between electronicdegrees of freedom such as the spin of individual electrons on a semiconductor chip and, thus, provide essential connectivity for a quantumdevice. Electron spins in semiconductor quantum dots have reached very long coherence times and allow for fast quantum gate operationswith increasing fidelities. We summarize recent progress and theoretical models that describe superconducting–semiconducting hybrid quan- tum systems, explain the limitations of these systems, and describe different directions where future experiments and theory are headed. Published under license by AIP Publishing. https://doi.org/10.1063/5.0004777 Quantum dots (QDs) are nanostructures hosted in semiconduc- tors where a few electrons can be electrostatically trapped in discrete states. Therefore, QDs allow us to access and control the quantum nature of single electrons and interactions between them. 1Since the first measurements of few-electron phenomena in lateral gated QDs, the focus of applications of these systems has shifted from single spin- tronics toward quantum information science, as originally envisionedby Loss and DiVincenzo. 2Beyond this first proposal, which employs the electron spin as a quantum bit (qubit), the field has developed both theoretically and experimentally as gate and memory fidelitieshave increased and more complex but robust alternative implementa- tions of spin qubits have been demonstrated, such as a spin qubit defined with a pair of electrons in two QDs 3(singlet-triplet qubit) and three QDs filled with one, two, and three electrons.4–6 The demonstrated long spin coherence times of electrons in silicon7have motivated a change of trend from the traditional host material GaAs to silicon-based QDs. Among the advantages, silicon offers an almost nuclear-spin-free environment for the electronicspins and a significantly reduced spin–orbit interaction, main sources of decoherence in GaAs QDs. 1,8,9An impressive series of promising quantum information experiments have been realizedwith silicon spin qubits, including high fidelity single-qubit gates, 10–13two-qubit gates,14–18and quantum non-demolitionmeasurements,19,20but scaling these QD systems to large numbers of qubits is still challenging, due to the large number of voltage gates and the lack of connectivity due to the intrinsically short- range exchange interaction. Moreover, since silicon is an indirectbandgap semiconductor, it comes with the additional—and often obstructive—valley degree of freedom, which is not yet well under- stood given the complexity of the heterostructures. Although thereare some measurements and estimations of valley features includ- ing the valley splitting, 21,22for scalable spin qubit platforms based on silicon QDs,23–25an accurate characterization of all valley fea- tures that could affect the fidelity of the computation is desirable. Impressive progress toward overcoming these challenges is occurring thanks to the development of superconductor–semiconduc-tor hybrid systems, 26–38where semiconductor QDs are coherently coupled to superconducting cavities (see Fig. 1 ). Hybrid systems mimic atomic cavity quantum electrodynamics (QED) systems, in which coherent interactions and quantum superpositions of light and matter were successfully demonstrated.39–41In fact, the hybrid systems of interest here are more similar to the so-called circuit QED systems where superconducting qubits are coupled to superconducting cavities.42,43An effective implementation of the cavity photons as mediators would allow one to simplify the QD qubit architecture and increase its connectivity. Appl. Phys. Lett. 116, 190502 (2020); doi: 10.1063/5.0004777 116, 190502-1 Published under license by AIP PublishingApplied Physics Letters PERSPECTIVE scitation.org/journal/aplAn ensemble of spins can be resonantly coupled to the field in a superconducting cavity via its large effective magnetic dipole,44–46but the coupling of the tiny magnetic dipole of a single electron to a cavity remains difficult. Different mechanisms and techniques to couple single-electron spin qubits26,47–49and multi-electron spin qubits27,50–52 to superconducting cavities have been theoretically proposed.However, despite their differences, all these mechanisms imply usingelectronic systems with more than the two quantum levels required f o rt h eq u b i ti t s e l f ,t h e r e b ye n d o w i n gt h ee l e c t r o ns p i nw i t ha ne f f e c - tive electric dipole that interacts strongly with the cavity electric field. First experimental observations of signatures of single-spin to cavity coupling 32,53were recently followed by demonstrations of spin–pho- ton coupling in the strong coupling regime with single-spin54,55and three-spin56qubits. In the strong coupling regime of cavity QED, the matter-light coupling gexceeds both the photon loss rate jfrom the cavity and the decoherence rate cin the two-level matter system (atom or qubit). This means that energy can be exchanged multiple times in a quantum-coherent manner between light and matter, which is typi- cally a prerequisite for the use of a cavity as a long-distance mediatorfor quantum information. Moreover, these mature superconductor– semiconductor hybrid systems recently opened a new way to measure QD valley features. 57,58 In the following, we briefly introduce the theory developed to predict signatures of the interaction between superconducting cavitiesand multi-level electronic systems in the cavity transmission, which is a generalized type of input–output theory. 57,59–61Then, we show how this theory can be applied to the interface between a single electron spin and cavity photons54,60and to high-resolution valley spectros- copy.57,58,62Figure 2 represents a general silicon double QD nano- structure with the position, spin, and valley degrees of freedom. The electrostatic detuning /C15between the left and right QDs and the intra(inter)-valley tunnel coupling tc(t0 c) can be controlled externally. One can also apply an external magnetic field and use micromagnets to introduce (static) magnetic field gradients along the double QD axis. For the purpose of spin–photon coupling, it is desirable for the valley splitting EL;Rto exceed the molecular (charge qubit) level split- tingffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C152þ4t2 cp as well as the Zeeman splitting due to the external magnetic field Bz. Then, a gradient of the transverse magnetic field component along the double QD axis (z) (such that the magnetic field at the center of the left and right QDs is Bz^z6bx^x) will induce spin- charge mixing and an effective interaction between the spin and thecavity electric field. For valley spectroscopy, the valley and charge qubit splitting should be comparable, with the spin states either degenerateor strongly detuned. We briefly review a theoretical framework that has allowed the quantitative prediction of the electromagnetic response of a supercon-ducting cavity coupled to an embedded electronic semiconductor sys-tem. The response of a QD-cavity system to a weak microwave probecan be determined using input–output theory, 59,63at r e a t m e n tt h a t enables the calculation of the fields ( bout 1and bout 2inFig. 1 ) emerging from the cavity ports, given the incoming fields ( bin 1,w h i l e bin 2is possi- ble but will not be considered here). Given a cavity frequency xcand a general system-cavity interaction Hamiltonian, HI¼gcdðaþa†Þ, mediated by the electric dipole operator d¼P n;mdn;mjnihmj,w h i c h describes transitions between QD eigenstates jniandjmi,w i t h HQDjni¼Enjni,32,57the quantum Langevin equation for the cavity operator areads _a¼/C0 ixca/C0j 2aþffiffiffiffiffij1pbin 1/C0igcd; (1) where jiare the cavity decay rates at the two ports and j¼j1þj2 þjintthe total photon loss rate, with jintbeing the intrinsic losses not related to the cavity ports. In the weak driving regime, we can assume that the electronic system remains near the thermal state, as it would in the absence ofany QD-cavity coupling, g c¼0, such that pnis the equilibrium popula- tion of state jni, which may be given by a Boltzmann distribution, pn¼e/C0En=kBT=ðP ne/C0En=kBTÞ, or by the solution of the corresponding rate equations in the case of a transport setup.58The evolution of the expectation value of the operators rnm¼jnihmjto first order in the coupling gcreads h_rnmi¼/C0 iðEm/C0EnÞhrnmi/C0X n0m0cnm;n0m0hrn0m0i /C0igcðhaiþh a†iÞdmnðpn/C0pmÞ: (2) Here, we have introduced decoherence processes via the matrix ele- ments cnm;n0m0. In frequency space, this constitutes, in general, a system of coupled linear equations for the susceptibilities, defined as FIG. 1. Schematic representation of a superconductor–semiconductor hybrid sys- tem. The superconducting microwave cavity (in gray) can be probed with micro- wave fields b1(b2) from port 1 (2) that is coupled to the center conductor (c) with a coupling rate j1(j2). A single electron (red dot) in the embedded semiconductor double QD (yellow) interacts with the cavity electric field E(blue curve) via the elec- tric dipole coupling.FIG. 2. Low-energy levels of a silicon double QD nanostructure with position (L,R), spin ( ",#), and valley degrees of freedom. Here, ELðRÞdenotes the left (right) QD valley splitting, /C15the detuning between the left and right QD ground state energy, and tc(t0 c) the intra(inter)-valley tunnel coupling. Inside the gray oval, more details on the spin sublevels of the lower valley states are given. A micromagnet can induce a different magnetic field direction atthe center of each QD, leading to canted spin quantization axes in thetwo QDs.Applied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 116, 190502 (2020); doi: 10.1063/5.0004777 116, 190502-2 Published under license by AIP PublishinghrnmðxÞi ¼vnmðxÞ½haðxÞi þ h a†ð/C0xÞi/C138.60,61In the simplest case of an electron in an aligned double QD ( /C15¼0) subject to relaxation and pure dephasing, the susceptibilities read v10ð01ÞðxÞ¼gcðp0/C0p1Þ 2tc7x7icc; (3) where j0iis the ground state and j1ithe excited state and ccis the total decoherence rate. Finally, for a cavity with a high quality factor probed close to resonance and for a sufficiently small coupling j;jx/C0xcj; f gcP n;mdnmvnmðxÞ/C28xcg,61the cavity transmission reads AðxÞ¼hbout 2i hbin 1i¼/C0iffiffiffiffiffiffiffiffiffiffij1j2p xc/C0x/C0ij=2þgcX nmdnmvnmðxÞ:(4) Depending on the level structure and the driving frequency, some- times, it is useful to simplify this expression accounting only for the transitions that contribute most to the response.57,60 The input–output theory has been generalized to periodically driven systems61,64and to more complex systems with vibrational degrees of freedom.65,66 The strong coupling regime for the interaction between a qubit and a cavity implies a coupling rate glarger than the decoherence rates of both the qubit ( c) and the photon ( j=2). The experimental achieve- ment of strong coupling ( g/C29c;j=2) between a spin qubit in a QD and a superconducting cavity54–56,67confirmed that the concept of a spin-based quantum computer with photon-mediated interactions isfeasible. The demonstrations in Refs. 54,55,a n d 67treat a single elec- tron spin qubit delocalized within a double QD and under the influ- ence of a magnetic field gradient perpendicular to the mainquantization axis of the spin, while the demonstration 56does not require magnetic fields but employs an exchange-only (three-electron) spin qubit in a triple QD. In both cases, the spin qubit acquires an elec- tric dipole moment that interacts with the cavity electric field. In thesingle electron case, the strength of this dipole coupling can be tuned by controlling the energy difference between the charge qubit tunnel splitting 2 t cand the magnetic Zeeman splitting glBBand allows for a compromise between a charge qubit with a short coherence that cou-ples strongly to cavity photons and a much protected pure spin qubit with negligible coupling to cavity photons. The coupling strength of the low-energy spin qubit to the cavity can be calculated exactly by asimple diagonalization of the double QD electronic Hamiltonian, and for a symmetric double QD ( /C15¼0), it reads g r¼gcsinð/þ=2 þ//C0=2Þ, with the spin-charge mixing angles /6¼arctan ½glBbx= ð2tc6glBBzÞ/C138 2 ½ 0;pÞ.68It turns out that there is an optimal point for coherent spin–photon coupling in terms of the relation between the interdot tunnel coupling and the externally applied magnetic field.60,69 This is summarized in Fig. 3(a) , where we show the coupling strength grof the low-energy spin qubit to the cavity as a function of tc(for /C15 ¼0) and the ratio between the coupling strength grand the total decoherence rate crþj=2, which has a maximum at the optimal point.68The dotted black line corresponds to the same ratio for a charge qubit. This comparison is valid under the assumption that the decoherence rate of the spin qubit cris dominated by the effects of the hybridization with charge, where cc¼1=T2cis the total charge deco- herence rate, the inverse of the charge decoherence time T2c.I nt h e middle region around 2 tc/C2525leV, the advantage obtained by usingthe electron spin with its long coherence time overcompensates the concomitant loss in spin coherence due to spin-charge hybridization. Spin–cavity interaction can be probed by tuning the spin qubit into resonance with the cavity mode, injecting a microwave tone into the cavity, and observing its transmission coefficient. In Fig. 3(b) ,w e show the cavity transmission coefficient Aas calculated using inpu- t–output theory, as a function of the detuning D0between the probe and cavity frequencies, predicting a well-resolved vacuum Rabi split- ting of the cavity resonance peak that was reported in Refs. 54and55, hallmarking the strong coupling between a single electron spin qubit and a cavity photon. The asymmetry between the two peaks, more apparent as gcincreases, is due to the presence of a third energy level and the interplay between the contributions v01andv02,60where j0iis the ground state and j1iandj2ithe first and second excited states. While pioneering works harnessed the magnetic field gradient generated by a micromagnet to electrically drive spin rotations on an electron spin situated in a single QD,12,70,71recent studies have dem- onstrated that a double QD configuration with aligned energy levels(/C15¼0) allows for low-power electric dipole spin transitions 72because in this “flopping mode,” the electron samples a larger magnetic field range and has a larger electric dipole. Also, in this mode of operation,the cavity-assisted spin readout has been theoretically optimized, with fidelities in the range of 80 –95% in a few lsb e i n gw i t h i nr e a c h . 73 Moreover, this configuration provides the spin qubit with “sweet spots,” i.e., points in the parameter space where the qubit is naturally protected from charge detuning fluctuations.74–77 The coupling to superconducting cavities has also provided high- fidelity readout of a two-electron spin state in a double QD.78The so- called singlet-triplet qubit, defined with two electrons in a double QD, can be operated in different regimes such that the nature of the cou-pling to the cavity can change from a standard transverse coupling 27,50 to a longitudinal coupling.79,80Although the strong-coupling regime to a cavity photon has not yet been demonstrated, recent theory pro-gress in identifying sweet spots, 81together with ongoing work to improve experimental devices, should make this possible. The experiments and theory discussed so far rely on large valley splitting, such that the valley degree of freedom barely affects the spindynamics. However, it is worth mentioning that within a QDFIG. 3. (a) Coupling strength gr(solid purple line) as a function of the tunnel split- ting 2 tcfor a fixed magnetic field profile; glBBz¼24leV and glBbx¼62leV. Also shown are the ratios between the coupling and decoherence for the spin(dashed orange line) and charge (dotted black line) qubit, for g c=2p¼50 MHz ; cc=2p¼5 MHz, and j=2p¼1:5 MHz. (b) Vacuum Rabi splitting peaks in the cavity transmission Aas a function of detuning D0¼x/C0xc, indicating strong spin–photon coupling. The two lines correspond to different values of the charge-photon coupling g c=2p¼f40;80gMHz.Applied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 116, 190502 (2020); doi: 10.1063/5.0004777 116, 190502-3 Published under license by AIP Publishingnanostructure, the nature of the low lying valley states, e.g., in silicon- based systems, may change from one QD to the other,57which results in inter-valley tunnel coupling and, therefore, the possibility to define a valley-orbit qubit that has been proven to couple strongly to the cavity photons.82It was theoretically predicted57and experimentally confirmed58that the low lying valley features, not only valley splitting but also intra- and inter-valley tunnel couplings, of a few electron sili- con QD systems are accessible in a hybrid circuit QED system since they generate a fingerprint on the cavity transmission. This complete information on valley features without the need of a magnetic field makes this scheme attractive in comparison to conventional magneto- spectroscopic approaches.21,83 What are the prospects of hybrid superconductor–semiconductor systems for quantum technology? Once the strong coupling between different spin qubits and superconducting cavities had been demon- strated, another important challenge emerged: to demonstrate spin qubit interactions mediated by cavity photons, in a similar way as previously achieved for superconducting qubits84,85and double QD charge qubits.37,86,87An important milestone in this direction was to tune two spin qubits simultaneously into resonance with the cavity and observe a collectively enhanced splitting in a transmission experiment.88 The advantage of using the spin rather than the charge is twofold: (i) the spin–cavity coupling can be turned off by increasing the tunnel coupling tc, therefore maintaining the qubit in a sweet spot protected from charge noise and (ii) the spin-qubit approach holds the potential of reducing the spin-charge mixing, with the corresponding reduction on spin–photon coupling grand spin qubit linewidth cr,s u c ht h a t eventually the condition cr/C28j=2, for which cavity-mediated two- qubit gates and readout fidelities are maximized for the device, is fulfilled.68This optimization demands a relatively small degree of spin-charge mixing, in order to make the spin decoherence rate com- parable to the Purcell relaxation rate. Therefore, experiments that attempt to demonstrate this effect would greatly benefit from the use of isotopically purified silicon.7 Recent theory work concludes that two-qubit gates mediated by cavity photons are capable of reaching fidelities exceeding 90%, even i nt h ep r e s e n c eo fc h a r g en o i s ea tt h el e v e lo f2 leV.68Since the fidelity is limited by the cooperativity C¼g2 r=crj, improvements are possible via increasing the double QD-cavity coupling gcor reducing the spin qubit and/or photon decoherence rates crandj.T oi n c r e a s e the coupling rate, superconducting cavities with higher kinetic induc- tance, which are to some extent resilient to the magnetic field, are available.89,90Improvements in the photon decay rate are possible via Purcell filters and improved cavity designs if one relies on separate superconducting cavities for readout or gate-based readout.78,91–95 Reducing the spin qubit decoherence rate may be the most challeng-ing, but one could try to reduce phonon emission 96and work at high- order sweet spots to reach a stronger protection against charge noise.77,81 The use of hybrid architectures, embedding semiconductor qubits in superconducting cavities, could potentially be an issue con- cerning the miniaturization and scaling. In this context, to truly har- ness the small size of the semiconductor qubits, one could benefit from the recent advances in the fabrication of QD arrays to increase the size of the computing nodes.23,97–103To this end, it is important to investigate short-distance coupling between spin qubits that have beenproven to couple to cavity photons.104Moreover, QD arrays allow for the exploration of new proposed qubits that couple to cavities but are more protected from decoherence such as the quadrupolar exchange only spin qubit.105 Eventually, for optimally controlled operations and for large- scale devices based on silicon, it will be necessary to have a micro- scopic understanding and control of the valley features.83,106 Alternatively, some researchers are considering a shift from the con-duction to the valence band since holes in silicon and germanium reside in a single non-degenerate valley. 107–109Interestingly, holes also have a relatively strong spin–orbit interaction, which is particularly pronounced in germanium, an effect that could substitute the external micromagnets.110–112 Recent works have explored further superconducting–semicon- ducting hybrid quantum systems containing also superconductingqubits. They employ a general circuit QED architecture to demon- strate a coherent interface between semiconductor and superconduct- ing qubits. 113,114In the future, superconducting cavities may, on the one hand, act as connectors between like qubits and, on the other hand, bridge between vastly different quantum systems. This work was supported by ARO through Grant No. W911NF-15-1-0149 and the DFG through the Collaborative Research Center SFB 767. DATA AVAILABILITY Data sharing is not applicable to this article as no new data were created or analyzed in this study. REFERENCES 1R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K. Vandersypen, Rev. Mod. Phys. 79, 1217 (2007). 2D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998). 3J. Levy, Phys. Rev. Lett. 89, 147902 (2002). 4J. Kyriakidis and G. Burkard, Phys. Rev. B 75, 115324 (2007). 5Z. Shi, C. B. Simmons, J. R. Prance, J. K. Gamble, T. S. Koh, Y.-P. Shim, X. Hu, D. E. Savage, M. G. Lagally, M. A. Eriksson, M. 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5.0007528.pdf

Appl. Phys. Lett. 116, 222403 (2020); https://doi.org/10.1063/5.0007528 116, 222403 © 2020 Author(s).Chiral-anomaly induced large negative magnetoresistance and nontrivial π-Berry phase in half-Heusler compounds RPtBi (R=Tb, Ho, and Er) Cite as: Appl. Phys. Lett. 116, 222403 (2020); https://doi.org/10.1063/5.0007528 Submitted: 15 March 2020 . Accepted: 19 May 2020 . Published Online: 02 June 2020 Jie Chen , Hang Li , Bei Ding , Enke Liu , Yuan Yao , Guangheng Wu , and Wenhong Wang ARTICLES YOU MAY BE INTERESTED IN The accurate measurement of spin orbit torque by utilizing the harmonic longitudinal voltage with Wheatstone bridge structure Applied Physics Letters 116, 222402 (2020); https://doi.org/10.1063/1.5145221 Enhancing the soft magnetic properties of FeGa with a non-magnetic underlayer for microwave applications Applied Physics Letters 116, 222404 (2020); https://doi.org/10.1063/5.0007603 Lateral domain wall oscillations in IMA/PMA bilayered nano-strips driven by a perpendicular current: A type of domain wall based oscillators Applied Physics Letters 116, 222405 (2020); https://doi.org/10.1063/5.0007771Chiral-anomaly induced large negative magnetoresistance and nontrivial p-Berry phase in half-Heusler compounds RPtBi (R 5Tb, Ho, and Er) Cite as: Appl. Phys. Lett. 116, 222403 (2020); doi: 10.1063/5.0007528 Submitted: 15 March 2020 .Accepted: 19 May 2020 . Published Online: 2 June 2020 JieChen,1,2Hang Li,1,2 BeiDing,1,2Enke Liu,1 Yuan Yao,1 Guangheng Wu,1and Wenhong Wang1,a) AFFILIATIONS 1State Key Laboratory for Magnetism, Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 2University of Chinese Academy of Sciences, Beijing 100049, China a)Author to whom correspondence should be addressed: wenhong.wang@iphy.ac.cn ABSTRACT We report on the observation of a large negative magnetoresistance (MR) with magnitudes of /C067%, /C045%, and /C031% in antiferromagnetic half-Heusler compounds TbPtBi, HoPtBi, and ErPtBi, respectively. It is found that with increasing temperature, the values of the negativeMR vary smoothly and persist well above their Neel temperature T N. Besides the negative MR effects, we have further observed a nontrivial Berry phase ( /C24p) extracted from Shubnikov–de Haas oscillation in HoPtBi. These results together with band structure calculations unambig- uously give evidence of the chiral anomaly effect and are valuable for understanding the Weyl fermions in magnetic lanthanide half-Heusler compounds. Published under license by AIP Publishing. https://doi.org/10.1063/5.0007528 Topological semimetals, such as Dirac and Weyl semimetals, are discovered quantum materials that have been attracting increasing attention recently as they possess a rich diversity of fundamental prop-erties as well as great potential for applications. 1–5One of the most intriguing transport signatures in these topological semimetals is the chiral anomaly induced negative magnetoresistance, MR¼½qBðÞ/C0q0ðÞ/C138=q0ðÞ, with unusual anisotropy being with respect to the angle between the electrical current and an applied mag- netic field. So far, large negative MR effects have been observed in both Dirac and Weyl semimetals such as Na 3Bi,6,7WTe 2,8Cd3As2,9,10 TaAs,11TaP,12and Co 3Sn2S2,13which serves as important experimen- tal evidence for chiral anomaly. Particular interest for searching topological semimetals has recently been focused on the magnetic lanthanide half-Heusler com-pounds RPtBi (R ¼rare-earth metal) since they possess a rich diversity of physical properties depending on R. Although considerable efforts were made, only a few of them, including GdPtBi 14–16and YbPtBi,17 were identified to be a Weyl semimetal by observing the chiral anom- aly induced negative MR. Compared to nonmagnetic half-Heusler compounds, such as ScPtBi,18YPtBi,19and LuPtBi,19,20the strongly localized 4f-electron in these magnetic lanthanide half-Heusler com- pounds may play an important role in forming Weyl points.21,22Moreimportantly, Hirschberger et al.15have pointed out that the same mechanism with GdPtBi can be generalized to other antiferromagnetic (AFM)-RPtBi half-Heusler compounds since these materials have asimilar electronic structure with a quadratic band (see Fig. S2, supple- mentary material ). Meanwhile, a small negative magnetoresistance has been observed in TbPtBi. 23TbPtBi, HoPtBi, and ErPtBi have an anti- ferromagnetic ground state, and the Neel temperature T Nis at 3.4 and <2 K. The nature of these antiferromagnetic RPtBi half-Heusler com- pounds is still under hot debate and much material is needed to uncover their underlying Weyl physics. Here, in this Letter, we present the evidence for magnetic field- induced Weyl physics, namely, a chiral anomaly induced large nega- tive MR in three AFM half-Heusler compounds, TbPtBi, HoPtBi, andErPtBi. The values of the negative MR are found to be /C067% (2.5 K), /C045% (3 K), and /C031% (3 K) in TbPtBi, HoPtBi, and ErPtBi, respec- tively. The regular variety of negative MR with localized 4f-electrons in these magnetic lanthanide half-Heusler compounds indicates a widerange of tunability. More importantly, a nontrivial Berry phase ( /C24p) extracted from Shubnikov–de Haas oscillation is revealed clearly in HoPtBi. The results might have a significant impact in the future ofthe general understanding of physics and technology of Weyl semimetals. Appl. Phys. Lett. 116, 222403 (2020); doi: 10.1063/5.0007528 116, 222403-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplSingle crystals of TbPtBi, HoPtBi, and ErPtBi were grown by a solution growth method from a Bi flux.24Magnetoresistance was mea- sured using a Quantum Design Physical Property Measurement System (PPMS) with a standard six-point probe method. An aberration-corrected scanning transmission electron microscope(STEM) measurement was performed using a JEOL ARM200F (JEOL, Tokyo, Japan) transmission electron microscope operated at 200 keV. The microscope was equipped with a CEOS (CEOS, Heidelberg, Germany) probe aberration corrector. The electronic band structures were determined using the WIEN2k code. 25Perdew–Burke–Ernzerhof Generalized Gradient Approximation (PBE-GGA) is used for the cal- culation of the exchange correlation potentials.26A large exchange parameter U eff¼0.6 Ry was applied to rear-earth elements (Tb, Ho, and Er), which would shift the 4f electrons far away from the Fermi level E F. Figure 1(a) shows the typical x-ray diffraction (XRD) pattern for the single crystal of HoPtBi; the two peaks (002) and (004) indicate a (001) plane, and the normal direction is [001]. The inset of Fig. 1(a) shows the optical image of HoPtBi single crystals. Generally, the (111) plane usually shows a triangle or hexagon. Here, we choose HoPtBi as the typical example to investigate using a high-resolution scanning transmission electron microscope (STEM) along the (110) plane. Asshown in Fig. 1(b) , the atoms arrange orderly that three atoms (Ho, Pt, and Bi) and one vacancy form a period along the [001] direction. All these indicate the high quality single crystals. In Weyl semimetals, nonconservation of chiral charge accounts for the chiral anomaly. As the schematic diagram shows in Fig. 1(c) , the Weyl states are quan- tized into Landau levels (LLs) under an intense Bfield. A pair of zero LLs have opposite chirality, and a charge imbalance will be generatedwhen a paralleled magnetic field and electric field are applied. The imbalanced charges will contribute to conductivity and a clear negative MR, where a strong dependence of the angle between the applied mag- netic field and current can be observed. We have carried out extensive measurements of the angular and magnetic field dependence of longitudinal resistivity q xxfor totally eight well-oriented samples along either [010] or [110] (see Table SI,supplementary material ).Figures 1(d)–1(f) show the typical MR curves of RPtBi (Tb, Ho, and Er) measured at 2.5 K, 3 K, and 2 K forselected values of h,w h e r e his the angle between Band Idirections. Forh¼0 /C14(B//I), the signature of Weyl fermions in transport experi- ment is clearly observed for all three samples. The negative MR forTbPtBi-#1, HoPtBi-#1, and ErPtBi-#1 reaches up to /C067%, /C045%, and/C031%, respectively. As we expected, the negative MR only per- sisted in a very narrow angle range. As hdeviates from 0 /C14, i.e., tilting B away from I, the negative MR goes away rapidly. Moreover, we found that the field dependence of the negative MR was different for thethree samples. In B<3T, the W-shaped MR of TbPtBi-#1 originates from the quantum interference effect. 24With the increase in B (B<9T), the negative value does not show any indication of satura- tion, while the negative MR of HoPtBi-#1 and ErPtBi-#1 shows a bell-shaped profile at low fields, which is similar to that observed in thefield-induced Weyl semimetal GdPtBi. 15In fact, similar results were obtained for all measured eight samples. These features strongly indi- cate that the negative MR in RPtBi (R ¼Tb, Ho, Er) arises from the chiral anomaly. Besides the chiral anomaly, the magnetic order,27weak localiza- tion effect,28and current jetting effect6can also induce a negative MR. The estimation of the effect of magnetic order on negative MR is car-ried out by picking up TbPtBi as the typical sample and roughly esti-mating 29the drop qsd/C25qxx(4 K, paramagnetism) /C0qxx(2.5 K, antiferromagnetism) ¼0.4247 m Xcm, where qsdis resistivity due to spin-disorder scattering in TbPtBi. The magnitude of negative MR forthe spin order can reach /C04% at 5 K under 9 T, which is much lower than/C067%, and thus, the magnetic effect as the origin of large nega- tive MR can be excluded. Moreover, the weak localization effect canalso be excluded because it usually occurs in a weak magnetic field. Toexclude the possibility reason for the current jetting effect, 6,30sample HoPtBi-#2 with eight small voltage contact pads [Fig. S3(a),supplementary material ] was prepared. Similar MR profiles for four pairs of nearest neighbors and a normal pair strongly indicate that thenegative MR is intrinsic, rather than caused by the inhomogeneouscurrent distribution. We have further measured the field dependence of MR for the three samples at different temperatures for h¼0 /C14(B//I). We found that, as shown in Figs. 2(a)–2(c) , with the increasing temperatures, the value of negative MR decreases gradually and disappears at about 50 K. On the other hand, the bell-shaped MR profiles for HoPtBi-#1 FIG. 1. (a) X-ray diffraction pattern and the optical image (inset) of HoPtBi single crystals. (b) Atomic-resolution image of the HoPtBi sample. The inset shows the atomic arrangement viewed from the [110] direction. (c) The schematic diagram of negative MR in the Weyl system under the B//E-field. (d)–(f) The MR curves of RPtBi (Tb, Ho, Er) measured at different hvalues, where his the angle between the magnetic field Band current directions Iin the polar plane x–z (see the inset). FIG. 2. Magnetic field dependence of MR at various temperatures for TbPtBi (a), HoPtBi (b), and ErPtBi (c) with B//I. Temperature dependence of the qxx(B) under different magnetic fields for TbPtBi (d), HoPtBi (e), and ErPtBi (f).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 222403 (2020); doi: 10.1063/5.0007528 116, 222403-2 Published under license by AIP Publishingand ErPtBi-#1 remain resolvable at about 40 K and 15 K, respectively. Figures 2(d)–2(f) show the temperature dependence of the resistivity qxx(B)u n d e r B//If o rt h e s et h r e es a m p l e s .A sn o t e di n Fig. 2(d) ,f o r example, the negative MR of TbPtBi-#1 persists up to 100 K, whichincludes the influence of the weak localization effect. But the tempera-ture intervals that host chiral anomaly contribution are about up to 50 K, 40 K, and 15 K for TbPtBi-#1, HoPtBi-#1, and ErPtBi-#2, respec- tively. The overall temperature-dependent MR behavior highlights themagnetic lanthanide half-Heusler compounds as a candidate platformfor exploring Weyl fermions at both low and elevated temperatures.Based on the magnetotransport and magnetization data, we can draw a phase diagram (Fig. S4, supplementary material ) that shows how the various physical properties of RPtBi (Tb, Ho, and Er) depend on thetemperature and applied field. We sketch the regions for the negativeMR, positive MR, AFM, and paramagnetic states based on field- andtemperature-dependent transport measurements. Quantum oscillations, superposed on the negative MR, are an important fingerprint of a Weyl semimetal with a chiral anomaly. 31In Figs. 3(a) and3(b), we show the obvious quantum oscillations super- posed on the negative MR for ErPtBi and HoPtBi, respectively. In theconventional material with a topological trivial band structure, the Berry Phase U B¼0. However, in topological materials with linear energy dispersion, the corresponding Fermi pockets should possess anontrivial p-Berry phase. p-Berry phase is, therefore, a valuable crite- rion for identifying topological Weyl materials. 32–34InFig. 3(c) ,w e show the plots of d2qxx/dB2as a function of the inverse magnetic field1/Bfor HoPtBi at different temperatures. The amplitude of the SdH oscillations decreases with increasing T, and the oscillations can- not be observed for T>10 K. According to the Onsager relation, F¼/C22h=2pe ðÞ AF, where F is the frequency, eis the electronic charge, /C22h is the reduced Planck constant, and the cross section of the Fermi sur-face A Fis 0.0038 A ˚/C02. By assuming a circular cross section, the corre- sponding wave vector value KFis estimated to be 0.035 A ˚/C01,w h i c hi s derived from equation, K2 F¼2eF=/C22h. This indicates that HoPtBi hosts an extremely small Fermi surface. We plotted the Landau index nas a function of 1/ BinFig. 3(d) . The maxima of d2qxx/dB2are defined as integer indices, and the minima of d2qxx/dB2are defined as half- integer indices. According to the Lifshitz–Kosevich (LK) formula, thephase factor was defined as cos 2 pF=Bþc/C0d ðÞ½/C138 . The phase factor is related to the Berry phase through c¼1=2/C0/ B=2p,w h e r e UBis the Berry phase. dis the phase shift, which is determined by the dimen- sionality of the Fermi surface. For the three-dimensional (3D) case,thedvalue for the hole pocket is þ1/8, corresponding to the electron pocket, whose dis/C01/8. 35Hall resistivity of HoPtBi shows a linear behavior, which means one carrier model with a hole pocket (Fig. S5,supplementary material ). By considering 3D hole pocket in HoPtBi, the Berry phase U Bis equal to (1.11 60.01)p, which is very close to thep-Berry phase. To confirm the p-Berry phase in HoPtBi, the SdH oscillation extracts from the Hall signal also shown in Fig. 3(d) .T h e measurement close to the quantum limit is also provided in Figs. 3(e) and3(f); the almost same intercept value n0¼/C00.18 further confirms the nontrivial p-Berry phase in HoPtBi. For comparison, in Fig. 4 , we show the large negative MR observed in current typical Weyl semimetals and Dirac semimetals.Remarkably, in these magnetic lanthanide half-Heusler compoundsRPtBi, with the atomic number of R decreasing, the values of negativeMR increase quickly. The regular variation of negative MR implies that the Weyl physics was effectively modified by the rear-earth ele- ment R. In addition, unlike the nonmagnetic Weyl and Dirac semime-tals, these magnetic lanthanide half-Heusler compounds with strongspin-orbit coupling (SOC) host magnetic ordered ground states (TableSII,supplementary material ). Therefore, in addition to the chiral anomaly induced large negative MR and nontrivial p-Berry phase, other signatures of Weyl fermions, such as the large intrinsic anomaly Hall effect 16,17and topological Hall effect,17are also observed in these magnetic lanthanide RPtBi compounds. FIG. 3. Magnetic field dependence of magnetoresistance qxxfor ErPtBi (a) and HoPtBi (b) measured at T¼2 K with different titled angles h. (c) and (e) show the periodic SdH oscillations d2qxx/dB2as a function of the inverse magnetic field 1/ B in HoPtBi under magnetic fields up to 13T and 31T, respectively. (d) and (f) showthe Landau level (LL) indices extracted from corresponding SdH oscillation. Theinset of (d) shows fast Fourier transform (FFT) spectra of SdH oscillations. FIG. 4. Comparison of the negative MR of topological semimetals, including half- Heusler RPtBi family, typical Dirac semimetal, and Weyl semimetals. The data aretaken from Refs. 7–9,11–13 ,15,17,23, and 36–39 .Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 222403 (2020); doi: 10.1063/5.0007528 116, 222403-3 Published under license by AIP PublishingIn summary, we report the large negative MR induced by the chi- ral anomaly in three antiferromagnetic half-Heusler compounds,TbPtBi, HoPtBi, and ErPtBi. We excluded other factors like magneticorder, current jetting, and weak localization effect by transport mea- surements. Apart from the negative MR, we have further observed a nontrivial Berry phase ( /C24p) extracted from Shubnikov–de Haas oscil- lation, which is obtained in HoPtBi. Our results also provide a way todiscover magnetic field-induced Weyl physics in a large range of mag-netic lanthanide half-Heusler compounds, which can be utilized forfuture discovery of topological materials, which is of urgent currentinterest. See the supplementary material for the crystal structure of RPtBi and a sketch of Weyl node formation, band structures, current jettingeffect, phase diagram of chiral anomaly induced negative MR, Hallresistivity of HoPtBi, list of samples measured, and chiral anomalyinduced negative MR in typical topological materials. This work was supported by the National Key R&D Program of China (Grant No. 2017YFA0206303), the National NaturalScience Foundation of China (Nos. 11974406 and 11874410), andthe Fujian Innovation Academy, Chinese Academy of Sciences. Aportion of this work was performed on the Steady High MagneticField Facilities, High Magnetic Field Laboratory, Chinese Academyof Science. DATA AVAILABILITY The data that support the findings of this study are available within this article. REFERENCES 1B. Bradlyn, J. Cano, Z. Wang, M. G. Vergniory, C. Felser, R. J. Cava, and B. A. Bernevig, Science 353, aaf5037 (2016). 2N. P. Armitage, E. J. Mele, and A. Vishwanath, Rev. Mod. 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5.0004997.pdf

NWChem: Past, present, and future Cite as: J. Chem. Phys. 152, 184102 (2020); https://doi.org/10.1063/5.0004997 Submitted: 17 February 2020 . Accepted: 07 April 2020 . Published Online: 11 May 2020 E. Aprà , E. J. Bylaska , W. A. de Jong , N. Govind , K. Kowalski , T. P. Straatsma , M. Valiev , H. J. J. van Dam , Y. Alexeev , J. Anchell , V. Anisimov , F. W. Aquino , R. Atta-Fynn , J. Autschbach , N. P. Bauman , J. C. Becca , D. E. Bernholdt , K. Bhaskaran-Nair , S. Bogatko , P. Borowski , J. Boschen , J. Brabec , A. Bruner , E. Cauët , Y. Chen , G. N. Chuev , C. J. Cramer , J. Daily , M. J. O. Deegan , T. H. Dunning , M. Dupuis , K. G. Dyall , G. I. Fann , S. A. Fischer , A. Fonari , H. Früchtl , L. Gagliardi , J. Garza , N. Gawande , S. Ghosh , K. Glaesemann , A. W. Götz , J. Hammond , V. Helms , E. D. Hermes , K. Hirao , S. Hirata , M. Jacquelin , L. Jensen , B. G. Johnson , H. Jónsson , R. A. Kendall , M. Klemm , R. Kobayashi , V. Konkov , S. Krishnamoorthy , M. Krishnan , Z. Lin , R. D. Lins , R. J. Littlefield , A. J. Logsdail , K. Lopata , W. Ma , A. V. Marenich , J. Martin del Campo , D. Mejia-Rodriguez , J. E. Moore , J. M. Mullin , T. Nakajima , D. R. Nascimento , J. A. Nichols , P. J. Nichols , J. Nieplocha , A. Otero-de-la-Roza , B. Palmer , A. Panyala , T. Pirojsirikul , B. Peng , R. Peverati , J. Pittner , L. Pollack , R. M. Richard , P. Sadayappan , G. C. Schatz , W. A. Shelton , D. W. Silverstein , D. M. A. Smith , T. A. Soares , D. Song , M. Swart , H. L. Taylor , G. S. Thomas , V. Tipparaju , D. G. Truhlar , K. Tsemekhman , T. Van Voorhis , Á. Vázquez-Mayagoitia , P. Verma , O. Villa , A. Vishnu , K. D. Vogiatzis , D. Wang , J. H. Weare , M. J. Williamson , T. L. Windus , K. Woliński, A. T. Wong , Q. Wu , C. Yang , Q. Yu , M. Zacharias , Z. Zhang , Y. Zhao , and R. J. Harrison COLLECTIONS Paper published as part of the special topic on Electronic Structure Software Note: This article is part of the JCP Special Topic on Electronic Structure Software. ARTICLES YOU MAY BE INTERESTED IN Recent developments in the general atomic and molecular electronic structure system The Journal of Chemical Physics 152, 154102 (2020); https://doi.org/10.1063/5.0005188 eT 1.0: An open source electronic structure program with emphasis on coupled cluster and multilevel methods The Journal of Chemical Physics 152, 184103 (2020); https://doi.org/10.1063/5.0004713 ReSpect: Relativistic spectroscopy DFT program package The Journal of Chemical Physics 152, 184101 (2020); https://doi.org/10.1063/5.0005094J. Chem. Phys. 152, 184102 (2020); https://doi.org/10.1063/5.0004997 152, 184102The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp NWChem: Past, present, and future Cite as: J. Chem. Phys. 152, 184102 (2020); doi: 10.1063/5.0004997 Submitted: 17 February 2020 •Accepted: 7 April 2020 • Published Online: 11 May 2020 E. Aprà,1 E. J. Bylaska,1 W. A. de Jong,2 N. Govind,1 K. Kowalski,1,a) T. P. Straatsma,3 M. Valiev,1 H. J. J. van Dam,4Y. Alexeev,5 J. Anchell,6V. Anisimov,5F. W. Aquino,7R. Atta-Fynn,8J. Autschbach,9 N. P. Bauman,1 J. C. Becca,10D. E. Bernholdt,11 K. Bhaskaran-Nair,12S. Bogatko,13P. Borowski,14J. Boschen,15 J. Brabec,16A. Bruner,17E. Cauët,18Y. Chen,19G. N. Chuev,20 C. J. Cramer,21 J. Daily,1M. J. O. Deegan,22 T. H. Dunning, Jr.,23 M. Dupuis,9 K. G. Dyall,24G. I. Fann,11S. A. Fischer,25A. Fonari,26,b)H. Früchtl,27 L. Gagliardi,21 J. Garza,28N. Gawande,1S. Ghosh,29,c) K. Glaesemann,1A. W. Götz,30 J. Hammond,6 V. Helms,31 E. D. Hermes,32K. Hirao,33S. Hirata,34 M. Jacquelin,2L. Jensen,10 B. G. Johnson,35 H. Jónsson,36 R. A. Kendall,11M. Klemm,6 R. Kobayashi,37 V. Konkov,38S. Krishnamoorthy,1M. Krishnan,19 Z. Lin,39R. D. Lins,40R. J. Littlefield,41A. J. Logsdail,42 K. Lopata,43 W. Ma,44A. V. Marenich,45,d) J. Martin del Campo,46 D. Mejia-Rodriguez,47 J. E. Moore,6J. M. Mullin,48T. Nakajima,49 D. R. Nascimento,1 J. A. Nichols,11P. J. Nichols,50J. Nieplocha,1A. Otero-de-la-Roza,51B. Palmer,1A. Panyala,1 T. Pirojsirikul,52 B. Peng,1 R. Peverati,38 J. Pittner,53L. Pollack,54R. M. Richard,55 P. Sadayappan,56G. C. Schatz,57 W. A. Shelton,58D. W. Silverstein,59D. M. A. Smith,6T. A. Soares,60 D. Song,1 M. Swart,61 H. L. Taylor,62 G. S. Thomas,1V. Tipparaju,63D. G. Truhlar,21 K. Tsemekhman,64T. Van Voorhis,65Á. Vázquez-Mayagoitia,5 P. Verma,66 O. Villa,67A. Vishnu,1K. D. Vogiatzis,68D. Wang,69 J. H. Weare,70M. J. Williamson,71 T. L. Windus,72K. Woli ´nski,14A. T. Wong,73Q. Wu,4 C. Yang,2Q. Yu,74M. Zacharias,75Z. Zhang,76 Y. Zhao,77 and R. J. Harrison78 AFFILIATIONS 1Pacific Northwest National Laboratory, Richland, Washington 99352, USA 2Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 3National Center for Computational Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA 4Brookhaven National Laboratory, Upton, New York 11973, USA 5Argonne Leadership Computing Facility, Argonne National Laboratory, Argonne, Illinois 60439, USA 6Intel Corporation, Santa Clara, California 95054, USA 7QSimulate, Cambridge, Massachusetts 02139, USA 8Department of Physics, The University of Texas at Arlington, Arlington, Texas 76019, USA 9Department of Chemistry, University at Buffalo, State University of New York, Buffalo, New York 14260, USA 10Department of Chemistry, The Pennsylvania State University, University Park, Pennsylvania 16802, USA 11Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA 12Washington University, St. Louis, Missouri 63130, USA 134G Clinical, Wellesley, Massachusetts 02481, USA 14Faculty of Chemistry, Maria Curie-Skłodowska University in Lublin, 20-031 Lublin, Poland 15Department of Chemistry, Iowa State University, Ames, Iowa 50011, USA 16J. Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, 18223 Prague 8, Czech Republic 17Department of Chemistry and Physics, University of Tennessee at Martin, Martin, Tennessee 38238, USA 18Service de Chimie Quantique et Photophysique (CP 160/09), Université libre de Bruxelles, B-1050 Brussels, Belgium 19Facebook, Menlo Park, California 94025, USA 20Institute of Theoretical and Experimental Biophysics, Russian Academy of Science, Pushchino, Moscow Region 142290, Russia 21Department of Chemistry, Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455, USA 22SKAO, Jodrell Bank Observatory, Macclesfield SK11 9DL, United Kingdom J. Chem. Phys. 152, 184102 (2020); doi: 10.1063/5.0004997 152, 184102-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 23Department of Chemistry, University of Washington, Seattle, Washington 98195, USA 24Dirac Solutions, Portland, Oregon 97229, USA 25Chemistry Division, U. S. Naval Research Laboratory, Washington, DC 20375, USA 26School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, Georgia 30332, USA 27EaStCHEM and School of Chemistry, University of St. Andrews, St. Andrews KY16 9ST, United Kingdom 28Departamento de Química, División de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana-Iztapalapa, Col. Vicentina, Iztapalapa, C.P. 09340 Ciudad de México, Mexico 29Department of Chemistry, Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 5545, USA 30San Diego Supercomputer Center, University of California, San Diego, La Jolla, California 92093, USA 31Center for Bioinformatics, Saarland University, D-66041 Saarbrücken, Germany 32Combustion Research Facility, Sandia National Laboratories, Livermore, California 94551, USA 33Next-generation Molecular Theory Unit, Advanced Science Institute, RIKEN, Saitama 351-0198, Japan 34Department of Chemistry, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA 35Acrobatiq, Pittsburgh, Pennsylvania 15206, USA 36Faculty of Physical Sciences, University of Iceland, Reykjavík, Iceland and Department of Applied Physics, Aalto University, FI-00076 Aalto, Espoo, Finland 37ANU Supercomputer Facility, Australian National University, Canberra, Australia 38Chemistry Program, Florida Institute of Technology, Melbourne, Florida 32901, USA 39Department of Physics, University of Science and Technology of China, Hefei, China 40Aggeu Magalhaes Institute, Oswaldo Cruz Foundation, Recife, Brazil 41Zerene Systems LLC, Richland, Washington 99354, USA 42Cardiff Catalysis Institute, School of Chemistry, Cardiff University, Cardiff, Wales CF10 3AT, United Kingdom 43Department of Chemistry, Louisiana State University, Baton Rouge, Louisiana 70803, USA 44Institute of Software, Chinese Academy of Sciences, Beijing, China 45Department of Chemistry, Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455, USA 46Departamento de Física y Química Teórica, Facultad de Química, Universidad Nacional Autónoma de México, México City, Mexico 47Quantum Theory Project, Department of Physics, University of Florida, Gainesville, Florida 32611, USA 48DCI-Solutions, Aberdeen Proving Ground, Maryland 21005, USA 49Computational Molecular Science Research Team, RIKEN Center for Computational Science, Kobe, Hyogo 650-0047, Japan 50Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 51Departamento de Química Física y Analítica, Facultad de Química, Universidad de Oviedo, 33006 Oviedo, Spain 52Department of Chemistry, Prince of Songkla University, Hat Yai, Songkhla 90112, Thailand 53J. Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, v.v.i., 18223 Prague 8, Czech Republic 54StudyPoint, Boston, Massachusetts 02114, USA 55Ames Laboratory, Ames, Iowa 50011, USA 56School of Computing, University of Utah, Salt Lake City, Utah 84112, USA 57Department of Chemistry, Northwestern University, Evanston, Illinois 60208, USA 58Cain Department of Chemical Engineering, Louisiana State University, Baton Rouge, Louisiana 70803, USA 59Universal Display Corporation, Ewing, New Jersey 08618, USA 60Dept. of Fundamental Chemistry, Universidade Federal de Pernambuco, Recife, Brazil 61ICREA, 08010 Barcelona, Spain and Universitat Girona, Institut de Química Computacional i Catàlisi, Campus Montilivi, 17003 Girona, Spain 62CD-adapco/Siemens, Melville, New York 11747, USA 63Cray Inc., Bloomington, Minnesota 55425, USA 64Gympass, New York, New York 10013, USA 65Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 661QBit, Vancouver, British Columbia V6E 4B1, Canada 67NVIDIA, Santa Clara, California 95051, USA J. Chem. Phys. 152, 184102 (2020); doi: 10.1063/5.0004997 152, 184102-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 68Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996, USA 69College of Physics and Electronics, Shandong Normal University, Jinan, Shandong 250014, China 70Department of Chemistry and Biochemistry, University of California San Diego, La Jolla, California 92093, USA 71Department of Chemistry, Cambridge University, Lensfield Road, Cambridge CB2 1EW, United Kingdom 72Department of Chemistry, Iowa State University and Ames Laboratory, Ames, Iowa 50011, USA 73Qwil, San Francisco, California 94107, USA 74AMD, Santa Clara, California 95054, USA 75Department of Physics, Technical University of Munich, 85748 Garching, Germany 76Stanford Research Computing Center, Stanford University, Stanford, California 94305, USA 77State Key Laboratory of Silicate Materials for Architectures, International School of Materials Science and Engineering, Wuhan University of Technology, Wuhan 430070, China 78Institute for Advanced Computational Science, Stony Brook University, Stony Brook, New York 11794, USA Note: This article is part of the JCP Special Topic on Electronic Structure Software. a)Author to whom correspondence should be addressed: karol.kowalski@pnnl.gov b)Current address: Schrödinger, Inc., New York, NY 10036, USA. c)Current address: Max Planck Institute für Kohlenforschung, 45470 Mülheim an der Ruhr, Germany. d)Current address: Gaussian, Inc., Wallingford, CT 06492, USA. ABSTRACT Specialized computational chemistry packages have permanently reshaped the landscape of chemical and materials science by providing tools to support and guide experimental efforts and for the prediction of atomistic and electronic properties. In this regard, electronic structure packages have played a special role by using first-principle-driven methodologies to model complex chemical and materials processes. Over the past few decades, the rapid development of computing technologies and the tremendous increase in computational power have offered a unique chance to study complex transformations using sophisticated and predictive many-body techniques that describe correlated behavior of electrons in molecular and condensed phase systems at different levels of theory. In enabling these simulations, novel parallel algorithms have been able to take advantage of computational resources to address the polynomial scaling of electronic structure methods. In this paper, we briefly review the NWChem computational chemistry suite, including its history, design principles, parallel tools, current capabilities, outreach, and outlook. Published under license by AIP Publishing. https://doi.org/10.1063/5.0004997 .,s I. INTRODUCTION The NorthWest Chemistry (NWChem) modeling software is a popular computational chemistry package that has been designed and developed to work efficiently on massively parallel processing supercomputers.1–3It contains an umbrella of modules that can be used to tackle most electronic structure theory calculations being carried out today. Since 2010, the code is distributed as open-source under the terms of the Educational Community License version 2.0 (ECL 2.0). Electronic structure theory provides a foundation for our understanding of chemical transformations and processes in com- plex chemical environments. For this reason, accurate electronic structure formulations have already permeated several key areas of chemistry, biology, biochemistry, and materials sciences, where they have become indispensable elements for building synergies between theoretical and experimental efforts and for predictions. Over the past few decades, intense theoretical developments have resulted in a broad array of electronic structure methods and their implementations, designed to describe structures, interac- tions, chemical reactivity, dynamics, thermodynamics, and spec- tral properties of molecular and material systems. The success ofthese computational tools hinges upon several requirements regard- ing the accuracy of many-body models, reliable algorithms for dealing with processes at various spatial and temporal scales, and effective utilization of ever-growing computational resources. For instance, the predictive power of computational chemistry requires sophisticated quantum mechanical (QM) approaches that system- atically account for electronic correlation effects. Therefore, the design of versatile electronic structure codes is a major undertak- ing that requires close collaboration between experts in theoretical and computational chemistry, applied mathematics, and computer science. NWChem,2–8like other widely used electronic structure pro- grams, was developed to fully realize the potential of computational modeling to answer key scientific questions. It provides a wide range of capabilities that can be deployed on supercomputing platforms to solve two fundamental equations of quantum mechanics9–11—time- independent and time-dependent Schrödinger equations, H∣Ψ⟩=E∣Ψ⟩, (1) i̵h∂∣Ψ⟩ ∂t=H∣Ψ⟩, (2) J. Chem. Phys. 152, 184102 (2020); doi: 10.1063/5.0004997 152, 184102-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp and a fundamental equation of Newtonian mechanics, miai=Fi, (3) where forces Fiinclude information about quantum effects. Given the breadth of electronic structure theory, it does not come as a surprise that Eqs. (1) and (2) can be solved using var- ious representations of quantum mechanics employing wavefunc- tions (| Ψ⟩), electron densities ( ρ(⃗r)), or self-energies [ Σ(ω)], which comprise the wide spectrum of NWChem’s functionalities to com- pute the electronic wavefunctions, densities, and associated proper- ties of molecular and periodic systems. These functionalities include Hartree–Fock12–15self-consistent field (SCF) and post-SCF cor- related many-body approaches that build on the SCF wavefunc- tion to tackle static and dynamic correlation effects. Among cor- related approaches, NWChem offers second-order Möller–Plesset perturbation theory; single- and multi-reference (MR), ground- and excited-state, and linear-response (LR) coupled-cluster (CC) theories; multi-configuration self-consistent field (MCSCF); and selected and full configuration interaction (CI) codes. NWChem provides extensive density functional theory16–18(DFT) capa- bilities with Gaussian and plane wave basis set implementa- tions. Within the Gaussian basis set framework, a broad range of DFT response properties, ground and excited-state molecular dynamics (MD), linear-response (LR), and real-time (RT) time- dependent density functional theory (TDDFT) are available. The plane wave DFT implementations offer the capability to run scalable ab initio and Car–Parrinello molecular dynamics19and band structure simulations. The plane wave code supports both norm-conserving20–22and projector augmented wave (PAW)23 pseudopotentials. For all DFT methods outlined above, both analytical or numer- ical gradients and Hessians are available for geometry optimization and vibrational analysis. Additionally, NWChem is capable of per- forming classical molecular dynamics (MD) simulations using either AMBER or CHARMM force fields. Through its modular design, the ab initio methods can be coupled with the classical MD to perform mixed quantum-mechanics and molecular-mechanics simulations (QM/MM). Various solvent models and relativistic approaches are also available, with the spin–orbit (SO) contribution only being sup- ported at the Hartree–Fock (HF) and DFT levels of theory and asso- ciated response properties. The NWChem functionality described is only a subset of its full capabilities. We refer the reader to the NWChem website8to learn about the full suite of functionalities available to the user community. Currently, NWChem is developed and maintained primarily by researchers at the Department of Energy (DOE) Pacific North- west National Laboratory (PNNL), with help from researchers at other research institutions. It has a broad user base, and it is being used across the national laboratory system and through- out academia and industry around the world. In this paper, we provide a high-level overview of NWChem’s core capabil- ities, recent developments in electronic methods, and a short discussion of ongoing and future efforts. We also illustrate the strengths of NWChem stemming from the possibility of seam- less integration of methodologies at various scales and review sci- entific results that would not otherwise be obtainable without using its highly scalable implementations of electronic structure methods.II. BRIEF HISTORY The NWChem project1–7,24,25started in 1992. It was originally designed and implemented as part of the construction project asso- ciated with the EMSL user facility at PNNL. Therefore, the software project started around four years before the EMSL computing center was up and running. This raised challenges for the software devel- opers working on the project, such as predicting the features of future hardware architectures and how to deliver high performing software while maintaining programmer productivity. Overcoming these challenges led to a design effort that strove for flexibility and extensibility, as well as high-level interfaces to functionality that hid some of the hardware issues from the chemistry software application developer. Over the years, this design and implementation have suc- cessfully advanced multiple science agendas, and NWChem’s exten- sive code base of more than 2 ×106lines provides high-performance, scalable software code with advanced scientific capabilities that are used throughout the molecular sciences community. NWChem is an example of a co-design effort harnessing the expertise of researchers from multiple scientific disciplines to pro- vide users with computational chemistry tools that are scalable both in their ability to treat large scientific computational chemistry prob- lems efficiently and in their use of computing resources from high- performance parallel supercomputers to conventional workstation clusters. In particular, NWChem has been designed to handle ●biomolecules, nanostructures, interfaces, and solid-state, ●chemical processes in complex environments, ●hybrid quantum/classical simulations, ●ground and excited-states and non-linear optical properties, ●simulations of UV–Vis, photo-electron, and x-ray spectro- scopies, ●Gaussian basis functions or plane waves, ●ab initio molecular dynamics on the ground and excited states, and ●relativistic effects. The scalability of NWChem has provided a computational plat- form to deliver new scientific results that would be unobtainable if parallel computational platforms were not used. For example, NWChem’s implementation of a non-orthogonally spin adapted coupled-cluster single double triple [CCSD(T)] method has been demonstrated to scale to 210 000 processors available at the Oak Ridge National Laboratory’s (ORNL) Leadership Computing Facil- ities,26–28whereas the plane wave DFT code has been able to utilize close to 100 000 processor cores on NERSC’s Cray-XE6 supercom- puter.29Although implemented only for the perturbative part of coupled-cluster with singles and doubles (CCSD)30and triples cor- rection [CCSD(T)],31NWChem was one of the first computational chemistry codes to have been ported to utilize graphics processing units (GPUs).32Several parts of the code have also been rewrit- ten to take advantage of the Intel Xeon Phi family of processors— good scalability and performance have been demonstrated for the ab initio molecular dynamics (AIMD) plane wave DFT code on the most recent Knights Landing version of the processor.33,34The non- iterative triples part of the CCSD(T) method has been demonstrated to scale to 55 200 Intel Phi threads and 62 560 cores through concur- rent utilization of central processing unit (CPU) and Intel Xeon Phi Knights Corner accelerators.35 J. Chem. Phys. 152, 184102 (2020); doi: 10.1063/5.0004997 152, 184102-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp III. DESIGN PRINCIPLES NWChem has a five-tiered modular architecture. The first tier is the Generic Task Interface . This interface (an abstract program- ming interface, not a user interface) serves as the mechanism that transfers control to the different modules in the second tier, which consists of the Molecular Calculation Modules . The molecular cal- culation modules are the high-level programming modules that accomplish computational tasks, performing particular operations using the specified theories defined by the user in the input file. These independent modules of NWChem share data only through a disk-resident database, which allows modules to share data or to share access to files containing data. The third tier consists of the Molecular Modeling Tools . These routines provide basic chemical functionality such as symmetry, basis sets, grids, geometry, and inte- grals. The fourth tier is the Software Development Toolkit , which is the basic foundation of the code. The fifth tier provides the Utility Functions needed by nearly all modules in the code. These include functionality such as input processing, output processing, and timing. The Generic Task Interface controls the execution of NWChem. The flow of control proceeds in the following steps: 1. Identify and open the input file. 2. Complete the initialization of the parallel environment. 3. Process start-up directives. 4. Summarize start-up information and write it to the output file. 5. Open the run-time database. 6. Process the input sequentially (ignoring start-up directives), including the first task directive. 7. Execute the task. 8. Repeat steps 6 and 7 until reaching the end of the input file or encountering a fatal error condition. The input parser processes the user’s input file and translates the information into a form meaningful to the main program and the driver routines for specific tasks. As mentioned in step 5 of the task flow control, NWChem makes use of a run-time database to store the main computational parameters. This is in the same spirit of check-pointing features available in other quantum chemistry codes. The information stored in the run-time database can be used at a later time in order to restart a calculation. Restart capabilities are available for most modules. For example, SCF generated files (run-time database and molecular orbitals) can be used either to continue a geometry optimization or to compute molecular properties. The important second and fourth tiers are discussed as part of Secs. IV–VI. IV. PARALLEL TOOLS The design and early development of the Global Arrays36–39 (GA) toolkit occurred in the same period when the NWChem project started. The GA toolkit, which is the central component of theSoftware Development Toolkit , was adopted by the NWChem developers as the main approach for the parallelization of the dense matrices present in quantum chemistry methods that make use of local basis functions. In current computer science parlance, global arrays can be viewed as a Partitioned Global Address Space (PGAS) model that provides a high level of abstraction for the programmerto the dense distributed arrays. In contrast to message passing con- structs such as Message Passing Interface (MPI), where the devel- oper has to worry about coordinating send and receive operations, the use of global arrays in NWChem requires the so-called single- sided functions (e.g., put, get, and accumulate) to manipulate data structures in a single operation. The choice of the distribution model for sharing a given global array among the memory available to the processes in use plays a crucial role in efficient parallelization at large scale. The GA toolkit has been ported to a variety of parallel computer architectures. The porting process has focused in the past in opti- mizing the ARMCI40library. The Aggregate Remote Memory Copy (ARMCI) library optimizes performance by fully exploiting network characteristics such as latency, bandwidth, and packet injection rate through the use of low-level network protocols (e.g., Infiniband verbs). More recent porting options make use of either ComEx41 or the ARMCI-MPI42communication runtimes. Both ComEx and ARMCI-MPI make use of MPI libraries, instead of low-level network protocols, albeit with different approaches. V. MAIN METHODOLOGIES In this section, we describe the key methods that comprise the Molecular Calculation Modules . We first describe the Gaus- sian basis HF and DFT implementations for molecular systems. This is followed by the post-SCF wavefunction-based perturba- tive (MP2), multi-configuration SCF, and high accuracy (coupled- cluster theory) approaches for molecules, including the tensor contraction engine (TCE). Molecular response properties and rela- tivistic approaches are then described. The plane wave based DFT implementation for Car–Parrinello molecular dynamics and peri- odic condensed phase systems is described next, followed by classical molecular dynamics and hybrid methods. A. Hartree–Fock The NWChem SCF module computes closed-shell restricted Hartree–Fock (RHF) wavefunctions, restricted high-spin open- shell Hartree–Fock (ROHF) wavefunctions, and spin-unrestricted Hartree–Fock (UHF) wavefunctions. The Hartree–Fock equations are solved using a conjugate-gradient method with an orbital Hes- sian based preconditioner.43 The most expensive part to compute in the SCF code is the two-electron contribution to the matrix element of the Fock oper- ator (resulting from the sum of Coulomb and exchange opera- tors). To compute these matrix elements, NWChem developers have implemented parallel algorithms using either a distributed data approach44(where the Fock matrix is distributed among the aggre- gate memory of the processes involved in the calculation) or a repli- cated data approach (where an entire copy of the Fock matrix is stored in the memory of each process). Several options are available for the initial guess of the SCF calculations. The default choice uses the eigenvectors of a Fock- like matrix formed from a superposition of the atomic densities. Other options include the use of eigenvectors of the bare-nucleus Hamiltonian or the one-electron Hamiltonian, the projections of the molecular orbital from a smaller basis to a larger one, or molecular orbitals formed by superimposing the orbitals of fragments of the J. Chem. Phys. 152, 184102 (2020); doi: 10.1063/5.0004997 152, 184102-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp molecule being studied. Symmetry can be used to speed up the Fock matrix construction via the petite-list algorithm. Molecular orbitals are symmetry adapted as well in NWChem. The resolution of the identity (RI) four-center, two-electron integral approximation has also been implemented.45 In order to avoid full matrix diagonalization, the SCF pro- gram uses a preconditioned conjugate gradient (PCG) method that is unconditionally convergent. Basically, a search direction is gener- ated by multiplying the orbital gradient (the derivative of the energy with respect to the orbital rotations) by an approximation to the inverse of the level-shifted orbital Hessian. In the initial iterations, an inexpensive one-electron approximation to the inverse orbital Hessian is used. Closer to convergence, the full orbital Hessian is used, which should provide quadratic convergence. For both the full or one-electron orbital Hessians, the inverse-Hessian matrix- vector product is formed iteratively. Subsequently, an approximate line search is performed along the new search direction. Both all-electron basis sets and effective core potentials (ECPs) can be used. Effective core potentials are a useful means of replac- ing the core electrons in a calculation with an effective potential, thereby eliminating the need for the core basis functions, which usu- ally require a large set of Gaussians to describe them. In addition to replacing the core, they may be used to represent relativistic effects, which will be discussed later. B. Density functional theory The NWChem DFT module for molecular systems uses a Gaussian basis set to compute closed- and open-shell densi- ties and Kohn–Sham orbitals in the local density approximation (LDA), generalized gradient approximation (GGA), τ-dependent and Laplacian-dependent meta-generalized gradient approximation (meta-GGA), any combination of local and non-local approxima- tions (including exact exchange and range-separated exchange), and asymptotically corrected exchange-correlation potentials. NWChem contains energy-gradient implementations of most exchange- correlation functionals available in the literature, including a flex- ible framework to combine different functionals. However, sec- ond derivatives are not supported for meta-functionals, and third derivatives are supported only for a selected set of function- als. For a detailed description, we refer the reader to the online documentation.46 The DFT module reuses elements of the Gaussian basis SCF module for the evaluation of the Hartree–Fock exchange and the Coulomb matrices by using 4-index 2-electron electron repulsion integrals; the formal scaling of the DFT computation can be reduced by choosing to use auxiliary Gaussian basis sets to fit the charge density47and use 3-index 2-electron integrals instead. The DFT module supports both the distributed data approach and the mirrored array48approach for the evaluation of the exchange-correlation potential and energy. The mirrored array option, used by default, allows the calculation to hide network communication overhead by replicating the data between processes belonging to the same network node. In analogy with what is available in the SCF module, the DFT module can perform restricted closed-shell, unrestricted open-shell, and restricted open-shell calculations. However, in contrast to the SCF module that uses PCG to solve the SCF equation, the DFTmodule implements diagonalization with parallel eigensolvers.49–54 DIIS (direct inversion in the iterative subspace or direct inversion of the iterative subspace),55level-shifting,56,57and density matrix damping can be used to accelerate the convergence of the iterative SCF process. Another technique that can be used to help SCF con- vergence makes use of electronic smearing of the molecular orbital occupations by using a gaussian broadening function following the prescription of Warren and Dunlap.58Additionally, calculations with fractional numbers of electrons can be performed to analyze the behavior of exchange-correlation functionals and their impact on molecular excited states and response properties.59–66 The Perdew and Zunger67method to remove the self- interaction contained in many exchange-correlation functionals has been implemented68within the Optimized Effective Poten- tial (OEP) method69,70and within the Krieger–Li–Iafrate (KLI) approximation.71,72 The asymptotic region of the exchange-correlation poten- tial can be modified by the van-Leeuwen–Baerends exchange- correlation potential that has the correct −1 rasymptotic behav- ior. The total energy is then computed using the definition of the exchange-correlation functional. This scheme is known to tend to over-correct the deficiency of most uncorrected exchange- correlation potentials73,74and can improve TDDFT-based exci- tation calculations, but it is not variational. A variationally consistent approach to address this issue is via range-separated exchange-correlation functionals and the recently developed nearly correct asymptotic potential or NCAP,75which are implemented in NWChem. To describe dispersion interactions, both the exchange-hole dipole moment (XDM) dispersion model76and Grimme’s DFT-D3 dispersion correction (both zero-damped and BJ-damped variants) for DFT functionals77,78are available. In many cases, one can obtain reasonably accurate non-covalent interaction energies at van der Waals distances with meta-functionals in NWChem even without adding extra dispersion terms.79 Numerical integration is necessary for the evaluation of the exchange-correlation contribution to the density functional when Gaussian basis functions are used. The three-dimensional molecu- lar integration problem is reduced to a sum of atomic integrations by using the approach first proposed by Becke.80NWChem implements a modification of the Stratmann algorithm,81where the polynomial partition function wA(r) is replaced by a modified error function erf n (where ncan be 1 or 2), wA(r)=∏ B≠A1 2[1−erf(μ′ AB)], μ′ AB=1 αμAB (1−μ2 AB)n, μAB=rA−rB ∣rA−rB∣. The default quadrature used for the atomic centered numeri- cal integration is an Euler–MacLaurin scheme for the radial com- ponents (with a modified Mura–Knowles82transformation) and a Lebedev83scheme for the angular components. On top of the petite-list symmetry algorithm used in the same fashion as in the SCF module, the evaluation of the exchange- correlation kernel incurs additional time savings when the molecular J. Chem. Phys. 152, 184102 (2020); doi: 10.1063/5.0004997 152, 184102-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp symmetry is a subset of the Ohpoint group, exploiting the octahedral symmetry of the Lebedev angular grid. NWChem also has an implementation of a variational treat- ment of the one-electron spin–orbit operator within the DFT frame- work. Calculations can be performed either with an all-electron rel- ativistic approach [for example, zeroth order regular approximation (ZORA)] or with an ECP and a matching spin–orbit (SO) potential. Other capabilities built on the DFT module include the electron transfer (ET),84,85constrained DFT (CDFT),86–88and frozen density embedding (FDE)89–91modules, respectively. 1. Time-dependent density functional theory a. Linear-response time-dependent density functional the- ory. NWChem supports a spectrum of single excitation theories for vertical excitation energy calculations, namely, configuration interaction singles (CIS),92time-dependent Hartree–Fock [TDHF or also known as random-phase approximation (RPA)], time- dependent density functional theory (TDDFT),93–95and Tamm– Dancoff approximation96to TDDFT. These methods are imple- mented in a single framework that invokes Davidson’s trial vector algorithm (or its modification for a non-Hermitian eigenvalue prob- lem). An efficient special symmetric Lanczos algorithm and a kernel polynomial method have also been implemented.97 In addition to valence vertical excitation energies, core-level excitations98and emission spectra99,100can also be computed. Ana- lytical first derivatives of vertical excitation energies with a selected set of exchange-correlation functionals can also be computed,101 which allows excited-state optimizations and dynamics. Origin- independent optical rotation and rotatory strength tensors can also be calculated with the LR-TDDFT module within the gauge includ- ing atomic orbital (GIAO) basis formulation.62,102–104Extensions to compute excited-state couplings are currently underway and will be available in a future release. b. Real-time time-dependent density functional theory. Real- time time-dependent density functional theory (RT-TDDFT) is a DFT-based approach to electronic excited states based on integrat- ing the time-dependent Kohn–Sham (TDKS) equations in time. The theoretical underpinnings, strengths, and limitations are similar to traditional linear-response (LR) TDDFT methods, but instead of a frequency domain solution to the TDKS equations, RT-TDDFT yields a full time-resolved, potentially non-linear solution. Real-time simulations can be used to compute not only spectroscopic prop- erties (e.g., ground and excited-state absorption spectra and polar- izabilities)98,105–108but also the time and space-resolved electronic response to arbitrary external stimuli (e.g., electron charge dynam- ics after laser excitation)105,109and non-linear spectroscopies.110,111 RT-TDDFT has the potential to be efficient for computing spectra in systems with a high density of states112as, in principle, an entire absorption spectrum can be computed from only one dynamics simulation. This functionality is developed on the Gaussian basis set DFT module for both restricted and unrestricted calculations and can be run with essentially any combination of basis set and exchange- correlation functional in NWChem. A number of time propagation algorithms have been implemented113within this module, with the default being the Magnus propagator.114Unlike LR-TDDFT, whichrequires second derivatives, RT-TDDFT can be used with all the functionals since only first derivatives are needed for the propa- gation. The current RT-TDDFT implementation assumes frozen nuclei and no dissipation. 2. Ab initio molecular dynamics This module leverages the Gaussian basis set methods to allow for seamless molecular dynamics of molecular systems. The nuclei are treated as classical point particles, and their motion is integrated via the velocity Verlet algorithm.115,116In addition to being able to perform simulations in the microcanonical ensemble, we have implemented several thermostats to control the kinetic energy of the nuclei. These include the stochastic velocity rescaling approach of Bussi, Donadio, and Parrinello,117Langevin dynamics accord- ing to the implementation of Bussi and Parrinello,118the Berendsen thermostat,119and simple velocity rescaling. The potential energy surface upon which the nuclei move can be provided by any level of theory implemented within NWChem, including DFT, TDDFT, MP2, and the correlated wavefunction methods in the TCE module. If analytical gradients are implemented for the specified method, these are automatically used. Numerical gradients will be used in the event that analytical gradients are not available at the requested level of theory. This module has been used to demonstrate how the molecular dynamics based determination of vibrational properties can complement those determined through normal mode analysis, therefore allowing to achieve a deeper under- standing of complex dynamics and to help interpret complex exper- imental signatures.120Extensions to include non-adiabatic dynam- ics have been implemented in a development version and will be available in a future release. C. Wavefunction formulations The wavefunction-based methods play a special role in all elec- tronic structure packages. Their strengths originate in the possibility of introducing, using either various orders of perturbation theory or equivalently through the linked cluster theorem (see Refs. 121 and 122) various ranks of excitations, a systematic hierarchy of elec- tron correlation effects. NWChem offers implementations of several correlated wavefunction approaches, including many-body pertur- bation theory (MBPT) approaches and coupled-cluster methods. 1. Perturbative formulations a. MP2. Three algorithms are available in NWChem to compute the Møller–Plesset (or many-body) perturbation theory second-order correction123to the Hartree–Fock energy (MP2). They vary in capability, the size of the system that can be treated, and the use of other approximations: ●Semi-direct MP2 is recommended for most large applica- tions on parallel computers with significant disk I/O capa- bility. Partially transformed integrals are stored on disk, multi-passing as necessary. RHF and UHF references may be treated including the computation of analytic derivatives. The initial semi-direct code was later modified to use aggre- gate memory instead of disk to store intermediate, therefore not requiring any I/O operation. J. Chem. Phys. 152, 184102 (2020); doi: 10.1063/5.0004997 152, 184102-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp ●Fully direct124MP2. This is of use if only limited I/O resources are available (up to about 2800 functions). Only RHF references and energies are available. ●The resolution of the identity (RI) approximation MP2 (RI- MP2)125uses the RI approximation and is, therefore, only exact in the limit of a complete fitting basis. However, with some care, high accuracy may be obtained with relatively modest fitting basis sets. An RI-MP2 calculation can cost over 40 times less than the corresponding exact MP2 cal- culation. RHF and UHF references with only energies are available. 2. Multi-configurational self-consistent field (MCSCF) A large-scale parallel multi-configurational self-consistent field (MCSCF) method has been developed in NWChem by the integra- tion of the serial LUCIA program of Olsen.126,127The generalized active space approach is used to partition large configuration inter- action (CI) vectors and generate a sufficient number of nearly equal batches for parallel distribution. This implementation allows the execution of complete active space self-consistent field (CASSCF) calculations with non-conventional active spaces. An unprecedented CI step for an expansion composed of almost one trillion Slater determinants has been reported.127 3. Coupled-cluster theory The coupled-cluster module of NWChem contains two classes of implementations: (a) parallel implementation of the CCSD(T) formalism31for closed-shell systems and (b) a wide array of CC formalisms for arbitrary reference functions. The latter class of implementations automatically generated by the Tensor Con- traction Engine128,129is an example of a successful co-design effort. a. Closed-shell CCSD(T). The coupled-cluster method was introduced to chemistry by ˇCížek130(see also Ref. 131) and is a post-Hartree–Fock electron correlation method. Development of the canonical coupled-cluster code in NWChem commenced in 1995 under a collaboration with CRAY Inc. to develop a massively parallel coupled-cluster program designed to run on a CRAY T3E. Full details of the implementation are given in the work of Kobayashi and Rendell.132 The coupled-cluster wavefunction is written as an exponential of excitation operators acting on the reference determinant, ∣ΨCC⟩=eT∣Φ⟩, (4) where T=T1+T2+⋯is a cluster operator represented as a sum of its many-body components, i.e., singles T1, doubles T2, etc., and |Φ⟩is the so-called reference function (usually chosen as a Hartree– Fock determinant). In practical applications, the above sum is trun- cated at some excitation rank. For example, the CCSD method30is defined by including singles and doubles, i.e., T≃T1+T2. Intro- ducing the exponential Ansatz (4) into the Schrödinger equation, premultiplying both sides by e−T, using the Hausdorff formula, and projecting onto the subspace of excitation functions give a set of coupled non-linear equations that are solved iteratively to yield thecoupled-cluster energy and amplitudes. For example, for the CCSD formulation, one obtains ⟨Φ∣(HNeT1+T2)C∣Φ⟩=ΔECCSD , (5) ⟨Φa i∣(HNeT1+T2)C∣Φ⟩=0, (6) ⟨Φab ij∣(HNeT1+T2)C∣Φ⟩=0, (7) where HNis the electronic Hamiltonian in the normal product form ( HN=H−⟨Φ|H|Φ⟩), subscript Crepresents a connected part of a given operator expression, and ΔECCSD is the CCSD cor- relation energy. The closed-shell CCSD implementation employs the optimized form of the CC equations discussed by Scuseria et al. ,133as was programmed in the TITAN program.134The nature of the CRAY T3E hardware required significant rewriting of earlier coupled-cluster algorithms to take into account the limited mem- ory available per core (8 MW) and the prohibitive penalty of I/O operations. Of the various four indexed quantities, those with four occupied indices were replicated in the local memory (i.e., the mem- ory associated with a single core) and those with one or two virtual indices were distributed across the global memory of the machine (i.e., the sum of the memory of all the processors) and accessed in computational batches. The terms involving integrals with three and four virtual orbital indices still proved too costly for the available memory and to circumvent this problem, these terms were evaluated in a “direct” fashion. This structure distinguishes NWChem from most other coupled-cluster programs. Thus, to make effective use of the available memory, as much as possible should be allocated, by using global arrays, with the bare minimum for the arrays replicated in local memory. The canonical CCSD implementation in NWChem also contains the perturbative triples correction, denoted as (T), of Raghavachari et al.31This correction is an estimate from Møller– Plesset perturbation theory123and evaluates the triples contribution to MP4 using the optimized cluster amplitudes at the end of a CCSD calculation. The CCSD(T) method is commonly referred to as the gold standard for ab initio electronic structure theory calculations. Its computational cost scales as n7, making it considerably more expensive than a CCSD calculation. However, the triples are non- iterative and only require two-electron integrals with at most three virtual orbital indices, hence avoiding the previous memory and I/O issues, and so, the correction was easily adapted from the “aijkbc algorithm” of an earlier work by Rendell et al.135 In recent years, a great deal of effort was invested to enhance the performance of the iterative and non-iterative parts of the CCSD(T) workflow. Performance tuning of the iterative part resulted in scal- ing the code up to 223 200 processors of the ORNL Jaguar com- puter.26,136Significant speed-ups for the CCSD iterative part were achieved by introducing efficient optimization techniques to alle- viate the communication bottlenecks caused by a copious amount of communication requests introduced by a large class of low- dimensionality tensor contractions. This optimization provided a significant twofold to fivefold performance increase in the CCSD iteration time depending on the problem size and available mem- ory and improved the CCSD scaling to 20 000 nodes of the NCSA Blue Waters supercomputer.137 J. Chem. Phys. 152, 184102 (2020); doi: 10.1063/5.0004997 152, 184102-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp b. Tensor contraction engine and high-accuracy formulations. NWChem implements a large number of high-rank electron- correlation methods for the ground, excited, and electron- detached/attached states as well as for molecular properties. The underlying Ansätze span configuration interaction (CI), coupled- cluster (CC), many-body perturbation theories (MBPT), and vari- ous combinations thereof. A distinguishing feature of these imple- mentations is their uniquely forward-looking development strategy. These parallel-executable codes, as well as their formulations and algorithms, were computer-generated by the symbolic algebra pro- gram138called the Tensor Contraction Engine (TCE).128TCE was one of the first attempts to provide a scalable tensor library for par- allel implementations of many-body methods, which extends the ideas of automatic CC code generation introduced by Janssen and Schaefer,139Li and Paldus,140and Nooijen and Lotrich.141,142 The merits of such a symbolic system are many: (1) It expe- dites otherwise time-consuming and error-prone derivation and programming processes, (2) it facilitates parallelization and other laborious optimizations of the synthesized programs, (3) it enhances the portability, maintainability, extensibility and thus the lifes- pan of the whole program module, and (4) it enables new or higher-ranked methods to be implemented and tested rapidly, which are practically impossible to write manually. TCE is, there- fore, one of the earliest examples139of an expert system that lifts the burden of derivation/programming labor so that compu- tational chemists can focus on imagining new Ansatz —a devel- opment paradigm embraced quickly by other chemistry software developers.143–145 The working equations of an ab initio electron-correlation method are written with sums-of-products of matrices, whose ele- ments are integrals of operators in the Slater determinants. For many methods, the matrices have the general form146 ⟨Φi∣ˆL† jˆHexp(ˆTk)ˆRl∣Φm⟩C/L, (8) where Φiis the whole set of the i-electron excited (or electron- detached/attached) Slater determinants, ˆHis the Hamiltonian oper- ator, ˆTkis ak-electron excitation operator, ˆRlis an l-electron exci- tation (or electron detachment/attachment) operator, and ˆL† jis a j-electron de-excitation (or electron detachment/attachment) oper- ator. Subscript “C/L” means that the operators can be required to be connected and/or linked diagrammatically. For example, the so- called T2-amplitude equation of coupled-cluster singles and doubles (CCSD) is written as 0=⟨Φ2∣ˆHexp(ˆT1+ˆT2)∣Φ0⟩C. (9) With the Ansatz of a method given in terms of Eq. (8), TCE (1) evaluates these operator-determinant expressions into sums-of- products of matrices (molecular integrals and excitation amplitudes) using normal-ordered second quantization and Wick’s theorem, (2) transforms the latter into a computational sequence (algorithm), which consists in an ordered series of binary matrix multiplications and additions, and (3) generates parallel-execution programs imple- menting these matrix multiplications and additions, which can be directly copied into appropriate directories of the NWChem source code and which are called by a short, high-level driver subroutine humanly written (see Fig. 1). FIG. 1 . A schematic representation of the TCE workflow (see the text for details). In step (2), TCE finds the (near-)minimum cost path of eval- uating sums-of-products of matrices by solving the matrix-chain problem (defining the so-called “intermediates”) and by perform- ing common subexpression elimination and intermediate reuse. In step (3), the computer-generated codes perform dynamically load-balanced parallel matrix multiplications and additions, taking advantage of spin, spatial, and index-permutation symmetries. The parallelism, symmetry usage, and memory/disk space management are all achieved by virtue of TCE’s data structure: every matrix (molecular integrals, excitation amplitudes, intermediates, etc.) is split into spin- and spatial-symmetry-adapted tiles, whose sizes are determined at runtime so that the several largest tiles can fit in the core memory. Only symmetrically unique, non-zero tiles are stored gapless (with their storage addresses recorded in hash tables, which are also auto-generated by TCE) and used in parallel tile-wise multi- plications and additions, which are dynamically distributed to idle processors on a first-come, first-served basis. NWChem’s parallel middleware, especially global arrays, was essential for making the computer-generated parallel codes viable. J. Chem. Phys. 152, 184102 (2020); doi: 10.1063/5.0004997 152, 184102-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TCE is a part of the NWChem source-code distribution, and a user is encouraged to implement their own Ansätze into high- quality parallel codes. Therefore, TCE has paved the way for quick development of various implementations of coupled-cluster meth- ods that would take disproportionately longer time if hand-coded. Additionally, TCE provided a new testing ground for several novel parallel algorithms for accurate many-body methods and has been used to generate a number of canonical implementations of sin- gle reference CC methods for ground- and excited-state calculations for arbitrary reference functions including RHF, ROHF, UHF, and multi-reference cases. In the following, we listed basic components of the TCE infrastructure in NWChem: ●various perturbative methods ranging from second [MBPT(2)/MP2] to fourth-order [MBPT(4)/MP4] of Möller– Plesset perturbation theory; ●single reference iterative (CCD,130CCSD,30CCSDT,147–149 and CCSDTQ150,151) and non-iterative [CCSD(T),31CR- CCSD(T),152LR-CCSD(T),153CCSD(2),154–156CCSD(2) T,156 and CCSDT(2) Q]156CC approximations for ground-state calculations; ●single reference iterative (EOMCCSD,157,158 EOMCCSDT,159,160and EOMCCSDTQ146,161) and non- iterative [CR-EOMCCSD(T)162] equation-of-motion CC (EOMCC) approximations163for excited-state calculations; ●ionization-potential and electron-affinity EOMCC (IP/EA- EOMCC) methods;164–170 ●linear-response CC (LR-CC) methods for calculating static and frequency-dependent polarizabilities and static hyper- polarizabilities at the CCSD and CCSDT levels of approxi- mation;171and ●state-specific multi-reference CC (MR-CC) methods for quasi-degenerate systems.172–178 The TCE infrastructure has also been used in exploring new paral- lel algorithms and algorithms for emerging computer architectures. The most important examples include ●parallel algorithms for excited-state CR-EOMCCSD(T) cal- culations with demonstrated scalability across 210 000 cores of the Jaguar Cray XT5 system at the Oak Ridge Leadership Computing Facility (OLCF),28 ●new CC algorithms for GPU and Intel MIC architectures (single-reference CC and MR-CC theories),32,34,35,179,180 ●new algorithms for multi-reference CC methods utilizing processor groups and multiple levels of parallelism (the so- called reference-level of parallelism of Refs. 181 and 182) with demonstrated scalability across 80 000 cores of the Jaguar Cray XT5 system,182and ●new execution models for the iterative CCSD and EOM- CCSD models.28 With TCE, one can perform CC calculations for closed- and open-shell systems characterized by 1000–1300 orbitals. Some of the most illustrative examples of TCE calculations are (1) static and frequency-dependent polarizabilities for the C 60molecule,183 excited state simulations for π-conjugated chromophores,184and IP-EOMCCSD calculations for ferrocene with explicit inclusion of solvent molecules. One cutting edge application of TCE CC was theearly application of EOMCC methodologies in excited-state stud- ies of functionalized forms of porphyrin.28Additionally, TCE has also served as a development platform for early implementations of the coupled-cluster Green’s function formalism.185–188The TCE development has since been followed by several other efforts toward enabling scalable tensor libraries. This includes Super Instruc- tion Assembly Language (SIAL),144,189Cyclop Tensor Framework (CTF),190TiledArray framework,191and Libtensor,192which have been used to develop scalable implementations of CC methods. D. Relativistic methods Methods that include treatment of relativistic effects are based on the Dirac equation,193which has a four-component wavefunc- tion. The solutions to the Dirac equation describe both positrons (the “negative energy” states) and electrons (the “positive energy” states), as well as both spin orientations and hence the four compo- nents. The wavefunction may be broken down into two-component functions traditionally known as the large and small components; these may further be broken down into the spin components.194–197 The implementation of approximate all-electron relativistic methods in quantum chemical codes requires the removal of the negative energy states and the factoring out of the spin-free terms. Both of these may be achieved using a transformation of the Dirac Hamiltonian known in general as a Foldy–Wouthuysen (FW) transformation. Unfortunately, this transformation cannot be rep- resented in a closed form for a general potential and must be approximated. One popular approach is the Douglas and Kroll198 method developed by Hess.199,200This approach decouples the pos- itive and negative energy parts to second-order in the external potential (also to the fourth-order in the fine structure constant, α). Other approaches include the zeroth order regular approxima- tion (ZORA),201–204modification of the Dirac equation by Dyall,205 which involves an exact FW transformation on the atomic basis set level,206,207and the exact 2-component (X2C) formulation, which is a catch-all for a variety of methods that arrive at an exactly decou- pled two-component Hamiltonian using matrix algebra.197,208–211 NWChem contains released implementations of the DKH, ZORA, and Dyall approaches, while the X2C method is available in a development version.209,211 Since these approximations only modify the integrals, they can, in principle, be used at all levels of theory. At present, the Douglas– Kroll, ZORA and X2C implementations can be used at all levels of theory, whereas Dyall’s approach is currently available at the Hartree–Fock level. 1. Douglas–Kroll approximation NWChem contains three second-order Douglas–Kroll approx- imations termed FPP, DKH, and DKHFULL. The FPP is the approx- imation based on free-particle projection operators,199whereas the DKH and DKFULL approximations are based on external-field pro- jection operators.200The latter two are considerably better approxi- mations than the former. DKH is the Douglas–Kroll–Hess approach and is the approach that is generally implemented in quantum chemistry codes. DKFULL includes certain cross-product integral terms ignored in the DKH approach (see Ref. 212). The third-order Douglas–Kroll approximation (DK3) implements the method by Nakajima and Hirao.213,214 J. Chem. Phys. 152, 184102 (2020); doi: 10.1063/5.0004997 152, 184102-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 2. Zeroth order regular approximation (ZORA) The spin-free and spin–orbit versions of the one-electron zeroth order regular approximation (ZORA) have been imple- mented. Since the ZORA correction depends on the potential, it is not gauge invariant. This is addressed by using the atomic approxi- mation of van Lenthe and co-workers.215,216Within this approxima- tion, the ZORA corrections are calculated using the superposition of densities of the atoms in the system. As a result, only intra-atomic contributions are involved, and no gradient or second derivatives of these corrections are needed to be calculated. In addition, the cor- rections need only to be calculated once at the start of the calculation and are stored. The ZORA approach is implemented in two ways in NWChem, one where the ZORA potential components are directly computed on an all-electron grid204and a second approach where the ZORA potential is computed using the model potential approach due to van Wüllen and Michauk.217,218 3. Dyall’s modified Dirac Hamiltonian approximation The approximate methods described in this section are all based on Dyall’s modified Dirac Hamiltonian. This Hamiltonian is entirely equivalent to the original Dirac Hamiltonian, and its solutions have the same properties. The modification is achieved by a transforma- tion on the small component. This gives the modified small com- ponent the same symmetry as the large component. The advantage of the modification is that the operators now resemble those of the Breit–Pauli Hamiltonian and can be classified in a similar fashion into spin-free, spin–orbit, and spin–spin terms. It is the spin-free terms that have been implemented in NWChem, with a number of further approximations. Negative energy states are removed by a normalized elimination of the small component (NESC), which is equivalent to an exact Foldy–Wouthuysen (EFW) transformation. Both one-electron and two-electron versions of NESC (NESC1E and NESC2E, respectively) are available, and both have analytic gradients.205–207 E. Molecular properties A broad array of simple and response-based molecular prop- erties can be calculated using the HF and DFT wavefunctions in NWChem. These include natural bond analysis, dipole, quadrupole, octupole moments, Mulliken population analysis and bond order analysis, Löwdin population analysis, electronic couplings for elec- tron transfer,84,85Raman spectroscopy,219,220electrostatic potential (diamagnetic shielding) at nuclei, electric field and field gradient at nuclei, electric field gradients with relativistic effects,221electron and spin density at nuclei, GIAO-based nuclear magnetic resonance (NMR) properties such as shielding, hyperfine coupling (Fermi- Contact and spin-dipole expectation values), indirect spin–spin cou- pling,222–224G-shift,225EPR, paramagnetic NMR parameters,226,227 and optical activity.102,103,228,229Note that only linear-response is supported for single frequency, electric field, and mixed electric- magnetic field perturbations. Ground state and dynamic dipole polarizabilities for molecules can be calculated at the CCSD, CCSDT, and CCSDTQ levels using the linear-response formalism.230For additional information, we refer to the reader to the online manual.8F. Periodic plane wave density functional theory The NWChem plane wave density functional theory (NWPW) module contains two programs: ●PSPW—a pseudopotential and projector augmented (PAW) plane wave Γ-point code for calculating molecules, liquids, crystals, and surfaces and ●BAND—a pseudopotential plane wave band structure code for calculating crystals and surfaces with small band gaps (e.g., semi-conductors and metals). These programs use a common infrastructure for carrying out oper- ations related to plane wave basis sets that are parallelized with the MPI and OpenMP libraries29,33,34,231–235The NWPW module can be used to carry out many different kinds of simulations. In addi- tion to the standard simulations implemented in other modules, e.g., energy, optimize, and frequency, there are additional capabil- ities specific to PSPW and BAND that can be used to carry out NVE and NVT236Car–Parrinello19and Born–Oppenheimer molecular dynamics simulations, hybrid ab initio molecular dynamics molec- ular and molecular mechanics (AIMD-MM) simulations,234,237 Gaussian/Fermi/Marzari–Vanderbilt smearing, Potential-of-Mean- Force (PMF)238/Metadynamics239,240/Temperature-Accelerated- Molecular-Dynamics (TAMD)241,242/Weighted-Histogram-Analysis- Method (WHAM)243free energy simulations, AIMD-EXAFS simu- lations using open source versions of the FEFF software244–246that have been parallelized, electron transfer calculations,247unit cell optimization, optimizations with space-group symmetry, Monte- Carlo NVT and NPT simulations, phonon calculations, simulations with spin–orbit corrections, Wannier248and rank reducing density matrix249localization calculations, Mulliken250and Blöchl251charge analysis, Gaussian cube file generation, periodic dipole and infrared (AIMD-IR) simulations, band structure plots, and density of states. Calculations can also be run using a newly developed i-PI252inter- face, and more direct interfaces to ASE,253nanoHUB,254and EMSL Arrows255simulation tools are currently being implemented. A variety of exchange-correlation functionals have been imple- mented in both codes, including the local density approxima- tion (LDA) functionals, generalized gradient approximation (GGA) functionals, full Hartree–Fock and screened exchange, hybrid DFT functionals, self-interaction correction (SIC) functionals,256 localized exchange method, DFT + U method, and Grimme dis- persion corrections,77,78as well as recently implemented vdW dispersion functionals,257and meta-generalized gradient approxi- mation (metaGGA) functionals. The program contains several codes for generating pseudopotentials, including Hamann20and Troulier– Martins,21and PAW23potentials. These codes have the option for generating potentials with multiple projectors and semi-core cor- rections. It also contains codes for reading in HGH,258GTH,259 and norm-conserving pseudopotentials in the CPI and TETER for- mats. Codes for reading Optimized Norm-Conserving Vanderbilt (ONCV) pseudopotentials260,261and USPP PAW potentials will become available in future releases of NWChem. The pseudopotential plane wave DFT methods implemented in NWChem are a fast and efficient way to calculate molecu- lar and solid-state properties using DFT.16,17,19,29,235,262–270In these approaches, the fast varying parts of the valence wavefunctions inside the atomic core regions and the atomic core wavefunc- J. Chem. Phys. 152, 184102 (2020); doi: 10.1063/5.0004997 152, 184102-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Illustration of the atomic core and interstitial regions in a valence wave- function. Bonding takes place in the interstitial region and the atomic core regions change very little from molecule to molecule. Figure from Bylaska et al.234 tions are removed and replaced by pseudopotentials.20–22,271–274 Pseudopotentials are chosen such that the resulting pseudoatoms have the same scattering properties as the original atoms. The ratio- nale for this approach is that the changes in the electronic structure associated with making and breaking bonds only occur in the inter- stitial region outside the atomic core regions (see Fig. 2). Therefore, removing the core regions should not affect the bonding of the sys- tem. For this approach to be useful, it is necessary for the pseudopo- tentials to be smooth in order for plane wave basis sets to be used. As the atomic potential becomes stronger, the core region becomes smaller and the pseudopotential grows steep. As a result, the pseu- dopotential can become very stiff, requiring large plane wave basis sets (aka cutoff energies), for the first-row transition metals atoms, the lanthanide atoms, and toward the right-hand side of the periodic table (fluorine). The projected augmented plane wave method (PAW)23,232,275–277 is another related approach that removes many of the problems of the somewhat ad hoc nature of the pseudopotentials approach. How- ever, in the PAW approach, instead of discarding the rapidly varying parts of the electronic functions, these are projected onto a local basis set (e.g., a basis of atomic functions) and no part of the elec- tron density is removed from the problem. Another key feature of PAW is that by maintaining a local description of the system, the norm-conservation condition (needed for proper scattering from the core) can be relaxed, which facilitates the use of smaller plane wave basis sets (aka cutoff energies) then for many standard pseu- dopotentials. Historically, the PAW method was implemented as a separate program in the NWPW module, rather than being fully integrated into the PSPW and BAND codes. This separation signif- icantly hindered its development and use. As of NWChem version 6.8 (released in 2017), the PAW approach has been integrated into the PSPW code, and it is currently being integrated into the BAND code. It will become available in future releases of NWChem. In recent years, with advances in High-Performance Com- puting (HPC) algorithms and computers, it is now possible to run AIMD simulations up to ∼1 ns for non-trivial system sizes. As a result, it is now possible to effectively use free- energy methods with AIMD and AIMD/MM approaches. Free energy approaches are useful for simulating reactions where tra- ditional quantum chemistry approaches can be difficult to use and often require the expertise of a very experienced quan- tum chemist, e.g., reactions that are complex with concerted or multi-step components and/or interact strongly with the solvent. Recent examples include solvent coordination and hydrolysis of actinides metals197,278–281(see Fig. 3), hydrolysis of explosives,234and FIG. 3 . Snapshots from a metadynamics simulation of the hydrolysis of the U4+ aqua ion.278During the simulation, a proton jumps from a first shell water molecule to a second shell water molecule and then subsequently to other water molecules via a Grotthuss mechanism. ion association in AlCl 3.237To help users learn how to use these new techniques, we developed a tutorial on carrying out finite temperature free energy calculations in NWChem.282 The NWPW module continues to be actively developed. There are on-going developments for RPA and GW-RPA methods, an elec- tron transfer MCSCF method, Raman and Mössbauer spectroscopy, and a hybrid method that integrates classical DFT283into ab initio molecular dynamics (AIMD-CDFT). In addition to these develop- ments, we are actively developing the next generation of plane wave codes as part of the NWChemEx project. These new codes, which are being completely written from scratch, will contain all the features currently existing in the NWPW module. Besides implementing fast algorithms to use an even larger number of cores and new algorithms to run efficiently on GPUs, it includes a more robust infrastructure to facilitate the implementation of an O(N) DFT code based on the work of Osei-Kuffuor and Fattebert.284 G. Optimization, transition state, and rate theory approaches A variety of drivers and interfaces are available in NWChem to perform geometry minimization and transition state optimizations. The default algorithms in NWChem for performing these optimiza- tions are quasi-Newton methods with line searches. These methods J. Chem. Phys. 152, 184102 (2020); doi: 10.1063/5.0004997 152, 184102-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp are fairly robust, and they can be used to optimize molecules, clus- ters, and periodic unit cells and surfaces. They can also be used in conjunction with both point group and space group symme- tries, excited state TDDFT surfaces, and with a variety of exter- nal fields, such as external point charges, COnductor-like Screen- ing MOdel (COSMO),285or Solvation Model based on Density (SMD).286The default methods also work seamlessly with electronic structure methods that do not have nuclear gradients implemented by automatically using finite difference gradients. NWChem also contains default methods for calculating harmonic vibrational fre- quencies and phonon spectra for periodic systems. These meth- ods are able to make use of analytic Hessians if they are available; otherwise, a finite difference approach is used. A vibrational self- consistent field287(VSCF) method is also available in NWChem, and it can be used to calculate anharmonic contributions to specified vibrational modes. There is also an interface called DIRDYVTST288 that uses NWChem to compute energies, gradients, and Hessians for direct dynamics calculations with POLYRATE.289 A variety of external packages, such as ASE253,290and Sella,291,292can also be used for finding energy minima, saddle points on energy surfaces, and frequencies using either python scripting or a newly developed i-PI252interface. Python programs may be directly embedded into the NWChem input and used to control the execution of NWChem. The python scripting language provides useful features, such as variables, conditional branches, and loops, and is also readily extended. Other example applications for which it could be used include scanning potential energy surfaces, com- puting properties in a variety of basis sets, optimizing the energy with respect to parameters in the basis set, computing polarizabil- ities with a finite field, simple molecular dynamics, and parallel in time molecular dynamics.293 NWChem also contains an implementation of the nudged elas- tic band (NEB) method of Jónsson and co-workers294–297and the zero-temperature string method of Vanden Eijden et al.298Both these methods can be used to find minimum energy paths. Cur- rently, a quasi-Newton algorithm is used for the NEB optimization. A better approach for this kind of optimization is to use a non- linear multi-grid algorithm, such as the Full Approximation Scheme (FAS).299A new implementation of NEB based on FAS is available on Bitbucket,300and an integrated version will be soon available in NWChem. H. Classical molecular dynamics The integration of a molecular dynamics (MD) module in NWChem enables the generation of time evolution trajectories based on Newton’s equation of motion of molecular systems in which the required forces can originate from a classical force field, any implemented quantum mechanical method for which spatial derivatives have been implemented, or hybrid quantum mechan- ical/molecular mechanical (QM/MM) approaches. The method is based on the ARGOS molecular dynamics software, originally designed for vector processors301but later redesigned for massively parallel architectures.25,302–304 1. System preparation The preparation of a molecular system is done by a separate prepare module that reads the molecular structure and assembles atopology from the databases with parameters for the selected force field. The topology file contains all static information for the sys- tem. In addition, this module generates the so-called restart file with all dynamic information. The prepare module has a wide range of capabilities that include the usual functions of placing counter-ions and solvation with any solvent defined in the database. The prepare module is also used to define Hamiltonian changes for free energy difference calculations and the definition of those parts of the molec- ular systems that will be treated quantum-mechanically in QM/MM simulations. Some of the more unique features include setting up a system for quantum mechanically derived proton hopping (QHOP) simulations305,306and the setup of biological membranes from a single lipid-like molecule. This last capability has been successfully used for the first extensive simulation studies of complex asymmet- ric lipopolysaccharide membranes of Gram-negative microbes307–311 and their role in the capture of recalcitrant environmental heavy metal ions,312microbial adhesion to geochemical surfaces,313–316and the structure and dynamics of trans-membrane proteins including ion transporters (Fig. 4).317–319 2. Force fields The force field implemented in NWChem consists of harmonic terms for bonded, angle and out of plane bending interactions, and trigonometric terms for torsions. Non-bonded van der Waals and electrostatic interactions are represented by Lennard-Jones and Coulombic terms, respectively. Non-bonded terms are evaluated using charge groups and subject to a user-specified cutoff radius. Electrostatic interaction corrections beyond the cutoff radius are estimated using the smooth particle mesh Ewald method.320Param- eter databases are provided for the AMBER321and CHARMM322 force fields. Even for purely classical MD simulations, the integration with the electronic structure methods provides a convenient way of deter- mining electrostatic parameters for missing fragments in standard force field databases through the use of restrained electrostatic FIG. 4 . The NWChem MD prepare utility facilitates the setup of trans-membrane proteins in complex asymmetric membrane environments in a semi-automated procedure. The top views of step 1 in which membrane lipopolysaccharide molecules with the necessary counter ions are placed on a rectangular grid around a trans-membrane protein in which each membrane lipid molecule is randomly rotated around the principal molecular axis (left panel), step 2 in which each clus- ter of a lipid molecule is translated toward the center of the transmembrane protein such that no steric clashes occur (center panel), and step 3 in which the system is equilibrated using strict restraint potentials to keep the lipid molecules aligned along the normal of the membrane and the lipid head groups in the plane of the membrane (right panel). After this procedure, the system would be solvated and equilibrated while slowly removing the positional restraint potentials. J. Chem. Phys. 152, 184102 (2020); doi: 10.1063/5.0004997 152, 184102-13 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp potential fitting323,324to which a variety of additional constraints and restraints can be applied. 3. Simulation capabilities Ensemble types available in NWChem are NVE, NVT, and NPT using the Berendsen thermostat and barostat.119Newton’s equations of motion are integrated using the standard leap-frog Ver- let or velocity Verlet algorithms. A variety of fundamental properties are evaluated by default during any molecular dynamics simulation. Parallel execution time analysis is available to determine the parallel efficiency. The MD module has extensive free energy simulation capa- bilities,325–330which are implemented in the so-called multi- configuration approach. For each incremental change in the Hamil- tonian to move from the initial to the final state, sometimes referred to as a window, a full molecular simulation is carried out. This allows for a straightforward evaluation of statistical and system- atic errors where needed, including a correlation analysis.331Based on the ARGOS code,301it has some unique features, such as the separation-shifted scaling technique to allow atoms to appear from or disappear to dummy atoms.332One of the advantages of the inte- gration of MD into the electronic structure methods framework in NWChem is the ability to carry out hybrid QM/MM simulations (discussed in Sec. VI). The preparation of molecular systems for the MD module allows for flexibly specifying parts of the molecular sys- tem to be treated by any of the implemented electronic structure methods capable of evaluating positional gradients. A unique feature in the NWChem MD module is the optional specification of protonatable sites on both solute and solvent molecules. Pairs of such sites can dynamically change between protonated or unprotonated state, effectively exchanging a pro- ton. Transitions are governed by a Monte Carlo type stochastic method to determine when transitions occur. This so-called QHOP approach was developed by the research group of Helms.306 4. Analysis capabilities The NWChem MD capability includes two analysis modules. The original analysis module, analyze , analyzes trajectories in a way that reads individual structures one time step at a time and dis- tributes the data in a domain decomposition fashion as in the molec- ular simulation that generated the data. The second data-intensive analysis module, diana , reads entire trajectories and distributes the data in the time domain. This is especially effective for analyses that require multiple passes through a trajectory but requires the avail- ability of potentially large amounts of memory.333,334An example of such analyses is the essential dynamics analysis, a principal compo- nent analysis (PCA) based calculation to determine the dominant motions in molecular trajectories. 5. Parallel implementation strategy The most effective way of distributing a system with large num- bers of particles is through the use of domain decomposition of the physical space. The implementation in NWChem, facilitated through the use of the Global Arrays (GA) toolkit, partitions the simulation space into rectangular cells that are assigned to differentprocesses’ ranks or threads. Each of these ranks carries out the cal- culation of intra-cell atomic energies and forces of the cells assigned. Inter-cell energies and forces are evaluated by one of the ranks that was assigned one or the other of the cell pairs. Two load balancing methods have been implemented in NWChem, both based on measured computation time. In the first one, the assignment of inter-node cell pair calculations is redefined such that assignments move from the busiest node to the less busy node. This scheme requires minimal additional communication, and since only two nodes are involved in the redistribution of work, the communication is local, i.e., node to node. In the second scheme, the physical size of the most time-consuming cell is reduced, while all other cells are made slightly larger. This scheme requires com- munication and redistribution of atoms on all nodes. In practice, the first scheme is used until performance no longer improves, after which the second scheme is used once followed by returning to use the first scheme. This approach has been found to improve load balancing even in systems with a very asymmetric distribution of computational intensity.335 VI. HYBRID METHODS We define hybrid methods as those coupling different levels of description to provide an efficient calculation of a chemical sys- tem, which otherwise may be outside the scope of conventional single-theory approaches. The physical motivation for such methods rests on the observation that, in the majority of complex chemi- cal systems, the chemical transformation occurs in localized regions surrounded by an environment, which can be considered chem- ically inert to a reasonable approximation. Since hybrid methods require the combination of multiple theoretical methods in a sin- gle simulation, the diversity of simulation methodologies available in NWChem makes it a platform particularly apt for this purpose. One common example involves chemical transformations in a bulk solution environment, forming the foundations of a wide variety of spectroscopic measurements (UV–vis, NMR, EPR, etc.). The reactive region, referred to as the “solute,” involves elec- tronic structure degrees of freedom and thus requires the quantum- mechanical (QM) based description, such as DFT or more com- plex wavefunction methods. In the conventional approach, such QM description would be necessarily extended to the entire sys- tem, making the problem a heroic computational task. In a hybrid approach, the treatment of a surrounding environment (“solvent”) would be delegated to a much simpler description, such as the continuum model (CM). The latter is supported in NWChem via two models—COSMO285(COnductor-like Screening MOdel) and SMD286(Solvation Model based on Density). The resulting QM/CM approaches are particularly well suited for accurate and efficient calculation of solvation free energies, geometries in solution, and spectroscopy in solution. The SMD model employs the Poisson equation with a non-homogeneous dielectric constant for bulk elec- trostatic effects and solvent-accessible-surface tensions for cavita- tion, dispersion, and solvent-structure effects, including hydrogen bonding. For spectroscopy in solution, the Vertical Excitation (or Emission) Model (VEM) has also been implemented for calculat- ing the vertical excitation (absorption) or vertical emission (fluores- cence) energy in the solution according to a two-timescale model of solvent polarization.336 J. Chem. Phys. 152, 184102 (2020); doi: 10.1063/5.0004997 152, 184102-14 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp For systems where an explicit solvation environment treat- ment is needed (for example, heterogeneous systems such as a protein matrix), NWChem provides a solution in terms of the combined quantum mechanics/molecular mechanics (QM/MM) approach.337,338Here, the environment is described at the classical molecular mechanics level. This offers more fidelity compared with a continuum solvent description, while still keeping the computa- tional costs down. The total energy of the system in the QM/MM approach can be represented as a sum of the energies corresponding to QM and MM regions, E(r,R;ψ)=Eqm(r,R;ψ)+Emm(r,R), (10) whereψdenotes electronic degrees of freedom and r, and Rrefer to nuclear coordinates of QM and MM regions, respectively. The QM energy term can be further decomposed into internal and external parts, Eqm[r,R;ψ]=Eint qm[r;ψ]+Eext qm[r,R;ρ], (11) whereρis the electron density. As a generic module, the QM/MM implementation can uti- lize any of the Gaussian basis set based QM modules available in NWChem and supports nearly all the task functionalities. The cal- culation of QM energy remains the main computational expense in the QM/MM approach. This issue is more pronounced compared with the continuum coupling case, because of the additional atom- istic degrees of freedom associated with the MM description. The latter comes into play because any change in the MM degrees of freedom will, in general, trigger the recalculation of the QM energy [Eqm(r,R;ψ)]. To alleviate these issues during the optimization, the QM/MM module offers the option of alternating relaxation of QM and MM regions. During the latter phase, the user may uti- lize an approximation where the QM degrees of freedom are kept frozen until the next cycle of QM region relaxation, offering sig- nificant computational savings. A similar technique can be utilized in the dynamical equilibration of the MM region and calculations of reaction pathways and free energies. In addition to the native MD module, the NWChem QM/MM module can also utilize the external AMBER MD code339for running the classical part of the calculations. In this case, QM/MM simulations involve two separate NWChem and AMBER calculations with data exchange mediated through files written to the disk. Additionally, the QM/MM capability in NWChem has resulted in the development and refinement of force-field parameters, which can, in turn, be used in classical molecular dynamics simula- tions. Over the last two decades, classical parameters obtained using NWChem have been employed to address the underlying mechanisms of a variety of novel complex biological systems and their interactions (e.g., lipopolysaccharide membranes, carbohy- drate moieties, mineral surfaces, radionuclides, and organophos- phorous compounds),307,308,311–313,340–344which has led to a signifi- cant expansion of the database of AMBER- and Glycam-compatible force fields and the GROMOS force field for lipids, carbohydrates, and nucleic acids.345–351 For cases where a classical description of the environment is deemed insufficient, NWChem offers an option to perform an ONIOM type calculation.352The latter differs from QM/MM in thatthe lower level of theory is not restricted to its region but also encom- passes regions from all the higher levels of description. For example, in the case of the two-level description, the energy is written as E(R)=EL(R)+(EH(RH)−EL(RH)), (12) where subscripts Hand Lrefer to high and low levels of theory, respectively. The high-level treatment is restricted to a smaller por- tion of the system ( RH), while the low level of theory goes over the entire space ( R). The second term in the above equation takes care of overcounting. The NWChem ONIOM module implements two- and three-layer ONIOM models for use in energy, gradient, geome- try optimization, and vibrational frequency calculations with any of the pure QM methods within NWChem. A new development in hybrid method capabilities of NWChem involves classical density functional theory (cDFT).353–355The latter represents a classical variant of electronic structure DFT, where the main variable is the classical density of the atoms.356,357Conceptu- ally, this type of description lies between continuum and classical force field models, providing orders of magnitude improvements over classical MD simulations. The approach is based on incorporat- ing important structural features of the environment in the form of classical correlation functions. This allows for efficient and reliable calculations of thermodynamical quantities, providing an essential link between the electronic structure description at the atomistic level and phenomena observed at the macroscopic scale. VII. PARALLEL PERFORMANCE The design and development of NWChem from the outset was driven by parallel scalability and performance to enable large scale calculations and achieve fast time-to-solution by using many CPUs where possible. The parallel tools outlined in Sec. IV provided the programming framework for this. The advent of new architectures such as the GPU358platforms has required the parallel coding strategy within NWChem to be revisited. At present, the coupled-cluster code within TCE can uti- lize both the CPU and GPU hardware at a massive scale.32,359The emergence of many-core processors in the last ten years provided the opportunity for starting a collaborative effort with Intel corpora- tion to optimize NWChem on this new class of computer architec- ture. As part of this collaboration, the TCE implementation of the CCSD(T) code was ported to the Intel Xeon Phi line of many-core processors35using a parallelization strategy based on a hybrid GA- OpenMP approach. The ab initio plane wave molecular dynamics code (Sec. V F) has also been optimized to take full advantage of these Intel many-core processors.33,231 In the rest of this section, we will discuss the parallel scalability and performance of the main capabilities in NWChem. A. Gaussian basis density functional theory In Fig. 5, we report the parallel performance of the Gaussian basis set DFT module in NWChem. This calculation involved per- forming a PBE0 energy calculation (four SCF iterations in direct mode) on the C 240molecule with the 6-31G∗basis set (3600 basis functions) without symmetry. These calculations were performed on the Cascade supercomputer located at PNNL. J. Chem. Phys. 152, 184102 (2020); doi: 10.1063/5.0004997 152, 184102-15 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . C240DFT benchmark. B. Time-dependent density functional theory In Fig. 6, we report the parallel performance of the Gaus- sian basis set LR-TDDFT module in NWChem. This calculation involved computing 100 excitation energies, requiring 11 David- son iterations, for the Au 20molecule surrounded by a matrix of 80 Ne atoms360(1840 basis functions) with D2symmetry using the B3LYP functional. These calculations were performed on the Cascade supercomputer located at PNNL. C. Closed-shell CCSD(T) The parallel implementation of the CCSD(T) approach by Kobayashi and Rendell,132employing the spin adaptation scheme based on the unitary group approach (UGA)133within NWChem, was one of the first scalable implementations of the CC formal- ism capable of taking advantage of several hundred processors. This implementation was used in simulations involving tera- and peta- scale architectures where chemical accuracy is required to describe ground-state potential energy surfaces. One of the best illustrations FIG. 6 . LR-TDDFT benchmark for the Au 20molecule in a neon matrix.of the performance of the CCSD(T) implementation is provided by calculations for water clusters.26In the largest calculation, (H 2O)24, a sustained performance of 1.39 PetaFLOP/s (double precision) on 223 200 processors of ORNL’s Jaguar system was documented. This impressive performance was mostly attributed to the (T)-part char- acterized by n3 on4 unumerical overhead (where noand nurefer to the total numbers of correlated occupied and virtual orbitals) and its relatively low communication footprint. D. Tensor contraction engine The TCE has enabled parallel CC/EOMCC/LR-CC cal- culations for closed- and open-shell systems characterized by 1000–1300 orbitals. Some of the most illustrative examples include calculations for static and frequency-dependent polarizabilities for polyacenes and C 60molecule,183,361excited state simulations for π- conjugated chromophores,184and IP-EOMCCSD calculations for carbon nanotubes.362A good illustration of the scalability of the TCE module is provided by the application of GA-based TCE implemen- tations of the iterative (CCSD/EOMCCSD) and non-iterative [CR- EOMCCSD(T)] methods in studies of excited states of β-carotene363 and functionalized forms of porphyrin28[see Figs. 7(a) and 7(b), respectively]. While non-iterative methods are much easier to scale across a large number of cores [Fig. 7(b)], scalability of the iterative CC methods is less easy to achieve. However, using early task-flow algorithms for TCE CCSD/EOMCCSD methods,28it was possible to achieve satisfactory scalability in the range of 1000–8000 cores. E. Recent implementation of plane wave DFT AIMD for many-core architectures The very high degree of parallelism available on machines with many-core processors is forcing developers to carefully revisit the implementation of their programs in order to make use of this hardware efficiently. In this section, after a brief overview of the computational costs and parallel strategies for AIMD, we present our recent work33on adding thread-level parallelism to the AIMD method implemented in NWChem.3,29,364 The main computational costs of an energy minimization or AIMD simulation are the evaluation of the electronic gradient δEtotal/δψ∗ i=Hψiand algorithms used to maintain orthogonality. These costs are illustrated in Fig. 8. Due to their computational com- plexity, the electron gradient Hψiand orthogonalization need to be calculated as efficiently as possible. The main parameters that deter- mine the cost of a calculation are Ng,Ne,Na, and Nproj, where Ng is the size of the three-dimensional FFT grid, Neis the number of occupied orbitals, Nais the number of atoms, Nprojis the number of projectors per atom, and Npackis the size of the reciprocal space. The evaluation of the electron gradient (and orthogonality) contains three major computational pieces that need to be efficiently parallelized: ●applying V Hand Vxc, involving the calculation of 2 Ne3D FFTs; ●calculating the non-local pseudopotential ,VNL, dominated by the cost of the matrix multiplications W=PTYand Y2= PW, where Pis an Npack×(Nproj⋅Na) matrix, Yand Y2are Npack×Nematrices, and Wis an ( NprojNa)×Nematrix; J. Chem. Phys. 152, 184102 (2020); doi: 10.1063/5.0004997 152, 184102-16 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 7 . Benchmark EOMCC scalability tests for (a) β-carotene and (b) free-base porphyrin (FBP) fused coronene. Timings for CR-EOMCCSD(T) approach for the coronene fused free-base porphyrin in the AVTZ basis set were determined from calculations on the ORNL’s Jaguar Cray XT5 computer system. FIG. 8 . Operation count of Hψiin a plane wave DFT simulation. Figure from Ref. 231.●enforcing orthogonality , where the most expensive matrix multiplications are S=YTYand Y2=YS, where Yand Y2 areNpack×Nematrices and Sis an Ne×Nematrix. In this work, Lagrange multiplier kernels are used for maintaining orthogonality of Kohn–Sham orbitals.29,365–368 In Fig. 9, the timing results for a full AIMD simulation of 256 water molecules on 16, 32, 64, 128, 256, and 1024 KNL nodes are shown. The “Cori” system at NERSC was used to run this bench- mark. This benchmark was taken from Car–Parrinello simulations of 256 H 2O with an FFT grid of Ng= 1803(Ne= 2056) using the plane wave DFT module (PSPW) in NWChem. In these timings, the number of threads per node was 66. The size of this bench- mark simulation is about four times larger than many mid-size AIMD simulations carried out in recent years, e.g., in recent work by Bylaska and co-workers.279,280,369–372The overall timings show strong scaling up to 1024 KNL nodes (69 632 cores) and the tim- ings of the major kernels, the pipelined 3D FFTs, non-local pseu- dopotential, and Lagrange multiplier kernels all displayed significant speedups. F. Classical molecular dynamics The molecular dynamics module in the current NWChem release is based on the distribution of cells over available ranks in the calculation. Simulations exhibit good scalability when cells only require communication with immediately neighboring cells. When the combination of the cell size and cutoff radius is such that interactions with atoms in cells beyond the immediate neigh- bors are required, performance is significantly affected. This limits the number of ranks that can effectively be used. For example, a system with 500 000 atoms will only scale well up to 1000 ranks. In future implementations, the cell–cell pair-list will be distributed over the available ranks. While this leads to additional communi- cation for ranks that do not “own” a cell, the implementation of a new communication scheme that avoids global communication has been demonstrated to improve scalability by at least an order of magnitude.304 FIG. 9 . Scalability of major components of an AIMD step on the Xeon Phi partition for a simulation of 256 H 2O molecules. Figure from Bylaska et al.33 J. Chem. Phys. 152, 184102 (2020); doi: 10.1063/5.0004997 152, 184102-17 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp VIII. OUTREACH Given the various electronic structure methods available in NWChem, it does not come as a surprise that many of these func- tionalities have been integral to various projects focused on exten- sions of quantum chemical capabilities to exa-scale architectures and emerging quantum computing (see Fig. 10 for a pictorial represen- tation of recent developments). In the following, we describe several examples of such a synergy. A. Interfacing with other software Over the years, many open-source and commercial developers have been using NWChem as a resource for their capability develop- ment and building add-on tools to increase the code’s usability. Var- ious open-source and commercial platforms provide user interface capabilities to setup and analyze the results of calculations that can be performed with NWChem.253,255,373–380NWChem initially devel- oped its own graphical user interface called the Extensible Compu- tational Chemistry Environment,381which is currently supported by a group of open-source developers. In addition, multiple codes use quantities from the NWChem simulation, such as wavefunctions as the input for the calculation of additional properties not directly available in the code.382–391NWChem is able to export electrostatic potential and charge densities with the Gaussian cube format392and can use the Molden format393to write or read molecular orbitals. This allows codes394–398to utilize NWChem’s data to, for example, display charge densities and electrostatic potentials. NWChem can also generate AIM wavefunction files that have been used by a variety of codes to calculate various properties.76,399–401Recently, NWChem has also been interfaced with the SEMIEMP code,402which can be used to perform real-time electronic dynamics using the INDO/S Hamiltonian.403,404 B. Common component architecture It is an attractive idea to encapsulate complex scientific applica- tions as components with standardized interfaces. The components FIG. 10 . A “connected diagram” describing ongoing efforts to extend computational chemistry models to exa-scale and quantum computing. In each case, NWChem provides a testing and development platform. A significant role in these projects is played by Tensor Algebra for the Many-body Methods (TAMM) library. The ECC acronym stands for the Exa-scale Catalytic Chemistry project supported by BES.411The QDK-NWChem interface with the libDUCC library is used for downfolding electronic Hamiltonians.412interact only through these well-defined interfaces and can be com- bined into full applications. The main motivation is to be able to reuse and swap components as needed and seamlessly create com- plex applications. There have been a few attempts to introduce this approach to the scientific software development community. The most notable DOE-led effort was the Common Component Archi- tecture (CCA) Forum,405which was launched in 1998 as a scien- tific community effort to create components designed specifically for the needs of high-performance scientific computing. A more recent development is the rise of Simulation Development Environ- ment (SDE) framework,406which has features that are related to the components of CCA. NWChem developers have participated in CCA and SDE effort resulting in the creation of the NWChem component. As an example, the NWChem CCA component was used in the build- ing applications for molecular geometry optimization from mul- tiple quantum chemistry and numerical optimization packages,407 combination of multiple theoretical methods to improve multi-level parallelism,408demonstration of multi-level parallelism,409and stan- dardization of integral interfaces in quantum chemistry.410In the end, the CCA framework was too cumbersome to use for devel- opers, requiring significant efforts to develop interfaces and mak- ing components to work together. It resulted in the retirement of CCA Forum in 2010, but the work done on standardization of inter- faces is continuing to benefit the quantum chemistry community to this day. C. NWChemEx The NWChemEx project is a natural extension of NWChem to overcome the scalability challenges associated with the migration of the current code base to exa-scale platforms. NWChemEx is being developed to address two outstanding problems in advanced biofu- els research: (i) development of a molecular understanding of proton controlled membrane transport processes and (ii) development of catalysts for the efficient conversion of biomass-derived intermedi- ates into biofuels, hydrogen, and other bioproducts. Therefore, the main focus is on enabling scalable implementations of the ground- state canonical CC formalisms utilizing the Cholesky decomposed form of the two-electron integrals,413–418as well as linear scaling CC formulations based on the domain-based local pair natural orbital CC formulations (DLPNO-CC)419–421and embedding methods. D. Scalable predictive methods for excitations and correlated phenomena (SPEC) The main focus of the SPEC software project is to provide the users with a new generation of methodologies to simulate excited states and excited-state processes using existing peta- and emerging exa-scale architectures. These new capabilities will play an important role in supporting the experimental efforts at light source facilities, which require accurate and reliable modeling tools. The existing NWChem capabilities are being used to verify and validate SPEC implementations, including excitation energy, ionization potential, and electron affinity variants of the EOMCC theory as well as hierarchical Green’s function formulations ranging from the lower order GW + Bethe–Salpeter equation (GW + BSE)422to hierarchi- cal coupled-cluster Green’s function (GFCC) methods185–188,423and multi-reference CC methods. J. Chem. Phys. 152, 184102 (2020); doi: 10.1063/5.0004997 152, 184102-18 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp E. Quantum information sciences Quantum computing offers the promise of not only overcom- ing exponential computational barriers of conventional computing but also achieving the ultimate level of accuracy in studies of chal- lenging processes involving multi-configurational states in catalysis, biochemistry, photochemistry, and materials science to name only a few areas where quantum information technologies can lead to the transformative changes in the way how quantum simulations are performed. NWChem, with its computational infrastructure to characterize second-quantized forms of electronic Hamiltonians in various basis sets (Gaussian and plane waves) and with wavefunction methodologies to provide an initial characterization of the ground- and excited-state wavefunctions, can be used as a support platform for various types of quantum simulators. The recently developed QDK-NWChem interface424(QDK designates Quantum Develop- ment Kit developed by Microsoft Research team) for quantum sim- ulations and libraries for CC downfolded electronic Hamiltonians for quantum computing412are good illustrations of the utilization of NWChem in supporting the quantum computing effort. IX. CONCLUSIONS The NWChem project is an example of a successful co-design effort that harnesses the expertise and experience of researchers in several complementary areas, including quantum chemistry, applied mathematics, and high-performance computing. Over the last three decades, NWChem has evolved into a code that offers a unique combination of computational tools to tackle complex chemical processes at various spatial and time scales. In addition to the development of new methodologies, NWChem is continuously upgraded with new algorithms to take advantage of emerging computer architectures and quantum infor- mation technologies. We believe that the community model of NWChem will continue to spur exciting new developments well into the future. SUPPLEMENTARY MATERIAL See the supplementary material for tutorial slides showing examples of NWChem input files. ACKNOWLEDGMENTS The authors acknowledge the important contributions to the NWChem project of the following deceased researchers: Ricky A. Kendall, Jarek Nieplocha, and Daniel W. Silverstein. We would like to thank people who have helped the progress of the NWChem project by providing valuable feedback, scien- tific direction, suggestions, discussions, and encouragement. This (incomplete) list includes A. Andersen, J. Andzelm, R. Baer, A. Bick, V. Blum, D. Bowman, E. A. Carter, A. Chakraborty, G. Cisneros, D. Clerc, A. J. Cohen, L. R. Corrales, A. R. Felmy, A. Fortunelli, J. Ful- ton, D. A. Dixon, P. Z. El-Khoury, D. Elwood, G. Fanourgakis, D. Feller, B. C. Garrett, B. Ginovska, E. Glendening, V. A. Glezakou, M. F. Guest, M. Gutowski, M. Hackler, M. Hanwell, W. L. Hase, A. C. Hess, W. M. Holden, C. Huang, W. Huang, E. S. Ilton, E. P. Jahrman, J. Jakowski, J. E. Jaffe, J. Ju, S. M. Kathmann, R. Kawai, M. Khalil, X. Krokidis, L. Kronik, P. S. Krsti ´c, R. Kutteh, X. Long, A. Migliore, E. Miliordos, M. Nooijen, J. Li, L. Lin, S. Liu, N. Maitra, M. Malagoli,B. Moore II, P. Mori, S. Mukamel, C. J. Mundy, V. Meunier, V. Murugesan, D. Neuhauser, S. Niu, R. M. Olson, S. Pamidighantam, M. Pavanello, M. P. Prange, C. D. Pemmaraju, D. Prendergast, S. Raugei, J. J. Rehr, A. P. Rendell, R. Renslow, T. Risthaus, K. M. Rosso, R. Rousseau, J. R. Rustad, G. Sandrone, G. K. Schenter, G. T. Seidler, H. Sekino, P. Sherwood, C. Skylaris, H. Song, L. Subra- manian, B. G. Sumpter, P. Sushko, S. Tretiak, R. M. Van Ginhoven, B. E. Van Kuiken, J. H. van Lenthe, B. Veeraraghavan, E. Vorpagel, X. B. Wang, J. Warneke, Y. A. Wang, J. C. Wells, S. S. Xantheas, W. Yang, J. Zador, and Y. Zhang. The core development team acknowledges support from the following projects at the Pacific Northwest National Laboratory. Pacific Northwest National Laboratory is operated by Battelle Memorial Institute for the U.S. Department of Energy under Con- tract No. DE-AC05-76RL01830: (i) Environmental and Molecular Sciences Laboratory (EMSL), the Construction Project, and Oper- ations, the Office of Biological and Environmental Research, (ii) the Office of Basic Energy Sciences, Mathematical, Information, and Computational Sciences, Division of Chemical Sciences, Geo- sciences, and Biosciences (CPIMS, AMOS, Geosciences, Heavy Ele- ment Chemistry, BES Initiatives: CCS-SPEC, CCS-ECC, BES-QIS, BES-Ultrafast), (iii) the Office of Advanced Scientific Computing Research through the Scientific Discovery through Advanced Com- puting (SciDAC), Exascale Computing Project (ECP): NWChemEx. Additional funding was provided by the Office of Naval Research, the U.S. DOE High Performance Computing and Communications Initiative, and industrial collaborations (Cray, Intel, Samsung). The work related to the development of QDK-NWChem inter- face was supported by the “Embedding Quantum Computing into Many-body Frameworks for Strongly Correlated Molecular and Materials Systems” project, which is funded by the U.S. Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences, the Division of Chemical Sciences, Geosciences, and Biosciences, and Quantum Algorithms, Software, and Architectures (QUASAR) Initiative at Pacific Northwest National Laboratory (PNNL). S. A. Fischer acknowledges support from the U.S. Office of Naval Research through the U.S. Naval Research Laboratory. A. J. Logsdail acknowledges support from the UK EPSRC under the “Scalable Quantum Chemistry with Flexible Embedding” (Grant Nos. EP/I030662/1 and EP/K038419/1). A. Otero-de-la-Roza acknowledges support from the Spanish government for a Ramón y Cajal fellowship (No. RyC-2016-20301) and for financial support (Project Nos. PGC2018-097520-A-100 and RED2018-102612-T). J. Autschbach acknowledges support from the U.S. Depart- ment of Energy, Office of Basic Energy Sciences, Heavy Element Chemistry program (Grant No. DE-SC0001136, relativistic meth- ods & magnetic resonance parameters) and the National Science Foundation (Grant No. CHE-1855470, dynamic response methods). D. Mejia Rodriguez acknowledges support from the U.S. Department of Energy (Grant No. DE-SC0002139). D. G. Truhlar acknowledges support from the NSF under Grant No. CHE–1746186. E. D. Hermes was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Chemical Sci- ences, Geosciences and Biosciences Division, as part of the Com- putational Chemistry Sciences Program (Award No. 0000232253). Sandia National Laboratories is a multimission laboratory managed J. Chem. Phys. 152, 184102 (2020); doi: 10.1063/5.0004997 152, 184102-19 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under Contract No. DE-NA0003525. Z. Lin acknowledges support from the National Natural Science Foundation of China (Grant Nos. 11574284 and 11774324). J. Garza thanks CONACYT for support under the Project No. FC-2016/2412. K. Lopata gratefully acknowledges support by the U.S. Depart- ment of Energy, Office of Science, Basic Energy Sciences, Early Career Program, under Award No. DE-SC0017868 and the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-SC0012462. L. Gagliardi and C. J. Cramer acknowledge support from by the Inorganometallic Catalyst Design Center, an Energy Fron- tier Research Center funded by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES) (DE- SC0012702). They acknowledge the Minnesota Supercomputing Institute (MSI) at the University of Minnesota for providing com- putational resources. This work used resources provided by EMSL, a DOE Office of Science User Facility sponsored by the Office of Biological and Envi- ronmental Research and located at the Pacific Northwest National Laboratory (PNNL) and PNNL Institutional Computing (PIC). PNNL is operated by Battelle Memorial Institute for the United States Department of Energy under DOE Contract No. DE-AC05- 76RL1830. The work also used resources provided by the National Energy Research Scientific Computing Center (NERSC), a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. This work also used resources provided by the Oak Ridge Lead- ership Computing Facility (OLCF) at the Oak Ridge National Lab- oratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. This manuscript was authored, in part, by UT–Battelle, LLC, under Contract No. DE-AC05-00OR22725 with the U.S. Depart- ment of Energy (DOE). The U.S. government retains and the pub- lisher, by accepting the article for publication, acknowledges that the U.S. government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for U.S. government pur- poses. The DOE will provide public access to these results of feder- ally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan). 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1.5142495.pdf

Appl. Phys. Lett. 116, 192402 (2020); https://doi.org/10.1063/1.5142495 116, 192402 © 2020 Author(s).Distinct domain reversal mechanisms in epitaxial and polycrystalline antiferromagnetic NiO films from high-field spin Hall magnetoresistance Cite as: Appl. Phys. Lett. 116, 192402 (2020); https://doi.org/10.1063/1.5142495 Submitted: 13 December 2019 . Accepted: 26 April 2020 . Published Online: 11 May 2020 Motoi Kimata , Takahiro Moriyama , Kent Oda , and Teruo Ono ARTICLES YOU MAY BE INTERESTED IN Manipulation of the zero-damping conditions and unidirectional invisibility in cavity magnonics Applied Physics Letters 116, 192401 (2020); https://doi.org/10.1063/5.0006363 The magnetic, electronic, and light-induced topological properties in two-dimensional hexagonal FeX 2 (X = Cl, Br, I) monolayers Applied Physics Letters 116, 192404 (2020); https://doi.org/10.1063/5.0006446 Magnetization switching induced by magnetic field and electric current in perpendicular TbIG/Pt bilayers Applied Physics Letters 116, 112401 (2020); https://doi.org/10.1063/1.5140530Distinct domain reversal mechanisms in epitaxial and polycrystalline antiferromagnetic NiO films from high-field spin Hall magnetoresistance Cite as: Appl. Phys. Lett. 116, 192402 (2020); doi: 10.1063/1.5142495 Submitted: 13 December 2019 .Accepted: 26 April 2020 . Published Online: 11 May 2020 Motoi Kimata,1,a) Takahiro Moriyama,2 Kent Oda,2and Teruo Ono2 AFFILIATIONS 1Institute for Materials Research, Tohoku University, Sendai, Miyagi 980-8577, Japan 2Institute for Chemical Research, Kyoto University, Gokasho, Uji, Kyoto, 611-0011, Japan a)Author to whom correspondence should be addressed: motoi.kimata@imr.tohoku.ac.jp ABSTRACT Magnetic-field-induced domain reversal mechanisms of antiferromagnetic (AF) NiO thin films are investigated by spin Hall magnetoresistance (SMR) measurements. The field strength dependence of SMR amplitudes is measured in epitaxial and polycrystalline NiO films. From the field-dependent behavior of SMR amplitude, two distinct domain reversal mechanisms are found for those NiO films. In theepitaxial films, the conventional monodomain formation against the destressing field due to the magneto-elastic coupling is observed. On theother hand, the thermally assisted domain reversal is dominant in the polycrystalline films. Based on our thermal assisted model, the effectivevalues of domain pinning potential and the number of spins contributing to domain reversal in polycrystalline films are determined. These values are quite important to design AF spin memory devices. This study contributes to building a method to determine the key parameters in AF spintronics with polycrystalline thin films, which are free from the lattice mismatching problem. Published under license by AIP Publishing. https://doi.org/10.1063/1.5142495 Antiferromagnetic (AF) materials are recently attracting much attention for spintronic applications due to their advantages comple-mentary to conventional ferromagnets. AF materials generally showsmall magnetic susceptibility and fast spin dynamics due to the com-pensated spin structures with a strong exchange interaction. Thus, theAF spintronic devices have properties such as small stray field, toler- ance for the external magnetic field, and fast spin dynamics for high- speed operations. 1–3Also, more recently, the spintronic functionalities caused by the characteristic spin and electronic structures in non-collinear AF materials have been reported. 4,5Among many challenges to realizing AF spintronics, one of the key techniques is writing andreading processes of compensated AF spins. This technique wasrecently demonstrated by using spin transfer torque from the stag-gered crystal structure or sandwich type layered structures. 6–8In these cases, the AF magnetic moments are switched by an effective field from the current induced spin–orbit torques (SOTs). On theother hand, it is important to understand the AF spin response toan external magnetic field, which is expected to be different fromthe response to SOT, for revealing the overall feature of the spindynamics and the domain reversal mechanisms especially in theform of AF thin films.In general, spins respond to an external magnetic field to mini- mize the total energy of the system. AF materials generally undergothe so-called spin-flop state (the antiparallel sublattice spins becomeperpendicular to the magnetic field) to reduce the total energy of thesystem with respect to the magnetic field. 9,10However, in thin film samples, which are perhaps more important for spintronic applica-tions, the situation is much more complicated since the additionalcontributions from the substrate and grain boundaries will largely affect the magnetic properties. Indeed, in strained epitaxial films of NiO, the destressing field due to the elastic coupling between NiO andthe substrate material is essential for the magnetic domain rever-sal. 11–13Compared with the well-ordered epitaxial film, the under- standing of polycrystalline films is also important to expand materialchoices in AF spintronics since these materials are free from the latticematching problem. However, the study of spin dynamics in AF poly-crystalline films is still lacking. Also, the methods to determine thefundamental parameters for designing AF spintronic devices, such asthe pinning potential and number of spins responsible for domainnucleation, have not been established yet. To detect the magnetic-field response of AF thin films, we have used the spin Hall magnetoresistance (SMR) as a probe to measure the Appl. Phys. Lett. 116, 192402 (2020); doi: 10.1063/1.5142495 116, 192402-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplAF spin orientation. SMR is a magnetoresistance caused by the spin current transmission and reflection at the interface of a heavy metal and a magnetic material. The magnitude of longitudinal and trans- verse resistances varies depending on the relative orientation between magnetization and spin accumulation.14,15When the magnetization is parallel to the charge current (the spin accumulation vector is always perpendicular to the flow direction of charge current in the framework of the spin Hall effect), the magnetization and the spin accumulation vector are perpendicular to each other. In this case, the accumulated spins at the interface are absorbed as a spin transfer torque, and thus,t h er e s i s t a n c eo ft h eh e a v ym e t a ll a y e ri sl a r g e rd u et ot h ea d d i t i o n a l energy dissipation at the interface. On the other hand, when the charge current is perpendicular to the magnetic field, the spin accumulation vector is parallel to the magnetization. In this case, the accumulated spins at the interface are not absorbed to the magnetic layer but directly reflected. Therefore, the resistance of the Pt layer is lower than the former case. Due to these mechanisms, the longitudinal (trans-verse) resistance shows the cos 2h(sin2h) dependence with the field angle hmeasured from the current direction. These principles are also applicable for the antiferromagnet/heavy metal layers. When antiferro- magnetic magnetization undergoes the spin-flop state, where the sub- lattice magnetization is perpendicular to the external magnetic field, thehdependence of the SMR shifts by 90/C14compared with the case of the ferromagnet/heavy metal bilayer, i.e., negative SMR.11This feature is indeed observed recently at the interface between the AF/heavy metal interface including thin film systems, such as NiO/Pt,11–13 Cr2O3/Ta,16anda-Fe2O3/Pt.17Based on these recent studies, the nega- tive SMR feature is recognized as a direct fingerprint of the spin flop magnetic structure of the antiferromagnetic thin films. In this Letter, we will report distinct magnetic-field-induced domain reversal mechanisms in epitaxial and polycrystalline NiO films probed by SMR. NiO is a typical AF material with rock-salt type crys- tal structure. The magnetic easy plane is (111), and nearly free rotation of the magnetic moments is expected in this plane compared withother directions. The domain reversal mechanisms are investigated based on the field dependence of the amplitude of SMR. In the epitax- ial films, we observed a quadratic magnetic field dependence of SMR amplitude, which can be explained by the conventional mechanism where the destressing energy from the substrate and Zeeman energy play an essential role. 12,13On the other hand, in the polycrystalline films, the field dependence of SMR amplitude cannot be understoodby the conventional mechanism: the domain reversal mechanism is different from the case for the epitaxial films. To explain this result, we will propose the thermally assisted domain reversal mechanism in AF thin films. Also, from the detailed analysis using a thermal excitation model, we have estimated the effective height of pinning potential for AF magnetic domains and the effective number of spins contributing to the domain reversal. These parameters would be key parameters todesign efficient AF spintronic devices. We also pointed out that the number of spins for domain reversal in the present polycrystalline film is much smaller than that contained in a NiO crystallite. This fact sug- gests that the domain wall nucleation in polycrystalline NiO films take place at the boundaries of microscopic crystallites. In this study, we used a Pt(4 nm)/NiO(10 nm)/Pt(4 nm) tri- layered structure, which is identical to the recently reported spin tor- que switching AF NiO devices. 7All the magnetic and metallic layers are deposited by magnetron sputtering on MgO (111) and thermallyoxidized Si/SiO 2substrates. The films are then patterned into a Hall cross shape with a channel width of 5 lm by photolithography and Ar dry etching. Figures 1(a) and1(b) show the XRD patterns of present films on MgO (111) and Si/SiO 2substrates, respectively. For the MgO substrate, a sharp and intense peak corresponding to NiO (111) impli-cates a high quality epitaxially grown NiO on MgO (111). TheRHEED measurements (not shown) 7indicate the orientation relation- ship of MgO(111)[1-10]//Pt(111)[1-10]//NiO(111)[1-10]. As showninFig. 1(b) , the NiO (111) peak was also observed even in the film on the Si/SiO 2substrate but shows a broad linewidth and weak intensity. This means that NiO on SiO 2is oriented to the (111) plane with poor crystallinity and/or the orientation within the (111) plane. This sug-gests that the film on a Si/SiO 2substrate is the (111) oriented polycrys- tal. From the XRD linewidths, the crystallite size of the polycrystallinefilms is estimated to be 7.3 nm by using Scherrer’s equation. This valueis comparable to the previously reported crystallite size of the NiOfilm on a grass substrate. 18On the other hand, XRD linewidths for the epitaxial film are as sharp as the residual linewidth of the instrument,which only leads to the lower bound estimation of the crystallite sizeof 100 nm. For the SMR experiments, longitudinal and transverseresistances were measured with a constant current excitation of 1 mA.Magnetotransport measurements are performed with a superconduct-ing or a hybrid magnet at the high field laboratory, IMR, TohokuUniversity. As shown in Figs. 1(c) and1(d),t h em a g n e t i cfi e l dw a sr o t a t e d within the basal plane of the Hall cross device, and the field angle h was measured from the current direction. In Fig. 1(d) , the AF sublat- tice magnetic moments are schematically shown as M AandMB,w h i c h are perpendicular to the external magnetic field in the spin flop state.Since the SMR is related to the relative angle between the sublatticemoment and the spin accumulation vector, the phase of the SMRcurve is a measure of dominant ferromagnetic or AF magneticdomains in the NiO film. So, if the AF spin flop configuration is domi- nant, the phase of the SMR curve is shifted by 90 /C14compared with the typical ferromagnetic case.11This feature is indeed observed in our FIG. 1. (a) and (b) XRD patterns for Pt/NiO/Pt layers on MgO (111) and Si/SiO 2 substrates, respectively. (c) Schematic of the measured Hall cross device and mea- surement setup. The layered structure consists of Pt(4)/NiO(10)/Pt(4) for epitaxialand polycrystalline films. (d) The antiferromagnetic sublattice moments are illus-trated as M AandMBin a spin-flop phase.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 192402 (2020); doi: 10.1063/1.5142495 116, 192402-2 Published under license by AIP Publishingpresent devices as shown in Fig. 2 .Figures 2(a) and2(b)show the lon- gitudinal (DRL RL/C17DqL q) and transverse (DRT RL=5/C17DqT q)S M Rs i g n a l sf o r B¼27 T and T¼298 K, respectively. Here, DRLandDRTare the oscillation parts of longitudinal and transverse resistances with respecttoh, respectively, and R Lis the longitudinal resistance. To calculate transverse SMR, the geometrical factor 5 originating from the design of present Hall cross is considered.11For longitudinal (transverse) SMR, the resistance takes minimum at h¼0/C14(45/C14), and these mini- mum positions are 90/C14shifted from the case of ferromagnetic SMR. This indicates that the dominant magnetic domain of the present NiOfilm is AF. Also, the AF order of the present film at room temperatureis consistent with the previously estimated Neel temperatures of NiOfilms, 19–21i.e., the Neel temperature of 10 nm-thick NiO is sufficiently higher than room temperature. The smaller amplitude of longitudinalSMR than that of transverse SMR might be caused by the spatial inho-mogeneity of spin mixing conductance in the long channel and/or aparasitic contact resistance. 11 Figures 3(a) and 3(b) show the magnetic field dependence of transverse SMR up to 27 T for epitaxial [ Fig. 3(a) ] and polycrystalline [Fig. 3(b) ] NiO film devices, respectively. For the epitaxial sample, the negative SMR amplitude is prominent above 5 T and shows smootht h es i n u s o i d a lf u n c t i o nw i t h o u tt h ec h a n g eo fp h a s e .T h i si n d i c a t e s that the SMR signal is dominated by the spin-flop sublattice moments, and magnetic domains coherently rotate with the field direction. Also,as shown in Fig. 3(c) , the monotonically increasing behavior of SMR amplitude shows that the ratio of the spin flop phase graduallyincreases up to the highest measured magnetic field, which is muchhigher than the spin flop field reported in bulk NiO ( /C245T ) . 22In con- trast to the epitaxial sample, the SMR signal is hard to observed below10 T in the polycrystalline sample, and a clear negative SMR signal isonly observed in the field range for B >15 T, i.e., the SMR amplitude is much smaller than that of epitaxial films [see Figs. 3(a) and3(b)].Moreover, the shape of the SMR curve is not a smooth sinusoidal as typically can be seen for the data at 15 and 20 T in Fig. 3(b) .T h e s e results suggest that the domains of the polycrystalline film are notsmoothly redistributed, indicating the presence of strong pinningpotential at the defects and/or crystallite boundaries. Note that the pre- sent magnetoresistance effect is quite sensitive to the difference of the underlying NiO layer, i.e., epitaxial or polycrystalline films. This factindicates that the observed magnetoresistance does not originate fromthe bulk properties of Pt, but it is dominated by that of the NiO filmthrough the interface. Next, we will discuss the difference of domain reversal mecha- nisms for epitaxial and polycrystalline films. The main panel of Fig. 4 shows the field strength dependence of SMR amplitude with model fit-tings. In the epitaxial films (solid circles), the field dependence of SMRamplitude in the low field region ( B</C2410 T) almost follows B 2 (dashed line in the figure), which is well explained by the model pre- dicted in the previous report.12,13In this model, the quadratic mag- netic field dependence is attributed to the magnetic domainredistribution by competition of magneto-elastic and Zeeman energyin the multidomain state. However, the SMR amplitude deviates from the quadratic field dependence and shows tendency to saturate in the high field region close to 27 T as can be seen for #2 in Fig. 4 .I nt h ep r e - vious reports, the saturation of SMR amplitude was indeed observedin sufficiently high fields ( /C245 T for the bulk single crystal sample 11and /C2413 T for the Al 2O3based epitaxial film with a thickness of 120 nm12). In this saturation region, it is considered that the magnetic domain of NiO undergoes a monodomain state, where all the mag-netic moments coherently rotate with field rotation. 12The gradual crossover to the saturation regime might be caused by a weak pinningof domain walls, which is not considered in the model. 12On the other hand, the SMR amplitude in the present epitaxial NiO still increases even at the highest field of 27 T. This means that the saturation field ofthe present NiO film is higher than that in the previous studies. 11,12 The difference of the saturation field is interpreted as the difference ofthe destressing field, which is caused by the elastic clamping between NiO and the substrate material. Thus, the present non-saturation FIG. 2. Typical (a) longitudinal and (b) transverse SMR curves for B¼27 T and T¼298 K. The solid lines are fits using cos2hand sin 2 hfor (a) and (b), respectively. FIG. 3. The field strength dependence of SMR curves for (a) epitaxial and (b) poly- crystalline NiO films. In epitaxial NiO, the SMR curves show a smooth sinusoidal curve for the whole magnetic field range. On the other hand, in the polycrystallineNiO film, the SMR curves are slightly distorted typically seen in the curves atB¼15 T and 20 T in Fig. 3(b) . The black lines in (a) and (b) indicate sinusoidal fits for SMR data at 27 T. (c) The magnetic field dependence of SMR amplitudes for epitaxial (samples #1 and #2) and polycrystalline (samples #3 and #4) NiO films.The samples #1 and #2 (also #3 and #4) are distinct devices fabricated from epitax-ial (and polycrystalline) films with the same batch, respectively.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 192402 (2020); doi: 10.1063/1.5142495 116, 192402-3 Published under license by AIP Publishingbehavior of SMR amplitude indicates the large destressing field of the present sample. The lattice constants of MgO (0.421 nm) and singlecrystalline Pt (0.392 nm), which are located under the NiO layer, arerather different from that of bulk NiO (0.4177 nm), and indeed, thislattice mismatching is larger than that between Al 2O3(0.419 nm) and NiO used in the previous study.24Also, since the thickness of the pre- sent NiO layer (10 nm) is thinner than that in the previous study (120 nm),11the effect of lattice mismatch dominantly affects the mag- netic properties. These reasons are likely to induce a relatively high sat-uration field of the present epitaxial samples. Nevertheless, theq u a d r a t i cfi e l dd e p e n d e n c eo fS M Ra m p l i t u d es h o w st h a tt h em o d e lwith Zeeman energy and destressing field is still applicable in the lowfield behavior of the epitaxial samples. In contrast to the case of the epitaxial film, the field dependence of SMR amplitude in the polycrystalline film (#3 and #4) is not repro-duced by the quadratic dependence (shown in the broken lines in themain panel of Fig. 4 ). This indicates that the domain redistribution mechanism in the polycrystalline film is different from that in the epi-taxial films. 12,13Indeed, the size of the crystallite for the present poly- crystal film (7.3 nm) is much smaller than the domain wall widthreported in bulk NiO (few hundred nanometers), 25and thus, the domain reversal mechanism is expected to be different from that inthe single crystalline samples including epitaxial films, i.e., the strongpinning effect at the crystallite boundaries should be more prominentin the polycrystalline film. A sc a nb es e e ni nt h ei n s e to f Fig. 4 , the field dependence of SMR amplitude for polycrystalline films is almost straight on a logarithmicscale, suggesting that the SMR amplitude is dominated by the thermalactivation mechanism. Since many crystallite boundaries are expected in the polycrystalline film, this activation energy likely originates fromthe domain wall pinning potential at the boundaries. Thus, we intro- duced a thermal activation law with field-reduced activation energy with an effective pinning potential to explain the field dependence ofSMR for the polycrystalline NiO film, which can be written asDq T=q¼q0exp½/C0ðU0/C0ng2lBBÞ=kBT/C138.H e r e , q0is an exponential pre-factor, which should be equal to the saturation SMR amplitude inthe high-field monodomain state with U 0¼ng2lBB(the factor 2 cor- responds to spin Sexpected in Ni2þions). U0is the zero-field effective pinning potential of the domain wall, and n,g,a n d lB, are the number of spins contributing to the magnetic domain reversal, g-factor, and Bohr magneton, respectively. In this model, we assume the linear reduction of domain wall pinning potential by the Zeeman energy of the magnetic domain wall.23The saturation of SMR amplitude is not observed in this study so that the saturation SMR amplitude isassumed to be 4–7.6 /C210 /C04, which is reported for some NiO(111)/Pt interfaces.11,12Then, we can obtain the values of U0andn. Although the ambiguity of q0affects the determination of U0, we can obtain these parameters as follows: U0=kB¼1780–1970 K and n¼14.7 for sample #3 and U0=kB¼2080–2270 K and n¼18.2 for sample #4. This analysis allows us to estimate the effective number of spins con-tributing to the domain reversal of polycrystalline NiO films as approximately 15–18. In contrast, the number of spins included in the crystallite is estimated to be about 2800 if a sphere-like crystallite isassumed. Thus, our analysis suggests that the domain reversal(or nucleation) in the polycrystal film is induced by much smallerspins than that inside the crystallite. Although the source of minorityactive spins in the polycrystal film is not clear at the moment, a possi-ble origin is a small number of uncompensated spins expected at theboundaries between multiple crystallites. A small number of uncom-pensated spins are indeed observed at the interface of AF magneticlayers. 26In other words, the domain reversal in the polycrystalline NiO film is likely dominated by the domain nucleation and the strong pinning at the boundaries between nanoscale NiO crystallites. In summary, we have performed SMR experiments of Pt/NiO/Pt multilayers with epitaxial and polycrystalline NiO layers. From themagnetic field dependence of the SMR amplitude, domain reversalmechanisms for each layer are distinguished. Specifically, for the poly-crystalline film, we propose the thermally excited domain reversalmechanism, which is dominated by domain nucleation and pinning atthe boundaries of nanoscale crystallites. Also, from the analysis based on our model, the effective number of spins contributing to the AF domain reversal for the polycrystalline film is estimated to be about15–18. This value is much smaller than that inside the polycrystallinecrystallite, suggesting that the major origin of the domain nucleationin the polycrystalline film is crystallite boundaries. Our observationdemonstrates that the field-induced SMR study is an efficient methodto determine the key spintronic parameters of polycrystalline AF thinfilms and will be applicable to broad AF materials. We are grateful to K. Fujiwara and A. Tsukazaki for the usage of AFM and fruitful discussions. This work was supported by JSPSKAKENHI Grant Nos. 19K03736, 19K21972, 17H04924,17H05181, and 15H05702. The high field measurements wereperformed at the High Field Laboratory for SuperconductingMaterials, Institute for Materials Research, Tohoku University, FIG. 4. The field strength dependence of SMR amplitudes with model fittings. In the epitaxial samples (#1 and #2 with the left axis), the SMR amplitude for the low field region ( B/H1135110 T) is well explained by the B2dependence as indicated by the dashed lines. The fitting curve of sample #2 is obtained from the data for B/H1101710 T since the SMR amplitudes deviate from the B2dependence in the high field region. On the other hand, in the polycrystalline samples, (#3 and #4 with the right axis), the field dependence of SMR follows the thermally assisted domain reversal modelDq T=q¼q0exp½/C0ðU0/C0ng2lBBÞ=kBT/C138(solid lines) rather than the B2depen- dence (dashed line). The error bars of SMR amplitude estimated from the uncer- tainty of the sinusoidal fit are included in symbols, and multiple data points at the same field strength were obtained from distinct angle scan measurements. Theinset shows the logarithmic plot for polycrystalline samples.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 192402 (2020); doi: 10.1063/1.5142495 116, 192402-4 Published under license by AIP Publishingunder Proposal Nos. 19H0042 and 19H0418. We also acknowledge the generous support from the Center for Spintronics ResearchNetwork (CSRN) at the Graduate School of Engineering Science,Osaka University. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 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. 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Bauer, “Theory of spin Hall magnetoresistance,” Phys. Rev. B 87, 144411 (2013). 16Y. Ji, J. Miao, Y. M. Zhu, K. K. Meng, X. G. Xu, J. K. Chen, Y. Wu, and Y. Jiang, “Negative spin Hall magnetoresistance in antiferromagnetic Cr 2O3/Ta bilayer at low temperature region,” Appl. Phys. Lett. 112, 232404 (2018). 17R. Lebrun, A. Ross, O. Gomonay, S. A. Bender, L. Baldrati, F. Kronast, A. Qaiumzadeh, J. Sinova, A. Brataas, R. A. Duine, and M. Kl €aui, “Anisotropies and magnetic phase transitions in insulating antiferromagnets determined by a spin-Hall magnetoresistance probe,” Commun. Phys. 2, 50 (2019). 18H.-L. Chen, Y.-M. Lu, and W.-S. Hwang, “Characterization of sputtered NiO thin films,” Surf. Coat. Technol. 198, 138–142 (2005). 19D. Hou, Z. Qiu, J. Barker, K. Sato, K. Yamamoto, S. Ve /C19lez, J. M. Gomez-Perez, L. E. Hueso, F. Casanova, and E. Saitoh, “Tunable sign change of spin Hallmagnetoresistance in Pt/NiO/YIG structures,” Phys. Rev. Lett. 118, 147202 (2017). 20J. St€ohr, A. Scholl, T. J. Regan, S. Anders, J. L €uning, M. R. Scheinfein, H. A. Padmore, and R. L. White, “Images of the antiferromagnetic structure of aNiO(100) surface by means of x-ray magnetic linear dichroism spectromicroscopy,” Phys. Rev. Lett. 83, 1862 (1999). 21D. Alders, L. H. Tjeng, F. C. Voogt, T. Hibma, G. A. Sawatzky, C. T. Chen, J. Vogel, M. Sacchi, and S. Iacobucci, “Temperature and thickness dependence of magnetic moments in NiO epitaxial films,” Phys. Rev. B 57, 11623 (1998). 22F. L. A. Machado, P. R. T. Ribeiro, J. Holanda, R. L. Rodr /C19ıguez-Su /C19arez, A. Azevedo, and S. M. Rezende, “Spin-flop transition in the easy-plane antiferro-magnet nickel oxide,” Phys. Rev. B 95, 104418 (2017). 23S. Lemerle, J. Ferr /C19e, C. Chappert, V. Mathet, T. Giamarchi, and P. L. Doussal, “Domain wall creep in an ising ultrathin magnetic film,” Phys. Rev. Lett. 80, 849 (1998). 24L. C. Bartel and B. Morosin, “Exchange striction in NiO,” Phys. Rev. B 3, 1039 (1971). 25K. Arai, T. Okuda, A. 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5.0009504.pdf

AIP Advances 10, 055118 (2020); https://doi.org/10.1063/5.0009504 10, 055118 © 2020 Author(s).Structural and magnetic behavior of Cr2Co(1−x)CrxAl inverse Heusler alloys Cite as: AIP Advances 10, 055118 (2020); https://doi.org/10.1063/5.0009504 Submitted: 31 March 2020 . Accepted: 29 April 2020 . Published Online: 18 May 2020 Manisha Srivastava , Guru Dutt Gupt , Priyanka Nehla , Anita Dhaka , and R. S. Dhaka COLLECTIONS Paper published as part of the special topic on Chemical Physics This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN Precision current measurement with thermal-drift-minimized offset current for single- parameter electron pumps based on gate-switching technique AIP Advances 10, 045332 (2020); https://doi.org/10.1063/5.0002587 Interface defect engineering for high-performance MOSFETs with novel carrier mobility model: Theory and experimental verification AIP Advances 10, 055020 (2020); https://doi.org/10.1063/5.0005813 On the redistribution of charge in La 0.7Sr0.3CrO3/La0.7Sr0.3MnO 3 multilayer thin films AIP Advances 10, 045113 (2020); https://doi.org/10.1063/1.5140352AIP Advances ARTICLE scitation.org/journal/adv Structural and magnetic behavior of Cr 2Co(1−x)CrxAl inverse Heusler alloys Cite as: AIP Advances 10, 055118 (2020); doi: 10.1063/5.0009504 Submitted: 31 March 2020 •Accepted: 29 April 2020 • Published Online: 18 May 2020 Manisha Srivastava,1Guru Dutt Gupt,1Priyanka Nehla,1Anita Dhaka,2and R. S. Dhaka1,a) AFFILIATIONS 1Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India 2Department of Physics, Sri Aurobindo College, University of Delhi, Malviya Nagar, New Delhi 110017, India a)Author to whom correspondence should be addressed: rsdhaka@physics.iitd.ac.in ABSTRACT We report the structural and magnetic behavior of single phase inverse Heusler alloys Cr 2Co(1−x)CrxAl (x= 0, 0.2, and 0.4) by using x-ray diffraction (XRD), Raman spectroscopy, isothermal magnetization, and magnetic susceptibility measurements. Interestingly, the Rietveld refinement of XRD data with space group I ¯4m2 reveal a tetragonal distortion with a c/a ratio around 1.38 in these inverse Heusler structures. The bulk compositions have been confirmed by energy dispersive x-ray spectroscopy measurements. The active Raman mode F 2gis observed at 320 cm−1, which confirms the X-type Heusler structure, as the A2 and B2 type structures are known to be not Raman active. The area of F 2gmode decreases with an increase in the Cr concentration, which indicates that the origin of this mode is due to Co vibrations. The isothermal magnetization data confirm the magnetic moment to be close to zero ( ≤0.02 μB/f.u.) at≈70 kOe and the negligible coercive field suggests the fully compensated ferrimagnetic nature of these samples. The susceptibility behavior indicates irreversibility between zero-field and field-cooled curves and complex magnetic interactions at low temperatures. ©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/10.1063/5.0009504 .,s INTRODUCTION The full Heusler alloys are X2YZtype ternary intermetallic compounds having XandYas transition metals and Zas a main group element or an sp-element.1In recent years, many Co-based Heusler alloys (HAs) have been studied extensively due to their structural stability and half-metallic ferromagnetic (HMF) nature, and they have an advantage of possessing high Curie temperatures (TC)2,3and also exhibit the spin gapless semiconducting proper- ties.4,5In the family of CoCr based full-Heusler alloys, Co 2CrAl has particularly attracted huge attention due to its true half-metallic fer- romagnetic (HMF) nature with 3 μB/f.u.magnetic moment as per the band structure calculations and the Slater–Pauling rule.6,7It is interesting to note here that Luo et al. studied the effect of Cr sub- stitution on the Co site and showed a transition from the HMF state to the HM fully compensated ferrimagnetic (FCF) state.8More- over, the appreciable value of spin polarization of about 68% was also reported for Cr 2CoAl.9This is found to be the most impor- tant material because by changing the Cr concentration, we can tune the magnetic moment from 3 μB/f.u. to zero without destroying thehalf-metallicity.8These compounds show the largest value of spin density among all known ferromagnetic half-metals so far. Inter- estingly, few Heusler alloys show the half-metallic fully compen- sated ferrimagnetic behavior.10These materials act as half-metals as well as gapless semiconductors. In 2013, Meinert et al. theoret- ically investigated Cr 2CoAl Heusler alloys as completely compen- sated half-metallic ferrimagnets.11The reason for the low moment (0.01 μB/f.u.) is the internal spin compensation of the transition met- als placed at different sites (Cr1: 1.36 μB, Cr2: −1.49 μB, Co: 0.30 μB) and the high curie temperature of these materials is because of their strong local moments.11Moreover, there is also a possibility of exhibiting low magnetic moments by these alloys due to the anti- ferromagnetic orientation of Cr–Cr neighboring elements.11,12The main advantage of these alloys is that they develop a very low stray field or demagnetizing field due to which they can be utilized in the fabrication of spin-torque based devices.13By calculating the lowest energy configuration, Meinert et al. also demonstrated that this alloy is stable as compared to their elemental constituents but not stable with respect to their binary phases; thus, it easily decomposes into CoAl and Cr phases.11On the other hand, it was reported that due AIP Advances 10, 055118 (2020); doi: 10.1063/5.0009504 10, 055118-1 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv to the negative formation energy, Cr 2CoAl can possibly stabilize in the inverse (X-type) Heusler structure.9 In this paper, we report the structural and magnetic proper- ties of single phase Cr 2Co(1−x)CrxAl (x= 0, 0.2, and 0.4) inverse Heusler alloys. The bulk compositions are confirmed by energy dis- persive x-ray spectroscopy measurements. The Rietveld refinement of x-ray diffraction (XRD) data of space group F ¯43m confirms the X-type Heusler structure. More interestingly, our analysis with space group I ¯4m2 reveals a tetragonal distortion in these samples. Interest- ingly, the vibrations of Raman modes with different compositions are observed. The active Raman mode F 2gis found to be at about 320 cm−1, which confirms the X-type Heusler structure, as the A2 and B2 type structures are known to be not Raman active. The area of the F 2gmode decreases with an increase in the Cr concentration, which indicates that the origin of this mode is due to Co vibrations. The magnetic analysis confirms zero moment confirming the fully compensated ferrimagnetic nature and complex interactions at low temperatures. EXPERIMENTAL DETAILS The polycrystalline samples of Cr 2Co(1−x)CrxAl (x= 0, 0.2, and 0.4) were prepared by arc melting using a vacuum furnace fromCENTORR Vacuum Industries, USA. In order to reduce the oxygen partial pressure, the chamber was evacuated and flushed with highly pure argon. This process was repeated few times and a small piece of Ti metal is melted (which act as getter pump of oxygen) before melt- ing the sample. The appropriate quantities of the constituent ele- ments of 99.99% purity (from Sigma–Aldrich) were arc melted in an inert argon atmosphere. The ingot was flipped and melted 4–5 times to ensure the homogeneity. For further homogenization and larger grain size samples, the ingot materials were wrapped in Mo foils and sealed in evacuated quartz ampules and then annealed at 1173 K for five days using a high temperature box furnace from Nabertherm, GmbH, Germany. The samples were finally quenched in ice water to obtain the highest degree of chemically ordered phase. The bulk compositions have been confirmed by the energy dispersive x-ray spectroscopy measurements. We used x-ray diffraction (XRD) with Cu K α(λ= 1.5406 Å) radiation for the structure study and ana- lyzed the XRD data by Rietveld refinement using the FULLPROF package, where the background was fitted by linear interpolation between the data points. The Raman measurements were carried out at room temperature using a LabRAM HR evolution Horiba spec- trometer. A He–Cd laser with 325 nm excitation wavelength, 1200 lines per mm grating, and 1 mW laser power was used. The magnetic measurements are performed using the VSM mode in a FIG. 1 . The room temperature XRD patterns (a) fitted with space group F ¯43mand (b)–(d) fitted with space group I ¯4m2; all the insets are the enlarged view of the most intense peak for the sake of clarity of the tetragonal nature. The XRD patterns (open circles) and the Rietveld refinement (black line) of Cr 2Co1−xCrxAl (x= 0, 0.2, and 0.4) with the difference profile (blue line) and Bragg peak positions (short vertical bars, green). AIP Advances 10, 055118 (2020); doi: 10.1063/5.0009504 10, 055118-2 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv physical properties measurements system (PPMS EVERCOOL-II) from Quantum design. RESULTS AND DISCUSSION In the X 2YZ type full-Heusler alloys, the unit cell consists of four interpenetrating fccsublattices with the positions A (0, 0, 0), B (0.25, 0.25, 0.25), C (0.5, 0.5, 0.5), and D (0.75, 0.75, 0.75). The perfectly ordered L21-type structure, with space group Fm ¯3m, con- sists of Xatoms at A and C sites, Yatoms at the B site, and Zatoms at the D site. If weobserve diagonally from any atom, the sequence of the sites is found to be X–Y–X–Z.1,14InX2YZcompounds, if the atomic number of the Y element is higher than that of the X element (as the case in the present study), an inverse Heusler struc- ture (also called X-type structure with the F ¯43m space group) is observed.15In this structure, half of the Xatoms are replaced by Yatoms so that the positions of the atoms change as Xatoms at the A and B sites, Yatoms at the C site, and Zatoms at D site. In this case, the diagonal sequence of atoms changes as X–X–Y–Z, i.e., the difference between the positions of two Xatoms in a particu- lar direction is1 2and1 4in regular and inverse Heusler structures, respectively.1,14,15In Figs. 1(a)–1(d), we present the room tempera- ture XRD patterns of Cr 2Co1−xCrxAl (x= 0, 0.2, and 0.4) recorded in the 2 θrange of 20○to 90○and they are identified to be in a sin- gle phase. The Rietveld refinement of the XRD pattern with cubic space group F ¯43m(No. 216) confirms the X-type Heusler structure and the obtained lattice constant value is a= 5.784 Å for the x= 0 sample, as shown in Fig. 1(a), which is consistent with the previously reported experimental value ( a= 5.794 Å) in Ref. 16. On the other hand, the calculated value of the lattice constant of Cr 2CoAl is 5.72 Å as reported in Ref. 17. We find the fitting reasonably good; how- ever, an asymmetry/distortion toward higher 2 θis clearly visible in the zoomed-in view of the 220 reflection [see the inset of Fig. 1(a)], which indicates the possibility of further improvement in the fitting. Therefore, we again fitted the XRD pattern of the x= 0 sample with space group I ¯4m2 (No. 119), i.e., the inverse tetragonal Heusler structure, which was found to be a better fit, shown in Fig. 1(b) and the inset therein. Moreover, for the x= 0.2 and 0.4 samples, the XRD patterns, fitted with space group I ¯4m2, are shown in Figs. 1(c) and 1(d) and the respective insets are the enlarged view of splitting of the most intense 220 reflections. The Wyckoff positions of the atoms of the Cr 2CoAl alloy in the inverse tetragonal structure are found to be as follows: Cr atoms occupy two different sites as Cr1 at 2b (0, 0, 0.5) and Cr2 at 2c (0, 0.5, 0.25). The Co and Al atoms occupy the sites 2d (0, 0.5, 0.75) and 2a (0, 0, 0), respectively.12 In order to find a best fit, we had to include the disor- ders between Cr and Al,18,19which are found to be about Cr1/Al (55%/45%) and Cr2/Al (45%/55%) at 2b and 2a sites, respectively, for all the samples in the inverse tetragonal phase. The lattice con- stant values obtained from the XRD patterns and the tetragonality ratios ( c/a) are summarized in Table I for all three samples, which agree with the reported values for Cr 2CoGa in Ref. 18 We found an increment in the lattice parameters with the substitution of Cr at the site of Co, which is expected, as the atomic radius of Cr (128 pm) is larger than that of Co (125 pm). In Fig. 2, we show the Raman spectroscopy data of Cr2Co(1−x)CrxAl (x= 0, 0.2, and 0.4) measured with 325 nmTABLE I . The lattice parameters of Cr 2Co(1−x)CrxAl in the tetragonal phase with the I¯4m2space group. x→ 0 0.2 0.4 a(Å) 4.085 4.119 4.169 c(Å) 5.679 5.682 5.698 c/a 1.39 1.38 1.37 excitation wavelength at room temperature. In general, it is not easy to observe the Raman signal from the metallic samples; however, it is interesting to note that Zayak et al. predicted about the experi- mental evidence of Raman vibrational modes in metallic Heusler alloys.20The authors show that for the Heusler alloys, the optical modes are split into three well separated triply degenerate modes (F2g+ 2F 1u), where F 2ghas been found to be Raman active, whereas two F 1umodes are IR active.20Interestingly, we observed the most intense Raman mode ( F2g) at a wavenumber ≈320 cm−1, which is in agreement with the value reported in the literature.21,22It should be noted that Zhan et al. used Raman scattering to study the Co 2FeAl Heusler compound across the Curie temperature and observed that the intensity of the Raman signal strongly depends on Co atom vibrational modes. In the present study, we found that the inten- sity of the F2gpeak decreases with a decrease in the amount of Co, which indicates that the origin of this mode is due to the vibrations of Co atoms in the lattice. We have fitted the F2gpeak for all three FIG. 2 . Raman spectra of Cr 2Co(1−x)CrxAl (x= 0, 0.2, and 0.4) measured at room temperature with 325 nm excitation wavelength. The inset shows the plot of the area of F 2gmode (calculated by fitting the peaks with the Gaussian function) as a function of the Cr concentration. AIP Advances 10, 055118 (2020); doi: 10.1063/5.0009504 10, 055118-3 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv samples with the Gaussian function and then plotted the area as a function of the Cr concentration, as shown in the inset of Fig. 2. This clearly indicates the decrease in the Raman signal with an increase in the Cr concentration. Also according to the literature,20,21the exis- tence of Raman modes indicates the presence of the X-type structure because A2 and B2 type structures are not Raman active. We have also observed two broad modes at around 800 cm−1and 1025 cm−1, which are consistent with those reported in Ref. 23 for the similar Heusler alloys. Furthermore, in order to understand the magnetic properties and moment of Cr 2CoAl, we have measured isothermal magneti- zation vs magnetic field (M–H) and magnetic susceptibility χvs temperature (M −T). Figures 3(a) and 3(b) show the M–H curves of Cr 2Co(1−x)CrxAl (x= 0 and 0.4) measured at 300 K and 50 K, respectively, which clearly indicate a very small moment of the order of 10−3μB/f.uand the nonsaturating behavior of magnetization up to±50 kOe for both the samples. It has been reported that theformation of the impure CoAl phase, which is paramagnetic in nature, can be responsible for the linear behavior in the M–H curves.11However, the absence of this CoAl phase in the XRD data of our samples rules out this possibility. Our results suggest an antiferromagnetic coupling between the Cr–Cr neighboring atoms, consistent with Ref. 9. On the other hand, at 300 K, a small finite hysteresis has been observed for the x= 0 sample (50 Oe) but not for the x= 0.4 sample, as clearly seen in the zoomed-in view in the inset of Fig. 3(a). In addition, at 50 K, the values of coercivity were found to be 200 Oe and 100 Oe for the x= 0 and 0.4 sam- ples, respectively [inset of Fig. 3(b)], which suggests the presence of ferromagnetic order. It has been shown that the partial density of states for minority (majority) spins are located above (below) the Fermi level for Cr1, whereas for Cr2, it is reversed, which implies that their spin moments are in antiparallel configuration.18The observed magnetic behavior in Figs. 3(a) and 3(b) is consistent, as Co at the Y site couples ferromagnetically with the nearest neighbor Cr and FIG. 3 . Isothermal magnetization Mof Cr 2Co(1−x)CrxAl (x= 0 and 0.4) samples as a function of the applied magnetic field Hmeasured at 300 K (a) and 50 K (b), where the respective insets show zoomed-in view in the range from −0.55 kOe to +0.55 kOe near the origin. (c) and (d) Temperature dependent zero-field-cooled (ZFC) and field-cooled (FC) magnetization data of the x= 0 and 0.4 samples, respectively, measured at a magnetic field of 100 Oe. Insets in (c) and (d) present the FC derivative plots to make the compensation temperatures clearly visible. AIP Advances 10, 055118 (2020); doi: 10.1063/5.0009504 10, 055118-4 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv antiferromagnetically with the second nearest neighbor Cr, and the two distinct Cr sites couple ferrimagnetically.17This type of alignment is attributed to a competition between the intra-atomic exchange splitting of the magnetic atom dstates and the inter-atomic covalent interaction of dstates from atoms at different sites.24This also agrees with the theoretical prediction of the half-metallic fully compensated ferrimagnetic nature in the Cr 2CoAl sample.11,12 In Figs. 3(c) and 3(d), we show the magnetic susceptibility χ as a function of the temperature in both zero-field-cooled (ZFC) and field-cooled (FC) modes for the x= 0 and 0.4 samples, respec- tively, measured in the temperature range of 10–380 K and at 100 Oe magnetic field. For the x= 0 sample, the moment is fairly con- stant and close to zero, i.e., the sample is in the fully compensated ferrimagnetic state above 200 K, which is defined a compensation temperature, as marked in the derivative plot shown in the inset of Fig. 3(c). Interestingly, the moment increases significantly below ≈200 K and the irreversible behavior between FC and ZFC moments is clearly evident. Furthermore, the ZFC curve shows a down hump at≈50 K and below, and around 25 K, there is a sharp increase in the moment for both FC and ZFC modes. The behavior of temperature dependent magnetization for the x= 0 sample is in good agreement with the one reported in Ref. 16. On the other hand, it is important to note here that Jamer et al. attributed the low temperature para- magnetic behavior to the presence of an extra CoAl phase in their sample,16which clearly is not the case in the present study as evident in our XRD analysis shown in Fig. 1. With an increase in the Cr con- centration at the Co site, i.e., for the x= 0.4 sample in Fig. 3(d), the compensation temperature shifted to around 220 K, which is clearly visible in the derivative plot shown in the inset. Below this temper- ature, we observe bifurcation in ZFC and FC curves and a consis- tent increase in the moment until the lowest measured temperature. This magnetization behavior for both the samples suggests complex magnetic interactions at low temperatures, which motivates us to study these materials using neutron diffraction for further under- standing of these interactions and the role/possibility of the atomic disorder.2Though the Curie temperature (T C) is expected to be at around 700 K,16the magnetization measurement at a high temper- ature requires the exact values of T Cof these materials.25Interest- ingly, the nature of the competing magnetic behavior is clear from the irreversibility in ZFC–FC curves, where the fully compensated feature is very sensitive to the compositions.26 Note that the half-metallic full-Heusler alloys follow the Slater– Pauling behavior, i.e., the total spin magnetic moment per formula unit, M t, inμBscales with the total number of valence electrons, Z t, following the rule: M t= Z t−24.15In the case of the fully compen- sated ferrimagnet (FCF), the total spin magnetic moment should be zero similar to Cr 2CoGa and Fe 2VGa, which have exactly 24 valence electrons, and the ground state of Cr 2CoAl is found to be ferrimag- netic in the inverse Heusler structure.11However, the ferrimagnetic state of the Cr 2CoAl sample arises due to the antiferromagnetic coupling of Cr–Cr atoms of inequivalent nearest neighbors, and hence it exhibits the fully compensated ferrimagnetic nature.11In a more precise way, for these types of alloys, there is a competition between the magnetic states of atoms, which decides whether the alignment of the moments should be ferromagnetic or antiferromag- netic. Therefore, Cr 2CoAl possess almost vanishing total spin mag- netic moments in the FCF states because of the antiferromagnetic alignment of the Cr–Cr inequivalent nearest neighboring atoms dueto the direct interaction between dstates.9Moreover, the Curie tem- perature of these alloys is expected to be high,27which makes them the most promising candidates for devices. CONCLUSION In summary, we successfully prepared single phase Cr 2Co(1−x) CrxAl (x= 0, 0.2, and 0.4) and investigated the structural and magnetic behavior by using x-ray diffraction (XRD), Raman spec- troscopy, isothermal magnetization, and magnetic dc susceptibility measurements. The Rietveld refinement of the XRD patterns con- firms the X-type Heusler structure, and interestingly, a tetragonal distortion (space group I ¯4m2) has been observed in these samples where the c/a value is found to be around 1.38. The active Raman mode F 2gis found to be at about 320 cm−1, which confirms the X- type Heusler structure, as the A2 and B2 type structures are known to be not Raman active. The area of F 2gmode decreases with an increase in the Cr concentration, which suggests that the origin of this mode is due to Co vibrations. The susceptibility data show an irreversible behavior between ZFC and FC curves, which indicate complex magnetic interactions. The magnetization data show that the magnetic moment is close to zero at around 70 kOe, which con- firms the fully compensated ferrimagnetic nature of these inverse Heusler alloys. ACKNOWLEDGMENTS This work was financially supported by the BRNS through the DAE Young Scientist Research Award to RSD with the project sanction Grant No. 34/20/12/2015/BRNS. M.S., G.D.G., and P.N. acknowledge the MHRD, India for the fellowship through IIT Delhi. Authors acknowledge various experimental facilities at IIT Delhi like XRD and PPMS EVERCOOL-II at Physics department; the glass blowing section, SEM, and EDX at central research facility (CRF); and Raman spectroscopy at nano research facility (NRF). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1C. Felser and A. Hirohata, Heusler Alloys: Properties, Growth, Applications , Springer Series of Materials Science Vol. 222 (Springer, 2016), p. 1, ISBN: 978- 3-319-21449-8. 2P. Nehla, Y. Kareri, G. D. Gupt, J. Hester, P. D. Babu, C. Ulrich, and R. S. Dhaka, “Neutron diffraction and magnetic properties of Co 2Cr1−xTixAl Heusler alloys,” Phys. Rev. B 100, 144444 (2019). 3P. Nehla, V. K. Anand, B. Klemke, B. Lake, and R. S. Dhaka, “Magnetocaloric properties and critical behavior of Co 2Cr1−xMn xAl Heusler alloys,” J. Appl. Phys. 126, 203903 (2019). 4D. Rani, Enamullah, L. Bainsla, K. G. Suresh, and A. Alam, “Spin-gapless semiconducting nature of Co-rich Co 1+xFe1−xCrGa,” Phys. Rev. B 99, 104429 (2019). 5S. Ouardi, G. H. Fecher, C. Felser, and J. Kubler, “Realization of spin gapless semiconductors: The Heusler compound Mn 2CoAl,” Phys. Rev. Lett. 110, 100401 (2013). 6J. Kübler, G. H. Fecher, and C. Felser, “Understanding the trend in the Curie tem- peratures of Co 2-based Heusler compounds: Ab initio calculations,” Phys. Rev. B 76, 024414 (2007). AIP Advances 10, 055118 (2020); doi: 10.1063/5.0009504 10, 055118-5 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv 7S. Wurmehl, H. C. Kandpal, G. H. Fecher, and C. Felser, “Valence electron rules for prediction of half-metallic compensated-ferrimagnetic behaviour of Heusler compounds with complete spin polarization,” J. Phys.: Condens. Matter 18, 6171 (2006). 8H. Luo, L. Ma, Z. Zhu, G. Wu, H. Liu, J. Qu, and Y. Li, “ Ab initio study of Cr substitution for Co in the Heusler alloy Co 2CrAl: Half–metallic and adjustable magnetic moments,” Physica B 403, 1797 (2008). 9M. Singh, H. S. Saini, J. Thakur, A. H. Reshak, and M. K. Kashyap, “Tuning Fermi level of Cr 2CoZ ( Z= Al and Si) inverse Heusler alloys via Fe-doping for maximum spin polarization,” J. Magn. Magn. Mater. 370, 81 (2014). 10I. Galanakis, K. Ozdogan, E. Sasioglu, and B. Aktas, “ Ab initio design of half- metallic fully compensated ferrimagnets: The case of Cr 2MnZ (Z = P, As, Sb, and Bi),” Phys. Rev. B 75, 172405 (2007). 11M. Meinert and M. P. Geisler, “Phase stability of chromium based compen- sated ferrimagnets with inverse Heusler structure,” J. Magn. Magn. Mater. 341, 72 (2013). 12H.-S. Jin and K.-W. Lee, “Stability of room temperature compensated half- metallicity in Cr-based inverse-Heusler compounds,” Curr. Appl. Phys. 19, 193 (2019). 13J. Finley, C.-H. Lee, P. Y. Huang, and L. Liu, “Spin–orbit torque switch- ing in a nearly compensated Heusler ferrimagnet,” Adv. Mater. 31, 1805361 (2019). 14T. Graf, C. Felser, and S. S. P. Parkin, “Simple rules for the understanding of Heusler compounds,” Prog. Solid State Chem. 39, 1 (2011); T. Graf, S. Parkin, and C. Felser, “Heusler compounds—A material class with exceptional properties,” IEEE Trans. Magn. 47, 367 (2011). 15S. Skaftouros, K. Özdo ˘gan, E. ¸ Sa¸ sıo ˘glu, and I. Galanakis, “Search for spin gap- less semiconductors: The case of inverse Heusler compounds,” Appl. Phys. Lett. 102, 022402 (2013); “Generalized Slater–Pauling rule for the inverse Heusler compounds,” Phys. Rev. B 87, 024420 (2013). 16M. E. Jamer, L. G. Marshall, G. E. Sterbinsky, L. H. Lewis, and D. Heiman, “Low- moment ferrimagnetic phase of the Heusler compound Cr 2CoAl,” J. Magn. Magn. Mater. 394, 32 (2015).17S. K. Mohanta, Y. Tao, X. Yan, G. Qin, V. Chandragiri, X. Li, C. Jing, S. Cao, J. Zhang, Z. Qiao, H. Gu, and W. Ren, “First principles electronic structure and magnetic properties of inverse Heusler alloys X 2YZ (X=Cr; Y=Co, Ni; Z=Al, Ga, In, Si, Ge, Sn, Sb),” J. Magn. Magn. Mater. 430, 65 (2017). 18H. Luo, L. Yang, B. Liu, F. Meng, and E. Liu, “Atomic disorder in Heusler alloy Cr2CoGa,” Physica B 476, 110 (2015). 19B. Deka, R. Modak, P. Paul, and A. Srinivasan, “Effect of atomic disorder on magnetization and half-metallic character of Cr 2CoGa alloy,” J. Magn. Magn. Mater. 418, 107 (2016). 20A. T. Zayak, P. Entel, K. M. Rabe, W. A. Adeagbo, and M. Acet, “Anomalous vibrational effects in nonmagnetic and magnetic Heusler alloys,” Phys. Rev. B 72, 054113 (2005). 21Z. Zhan, Z. Hu, K. Meng, J. Zhao, and J. Chu, “Temperature dependent phonon Raman scattering of Heusler alloy Co 2Mn xFe1−xAl/GaAs films grown by molecular-beam epitaxy,” RSC Adv. 2, 9899 (2012). 22M. Zhai, S. Ye, Z. Xia, F. Liu, C. Qi, X. Shi, and G. Wang, “Local lattice distor- tion effect on the magnetic ordering of the Heusler alloy Co 2FeAl 05Si0.5film,” J. Supercond. Novel Magn. 27, 1861 (2014). 23P. Nehla, C. Ulrich, and R. S. Dhaka, “Investigation of the structural, electronic, transport and magnetic properties of Co 2FeGa Heusler alloy nanoparticles,” J. Alloys Compd. 776, 379 (2019). 24J. Kübler, A. R. William, and C. B. Sommers, “Fortnation and coupling of magnetic moments in Heusler alloys,” Phys. Rev. B 28, 1745 (1983). 25P. V. Midhunlal, J. A. Chelvaneb, D. Prabhu, R. Gopalan, and N. Harish Kumar, “Mn 2V0.5Co0.5Z (Z = Ga, Al) Heusler alloys: High T Ccompensated P-type ferri- magnetism in arc melted bulk and N-type ferrimagnetism in melt-spun ribbons,” J. Magn. Magn. Mater. 489, 165298 (2019). 26R. Stinshoff, A. K. Nayak, G. H. Fecher, B. Balke, S. Ouardi, Y. Skourski, T. Nakamura, and C. Felser, “Completely compensated ferrimagnetism and sub- lattice spin crossing in the half-metallic Heusler compound Mn 1.5FeV 0.5Al,” Phys. Rev. B 95, 060410 (2017). 27I. Galanakis and E. ¸ Sa¸ sıo ˘glu, “High-T Chalf metallic fully-compensated ferri- magnetic Heusler compounds,” Appl. Phys. Lett. 99, 052509 (2011). AIP Advances 10, 055118 (2020); doi: 10.1063/5.0009504 10, 055118-6 © Author(s) 2020

5.0007512.pdf

Appl. Phys. Lett. 116, 262103 (2020); https://doi.org/10.1063/5.0007512 116, 262103 © 2020 Author(s).Strain dependence of Auger recombination in 3μm GaInAsSb/GaSb type-I active regions Cite as: Appl. Phys. Lett. 116, 262103 (2020); https://doi.org/10.1063/5.0007512 Submitted: 13 March 2020 . Accepted: 16 June 2020 . Published Online: 30 June 2020 Kenneth J. Underwood , Andrew F. Briggs , Scott D. Sifferman , Varun B. Verma , Nicholas S. Sirica , Rohit P. Prasankumar , Sae Woo Nam , Kevin L. Silverman , Seth R. Bank , and Juliet T. Gopinath ARTICLES YOU MAY BE INTERESTED IN Surface control and MBE growth diagram for homoepitaxy on single-crystal AlN substrates Applied Physics Letters 116, 262102 (2020); https://doi.org/10.1063/5.0010813 Direct evidence of hydrogen interaction with carbon: C–H complex in semi-insulating GaN Applied Physics Letters 116, 262101 (2020); https://doi.org/10.1063/5.0010757 Defect properties of Sb 2Se3 thin film solar cells and bulk crystals Applied Physics Letters 116, 261101 (2020); https://doi.org/10.1063/5.0012697Strain dependence of Auger recombination in 3lm GaInAsSb/GaSb type-I active regions Cite as: Appl. Phys. Lett. 116, 262103 (2020); doi: 10.1063/5.0007512 Submitted: 13 March 2020 .Accepted: 16 June 2020 . Published Online: 30 June 2020 Kenneth J. Underwood,1,a) Andrew F. Briggs,2 Scott D. Sifferman,2 Varun B. Verma,3Nicholas S. Sirica,4 Rohit P. Prasankumar,4 Sae Woo Nam,3Kevin L. Silverman,3Seth R. Bank,2 and Juliet T. Gopinath1,5 AFFILIATIONS 1Department of Physics, University of Colorado Boulder, Boulder, Colorado 80309, USA 2Department of Electrical and Computer Engineering, University of Texas Austin, Austin, Texas 78758, USA 3National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305, USA 4Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 5Department of Electrical, Computer, and Energy Engineering, University of Colorado Boulder, Boulder, Colorado 80309, USA a)Author to whom correspondence should be addressed: kenneth.underwood@colorado.edu ABSTRACT We differentiate the effect of strain induced by lattice-mismatched growth from strain induced by mechanical deformation on cubic nonra- diative Auger recombination in narrow-gap GaInAsSb/GaSb quantum well (QW) heterostructures. The typical reduction in the Auger coeffi- cient observed with lattice-mismatched growth appears to be due to the concomitant compositional change rather than the addition ofstrain, with implications for mid-IR semiconductor laser design. We induced a range of internal compressive strain in five samples from/C00.90% to /C02.07% by varying the composition during the growth and mechanically induced a similar range of internal strain in analogous quantum well membrane samples. We performed time-resolved photoluminescence and differential reflectivity measurements to extract the carrier recombination dynamics, taken at 300 K with carrier densities from 2 :7/C210 18cm/C03to 1:4/C21019cm/C03. We observed no change with strain in the cubic Auger coefficient of samples that were strained mechanically, but we did observe a trend with strain in samples thatwere strained by the QW alloy composition. Measured Auger coefficients ranged from 3 :0/C210 /C029cm6s/C01to 3:0/C210/C028cm6s/C01. Published under license by AIP Publishing. https://doi.org/10.1063/5.0007512 The mid-infrared (mid-IR) spectral region from 2 to 5 lmi so f significant technological interest due to applications in trace gas sens- ing (CH 4,C O 2,S O 2, etc.),1,2disease recognition and treatment,3free space communications,4and defense.5Semiconductor lasers, such as intersubband quantum cascade lasers (QCLs), type-II interband cas-cade lasers (ICLs), and type-I diode lasers, can generate mid-IR light at low cost with high wall-plug efficiency. 6Type-I diode lasers in the mid-IR, typically composed of compressively strained GaInAsSbquantum wells (QWs) with barriers of GaSb or lattice-matchedAlGaAsSb or AlGaInAsSb, have achieved comparable performance toQCLs and ICLs below 3 lm. 7Performance beyond 3 lm with such lasers has been plagued by issues with carrier capture, free-carrier absorption, and Auger recombination, with as much as 80% of excited carriers at threshold lost to Auger recombination.8Reducing such nonradiative recombination is critical for achieving high efficiencies.Tailoring the alloy composition of the barriers in such devices allowsresearchers to increase hole capture and can improve device perfor- mance, but such adjustments tend to also increase free carrierabsorption. 9Using GaSb barriers avoids this issue, but the reduced hole confinement still results in poor lasing efficiency.10Due to the type-I geometry, increasing the quantum well Sb percentage canincrease hole confinement without requiring complex barrier compo-sitions. 6This increased Sb percentage also increases the compressive strain, which should decrease Auger losses, as compressive strain breaks the heavy-hole–light-hole (HH–LH) degeneracy and lowers theeffective HH mass. This lower mass decreases the hole quasi-Fermilevel at threshold, reducing the carrier density required for gain and proportionately reducing losses from Auger. 11,12Improvements in laser performance have been observed in such highly strained mid-IRGaInAsSb/GaSb devices. 13 The reduced effective HH mass not only will decrease the carrier density at threshold but could also decrease the Auger coefficient itself, with a strong impact on the CHHS Auger process [one conductionband (CB) state, two HH states, and one spin–orbit (SO) state], whichdominates for near-IR semiconductors. 14Recent work strongly sug- gests that CCCH is the driving factor for parasitic Auger loss in Appl. Phys. Lett. 116, 262103 (2020); doi: 10.1063/5.0007512 116, 262103-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplGaInAsSb/GaSb mid-IR devices,15,16which could alter the impact of mechanical strain on the Auger coefficient for such narrow-gap GaSb- based devices. In this paper, we differentiate between the effect changes in strain due to the alloy composition and changes in strain due toexternal stress have on Auger recombination in narrow-gap, highly strained GaInAsSb/GaSb active regions. We accomplish this using time-resolved carrier decay measurements of epitaxially strain-varied samples and mechanically strain-varied samples. We observe a notable performance difference between the two strain-adjustment techniques. The heterostructures under test [shown in Fig. 1(b) ] consisted of a stack of four 10-nm GaInAsSb wells surrounded by 20-nm GaSb barriers, sandwiched between /C24200-nm GaSb spacing layers, emitting with peak photoluminescence (PL) at about 3 lm. Each sample was coated with a Ti/Au reflective backing layer. In the first set of samples, chips with varying As concentrations in the QWs were grown, induc- ing internal compressive strain ranging from /C00.90% to /C02.07%. These chips were bonded to rigid Mo heat sinks before selectively etch- ing away the GaSb substrate and the lattice-matched InAsSb etch release layer (see the supplementary material ). In the second set of samples, we grew chips with the same layer structure with an As con- centration in the QWs that induced about /C02% internal compressive strain, then bonded the chips to a flexible film, and etched away the GaSb substrate. The final sample was a 250 nm thick QW region with reflective backing, which could flex and stretch with the membranewithout disadherence or cracking. In testing, we biaxially stretched the flexible membrane, reducing the internal compressive strain in the QWs by mechanically applying external tensile strain. We performed time-resolved photoluminescence (TRPL) and differential reflection pump probe (PP) measurements of the above samples in a range of applied strains. We conducted TRPL studies with direct detection in the mid-infrared and performed differential reflection PP measurements on the epitaxially strain-varied samples. In both sample sets, we performed measurements in a range of excitedcarrier densities and simultaneously monitored the PL emission spec- trum vs time, pump fluence, and applied strain. For the stretched sam- ples, the peak emission wavelength increases with applied strain asshown in Fig. 1(a) , providing a useful metric for calibrating the biaxial stretcher to the internal strain of the sample. Given the likelihood of anisotropic stretching, the peak PL wavelength was monitored for each stretching position and calibrated to the internal strain using eight-band k/C1psimulations. As the stretching process is destructive for the membrane samples, we mounted and etched pieces of the same wafer on several different membranes, each of slightly different sizes and etch quality. Taken as a set, the samples approximate the behavior of the QW material under mechanical stress. TRPL measurements were taken using a mid-infrared PL spec- troscopy system, with a commercial mode-locked Ti:sapphire source emitting /C241Wo f <100 fs pulses at 800 nm with an 80 MHz repetition rate used for carrier excitation. The laser was focused onto the samples using a 15 mm focal length off-axis gold-coated parabolic mirror and pulse-picked down by 10 /C2to decrease the average power below the onset of melting/chip damage, which allows complete carrier relaxa- tion between successive pulses, enabling us to measure a time window /C24100 ns wide after the excitation pulse. A fast TeO 2acousto-optic modulator (AOM) was used for pulse picking and provided /C2450 dB of extraneous pulse suppression outside of the /C2410 ns electronic pass window. PL from 1 to 4 lm was collected by the off-axis parabolic mir- ror and filtered to remove the reflected pump light. The PL was then fiber coupled with ZBLAN patch cable into a He sorption fridge kept at <1 K containing a WSi superconducting nanowire single-photon detector (SNSPD), with short time resolution ( <200 ps). Further detec- tor performance is described in Ref. 18. A characteristic TRPL mea- surement of an epitaxially strained sample is shown in Fig. 2(a) .W efi t these decay curves with a cubic recombination rate differential equation solved using the Runge–Kutta method, which we then convolved with the instrument response to obtain our decaying peak function. In the PP measurements, /C24200 fs pulses from a Yb fiber laser at 1.04lm excited electron–hole pairs in samples and /C24140 fs pulses from a synchronously pumped optical parametric oscillator probed the reflectivity at the peak of the PL, /C242.9–3.0 lm. Representative data from the pump-probe measurements are shown in Fig. 2(b) ,a l o n g with a fit with a degenerate cubic carrier recombination rate equation using the same coefficients as for the TRPL. In order to interpret data such as shown in Fig. 2 ,w em u s tc o n - vert these signals into excited carrier densities. The PL intensity is expressed as a function of carrier density JPL¼CBðniþDnÞ ðpiþDpÞ,w i t h niandpibeing the intrinsic electron and hole densi- ties,DnandDpthe excess electron and hole densities, Bthe radiative coefficient, and Cthe geometric collection factor.19From eight-band k:pcalculations, it is found that intrinsic carrier populations should be well below excess carrier levels (assuming minimal doping in wells and barriers), and so niþDn/C25Dn;w ea s s u m e Dn¼Dp,w h i c hi sr e a - sonable for primarily band–band generation and recombination.20 Therefore, the PL equation becomes JPL¼CBn2. Normalizing the PL intensity to the t¼0 value eliminates the collection factor and radia- tive term, reducing the PL intensity to solely a function of the carrier density, JPL;norm¼n2=n02, with peak excited carrier density n0 ¼FR 2Ephottð1/C0e/C0aTÞ,w i t h Fthe pulse fluence, athe absorption, Rthe reflectivity, Ephotthe pump photon energy, Tthe GaSb spacer thick- ness, and tthe combined well width.21 In the case of the PP measurements, conversion from PP signal DR=Rto carrier density requires knowledge of the instantaneous car- rier density for a given DR=R, which is only available at time t¼0. FIG. 1. (a) Peak photoluminescence wavelength with applied strain compared to the simulated shift in the bandgap expected for the given applied strain. The bandgap shift is calculated using eight-band k/C1psimulations.17(b) Sample layer structure. The Ti/Au reflector layer is bonded to either a flexible membrane or inflex-ible heat sink. The region below the dashed line is etched away prior to themeasurement.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 262103 (2020); doi: 10.1063/5.0007512 116, 262103-2 Published under license by AIP PublishingWe use the same equation as above to find n0and then fit a phenome- nological performance function fto the carrier density-reflectivity curve, as shown in the inset of Fig. 2(b) .W ec a nu s et h e n¼fðDR=RÞ relationship to transform DR=Rinto a time-dependent normalized carrier density,19directly comparable to the TRPL result. With both measurements in terms of excited carrier density, we can fit both the TRPL and PP results to a recombination rate, d dtn n0/C18/C19 ¼An n0þBn0n2 n02þCn02n3 n03; (1) with Athe Shockley–Read–Hall (SRH) coefficient, Bthe radiative coef- ficient, and Cthe effective Auger coefficient. Precise measurements of the cubic contributions to recombina- tion decay can be achieved if the Auger lifetime approaches the SRHlifetime ( Cn 2/C24A). However, given the short SRH lifetime and low Auger coefficient of these heterostructures (as well as the narrow bandgap), we needed to excite our samples with high carrier densities,>10/C2beyond the degenerate carrier density limit, above carrier con- centrations typical of diode laser thresholds. 22For nondegenerateexcitation densities, Auger processes are well described by Boltzmann statistics, but exciting as high as we have necessitates treating theAuger processes using Fermi statistics. 22,23Measurements with excited carrier densities higher than calculated diode laser thresholds werecombined with measurements at lower excited carrier densities,obtaining a continuous convergence curve as the carrier concentra-tions excited the degenerate regime. Note that ABC coefficients fromTRPL data were extracted in a time window where carrier density wassignificantly reduced from the degenerate peak (beginning /C241.5 ns after zero delay) in part due to the finite impulse response time of theSNSPD. This reduced the excited carrier density for ABC fitting byapproximately 11 dB, into the regime of typical narrow-gap GaSb- based laser operation ( n/C2410 12cm/C02in the well24). In the nondegenerate case, the Auger coefficient is presumed to be independent of carrier density. However, in the degenerate case, theAuger coefficient is itself nonlinear with carrier density and contrib- utes to the linear, quadratic, and cubic recombination rates. Thus, the Auger recombination rate is given by CðnÞn X.25TheXterm can range from 1 to 3, and so the Auger term can be approximated by CðnÞ ¼C1nþC2n2þC3n3. Thus, the net recombination rate becomes d dtn n0/C18/C19 ¼ðAþC1Þn n0þðBþC2Þn0n2 n02þC3n02n3 n03:(2) In this case, the linear and quadratic Auger rates cannot be unambigu- ously distinguished from the SRH and radiative terms, respectively, but the cubic Auger C3term is separable. It should be noted that while the cubic contribution to the recombination rate is unambiguouslydiscriminable from SRH and radiative contributions, there could stillbe cubic contributions to the decay from other effects, notably carrierleakage, which can play a significant role in devices designed to operateabove 3 lm and could contribute cubically to the decay. 26,27Such effects are not discriminated here although efforts to distinguishbetween effective cubic Auger recombination (including carrier leak-age) and pure Auger recombination through temperature-dependentmeasurements are still ongoing. While the linear and quadratic decay coefficients are independent of carrier density, the cubic Auger term is not. C 3values for different carrier densities fit well with a convergence equation,28 C3ðn0Þ¼C0 3 1þðn0=nCÞ; (3) allowing us to extract a low carrier density cubic Auger rate C0 3for each sample, along with SRH and radiative contributions. A fit of thedegenerate cubic Auger coefficients vs n 0to convergence equation (3) for a representative sample measured with TRPL is shown in Fig. 3 . For each epitaxially strain-varied sample, we fit with this rate equationfor both pump-probe and TRPL data to confirm performance. The values of the nondegenerate-carrier-density cubic Auger C 0 3, effective radiative, and effective SRH coefficients are shown in Fig. 4 . We observe that there is an obvious minimum in the Auger coefficientfor samples grown with epitaxially varying strain [ Fig. 4(a) ]. However, there does not appear to be any significant trend in the Auger coeffi-cient measured for samples with mechanically varying strain [ Fig. 4(d) ], a n dr e s p o n s ev se x t e r n a ls t r e s si sq u i t efl a ta c r o s sar a n g eo fi n t e r n a lstrains/peak emission wavelengths. When a QW sample is grownwith/C01.69% internal strain, the Auger coefficient is /C240.42/C260.06 FIG. 2. Representative time-resolved measurements of an epitaxially strain-varied sample, by time-resolved (a) photoluminescence and (b) differential reflectivity pump-probe (PP). The inset in (b) shows the phenomenological fit between the PP peak value and known instantaneous peak carrier density n0. This functional depen- dence is used for converting the PP signal to an excited carrier density, necessaryfor fitting Auger recombination equations. Satellite peaks in (a) are due to imperfect AOM pulse-picking.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 262103 (2020); doi: 10.1063/5.0007512 116, 262103-3 Published under license by AIP Publishingthat of a sample grown with /C00.9% strain and /C240.27/C260.05 that of one grown with /C02.07% strain. When a membrane sample is stretched to /C01.69%, the average Auger coefficient is 1.12 /C260.14 that of a sample stretched to /C00.9% strain and 1.06 /C260.15 that of the one stretched to /C02.02% strain, with effectively no change. Note that there are minor structural differences between the epitaxiallystrain-varied samples and mechanically strain-varied samples, wherethe membrane samples only had one GaSb spacer region, on thegold mirror side of the QWs. This slight difference in the barrierregions will introduce error when comparing one sample set withanother, which however, should not impact the trend observed ineach set. Discussions of these structural differences are included inthesupplementary material . The radiative coefficient measured for both sets of samples and in both experimental systems is consistently low, /C2410 /C011/C010/C012 cm3/s [Figs. 4(b) and4(e)], in agreement with values reported for other sources for similar strained QW systems.29The SRH coefficient <108 1/s for both sample sets suggests good growth quality, and the consis- tency vs applied strain in the membrane suggests that the material sys- tem is not damaged on the scale of the diffusion length throughout thestretching process. A notable effect of using binary GaSb barriers rather than AlGaAsSb or AlGaInAsSb barriers is a reduced valence band offset, even at pronounced compressive strain. This could increase the contri-bution of hole leakage to the cubic recombination term, particularlyfor small compressive strain at large As percentage in the well(>20%). Calculations based on the work of Chuang 30(see the supple- mentary material ) suggest that the reduced hole confinement com- mensurate with increased As percentage is also present for increasedexternal strain and, thus, should contribute to the cubic coefficientsextracted from the membrane samples as well. Further work calculat-ing the impact of the valence band offset and carrier effective masseson cubic recombination in such devices is ongoing. These measurements indicate that the reduction in the Auger coefficient of a narrow-gap strained heterostructure is predominatelydue to the alloy compositional change in the well, rather than due tomechanical stress. The results advance the notion that primarilymechanical descriptions of Auger strain tuning are insufficient, partic- ularly for narrow band GaSb-based devices and large amounts of internal strain. Reduced hole confinement may also play a significantrole in these measurements, but does not appear to have a direct rela-tionship given the difference in the performance of externally andinternally applied strain. Changing the relative spacing of the SO, LH, HH, and conduction bands by alloy concentration plays a more signif- icant role in reducing the Auger coefficient than reducing the HHmass for such systems, which has potent implications for future mid-IR device designs. See the supplementary material for details on sample growth, membranes, the membrane-stretcher, and the PP system. We acknowledge helpful technical support from Lange Simmons and Dr. Thinh Bui, as well as the support of the National Science Foundation (Grant Nos. DMR 1508783 and 1508603); the FIG. 3. Cubic Auger coefficients of a representative sample measured in a range of nominal excited carrier densities, all above the degenerate limit. The coefficientsare fit to Eq. (3), from which we extract a low-carrier-density cubic Auger coefficient C 0 3for each sample. FIG. 4. Recombination coefficients for the epitaxially strained samples, (a) Auger, (b) radiative, and (c) SRH and for a mechanically strained membrane sample, (d) Auger, (e) radiative, and (f) SRH. The membrane results are plotted against thepredicted internal strain found using the peak PL emission wavelength as inFig. 1(a) , where increasing mechanical strain would decrease the magnitude of the internal strain, and so the right-hand side of the plots corresponds to the highest degree of stretching. Epitaxially strained samples are plotted against calculatedstrain for their alloy percentages, with atomic spacing confirmed by high-resolutionX-ray diffraction (HRXRD). The internal strain of the mechanically strained samples is the sum of the built-in strain from epitaxial growth and the strain from mechanical deformation (calculated as DL=L), where increasing mechanical strain decreases the magnitude of the internal strain.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 262103 (2020); doi: 10.1063/5.0007512 116, 262103-4 Published under license by AIP PublishingCenter for Integrated Nanotechnologies (CINT) (No. 2018BC0091); the Defense Advanced Research Projects Agency (DARPA) (No.W911NF-15-1-0621); and the Air Force Office of ScientificResearch (AFOSR) (No. FA9550-15-1-0506). DATA AVAILABILITY The data that support the findings of this study are available within this article. REFERENCES 1D. Popa and F. Udrea, Sensors 19, 2076 (2019). 2P. Werle and A. Popov, Appl. Opt. 38, 1494–1501 (1999). 3R. Waynant, I. Ilev, and I. Gannot, P h i l o s .T r a n s .R .S o c .B 359, 635 (2001). 4Y. Su, W. Wang, X. Hu, H. Hu, X. 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Olesberg, and T. F. Boggess, Phys. Rev. B 59, 5745–5750 (1999). 29B. V. Olson, E. Kadlec, J. K. Kim, J. F. Klem, S. D. Hawkins, E. A. Shaner, and M. Flatt /C19e,Phys. Rev. Appl. 3, 044010 (2015). 30S. L. Chuang, Physics of Photonic Devices (John Wiley & Sons, 2009).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 262103 (2020); doi: 10.1063/5.0007512 116, 262103-5 Published under license by AIP Publishing

10.0000905.pdf

Rolling along a square path: The dynamics of biased balls Michael S. Wheatland , Rodney C. Cross , Andrew Ly , Michael Sacks , and Karl Smith Citation: American Journal of Physics 88, 465 (2020); doi: 10.1119/10.0000905 View online: https://doi.org/10.1119/10.0000905 View Table of Contents: https://aapt.scitation.org/toc/ajp/88/6 Published by the American Association of Physics TeachersRolling along a square path: The dynamics of biased balls Michael S. Wheatland,a)Rodney C. Cross,b)Andrew Ly,Michael Sacks, and Karl Smith School of Physics, The University of Sydney, New South Wales 2006, Australia (Received 11 December 2019; accepted 15 February 2020) A biased ball rolled and spun on a horizontal surface exhibits interesting dynamics. We investigate the motion of a truncated billiard ball, via experiments, analytical methods, and numericalsolutions of the equations of motion for a biased sphere rolling without slipping. Solutions are identified where the center of mass moves in a circular or a square path, and we investigate other quasi-periodic motions of the ball. VC2020 American Association of Physics Teachers . https://doi.org/10.1119/10.0000905 I. INTRODUCTION Biased balls rolled and spun on a horizontal surface exhibit interesting and sometimes curious, unexpected dynamics. For example, a biased ball can roll along a circu- lar path, but it can also roll along a square or triangular path.The torque produced by an offset center of mass causes abiased ball to depart from a straight line when rolling. Afamiliar example is provided by a lawn bowl, which gener- ally rolls in a leisurely arc, but may also wobble as it rolls. 1–3 In general, the motion involves rotations about three axes: a spin about the symmetry axis of the bowl; a slow precession(the rotation of the symmetry axis about the vertical as theball curves); and a nutation (the wobble of the axis up and down). The bowl rolls without slipping, and so the point of contact with the ground is instantaneously at rest. A second example of the motion of a biased rolling ball is provided by hurricane balls, which consist of two bearingballs joined together. 4When the balls are spun rapidly on a surface, they quickly achieve a steady state in which one ballis rolling without slipping, the center of mass is at rest, and the second ball is lifted off the surface. In this case, the cen- ter of mass of the system is the point at which the balls arejoined, and so it is at the edge of the lower, rolling ball. Thecenter of mass of the lower ball, which can be considered abiased rolling ball, moves in a circle. The dynamics of biased balls have previously also been analysed theoretically and experimentally. Studies have looked at tippe tops, spheroids, 5and a small ball connected to a larger ball.6In this paper, we will focus on a truncated sphere: a sphere with a part removed by a taking a planarslice through the sphere. The motion of a rolling truncated sphere has previously been investigated theoretically, under the restrictive assumption of zero spin about a vertical axis7 (a “rubber body” model). In this paper, we provide a moregeneral description of the motion of a rolling truncatedsphere. The present investigation was motivated in part by some interesting observations concerning the dynamics of a metal ring spun about a vertical axis. Several authors have reported a surprising effect, where the center of mass rotates initiallyin a prograde sense about a remote vertical axis and thenchanges direction to rotate in a retrograde sense before itcomes to a stop. 8,9The authors provided numerical solutions but were not able to explain the effect in simple terms. In the present paper, we investigate experimentally the motion of a biased ball under conditions where the ballrotates about its axis of symmetry and also rotates about avertical axis through its center of mass, G. Rotation aboutthe symmetry axis is generated as a result of rolling motion, while an arbitrary rotation frequency about a vertical axiscan be imposed by spinning the ball in that manner, a tech- nique commonly employed in ten-pin bowling. 10,11In ten-pin bowling, the center of mass generally rotates at lowfrequency about a remote vertical axis in a prograde sense, with a large radius of curvature. That is, the direction of rota- tion about the remote axis is in the same direction as rotationabout the vertical axis through G. In our case, a billiard ball was biased by removing a portion of the ball, with the result that the ball curved in a retrograde sense. That is, the direc-tion of rotation about the remote axis was in the opposite direction to that about the vertical axis through G. The radius of curvature was relatively small since the ball was launchedat low speed. A wide range of trajectories can be generated by varying the spin about the vertical axis and also by varying the incli- nation of the axis of symmetry. A representative sample oftrajectories is presented below, including the case that the ball rolls along an almost square path. The trajectories were calculated numerically and are compared with the experi-mental results obtained with the truncated billiard ball. One particular mode is omitted from discussion in this paper since it has been described elsewhere and also because itrequires slipping. 12That is, if a truncated ball is spun at suffi- ciently high speed, it can completely invert like a tippe top. II. MODEL Here, we present a model for a rolling axisymmetric biased ball, following an approach used to describe a lawn bowl.1,2Figure 1illustrates the ball geometry. A set of mov- ing axes Gngfis chosen so that Gis the center of mass of the ball, gis aligned with the axis of symmetry of the ball, and the axis nis always horizontal. In Fig. 1, the n-axis is directed into the page. The axes Gngfare aligned with and move with the ball, but are not fixed in the ball: The ball rotates about the g-axis as it rolls. The ball has radius a, and the center of mass is offset by a distance cfrom the center. The position and orientation of the ball in a frame O xyzfixed in the rolling surface is then defined by the position ( x,y,z) ofG, and the Euler angles h,v, and /, where his the angle between fand z,vis the angle between the xand n directions, and /is an azimuthal angle around g, measured from n. From Fig. 1, we can see that the angular velocity xof the ball is given by x¼/C0 _h^nþ _//C0_vsinh/C0/C1 ^gþ_vcosh^f: (1) 465 Am. J. Phys. 88(6), June 2020 http://aapt.org/ajp VC2020 American Association of Physics Teachers 465The axes Gngfcoincide with the principal axes of the ball, so the angular momentum of the ball is L¼Axn^nþBxg^gþAxf^f; ¼/C0A_h^nþB_//C0_vsinh/C0/C1 ^gþA_vcosh^f; (2) where Bis the moment of inertia for rotation about the sym- metry axis, and Ais the second principal moment of inertia. We can transform a vector with components in n,g, and f to a vector with components in x,y, and zby multiplying by the matrix K¼cosv/C0coshsinv/C0sinhsinv sinv coshcosv sinhcosv 0 /C0sinh cosh0 B@1 CA; (3) so that, e.g., the vector Kxhas components ðxx;xy;xzÞ, where x¼ðxn;xg;xfÞ. We assume that the ball rolls without slipping, which implies vþKx/C2rGP¼0; (4) where v¼ð_x;_y;_zÞis the center of mass velocity and rGP¼/C0c^g/C0a^z; ¼ccoshsinv;/C0ccoshcosv;/C0aþcsinh ðÞ (5) is the vector from the center of mass Gto the point Pin con- tact with the surface. Equation (4)is a kinematic relationship, which defines the motion of the ball’s center of mass in terms of the Eulerangles. From Eqs. (1)and(3)–(5) , it follows that _x¼a_//C0c_v/C0/C1 coshcosv/C0a/C0csinh ðÞ _hsinv; (6) _y¼a_//C0c_v/C0/C1 coshsinvþa/C0csinh ðÞ _hcosv; (7) _z¼/C0ccosh_h: (8) The last equation can be directly integrated to givez¼a/C0csinh; (9) using the condition z¼awhen h¼0. Again using the fact that the axes Gngfcoincide with the principal axes of the ball, the kinetic energy Tis given by T¼1 2Ax2 nþ1 2Bx2 gþ1 2Ax2 fþ1 2M_x2þ_y2þ_z2/C0/C1 ; (10) where Mis the ball’s mass. Using Eqs. (1)and(6)–(8) leads to T¼1 2A_h2þ_v2cos2h/C16/C17 þ1 2B_//C0_vsinh/C0/C12 þ1 2Ma _//C0c_v/C0/C12cos2hþa2/C02acsinhþc2 ðÞ _h2hi : (11) Also, the potential energy of the ball is V¼mg a /C0csinh ðÞ : (12) The equations of motion may be obtained using the Lagrangian method with the non-holonomic constraintsimplied by the rolling conditions Eqs. (6)–(8) . The details are given in Appendix A . The results are AþMa 2/C02acsinhþc2 ðÞ ½/C138 €h/C0Mac cosh_h2 /C0B/C0A ðÞ sinhþMc a /C0csinh ðÞ/C2/C3 cosh_v2 þBþMa a /C0csinh ðÞ ½/C138 cosh_/_v/C0Mgc cosh¼0; (13) AþMc2 ðÞ cos2hþBsin2h ½/C138 €v/C0BsinhþMac cos2h ðÞ €/ þ2B/C0A/C0Mc2 ðÞ sinhþMac/C2/C3 cosh_h_v /C0B/C0Mac sinh ðÞ cosh_h_/¼0; (14) /C0BsinhþMac cos2h ðÞ €vþBþMa2cos2h ðÞ €/ /C0B/C02Mac sinhþMa2 ðÞ cosh_h_v /C0Ma2sinhcosh_h_/¼0: (15) Equations (13)–(15) together with Eqs. (6)–(8) describe the motion of the ball. The force acting on the ball (for general motion) is F¼Mð€x;€y;€zÞ, which may be calculated from Eqs. (6)–(8) , €x¼a€//C0c€v/C0/C1 coshcosv/C0a_//C0c_v/C0/C1 /C2sinhcosv_hþcoshsinv_v/C0/C1 (16) þccoshsinv_hðÞ2/C0a/C0csinh ðÞ sinv€hþcosv_h_v/C0/C1 ; (17) €y¼a€//C0c€v/C0/C1 coshsinvþa_//C0c_v/C0/C1 /C2/C0 sinhsinv_hþcoshcosv_v/C0/C1 (18) /C0ccoshcosv_hðÞ2þa/C0csinh ðÞ cosv€h/C0sinv_h_v/C0/C1 ; (19) €z¼c½sinh_hðÞ2/C0cosh€h/C138: (20) Similarly, the net torque on the ball may be calculated asFig. 1. Geometry of the model for a rolling biased ball. We consider axes n, g, and fthrough the center of mass Gof the ball, which move with the ball. The axis gis aligned with the symmetry axis of the ball, and the axis n, which is directed into the page in the figure, remains horizontal. 466 Am. J. Phys., Vol. 88, No. 6, June 2020 Wheatland et al. 466s¼rGP/C2Fx^xþFy^yþN^z/C0/C1; (21) where N/C0Mg¼Fz. This gives sx¼/C0mgþ€z ðÞ ccoshcosv/C0m€y/C0aþcsinh ðÞ ;(22) sy¼m€x/C0aþcsinh ðÞ /C0mgþ€z ðÞ ccoshsinv; (23) sz¼mccosh€ysinvþ€xcosv ðÞ : (24) III. STEADY-STATE SOLUTIONS Steady-state solutions correspond to €h¼€v¼€/¼0: (25) If the ball rolls with a constant inclination of the axis to the vertical ( _h¼0), then we have h¼hc;v¼Xct;/¼xct; (26) where hc;Xc,a n d xcare constant and where we have chosen v¼/¼0 at time t¼0. The angular velocity Xc represents the rate at which the moving frame Gngf aligned with the ball rotates about a vertical axis throughthe ball’s center of mass, and x ci st h er a t ea tw h i c ht h eb a l l rotates about the g-axis as it rolls (see Fig. 1). With these choices, Eqs. (14) and(15) become trivial, and Eq. (13) reduces to /C0B/C0A ðÞ sinhcþMc a /C0csinhc ðÞ/C2/C3 X2 c þBþMa a /C0csinhc ðÞ ½/C138 xcXc/C0Mgc ¼0; (27) assuming cos hc6¼0. We can also rewrite this equation as xc¼MgcþB/C0A ðÞ sinhcþMc a /C0csinhc ðÞ/C2/C3 X2 c BþMa a /C0csinhc ðÞ ½/C138 Xc:(28) The rolling conditions [Eqs. (6)–(8) ] imply _x¼vccosXctðÞ ; (29) _y¼vcsinXctðÞ ; (30) _z¼0; (31) where vc¼axc/C0cXc ðÞ coshc: (32) Equations (29)–(31) may be integrated to give x/C0x0¼RsinXctðÞ ; (33) y/C0y0¼/C0RcosXctðÞ ; (34) z¼a/C0csinh; (35) where x0andy0are the integration constants and R¼vc=Xc: (36) Equations (33)–(34) describe uniform circular motion in x andycentered on ( x0,y0)w i t hr a d i u s R. The ball rolls in a circle with a constant inclination of the spin axis tothe vertical, with the sense of rotation determined by the sign of Xc. The spin frequency xcmay be eliminated between Eqs. (27)and(36)to give X2 c¼Mgac coshc BRþccoshc ðÞ þMaR a /C0csinhc ðÞ /C0aB/C0AðÞ sinhccoshc; (37) and then Eq. (36)implies xc¼Xc aR coshcþc/C18/C19 : (38) From Eqs. (22)–(24) , the torque is given by sz¼0 and sx¼sccosXctðÞ ;sy¼scsinXctðÞ ; (39) where sc¼mcoshc/C0gcþa/C0csinhc ðÞ axc/C0cXc ðÞ Xc/C2/C3 : (40) The torque is constant in magnitude, horizontal, and parallel or anti-parallel to the velocity of the ball. The torque induces precessional motion, with the result that the center of mass follows a circular path, the torque being equal to the rate of change of angular momentum. If the center of mass is at rest then R¼0. In that case, Eq. (36)implies axc¼cXc; (41) which is the no-slip condition for a stationary center of mass. In the game of lawn bowls, hcis usually zero, in which case Eqs. (37) and(38) simplify to give ðRþcÞXc¼axc and X2 c¼Mgac BRþc ðÞ þMa2R: (42) IV. NUMERICAL METHOD For numerical solution, the equations of motion are writ- ten in the form d u=dt¼f, where u¼x;y;h;v;/;_h;_v;_//C0/C1 (43) and f¼ _x;_y;_h;_v;_/;€h;€v;€//C0/C1 : (44) The equations are also non-dimensionalised by dividing lengths bya,t i m e sb yffiffiffiffiffiffiffiffi a=gp , and moments of inertia by Ma2.I nt h i s form, the equations are solved for the dependent variables uat a set of time steps using fourth order Runge–Kutta.13The acceleration terms (the second derivatives of the Euler angles) infare evaluated at each timestep using Eqs. (13)–(15) , includ- ing solving a linear system of equations at each time step. Further details are given in Appendix B . The code implementing the numerical solution requires initial conditions hðt¼0Þ¼h0;_hðt¼0Þ¼ _h0;_vðt¼0Þ ¼_v0, and _/ðt¼0Þ¼ _/0. The initial values of the angles v 467 Am. J. Phys., Vol. 88, No. 6, June 2020 Wheatland et al. 467and/can be assumed to be zero although a non-zero vcan be used to rotate the trajectory of the ball in the xy-plane. The time step is chosen so that it is initially less than2p=_h 0;2p=_v0, and 2 p=_/0, in the case that the initial angular velocities are non-zero. The code integrates for a chosentotal time, and it produces a visualisation of the motion ofthe ball. As a check on the numerical calculation, we evaluate the total energy E¼TþVusing Eqs. (11) and(12) and ensure that it is close to being conserved during the integration time. The forces and torques acting on the ball are also calculated, using Eqs. (16)–(19) and(22)–(24) , respectively. V. EXPERIMENTAL METHOD The top 12 mm of a two inch (50.8 mm) diameter billiard ball was removed to shift the center of mass by 2.9 mm fromthe geometric center of the ball, as shown in Fig. 2. The mass of the ball was thereby reduced to 98.4 g, givingA¼2:39/C210 /C05kg m2andB¼2:69/C210/C05kg m2. A thin wire probe was inserted in the center of the flat section sothat the center of mass could be located by extrapolatingthe measured coordinates of the top and bottom of the probe. When viewed from above, the apparent length of the probe was used to calculate the angle of inclination of the probe fromthe vertical, and the angular displacement of the probe in thehorizontal plane was used to calculate the precession frequencyX¼_vof the ball. The ball was launched by hand at low speed on an accu- rately horizontal and flat granite surface measuring 30 cm/C230 cm, and was filmed from above at 300 fps with a Casio EX-F1 camera mounted on a tripod. The video was analysedwith Tracker motion analysis software to measure the trajec-tory and speed of the center of mass in the horizontal plane,as well as the inclination angle, h, and the precession fre- quency, X. We did not attempt to measure the spin frequency x¼_/. Typical results are shown in four supplementary videos. 14 VI. EXPERIMENTAL AND NUMERICAL RESULTS A. Lawn bowl mode A relatively simple result was obtained by launching the ball at low speed with the axis of symmetry approximatelyhorizontal and without imparting any deliberate rotationabout a vertical axis. That is, the usual method of launching a ball in lawn bowls. The result, shown in Fig. 3, corresponds to the steady-state circular motion solution described in Sec. III. The center of mass follows a circular path of radius R¼0.073 m, completing one orbit in time T¼1.95 s at an average speed v¼0.235 m/s and at angular velocity jXj ¼2p=T¼3:22 rad/s. The average angle of inclination, h, was 24 /C14, giving a theoretical precession frequency (from Eq.(37))o fjXj¼3:23 rad/s. The circular motion solution provides a simple test for our code. In Fig. 4, we show the numerical integration of the equations of motion for initial values h0¼24/C14;_h0¼0; _v0¼/C03:23 rad/s, and with _/0/C25/C010:16 rad/s, evaluated using Eq. (28). The initial position is taken to be x¼0, y¼/C0R, with Rgiven in Eqs. (32) and(36) usingXc¼_v0. As expected, these initial conditions produce a steady state with the ball rolling clockwise in a circle centered on the ori- gin, and moreover, reproduce the experimental results shownin Fig. 3. Figure 4(a)shows the result of the numerical calcu- lation. A visualisation of the solution, which animates the rolling ball during the motion, is provided as supplementarymaterial. 14The red curve in Fig. 4(a)is the path of the center of mass, the blue curve is the path of the point at the center of the flat part of the truncated ball, and the black curvecorresponds to the path of the tip of the probe. Figure 4(b) shows the components of the net force on the ball (upper) and the components of the torque (lower). The static frictionforce and the torque are both constant, horizontal vectorswhich rotate with the ball, so their components show sine- cosine variation. The friction force provides the centripetal force for the circular motion of the center of mass, so it isalways radially inwards. The torque is in the direction of the instantaneous velocity, as expected from Eqs. (39)–(40) , and causes the continuous change in the angular momentumvector—which is directed radially outwards, for clockwise rolling—needed for the ball to roll in a circle. B. Approximately straight trajectories If the ball in Fig. 3is spun about a vertical axis when it is launched in a horizontal direction, then one might expect Fig. 2. Truncated billiard ball with probe.Fig. 3. Lawn bowl mode result for the truncated b illiard ball showing the tra- jectories of the tip and base of the probe and the extrapolated coordinates of the center of mass, G, in the horizontal plane. See Supplementary video 8853-3. 468 Am. J. Phys., Vol. 88, No. 6, June 2020 Wheatland et al. 468that it will follow a relatively straight line path on average. If the ball curves initially to the right then it will subsequently curve to the left when the ball rotates to face the oppositedirection, as indicated in Fig. 5. That type of behaviour was observed experimentally when the axis of symmetry washorizontal, although the axis did not remain perfectly hori-zontal throughout the motion. Instead, hincreased and decreased periodically, by about 10 /C14, with the result that the ball followed a curved path with a large radius of curvature. A typical experimental result is shown in Fig. 6. In this case, there are five precession cycles in 1.35 s, so the averageperiod is 0.27 s and the average precession frequency is X ¼2p=0:27¼23 rad/s. The angle of the spin axis was observed to vary between about 5 /C14below the horizontal, and 20/C14above. If the spin direction was reversed, the ball curved in the opposite direction. In both cases, the ball curved in aretrograde sense. To reproduce the approximately straight-line motion shown in Fig. 6with the code, we chose the initial conditions _v 0¼23 rad/s, h0¼/C05/C14, and _h0¼0. The initial spin fre- quency _/0needs to be guessed. With the choice _/0¼/C03rad/s we obtain the result shown as panel (a) in Fig. 7.W e have chosen v0¼190/C14to make the overall orientation approximately match Fig. 6. The numerical solution has h varying between /C05/C14and about 25/C14. Panel (b) of Fig. 7 shows the components of the net force on the center of mass of the ball, and the components of the torque about the center Fig. 4. Numerical solution for the lawn bowl mode. Panel (a) shows the tra- jectory (inner circle: center of mass; middle circle: center of flat section of ball; outer circle: tip of probe), and panel (b) shows the time variation of the net force and torque on the ball. Fig. 5. Assumed path of the truncated ball when it is rotating about a vertical axis, viewed from above.Fig. 6. Approximately straight trajectory showing the paths of the tip andbase of the probe and the extrapolated coordinates of the center of mass, G, in the horizontal plane. See Supplementary video 8899-1. Fig. 7. Numerical solution for nearly straight-line motion. Panel (a) showsthe trajectory, and panel (b) shows the time variations of the net force and torque on the ball. 469 Am. J. Phys., Vol. 88, No. 6, June 2020 Wheatland et al. 469of mass. A basic question concerning the motion shown in Fig.6is whether the ball is rolling without slipping. The numerical solution can provide insight. Panel (b) of Fig. 7indicates that the frictional forces in the numerical solution are less than about 0 :05Mgin magnitude. This is much smaller than the expected maximum static friction force for the billiard ball on the granite surface, which sug- gests that the ball is always rolling. Also, panel (b) of Fig. 7 shows that the torque vector is rotating in the x-yplane as theball precesses. The symmetry of the variation of the horizon- tal torque accounts for the nearly linear motion of the ball. C. Approximately circular trajectories If the initial value of his significantly less than 90/C14, say about 40/C14, then the large radius of curvature trajectory in Fig. 6changes to a small radius of curvature trajectory, as shown in Fig. 8and in supplementary video 8855-1. Large Fig. 8. Two different trajectories (a) and (d) showing the paths of the tip and base of the probe and the extrapolated coordinates of the center of mass, G, in the horizontal plane. The measured variations of handXfor trajectory (a) are shown in (b) and (c). The corresponding graphs of handXfor trajectory (d) are shown in (e) and (f). 470 Am. J. Phys., Vol. 88, No. 6, June 2020 Wheatland et al. 470variations in hare observed during each precession cycle, as well as large variations in X. The center of mass rotates in an approximately circular path, in a retrograde sense. However, the local radius of curvature of the path followed by G changes during each precession cycle, in a manner similar to that shown in Fig. 5, depending on whether the flat face of the ball is facing away from or towards the center of the cir- cular path. To reproduce this motion with the code, we can choose initial conditions based on the plots of handXversus time in Fig. 8. For example, to attempt to reproduce the case shown in Figs. 8(a)–8(c) , we choose h0¼20/C14, and _v0¼10 rad/s. After some numerical experimentation, we find that _/0¼/C02:9 rad/s gives the motion shown in panel (a) of Fig. 9, for an integration time of 3.6 s. The time variation of _v andh(as well as _hand _/) is shown in panel (b). There is a reasonable match to the data shown in Fig. 8. For the case shown in Figs. 8(d)–8(f) , we choose h0¼40/C14,_h0¼0, and _v0¼5 rad/s. Again, we need to guess the spin frequency, and we find that _/0¼/C07:5 rad/s gives the motion shown in panel (c) of Fig. 9, for an integration time of 2.95 s. The variation of the angular speeds _v;_h, and _/, as well as of the angle his shown in panel (d) of Fig. 9. Once again there is good correspondence with the data in Fig. 8. Some insight into the approximately circular center-of- mass motion shown in Fig. 9follows from an inspection of the torque. Figure 10compares the variation of the torque inthe lawn bowl mode shown in Fig. 4(a)(top panel) with the variation of the torque in the nearly circular case shown inFig. 9(c) (lower panel). The total time for each plot corre- sponds to one period of the circular/approximately circularmotion. In the lawn bowls case, the torque is horizontal, in the direction of motion, and constant in magnitude [Eqs. (39) and (40)]. The torque causes the change in angular Fig. 9. Numerical solutions with approximately circular trajectories. Panels (a) and (c) show the two trajectories, and panels (b) and (d) show the co rresponding time variations of the angular speeds and of h. Fig. 10. Variation of torque with time in the lawn bowl mode shown in Fig. 4(a) (top panel) and in the nearly circular trajectory shown in Fig. 9(c) (lower panel). 471 Am. J. Phys., Vol. 88, No. 6, June 2020 Wheatland et al. 471momentum needed for the ball to roll in a circle. In the nearly circular motion case, the average variation in the tor- que has the same pattern, but superimposed on this average variation are oscillations corresponding to the precession ofthe rolling ball. D. Low Xtrajectories: Rolling along a square Even though the paths of the center of mass are approxi- mately circular in Fig. 8, results at low Xcan have distinctly non-circular paths, as demonstrated in Fig. 11. In particular, Fig.11(a) presents the unexpected result that, for certain ini- tial choices of Xandh, it is possible for the biased sphere to roll along a nominally square path! During the motion h varies from about 0 /C14to about 90/C14during each precession cycle, with the extreme value achieved at the corners of the square. A supplementary video makes this clear. Figure11(b) also shows that, for other initial conditions on the motion, the trajectory of the center of mass can also be approximately triangular. During these motions, the loopsfollowed by the tip of the probe can be relatively large [as they are in Fig. 8(a)] or simple cusp points [as in Fig. 8(b)]. Numerical solutions which approximately match these results are shown in Fig. 12. Although the precession fre- quency is low on average over the motion, the spin frequen-cies _vand _/achieve large values at the corners of the shapes, and the angle hchanges rapidly at these locations. The initial conditions producing these trajectories are givenin the caption to the figure. VII. CONCLUSION A biased ball that rolls without slipping on a horizontal surface can do so in a wide variety of ways. The simplestinvolves steady state motion in a circular path, where the axis of symmetry remains approximately horizontal, corre- sponding to a lawn bowl mode. The motion can be describedanalytically and can be understood intuitively from the fact that the rate of change of angular momentum is equal to the torque acting about the center of mass.Fig. 11. Trajectories in the horizontal plane, observed at relatively low values of X, which demonstrate rolling around a square, and rolling around a triangle. Data points for the probe tip are shown at intervals of 0.04 s. See Supplementary video 9466-7. Fig. 12. (a) Rolling around a square, and (b) rolling around a triangle. The initial conditions are: (a) h0¼/C05/C14,_h0¼0;_v0¼4:05 rad/s, and _/0¼/C03 rad/s; (b)h0¼/C010/C14,_h0¼0;_v0¼2:3 rad/s, and _/0¼/C02:5 rad/s. 472 Am. J. Phys., Vol. 88, No. 6, June 2020 Wheatland et al. 472If the lawn bowl mode is modified by imposing spin about a vertical axis through the center of mass, then the result is also intuitive. That is, the ball curves alternatively to the left and then to the right due to the alternating torque. As a result,the center of mass follows a path that is relatively straight, but it curves slightly in a retrograde sense. If the axis of symmetry is not horizontal, then motion of the ball is more complex and so are the relevant equations of motion. In general, the axis of symmetry oscillates up and down in a vertical plane while simultaneously precessingabout a vertical axis through the center of mass. Furthermore, the center of mass follows a path that is approximately circu- lar if the average precession frequency is large, but which canbe approximately triangular or square or many-sided if the average precession frequency is relatively low. The center of mass rotates in a retrograde sense, opposite to the direction ofrotation of the precessing ball. The average precession fre- quency is easily controlled experimentally, simply by spin- ning the ball about a vertical axis at any desired speed. Similarly, the initial angle of inclination of the axis of sym- metry is easily controlled, but the subsequent angle is deter-mined by the precession frequency and the rolling condition. The ability of the ball to roll along a square path is a curi- ous result, which should appeal to all students of physics. More generally, the motion of a rolling biased ball presents awonderful topic for student projects—and in fact, this article originated in a semester-long undergraduate project. On the theory side, the description of the motion introduces studentsto Euler angles, and the derivation of the equations of motion using the Lagrangian approach (or directly using forces and torques) is a problem that is challenging but accessible to junior undergraduates. The experimental investigation of the motion tests laboratory skills, has scope for creativity, andallows students to directly connect the theory with the unex- pected behaviour of the ball. ACKNOWLEDGMENT The authors acknowledge discussions with Y. Shimomura. APPENDIX A: DERIVATION OF THE EQUATIONS OF MOTION We use a Lagrangian approach to derive the equations of motion, following Brearley. 2However, our derivation is exact, whereas Brearley made approximations relevant for a lawn bowl. Equations (6)–(8) represent constraint relations for the motion. Variations in the coordinates are related by dx/C0acoshcosvd / þccoshcosvd v þa/C0csinh ðÞ sinvd h¼0; (A1) dy/C0acoshsinvd/þccoshsinvd v /C0a/C0csinh ðÞ cosvd h¼0; (A2) dzþccoshd h¼0; (A3) which represent a set of constraints X6 r¼1Arjdqrðj¼1;2;3Þ; (A4) where the qrdenote the six coordinates x,y,z,h,v, and /.The Lagrange equations may be written2,15as d dt@T @_qr/C18/C19 /C0@T @qr/C0QrþX3 j¼1kjArj¼0; where r¼x;y;z;h;v;/: (A5) The term Qrdenotes the generalised force corresponding to coordinate r, andP3 j¼1kjArjrepresents the forces associated with the constraints. The kjare Lagrange multipliers, and the Arjare defined in Eq. (A4). The constraint forces do no work: The only generalised force (which does work) is Qh¼/C0@V @h¼Mgcosh: (A6) The six Lagrange equations together with the three con- straint equations represent nine equations in the nineunknowns (the generalized coordinates plus the Lagrange multipliers). The Lagrange equations are M€xþk 1¼0; (A7) M€yþk2¼0; (A8) M€zþk3¼0; (A9) A€hþAsinhcosh_v2þB_//C0_vsinh/C0/C1 cosh_v/C0Mgccosh þa/C0csinh ðÞ k1sinv/C0k2cosv ðÞ þ k3ccosh¼0; (A10) Acos2hþBsin2h ðÞ €v/C0Bsinh€/þ2B/C0A ðÞ sinhcosh_h_v /C0Bcosh_h_/þk1cosvþk2sinv ðÞ ccosh¼0;(A11) /C0Bsin€vþB€//C0Bcosh_h_v /C0acoshk 1cosvþk2sinv ðÞ ¼0: (A12) Eliminating k1,k2, and k3using Eqs. (A1)–(A3) gives the three equations describing the evolution of the Euler angles, namely, Eqs. (13)–(15) . APPENDIX B: FORM OF THE EQUATIONS OF MOTION FOR NUMERICAL SOLUTION To solve the equations of motion numerically, we require the angular accelerations €h;€v, and €/. The accelerations are evaluated from the other dependent variables using Eqs. (13)–(15) in the form €h¼cosh AþMa2/C02acsinhþc2 ðÞ /C2Mac _h2þB/C0A ðÞ sinhþMca0½/C138 _v2n /C0BþMaa0ðÞ _/_vþMgco ; (B1) where a0¼a/C0csinh, together with Cg¼h; (B2) where 473 Am. J. Phys., Vol. 88, No. 6, June 2020 Wheatland et al. 473C¼AþMc2 ðÞ cos2hþBsin2h/C0Bsinh/C0Maccos2h /C0Bsinh/C0Maccos2h BþMa2cos2h ! ; (B3) g¼€v €/; ! ; (B4) and h¼cosh/C02B/C0A/C0Mc2 ðÞ sinhþMac ½/C138 _h_v þB/C0Macsinh ðÞ _h_/ B/C02MacsinhþMa2 ðÞ _h_v þMa2sinh_h_/0 BBBB@1 CCCCA:(B5) The linear system of Eqs. (B2)–(B5) is solved at each time step. As mentioned in Sec. IV, all equations are implemented in code in a non-dimensional form, with lengths divided by a,t i m e s divided byffiffiffiffiffiffiffiffi a=gp , and moments of inertia divided by Ma 2. When h¼90/C14, the matrix system Eqs. (B2)–(B5) is singu- lar. In this configuration, the axes gandzused to describe _/ and _vare parallel, so the chosen axes and angles cannot instantaneously describe three-dimensional rotation. In prac-tice, this is not a significant problem: All of the solutionsdepicted in this article avoid h¼90 /C14. a)Electronic mail: michael.wheatland@sydney.edu.au b)Electronic mail: rodney.cross@sydney.edu.au 1M. N. Brearley and B. A. Bolt, “The dynamics of a bowl,” Q. J. Mech. Appl. Math. 11, 351–363 (1958).2M. N. Brearley, “The motion of a biased bowl with perturbing projection conditions,” Math. Proc. Cambridge Philos. Soc. 57, 131–151 (1961). 3Rod Cross, “The trajectory of a ball in lawn bowls,” Am. J. Phys. 66, 735–738 (1998). 4David P. Jackson, David Mertens, and Brett J. Pearson, “Hurricane balls:A rigid-body-motion project for undergraduates,” Am. J. Phys. 83, 959–968 (2015). 5H. K. Moffatt, Y. Shimomura, and M. Branicki, “Dynamics of an axi-symmetric body spinning on a horizontal surface. I. Stability and thegyroscopic approximation,” Proc. R. Soc. London A 460, 3643–3672 (2004). 6Rod Cross, “Spin experiments with a biased ball,” Eur. J. Phys. 40, 055003 (2019). 7A. Kilin and E. Pivovarova, “The rolling motion of a truncated ball with-out slipping and spinning on a plane,” Regul. Chaotic Dyn. 22, 298–317 (2017). 8M. A. Jalali, M. S. Sarebangholi, and M.-R. Alam, “Terminal retrogradeturn of rolling rings,” Phys. Rev. E 92, 032913 (2015). 9A. V. Borisov, A. A. Kilin, and Y. L. Karavaev, “Retrograde motion of a rolling disk,” Phys. Usp. 60, 931–934 (2017). 10Cliff Frohlich, “What makes bowling balls hook?,” Am. J. Phys. 72, 1170–1177 (2004). 11Kevin King, N. C. Perkins, Hugh Churchill, Ryan McGinnis, Ryan Doss, and Ron Hickland, “Bowling ball dynamics revealed by miniature wirelessMEMS inertial measurement unit,” Sports Eng. 13, 95–104 (2011). 12Rod Cross, “A hemispherical tippe top,” Eur. J. Phys. 41, 025001 (2020). 13W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling,Numerical Recipes in C , 2nd ed. (Cambridge U. P., New York, 1992). 14See supplementary material at http://dx.doi.org/10.1119/10.0000905 for movies of the motion of the truncated billiard ball filmed at 300 fps. Thevisualisations produced by the code, as well as the supplementary videos,are at <http://www.physics.usyd.edu.au/ /C24wheat/biased-ball/ >. 15E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies , 2nd ed. (Cambridge U. P., Cambridge, UK, 1917. 474 Am. J. Phys., Vol. 88, No. 6, June 2020 Wheatland et al. 474

5.0001808.pdf

AIP Conference Proceedings 2220 , 090020 (2020); https://doi.org/10.1063/5.0001808 2220 , 090020 © 2020 Author(s).Effect of SHI irradiation on electronic structure and electrical transport in LaCoO3 thin film Cite as: AIP Conference Proceedings 2220 , 090020 (2020); https://doi.org/10.1063/5.0001808 Published Online: 05 May 2020 Ashok Kumar , Vinod Kumar , Dinesh Shukla , Rajesh Kumar , Ram Janay Choudhary , and Ravi Kumar ARTICLES YOU MAY BE INTERESTED IN Influence of transition metal doping on the structural and transport properties of LaCoO 3 cobaltite AIP Conference Proceedings 2220 , 040011 (2020); https://doi.org/10.1063/5.0002710Effect of SHI Irradiation on Electronic Structure and Electrical Transport in LaCoO 3 Thin Film Ashok Kumar1), Vinod Kumar2, a), Dinesh Shukla3), and Rajesh Kumar1), Ram Janay Choudhary3) and Ravi Kumar4) 1Department of Physics, National Institute of Technology, Hamirpur (HP), India. 2Department of Physics, BCET Gurdaspur, (Punjab), India. 3UGC CSR -DAE Indore (MP), India. 4Department of Material Science & Engineering, National Institute of Technology, H amirpur (HP), India. a)Corresponding author: kumarvinodphy@gmail.com Abstract. In this report, we have investigated the effect of SHI on the electronic structure and electrical transport properties of strained L aCoO 3 (LCO) thin films deposited on SrTiO 3 (STO) substrate. XANES study revealed changes in electronic distribution and transition metal (TM) 3d -O2p ion hybridization in SHI irradiated thin film. Both pristine, as well as irradiated LCO thin film, have been found to have semiconducting behavior in the whole temperature range. Further, SHI irradiation observed to increase the electrical resistivity of thin -film, a possible cause for this has been discussed in the light of XANES result Keywords: Strongly correlated electron systems, hybridization, Swift Heavy Ion, etc. INTRODUCTION Strongly correlated electron systems (SCESs) include a class of materials that show unusual electronic and magnetic properties i.e. high temperature (T c) superconductivity, col ossal magneto -resistance, metal -insulator transitions, etc. Strongly correlated electrons bear multiple degrees of freedom from the charge, orbital and spin sector. Strong interactions inside these sectors and complex interplay between the charge, spin and orbital degrees of freedom give rise to a rich variety of different electronic and magnetic properties that can be controlled by utilizing cooperative response of strongly correlated electrons to external parameters such as electric field, magnetic fields , temperature and pressure. Among SCESs, rare earth cobaltites are studied extensively due to the presence of an extra degree of freedom i.e. different spin states of cobalt ions. Cobalt ions can exist in low spin (LS), intermediate spin (IS) and high spin (HS). Due to these uncommon spin states, the spin crossover arises in cobalt ions as a result of variation in parameters like temperature, external or internal pressure and doping of hole or electron [ 1-6]. The magnetic properties of LCO, in bulk and thin -film form have been studied extensively. Bulk, LCO has non -magnetic (with L S) behaviour at low temperature, while recent studies reveal that biaxial tensile strain stabilizes the insulating ferromagnetic (FM) ground state in LCO thin film [ 7-10]. Bulk as well as thin -films of LCO exhibit semiconducting behavior with temperature [ 11]. SHI irradiation provides an opportunity to amend the properties of materials in a precise manner on the microscopic scale. SHI irradiations produce numerous kinds of defects such as point defect and columnar amorphization which in turn can change the material properties [ 12-15]. The improved hyb ridization strength of O 2p and Ni 3d due to SHI irradiation has also has been observed to change the insulating feature of pristine LaNiO 3 film into the metallic character [ 16]. In this report, we have investigated the effect of SHI irradiation on the electronic structure and the transport properties of LCO thin film. 3rd International Conference on Condensed Matter and Applied Physics (ICC-2019) AIP Conf. Proc. 2220, 090020-1–090020-4; https://doi.org/10.1063/5.0001808 Published by AIP Publishing. 978-0-7354-1976-6/$30.00090020-1EXPERIMENTAL DETAIL Bulk LCO synthesized, followed by XRD characterization for phase purity of the m aterial has been performed. LCO epitaxial films were grown using the PLD technique on single -crystal STO (100) substrate at an oxygen pressure of 300 m torr with the energy of 220 mJ. The substrate temperature was kept at 700 ℃. After deposition, the films were cooled to room temperature at an oxygen pressure of 400 torr and at the rate of 30 ℃ per minute. For irradiation and further study, the film was cut into equal parts to maintain the growth conditions uniform for all the films. One part of the film was retained as pristine and another was irradiated with 200 MeV Ag+15 ion (SHI) beam using 15UD tandem accelerator (at Inter -University Accelerator Center, New Delhi) with a fluence of 5x1011 ions/cm2. The pristine, as well as irradiated LCO, were characteri zed by powder x -ray diffraction (XRD) using Cu Kα radiation at UGC -DAE CSR Indore. Soft x -ray absorption spectroscopy (XAS) measurements were carried out at BL -01 beam -line of Indus - 2 synchrotron radiation source (at RRCAT Indore) to probe the films electr onic structure [ 17]. The electrical resistivity (dc) measurements were carried in the temperature range from 100 to 300 K by four -probe method at UGC -DAE CSR LAB at Indore. RESULTS AND DISCUSSI ON X-ray absorption near -edge structure (XANES) studies of L 3,2 and O K -edge gives the information of delicate change in the electronic state for the material, like oxidation state, spin state. Normalized L 3,2 of Co edge in XANES spectra for pristine and irradiated LCO film is shown in Error! Reference source not found .. In the Co L 3 edge spectra of pristine as well as irradiated LCO films show dominant peaks around 779.5 eV [ 18]. No pronounced shoulder before the main peaks implies that only Co+3 ions exist in our pristine and irradiated LCO films i.e. Co is present in +3 charge state. FIGURE 1. Normalized spectra for Co -L3,2 edge of pristine and irradiated LCO thin films irradiated with a dose of 5×1011 ions/cm2 at 300 K temperature. The norma lized O K -edge XANES spectra for pristine and irradiated LCO film has shown in Figure 2 (a). The features are seen in between 526.5 and 531.5 eV (highlighted by the green background) in O K -edge XANES spectra appear due to the hybridization between unoccup ied O 2p and Co 3d orbitals. The enlarged view of the region between 526.5 and 531.5 eV is shown in Error! Reference source not found. (b). In Error! Reference source not found. (b), peak A appears at point 528 eV, which has been assigned to the transitio n of an electron from O 1s to O 2p-Co3d unoccupied hybridized t 2g states, while the transit ion of electron from O 1s to O 2p-Co3d unoccupied hybridized e g states have been assigned to the peak at point B around 531 eV. In O K -edge spectrum of the irradiated LCO film, it can be clearly seen that after irradiation with 200 MeV Ag+15 (SHI) ions, the intensity at point A has redu ced, while at B it has increased. These changes in intensity at points A & B indicate that the occupancy at t 2g states has increased whereas at e g states it has reduced. This may be due to the changes in stress/strain by SHI irradiation which leads to the electronic redistribution between t 2g & e g states. Such types of electronic redistribution may lead to the changes in electrical properties also. 090020-2 FIGURE 2 . Normalized O K -edge XANES spectra for pristine and irradiated LCO film with a dose of 5×1011 ions/ cm2. (b) Enlarged view of O K -edge XANES spectra for pristine and irradiated LCO film for energy range 526 to 535 eV. FIGURE 3 . Resistivity versus temperature plot of pristine LCO film (a) and irradiated LCO film samples. To observe such changes we measured the electrical resistivity of pristine and irradiated films as a function of temperature, which is presented in Error! Reference source not found. . Both pristi ne, as well as irradiated LCO films, exhibit semiconducting behavior in the measured temperature range. From the figure, we can observe that resistivity of the film has enhanced after irradiating with a fluence of 5 x 1011 ions/cm2. This increase in resist ivity of the irradiated film can be understood in the light of XANES results as discussed above. As it is known, that for most of these perovskites, t 2g states are localized while the conduction band is formed by the e g states [ 20-22]. Further, the bandwidth is depending on the extent of hybridization/overlapping of 3d -O2p orbit als, which is substantially higher for the e g derived band than that of t 2g [22, 22]. So in the present case, the e g and t 2g electronic redistribution suggest the reduction of hybridization/overlapping of Co3d -O2p states, consequently leads to the reduction in bandwidth and hence observed increase in resistivity. So SHI irradiatio n observed to tailor the electrical transport properties of LCO thin films by bringing about changes in electronic structure & 3d -O2p hybridization. CONCLUSION In this report, we have studied the effect of Ag+15 ions (SHI) on electronic structure and elect rical transport of LCO thin film. LCO thin film was deposited on (STO) single crystalline substrate using PLD. SHI irradiation -induced changes in stress/strain observed to affect the electronic structure and Co3d-O2p hybridization in the thin film. Both pristine and irradiated LCO thin films show the semiconducting behavior in the measured temperature range. Further, 090020-3SHI irradiated thin -film has been found to have increased in resistivity in the whole temperature range, which has been attributed to the chan ges in electronic structure and hybridization upon irradiation. ACKNOWLEDGMENTS We thanks to D. M. Phase and M. Gupta for providing XAS characterization facility at RRCAT Indore (M.P.) , India. We also thank Fouran Singh and director IUAC New Delhi, India for allowing the use of their facilities. REFERENCES 1. P. M. Raccah and J. B. Goodenough, Phys. Rev. 155(3), (1967 ). 2. A. Podlesnyak, S. Streule, J. Mesot, M. Medarde, E. Pomjakushina, K. Conder, A. T anaka, M. W. Haverkort, and D. I. Khomskii, Physical Review Letters 97, 247208 (2006). 3. M. A. Korotin, S. Yu. Ezhov, I. V. Solovyev, and V. I. Anisimov, Phys. Rev. 54(8), 15 August 1996 -II. 4. J.S. Zhou, J.Q. Yan, and J. B. Goodenough, Phys. Rev. B 71, 220103 (R) (2005). 5. D. M. Sherman, Advances in Physical Geochemistry, edited by S. K. Saxena Springer -Verlag, Berlin, (1988). 6. P. G. Radaelli and S. W. Cheong, Phys. Rev. B 66, 094408 (2002). 7. D. Fuchs, E. Arac, C. Pinta, S. Schuppler, R. Schneider, an d H. V. Lohneysen, Physical Review B 77, 014434 (2008). 8. D. Fuchs, C. Pinta, T. Schwarz, P. Schweiss, P. Nagel, S. Schuppler, R. Schneider, M. Merz, G. Roth, and H. v. Lohneysen, Physical Review B 75, 144402 (2007). 9. J. W. Freeland, J. X. Ma, and J. Shi, Appl. Phys. Lett. 93, 212501 (2008). 10. A. Herklotz, A. D. Rata, L. Schultz, and K. Dorr, Physical Review. B 79, 092409 (2009). 11. Y. Li, S. J. Peng, D. J. Wang, K. M. Wu, and S. H. Wang, AIP Advances 8, 056317 (2018). 12. R. N. Parmar, J. H. Markna, D. G. Kuberkar, Ra vi Kumar, D. S. Rana, Vivas C. Bagve, and S. K. Malik, Applied Physics Letters 89, 202506 (2006). 13. M.S. Sahasrabudhe, D. N. Bankar, A.G. Banpurkar, S.I. Patil, K.P. Adhi, Ravi Kumar, Nuclear Instruments, and Methods in Physics Research B 263, 407–413 (2007) . 14. D. K. Shukla,1, Ravi Kumar, S. Mollah, R. J. Choudhary, P. Thakur, S. K. Sharma, N. B. Brookes, and M. Knobel, Physical Review B 82, 174432 (2010). 15. P. Kaur, K. K. Sharma, R. Pandit, R.J.Choudhary, Ravi Kumar, Journal of Magnetism and Magnetic Materials 398, 220–229 (2016) . 16. Y. Kumar, A. Pratap Singh, S.K. Sharma, R.J. Choudhary, P. Thakur, M. Kn obel, N.B. Brookes, Ravi Kumar, Thin Solid Films 619, 144–147 (2016) . 17. D.M. Phase, M. Gupta, S. Potdar, L. Behera, R. Sah, A. Gupta, Development of soft X -ray polarized light beamline on Indus -2 synchrotron radiation source, AIP Conf. Proc. 1591 685–686 (2014) . 18. M. Abbate and J. C. Fuggle, Phys. Rev. B 47(24), (1993). 19. M. Merz, P. Nagel, C. Pinta, A. Samartsev, H. V. Lohneysen, M. Wissinger, S. Uebe, A. Assmann, D. Fuchs, and S. Schuppler, Physical Review B 82, 174416 (2010). 20. T. Wolfram and S. Ellialrioglu, Electronic and Optical Properties of D -Band Perovskites , Cambridge University Press. 21. . B. Goodenough and P. M. Raccah , Journal of Applied Physics 36, 1031 (1965). 22. M. Topsakal , C. Leighton , and R. M. Wentzcovitch , Journal of Applied Physics 119, 244310 (2016). 23. J. Suntivich, W. T. Hong, Y. L. Lee, J. M. Rondinelli, W. Yang, J. B. Goodenough, B. Dabrowski, J. W. Freeland, and Y. S. Horn, J. Phys. Chem. C 118, 1856 −1863 (2014 ). 090020-4

1.5145070.pdf

J. Appl. Phys. 128, 035902 (2020); https://doi.org/10.1063/1.5145070 128, 035902 © 2020 Author(s).Structural, elastic, vibrational, thermophysical properties and pressure- induced phase transitions of ThN2, Th2N3, and Th3N4: An ab initio investigation Cite as: J. Appl. Phys. 128, 035902 (2020); https://doi.org/10.1063/1.5145070 Submitted: 16 January 2020 . Accepted: 04 July 2020 . Published Online: 17 July 2020 B. D. Sahoo , K. D. Joshi , and T. C. Kaushik ARTICLES YOU MAY BE INTERESTED IN On the elastic anisotropy of the entropy-stabilized oxide (Mg, Co, Ni, Cu, Zn)O compound Journal of Applied Physics 128, 015101 (2020); https://doi.org/10.1063/5.0011352Structural, elastic, vibrational, thermophysical properties and pressure-induced phase transitions of ThN 2,T h 2N3, and Th 3N4:A nab initio investigation Cite as: J. Appl. Phys. 128, 035902 (2020); doi: 10.1063/1.5145070 View Online Export Citation CrossMar k Submitted: 16 January 2020 · Accepted: 4 July 2020 · Published Online: 17 July 2020 B. D. Sahoo,1,a) K. D. Joshi,1,2and T. C. Kaushik1,2 AFFILIATIONS 1Applied Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, India 2Homi Bhabha National Institute, Mumbai 400094, India a)Author to whom correspondence should be addressed: bdsahoo@barc.gov.in ABSTRACT The structural, electronic, elastic, lattice dynamical properties and pressure-induced phase transitions in ThN 2,T h 2N3, and Th 3N4have been investigated through density functional theory based electronic band structure calculations. Our theoretical calculations on ThN 2 reveal the monoclinic structure (C2/m space group) at 0 GPa instead of the previously reported cubic (Fm /C223m spatial crystal symmetry) phase [K. O. Obodo and N. Chetty, J. Nucl. Mater. 440, 229 (2013)]. More refined calculations on enthalpy of formation reveal that this ground state C2/m phase of ThN 2transforms to an orthorhombic structure (Pnma symmetry) at a pressure of ∼7G P a . I n a g r e e m e n t with experimental observations, we predict the La 2O3-type trigonal structure (P /C223ml symmetry) in Th 2N3at ambient conditions, which is further predicted to transform to an initial monoclinic structure again at ∼62 GPa. Our theoretical results also agree with the experi- ment regarding the rhombohedral structure (R /C223m symmetry) of Th 3N4revealed at 0 GPa, which, at ∼37 GPa, is predicted to transform to an another rhombohedral structure with reduced space group symmetry of R /C223. The predicted structural phases are further substanti- ated with the mechanical and dynamical stability criteria in the pressure regime of their structural stability. Furthermore, the electronicband structure calculations at zero pressure suggest that with limited density of states above Fermi energy, ThN 2and Th 2N3exhibit semi-metallic characteristics, whereas a bandgap of ∼1.44 eV in Th 3N4makes it a semiconductor. The semiconducting nature of Th 3N4 ceases at a transition pressure of ∼62 GPa. Published under license by AIP Publishing. https://doi.org/10.1063/1.5145070 I. INTRODUCTION The actinide nitrides have attracted the attention of the scien- tific community for the last few decades.1–18These materials display interesting physics owing to the unique behavior of 5f electrons of the actinide counterpart arising due to strong electron –electron correlations.13–15Apart from this, these materials, e.g., nitrides of thorium, find applications in nuclear industry as fuel in fourth gen- eration reactors due to good thermo-physical properties including high melting points, high metal density, high thermal conductivity,low creep rate, good corrosion resistance, etc. 1–6,9–11It is also well known that the fuel cycle based on thorium has the advantage ofproduction of the small amount of plutonium and other long lived minor actinides as compared to the uranium fuel cycle.Recent calculations of Zhou et al. 14predicted the phase dia- grams of the experimentally known stoichiometries of UN 2and U2N3. At zero temperature and zero pressure, they found that the experimentally observed CaF 2-type UN 2transforms into another new I4 1/amd-type UN 2, and this transformation is related to dynamical instability originated from the Peierls mechanism. Alsotwo new stable high pressure phases of UN 2and U 2N3have been predicted by these authors. To our knowledge, among actinide nitrides, the phase diagram of thorium nitrides at low temperature and high pressure is rarely explored. Until now, ThN, Th 2N3, and Th 3N4are the only three ordered compounds that have been experimentally reported toexist. 11–14Moreover, ThN has been investigated through firstJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 035902 (2020); doi: 10.1063/1.5145070 128, 035902-1 Published under license by AIP Publishing.principle calculations for ground state properties and pressure-induced phase transition.4,15,18–20Th2N3and Th 3N4have been investigated for crystal structure, electronic properties, andelastic properties at ambient conditions; 6however, their response under high pressure still remains unexplored. Furthermore, in a theo-retical study, Obodo and Chetty 5predicted the existence of ThN 2in the CaF 2-type (space group Fm /C223m) crystal structure at zero pressure. In view of the importance of thorium based nitrides in nuclear industry, we have carried out detailed first principle basedelectronic band structure calculations on three of these nitrides(ThN 2,T h 2N3, and Th 3N4) to determine the ground state proper- ties and to analyze structural stability under high pressure up to 100 GPa. Apart from this, we have examined the elastic and latticedynamical stabilities of these compounds. These systems arealready reported to be non-magnetic in previous studies; 5,6hence, present calculations are performed for the non-magnetic case only without spin –orbit coupling. Also we find that the density func- tional theory (DFT) + U calculations do not describe the ambientstructure properly, e.g., the equilibrium volume and the bulkmodulus of the ambient structure do not compare as well with theexperiment as with simple GGA. A similar finding is reported in other theoretical works for ThN, ThN 2,T h 2N3and Th 3N4.4–6 Unlike that for the other light actinide members, in Th, the f elec- tron population is significantly small but plays a decisive role instabilizing the fcc phase at ambient condition. This small f- population does not significantly alter the results if treated within GGA or GGA + U. II. THEORETICAL METHOD The structural prediction technique utilized by us is based on the global minimization of free energy surfaces using an evolution- ary structure search algorithm as implemented in the USPEXcode. 21–23The search of stable crystal structures for ThN 2,T h 2N3, and Th 3N4has been carried out at 0, 20, 50, and 100 GPa with sim- ulation cells containing up to four formula units. The search of the lowest enthalpy structure at these pressures has been carried outusing a pseudopotential with the valence electron configuration of6s 26p66d15f17s2for Th and 2s22p3for N. The necessary structural relaxation at each pressure has been performed using the Vienna Ab initio Simulation (VASP)24–26package, with the all electron projector augmented wave (PAW) method within the framework ofdensity functional theory. 27,28The exchange correlation potential has been treated within a generalized gradient approximation(GGA) of Perdew –Burke –Ernzerhof (PBE) type. 29The first genera- tion of USPEX search started with an initial population of 60 ran- domly generated structures. The enthalpy for each structure hasbeen calculated by the VASP code and then used by USPEX toselect the “best ”structures of the generation (lowest enthalpy per atom). After the first generation, USPEX applied a number of mutation operations to these “best ”structures in order to produce 60 new structures that represent the next generation. Standard per-centages of 50, 10, 20, 10, and 10, respectively, have been used formutation operations carried out by USPEX for hereditary, random, soft-mutation, lattice mutation, and transmutation operators. The production of best structures from previous generations and theuse of mutational operators in them ensured the lowest enthalpy structure throughout the search process. After exploring the least enthalpy structures at a few pressures using USPEX, calculations with fine steps of hydrostatic compres-sions have been carried out on these plausible phases. A kineticenergy cutoff of 800 eV has been chosen to expand the wave func- tion into plane waves to ensure that the results are consistent with the energy convergence criteria of 10 −6eV. The Monkhorst –Pack k-grid spacing mesh of 2 π× 0.03 Å−1has been selected to ensure that the total energy calculations are within the tolerance of betterthan 1 meV/atom. 30 Since mechanical stability criteria should be satisfied for a stable structure, calculations of elastic moduli have, thus, beencarried out on stable structures of ThN 2,T h 2N3, and Th 3N4in dif- ferent pressure regimes. The stress –strain approach available in the VASP code has been used for this purpose.31,32Apart from elastic stability, the lattice dynamical stability also needs to be satisfied for a structure to be stable. For this purpose, phonon spectra have beencalculated through lattice dynamical simulations performed usingthe Phonopy code 33that is based on a small displacement method with a supercell approach. The supercell size for ThN 2used in the present calculation is 2 × 2 × 2 for the cubic structure (Fm /C223m), 2 × 2 × 4 for the monoclinic structure (C2/m), and 2 × 2 × 3 for theorthorhombic (Pnma) structure. For Th 2N3, the cell size was 3 × 3 × 3 for both P /C223ml and C2/m phases, whereas for Th 3N4,i tw a s 3×3×1 f o r R /C223 m a n d 1×1×2 f o r R /C223 phases. For determination of the forces due to small atomic displacements, a 4 × 4 × 4 Monkhorst – Pack mesh30has been used to sample the Brillouin zone (BZ). III. RESULTS AND DISCUSSIONS As stated above, we have carried out extensive structural inves- tigations on ThN 2,T h 2N3, and Th 3N4systems for selected pres- sures of 0 GPa, 20 GPa, 50 GPa, 80 GPa, and 100 GPa at 0 K usingthe evolutionary structure search algorithm method implemented in the USPEX code. 21–24The evolution of the structure for ThN 2at different pressures implied that the monoclinic phase having spacegroup C2/m is stable at 0 GPa, whereas at higher pressures, theorthorhombic structure having space group Pnma appears as thelowest enthalpy phase. For Th 2N3, the experimentally observed La2O3-type trigonal structure with space group P /C223ml exhibits the lowest enthalpy at 0 GPa, 20 GPa, and 50 GPa; however, at higherpressures of 80 GPa and 100 GPa, a monoclinic phase having spacegroup C2/m emerges as the lowest enthalpy structure. In the caseof Th 3N4, the evolutionary method reproduces the experimentally observed rhombohedral structure with space group R /C223m as the stable phase at 0 GPa and 20 GPa, whereas at elevated pressures of50 GPa, 80 GPa, and 100 GPa, another rhombohedral type structurehaving space group R /C223 appears as the lowest enthalpy structure. To find the exact transition pressure for these phases, the total energy is calculated for several hydrostatic compressions in refined simula- tions. Figure 1 presents the unit cells of phases found to be stable at zero pressure and at high pressure in ThN 2,T h 2N3, and Th 3N4, respectively, from present calculations. Tables I andIIlist the opti- mized lattice parameters and internal free parameters, respectively, for these phases.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 035902 (2020); doi: 10.1063/1.5145070 128, 035902-2 Published under license by AIP Publishing.The enthalpy of formation of these three compounds has also been calculated employing the relation ΔH(Th xN1/C0X)¼H(Th xN1/C0X)/C0[x/C2H(Th) þ(1–x)/C2H(N2)/2], (1) where x is the concentration of Th, ΔHis the enthalpy of formation per atom, and His the calculated enthalpy per stoichiometric unit for each compound. For obtaining the enthalpy of formation ofThN 2, the energy of all the stable phases of Th, N 2,T h 2N3, and Th3N4have been taken into consideration at each pressure. Figure 2 depicts the enthalpy of formation of ThN 2with respect to its constituents Th and N 2and also with respect to other stoichiometriesTh 2N3and Th 3N4. As seen in the figure, at 0 GPa, though ThN 2is stable with respect to Th + N 2and ThN+1/2N 2, decomposition is observed to occure in Th 2N3and Th 3N4. This is the reason why ThN 2does not exist in the experimental phase diagram of the Th –N system at ambient conditions. However, at 3.5 GPa and above, the C2/m phase of ThN 2becomes stable with respect to all its constituents as well as for Th 2N3and Th 3N4.A s shown in Table III , the enthalpy of formation for the Th –N-based compounds is negative, and it is lowest for Th 3N4. Figure 3 depicts the enthalpy relative to that of the C2/m phase at various hydrostatic pre ssures for the competing phases in ThN 2, derived from present evolution simulations as well as those predicted by other studies in similar systems such as the Fm/C223m phase in ThN 2,5,6the Pnma phase, the C2/c phase, and the I 4/mmm phase in ThC 2.34At 0 GPa, the C2/m phase is ener- getically favorable over the previously predicted5CaF 2-type cubic phase (space group Fm-3 m). It may also be noticed from the figure that the C2/m phase has the lowest enthalpy up to 7 GPa, and above this pressure, it is the Pnma phase that becomes lowerin enthalpy than the C2/m phase causing a phase transition at7 GPa. This is the first theoretical prediction on the existence of ThN 2and occurrence of pressure-induced phase transition in this material. Figure 4 displays the enthalpies of various predicted struc- tures in ThN 2,T h 2N3,a n dT h 3N4as a function of pressure upto 100 GPa. In ThN 2, as mentioned above, the C2/m structure transforms to the Pnma phase that remains stable upto 100 GPa, the maximum pressure explored in the present work. For Th 2N3, the ambient La 2O3-type trigonal phase transforms to a new monoclinic phase (space group C2/m, two formula units percell) around ∼62 GPa. In Th 3N4, however, the experimentally reported R /C223m (three formula units per cell) rhombohedral phase undergoes a transition to another similar type phase at ∼37 GPa. The new rhombohedral phase has space group symmetry R /C223with six formula units per cell. It may also be noticed that the sym-metry of the ground state structure enhances as one goes from ThN 2to Th 3N4, indicating that the increase in thorium to nitro- gen ratio in these compounds leads to a more symmetric crystalstructure at zero pressure. Recently, similar high pressure phasehas been predicted for UN 2and U 2N3using first principle calculations.14 InFig. 5 , the 0 K isothermal pressure as a function of com- pression has been plotted for the three compounds. It may benoticed from the figure that for all the three compounds, thereexists a volume discontinuity at the corresponding phase transition pressure and found to be 10.96%, 10.32%, and 6.45%, respectively, for ThN 2,Th2N3, and Th 3N4. The discontinuous volume change at transition pressure in these compounds indicates that the corre-sponding phase transitions are of first order nature. Furthermore, the elastic stability of ThN 2,T h 2N3, and Th 3N4 has been analyzed at different pressures by calculating their elasticmoduli. The values of elastic constants determined at zero pressurefor ThN 2,T h 2N3, and Th 3N4are listed in Table IV along with FIG. 1. The unit cells of optimized structures for ThN 2,T h 2N3, and Th 3N4.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 035902 (2020); doi: 10.1063/1.5145070 128, 035902-3 Published under license by AIP Publishing.those reported in other works.5,6To analyze the elastic stability of phases stable at different pressure regimes, the following elasticstability conditions 35have been examined: For cubic (Fm /C223m) structure: c11.0, c 44.0, c 11.c12,( c 11þ2c12).0: (2)For Trigonal (P /C223m1) phase: c44.0, c 11.c12,( c 11þ2c12)c33.2c2 13: (3) For Rhombohedral (R /C223m) phase: c44.0, c 11.c12,( c 11þ2c12)c33.2c2 13,c44c66.2c2 14:(4)TABLE I. The calculated and experimental lattice constants for ThN 2,T h 2N3, and Th 3N4. Compound Space group Pressure (GPa)a (Å) b (Å) c (Å) β(deg) Theory Exp. Theory Exp. Theory Exp. Theory Exp. ThN 2 C2/m 0 6.909 3.990 6.904 96.93 C2/c 0 6.375 4.472 6.234 102.39 I4/mmm 0 3.572 3.572 6.252 90.00 Fm/C223m 0 5.680 5.680 5.680 90.00 5.66055.66055.660590.00 Pnma 10 7.923 4.943 3.948 90.00 Th2N3 P/C223ml 0 3.921 3.87513,143.921 3.87513,146.135 6.17513,1490.00 90.00 3.91363.91366.161690.00 C2/m 80 3.976 4.325 6.329 98.207 Th3N4 R/C223m 0 3.882 3.87313,143.882 3.87313,1427.476 27.38513,1490.00 90.00 3.88263.882627.452690.00 R/C223 60 10.618 10.618 5.176 90.00 TABLE II. The theoretically calculated crystallographic position for ThN 2,T h 2N3, and Th 3N4. Compound Space group Pressure (GPa) Element Wyckoff positions x y z ThN 2 C2/m 0 Th 4i 0.1957 0.0000 0.2838 N1 4i 0.1513 0.0000 0.6341N2 4i 0.4378 0.0000 0.9407 C2/c 0 Th 4i 0.0000 0.2051 0.2500 N 8f 0.3003 0.1302 0.0524 I4/mmm 0 Th 2a 0.0000 0.0000 0.0000 N 4e 0.0000 0.0000 0.6261 Fm/C223m 0 Th 4b 0.5000 0.5000 0.5000 N 8c 0.2500 0.2500 0.2500 Pnma 10 Th 4c 0.8819 0.2500 0.7243 N1 4c 0.5318 0.2500 0.2805 N2 4c 0.1722 0.2500 0.3954 Th 2N3 P/C223ml 0 Th1 2d 0.3333 0.6666 0.2416 N1 2d 0.3333 0.6666 0.6430 N3 1a 0.0000 0.0000 0.0000 C2/m 80 Th 4i 0.9527 0.0000 0.7328 N1 2d 0.0000 0.5000 0.5000N2 4i 0.4027 0.0000 0.9188 Th 3N4 R/C223m 0 Th1 3b 0.0000 0.0000 0.5000 Th2 6c 0.0000 0.0000 0.2783 N1 6c 0.0000 0.0000 0.1230N2 6c 0.0000 0.0000 0.3674 R/C223 (148) 60 Th 18f 0.7959 0.9734 0.2310 N1 18f 0.7120 0.1245 0.0210 N2 3a 0.0000 0.0000 0.0000Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 035902 (2020); doi: 10.1063/1.5145070 128, 035902-4 Published under license by AIP Publishing.For monoclinic (C2/m) structure: c11.0;c22.0;c33.0;c44.0;c11þc22þc33 þ2ðc12þc13þc23Þ.0;c55.0;c66.0; c22þc33/C02c23.0;c33c55/C0c2 35.0;c44c66/C0c2 46.0; ½c22ðc33c55/C0c2 35Þþ2c23c25c35/C0c2 23c55/C0c2 25c33/C138.0; {2[c15c25(c33c12/C0c13c23)þc15c35(c22c13/C0c12c23) þc25c35(c11c23/C0c12c13)] /C0[c2 15(c22c33/C0c2 23)þc2 25(c11c33/C0c2 13)þc2 35(c11c22/C0c2 12)] þc55(c11c22c33/C0c11c2 23/C0c22c2 13/C0c33c2 12þ2c12c13c23)}.0:(5) For orthorhombic (Pnma) phase: c11.0;c22.0;c33.0;c44.0;c55.0;c66.0, (c11/C0c12).0; (c11þc22/C02c12).0; (c11þc33/C02c13).0, (c22þc33/C02c23).0;c11þc22þc33þ2(c12þc13þc23).0:(6)The examination of elastic stability conditions in ThN 2indi- cated that the previously predicted6Fm/C223m phase at ground state is the mechanically unstable phase due to negative shear modulus(C 11−C12)/2 (as can be seen by substituting the value of C 11and C12from Table IV at zero pressure, supporting our static lattice cal- culations. Additionally, we find that the presently predicted C2/m phase at 0 GPa in this compound fulfills all the above mentioned born stability criteria, supporting its existence at ground state.Similar analysis for trigonal (P /C223ml) and rhombohedral (R /C223m) phases, predicted as ground state structures for Th 2N3and Th 3N4, respectively, also confirm the elastic stability of these phases at zero pressure. The elastic stability conditions have also been tested forhigh pressure phases like Pnma, C2/m, and R /C223 to check the mechanical stability of these phases in their respective stabilitypressure regime. Figure 6 presents the pressure dependence of the elastic constants for various stable phases of ThN 2,T h 2N3and Th3N4. The elastic stability conditions have been examined for dif- ferent phases by employing the elastic constants displayed in Fig. 6 (orTable IV ) into the stability criterion corresponding to each phase of these compounds. It has been found that the entire crite- rion required for elastic stability is fulfilled by structures lowest in enthalpy in respective pressure regimes. The brittle or ductile nature of ThN 2,T h 2N3, and Th 3N4has also been investigated from single crystal elastic constants using thephenomenological behavior of the bulk modulus to the shear modulus ratio. Since the shear module (G) is proportional to the resistance against plastic deformation and fracture strength is pro-portional to the bulk modulus (B) of the material, the B/G ratio isa measure of the material plastic range and its high value corre- sponds to high malleability. Pugh 36suggested a critical value of 1.75 for the separation of two regimes. To understand the nature of FIG. 2. The decomposition enthalpy curves per formula unit as a function of pressure relative to the predicted C2/m structure for ThN 2. TABLE III. The computed enthalpy of formation of ThN 2,T h 2N3, and Th 3N4at 0G P a . Compound Space group Present calculations Others ThN 2 C2/m −3.6369 ThN 2 Fm/C223m −1.6919 −1.5905 Th2N3 P/C223ml −7.3445 −6.8906 Th3N4 R/C223m −11.3445 −11.3806 FIG. 3. Per formula unit enthalpy of various phases relative to the C2/m struc- ture as a function of pressure for ThN 2.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 035902 (2020); doi: 10.1063/1.5145070 128, 035902-5 Published under license by AIP Publishing.FIG. 4. The relative enthalpy per formula unit as a function of pressure upto 100 GPa for ThN 2,T h 2N3, and Th 3N4. FIG. 5. The equation of state and volume collapse at transition pressure for ThN 2,T h 2N3, and Th 3N4. TABLE IV . Single crystal elastic constants of ThN 2,T h 2N3, and Th 3N4at 0 GPa. ThN 2(C2/m) ThN 2(Fm /C223m) ThN 2(Pnma) ThN 2(p/C223ml) Th 2N3(C2/m) Th 3N4(R/C223m) Th 3N4(R/C223) 0 GPa 0 GPa 10 GPa 0 GPa 80 GPa 0 GPa 60 GPa C11(GPa) 227.66 96.80 253.92 248.30 565.45 275.94 490.92 C12(GPa) 106.43 180.19 103.67 152.27 236.41 163.80 255.65 C13(GPa) 76.73 81.05 105.25 262.74 89.30 196.50 C33(GPa) 177.81 375.79 182.97 672.00 170.77 330.86 C44(GPa) 53.59 80.33 66.79 72.21 28.40 79.15 104.80 C66(GPa) 44.25 99.69 252.69 C22(GPa) 219.20 389.43 506.00 C55(GPa) 78.96 121.81 159.05 C23(GPa) 66.14 101.90 112.97 C15(GPa) −4.63 −0.85 C25(GPa) −2.75 −26.41 C35(GPa) −15.86 22.82 C46(GPa) −0.06 0.512 C14(GPa) 38.87 31.68 17.04Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 035902 (2020); doi: 10.1063/1.5145070 128, 035902-6 Published under license by AIP Publishing.ThN 2,T h 2N3, and Th 3N4, from this aspect, the polycrystalline shear modulus has been derived by taking the arithmetic mean of the shear modulus calculated from the Voigt37and Reuss38approx- imations as discussed elsewhere.39The B/G values so derived and listed in Table V are obtained to be 2.06 for ThN 2, 2.62 for Th 2N3, and 2.44 for Th 3N4, respectively, indicating that all these three compounds satisfy the Pugh ’s criteria of ductility. Furthermore, fol- lowing the procedure described in Ref. 40, the shear velocity (V t), longitudinal velocity (V l), mean velocity (V m), Young ’s modulus (E), and Poisson ratio ( μ) have been calculated at zero pressure using theoretical elastic constants. These velocities are then utilized to derive the Debye temperature ( θD) at zero pressure. The lattice dynamic stability is also an essential requirement for a structure to be stable thermodynamically. For ThN 2,w eh a v eexamined the lattice dynamic stability of the previously predicted5 Fm/C223m structure at zero pressure. Though the author in Ref. 5 report that Fm /C223m structure has positive phonon frequency, but present computations suggest that the phonon frequency along[110] directions becomes imaginary ( Fig. 7 ). It appears that Obodo and Chetty 5have investigated the phonon frequency only along [100] as the frequency spectrum along the [110] direction is not reported by them. The phonon spectrum determined in presentcalculations rules out the existence of the previously predicted 5 Fm/C223m structure in ThN 2at zero pressure. Figure 8 displays the phonon spectrum of ThN 2,T h 2N3, and Th 3N4at various pressures. For ThN 2, the phonon spectrum is calculated for the C2/m phase at 0 GPa and the Pnma structure at 10 GPa. Similarly, for Th 2N3, the P /C223ml phase is considered at zero pressure while the C2/m FIG. 6. Theoretically calculated single crystal elastic constants of ThN 2,T h 2N3, and Th 3N4at various pressures. TABLE V . Polycrystalline elastic modulus, sound velocity, and Debye temperature for ThN 2,T h 2N3, and Th 3N4at 0 GPa. Compounds Space group B (GPa) G (GPa) B/G E (GPa) μ Vl(m/s) V t(m/s) V m(m/s) θD(K) ThN 2 C2/m 108.53 52.68 2.06 136.03 0.29 4325 2348 2617 314.5 Th2N3 p/C223ml 152.01 57.93 2.62 154.20 0.33 4723 2374 2660 311.0 Th3N4 R/C223m 147.92 60.50 2.44 159.73 0.32 4678 2406 2692 309.4Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 035902 (2020); doi: 10.1063/1.5145070 128, 035902-7 Published under license by AIP Publishing.structure is taken at 80 GPa. For Th 3N4, the R /C223m phase is consid- ered at 0 GPa and the R /C223 phase at 60 GPa. As indicated by the fre- quency spectrum, the C2/m in ThN 2,P/C223m1 in Th 2N3, and R /C223m in Th3N4are dynamically stable at zero pressure. However, the Pnma phase in ThN 2, C2/m in Th 2N3, and R /C223i nT h 3N4are dynamically stable at 10 GPa, 80 GPa, and 60 GPa, respectively. Furthermore, asdisplayed in Fig. S1 in the supplementary material , the dynamical stability of the C2/m phase in ThN 2, the P /C223ml phase in Th 2N3, and the R /C223m phase in Th 3N4persists up to transition pressures. These results of lattice dynamic simulations support the results of thestatic lattice calculations. It may also be noticed from Fig. 8 that for both ThN 2and Th 2N3, the highest frequency bands havingdispersive nature at zero pressure flatten at high pressures well above the transition pressures in these materials. In Th 3N4, however, the phonon branches are so dispersive that no clear bandgap is seen to exist. Next, in Fig. 9 , the projected phonon density of states (DOS) has also been plotted for all the three materials at ambient as wellas high pressures. As shown in the figure, the phonon spectrum of the C2/m structure for ThN 2comprises three bands. The lowest frequency band extending up to ∼8 THz dominantly consists of vibrations from Th atoms and partly overlaps with another low fre-quency band extending up to 20 THz due to N atoms, whereas the highest frequency band due to N atom is well separated from these. FIG. 7. Theoretically determined phonon spectra of ThN 2in Fm /C223m phase at 0 GPa from present study andby Obodo and Chetty .5 FIG. 8. Theoretically determined phonon spectra of ThN 2,T h 2N3, and Th3N at various pressures.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 035902 (2020); doi: 10.1063/1.5145070 128, 035902-8 Published under license by AIP Publishing.At high pressure of 10 GPa, the first two bands in the Pnma phase of this material show more or less same character as displayed bythe C2/m phase at 0 GPa; however, the highest frequency band shifts toward lower frequency with center at ∼32 THz and com- pared to 42 THz in the C2/m phase. Also, the peak in density ofstates for the highest frequency band enhances to significantlyhigher value of ∼4. Like that for ThN 2,i nT h 2N3also, the phonon density of state for ground state P /C223ml shows three bands with lowest frequency band dominated by Th vibrations, while two higher frequencybands are dominated by N vibrations. At pressure of 80 GPa, thephonon density of states in the C2/m phase (stable phase at thispressure) also displays three bands but with different features. For example, all the three bands are relatively well separated with first two low energy bands wider than those in the zero pressure P /C223ml phase. The third band is highly localized in energy. Unlike that for ThN 2and Th 2N3, the phonon DOS spectrum of Th 3N4at zero pressure in the R /C223m phase as well as at high pres- sure (60 GPa) in the R /C223 phase, the bands do not show well resolved gap. Also, the three band structures at zero pressure get reduced totwo band structure at high pressure with the high frequency bandsignificantly broadened due to merger of two zero pressure bands. The dominance of acoustic and low frequency optical modes by phonons from Th atoms and dominance of high frequency opticalmodes by vibrations from N atoms in all the cases for these three compounds is associated with relatively large mass of Th atoms ascompared to that of N atoms. Furthermore, each unit cell of ThN 2in the zero pressure C2/m phase contains six atoms providing 18 normal modes. Threeof these are acoustic and the remaining 15 are optical modes. Nowas the point group of ThN 2is C 2h, following are the irreducible rep- resentations of the vibrational modes in the center of the BZ, Γaco¼Auþ2Bu, (7) Γopt¼3Agþ6Bgþ2Auþ4Bu, (8) ΓRAMAN ¼6Agþ3Bg, (9) ΓIR¼2Auþ4Bu: (10) Among 15 optical modes, six are infrared active while nine modes are Raman active. The calculated phonon frequencies at0 GPa and 5 GPa and the assignment of infrared-active andRaman-active modes at Γpoint are listed in Table VI . Similarly, the unit cell of the P /C223ml phase in Th 2N3contains five atoms, thereby having 15 normal modes. Considering the point group symmetry FIG. 9. The total and projected phonon densities of states at zero and high pressures for ThN 2,T h 2N3, and Th 3N4.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 035902 (2020); doi: 10.1063/1.5145070 128, 035902-9 Published under license by AIP Publishing.of D 3dassociated to the P /C223ml space group, the irreducible represen- tations of the vibrational modes in the center of the BZ is expressedas follows: Γ aco¼A2uþEu, (11) Γopt¼2A1gþ2Egþ2A2uþ2Eu, (12) ΓRAMAN ¼2A1gþ2Eg, (13) ΓIR¼2A2uþ2Eu: (14) Here, among 12 optical modes, six are infrared active and the remaining six modes are Raman active. The calculated phonon fre- quencies at various pressures and the assignment of infrared-activeand Raman-active at Γpoint are shown in Table VII . For Th 3N4, the R /C223m unit cell contains 21 atoms, providing 63 normal modes that have following irreducible representations: Γaco¼A2uþEu, (15) Γopt¼10A 1gþ10E gþ10A 2uþ10E u, (16) ΓRAMAN ¼10A 1gþ10E g, (17) ΓIR¼10A 2uþ10E u: (18) Out of 60 optical modes 30 are infrared active and 30 modes are Raman active. The calculated phonon frequencies with the assignment of infrared-active and Raman-active modes at Γpoint are provided in Table VIII . To describe all above modes of molecu- lar vibrations, group theory has been applied in which one-dimensional (a non-degenerate mode) representations are denoted by A or B, two-dimensional representations (a doubly degenerate) by E, and three-dimensional representations (a triple degenerate)by T. If a one-dimensional vibration mode is symmetric withrespect to the principal symmetry axis C n, the mode is denoted by A. If the vibration mode is anti-symmetric with respect to the prin- cipal symmetry axis C n, it is denoted by B. On the other hand,irrespective of the degeneracy of the mode, a subscript 1 or 2 is used in the respective letter d epending on whether the mode is symmetric or anti-symmetric with respect to a rotation around a twofold axis (C 2)n o r m a lt oC n. Another subscript “g”or“u” accompanies to 1 or 2 depending on whether the mode is sym-metric or anti-symmetric with respect to the inversion of coor-dinate. 41,42In all the three compounds, to our knowledge, no experimental data exist in open l iterature; hence, no comparison could be made. Our results could still be useful inputs forvarious studies. In order to understand the electronic origin of structural phase transitions, we have carried out an analysis of electronic structures in all stable phases for which corresponding density of states are displayed in Fig. 10 . The electronic DOS of the low sym- metry C2/m phase of ThN 2comprises three well separated bands with the narrow lowest energy band consisting of mostly p-DOS from nitrogen lies well below the Fermi level. The second band extending from −4e V t o ∼0.5 eV crosses the Fermi level, domi- nantly consists of a mixer of p-DOS from nitrogen and d-DOSfrom thorium, and governs the metallic character of this phase. Thehighest energy third band is ∼2 eV away from Fermi energy and consists mainly of f-DOS of thorium with some contribution from d-electrons. In an attempt to understand the occurrence of the lowsymmetry C2/m structure for ThN 2in place of a high symmetry Fm/C223m phase predicted earlier, we have analyzed the electronic density of states (DOS) of the Fm /C223m structure also and compared with that of the C2/m phase. The electronic DOS for this structure consists of two bands with one extending from ∼−4e Vt o ∼0.5 eV and the well separated band lying in the energy range of 4 eV to∼8 eV. Like that for the C2/m phase, the former consists of mainly p-DOS from nitrogen and some contribution from d-DOS of thorium, whereas the latter is mostly due to f-DOS of Th with some contribution from d-DOS. The comparison of the DOS spec-trum of C2/m and Fm /C223m phases reveals that (i) the DOS spectrum splits into three bands in the C2/m phase as compared to two bands in the Fm /C223m structure due to lowered symmetry and (ii) the density of states around Fermi energy are significantly lower in theTABLE VI. Raman and IR phonon frequencies (THz) of ThN 2at various pressures. Raman active IR active Pressure → Modes ↓ 0G P a 5G P aPressure → Modes ↓ 0G P a 5G P a Ag 2.82 3.00 B g 2.18 0.18 Ag 4.58 5.05 B g 5.46 3.39 Ag 9.56 9.03 B g 13.22 14.00 Ag 12.87 13.58 A u 5.01 5.72 Ag 13.77 14.48 A u 11.76 12.66 Ag 42.83 45.99 B u 6.37 6.94 Bu 7.13 8.07 Bu 11.34 12.13 Bu 14.11 14.93TABLE VII. Raman and IR phonon frequencies (THz) of Th 2N3at various pressures. Raman active Pressure → Modes ↓ 0 GPa 4.8 GPa 10.9 GPa 17.6 GPa A1g 12.28 12.99 13.47 14.82 A1g 4.68 4.90 5.24 5.48 Eg 12.42 12.78 13.57 13.98 Eg 2.60 2.65 2.74 2.80 IR active 0 GPa 48 GPa 10.9 GPa 17.6 GPa A2u 13.33 13.74 14.10 14.45 A2u 5.44 6.28 6.90 7.49 Eu 12.26 12.98 13.79 14.15 Eu 5.92 7.03 7.67 8.27Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 035902 (2020); doi: 10.1063/1.5145070 128, 035902-10 Published under license by AIP Publishing.C2/m phase as compared to that for the Fm /C223m phase. The overall effect is to reduce the total energy of the C2/m structure as com-pared to that of the Fm /C223m phase in ThN 2. Another aspect, as already discussed by Modak and Verma15and Sahoo et al.,4is that at 0 GPa, the increase in mixing of d and f states of Th and p states of nitrogen in the Fm /C223m structure adjusts the attractive and repul- sive forces between nearest neighbors such that this structure getsdestabilized, causing elastic phonon softening. For the high pres-sure Pnma phase, the total density of states around the Fermi level displays a wide minimum with significantly small ( ∼0.05 states / eV-f.u) densities, indicating that this phase could be a poor metalor a semi-metal. In Th 2N3, the DOS structure for ambient condition trigonal phase (P /C223ml) displays two bands, well separated from each other and the lower band extends from −4e V t o ∼0.3 eV giving it a metallic character governed mainly by the N-2p states, with little contributionfrom Th-d states. The high pressure C2/m structure also has metallicnature and the only difference between two structures is that in thecase of trigonal phase, there is a gap of ∼2 eV between the valence and conduction bands, but there is no gap for the C2/m phase. As far as Th 3N4is concerned at ambient condition, the DOS spectrum for stable R /C223m phase has two bands with a bandgap of 1.44 eV, which makes this compound an insulator. Furthermore, the top ofthe valence band is determined mainly by the N-2p states and the bottom of the conduction band is determined mainly by the hybridi- zation of the thorium d and f states. This theoretically derivedbandgap is in good agreement with the experimentally reported valueof 1.7 eV 3and the theoretical value of 1.59 eV.5,6Furthermore, it may be noticed that the DOS spectrum of high pressure R /C223 structure at 62 GPa indicates the metallic nature of Th 3N4at high pressure.Considering the usefulness of thermodynamic properties for application in nuclear industry, the high temperature behavior ofThN 2,T h 2N3, and Th 3N4has also been examined through a quasi- harmonic approximation (QHA) based theoretical technique.40 This method uses the phonon density of states obtained from lattice dynamic calculations for determination of various thermo- physical properties. We have computed the Helmholtz free energyF(V,T) using QHA 40as follows: F(V,T)¼Fstatic(V)þFvib(V,T) ¼Ec(V,0 )þEvib(V,T)/C0TSvib(V,T): (19) Fvib(V,T),Evib(V,T), and Svib(V,T),respectively, are the free energy, internal energy, and entropy contribution from phonons. Furthermore, Fvib(V,T) can be expressed as Fvib(V,T)¼X q,j1 2/C22hωj(q,V)þkBTX q,jln 1/C0exp/C0/C22hωj(q,V) kBT/C18/C19 /C20/C21 , (20) Svib(V,T)¼/C0 kBX q,nln 1/C0exp/C0/C22hωn(q,V) kBT/C18/C19 /C20/C21 þX q,nkB/C22hωn(q,V) kBTexp /C0/C22hωn(q,V) kBT/C18/C19 /C01/C20/C21/C01 :(21) Table IX lists the room temperature values of various physical quantities such as thermal expansion and heat capacity of theseTABLE VIII. Raman and IR phonon frequencies (THz) of Th 3N4at various pressures. Raman active IR active Modes 0 GPa 19 GPa 37 GPa Modes 0 GPa 19 GPa 37 GPa A1g 1.47 1.53 1.72 A 2u 1.38 1.52 1.65 A1g 2.77 3.20 2.27 A 2u 2.74 3.20 3.37 A1g 4.47 6.20 7.23 A 2u 3.40 3.71 4.02 A1g 4.54 6.36 7.43 A 2u 4.56 6.37 7.45 A1g 9.38 11.13 13.06 A 2u 5.53 7.59 8.76 A1g 13.73 15.14 15.83 A 2u 9.40 11.18 13.05 A1g 13.81 15.15 15.83 A 2u 13.75 15.14 15.98 A1g 13.99 16.48 16.99 A 2u 14.14 16.51 16.99 A1g 14.18 16.52 17.01 A 2u 15.61 18.69 24.27 A1g 16.12 18.94 24.63 A 2u 16.01 18.70 24.56 Eg 0.97 1.11 1.31 E u 0.85 1.11 1.29 Eg 1.61 2.04 2.31 E u 1.58 2.03 2.30 Eg 2.73 3.07 3.19 E u 1.84 2.40 2.85 Eg 2.79 3.28 3.40 E u 2.80 3.30 3.41 Eg 5.74 8.28 9.88 E u 5.64 8.09 9.57 Eg 8.62 10.80 12.72 E u 5.72 8.18 9.81 Eg 8.67 10.94 12.91 E u 8.62 10.83 12.78 Eg 13.19 14.91 15.32 E u 12.96 14.53 15.02 Eg 14.27 16.56 17.01 E u 13.19 15.12 17.16 Eg 14.48 16.58 17.16 E u 14.48 16.59 21.46Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 035902 (2020); doi: 10.1063/1.5145070 128, 035902-11 Published under license by AIP Publishing.materials. Figure 11 displays the vibrational part Fvibof Helmholtz free energy along with heat capacity at constant volume (C v) and vibrational entropy Svibdetermined from phonon frequencies. As is evident from the figure, for all the three materials, Svibincreases monotonically with increasing temperature. The values of vibra- tional entropy Svibat 300 K are found to be ∼24.89 J/K/mol, 27.78 J/K/mol, and 27.65 J/K/mol ( Table IX ), respectively, for ThN 2,T h 2N3, and Th 3N4. It is the monotonic increase in Svib, which causes Fvibto decrease with temperature. Figure 12 depicts the temperature variation of the thermal expansion coefficient. Various features exhibited by this include strong dependence upon TABLE IX. The calculated thermophysical properties of ThN 2,T h 2N3, and Th 3N4at 0G P a . CompoundSpace groupα0 (×10−5K−1)Cv(J/ K/mol)CP(J/ K/mol)Svib(J/ K/mol) ThN 2 C2/m 2.243 20.210 20.489 24.895 Th2N3 P/C223ml 4.426 21.488 22.385 27.780 Th3N4 R/C223m 3.612 21.186 21.808 27.651 FIG. 10. The total and projected densities of states (DOS) for ThN 2,T h 2N3, and Th 3N4at various pressures. FIG. 11. T emperature dependence of the vibrational part of Helmholtz free energy Fvib, vibrational entropy Svib, and specific heat at constant volume C vof ThN 2,T h 2N3, and Th 3N4.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 035902 (2020); doi: 10.1063/1.5145070 128, 035902-12 Published under license by AIP Publishing.temperature for small temperatures ( T/θD< < 1) and a weaker dependence upon temperature at higher temperature range of300 K –1000 K. Additionally, we have plotted the temperature dependence of the theoretically derived heat capacity at constantpressure (C p)i nFig. 13 . As expected, the increase in C pwith tem- perature slows down for T>θDand starts approaching the Dulong –Petit limit asymptotically as shown in Fig. 13 . All these theoretically derived thermo-physical properties may need experi-mental verification.IV. SUMMARY Using the ab initio pseudopotential technique, the pressure- induced phase transition along with elastic, lattice dynamic, andelectronic properties of ThN 2,T h 2N3, and Th 3N4has been investi- gated. Our analysis of enthalpy calculated at various pressures sug- gests the structural transition from the C2/m to the Pnma phase at∼7 GPa in ThN 2, the P /C223ml to the C2/m phase at ∼63 GPa in Th2N3, and the R /C223m to the R /C223 phase in Th 3N4at∼37 GPa. All these pressure-induced phase transitions are of first order in nature, with a volume discontinuity of ∼10.96%, ∼10.32%, and ∼6.45%, respectively. These pressure-induced structural transitions suggested by enthalpy analysis are further substantiated by exami-nation of elastic and lattice dynamic stability. In ThN 2, on the basis of elastic and lattice dynamic stability analyses, the present work rules out the existence of the CaF 2-type cubic phase (space group Fm/C223m) predicted in previous studies.5Furthermore, the stability of the lower symmetry C2/m phase over the higher symmetry Fm /C223m structure in ThN 2at zero pressure is explained qualitatively by employing an electronic band structure aspect. The theoretically derived electronic density of the state spectrum for various phases, in the present study, suggests that ThN 2and Th 2N3are metallic due to hybridization of thorium d and f states with nitrogen 2p, whereasTh 3N4is insulating with a bandgap of ∼1.44 eV. The physical prop- erties of these materials derived in this work could be useful in various basic understandings as well as technological applications. SUPPLEMENTARY MATERIAL See the supplementary material for dynamical stability of dif- ferent phases of ThN 2,T h 2N3, and Th 3N4. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1R. Didchenko and F. P. Gortsema, Inorg. Chem. 2, 1079 (1963). 2S. Aronson and A. B. Auskern, J. Phys. Chem. 70(12), 3937 (1966). 3T. Gouder, L. Havela, L. Black, F. Wastin, J. Rebizant, P. Boulet, D. Bouexière, S. Heathman, and M. Idiri, J. Alloys Compd. 336, 73 (2002). 4B. D. Sahoo, K. D. Joshi, and T. C. Kaushik, J. Nucl. Mater. 521, 161 (2019). 5K. O. Obodo and N. Chetty, J. Nucl. Mater. 440, 229 (2013). 6K. O. Obodo and N. Chetty, Solid State Commun. 193, 41 (2014). 7P. Bagla, Science 350, 726 (2015). 8R. W. Grimes and W. J. Nuttall, Science 329, 799 (2010). 9H. Tagawa, J. Nucl. Mater. 51, 78 (1974). 10H. Tagawa and N. Masaki, J. Inorg. Nucl. Chem. 36, 1099 (1974). 11L. Gerward, J. Staun Olsen, U. 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Kresse, J. Furthmuller, and J. Hafner, Europhys. Lett. 32, 729 (1995); A. van de Walle and G. Ceder, Rev. Mod. Phys. 74, 11 (2002). 32Y. Le Page and P. Saxe, Phys. Rev. B 65, 104104 (2002).33A. Togo, F. Oba, and I. Tanaka, Phys. Rev. B 78, 134106 (2008). 34Y. Guo, C. Yu, J. Lin, C. Wang, C. Ren, B. Sun, P. Huai, R. Xie, X. Ke, Z. Zhu, and H. Xu, Sci. Rep. 7, 45872 (2017). 35Z.-J. Wu, E.-J. Zhao, H.-P. Xiang, X.-F. Hao, X. Liu, and J. Meng, J. Meng Phys. Rev. B 76, 054115 (2007). 36S. F. Pugh, Philos. Mag. 45, 823 (1954). 37W. Voigt, Lehrbuch der Kristallphysik (Taubner, Leipzig, 1928). 38A. Reuss, ZAMM - Zeitschrift Für Angew. Math. Und Mech. 9, 49 (1929). 39D. Mukherjee, B. D. Sahoo, K. D. Joshi, S. C. Gupta, and S. K. Sikka, J. Appl. Phys. 109, 103515 (2011). 40B. D. Sahoo, D. Mukherjee, K. D. Joshi, and T. C. Kaushik, J. Appl. Phys. 120, 085902 (2016). 41J. E. Rodrigues, D. M. Bezerra, R. C. Costa, P. S. Pizani, and A. C. Hernandes, J. Raman Spectrosc. 48, 1243 (2017). 42I. I. Leonidov, V. P. Petrov, V. A. Chernyshev, A. E. Nikiforov, E. G. Vovkotrub, A. P. Tyutyunnik, and V. G. Zubkov, J. Phys. Chem. C 118, 8090 (2014).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 035902 (2020); doi: 10.1063/1.5145070 128, 035902-14 Published under license by AIP Publishing.

5.0008432.pdf

J. Chem. Phys. 152, 180901 (2020); https://doi.org/10.1063/5.0008432 152, 180901 © 2020 Author(s).Essentials of relativistic quantum chemistry Cite as: J. Chem. Phys. 152, 180901 (2020); https://doi.org/10.1063/5.0008432 Submitted: 23 March 2020 . Accepted: 19 April 2020 . Published Online: 13 May 2020 Wenjian Liu ARTICLES YOU MAY BE INTERESTED IN eT 1.0: An open source electronic structure program with emphasis on coupled cluster and multilevel methods The Journal of Chemical Physics 152, 184103 (2020); https://doi.org/10.1063/5.0004713 Recent developments in the general atomic and molecular electronic structure system The Journal of Chemical Physics 152, 154102 (2020); https://doi.org/10.1063/5.0005188 Large scale and linear scaling DFT with the CONQUEST code The Journal of Chemical Physics 152, 164112 (2020); https://doi.org/10.1063/5.0005074The Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp Essentials of relativistic quantum chemistry Cite as: J. Chem. Phys. 152, 180901 (2020); doi: 10.1063/5.0008432 Submitted: 23 March 2020 •Accepted: 19 April 2020 • Published Online: 13 May 2020 Wenjian Liua) AFFILIATIONS Qingdao Institute for Theoretical and Computational Sciences, Shandong University, Qingdao, Shandong 266237, People’s Republic of China a)Author to whom correspondence should be addressed: liuwj@sdu.edu.cn ABSTRACT Relativistic quantum chemistry has evolved into a fertile and large field and is now becoming an integrated part of mainstream chemistry. Yet, given the much-involved physics and mathematics (as compared with nonrelativistic quantum chemistry), it is still necessary to clean up the essentials underlying the relativistic electronic structure theories and methodologies (such that uninitiated readers can pick up quickly the right ideas and tools for further development or application) and meanwhile pinpoint future direc- tions of the field. To this end, the three aspects of electronic structure calculations, i.e., relativity, correlation, and QED, will be highlighted. Published under license by AIP Publishing. https://doi.org/10.1063/5.0008432 .,s I. INTRODUCTION: ABC OF RELATIVISTIC QUANTUM MECHANICS As one of the two pillars of modern physics, the theory of spe- cial relativity was founded by Einstein in 1905.1Among others, the best known energy–mass relation E=γmc2(1) is most relevant for our purpose. Here, mis the rest mass of a par- ticle moving with velocity v, whileγ=(1−v2 c2)−1/2is the Lorentz factor, with cbeing the constant velocity of light. This relation can be converted to E2=c2p2+m2c4(2) via the very definition of the momentum p=γmv. In fact, the energy–momentum relation (2) is more fundamental than the energy–mass relation (1), since the former applies to both mas- sive and massless particles, whereas the latter applies only to massive particles. Moreover, relation (1) is merely the positive- energy part of the square root of the right-hand side of rela- tion (2), i.e., E=±√ c2p2+m2c4. In view of the correspondence principle, i.e., E→i̵h∂ ∂t,p→−i̵h∇, (3)relation (2) can be mapped, as done by Klein and Gordon in 1926,2,3 to a first-quantized wave equation, (1 c2∂2 ∂t2−∇2+k2)ψKG(x)=0, k=mc ̵h,x=rt, (4) which treats space and time on the same footing and is man- ifestly Lorentz covariant. However, it is second order in time, which is fundamentally different from the Schrödinger equation that is first order in time. As a result, the norm-conserving den- sity,4ρKG=i̵h 2mc2(ψ∗ KG∂ψKG ∂t−ψKG∂ψ∗ KG ∂t), is not positive defi- nite (since ψKGand∂ψKG ∂tare independent of each other and can have arbitrary values at a given time t) and hence cannot be interpreted as a probability density. Because of this, the Klein– Gordon equation (4) was not regarded to be physically meaning- ful until Pauli and Weisskopf5recognized through a theoretical exercise6that it is a relativistic wave equation for spin-0, charged, and massive particles, which were discovered to be π+andπ− mesons in the late 1940s (NB: ρKGmultiplied by charge qcan be reinterpreted as a charge density, which can be either posi- tive or negative). A relativistic first-quantized wave equation that is first order in time and in space was first proposed by Dirac in 19287,8by noticing that the energy–momentum relation (2) can be written as J. Chem. Phys. 152, 180901 (2020); doi: 10.1063/5.0008432 152, 180901-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp E2=D2 0,D0=cα⋅p+βmc2=D† 0, (5) =c2∑ i=x,y,zα2 ip2 i+β2m2c4+∑ i=x,y,z[β,αi]+pimc3 +c2 23 ∑ i≠j[αi,αj]+pipj, (6) =c2p2+m2c4, (7) provided that the following conditions hold: β†=β,α† i=αi,β2=α2 i=1,[β,αi]+=[αi,αj]+=0(i≠j). (8) It is then not a difficult math to figure out the explicit, lowest- dimensional matrix expressions for βandα, β=(1 0 0−1),α=(0σ σ0), (9) whereσis the vector of the 2 ×2 Pauli spin matrices, σx=(0 1 1 0),σy=(0−i i0),σz=(1 0 0−1). (10) Recognizing the function D0(5) as the Hamiltonian, the correspon- dence principle (3) then leads immediately to the free-particle Dirac equation i̵h∂ ∂tψ(x)=D0ψ(x). (11) At variance with the scalar form of ψKG, the wave function ψof the Dirac equation (11) is a bispinor with four components, i.e., ψ(x)=(ψL(x) ψS(x))=⎛ ⎜⎜⎜⎜ ⎝ψLα(x) ψLβ(x) ψSα(x) ψSβ(x)⎞ ⎟⎟⎟⎟ ⎠. (12) It can readily be checked that each of the four components satisfies the Klein–Gordon equation (4). It can also be shown that the density ρ=ψ†ψis positive definite. Moreover, the appearance of the Pauli spin matrices (10) in the Dirac αmatrix (9) reveals that the Dirac equation (11) is a relativistic wave equation for spin-1 2particles such that the components of the wave function (12) can be labeled by the αandβspins. As a matter of fact, the Dirac equation (11) can directly be obtained from the Klein–Gordon equation (4) by decomposing the latter into two coupled first-order equations. Following Kramers,9,10 this can proceed by rewriting Eq. (4) as (i̵h∂ ∂t−cσ⋅p)(i̵h∂ ∂t+cσ⋅p)ψKG(x)=m2c4ψKG(x), (13) where the identity p2= (σ⋅p)2has been used. Further replacing ψKG with spinor ψland(i̵h∂ ∂t+cσ⋅p)ψlwith mc2ψr, we obtain i̵h∂ ∂tψW(x)=DW 0ψW(x), (14)where DW 0=(cσ⋅pmc2 mc2−cσ⋅p),ψW(x)=(ψr(x) ψl(x)). (15) Equation (14) is known as the Dirac equation in the Weyl rep- resentation. By further carrying out the unitary transformation, UW=1√ 2(1 1 1−1)=U−1 W, (16) the Dirac equation in the standard representation (11) can be recov- ered. At this moment, it is worth mentioning that, although electron spin appears naturally in the Dirac equation, it is not a relativistic quantity since it appears also in the Lévy-Leblond equation,11 i̵h∂ ∂t(1 0 0 0)(ψL(x) ϕL(x))=(0σ⋅p σ⋅p−2m)(ψL(x) ϕL(x)), (17) which is just the nonrelativistic limit (NRL)12of the Dirac equa- tion (11) and can be introduced a priori by means of a spinor representation of the (nonrelativistic) Galilei group. For an electron ( q=−e) moving in an external electromagnetic field characterized by the vector potential Aextand scalar potential ϕext, the following minimal coupling relations p→π=p−qAext,i̵h∂ ∂t→i̵h∂ ∂t−qϕext (18) for electromagnetic interaction can be invoked so as to obtain i̵h∂ ∂tψ(x)=Dψ(x), (19) D=cα⋅(p−qAext)+βmc2+qϕext. (20) If the external field is static, suffice it to consider the following eigenvalue problem: Dψp(r)=εpψp(r), (21) which has three branches of solutions if the external field arises from a net positive charge distribution: positive-energy continuum, discrete positive-energy bound states, and negative-energy contin- uum, as illustrated by the left panel of Fig. 1. The gap ( ΔE) between the lowest positive-energy level ϵ1sand the top edge ( −mc2) of the negative-energy continuum can be calculated as ΔE=ϵ1s−(−mc2)=mc2√ 1−(Z/c)2+mc2=2mc2fZ, 1 2<fZ=1 2(1 +√ 1−(Z/c)2)<1,(22) whereϵ1sis the ground state energy of the Dirac equation for a one- electron atom of nuclear charge Z. It is seen that the gap is indeed very large (e.g., fZ≈0.91 for Hg79+). It is not much changed for many-electron systems ( fZ≈0.92 for Hg). The existence of an empty negative-energy continuum was extremely troublesome in the early days of relativistic quantum mechanics, for it implied that no atom would be stable! For instance, it can be estimated4that, in the presence of a radiative field (which J. Chem. Phys. 152, 180901 (2020); doi: 10.1063/5.0008432 152, 180901-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp FIG. 1 . Energy spectra of the electron (left) and positron (right) Dirac equations related by charge conjugation (CC). always exists in reality), the electron in the ground state of the hydro- gen atom can fall down to the top of the negative-energy continuum in less than one nanosecond and it can even trigger a radiation catas- trophe via continuous radiative transitions. To resolve this apparent untruth, Dirac13proposed in 1930 that all states of negative energy be filled such that transitions of electrons to negative-energy states are forbidden by virtue of the Pauli exclusion principle. Since a hole left by exciting an electron from the filled negative-energy sea has a positive energy and the same mass but opposite charge as the leaving electron, it was interpreted by Dirac in 193114as an anti- electron (positive electron/positron). Although highly controversial, this bold prediction was confirmed by experiment just one year later.15Notwithstanding such a big triumph, Dirac’s hole theory has a number of defects:16 (1) It is asymmetric with respect to electrons and positrons. (2) It characterizes a positron as a virtual hole rather than a real particle. (3) It involves an infinite negative electric charge filling the whole space even if only one electron is under consideration. (4) It has to assume that the negative-energy electrons do not generate any potential acting on the positive-energy electrons. Otherwise, no nuclear charge could generate an enough attraction to compensate the infinitely repulsive potential. (5) It does not, in a strict sense, explain the stability of a positive- energy electron: being infinitely large, the sea can always accept infinitely many electrons. In other words, the Pauli exclusion principle does not really hold for a system of an infinite number of fermions. (6) It does not apply to spin-0 particles [described by the Klein– Gordon equation (4)], which do not satisfy an exclusion principle. The above problems associated with the negative-energy con- tinuum drove the pioneers of quantum mechanics to formulate a quantum field theory for electrodynamics (QED) through a particu- lar second quantization of the Dirac matter field and electromagnetic field, where all particles are of positive energy (see Ref. 6 for a histori- cal review of the early days of QED). To see how this can be achieved, we first take a look at the charge-conjugation transformation, ˆC=CβˆK0,C=−iαy, (23)of the Dirac equation (21): taking the complex conjugate ( ˆK0) followed by multiplying Cβfrom the left leads to [cα⋅(p+qAext)+βmc2−qϕext]ˆCψp(r)=ˆCψp(r)ϵC p,ϵC p=−ϵp. (24) The manipulation is facilitated by making use of the following identities: C†=CT=C−1=−C,Cβ=−βC,Cα∗=−αC,C†αC=−αT. (25) To unify the notion, we define ψC ˜p(x)=ψC ˜p(r)e−iϵ˜pt=ˆCψp(x),ψC ˜p(r)=Cβψ∗ p(r),ϵ˜p=−ϵp<0, (26a) ψC p(x)=ψC p(r)e−iϵpt=ˆCψ˜p(x),ψC p(r)=Cβψ∗ ˜p(r),ϵp=−ϵ˜p>0. (26b) That is, apart from its apparent action, charge conjugation will also interchange the indices of the argument as p↔˜psuch thatϵp>0 andϵ˜p<0 always hold. Another example is ˆC[apψp]=Cβ(apψp)†T =Cβ(apψ∗ p)=a˜pψC ˜p. It is clear that, for the same time-independent external field [ϕext(r),Aext(r)], ifψp(x)=ψp(r)e−iϵptis a stationary state of the Dirac equation for an electron ( q=−1) of positive energy ϵp, ψC ˜p(x)=ˆCψp(x)=ψC ˜p(r)e−iϵ˜ptwill then be a stationary state of the Dirac equation for a positron ( q= +1) of negative energy ϵ˜p=−ϵp. Likewise, if ψ˜p(x)=ψ˜p(r)e−iϵ˜ptis an electronic negative-energy state (NES), ψC p(x)=ˆCψ˜p(x)=ψC p(r)e−iϵptwill be a positronic positive-energy state (PES; ϵp=−ϵ˜p; cf. the right panel of Fig. 1). Note, in particular, that the probability density of a negative-energy electronψ˜p(r)e−i∣ϵ˜p∣(−t)is indistinguishable from that of a positive- energy positron ψC p(r)e−i∣ϵ˜p∣t, i.e.,∣ψC p(r)∣2=∣ψ˜p(r)∣2. As such, a negative-energy electron propagating backward in time can be regarded as the mirror image of a positive-energy positron propa- gating forward in time. Therefore, the Dirac matter field should be quantized as ˆϕ(x)=bpψp(x)+b˜pψ˜p(x), (27a) bp∣0⟩=b˜p∣0⟩=0, bp=b† p,b˜p=b† ˜p,ϵp>0,ϵ˜p<0, (27b) in the interaction picture and the Einstein summation conven- tion over repeated indices, in order for the field to comprise only of positive-energy particles: bpannihilates an electron of positive energyϵp, whereas b˜pcreates a positron of positive energy ∣ϵ˜p∣=−ϵ˜p. Since any operator must be expanded in a complete (orthonormal) basis spanned by the PES and NES of the same Dirac equation (21), the amplitude companying b˜pcan only be the electronic NES ψ˜p(r)e−iϵ˜ptinstead of the corresponding positronic PES ψC p(r)e−i∣ϵ˜p∣t [NB: in the presence of an external field, {ψC p(r)}are even not orthogonal to the electronic PES { ψq(r)}, i.e., the inner products ⟨ψC p∣ψq⟩are generally nonzero]. On the other hand, charge conser- vation dictates that the operator b˜p(instead of b˜p) must accompany bp. Both bpand b˜pincrease the charge of a state by one unit; bp J. Chem. Phys. 152, 180901 (2020); doi: 10.1063/5.0008432 152, 180901-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp does this by destroying an electron, whereas b˜pdoes this by creat- ing a positron. Thus, the field operator ˆϕ(x)always increases one unit of charge. Similarly, the field operator ˆϕ†(x)always decreases one unit of charge. Therefore, the operator ˆϕ†(x)ˆϕ(x)conserves the charge. Had b˜pbeen chosen to accompany bp, the operator ˆϕ†(x)ˆϕ(x)would not conserve the charge: it would include terms such as bpb˜pand b˜pbpthat decrease and increase two units of charge, respectively. The particular form (27) for the quantized Dirac matter field is the very first cornerstone of QED. It can actually be rewritten as16 ˆϕ(x)=apψp(x)+a˜pψ˜p(x), (28a) ap∣0e−;Ne−⟩=a˜p∣0e−;Ne−⟩=0, ap=a† p, a˜p=a† ˜p,ϵp>0,ϵ˜p<0,(28b) by replacing the genuine vacuum |0 ⟩with the physical vacuum ∣0e−;Ne−⟩consisting of zero positive-energy electrons and Ne−(→∞) negative-energy electrons. That is, the particle–hole picture bp=ap,bp=ap,b˜p=a˜p,b˜p=a˜p,ϵp>0,ϵ˜p<0 (29) implied in Eq. (27a) is merely a mathematical operation and is only convenient for pictorial interpretation in terms of diagrams, where the expression (28a) is more convenient for algebraic manipula- tions.16At first glance, we have just gone back to the filled Dirac pic- ture such that the aforementioned problems associated with Dirac’s hole theory would arise again. However, the picture can be reversed: it is perfectly legitimate to quantize the Dirac matter field in terms of the solutions of the positron Dirac equation (24), ˆϕC(x)=dpψC p(x)+d˜pψC ˜p(x), (30a) dp∣0⟩=d˜p∣0⟩=0, dp=d† p,d˜p=d† ˜p,ϵp>0,ϵ˜p<0, (30b) where dpannihilates a positron of positive energy ϵp, whereas d˜p creates an electron of positive energy ∣ϵ˜p∣. The charge-conjugation transformation of ˆϕ(x)(27) leads to17,18 ˆϕC(x)=Cβˆϕ†T(x) (31) =Cβ[bpψ∗ p(x)]+Cβ[b˜pψ∗ ˜p(x)] (32) =b˜pψC ˜p(x)+bpψC p(x), s.t.ϵp>0,ϵ˜p<0. (33) By comparing Eq. (33) with Eq. (30a), we obtain dp=bp,d˜p=b˜p,ϵp>0,ϵ˜p<0. (34) That is, the dand btypes of annihilation and creation processes are the same, although their carriers are different (positrons vs elec- trons). This is more transparent17for the case of free particles for which Eqs. (21) and (24) are identical [i.e., ψC p(x)=ψp(x)and ψC ˜p(x)=ψ˜p(x)] such that it is immaterial to interpret which set of the PES and NES as electrons or positrons. Moreover, just like Eq. (28), Eq. (30) can be rewritten asˆϕC(x)=cpψC p(x)+c˜pψC ˜p(x), (35a) cp∣0e+;Ne+⟩=c˜p∣0e+;Ne+⟩=0, cp=c† p,c˜p=c† ˜p,ϵp>0,ϵ˜p<0. (35b) Now the vacuum is ∣0e+;Ne+⟩in lieu of the original |0 ⟩. Since the two types of (second) quantization of the same Dirac matter field are equivalent, they can simply be averaged with an equal weight. To show how this can be done, let us look at the four-current operators for electrons and positrons, ˆjμ e−(x)=−eˆϕ†(x)cαμˆϕ(x),αμ=(c−1,α),e=+1, (36) =−e{ˆϕ†(x)cαμˆϕ(x)}−e⟨0;Ne−∣ˆϕ†(x)cαμˆϕ(x)∣0;Ne−⟩(37) =−e{ˆϕ†(x)cαμˆϕ(x)}−eψ† ˜p(r)cαμψ˜p(r), (38) ˆjμ e+(x)=eˆϕC†(x)cαμˆϕC(x) (39) =e{ˆϕC†(x)cαμˆϕC(x)}+e⟨0;Ne+∣ˆϕC†(x)cαμˆϕC(x)∣0;Ne+⟩ (40) =e{ˆϕC†(x)cαμˆϕC(x)}+eψC† ˜p(r)cαμψC ˜p(r), (41) where the first terms of Eqs. (37), (38), (40), and (41) are normal ordered with respect to ∣0;Ne−⟩and∣0;Ne+⟩, respectively. By means of the relation (31), the first term of Eq. (41) can be written as e{ˆϕC†(x)cαμˆϕC(x)}=ec{ˆϕTβC†αμCβˆϕ†T(x)} (42) =ec{ˆϕT γ(x)(αμ)T γρˆϕ†T ρ(x)} (43) =e{cαμˆϕ(x)ˆϕ†(x)} (44) =−e{ˆϕ†(x)cαμˆϕ(x)}, (45) where the normal ordering is now taken with respect to ∣0;Ne−⟩. Likewise, the second term of Eq. (41) can be written as eψC† ˜p(r)cαμψC ˜p(r)=eψ† p(r)cαμψp(r) =e⟨0;Ne−∣cαμˆϕ(x)ˆϕ†(x)∣0;Ne−⟩. (46) Therefore, ˆjμ e+(x)(39) can be written as ˆjμ e+(x)=ecαμˆϕ(x)ˆϕ†(x) (47) =−e{ˆϕ†(x)cαμˆϕ(x)}+e⟨0;Ne−∣cαμˆϕ(x)ˆϕ†(x)∣0;Ne−⟩. (48) The four-current operator averaged over electrons and positrons then reads J. Chem. Phys. 152, 180901 (2020); doi: 10.1063/5.0008432 152, 180901-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp ˆjμ(x)=1 2(ˆjμ e−(x)+ˆjμ e+(x)) (49) =−1 2e[ˆϕ†(x),cαμˆϕ]=−1 2ecαμ γρ[ˆϕ† γ(x),ˆϕρ(x)] (50) =−e{ˆϕ†(x)cαμˆϕ}+jμ vp(r), (51) jμ vp(r)=−e⟨vac∣1 2[ˆϕ†(x),cαμˆϕ(x)]∣vac⟩ (52) =−1 2e[ψ† ˜p(r)cαμψ˜p(r)−ψ† p(r)cαμψp(r)]. (53) Note in passing that the vacuum |vac ⟩in Eqs. (51) and (52) can either be |0⟩along with the definition (27) or be ∣0e−;Ne−⟩along with the definition (28). The zero component of jμ vp(r)(53), i.e., the vacuum densityρvp(r), reads ρvp(r)=−e⟨vac∣1 2[ˆϕ†(x),ˆϕ(x)]∣vac⟩ (54) =−1 2e[n−(r)−n+(r)], (55) n+(r)=ψ† p(r)ψp(r)=ψC† ˜p(r)ψC ˜p(r), (56) n−(r)=ψ† ˜p(r)ψ˜p(r)=ψC† p(r)ψC p(r), (57) where n+(r) and n−(r) are the number densities of the PES and NES of the electron Dirac equation (21), respectively. By virtue of the identity n++n−=¯n++¯n−=2¯n−, with ¯n+and ¯n−(=¯n+) being the free-particle number densities, we have ρvp(r)=−e[n−(r)−¯n−(r)], (58) which is clearly the charge polarization of the vacuum. Moreover, Eq. (55) reveals that the NESs of the electron Dirac equation (21) are all occupied by electrons e−with charge −1 (i.e., filled Dirac sea of electrons), whereas the PES by positrons e+with charge +1. As shown above, the latter arises actually from the filled Dirac sea of positrons (i.e., ∣0e+;Ne+⟩), as a direct consequence of charge conju- gation. Therefore, the genuine vacuum |0 ⟩can be viewed19as the superposition of ∣0e−;Ne−⟩and∣0e+;Ne+⟩: the electrons and positrons annihilate each other spontaneously so as to leave an empty vac- uum. In other words, the original hole theory of Dirac13for rela- tivistic electrons should be generalized to “charge-conjugated hole theory” or simply “extended hole theory.” This feature is incor- porated automatically into the symmetrized four-current operator (50) introduced first by Schwinger in 1951.20Given its great impor- tance, the expression (50) should be viewed as another corner- stone of QED. Note in passing that, in the free-particle (FP) rep- resentation, jμ vp(r)(52) vanishes pointwise, thereby leading to ˆjμ fp(x) =−e{ˆϕ†(x)cαμˆϕ}.Last but not least, it is worth mentioning that the commuta- tor form of contraction (54) is just a special case of the equal-time contraction (ETC)20of fermion operators, A(t)B(t)=⟨vac∣T[A(t)B(t)]∣vac⟩ (59) ≜1 2⟨vac∣T[A(t)B(t′)]∣vac⟩∣t′−t→0± (60) =⟨vac∣1 2[A(t),B(t)]∣vac⟩, (61) which is symmetric in time. That is, the two expressions A(t)B(t) and−B(t)A(t) obtained by letting t′approach tfrom the past and future are both considered and averaged here. Equation (61) is fundamentally different from the following ETC: A(t)B(t)=lim η→0+⟨0;˜0∣T[A(t)B(t+η)]∣0;˜0⟩, (62) which is asymmetric in time and holds only in the NRL. Note that the ETC (61) is only implicit in the Feynman fermion prop- agator19such that its importance is often overlooked in the lit- erature. Instead, we should regard it as an essential ingredient to distinguish relativistic from nonrelativistic quantum mechan- ics. As a time-independent analog of the ETC (61), the charge- conjugated contraction (CCC) of fermion operators was also introduced,21 apaq=⟨0;Ne−∣1 2[ap,aq]∣0;Ne−⟩,p,q∈PES, NES (63) =1 2⟨0;Ne−∣a˜pa˜q∣0;Ne−⟩∣ϵ˜p<0,ϵ˜q<0 −1 2⟨0;Ne−∣aqap∣0;Ne−⟩∣ϵp>0,ϵq>0 (64) =−1 2δp qsgn(ϵq),p,q∈PES, NES, (65) which distinguishes from the standard contraction apaq=⟨0;Ne−∣apaq∣0;Ne−⟩,p,q∈PES, NES (66) =δ˜p ˜qn˜q. (67) Although the introduction of CCC (65) looks very trivial, it is a key ingredient in a time-independent Fock space formulation of relativistic quantum mechanics. In particular, it allows us to con- struct19,21an effective QED (eQED) Hamiltonian in a bottom-up fashion (i.e., without ever recourse to QED, a time-dependent per- turbation theory). In contrast, the standard contraction (67), the time-independent analog of Eq. (62), will result in wrong, non- relativistic type of potential energy expressions even for relativis- tic operators. As an illustration, we look at the number operator, which reads J. Chem. Phys. 152, 180901 (2020); doi: 10.1063/5.0008432 152, 180901-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp ˆN=ap p+a˜p ˜p(68) ={ap p}+{a˜p ˜p}+⟨0;Ne−∣1 2[ap,ap]∣0;Ne−⟩ +⟨0;Ne−∣1 2[a˜p,a˜p]∣0;Ne−⟩ (69) ={ap p}+{a˜p ˜p}−1 2δp p+1 2δ˜p ˜p(70) ={ap p}+{a˜p ˜p} (71) according to Eq. (65) but reads ˆN={ap p}+{a˜p ˜p}+Ne− (72) according to Eq. (67), with Ne−→∞in line with the filled Dirac sea. It can readily be checked that the correct (71) and incorrect (72) results can also be obtained by using the contractions (65) and (67), respectively, in terms of the b-operators (29) and the associated vacuum |0 ⟩. Having discussed pedagogically the basics of QED [including first quantization (19) of special relativity (2), second quantization of the Dirac field (27) or equivalently (28), extended hole theory, sym- metrized 4-current (50), equal-time contraction (61), and charge- conjugated contraction (65)], we just comment briefly on the appli- cations of QED. Undoubtedly, QED is the most accurate theory ever designed in physics. For instance, the anomalous magnetic moment, (g−2)/2, of the electron has been determined to the 11th deci- mal place,22which leads further to improved values for the electron mass23and the fine structure constant.24However, the situation is very different for bound states of many-electron systems25for which QED is computationally too expensive: the more the electrons, the higher the order of perturbation and hence the more the Feynman diagrams required to achieve high precision. Because of this, rela- tivistic QED has thus far been applied successfully only to single ions of at most 5 electrons (see Refs. 26 and 27 for recent reviews). As for molecular systems, only nonrelativistic QED has been applied to the lightest molecules (e.g., H 2,28D2,29and HD30). Hence, the question is how to account for QED effects in heavy atoms and molecules. To show relevance of this question, we just quote a few results here: (1) according to the rough estimates,31the leading- order QED (Lamb shift) effects can be as large as 1 kcal/mol in chemical processes involving heavy elements. (2) According to the most recent and to date most accurate relativistic calculations32of the first ionization potential (IP) and electron affinity (EA) of the gold atom, QED effects are roughly the same as electron correlation beyond the gold standard CCSD(T) (coupled-cluster with singles and doubles and perturbative triples). (3) As for core properties such as the K-edge electron spectra, QED effects become signifi- cant already for the third row of the periodic table.33It is therefore clear that we do need a feasible relativistic QED approach for the electronic structure and spectroscopies of heavy atoms and even molecules. It is also clear that, at variance with the “relativity-QED then correlation” paradigm of QED, we should think of something like “first relativity then correlation and finally QED.”34Such effec- tive QED (eQED) approaches19,21,35,36do exist, which will be dis- cussed in Sec. III. Before this, we need to know how to solve the time-independent Dirac equation (21) via a finite basis expansion(see Sec. II). After having presented the eQED Hamiltonians in Sec. III, we will discuss in Sec. IV the correlation problem of NES (or virtual positrons) as well as a relativistic theory of real positrons. Section V is devoted to a summary of no-pair relativistic Hamiltoni- ans, whereas Sec. VI to the no-pair correlation problem. The account will be closed with perspectives in Sec. VII. II. THE MATRIX DIRAC EQUATION For brevity, consider first the Dirac equation for an electron moving in a local potential V, (V cσ⋅p cσ⋅pV−2c2)(ψL p ψS p)=(ψL p ψS p)ϵp, (73) where the rest-mass energy mc2has been subtracted to align the energy scale to that of the Schrödinger equation. Early attempts37 to solve this equation in a basis expansion were plagued by the occurrence of matrix eigenvalues in the forbidden region between the lowest positive-energy and the highest negative-energy operator eigenvalues. This phenomenon is usually called variational collapse and is often traced back to the lack of a lower-bound property of the Dirac operator. Actually, the “variational collapse” is due to the fact that inappropriately chosen basis sets are unable to describe the kinetic energy correctly and to guarantee the correct NRL.38It can be removed rigorously via the minimax principle39,40without the need to impose a lower-bound property on the matrix representation of the Dirac equation (73). On the practical side, several prescrip- tions have been proposed to construct suitable basis sets, includ- ing restricted kinetic balance (RKB),41unrestricted kinetic balance (UKB)42[NB: the acronyms RKB and UKB were first coined by Dyall and Fægri43], dual kinetic balance (DKB),44and inverse kinetic balance (IKB).45According to the thorough formal and numerical analyses,45the following conclusions can be drawn: (I) RKB is the least adequate condition for constructing the small-component spinor basis {fμ}2NL μ=1directly from the large-component set {gμ}2NL μ=1, viz., fμ=α 2σ⋅pgμ,α=c−1,μ=1,..., 2NL. (74) Why this is the case can best be understood in terms of the modified Dirac equation46,47 DMψM p=SMψM pϵp, (75) TM=(1 0 0α 2σ⋅p), (76) DM=T† MDTM=(V T Tα2 4σ⋅pVσ⋅p−T), (77) SM=T† MTM=(1 0 0α2 2T), (78) ψM p=T−1 Mψp=(ψL p ϕL p), (79) ψS p=α 2σ⋅pϕL p. (80) J. Chem. Phys. 152, 180901 (2020); doi: 10.1063/5.0008432 152, 180901-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp It has been proven48that the large ( ψL p) and pseudo-large (ϕL p) components must be expanded in the same spinor basis {gμ} in order to guarantee the correct NRL, a prerequisite to ensure that the energies of the PES are correct to O(c−2). Relation (80) then implies immediately the RKB (74). The expansion of ψpin the RKB basis (74) and that of ψM pin the {gμ} basis, i.e., ψp=(ψL p ψS p)=(gμAμp 0)+(0 fμBμp), (81) ψM p=(ψL p ϕL p)=(gμAμp 0)+(0 gμBμp), (82) give rise to the same matrix Dirac equation (V T Tα2 4W−T)(Ap Bp)=(S0 0α2 2T)(Ap Bp)ϵp, (83) where the individual matrices are all of dimension 2 NL, with the elements being Vμν=⟨gμ∣V∣gν⟩,Tμν=⟨gμ∣p2 2∣gν⟩, Wμν=⟨gμ∣σ⋅pVσ⋅p∣gν⟩,Sμν=⟨gμ∣gν⟩. (84) Equation (83) is therefore of dimension 4 NLwith 2 NLPES and 2 NLNES, which are separated by ca. 2 mc2≈1 MeV. When solving equation (83) iteratively, the energetically lowest PESs are chosen to be occupied in each iteration cycle so as to avoid variational collapse. While the rotations between the occupied and unoccupied PES lower the total energy, those between the occupied PES and unoccupied NES increase the total energy, although, to a much lesser extent. The following points concerning RKB still deserve to be highlighted. (a) The RKB condition does not provide full variational safety because the NESs are in error of O(c0).45 Depending very much on the construction of the large- component basis, some bounds failures (or prolapse49) of O(c−4) may occur. Nevertheless, such bound failures will diminish when approaching to the basis set limit at a rate that is not much different from the nonrelativistic counter- part.50(b) It turns out that the use of spherical Gaussians with principal quantum number nlarger than the angu- lar momentum lplus one leads to terrible variational col- lapse,45although such functions are valid in the nonrelativis- tic case. Therefore, the use of spherical Gaussians subject to the restriction n=l+ 1 (i.e., 1 s, 2p, 3d, 4f, 5g) is not merely a matter of economy but also a must. (II) IKB is the charge-conjugated version of RKB. It guarantees the correct NRL for the NES instead of the PES. Because of this, it requires basis functions that are very different from the standard ones and is therefore only of conceptual interest rather than of practical usage. (III) DKB combines the good of both RKB and IKB and even pro- vides full variational safety.45However, such an advantage islargely offset by its complicated nature and doubled num- ber of integrals compared to RKB. It is therefore recom- mended only for calculations of tiny quantities (e.g., QED and parity non-conserving effects), where the complexity of DKB is only minor compared to the high precision to be achieved. (IV) UKB is not uniquely defined. A scalar UKB basis does not transform as the basis of irreducible representations of dou- ble point groups or of time-reversal symmetry, although not a serious problem. More problematic is that an UKB basis often suffers from severe linear dependence. Moreover, UKB does not offer a faster convergence to the basis set limit than RKB. In short, RKB is the right choice for discretizing the Dirac equation in the absence of external magnetic fields. Since RKB is also a built- in condition for two-component relativistic theories,48it should be regarded as a cornerstone of relativistic quantum chemistry. In the presence of external magnetic fields, RKB can be generalized to Z=(Z11Z12 Z21Z22), (85) which leads to a most general expansion of ψp, ψp=Z˜ψp,˜ψp=(gμAμp gμBμp) =(Z11gμAμp+Z12gμBμp Z21gμAμp+Z22gμBμp). (86) Specific examples for the Zoperator (85) can be found from Refs. 35 and 51 and are not repeated here. III. THE eQED HAMILTONIAN Given the one-electron Dirac operator, the question is how to construct a relativistic many-electron Hamiltonian. The com- mon practice is to add in simply the Coulomb interaction. Since the instantaneous Gaunt and Breit interactions can also be derived in a semiclassical manner,52they can likewise be included, thereby leading to the Dirac–Coulomb (DC)/Dirac–Coulomb– Gaunt(DCG)/Dirac–Coulomb–Breit (DCB) Hamiltonian H=N ∑ i=1D(i)+1 2N ∑ i≠jV(rij), (87) D=cα⋅p+(β−1)mc2−NA ∑ AZA ∣RA−r∣, (88) V(r12)=VC(r12)+VB(r12), (89) VC(r12)=1 r12, (90) VB(r12)=VG(r12)+Vg(r12), (91) VG(r12)=−αi⋅αj r12, (92) Vg(r12)=α1⋅α2 2r12−(α1⋅r12)(α2⋅r12) 2r3 12. (93) J. Chem. Phys. 152, 180901 (2020); doi: 10.1063/5.0008432 152, 180901-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp On the formal side, the DCB Hamiltonian should be adopted as it is correct to O(α2), whereas so is neither DC nor DCG. Yet, on the practical side, the DCG Hamiltonian is more appealing, for it describes all inter-electronic spin-same-orbit, spin-other-orbit, orbit–orbit, and spin–spin interactions of O(α2)and is computa- tionally cheaper than DCB. That is, the difference between DCB and DCG is merely a scalar gauge term Vgthat is of minor impor- tance but leads to complicated integrals. As such, the DC and DCG Hamiltonians have been the major basis of relativistic quantum chemistry for molecular chemistry and physics. However, unlike the Schrödinger–Coulomb (SC) Hamiltonian that has well-defined mathematical and spectral properties, such ad hoc relativistic Hamil- tonians have serious problems.16,21Without going into details, suf- fice it to say here that such first-quantized Hamiltonians violate a fundamental law of relativistic quantum mechanics, viz., it is the charge instead of the number of particles that is conserved. Therefore, it is pointless to solve the DC/DCG/DCB equation HΨ =EΨexactly, unless one is interested in its mathematical solutions. Instead, to conserve the number of electrons, it is only consistent to adopt the no-pair approximation (NPA) from the outset, regardless of the existence of bound states or not. An immediate consequence is that the resulting energy Enpis not unique but is always depen- dent on how the projection operator is defined. Since the projector can only be defined in terms of the PES of some effective poten- tial, it can be said that Enpis always potential dependent, a situation that is very different from the FCI (full configuration interaction) solution of the Schrödinger equation. Even though such ambiguity can be largely removed by optimizing the potential/projector at a correlated level (e.g., no-pair full multiconfiguration self-consistent field including orbital rotations to the unoccupied NES53), how to account for the (dynamic) correlation of NES still remains to be resolved. This requires a “with-pair relativistic Hamiltonian” in the first place. As emphasized in the Introduction, the correct description of relativistic electrons must be done via second quantization. More specifically, it is the “extended hole theory,” the field Dirac picture coupled with charge conjugation, that is the proper tool for con- structing many-electron relativistic Hamiltonians. To begin with, aprimitive second quantization of the Dirac matter field can be introduced, viz., ˆϕ(r)=apψp(r),ap∣vac⟩=0, p∈PES, NES, (94) where the spinors are eigenfunctions of the following effective Dirac equation: (D+U)ψp=ϵpψp, (95) with Ubeing some local or nonlocal screening potential. The term “primitive” here means that this form of second quantiza- tion does not distinguish the empty from the filled Dirac picture. This gives rise to the following normal-ordered, second-quantized DC/DCG/DCB Hamiltonian H=Dq pap q+1 2grs pqapq rs,p,q,r,s∈PES, NES, (96) Dq p=⟨ψp∣D∣ψq⟩,grs pq=⟨ψpψq∣V(r12)∣ψrψs⟩, (97) ap q=apaq,apq rs=apaqasar. (98)The filled Dirac picture can be realized in a finite basis representation by setting the Fermi level below the energetically lowest of the ˜N (=Ne−) occupied NES. The physical energy of an N-electron state can be calculated16as the difference between those of states Ψ(N;˜N) andΨ(0;˜N), E=⟨Ψ(N;˜N)∣H∣Ψ(N;˜N)⟩−⟨Ψ(0;˜N)∣H∣Ψ(0;˜N)⟩, (99) provided that the charge-conjugation symmetry is incorporated properly. To do so, we first shift the Fermi level just above the top of the NES. This amounts to normal ordering the Hamiltonian H (96) with respect to the non-interacting vacuum |0; ˜N⟩(=∣0e−;Ne−⟩) of zero positive energy electrons and ˜Nnegative-energy electrons. Here, the CCC (65) of fermion operators21must be invoked so as to obtain ap q={apaq}n+⟨0;˜N∣1 2[ap,aq]∣0;˜N⟩,p,q∈PES, NES, (100) ={apaq}n−1 2δp qsgn(ϵq),p,q∈PES, NES, (101) Dq pap q=Dq p{apaq}n+C1n,C1n=−1 2Dp psgn(ϵp), (102) where the subscript nof the curly brackets emphasizes that the nor- mal ordering is taken with respect to the reference |0; ˜N⟩. More specifically, {apaq}n=⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩apaq,ϵp>0,ϵq>0 apaq,ϵp>0,ϵq<0 apaq,ϵp<0,ϵq>0 −aqap,ϵp<0,ϵq<0.(103) By applying the relation (65) repeatedly, we obtain apq rs={apq rs}n−1 2{δp raq ssgn(ϵr)+δq sap rsgn(ϵs) −δq rap ssgn(ϵr)−δp saq rsgn(ϵs)}n +1 4(δp rδq s−δq rδp s)sgn(ϵr)sgn(ϵs), (104) and hence, 1 2grs pqapq rs=1 2grs pq{apq rs}n+Qq p{ap q}n+C2n, (105) Qq p=˜Qq p+¯Qq p=−1 2¯gqs pssgn(ϵs), (106) ˜Qq p=−1 2gqs pssgn(ϵs), (107) ¯Qq p=1 2gsq pssgn(ϵs), (108) C2n=1 8¯gpq pqsgn(ϵp)sgn(ϵq)=−1 4Qp psgn(ϵp). (109) Note that the implicit summations in C1n(102), ˜Q(107), ¯Q(108), and C2n(109) include all the PES and NES, whether occupied J. Chem. Phys. 152, 180901 (2020); doi: 10.1063/5.0008432 152, 180901-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp or not. The Hamiltonian H(96) in the filled Dirac picture can then be written as H=HQED a +Cn, (110) HQED a=HFS a+Qq p{ap q}n, (111) HFS a=Dq p{ap q}n+1 2grs pq{apq rs}n, (112) Cn=C1n+C2n=⟨0;˜N∣H∣0;˜N⟩ =−1 2Dp psgn(ϵp)−1 4Qp psgn(ϵp). (113) HQED a (111) is just the desired “with-pair relativistic Hamiltonian” or simply effective QED (eQED) Hamiltonian, whereas HFS ais the so-called Fock space Hamiltonian advocated by Kutzelnigg54(see also Ref. 52), which missed, by construction, the vacuum polariza- tion (VP) ˜Q(107) [see Fig. 2(b)] and electron self-energy (ESE) ¯Q (108) [see Fig. 2(c)]. If wanted, HQED a (111) can also be expressed in terms of the b-operators in view of the relations (29), viz., HQED b=(D+Q)q p{bpbq}+(D+Q)q ˜p{b˜pbq}+(D+Q)˜q p{bpb˜q} +(D+Q)˜q ˜p{b˜pb˜q}+1 4¯grs pq{bpbqbsbr}+1 2¯grs ˜pq{b˜pbqbsbr} +1 2¯g˜rs pq{bpbqbsb˜r}+1 4¯grs ˜p˜q{b˜pb˜qbsbr} +1 4¯g˜r˜s pq{bpbqb˜sb˜r}+¯g˜rs ˜pq{b˜pbqbsb˜r}+1 2g˜rs ˜p˜q{b˜pb˜qbsb˜r} +1 2¯g˜r˜s p˜q{bpb˜qb˜sb˜r}+1 4¯g˜r˜s ˜p˜q{b˜pb˜qb˜sb˜r}, (114) where the normal ordering is taken with respect to |0 ⟩. Note that the eQED Hamiltonian (111) and (114) can also be obtained by a diagrammatical procedure.17,19Had the standard contraction (67) of fermion operators been taken, we would obtain the following “configuration space” (CS) Hamiltonian: HCS a=HFS a+¯gq˜j p˜j{ap q}n. (115) At variance with the Qpotential (106), the potential ¯gq˜j p˜jhere arises from the occupied NES {ψ˜j}alone [which is also a conventional FIG. 2 . Diagrammatical representation of the (a) one-photon exchange, (b) vacuum polarization, and (c) electron self-energy.interpretation of Fig. 2(b)]. It is infinitely repulsive, leading to that no atom would be stable. This is of course plainly wrong. Finally, the proper evaluation of the Qpotential (106) should be discussed. The Coulomb-only ˜Qterm (107) is the full vacuum polar- ization55due to the polarization density ρvp(55). In practice, it can be split into the Uehling56and Wichmann–Kroll57terms, which can then be evaluated with the analytic formulae.58,59The ESE term ¯Q (108) is more difficult to handle. In addition to the Coulomb inter- action, the transverse-photon contribution should also be included. In the Coulomb gauge adopted here, the transverse part of the ESE reads60 (¯QT)q p=1 2⟨p∣ΣC T(ϵp)+ΣC T(ϵq)∣q⟩, (116) ⟨p∣ΣC T(ϵp)∣q⟩=⟨ps∣∫∞ 0cdkfC T(k,r1,r2) ϵp−ϵs−(ck−iγ)sgn(ϵs)∣sq⟩, (117) fC T(k,r1,r2)=sin(kr12) πr12[α1⋅α2−(α1⋅∇1)(α2⋅∇2) k2]. (118) Therefore, the total ESE (still denoted as ¯Q) can be written in a symmetric form ¯Qq p=1 2⟨p∣ΣC C+ΣC T(ϵp)+ΣC T(ϵq)∣q⟩,⟨p∣ΣC C∣q⟩=gsq pssgn(ϵs). (119) It has recently been shown that the ESE (119) can be fitted into a simple and accurate semilocal model operator for each atom.61,62 Therefore, the VP-ESE (Lamb shift) can readily be included in the mean-field treatment so as to account for screening effects on the VP-ESE automatically. IV. APPLICATION OF THE eQED HAMILTONIAN In this section, the occupied PES and NES are to be denoted by {i,j,...} and{˜i,˜j,...}, respectively, whereas the unoccupied PES by {a,b,...}. Unspecified orbitals are denoted as { p,q,r,s}. When necessary, the NES will explicitly be designated by {˜p,˜q,˜r,˜s}. A. The second-order QED energy of an N-electron system The eQED Hamiltonian HQED a (111) or HQED b(114) can be employed in the Bloch equation to determine the wave operators order by order. The resulting energy expressions are in full agree- ment with those obtained by the S-matrix formulation of QED.21 However, the procedure treating all the PES as particles is rather involved. It is more expedite16to calculate the physical energy according to Eq. (99) by treating the occupied PES also as holes. That is, to calculate the first term on the right-hand side of Eq. (99), the eQED Hamiltonian HQED a (111) can be further normal-ordered with respect to the non-interacting reference | N;˜N⟩, the zero order ofΨ(N;˜N). Since the normal ordering is now taken with respect to the occupied PES alone, the standard contraction of Fermion operators, e.g., apaq=⟨N;˜N∣{apaq}n∣N;˜N⟩=⟨N; 0∣apaq∣N; 0⟩=δp qnq,ϵq>0, (120) J. Chem. Phys. 152, 180901 (2020); doi: 10.1063/5.0008432 152, 180901-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp should be invoked. More specifically, Dq p{ap q}n=Dq p{ap q}F+Di i, (121) Qq p{ap q}n=Qq p{ap q}F+Qi i, (122) 1 2grs pq{apq rs}n=1 2grs pq{apq rs}F+(VHF)q p{ap q}F+1 2(VHF)i i. (123) Consequently, we have35 H=HQED F +CF, (124) HQED F=fq p{ap q}F+1 2grs pq{apq rs}F, (125) fq p=(fe)q p+Qq p, (126) (fe)q p=Dq p+(VHF)q p,(VHF)q p=¯gqj pj, (127) CF=Cn+E[1]=(D+VHF+Q)i i−(1 2D+1 4Q)p psgn(ϵp), (128) E[1]=E[1] np+Qi i, (129) E[1] np=(D+1 2VHF)i i. (130) To facilitate the use of many-body perturbation theory (MBPT) for electron correlation, the Hamiltonian (124) can further be parti- tioned as H=H0A+V0A+V1A+V2A, (131) H0A=ϵp{ap p}F+N ∑ iϵi−1 2ϵpsgn(ϵp), (132) V0A=(Q−U+1 2VHF)i i+(1 2U−1 4Q)p psgn(ϵp), (133) V1A=(V1A)q p{ap q}F,(V1A)q p=(Q+VHF−U)q p, (134) V2A=1 2grs pq{apq rs}F, (135) where the appearance of the counter potential −Uq p{ap q}=−Uq p{ap q}n+1 2Up psgn(ϵp) =−Uq p{ap q}F−Ui i+1 2Up psgn(ϵp) (136) is due to the fact that the general mean-field equation (95) has been employed to determine the spinors and energy levels. As for the second term of Eq. (99), the Hamiltonian (110) can be partitioned as H=H0B+V0B+V1B+V2B, (137) H0B=ϵp{ap p}n−1 2ϵpsgn(ϵp), (138) V0B=(1 2U−1 4Q)p psgn(ϵp), (139) V1B=(V1B)q p{ap q}n,V1B=Q−U, (140) V2B=1 2grs pq{apq rs}n. (141)Following the standard MBPT, we obtain immediately E(0)=−1 2ϵpsgn(ϵp−ϵF)+1 2ϵpsgn(ϵp)=N ∑ iϵi, (142) E(1)=V0A−V0B=(1 2VHF−U+Q)i i, (143) E(2)=E(2) 1+E(2) 2, (144) E(2) 1=⎡⎢⎢⎢⎢⎣(V1A)a i(V1A)i a ϵi−ϵa+(V1A)a ˜i(V1A)˜i a ϵ˜i−ϵa⎤⎥⎥⎥⎥⎦ −⎡⎢⎢⎢⎢⎣(V1B)i ˜i(V1B)˜i i ϵ˜i−ϵi+(V1B)a ˜i(V1B)˜i a ϵ˜i−ϵa⎤⎥⎥⎥⎥⎦(145) =E(2) FS,1+E(2) Q,1, (146) E(2) FS,1=(VHF−U)a i(VHF−U)i a ϵi−ϵa+(VHF−U)a ˜i(VHF−U)˜i a ϵ˜i−ϵa −Ua ˜iU˜i a ϵ˜i−ϵa−Ui ˜iU˜i i ϵ˜i−ϵi, (147) E(2) Q,1=(VHF−U)a iQi a+Qa i(VHF−U)i a+Qa iQi a ϵi−ϵa +(VHF)a ˜iQ˜i a+Qa ˜i(VHF)˜i a ϵ˜i−ϵa−Qi ˜iQ˜i i−Ui ˜iQ˜i i−Qi ˜iU˜i i ϵ˜i−ϵi, (148) E(2) 2=1 4¯gab mn¯gmn ab ϵm+ϵn−ϵa−ϵb∣m,n=i,j,˜i,˜j −1 4¯gpq ˜i˜j¯g˜i˜j pq ϵ˜i+ϵ˜j−ϵp−ϵq∣p,q=i,j,a,b (149) =⎡⎢⎢⎢⎢⎢⎣1 4¯gab ij¯gij ab ϵi+ϵj−ϵa−ϵb+1 2¯gab i˜j¯gi˜j ab ϵi+ϵ˜j−ϵa−ϵb⎤⎥⎥⎥⎥⎥⎦ −⎡⎢⎢⎢⎢⎢⎣1 4¯gij ˜i˜j¯g˜i˜j ij ϵ˜i+ϵ˜j−ϵi−ϵj+1 2¯gia ˜i˜j¯g˜i˜j ia ϵ˜i+ϵ˜j−ϵi−ϵa⎤⎥⎥⎥⎥⎥⎦. (150) The first and second terms of E(2) 1(145) and E(2) 2(149) arise from theΨ(N;˜N) and Ψ(0;˜N) states, respectively. The one-body E(2) 1 (145) can further be decomposed into two terms, E(2) FS,1(147) and E(2) Q,1(148). Both E(2) FS,1(147) and E(2) 2(150) arise from the Fock space Hamiltonian54HFS a(112), while E(2) Q,1(148) is due exclusively to the VP and ESE [NB: E(2) Q,1and the Qterm in E(1)will not show up if theQpotential is included in the mean-field equation (73)]. The two terms of E(2)(144) can be represented by the same Goldstone-like diagrams shown in Fig. 3. It is just that the particles and holes, as well as the one-body potential, are interpreted differently. Note that the frequency-dependent Breit interaction J. Chem. Phys. 152, 180901 (2020); doi: 10.1063/5.0008432 152, 180901-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp FIG. 3 . Diagrammatical representation of the second-order QED energy. (a) Anti- symmetrized two-body; (b) one-body. For the Ψ(N;˜N) state, the particles (up- going lines) and holes (down-going lines) are { a,b} and {i,j,˜i,˜j}, respectively, and the one-body potential represented by the square is V1A. For the Ψ(0;˜N) state, the particles and holes are { a,b,i,j} and {˜i,˜j}, respectively, and the one- body potential is V1B. A global negative sign should be inserted to the terms of Ψ(0;˜N). VT(ω,r12)=−α1⋅α2cos(∣q∣r12) r12,ω=qc +[(α1⋅∇1),[(α2⋅∇2),cos(∣q∣r12)−1 q2r12]](151) must be employed to account for the contribution of NES to cor- relation, which would otherwise be severely overestimated if the frequency dependence is neglected.63This amounts to replacing the integrals grs pqinE(2) 2(150) with grs pq=⟨pq∣VC(r12)+1 2VT(ϵr−ϵp,r12)+1 2VT(ϵs−ϵq,r12)∣rs⟩. (152) Here, we should recall again the most recent and to date most accu- rate relativistic calculations32of the IP and EA of Au: the calcu- lated IP (9.2288 eV) deviates from the experimental one (9.2256 eV) somewhat larger than the corresponding EA (calculated 2.3072 vs experimental 2.3086 eV). This is counterintuitive, since IPs can be usually calculated more accurately than EAs. There could be two possible reasons for this: (a) the no-pair correlation still needs to be improved and (b) the missing contribution of NES to correlation has to be taken into account. Even if the contribution of NES to correla- tion is not the reason for such discrepancy, it is certainly important for core properties. It also deserves to be mentioned that the same E(2) 2(149) would correspond to the seven Feynman diagrams shown in Fig. 4, if the occupied PES are to be treated as particles instead of holes as in Fig. 3(a). The first three and the next four of these Feynmandiagrams are usually called non-radiative and radiative contribu- tions in QED, but all of which are contributions to electron cor- relation in the present context. It is now clear that only the Q potential (106) entering the eQED Hamiltonian (111) and (114) arises from the unconventional contraction (65) of fermion oper- ators, whereas the treatment of electron correlation follows stan- dard many-body theories in conjunction with the filled Dirac picture. This is because the nth order correlation energy E(n) =⟨vac|VΩ(n−1)|vac⟩arises from the (full) contraction between the first-order fluctuation potential Vand the ( n−1)th-order wave operator Ω(n−1)that are already normal ordered separately and hence originate from “different times.” In the parlance of diagrams, all the terms herein refer to reducible Feynman diagrams (see Fig. 4). In other words, only those irreducible Feynman diagrams that go beyond the eQED Hamiltonian (which is, by definition, linear in the two-particle interaction involving only one-photon exchange) must be treated via full QED, a time-dependent perturbation theory. Finally, it is instructive to compare the second-order QED energy E(2)(144) with that of the configuration space approach,16 E(2) CS=E(2) CS,1+E(2) CS,2, (153) E(2) CS,1=(VHF−U)a i(VHF−U)i a ϵi−ϵa+(VHF−U)˜i i(VHF−U)i ˜i ϵi−ϵ˜i, (154) E(2) CS,2=1 4¯gab ij¯gij ab ϵi+ϵj−ϵa−ϵb+1 4¯g˜i˜j ij¯gij ˜i˜j ϵi+ϵj−ϵ˜i−ϵ˜j+1 2¯ga˜j ij¯gij a˜j ϵi+ϵj−ϵa−ϵ˜j. (155) It is seen that E(2) CS,1and E(2) CS,2agree with E(2) FS,1(147) and E(2) 2 (150), respectively, only in the first terms involving solely the PES but are very different from the latter in the terms involving the NES. In particular, the denominator of the last term of Eq. (155) can be zero (e.g., ϵa=ϵi+ |X| andϵ˜j=ϵj−∣X∣). Since there exists an infinite number of such “+ −” intermediates, this prob- lem has been termed continuum dissolution.64As shown here, it is purely an artifact due to the underlying empty Dirac picture. As such, only no-pair projected wave functions are acceptable in the configuration space formulation. Efforts65to solve exactly the DC/DCG/DCB equation HΨ=EΨare then purely mathematical exercises. Since there is no analytic Hamiltonian in Fock space as well, the term “exact (analytic) relativistic wave function” is simply meaningless.66 FIG. 4 . [(a)–(g)] Feynman diagrams for E(2) 2. J. Chem. Phys. 152, 180901 (2020); doi: 10.1063/5.0008432 152, 180901-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp B. Mean-field theory of real positrons As an another application of the eQED Hamiltonian (111), we present here a mean-field theory for a system of Nelectrons and ˜M positrons. The energy up to first order reads E[1] ep=⟨N;˜N∣A†HQED aA∣N;˜N⟩,A=Π˜M ˜ia˜i (156) =⎡⎢⎢⎢⎢⎣N ∑ i=1(D+Q)i i+1 2N ∑ i,j=1¯gij ij⎤⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎣−˜M ∑ ˜i=˜1(D+Q)˜i ˜i +1 2˜M ∑ ˜i,˜j=˜1¯g˜i˜j ˜i˜j⎤⎥⎥⎥⎥⎦−N ∑ i=1˜M ∑ ˜j=˜1¯gi˜j i˜j. (157) Use of Wick’s theorem for expressing products of normal-ordered operators as a linear combination of contracted ones has been made when going from Eqs. (156) to (157). The first and sec- ond terms are the average energies of the Nelectrons and of the ˜Mpositrons, respectively, whereas the third, cross term represents their mutual interaction. The negative sign in the second and third terms results from the normal ordering (103) implicit in HQED a (111) and can be understood as a negative occupation number (n˜i=−1) of the hole arising from the ionization a˜i|N;˜N⟩. The other occupied NES not involved in the ionization Π˜M ˜ia˜i∣N;˜N⟩has been normal-ordered away and can therefore be viewed as unoccupied, just like the unoccupied PES. As such, the expression (157) can be written as E[1] ep=∑ knk(D+Q)k k+1 2∑ k,lnknl¯gkl kl,k,l∈PES, NES, (158) by assigning an occupation number nkto each orbital ψk:nkis zero for the unoccupied PES and NES, + 1 for the Noccupied PES, and −1 for the ˜Moccupied NES. Formally, this agrees with the empty Dirac picture. However, such agreement between the empty and filled Dirac pictures holds only at the mean-field level but not at the correlated level (see Sec. IV A). More generally, such agreement holds for all one-body but not for any two-body operators.16 To minimize the energy E[1] ep(158) subject to the orthonormal conditions, we can introduce the following canonical Lagrangian: L=E[1] ep−∑ knk[⟨ψk∣ψk⟩−1]ϵk,k∈PES, NES. (159) The conditionδL δψ† i=0 then gives rise to fni∣ψi⟩=ϵini∣ψi⟩,i∈PES, NES, (160) where f=D+Q+∑ knk¯gk⋅ k⋅,k∈PES, NES, (161) fq p=(D+Q)q p+∑ knk¯gkq kp,k∈PES, NES. (162) As it stands, Eq. (160) determines only the occupied PES and NES, which can be extended to the unoccupied ones (which are arbitrary anyway), viz.,f∣ψp⟩=ϵp∣ψp⟩,p∈PES, NES. (163) The energetically lowest PES and highest NES are to be occupied in each iteration. Some remarks are in order: (a) the cross, exchange term −∑i˜jg˜ji i˜j vanishes in the NRL, meaning that electrons and positrons are dis- tinguishable particles in the nonrelativistic world such that their mutual anti-symmetrization is no longer required. In other words, only QED treats electrons and positrons on an equal footing. (b) If the VP-ESE term Qis neglected, the present mean-field theory of electrons and positrons will reduce to that formulated by Dyall in a different way.67 V. NO-PAIR RELATIVISTIC HAMILTONIANS There have been a number of comprehensive reviews35,48,68–71 on the no-pair relativistic Hamiltonians. Therefore, only a brief sum- mary of the essentials is necessary here. The no-pair relativistic Hamiltonians can be classified into four-component (4C), quasi- four-component (Q4C), and two-component (2C) ones, the last of which can further be classified into approximate (A2C) and exact (X2C) two-component ones. A. Four-component First of all, confining the orbital indices of HQED a (111) only to PES leads to the following no-pair QED Hamiltonian: HQED +=Dq pap q+Qq pap q+1 2grs pqapq rs,p,q,r,s∈PES (164) =(fe+Q)q p{ap q}N+1 2grs pq{apq rs}N+E[1],p,q,r,s∈PES, (165) with(fe)q pand E1defined in Eqs. (127) and (129), respectively. Here, the subscript Nindicates that the normal ordering is taken with respect to ∣N;˜0⟩. The HQED + Hamiltonian (164), along with ˜Q(107), ¯Q(119), and grs pq(152), was already obtained by Shabaev36but in a top-down fashion. The aforementioned potential dependence in the calculated energies can be removed by introducing the following correction:72 E(2) PC=(VHF)i ˜iU˜i i+Ui ˜i(VHF)˜i i−Ui ˜iU˜i i ϵ˜i−ϵi, (166) where Uis the potential in Eq. (73). One then has a potential- independent no-pair QED (PI-QED) Hamiltonian21 HPI-QED +=(fe+Q−U)q p{ap q}N+1 2grs pq{apq rs}N +E[1]+E(2) PC,p,q,r,s∈PES. (167) Neglecting the Qterm in HPI-QED + leads to the potential-independent no-pair DCB (PI-DCB) Hamiltonian J. Chem. Phys. 152, 180901 (2020); doi: 10.1063/5.0008432 152, 180901-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp HPI-DCB +=(fe−U)q p{ap q}N+1 2grs pq{apq rs}N+E[1] NP+E(2) PC, p,q,r,s∈PES. (168) Further neglecting the Qterm in HPI +leads to the standard no-pair DCB Hamiltonian HDCB +=(fe)q p{ap q}N+1 2grs pq{apq rs}N+E[1] np,p,q,r,s∈PES, (169) which has been the basis of “no-pair relativistic quantum chemistry.” B. Quasi-four-component The previous no-pair four-component approaches first gener- ate both PES and NES at the mean-field level but then discard the NES at a correlated level. The question is how to avoid the NES from the outset. Actually, this can be done in two different ways. One is to retain the aesthetically simple four-component structure butfreeze the NES, while the other is to remove the NES so as to obtain a two-component approach. While the former employs the untransformed Hamiltonian and introduces approximations from the very beginning, the latter invokes an effective Hamiltonian and has to introduce suitable approximations at a later stage. Note that in each case, the approximations introduced to the Hamilto- nians are orders of magnitude smaller than other sources of errors (e.g., incompleteness in the one- and many-particle bases) and are therefore hardly “approximate.” Moreover, since the two paradigms stem from precisely the same physics, they should be made fully equivalent. To realize the first paradigm, we first take a look at the S/Lratio between the small and large components of a PES ψi, which can be obtained from the second row of Eq. (73), ψS i=α 2Riσ⋅pψL i, (170) Ri(r)=[1 +α2 2(ϵi−V(r))]−1 α→0→1. (171) The major effect of σ⋅pis to change the parity of the large com- ponent to that of the small component. Hence, the S/Lratio is determined mainly by the Ri(r) operator (171). As can be seen from Fig. 5, the effect of Ri(r) is extremely short ranged: each Ri(r) becomes just a constant factor beyond a small radius rc(ca. 0.05 a.u., roughly the radii of 2 sand 2 p). Imagine that we have first solved the (radial) Dirac equation for each isolated (spherical and unpolarized) atom and thus obtained the corresponding atomic 4- spinors (A4S) { φμ}. Then, the atoms are brought together to syn- thesize the molecule. While both the large and small components ofφμwill change, the S/Lratio will not!.73,74The mathematical realization75of such a physical picture is to expand the molec- ular 4-spinors (M4S) ψiin the basis only of positive-energy A4S {|φ+,μ⟩}, viz., ∣ψi⟩=∑ μ∣φ+,μ⟩Cμi=∑ μ(∣φL +,μ⟩ ∣φS +,μ⟩)Cμi, (172) FIG. 5 . The Ri(r) operator (171) with V=VN+VH+VLDAas a function of the distance from the position of Rn. The radial expectation values of 1 s1/2, 2s1/2, 2p1/2, 2p3/2, and 3 s1/2are 0.015, 0.063, 0.051, 0.060, and 0.163 a.u., respectively. which gives rise to the following projected four-component (P4C) approach:75 hP4C +C=SP4C +Cϵ, (173) (hP4C +)μν=⟨φL +,μ∣V∣φL +,ν⟩+⟨φS +,μ∣cσ⋅p∣φL +,ν⟩ +⟨φL +,μ∣cσ⋅p∣φS +,ν⟩+⟨φS +,μ∣V−2mc2∣φS +,ν⟩, (174) (SP4C +)μν=⟨φL +,μ∣φL +,ν⟩+⟨φS +,μ∣φS +,ν⟩. (175) The dimension of hP4C + is 2NLinstead of 4NL. That is, the molecular NES are excluded completely. Physically, this amounts to neglecting rotations between the PES and NES of the iso- lated atoms, a kind of polarization on the atomic vacua induced by the molecular field. As molecular formation is a very-low energy process, its O(c−4)perturbation on the vacuum intro- duces no discernible errors at all.73,75,76By further introducing a “model small component approximation” (MSCA), a quasi-four- component (Q4C) approach76can be obtained, which is four- component in structure but is computationally very much like a two-component approach. Without going into further details (see Refs. 48 and 71 for the matrix elements fQ4C pq offQ4C +), we now have the following second-quantized, normal-ordered many-electron Hamiltonian HQ4C +=EQ4C ref+fQ4C pq{ap q}+1 2grs pq{apq rs}. (176) Q4C shares precisely the same integral transformation and correla- tion treatment as two-component approaches48,71but does not suffer from picture-change errors (PCE),77which otherwise plague two- component approaches. Moreover, the model spectral form61,62of theQpotential (106) can readily be incorporated into fQ4C pq, thereby leading to an QED@Q4C approach. J. Chem. Phys. 152, 180901 (2020); doi: 10.1063/5.0008432 152, 180901-13 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp C. Two-component By definition, a two-component relativistic theory is to trans- form away the positronic degrees of freedom of the Dirac operator so as to obtain a Hamiltonian that describes only electrons. This can be done with either unitary transformation or elimination of the small component (ESC). However, neither route can be done in closed form, except for the trivial free-particle case. As such, only approxi- mate two-component (A2C) operator (analytic) Hamiltonians such as the Breit–Pauli Hamiltonian and the zeroth-order regular approx- imation (ZORA)78,79can be obtained in this way. The situation is changed dramatically when going to the matrix formulation, where the exact decoupling is readily achieved. In essence, the matrix for- mulation amounts to block-diagonalizing the matrix Dirac equa- tion (83), which can be done in one step,80–83two steps,84–86and multiple steps.87–89The three types of formulations share the same decoupling condition and differ only in the renormalization.48There exist even closed mapping relations among three formulations.48 Since the initio free-particle transformation invoked in the two-step and multiple-step formulations is only necessary for finite orders90,91 but not for infinite order, it is clear that it is the one-step formulation that should be advocated. This approach has been coined “exact two- component” (X2C).92For generality, we extend Eq. (83) to a generic eigenvalue problem hC=MCE, (177) h=(h11h12 h21h22)=h†,M=(S110 0 S 22)=M†, C=(A+A− B+B−). (178) To decouple the PES and NES, we first introduce the formal relations B+=XA +,A−=˜XB− (179) between the small- and large-component coefficients for the PES and NES, respectively. The following unitary transformation matrix UX can then be introduced,48 UX=ΩNΩD,ΩN=(R† +0 0 R† −),ΩD=(I X† ˜X†I), (180) where81 R+=(S−1 11˜S+)−1 2=S−1 2 11(S−1 2 11˜S+S−1 2 11)−1 2S1 2 11, (181) R−=(S−1 22˜S−)−1 2=S−1 2 22(S−1 2 22˜S−S−1 2 22)−1 2S1 2 22, (182) ˜S+=S11+X†S22X, (183) ˜S−=S22+˜X†S11˜X. (184) The requirement that UXMU† X=Mleads to ˜X=−S−1 11X†S22, (185) meaning that ˜Xis determined directly by X, which is further deter- mined by (UXhU† X)21=0, viz.,h21+h22X=S22XS−1 11LUESC + ,LUESC +=h11+h12X, (186) =S22X˜S−1 +LNESC + . (187) TheUX-transformation of Eq. (177) then yields (UXhU† X)CX=(fX2C +0 0 fX2C −)CX=MC XE, (188) CX=(U† X)−1C=M−1UXMC=(C+0 0 C−). (189) The upper-left block of Eq. (188) defines the equation for the PES, fX2C +C+=S11C+E+, (190) fX2C +=R† +LX +R+, X=NESC, SESC, (191) LNESC +=h11+h12X+X†h21+X†h22X, (192) LSESC +=1 2(˜S+S−1 11LUESC + +c.c.), (193) C+=R−1 +A+. (194) Here, the acronyms UESC, NESC, and SESC refer to the unnor- malized, normalized,93and symmetrized73eliminations of the small component, respectively. Equation (187) arises from Eq. (186) via the relation S−1 11LUESC +=˜S−1 +LNESC + (because LUESC +A+=S11A+E+ andLNESC +A+=˜S+A+E+), whereas Eq. (193) arises from LSESC + =1 2(LNESC + +LNESC +)and the decoupling condition (186). Like- wise, the lower-right block of Eq. (188) defines the equation for the NES, fX2C −C−=S22C−E−, (195) fX2C −=R† −LX −R−, X=NESC, SESC, (196) LNESC −=h22+h21˜X+˜X†h12+˜X†h11˜X, (197) LSESC −=1 2(˜S−S−1 22LUESC − +c.c.),LUESC −=h22+h21˜X, (198) C−=R−1 −B−. (199) It can be proven94thatC+(C−) is closest to A+(B−) in the least- squares sense. The above manipulation can further be extended to include magnetic fields as well.35,71Moreover, at variance with the explicit expression (191), fX2C + can also be constructed on the fly by an orthonormalization and back-transformation procedure.83The fol- lowing remarks are still in order: (1) The one-step matrix formulation of two-component rela- tivistic theories was initiated by Dyall93in 1997. However, the proper formulation of the (energy-independent) decou- pling condition (186) and (187)80and the correct renor- malization (181)81were found only later on. It was also found48that the same results can be obtained by converting the Foldy–Wouthuysen (FW) Hamiltonian95(which has no closed form though) directly into matrix form in terms of the RKB basis. That is, the matrix and operator (more precisely J. Chem. Phys. 152, 180901 (2020); doi: 10.1063/5.0008432 152, 180901-14 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp operator-like) formulations of X2C are identical, as should be. The situation is different for finite-order A2C approaches. To see this, we look at the ZORA equation,78,79 (V+TZORA)ψZORA p=ψZORA pϵZORA p , (200) TZORA=σ⋅p1 2−α2Vσ⋅p. (201) In view of the identity 1/(2 −α2V)×(2−α2V) = 1, the matrix elements of TZORAcan be calculated as ⟨σ⋅pgμ∣1 2−α2V∣σ⋅pgρ⟩[(2T)−1]ρσ⟨σ⋅pgσ∣2−α2V∣σ⋅pgν⟩ =2Tμν, (202) which leads to TZORA=TXZORA,XZORA=(T−α2 4W)−1T. (203) Therefore, the matrix representation of the ZORA equa- tion (200) reads (V+TXZORA)AZORA=SAZORAϵZORA. (204) Thanks to the use of the resolution of the identity (RI) in terms of the { σ⋅pgμ} basis in Eq. (202), the matrix ZORA equation (204) agrees with the operator ZORA equation (200) only when the basis { gμ} is complete. This is totally differ- ent from the matrix counterpart (i.e., X2C) of the (non- expanded) FW Hamiltonian,95where the use of the same RI is notan approximation but only a formal step.48Since Eq. (204) is never used in practice, the commonly called ZORA (also the infinite-order regular approximation96) is a genuine ana- lytic relativistic theory. In contrast, other relativistic theories, whether finite-order90,91or infinite-order,80–89are all alge- braic. However, the analyticity of a relativistic Hamiltonian should not be celebrated simply because only Fock space is the right framework for relativistic quantum mechanics, which gives rise to only algebraic relativistic many-electron Hamiltonians (see Sec. III). (2) The eigenvalue equation (190) and the decoupling condition (186) and (187) are coupled and have to be solved itera- tively.80,82The so-obtained results agree with those by the parent matrix Dirac equation (83) up to machine accuracy, thereby justifying the name “exact two-component.”92How- ever, the computation is much more expensive than solving Eq. (83) directly, even for a one-electron system. Therefore, a suitable approximation to Xmust be found in order to make X2C practical. To this end, we take a look at the matrix presentation of the key relation (170) in a RKB basis (74) (without caring for the inherent singularities97), viz., B+,i=1 2T−1R(i)A+,i,R(i) μν=⟨gμ∣σ⋅pRiσ⋅p∣gν⟩, (205) =U(i)A+,i, (206) where U(i)is the energy/state-dependent equivalent93of the state-universal X. Since the Ri(r) operator is extremelyshorted ranged [see Fig. (5)], it can be envisaged that the molecular U(i)(and hence X) should be strongly block- diagonal in atoms. As shown in Fig. 6, this is indeed the case. Note, in particular, that to enhance the interatomic interac- tion, we have set the interatomic distance of Au 2to 1.5 Å, which is much shorter than the equilibrium distance of 2.47 Å. Therefore, a general deduction is that the molecular X can be well approximated as the superposition of the atomic ones,73,74,76,98 X=⊕ ∑ FXF, (207) which stays in the same spirit as P4C (see Point 3 below). Here, each atomic XFcan, in view of the very definition (179), be obtained by solving the (radial) Dirac equation for a neutral or ionic spherical and unpolarized configuration. The atomic approximation to Xworks very well not only for ground state energies of molecular systems73,76but also for electric99–104and magnetic105,106response properties, analytic energy gradients and Hessian,107and periodic systems.108 In contrast, the widely used approximation X1eobtained by diagonalizing the one-electron Dirac matrix is not accu- rate enough for nuclear magnetic shielding and cannot be applied to periodic systems. There have been attempts109–111 to approximate the renormalization matrix R+(181) also as the superposition of the atomic ones. Since R+is much less local than X, such an approximation does introduce discernible errors.112Nevertheless, such errors are tolerable for large systems in view of the dramatic gain in computa- tional efficiency (especially in gradient and Hessian calcula- tions107). The atomic approximation to X(and R+) can be obviously generalized to a diatomic (fragmental) approxima- tion,73,76which is of course only necessary if one is interested in highly distorted molecular systems in which two heavy FIG. 6 . Distribution of the matrix elements of X=B+A† +(A+A† +)−1(179) for Au 2 at a distance of 1.5 Å. Dirac–Hartree–Fock (DHF) result with the uncontracted ANO-RCC basis set (594 functions for each atom). J. Chem. Phys. 152, 180901 (2020); doi: 10.1063/5.0008432 152, 180901-15 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp atoms are very close to each other in distance. It is of par- ticular interest to note that the atomic/fragmental approxi- mation to both XandR+(i.e., the X2C/AU Ansatz defined in Ref. 107) allows one to interpret48X2C as a seamless bridge between the Dirac and Schrödinger equations because it can treat the heavy and light atoms in the system relativisti- cally and nonrelativistically, respectively, unlike that the Dirac (Schrödinger) equation treats the whole system relativistically (nonrelativistically). (3) LNESC + (192) is closely related to hP4C +(174). To see this, we assume that the A4S {| φ+,μ⟩} in the latter are further expanded in a RKB basis, viz., ∣φ+,μ⟩=∑ λ∈K(gλaK,λμ α 2σ⋅pgλbK,λμ),∀μ∈K;bK=XKaK, (208) for each atom K. We then have hP4C +=a†LNESC +a,a=⊕ ∑ KaK, (209) SP4C +=a†˜S+a, (210) where LNESC + in Eq. (209) and ˜S+in Eq. (210) have adopted the atomic approximation (207) to X. It is hence clear that P4C is just NESC, provided that the atomic-natural-spinor- type generally contracted RKB basis and the atomic approx- imation (207) to Xare used in both cases. However, P4C75 and NESC93were introduced in completely different ways in the same year though. Unlike NESC, P4C is not lim- ited to the RKB condition. Rather, it can also adopt, e.g., numerical A4S. (4) All physical operators are subject to the same transforma- tion going from the Dirac equation to a two-component the- ory. Neglecting this will result in PCE.77This can readily be done in the case of X2C, thanks to the simple relations A+=R+C+andB+=XA +. Moreover, the MSCA (which takes care of both scalar and spin–orbit one-centered two- electron picture-change effects) underlying Q4C76can also be employed in X2C. A more dramatic simplification of X2C is to assemble the two-electron spin–orbit part of fX2C + from DHF calculations of spherically averaged atomic configura- tions and then neglect all molecular relativistic two-electron integrals.113All in all, the second-quantized, normal-ordered, and PCE-corrected many-electron X2C Hamiltonian can be written as73 HX2C +=EX2C ref+fX2C pq{ap q}+1 2grs pq{apq rs}. (211) Note in passing that if the model spectral form61,62of the Qpotential (106) is included in h(178), we would obtain automatically a QED@X2C approach.35 At this stage, it should have been clear that the Q4C and X2C formalisms render no-pair four- and two-component relativis- tic calculations completely identical in all aspects of simplicity,accuracy, and efficiency at both the mean-field and correlated levels (a point that was observed more than a decade ago73). D. Spin-separated two-component There are various situations where one would like to treat spin- free (sf) and spin-dependent (sd) relativistic effects separately. For instance, the terms “intersystem crossing” and “multistate reaction” are both rooted in the perturbative treatment of spin–orbit coupling (SOC). In addition to SOC, spin-dependent interactions include also spin–spin coupling (SSC). While SOC contains both one- and two- body terms, SSC is purely a two-body operator arising from the spin separation of the Gaunt interaction114and should be taken into account in calculations of magnetic properties.115,116Here, we out- line briefly how to extract SOC from the X2C Hamiltonian. The very first issue lies in that fX2C +(191) is defined only in matrix form such that the Dirac identity (σ⋅A)B(σ⋅C)=A⋅(BC)+𝕚σ⋅[A×(BC)] (212) cannot be used. However, we can start with the partitioning of the Dirac matrix (83) into a scalar and a spin–orbit term, (V T Tα2 4W−T)=(V T Tα2 4Wsf−T)+(0 0 0α2 4Wsd), (213) where (Wsf)μν=⟨gμ∣p⋅Vp∣gν⟩,(Wsd)μν=⟨gμ∣iσ⋅(pV×p)∣gν⟩. (214) The first, spin-free term can be block-diagonalized in the same way as before so as to obtain hX2C +,sf=R† +,0(V+TX 0+X† 0T+X† 0[α2 4Wsf−T]X0)R+,0, (215) where p,q,r, and srefer to real-valued spin orbitals. Allying the spin- freeU0transformation [cf. Eq. (180)] to the second term of Eq. (213) leads to ⎛ ⎝α2 4R† +,0X† 0WsdX0R+,0α2 4R† +,0X† 0WsdR−,0 α2 4R† −,0WsdX0R+,0α2 4R† −,0WsdR−,0⎞ ⎠, (216) where the upper-left block is just the first-order SOC (to be denoted as so-DKH1), h(1) SO,1e=α2 4R† +,0X† 0WSOX0R+,0. (217) Higher-order SOC can readily be obtained117by carrying out fur- ther DKH-type unitary transformations that eliminate at each step the lowest-order odd terms in Wsd. In particular, the so-DKH2 and so-DKH3 operators h(n) SO,1ecan be constructed essentially for free (see Ref. 94 for the explicit expressions) because all necessary J. Chem. Phys. 152, 180901 (2020); doi: 10.1063/5.0008432 152, 180901-16 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp quantities are already available after constructing hX2C +,sf. As for the two-electron SOC, a mean-field approximation to the first-order terms is sufficient,94 f(1) SO,2e=α2 4R† +,0[GLL SO+GLS SOX0+X† 0GSL SO+X† 0GSS SOX0]R+,0, (218) where GXY SO(X,Y∈{L,S}) are the matrices of the effective one- electron operators GXY SO, GXY SO=𝕚σ⋅gXY=𝕚∑ lσlgXY,l,X,Y∈{L,S},l∈{x,y,z}, (219) gLL,l μν=−∑ λκ2Kl λμ,κνPSS λκ, (220) gLS,l μν=−∑ λκ(Kl μλ,κν+Kl λμ,κν)PLS λκ, (221) gSS,l μν=−∑ λκ2(Kl μν,κλ+Kl μν,λκ−Kl μλ,νκ)PLL λκ, (222) Kl μν,κλ=∑ mnεlmn(μmν∣κnλ) =−Kl κλ,μν,μm=∂mμ,l,m,n∈{x,y,z}, (223) PLL=R+,0PR† +,0,PLS=PLLX† 0,PSS=X0PLLX† 0. (224) Here,κ,λ,μ,νrefer to atomic (Gaussian) spin orbitals and εlmnis the Levi–Civita symbol, while P=1 2(Pα+Pβ)is the spin-averaged molecular density matrix, with PαandPβbeing the converged sf-X2C-ROHF/ROKS (restricted open-shell Hartree–Fock/Kohn– Sham) spin density matrices. The terms in Eq. (221) and the first two terms in Eq. (222) arise from the Coulomb interaction and represent the so-called spin-same-orbit coupling, whereas the term (220) and the third term of Eq. (222) originate from the Gaunt interaction and hence represent the spin-other-orbit cou- pling.94In view of the short-range nature of SOC, a one-center approximation to the integrals Kl μν,κλ(223) can further be invoked. In this case, only the atomic blocks of the molecular density matrix PXYcontribute to GXY SO. Yet, f(1) SO,2eis still a full matrix. If wanted, the SSC114can readily be added to f(1) SO,2e. The second- quantized, normal-ordered, spin-separated X2C Hamiltonian then reads HX2CSOC +,n=Eref+Hsf+H[n] sd,n=1 or 3, (225) Hsf=[hX2C +,sf]q p{ap q}+1 2grs pq{apq rs}, (226) H[n] sd=[h[n] SO,1e+f(1) SO,2e]q p{ap q}, (227) which is the simplest variant in the whole family of spin-separated X2C Hamiltonians118(NB: [ n] denotes up to nth order). It has been combined with both spin-adapted open-shell time-dependent density functional theory119–121and equation-of-motion coupled cluster for calculating fine structures of electronically excited states.102,103 Note in passing that, if the decoupling matrix X0(179) and the renormalization matrix R+,0(181) are set to identity in both h(1) SO,1e(217) and f(1) SO,2e(218) (i.e., so-DKH1), H(1) sd(227) will reduce to the Breit–Pauli spin–orbit Hamiltonian (so-BP). Hence, H(1) sdcan be understood as a bracketed (stabilized) so-BP. While so-BP can only be used as a first-order perturbation operator on top of the non- relativistic problem, H[n] sdis bounded from below and can hence be treated variationally. On the other hand, if X0andR+,0inh(1) SO,1e(217) andf(1) SO,2e(218) are both set to the free-particle counterparts, H(1) sd (227) will reduce to the original mean-field so-DKH1.122Moreover, H[3] sdis extremely accurate for both core and valence states117and can therefore be regarded as an equivalent of the non-perturbative SOX2CAMF operator.113 The various Hamiltonians discussed so far, including HQED a (111), HPI-QED + (167), HQED + (164), HPI-DCB + (168), HDCB + (169), HQ4C + (176), HX2C +(211), HX2CSOC +,n (225), and those A2C and nonrelativistic ones, share the same generic form H=Eref+fq p{ap q}+1 2grs pq{apq rs}. (228) It is just that the Fockian operator fhas to be interpreted differ- ently. A complete and continuous “Hamiltonian Ladder” can then be depicted.34,35The following points deserves to be emphasized again: (a) Relativistic Hamiltonians can only be formulated in Fock space, whereas all first-quantized relativistic Hamiltonians suffer from contaminations of NES. (b) HQED a (111) is the most accurate relativistic many-electron Hamiltonian and serves as the basis of the emerging field of “molecular QED.” (c) Under the NPA, four- and two-component approaches are fully equivalent in all aspects of simplicity, accuracy, and effi- ciency. Therefore, one should speak of “four-component and two-component equally good,” instead of “four-component good and two-component bad” or “two-component good and four-component bad.” (d) X2C is computationally the same but is much simpler and more accurate than A2C. As such, A2C should be regarded as outdated. (e) sf-X2C+so-DKH1 is computationally the same but is more accurate than NR+so-BP. As such, NR+so-BP should be regarded as outdated. VI. NO-PAIR CORRELATION Having discussed extensively the QED and relativistic many- electron Hamiltonians, we comment briefly on the correlation prob- lem. Due to the large gap between the NES and PES, a second- order treatment of the NES is sufficient (see Sec. IV A). There- fore, the major challenge still resides in the no-pair correlation within the manifold of PES. In this regard, like the nonrelativis- tic case, one has to face two general issues, i.e., the slow basis-set convergence and the strong correlation problem. The former can only be improved by the so-called explicitly correlated methods. However, relativistic explicit correlation is plagued by two concep- tual points: (a) no-pair projected or second-quantized relativistic J. Chem. Phys. 152, 180901 (2020); doi: 10.1063/5.0008432 152, 180901-17 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp FIG. 7 . (a) Number of relativistic arti- cles per year. (b) Distributions of rel- ativistic articles in journals. JACS: J. Am. Chem. Soc.; IJQC: Int. J. Quan- tum Chem.; PCCP: Phys. Chem. Chem. Phys.; TCA: Theor. Chem. Acc.; JCTC: J. Chem. Theory Comput. Hamiltonians are simply incompatible with explicit correlation due to the lack of analytic operators. (b) The fact123that the two limits c→∞and r12→0 do not commute makes how to apply the correlation factor f12(which itself is a complicatedquantity66) an open question. Rather unexpectedly, although the small-component ψS pof a PES is indeed smaller (albeit in the mean) than the large-component ψL pby a factor of c−1, the small– small component ΨSS(r1,r2) of a two-electron wave function Ψ(r1, J. Chem. Phys. 152, 180901 (2020); doi: 10.1063/5.0008432 152, 180901-18 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp r2) is of the same order of magnitude as the large–large compo- nentΨLL(r1,r2) at the coalescence point.66This means simply that there is no obvious argument to favor the incorporation of the cor- relation factor f12in a way that is in line with “first c→∞and then r12→0” or “first r12→0 and then c→∞.” These issues have recently been scrutinized in depth.66,124–126Since there are no new numerical results thereafter, we do not repeat the discussions here. However, it does deserve to be mentioned that the short- range density-functional type of corrections for basis-set incom- pleteness127–129is highly promising not only because of its simplicity but also because it is rooted in second quantization and is hence compatible with relativistic Hamiltonians. The remaining issue is to develop suitable short-range relativistic density functionals for this purpose.130 Compared to the slow basis-set convergence problem, the strong correlation problem is even more intricate in practice. A sys- tem is characterized as strongly correlated if a qualitative description already requires a multiconfigurational wave function. The main issue here lies in that the static and dynamic components of elec- tron correlation are often strongly entangled and even interchange- able. The situation is further worsened by SOC. Although a num- ber of relativistic schemes have been developed in the past,112,131–148 approaches that can provide a balanced and self-adaptive description of the static and dynamic components of correlation still remain to be formulated. It is believed that the ultimate way is to introduce some selection procedure that can adapt to the variable static corre- lation automatically and meanwhile can be terminated at a stage at which the residual dynamic correlation can well be described by a low-order approach. This leads naturally to “selected configuration interaction plus second-order perturbation theory” (sCIPT2), a very old idea that can be traced back to the end of 1960s and has recently been revived in various ways (see Ref. 149 for a recent review). Such approaches are most suited for relativistic calculations because of the following reasons: 1. A compact yet high-quality variational space can readily be determined by an iterative selection procedure, thereby avoid- ing problems149inherent in the scenario of complete active space (CAS). For instance, the size of the CAS would be doubled in the presence of SOC so as to limit severely the applicability of CAS-based four- or two-component relativis- tic correlation methods. This problem can only be resolved by selection. 2. The selection of important configurations is particularly effec- tive for SOC, thanks to the short-range nature of SOC. This had better be combined with a local representation150–153from the outset. 3. Unlike nonlinear wave function Ansätze, the symmetry adap- tation of CI wave functions can readily be achieved by means of the spin-dependent unitary group approach.154,155 4. SOC is strongly dominated by the one-body terms such that a second-order perturbative treatment of dynamic correla- tion, on top of a well-controlled variational space, should be sufficient. An X2C-based heat bath CI version156of sCIPT2 has just been realized, showing great promises although SOC is included therein only at the correlated level but not at the orbital level. Thecombination of the QED@Q4C or QED@X2C Hamiltonian with the recently proposed iCIPT2149(iterative CI157with selection plus second-order perturbation158,159) should be even more promising because iCIPT2 is spin-symmetry adapted and has the capability of targeting directly high-lying excited states that have little or even no overlap with the low-lying ones.160,161 VII. SUMMARY Ironically, in history, just one year after he proposed the famous relativistic equation of motion,7,8Dirac himself stated162that “rela- tivistic effects are of no importance in the consideration of atomic and molecular structure and ordinary chemical reactions.” Unfor- tunately, such a naive point of view lasted for nearly half a cen- tury until the mid-1970s when relativistic effects163were found to be indeed very important for electronic structure, sometimes even for light atoms. Since then the field of relativistic quantum chem- istry has witnessed fast development, especially in the last 15 years, as evidenced by nearly 20 000 relativistic articles164(see Fig. 7) as well as more than 10 relativistic books.52,165–174With the advent of powerful computational software,75,175–188it can be envisaged that relativistic quantum chemistry will play an increasingly important role in the exploration of molecular science. Apart from further improvement in the computational efficiency, the most important and urgent methodological developments include (1) the combina- tion of the QED@Q4C/X2C and sf-X2C+so-DKH1 Hamiltonians with sophisticated, symmetry-adapted wave function-based no-pair correlation methods (e.g., iCIPT2) for high-precision calculations of the electronic structure and (2) full implementation of the eQED Hamiltonian to establish the field of “molecular QED” for ultrahigh- precision calculations of spectroscopic parameters. 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Appl. Phys. Lett. 116, 222904 (2020); https://doi.org/10.1063/5.0006638 116, 222904 © 2020 Author(s).Preparation and characterization of a flexible ferroelectric tunnel junction Cite as: Appl. Phys. Lett. 116, 222904 (2020); https://doi.org/10.1063/5.0006638 Submitted: 04 March 2020 . Accepted: 19 May 2020 . Published Online: 03 June 2020 Ruonan Li , Yeming Xu , Jiamei Song , Peng Wang , Chen Li , and Di Wu COLLECTIONS This paper was selected as Featured Preparation and characterization of a flexible ferroelectric tunnel junction Cite as: Appl. Phys. Lett. 116, 222904 (2020); doi: 10.1063/5.0006638 Submitted: 4 March 2020 .Accepted: 19 May 2020 . Published Online: 3 June 2020 Ruonan Li,Yeming Xu,Jiamei Song, Peng Wang, Chen Li,a)and Di Wua) AFFILIATIONS National Laboratory of Solid State Microstructures, Department of Materials Science and Engineering, Jiangsu Key Laboratory for Artificial Functional Materials, and Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, People’s Republic of China a)Authors to whom correspondence should be addressed: chenli@nju.edu.cn anddiwu@nju.edu.cn ABSTRACT In this work, we propose a flexible ferroelectric tunnel junction (FTJ) with a nanometer-thick single-crystalline BaTiO 3barrier prepared by exfoliating and transferring epitaxial BaTiO 3thin films onto flexible poly(styrenesulfonate)-doped poly(3,4-ethylenedioxythiophene) (PEDOT:PSS) conductive electrodes using a water-soluble Sr 3Al2O6sacrificial layer. The transferred freestanding BaTiO 3films remain single crystalline and exhibit clear ferroelectric hysteresis, no matter being flat or bent. A combined piezoelectric force microscopy and conductiveatomic force microscopy measurement reveals that the Pt/BaTiO 3/PEDOT:PSS FTJ shows a clear polarization direction modulated tunneling resistance. By using x-ray photoelectron spectroscopy, the polarization direction-dependent electrostatic potential profile of this flexible FTJ has been reconstructed, consistent with the observed resistance modulation. Published under license by AIP Publishing. https://doi.org/10.1063/5.0006638 Recently, ferroelectric tunnel junctions (FTJs) have attracted much attention due to their potential applications in high-density and low-power-consumption memory devices.1Usually, FTJs are com- posed of two metallic electrodes separated by a nanometer-thick ferro-electric barrier. If the two electrodes are asymmetric, switching the polarization in the barrier toggles the average barrier height between two values, resulting in two nonvolatile tunneling resistance values for nondestructive readout memory applications. 2–4Considerable atten- tion has been paid to control the tunneling resistance by modulatingthe energy profile of the barrier. For example, Garcia et al. reported a FTJ using super-tetragonal BiFeO 3with a large polarization as the bar- rier to achieve a large electroresistance.5Wen et al. proposed a metal/ ferroelectric/semiconductor FTJ by modulating both the height and the effective width of the barrier.6On the other hand, flexible electron- ics are desirable for wearable applications.7–9One of the key challenges for developing such flexible electronics is to prepare light-weight, fold- able, and reliable memory elements.10Therefore, FTJs composed of flexible barriers and electrodes have been proposed. Poly-vinylidene fluoride-trifluoroethylene (PVDF-TrFE) is a well-known ferroelectric polymer and has been naturally used in flexible FTJs as the barrier. However, it is hard to prepare such ferroelectric polymer thin films with an atomic scale roughness, which is crucial for memory functionsof FTJs. 11One alternative method is to deposit ferroelectric thin films on an inorganic flexible substrate, e.g., mica. For example, Jie et al. suc- cessfully deposited high-quality PbZr 0.2Ti0.8O3(PZT) films on KAl 2(AlSi 3O10)(OH) 2mica via van der Waals epitaxy for flexible devi- ces.12Recent breakthroughs in exfoliating epitaxial perovskite oxide thin films from lattice-matched substrates provide yet another alterna-tive. 13–15Selective etching of a sacrificial buffer layer is used to obtain freestanding films. Very recently, using a water-soluble Sr 3Al2O6 (SAO) sacrificial layer, which is lattice matched with SrTiO 3(STO) frequently used as substrates for perovskite oxide deposition, Lu et al. obtained free-standing La 0.7Sr0.3MnO 3/STO thin films millimeter in size by exfoliation from STO substrates in water.15Further studies show that SAO can serve as a stable template for subsequent deposi- tion and exfoliation of BiFeO 3films with only a few unit cells in thick- ness.16These advances suggest the possibility to transfer ferroelectric thin films deposited in a conventional way to desired flexible substratesto fabricate flexible ferroelectric devices such as FTJs. Here, we report the preparation of a flexible FTJ by exfoliating and transferring a BaTiO 3(BTO) thin film onto a poly(styrenesulfo- nate)-doped poly(3,4-ethylenedioxythiophene) (PEDOT:PSS) polymer electrode deposited on a flexible polyethylene terephthalate (PET) sub-strate. Memory functions and band alignment of the Pt/BTO/PEDOT:PSS FTJs are measured and discussed. Appl. Phys. Lett. 116, 222904 (2020); doi: 10.1063/5.0006638 116, 222904-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplBTO/SAO heterostructures were deposited on (001)-oriented single-crystalline STO substrates by pulsed laser deposition (PLD) using a KrF excimer laser (Coherent COMPexPro 205F). The layer- by-layer growth was monitored in situ by reflection high energy elec- tron diffraction (RHEED). Prior to the deposition, STO substrates were etched by NH 4F buffered HF solution and then annealed at 950/C14C for 60 min in flowing O 2to form a TiO 2-terminated step- terrace surface. The sacrificial SAO layer was first deposited with a laser energy density of 1.0 J/cm2at 4 Hz. The temperature and O 2pres- sure are maintained at 700/C14C and 0.01 mbar, respectively. Then, BTO films were epitaxially deposited with a laser energy density of 2.5 J/cm2 at 2 Hz, keeping the temperature and O 2pressure at 750/C14Ca n d 5/C210/C03mbar, respectively. The thickness of the BTO film was con- trolled at 12 unit cells (u.c.) by counting the number of RHEED inten- sity oscillations. PEDOT:PSS (AGFA-Gevaert Orgacon EL-3040)conductive polymer (solid content: 4%, viscosity >8000 mPa s) elec- trodes were deposited on flexible PET substrates (8 /C28m m 2)b yr e p e t - i t i v e s p i n - c o a t i n g a t 1 5 0 0r p m f o r 1 5s , f o l l o w e d b y 6 0 0 0r p m f o r 4 0s and then 1500 rpm for 20 s. Afterward, the films were baked at 80/C14C for 30 min. To obtain the flexible FTJ, the transfer process of the ultrathin BTO film is quite important. Taking the BTO/PEDOT:PSS structure as an example, a commercial polydimethylsiloxane (PDMS) film was put on the surface of a BTO/SAO/STO sample. Then, the PDMS/ BTO/SAO/STO sample was immersed into de-ionized water for one hour to completely dissolve the soluble SAO layer. The exfoliated PDMS/BTO bilayer was transferred to a prepared PEDOT:PSS/PET substrate. Finally, the sample was baked at 80/C14C for 10 min to detach the PDMS film and obtain the desired BTO/PEDOT:PSS/PET hetero- structure. More details are provided in the supplementary material . The crystal structure of the thin BTO film was characterized by x-ray diffraction (XRD) using Cu K aradiation (Bruker D8 Discover). High resolution transmission electron microscopy (HRTEM) and selected area electron diffraction (SAED) of the transferred BTO film were performed using a FEI Tecnai F20 transmission electron microscope. The surface morphology, polarization reversal, and electroresist- ance properties were characterized using a Cypher-ES scanning probe microscope (Asylum Research). Hysteresis loops were measured in dual frequency resonant tracking mode with a triangular pulse of 6.0 V in amplitude and a detecting ac voltage applied to Pt coated can- tilever probes (Nanoworld EFM). Phase and amplitude images werecollected in single-frequency Piezoelectric Force Microscopy (PFM) mode. Current mapping and local I–V characteristics were recorded using conductive-diamond-coated cantilever probes (Nanoworld CDT-NCHR) in the ORCA module. X-ray photoelectron spectroscopy (XPS, K-alpha) was employed to determine the band alignment of the Pt/BTO/PEDOT:PSS FTJs byusing Al K aradiation (1486.6 eV) as the excitation source. Slight Ar ion beam etching was used to clean the sample surface. For XPS mea- surements, a thin Pt layer (5 nm in thickness) was sputter-deposited on the transferred BTO surface. Figure 1(a) shows the surface morphology of a BTO film depos- ited on SAO buffered STO substrates. The thickness of the BTO film is 12 u.c. as indicated by the clear oscillation in RHEED intensity (see Fig. S1 in the supplementary material ). The root mean square rough- ness over a 3 /C23lm 2area is about 89 pm, indicating an atomicallysmooth surface. The step-terrace surface with a step height of about 0.4 nm implies high-quality growth of both SAO and BTO layers. Figure 1(b) shows the XRD pattern of a BTO film transferred to a sili- c o nw a f e r ,i nc o m p a r i s o nw i t ht h a to ft h ea s - d e p o s i t e dB T O / S A O / STO heterostructure. The (00 l) diffraction peaks of the transferred BTO shift obviously toward higher diffraction angles, indicating asmaller out-of-plane lattice parameter due to the release of the com- pressive epitaxial strain imposed by the STO substrate. The obtained out-of-plane lattice parameter of the transferred BTO is 0.402 nm, ina g r e e m e n tw i t ht h a to ft h eB T Os i n g l ec r y s t a l . 17Figures 1(c) and1(d) show a high resolution TEM plain view image of an exfoliated BTO film and the corresponding SAED pattern, respectively. The atoms arearranged in a square lattice, with a period about 0.396 nm in two per- pendicular directions. This period is also in good agreement with the {001} lattice spacing of BTO single crystals, as expected. 17The SAED pattern shows bright spots with a clear fourfold symmetry, in agree- ment with the diffraction pattern of the [001] zone axis. Previous reports reveal that a water-rich environment may have some effects onthe structure of BaTiO 3. For example, exposure to high water pressures may result in hydroxylation and partial hydrolysis of the first BaO layer and lead to surface oxygen vacancies and formation of a thin dis-ordered hydroxide layer. 18Ab initio calculations revealed that water molecules absorbed on the BaTiO 3surface may induce Ti displace- ments in the TiO 2plane and an in-plane domain switching.19 Structural degradation in the transfer process and their effects on elec- trical properties are worth of further studies. Organic polymer PEDOT:PSS exhibits high electrical conductiv- ity and has been widely used in antistatic coatings, solid electrolytic capacitors, and organic LEDs.20,21PEDOT:PSS thin films were spin- coated on flexible PET substrates. Then, BTO films, 12 u.c. in thick-ness, were transferred onto the PEDOT:PSS electrode. The PFM measurement was performed to examine the ferroelectric properties of FIG. 1. (a) Surface morphology of the 12 u.c. thick BTO thin film deposited on SAO/STO, as demonstrated by an AFM image over a 3 /C23lm2area; (b) XRD pat- terns of a BTO film transferred on a silicon wafer and a BTO/SAO/STO heterostruc-ture; (c) HRTEM image of a 12 u.c. thick freestanding BTO film supported on a holy carbon covered TEM grid; (d) SAED pattern of the freestanding BTO film.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 222904 (2020); doi: 10.1063/5.0006638 116, 222904-2 Published under license by AIP Publishingthe transferred BTO, using a Pt-coated cantilever. Representative PFM loops are plotted in Fig. 2(b) . The butterfly-like amplitude hysteresis and the 180/C14phase change at 66 V suggest a ferroelectric switching. The out-of-plane phase and amplitude images of two adjacent domains written with þ5a n d/C05 V, respectively, are shown in Figures 2(c)and2(d). The clear domain boundary in the amplitude image and the 180/C14phase change in the phase image indicate that ferroelectric properties of the 12 u.c. BTO film are maintained after the transfer. The polarization is unswitched by applying a positive voltage. Thisalso indicates that the spontaneous polarization of the as-transferred BTO film points downward. However, it has been reported that some non-ferroelectric materials might also exhibit such PFM loops andimages due to charge injection or electrostatic forces on the tip. 22In order to confirm the ferroelectricity in the transferred BTO film, a series of AC signals with various amplitudes were used to measure thehysteresis loop. As shown in Fig. S4 in the supplementary material , the shape of the hysteresis loop depends strongly on the amplitude of the AC signal and become slimmer with the increase in the detectingsignal amplitude. This behavior is consistent with previous reports and supports that the observed hysteresis loop can be ascribed to ferroelec- tric polarization reversal in the transferred BTO film. 23 PFM measurements on transferred BTO films are also performed in different bending states [curvature radii r(cm)]), as shown schemat- ically in Figs. 3(a) and3(b).A ss h o w ni n Fig. 3(c) , butterfly hysteresis can be observed in all bending situations. However, compared to theflat sample, the coercive field of BTO decreases as the bending curva- t u r ei n c r e a s e sn om a t t e rt h ec u r v a t u r ei sp o s i t i v eo rn e g a t i v e .T h i s agrees with the results in other reports of similar ferroelectric filmssuch as the local coercive fields of the PZT freestanding film decreasing slowly under bending. 12It is clear that the transferred BTO films on PEDOT:PSS electrodes exhibit robust ferroelectric properties, even atlarge bending curvatures. XPS was used to measure the band alignment of the Pt/BTO/ PEDOT:PSS FTJs with BTO polarization pointing to or opposite tothe Pt electrode. 24All the core-level peak positions were fitted to the Voigt (mixed Lorentz–Gaussian) line with a Shirley background. Thefitting uncertainty of the core-level peak position is 0.05 eV. As shown in the supplementary material , the valence band offset (VBO) of the Pt/BTO interface can be calculated as VBO¼EPt4f7=2/C0EF/C0/C1/C0ETi2p3=2/C0EVBM/C0/C1 /C0EPt4f7=2/C0ETi2p3=2/C0/C1 Pt=BTO: (1) First, we measure the band structure of Pt/BTO/PEDOT:PSS with BTO polarization pointing opposite to the Pt electrode ( Fig. 4 ). The energy difference EPt4f7=2/C0EF, obtained from a Pt/BTO/STO sample, is 71.8460.05 eV, as shown in Fig. S5 in the supplementary material . The energy separation between the Ti2p3=2orbital and the valence band maximum (VBM), i.e., ETi2p3=2/C0EVBM, is 455.88 60.05 eV, and the binding energy difference between the Ti2p3=2andPt4f7=2core levels is 387.0560.05 eV. Thus, the VBO at the Pt/BTO interface can be esti- mated to be 3.01 60.05 eV.25,26 The VBO at the PEDOT:PSS/BTO interface can be measured fol- lowing the same procedure. Usually, the S2p3=2peak in PEDOT:PSS FIG. 2. (a) Schematic of the scanning probe measurement setup; (b) local PFM phase and amplitude hysteresis loops of BTO films transferred on the PEDOT:PSSelectrode; (c) and (d) out-of-plane PFM amplitude and phase images, respectively,of ferroelectric domains written with 65.0 V. FIG. 3. (a) and (b) Schematic of the bending measurement for the transferred BTO films on PEDOT:PSS/PET; (c) PFM amplitude and phase hysteresis loops of BTO/ PEDOT:PSS in different bending states. FIG. 4. XPS spectra of the Pt electrode reference, BTO thin film reference, Pt/BTO interface, PEDOT: PSS layer reference, and PEDOT:PSS/BTO interface.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 222904 (2020); doi: 10.1063/5.0006638 116, 222904-3 Published under license by AIP Publishingconsists of spin-split doublets, S2p1=2and S2p3=2.27The peak at lower binding energy is related to sulfur in PEDOT, while the higher bindingenergy peak corresponds to that in PSS. According to Fig. S7 in the sup- plementary material , the energy difference between the E S2p3=2orbital and the VBM of PEDOT:PSS is 161.55 60.05 eV. As for in the PEDOT:PSS/BTO, the energy difference equals 296.39 60.05 eV. Thus, the VBO at the BTO/PEDOT:PSS interface can be estimated to be2.0660.05 eV. Considering that the epitaxial strain in the BTO film has been released, we take the bandgap (Eg) value of bulk BTO, 3.5 eV, toestimate the band offset. 25The conduction band offset (the potential bar- rieru) can be calculated as u¼Eg-VBO, resulting in 0.49 and 1.44 eV at the Pt/BTO and BTO/PEDOT:PSS interfaces, respectively. Then,another BTO film was transferred onto the PEDOT:PSS electrode withBTO polarization pointing opposite to the electrode. In order to achievethis, the BTO/SAO/STO sample (with BTO polarized upward) wasdirectly put onto the surface of the PEDOT:PSS/PET substrate with theBTO film facing the PEDOT:PSS electrode. Then, the whole sample wasimmersed into de-ionized water to dissolve the SAO layer. After theSTO substrate was removed, the BTO/PEDOT:PSS/PET sample withBTO polarized opposite to the PEDOT:PSS electrode was obtained. Thebarrier heights were determined to be 0.2 eV at the Pt/BTO interfaceand 2.12 eV at the BTO/PEDOT:PSS interface, respectively, shown insupplementary material IV. The potential profiles with two opposite BTO polarization directions are plotted in Fig. 5 for comparison. Since the conductivity of PEDOT:PSS is as large as 460 620 S/cm, close to that of the graphite electrode, it is treated as a metal and the barrier in the electrode is omitted. 28,29It is clear that the polarization direction in the transferred BTO switches the barrier height at the two interface andtoggles the average barrier height in two values, as predicted by the FTJtheory proposed by Tysmbal and coworkers. 3,30 Based on the polarization-direction-dependent potential profile shown in Fig. 5 , the TER effect is expected. Figures 6(a) and6(b)pre- sent C-AFM current mapping over a 1.5 /C21.5lm2area (indicated by a dashed square in red) with BTO polarization switched downward and upward, respectively. The unambiguous current change after thepolarization reversal suggests different tunneling resistance, in agree-ment with the reconstructed potential profiles revealed by XPSmeasurements discussed above. Figure 6(c) shows the C-AFMcurrent–voltage (I–V) curves of the Pt-tip/BTO/PEDOT:PSS FTJ with BTO polarization switched to opposite directions by applying 63.5 V pulses. It is evident that the conductivity is larger when the polariza- tion points to the PEDOT:PSS electrode with a TER ratio of /C2410. With BTO polarization reversal, the potential changes at the Pt/BTO and BTO/PEDOT:PSS interfaces are 0.29 and 0.68 eV, respectively. The average barrier height is smaller with 0.2 eV in downward polari- zation, resulting in higher conductivity. In summary, we have shown that single-crystalline BTO films, which can be exfoliated from STO substrates by dissolving the water-soluble SAO sacrificial layer, become flexible while keeping the polarization switchable. By transferring the exfoliated BTO film on PEDOT:PSS polymer electrodes as the barrier of a Pt/BTO/PEDOT:PSS FTJ, clear TER can be observed as modu- lated by the polarization of BTO. The potential profile measured by XPS as a function of BTO polarization direction is in good agreement with the TER. These demonstrate that exfoliating epi- taxial thin films with a SAO sacrificial layer has potential for application in flexible electronics requiring functional oxides with atomic scale smoothness. See the supplementary material for much information about the exfoliation process, characterization, and band alignment measurements. This work was financially supported by the Natural Science Foundation of China (Nos. 51725203, 51702153, 51721001, U1932115, and 11874199) and the Natural Science Foundation of Jiangsu Province (No. BK20160627). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. FIG. 5. The band diagram of the Pt/BTO/PEDOT:PSS FTJ with the BTO polarization pointing toward and opposite to the PEDOT:PSS electrode. FIG. 6. Current mapping of the Pt/BTO/PEDOT:PSS FTJ, acquired before (a) and after (b) the area indicated by the red dashed line is switched by applying 4.0 V; (c) current–voltage characteristics as a function of the BTO polarization direction.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 222904 (2020); doi: 10.1063/5.0006638 116, 222904-4 Published under license by AIP PublishingREFERENCES 1J. F. Scott, Science 315, 954 (2007). 2A. Gruverman, D. Wu, H. Lu, Y. Wang, H. W. Jang, C. M. Folkman, M. Y. Zhuravlev, D. Felker, M. Rzchowski, C.-B. Eom, and E. Y. Tsymbal, Nano. Lett. 9, 3539 (2009). 3E. Y. Tsymbal and H. Kohlstedt, Science 313, 181 (2006). 4P. W. M. Blom, R. M. Wolf, J. F. M. Cillessen, and M. P. C. M. Krijn, Phys. Rev. Lett. 73, 2107 (1994). 5H. Yamada, V. Garcia, S. Fusil, S. Boyn, M. Marinova, A. Gloter, S. 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4.0000018.pdf

Struct. Dyn. 7, 034304 (2020); https://doi.org/10.1063/4.0000018 7, 034304 © 2020 Author(s).Structural dynamics of incommensurate charge-density waves tracked by ultrafast low-energy electron diffraction Cite as: Struct. Dyn. 7, 034304 (2020); https://doi.org/10.1063/4.0000018 Submitted: 07 April 2020 . Accepted: 13 April 2020 . Published Online: 22 June 2020 G. Storeck , J. G. Horstmann , T. Diekmann , S. Vogelgesang , G. von Witte , S. V. Yalunin , K. Rossnagel , and C. Ropers COLLECTIONS This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN Nanoscale diffractive probing of strain dynamics in ultrafast transmission electron microscopy Structural Dynamics 5, 014302 (2018); https://doi.org/10.1063/1.5009822 Imaging phonon dynamics with ultrafast electron microscopy: Kinematical and dynamical simulations Structural Dynamics 7, 024103 (2020); https://doi.org/10.1063/1.5144682 Ultrafast electron imaging of surface charge carrier dynamics at low voltage Structural Dynamics 7, 021001 (2020); https://doi.org/10.1063/4.0000007Structural dynamics of incommensurate charge-density waves tracked by ultrafast low-energy electron diffraction Cite as: Struct. Dyn. 7, 034304 (2020); doi: 10.1063/4.0000018 Submitted: 7 April 2020 .Accepted: 13 April 2020 . Published Online: 22 June 2020 G.Storeck,1,a) J. G.Horstmann,1 T.Diekmann,1S.Vogelgesang,1G.von Witte,1 S. V. Yalunin,1 K.Rossnagel,2,3 and C. Ropers1,4,b) AFFILIATIONS 14th Physical Institute, Solids and Nanostructures, University of G €ottingen, 37077 G €ottingen, Germany 2Institute of Experimental and Applied Physics, Kiel University, 24098 Kiel, Germany 3Ruprecht Haensel Laboratory, Deutsches Elektronen-Synchrotron DESY, 22607 Hamburg, Germany 4Max Planck Institute for Biophysical Chemistry (MPIBPC), G €ottingen, Am Fassberg 11, 37077 G €ottingen, Germany a)gero.storeck@uni-goettingen.de b)Author to whom correspondence should be addressed: cropers@gwdg.de ABSTRACT We study the non-equilibrium structural dynamics of the incommensurate and nearly commensurate charge-density wave (CDW) phases in 1T-TaS 2. Employing ultrafast low-energy electron diffraction with 1 ps temporal resolution, we investigate the ultrafast quench and recovery of the CDW-coupled periodic lattice distortion (PLD). Sequential structural relaxation processes are observed by tracking the intensities of main lattice as well as satellite diffraction peaks and the diffuse scattering background. Comparing distinct groups of diffraction peaks, wedisentangle the ultrafast quench of the PLD amplitude from phonon-related reductions of the diffraction intensity. Fluence-dependent relax-ation cycles reveal a long-lived partial suppression of the order parameter for up to 60 ps, far outlasting the initial amplitude recovery andelectron-phonon scattering times. This delayed return to a quasi-thermal level is controlled by lattice thermalization and coincides with the population of zone-center acoustic modes, as evidenced by a structured diffuse background. The long-lived non-equilibrium order parameter suppression suggests hot populations of CDW-coupled lattice modes. Finally, a broadening of the superlattice peaks is observed at high flu-ences, pointing to a non-linear generation of phase fluctuations. VC2020 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/10.1063/4.0000018 I. INTRODUCTION The spontaneous breaking of a continuous symmetry is a funda- mental concept of physics with broad relevance in such diverse areas as particle physics,1cosmology,2,3and condensed matter physics.4,5 An essential consequence of this symmetry breaking is the emergenceof new amplitude and phase excitations of the fields considered, exem- plified in the Higgs mechanism 6and massless Nambu–Goldstone bosons,7,8respectively. Moreover, the degenerate ground state of such systems allows for non-trivial topological states, as in the case of mag- netic vortices.4 Electron–lattice interaction is an important source of symmetry breaking in solids, most prominently in superconductivity and the for- mation of charge-density wave (CDW) phases.9–12Specifically, CDWs constitute a periodic modulation of the charge density by electron–holepairing,12coupled to a periodic lattice distortion (PLD)13–15and an electronic gap.16–19The emergence, correlations, and fluctuations of symmetry-broken CDW states can be revealed in the time domainby ultrafast measurement techniques. In this way, quenches of theelectronic gap coupled to coherent amplitude oscillations, 20–26light- induced PLD dynamics,27–30and phase transitions have been investi- gated.20,31,32In particular, ultrafast structural probes trace changes of structural symmetry33,34and long-range ordering following a phase transformation.35,36 However, while the initial quench and coherent amplitude dynamics of CDW systems following short-pulsed excitation are rather well characterized,20,21,23–26the subsequent paths to thermal equilibrium, including the roles of different collective modes in re- establishing a thermal CDW state, are far less understood. In Struct. Dyn. 7, 034304 (2020); doi: 10.1063/4.0000018 7, 034304-1 VCAuthor(s) 2020Structural Dynamics ARTICLE scitation.org/journal/sdyparticular, a sensitive structural probe is required to study the interplay of CDW-coupled excitations and regular phonons. Here, we employ ultrafast low-energy electron diffraction (ULEED), a recently developed surface-sensitive structuralprobe, 35,37–39to give a comprehensive account of the non-equilibrium structural dynamics of the incommensurate (IC) charge-density wave phases at the surface of 1 T-TaS 2. Harnessing the sensitivity of ULEED to the out-of-plane periodic lattice displacements of the sulfur atoms, we isolate the dynamics of an optically induced amplitude quench from a multi-stage excitation of phonons. Following a rapid partialrecovery, we observe a surprisingly long-lived non-thermal amplitudesuppression that equilibrates only after approximately 60 ps. Energy transfer to acoustic phonons is required to re-establish a thermal value of the PLD amplitude, suggesting that transient populations of collec-tive CDW modes have a lasting impact on the structural order parameter. II. MATERIALS SYSTEM AND EXPERIMENTAL APPROACH In this work, we study one of the most prominent CDW sys- tems, 1 T-TaS 2, which is part of the class of transition metal dichalco- genides. The atomic structure of this material consists of weakly interacting S–Ta–S trilayers,40,41in which the tantalum atoms are octahedrally coordinated between the sulfur atoms [ Fig. 1(a) ]. This compound has attracted much attention for its various CDW phases,12,40–42excitations21,24–27,43[Fig. 1(c) ], and correlation effects,44–46serving as a model system to study, for example, Peierls-vs Mott-type metal–insulator transitions,23,47pressure-induced super- conductivity in coexistence with CDWs,48transitions to metastable “hidden” CDW states,32,49the emergence of complex orbital tex- tures,50or quantum spin liquid behavior.51 The material exhibits multiple temperature-dependent phases [Fig. 1(f) ] with characteristic lattice deformations coupled to electronic structure changes.40,47,52Starting from a metallic phase with an undis- torted trigonal structure [ Fig. 1(a) ] above 543 K, the system undergoes a sequence of CDW transitions, forming a commensurate (C) (Mott-insulating) state below 187 K. At intermediate temperatures, two incommensurate phases are found, namely the so-called “nearly commensurate” (NC) phase (187–353 K), exhibiting commensuratepatches separated by discommensurations, 41,53–55and a homogeneous, fully incommensurate (IC) structure [ Fig. 1(b) ] between 353 K and 543 K. The periodic lattice distortions in these phases are characterized by primarily in-plane and out-of-plane displacements of the tantaluma n ds u l f u ra t o m s ,r e s p e c t i v e l y[ Figs. 1(b) and1(c)]. Ultrafast transi- tions between and manipulation of these phases, as well as their collec- tive modes [ Fig. 1(d) ], have been observed in various diffraction and spectroscopy studies. 20,21,23–27,33,35,43,56–61 In our experiments, we employ pulses of electrons at low ener- gies, typically in the range of 40–150 eV, to probe the structuralevolution of the NC and IC states in backscattering diffraction.ULEED 35,37–39allows us to trace the changes of the diffraction pattern in the time domain, following intense fs-laser illumination [red pulse inFig. 1(e) ]. In this optical-pump/electron-probe scheme, excitation and relaxation processes are sampled by varying the time delay t FIG. 1. Materials system and experimental setup. (a) Layered transition metal dichalcogenide 1 T-TaS 2exhibiting a trigonal crystal structure in the high-temperature phase (green lines: octahedral 1 T-coordination; red: unit cell). (b) Top view of the incommensurate (IC) CDW phase illustrating the charge density (green), distorted lattice (black dots: Ta atoms, displacements exaggerated), and superstructure unit cell (orange). (c) Side view of a single S–Ta–S trilayer, illustrating the out-of-pl ane periodic lattice displacements of the sulfur atoms (exaggerated). (d) 1D sketch of CDW amplitude and phase excitations and corresponding lattice fluctuations. (e) Schematic of the e xperimental setup, showing ultrafast LEED in a backscattering geometry. Ultrashort electron pulses (green) from a nanofabricated electron gun probe the dynamical evo lution of the laser-excited surface structure. (f) Temperature-dependent CDW phases. (g) Achieved electron pulse duration of 1 ps (see Appendix A for details). (h) Scanning electron micrograph of the miniaturized electron gun.Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 7, 034304 (2020); doi: 10.1063/4.0000018 7, 034304-2 VCAuthor(s) 2020between the optical pump pulse (red) and the photoemission pulse (blue) generating the electron probe (green). Reducing electronpulse broadening by short propagation lengths, a miniaturizedelectron gun [ Fig. 1(h) ] 38allows for a temporal resolution of 1 ps [Fig. 1(g) ]. Further experimental details are provided in Appendix A (Fig. 7 ). To facilitate the discussion, we focus the presentation on the response of the IC phase, which has not been studied by ultrafast dif- fraction, and provide a comprehensive dataset of related observationsfor the NC phase in Appendix B . The IC phase exhibits a triple-Q CDW/PLD, with lattice dis- placements for each unit-cell atom of the form 62,63 uðLÞ¼X i¼1;2;3AisinðQi/C1LþuiÞ (1) for lattice sites L, CDW wavevectors Qi,a n dp h a s e s ui.T h eC D W / PLD texture of a “dot-lattice” arises for the phasing conditionP iui¼0, and for symmetry reasons, the individual plane wave com- ponents share a common amplitude A¼jAij. The PLD at a wave- length kIC¼3:53a(a: lattice constant) leads to characteristic arrangements of satellite peaks64,65around the main lattice diffraction spots, seen in the ULEED pattern displayed in Fig. 2(c) .A st h eI Cs t a t e wave vectors are collinear to the lattice vectors, the satellites are located on the lines connecting the main reflexes. Due to the harmonic (and weak) structural modulation,53,54only first-order satellites are observed, with an intensity64Isat/C24jJ1ðs/C1AiÞj2/C24A2(s: scattering vector). We note that in this energy range, LEED is a very efficientstructural probe of the PLD because (i) backscattering diffraction isdominated by the sulfur sublattice and (ii) the large out-of-plane momentum transfer enhances the sensitivity to out-of-plane displacements. We study the excitation and relaxation of the IC and NC phases, without driving the system across a phase transition. 29,33,35,36,53,62Thedynamics of this incommensurate Peierls system can be discussed based on a simplified picture of three coupled subsystems, namely, theelectronic system exhibiting a gapped band structure [ Fig. 2(a) ,t o p ] , the collective amplitude and phase excitations around the symmetry- broken CDW ground state (center), 12and the ordinary lattice modes far from the CDW wavevector in reciprocal space, i.e., regular pho- nons (bottom). It is widely established that electronic excitation by an ultrashort laser pulse induces a carrier population above the bandgap, whichresults in a quench of the CDW/PLD amplitude that recovers upon carrier cooling by electron-phonon scattering. 24,27,43The correspond- ing sequence of relaxation processes involving the three subsystemscauses characteristic changes to the diffraction intensities of the satel- lite peaks and the main peaks (intensity I main). Specifically, for small PLD amplitudes, the peak intensities are expected to scale as64–67 Isat/C24A2e/C02Wue/C02Ws; (2) Imain/C241/C0csA2/C0/C1 e/C02W s: (3) These expressions reflect that a light-induced quench of the mean PLD amplitude Awill lead to a redistribution of intensity from the satellites to the main peaks.27,28,68Different main reflexes are sensi- tive to the PLD to a varying degree, which requires the introduction of the factor csthat depends on the momentum transfer s. Inelastic scat- tering by generated phonons transfers intensity from the reflexes to adiffuse background [ Fig. 2(b) ], 69–72leading to a peak suppression by a Debye–Waller factor exp ð/C02WsÞ.67,69The general form of the expo- nent67,69Ws/C24P phðs/C1uphÞ2sums over the momentum transfer pro- jected onto phonon displacements uphin various branches. According to Overhauser,64phase fluctuations result in the additional “‘phason Debye–Waller factor” e/C02Wu¼e/C0hu2i, which only affects the satellite spots and also causes diffuse scattering in the vicinity of the satellite peaks.66,73Finally, dislocation-type topological defects in the CDW FIG. 2. Dynamics and excitations in CDW systems influencing diffraction. (a) Electron and lattice subsystems (right) governing CDW dynamics. Gapped band st ructure (top, left), symmetry broken CDW state with phase and amplitude excitations (middle), and non-CDW phonons (bottom). (b) Changes in average amplitude and a ll lattice excitations (CDW and non-CDW) lead to a redistribution of intensity in the electron diffraction pattern. (c) Diffraction pattern of the IC phase of 1 T-TaS 2showing main lattice reflexes and first-order PLD-induced satellites (integration time: 90 s; electron energy: 100 eV). (d) Time-dependent measurement of reflexes [blue and red circl es in (c)] and diffuse back- ground (fluence F¼2.5 mJ/cm2). The main lattice signal is averaged over (10) and ( /C01 1) spots (blue), the satellite signal over several reflexes. Curves are normalized to the signal at negative times.Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 7, 034304 (2020); doi: 10.1063/4.0000018 7, 034304-3 VCAuthor(s) 2020may broaden the superlattice peaks and also reduce the PLD in the dislocation core.35,59 III. RESULTS AND ANALYSIS Our ULEED experiments directly show the characteristic diffrac- tion changes mentioned above: in the exemplary data displayed in Fig. 2(d), a main lattice peak (blue) exhibits a transient intensity increase after the pump pulse, before experiencing an initially rapid and then slowed suppression to a minimum at t¼60 ps. The satellite peaks (red), on the other hand, are first suppressed, before approaching a similar trend as the main peak beyond approximately 10 ps. Both the satellite and main peak intensities are significantly reduced by phonon populations.69These are evident in the diffuse background (gray), which mirrors the suppression of the reflexes, with a step-like increase in the first ps and a slower rise to a maximum at the delay of 60 ps. The initial step can be interpreted as the excitation of a broad popula- tion of optical and acoustic phonons on the timescale of electron–pho- non energy relaxation ( <1 ps),26while the slower timescale corresponds to phonon–phonon equilibration74and the population of low-energy acoustic modes. LEED intensities are rather sensitive to the large amplitudes of low-frequency modes, particularly those with out-of-plane polarization. Specifically, phonon modes with out-of- plane displacements uphhave a more pronounced Debye–Waller fac- tor due to the backscattering geometry with a primarily out-of-plane scattering vector of the electron. In addition, these modes exhibit com- paratively slow phase velocities, as is typical for layered van der Waals materials.75Thus, the prominent main lattice suppression evolving over tens of picoseconds primarily stems from the increasing popula- tion of low frequency acoustic modes modulating the layer distance. These strong Debye–Waller factors complicate an analysis of the temporal evolution of the amplitude quench. On the other hand, our experimental data show that different reflexes share a common phonon-induced peak suppression. In Secs. III A andIII B, we pursue two approaches of disentangling the dynamics of the structural order parameter from the phonon population, exploiting the different sensi- tivities of two inequivalent classes of main lattice reflections to the PLD (Sec. III A) and the direct sensitivity of the satellite reflexes to the PLD (Sec. III B). A. Amplitude analysis based on main lattice reflexes Concerning the time-dependent peak intensity, the main reflexes fall into two different groups. Whereas all five visible main peaks show a suppression opposite to the increase in the diffuse background [Fig. 3(a) ], we find that the transient amplitude signal is prominent only in the (1 0) and ( /C01 1) peaks, while it is largely absent in the (01), (/C010), and (1 /C01 )p e a k s[ s e ea l s od i f f e r e n c em a p si n Fig. 3(b) ].76 These two groups of peaks are crystallographically distinct, and the peaks within each group are equivalent in the effective threefold sym- metry of the 1T structure.77The different sensitivities of the peak intensities to the PLD are a particular feature of LEED, as described in the following. In the electron energy range of 70–110 eV, diffraction intensities are mainly governed by scattering from sulfur atoms due to large atomic scattering factors.41,77As a result of the CDW-induced con- traction of the tantalum sublattice, the sulfur atoms predominantly exhibit out-of-plane displacements. In backscattering, the opposing directions for the displacements in the upper and lower sulfur layerswithin each S–Ta–S trilayer41,77[Fig. 1(c) ] lead to an interference with enhanced or suppressed sensitivity of the two groups of main latticepeaks to the lattice distortion. This feature is expected in all CDW phases of 1 T-TaS 2, which share the phasing condition mentioned above [compare Fig. 1(b) ]. Experimentally, we found the same trend FIG. 3. Amplitude dynamics of the PLD obtained from main lattice reflexes. (a) Time-dependent intensity of visible main lattice reflexes and integrated backgroundintensity, for a fluence of F¼3:8 mJ/cm2. Two inequivalent classes of spot groups are found, featuring a strong (dark blue) and a weak (light blue) sensitivity to the amplitude quench. (b) Sketch of the IC diffraction pattern, and parts of the differ-ence diffraction image ( I t¼1ps/C0It<0) around the (10) and (01) main reflexes (insets). (c) Schematic comparison of peak intensities in the spot groups. The red area highlights different sensitivities to the PLD. (d) Extracted PLD amplitude quench and relaxation (see also Appendix A ) for three fluences, showing a rapid and a slower relaxation component (time constants from a biexponential fit (blackline) to the highest fluence data: (2.8 60.3) ps and (96 63) ps).Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 7, 034304 (2020); doi: 10.1063/4.0000018 7, 034304-4 VCAuthor(s) 2020in experiments on the NC phase (see Appendix B ), which exhibits different wavevectors but the same phasing between the three CDWs.In order to further corroborate these findings and consider theimportance of multiple scattering in LEED, we conducted dynamicalLEED simulations for a PLD of varying amplitudes and as a func- tion of the electron beam energy (see Appendix F ). In these simula- tions, for computational reasons, the commensurate modulationwas employed, taking quantitative displacements from a recentLEED reconstruction. 77Importantly, the dynamical LEED simula- tions qualitatively reproduce our experimental findings of different sensitivities to the PLD by the two groups of main lattice peaks. Moreover, the simulations predict an energy-dependent andstrongly reduced PLD sensitivity at an electron energy of 80 eV.Indeed, experiments at this lower energy show that the transientincrease in the main peak is generally much weaker (see additionaldata in Appendix C ). We employ these different sensitivities to the PLD to derive a phonon-corrected amplitude signal. Specifically, we remove the phonon-induced Debye–Waller suppression by normalizing the inten- sity of the PLD-sensitive peaks to that of the weakly sensitive peaks[Fig. 3(c) ;s e eAppendix A for details]. The resulting phonon-corrected amplitude suppression is displayed in Fig. 3(d) for three pump fluen- ces. In each case, the amplitude exhibits a rapid initial quench (withinour temporal resolution) and a recovery with an exponential time con- stant of about 3 ps. The re-establishment of the amplitude is, however, incomplete, slowing down considerably beyond 4 ps and lasting wellinto the range of tens to one-hundred picoseconds. B. Amplitude analysis based on satellite reflexes We now aim at characterizing the evolution of the mean ampli- tude based on the satellite peak intensities, again removing a time-dependent phonon Debye–Waller factor. To this end, we compare the intensities of the main peaks with weak PLD sensitivity to the satellite peaks. In Fig. 4(a) , we plot the logarithm of these intensities (normal- ized to the signal at t<0), divided by the fluence. For all three fluen- ces, the traces of the main lattice peaks collapse to a single universalcurve (blue), illustrating the phonon-induced Debye–Waller suppres-sion W sand its proportionality to fluence. The satellite peaks show a non-exponential fluence dependency in their suppression and recov- ery. At low fluences, however, where only a minor amplitude quenchis induced, the satellite peak suppression closely follows that of themain peaks. We use this information to derive a phonon-correctedamplitude signal from the satellite peaks (see Appendix A ).Figure 4(b) shows the resulting amplitude evolution. For this graph, the satel- lite intensities were integrated over circular masks in the diffractionpattern (width of Dk sat¼0:36 ˚A/C01), therefore including also electrons scattered by a small angle from the reflex. We find a very similarbehavior as from the main peak analysis (see Sec. III A), namely, a rapid and fluence-dependent quench, a fast initial recovery, and a rather persistent partial suppression, and we therefore consider this quantity as representative for the evolution of the amplitude A. A somewhat different curve is obtained by utilizing not the area- integrated intensity but the maximum intensity on top of the diffrac-tion spot [bottom graph in Fig. 4(b) ]. Whereas the maximum and integrated intensities behave similarly at low fluence, at the highest flu-ence, the suppression of the maximum intensity exceeds that of theintegrated intensity (gray curve from integrated intensity shown againfor comparison). Moreover, the recovery of the maximum proceeds more gradually than the integrated intensity. The difference between the evolution of the integrated and maxi- mum intensities implies a change in the diffraction peak shape, whichFIG. 4. Amplitude dynamics of the PLD obtained from satellite reflexes. (a) Logarithm of the normalized main lattice and satellite peak intensities (mean value),divided by fluence, vs time delay. While the main peak intensities (blue) collapse to a single curve due to the exponential (in fluence) Debye–Waller-type suppression, the satellite intensities (red) show a strong fluence-dependent behavior for earlytimes, before converging for long time delays. (b) Phonon-corrected PLD amplitudeobtained from integrated (top) and maximum (bottom) satellite intensities. (c) Fluence-dependent azimuthal spot width rvs time.Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 7, 034304 (2020); doi: 10.1063/4.0000018 7, 034304-5 VCAuthor(s) 2020is analyzed in Fig. 4(c) . Plotting the azimuthal width of the diffraction peak, we find a significant time-dependent broadening for the highestfluence. This effective broadening may be a result of several phenomena: (i) diffuse scattering to the wings of the peak by low-energy phase exci-tations 14will suppress the reflex maximum via the phason Debye–Waller factor exp ð/C02WuÞwhile largely maintaining the inte- grated intensity. (ii) An overall peak broadening from reduced correla- tion lengths will arise from the generation of CDW dislocation-typetopological defects. 35,59Except for the amplitude suppression in the dislocation core, this broadening also preserves the integrated inten-sity. At this point, we cannot rule out either scenario, and a moredetailed spot profile analysis or higher momentum resolution may be required to further elucidate the different contributions. C. Non-equilibrium dynamics of the amplitude The incomplete recovery and persistent suppression of the PLD amplitude, independently obtained from the main [ Fig. 3(d) ]a n ds a t - ellite [ Fig. 4(b) ] reflexes, warrant further investigation. This implies that the system is either thermalized at a higher temperature withreduced equilibrium amplitude 28or, alternatively, that non- equilibrium dynamics inhibit the recovery of the order parameter. It was previously suggested for the NC phase that the rapid recoveryresults in a thermalized system at elevated temperature. 27Specifically, this would entail equilibrium between the electronic and differentstructural degrees of freedom after approximately 4 ps. As shown in the following, we have evidence for a sustained non- thermal suppression of the order parameter. In Fig. 5 , we consider in more detail the path to thermal equilibrium. An instructive depictionis obtained by plotting the main and satellite intensities against each other, resulting in cyclic trajectories in a two-dimensional plane [ Fig. 5(a)], traced out over time in a clockwise fashion. At long delays (beyond 100 ps), the curves for all fluences follow a universal path(dashed line) representing a thermalized system at elevated tempera-tures, cooling down. Different trajectories reach the same combinationof intensities at different times. For instance, the high-fluence trajec- tory exhibits the same combination of intensity suppressions at 1500 ps as the intermediate fluence at a somewhat earlier time of 290 ps[black circle in Fig. 5(a) ]. Once the trajectory reaches this line, the sur- face is in local thermal equilibrium, characterized by a single tempera-ture, and the satellite peak suppression is composed of aDebye–Waller factor and a thermal reduction of the amplitude. The further progression of the system, i.e., its cooling, is governed by ther- mal diffusion to the bulk. All points displaced from the dashed line represent deviations from a thermal state, with the distance being a very sensitive measureof the structural non-equilibrium. For example, within the first pico-second after the excitation (dark segments of the curves), the rapidquench of the order parameter causes a reduction of satellite intensityand a moderate enhancement of the main lattice signal, with a fluence-dependent maximum displacement from thermal equilibrium (the corresponding curves for the main peaks insensitive to the ampli-tude are found in Appendix D ). The recovery to the thermal state now proceeds through various stages and in a fluence-dependent manner.After about 4 ps (see marks), the fast component of the amplituderecovery is completed [cf. Fig. 3(d) , compare also Ref. 74]. 78However, the system remains far from the equilibrium state, i.e., exhibits alower-than-thermal satellite intensity. Interestingly, for all curves, a surprisingly long time of approximately 60 ps is required to reach thethermal state. This depiction directly shows that the persistent ampli-tude suppression discussed in Figs. 3(d) and 4(b) is, in fact, not of a thermal nature and that we have a pronounced deviation from equilibrium between the degrees of freedom affecting the diffractionintensities. To identify the origin of this long-lived amplitude suppression, we first note that the time at which the system reaches a thermalamplitude nearly coincides with the strongest suppression of the mainlattice peaks. As this time also corresponds to the maximum intensityof the diffuse background [cf. Figs. 2(d) and3(a)], the full equilibration (a) (b) FIG. 5. Path to equilibrium. (a) Intensities of satellite and main peaks (with PLD sensitivity) plotted against each other, leading to cyclic trajectories in a 2D planewith varying sizes. Note that all curves reach a common equilibrium line after approximately 60 ps. The gray color scale highlights certain time intervals (dark gray: 0–1 ps, medium, gray: 1–60 ps, and light gray: 60–1500 ps). The same com-bination of intensity suppressions is found for different fluences at different times(the black circle corresponds to 1500 ps/290 ps at high/intermediate fluence). (b) Ratio of time-integrated frames exhibits prominent pedestals around diffraction peaks, pointing to an enhanced acoustic phonon population on the equilibrium line.Late frames (dark magenta in the inset, t¼790;…;1500 ps) are divided by early frames (light magenta in the inset, t¼4:5;…;10 ps).Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 7, 034304 (2020); doi: 10.1063/4.0000018 7, 034304-6 VCAuthor(s) 2020of lattice fluctuations appears to be critical in controlling the structural order parameter. In particular, this lattice equilibration induces a sig- nificant increase in diffuse background intensity around main lattice peaks [difference image in Fig. 5(b) ], directly pointing to the excitation of low-energy acoustic modes near the center of the Brillouin zone. IV. DISCUSSION Taken together, these observations indicate the sequence of relax- ation processes illustrated in Fig. 6(b) , which can be related to the intensity curves [ Fig. 4(a) ] and the cyclic trajectories introduced above [simplified sketch in Fig. 6(a) ]. Within the first picosecond, the optical excitation of the electronic system leads to a CDW amplitude quench and a strong deformation of the potential energy landscape [see insets inFig. 6(b) ], which triggers cooperative motion of the lattice toward its unmodulated state, including the excitation of coherent amplitude modes (stage 1).26,43Facilitated by the generation of high-energy lattice modes, the electron system cools down within few ps (stage 2), and as a result, the electronic potential and amplitude partially recover. The remaining PLD suppression in the following stage 3 strongly indicates a substantial population of CDW-coupled lattice excitations [ Fig. 6(b) , red filling in the bottom inset], such as amplitudons, phasons, and possible dislocation-type topological defects. Remaining non-thermal electronic excitations, on the other hand, can be largely ruled out at these late times, based on results from time- and angle-resolved photo- emission spectroscopy.23,24,32,43,79While it is known that phonon equilibration may take tens of picoseconds,80,81the present observa- tions are significant in the sense that the persistent structural non- equilibrium is found to directly lead to an amplitude suppression via long-lived CDW-coupled excitations.Both amplitude and phase modes are expected to be rather effi- ciently excited by the optical pump, either directly by the deformationof the electronic potential (amplitude modes) 25or by electron lattice scattering between gap regions [ Fig. 6(d) ]. Specifically, Fermi surface nesting is expected to result in a high probability of scattering events with a momentum transfer around the CDW wavevector Q[Fig. 6(d) , left]. Subsequent cooling of the carrier temperature below the energyscale of the electronic gap will effectively suppress these inelastic scat-tering pathways [ Fig. 6(d) , right] and decouple the subsystems [ Fig. 6(c)], contributing to the persistent amplitude suppression in stage (3). Full lattice thermalization and the excitation of zone-center acoustic modes are then only achieved after 60 ps, from which point the equili-brated system cools down (stage 4). Let us consider the possible roles of different CDW excitations in the long-lived amplitude suppression, namely, amplitudons, phasons,and CDW dislocation defects. Spatiotemporal variations of the ampli-tude and phase affect the observable value of A. Specifically, amplitu- dons represent amplitude oscillations DAaround an equilibrium amplitude A 0, leading to an observed average value of hA0þDAi.B y an anharmonicity of the electronic potential, these oscillations becomeasymmetric, and a high population of amplitudons can reduce thevalue of A. In the case of phasons, despite early theoretical and experi- mental work, 13,14,64,65,82–86a unifying picture has not been reached, and recent assignments of their contribution in diffraction studies range from largely negligible68to dominant.73While our results do not definitely resolve this issue, the redistribution of scattering inten-sity near the satellite peaks suggests significant spatial or spatiotempo-ral phase distortions. CDW dislocation defects should also be considered as a possible cause for the long-lived order parameter suppression, as they have FIG. 6. Linking relaxation pathways to CDW/PLD dynamics. (a) and (b) Simplified sketches of Figs. 2(d) and 4(a), respectively, highlighting four phases of the relaxation pro- cess observed in the data. (c) Illustration of the sequential relaxation process and the excitation levels of the three subsystems. The color shade re presents the energy con- tent/temperature, and black arrows indicate energy flow. (d) Simplified electronic band structure and populations (saturation of the orange line) fo r high (left) and low (right) electronic temperatures. Arrows indicate electron-lattice scattering processes. Scattering between gap regions (momentum transfer Q) is suppre ssed for reduced electronic temperatures.Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 7, 034304 (2020); doi: 10.1063/4.0000018 7, 034304-7 VCAuthor(s) 2020been observed as a consequence of phase transitions, e.g., in 1 T-TaS 235 or LaTe 3.59,87T h ef a c tt h a tw efi n das i g n i fi c a n ts p o tb r o a d e n i n go ft h e satellites [ Fig. 4(c) ] most strongly at high fluences suggests a non-linear dependence of phase fluctuations. This would be consistent with either CDW dislocations generated by critical phase fluctuations or a paramet- ric decay of amplitudons into phase modes, as previously proposed.22 V. CONCLUSIONS The impact of fluctuations on symmetry breaking transitions has long been considered, for example, in the Peierls instability.88–91 Providing a time-domain view of the structural relaxation pathways, the present measurements highlight the impact of long-lived structural non-equilibrium to the order parameter. The general mechanism of amplitude suppression by CDW- coupled modes should apply also to other phases and systems. Indeed, measurements in the NC phase feature a similar behavioras the IC phase ( Appendix B ). Despite differences in symmetry, CDW wavevectors, and electronic gaps, both phases exhibit closely related amplitude and phase excitations, as pointed out byNakanishi and Shiba. 92 Relevant further questions pertain to the possible mechanisms of generating phasons and dislocation-type topological defects, as well as their coupling to regular lattice modes. Also, the link between fluctua-tion modes and the creation and relaxation of metastable states 32,49 and the influence of partial and full commensurability in differentCDW states call for further investigation. Additional insights may begained by investigating the ultrafast phase transitions between differ- ent CDW states 35,36,56and the populations of amplitude and phase modes in the nascent state after transition. Considering methodical aspects, this work represents the first comprehensive study employing ULEED with a temporal resolution of 1 ps. Future investigations using ULEED will enable a quantitativeanalysis of the three-dimensional structural evolution based on time- and energy-dependent diffraction. Moreover, the method is applicableto a wide variety of other surface systems and low-dimensional struc- tures, harnessing its strengths of high momentum resolution, efficientscattering, and enhanced sensitivity to lattice fluctuations. ACKNOWLEDGMENTS This work was funded by the European Research Council (ERC Starting Grant “ULEED,” ID: 639119) and the Deutsche Forschungsgemeinschaft (No. SFB-1073, Project A05). Wegratefully acknowledge insightful discussions with H. Schwoerer, B.Siwick, J. D. Axe, and T. Aslanyan. We thank L. Hammer for introducing us to the dynamical LEED computations. Furthermore, we thank K. Hanff for help with sample preparation. The data that support the findings of this study are available from the corresponding author upon reasonable request. APPENDIX A: METHODS 1. Experimental details Here, we briefly describe our experimental ULEED apparatus (Fig. 7 ). The time-resolved measurements are conducted in an ultra-high vacuum chamber (base pressure p¼5/C210/C011mbar) into which samples are transferred via a load-lock system andcleaved in situ . Inside the chamber, the electron source (micro- gun 38) and a microchannel plate detector are mounted. A cooled CMOS camera records the detected electron diffraction patternsfrom outside the UHV chamber. A femtosecond laser system (Amplifier, NOPA and OPA) gen- erates three femtosecond laser beams of different center wave- lengths. An ultraviolet beam (center wavelength of 400 nm) is focused on a nanometric tungsten needle that is embedded insidethe microgun 38generating ultrashort electron pulses via two- photon photoemission. An electrostatic lens assembly controls the FIG. 7. Schematic of the ultrafast LEED setup.Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 7, 034304 (2020); doi: 10.1063/4.0000018 7, 034304-8 VCAuthor(s) 2020collimation of the electron beam having an energy range of 40–100 eV. With a gun front diameter of 80 lm and a working dis- tance of around 150 lm, we achieve a temporal resolution of 1 ps and an electron beam diameter of approximately 10 lm at the sam- ple. An upper limit of the technique’s temporal resolution isobtained by the derivative of the fastest intensity change in ourdataset (suppression of the satellite peak at 100 eV) shown in Fig. 1(g). An infrared beam (center wavelength, 1030 nm) optically excites the sample at specific times controlled by an automated lin-ear delay stage. In order to heat the 1 T-TaS 2sample and stabilize it in the IC phase slightly above T/C25353 K, a third beam (800 nm) is aligned collinearly with the infrared beam. This pulse is delayed byabout 3 ns with respect to the electron and 1030 nm pulses (i.e., itarrives 10 ls before the next pulses) and thus leads to an average increase in sample temperature.2. Data analysis The recorded LEED images are preprocessed to correct for minor drifts in-between measurement runs and for distortionscaused by local electromagnetic fields and the projection to a flat MCP detector. In order to obtain time curves [ Fig. 2(d) ]f r o mt h es t a c k so f diffraction patterns, we process the data in a sequence of opera-tions. First, a binary circular mask is laid on top of each individ-ual reflex [ Fig. 2(c) ; blue and red circle, diameter Dk main¼0:6˚A/C01andDksat¼0:36 ˚A/C01,r e s p e c t i v e l y ]f o re a c h time delay. Second, we fit 2D Cauchy distributions (background:slope and constant offset) to the satellite reflexes and 2D pseudoVoigt profiles (background: slope and constant offset) to the mainlattice reflexes, to determine a background profile and subtract it from each spot: FIG. 8. Measurements in the NC phase for an electron energy of 100 eV. (a) Top view of the nearly commensurate (NC) CDW phase illustrating the charge density (gr een), distorted lattice (black dots, displacements exaggerated), and superstructure unit cell (orange). (b) Diffraction pattern of the NC phase of 1 T-TaS 2showing main lattice reflexes and several orders of PLD induced satellites (integration time: 90 s). (c) Time-dependent measurement of reflexes [blue and red circles in (b)] and dif fuse background (for three fluences). The main lattice signal is averaged over the (10) and ( /C01 1) spots (blue), the satellite signal over several reflexes. Curves are normalized to the signal at negative times. (d) Time-dependent intensity of visible main lattice reflexes and integrated background intensity, for a fluence of F¼2:0 mJ/cm2. Two inequivalent classes of spot groups are found, featuring a strong (dark blue) and a weak (light blue) sensitivity to the amplitude quench. (e) Extracted PLD amplitude quench and re laxation for three fluen- ces, showing a rapid and a slower relaxation component [time constants from a biexponential fit (black line) to the highest fluence data: 1.3 ps and 88.5 p s]. (f) Main lattice peak intensities vs satellite peak intensities, leading to cyclic trajectories in a 2D plane with varying sizes. Following a two-stage relaxation, a ll curves reach a common equilib- rium line after approximately 60 ps. The gray color scale highlights certain time intervals (dark gray: 0–1 ps, medium, gray: 1–60 ps, and light gray: 6 0–1500 ps).Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 7, 034304 (2020); doi: 10.1063/4.0000018 7, 034304-9 VCAuthor(s) 2020Cðx;yÞ¼A 2pr1r2/C11þx r1/C18/C192 þy r2/C18/C192 !/C03=2 þða/C1xþb/C1yÞþC; (A1) PVðx;yÞ¼A/C11þx r1/C18/C192 þy r2/C18/C192 !/C03=2 þB/C1e/C0x=r3ðÞ2/C0x=r4ðÞ2þða/C1xþb/C1yÞþC: (A2) Here, the x and y axes correspond to the azimuthal and radial directions (with respect to the main peak) for a given spot. Third,from the background-corrected segment, the average and the maxi-mum intensity (average over brightest 4% within a mask) are deter-mined for each reflex within the mask. The remaining intensityoutside the circular masks forms the integrated background. For animproved signal-to-noise ratio, several spot curves are averaged, i.e.,the satellite curves represent the mean of the 11 brightest reflexes. Furthermore, from the 2D fit functions, we obtain the azimuthal (r 1) and radial ( r2) widths for each reflex [ Fig. 4(b) ]. a. Debye–Waller–corrected amplitude signal Next, we describe the separation of the amplitude-quench- related intensity changes from Debye–Waller–type peak suppres- sion for the main lattice [ Fig. 3(d) ] and satellite reflexes [ Fig. 4(b) ]. The dynamical LEED simulations indicate that there can be considerable differences in the coefficients cs[Eq. (3)]. Empirically, we find that for the IC phase, the [(0 1), ( /C01 0), (1 /C01)] peaks show a negligible influence of the initial quench ( cs/C250). In Fig. 3(a) , for each fluence, light blue ( cs/C250) and dark blue curves ( cs>0) are averaged ( Inon/C0ampandIamp). We now use the two peak groups to extract the temporal evolution of Aby removing the time- FIG. 9. Measurements in the IC phase for an electron energy of 80 eV. (a) Diffraction pattern of the IC phase of 1 T-TaS 2showing main lattice reflexes and first-order PLD- induced satellites (integration time: 90 s). (b) Time-dependent measurement of reflexes [blue and red circles in (a)] and diffuse background (for thr ee fluences). The main lattice signal is averaged over the (10) and ( /C01 1) spots (blue), the satellite signal over several reflexes. Curves are normalized to the signal at negative times. (c) Time-dependent intensity of visible main lattice reflexes and integrated background intensity, for a fluence of F¼3.8 mJ/cm2. Two inequivalent classes of spot groups are found, but none shows a strong dependence on the amplitude quench. (d) Main lattice intensity vs satellite peak intensity, leading to cyclic trajectories in a 2D plan e with varying sizes. Note that all curves reach a common equilibrium line after approximately 60 ps. The gray color scale highlights certain time intervals (dark gray: 0–1 ps, m edium, gray: 1–60 ps, and light gray: 60–1500 ps).Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 7, 034304 (2020); doi: 10.1063/4.0000018 7, 034304-10 VCAuthor(s) 2020dependent Debye–Waller suppression e/C02Wsfrom the intensity of the peaks sensitive to the PLD, with a constant factor C1 ¼0:81 that accounts for the slightly different Wsof these peaks: Iratio;main ;F¼Iamp;F Inon/C0amp;F1/C18/C19 F F1/C1C1 ¼1/C0csA2: (A3) The value of C1was determined by the main peak suppression at long delays (beyond 1 ns) and the lowest fluence ( F1¼1:3 mJ/ cm2), for which a negligible amplitude change is expected. The value of csfor the amplitude-sensitive peaks is determined from the A2-intensity dependence of the phonon-corrected satellite peaks [Eq. (3)], evaluated at maximum suppression of the lowest fluence. From the satellite reflexes (11 brightest spots), the corrected amplitude is obtained similarly using the Debye–Waller–dominated main lattice curve Inon/C0amp;F1for the lowest fluence with the factor C2¼1:2: Iratio;sat;F¼Isat;F Inon/C0amp;F1/C18/C19 F F1/C1C2 ¼A2: (A4) b. Fitting of time constants The fit function in Fig. 3(d) is based on a step-like decrease fol- lowed by two exponential relaxations, SðtÞ¼1/C0hðt/C0t0Þ/C1ð /C0 A1þA2/C1ð1/C0e/C0ðt/C0t0Þ=sÞÞ þA3/C1ð1/C0e/C0ðt/C0t0Þ=s2Þ; (A5) where his the Heaviside function, t0is time zero, A1,A2, and A3are the amplitudes, and s1ands2time constants. The complete fit func- tion is the convolution of S(t) with a Gaussian (FWHM of 1 ps) cor- responding to the temporal resolution in our experiment. APPENDIX B: DATA FOR THE NEARLY COMMENSURATE (NC) PHASE Figure 8 displays the analysis discussed above applied to the nearly commensurate phase. Similar features are found in the pump–probe curves for the main and satellite diffraction peaks, as well as the back-ground [ Fig. 8(c) ], the long-lived amplitude suppression [ Figs. 8(d) and 8(e)], and the relaxation cycles [ Fig. 8(f) ]. In particular, the two-stage amplitude relaxation process (first stage up to 4 ps and second stage upto 60 ps) is very pronounced at all fluences. APPENDIX C: DATA AT 80 EV ELECTRON ENERGY Figure 9 shows additional data recorded in the IC phase at 80 eV energy. The main lattice peaks show a much weaker depen- dency on the PLD amplitude [ Figs. 9(b)–9(d) ]. APPENDIX D: RELAXATION CYCLES FOR MAIN PEAKS ( 21 1), (0 1), AND (1 21) Figure 10 shows the relaxation cycles in the IC phase as in Fig. 5(a), using the intensities of the main lattice peaks ( /C01 1), (0 1), and (1 –1) without sensitivity to the PLD amplitude.APPENDIX E: IMPACT OF CDW DEFECTS ON PEAK WIDTH Here, we argue that our data rules out a linear scaling of CDW defect density with fluence and is only consistent with a non-linearor threshold behavior. Assuming a linear relation of the defect den- sity with the fluence n/C24Fand a correlation length L/C241=ffiffiffinp(see Ref. 35), the defect-induced broadening should scale as r top/C241=L/C24ffiffiffinp/C24ffiffiffi Fp . A doubling of the normalized peak widths rtot¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 0þr2 topq with respect to the instrument resolution r0for the highest fluence would then imply considerably higher broaden- ing values for lower fluences ( rtot;2:5¼1:7 and rtot;1:3¼1:4) than observed in our measurement [ Fig. 4(c) ]. Experimentally, we find maximum broadening values at t/C251p so f rtot;3:8/C252, rtot;2:5/C251:3, and rtot;1:3/C251 for the highest, intermediate and low- est fluence, respectively [see Fig. 4(c) ]. Thus, we infer that the den- sity of topological defects does not scale linearly with fluence. APPENDIX F: DYNAMICAL LEED COMPUTATION We performed dynamical LEED simulations on the commen- surate CDW phase of 1 T-TaS 2varying the atomic displacements of sulfur and tantalum continuously from the undistorted structuretoward the C-phase structure recently reconstructed. 77We are aware that the C phase is a simple approximation for the descrip-tion of the high-temperature incommensurate CDW phase.However, it exhibits the same crucial feature of opposing sulfur dis-placements that we believe is responsible for the different sensitivi- ties of the main lattice peaks. Also, dynamic LEED calculations involve high computational effort, in particular for large unit cellsnecessary for incommensurate structures. The obtained dataFIG. 10. Main lattice peaks without amplitude features vs satellite peak intensities, leading to cyclic trajectories in a 2D plane with varying sizes. All curves reach a common equilibrium line after approximately 60 ps. The gray color scale highlightscertain time intervals (dark gray: 0–1 ps, medium, gray: 1–60 ps, and light gray:60–1500 ps).Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. Dyn. 7, 034304 (2020); doi: 10.1063/4.0000018 7, 034304-11 VCAuthor(s) 2020contain PLD-amplitude- and energy-dependent scattering intensi- ties for main lattice and CDW satellite spots. In the following, wefocus on main lattice diffraction intensities. In the electron energy range of 70–140 eV, the diffraction intensity is mainly determined by scattering from sulfur atoms, explaining the strong dependence from the PLD amplitude of sulfur atoms [ Fig. 11(a) ]. Figure 11(b) shows PLD dependent intensities for electron ener- gies of 100 and 80 eV, each normalized to the intensity value for zerodistortion (metal structure). The PLD amplitude range is adapted tothe expected values realized in the incommensurate phase, 40which are assumed to be considerably smaller ( /C2430% of PLD amplitude of thecommensurate low-temperature phase). In this range for 100 eV, we can show that there are two groups of main lattice spots that respond differently upon PLD changes, whereas for 80 eV, all intensities follow a common curve. Moreover, the magnitude of the relative intensity changes approximately matches the observed ones in the experiment. The curves within a group of main lattice peaks [(1 0), ( /C011 ) ]a n d [(/C010 ) ,( 01 ) ,( 1– 1 ) ]c o i n c i d es i n c et h es i m u l a t i o ni sp e r f o r m e da t normal incidence. Figure 11(c) shows energy-dependent intensity curves for two main lattice peaks contained in one of the two groups (light and dark blue), each for zero PLD and 30% PLD amplitude of the com- mensurate low-temperature phase. The ratio of spectra for each FIG. 11. Dynamical LEED simulations. (a) Normalized intensity of the main lattice reflex (1 0) as a function of sulfur and tantalum displacement for an electron energy of 100 eV. Enhanced scattering of sulfur atoms results in a much stronger dependence on the sulfur atom displacements. (b) Normalized intensities of mai n lattice spots for elec- tron energies of 80 and 100 eV as a function of the fraction of the maximum commensurate PLD amplitude. The diffraction reflexes split up into two spot gro ups. Light and dark blue curves coincide, respectively, due to the normal incidence of the electron beam. (c) LEED spectra (top) for both groups (light and dark blue) for vanishing (points) and finite (dash points) distortion. The percentage refers to the amplitude of the commensurate PLD in the low-temperature phase. The intensity ratio (bottom) illustrates the energy-dependent sensitivity between reflex groups.Structural Dynamics ARTICLE scitation.org/journal/sdy Struct. 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Phys. Plasmas 27, 072901 (2020); https://doi.org/10.1063/5.0010050 27, 072901 © 2020 Author(s).On Beltrami states near black hole event horizon Cite as: Phys. Plasmas 27, 072901 (2020); https://doi.org/10.1063/5.0010050 Submitted: 06 April 2020 . Accepted: 15 June 2020 . Published Online: 01 July 2020 Chinmoy Bhattacharjee , and Justin C. Feng ARTICLES YOU MAY BE INTERESTED IN Variational formulation of plasma dynamics Physics of Plasmas 27, 022110 (2020); https://doi.org/10.1063/1.5139315On Beltrami states near black hole event horizon Cite as: Phys. Plasmas 27, 072901 (2020); doi: 10.1063/5.0010050 Submitted: 6 April 2020 .Accepted: 15 June 2020 . Published Online: 1 July 2020 Chinmoy Bhattacharjee1,a) and Justin C. Feng2 AFFILIATIONS 1Department of Physics, New York Institute of Technology, Old Westbury, New York 11586, USA 2CENTRA, Departamento de F /C19ısica, Instituto Superior T /C19ecnico—IST, Universidade de Lisboa—UL, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal a)Author to whom correspondence should be addressed: usa.chinmoy@gmail.com ABSTRACT In this article, we study Beltrami equilibria for plasmas near the horizon of a spinning black hole and develop a framework for constructing the magnetic field profile in the near horizon limit for Clebsch flows in the single-fluid approximation. We find that the horizon profile forthe magnetic field is shown to satisfy a system of first-order coupled ODEs dependent on a boundary condition for the magnetic field. Forstates in which the generalized vorticity vanishes (the generalized “superconducting” plasma state), the horizon profile becomes independent of the boundary condition and depends only on the thermal properties of the plasma. Our analysis makes use of the full form for the time- independent Ampe `re’s law in the 3 þ1 formalism, generalizing earlier conclusions for the case of vanishing vorticity, namely, the complete magnetic field expulsion near the equator of an axisymmetric black horizon assuming that the thermal properties of the plasma are symmet-ric about the equatorial plane. For the general case, we find and discuss additional conditions required for the expulsion of magnetic fields atgiven points on the black hole horizon. We perform a length scale analysis, which indicates the emergence of two distinct length scales char- acterizing the magnetic field variation and the strength of the Beltrami term, respectively. Published under license by AIP Publishing. https://doi.org/10.1063/5.0010050 I. INTRODUCTION Magnetic fields play an important role in the formation and evo- lution of many astrophysical compact objects such as white dwarfs,neutron stars, and black holes. 1–11Black holes are of particular interest because they possess a horizon, distinguishing them from other typesof compact objects; the presence of a horizon can, in principle, lead to qualitatively different behaviors in a surrounding plasma. 12–20One might expect the extreme conditions near the black hole horizon tolead to extreme conditions in the plasma itself so that the large scalemagnetic field structure is dominated by plasma dynamics in the vicin-ity of the horizon. Even in the absence of a horizon, a strong coupling between the magnetic field and plasma dynamics is already relevant for manysituations such as plasma confinement, solar physics, laser–plasmainteraction, and magnetic reconnection. An interesting example is thetendency of a magnetized plasma to evolve toward equilibrium states characterized by the ordered large scale magnetic field and flow struc- ture. One such state, first derived in the context of single-fluid magne-tohydrodynamics, ~r/C2 ~B¼K~B, is known as a “relaxed state,” where ~BandK, respectively, denote the magnetic field and a Lagrange multi- plier. 21,22Later, these states were extended by incorporating multi- species effects in the classical and relativistic plasmas.23–26Thedefining characteristic of these equilibrium states is the alignment of a more general physical quantity known as vorticity ~X¼~Bþm=q~r /C2~v, with the flow field ~v, in particular, the generalized Beltrami condi- tion~X¼k~v(where kis a Lagrange multiplier), which is an equilib- rium solution of the time-independent vorticity equation.27It should be noted here that ~X¼~r/C2 ~P,w h e r e ~P¼~Aþm=q~vis the canoni- cal momentum. The full equilibrium also requires the simultaneous satisfaction of the Bernoulli condition signifying homogeneous energydistribution. One might expect steady-state flows in the near horizon limit for astrophysical black holes, which slowly accrete plasma, and it is rea-sonable to model such plasma flows as force-free states; the existence and implications of a force-free region near the black hole horizon have been extensively studied in theoretical and numerical frame-works. 1,20,28–30Similarly, it is physically reasonable to expect a broader class of Beltrami equilibria, which incorporates the inertia of plasma constituents in the near horizon limit. In this paper, we adopt thevortical formalism of plasma dynamics to investigate these types of equilibria. On the other hand, these Beltrami states can also terminate because of the changes (via reconnection or other non-ideal effects) in field topology. For example, in solar physics, a class of states known as Phys. Plasmas 27, 072901 (2020); doi: 10.1063/5.0010050 27, 072901-1 Published under license by AIP PublishingPhysics of Plasmas ARTICLE scitation.org/journal/phpdouble Beltrami (DB) states, the superposition of two force free states, is shown to have the characteristics similar to those of active regions inthe solar corona. 31–33The breakdown of these DB states in those active regions can lead to catastrophic eruptive events such as coronal mass ejections (CMEs).33In general, the DB equilibrium has more energy available to cause eruptive events than a force-free linear MHD state,and the critical energy of the equilibrium is determined by invariantsof the system. If the plasma flow near the horizon of a black hole satis-fies the Beltrami condition, then the eventual termination of theBeltrami equilibrium may lead to similar eruptive events with observ-able signatures—such events and their signatures, while interesting,are beyond the scope of the present work and are left for future investi-gation. It should be mentioned that we regard such phenomena to beindependent of the jet-producing mechanisms explored in the litera-ture, such as the well known Blandford–Znajek mechanism. 20 To investigate Beltrami equilibria in the near horizon limit of a Kerr black hole, we employ the electrovortical formalism described in Refs. 24and34, which was later extended to curved spacetime in Refs. 4and27. The electrovortical formalism reorganizes the equations of a charged fluid into the form qUlMl/C23¼0. The electrovortical ten- sorMl/C23is an antisymmetric rank-two tensor constructed from the electromagnetic field strength tensor Fl/C23and the antisymmetrized derivatives of the “temperature-transformed” fluid flow field GUl, where Ulis the fluid four-velocity. The thermodynamic factor G0is a function of temperature Tand entropy r. This is similar to the ideal Ohm’s law in MHD; one can see the force-free structure when writingthe covariant equation of motion in its three-dimensional form C~E Gþ~U/C2~XG¼0; (1) ~U/C1~EG¼0; (2) where ~EG;~XG,Care the generalized electric field, generalized mag- netic field (generalized vorticity), and Lorentz factor, respectively.34–36 What is remarkable about this formalism is that the complicateddynamics of hot relativistic plasmas in curved spacetime has the famil-iar force-free MHD state, if expressed in suitably constructed variables.Equation (1)also indicates that the plasma is frozen to the generalized magnetic field lines and that the generalized helicity H¼h ~P/C1~Xiis a complete invariant of the model for arbitrary thermodynamics. As aresult, this formalism yields a broader class of equilibrium states,which is inaccessible to traditional fluid theories. 35,37,38 Recently, one such equilibrium state, obtained by setting ~XG¼0 in Eq. (1), has revealed a “Meissner”-like effect, in which the magnetic field is completely expelled from the equator of a black hole event hori- zon.36This equilibrium state also displays a rich interplay between plasma dynamics and general relativity by satisfying the Bernoulli con-dition ~E G¼0, which represents the homogeneity of the total energy. Since this result is partly a consequence of the spacetime geometrynear the horizon, it is natural to ask whether the expulsion of the mag-netic field from the event horizon in a black hole is present in a moregeneral equilibrium state. In this article, we develop a general framework for studying Beltrami equilibria in the near horizon limit of a spinning (Kerr) blackhole. In particular, we construct a series expansion that we use toobtain expressions for the magnetic field profile on the horizon, valid for arbitrary thermodynamics. We study the general properties of these profiles, identifying conditions for magnetic field expulsion atvarious points on the horizon. We also generalize the result of Ref. 36; our present analysis includes a previously neglected term in Ampe `re’s law—eliminating an implicit assumption—and we find that the Meissner-like effect at the equator of the horizon for the Beltrami states ~X G¼0 is still present even when this term is included. We also provide the correct version of the generalized Bernoulli’s condition for plasmas near a rotating black hole. II. KERR GEOMETRY Here, we describe the Kerr geometry for an uncharged, rotating black hole and choose c¼1. The metric components gl/C23can be read off from the line element ds2¼gl/C23dxldx/C23,w h i c h ,i nB o y e r – L i n d q u i s t coordinates ðt;r;h;/Þ(rbeing a radial coordinate, and h;/being the angular coordinates on a spheroid), takes the form:39,40 ds2¼/C0 1/C02GMr R/C18/C19 dt2þR Ddr2þRdh2/C04GMra sin2h Rdtd/ þr2þa2þ2GMra2 Rsin2h/C18/C19 sin2hd/2; (3) where Mis the mass, ais the spin parameter, and R:¼r2þa2cos2h D:¼r2/C02GMrþa2:(4) The event horizon of the black hole is located at one of the roots of D, in particular, the root given by the following expression: rH¼GMþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G2M2/C0a2p : (5) As is well known, the Kerr metric becomes singular at the horizon in Boyer–Lindquist coordinates; in particular, grr/1=D,w h i c hd i v e r g e s in the limit D!0. For our analysis, it will be convenient to work in terms of the metric components rather than their explicit expressions, as given in the line element. We begin with the form of a stationary and axisym- metric spacetime41,42 ds2¼/C0A2dt2þ2b/dt d/þh2 1dr2þh2 2dh2þh2 3d/2:(6) The quantities A2,b/;h2 1;h2 2,a n d h2 3are the functions of randhonly, and all correspond to the appropriate components of the Kerr metric (3). It is helpful to relate these quantities to the ADM 3 þ1f o r m a l - ism,43–47in which spacetime is foliated by a family of three- dimensional spacelike hypersurfaces such that each hypersurface Rtis defined by a constant value for the time coordinate t.T h em e t r i ci s decomposed into the ADM variables a(the lapse function), bi(the shift vector), and cij(the induced metric), which can be identified from the relations gtt¼/C0A2¼/C0a2þcijbibjand g0i¼cijbj.T h e explicit expression for ais a¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DR a2þr2 ðÞ a2/C0Dþr2 ðÞ þDRs : (7) It is helpful to define the quantity nl¼/C0agl/C23r/C23t,w h i c hi st h eu n i t normal vector to Rt. One can interpret nlto be the four-velocity for observers whose spatial frames are tangent to the surfaces of constant t, termed zero angular momentum observers (ZAMOs). One canPhysics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 27, 072901 (2020); doi: 10.1063/5.0010050 27, 072901-2 Published under license by AIP Publishingdecompose the four-velocity with respect to this frame, with U0¼C=a;~U¼C~V,w h e r e C¼1=ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C0V2p is the Lorentz factor. Since we will be considering Beltrami states in the near horizon limit, it is appropriate to establish the scaling of geometric quantitiesas one approaches the horizon. Note that in the near horizon limitr!r H, the Lapse function avanishes, and the metric component h1¼ffiffiffiffiffiffiffiffiffi R=Dp diverges. We find the following behavior for aandh1 and their derivatives in the near horizon limit by expanding in powers ofs:¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jr/C0rHj=rHp ;WLOG ,w es e t rHto unity. Noting that @rð/C1Þ ¼ ð 1=2sÞ@sð/C1Þ, one finds to leading order a¼a1ðÞsþOs3ðÞ; @ra¼a1ðÞ 2sþOs2ðÞ; @ha¼a0 1ðÞsþOs3ðÞ; h1¼h1/C01ðÞ sþOsðÞ; @rh1¼/C0h1/C01ðÞ 2s3þOs/C01ðÞ ; @hh1¼h0 1/C01ðÞ sþOsðÞ;(8) where all coefficients are functions of h, and the order of the coefficient appears in the parentheses (so that h1;ðIÞis the coefficient for the sI t e r m ) .O n ec a ns e et h a tt h eq u a n t i t y ah1/C24að1Þh1ð/C01Þremains finite as one approaches the horizon. Another important quantity to con- sider is b/:¼b/=h2 3, which scales in the following manner: b/¼b0þb/ ð2Þs2þOðs4Þ; (9) where b0is a constant. Though b/does not vanish on the horizon, @hb//C24Oðs2Þvanishes, and one finds that the combination h2 1@hb/ remains finite at the horizon. It turns out that in the near horizon limit, it suffices to consider only these general properties of ah1andb/. One can also establish the scaling behavior for the motion of matter in the near horizon limit. The Kerr metric admits two Killingvectors g¼@=@tand w¼@=@/with components g l¼dl 0and wl¼dl 3. Lowering the indices, one obtains gl¼gl0; wl¼gl3:(10) Killing vectors can be used to define48the energy Eand angular momentum Lfor a fluid particle with four-velocity Ul E:¼/C0mglUl¼ma2/C0b2 /=h2 3/C16/C17 U0/C0mb/U3; L:¼mwlUl¼mb/U0þmh2 3U3:(11) Eliminating U3, one may write Lþh2 3E b/¼h2 3a2 b/U0: (12) For finite energy and angular momentum, the left-hand side must be finite at the horizon, and since a2/C24a2 ð1Þs2, then to leading order, U0 must scale as U0/C24U0 ð2Þ=s2.F r o mt h ee x p r e s s i o n U0¼C=a, it follows thatCscales in the following way:C/C24Cð/C01Þ s; (13) implying that the fluid velocity reaches the speed of light at the hori- zon. Note that this scaling behavior depends only on the demand that the energy and angular momentum for the fluid particles are finite. Physically, one expects the Lorentz factor Cto diverge, since the four- velocities of ZAMOs become null at the horizon, so infalling matter will appear to be moving at velocities close to the speed of light for near horizon ZAMOs. III. PLASMA DYNAMICS IN CURVED SPACETIME: GRAND GENERALIZED VORTICITY A. Covariant electrovortical formalism The dynamics of a multi-species ideal plasma is summarized in the expression rlTl/C23¼qUlFl/C23,w h e r e Tl/C23¼hUlU/C23þpgl/C23and h¼pþq.H e r e , pandqare the respective pressure and plasma den- sity. The dynamical equations may be rewritten in the standard form mnU/C23r/C23GUlðÞ ¼qnFlbUb/C0rlp; (14) where the quantities m,q,a n d nare the respective mass, charge, and number density for the constituent particles of the fluid. The fluid four-velocity for each species may be written as Ul¼dxl=ds,w h e r e sis the proper time for a fluid element. The thermodynamic factor G is given by the expression h¼mnG,w i t h handqbeing the respective enthalpy and mass density of the fluid. We assume pressure p¼nkT and a local Maxwellian distribution function for which the corre- sponding thermodynamics factor has the form G¼ K3ðmc2=kbTÞ= K2ðmc2=kbTÞ,w h e r e Kjis the modified Bessel function of order j,a n d kbis the Boltzmann constant. Though one typically requires a multi- fluid model to fully describe the behavior of a plasma, we explore a simplified model (which is nonetheless still more general than that of MHD) in which the behavior of a quasineutral plasma is described by the dynamics of an effective single charged fluid in a neutralizing back- ground, assuming that the motion of the effective fluid does not differ strongly from the bulk motion. In a low density and low collisionality plasma, where the species have different thermodynamics, it may be important to analyze the dynamics of single species separately in a neutralizing bulk plasma.49 The electrovortical formalism is based on the following observations: •An anti-symmetric flow field tensor can be obtained by incorporat- ing the temperature into the flow, expressed as Sl/C23:¼rlðGU/C23Þ /C0r/C23ðGUlÞ.E q u a t i o n (14) c a nt h e nb er e w r i t t e na s qUlMl/C23¼Tr/C23r; (15) where Ml/C23¼Fl/C23þðm=qÞSl/C23is the electrovortical tensor, and the entropy density rfor the fluid obeys Tr/C23r¼ðmnr/C23G /C0r/C23pÞ=n.24 •A relativistic perfect fluid is isentropic, U/C23r/C23r¼0; (16) the entropy density rbeing constant along a flow line. Equation (15) for plasma dynamics can be recast in a source-free form by defining the following electrovortical potential Pl:Physics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 27, 072901 (2020); doi: 10.1063/5.0010050 27, 072901-3 Published under license by AIP PublishingPl¼Alþm qGUlþrrlQ; (17) where Alis the usual electrodynamical four-potential and Qis a sca- lar, which is defined by the expression U/C23r/C23Q¼ T=q; (18) and an appropriate set of initial data for Q, which is determined by the invariants of the vortical dynamics such as circulation, helicity, etc. The covariant equation of motion Eq. (15)may then be written36 qUlMl/C23¼0; (19) where we define the grand electrovortical tensor Ml/C23to be Ml/C23¼rlP/C23/C0r/C23Pl: (20) It is straightforward to show [keeping in mind Eq. (18)forQ]t h a to n e can recover Eq. (15)from Eq. (19). The source-free formalism summa- rized in Eqs. (17)–(20) is referred to as the grand generalized vortical formalism in the literature, and we use the modifier “grand” to refer to quantities constructed from the grand electrovortical tensor Ml/C23. In general, flow fields Ulsatisfying Eq. (18) can be written in the form TUl¼/C0qrlQþ bl; (21) where the vector blis orthogonal to Ul.50To simplify the analysis, we require bl¼0, so that TUl¼/C0qrlQ.O n em a yr e c o g n i z et h i s Clebsch flow restriction to be the requirement the flow ishypersurface-orthogonal (i.e., irrotational); the analysis of equilibriafor more general flows ( b l6¼0) will be explored in a forthcoming article. B. 3þ1 decomposition of the electrovortical formalism It will be useful to rewrite the covariant formulas of the electro- vortical formalism in terms of more familiar three-dimensional varia- bles. We do this by rewriting Eq. (19)using the 3 þ1 ADM formalism discussed earlier—this decomposition of the electrovortical equationsis discussed in detail in Refs. 27and 35.E q u a t i o n (19) is split into space and time components; the spatial components, which form thethree-dimensional equation of motion, are obtained by applying the projection operator c l/C23¼dl /C23þnln/C23; one obtains Cða~EGþ~b/C2~XGÞþ~u/C2~XG¼0; (22) where the grand generalized electric field ~EGand vorticity ~XGare given by the respective equations: ~EG¼~E/C0m aq~rðaTG0CÞ/C0m q2r/C1ð G0T~UÞþ2 3KG0T~U/C20/C21 /C0m qa@tðG0T~UÞ/C0L ~bðG0T~UÞ/C16/C17 ; (23) ~XG¼~r/C2 ~PG¼~r/C2 q~AþmG0T~U/C0/C1 ; (24) whereC¼1=ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C0V2p is the Lorentz factor, flow velocities ~U¼C~V and~u¼C~v;~Ais the vector potential, and G0¼ð G =T/C0r=mÞis the modified thermodynamic factor. Here, ~bis the shift vector, L~bis the Lie derivative with respect to ~b;ris a trace-free rank-2 tensor (with components rijformed from @tcijandbi) called the shear tensor, andKis the mean curvature for Rt. The dynamics of a hot, relativistic mag- netized plasma as expressed by Eq. (22) has a similar structure of the ideal Ohm’s law in ideal MHD qUlFl/C23¼0. It is clear that plasma inertia plays a critical role in the electrovortical dynamics and the char-acteristics of any equilibria near the Kerr black hole will be fundamen-tally different. IV. THE BELTRAMI EQUILIBRIUM A. Beltrami equilibria in axisymmetric spacetimes Equation (22) is trivially satisfied if, in the stationary state, the combination ~E Gþ~b/C2~Xequals zero and the grand generalized vor- ticity ~XG(GGV) is parallel to the coordinate flow velocity ~u:¼a~U /C0C~b a~EGþ~b/C2~X¼~rW¼0; (25) ~XG¼q~BþmT~rG0/C2~U¼l^n~u; (26) where the magnetic field is ~B¼~r/C2 ~A,lis the separation constant (inverse length), and Wcontains all the gradient forces. We have used the vector identity ~r/C2 ~rQ ¼ 0i nE q . (26). The first equation is a general relativistic Bernoulli’s condition in Kerr spacetime signifyingthe homogeneity of the total energy, and the second equation is theBeltrami condition. It should be noted here that Eq. (25) follows from the steady-state generalized Faraday’s law for electrovortical variableswith the condition ~r/C1~b¼0a n d c ij_cij¼0.35T h ea p p e a r a n c eo f ~b (implicit in ~u¼a~U/C0C~b) on the right-hand side of the second equation (26) follows from the condition that the plasma is sta- tionary with respect to the Killing vector @=@t.T h es e p a r a t i o nc o n - stant lplays the role of a Lagrange multiplier if one were to derive Eq. (26) via a constrained energy minimization principle—the separation constant is related to the invariants of the system suchas energy, helicity, etc. 51The helicity is useful in understanding the astrophysical dynamos, solar wind, and fusion, as well as in deter-mining the conditions for the loss of Beltrami equilibria that leadto eruptive events. 31,32When satisfied, Eqs. (25) and (26) consti- tute a class of plasma states known as Beltrami–Bernoulli equilib-ria. The factors in the RHS of Eq. (26) satisfy the continuity equation ~r/C1ð ^n~uÞ¼0, where ^nis the density envelope of the plasma species and also ensures that ~r/C1~X G¼0. Since we seek solutions corresponding to a steady-state charge neutral Maxwell-fluid system (for example, an electron plasma withions as the neutralizing background), Eq. (26) should be coupled with the steady-state Ampe `re’s law 46,52 ~r/C2ð a~BÞ¼4pnqa~U/C0£b~E: (27) The term £ b~Ein Ampe `re’s law is often implicitly assumed to vanish. However, one should not neglect it when considering plasma dynam-ics in the near horizon limit for generic flows. 53The Lie derivative term may be written in terms of partial derivatives as £bEi¼bj@jEi/C0Ej@jbi: (28) Since the shift vector ~bis aligned with the /direction, the first term vanishes by axisymmetry. However, the second term does not, in gen-eral, vanish. If the plasma is assumed to be quasineutral in the comov-ing frame of the effective fluid, then the electric field ~E¼/C0~U/C2~B=C, so that Ampe `re’s law takes the form:Physics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 27, 072901 (2020); doi: 10.1063/5.0010050 27, 072901-4 Published under license by AIP Publishing~r/C2ð a~BcÞ¼^n k2a~U/C01 Cð~U/C2~BcÞ/C1~@hi ~b; (29) where the fields have been normalized in terms of the cyclotron fre- quency Bc¼q=mB;n¼^nn0,w h e r e n0is the average density, and the skin depth k¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pn0q2=mp , associated with some average density, is an intrinsic length scale of the dynamics. Since Eq. (29) is an algebraic equation for ~U, one can, in principle, solve it for the components of ~U and use it in Eq. (26) to obtain an equation for the magnetic field ~B. One property that simplifies the analysis is the fact that the last termon the RHS of Eq. (29)contributes only to the /component; the rand hcomponents of ~Ucan then be obtained independently of the /com- ponent. An explicit calculation reveals that ~U, obtained in this way, sat- isfies the steady-state continuity equation ~r/C1ð ^n~uÞ¼0. To simplify the results, it is helpful to define the rescaled compo- nents of the magnetic field B r:¼ah1Br; Bh:¼ah2Bh; B/:¼ah3B/;(30) where Br,Bh;B/form the orthonormal basis components of ~B.T h e n , Ampe `re’s law yields the following components of ~Uin the orthonor- mal basis: Ur¼qk2Ch1ah3@hB/ W/C20/C21 ; (31) Uh¼/C0qk2Ch2ah3@rB/ W/C20/C21 ; (32) U/¼/C0qk2aCh2 3 Wh3 3k2qBh@rb//C0Br@hb//C0/C1 þW/C16/C17 /C2h3k2qB/ðh2 1@hb/Þ@hB/þðh2 2@rb/Þ@rB//C16/C17h þW@hBr/C0@rBh ðÞi ; (33) where the following quantity has been defined (note that, assuming the number density remains finite, these quantities remain finite in thenear horizon limit): W:¼4pm^nh 2h2 3a2Ch1: (34) Plugging these results into (25), the rescaled magnetic field compo- nents become Br¼ah1h3k2l^nCa2h1@hB/ W/C0mTG0 ;h qh2U/ ! ; (35) Bh¼ah2h3k2l^nCa2h2@rB/ W/C0mTG0 ;r qh1U/ ! ; (36) B/¼k2ma2Ch2 3Th21G0 ;h@hB/þh2 2G0 ;r@rB/hi h1h2W þl^nah2 3aU//C0Cb//C16/C17 q; (37)where U/is given in (33), and we have assumed that the thermody- namic potentials (in particular, G0) depend only on randhand have defined G0 ;h¼@G0=@handG0 ;r¼@G0=@r. Equations (35)–(37) form the complete description of Beltrami states of an ideal plasma in a spacetime given by the line element (6). Next, we analyze the characteristics of these equations in the near hori- zon limit of a black hole. B. Near horizon limit We now consider the behavior of magnetic fields in the near horizon limit. As discussed in Ref. 52, the tangential components of the magnetic fields BhandB/(in the orthonormal frame) diverge in the near horizon limit—this is attributed to the fact that the four- velocities of ZAMOs become null at the horizon. On the other hand, the rescaled components Br;Bh,a n dB/remain finite or are zero in the near horizon limit; for this reason, it is appropriate to work in terms of these components (which are lowered-index coordinate basis elements). To perform the near horizon analysis, we expand in s¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jr/C0rHj=rHp .F o rs o m ef u n c t i o n Qðr;hÞ,t h ee x p a n s i o nw i l lb e denoted in the following manner: Q¼X IQðIÞsI; (38) where QðIÞare the functions of honly. In general, the leading order terms can have inverse powers of s( w ei n c l u d et h en e g a t i v ev a l u e s forI), and the linear terms in scan yield divergent radial derivatives at the horizon by virtue of @rð/C1Þ ¼ ð 1=2sÞ@sð/C1Þ. To simplify the analysis, the thermodynamic potentials and their derivatives are assumed to be finite in the near horizon limit, meaning that we exclude inverse and odd powers of s. Equations (31)–(33) c a nb eu s e dt op l a c et h el i m i t so nt h el e a d i n g order and odd powers for the magnetic field. First, recall that ~U¼C~V and that (as shown earlier) V2!1 at the horizon for finite energy and angular momentum. Then, making use of (8)and(9),a n dt h ef a c t thatah1h2 1@hb/are finite at the horizon, we find from Eqs. (31)–(33) that the radial derivatives @rBh;@rB/and the derivative @hBrcan diverge no faster than s/C01, which implies that the smallest possible term of odd power is O(s). We are now in a position to examine the leading order behavior for Eq. (36) forBh. From the considerations discussed in the preced- ing paragraph, one can show that the terms within the parentheses of Eq.(36) cannot have inverse powers of sto the leading order. The overall factor of aimpliesBh!0 on the horizon, or that the coeffi- cients in the expansion for BhsatisfyBhðIÞ¼0f o r I/C200. Now, we consider the expansion of Eqs. (35)–(37) ins;i n principle, one can solve these equations in the near horizon limit by demanding that the equations hold to each order in s. The full expansion, performed in Mathematica , is rather com- plicated and will not be presented here in full. However, we will describe some relevant features of the expansion. Upon Eq. (35) forBr, we find that the leading order term is Oðs/C01Þ, which diverges at the horizon. This divergent term can be removed with the condition Bhð1Þ¼qk2h2ð0Þb/ ð2ÞB/ð0ÞB/ð1Þ 4mp^nð0Þh3ð0Þa2 ð1Þh1ð/C01ÞCð/C01Þ; (39)Physics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 27, 072901 (2020); doi: 10.1063/5.0010050 27, 072901-5 Published under license by AIP Publishingwhich can be satisfied if Bhð1Þ¼B/ð1Þ¼0. When expanding Eq. (36) forBh,w efi n dt h a tt h efi r s t - o r d e r[ O(s)] term implies Bhð1Þ¼0{ a s argued earlier, the zeroth-order [ Oðs0Þ]t e r mi m p l i e s Bhð0Þ¼0}. The left-hand side of Eq. (39) vanishes, which implies that either B/ð0Þ¼0o rB/ð1Þ¼0. If the fluid is flowing into the horizon (as one might expect), then Eq. (31) indicates that a nonzero component Vr requires @hB/6¼0, which, in turn, implies B/ð0Þ6¼0. It follows that B/ð1Þ¼0, meaning that @rB/must be finite at the horizon. This, combined with Eq. (32), implies that the fluid velocity ~Vmust lie in ther-/plane; since ~Balso lies in the r-/plane, one can show that £~b~E!0 at the horizon, as was assumed in Ref. 36. The expression for B/ð0Þmay be obtained from the Oðs0Þterm in Eq. (37) forB/, which yields a differential equation for B/ð0ÞðhÞ ¼XðhÞof the form @XðhÞ @h¼KðhÞþUðhÞXðhÞ; (40) which admits the solution XðhÞ¼expðh h0KðxÞdx() /C2C1þðh h0UðyÞ expðz h0KðzÞdz() dy2 6643 7752 6643 775; (41) where KðhÞ¼ 4plb/ ð0Það1ÞCð/C01Þh2 2ð0Þh2 3ð0Þ^n2 ð0Þ k2qTð0Þ@hG0 ð0Þ; UðhÞ¼4ph2 2ð0Þ^nð0Þ k2Tð0Þ@hG0 ð0Þ:(42) Expanding (37) toO(s), making use of the results Bhð1Þ¼B/ð1Þ¼0, yields the constraint l^nð0ÞCð0Þ¼0, which can be satisfied if l¼0o r Cð0Þ¼0; for general l,w es e t Cð0Þ¼0; we remind the reader that the leading order behavior in Ccomes from Cð/C01Þ,n o tCð0Þ. A first-order differential equation for Brð0Þcan be obtained from theOðs0Þterm in Eq. (35),w h i c hi ss i m i l a ri nf o r mt oE q . (41),b u t now depends on Bhð2ÞandB/ð2Þ. In order to obtain expressions for Bhð2ÞandB/ð2Þ,E q s . (36) and(37) must be expanded to Oðs2Þand the equation ~r/C1~B¼0 must be expanded to O(s). The Oðs2Þterm in Eq.(37)yields an algebraic constraint: qBhð2Þ¼k2lqað1ÞG0 ð2Þh2ð0Þ@hB/ð0Þ 4pmhð/C01Þh3ð0Þ@hG0 ð0Þ/C0qG0 ð2Þh2 2ð0ÞBrð0Þ h2 ð/C01Þ@hG0 ð0Þ /C0k2lqað1Þh2ð0ÞB/ð2Þ 4pmhð/C01Þh3ð0Þ: (43) The Oðs0Þterm in Eq. (35), combined with the O(s)t e r m si n ~r/C1~B¼0 [the Oðs0Þterm yields Brð1Þ¼0] and Oðs2Þterms in Eq.(37), yields a system of linear first-order coupled ordinary differen- tial equations (ODEs) of the form @h~Z¼~nþN/C1~Z; (44) where N¼NðhÞis a 3 /C23 matrix, and the components of ~Zare defined asZ1:¼Brð0Þ; Z2:¼Bhð2Þ; Z3:¼B/ð2Þ:(45) It is known that a system of the form (44) yields unique solutions for the initial value problem. However, the equation is not homogeneous, as the components of the vector ~n¼~nðhÞdo not vanish. The explicit components have the following form: n1¼/C0lað1Þhð/C01Þh2ð0Þ^nð0Þ@hB/ð0Þ mh 3ð0ÞTð0Þ@hG0 ð0Þ /C0k2q@hb/ ð2Þhð/C01ÞB/ð0Þ@hB/ð0Þ 4pma2 ð1ÞCð/C01Þh2ð0Þh3ð0Þ^nð0Þ; n2¼/C0h2 2ð0ÞBrð2Þ h2 ð/C01Þ; n3¼@hB/ð0Þ/C0@hG0 ð2Þ @hG0 ð0Þþ^nð2Þ ^nð0Þ/C0Tð2Þ Tð0Þ ! þ2h2ð2Þ@hB/ð0Þ h2ð0Þ þ4pl^n2 ð0Þh2 2ð0Þh2 3ð0Það1Þb/ ð0ÞCð/C01Þ k2qTð0Þ@hG0 ð0Þb/ ð2Þ b/ ð0Þþ^nð2Þ ^nð0Þ2 4 þ2h3ð2Þ h3ð0ÞþCð1Þ Cð/C01Þþað3Þ að1Þ3 5:(46) Note that n2depends on Brð2Þ, which may be interpreted as the value of@rBrevaluated on the horizon, since Brð1Þ¼0 (which is obtained from the zeroth-order term in ~r/C1~B¼0). To specify Brð2Þ,o n em u s t continue the expansion to higher powers in s, which will depend on higher-order coefficients; in this sense, @rBrcaptures the dependence of the horizon profile on the behavior of the magnetic field far from the horizon, at least for l6¼0 (the l¼0 case will be discussed later). It is therefore appropriate to regard Brð2Þas a boundary condition for the magnetic field at the horizon. For completeness, the nontrivial components of the matrix N¼ NðhÞare written below: N11¼4ph2 2ð0Þ^nð0Þ k2Tð0Þ@hG0 ð0Þ; N13¼k2qb/ ð2Þh2ð0ÞB/ð0Þ 4pma2 ð1ÞCð/C01Þhð/C01Þh3ð0Þ^nð0Þ; N21¼h1ð1Þh2 2ð0Þ h3 ð/C01Þþað3Þh2 2ð0Þ að1Þh2 ð/C01Þ/C0h3ð2Þh2 2ð0Þ h2 ð/C01Þh3ð0Þ/C0h2ð2Þh2ð0Þ h2 ð/C01Þ; N22¼@hað1Þ að1Þþ@hh2ð0Þ h2ð0Þ/C0@hh3ð0Þ h3ð0Þ/C0@hh1ð/C01Þ hð/C01Þ; N32¼/C04plað1Þhð/C01Þh2ð0Þh3ð0Þ^n2 ð0Þ k2mG0 ð2ÞT2 ð0Þ@hG0 ð0Þ; N33¼G0 ð2Þh2 2ð0Þ h2 ð/C01Þ@hG0 ð0Þ/C0l2a2 ð1Þh2 2ð0Þ^n2 ð0Þ m2G0 ð2ÞT2 ð0Þ@hG0 ð0Þþ4ph2 2ð0Þ^nð0Þ k2Tð0Þ@hG0 ð0Þ;(47) with the remaining components being N12¼1;N23¼0;N31¼1. One can specify the horizon profile in the l¼0c a s ew i t h o u t specifying Brð2Þ.N o t et h a t N32¼0w h e n l¼0, so that the equationPhysics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 27, 072901 (2020); doi: 10.1063/5.0010050 27, 072901-6 Published under license by AIP PublishingforB/ð2Þdecouples from Bhð2Þ, and has the form given in Eq. (40), with X¼B/ð2Þand the coefficients K¼n3;U¼N33evaluated at l¼0. One may then solve the ODE for Bhð2Þfirst and then use the constraint (43) to eliminate the dependence on Bhð2Þin the ODE forBrð0Þ. The resulting equation also has the form of Eq. (40),w i t h X¼Brð0Þand the coefficients K¼n1(evaluated at l¼0) and UðhÞ¼N11þN13/C0G0 ð2Þh2 2ð0ÞBrð0Þ h2 ð/C01Þ@hG0 ð0Þ: (48) The horizon profile in the l¼0 case can be obtained in a manner independent of the behavior of the magnetic field far from the horizon.It should be noted that the profile depends on the horizon profile of the fluid potentials G 0;^n, and also their normal derivatives G0 ð2Þ;^nð2Þ [also note the dependence on Cð2Þ] can be determined by the Bernoulli condition along with an appropriate equation of state. One can still extract some general properties of the magnetic field without specifying the equation of state. As discussed in Ref. 36,o n e can have G0 ;h¼0 at various points on the horizon. In particular, this condition may be satisfied at the equator of the horizon under theassumption that the thermal properties of the fluid are symmetricabout the equatorial plane or at any value of hwhere G 0has a local maximum or minimum. We also add that regularity conditions on the axis of the black hole require G0 ;h¼0m u s th o l da tt h ep o l e s ,t h o u g h one must also account for the fact that h3ð0Þ¼0a tt h ep o l e sa sw e l l . At points on the horizon where G0 ;h¼0, Eqs. (35) and(37) yield the following expressions: Brð0Þ¼lk2að1Þhð/C01Þ@hB/ð0Þ 4pmh 2ð0Þh3ð0Þ; B/ð0Þ¼/C0 ð l=qÞb0að1ÞCð/C01Þh2 3ð0Þ^nð0Þ:(49) Ifl¼0, the magnetic field is completely expelled at points on the horizon where G0 ;h¼0, recovering the result in Ref. 36. and generaliz- ing it to the case where £ ~b~E6¼0 (though as in that case, we still con- sider the Clebsch flow). In the more general l6¼0 case, we argue that the extrema of B/ð0Þcannot all coincide with G0.A tt h ee x t r e m a , @hB/ð0Þ¼0, which, combined with the expression for Urin Eq. (31), implies that the fluid velocity is tangent to the horizon. One might expect such behavior forplasma elements to be unphysical near the horizon since there are nocircular orbits close to the horizon for finite 1 /C0a(abeing the spin parameter) and there is no physical mechanism that can prevent theplasma elements from falling in; one, therefore, expects the density ^n ð0Þ¼0 to vanish at these points. However, this implies that B/ð0Þ must vanish as well; if the extrema for B/ð0Þall coincide with an extrema of G0,t h e nE q . (49) holds for all extrema of B/ð0Þand follows from physical considerations that B/ð0Þ¼0 everywhere on the hori- zon. However, this would imply that Ur¼0, so no plasma falls into the black hole. This indicates that there must exist at least one extre-mum for B /ð0Þthat does not coincide with an extremum of G0.T h i s does not, of course, exclude the possibility that some extrema for B/ð0Þ can coincide with an extremum of G0, and in the instances where this occurs, one has a complete expulsion of the magnetic field at thesecoincident points on the horizon. Though the Beltrami equilibria we have described here permit nonvanishing magnetic fields (in which energy can be stored), astability analysis is needed to determine whether these Beltrami equi- libria can be used to describe eruptive events similar to those observedin solar plasmas. In particular, whether such equilibria are stable ormetastable depends on the exact relationship between helicity andtotal energy, which has not been discussed in the present analysis. Such an analysis will be left for future work. C. Length scale analysis The main distinction between the electrovortical formalism and that of GRMHD is that the electrovortical formalism contains multiplelength scales that allow one to describe in greater detail the depen-dence of the macroscopic dynamics on the microphysical properties ofthe plasma. To illustrate this, we extend the length scale analysis pre- sented in Ref. 34to the GR Beltrami states. The generalized supercon- ducting state ~X G¼0 corresponds to the limit l¼0 or the vanishing of the length scale LB¼lk2, which, in turn, corresponds to the scale at which the Beltrami term in the equations becomes important. Bydefining ^k2¼fG0,w em a yw r i t eE q s . (36)and(37),a s 1 ^k2¼1 h1@lnG0 @r/C18/C19 1 h1@lnBðh;/Þ @r/C18/C19 ¼1 LgLmag; (50) where 1 =Lg¼1=h1ð@lnG0=@rÞand 1 =Lmag¼1=h1ð@lnBðh;/Þ=@rÞ. To keep our calculation simple, we are considering the limit whenh¼p=2 and the thermodynamic factor is symmetric about the equi- torial plane. Therefore, the magnetic field variation occurs on a hybridlength scale L mag¼^k2=Lg, which is a combination of a modified skin depth and a thermal gradient scale. For Lg>^k, the magnetic field can vary at a length scale shorter than the skin depth, which one might expect for relativistically hot plasmas. In general, the skin depth char-acterizes the kinetic–magnetic reservoir of energy arising from micro-scale physics, which can drive the dynamo and reverse-dynamomechanisms. 34,54,55 On the other hand, when LB6¼0 [in our case, Eq. (26)], there are two intrinsic length scales LgandLB(associated with the magnetic field structures) in this formalism, which do not appear in GRMHD. Wenote here that the generalized helicity in this formalism is a topological invariant if the divergence of helicity four vector K lis zero, i.e., rlKl¼0. Here, we define the four helicity as Kl¼M/C3l/C23P/C23,w h e r e M/C3l/C23is the dual of the electrovortic field tensor Ml/C23. One can com- pute the divergence as rlKl¼1 2Ml/C23M/C3l/C23¼/C02~XG/C1~EG¼0; (51) w h e r ew eh a v eu s e dE q . (22)to compute the last equality. The corresponding species helicity is written as h¼/C0ð RtnlKlffiffifficpd3x¼ð Rt~PG/C1~XGffiffifficpd3x; (52) where ~P¼~Aþm=qG~Uþr~rQis the generalized three- momentum, nlis the four-velocity of the Eulerian observers (the unit normal vector to constant tsurfaces Rt)a n d c¼detðcijÞ.T h e Beltrami condition, when substituted from Eq. (26) into Eq. (52), determines the Beltrami length scale LBcompletely. One can, therefore, relate LBto the profile-modified skin depth ^k2, the thermal gradient Lg, and helicity Hnear the black hole event horizon. One might recognize the similarity to the Taylor relaxed statePhysics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 27, 072901 (2020); doi: 10.1063/5.0010050 27, 072901-7 Published under license by AIP Publishingin MHD where the length scale is fully determined by the ratio between the magnetic helicity and magnetic energy. It should be noted that for superconducting solution ~XG¼0, the generalized helicity is identically zero since the Beltrami length scale vanishes. Finally, weremark that the generalized helicity is useful for establishing the crite-ria for the termination of Beltrami equilibria in solar physics 31,32and may be similarly useful for investigating the stability of Beltrami equi- libria in black hole spacetimes. V. SUMMARY AND DISCUSSION In this work, we have described a single-fluid Beltrami state for an ideal plasma with the Clebsch flow surrounding a rotating blackhole. In particular, we have presented a framework for characterizingthe behavior of magnetic fields near black hole horizons, valid inBoyer–Lindquist coordinates in which the metric components become singular at the horizon. The Beltrami condition in rotating black holes dictates the alignment of the generalized vorticity and coordinate flowvelocity, which is completely different from the non-rotating blackholes. The inherent rotation of spacetime also fundamentally alters the generalized Bernoulli’s condition, which indicates the balances among different potential forces. We have demonstrated how one can obtain the magnetic field profile at the horizon, given the profiles for the fluid quantities G 0,T, and ^n. In particular, we find that the tangential profile for the magnetic field at the horizon can be obtained by expanding Eqs. (35)–(37) to zeroth order in s¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jr/C0rHj=rHp ,w i t hBh¼0, andB/given by Eqs. (41) and(42). The horizon profile for the radial component Br requires expanding Eqs. (35)–(37) to second order in sand~r/C1~B¼0 to zeroth order; in doing so, we find that given the radial derivative @rBrat the horizon, Brcan be obtained from the system of equations described in Eqs. (43)–(45) . We also find that the radial derivatives of t h em a g n e t i cfi e l da tt h eh o r i z o nm u s tb efi n i t e .I nt h e l¼0 case, the horizon profile can be obtained without specifying @rBrat the hori- zon, so that the system Eqs. (43)–(45) forBrdepends only on the pro- files for the fluid quantities and the other components of the magneticfield at the horizon. We have also described some general features of the horizon pro- file and have extended the analysis of Ref. 36for the generalized super- conducting states (in the sense of vanishing generalized vorticity) to a more general class of flows in which £ ~b~Eis not assumed to be zero a priori (though this holds at the horizon), finding that the expulsion of magnetic fields at the maxima of the thermal factor G0on the horizon holds for these more general flow conditions. We have also demon- strated for more general Beltrami states satisfying l6¼0, one can also have the magnetic field expulsion at points where the extrema of thehorizon profiles for G 0andB/coincide and have argued that B/ must possess extrema that do not coincide with those of G0. Of course, a complete account of the magnetic field in the near horizon limit requires an expansion to higher order ins¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jr/C0r Hj=rHp , at least to the point needed to determine Br. Furthermore, knowledge of the thermal profiles for the fluid, in partic- ular, the form of G0;^n,Tand their radial derivatives at the horizon must be supplied to obtain an explicit profile for the magnetic field; inprinciple, this can be obtained by specifying an appropriate equationof state. This is because the Beltrami states are essentially solutions tothe fluid equations—the magnetic field profiles we obtained in this way are those that correspond to a Clebsch flow Beltrami state for allthermodynamics. The most remarkable aspect of this formalism is the emergence of two length scales, i.e., thermodynamics modified intrin- sic length L magand Beltrami length LB, which is foreign to the existing single- or multi-fluid plasma models. 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1.5144974.pdf

J. Chem. Phys. 152, 174107 (2020); https://doi.org/10.1063/1.5144974 152, 174107 © 2020 Author(s).Toward DMRG-tailored coupled cluster method in the 4c-relativistic domain Cite as: J. Chem. Phys. 152, 174107 (2020); https://doi.org/10.1063/1.5144974 Submitted: 13 January 2020 . Accepted: 06 April 2020 . Published Online: 05 May 2020 Jan Brandejs , Jakub Višňák, Libor Veis , Mihály Maté , Örs Legeza , and Jiří Pittner COLLECTIONS Paper published as part of the special topic on Collection ARTICLES YOU MAY BE INTERESTED IN QMCPACK: Advances in the development, efficiency, and application of auxiliary field and real-space variational and diffusion quantum Monte Carlo The Journal of Chemical Physics 152, 174105 (2020); https://doi.org/10.1063/5.0004860 A collocation-based multi-configuration time-dependent Hartree method using mode combination and improved relaxation The Journal of Chemical Physics 152, 164117 (2020); https://doi.org/10.1063/5.0006081 Recent developments in the general atomic and molecular electronic structure system The Journal of Chemical Physics 152, 154102 (2020); https://doi.org/10.1063/5.0005188The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Toward DMRG-tailored coupled cluster method in the 4c-relativistic domain Cite as: J. Chem. Phys. 152, 174107 (2020); doi: 10.1063/1.5144974 Submitted: 13 January 2020 •Accepted: 6 April 2020 • Published Online: 5 May 2020 Jan Brandejs,1,2,a) Jakub Viš ˇnák,1,2,3,b)Libor Veis,1,c) Mihály Maté,4,5,d)Örs Legeza,4,e)and Ji ˇrí Pittner1,f) AFFILIATIONS 1J. Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, v.v.i., Dolejškova 3, 18223 Prague 8, Czech Republic 2Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic 3Czech Academic City in Erbil, Yassin Najar Street, Kurani Ankawa, Erbil, Kurdistan, Region of Iraq 4Strongly Correlated Systems “Lendület” Research Group, Institute for Solid State Physics and Optics, MTA Wigner Research Centre for Physics, Konkoly-Thege Miklós út 29-33, H-1121 Budapest, Hungary 5Department of Physics of Complex Systems, Eötvös Loránd University, Pf. 32, H-1518 Budapest, Hungary a)Electronic mail: jan.brandejs@jh-inst.cas.cz b)Electronic mail: jakub.visnak@jh-inst.cas.cz c)Electronic mail: libor.veis@jh-inst.cas.cz d)Electronic mail: mate.mihaly@wigner.mta.hu e)Electronic mail: legeza.ors@wigner.mta.hu f)Author to whom correspondence should be addressed: jiri.pittner@jh-inst.cas.cz ABSTRACT There are three essential problems in computational relativistic chemistry: Electrons moving at relativistic speeds, close lying states, and dynamical correlation. Currently available quantum-chemical methods are capable of solving systems with one or two of these issues. However, there is a significant class of molecules in which all the three effects are present. These are the heavier transition metal com- pounds, lanthanides, and actinides with open d or f shells. For such systems, sufficiently accurate numerical methods are not available, which hinders the application of theoretical chemistry in this field. In this paper, we combine two numerical methods in order to address this challenging class of molecules. These are the relativistic versions of coupled cluster methods and the density matrix renormalization group (DMRG) method. To the best of our knowledge, this is the first relativistic implementation of the coupled cluster method externally corrected by DMRG. The method brings a significant reduction of computational costs as we demonstrate on the system of TlH, AsH, and SbH. Published under license by AIP Publishing. https://doi.org/10.1063/1.5144974 .,s I. INTRODUCTION At the turn of the millennium, the density matrix renormal- ization group (DMRG) method1was introduced to the quantum- chemical community,2–4and since then, it has seen a large surge in the use for multireference systems. The biggest advantage of the DMRG method is its capability to treat large active spaces, and cur- rent implementations can go to about 50 active space spinors.5,6 However, a major drawback of DMRG is its inability to capture dynamical correlation, since it cannot include all virtual spinors. This correlation has a strong influence on the target systems ofthis project, which thus aims to address this problem. The DMRG method is already well established and computational chemists started to use it; however, the methods for treating the dynami- cal correlations on top of DMRG are still in the pioneering stage. Past efforts were either based on second order perturbation theory,7 internally contracted MRCI (multireference configuration interac- tion),8random phase approximation,9canonical transformation method,10or the perturbation theory with matrix product states (MPS).11 Our group has followed a different pathway to deal with the dynamical correlation: the coupled cluster (CC) method externally J. Chem. Phys. 152, 174107 (2020); doi: 10.1063/1.5144974 152, 174107-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . Spin–orbit splitting in AsH and SbH. corrected by DMRG.12As the name suggests, this is a combination of DMRG and the coupled cluster (CC) method. The CC method is known for its ability to describe dynamical correlation. In the exter- nally corrected approach, first a DMRG calculation is performed on the strongly correlated active space, keeping the rest of the sys- tem fixed. This accounts for the static correlation. The second step is the CC analysis of the matrix product state (MPS) wave func- tion, obtained from DMRG. Then, a CC calculation is performed on the rest of the system, keeping, in turn, the active space ampli- tudes fixed, which captures the dynamical correlation. Already the simplest version thereof, the tailored CCSD (CC with single and double excitations) approach,13,14yields very promising results.12 Remarkably, all previous approaches based on the use of DMRG output in another method have so far been non-relativistic, leav- ing the relativistic domain unexplored. This is the focus of this paper. First, we demonstrate the capabilities of our relativistic 4c-TCCSD implementation on the example of the thallium hydride (TlH) molecule, which has become a standard benchmark molecule for relativistic methods and most importantly large-scale DMRG and up to CCSDTQ (CC with single, double, triple, and quadruple excitations) results are available.15It should be noted that DMRG is best suited for static-correlation problems, while TlH is domi- nated mostly by dynamic correlation, for which CC approaches are excellent. In order to study the behavior of the TCCSD method for more multireference systems, we performed tests on AsH and SbH molecules. The multireference character in the AsH and SbH ground state arises from the fact that two determinants are needed to describe the M s= 0 triplet component in the spin-free case and therefor is somewhat “artificial.” Figure 1 depicts the π2 1/2andπ2 3/2 determinants arising from Ms= 0 triplet component’s determinants due to the spin–orbit splitting for AsH and SbH. The heavier the atom is, the greater the splitting, and then, π2 1/2is more dominant and the X0+ground state becomes less multireference. Hence, we expect AsH to be of stronger multireference nature than SbH. II. THEORY Present-day relativistic calculations are often carried out within the no-pair approximation, where the Dirac–Coulomb Hamiltonian is embedded by projectors eliminating the troublesome negative- energy solutions, which yields a second quantized Hamiltonian formally analogous to the non-relativistic case, H=∑ PQhQ Pa† PaQ+1 4∑ PQRS⟨PQ∥RS⟩a† Pa† QaSaR, (1)where the indices P,Q,R, and Srun over the positive-energy four- component spinors spanning the one-electron basis. The barred spinors (φ¯p) and unbarred spinors ( φp) form Kramers pairs related to each other by action of the time-reversal operator K, Kφp=φ¯p, Kφ¯p=−φp. (2) The Kramers symmetry replaces the spin symmetry in the non- relativistic theory; in particular, MSis not a good quantum num- ber and MKprojection is defined instead, which is 1/2for unbarred spinors (A) and −1/2 for spinors with barred indices (B). The capital indices in (1) run over both spinors of a Kramers pair. In contrast to the non-relativistic case, the Hamiltonian (1) is, in general, not block-diagonal in MK. Since each creation or annihilation oper- ator in (1) changes MKby±1/2, the Hamiltonian couples states with |ΔMK|≤2. Moreover, the index permutation symmetry of the 2e-integrals in (1) is lower than in the non-relativistic case. The Dirac program16employs a quaternion symmetry approach that combines the Kramers and binary double group sym- metry ( D∗ 2hand subgroups).17The double groups can be sorted into three classes based on the application of the Frobenius–Schur indica- tor to their irreducible representations: “real groups” ( D∗ 2h,D∗ 2, and C∗ 2v), “complex groups” ( C∗ 2h,C∗ 2, and C∗ s), and “quaternion groups” (C∗ iandC∗ 1).18Generalization of non-relativistic methods is sim- plest in the “real groups” case, where the integrals are real-valued and the ones with odd number of barred (B) indices vanish. In prac- tice, it means that additional “spin cases” of integrals (AB|AB) and (AB|BA) (in Mulliken notation) have to be included. For the com- plex groups, the integrals are complex-valued, but still, only integrals with an even number of barred indices are non-zero. Finally, in the remaining case of “quaternion groups,” all the integrals have to be included and are complex-valued.18,19 The idea of externally corrected coupled cluster methods is to take information on the static correlation from some non-CC exter- nal source and to include it into the subsequent CC treatment.20The conceptually simplest approach is the tailored CC (TCC) method proposed by Bartlett et al. ,13,21–23which uses the split-amplitude ansatz for the wave function introduced by Piecuch et al. ,24,25 ∣Ψ⟩=eTexteTcas∣Φ⟩, (3) where Tcascontaining amplitudes with all active indices is “frozen” at values obtained from the complete active space configuration inter- action (CASCI) or, in our case, from DMRG. The external cluster operator Textis composed of amplitudes with at least one index outside the CAS space. Another way to justify this ansatz is the for- mulation of CC equations based on excitation subalgebras recently introduced by Kowalski.26The simplest version of the method trun- cates both Tcasand Textto single and double excitations. Since there is a single-determinantal Fermi vacuum, the excitation oper- ators TextandTcascommute, which keeps the method very simple. TCC can thus use the standard CCSD solver, modified to keep the amplitudes from Tcasfixed. Thanks to the two-body Hamiltonian, tailored CCSD energy with Text= 0 and Tcasfrom CASCI (com- plete active space configuration interaction) reproduces the CASCI energy. In the limit of CAS space, including all MOs, TCC thus J. Chem. Phys. 152, 174107 (2020); doi: 10.1063/1.5144974 152, 174107-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp recovers the full configuration interaction (FCI) energy. In gen- eral, error bound valid for tensor network state TCC (TNS-TCC) methods is derived.27 In Refs. 12 and 28, we have described how to obtain Tcasfrom the DMRG wave function using concepts of quantum information theory29in the non-relativistic case, yielding the DMRG–TCCSD method. The DMRG method30is a procedure that variationally optimizes the wave function in the form of the matrix product state (MPS) ansatz.5The quantum chemical version of DMRG (QC-DMRG)31–36eventually converges to the FCI solution in a given orbital space, i.e., to CASCI. The practical version of DMRG is the two-site algorithm, which provides the wave function in the two-site MPS form,5 ∣ΨMPS⟩=∑ {α}Aα1Aα2⋯Wαiαi+1⋯Aαn∣α1α2⋯αn⟩, (4) whereαi∈{|0⟩, |↓⟩, |↑⟩, |↓↑⟩}, and for a given pair of adjacent indices [i, (i+ 1)], Wis a four index tensor, which corresponds to the eigenfunction of the electronic Hamiltonian expanded in the tensor product space of four tensor spaces defined on an ordered orbital chain, so called left block (Mldimensional tensor space), left site (four dimensional tensor space of the ith orbital), right site [four dimensional tensor space of the ( i+ 1)th orbital], and right block (Mrdimensional tensor space). When employing the two-site MPS wave function [Eq. (4)] for the purposes of the TCCSD method, the CI expansion coefficients ca iandcab ijfora,b,i,j∈CAS can be efficiently calculated by con- tractions of MPS matrices.37,38We would like to note that using the two-site DMRG approach, in practice, means using the wave- function calculated at different sites, and it can only be employed together with the dynamical block state selection (DBSS) procedure4 assuring the same accuracy along the sweep. Alternatively, one can use the one-site approach in the last sweep.39 Once the CI coefficients ca iand cab ijhave been obtained, the standard CC analysis is performed to convert them to the CC amplitudes, T(1) CAS=C(1), (5) T(2) CAS=C(2)−1 2[C(1)]2. (6) The generalization of the DMRG–TCCSD method to the rela- tivistic 4c case has to consider several points. First of all, the addi- tional integral classes with nonzero ΔMKhave to be implemented in the DMRG Hamiltonian.15,40Second, there will be more CI coef- ficients and, subsequently, CC amplitudes to be obtained from the MPS wave function, corresponding to excitations with nonzero ΔMK. Finally, except for the “real groups,” the DMRG procedure has to work with complex matrices, and the resulting cluster amplitudes will also be complex-valued. In the present work, we have selected numerical examples with “real groups” symmetry, while the complex generalization of the DMRG code is in progress. III. COMPUTATIONAL DETAILS In the present work, we have used the two-site DMRG vari- ant together with DBSS through the course of the whole DMRG procedure and obtained the CC amplitudes from the resultingtwo-site form of the MPS wave function. For all systems, DMRG cal- culations were performed exclusively with the QC-DMRG-Budapest program.41The Dirac program package16was used for the remain- ing relativistic calculations, whereas the Orca program was used for remaining non-relativistic calculations. Orbitals and MO integrals were generated with the Dirac program package.16We used the Dirac–Coulomb Hamiltonian and triple-zeta basis sets for the heav- ier of the two atoms (cv3z for Tl, As, and Sb and cc-pVTZ for F) as well as for hydrogen (cc-pVTZ), which include core-correlating functions for the heavier atom. Initialization of DMRG, i.e., opti- mal ordering of spinors, was set up, as discussed in Ref. 35. The numerical accuracy was controlled by DBSS,4keeping up to thou- sands of block states for the a priori set quantum information loss thresholdχ= 10−6. A. Comparison with non-relativistic TCC In order to compare with the non-relativistic version of the TCCSD method, the system of hydrogen fluoride was chosen as it is biatomic with light nuclei. CC-pVTZ basis was used at the internuclear distance of 0.8996 Å. Consistent methods should exhibit a constant shift of relativis- tic and non-relativistic energy ΔE=Erel−Enonrel , given by a different Hamiltonian. Table I shows that TCCSD is consistent with RHF, CCSD, and DMRG methods in terms of ΔEup to a millihartree. B. TlH We have used the computational protocol of Ref. 15 for direct comparison with their energies. C⋆ 2vdouble group symmetry with real irreps was assumed. The 4c-RHF energy was −20 275.416 61 E h. We used MP2 natural spinors (NS) from the Dirac program16as the spinor basis for electron-correlation calculations, correlating the Tl 5s, 5p, 4f, 5d, 6s, 6p, and H 1s electrons, while keeping the remain- ing core electrons of Tl frozen. Using uncontracted basis, a virtual spinor threshold was set at 135 E h. The resulting space (14, 47) was chosen by ordering MP2 NS by their occupations and taking those with values between 1.98 and 0.001. In this space, the 4c-TCCSD was performed, with DMRG calculations in the procedure limited to subspaces of (14, 10), (14, 14), (14, 17), (14, 25), and (14, 29), with spinors sorted by MP2 occupations. Figure 2 shows a scheme of embedded active spaces used in the procedure. C. AsH and SbH Since SbH is a heavier homolog of AsH (which is itself a homolog of nitrene, NH), the procedure was very similar for both TABLE I . Comparison of energies of the HF molecule obtained from relativistic and non-relativistic methods. The rightmost column shows the difference between the 4c-relativistic energy Ereland non-relativistic energy Enonrel . Method Erel(Eh) Enonrel (Eh) ΔE(Eh) RHF −100.149 72 −100.058 46 −0.091 26 DMRG(6,6) −100.158 68 −100.067 37 −0.091 30 CCSD −100.423 22 −100.331 72 −0.091 50 TCCSD(6,6) −100.424 18 −100.332 46 −0.091 72 J. Chem. Phys. 152, 174107 (2020); doi: 10.1063/1.5144974 152, 174107-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Schematic depiction of active spaces used in the 4-TCCSD procedure for TlH. of them. In contrast, with TlH, instead of MP2 NS, we used average- of-configuration SCF spinors. The 4c-SCF energy was −2260.042 61 Ehfor AsH and −6481.107 75 E hfor SbH. The DMRG calculation in the 4c-TCCSD procedure was limited to subspaces of (16, 14) and (16, 23). Dominant contributions to active spaces for the AsH active space (16, 14) are As (3d, 4s, 4p, 5s, 5p and H: 1s, and for active space (16, 23), we add As: 4d and H: 2s, 2p to the former. Ener- gies of MOs are from −2.1 E hto +0.25 E hand +1.01 E hfor (16, 14) and (16, 23), respectively (for internuclear distance 1.52 Å). Domi- nant contributions to active spaces for SbH are analogous, except for principal quantum numbers, which are higher by 1. IV. RESULTS AND DISCUSSION Once we reproduced the MP2 and CCSD energy of TlH in equi- librium geometry from Ref. 15, we applied the 4c-TCCSD method. Obtained energies and their respective deviations from the refer- ence CCSDTQ calculation15are listed in Table II. In the case of the optimal selection of active space of 14-spinors, the TCCSD method improved the CCSD energy by 4.94 mE H. While TCCSD intro- duces only a minor computational cost increase over CCSD, it cuts the energy error in half. This shows the practical advantage of the method. The energy obtained by TCCSD is comparable even with large-scale DMRG in the full CAS(14, 47). As we can see from the high accuracy of the 4c-CCSD(T) energy, the system does not exhibit a considerable multireference TABLE II . Total electronic energy and energy differences ΔEel(in mE h) for various methods with respect to the 4c-CCSDTQ(14, 47) reference energy of −20 275.840 242 33 E h15for TlH at the experimental equilibrium internuclear dis- tance 1.872 Å. Method Eel(Eh) ΔEel(mE h) 4c-MP2(14, 47) −20 275.853 72 −13.49 4c-CCSD(14, 47) −20 275.829 66 10.58 4c-CCSD(T) (14, 47) −20 275.840 56 −0.32 4c-DMRG(14, 47)a,15−20 275.837 67 2.57 4c-TCCSD(14, 10) −20 275.830 42 9.83 4c-TCCSD(14, 11) −20 275.831 70 8.54 4c-TCCSD(14, 12) −20 275.832 57 7.67 4c-TCCSD(14, 13) −20 275.833 29 6.95 4c-TCCSD(14, 14) −20 275.834 30 5.94 4c-TCCSD(14, 15) −20 275.832 24 8.00 4c-TCCSD(14, 16) −20 275.829 02 11.23 4c-TCCSD(14, 17) −20 275.824 05 16.19 a4c-DMRG(14, 47)[4500, 1024, 2048, 10−5] (see Ref. 15).character. Therefore, even a rather small CAS of 14 spinors is sufficient for a good description of the system. As shown on the chart in Fig. 3(a), TCCSD significantly improves the DMRG energy toward FCI, even for the smallest CAS space. In fact, further enlarging of CAS over the size of 14 spinors is counterproductive. Although the TCC must reproduce the FCI energy when CAS is extended to all spinors, the TCC energy does not approach this limit monotonically.42The obvious reason is that the “frozen” TCAS amplitudes cannot reflect the influence of the dynamical correlation on the external space back on the active CAS space; therefore, extending CAS space first exacerbates the results. Unfortunately, more detailed understanding of this highly nonlin- ear behavior is still an open problem. Despite a considerable effort, so far, no quantities able to predict the optimal CAS-EXT split a priori have been identified.42Nevertheless, we have found that an error minimum can be obtained by sweeping through the entire FIG. 3 . Equilibrium energy of TlH calculated using the 4c-TCCSD and 4c-DMRG methods with different sizes of DMRG active space, as given in Table II. (a) Com- parison of TCCSD and DMRG methods. The horizontal solid line represents the “FCI-limit” from the large 4c-DMRG(14, 47)15calculation. (b) Details of 4c-TCCSD energies. J. Chem. Phys. 152, 174107 (2020); doi: 10.1063/1.5144974 152, 174107-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE III . Spectroscopic constants of205TlH obtained from 4c-TCCSD, compared with calculations and experimental work from the literature. The spectroscopic con- stants have been evaluated from the potential energy curve fit with two different methodologies. In the case of the TWOFIT methodology, the number of points has been selected according to mean displacement in the harmonic ground state criterion. In the case of VIBANAL methodology, a wider symmetric interval around equilibrium geometry has been selected. In all cases, internuclear separation axis sampling was chosen to be 0.02 Å. Method re(Å)ωe(cm−1)ωexe(cm−1) Expt.a,431.872 1391 22.7 4c-DMRG(14, 47)151.873 1411 26.6 4c-CCSD(14, 47)151.871 1405 19.4 4c-TCCSD(14, 10)bDBSS 1.874 1404 24.6 4c-TCCSD(14, 10)bM = 512 1.874 1403 23.4 4c-TCCSD(14, 14)bDBSS 1.869 1412 22.6 4c-TCCSD(14, 14)cDBSS 1.869 1411 20.1 4c-TCCSD(14, 14)bM = 512 1.869 1411 22.6 4c-TCCSD(14, 14)cM = 512 1.869 1411 20.3 4c-TCCSD(14, 17)bDBSS 1.859 1426 20.5 4c-TCCSD(14, 17)cDBSS 1.859 1426 17.5 4c-TCCSD(14, 17)bM = 512 1.859 1428 22.4 4c-TCCSD(14, 17)cM = 512 1.859 1428 29.8 aGRECP spin–orbit MRD-CI (see Ref. 43). bTWOFIT fourth order polynomial. cVIBANAL 10thorder polynomial. R min–Rmax(Å): 1.64–2.20 for (14, 14), 1.70–2.04 for (14, 17) DBSS, and 1.72–2.00 for (14, 17) M = 512. orbital space with low cost DMRG calculations, and this mini- mum does not shift or shifts only a little when more accurate cal- culations are performed. Therefore, in practice, the optimal CAS size related to the energy minimum is usually independent of M and can be determined with low bond dimension (M) DMRG calculations.42 As demonstrated by the chart in Fig. 3(b), the optimal CAS size is 14 spinors for the equilibrium energy calculation. This CAS size FIG. 4 . Dissociation curve of TlH.TABLE IV . Three configurations with the highest coefficients (in absolute values) for TlH, SbH, and AsH in the equilibrium internuclear distance, as generated by DMRG with the active space of 14 spinors. In each case, the CC amplitudes were generated with respect to the closed shell reference listed in the first row for given system. Here, 2 is for a doubly occupied Kramers pair and 0 is for an empty Kramers pair. Determinant Coefficient TlH 2 2 2 2 2 2 2 0 0 0 0 0 0 0 Reference 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0.978 62 2 2 2 2 2 2 0 2 0 0 0 0 0 0 0.059 28 2 2 2 2 2 2 0 0 2 0 0 0 0 0 0.055 40 SbH 2 2 2 2 2 2 2 2 0 0 0 0 0 0 Reference 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0.825 18 2 2 2 2 2 2 2 0 2 0 0 0 0 0 0.547 48 2 2 2 2 2 2 0 2 0 2 0 0 0 0 0.048 72 AsH 2 2 2 2 2 2 2 2 0 0 0 0 0 0 Reference 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0.756 73 2 2 2 2 2 2 2 0 2 0 0 0 0 0 0.640 48 2 2 2 2 2 2 0 2 0 2 0 0 0 0 0.041 23 is optimal not only for energies but also for the calculation of spec- troscopic properties, including the low bond dimension calculations with M= 512 (see Table III). The obtained spectroscopic properties of TlH are listed in Table III and the corresponding dissociation curve is depicted in Fig. 4. Even for a small active space of 10 spinors, TCCSD shows an agreement with the experiment comparable with the large DMRG(14, 47) calculation. For the 14-spinor space, spectroscopic constants obtained by TCCSD exhibit the best agreement with the experiment, thus being consistent with the lowest energy single point result of the 14-spinor space in Table II. Moreover, TCCSD based on DMRG with M = 512 states or on DMRG with DBSS yields very similar results, indicating that the underlying DMRG is well con- verged. However, for 17-spinors, there is a bigger difference and M = 512 might not be accurate enough. This is in accordance with the previous findings of dynamical correlation effects. In order to further assess the feasibility of the method for mul- tireference systems, we studied AsH and SbH molecules. Table IV compares the determinants with highest coefficients as calculated by 4c-DMRG with the space of (14, 14) for TlH and (16, 14) for AsH and SbH. The coefficients confirm that the need arises for a multireference description of the ground state of AsH and that SbH is between AsH and TlH in terms of its multireference character. TABLE V . Total energy for various methods for AsH at the experimental equilibrium internuclear distance45of 1.5343 Å. Method Eel(Eh) 4c-MP2(16, 81) −2260.532 27 4c-CCSD(16, 81) −2260.532 41 4c-TCCSD(16, 14) −2260.549 45 4c-CCSD(T)(16, 81) −2260.551 33 J. Chem. Phys. 152, 174107 (2020); doi: 10.1063/1.5144974 152, 174107-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE VI . Spectroscopic constants of75AsH obtained from 4c-TCCSD, compared with calculations and experimental work from the literature. Method re(Å)ωe(cm−1)ωexe(cm−1) Expt.441.523 2156 39.2 4c-SCFa1.513 2382 33.6 4c-MP2(16, 81)a1.503 2256 35.0 4c-DMRG(16, 14)aDBSS 1.545 2051 42.2 4c-CCSD(16, 81)a1.505 2281 40.0 4c-TCCSD(16, 14)bDBSS 1.517 2154 43.9 4c-TCCSD(16, 14)aDBSS 1.517 2172 39.6 4c-CCSD(T) (16, 81)a1.521 2146 40.6 aTWOFIT fourth order polynomial. bVIBANAL 10thorder polynomial. The equilibrium energies of AsH in Table V show that 4c-TCCSD improved the 4c-CCSD by 17 mE h, with just 3 mE h difference from 4c-CCSD(T). In this case, a more accurate theo- retical reference energy is unavailable, and therefore, we turned to spectroscopic constants, as shown in Table VI, to enable a com- parison with accurate IR spectra obtained by CO laser magnetic resonance in Ref. 44. The respective potential curve is plotted in Fig. 5. The comparison of both the internuclear distance reand the vibrational constant ωeshows a clear advantage of 4c-TCCSD over 4c-CCSD in this case, which we attribute to the multiref- erence nature of AsH. Despite the multireference character, the 4c-CCSD(T) with perturbative triples still prevails over 4c-TCCSD. This shows that at this range of internuclear distances, the mul- tireference character is not strong enough to cause CCSD(T) to fail. Nevertheless, in AsH, it is strong enough that TCCSD pro- vides a major improvement over CCSD at the same computational scaling. Calculated equilibrium energies for SbH are listed in Table VII. Compared with the higher order methods, the 4c-MP2 method outputs lower energy, which might be due to a failure to describe FIG. 5 . Detail of the potential curve of AsH near the equilibrium internuclear distance.TABLE VII . Total energy for various methods for SbH at the equilibrium internuclear distance46of 1.701 87 Å. Method Eel(Eh) 4c-MP2(16, 81) −6481.696 51 4c-CCSD(16, 81) −6481.671 08 4c-TCCSD(16, 14) −6481.683 80 4c-CCSD(T)(16, 81) −6481.693 15 the multireference character of this system. As with the previous sys- tems, the TCCSD energy is between CCSD and CCSD(T). However, in constrast, with TlH, where static correlation was not essential for a good description, here it is not clear if CCSD(T) is a good bench- mark, since for multireference systems, CCSD(T) tends to output TABLE VIII . Spectroscopic constants of121SbH obtained from 4c-TCCSD compared with calculations and experimental work from the literature. Method re(Å)ωe(cm−1)ωexe(cm−1) Expt.461.702 Expt.441.711 1897 Expt.471923 34.2 4c-SCFa1.705 2024 28.0 4c-SCFb1.704 2043 38.1 4c-MP2(16, 81)a1.693 2009 30.2 4c-DMRG(16, 14)aM = 2200 1.737 1839 35.3 4c-CCSD(16, 81)a1.706 1945 32.6 4c-TCCSD(16, 14)aM = 2200 1.706 1937 36.4 4c-CCSD(T)(16, 81)a1.710 1916 35.3 aTWOFIT fourth order polynomial. bVIBANAL 10thorder polynomial. FIG. 6 . Detail of the potential curve of SbH near the equilibrium internuclear distance. J. Chem. Phys. 152, 174107 (2020); doi: 10.1063/1.5144974 152, 174107-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp too low energy. Unfortunately, we miss a more accurate calculation to compare with; hence, we again turn to spectroscopic constants, which are listed in Table VIII, with the respective potential curve in Fig. 6. Considering the calculated internuclear distance, MP2 and DMRG are inaccurate, as they were for AsH. Oddly enough, plain DMRG with a small active space is in both cases very accurate for ωeandωexe. Methods from CC theory succeeded in describing re, ωe, andωexe, and TCCSD again outputs values between CCSD and CCSD(T). Compared with CCSD, TCCSD improved the vibrational constantωe. V. CONCLUSION We have implemented the relativistic tailored coupled clusters method, which is capable of treating relativistic, strongly correlated systems both in terms of static and dynamical correlation. The aim was to show that compared with the previously published calcu- lations, we can obtain results of equal quality with much smaller active space, i.e., at a fraction of computational cost. The results presented are promising. Even with a small active space, the new method showed comparable performance for TlH to DMRG with large CAS(14, 47). The optimal CAS size related to the energy min- imum was determined with low cost DMRG calculations. The cal- culated spectroscopic properties of TlH are in agreement with the experimental values within the error bounds. The comparison with experimental spectroscopic constants for AsH, which has a stronger multireference character, has shown that TCCSD is able to describe such systems more accurately that CCSD, with a computational cost lower than CCSD(T). ACKNOWLEDGMENTS We thank the developers of the Dirac program, in particular, Dr. Lucas Visscher and Dr. Stefan Knecht for providing access to the development version of the code and for helpful discussions. The work of the Czech team has been supported by the Czech Science Foundation (Grant No. 18-24563S). Ö. Legeza has been supported by the Hungarian National Research, Development and Innovation Office (NKFIH) through Grant No. K120569 and by the Hungar- ian Quantum Technology National Excellence Program (Project No. 2017-1.2.1-NKP-2017-00001). The development of the relativistic DMRG libraries was supported by the Center for Scalable and Pre- dictive methods for Excitation and Correlated phenomena (SPEC), which is funded as part of the Computational Chemical Sciences Program by the U.S. Department of Energy (DOE), Office of Sci- ence, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences at Pacific Northwest National Labo- ratory. M. Máté was supported by the ÚNKP-19-3 New National Excellence Program of the Ministry for Innovation and Technology. Mutual visits with the Hungarian group have partly been supported by the Hungarian–Czech Joint Research (Project No. MTA/19/04). Part of the CPU time for the numerical computations was supported by the Ministry of Education, Youth and Sports from the Large Infrastructures for Research, Experimental Development and Inno- vations project “IT4Innovations National Supercomputing Center – LM2015070.” Access to computing and storage facilities owned by parties and projects contributing to the National Grid Infrastruc- ture MetaCentrum provided under the program “Projects of LargeResearch, Development, and Innovations Infrastructures” (CESNET LM2015042) is appreciated. REFERENCES 1S. R. White, Phys. Rev. Lett. 69, 2863 (1992). 2S. R. White and R. L. Martin, J. Chem. Phys. 110, 4127 (1999). 3G. K.-L. Chan and M. Head-Gordon, J. Chem. Phys. 116, 4462 (2002). 4Ö. Legeza, J. Röder, and B. A. Hess, Phys. Rev. B 67, 125114 (2003). 5U. Schollwöck, Ann. Phys. 326, 96 (2011). 6R. Olivares-Amaya, W. Hu, N. Nakatani, S. Sharma, J. Yang, and G. K.-L. Chan, J. Chem. Phys. 142, 034102 (2015). 7Y. Kurashige and T. Yanai, J. Chem. Phys. 135, 094104 (2011). 8M. Saitow, Y. Kurashige, and T. Yanai, J. Chem. Phys. 139, 044118 (2013). 9S. Wouters, N. Nakatani, D. Van Neck, and G. K.-L. Chan, Phys. Rev. B 88, 075122 (2013). 10T. Yanai and G. K.-L. Chan, J. Chem. Phys. 124, 194106 (2006). 11J. Ren, Y. Yi, and Z. Shuai, J. Chem. Theory Comput. 12, 4871 (2016). 12L. Veis, A. Antalík, J. Brabec, F. Neese, Ö. Legeza, and J. Pittner, J. Phys. Chem. Lett.7, 4072 (2016). 13T. Kinoshita, O. Hino, and R. J. Bartlett, J. Chem. Phys. 123, 074106 (2005). 14O. Hino, T. Kinoshita, G. K.-L. Chan, and R. J. Bartlett, J. Chem. Phys. 124, 114311 (2006). 15S. Knecht, Ö. Legeza, and M. Reiher, J. Chem. Phys. 140, 041101 (2014). 16T. Saue et al. , DIRAC, A relativistic ab initio electronic structure program (2018), http://www.diracprogram.org. 17T. Saue and H. J. A. Jensen, J. Chem. Phys. 111, 6211 (1999). 18K. G. Dyall and K. Fægri, Jr., Introduction to Relativistic Quantum Chemistry (Oxford University Press, 2007). 19J. Thyssen, “Development and applications of methods for correlated relativis- tic calculations of molecular properties,” Ph.D. thesis, Univeristy of Southern Denmark, 2001. 20X. Li and J. Paldus, J. Comput. Phys. 107, 6257 (1997). 21D. I. Lyakh, V. F. Lotrich, and R. J. Bartlett, Chem. Phys. Lett. 501, 166 (2011). 22A. Melnichuk and R. J. Bartlett, J. Comput. Phys. 137, 214103 (2012). 23A. Melnichuk and R. J. Bartlett, J. Comput. Phys. 140, 064113 (2014). 24P. Piecuch, N. Oliphant, and L. Adamowicz, J. Chem. Phys. 99, 1875 (1993). 25P. Piecuch and L. Adamowicz, J. Chem. Phys. 100, 5792 (1994). 26K. Kowalski, J. Chem. Phys. 148, 094104 (2018). 27F. M. Faulstich, A. Laestadius, Ö. Legeza, R. Schneider, and S. Kvaal, SIAM J. Numer. Anal. 57, 2579 (2019). 28L. Veis, A. Antalík, J. Brabec, F. Neese, Ö. Legeza, and J. Pittner, J. Phys. Chem. Lett.8, 291 (2017). 29Ö. Legeza and J. Sólyom, Phys. Rev. B 68, 195116 (2003). 30U. Schollwöck, Rev. Mod. Phys. 77, 259 (2005). 31Ö. Legeza, R. Noack, J. Sólyom, and L. Tincani, in Computational Many-Particle Physics , Lecture Notes in Physics Vol. 739, edited by H. Fehske, R. Schneider, and A. Weisse (Springer Berlin Heidelberg, 2008), pp. 653–664. 32K. H. Marti and M. Reiher, Z. Phys. Chem. 224, 583 (2010). 33G. K.-L. Chan and S. Sharma, Ann. Rev. Phys. Chem. 62, 465 (2011). 34S. Wouters and D. Van Neck, Eur. Phys. J. D 68, 272 (2014). 35S. Szalay, M. Pfeffer, V. Murg, G. Barcza, F. Verstraete, R. Schneider, and Ö. Legeza, Int. J. Quantum Chem. 115, 1342 (2015). 36T. Yanai, Y. Kurashige, W. Mizukami, J. Chalupský, T. N. Lan, and M. Saitow, Int. J. Quantum Chem. 115, 283 (2015). 37G. Moritz and M. Reiher, J. Chem. Phys. 126, 244109 (2007). 38K. Boguslawski, K. H. Marti, and M. Reiher, J. Chem. Phys. 134, 224101 (2011). J. Chem. Phys. 152, 174107 (2020); doi: 10.1063/1.5144974 152, 174107-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 39D. Zgid and M. Nooijen, J. Chem. Phys. 128, 144115 (2008). 40S. Battaglia, S. Keller, and S. Knecht, J. Chem. Theory Comput. 14, 2353 (2018). 41Ö. Legeza, L. Veis, and T. Mosoni, QC-DMRG-Budapest, a Program for Quantum Chemical DMRG Calculations. 42F. M. Faulstich, M. Máté, A. Laestadius, M. A. Csirik, L. Veis, A. Antalik, J. Brabec, R. Schneider, J. Pittner, S. Kvaal, and Ö. Legeza, J. Chem. Theory Comput. 15, 2206 (2019).43A. V. Titov, N. S. Mosyagin, A. B. Alekseyev, and R. J. Buenker, Int. J. Quantum Chem. 81, 409 (2001). 44K. D. Hensel, R. A. Hughes, and J. M. Brown, J. Chem. Soc., Faraday Trans. 91, 2999 (1995). 45K. Balasubramanian, Chem. Rev. 89, 1801 (1989). 46M. Beutel, K. D. Setzer, O. Shestakov, and E. H. Fink, J. Mol. Spectrosc. 179, 79 (1996). 47R. D. Urban, K. Essig, and H. Jones, J. Chem. Phys. 99, 1591 (1993). J. Chem. Phys. 152, 174107 (2020); doi: 10.1063/1.5144974 152, 174107-8 Published under license by AIP Publishing

5.0008102.pdf

J. Appl. Phys. 128, 073908 (2020); https://doi.org/10.1063/5.0008102 128, 073908 © 2020 Author(s).Electronic structure and magnetic exchange interactions in Zn diluted CuFe2O4 magneto- ceramics Cite as: J. Appl. Phys. 128, 073908 (2020); https://doi.org/10.1063/5.0008102 Submitted: 19 March 2020 . Accepted: 06 August 2020 . Published Online: 20 August 2020 Suchit Kumar Jena , Deep Chandra Joshi , Zhuo Yan , Yajun Qi , Sayandeep Ghosh , and Subhash Thota Electronic structure and magnetic exchange interactions in Zn diluted CuFe 2O4 magneto-ceramics Cite as: J. Appl. Phys. 128, 073908 (2020); doi: 10.1063/5.0008102 View Online Export Citation CrossMar k Submitted: 19 March 2020 · Accepted: 6 August 2020 · Published Online: 20 August 2020 Suchit Kumar Jena,1 Deep Chandra Joshi,1 Zhuo Yan,2Yajun Qi,2 Sayandeep Ghosh,1 and Subhash Thota1,a) AFFILIATIONS 1Department of Physics, Indian Institute of Technology Guwahati, 781039 Assam, India 2School of Materials Science and Engineering, Hubei University, Wuhan, Hubei 430062, China a)Author to whom correspondence should be addressed: subhasht@iitg.ac.in ABSTRACT We report a detailed study on the electronic structure and temperature (1 :9K/C20T/C20900 K) dependence of magnetization in Zn diluted cuprospinel [Cu 1/C0xZnxFe2O4(0/C20x/C200:6)]. The electronic structure determined from the x-ray photoelectron spectroscopy and Rietveld analysis of the x-ray diffraction patterns reveals the structure to be (Cu (1/C0x)=5ZnxFe4(1/C0x)=5)A[Cu 4(1/C0x)=5Fe2/C04(1/C0x)=5]BO4. Beyond a specific dilution limit (0 :05/C20xp/C200:1), a sudden phase-change from tetragonal ( I41=amd) to cubic ( Fd3m) is noticed with an alteration in the A–O–A (3.29%) bond angle and A –O bond length (0.67%). Our analysis shows that all these compounds order ferrimagnetically below the Néel temperature ( TFN) due to dissimilar site-specific magnitudes of spins, yet, they undergo a second transition at low temperatures T1/difference66 K with asymptotic Curie temperature TA(¼C=χ0) as high as /C0547:2 K for the undoped case. Dilution with Zn cause quadratic decay ( b2x2þb1xþyo) of the ferrimagnetic ordering temperature from 743 K to 370.5 K for x¼0 and 0.6, respectively. On the contrary, a significant increase in the saturation magnetization ( MS) was observed with increasing xuntil the critical composition xc/difference0:4 beyond which MSdecreases continuously ( MS¼1:64μBand 4 :73μBforx¼0 and 0.4, respectively). From the temperature dependence of inverse paramagnetic susceptibility [ χ/C01(T.TFN)] data and Néel ’s expression for ferrimagnets, we evaluated the molecular field constants and exchange interactions ( J) between the tetrahedral A- and octahedral B-sites. A systematic compositional dependence of this analysis yields that JAB(/difference25kBforx¼0) is the dominant exchange interaction in comparison to JBBandJAA; however, JABdecreases significantly with increasing the composition ( JAB/difference/C05:5kBforx¼0:6). The isothermal magnetization data and law of approach to saturation analysis reveals that the investigated system possesses very high anisotropy field HK/C215:5 kOe with cubic anisotropy constant K1/C211:6/C2106erg=cc at xc. Published under license by AIP Publishing. https://doi.org/10.1063/5.0008102 I. INTRODUCTION Copper ferrite (CuFe 2O4), often called cuprospinel, is a well- studied system in the literature because of its tunable tetragonal tocubic crystal structure transition ( T *) owing to the Jahn –Teller effect below its ferrimagnetic ordering ( TFN/difference507+20/C14C, although significant disparities in the TFNvalues are found in the literature).1–4Numerous methods are implemented by researchers to tune the degree of tetragonality at room temperature, such asquenching to low temperatures, slowly cooling from sinteringpoint, applying high pressure, and varying the chemical stoichiom- etry by replacing the B- and A-site cations with other transition metals. 5–14In all these cases, the proportion of divalent copperatoms on the tetrahedral A-sites to that of the octahedral B-sites decides the stability of the tetragonal structure.15On the other hand, these spinels under reduced dimensions play an important role in industrial applications such as filters in electromagneticinterference, gas-sensors, 16,17catalysts,18–20microwave devices,21 hydrogen energy,22and magneto-electronics.23Several reports sug- gested the following cationic distribution in this compound: (Cu xFe1/C0x)A[Cu 1/C0η/C0xFe1þηþx]BO4with the degree of inversion x lying between 0 (inverse-spinel) and 1 (normal-spinel).14,24,25 It is a very challenging task to maintain the stoichiometric proportion of the cations in this compound using conventional heat-treatment conditions at the synthesis level because of theJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 073908 (2020); doi: 10.1063/5.0008102 128, 073908-1 Published under license by AIP Publishing.formation of CuO as secondary phase due to which one always observes deviation from exact stoichiometry leading to η/difference0:04+0:01.26Thermal kinetics of Cu are quite inimitable in spinel lattice, for example, Cu2þexhibits a high migration rate above 400/C14C (with low activation energy E A/C200:1 eV) so that they swiftly reorder in the lattice leading to fractional inverse-spinel structure (approximately 0 :85 Cu2þions on the octahedral sites).27 Thus, the rate of cooling (either rapid or slow) plays an important role in deciding the crystal structure in the bulk form whether it iscubic or tetragonal. Moreover, the oxygen deficiency in this systemis quite unfavorable to the formation of tetragonal structure, which in turn affects the phase transition temperature and magnetic ordering. 28Nevertheless, the end compound x¼1 (ZnFe 2O4) exhibits a normal spinel structure and low-temperature antiferro-magnetic ordering with Néel temperature of /difference10 K. 29–34In this compound, both A- and B-sites contain divalent Zn and trivalent Fe ions with a general formula (ZnFe) x[ZnFe] 2/C0xO4, in which the magnetic ordering strongly depends on the stoichiometry/state ofchemical order and the site occupancy of cations. 34–37The cations at B-sites are positioned at the junctions of the tetrahedron, inwhich each corner is shared by two tetrahedral. If we consider the tetrahedron as a single entity, then the B-sites of the spinel struc- ture can be described as fcc configuration of molecules similar tothe pyrochlore structure and some intermetallic Laves-phases. Insuch a case, one can expect unusual magnetic behavior in the system such as geometrical frustration, spin-liquid state, Griffiths phase, reentrant spin-glass behavior with unusual ground states,negative magnetization, and bipolar exchange-bias. 38–43Although the physical properties of Zn-substituted CuFe 2O4have been exten- sively investigated, studies related to its compositional dependence of magnetic ordering temperature, exchange interactions, and anisotropy studies in bulk Cu 1/C0xZnxFe2O4system are still lacking in the literature. The majority of today ’s research activity on this system is intensive on the nanostructures and thin films forms of the com- pounds to explore their potential applications in industries, espe- cially in the renewable energy sector.2,44,45Also, a systematic correlation of the change in the electronic and crystal structure ofbulk grain sized Cu 1/C0xZnxFe2O4magneto-ceramics is scarce in the literature. Recent numerical calculations by Salmi et al. reported the variation in the exchange interactions [ JAA(x),JBB(x), and JAB(x)], long-range ordering temperature, and critical exponent ( γ) for various compositions of Cu 1/C0xZnxFe2O4using the high- temperature series expansion studies linked with the Padé approxi- mation.46Therefore, in the current work, we focus our study on magnetic behavior and exchange interactions between the tempera-ture range 1.9 K and 900 K of various levels of Zn diluted CuFe 2O4 bulk grain size compounds under different measurement protocolsalong with a systematic correlation of these results with the crystal structure, electronic properties, and morphology. Detailed method- ology, experimental results and their discussion, and the analysisare presented below. II. MATERIALS AND METHODOLOGY The standard solid-state-reaction method was employed to synthesize the bulk samples of Cu 1/C0xZnxFe2O4. For this, we usedcopper oxide (CuO), zinc oxide (ZnO), and ferric oxide (Fe 2O3)a s precursors. First, stoichiometric proportions of these compounds are mixed in a ball-milling machine using a tungsten-carbide jarand ethanol as milling medium. Milling was carried out by usingtungsten-carbide balls of 10 mm diameter in which the ratio of theweight of the balls to powder is maintained as 5:1. Adequate amount of ethanol was also added as the milling medium so that the balls would rotate (with 150 rpm speed for 10 h) freely withoutany friction. The solution was then dried in an oven and calcinedat 900 /C14C for 4 h in air. The calcined powder was made into 13 mm diameter pellets using a hydraulic press and sintered at 1050/C14C for 8 h in air. These pellets were further reground and sintered at the same temperature for multiple times to avoid the formation of anysecondary phases. This procedure is employed for the synthesis ofalmost all the compositions; however, Cu rich samples are heattreated at slightly lower temperatures. The crystal structure and phase purity of the compounds are examined by performing the x-ray diffraction (XRD) experiments using Rigaku based x-ray dif-fractometer (Model: TTRAX III) with Cu K αradiation (λ¼1:540 56 A /C14). In order to investigate the microstructure of the samples, we used Field Emission Scanning Electron Microscope, FESEM (Model: Sigma-Zesis with extra high tension EHT 2 KV) and a Transmission Electron Microscope (TEM, JEOL JEM-2100operated under secondary electron mode with 200 keV potential).These studies reveal the bulk grain size of the investigated composi- tion with an average grain size of /difference15:768μm(x¼0:4). The selected area electron diffraction (SAED) pattern further supportsthe formation of cubic crystal structure consistent with the x-raydiffraction studies. The SAED pattern for the composition x¼0:4 agrees well with the cubic phase determined from the x-ray diffrac- tion studies. The bright-field, high resolution transmission electron micrograph (HRTEM) reveals the uniform lattice spacing of/difference3:28+0:01 A /C14consistent with the interplanar spacing ( d220) determined from the x-ray diffraction analysis discussed below.Electronic structure and elemental analysis was performed by x-ray photoelectron spectroscopy (XPS) from Thermo Fisher Scientific 250Xi and Energy Dispersive Spectroscopy, EDAX from Zeiss(Model GEMINI 300). Temperature dependence of the magnetiza-tion measurements are performed using a physical property mea-surement system, PPMS from Quantum Design (Model: DynaCool), which is capable of reaching the low temperatures down to 1.9 K from 300 K and a maximum dc-magnetic field of90 kOe. For the high-temperature (until 900 K) magnetization mea-surements, we used VSM oven accessory separately. III. RESULTS AND DISCUSSION Figure 1 shows the XPS spectra (photoelectron intensity vs binding energy) of all the individual elements for x¼0:4 composi- tion. Here, the adventitious carbon (C-1 speak at 284.8 eV) was used as a charge reference to calibrate the Cu-2 p, Zn-2 p, and Fe-2 p core level intensity spectra. The Zn-2 pcore level spectrum [ Fig. 1(a) ] exhibits two sharp symmetrical peaks centered at 1020.96 eV and1044.03 eV, without any signature of satellite peaks. The bindingenergy separation between these peaks is Δ/difference23:06 eV confirms the divalent oxidation state of Zn. 47On the other hand, the Cu-2 pcore level spectrum is deconvoluted into four peaks: two main peaksJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 073908 (2020); doi: 10.1063/5.0008102 128, 073908-2 Published under license by AIP Publishing.centered at 933.65 eV and 953.56 eV and two satellite peaks at 941.1 eV and 961.48 eV [ Fig. 1(b) ]. The observed value of spin – orbit splitting between two main peaks is approximately 19.91 eV, confirming the divalent oxidation state of Cu, which is further sup-ported by the observation of the broad satellite peak at 941.1 eV. 48 Figure 1(c) shows the Fe-2 pcore level spectrum that consists of minimum of six peaks to deconvolute. The four main peaks are centered at 710.10 eV(P 1), 712.3 eV(P 2), 719.01 eV(P 3), and 724.6 eV(P 4), and two broad satellite peaks are located at 714.6 eV (S1) and 732.46 eV(S 2). The peak centered around 710 eV in the 2p3=2core level regime can be assigned to Fe2þ, while the peak at 712 eV is associated with Fe3þ.49,50Moreover, the observed values of spin –orbit splitting Δ(P3/C0P1)/difference9 eV and Δ(P4/C0P2)/difference12 eV further confirms the mixed (divalent and trivalent) oxidation stateof“Fe.”These results are consistent with the results obtained from the EDAX measurements. Figure 2 shows the XRD patterns recorded at room tempera- ture for various compositions of Cu 1/C0xZnxFe2O4(0/C20x/C200:6) system along with the corresponding Rietveld refinement dataobtained using the FullProf program. The XRD pattern forundoped and moderately diluted Zn compositions ( x¼0 and x¼0:05) corresponds to the tetragonal crystal structure of space group I4 1=amd with lattice parameters a¼8:243 A/C14,b¼8:243 A/C14, and c¼8:699 A/C14. The degree of tetragonal distortion abruptly decreases as the Zn dilution level increases in CuFe 2O4.F o r 0:1/C20x/C200:6, all the compounds exhibit a change in the crystal structure from the tetragonal to cubic phase of space group Fd3m.Figure 3 depicts the variation of the lattice parameters ( aandc)a s a function of composition ( x), which divulges a step increase in the lattice parameter above 10 atomic percent of the Zn level beyond which it remains almost constant at 8.424 Å up to x¼0:6. The bond angles between the ions A –O–B and A –O–A follows a completely opposite trend, and the A –O bond length was found to remain constant at /difference2A/C14(Fig. 4 ). Such variations occurring in the unit cell dimensions are consistent with different ionic radius ofthe Cu 2þ(0:57 A/C14), Zn2þ(0.6 Å), and Fe3þ(0.49 Å) ions on A-sites as compared to the B-site occupied cations Cu2þ(0.73 Å), Fe2þ (0.78 Å), and Fe3þ(0.55 Å). Nevertheless, it is very interesting to note that the trivalent Fe occupies the tetrahedral A-sites with either low spin or high spin states for different compositions ( x) along with part of divalent Cu, whereas both the divalent and triva-lent Fe along with most of divalent Cu reside on the octahedralB-sites and the divalent Zn occupies only the tetrahedral A-sites forming a mixed spinel structure in this interesting spinel. Table I summarizes all the parameters obtained from the refinementprocess. Figures 5(a) –5(c) show the temperature dependence of dc-magnetic susceptibility [ χ(T)] recorded under zero-field-cooled (ZFC) and field-cooled (FC, H dc@500 Oe) protocols. In these mea- surements, the data were recorded under warming condition from1.9 K to 300 K after cooling it from the room temperature. Bothχ ZFC and χFCcurves exhibit bifurcation below Tirr/difference300 K (83.7 K), for x¼0( 0:4). Such thermal irreversibility is prominent at low-measuring fields which occurs due to the magneto- FIG. 1. X-ray photoelectron spectra of (a) Zn-2 p, (b) Cu-2 p, (c) Fe-2 p, and (d) O-1sfor the composition x¼0:4.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 073908 (2020); doi: 10.1063/5.0008102 128, 073908-3 Published under license by AIP Publishing.crystalline anisotropy, beyond Tirrthe magnetic moment gradually decreases and approaches a minimum of 5.66 emu/mol Oe (21.31 emu/mol Oe) for x¼0( 0:4), respectively at 300 K. Moreover, upon close examination of χZFC(T) one can notice a fine transition at T1¼66 K (27 K) below the maximum in χZFCat TP¼194 K (68 K) for x¼0( 0:4); however, as the compositions increase both TPand T1gradually smears-off [ Fig. 5(c) ]. This system shows ferrimagnetic behavior similar to the parent com- pound CuFe 2O4due to the unequal magnetic moment of the mag- netic cations. In order to precisely estimate the exact magneticordering temperature, we performed the high-temperature mag-netic measurements separately under the warming condition from 300 K to 900 K in the presence of the external magnetic field (500 Oe). Figure 5(e) shows the χ(T) plot of x¼0:4, where the sus- ceptibility drops to zero across 550 K. In order to estimate theordering temperature precisely, we plotted the differential magnetic susceptibility of the function χT with respect to T because the tem- perature dependence of @(χT)/@T is analogous to the thermal vari- ation of the heat capacity C P(/differenceA@(χT)=@T). Also, typically, the order to disorder transition in ferri/antiferromagnets occurs at a FIG. 2. X-ray diffraction spectra and their corresponding Rietveld refinement patterns of various compositions (a) x¼0, (b) x¼0:05, (c) x¼0:1, (d) x¼0:2, and (e) x¼0:6o fC u 1/C0xZnxFe2O4. FIG. 3. Composition variation of the lattice parameters ( aandc) obtained from the Rietveld refinement patterns of Cu 1/C0xZnxFe2O4. FIG. 4. Variation of the bond angle between the cations A –O–B (blue sphere) and A –O–A (red sphere) plotted on the LHS scale and bond length A –O plotted on RHS scale as a function of the composition of Cu 1/C0xZnxFe2O4. TABLE I. The list of crystallographic parameters obtained from the XRD data and their corresponding Rietveld refinement data for different compositions ofCu 1−xZnxFe2O4. Composition abc A–OA –O–AA –O–B (x) (Å) (Å) (Å) (Å) (deg) (deg) 0 8.243 8.243 8.699 2.032 98.02 123.71 0.05 8.287 8.287 8.613 2.028 98.03 123.870.1 8.395 8.395 8.395 2.015 94.9 121.80.2 8.405 8.405 8.405 2.011 93.6 119.670.4 8.414 8.414 8.414 2.051 92.97 116.49 0.6 8.424 8.424 8.424 2.015 95.33 121.39Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 073908 (2020); doi: 10.1063/5.0008102 128, 073908-4 Published under license by AIP Publishing.temperature few percent lower than the usual transition. Thus, instead of considering the minimum in the χ(T) plot, we took the temperature corresponding to the minimum in the @(χT)=@Tvs T plot as the exact ordering temperature. The LHS and RHS scales ofFigs. 5(d) –5(f) show the temperature variation of χand @(χT)=@T, respectively, whereas insets in these figures depict the variation of χT vs T. This analysis yields a sharp minimum in derivative plots at 534.3 K corresponding to the ferrimagnetic Néel temperature(T FN) of the investigated system for x¼0:4, which is less than the undoped case 743 K [ Fig. 5(d) ]. For x¼0:6, as shown in Fig. 5(f) , still lower TFN(¼370:5 K) was observed. In order to estimate the effective magnetic moment μeffof the compound, we have plotted the thermal variation of inverse para-magnetic susceptibility χ /C01(T) obtained from the χZFC(T) data in the paramagnetic regime ( T.TFN). Accordingly, the χ/C01(T) (for T.TFN) data are fitted with the Curie –Weiss law χ¼[C=(Tþθ)] with C¼Nμ2 eff=3kB,μ2¼g2J(Jþ1)μ2 B,θisthe Curie –Weiss temperature, gis the Landé factor, and J is the total angular momentum. For x¼0:4, the magnitudes of C(¼3:318 emu K mol/C01Oe/C01) and θ(¼426:8 K) are evaluated from the slope and intercepts of the χ/C01(T) line. Using the magni- tudes of g¼2,C, and θwe have evaluated the μeff¼5:15μB=f:u:, which is consistent with the theoretically estimated value of 5:02μB=f:u: according to the cationic distribution (Cu 0:12Zn0:40Fe0:48)A[Cu 0:48Fe1:52]BO4.2,27,51–55For a precise under- standing of the exchange interactions, we fitted the χ/C01vs T data (for T.TFN) with the Néel ’s expression for ferrimagnets as follows: (1=χ)¼(T=C)þ(1=χ0)/C0[σ0=(T/C0Θ)]: (1) The scattered symbols shown in Fig. 6 signify the experimental χ/C01vs T data, whereas the solid lines represent their corresponding fits (for x¼0 and 0.6). In Eq. (1),Crepresents the Curie constant, FIG. 5. T emperature variation of the dc-magnetic susceptibility χ(T) mea- sured under both zero-field-cooled (ZFC) and field-cooled (FC) conditions for different compositions (a) x¼0, (b) 0.4, and (c) 0.6 between the tem-peratures 1.9 K and 300 K [insets of (a) and (b) clearly depict the low tempera- ture transitions T Pand T1]. (d) –(f) show the χ(T) plots (LHS scale) recorded at high temperatures from 300 K to 900 K under warming condi- tions. However, the RHS scale showsthe temperature dependence of @(χT)/ @T , and insets represent temperature variation of χT.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 073908 (2020); doi: 10.1063/5.0008102 128, 073908-5 Published under license by AIP Publishing.Θis the Curie –Weiss temperature, σ0and χ0are constants, and TA¼C=χ0is the asymptotic Curie temperature ( /C0547:2 K and /C0315:8 K for the compositions x¼0 and x¼0:6, respectively). Usually, the asymptotic Curie temperature represents the strengthof the magnetic exchange coupling between the tetrahedral A-sitespins and the remaining cations occupying the octahedral B-sites.Using the fitted parameters obtained from the Néel ’s expression for ferrimagnets, we have determined the molecular field constants and exchange constants ( J AA,JBB, and JAB). A detailed composition dependent analysis of exchange interaction [J(x)] reveals thatJ AB(/difference25kBforx¼0) is ferromagnetic in nature, and it is the main dominant exchange interaction in comparison to the other two exchange constants ( JAA,JBB), though, JABdecreases significantly with increasing the composition ( JAB/difference/C05:5kBfor x¼0:6) (Fig. 7 ). At very high compositions, these interactions are antiferro- magnetic in nature ( /C0JAB) consistent with the AFM ordering of the end compound ZnFe 2O4(x¼1).Table II lists all the physical parameters obtained from the Néel fits of the χ/C01(T) data mea- sured under the ZFC condition for all the six compositions. Withan increase in x, the A-site gets diluted, hence magnetization of A sublattice becomes too small. Also, the interaction between B cations becomes weaker due to competing exchange interaction on B-sites, finally leading to a canted spin structure. 70Nevertheless, for very dilute dispersion of Zn ( x/C200:05), the system stables in the next lower symmetry tetragonal crystal structure of space groupI4 1=amd, which influences the exchange interactions significantly. Figure 8 shows the variation of ordering temperature and the effective magnetic moment plotted as a function of composi-tion [ T FN(x) and μeff(x)]. We noticed continuous decrease ofTFNwith increasing the Zn content following the trend TFN¼yoþb1xþb2x2with intercept yo¼740:42 K and the con- stants b1¼/C0306:73,b2¼/C0517:84. However, the magnitudes ofμeffincreases progressively from 3 :2μB=f.u. to 5 :15μB/f.u., with increasing the xfrom 0.05 to 0.4, respectively. In the case of tetrag- onally distorted compositions, μeffessentially decreases with increasing x, which is apparent due to the increase of the non- magnetic Zn ions. For the compositions with the stable cubic struc-ture, μ effincreases initially and then decreases beyond xc(¼0:4). Such a variation is consistent with the anisotropy calculations dis- cussed below. Figure 9 shows the magnetization vs field ( M–H) magnetic hysteresis loops of the composition x¼0:4 measured at different temperatures between 2 K and 300 K. These loops quickly saturate with the negligible coercive field. In order to probe the anisotropyof the system, we measured the isotherms of M–Hat different tem- peratures (as shown in the inset of Fig. 9 ) and employed the law of approach to saturation (LAS) technique. 56–58The virgin magnetiza- tion isotherm curves obtained from the experiments are fitted with FIG. 7. Composition dependence of the exchange constants JABfor the system Cu1/C0xZnxFe2O4. TABLE II. The list of parameters obtained after fitting the Néel ’s expression [Eq. (1)] for ferrimagnets with the temperature dependence of inverse paramagnetic suscepti- bility data.The parameter Jrepresents the exchange constant, whereas C, μeff, and Θdenotes Curie constant, effective magnetic moment, and Curie –Weiss temperature, respectively. σ0andχ0are the constants. C1 / χ0 σ0 Θ μeff JAB/kB x (emu K mol Oe)(mol Oe emu)(mol Oe K emu) (K) ( μB/f.u.) (K) 0 1.83 −299 46.17 867 3.82 25 0.05 2.01 −248.78 30.86 831.5 4.01 23 0.1 2.42 −160 409.15 800.12 4.39 12 0.2 3.1 −107.07 87.08 787.63 4.97 14 0.4 4.21 −72 186.74 734.74 5.80 5 0.6 3.76 −84 488.4 625.68 5.48 −5.5 FIG. 6. T emperature dependence of inverse paramagnetic susceptibility χ/C01(T) and the corresponding fitting (solid red lines) to the Néel ’s expression for ferri- magnets for the compositions x¼0 and 0.6.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 073908 (2020); doi: 10.1063/5.0008102 128, 073908-6 Published under license by AIP Publishing.the mathematical expression of LAS [Eq. (2) given below]. According to LAS, near the saturation magnetization ( MS), the magnetic moment of the samples can be expressed as follows: M¼MS1/C0a H/C0b H2/C18/C19 þχH: (2) In the above equation, the term a/His linked with the structural defects, whereas the magneto-crystalline anisotropy of the materialis defined by the b/H2term, and the last term χHrepresents the paramagnetic behavior of the system. The value of bin Eq. (2)is given as b¼8 105K2 1 μ20M2 S, where K1represents the cubic anisotropy constant and MSbeing the saturation magnetization. The corre- sponding anisotropy field HKfor a cubic crystal with easy direction along the [100] direction has been calculated using the relation HK¼2K1=μ0MS. The solid lines in the inset of Fig. 9 represent the best fit obtained using Eq. (2)to the experimental data points shown as scattered symbols. At T¼2 K, the magnitude of K1and HKare 1 :616/C2106erg=cc and 5.49 kOe, respectively, which decreases drastically with increasing the temperature. From Fig. 10 , we noticed that the temperature variation of K1andHKparameters decrease monotonically with increasing the temperature. Thesemagnitudes of K 1andHKare consistent with the previously reported data on and slightly larger than the case of nanoparticles, for example, K1/difference1:8/C2105erg=cc (at T¼4:2K ) f o r C u F e 2O4nano- structures59and two orders greater than ( K1¼7:2/C2104erg=cc, HK¼1:97 kOe) the case of magnesium ferrites aerogels.37 Interestingly, the results obtained from the current study are compara- ble to K1¼2:23/C2106erg=cc reported by Rondinone et al. for CoFe 2O4nanoparticles of size 8.5 nm using Mössbauer spectroscopy.60In the case of Zn-substituted cobalt ferrite, nanoparti- cles coated with triethylene glycol K1/difference7:1/C2104erg=cc and HK/difference1:9 kOe, which are significantly lower than the current results.61 Figure 11 shows the temperature variation of the saturation magnetization MS(T) of x¼0:4, which shows a decreasing trend with increasing the temperature. Normally, the spontaneous mag- netization in spinel ferrites arises due to the difference in the mag- netic moments of cations, which are distributed in the octahedraland tetrahedral sites. The saturation magnetization ( M S) is found to be 2 :86μB/f.u. at 300 K, which increases to 4 :73μB/f.u. at 2 K. In Fig. 11 , the dotted line represents linear extrapolation of the experi- mental data, which intersects the temperature axis at 767.74 K ( T*); FIG. 8. Compositional variation of the Néel temperature TFN(x) and effective magnetic moment μeff(x). The red line is the quadratic fitting to the TFNvalues obtained for different compositions. FIG. 9. Magnetization vs field ( M–H) hysteresis loops recorded at different tem- peratures (2 K /C20T/C20300 K) for the composition x¼0:4 under the ZFC condi- tion. The inset shows the M–Hisotherms recorded in the first quadrant (from 0 to 90 kOe) at different temperatures. FIG. 10. T emperature variation of the cubic anisotropy parameter, K1(T), and the anisotropy field HK(T) for the composition x¼0:4.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 073908 (2020); doi: 10.1063/5.0008102 128, 073908-7 Published under license by AIP Publishing.this value is well above the TFN(¼534 K), representing that high- field susceptibility is still non-zero till T*. Although the nonlinear behavior of M–Hloops disappear due to the increasing thermal fluctuations above TFN, which is typical to the paramagnets, the magnetic moment does not approach exactly to zero value until themeasurement temperature approaches T *.62Forx¼0:4, the mag- nitude of MSis found to be /difference16:08/C2103emu =mol (67 emu/g) at room temperature, which is higher than that of the ball milled pre-pared nanostructured CuFe 2O4(26.3 emu/g),63bulk CuFe 2O4 (33.1 emu/g),64nanostructured CuFe 2O4(37:98 emu =g),65and CuFe 2O4nanorods (8.35 emu/g).66For a detailed understanding of the role of Zn dilution on the low temperature magnetic behavior,we measured the virgin M–Hisotherms (as shown in Fig. 12 )a t T¼2 K for various compositions in the range 0 /C20x/C200:6:It has been observed that the magnitude of M Sincreases progressively with increasing xfrom x¼0t o x¼0:4 as shown in Fig. 13 [for T¼2 K (300 K) MSincreases from 1.64 (1.41) μB/f.u. to 4.73 (2.86) μB/f.u.]. Such an increase in MSwith the Zn content in CuFe 2O4may be ascribed due to the increase of divalent Fe frac- tion at the B-site and a significant increase in the canting angle (tri- angular Yafet –Kittle angle) between the magnetic ions.67But beyond 40 at. %, it decreases due to competing exchange interactionamong the canted spin arrangements on the B-site. 70This variation is in well agreement with the variation of exchange constants withcomposition. Interestingly, the compositional dependence of anisotropy field H K(x) and cubic anisotropy constant K1(x) exhibits very higher magnitudes /difference1:6/C2106erg=cc and 5.5 kOe, respectively, for xc(atT¼2 K), but falls beyond it, which is in consonance with the compositional variations of MSandμeff. Following a tiny devi- ated approach in calculating b[in Eq. (2)] for tetragonal systems, FIG. 11. T emperature dependence of saturation magnetization MS(T) obtained from the isothermal magnetization curves (measured till H ¼90 kOe) for the composition x¼0:4. The dotted line represents the linear extrapolation. FIG. 12. Isothermal magnetization ( M–H) curves recorded at 2 K for different compositions (0 /C20x/C200:6) measured in the first quadrant (from 0 to 90 kOe) under the ZFC condition. FIG. 13. Composition dependence of the saturation magnetization MS(x) for T¼2 K and 300 K [Fig. (a)], cubic anisotropy constant K1(x) [LHS scale in Fig. (b)] and the anisotropy field HK(x) forT¼2 K [RHS scale in Fig. (b)].Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 073908 (2020); doi: 10.1063/5.0008102 128, 073908-8 Published under license by AIP Publishing.the obtained results are consistent with the fact that any change in the crystal structure (from tetragonal to cubic crossover in the present case) is accompanied by a large change in the magneto-crystalline anisotropy of the system. 68,69Similar trend has been observed in the case of Zn diluted CuFe 2O4thin films and other Zn diluted systems.70–73It is well known that in cubic crystals, the higher order anisotropy constants are negligible because of the crystal-symmetry, in turn, leading to a smaller value of anisotropyenergy and hence anisotropy constant as compared to the lowersymmetry-crystals such as tetragonal structure, which is clearlyreflected in our measurements across x p. The compositional varia- tions of K1and HKin the present case are in line with that of μeff andMS. The high value of K1in the vicinity of xccan be attributed to the most stable canted spin configuration of the system. IV. SUMMARY The important results presented in this work on the Cu1/C0xZnxFe2O4(0/C20x/C200:6) system are summarized as follows: (i) The electronic structure probed by the x-ray photoelectron spectroscopy reveals the divalent oxidation state of both Cuand Zn in which the 2 porbitals exhibit doublet correspond- ing to 2 p 3=2and 2 p1=2with spin –orbit splitting Δ/difference19:91 eV and 23.06 eV, respectively, whereas the core level spectrum ofFe-2pexhibits Δ/difference9 eV and 12 eV corresponding to the doublet 2 p 3=2and 2 p1=2of divalent and trivalent Fe, respectively. (ii) The crystal structure analysis provides evidence for morpho- tropic phase boundary between the compositions x¼0:05 and 0.10 across which the crystal structure changes from thetetragonal ( I4 1=amd) to cubic ( Fd3m) phase. This structural change leads to a step change in the bond angle (A –O–A and A–O–B) and slight change in the bond length (A –O). Nonetheless, under the dilute limit ( x/C200:05), the investi- gated system stabilizes in next lower symmetry space groupI4 1=amd and influences the magnetic exchange interactions significantly. (iii) Detailed temperature and field dependence of magnetization studies reveal that all the compositions orders ferrimagneticallyand undergoes transition at low temperatures, which smears-offas the composition increases. Ferrimagnetic ordering tempera- ture decays quadratically ( T FN¼b2x2þb1xþyo)w i t hi n c r e a s - ing composition. On the contrary, μeffand MSincrease from 3:59μB/f.u. ( x¼0) to 5 :15μB/f.u. ( x¼0:4) and 1 :64μB/f.u. (x¼0) to 4 :73μB/f.u. ( x¼0:4), respectively. (iv) Furthermore, magnetization results demonstrate that the ferromagnetic exchange interaction, JAB,b e t w e e nJ a h n – Teller Active Cu and Fe is dominant for lower and inter-mediate compositions as compared to J AAand JBB,w h e r e a s for higher compositions, JABbecomes negative, implying the onset of strong antiferromagnetic correlations at B - s i t e s .O nt h eb a s i so ft h ee v i d e n c eg a t h e r e df r o mX P S and magnetization data, we conclude that the system exhibitsthe following complex mixed spinel cationic distribution:(Cu (1/C0x)=5ZnxFe4(1/C0x)=5)A[Cu 4(1/C0x)=5Fe2/C04(1/C0x)=5]BO4. (v) Using the low temperature M–Hisotherms and the law of approach to saturation, we have estimated the cubicanisotropic constant ( K1) and anisotropy field ( HK) in which the structural change from tetragonal to cubic is accompa-nied by a large change in the magneto-crystalline anisotropy.Both the parameters K 1(x) and HK(x) exhibit decreasing trend with anomalous change across the critical composition xc¼0:4[K1/difference8:2/C2105erg=cc (x¼0) to 1 :6/C2106erg/cc (x¼0:4), and HK/difference7:9 kOe ( x¼0) to 5.5 kOe ( x¼0:4)]. For a particular composition, the temperature variation(2 K/C20T/C20300 K) of these parameters, K 1(T) and HK(T) show a continuously decreasing trend until 250 K and starts increasing beyond this point. ACKNOWLEDGMENTS S.K.J. acknowledges the FIST program of Department of Science and Technology, India, for partial support of this work (Grant Nos.SR/FST/PSII-020/2009 and SR/FST/PSII-037/2016). S.K.J. and S.G. acknowledge the Central Instrument Facility (CIF) of the Indian Institute of Technology Guwahati for partial support of this work.Z.Y. and Y.Q. thank the support from the National Natural ScienceFoundation of China (NNSFC, No. 11974104). S.K.J. acknowledgesDr. P. Pramanik, Department of Physics, IIT Guwahati, for help in calculating the molecular field and exchange constants using the high-temperature paramagnetic susceptibility data. DATA AVAILABILITY The data that support the findings of this study are available within the article. REFERENCES 1V. K. Lakhani and K. B. Modi, J. Phys. D Appl. Phys. 44, 245403 (2011). 2K. Verma, A. Kumar, and D. Varshney, Curr. Appl. Phys. 13, 467 (2013). 3I. Onyszkiewicz and J. Pietrzak, Phys. Status Solidi A 73, 641 (1982). 4H. Nagata, T. Miyadai, and S. Miyahara, IEEE Trans. Magn. 8, 451 (1972). 5H. Ohnishi and T. Teranishi, J. Phys. Soc. Jpn. 16, 35 (1961). 6T. Yamadaya, T. Mitui, and T. Okada, J. Phys. Soc. Jpn. 17, 1897 (1962). 7A. 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Mater. 393, 429 (2015).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 073908 (2020); doi: 10.1063/5.0008102 128, 073908-10 Published under license by AIP Publishing.

5.0022271.pdf

APL Mater. 8, 100901 (2020); https://doi.org/10.1063/5.0022271 8, 100901 © 2020 Author(s).Lead-free halide perovskite photovoltaics: Challenges, open questions, and opportunities Cite as: APL Mater. 8, 100901 (2020); https://doi.org/10.1063/5.0022271 Submitted: 19 July 2020 . Accepted: 23 September 2020 . Published Online: 05 October 2020 Vincenzo Pecunia , Luigi G. Occhipinti , Abhisek Chakraborty , Yiting Pan , and Yueheng Peng ARTICLES YOU MAY BE INTERESTED IN Understanding the interplay of stability and efficiency in A-site engineered lead halide perovskites APL Materials 8, 070901 (2020); https://doi.org/10.1063/5.0011851 Light emission from perovskite materials APL Materials 8, 070401 (2020); https://doi.org/10.1063/5.0019554 Toward high efficiency tin perovskite solar cells: A perspective Applied Physics Letters 117, 060502 (2020); https://doi.org/10.1063/5.0014804APL Materials PERSPECTIVE scitation.org/journal/apm Lead-free halide perovskite photovoltaics: Challenges, open questions, and opportunities Cite as: APL Mater. 8, 100901 (2020); doi: 10.1063/5.0022271 Submitted: 19 July 2020 •Accepted: 23 September 2020 • Published Online: 5 October 2020 Vincenzo Pecunia,1,a) Luigi G. Occhipinti,2,a) Abhisek Chakraborty,1Yiting Pan,3and Yueheng Peng1 AFFILIATIONS 1Jiangsu Key Laboratory for Carbon-Based Functional Materials & Devices, Joint International Research Laboratory of Carbon- Based Functional Materials and Devices, Institute of Functional Nano & Soft Materials (FUNSOM), Soochow University, 199 Ren’ai Road, Suzhou, 215123 Jiangsu, China 2Department of Engineering, University of Cambridge, 9 JJ Thomson Avenue, Cambridge CB3 0FA, United Kingdom 3School of Architecture, Soochow University, 199 Ren’ai Road, Suzhou, 215123 Jiangsu, China a)Authors to whom correspondence should be addressed: vp293@suda.edu.cn and lgo23@cam.ac.uk ABSTRACT In recent years, lead-free metal-halide perovskite photovoltaics has attracted ever-growing attention, in view of its potential to replicate the outstanding properties of lead-halide perovskite photovoltaics, but without the toxicity burden of the latter. Despite a research effort much smaller in scale than that pursued with lead-based perovskites, considerable progress has been achieved in lead-free perovskite photovoltaics, with the highest power conversion efficiencies now being in the region of 13%. In this Perspective, we first discuss the state of the art of lead- free perovskite photovoltaics and additionally highlight promising directions and strategies that could lead to further progress in material exploration and understanding as well as in photovoltaic efficiency. Furthermore, we point out the widespread lack of experimental data on the fundamental optoelectronic properties of lead-free halide perovskite absorbers (e.g., charge carrier mobility, defect parameters, Urbach energy, and the impact of dimensionality). All of this currently hampers a rational approach to further improving their performance and points to the need for a concerted effort that could bridge this knowledge gap. Additionally, this Perspective brings to the fore the manifold photovoltaic opportunities—thus far largely unexplored with lead-free perovskite absorbers—beyond single-junction outdoor photovoltaics, which may potentially enable the realization of their full potential. The exploration of these opportunities (tandem photovoltaics, indoor photovoltaics, and building-integrated and transparent photovoltaics) could energize the investigation of existing and new classes of lead-free perovskite absorbers beyond current paradigms and toward high photovoltaic performance. ©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/10.1063/5.0022271 .,s I. INTRODUCTION Over the past decade, lead-halide perovskites have reached prominence in photovoltaics and beyond,1–6delivering a tremen- dous rise in single-junction power conversion efficiency (PCE) (now greater than 25%)7through remarkably simple manufactur- ing processes. Apart from instability issues currently being tack- led,8their reliance on toxic lead is a fundamental limiting factor preventing lead-halide perovskite photovoltaics from reaching com- mercial maturity as an alternative to silicon-based photovoltaics. This has spurred researchers to look for alternative metal-halide perovskite absorbers with closely related properties yet lead-free.9 Prominent classes of such absorbers comprise tin-based andgermanium-based perovskites and derivatives, antimony-based and bismuth-based perovskite derivatives, and double perovskites. Con- siderable progress has been recently achieved in this area—with their highest single-junction PCE now at 13.2%—despite a research effort incomparably smaller in scale and spanning a much shorter time than lead-based perovskite photovoltaics research. This Perspective provides a timely fresh look at the sta- tus and prospects of the rapidly evolving area of lead-free per- ovskite absorbers for photovoltaics. It first surveys the main classes of lead-free metal-halide perovskite absorbers—tin-based, germanium-based, bismuth-based, and antimony-based and halide double perovskites (HDPs)—highlighting the most promising solu- tions explored to date. This is followed by a discussion of the most APL Mater. 8, 100901 (2020); doi: 10.1063/5.0022271 8, 100901-1 © Author(s) 2020APL Materials PERSPECTIVE scitation.org/journal/apm important challenges in lead-free perovskite photovoltaics, with a particular focus on photovoltaic efficiency and stability character- ization. General strategies that may enable these challenges to be overcome are also jointly discussed. Subsequently, this Perspective highlights key outstanding questions in the area of lead-free per- ovskite absorbers for photovoltaics, identifying as a priority area the detailed investigation of their charge transport and defect properties, as well as the role of dimensionality. Finally, this Perspective looks at the opportunities beyond single-junction solar harvesting that could realize the full photovoltaic potential of lead-free perovskites in the near future. II. STATUS OF LEAD-FREE PEROVSKITE PHOTOVOLTAICS In this section, we provide an overview of the status of photo- voltaics research based on lead-free perovskite absorbers, highlight- ing the most recent developments. While a very large number of lead-free perovskite absorbers have been pursued to date, our dis- cussion focuses on the material families that have attracted the most interest because of their photovoltaic potential, i.e., tin-based and germanium-based perovskites and derivatives, antimony-based and bismuth-based perovskite derivatives, and double perovskites (see Fig. 1 for their structures and constituents). A. Tin-based perovskites and derivatives A mainstream strategy for lead-free perovskite photovoltaics involves the replacement of lead with same-group tin.10The result- ing tin-based perovskites and derivatives have been central in lead- free perovskite photovoltaics, thus far delivering the highest PCE (13.24%).11 Since Sn2+has a similar ionic radius as Pb2+, tin allows the formation of ASnX 3perovskites (with A being a monovalent cationand X being a halide anion), which possess a three-dimensional (3D) structure analogous to that of the mainstream lead-based per- ovskites [Fig. 1(a)]. MASnI 3[MA: methylammonium, see Fig. 1(h)], FASnI 3[FA: formamidinium, see Fig. 1(h)], and CsSnI 3have been at the forefront of tin-based perovskite research. In addition to their structural and optoelectronic similarity with their lead-based coun- terparts, the attractiveness of these absorbers relates to their direct gaps in the region of 1.3 eV–1.4 eV [see Fig. 2(a) for their absorption spectra],12which are ideal for single-junction photovoltaics. The resulting solar cells have delivered excellent short-circuit current (in many instances >20 mA cm−2) but have generally suffered from low open-circuit voltage V oc(often in the range of 0.3 V–0.4 V), lead- ing to PCE values typically <7%.12This has been traced to several issues, most importantly (a) the inherent tendency of Sn2+to oxi- dize into Sn4+, leading to severe device instability and a large carrier background due to concurrent p-type doping,13,14and (b) a high defect density due to morphological imperfections.15An approach that has been widely pursued to mitigate the instability issue involves the use of reducing agents as additives.16–19This has led to PCE values up to ∼10% [Fig. 2(e)] and improved stability, the latter being, nonetheless, unsuitable for real-world applications18,19(e.g., a PCE reduction of 30% after hundreds of minutes in low relative humidity19). Most recently, PCE values up to 13.24% [Fig. 2(e)]— the highest to date for all lead-free perovskite absorbers— have been achieved through defect passivation and lattice-strain reduction.11,20,21 In order to overcome the issues faced by ASnX 3photovoltaics, a number of alternative tin-based approaches have been pursued based on the manipulation of the perovskite structure. One such approach involves tin-based vacancy-ordered (VO) double per- ovskites, which manifest superior stability but have achieved rather low PCE to date (see Sec. II D). Another notable route involves “hol- low” perovskites, in which medium-sized cations are used to pro- duce structural voids within a 3D perovskite structure [Fig. 1(b)].22 This has resulted in promising PCE values [ ∼7%, see Fig. 2(e)] and improved stability compared to additive-free ASnX 3.23Higher FIG. 1 . Lead-free halide perovskites: (a)–(f) common structures; common (g) inorganic and (h) organic constituents. APL Mater. 8, 100901 (2020); doi: 10.1063/5.0022271 8, 100901-2 © Author(s) 2020APL Materials PERSPECTIVE scitation.org/journal/apm FIG. 2 . Normalized absorbance of representative materials: (a) tin-based;18,20,23,25,33,34(b) germanium-based;29–31(c) antimony-based and bismuth-based;35–38(d) double perovskites.39–41The AM 1.5G spectrum is overlaid for the sake of comparison. (e) Single-junction PCE under AM 1.5G illumination vs optical gap of representative absorbers (solid symbols);11,18,20,21,23,25,29–32,35,37,39–50single-junction theoretical solar cell efficiency under AM 1.5G illumination in the Shockley–Queisser limit (line). PCE values have been achieved through dimensional manipula- tion, with a particularly promising route involving “2D/3D” per- ovskites, in which 3D domains are mixed with layered (i.e., quasi- two-dimensional) domains.24–26In fact, the bulky cations of the lay- ered phase have been identified as conducive to enhanced stability.27 PCE values in the region of 9% and up to 12.4% have been achieved with the latter approach24–26[Fig. 2(e)], along with a shelf-life of 3800 h for encapsulated devices.25 B. Germanium-based perovskites and derivatives The interest in germanium-based perovskites arises from the electronic similarity of Ge2+with Pb2+, which enables the formation of structurally 3D AGeX 3perovskites (with A being a monovalent cation and X being a halide anion).1In regard to single-junction photovoltaics, the most promising candidate has been identified in CsGeI 3, due to its direct gap of 1.63 eV [Fig. 2(b)].28The use of lighter halogens or alternative A-site cations leads to wider bandgaps [e.g., 2.0 eV for MAGeI 3, see Fig. 2(b)].28,29 Despite their high theoretical single-junction efficiency in the Shockley–Queisser limit (e.g., ≈30% for CsGeI 3), very few reports on germanium-based-perovskite photovoltaics have appeared to date. This can be traced to the pronounced instability arising from the tendency of Ge2+to oxidize into Ge4+.28The highest reported PCE is∼0.6% [Fig. 2(e)] and has been achieved through compositional engineering and morphological improvements of the perovskite layers.29,30 An alternative direction in Ge-based-perovskite photovoltaics involves absorbers in which germanium is alloyed with tin, lead- ing to ASn 1−xGexX3. A particularly promising result was obtained with CsSn 0.5Ge0.5I3, which delivered a PCE of 7.11% [Fig. 2(e)] along with improved stability with respect to the CsSnI 3case.31This was attributed to the enhanced stability resulting from alloying and to the passivating effect of the native germanium oxide formed at the perovskite surface and interfaces. Moreover, the trap-healing effectof germanium in FA 0.75MA 0.25Sn1−xGexI3was recently shown to deliver a PCE of 7.9% [Fig. 2(e)].32 C. Antimony-based and bismuth-based perovskite derivatives Antimony and bismuth have emerged as attractive for the development of lead-free perovskite absorbers due to their low tox- icity and the electronic similarity of their 3+ cations with Pb2+.51 A3B2X9absorbers (B3+= Sb3+or Bi3+) may come in two different phases: a dimer phase (0.5-dimensional, i.e., 0.5D, following Xiao et al. )52featuring isolated face-sharing metal-halide bi-octahedra [Fig. 1(c)] and a layered phase (2.5-dimensional, i.e., 2.5D, follow- ing Xiao et al. )52featuring planes of corner-sharing metal-halide octahedra [Fig. 1(d)].53Research efforts have primarily focused on 0.5D ternary iodide absorbers (A 3B2I9, with A = Cs+, MA+), which have bandgaps in the region of 2.1 eV–2.4 eV [Fig. 2(c)].54–56While this points to considerable potential for tandem photovoltaics (see Sec. IV), research on 0.5D Sb-based and Bi-based absorbers to date has narrowly focused on single-junction photovoltaics. While the majority of the early studies on 0.5D absorbers delivered rather mod- est PCE (<1%),54,56,57recent developments—building on the use of additives,35,58optimized transport layers,35or dedicated deposition protocols45,46—have led to significant performance improvement, with the PCE reaching 2.8% for 0.5D MA 3Sb2I9and 3.2% for 0.5D A3Bi2I9(A = Cs+, MA+) [Fig. 2(e)],35,45,46in some instances also with excellent device stability in air (e.g., non-encapsulated devices retaining 97% of their initial PCE over a period of 60 days).45 The higher dimensionality of 2.5D antimony-based and bismuth-based A 3B2X9absorbers makes them more attractive for photovoltaics, also in view of their narrower bandgaps [Fig. 2(c)] and smaller exciton binding energy and effective masses.55,59While mainstream MA 3B2I9and Cs 3B2I9(B3+= Sb3+, Bi3+) come in the 0.5D phase when deposited through conventional methods, 2.5D APL Mater. 8, 100901 (2020); doi: 10.1063/5.0022271 8, 100901-3 © Author(s) 2020APL Materials PERSPECTIVE scitation.org/journal/apm A3M2X9absorbers have been demonstrated through dedicated pro- cessing protocols,59–61or halide mixing,37,43,62or the use of smaller A-site cations such as Rb+.38,63A PCE of 1.4% has been achieved with 2.5D Rb 3Sb2I9[Fig. 2(e)], with both a mesoporous and a planar device structure.38,47Moreover, PCEs of 2.2% and 3.34% have been recently obtained with 2.5D Cs 3Sb2ClxI9−xand MA 3Sb2ClxI9−x, respectively [Fig. 2(e)].37,42 D. Halide double perovskites (HDPs) Cation-ordered halide double perovskites (CO-HDPs) have attracted significant attention in view of their nominally appealing three-dimensional structure. They have a general formula A 2BB′X6, where B and B′are a monovalent cation and a trivalent cation, respectively, alternating at the octahedra centers of a perovskite lat- tice [while A is a monovalent cation and X is a halide anion, see Fig. 1(e)].64Additionally, HDPs comprise vacancy-ordered (VO) embodiments, with a general formula A 2BX 6and featuring the alter- nation of a tetravalent cation B4+and a B-vacancy at the octahe- dra centers of a perovskite lattice [Fig. 1(f)].65Therefore, in con- trast to cation-ordered HDPs, vacancy-ordered HDPs are struc- turally quasi-zero-dimensional (quasi-0D), in consideration of the fact that neighboring [BX 6]2−octahedra are isolated from one another.66 Cation-ordered HDPs constitute a very broad material class, with more than 350 compounds synthesized to date and many more predicted.67Photovoltaics research has primarily focused on silver– bismuth HDPs such as Cs 2AgBiX 6(X = Cl or Br). Due to their large and indirect gaps (2 eV–2.3 eV) as well as the presence of obvious excitonic features [see Fig. 2(c)],65,68these absorbers do not represent the ideal choice for single-junction photovoltaics. For instance, the single-junction spectroscopic limited maximum effi- ciency (which quantifies the maximum achievable efficiency taking into account the magnitude of the absorption coefficient and the nature of the bandgap)69for Cs 2AgBiBr 6is∼8%.70In fact, despite their 3D structure, these cation-ordered HDPs are electronically 0D, as the orbitals contributing to their band edges are spatially iso- lated.52Their low-toxicity profile and 3D structure have nonethe- less prompted their investigation for photovoltaics, also encour- aged by their long carrier lifetimes ( ≈600 ns)48,71and outstanding stability in air (e.g., non-encapsulated devices showing no obvi- ous PCE degradation after 30 days in air).72The highest PCE val- ues in single-junction devices to date are in the region of 2.2%– 2.8% [Fig. 2(e)],40,48–50with the highest performance being achieved in combination with an organic interlayer that slightly increases the overall photon absorption beyond the onset of the perovskite absorber.49 Owing to the limitations of the Cs 2AgBiX 6(X = Cl, Br) sys- tem, research efforts in HDP photovoltaics have also been directed at the synthesis and assessment of alternative cation-ordered HDPs with narrower gaps. While using iodine as the halogen in Cs 2AgBiX 6 could potentially deliver in this direction, Cs 2AgBiI 6has been gen- erally dismissed due to thermodynamic instability considerations.73 However, recent progress in the synthesis of colloidal Cs 2AgBiI 6 nanocrystals with a bandgap of 1.75 eV indicates that Cs 2AgBiI 6 offers a promising opportunity.74An alternative approach that has attracted attention involves the substitution of bismuth with anti- mony. A representative compound of this class is Cs 2AgSbBr 6,which delivers an indirect gap of ∼1.64 eV, yet the PCE reported to date is rather poor (0.01%).75Another system that has been proposed is Cs 2AgInX 6, which has a direct gap, yet associated with an undesirable parity-forbidden transition.76 Vacancy-ordered HDP research has thus far focused on Cs2SnX 6and Cs 2TiBr 6. In contrast to the case of ASnX 3perovskites, tin is present in its stable 4+ oxidation state in Cs 2SnX 6compounds, thus making them more robust against degradation. With a direct gap of 1.3 eV–1.6 eV [Fig. 2(d)],12Cs2SnX 6absorbers inherently overcome the issues thus far affecting cation-ordered HDPs, and their Shockley–Queisser efficiency limit for single-junction photo- voltaics is greater than 25% [Fig. 2(e)].77,78The highest PCE achieved with Cs 2SnX 6to date amounts to 2.0% [Fig. 2(e)] and has been demonstrated along with promising stability in air (a reduction of only∼5% of the original PCE in encapsulated devices during a period of 50 days).39Nonetheless, the quasi-0D structural and elec- tronic nature of Cs 2SnX 6absorbers (with associated large effective masses)52may hamper further progress in their photovoltaic perfor- mance. Another attractive vacancy-ordered HDP is titanium-based Cs2TiBr 6, which has been shown to deliver a promising PCE of 3.3% [Fig. 2(e)] in single-junction devices along with good environmental stability (non-encapsulated devices retain 94% of their initial PCE after 14 days at 70○C in air and under ambient illumination), build- ing on a quasi-direct gap of 1.8 eV [Fig. 2(d)] and a carrier diffusion length>100 nm.41While these results have been recently brought into question,79research on the vacancy-ordered A 2TiX 6system is still at its infancy, and many such compounds have never been syn- thesized to date—which points to the need for further investigation of this area. III. CHALLENGES AND OPEN QUESTIONS Despite the considerable advances in recent years (Sec. II), further progress in both photovoltaic efficiency and stability is needed for lead-free perovskite photovoltaics to approach commer- cial exploitation. In the following (Secs. III A and III B), we provide our perspective on both of these aspects, discussing the associated challenges and potential solutions, wherever relevant. Subsequently, we highlight some key open questions concerning charge trans- port (Sec. III C), defect tolerance (Sec. III D), and dimensionality (Sec. III E), whose investigation is critical in order to catalyze further progress in lead-free perovskite photovoltaics. A. Photovoltaic efficiency Tin-based perovskites motivate further efforts in their explo- ration for use in single-junction solar cells. While they have achieved the highest photovoltaic performance to date (13.24%) of all lead- free perovskites, their bandgap values point to a single-junction Shockley–Queisser efficiency limit of ∼32% [Fig. 2(e)], thereby sug- gesting considerable scope for improvement. Importantly, the short- circuit current of the best performing tin-based perovskite solar cells to date is consistently close to the Shockley–Queisser limit (Fig. 3), indicating efficient photocarrier generation and collection at the contacts. The gap between the reported performance of ASnX 3cells and their Shockley–Queisser limit can be traced to a V ocdeficit of ∼0.4 V, as shown in Fig. 3.11This V ocdeficit can be attributed to a high defect density, which leads to non-radiative recombination—as APL Mater. 8, 100901 (2020); doi: 10.1063/5.0022271 8, 100901-4 © Author(s) 2020APL Materials PERSPECTIVE scitation.org/journal/apm FIG. 3 . Reported short-circuit current (left) and open-circuit voltage (right) of representative lead-free perovskite solar cells under AM 1.5G illumina- tion.11,18,20,21,23,25,29–32,35,37,39–50The short-circuit current and open-circuit voltage in the Shockley–Queisser limit (for single-junction devices under AM 1.5G illumination) are also shown for the sake of comparison. also confirmed by the fact that the best performing devices have an ideality factor 1 <n<2.11All of this indicates that the outstanding challenge in ASnX 3photovoltaics research concerns the develop- ment of refined defect passivation protocols. Based on recent devel- opments, it can be envisaged that this should not be pursued through standalone strategies, but instead through a holistic approach. Defect passivation protocols should combine the compositional and mor- phological optimization of the photoactive layer (through A-site cation mixing and additive incorporation), the enhancement of its stability against oxidation, and the use of a dedicated device stack that minimizes interfacial recombination. In particular, 3D systems with mixed cations and mixed 2D/3D systems are highly promis- ing in order to achieve low defect density and good stability, pro- vided that additives are also used to aid film formation and enhance crystallinity. In addition to non-radiative losses, future efforts in tin-based perovskite research should also be directed at ascertaining and tack- ling the V ocdeficit associated with their energetic disorder. For the sake of illustration, Jiang et al. recently reported that their opti- mized tin-based perovskite films (which deliver one of the highest PCEs, i.e., 12.4%, and a record-high V ocof 0.94 V) have an Urbach energy of 65 meV,25which is expected to result in a V ocdeficit of several hundreds of meV.2It is worth noting that significantly lower Urbach energy values (16 meV–32 meV) were observed from CsSnI 3−xBrx, albeit in association with a much lower V oc.80Con- sidering that energetic disorder has been seldom reported in the tin-based perovskite literature, it is apparent that an important pri- ority is to explore the impact of processing and composition on the energetic disorder and the associated V ocdeficit of tin-based perovskites. Germanium-based AGeX 3perovskites have thus far delivered the lowest PCE of all lead-free perovskite systems discussed herein [Fig. 2(e)]. Their particularly low performance is jointly due to sig- nificant losses in both V oc(Voc≈0.5 V) and J sc(Jsc≈2 mA/cm2) (Fig. 3). The particularly low efficiency of germanium-based AGeX 3 perovskite solar cells can be traced to the presence of deep lev- els,29,81in addition to their rapid degradation due to oxidative insta- bility. All of this indicates a considerable analogy with additive- free ASnX 3perovskites, suggesting that future efforts in AGeX 3 photovoltaics may require the development of material stabilizationprotocols as well as defect-healing strategies based on additives and compositional engineering. Tin–germanium perovskites have achieved much higher effi- ciencies than the germanium-only system (i.e., AGeX 3). Their PCE values (∼7%) are significantly lower than the Shockley–Queisser limit, however [Fig. 2(e)]. Their J scis in a similar range as the tin-based counterparts (Fig. 3), thereby leaving some scope for improvement. However, the major challenge with tin–germanium perovskites to date is their V ocdeficit, which is greater than 0.5 V (Fig. 3). Considering that the recently reported increase in the effi- ciency of tin–germanium perovskite solar cells has been traced to a stability enhancement against oxidation as well as to trap passi- vation,31,32it can be foreseen that tin–germanium perovskite solar cells could potentially deliver higher efficiency through the adoption of a holistic approach to trap passivation and stabilization (along similar lines as pursued with ASnX 3). In addition to non-radiative recombination losses, however, the reported large Urbach energy (in the region of 50 meV for polycrystalline thin films)82points to the urgency of also reducing the energetic disorder as a means of boosting V ocand the overall PCE. Despite the recent progress in antimony-based and bismuth- based A 3B2X9perovskite derivatives, their photovoltaic perfor- mance is still appreciably lower than their Shockley–Queisser limit—with no significant difference between antimony-based and bismuth-based absorbers [Fig. 2(e)]. Their J scis significantly lower than the Shockley–Queisser limit (Fig. 3), indicating significant losses in charge generation and/or collection, notwithstanding the widespread use of a mesoporous device structure. However, in one particular instance in which a 2.5D system (Rb 3Sb2I9) was employed, a J scup to≈50% of the Shockley–Queisser limit was reached (Fig. 3), remarkably with a planar device structure.38In terms of V oc, the gap between the reported values and the corre- sponding Shockley–Queisser limit is significantly more severe (often >1 V), with 0.5D bismuth-based systems performing better than 0.5D and 2.5D antimony-based systems (Fig. 3). Additionally, while 2.5D systems have been generally regarded as superior (e.g., they can provide improved charge transport within their sheets of octa- hedra and are expected to be more defect-tolerant),55,59,83interest- ingly, such systems have only been investigated in very few instances. Notably, all 2.5D antimony-based and bismuth-based embodiments APL Mater. 8, 100901 (2020); doi: 10.1063/5.0022271 8, 100901-5 © Author(s) 2020APL Materials PERSPECTIVE scitation.org/journal/apm have been explored with their sheets of octahedra either predomi- nantly parallel to the substrate or randomly aligned,37,38,42,43,47,55,59,63 likely resulting in enhanced recombination losses. Therefore, it can be envisaged that their efficiency could be significantly improved by further exploring 2.5D A 3B2X9systems—in particular by develop- ing deposition strategies for the realization of films with the sheets of octahedra aligned in the out-of-plane direction.37,38Addition- ally, while the energetic disorder has been scarcely characterized in bismuth-based and antimony-based absorbers, Urbach energy val- ues in the region of 60 meV or larger have been reported for some of these materials.56,63This indicates the future efforts in reducing the V ocdeficit should also aim at improving composition, deposi- tion conditions, and device structures toward the minimization of the energetic disorder. The efficiency of cation-ordered Cs 2AgBiX 6HDPs thus far has been well below the Shockley–Queisser limit for single-junction cells [Fig. 2(e)]. This is primarily determined by a considerably low J sc, which is often less than 1/3 of the Shockley–Queisser limit, while the V ocdeficit has been comparatively moderate (in some instances of≈0.6 V) (Fig. 3). The J scdeficit can be traced to the indirect gap of the cation-ordered double perovskites explored to date, which leads either to a suboptimal light collection for particularly thin films or to an inefficient charge collection for thicker films intended to boost light absorption. Additionally, their absorption spectra man- ifest obvious excitonic effects near the absorption edge [Fig. 2(d)], which may further reduce the photogeneration efficiency. Therefore, the search is still open for alternative cation-ordered HDPs inher- ently suitable for single-junction photovoltaics—especially with a direct gap and with a higher electronic dimensionality than elec- tronically 0D Cs 2AgBiX 6—confirming that this area is at its infancy. Considering the wealth of additional absorbers that have been pre- dicted (some of which are expected to have desirable optoelectronic properties)76but have not been explored to date, further progress in cation-ordered HDPs requires first and foremost the synthe- sis and characterization of such materials. Furthermore, despite their smaller direct gaps, vacancy-ordered HDPs have also deliv- ered PCE and J scvalues much lower than the Shockley–Queisser limit [Figs. 2(e) and 3]. While research on vacancy-ordered HDPs is still in its nascent stage and further optimization of the synthe- sis and deposition conditions may be needed to boost their photo- voltaic performance, current indications of their defect-intolerance and poor charge transport may ultimately limit the scope of these efforts.66 Regardless of the specific lead-free perovskite considered, pho- tovoltaic performance is influenced considerably by the morphol- ogy of the photoactive layer. This aspect is of primary importance because the photoactive layers employed in lead-free perovskite photovoltaics are generally polycrystalline and typically deposited through solution-based methods. Control of the processing con- ditions (e.g., solvent selection, solution concentration, and anneal- ing temperatures) and the specific methodology adopted (e.g., one- step spin-coating, antisolvent processing, solvent-vapor annealing) are, therefore, key to obtaining, first of all, compact and uniform films, given that pinholes are detrimental to photovoltaic efficiency due to their shunting effect.84Additionally, grain boundaries are to be minimized, as they provide a barrier to charge transport and may also contain a high density of defect states acting as recombination centers. The deposition of high-quality films (i.e.,compact and with grain size in the region of 1 μm) of tin-based perovskites requires considerable efforts (e.g., solvent engineering and the identification and optimization of suitable additives),18,20 given the fast crystallization rate of such perovskites during solution processing.12This endeavor has allowed tin-based perovskites to reach a film quality approaching that of state-of-the-art lead-based perovskites,85,86thereby playing an important role in enabling them to deliver the highest efficiencies among lead-free perovskites.18,44 By contrast, the film morphology of other classes of lead-free per- ovskites is yet to approach similar levels. For instance, in the case of double perovskite Cs 2AgBiBr 6, progress in photovoltaic effi- ciency has been achieved through films with grain size in the region of 400 nm, which were obtained through the optimization of antisolvent processing and high-temperature annealing.40In the case of Bi-based and Sb-based A 3B2X9absorbers, research efforts have primarily relied on films with suboptimal morphology, with grain size often much less than 100 nm and/or imperfect cover- age.35,37,45,46While the challenge of depositing films of Bi-based and Sb-based A 3B2X9absorbers without a large number of pin- holes has been addressed through several methods (e.g., dissolution– recrystallization,46solvent-vapor annealing,37vapor assisted solu- tion process45), the grain size of the resultant films is well below 1 μm. As illustrated by the work of Li et al. for the case of Rb 3Sb2I9,38 boosting the grain size of such perovskite derivatives has a consid- erable impact on the photovoltaic performance and requires ded- icated processing protocols (e.g., reduced-supersaturation anneal- ing and high-temperature vapor annealing) that strike a balance between crystallization and nucleation rates. This generally repre- sents an outstanding challenge in cation-ordered double perovskites and Bi-based and Sb-based A 3B2X9photovoltaics. Therefore, it can be envisaged that a considerable enhancement in their photovoltaic efficiency could be realized through the development of process- ing protocols delivering films with grain size in the micrometer range. As a final point on the challenges thus far encountered with regard to the photovoltaic efficiency of lead-free perovskite absorbers, interfacial recombination and inefficient extraction at the transport layers and contacts may also be currently limiting their photovoltaic performance—in addition to the bulk recombi- nation properties discussed earlier. This relates to the choice of transport layers, which are typically drawn from lead-based per- ovskite research, hence may be suboptimal for lead-free perovskite absorbers. Therefore, further progress in the photovoltaic efficiency of lead-free perovskites will also require the investigation of interfa- cial recombination effects and the exploration and optimization of dedicated charge transport layers. B. Stability The search for inherently stable absorbers—beyond the limi- tations of mainstream lead-halide perovskites—has been an impor- tant driving force in the exploration of lead-free halide per- ovskites. While several solutions have been developed over time to improve the stability of inherently unstable Sn2+-based absorbers (see Sec. II A), many double perovskites as well as bismuth-based and antimony-based A 3B2X9absorbers have demonstrated inher- ent stability, based on indications from both thin-film proper- ties and device performance (see Secs. II C and II D). In fact, APL Mater. 8, 100901 (2020); doi: 10.1063/5.0022271 8, 100901-6 © Author(s) 2020APL Materials PERSPECTIVE scitation.org/journal/apm an important remaining challenge in the development of stable lead-free perovskite solar cells suitable for commercialization (aside from the obvious efficiency requirements) arises from the diver- sity of approaches that have been used to date to characterize their photovoltaic stability. This has been particularly limiting because it has prevented (a) the benchmarking of the photovoltaic sta- bility of the manifold lead-free absorbers explored to date; (b) the identification of the best candidates for highly stable lead- free perovskite photovoltaics; and (c) the assessment of the gap that still needs to be bridged for lead-free perovskite photovoltaics to approach commercialization. In light of the standardized pro- tocols for the stability characterization of halide perovskites that have been proposed recently,87it is therefore highly desirable that future studies on lead-free perovskite photovoltaics may assess sta- bility through these protocols. Indeed, this would pave the way for a systematic understanding of stability issues in lead-free per- ovskite absorbers, ultimately enabling the optimization of systems with a promising stability profile toward commercialization-ready levels. C. Defect parameter characterization and experimental screening for defect tolerance A guiding light in the search for lead-free perovskite absorbers with promising optoelectronic potential has been the develop- ment of absorbers that may have a defect-tolerant character, i.e., insensitivity to the defects inevitably present in low-temperature- deposited semiconductor thin films. In electronic terms, defect- tolerant semiconductors may present defect states that either are shallow and with small capture cross sections or fall within the energy bands.88Extrapolating from lead-halide perovskite studies, the search for defect-tolerant lead-free perovskite absorbers has tar- geted materials based on large, highly polarizable metal cations with ns2outer orbitals.88–90In this regard, computational studies have offered highly valuable indications on the compositions that are potentially most promising from the point of view of defect tol- erance.91,92However, conclusive evidence on the defect tolerance of the manifold classes of lead-free perovskite absorbers should be sought experimentally—as a means of validating the indica- tions drawn from computational studies as well as a tool to ratio- nally develop compositions and deposition strategies for higher photovoltaic performance. In fact, the experimental assessment of defects in lead-free perovskites has been primarily phenomenolog- ical to date, relying on techniques that provide only a measure of the total volumetric defect density (e.g., basic space-charge-limited- current characterization),40,60,93or on indicators that largely relate to defect tolerance but do not constitute a univocal measure of the same (e.g., photoluminescence lifetime).19,20,94By contrast, a thor- ough experimental evaluation of defect tolerance in lead-free per- ovskite absorbers—in terms of defect densities, characteristic ener- gies, and capture cross sections—has not been pursued to date. Importantly, the experimental assessment of the defect properties of lead-free perovskite absorbers would aid the identification of those that have high photovoltaic potential and concurrently offer rational criteria for the development of processing protocols for defect state passivation. Therefore, the investigation of experimen- tal approaches for the characterization of the defect parameters oflead-free perovskites is a key priority in lead-free perovskite photo- voltaics research. D. Charge transport characterization Charge transport plays an essential role in photoconver- sion, allowing photocarriers to be collected prior to undergoing recombination. Therefore, charge transport data on lead-free per- ovskite absorbers are of primary importance in order to rationally approach the optimization of their photovoltaic performance. How- ever, experimental charge transport data reported to date from lead- free perovskites are sparse and incomplete, ultimately preventing the identification of relevant trends and hampering the development of solutions for high-performance photovoltaics. Most experimental charge transport data on lead-free per- ovskites have been obtained through space-charge-limited current (SCLC) characterization. Importantly, SCLC data are typically pre- sented in the single-sweep mode (e.g., forward scan only), pre- venting the appraisal of any possible hysteretic effects and other non-idealities that may result from the large fields applied in such measurements and from the possible presence of mixed (electronic– ionic) conductivity. A recent study by Duijnstee et al. has pointed out that single-sweep SCLC transport characterization of halide per- ovskites may lead to a rather inaccurate assessment of their charge transport properties.95Additionally, the validation of SCLC data against the expected thickness dependence (e.g., with the inverse cube of the thickness, as per the Mott–Gurney law)—a tenet of robust SCLC transport analysis—is usually lacking. To ensure the accurate determination of charge transport data, it is therefore rec- ommended that future SCLC investigations on lead-free perovskite absorbers should adopt (a) a double-sweep routine (i.e., compris- ing both forward and reverse scans), including pulsed biasing if hysteretic effects are pronounced,95and (b) the validation of the measured SCLC data against the expected model dependence on the semiconductor layer thickness. Hall effect characterization is another approach that has also been widely adopted to experimentally determine the mobility of lead-free perovskite absorbers.11,21,32,37,38,63,77,96–102While its use with comparatively high-mobility and low-resistivity semiconductors is well-established, Hall effect characterization with a direct-current (DC, i.e., constant) magnetic field presents inherent challenges— and could be heavily impacted by measurement artifacts—when applied to high-resistivity, low-mobility ( <1 cm2V−1s−1) semicon- ductors.103,104Taken aside the case of moderate/high-conductivity tin–germanium-based and tin-based perovskites,32,77DC-magnetic- field Hall effect characterization has nonetheless been pursued also with moderate/high-resistivity bismuth-based and antimony-based perovskite derivatives.63,105Considering that high-resistivity materi- als may lead to considerable experimental errors in the Hall effect parameter extraction,104it is advisable that future studies based on DC-magnetic-field Hall effect experiments should validate their data by critically assessing whether the expected trends are verified (e.g., in regard to the dependence of the carrier density and the Hall mobility on the magnetic field). On the other hand, Hall effect char- acterization relying on a modulated (AC) magnetic field has been demonstrated to offer a viable route to the characterization of low- mobility materials, down to a mobility range of 0.001 cm2V−1s−1.103 Such an AC characterization has been successfully reported in a few APL Mater. 8, 100901 (2020); doi: 10.1063/5.0022271 8, 100901-7 © Author(s) 2020APL Materials PERSPECTIVE scitation.org/journal/apm instances in the recent literature on lead-free perovskites37,38,101,102 and has the potential to become an attractive and widespread route to the charge transport characterization of lead-free perovskite absorbers. E. Electronic dimensionality In the search for high-performance lead-free perovskite absorbers, electronic dimensionality has been recognized as a key determinant of photovoltaic potential.52Indeed, higher electronic dimensionality has been related to superior charge transport prop- erties, a higher defect tolerance, and reduced excitonic effects. Elec- tronic dimensionality is determined by the connectivity of the orbitals contributing to the conduction and valence band edges. In many cases, electronic dimensionality overlaps with structural dimensionality (i.e., the structural connectivity of the perovskite net- work), e.g., as in 3D ASnX 3and 0D Cs 2SnX 6. Along the same lines, bismuth-based and antimony-based A 2B3X9systems are electroni- cally 0.5D or 2.5D, depending on whether they are in their dimer or layered phase, respectively. Finally, in contrast to their structurally 3D character, cation-ordered double perovskites such as Cs 2AgBiX 6 are electronically 0D. All things considered, apart from the electronically 3D ASnX 3 system (and taken aside the AGeX 3system due to its inherent stability limitations), bismuth-based and antimony-based layered A2B3X9absorbers have the highest electronic dimensionality (2.5D) of all lead-free perovskites developed to date. This first highlights that the potential of antimony-based and bismuth-based A 3Sb2X9 absorbers should not be dismissed by narrowly referring only to their dimer-phase embodiments. It is of course puzzling that the PCE values of 2.5D A 3Sb2X9and A 3Bi2X9absorbers reported to date are, in fact, on par with or slightly lower than those of the 0.5D (dimer-phase) counterparts, as well as of the electronically 0D cation-ordered and vacancy-ordered double perovskites [Fig. 2(e)]. However, the literature on layered A 3Sb2X9and A 3Bi2X9reveals that their higher dimensionality compared to other classes of lead-free perovskites has not been truly exploited to date. Indeed, the pho- tovoltaic implementations of layered A 3B2X9demonstrated up to now have been realized with the planes of octahedra either pre- dominantly parallel to the substrate or randomly oriented. There- fore, an outstanding question that needs to be addressed in order to exploit the 2.5D dimensionality of layered antimony-based and bismuth-based A 2B3X9absorbers pertains to the development of deposition strategies for the favorable orientation of their planes of octahedra. An additional dimensionality-related question that still needs to be addressed concerns the development of halide double per- ovskites that are electronically 3D (cf. electronically 0D character of Cs2AgBiX 6). While some compositions with a low-toxicity profile that are expected to be electronically 3D have been proposed (e.g., indium–bismuth double perovskites),106,107to the best of our knowl- edge, no photovoltaic implementations based on such absorbers have been reported to date. Therefore, an important priority in halide double perovskite research pertains to the exploration of the photovoltaic properties of double perovskite embodiments that are electronically 3D, which are anticipated to enable a considerable increase in photovoltaic efficiency.IV. OPPORTUNITIES Lead-free halide perovskites are meant to address the toxicity concerns associated with their lead-based counterparts and concur- rently provide opportunities for green photovoltaics. While lead-free perovskite research has thus far focused narrowly on investigating their capability for single-junction outdoor solar harvesting,108in fact, lead-free perovskites are potentially suitable for other appli- cations in both indoor and outdoor photovoltaics (Fig. 4). Impor- tantly, this potential is still largely unexplored. In this section, we present these photovoltaic opportunities, highlighting promising areas for future investigations that could enable lead-free perovskites to deliver their full photovoltaic potential. A. Tandem photovoltaics Lead-free perovskites are highly promising for silicon– perovskite tandem cell configurations. In consideration of their bandgap of ≈2 eV, lead-free absorbers with significant potential for perovskite–silicon tandem cells are antimony-based and bismuth- based A 3B2X9perovskite derivatives, as well as cation-ordered dou- ble perovskites Cs 2AgBiX 6. The latter are particularly attractive also because of their comparatively large V ocof≈1 V under solar illu- mination. In regard to germanium-based perovskites, a potential route to tandem perovskite–silicon photovoltaics could be explored by pursuing wide-gap compositions through lighter halogens and/or alternative A-site cations.28,29In all of these cases, single or few layer graphene and 2D material electrodes may represent valid alterna- tives to the ITO top electrode and interconnection layer between the top perovskite and the bottom silicon cell, given the low trans- parency of ITO to near-infrared photons. B. Indoor photovoltaics The emergence of the Internet of Things (IoT)—with the rise of compact and energy-autonomous electronic devices such as wireless sensors and radio-frequency identification (RFID) tags—is expected to benefit from the availability of energy harvesters either in com- bination with or as an alternative to small energy storage devices (batteries, supercapacitors) in order to power the sensors and com- municate via standard wireless radio modules and active RFIDs. Small (few cm2) photovoltaic devices are expected to be commer- cialized for integration in IoT applications and represent a growing market opportunity for new technologies offering thin and flexi- ble form factors. Photovoltaic cells located indoors with no access to solar illumination operate by harvesting the energy emitted by artificial light sources, with an illumination intensity 3 orders of magnitude lower than solar illumination, and different radiation spectra in the visible range depending on the light source. The opti- mum bandgap of photovoltaic absorbers required to match the emis- sion spectra of both compact fluorescence lamps and white LED lamps is around 2.0 eV.109Importantly, this requirement matches the bandgap of many lead-free perovskites and derivatives (e.g., Cs2AgBiX 6, antimony-based and bismuth-based A 3B2X9). More- over, considering the comparatively low illumination levels relevant to indoor photovoltaics, the most promising absorbers are those that can deliver an ideality factor nclose to unity, which requires the minimization of the bulk and interfacial defect state concentra- tions.2,110All things considered, the wide-bandgap nature of many APL Mater. 8, 100901 (2020); doi: 10.1063/5.0022271 8, 100901-8 © Author(s) 2020APL Materials PERSPECTIVE scitation.org/journal/apm FIG. 4 . Photovoltaic opportunities for lead-free perovskites. lead-free perovskites points to their considerable potential as a green route to indoor photovoltaics and concurrently motivates future efforts toward deposition protocols enabling photoactive layers with a minimal concentration of recombination centers. C. Building-integrated photovoltaics and transparent photovoltaics The incorporation of transparent or semitransparent photo- voltaic devices in different parts of buildings (e.g., facades, roofs, windows) opens up the prospect of harvesting solar energy while serving the esthetic and functional needs of the end-users. Lead- free perovskites come with colors covering different regions of the visible range, thereby providing attractive opportunities for color- tunable building-integrated photovoltaics (BIPV) applications in architectural design. In regard to transparent photovoltaics (TPV) for window applications, lead-free perovskites with a direct gap of around 2.7 eV would be particularly attractive in order to ensure high average visible transparency (AVT). Additionally, the adoption of indirect-gap double perovskite absorbers (e.g., Cs 2AgBiX 6) pro- vides the opportunity to achieve high AVT with an optical gap of 2.0 eV, thereby concurrently allowing for the harvesting of visible light photons.111Finally, the full potential of lead-free perovskites for transparent photovoltaics could be realized by pursuing tandem configurations in combination with cells (e.g., organic) that absorb near-infrared photons. V. CONCLUSIONS Lead-free perovskites provide an attractive combination of perovskite-related optoelectronic properties with a generally low- toxicity profile. In recent years, considerable progress has been achieved in lead-free perovskite photovoltaics, notwithstanding aresearch effort of an incomparably smaller scale than that driv- ing lead-based perovskite photovoltaics. Tin-based perovskites have delivered by far the highest single-junction PCE ( ≅13%) of all lead-free perovskites to date, and it can be envisaged that they may become an attractive technology provided that their stabil- ity is further improved and their toxicity profile is fully assessed. Another promising single-junction photovoltaics technology is rep- resented by tin–germanium perovskites. Antimony/bismuth-based perovskite derivatives and double perovskites offer highly promis- ing stability indications, but they require further development at the materials, processing, and device levels to further boost their performance. In addition to general strategies that may lead to higher photo- voltaic performance, in this Perspective, we have discussed a num- ber of key open questions that pertain to the incomplete exper- imental assessment of the optoelectronic properties of lead-free perovskite absorbers. This involves first the widespread lack of Urbach energy data, which prevents detailed photovoltaic model- ing and analysis of the dominant V ocloss mechanism. Addition- ally, charge transport data are widely lacking, and the reliability of the charge transport data reported to date may require a criti- cal re-assessment. A detailed experimental defect characterization of lead-free perovskites—aiming at the quantitative evaluation of defect densities, energies, and capture cross sections—has not been pursued to date, limiting the assessment of their defect tolerance, a property that closely relates to their photovoltaic potential. The impact of dimensionality and crystallographic orientation has not been fully assessed and exploited. Furthermore, standardized device stability data from lead-free perovskite solar cells are generally lack- ing, preventing the rational identification of the absorbers that are most promising from a stability point of view. All of these ele- ments point to the need for a concerted effort toward a more in- depth experimental assessment of the optoelectronic properties and APL Mater. 8, 100901 (2020); doi: 10.1063/5.0022271 8, 100901-9 © Author(s) 2020APL Materials PERSPECTIVE scitation.org/journal/apm stability of lead-free perovskite absorbers. By enabling detailed insight, such an effort would allow the identification of promising structures, compositions, and deposition protocols, ultimately cat- alyzing further progress in photovoltaic performance and stability. While the bulk of the research on lead-free perovskites has nar- rowly focused on compositions and structures suitable for single- junction outdoor photovoltaics, this Perspective brings to the fore the manifold photovoltaic applications to which lead-free per- ovskites are also relevant. These involve tandem photovoltaics, indoor photovoltaics, and building-integrated and transparent pho- tovoltaics. Importantly, these areas feature spectral requirements different from those of single-junction outdoor photovoltaics and may potentially offer a better match with the properties of many classes of lead-free perovskites. Therefore, we believe that for the full photovoltaic potential of lead-free perovskites to be realized, efforts should be directed at photovoltaic applications for which these absorbers truly represent an opportunity. ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Grant Nos. 61950410759 and 61805166), the Jiangsu Province Natural Science Foundation (Grant No. BK20170345), the Collaborative Innovation Center of Suzhou Nano Science & Technology, the Priority Academic Program Develop- ment of Jiangsu Higher Education Institutions (PAPD), the 111 Project, and the Joint International Research Laboratory of Carbon- Based Functional Materials and Devices. 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5.0025455.pdf

J. Chem. Phys. 153, 124704 (2020); https://doi.org/10.1063/5.0025455 153, 124704 © 2020 Author(s).Origin of enhanced boric acid adsorption in light-burned magnesium oxide Cite as: J. Chem. Phys. 153, 124704 (2020); https://doi.org/10.1063/5.0025455 Submitted: 15 August 2020 . Accepted: 08 September 2020 . Published Online: 22 September 2020 Kiminori Sato , and Minori Kamaya ARTICLES YOU MAY BE INTERESTED IN Unsupervised search of low-lying conformers with spectroscopic accuracy: A two-step algorithm rooted into the island model evolutionary algorithm The Journal of Chemical Physics 153, 124110 (2020); https://doi.org/10.1063/5.0018314 Spontaneous polarization of thick solid ammonia films The Journal of Chemical Physics 153, 124707 (2020); https://doi.org/10.1063/5.0017853 Efficient evaluation of exact exchange for periodic systems via concentric atomic density fitting The Journal of Chemical Physics 153, 124116 (2020); https://doi.org/10.1063/5.0016856The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Origin of enhanced boric acid adsorption in light-burned magnesium oxide Cite as: J. Chem. Phys. 153, 124704 (2020); doi: 10.1063/5.0025455 Submitted: 15 August 2020 •Accepted: 8 September 2020 • Published Online: 22 September 2020 Kiminori Sato1,a) and Minori Kamaya2 AFFILIATIONS 1Department of Environmental Sciences, Tokyo Gakugei University, Koganei, Tokyo 184-8501, Japan 2Department of Applied Chemistry, Kogakuin University, Hachioji, Tokyo 192-0015, Japan a)Author to whom correspondence should be addressed: sato-k@u-gakugei.ac.jp ABSTRACT Light-burned magnesium oxide (MgO) possesses a high surface area and has attracted interest as a promising candidate for boron adsorption materials; however, the detailed molecular structures decisive for enhancing the adsorption performance have not yet been elucidated. Here, the origin of enhanced boric acid adsorption for the light-burned MgO is studied by multiple probes, including positronium (Ps) annihilation spectroscopy, Fourier transform infrared spectroscopy, and sorption experiments coupled with molecular simulations. The state-of-the-art technique of open space analysis using Ps revealed the detailed structure of the interfaces between MgO nanograins: ∼10 Å and∼30 Å open spaces, participating in the chemisorption of B(OH) 4−and BO 33−simultaneously with the physisorption of neutral B(OH) 3molecules. Furthermore, in addition to the fraction of open spaces, a proton quasi-layer formed on the interior surfaces of the above-mentioned angstrom-scale open spaces was identified to be attributable for enhancing both the chemisorption and physisorption. Published under license by AIP Publishing. https://doi.org/10.1063/5.0025455 .,s I. INTRODUCTION Boron, an essential chemical element for animals, human beings, plants, and bacteria, is ubiquitously and abundantly avail- able in the form of boric acid B(OH) 3and borate ions such as B(OH) 4−, resulting from rock weathering, seawater volatilization, and volcanic activity.1In addition to the above-mentioned natu- ral sources, numerous boron products are used in industrial and agricultural applications, whereby the use of boron is anthropogeni- cally broadened.2In general, boron compounds are highly soluble in water; the average concentrations in ground water and seawater are 0.3 mg/l–100 mg/l and ∼4.5 mg/l, respectively.3Owing to the widespread distribution of boron compounds in addition to their high-water solubility, boron toxicity risks human health, particu- larly via the oral intake of drinking water, which is a global pub- lic concern.4,5Toxicology studies report that high boron concen- trations in human beings cause the malfunction of cardiovascular, nervous, alimentary, and sexual systems, as summarized in other literature.6 Magnesium oxide (MgO) is an inorganic compound that natu- rally emerges as the mineral periclase in contact metamorphic rocks, which are a major component of refractory bricks.7Due to thecost effectiveness and environmental friendliness of MgO in mate- rial applications as well as production processes, MgO has long been considered as one of the potential sorbents suitable for anionic species such as borate ions.8The adsorption of boric acid on the sur- face of MgO has been discussed exclusively from the viewpoint of chemisorption of B(OH) 4−so far, since adsorption increases with the increasing concentration of B(OH) 4−that is obtained by the addition of OH−to B(OH) 3in an alkaline solution.9,10Accord- ing to the mechanism proposed on the basis of sorption isotherm data, the surface of MgO is hydrated, forming Mg(OH) 2in a gel state upon contact with the aqueous solution. B(OH) 4−predomi- nantly present in a solution of high pH is, thus, immobilized as a consequence of ligand exchange (LE) with OH−at the Mg(OH) 2 surface.9To enhance the adsorption performance in a material with a high surface area, various morphologies such as nanoparti- cles, nanocrystals, nanosheets, nanodisks, and nanofibers have been proposed.11 Light-burned MgO obtained by calcination at relatively low temperatures in the range of 700○C–1000○C12has a specific surface area higher than conventional MgO, low crystallinity with coor- dinatively unsaturated atoms, and is reactive in nature. One can speculate the presence of angstrom-scale open spaces specific to J. Chem. Phys. 153, 124704 (2020); doi: 10.1063/5.0025455 153, 124704-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp boron adsorption in the light-burned MgO, which is of high inter- est with respect to nanosurface chemistry in the interior of open spaces. However, angstrom-scale open spaces have not been well- studied for both the light-burned and conventional MgO because of the difficulty in probing the local atomic structures by diffrac- tion and scattering techniques. Thus, the interior structures of the angstrom-scale open spaces as the above-mentioned gel state partic- ipating in boric acid adsorption have not been clearly understood. In this work, the adsorption of boric acid for the light-burned MgO was comprehensively investigated using multiple probes, includ- ing positronium (Ps) annihilation spectroscopy, Fourier transform infrared (FT-IR) spectroscopy, and sorption experiment coupled with molecular dynamics (MD) calculations. The emergence mech- anism of the physisorption of neutral boric acid molecules in addi- tion to the chemisorption of borate ions on the interior surfaces of angstrom-scale open spaces is highlighted. II. EXPERIMENT A. Materials The powder samples of MgO ( >97% MgO, 0.91% CaO, 0.18% SO 2, 0.04% Fe 2O3, 0.06% Al 2O3, 0.38% B 2O3, and 1.45% SO 3) com- mercially available from Kanto Chemical. Co., Inc., Japan, referred to as conventional MgO hereafter, were used. Light-burned MgO (>97% MgO, 0.87% CaO, 0.17% SO 2, 0.04% Fe 2O3, 0.06% Al 2O3, 0.37% B 2O3, and 1.37% SO 3) was supplied by UBE Materials Indus- tries, Ltd., Japan. The values of the Brunauer–Emmett–Teller (BET) specific surface area evaluated by a nitrogen gas adsorption method are∼10 m2/g and∼170 m2/g for the conventional and light-burned MgO, respectively. NaCl-type cubic MgO single crystals with a lattice constant of a= 0.4213 nm and a mass density of ∼3.59 g/cm3, produced by SHINKOSYA Co., Ltd., Japan, were additionally studied for comparison. B. Sorption experiments A 100-ppm boron stock solution was prepared by dissolving boric acid B(OH) 3purchased from Kanto Chemical Co., Inc., Japan, in double distilled water. The 1 g amounts of conventional and light- burned MgO were added to 30 ml of this boron stock solution, and the value of pH was adjusted to be 10 by adding NaOH solution. The suspension was stirred at 100 rpm at 25○C. The removal rate for boric acid at each stirring time was determined from the analysis of the supernatant liquid after three days of equilibration using the azomethine H method. C. Fourier transform infrared (FT-IR) spectroscopy Attenuated total reflection (ATR) FT-IR spectra were measured using a Nicolet iS5 FT-IR spectrometer (Thermo Fisher Scientific, Inc., USA) equipped with an ATR device and a diamond crystal plate. Conventional and light-burned MgO were subjected to FT- IR spectroscopy after the following treatments: as-received, soaked in water at 25○C for 60 min and then dried (referred to as water treated), and soaked in the above-mentioned B(OH) 3stock solu- tion at 25○C for 60 min and then dried [referred to as B(OH) 3 stock-solution treated]. All FT-IR spectra were measured at room temperature with a resolution of 4 cm−1. The measurements wererepeated 100 times to improve statistical accuracy, and the final spec- tra were obtained by averaging them. OMNIC 8.2 software was used to display absorbance spectra by converting ATR data, where the absorption band at wavenumbers around 1000 cm−1is focused. D. Positronium (Ps) lifetime spectroscopy The sizes of angstrom-scale open spaces and their fractions were investigated by Ps annihilation lifetime spectroscopy. A frac- tion of energetic positrons injected into samples forms a bound state with an electron, Ps.13Singlet para -Ps ( p-Ps) with the spins of the positron and electron antiparallel and triplet ortho -Ps ( o-Ps) with parallel spins are formed at a ratio of 1:3. Hence, three states of positrons, p-Ps,o-Ps, and free positrons, exist in the samples. The annihilation of p-Ps results in the emission of two γ-ray photons of 511 keV with a lifetime of ∼125 ps. Free positrons are trapped by negatively charged parts such as polar elements13or defects14 and annihilated into two photons with a lifetime of ∼450 ps. The positron in o-Ps undergoes two-photon annihilation with one of the bound electrons with a lifetime of a few nanoseconds after local- ization in angstrom-scale open spaces. The last process is known aso-Ps pick-off annihilation and provides information on an open space size through the annihilation rate of o-Psλo-Ps(=1/τo-Ps) based on quantum mechanical models. The present MgO samples con- tain open spaces larger than ∼1 nm (see below). Thus, we employed the extended Tao–Eldrup model15–17including excited states in the calculation with a rectangular well characterized by sides a,b,cat temperature Tas λo−Ps=λA−λS−λT 4F(a,δ,T)F(b,δ,T)F(c,δ,T), (1) where F(x,δ,T)=1−2δ x+∑∞ i=11 iπsin(2iπδ x)exp(−βi2 x2kT) ∑∞ i=1exp(−βi2 x2kT), (2) with the parameter δ= 0.18 nm. Here, λS,λT, andλAare the sin- glet, triplet, and their averaged vacuum annihilation rates, respec- tively, and β=h2/16m= 0.188 eV nm2with the electron mass m. In the present work, the open space size was calculated assuming three-dimensional (3D) cubic shape, thus yielding cube side-length a. Conventional and light-burned MgO were subjected to Ps life- time spectroscopy after the following treatments: as-received, water- treated, B(OH) 3stock-solution-treated, and exposed to air for 6 months. A 10 μCi positron source (22Na), sealed in a thin foil of Kap- ton, was mounted in a sample–source–sample sandwich. Ps lifetime spectra were measured under vacuum conditions of ∼10−2Pa at the temperature of 300 K ±2. The measured spectra were numerically analyzed using the POSITRONFIT code.18 E. Positron-age-momentum correlation (AMOC) spectroscopy The interior surface of angstrom-scale open spaces was investi- gated by means of momentum distributions of o-Ps pick-off annihi- lation localized in open spaces. This momentum spectroscopy focus- ing on o-Ps pick-off annihilation photons is based on the principle that if the positron–electron annihilation accompanies a longitu- dinal momentum p, the resulting annihilation γrays are Doppler J. Chem. Phys. 153, 124704 (2020); doi: 10.1063/5.0025455 153, 124704-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp shifted from m0c2by±cp/2.19Here, m0and care the electron rest mass and velocity of light, respectively. Measurements of the Doppler shifts by γ-ray energy spectroscopy with a high-purity Ge detector make it possible to obtain information on the momen- tum distribution of the positron–electron annihilation pairs. The momentum of p-Ps annihilation has a narrow distribution because p-Ps self-annihilates after losing the initial energy. In contrast, the momentum distribution of free positrons is significantly broadened due to the large momenta of electrons participating in the annihi- lation. In addition, the momentum distribution of o-Ps is broad- ened because the pick-off annihilation is influenced by the electrons bound to the surrounding molecules.20,21Thus, the momentum distribution of o-Ps pick-off annihilation provides information on the elemental environment on the interior surfaces of open spaces. However, p-Ps, free positrons, and o-Ps all contribute to the over- all momentum distribution of the annihilation pairs in Ps-forming materials such as the present MgO. In this work, the momentum distribution of o-Ps pick-off annihilation was investigated by the time-resolved momentum measurements of positron–electron anni- hilation photons by means of positron-age-momentum correlation (AMOC) spectroscopy.22In the AMOC spectroscopy, the momen- tum distribution attributable to o-Ps pick-off annihilation can be extracted in the older positron age region above ∼2 ns, where p-Ps and positrons disappear. AMOC spectroscopy was conducted for the conventional and light-burned MgO after water treatment with a 10 μCi positron source (22Na) under a vacuum condition of ∼10−2Pa at the temperature of 300 K ±2. Taking the ratio of the central area over ( −3.6 to +3.6) ×10−3m0cto the total area of the momentum spectrum, o-Ps pick-off annihilation was parameterized. Hereafter, we call this parameter interior-surface parameter that was found to be sensitive to light elements on the surfaces of open spaces such as oxygen.23–25 F. Molecular simulations The molecular structure of boric acid B(OH) 3was initially opti- mized by the molecular orbital (MO) calculation using the Zerner- modified semiempirical INDO (Intermediate Neglect of Differen- tial Overlap) method known as ZIND in a package of SCIGRESS 2.8 (Fujitsu Ltd. Japan).26The physisorption behavior of boric acid optimized above in the vicinity of surface of Mg(OH) 2was then examined by molecular dynamics (MD) simulations. The LJDreid- ing model was employed as a potential function between boric acid and the Mg(OH) 2surface.27In this model, the potential energy Efor an arbitrary geometry of a molecule is expressed as E=EB+EA+ED+Ec+Ev. (3) Here, EB,ED, and ETare the potential parameters for bond stretch- ing, bond angle, and dihedral angle torsion, respectively. On the one hand, EcandEvare electrostatic energy with a Coulombic potential and van der Waals energy, respectively. The electrostatic energy with a Coulombic potential Ec=1 4πεqiqj rij(4) represents interactions between static atomic charges qiandqjat dis- tance rij, andεis the dielectric constant. The van der Waals energyis calculated with a standard 12-6 Lennard-Jones potential Ev(r)=4ε[(σ r)12 −(σ r)6 ], (5) modeling nonbounded interactions between pairs of atoms at dis- tance rtogether with the Lennard-Jones well depth ϵand distance at the Lennard-Jones minimum σ, respectively. The parameters ϵand σincluded in the MD software package were used. MD calculations were conducted at 300 K and 1 atm. III. RESULTS AND DISCUSSION Figure 1 shows the removal rates of boric acid for the con- ventional and light-burned MgO as a function of stirring time. The conventional and light-burned MgO exhibited removal rates of∼8 and∼40% at a stirring time of 0, respectively. The removal rates for the conventional MgO monotonously and slowly increase with the increasing stirring time, whereas the light-burned MgO yields a rapid increase of up to 90% at the stirring time of 20 min and remained constant thereafter. The results demonstrate that the uptake amount of B(OH) 3in aqueous solution for the light-burned MgO is significantly higher than that for the conventional one. This may be reasonable because the BET specific surface area for the light-burned MgO is one order of magnitude higher than that for the conventional one implying the presence of surface states responsible for boron adsorption. The rapid increase in the removal rates with stirring time to 20 min for the light-burned MgO seems to be physisorption behavior, which will be discussed in detail later. Figure 2 shows the results of FT-IR spectroscopy for (a) as-received conventional MgO, water treated, and B(OH) 3stock- solution treated, and (b) as-received light-burned MgO, water treated, and B(OH) 3stock-solution treated in which the wavenum- ber region from 950 cm−1to 3750 cm−1is focused. A broad absorp- tion band together with tiny signals can be observed at wavenumbers FIG. 1 . Removal rates of boric acid for the conventional (gray circles) and light- burned MgO (red squares) as a function of stirring time. J. Chem. Phys. 153, 124704 (2020); doi: 10.1063/5.0025455 153, 124704-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . FT-IR spectra for (a) as-received conventional MgO, water treated, and B(OH) 3stock-solution treated, and (b) as-received light-burned MgO, water treated, and B(OH) 3stock-solution treated. Adsorption peaks arising from carbon- ate species, hydroxyl groups, and boron compounds/ions are indicated by gray, blue, and red arrows, respectively. The schematic illustrations of B(OH) 4−, BO 33−, and B(OH) 3species, and protons attached on the surface of Mg(OH) 2, are shown in the insets. White, red, and green atoms correspond to proton, oxygen, and magnesium, respectively. of∼1120 cm−1for the conventional MgO, which could be attributed to the carbonate/carboxylate species originally present in the com- pound (see the gray arrow).28Intense and broad peaks appear around∼3700 cm−1and∼1420 cm−1in addition to the ∼1120 cm−1 peak for the conventional MgO soaked in water for 60 min. The appearance of an intense peak around ∼3700 cm−1is caused by the lattice vibration of OH due to the formation of Mg(OH) 2upon addi- tion of water, typically observed for hydrated MgO.29The broad peak around ∼1420 cm−1could be attributed to both the symmet- ric and asymmetric O–C–O stretching vibrations in the unidentate carbonate species, which is formed by the reaction of the Mg(OH) 2 surface with CO 2dissolved in solution.30Furthermore, the weak adsorption band appears around ∼3650 cm−1, which corresponds to the isolated hydroxyls resulting from the protonation of surface oxide anions O 2−with a low coordination number of 3 or 4, asillustrated in the inset of Fig. 2.31For the B(OH) 3stock-solution- treated MgO, additional small peaks resulting from the symmet- ric stretching vibration in bidentate carbonate and the asymmet- ric stretching vibration in polydentate carbonate32appear around ∼1300 cm−1and∼1450 cm−1due to the carbonation enhancement in alkaline solution (see gray arrows). A notable feature is the very small signal around ∼1025 cm−1, as indicated by the red arrow, which could be caused by the asymmetric stretching vibration of B(OH) 4− demonstrating the occurrence of chemisorption.33 The adsorption bands arising from the unidentate carbon- ate species already appear together with those of the carbon- ate/carboxylate species for the light-burned MgO (see gray arrows). The water-treated sample exhibits the peaks of carbonate species earlier than the conventional MgO (see gray arrows), implying that the light-burned MgO is more effectively hydrated than the conven- tional one and reacts with CO 2dissolved in solution. Furthermore, the adsorption peak of isolated hydroxyl around ∼3650 cm−1is more prominent for the light-burned MgO than for the conventional one, implying that a number of isolated hydroxyls are built on the Mg(OH) 2surface of the light-burned MgO (see blue arrows). Signif- icant adsorption bands emerge around ∼1410 cm−1and∼1260 cm−1 for the B(OH) 3stock-solution treated light-burned MgO, as indi- cated by red arrows. In addition, the signal of B(OH) 4−chemisorp- tion around ∼1000 cm−1grows along with the appearance of addi- tional peaks in this region, again caused by B(OH) 4−chemisorption. FT-IR studies at varying pH indicated that the absorption bands around∼1410 cm−1and 1260 cm−1originate from asymmetric B–O stretching vibrations in trigonal B(OH) 3and BO 33−relevant to the outer- and inner-sphere reactions, respectively.32It is reasonably inferred that the peaks observed at ∼1410 cm−1and 1260 cm−1result from the physisorption of neutral B(OH) 3and chemisorption of BO 33−, respectively. It is of interest that the adsorption signals of boric acid and borate ions are effectively observed for the B(OH) 3 stock-solution treated light-burned MgO. This implies that the light- burned MgO has a higher capacity for boron adsorption than the conventional one, in agreement with the results of the sorption experiments (see Fig. 1). Figure 3(a) shows the Ps lifetime spectra observed for the as- received conventional MgO, B(OH) 3stock-solution treated, water treated, and exposed to air for 6 months. The spectrum of MgO single crystals is presented as well for comparison. The Ps lifetime spectrum observed for the MgO single crystals decays more rapidly than those for the other samples, indicating the void structure in the burned magnesia samples. The Ps lifetime spectrum for the B(OH) 3 stock-solution treated MgO decays slower than that for the conven- tional MgO. The decay in the spectrum is, in turn, slightly accel- erated for the water-treated sample. The lifetime spectrum for the sample exposed to air for 6 months decays slightly more rapidly than that for the water-treated sample. Figure 3(b) shows the Ps life- time spectra for the as-received light-burned MgO, B(OH) 3stock- solution treated, water treated, and exposed to air for 6 months. In accordance with the BET specific surface area data, the Ps lifetime spectrum for the light-burned MgO exhibits a slow decay, indicat- ing a more porous structure than that of the conventional MgO. In contrast to the conventional MgO sample, the decaying tendency for the light-burned MgO sample soaked in the B(OH) 3stock solu- tion becomes more rapid. The water-treated sample without B(OH) 3 is identical to that of the B(OH) 3stock solution-treated sample. J. Chem. Phys. 153, 124704 (2020); doi: 10.1063/5.0025455 153, 124704-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Peak normalized Ps lifetime spectra observed for (a) as-received conven- tional MgO (black), B(OH) 3stock-solution treated (red), water treated (blue), and exposed to air for 6 months (green) together with that of the MgO single crys- tal (gray); and (b) as-received light-burned MgO (black), B(OH) 3stock-solution treated (red), water treated (blue), and exposed to air for 6 months (green). The Ps lifetime spectrum for the sample exposed to air for 6 months decays slightly more rapidly than that for the conventional MgO sample. The analyses of Ps lifetime spectra shown in Fig. 3 revealed the presence of two components of o-Ps lifetimes τ3and τ4regard- less of the sample treatment, signifying the presence of two kinds of open spaces. The o-Ps lifetimes τ3and τ4are∼3 ns and∼40 ns for the conventional and light-burned MgO, respectively, whereas the single component of positron lifetime ∼180 ps corresponding to the crystalline matrix is evaluated for the MgO single crystal. The appearance of o-Ps lifetimes demonstrates that the electron densi- ties are locally reduced in the conventional as well as light-burned MgO. The lowering of electron densities is in agreement with a mass density of ∼3.59 g/cm3for the MgO single crystal, which is higher than those of the present burned magnesia samples ( ∼0.4 g/cm3). It is expected that a high concentration of interfaces is introduced for the conventional and light-burned MgO, as often observed for oxide materials.34,35 Table I lists the sizes of open spaces aand their correspond- ing relative intensities Ievaluated for the conventional MgO (a) asTABLE I . Sizes of open spaces aand their corresponding relative intensities Ievalu- ated for the conventional MgO (a) as received, (b) B(OH) 3stock-solution treated, (c) water treated, and (d) exposed to air for 6 months, together with those of the light- burned MgO (e) as received, (f) B(OH) 3stock-solution treated, (g) water treated, and (h) exposed to air for 6 months. The error bars of a1,a2,I1, and I2are±0.12 Å, ±0.44 Å, 1%, and 1%, respectively. Treatment a1(Å) I1(%) a2(Å) I2(%) Conventional MgO (a) As received 10.3 1 32.6 3 (b) B(OH) 3stock-solution treated 8.5 5 29.2 2 (c) Water treated 9.2 4 25.5 1 (d) Exposure to air for 6 months 9.1 2 24.7 1 Light-burned MgO (e) As received 9.1 5 32.7 15 (f) B(OH) 3stock-solution treated 8.9 6 27.6 4 (g) Water treated 8.9 6 27.8 4 (h) Exposure to air for 6 months 8.3 6 25.4 1 received, (b) B(OH) 3stock-solution treated, (c) water treated, and (d) exposed to air for 6 months, together with those of the light- burned MgO (e) as received, (f) B(OH) 3stock-solution treated, (g) water treated, and (h) exposed to air for 6 months. The sizes of the small and large open spaces for the as-received conventional MgO are a1∼10.3 and a2∼32.6 Å, respectively, which are simi- lar to those of the as-received light-burned MgO with a1∼9.1 Å anda2∼32.7 Å, respectively. The relative intensities of small and large open spaces are I1∼1% and I2∼3% for the as-received con- ventional MgO, respectively, whereas the as-received light-burned MgO exhibits high values of I1∼5% and I2∼15%, for small and large, respectively, in agreement with the BET specific surface area. In light of the fact that the positron lifetimes in the crystalline matrix of the present MgO single crystal and Mg vacancy are ∼180 ps and ∼200 ps,36respectively, it is not probable that Ps formation occurs inside the MgO nanoparticles. Indeed, Ps lifetime spectroscopy con- ducted for MgO nanocrystals with a few tens of nanometers exhib- ited positron lifetimes at the grain boundary region, which is longer than 50 ns.37The small and large open spaces evaluated here could be thus located at interfaces between MgO nanograins, which is often observed for materials synthesized by burning and sinter- ing34,35as well as nanocrystalline materials.38It is noted that such open spaces cannot be obtained for the MgO single crystal, where the positron exclusively annihilates in the matrix of the MgO crystal (see above). It is advantageous that the local structure of the surface area examined by the gas adsorption method is further detailed by the present Ps lifetime spectroscopy. The sizes of small and large open spaces a1and a2for the B(OH) 3stock-solution treated MgO are ∼8.5 Å and∼29.2 Å, respec- tively, which are smaller than those for the as-received one, signify- ing the shrinkage of open spaces. The relative intensity I1increases up to∼5%, whereas the intensity I2decreases to ∼2% implying that the large open spaces are preferentially reduced, consequently increasing the apparent intensities of small open spaces. A similar J. Chem. Phys. 153, 124704 (2020); doi: 10.1063/5.0025455 153, 124704-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp tendency was obtained for the water-treated one, indicating that water-treatment has an influence solely on the open spaces. The above variations of open spaces upon water and B(OH) 3stock solu- tion treatments provide evidence that water molecules enter the angstrom-scale open spaces. The water molecules could allow hydra- tion on the interior surfaces of open spaces, forming Mg(OH) 2 and locally shrinking open space, as can be seen in the decrease in open space sizes for the water and B(OH) 3stock solution-treated samples. All parameters, I1,I2,a1, and a2decrease for the conven- tional MgO exposed to air for 6 months, indicating the shrinkages of two open spaces along with the decrease in their fractions. This demonstrates that water molecules in air enter the angstrom-scale open spaces, similarly to the case of solution treatment. Diffusion of water molecules into the angstrom-scale open spaces is also antic- ipated from the fact that the diffusion constant of water molecules through open spaces with a size of ∼7.4 Å in amorphous silica is ∼10−6cm2/s.24 The sizes of small and large open spaces a1anda2decrease for the B(OH) 3stock-solution treated light-burned MgO together with the increase in I1and decrease in I2due to the diffusion of solution molecules into open spaces and successive reactions on their inte- rior surfaces. The open space sizes and relative intensities for the water-treated sample are essentially identical to those of the B(OH) 3 stock solution-treated sample, indicating that little structural change is involved after the introduction of B(OH) 3. The sizes of small and large open spaces and the corresponding intensities decrease for the sample exposed to air for 6 months owing to the diffusion of water molecules in air into the angstrom-scale open spaces. It is noted that the variations of angstrom-scale open spaces caused by the diffusion and reaction of water and solution molecules become prominent for the light-burned MgO samples. Figure 4 shows the results of AMOC spectroscopy for the conventional and light-burned MgO after water treatment, where the interior-surface parameters in the positron-age region domi- nantly contributed by o-Ps pick-off annihilation are presented. The interior-surface parameters for the light-burned MgO are consis- tently higher than those for the conventional one in the whole positron-age region. The high parameters of the light-burned MgO indicate significant narrowing of the momentum distribution of o-Ps pick-off annihilation photons, demonstrating the distinct ele- mental environment on the interior surfaces of open spaces. As described in Sec. II, the element-specific interior-surface param- eter can sensitively differentiate light atoms such as oxygen that constitute the surface of open spaces in polymers.14,19,20,22Oxygen located on the surface of open spaces for the burned magnesia is a typical atom yielding Doppler broadening due to o-Ps pick- off annihilation with orbital electrons, leading to a decrease in the interior-surface parameter.39The lower interior-surface parameter for the conventional MgO is thus resulting from a greater fraction of oxygen atoms on the interior surfaces of open spaces, in other words, a smaller fraction of oxygen on the interior surfaces for the light-burned MgO. In principle, the protons tend to attach on the surface of MgO in an aqueous solution with a pH lower than the point of zero charge (pzc) of 12.440like the present case. Furthermore, the broken bonds and/or coordinatively unsaturated atoms are abundantly present on the surface of the light-burned MgO.41Naturally, such nano-surface sites associated with local structural imperfections exist on the FIG. 4 . Interior-surface parameters in the positron age region dominated by o-Ps pick-off annihilation for the conventional (black circles) and light-burned MgO (red squares) after water treatment. The schematic illustrations of protons attached and forming a proton quasi-layer on the surface of Mg(OH) 2are shown in the insets. White, red, and green atoms correspond to proton, oxygen, and mag- nesium, respectively. The proton in the Mg(OH) 2lattice is depicted in gray for clarification. interior surfaces of open spaces accepting protons. Increased pro- tons on the surface oxygen of Mg(OH) 2for the light-burned MgO are anticipated from the results of chemical bond studies by FT-IR spectroscopy (see Fig. 2). Of more importance is that AMOC spec- troscopy probes the two-dimensional spread of protons covering oxygen on the interior surfaces of open spaces for the light-burned MgO. This evidences a well-expanded protonic state on the interior surfaces, which is thereafter called the proton quasi-layer (see the schematic illustration in Fig. 4). FT-IR spectroscopy successfully revealed the significant chemisorption of B(OH) 4−and BO 33−together with the physisorp- tion of neutral B(OH) 3molecules for the light-burned MgO, while a small amount of B(OH) 4−chemisorption for the conventional MgO. The chemisorption of B(OH) 4−could be caused by the forma- tion of an inner-sphere surface complex involving LE with OH−in Mg(OH) 2followed by the surface hydration of MgO, which has been discussed as the immobilization process9[see schematic illustrations (a) and (b) in Fig. 5]. On the contrary, BO 33−chemisorption could be due to the formation of the inner-sphere surface complex without LE on the Mg(OH) 2surface [see illustration (c) in Fig. 5]. We can easily imagine that the high adsorption performance of the light-burned MgO arises from a high fraction of angstrom-scale open spaces (see Table I). It is, in addition to the fraction open spaces, noted that the positively charged proton quasi-layer on the interior surfaces of open spaces attract anionic borate species as B(OH) 4−and BO 33− with more frequent participation in the formation of inner-sphere surface complexes. The question as to why neutral B(OH) 3is physisorbed on the interior surfaces of angstrom-scale open spaces for the light-burned MgO is addressed by means of molecular simulations. Here, B(OH) 3 was moved in the direction of the Mg(OH) 2surface, where the total J. Chem. Phys. 153, 124704 (2020); doi: 10.1063/5.0025455 153, 124704-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . Schematic illustrations of chemisorption of B(OH) 4−[(a) and (b)] and BO 33−(c) on the interior surface of angstrom-scale open spaces. White, red, and green atoms correspond to proton, oxygen, and magnesium, respectively. The proton in the Mg(OH) 2lattice is depicted in gray for clarification. potential energy at each distance was calculated by MD simula- tion. The water confined in the angstrom-scale open spaces could exhibit a lower dielectric constant than bulk water of ∼80, since the surface-induced alignment of molecular dipoles makes it dif- ficult to reorient by applying an electric field. Local capacitance FIG. 6 . MD calculated total potential energies as a function of distance upon approaching oxygen in B(OH) 3toward the Mg(OH) 2surface with (red squares) and without (black circles) protons. Schematic illustrations of B(OH) 3in the vicinity of the MgO surface are presented in the insets. White, red, and green atoms corre- spond to proton, oxygen, and magnesium, respectively. The proton in the Mg(OH) 2 lattice is depicted in gray for clarification.measurements revealed that the dielectric constant of water confined between two atomically flat walls separated by ∼10 Å decreases down to∼2.42The optimized structure of B(OH) 3using MO calculations was conducted taking the medium with a dielectric constant of 2 into consideration. The MO calculations indicated that the partial charge of oxygen was −0.4176 e, but the molecular structure of B(OH) 3did not change much in the water system strongly confined in angstrom- scale open spaces; H–O–B bond angles are deviated by ∼0.02% in maximum from perfect symmetry. As revealed by the analyses of the elemental environment by AMOC spectroscopy, the proton quasi-layer is formed on the inte- rior surfaces of open spaces for the light-burned MgO. A proton with a charge of + eis thus tentatively given to oxygen on the sur- face of Mg(OH) 2in the MD calculations. Figure 6 shows the results of MD calculations upon approaching oxygen in B(OH) 3toward the Mg(OH) 2surface with and without a proton in which the total potential energies are plotted as a function of distance. The distance- dependent total potential energies reasonably show Lennard-Jones type curves, where the potential well around 0.3 nm is pronounced for the surface with proton. This implies that B(OH) 3physisorption observed for the light-burned MgO is caused by the proton quasi- layer on the interior surfaces of open spaces attracting negatively charged oxygen in B(OH) 3. The B(OH) 3physisorption preferen- tially could occur for the light-burned MgO at the beginning of the stirring time, thus exhibiting a rapid increase in the sorption curve in contrast to that of the conventional MgO (see Fig. 1). IV. CONCLUSIONS We succeeded in shedding light on the adsorption mecha- nism of boron compounds and/or borate ions upon contact with J. Chem. Phys. 153, 124704 (2020); doi: 10.1063/5.0025455 153, 124704-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp aqueous solution for the light-burned MgO and the conventional MgO. The adsorption performance of boric acid for the light-burned MgO examined by sorption experiments was found to be higher than that of MgO, which is in agreement with the results of the chemical bond study by FT-IR spectroscopy. Open space analysis using Ps coupled with the chemical bond study revealed the detailed structure of the interfaces between MgO nanograins participating in the chemisorption of B(OH) 4−and BO 33−along with physisorp- tion of neutral B(OH) 3molecules for two types of MgO: ∼10 Å and ∼30 Å open spaces. The fraction of angstrom-scale open spaces for the light-burned MgO is higher than that of the conventional one, contributing to the above born adsorption. Furthermore, the pro- ton quasi-layer is formed on the interior surfaces of angstrom-scale open spaces for the light-burned MgO because structural imper- fections such as broken bonds and/or coordinatively unsaturated atoms accept protons. The proton quasi-layer attracts anionic borate species as B(OH) 4−and BO 33−with more frequent participation in chemisorption via the formation of an inner-sphere surface complex with and without LE. Simultaneously, the proton quasi-layer on the interior surfaces of open spaces attracts negatively charged oxygen in B(OH) 3, bringing about the physisorption of neutral B(OH) 3. The present findings imply that the nanosurface state in the interior of angstrom-scale open spaces is taken into consideration along with their fraction for the future design of adsorption materials based on MgO. ACKNOWLEDGMENTS Fruitful discussions with Y. Kobayashi (Waseda University), K. Ito (National Institute of Advanced Industrial Science and Technol- ogy), and S. Tanaka (UBE Materials Industries, Ltd.) are gratefully appreciated. This work was partially supported by the Grants-in-Aid of the Ministry of Education, Culture, Sports, Science and Tech- nology of Japan (Grant Nos. 16K05394, 18K04884, 18KK0382, and 20K14372). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1J. L. Parks and M. Edwards, Crit. 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Y. Li, X. F. Li, C. Y. Cao, and Z. Tang, Physica B 407, 2665 (2012). 38K. Sato, H. Murakami, W. Sprengel, H.-E. Schaefer, and Y. Kobayashi, Appl. Phys. Lett. 94, 171904 (2009). 39K. Sato and T. Hatta, J. Chem. Phys. 142, 094307 (2015). 40G. A. Parks, Chem. Rev. 65, 177 (1965). 41A. Zecchina, S. Coluccia, G. Spoto, D. Scarano, and L. Marchese, J. Chem. Soc., Faraday Trans. 86, 703 (1990). 42L. Fumagalli, A. Esfandiar, R. Fabregas, S. Hu, P. Ares, A. Janardanan, Q. Yang, B. Radha, T. Taniguchi, K. Watanabe, G. Gomila, K. S. Novoselov, and A. K. Geim, Science 360, 1339 (2018). J. Chem. Phys. 153, 124704 (2020); doi: 10.1063/5.0025455 153, 124704-8 Published under license by AIP Publishing

5.0024265.pdf

Appl. Phys. Lett. 117, 112403 (2020); https://doi.org/10.1063/5.0024265 117, 112403 © 2020 Author(s).Formation of zero-field skyrmion arrays in asymmetric superlattices Cite as: Appl. Phys. Lett. 117, 112403 (2020); https://doi.org/10.1063/5.0024265 Submitted: 05 August 2020 . Accepted: 08 September 2020 . Published Online: 18 September 2020 Maxwell Li , Anish Rai , Ashok Pokhrel , Arjun Sapkota , Claudia Mewes , Tim Mewes , Marc De Graef , and Vincent Sokalski ARTICLES YOU MAY BE INTERESTED IN Spin–orbit torque-induced multiple magnetization switching behaviors in synthetic antiferromagnets Applied Physics Letters 117, 112401 (2020); https://doi.org/10.1063/5.0020925 Strong interface-induced spin-charge conversion in YIG/Cr heterostructures Applied Physics Letters 117, 112402 (2020); https://doi.org/10.1063/5.0017745 Reversible thermally controlled spontaneous magnetization switching in perovskite-type manganite Applied Physics Letters 117, 112404 (2020); https://doi.org/10.1063/5.0017506Formation of zero-field skyrmion arrays in asymmetric superlattices Cite as: Appl. Phys. Lett. 117, 112403 (2020); doi: 10.1063/5.0024265 Submitted: 5 August 2020 .Accepted: 8 September 2020 . Published Online: 18 September 2020 Maxwell Li,1,a) Anish Rai,2 Ashok Pokhrel,2 Arjun Sapkota,2 Claudia Mewes,2 Tim Mewes,2 Marc De Graef,1 and Vincent Sokalski1 AFFILIATIONS 1Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA 2Department of Physics and Astronomy/MINT Center, The University of Alabama, Tuscaloosa, Alabama 35487, USA a)Author to whom correspondence should be addressed: mpli@andrew.cmu.edu ABSTRACT We demonstrate the formation of metastable N /C19eel-type skyrmion arrays in Pt/Co/Ni/Ir multi-layers at zero-field following the ex situ application of an in-plane magnetic field using Lorentz transmission electron microscopy. The resultant skyrmion texture is found to depend on both the strength and misorientation of the applied field as well as the interfacial Dzyaloshinskii–Moriya interaction. To demonstrate theimportance of the applied field angle, we leverage bend contours in the specimens, which coincide with transition regions between skyrmionand labyrinth patterns. The subsequent application of a perpendicular magnetic field near these regions reveals the unusual situation whereskyrmions with opposite magnetic polarities are stabilized in close proximity. Published under license by AIP Publishing. https://doi.org/10.1063/5.0024265 Skyrmions are topologically protected, particle-like objects that were first proposed by Skyrme 1and later theoretically predicted to be stabilized in chiral magnets.2–4These magnetic skyrmions have gar- nered a great deal of attention since their theoretical prediction and subsequent experimental discovery.5–12Properties such as topological protection, small size, and high efficiency by which they can be manip-ulated with electric current make them promising candidates for usein future spintronic devices. Skyrmions with deterministic chirality arestabilized by the Dzyaloshinskii–Moriya interaction (DMI), which can be found in bulk magnetic materials lacking crystallographic inversion symmetry or in heavy metal/ferromagnet interfaces with large spi-n–orbit coupling where z-mirror symmetry is also broken. 13–15 Generally, skyrmions found in bulk materials have a Bloch-type configuration, whereas those found in magnetic multi-layers have a N/C19eel-type configuration consistent with C 1vsymmetry.16In thin films, the application of a perpendicular field is generally required tostabilize skyrmions, which evolve reversibly from a labyrinth domainpattern as the field is increased. 17 In addition to understanding properties supporting their forma- tion and controlling their size/mobility, methods for increasing theskyrmion density have also been investigated. 6,18Increased skyrmion density in thin films has been attributed to maximizing a criticalparameter, j¼pD=4ffiffiffiffiffiffiffiffiffiffi ffiAK effp,w h e r e Dis the DMI strength, which normalizes the reduction in the domain wall (DW) energy due to theDMI by the standard Bloch wall energy. Here, Ais the exchange stiff- ness and Keffis the effective magnetic anisotropy.3Previous works have demonstrated that temperature and in-plane fields assist in the formation of skyrmions.18–21However, in most cases, a perpendicular magnetic field was still required to prevent them from relaxing backinto a labyrinth structure. 18–20Field-free skyrmions have also been achieved through the application of voltage pulses, electric current pulses, or complex sample architecture.22–26 Historical work on uniaxial bubble materials considered the role of in-plane magnetic fields in breaking up labyrinth domain patterns,which we leverage to address open questions about the possible role of the DMI in this process and its impact on the skyrmion size and stability. 27–29Here, we study the formation of metastable skyrmion arrays at remanence after the application of an ex situ in-plane mag- netic field. To vary important material parameters such as thickness, magnetic anisotropy, and DMI strength, we use a tunable asymmetric superlattice based on [Pt/(Co/Ni) M/Ir] N,w h e r e MandNcan be inde- pendently varied.30The magnetic texture of these multi-layers is examined over a range of in-plane magnetic field magnitudes. Observations are compared to recent results on metastable skyrmions as well as those observed in magnetic bubble materials. Multi-layers of [Pt(0.5 nm)/(Co(0.2 nm/Ni(0.6 nm) M/Ir(0.5 nm)] N were deposited onto 10 nm thick amorphous Si 3N4membranes via magnetron sputtering in an Ar environment with a working pressure Appl. Phys. Lett. 117, 112403 (2020); doi: 10.1063/5.0024265 117, 112403-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplof 2.5 mTorr and a base pressure of <3:0/C210/C07Torr. An adhesion/ seed layer of Ta(3 nm)/Pt(3 nm) and a Ta(3 nm) cap are present in all films. Alternating gradient field magnetometry (AGFM) and vibrating sample magnetometry (VSM) were used to measure magnetic proper- ties (saturation magnetization, MS, and effective magnetic anisotropy Keff). M-H loops confirm perpendicular magnetic anisotropy (PMA) in these multi-layers (Fig. S1 in the supplementary material ). Although we find a monotonic increase in Keffwith increased M, the observed HK values from these films remain very similar. Broadband ferromagnetic resonance (FMR) spectroscopy was used to probe dynamic behavior and provides a more accurate measurement of the uniaxial magnetocrystalline anisotropy, Ku, which we find to be similar to the values calculated from M–H loops; these values as well as those calculated for damping constant, a, are tabled in the supplementary material . Fresnel-mode LTEM images of [Pt/(Co/Ni) M/Ir] Nmulti- layers were captured with an aberration-corrected FEI Titan G2 80–300 operated in Lorentz mode at room temperature.31Ad e f o c u sv a l u eo f /C02.0 mm was used to capture the out-of-focus images shown here. In the remnant state (after perpendicular saturation) for all sam- ples investigated, magnetic contrast depicting a labyrinth domain structure is only observed with the application of sample tilt in situ (Fig. 1 ). This indicates the presence of N /C19eel walls and, thus, an appre- ciable interfacial DMI to stabilize them, which is expected based on direct measurements of the DMI in this system.32We now examine conventional skyrmion formation through the application of a perpen- dicular magnetic field in situ by exciting the objective lens of the TEM. We note that in the presence of a sample tilt, an effective in-plane fieldis generated. However, the magnitude of this field is not expected to have a significant effect of the resultant domain structure as it is much smaller than HK. In a majority of these samples, instead of forming skyrmions, long worm-like domains form before annihilating at suffi-ciently large fields [ Figs. 2(d)–2(f) ]; it is only in samples with large N, such as M¼2;N¼20, that skyrmions are observed to form in such am a n n e r[ Figs. 2(a)–2(c) ]. This process is largely reversible as the lab- yrinth state is recovered after the field is removed. Next, in-plane magnetic fields are applied to these samples ex situ before their remnant states are imaged with LTEM. Each sample wasperpendicularly saturated and returned to remanence before in-plane fields were applied. Fresnel mode images reveal /C24100–200 nm diame- ter skyrmion arrays following the application of a 1.3 T field ( Fig. 3 ). These arrays were also observed in samples that did not form sky-rmions through the prior application of a perpendicular magnetic field. For the case of a 0.5 T field, which is well below the in-plane satu- ration field, Fresnel mode images revealed mixed stripe domains withsome skyrmions, especially, in the higher DMI (M ¼2) case ( Fig. 3 ). The domain widths of these stripe domains are observed to be smaller than those of labyrinth domains in the demagnetized state ( Fig. 1 )a n d are comparable to those of field-free skyrmions. The observed sky- rmion size from these micrographs was not clearly correlated with H applied beyond a critical value. However, the skyrmion size was observed to decrease with decreasing M(i.e., increasing DMI), which was not consistent with analytically calculated relationships between DMI strength and individual skyrmion size.33However, because the DMI decreases DW energy it can be reasoned that it suppresses the FIG. 1. Fresnel mode LTEM images of the as-prepared [Pt/(Co/Ni) M/Ir]Nwhere (a) and (d) M¼2;N¼20, (b) and (e) M¼3;N¼10, and (c) and (f) M¼5;N¼2; magnetic contrast is only observed with the presence of the sample tilt. The sample was tilted by 15/C14in (d)–(f). FIG. 2. Fresnel mode LTEM images of [Pt/(Co/Ni) M/Ir]Nwhere (a)–(c) M¼2; N¼20 and (d)–(f) M¼2;N¼10 in the presence of increasing perpendicular magnetic field applied in situ . Isolated N /C19eel skyrmions are highlighted with arrows in (c). A sample tilt of 15/C14is present in each image.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 112403 (2020); doi: 10.1063/5.0024265 117, 112403-2 Published under license by AIP Publishingcoarsening of individual skyrmions. In low DMI samples, the system would tend towards coarsening in order to reduce the DW volumelending toward the observed larger sizes/smaller densities. This is inturn consistent with the notion that increasing jincreases the resul- tant skyrmion density. 18Plots depicting observed skyrmion density and size with respect to Mcan be found in the supplementary material . Further decreasing the applied field strengths reveals laby- rinth domain structures at remnance. Our observations were com-pared with those from a symmetric [Co/Ni] 10.5sample where the DMI is expected to be negligible as confirmed by the observation of BlochDWs [Fig. S7(a)]. 30Following similar in-plane field treatment to that of the asymmetric samples, Fresnel mode LTEM images reveal the for-mation of field-free bubble arrays albeit with large numbers of verticalBloch lines along their circumferences [Fig. S7(b)]; these bubbles havediameters that are notably larger than those observed in asymmetricsamples with comparable thicknesses. This highlights the importanceof the DMI in reducing the skyrmion size and controlling chirality inthese field-free arrays. It has previously been pointed out that there is a reduced contri- bution of K effto DW energy upon application of in-plane fields, which serves in enhancing the formation of skyrmions.18,20This helps explain the initial formation of skyrmions in response to an in-planefield, but it remains an open question why the skyrmion arraysobserved here remain in the remnant state after its removal and whythey appear to require a particularly large applied field to form. In theera of bubble materials, the formation of field-free arrays through theapplication of in-plane magnetic fields had also been examined in gar-n e tfi l m sa tm u c hl a r g e rd i m e n s i o n s . 27–29Shimada found that stripe domains form bubble arrays in the remnant state after the applicationof an in-plane magnetic field due to the reduction of DW energy (similar to that noted previously) and consequent pinching to form bubble domains before they saturated. 29The resultant metastablebubble domain formation is akin to grain coarsening or coalescence of soap bubbles, where growth rapidly decreases as the smallest featuresare absorbed into larger ones. 34,35In our case, we speculate that a combination of the energy barrier preventing annihilation of 360/C14 transitions between skyrmions and pinning associated with our poly-crystalline thin film microstructure counteract the driving force for themagnetic texture to return to a labyrinth ground state. This is sup-ported by previous micromagnetic simulations where variations inlocal magnetocrystalline anisotropy, which simulates the disorderinherent to a polycrystalline film, promoted nucleation sites for sky-rmions. 18,36Irrespective of the finer details of the formation mecha- nism, one key observation made by Shimada that we draw attention toin this work is that a slight misalignment of the applied field withrespect to the film surface can greatly affect the resultant domainstructure. 27–29 To test the role of a misaligned in-plane magnetic field in our samples, we leverage bend contours associated with the TEM mem- branes. Fallon et al. have found that the flexible membrane can tilt as much as 15/C14, which is consistent with our observations.37Throughout each sample, transition regions between a labyrinth domain structureand a skyrmion array (noted by red lines in Fig. 4 ) were observed at the location of bends in the specimen. Although it would otherwise bereasonable to speculate that this magnetic texture could result frommagnetoelastic effects, the analyses provided by Shimada make clearthat the origin is actually due to variations in the applied field anglewith the sample surface. 29The regions that retain a labyrinth pattern are those which were significantly offset from the applied field direc-tion. The subsequent in situ application of a perpendicular magnetic field at these labyrinth regions did not lead to the formation of sky-rmions despite being surrounded by skyrmion arrays [ Figs. 4(b) and 4(c)]. Additionally, when the domain pattern of [ Fig. 4(a) ] is examined during the in situ application of a perpendicular magnetic field, we FIG. 3. Fresnel mode LTEM images of the remnant domain structure in [Pt/(Co/Ni) M/Ir]N, where top) M¼5;N¼4, middle) M¼3;N¼10, and bottom) M¼2;N¼20 following the application of an in-plane magnetic fields of 1.3, 1.0, and 0.5 T ex situ and perpendicular saturation (0.0 T). The sample was tilted by 15/C14in each image.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 112403 (2020); doi: 10.1063/5.0024265 117, 112403-3 Published under license by AIP Publishingfind the unusual circumstance where skyrmions of both polarities exist (i.e., the magnetization at their cores point in opposite directions) in close proximity. This is because the labyrinth regions enter the con- ventional transition to skyrmions, while the metastable skyrmionarrays have not yet begun to coalesce. In summary, metastable skyrmion arrays were observed at the remnant state in [Pt/(Co/Ni) M/Ir] Nmulti-layers after the ex situ appli- cation of a large in-plane magnetic field—an approach adopted fromprior work on bubble materials. 29The average skyrmion size was not strongly correlated with the strength of the field applied beyond 1 T but appeared inversely related to the interfacial DMI strength. Themetastability of these arrays likely originates from a combination ofthe DMI-reduced wall energy, which suppresses the driving force forcoarsening, and the energy barrier associated with annihilation of 360 /C14 transitions between skyrmions. We also speculate that intrinsic pin-ning of the polycrystalline microstructure contributes to array forma-tion. The origin of morphological transitions between labyrinths and skyrmion arrays throughout bent specimen membranes is confirmed to originate from variations in the applied field angle with respect tothe film plane emphasizing the importance of such an angle.Additionally, the simultaneous observation of skyrmions with oppositepolarities was seen at these bends. This simple method of formingdense, field-free skyrmion arrays, coupled with our highly tunable Pt/Co/Ni/Ir system, offers a means of investigating skyrmions out ofequilibrium for future development of spintronic devices. See the supplementary material for the detailed explanation of FMR measurements, values of the measured magnetic parameters, plots detailing skyrmion size/density, and additional Lorentz TEMimages. This work was financially supported by the Defense Advanced Research Project Agency (DARPA) program on TopologicalExcitations in Electronics (TEE) under Grant No. D18AP00011. The authors also acknowledge the use of the Materials CharacterizationFacility at Carnegie Mellon University supported by Grant No. MCF- 677785. M.L. would like to also acknowledge Michael Kitcher for assistance with the calculation of skyrmion schematics used. DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material . REFERENCES 1T. H. R. Skyrme, “A unified field theory of mesons and baryons,” Nucl. Phys. 31, 556–569 (1962). 2A. N. Bogdanov and D. A. Yablonskii, “Thermodynamically stable “vortices” in magnetically ordered crystals. The mixed state of magnets,” Sov. Phys. JETP68, 101–103 (1989). 3A. Bogdanov and A. Hubert, “Thermodynamically stable magnetic vortex states in magnetic crystals,” J. Magn. Magn. Mater. 138, 255–269 (1994). 4U. K. R €oßler, A. N. Bogdanov, and C. Pfleriderer, “Spontaneous skyrmion ground states in magnetic metals,” Nature 442, 797–801 (2006). 5S. M €uhlbauer, B. Binz, F. Jonietz, C. 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Skyrmions of a polarity opposite that of the field direction are highlighted with black arrows, whereas skyrmions with polarities parallel with thefield direction are enclosed by a red dotted line. The sample was tilted by 15/C14in each image. (d) Schematics depicting skyrmions with opposite polarities (e.g., blue points “down;” red points ‘up’).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 112403 (2020); doi: 10.1063/5.0024265 117, 112403-4 Published under license by AIP Publishing20S. Zhang, J. Zhang, Y. Wen, E. M. Chudnovsky, and X. Zhang, “Creation of a thermally assisted skyrmion lattice in Pt/Co/Ta multilayer films,” Appl. Phys. Lett. 113, 192403 (2018). 21Z. Qin, W. Wang, S. Zhu, C. Jin, J. Fu, Q. Liu, and J. Cao, “Stabilization and reversal of skyrmions lattice in Ta/CoFeB/MgO multilayers,” ACS Appl. Mater. Interfaces 10, 36556–36563 (2018). 22S. Woo, K. Litzius, B. Kr €uger, M.-Y. Im, L. Caretta, K. Richter, M. Mann, A. Krone, R. M. 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Graef, “Lorentz microscopy: Theoretical basis and image simulations,” in Magnetic Microscopy and Its Applications to Magnetic Materials , Experimental Methods in the Physical Sciences Vol. 36, edited by M. De Graef and Y. Zhu (Academic Press, 2000), Chap. 2. 32D. Lau, J. P. Pellegren, H. T. Nembach, J. M. Shaw, and V. Sokalski,“Disentangling factors governing Dzyaloshinskii domain wall creep in Co/Ni thin films using Pt xIr 1-xseedlayers,” Phys. Rev. B 98, 184410 (2018). 33X. S. Wang, H. Y. Yuan, and X. R. Wang, “A theory on skyrmion size,” Commun. Phys. 1, 31 (2018). 34K. L. Babcock and R. M. Westervelt, “Elements of cellular domain patterns in magnetic garnet films,” Phys. Rev. A 40, 2022 (1989). 35K. L. Babcock, R. Seshadri, and R. M. Westervelt, “Coarsening of cellular domain patterns in magnetic garnet films,” Phys. Rev. A 41, 1952 (1990). 36Y. Wang, J. W. Cao, and Q. F. Liu, “The formation process and structure of the skyrmion bubble lattice in magnetic multilayers,” J. Appl. 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5.0024103.pdf

J. Chem. Phys. 153, 114114 (2020); https://doi.org/10.1063/5.0024103 153, 114114 © 2020 Author(s).Nuclear magnetization distribution effect in molecules: Ra+ and RaF hyperfine structure Cite as: J. Chem. Phys. 153, 114114 (2020); https://doi.org/10.1063/5.0024103 Submitted: 04 August 2020 . Accepted: 02 September 2020 . Published Online: 17 September 2020 Leonid V. Skripnikov COLLECTIONS This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN Extension of the Fock-space coupled-cluster method with singles and doubles to the three-valence sector The Journal of Chemical Physics 153, 114115 (2020); https://doi.org/10.1063/5.0014941 Molecular second-quantized Hamiltonian: Electron correlation and non-adiabatic coupling treated on an equal footing The Journal of Chemical Physics 153, 124102 (2020); https://doi.org/10.1063/5.0018930 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 Nuclear magnetization distribution effect in molecules: Ra+and RaF hyperfine structure Cite as: J. Chem. Phys. 153, 114114 (2020); doi: 10.1063/5.0024103 Submitted: 4 August 2020 •Accepted: 2 September 2020 • Published Online: 17 September 2020 Leonid V. Skripnikov1,2,a) AFFILIATIONS 1Petersburg Nuclear Physics Institute Named By B.P. Konstantinov of National Research Centre “Kurchatov Institute”, Gatchina, Leningrad 188300, Russia 2Saint Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg 199034, Russia a)Author to whom correspondence should be addressed: skripnikov_lv@pnpi.nrcki.ru and leonidos239@gmail.com. URL: http://www.qchem.pnpi.spb.ru ABSTRACT Recently, the first laser spectroscopy measurement of the radioactive RaF molecule has been reported by Ruiz et al. [Nature 581, 396 (2020)]. This and similar molecules are considered to search for the new physics effects. The radium nucleus is of interest as it is octupole- deformed and has close levels of opposite parity. The preparation of such experiments can be simplified if there are reliable theoretical predictions. It is shown that the accurate prediction of the hyperfine structure of the RaF molecule requires to take into account the finite magnetization distribution inside the radium nucleus. For atoms, this effect is known as the Bohr–Weisskopf (BW) effect. Its magnitude depends on the model of the nuclear magnetization distribution which is usually not well known. We show that it is possible to express the nuclear magnetization distribution contribution to the hyperfine structure constant in terms of one magnetization distribution depen- dent parameter: BW matrix element for 1 s-state of the corresponding hydrogen-like ion. This parameter can be extracted from the accurate experimental and theoretical electronic structure data for an ion, atom, or molecule without the explicit treatment of any nuclear mag- netization distribution model. This approach can be applied to predict the hyperfine structure of atoms and molecules and allows one to separate the nuclear and electronic correlation problems. It is employed to calculate the finite nuclear magnetization distribution contri- bution to the hyperfine structure of the225Ra+cation and225RaF molecule. For the ground state of the225RaF molecule, this contribution achieves 4%. Published under license by AIP Publishing. https://doi.org/10.1063/5.0024103 .,s I. INTRODUCTION Theoretical predictions of hyperfine splittings in heavy atoms and molecules are of great importance for a wide scope of fun- damental physical applications.1They are used as a test of the accuracy of calculated characteristics that are required for the inter- pretation of experiments to search for effects of violation of spa- tial parity (P) or spatial parity and time-reversal (T) symmetries of fundamental interactions in atoms1–5(see also references therein) and molecules6–12or for the semi-empirical predictions.13–15The hyperfine splitting in spectra of highly charged ions can be used to test bound-state quantum electrodynamics (QED) in strong elec- tric and magnetic fields.16–22Accurate theoretical prediction is alsorequired to obtain the values of magnetic moments of short-lived isotopes.23–28 In many important cases, the largest uncertainty of the theo- retically predicted value of the hyperfine splitting comes from the nuclear part of the problem.29,30There are two important contribu- tions to the hyperfine splitting due to the finite nucleus size. The first one is caused by the finite change distribution inside the nucleus.31,32 It can be calculated with high accuracy as this distribution is known from experiments. The second finite nucleus contribution to the hyperfine splitting is caused by the distribution of magnetization inside the nucleus. In atoms, this effect is known as the Bohr– Weisskopf (BW) effect.33,34In most cases, it hardly can be calculated with high accuracy due to difficulties of ab initio prediction of the J. Chem. Phys. 153, 114114 (2020); doi: 10.1063/5.0024103 153, 114114-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp magnetization distribution within the nuclear structure theory in the case of heavy nuclei as well as the absence of corresponding experimental data. In Ref. 18, it was proposed to use the specific difference of hyperfine splittings in lithium-like and hydrogen-like bismuth to test bound-state QED in strong fields. It was shown that it is pos- sible to choose a linear combination of hyperfine splittings in such a way that the BW contributions cancel. In Ref. 30, it was pro- posed to combine experimental and theoretical data on the hyper- fine splitting for high lying electronic states of an atom to obtain the BW correction for the ground electronic state. The method is valid for nS 1/2, nP 1/2, and nP 3/2states of alkali–metal atoms and alkali–metal-like ions.30These methods are of great importance for atomic and highly charged ion experiments. For example, the accu- racy of the prediction of the hyperfine structure (HFS) constant of 133Cs is used to estimate the uncertainty of the prediction of the theoretical parameter that is used to interpret the experiment on the133Cs35atom in terms of fundamental parameters of P-violating forces. The strongest upper bounds on the electron electric dipole moment (T,P-violating property) have been established in the molecular beam experiment with diatomic ThO molecules.36 Another candidate is the RaF molecule. This molecule can be laser cooled.37This will increase the coherence time and, there- fore, enhance the sensitivity of such an experiment to T,P-violating effects.37–42An important feature of this molecule is that it pos- seses the deformed Ra nucleus with close levels of opposite par- ity.43–46Therefore, the T,P-violating effects such as the nuclear Schiff moment are strongly enhanced.43,44,47–49The most elaborated calcu- lation of the nuclear Schiff moment has been performed for the225Ra isotope.50 Recently, the first laser spectroscopy measurement of the RaF molecule has been performed.42It is expected that further experi- mental study of the molecule will allow one to measure magnetic dipole hyperfine constants.51Interpretation of such an experiment can be simplified if the accurate theoretical prediction of the HFS constants for RaF is available. Interpretation of the experiments with heavy-atom molecules to search for T,P-violating or P-violating effects requires high- precision calculations of a number of molecular constants that are required to express the observed effect in terms of fundamental properties of the nucleus and electron.4,11,52–58These constants can- not be measured. Therefore, to test the accuracy of their calcula- tion, one usually compares the theoretical value of the hyperfine structure constant with the experimental one. All the above char- acteristics are mainly determined by the behavior of the valence wavefunction of the molecule in the vicinity of the heavy-atom nucleus. However, there was no systematic attempt to take into account the BW contribution in such recent accurate calculations for the molecules of interest. In Ref. 59, there was an attempt to treat the BW contribution to the hyperfine structure of small molecules within the Gaussian distribution model of the nuclear magnetization using the density functional electronic structure calculations. In the present paper, we develop a method that can be employed to treat the finite nuclear magnetization distribution effect in atoms and molecules if the corresponding (nonzero) atomic or molecular data are known for one of the electronic states withsufficient accuracy. This method considers a general model of the magnetization distribution inside a nucleus, i.e., without using any particular model such as uniformly magnetized ball, Gaussian- distributed magnetization model, etc. The proposed method is applied to predict hyperfine structure constants for the225RaF molecule and the225Ra+cation in the ground and low-lying excited states within the four-component relativistic coupled cluster the- ory. The225Ra isotope has the nuclear spin equal to 1/2 and has only a magnetic dipole hyperfine structure, i.e., does not exhibit a quadruple one. This is important from the experimental point of view to unambiguously extract the magnetic hyperfine dipole interaction.51 II. THEORY For the nucleus with spin Iand the nuclear g−factor, the nuclear magnetic dipole moment is μ=gIμN, (1) where μN=e̵h 2mpcis the nuclear magneton, eis the elementary charge, ̵his the Planck constant, and mpis the proton mass. To treat the nuclear magnetization distribution inside the finite nucleus, one can use the following substitution:29,60,61 μ→μ(r)=μF(r). (2) The function F(r) takes into account the nuclear magnetization dis- tribution inside the finite nucleus. In the point magnetic dipole moment approximation, F(r) = 1. The expressions for different models can be found elsewhere.59–61 The hyperfine interaction of the magnetic moment μof a given nucleus with electrons is described by the following one-electron operator: hHFS=∑ iμ⋅[ri×αi] r3 iF(ri)=(μT), (3) where T=∑ i[ri×αi] r3 iF(ri), (4) αare Dirac’s matrices, α=(0σ σ0), (5) σare Pauli matrices, and riis the radius vector of electron iwith respect to the position of the considered nucleus. For a linear diatomic molecule, one can introduce the HFS constant A∥associated with the nucleus Kas A∥(K)=μK IΩ⟨ΨΩ∣∑ i(riK×αi r3 iK) zF(riK)∣ΨΩ⟩, (6) =μK IΩ⟨ΨΩ∣Tz(K)∣ΨΩ⟩, (7) J. Chem. Phys. 153, 114114 (2020); doi: 10.1063/5.0024103 153, 114114-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp where riKis the radius vector of electron iwith respect to the nucleus Kand Ω is the projection of the total electronic angular momentum on the internuclear axis z. For a molecule in the electronic state with |Ω| = 1/2, one introduces also the A/⊙◇⊞constant, A⊥(K)=μK IΩ⟨ΨΩ=+1/2∣∑ i ×(riK×αi r3 iK) +F(riK)∣ΨΩ=−1/2⟩, (8) where the V+component of some vector Vmeans V+=Vx+iVy. In the present paper, we assume that the relative phase of ΨΩ=+1/2 and ΨΩ=−1/2wavefunctions is chosen in such a way that the matrix ele- ment of the total electronic angular moment ⟨ΨΩ=+1/2∣J+∣ΨΩ=−1/2⟩is positive. Equation (6) can also be used to define the hyperfine struc- ture constant, A, of the atom in the electronic state with the total electronic angular momentum J and its projection MJ. For this, one should replace Ω by MJin Eq. (6). One can use the following expression for the hyperfine struc- ture constant,33,62 A=A(0)−ABW, (9) where A(0)is the HFS constant for the point nuclear magnetic moment [i.e., for F(r) = 1] and ABWis ABW=A(0)−A. (10) ABWgives contribution of the finite nuclear magnetization distribu- tion to the HFS constant. Note that for a more direct comparison with the experimental HFS values, one should also include a contri- bution of the QED effects to the hyperfine structure constant, AQED. However, such contributions are available only for a limited number of systems. For simplicity, this contribution has been omitted in the above equations but can be added to the right-hand side of Eq. (9). Function F(r) is notably different from 1 only inside the nucleus. Therefore, it follows from Eqs. (6) and (10) that ABW depends on the behavior of the wavefunction only in this region. One can also introduce the following operator: TBW=∑ i[ri×αi] r3 i(1−F(ri)). (11) Thus, ABW ∥(K)=μK IΩ⟨ΨΩ∣TBW z(K)∣ΨΩ⟩. (12) The operator (11) is zero outside the nucleus. The Dirac equation for radial components of the one-particle wavefunction can be written in the following form: ̵hc(dgnκ dr+1 +κ rgnκ)−(Enκ+mc2−V)fnκ=0, (13) ̵hc(dfnκ dr+1−κ rfnκ)+(Enκ−mc2−V)gnκ=0, (14)where gandfare radial functions of large and small components of the Dirac bispinor, κ= (−1)j+l+1/2(j+ 1/2), j is the total momentum of the electron, nis the principle quantum number, and Vis the nuclear potential. In the nuclear region, it is possible to neglect the binding energy Enκwith respect to the nuclear potential. Therefore, the wavefunctions with different nand the same κare proportional to each other in this region. This property is widely used in different applications.8,13,63–70 From the structure of the HFS Hamiltonian (4), one can see that its expectation value for a given bispinor for a system with a spherical symmetry is proportional to ∫∞ 0g f F(r)dr. At the same time, ABW∝∫∞ 0g f(1−F(r))dr≈∫Rnuc 0g f(1−F(r))dr, where Rnucis the radius of the nucleus. As only s1/2andp1/2functions have non-negligible amplitudes inside the nucleus, only these types of functions can notably contribute to ABW. In the nuclear region r∼Rnucof a heavy ion (atom), the abso- lute value of the nuclear potential V(r) is larger than the electron rest energy, and one can omit the mc2terms in addition to Enκin Eqs. (13) and (14).68From the simplified equations, consequently, in the region r∼Rnuc, the product gnκfnκdiffers from gn−κfn−κ only by a constant factor.68Fors1/2and p1/2functions, it can be also seen from the explicit analytical expressions that are given in Ref. 13. Thus, one obtains the following expression that connects matrix elements of the z-component of the operator (11), TBW z, over the 1 sj=1/2,mj=1/2and 2 pj=1/2,mj=1/2functions, ∫2p† 1/2,1/2TBW z2p1/2,1/2dr ≈β∫1s† 1/2,1/2TBW z1s1/2,1/2dr, (15) where βis the proportionality constant independent on the actual expression for the function F(r). The only suggestion is that [F(r)−1] is localized inside the nucleus. A similar idea has been used in Ref. 68 to introduce the specific difference of the electronic g-factors of hydrogen-like and boron-like ions of lead. Let us now consider a reduced one-particle density matrix ρ(r|r′) obtained from the correlation calculation of some molecule containing a heavy nucleus K and choose the origin at the position of K. One can write ρ(r∣r′)=∑ p,qρp,qφp(r)φ† q(r′),(16) where { φp} are molecular bispinors. The expectation value of some one-particle operator Xcan be calculated as ⟨X⟩=∑ p,qρp,q∫φ† qXφpdr. (17) In this paper, we are interested in the operator TBW(11), which is zero outside the nucleus. It means that for such operator X, one can perform integration in Eq. (17) inside the sphere of radius Rc≈Rnuc. In this region, one can re-expand { φp} in terms of some sufficiently complete set of basis functions ηnljm(r) centered at the nucleus K, J. Chem. Phys. 153, 114114 (2020); doi: 10.1063/5.0024103 153, 114114-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp φp(r)≈∑ nljmCp nljmηnljm(r),∣r∣≤Rc, (18) where Cp nljmare expansion coefficients. For example, one can take hydrogen-like functions for the ion with the nucleus K as a set of {ηnljm}. From the above consideration, such functions with a given set of l,j,mand different nare proportional to each other inside the nucleus. Therefore, for each combination of l,j,m, one can introduce some reference function ηljm(r) and write ηnljm(r)≈knljmηljm(r),∣r∣≤Rc, (19) where knljmis the proportionality coefficient. Functions ηljm(r) with different mand the same l,jdiffer only in their spin-angular part. Substituting Eqs. (18) and (19) in Eq. (17) and taking into account that the integration in Eq. (17) can be performed inside the sphere of radius Rcwith the center at the nucleus K, one obtains ⟨X⟩≈∑ ljm;l′j′m′Pljm,l′j′m′∫ ∣r∣<Rcη† l′j′m′Xηljmdr, (20) where Pljm,l′j′m′=∑ p,q,n,n′ρp,qCp nljmknljmCq∗ n′l′j′m′k∗ n′l′j′m′.(21) Pljm, ljm can be called the reduced occupation associated with the ref- erence function ηljm. Such parameters are introduced in the atoms in compound theory63(see also Refs. 8 and 71–73) as they can serve as some certain characteristics of an atom inside a compound. As it was noted above, inside the nucleus, only s1/2and p1/2 functions have non-negligible amplitudes and can contribute to ABW. Therefore, in the present case, one can take 1 s1/2and 2 p1/2func- tions of the hydrogen-like ion with the nucleus Kas reference func- tions ηljm(r). One can also take into account the following relation for the z-component of the operator (11) [and (4)]: ∫η† ljmTBW zηljmdr=−∫η† lj−mTBW zηlj−mdr. (22) Besides, off-diagonal matrix elements of the z-component of opera- tors (4) and (11) between s1/2andp1/2functions are zero. Introduc- ing also Ps=P1s1/2,1/2, 1s1/2,1/2−P1s1/2,−1/2, 1s1/2,−1/2, (23) Pp=P2p1/2,1/2, 2p1/2,1/2−P2p1/2,−1/2, 2p1/2,−1/2, (24) Bs=∫ ∣r∣≤Rnucη† 1s1/2,1/2TBW zη1s1/2,1/2dr, (25) Bp=∫ ∣r∣≤Rnucη† 2p1/2,1/2TBW zη2p1/2,1/2dr, (26) one obtains ⟨TBW z⟩≈Ps∫ ∣r∣≤Rnucη† 1s1/2,1/2TBW zη1s1/2,1/2dr +Pp∫ ∣r∣≤Rnucη† 2p1/2,1/2TBW zη2p1/2,1/2dr =PsBs+PpBp. (27)Now, we can rewrite Eq. (12) as ABW ∥(K)≈μK IΩ(PsBs+PpBp). (28) Using Eq. (15), it is possible to simplify Eq. (28), ABW ∥(K)≈μK IΩ(Ps+βPp)Bs (29) =ABW,s ∥(K)+ABW,p ∥(K), (30) where ABW,s ∥(K)=μK IΩPsBs, (31) ABW,p ∥(K)=μK IΩPpβBs. (32) Corresponding expression can be written for ABW ⊥(K)for the diatomic molecule in the Ω = 1/2 electronic state as well as ABW in the case of a single atom. Generalizations are also possible for polyatomic molecules. In some cases, it can be more convenient to introduce functions hljm(r), defined as hljm(r)=ηljm(r)θ(Rc−∣r∣), (33) where θ(Rc−|r|) is the Heaviside step function, θ(Rc−∣r∣)={1,∣r∣<Rc 0, otherwise. Using functions h(r), it is possible to calculate Pljm,l′j′m′as a mean value of the following operator: Rljm,l′j′m′=∣hljm⟩⟨hl′j′m′∣ ⟨hljm∣hljm⟩⟨hl′j′m′∣hl′j′m′⟩. (34) Constants PsandPpin Eq. (29) are determined by the elec- tronic structure of a heavy atom or a molecule under consideration and do not depend on the nuclear magnetization distribution model. In contrast, the Bsconstant is common for a given heavy atom, its ion, or a molecule, containing this atom. It directly depends on the magnetization distribution inside the nucleus under consideration. Therefore, if one has sufficiently accurate theoretical data for the electronic part of the problem of calculating the HFS constant for a given heavy atom, ion, or molecule containing this atom as well as accurate experimental data, it is possible to extract the Bsconstant value. It can be further used to predict the finite nuclear magnetiza- tion distribution effect associated with a given nucleus in any other system that contains this nucleus, i.e., an atom, ion, or molecule in any other electronic state. It can also be used to test predictions of the nuclear models using sufficiently accurate atomic data. Note that the suggested approach can be used to approximately calculate the specific difference parameter ξintroduced in Ref. 18 for lithium-like and hydrogen-like ions using Eq. (29). J. Chem. Phys. 153, 114114 (2020); doi: 10.1063/5.0024103 153, 114114-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp In the present paper, we have implemented a theoretical approach to calculate PsandPpconstants in Eq. (29). III. COMPUTATIONAL DETAILS All calculations have been performed within the Dirac– Coulomb Hamiltonian. To solve electronic many-body problems for the atom and molecule under consideration, we have used the coupled cluster with single, double, and perturbative triple clus- ter amplitudes method, CCSD(T).74The energy cutoff for virtual orbitals was set to 10 000 hartree in the correlation treatment. In Ref. 75, it was demonstrated that such an energy cutoff is important to ensure including functions that describe spin-polarization and correlation effects for inner-core electrons. To treat higher-order correlation effects for valence and outer-core electrons, we have employed the coupled cluster method with inclusion up to triple and perturbative quadruple cluster amplitudes, CCSDT(Q).76 To be able to include the most important basis functions with high ljin the four-component correlation calculation, we have used the method of constructing natural basis sets.77This method implies scalar–relativistic correlation calculation using a large basis set fol- lowed by diagonalization of atomic blocks of the one-particle den- sity matrix.77In the present paper, this method has been extended in the following way. For a given l(j), several natural basis func- tions with the largest occupations have been chosen. These func- tions are contracted, i.e., expanded in terms of a large number of primitive Gaussian-type functions. These contracted functions have been re-expanded in terms of a small number of primitive Gaussian-type functions. In the final calculation, we include these primitive Gaussians instead of original natural contracted basis functions. Such an approach allows one to have the additional flex- ibility of the basis set as well as use it in the four-component calculations. For atomic calculations, the following uncontracted Gaussian- type basis sets have been used. The LBas basis set consists of 38 s, 33p, 24d, 14f, 7g, 3h, and 2 ifunctions, which can be written as (38s, 33p, 24d, 14f, 7g, 3h, 2i). This basis set has been obtained by aug- mentation of the uncontracted Dyall’s AE3Z basis set78by 5s, 4p, 5d, and 1 fdiffuse functions. In addition, we have added 3 g, 2h, and 2 ifunctions generated using the method of constructing natural basis sets described above. To treat high-order correlation effects, we have used the SBas basis set that is equal to the uncontracted Dyall’s VDZ78basis set augmented by a few s−,p−, and d−type functions. To test basis set completeness, we have also employed the LBasExt basis set: (42s, 38p, 27d, 17f, 11g, 3h, 2i). This basis set has been obtained by the augmentation of the uncontracted Dyall’s AE4Z basis set78by 5s, 4p, 4d, and 1 fdiffuse functions and 3 g, 2h, and 2 inatural functions. For the calculation of HFS constants for the RaF molecule, we have used the LBasRaF basis set that corresponds to the LBas basis set on Ra and the uncontracted Dyall’s AETZ78basis set on F. The LBasExtRaF basis set corresponds to the LBasExt basis set for Ra and the Dyall’s AE4Z78basis set on F. Finally, we have used the SBasRaF basis set that corresponds to the SBas basis set for Ra and the aug- cc-PVDZ-DK basis set79,80for F. In calculations of the ground and first excited electronic states of RaF, we have used the calculated values of equilibriuminternuclear distances: R(Ra–F) = 4.23 bohrs for both states, obtained in this work. This distance is in agreement with the pre- vious study.40 In the electronic structure calculations, the Gaussian charge distribution model81has been employed. For molecular calculations, the DIRAC82and MRCC76,83codes have been used. To calculate Psand Ppparameters, the code developed in this work has been employed. The225Ra nucleus has a spin of I= 1/2 and a magnetic moment ofμ=−0.7338(15) obtained in the direct (atomic) measurement.84 IV. RESULTS AND DISCUSSION There is a strong uncertainty in calculating the BW correction for heavy atoms from first principles due to a rather limited knowl- edge of the nuclear magnetization distribution F(r). However, as it can be seen from Eq. (29), the BW contribution to the hyperfine structure constant of an atom or a molecule depends only on one matrix element Bsgiven by Eq. (25). The latter element depends onF(r) through Eq. (11). Thus, if there are accurate experimental data for the (nonzero) hyperfine structure constant for some system induced by a given nucleus as well as accurate enough electronic structure calculation of the HFS constant in the point magnetic dipole approximation, it is possible to extract the Bsconstant and use it for further predictions. Below, we perform this extraction from the atomic data for the225Ra+cation to accurately predict hyperfine structure constants for the225RaF molecule. A.225Ra+cation The hyperfine structure constant for the ground 7s2S1/2state of the225Ra+cation is known with high accuracy.85–87Moreover, for this system, even the QED contribution to the hyperfine struc- ture constant has been recently calculated in Ref. 3. Table I gives the calculated values of the hyperfine structure constant A(0)for the ground and excited electronic states of the225Ra+cation calculated in the point magnetic dipole approximation, i.e., without the BW correction. Calculated values are in good agreement with previous theoretical works.3,41,88–91 The leading contributions to the HFS constants given in Table I have been calculated within the four-component CCSD(T) approach using the LBas basis set. All 87 electrons of the Ra+ion have been included in the correlation treatment. For comparison, Table I gives also the values obtained within the Dirac–Hartree–Fock (DHF) TABLE I . Hyperfine structure constant (in MHz) for the ground and excited states of the225Ra+cation calculated in the point magnetic dipole approximation. Method 7s2S1/2 7p2P1/2 7p2P3/2 DHF −21976 −3657 −276 CCSD −29160 −5484 −459 CCSD(T) −28896 −5498 −463 Correlation correction −117 −31 0 Basis set correction 2 3 0 Total, electronic ( A(0)) −29012 −5526 −463 J. Chem. Phys. 153, 114114 (2020); doi: 10.1063/5.0024103 153, 114114-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp and CCSD approaches. Higher-order correlation effects have been estimated as a difference of the CCSDT(Q) and CCSD(T) values obtained within the SBas basis set and correlating 27 outer electrons of Ra+. The basis set correction given in Table I is the contribution of the basis set extension up to the LBasExt basis set. It has been calculated within the 59-electron CCSD(T) approach, i.e., 1 s. . .3d electrons have been excluded from the correlation treatment. The final value of the HFS constant calculated in the point mag- netic dipole approximation ( A(0)) for the ground state of225Ra+ given in Table I is in excellent agreement (within 0.03%) with the most recent previous theoretical value for this constant from Ref. 3. It can be seen that the basis set corrections for the ground and excited state HFS constants are small. This confirms the quality of the main LBas basis set. From the values given in Table I, one can expect that the theoretical uncertainty of the HFS constant, calcu- lated in the point magnetic dipole approximation for the ground state of225Ra+,A(0)[see Eq. (9)], is about 150 MHz or less than 1%. Finite nuclear magnetization distribution contribution has not been included in the uncertainty as it is related to the constant A (see below). Breit and QED effects, calculated in Ref. 3, contribute in total about 66 MHz. Taking this into account and using the calcu- lated value of A(0)from Table I and experimental value85ofA, one finds ABW(225Ra+, 7s2S1/2)=−1215 MHz. This is about an order of magnitude bigger than the QED and Breit effects and more than 4% of the total HFS constant. In Refs. 3, 88, 89, and 92, there were attempts to obtain ABW within the concrete nuclear magnetization distribution models for different isotopes of Ra. In the present work, we have not used any nucleus model to calculate ABWand extracted it from the accurate experimental and theoretical data for the ground state of225Ra+. According to calculations, 1 s. . .3delectrons of Ra contribute about 2.3% to the HFS constant A(0)of the ground state of225Ra+. 1s. . .4felectrons of Ra contribute about 4%. In Ref. 75, a similar cor- relation contribution to the enhancement factor of the T,P-violating scalar–pseudoscalar nucleus–electron interaction has been found for the Fr atom. It is interesting to note that while the inclusion of 1 s. . .4f electrons in the correlation treatment increases the absolute value of the HFS constant, the BW effect in the ground electronic state of 225Ra decreases it by 4%. Therefore, it is possible to obtain an “excel- lent agreement” (e.g., less than 1%) of the calculated HFS constant A(0)in the simplified calculation (with the exclusion of the BW con- tribution and correlation contribution of the core electrons) with the experimental HFS constant A. However, this will not mean that the atomic enhancement factor of the T,P-violating effect is calculated with such accuracy (less than 1%). As it was noted above, Bs(as well as β) constant does not depend on the electronic state of the system containing heavy nucleus under consideration. Therefore, it is possible to use Eq. (29) to express the BW contribution to the hyperfine structure constant of the system containing the225Ra nucleus in terms of ABW(225Ra+, 7s2S1/2) by cal- culating parameters PsandPpfor the225Ra+in the 7 s2S1/2state as well as for the system under consideration. Such calculations have been performed for the excited states of the225Ra+. It was found (seeTABLE II . BW contributions ABW,ABW,s, and ABW,pand the final values of the hyperfine structure constants (in MHz) for the ground and excited states of the 225Ra+cation. For the ground state, ABWhas been obtained as a difference between the theoretical value of the HFS constant calculated in the point magnetic dipole approximation and the experimental value taking into account QED and Breit effects. 7s2S1/2 7p2P1/2 7p2P3/2 −ABW,s1214 −5 3 −ABW,p1 80 0 −ABW1215 75 2 A(0)(see Table I) −29012 −5526 −463 Breit+QED,aRef. 3 66(23) . . . . . . Final −27731 −5451 −461 Experiment85–87−27731(13) −5446.0(7) −466.4(4.6) aExtracted from Ref. 3: Breit: −93 MHz; QED: 159(23) MHz; Electron+Breit: −29113 MHz. Table II) that ABW(225Ra+, 7p2P1/2)=−75 MHz. Using Eq. (9), we obtain A(225Ra+, 7p2P1/2)=−5451 MHz. This value is in a far better agreement with the experimental value than the value of A(0)(225Ra+, 7p2P1/2) given in Table I. Note that there can be some contribution from the QED (and Breit) effects. No such contributions have been calculated for the 7p2P1/2state of Ra+, but one can expect that they are small taking into account the cor- responding value for the ground state of Ra+(0.2%3). Table II gives the calculated values of ABWfor all the considered states of225Ra+. Table II also gives contributions ABW,sandABW,pdue to s1/2andp1/2 terms [see Eq. (29)]. It can be seen that for the 7s2S1/2state, the former contribution dominates, while for 7p2P1/2, the latter contri- bution dominates. The BW contribution to the HFS constant of the 7p2P3/2state is very small in contrast to the case of the 6p2P3/2state of the Tl atom.26 In the above treatment of the ABWconstant, no specific nuclear distribution model has been used. However, if we assume some nuclear magnetization distribution model by specifying F(r) in Eq. (2), then the ABWconstant can be calculated either (i) directly using Eq. (12) or (ii) in the way suggested by Eq. (29). In the latter case, the “nuclear part” reduces to just one matrix element Bsgiven by Eq. (25) on the 1 s1/2function of the corresponding hydrogen-like ion. In our case, one can consider the matrix element for the ground 1s-state of the hydrogen-like225Ra87+ion. Note that the approach (ii) should be valid not only for the 7s2S1/2state of Ra+but also for the 7p2P1/2state and other states. To check the accuracy of the approach (ii) numerically, we have considered the uniformly mag- netized ball model of radius Rm. In this model, F(r)=R3 m/r3for r≤RmandF(r) = 1 for r>Rm. Here, Rm∼Rnuccan be considered as a constant parameter of the model.26Within this model, we have found that the values of the ABWconstant calculated using the direct approach (i) and using the approach (ii) coincide within 0.4% for all the considered states of225Ra+, i.e., 7s2S1/2, 7p2P1/2, and 7p2P3/2. J. Chem. Phys. 153, 114114 (2020); doi: 10.1063/5.0024103 153, 114114-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp As it was noted above, ABW(7p2P1/2) is determined by the second term in Eq. (29) that uses property (15). This numerical test ver- ifies the suggested approach (ii), i.e., the use of Eq. (29). Thus, to calculate BW contributions, one needs to know the matrix element Bsgiven by Eq. (25) and perform corresponding electronic structure calculations of PsandPpgiven by Eqs. (23) and (24). B.225RaF molecule The ground X2Σ1/2as well as the first excited A2Π1/2electronic states of the RaF molecule qualitatively have one unpaired electron on the so-called non-bonding (atomic-like) orbital. This results in an almost diagonal Franck–Condon matrix element between these states. Due to this feature and the fact that the transition frequency between these states lies in the visible region, it is possible to laser cool this molecule.37,42 From the above consideration of the225Ra+cation, it is expected that the non-negligible contribution of the finite nuclear magnetization distribution to the hyperfine structure constants can be expected also for225RaF. This effect has not been accurately con- sidered for this molecule before. Note, however, that the need to con- sider this effect arises only when the calculation of the HFS constant in the point magnetic dipole approximation is accurate enough, i.e., its uncertainty is smaller than the effect under consideration. Table III gives the calculated values of the hyperfine struc- ture constant for the ground X2Σ1/2and first excited A2Π1/2elec- tronic states of the225RaF molecule. The main calculation of A(0) ∥has been performed within the four-component CCSD(T) approach using the LBasRaF basis set and correlating all 97 electrons. To calculate A(0) ⊥, 1s. . .3delectrons of Ra have been excluded from the correlation calculation. However, the correct- ing scaling factor for the treatment of these electrons (obtained from the A∥calculation) has been applied.93,94Higher-order correlation effects have been estimated as a difference of the TABLE III . Hyperfine structure constants A∥andA/⊙◇⊞(in MHz) for the ground X2Σ1/2 and excited A2Π1/2states of the225RaF molecule induced by the225Ra nucleus. Method X2Σ1/2 A2Π1/2 A∥ A/⊙◇⊞ A∥ A/⊙◇⊞ DHF −12048 −11670 −1638 −1235 CCSD −17814 −17148 −2848 −2173 CCSD(T) −17595 −16941 −2842 −2198 Correlation correction −134 −134 −48 −28 Basis set correction −31 −30a−4 −3a Vibrational correction −19 −18a−2 −2a Total, electronic ( A(0) ∥/⊥)−17780 −17123 −2896 −2230 −ABW,s ∥723 9 −ABW,p ∥7 34 −ABW ∥ /⊥ 730 720a44 26a Final −17049 −16403 −2852 −2204 aRescaled from A∥.CCSDT and CCSD(T) values obtained within the SBasRaF basis set and correlating 27 outer electrons of RaF. We have also calculated the vibrational correction to the considered parameters for the case of zero vibrational levels of the considered electronic states of the225RaF molecule using the approach described in Ref. 9. Finally, we have applied the basis set correction given in Table III. It is the contribution of the basis set extension up to the LBasEx- tRaF basis set. It has been calculated within the 69-electron CCSD(T) approach, i.e., 1 s. . .3delectrons of Ra have been excluded from the correlation calculation. One can expect that the uncertainties of the calculated HFS constants A(0) ∥andA(0) ⊥are of order 1% similar to the atomic case. The values of the ground state hyperfine constants calculated in the point magnetic dipole approximation are in good agreement with previous calculations,40,41but have much smaller uncertain- ties.95 To calculate ABW ∥, we have calculated PsandPpconstants for the225RaF molecule. The resulting value of ABW ∥as well as its con- tributions ABW,s ∥andABW,p ∥are given in Table III. It can be seen that the finite nuclear magnetization distribution effect ABW ∥contributes about 4% to the hyperfine structure of the ground state of the225RaF molecule and cannot be neglected in an accurate consideration. The leading contribution to ABW ∥comes from ABW, s ∥. A similar effect and its contributions have been found for the ground state of the225Ra+ cation above. For the first excited state A2Π1/2,ABW ∥contributes about 1.5%. Here, ABW, p ∥dominates. This is close to what was found for the excited 7p2P1/2state of225Ra+. This also correlates with the assumed atomic-like character of the ground and the first excited states of RaF. Note that the ABW, p ∥domination over ABW, s ∥is not so strong as in the atomic case of the 7p2P1/2state. The relative asymmetry in HFS constants A∥andA/⊙◇⊞is larger for the excited state 7p2P1/2than for the ground state. It indicates some deviation from the idealized atomic character of this state. For the ground state of the225RaF molecule, we have found that the correlation contribution of 1 s. . .4felectrons of Ra and 1 sof F is about 4% which is close by the absolute value but has an oppo- site sign to the finite nuclear magnetization distribution effect. Thus, the situation is similar to what was found for the case of the225Ra+ ground state HFS above. It is possible to obtain an excellent agree- ment of the calculated HFS constant A(0) ∥in the simplified calcula- tion with the exclusion of finite nuclear magnetization distribution contribution and electron correlation contribution of the core elec- trons with the experimental HFS constant A∥. However, this will not mean that the effective electric field acting on the electron electric dipole moment and other similar parameters of the T,P-violating effects are calculated with such good accuracy using the same wave- function. It should also be stressed that the proper correlation of the core electrons requires the use of a sufficiently high energy cutoff; see Fig. I in Refs. 9 and 75. V. CONCLUSIONS A theoretical method to treat the contribution of the finite nuclear magnetization distribution to the hyperfine structure J. Chem. Phys. 153, 114114 (2020); doi: 10.1063/5.0024103 153, 114114-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp constants of heavy atoms and molecules containing heavy atoms has been proposed. It is shown that the nuclear part of the prob- lem can be reduced to the treatment of just one matrix element over the hydrogen-like function. This matrix element can be calcu- lated using some nuclear structure model or it can be extracted by combining accurate experimental and theoretical data for an atom, (highly charged) ion, or molecule containing the heavy nucleus under consideration. An important feature of the formulated theory is that it is pos- sible to separate the electron correlation problem and the nuclear problem using Eq. (29). The accuracy of such separation (factoriza- tion) has been tested numerically for one of the nuclear magneti- zation models and is found to be very high, and the corresponding theoretical uncertainty is less than 1%. Most of the previous studies of the effects of the finite distri- bution of nuclear magnetization concerned atoms. The approach developed here helps extend this treatment to molecular physics. The method has been applied to study the hyperfine structure for the225Ra+ion and225RaF molecule in low-lying electronic states. In particular, it was found that the finite nuclear magnetization dis- tribution effect strongly (more than 4%) contributes to the HFS of the ground state of225RaF and should be taken into account in precise calculations. This effect has an opposite sign to the corre- lation contribution of the core electrons, but similar absolute value. Therefore, the neglect of both these effects can lead to an incorrect conclusion about the accuracy of the calculated wavefunction and uncertainties of other properties calculated with this wavefunction. This conclusion should be taken into account in most precise calcu- lations, e.g., in the field of the search of the P- or/and T,P-violation effects in molecules, etc. Predicted HFS constants for225RaF can be used to simplify the interpretation of further experiments with this molecule.51 ACKNOWLEDGMENTS The author is grateful to A. V. Titov, V. M. Shabaev, M. G. Kozlov, and T. A. Isaev for useful discussions. Electronic structure calculations have been carried out using computing resources of the federal collective usage center Complex for Simulation and Data Processing for Mega-science Facilities at NRC “Kurchatov Institute,” http://ckp.nrcki.ru/, and computers of Quantum Chemistry Lab at NRC “Kurchatov Institute” - PNPI. 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5.0013596.pdf

J. Appl. Phys. 128, 063904 (2020); https://doi.org/10.1063/5.0013596 128, 063904 © 2020 Author(s).Spin susceptibility, upper critical field, and disorder effect in superconductors with singlet–quintet mixing Cite as: J. Appl. Phys. 128, 063904 (2020); https://doi.org/10.1063/5.0013596 Submitted: 14 May 2020 . Accepted: 20 July 2020 . Published Online: 12 August 2020 Jiabin Yu , and Chao-Xing Liu ARTICLES YOU MAY BE INTERESTED IN Interfacial magnetic anisotropy and Dzyaloshinskii–Moriya interaction at two-dimensional SiC/Fe 4N(111) interfaces Journal of Applied Physics 128, 063903 (2020); https://doi.org/10.1063/5.0019092 Magnetic field modeling with surface currents. Part II. Implementation and usage of bfieldtools Journal of Applied Physics 128, 063905 (2020); https://doi.org/10.1063/5.0016087 Magnetic-field modeling with surface currents. Part I. Physical and computational principles of bfieldtools Journal of Applied Physics 128, 063906 (2020); https://doi.org/10.1063/5.0016090Spin susceptibility, upper critical field, and disorder effect in j¼3 2superconductors with singlet –quintet mixing Cite as: J. Appl. Phys. 128, 063904 (2020); doi: 10.1063/5.0013596 View Online Export Citation CrossMar k Submitted: 14 May 2020 · Accepted: 20 July 2020 · Published Online: 12 August 2020 Jiabin Yu and Chao-Xing Liua) AFFILIATIONS Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA Note: This paper is part of the Special Topic on Topological Materials and Devices. a)Author to whom correspondence should be addressed: cxl56@psu.edu ABSTRACT Recently, a new pairing state with the mixing between the s-wave singlet channel and the isotropic d-wave quintet channel induced by cen- trosymmetric spin –orbit coupling has been theoretically proposed in the superconducting materials with j¼3 2electrons [J. Yu and C.-X. Liu, Phys. Rev. B 98, 104514 (2018)]. In this work, we derive the expressions of the zero-temperature spin susceptibility, the upper critical field close to the zero-field critical temperature Tc, and the critical temperature with weak random non-magnetic disorders for the singlet – quintet mixed state based on the Luttinger model. Our study reveals the following features of the singlet –quintet mixing. (1) The zero- temperature spin susceptibility remains zero for the singlet –quintet mixed state if only the centrosymmetric spin –orbit coupling is taken into account and deviates from zero when the non-centrosymmetric spin –orbit coupling is introduced. (2) The singlet –quintet mixing can help enhance the upper critical field roughly because it can increase Tc. (3) Although the quintet channel is generally suppressed by the non-magnetic disorder scattering, we find the strong mixing between singlet and quintet channels can help to stabilize the quintet channel.As a result, we still find a sizable quintet component mixed into the singlet channel in the presence of weak random non-magnetic disor- ders. Our work provides the guidance for future experiments on spin susceptibility and upper critical field of the singlet –quintet mixed superconducting states and illustrates the stability of the singlet –quintet mixing against the weak random non-magnetic disorder. Published under license by AIP Publishing. https://doi.org/10.1063/5.0013596 I. INTRODUCTION Increasing research interest has recently been focused on the superconductivity in half-Heusler materials, including RPtBi(R¼La, Y, and Lu) and RPdBi (R ¼Er, Lu, Ho, Y, Sm, Tb, Dy, and Tm), due to their possible unconventional mechanism indi- cated by the low carrier density (10 18–1019cm/C03) compared with the critical temperature (0.5 –1.9 K), the power-law temperature dependence of London penetration depth observed in YPtBi, andthe large upper critical field. 2–13The low-energy excitation of these compounds in the normal phase has a total angular momentum j¼3 2given by the addition of12spin and angular momentum of p atomic orbitals ( l¼1), making half-Heusler superconductors (SCs) an intriguing platform to study superconductivity with j¼3 2 fermions.13,14Such j¼3 2fermions also exist in anti-perovskite materials15and the cold atom system.16,17The effective spin j¼3 2of electrons allows the spin of Cooper pairs to take four values, S¼ 0 (singlet), 1 (triplet), 2 (quintet), and 3 (septet), instead of only the singlet and the triplet for spin-1 2electrons. A variety of pairing states have been studied in such systems, including mixed singlet – septet pairing,13,14,18,19mixed singlet –quintet pairing,1,20s-wave quintet pairing,14,19,21,22d-wave quintet pairing,23,24odd-parity (triplet and septet) parings,23–26etc.24,27In particular, the mixing between the s-wave singlet and isotropic d-wave quintet channels proposed in Ref. 1is the first realistic proposal of the mixing between different spin channels that preserves the inversion sym-metry in solid state systems. The mixing is promising because it isinduced by the strong inversion-invariant “spin –orbit coupling (SOC) ”(the coupling between the “ 3 2-spin ”and the orbital) and the resultant topological nodal-line superconductivity (TNLS) is pro- tected by the non-trivial topological invariant.1Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063904 (2020); doi: 10.1063/5.0013596 128, 063904-1 Published under license by AIP Publishing.In this work, we study the spin susceptibility, the upper criti- cal field, and the non-magnetic disorder effect of such pairing mixing states and obtain the following results. The spin suscepti-bility is isotropic and approaches to a non-zero (zero) value as thetemperature decreases in the presence (absence) of the inversion-breaking SOC. The upper critical field near the zero-field critical temperature T cis isotropic and enhanced by the mixing, and its slope at Tcvaries significantly with the band structure. In the pres- ence of the non-magnetic random disorder, the critical tempera-ture and the quintet channel are suppressed, while the lattercannot be entirely suppressed due to the singlet –quintet mixing. Our results show several properties of the singlet –quintet mixed state that can be experimentally measured. The rest of the paper is organized as the following. We describe the model for the mixing between the s-wave singlet andisotropic d-wave quintet channels in Sec. II, address spin suscepti- bility in Sec. III, study the upper critical field in Sec. IV, discuss the disorder effect in Sec. V, and eventually conclude our work with the discussion about experiments in Sec. VI. II. MODEL HAMILTONIAN In this section, we first review the model without magnetic fields proposed in Ref. 1and then introduce the modification due to the external magnetic field. As mentioned in Sec. I, the low-energy excitation of the half-Heusler material in the normal phase has total angular momentum j¼ 3 2, meaning that we can choose the bases as ( j3=2i, j1=2i,j/C01=2i,j/C03=2i). With this choice, the effective non- interacting Hamiltonian is just the Luttinger model1,28–30that reads h(k)¼ξkΓ0þhSSOC(k)þhASOC(k), (1) where ξk¼1 2mk2/C0μwith μthe chemical potential is the parabolic band part and Γ0is the 4 /C24 identity matrix. In Eq. (1),hSSOC(k) is the symmetric SOC (SSOC) that consists of five d-orbital cubic harmonics giand five gamma matrices Γias hSSOC(k)¼c1X3 i¼1gk,iΓiþc2X5 i¼4gk,iΓi, (2) while hASOC is the anti-symmetric SOC (ASOC) that has the form hASOC¼2Cffiffiffi 3p(kxVxþkyVyþkzVz), (3) where { Γa,Γb}¼2δabΓ0, three Vi’s matrices are Vx¼1 2{Jx, J2 y/C0J2 z},Vy¼1 2{Jy,J2 z/C0J2 x}, and Vz¼1 2{Jz,J2 x/C0J2 y}, and J’sa r e j¼3=2 angular momentum matrices (see Appendix A for more details). Here, “symmetric ”(“anti-symmetric ”) means parity even (odd), and both SSOC and ASOC refer to the coupling between the“3/2-spin ”and the orbit degree of freedom of j¼3=2 fermions. h(k) in Eq. (1)hasO(3) point group symmetry for c 1¼c2, while c1=c2reduces O(3) to OhandC=0 further reduces Ohto Td.h(k) also has time-reversal (TR) symmetry γh*(/C0k)γy¼h(k), where γ¼/C0Γ1Γ3is the TR matrix. When C¼0, the combinationof inversion and TR symmetries make every band of h(k) doubly degenerate, resulting in two bands with dispersion ξ+(k)¼k2= (2m+)/C0μ. Here, we adopt m+¼mem+,em+¼1=(1+2mQ c), Qc¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 1Q21þc2 2Q22p ,Q1¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^g2 1þ^g2 2þ^g2 3p ,Q2¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^g2 4þ^g2 5p ,a n d ^gi¼gi=k2. In the following, we choose μ,0 for p-type carri- ers,14,25m,0 according to Ref. 31,a n d c1c2.0 for simplicity. Such choice guarantees m/C0to be always negative but leads to three parameter regimes depending on the sign of mþ: (I) mþ,0 (normal band structure), (II) mþ.0 (inverted band structure), and (III) the sign of mþbeing angular dependent1(seeFig. 1 ). In partic- ular, the ξþband in regime III forms a saddle point and its corre- sponding Fermi surface is unbounded, which is just an artifact of the Luttinger model. To overcome this issue, we introduce a momentum cut-off Λin this regime and only care about the Fermi surface inside Λsince the Luttinger model is only valid around the Γpoint.1Such a momentum cut-off is not included in regimes I and II since the Fermi surfaces are closed and finite. To study the superconductivity, we introduce a minimal O(3)-invariant attractive interaction in the s-wave singlet and iso- tropic d-wave quintet channels as HI¼1 2VX qV0P0(q)Py 0(q)þV1P1(q)Py 1(q)hi , (4) where P0(q)¼P kcy kþq 2(Γ0γ=2)(cy /C0kþq 2)T,P1(q)¼P kcy kþq 2(a2gk /C1Γγ=2)(cy /C0kþq 2)T,V0,0, and V1,0 stand for the attractive interaction in singlet and quintet channels, respectively, cy k¼(cy k,3 2,cy k,1 2,cy k,/C01 2,cy k,/C03 2) creates a j¼3 2fermion with wavevector k,Vis the volume, and ais the lattice constant.1The above attrac- tive interaction only applies to the electrons near the Fermi energy within the energy cut-off ϵc. Compared to Ref. 1, we include a non-zero qin the interaction term Eq. (4), which is essential for the study of the upper critical field. The mean-field gap functionderived from Eq. (4)reads Δ(k,q)¼Δ 0(q)Γ0γ 2þΔ1(q)a2gk/C1Γγ 2, (5) where Δ0(q)a n d Δ1(q) are order parameters in the singlet and FIG. 1. The solid lines in (a) –(c) plot the typical band structure in regimes I –III, respectively. The red dashed lines stand for the position of the chemical poten-tialμ. The dashed purple line in (c) shows the realistic band structure beyond the Luttinger model.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063904 (2020); doi: 10.1063/5.0013596 128, 063904-2 Published under license by AIP Publishing.quintet channels, respectively. Since the two channels belong to the same irreducible representation of O(3), the mixing between them in the linearized gap equation is allowed by the symmetryeven if O(3) is reduced to O horTdby the non-interacting Hamiltonian.1,32–38 To study spin susceptibility and upper critical field, a uniform magnetic field Bis required to couple to the electrons. We assume the magnetic field is small enough so that only thefirst order of B¼jBjis kept, which is suitable for the calculation of spin susceptibility but restric ts the study of the upper critical field at the temperature close to th e zero-field critical tempera- ture. Although the magnetic field does not influence the elec- tron –electron interaction, it changes the non-interacting Hamiltonian Eq. (1)via two effects: the Zeeman effect and the orbital effect. 39–46The Zeeman effect can be included by directly adding the following term to Eq. (1): hΓ8 Z¼2μB 3B/C1J, (6) where J¼(Jx,Jy,Jz) are angular momentum matrices for j¼3 2, μB¼e/C22h 2meis the Bohr magneton, eis the elementary charge, and me is the rest mass of the electron (see more details in Appendix A ). The orbital effect can be included by choosing the symmetric gauge for the vector potential A(r)¼B/C2r 2and adding the vector potential to the model h(k)þhΓ8 Zwith the Peierls substitution.39–46 As the proper choice of the bases can simplify further derivation, we first project h(k)þhΓ8 Zonto ξ+bands before performing the Peierls substitution, resulting in the projected h(k)þhΓ8 Zas Ξ+(k,B)¼ξ+(k)þCkp+(^k)/C1σþB/C1M+(^k), (7) where p+(^k)/C1σandM+(^k) are the corresponding 2 /C22 blocks of the projected2ffiffi 3p^k/C1Vand2μB 3Jonξ+bands, respectively, ^k¼k=k, p+(/C0^k)¼/C0p+(^k), and σ¼(σx,σy,σz) are Pauli matrices for the double degeneracy of each band, M+(/C0^k)¼M+(^k), and Tr[M+(^k)]¼0. In Eq. (7), we neglect the terms of orderCk 2Qck2 because the energy scale of SSOC near the Fermi surface is typi- cally much larger than that of ASOC, e.g., 2 Qck2 F/difference20 meV and CkF/difference4 meV for YPtBi13,14,25with kFbeing the magnitude of the Fermi momentum. Including the orbital effect in Eq. (7)via the Peierls substitution leads to the effective non-interacting Hamiltonian in the magnetic field, which reads Ξ+(K,B)¼h+(k)þB/C1M+(^k)þe /C22h∇kh+(k)/C1A(i∇k) (8) with K¼kþe /C22hA(i∇k)a n d h+(k)¼ξ+(k)þCkp+(^k)/C1σ. Equation (8), together with (4), forms the model that we use for the spin susceptibility and the upper critical field.III. SPIN SUSCEPTIBILITY The spin susceptibility χijcan be defined as χij¼@Mspin i @Bj/C12/C12/C12/C12/C12 B!0, (9) where Mspin iis the ith component of the magnetic moment gener- ated by the spins of conduction electrons.47The spin susceptibil- ity of a material in the superconducting phase χS ijis typically different from that in the normal metal phase χN ijdue to the for- mation of Cooper pairs. Such difference causes Knight shifts48,49 in nuclear-magnetic-resonance (NMR) experiments, which serves as an important experimental tool to identify the pairing form. In this section, we study the spin s usceptibility of the singlet – quintet mixed superconducting state. We first analyze the symmetry properties of χSand χN. According to the definition of χij(9), the spin susceptibility satisfies χij¼P i0,j0Rii0Rjj0χi0j0for any operation ^Rin the point group of the material, where Rii0represents the transformation of a pseudo- vector under ^R. The model considered here (1),(4)hasTdsymme- try, meaning that χNsatisfies Tdsymmetry. In the zero magnetic field limit,47,50,51we consider uniform order parameters in the superconducting phase, i.e., Eq. (5)is zero for q=0. Such pairing hasO(3) symmetry,1implying that χSis also Tdinvariant. Since Rii0belongs to T1irreducible representation of Tdgroup, χS ij¼χSδij and χN ij¼χNδijcan be derived from Schur ’s lemma.52Thus, χS ij andχN ijare isotropic, which simplifies our calculations. Following Ref. 50, the spin susceptibilities in superconducting phase and normal metal phase read χS χN¼1/C0VN0 βχNX λ,ωnðdΩ 4πθ(emλ)em3=2 λmz λ(^k)þ/C22mz λ(^k) 2πjdλ(kF,λ)j2 (jdλ(kF,λ)j2þω2 n)3=2( þmz λ(^k)/C0/C22mz λ(^k) 2πjdλ(kF,λ)j2 (jdλ(kF,λ)j2þω2 n)1=2(jdλ(kF,λ)j2þα2 λ(^k)þω2 n)) (10) and χN¼VN0X λðdΩ 4πθ(emλ)em3=2 λmz λ(^k), (11) respectively. Here, β¼1 kBT,kBis the Bolzmann constant, ωn¼(2nþ1)π=βis the fermionic Matsubara frequency, N0 ¼4π (2π)3jmjffiffiffiffiffiffiffiffiffi2mμp,mλ z(^k)¼Tr[Mλ z(^k)Mλ z(^k)],/C22mλ z(^k)¼Tr[Mλ z(^k)^pλ(^k)/C1 σMλ z(^k)^pλ(^k)/C1σ],^pλ(^k)¼pλ(^k)=pλ(^k),pλ(^k)¼jpλ(^k)j,dλ(k)¼Δ0 2 þλΔ1 2a2k2sgn(c1)fQ,Δ0,1are uniform order parameters in singlet and quintet channels, respectively, fQ¼(jc1jQ2 1þjc2jQ2 2)=Qc, αλ(^k)¼jCkF,λjpλ(^k), and the terms of order 1 =(βϵc),αλ=ϵc,jdλj=ϵc and ϵc=jμjare neglected (see Appendix B for more details). Moreover, since the pairing can only exist within the energy cut-offJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063904 (2020); doi: 10.1063/5.0013596 128, 063904-3 Published under license by AIP Publishing.ϵcof the attractive interaction and normally we have ϵc/C282QckF (the ratio is around 0.04 –0.4 for ϵc=kB¼10–100 K), we project the pairing onto ξ+bands and neglect the inter-band contribution. In the case where only one of the λ¼+bands is cut by the Fermi energy and Mλ(^k)¼μBσand the system is isotropic, Eqs. (10) and(11) match the results in Ref. 50. In particular, we focus on the zero-temperature limit of Eq.(10). One should be careful that T!0 limit and dλ!0 limit are not exchangeable, and dλ!0 limit, if needed, should be per- formed before T!0 limit since the later is not physically achiev- able. Although TNLS indicates dλcan be zero along some lines on the Fermi surface, such lines can be neglected in Eq. (10) since they do not cause any divergence and have zero measure in the surface integration. After summing over ωn, Eq. (10) at zero tem- perature reads χS χN/C12/C12/C12/C12 T!0¼1/C0VN0 χNX λðdΩ 4πθ(emλ)em3=2 λ /C2mz λ(^k)þ/C22mz λ(^k) 2þmz λ(^k)/C0/C22mz λ(^k) 2Jjdλ(kF,λ)j αλ(^k)/C18/C19"# , (12) where J(x)¼x2ffiffiffiffiffiffiffiffi 1þx2p ln (1þffiffiffiffiffiffiffiffi 1þx2p x). According to Eq. (12), a non- vanishingχS χN/C12/C12/C12 T!0comes from the ASOC term.50In the limit of zero ASOC, i.e., C!0 or equivalently αλ!0, we findχS χN/C12/C12/C12 T!0¼0 using J(x!þ1)¼1. On the other hand, if ASOC is much larger than the superconducting gap on the Fermi surfaceα λ/C29dλ,E q . (12) is simplified as χS χN/C12/C12/C12/C12 T!0¼1/C0VN0 χNX λðdΩ 4πθ(emλ)em3=2 λmz λ(^k)þ/C22mz λ(^k) 2(13)using J(x!0)¼0. This expression is generally non-zero. Figures 2(a) –2(c) show the behavior ofχS χN/C12/C12/C12 T!0as a function of the ratio between ASOC and pairing amplitude in regimes I –III, respectively. We find theχS χN/C12/C12/C12 T!0drops to zero for zero ASOC and approaches to the limit set by Eq. (13) (dashed lines in Fig. 2 ) when ASOC increases. We can also see thatχS χN/C12/C12/C12 T!0is not sensitive to SSOC, and Eq. (13) gives a slightly smaller value in regime II than those in regimes I and III. Based on this calculation, we arrive at the following conclu- sions. (1) Unlike the singlet –triplet mixing with a non-zeroχS χN/C12/C12/C12 T!0, zero-temperature spin susceptibility can be zero for singlet –quintet mixing. This is because the singlet –triplet mixing is from ASOC and the singlet –quintet mixing comes from SSOC, whileχS χN/C12/C12/C12 T!0is only sensitive to ASOC. This indicates that in some centrosymmet- ric SCs with j¼3=2 (e.g., anti-perovskite materials15), even if one measures a vanishing zero-temperature spin susceptibility, the pos- sibility of singlet –quintet mixing cannot be excluded. (2) In half-Heusler SCs, such as YPtBi, since the energy scale of ASOCnear the Fermi surface ( /difference4 meV) is much larger than the gap function of the similar order as k BTc/difference0:06 meV, a non-zero χS χN/C12/C12/C12 T!0is expected for the singlet –quintet mixed pairing. We notice that the situation here is similar to the case of other non- centrosymmetric SCs with “spin-1/2 ”electrons.50 IV. UPPER CRITICAL FIELD In this section, we study the upper critical field Bc,2, at which the superconductivity is destroyed by the external magnetic field,53 in our model. The upper critical field can be obtained by solving the linearized gap equation with a non-zero magnetic field. To do so, we need to first derive the single particle Green function basedon Eq. (8). Although Eq. (8)is derived with the infinitesimal FIG. 2. The solid lines in (a) –(c) plot the zero-temperature ( T¼0) spin susceptibility χS=χNas a function of the ratio between ASOC and pairing Cffiffiffiffiffiffiffiffi 2mμp =~Δ0in regimes I –III, respectively. The dashed lines show the zero-temperature spin susceptibility at large ASOC limit ( C!1), which is given by Eq. (13).eΔ1=eΔ0¼1:6 and c2¼2c1are chosen for every graph, where eΔ0¼sgn(c1)Δ0andeΔ1¼2mμa2Δ1.j2mjc1¼0:4,j2mjc1¼1:2, and j2mjc1¼0:6 are chosen for (a) –(c), respectively, and a finite momentum cut-off Λ=ffiffiffiffiffiffiffiffi 2mμp ¼3 is set for (c).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063904 (2020); doi: 10.1063/5.0013596 128, 063904-4 Published under license by AIP Publishing.magnetic field in Sec. II, it still holds even if the field is not infini- tesimal, since in half-Heusler materials, the energy scale of the Zeeman term ( μBB/difference0:1m e V f o r B/difference2Tas the typical zero- temperature upper critical field3–8,10) is much smaller than the energy gap between Γ8andΓ7bands ( jEΓ8/C0EΓ7j/difference1e V8,13,31,54) and the energy scale of SSOC near the Fermi surface (2Qck2 F/difference20 meV for YPtBi13,14,25). However, we still restrict Bto be small (though not infinitesimal) and only keep the leading order of B, which requires us to focus on the temperature Tclose to the zero-field critical temperature Tcfor which Bc,2is small enough. In addition, we neglect the ASOC, i.e., C¼0, for simplicity. As a result, we have the effective Hamiltonian for each band E+(K,B)¼ξ+(k)þB/C1M+(^k)þe /C22h∇kξ+(k)/C1A(i∇k), (14) which is just Eq. (8)with C¼0. Furthermore, the corresponding Green function G+(r1,r2,ωn) for each band reads G+(r1,r2,ωn)¼e/C0ie /C22hr1/C1A(r2)eGλ(r1/C0r2,ωn), (15) whereeGλ(r,ωn)¼1 VP keik/C1reGλ(k,ωn) and eG+(k,ωn)¼1 iωn/C0ξ+(k)þB/C1M+(k) (iωn/C0ξ+(k))2: (16) To the first order of B, it is clear that the orbital effect only appears in the phase factor of G+(r1,r2,ω) (see Appendix C for details). As mentioned above, the upper critical field is solved via line- arized gap equation,53which is derived from the superconducting Ginzburg –Landau free energy FSCto the second order of the order parameter. Typically, it is legitimate to choose 1 =(βϵc)/C281, ϵc=2Qck2 F/C281, and ϵc=jμj/C281, leading to the form of FSCas FSC¼/C0X a1ð d3r1 2eVa1jeΔa1(r)j2 /C01 2X a1,a2ð d3reΔ* a1(r)(eKa1a2 0þeKa1a2 1D2)eΔa2(r), (17) where a1,a2¼0, 1,eΔ0¼sgn(c1)Δ0,eΔ1¼2mμa2Δ1,eV0¼V0, eV1¼(2mμa2)2V1, eK0¼xN0 2y1y2 y2y3/C18/C19 , (18) eK1¼/C0N0 2β2μ 2mz1z2 z2z3/C18/C19 , (19) x¼ln (2e/C22γβϵc=π) with /C22γ¼0:577 ... Euler ’s constant, D¼/C0i∇rþe /C22h(B/C2r), and the expressions of y1,2,3and z1,2,3are shown in Appendix A .FSConly contains the orbital effect because the Zeeman effect in FSCcan only appear as the second order of B. In addition, FSCdoes not depend on the direction of Bsince D2is isotropic, meaning that the upper critical field is isotropic near thecritical temperature (see Appendix D for more details.) Performing the functional derivative on Eq. (17) with respect to eΔ* aleads to the linearized gap equation that reads 1 xeΔ0 eΔ1/C18/C19 ¼1 2λ0ey1λ0ey2 λ1ey2λ1ey3/C18/C19eΔ0 eΔ1/C18/C19 , (20) where λ0,1¼/C0N0eV0,1,ey1¼y1/C0β2μ2 2mμxz1l2,ey2¼y2/C0β2μ2 2mμxz2l2, ey3¼y3/C0β2μ2 2mμxz3l2, and l2¼4eB /C22h(nþ1 2)þk2 3is the eigenvalue of D2with n/C210 and k3being the component of the momentum in the direction of the magnetic field. The upper critical field Bc,2can be obtained by solving the above equation with a fixing tempera-tureTbelow T cand the solution gives Bc,2 B0¼T Tc/C01 α(βcϵc)2xc, (21) where B0¼8/C22hmμϵ2 c 2eμ2,Tcis given by1 lnTc T0/C18/C19 ¼/C04ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (λ0y1/C0λ1y3)2þ4λ0λ1y2 2q þλ0y1þλ1y3, (22) T0¼2e/C22γϵc πkB,βc¼1=(kBTc),xc¼ln (2e/C22γβcϵc π)¼ln (T0 Tc), and α¼/C0z1λ0/C0z3λ1 þ(/C0λ0z1þλ1z3)(y1λ0/C0y3λ1)/C04y2z2λ0λ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (y1λ0/C0y3λ1)2þ4y2 2λ0λ1q :(23) As a result, the slope /C0dBc,2=dTat the zero-field critical tempera- ture has the form /C0dBc,2=B0 dT=T0¼T0 (/C0α)(βcϵc)2xcTc¼1 /C0α2e/C22γ π/C18/C192Tc T01 xc: (24) Below we label X(α)¼1 /C0α2e/C22γ π/C0/C1 2for short. The temperature depen- dence of Bc,2for the pure singlet (quintet) channel can be obtained by setting λ1¼0(λ0¼0) in Eq. (21) (see Appendix E for more details). The blue line of Fig. 3(a) shows the slope RBT¼/C0dBc,2=dTat T¼Tcas a function of SSOC strength. The slope in the regimes I and III is significantly larger than that in the regime II, which can be attributed to the behavior of the functionTc T0xc, as shown by the red line of Fig. 3(a) and in Eq. (24). With increasing j2mc1j,w e find a peak of the slope RBTappearing in the regime I close to the I–III boundary, and then the slope drops rapidly to a dip around the point CinFig. 3(a) . Another small peak is found in regime III, and then the slope drops to almost zero due to the extremely small Tcin regime II. The behavior of the slope RBTis mainly determined by the functionTc T0xc[see the red line in Fig. 3(a) ] except when j2mc1jis tuned to the boundary between regimes I and III. The dif- ference there is attributed to the rapid decrease of the factor X(α)Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063904 (2020); doi: 10.1063/5.0013596 128, 063904-5 Published under license by AIP Publishing.when increasing j2mc1jtoward the I –III boundary, as shown in Fig. 6 ofAppendix E .Figures 3(b) –3(d) show the upper critical field of the pairing mixed state (blue lines) as a function of temper-atures (close to T c) for typical parameters in regimes I, II, and III, respectively. The upper critical fields for pure singlet pairing(orange lines) and quintet pairing (red lines) are also shown in these figures for comparison. It is clear that the mixing can increase the upper critical fields in Eq. (20). Moreover, we can see Bc,2of the quintet channel is larger than that of the singlet channel inregime III as shown in Fig. 3(d) while the opposite happens in regimes I and II [ Figs. 3(b) and3(c)], which coincides with the fact that the quintet channel can be dominant around regime III. In conclusion, the upper critical field B c,2close to the zero- field critical temperature Tcis isotropic and can be enhanced by the singlet –quintet mixing. The slope dBc,2=dTat the zero-field critical temperature is mainly determined by the zero-field critical temperature Tc. The slope is much larger in regimes I and III than that in regime II mainly due to its Tcdependence and reaches its maximum around the boundary between I and III as a result of theinterplay between the T candαdependence in Eq. (24). V. EFFECT OF RANDOM NON-MAGNETIC DISORDER In this section, we study the effect of weak random non- magnetic disorder on the singlet –quintet mixed SC. The non- magnetic disorder is included in the Hamiltonian as Hdis¼ð d3rV(r)cy rcr, (25) where cy ris the Fourier transformation of cy kand V(r) is the random potential describing the disorder scattering. The probabil- ity measure of the disorder configuration is chosen as P[V]¼exp/C01 2γ2 dð d3rV2(r)/C20/C21 , (26) and thereby the spatial correlation of V(r) is just the delta function V(r)V(r0) hidis¼γ2 dδ(r/C0r0), where γ2 dmeasures the strength of the disorder with larger γ2 dmeaning stronger disorder. In order to carry out the disorder average, we use the Replica trick55which, in our case, is equivalent to eliminating all fermionic loops inFeynman diagrams, as elaborated in Appendix F . We assume the disorder is weak:γ2 dNF jμj/C281 with NF¼N0y1the density of states at the Fermi energy without considering the spin index. In this case, we consider the self-energy correction [ Fig. 4(a) ] and vertex correc- tion [ Fig. 4(b) ] with the Born approximation, where all Feymann FIG. 3. (a) Shows the slope (blue) RBT¼/C0d(Bc,2=B0)=d(T=T0) at the zero- field critical temperature ( Tc) and the Tcfactor (red) of the slope as functions of SSOC j2mc 1j.I–III stand for the three regimes. (b) –(d) depict the upper critical field Bc,2at various temperatures for points A –C in (a), receptively. In (b) –(d), blue, orange, and red lines stand for mixed, singlet, and quintet channels, respectively, and the dashed parts are not precise since Eq. (21) is only suitable forTclose to Tc. The missing quintet channel in (c) is because it is too small. c2¼2c1is chosen for all figures, j2mc 1j¼ 0:4 for (b), j2mc 1j¼ 1:2 for (c), andj2mc 1j¼ 0:6 andΛ¼3ffiffiffiffiffiffiffiffi 2mμp for (d). The interaction parameter choices areλ1¼0:1λ0¼0:02 for (a) and the mixed channel in (b) –(d). The singlet (quintet) channel in (b) –(d) is determined by setting λ0(λ1) to be the same as the mixed channel and λ1(λ0) to be zero. FIG. 4. Feynman diagrams [(a) and (b)] show the equations of the exactpropagator and vertex via the replica trick with the Born approximation, respectively. The dashed line standsfor the disorder, and the solid singleline and the solid double lines are the bare and exact fermionic propagators, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063904 (2020); doi: 10.1063/5.0013596 128, 063904-6 Published under license by AIP Publishing.diagrams with crossed disorder lines are neglected since those terms have higher orders ofγ2 dNF jμj55(see Appendix F for the defini- tion of disorder lines). As a result, the linearized gap equation in the presence of the disorder reads (see Appendix G for details) ~Δ0 ~Δ1/C18/C19 ¼xλ0 2y1λ0 2y2 λ1 2y2λ1 2y3b1 ! ~Δ0 ~Δ1/C18/C19 , (27) where ϵc=jμj/C281,ϵc=(2Qck2 F)/C281,βϵc/C291, and 1 =(ϵcτd)/C281 are used. The disorder contribution only appears in the function b1, which is given by b1¼1þF(β 4πτd) xy2 2 y1y3/C01/C18/C19 , (28) withF(β 4πτd)¼Ψ(0)(β 4πτdþ1 2)/C0Ψ(0)(1 2) which is defined in Ref. 56, Ψ(0)(/C1/C1/C1) is the digamma function, and 1 =τd¼γ2 dπNF. The critical temperature Tcan be solved from Eq. (27)in the presence of disorderfor the mixed state as ln(T T0)¼/C04ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (λ0y1/C0λ1y3b1)2þ4λ0λ1y2 2q þλ0y1þλ1y3b1:(29) The critical temperature expression for the pure singlet (quintet) channel can be given by setting λ1(λ0)t ob ez e r o ,w h i c hg i v e s lnTs T0/C18/C19 ¼/C02 λ0y1(30) and lnTq T0/C18/C19 ¼/C02 λ1y3b1: (31) Equations (27) –(31) are the main results of this section and form the basis for our analysis of the disorder effect of singlet – quintet mixing pairing. The disorder scattering is controlled by a single function b1in the linearized gap equation (27) and it is FIG. 5. The impacts of weak non-magnetic random disorder on the critical temperature Tdis cin singlet (orange), quintet (red), and mixed (blue) channels in regimes I –III are shown in (a) –(c), respectively, and the pairing ratios as functions of disorder strength in regimes I –III are shown in (d) –(f), respectively. ( τdϵc)/C01measures the strength of disorder with larger ( τdϵc)/C01meaning stronger disorder. Due to the limitation of weak disorder scattering [( τdϵc)/C01/C281] for Eq. (27), we use the dashed line for a rela- tive large ( τdϵc)/C01.eΔ1=eΔ0stands for the pairing ratio between the quintet and singlet channels. The band structure parameter choices are c2¼2c1for all figures, j2mc 1j¼ 0:4 for (a) and (d), j2mc 1j¼ 1:2 for (b) and (e), and j2mc 1j¼ 0:6 andΛ¼3ffiffiffiffiffiffiffiffi 2mμp for (c) and (f ). λ1¼0:1λ0is chosen for (d) –(f) and the mixed channel of (a) –(c), with λ0¼0:2 for (d) and the mixed channel of (a), λ0¼5 for (e) and the mixed channel of (b), and λ0¼0:05 for (f ) and the mixed channel of (c). The missing quintet channel in (b) is because it is too small. The interaction parameters for the pure singlet(quintet) channel in (a) –(c) are given by choosing λ1(λ0) to be zero while λ0(λ1) is the same as the corresponding mixed channel.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063904 (2020); doi: 10.1063/5.0013596 128, 063904-7 Published under license by AIP Publishing.found (see Appendix G 4 )t h a t0 ,b1/C201w i t h b1¼1o n l y occuring either in the clean limit (1 =τd¼0) or for the isotropic case in regime II. By inspecting Eqs. (27) –(31),w ec a nd r a wt h e following conclusions. (1) The pure s-wave singlet channel is notinfluenced by the non-magnetic disorder under the condition thatwe neglect the inter-band scattering. This conclusion is consistent with and required by the Anderson theorem. 57(2) An interesting observation for Eq. (27) is that the disorder scattering (the b1 function) only appears in the d-wave quintet channel (the y3 term) due to the momentum dependence of d-wave function but the coupling between singlet and quintet channels is independent of the disorder scattering. By examining the derivation of Appendix G 4, we find that such behavior originates from the cancelation between the self-energy correction and the vertex cor-rection for the singlet –quintet coupling term, which is similar to the stable s-wave singlet pairing. Therefore, our calculation sug- gests that the cancelation, which appears for the s-wave singlet pairing, also works for the singlet –quintet coupling term, at least at the level of Born approximation. As shown in the following,such cancelation has a substantial influence on the pairing formin the disordered SCs. According to Eq. (27), if there is no mixing term (the y 2¼0 case), the quintet pairing is completely con- trolled by the y3b1term and a small value of b1from the disorder effect will greatly suppressed the quintet pairing. In contrast, for alarge y 2term, due to its independence of disorder scattering, a significant quintet pairing channel can still be induced through the mixing effect even if the value of b1is small. Therefore, the mixing effect stabilizes the quintet pairing channel against theweak non-magnetic disorder scattering. We further plot the calculated critical temperature with disorder T dis cas a function of disorder scattering strength in Figs. 5(a) –5(c) for singlet –quintet mixed pairing (blue lines), pure singlet pairing (orange lines), and pure quintet pairing (red lines) in the regimes I–III, respectively. We find the Tdis cfor the pure singlet pairing is always independent of disorder scattering, as expected, while the Tdis c for the pure quintet pairing and mixed pairing decays with increas- ing the disorder strength 1 =τd. The small decay magnitude is due to the weak scattering potential approximation1 τdϵc/C281t h a ti su s e di n our theory. Beside the critical temperatures, the expression of the pairing ratio can also be solved from Eq. (27) and reads eΔ1 eΔ0¼2 xλ0y2/C0y1 y2: (32) Since x, which depends on the critical temperature, increases as 1=τdincreases, the pairing ratio eΔ1=eΔ0would generally decrease if the disorder strength 1 =τdincreases with the isotropic system in regime II being the exception. The decreasing pairing ratio eΔ1=eΔ0 is shown in Figs. 5(d) –5(f) for regimes I –III, respectively. VI. CONCLUSION In this work, we studied the zero-temperature spin susceptibil- ity, the upper critical field near the zero-field critical temperatureTc, and the non-magnetic disorder scattering of the SCs with j¼3 2 fermions in the presence of the mixing between s-wave singlet and isotropic d-wave quintet channels. Our results show that the spinsusceptibility is isotropic due to the T dgroup symmetry and zero (non-zero) at zero temperature without (with) ASOC. As a result, the zero-temperature spin susceptibility given by the singlet – quintet mixing is zero in centrosymmetric SCs, e.g., anti-perovskite materials, 15but non-zero in non-centrosymmetric SC YPtBi due to the large energy scale of ASOC near the Fermi surface (/difference4 m e V )c o m p a r e dw i t ht h eg a pf u n c t i o n( kBTc/difference0:06 meV). The spin susceptibility can be mea sured in the NMR-Knight shift experiment.48,49Near Tcand without ASOC, it is found that the upper critical field is isotropic and enhanced by the pairingmixing. The slope /C0dB c,2=dTatTcvaries with the SSOC strength, and it is the largest in the intermediate region between regimes I and III and smallest in regime II. Finally, our results on therandom non-magnetic disorder effect demonstrate that thes-wave singlet channel as well as the singlet –quintet coupling in the linearized gap equation are not influenced by the weak disor- der within the Born approximation if neglecting the inter-bandscattering. This suggests that the singlet –quintet mixing, as well as the nodal-line superconductivity, found in Ref. 1are stable against the weak non-magnetic disorder scattering in real materi- als. Nevertheless, it is intriguing to ask whether the singlet – quintet mixing is stable against non-magnetic disorder witheffective SOC component. The Bogloiubov excitations for singlet – quintet pairing states may carry non-trivial spin structure inanalogy to the carriers in Luttinger semiconductors, 58,59which is worth future consideration. ACKNOWLEDGMENTS J.Y. thanks Yang Ge, Rui-Xing Zhang, Jian-Xiao Zhang, and Tongzhou Zhao for helpful discussions. C.-X.L. and J.Y. acknowl-edge the support from the Office of Naval Research (ONR) (GrantNo. N00014-15-1-2675 and Renewal No. N00014-18-1-2793) andthe U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES) under Award No. DE-SC0019064. APPENDIX A: CONVENTION AND EXPRESSIONS The Fourier transformation of creation operators in the con- tinuous limit reads c y r¼1ffiffiffiffi VpX ke/C0ik/C1rcy k, (A1) where Vis the total volume. The Fourier transformation of corre- sponding Grassmann field in the continuous limit reads /C22cτ,r¼1ffiffiffiffiffiffiβVpX ωn,keiωnτ/C0ik/C1r/C22ck,ωn, (A2) where β¼1=(kBT).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063904 (2020); doi: 10.1063/5.0013596 128, 063904-8 Published under license by AIP Publishing.The five d-orbital cubic harmonics read60 gk,1¼ffiffiffi 3p kykz, gk,2¼ffiffiffi 3p kzkx, gk,3¼ffiffiffi 3p kxky, gk,4¼ffiffi 3p 2(k2 x/C0k2 y), gk,5¼1 2(2k2 z/C0k2 x/C0k2 y):8 >>>>>>< >>>>>>:(A3) Thej¼ 3 2angular momentum matrices are30 Jx¼0ffiffi 3p 200ffiffi 3p 201 0 01 0ffiffi 3p 2 00ffiffi 3p 200 BBBBB@1 CCCCCA, (A4) J y¼0/C0iffiffi 3p 200 iffiffi 3p 20/C0i 0 0 i0/C0iffiffi 3p 2 00iffiffi 3p 200 BBBBB@1 CCCCCA, (A5) J z¼3 200 0 01 200 00/C01 20 00 0 /C03 20 BBB@1 CCCA: (A6) The five Gamma matrices are 60 Γ1¼1ffiffi 3p(JyJzþJzJy), Γ2¼1ffiffi 3p(JzJxþJxJz), Γ3¼1ffiffi 3p(JxJyþJyJx), Γ4¼1ffiffi 3p(J2 x/C0J2 y), Γ5¼1 3(2J2 z/C0J2 x/C0J2 y):8 >>>>>>>< >>>>>>>:(A7) The Luttinger Hamiltonian h(k) with C¼0 can be diagonal- ized by the unitary transformation 60 U(^k);(1þc2gk,5 k2Qc)Γ0þiP3 a¼1c1gk,a k2QcΓa5þic2gk,4 k2QcΓ45 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(1þc2gk,5 k2Qc)q D, with D¼1000 0010 0001 01000 BB@1 CCA:ForC¼0, this leads to U y(^k)h(k)U(^k)¼ξþ 000 0ξþ 00 00 ξ/C0 0 000 ξ/C00 BB@1 CCA: The expressions of jj,j ziwith j¼3=2 and jz¼+3=2,+1=2 in terms of the electron spin and atomic porbitals30are j3=2, 3=2i¼/C01ffiffiffi 2pjXþiYij "i, j3=2, 1=2i¼1ffiffiffi 6p(2jZij "i /C0 j XþiYij #i), j3=2,/C01=2i¼1ffiffiffi 6p(2jZij #i þ j X/C0iYij "i), j3=2,/C03=2i¼1ffiffiffi 2pjX/C0iYij #i, where jXi,jYi, andjZiare atomic porbitals and real. The expressions of y1,2,3,4,5 andz1,2,3in Eqs. (18) and(19) are y1¼X λðdΩ 4πθ(emλ)em3=2 λ, y2¼X λλðdΩ 4πθ(emλ)em5=2 λfQ, y3¼X λðdΩ 4πθ(emλ)em7=2 λf2 Q, y4¼X λλðdΩ 4πθ(emλ)em9=2 λf3 Q, y5¼X λðdΩ 4πθ(emλ)em11=2 λf4 Q, z1¼7ζ(3) 16π2X λðdΩ 4πθ(emλ)em3=2 λ(evλ z)2, z2¼7ζ(3) 16π2X λλðdΩ 4πθ(emλ)em5=2 λfQ(evλ z)2, z3¼7ζ(3) 16π2X λðdΩ 4πθ(emλ)em7=2 λf2 Q(evλ z)2,(A8) where fQ¼(jc1jQ2 1þjc2jQ2 2)=Qc,(evλ z)2¼(vλ z(kF,λ))22mμ=(μ2), vλ z(k)¼@kzξλ(k), and θ(...) is the Heaviside step function. To include the momentum cut-off in those y’s and z’s, just do the fol- lowing replacement: θ(emλ)!θ(emλ)θΛ2 2mμ/C0emλ/C18/C19 ¼θ1 emλ/C02mμ Λ2/C18/C19 , (A9) where the extra factor in the second expression is given by the fact that the momentum cut-off requires 2 mλμ/C20Λ2. As mentioned inJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063904 (2020); doi: 10.1063/5.0013596 128, 063904-9 Published under license by AIP Publishing.Sec.II, we neglect the momentum cut-off Λin regimes I and II, which is equivalent to taking Λ!1, and choose a finite value for Λonly in regime III. APPENDIX B: MORE DETAILS ON SPIN SUSCEPTIBILITY In this section, we derive Eqs. (10) and(11) following Ref. 50. The magnetic moment generated by the conduction electron spins has the following expression: Mspin¼1 Zð D/C22cDcX k,ωn/C22ck,ωn/C02μB 3βJck,ωne/C0S, (B1) where Z¼Ð D/C22cDce/C0S,/C22c,care Grassmann fields of the j¼3=2 fermion. The action Scontains two parts S¼SniþSΔ: the non- interacting part Sniand the pairing part SΔ. Below, we talk about these two parts carefully. According to Eq. (9), the derivation of χS,Nonly requires terms up to the first order of the infinitesimal uniform magnetic field. Therefore, Snicontains three parts Sni¼S0þSorb BþSZ B, where S0¼X k,ωn,λ/C22ψk,ωn,λ[/C0iωnþhλ(k)]ψk,ωn,λ (B2) is the non-magnetic part, Sorb B¼X k,ωn,λ/C22ψk,ωn,λ0 e/C22h∇khλ(k)/C1A(i∇k)/C20/C21 ψk,ωn,λ (B3) is the orbital part, SZ B¼X k,ωn,λ/C22ψk,ωn,λ[B/C1Mλ(^k)]ψk,ωn,λ (B4) is the Zeeman part, λ¼+and /C22ψk,ωn,λ,ψk,ωn,λare Grassmann fields corresponding to eigen-wavefunctions of ξλband. Now, we discuss the pairing part SΔ. The reason for using the pairing instead of the interaction is that we consider the infinitesi- mal magnetic field in the superconducting phase where the Cooper pairs are already formed. And, we neglect the change of orderparameter due to the magnetic field 47,50,51and only need to con- sider the uniform order parameters here. Moreover, since thepairing can only exist within the energy cut-off ϵ cof the attractive interaction and ϵc/C282QckF, we should also project the pairing onto ξ+bands and neglect the inter-band contribution. Therefore, SΔreads SΔ¼1 2X k,ωn/C22ψk,ωn,λΔλ(k)/C22ψT /C0k,/C0ωn,λ (B5) þ1 2X k,ωnψT /C0k,/C0ωn,λΔy λ(k)ψk,ωn,λ, (B6)where Δλ(k)¼Δ0n0 λ(k)þΔ1n1 λ(k) , and n0 +(k)¼+1 2iσy (B7) and n1 +(k)¼1 2k2a2sgn(c1)fQiσy (B8) are pairing matrices projected to ξ+bands. Now, we have S¼S0þSorb BþSZ BþSΔ. Clearly, Shave fermion parity symmetry for either of λ¼+subspace since they are decoupled. As a result, Eq. (B1) can be re-written as Mspin¼/C01 Zβð D/C22ψDψX k,ωn,λ/C22ψk,ωn,λMλ(^k)ψk,ωn,λe/C0S, (B9) where we neglect inter-band terms given by μBJbecause they are odd under the fermion parity for one λsubspace. By defining SM i¼P k,ωn,λ/C22ψk,ωn,λMλ i(^k)ψk,ωn,λ, we have the expression of spin susceptibility χij¼1 βSM i@(Sorb BþSZ B) @Bj/C28/C29 0/C01 βSM i/C10/C11 0@(Sorb BþSZ B) @Bj/C28/C29 0,( B 1 0 ) where hXi0¼1 Z0ÐD/C22ψDψXe/C0S0/C0SΔwith Z0¼ÐD/C22ψDψe/C0S0/C0SΔ.I n the following, we neglect the orbital contribution to χijas done in Refs. 47,51,a n d 50and choose the i,j¼zsince χijis isotro- pic. Eventually, the expression of spin susceptibility becomes χ¼1 β(hSM zSMzi0/C0hSM zi0hSM zi0), (B11) where@SZ B @Bj¼SM jis used. Next, we will derive Eqs. (10) and (11) from the expression presented above. In Nambu representation, S0þSΔis re-written as S0þSΔ¼X0 k,ωn,λ/C22Ψk,ωn,λ[/C0iωnþhBdG λ(k)]Ψk,ωn,λ, (B12) where hBdG λ(k)¼hλ(k)Δλ(k) Δy λ(k)/C0hT λ(/C0k)/C18/C19 , (B13) /C22Ψk,ωn,λ¼(/C22ψk,ωn,λ,ψT /C0k,/C0ωn,λ),Ψk,ωn,λ¼(ψT k,ωn,λ,/C22ψ/C0k,/C0ωn,λ)T,a n dt h e “0”on top ofPmeans only summing over half the region of (k,ωn) with the other half obtained by ( k,ωn)!/C0 (k,ωn). Define GBdG λ(k,ωn)¼[iωn/C0hBdG λ(k)]/C01. The expression of GBdG λ(k,ωn)Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063904 (2020); doi: 10.1063/5.0013596 128, 063904-10 Published under license by AIP Publishing.reads GBdG λ(k,ωn)¼Gλ(k,ωn) Fλ(k,ωn) Fy λ(k,ωn)/C0Gλ(/C0k,/C0ωn)/C18/C19 ,( B 1 4 ) where Gλ(k,ωn)¼Gλ,þ(k,ωn)þGλ,/C0(k,ωn)^pλ(^k)/C1σ,( B 1 5 ) Fλ(k,ωn)¼[Fλ,þ(k,ωn)þFλ,/C0(k,ωn)^pλ(^k)/C1σ]Δλ(k), (B16) Gλ,+(k,ωn)¼/C01 2iωnþEλ,þ(k) ω2 nþjdλ(k)j2þE2 λ,þ(k) +iωnþEλ,/C0(k) ω2 nþjdλ(k)j2þE2 λ,/C0(k)! ,( B 1 7 ) Fλ,+(k,ωn)¼/C01 21 ω2 nþjdλ(k)j2þE2 λ,þ(k) +1 ω2 nþjdλ(k)j2þE2 λ,/C0(k)! ,( B 1 8 ) andEλ,+(k)¼ξλ(k)+jCjkpλ(^k). Then, Z0can be expressed as Z0¼ð D/C22ΨDΨeP0 k,ωn,λ/C22Ψk,ωn,λ[GBdG λ(k,ωn)]/C01Ψk,ωn,λ: (B19) On the other hand, SM zis expressed in the Nambu representation as SM z¼X0 k,ωn,λ/C22Ψk,ωn,λWλ z(^k)Ψk,ωn,λ,( B 2 0 ) where Wλ z(^k)¼diag( Mλ i(^k),/C0[Mλ i(/C0^k)]T). Now, we can work out Eq. (B11) , χ¼/C01 βX0 k,ωn,λTr[GBdG λ(k,ωn)Wλ z(^k)GBdG λ(k,ωn)Wλ z(^k)]:(B21) Using /C0FT λ(/C0k,/C0ωn)¼Fλ(k,ωn), the equation can be simpli- fied into χ¼/C01 βX k,ωn,λTr[Mλ z(^k)Gλ(k,ωn)Mλ z(^k)Gλ(k,ωn)]/C16 /C0Tr[Mλ z(^k)Fλ(k,ωn)[Mλ z(/C0^k)]TFy λ(k,ωn)]/C17 : (B22)Using Tr[Mλ z(^k)Mλ z(^k)^pλ(^k)/C1σ]¼0, χ¼/C01 βX k,ωn,λmz λ(^k)(G2 λ,þ(k,ωn)þF2 λ,þ(k,ωn)jdλ(k)j2)h þ/C22mz λ(^k)(G2 λ,/C0(k,ωn)þF2 λ,/C0(k,ωn)jdλ(k)j2)i : (B23) The spin susceptibility χNin the normal state can also be obtained from Eq. (B22) by choosing zero value for the order parameters dλ(k)¼0. As a result, we can get Eq. (11) by neglecting terms of order 1 =(βϵc),αλ=ϵc, and ϵc=jμj. If the temperature is below Tcand the superconducting order parameters are not zero, Eq. (B22) gives the superconducting spin susceptibility χS. In this case, we can first subtract χSbyχNin order to exchange the sum of ωnwith the energy integration. Then, by neglecting terms of order 1 =(βϵc),αλ=ϵc,jdλj=ϵc, and ϵc=jμj,w e can get Eq. (10). APPENDIX C: NON-INTERACTING GREEN FUNCTION WITH MAGNETIC FIELD In this part, we derive Eq. (16) following Ref. 46. In the con- tinuous limit, the corresponding effective Green function for eachband satisfies the equation [iω n/C0Eλ(Kr1,B)]Gλ(r1,r2,ωn)¼δ(r1/C0r2), (C1) where Kr¼/C0i∇rþe /C22hA(r),ωn¼(2nþ1)π=βis the fermionic Matusbara frequency, λ¼+, and 1 =β¼kBT. Clearly, the Green function Gλ(r1,r2,ωn) is not translationally invariant. Define Gλ(r1,r2,ωn)¼e/C0ie /C22hr1/C1A(r2)eGλ(r1/C0r2,ωn), (C2) resulting that eGλ(r1/C0r2,ωn) satisfying a translationally invariant equation, [iωn/C0Eλ(Kr,B)]eGλ(r,ωn)¼δ(r) (C3) or equivalently [iωn/C0Eλ(K,B)]eGλ(k,ωn)¼1, (C4) witheGλ(r,ωn)¼1 VP keik/C1reGλ(k,ωn). Note that, the derivation shown above uses A(r)¼B/C2r 2,[r/C1k,r/C1A(i∇k)]¼0, and [r0/C1∇r,r0/C1A(r)]¼0. To solve Eq. (C4) analytically, we make another assumption thatBis sufficiently small so that we can treat the magnetic field dependence in the equation as a perturbation, as mentioned in the main text. It means BμB/C28kBTand /C22hωc/C28kBTfor each band, where the latter is for the field dependence in Kand /C22hωc¼2me jmλjBμB is the cyclotron frequency of the band under the magnetic field.46 Since the upper critical field approaches to zero as temperature approaches to the zero-field critical temperature Tc, that assump- tion restricts us to consider the temperature near Tcwhere the upper critical field is small.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063904 (2020); doi: 10.1063/5.0013596 128, 063904-11 Published under license by AIP Publishing.Finally, we solve Eq. (C4) to the first order of /C22hωc=(kBT) fol- lowing Ref. 46. Since /C22hωc=(kBT) is linear in B, we will directly use the order of Bto indicate the order of /C22hωc=(kBT). If B¼0, the zero-field Green function is easy to solve eG+ 0(k,ωn)¼1 iωn/C0ξ+(k): (C5)Since the Periels substitution is given by E+(K,B)¼ð d3rδ(r)E+(i∇r,B)e/C0ir/C1K, (C6) with E+(i∇r,B) obtained by replacing kinE+(k,B)b y i∇r, E+(K,B) to the first order of Bhas the following expression: E+(K,B)¼ð d3rδ(r)(ξ+(i∇r)þB/C1M+(i∇r))e/C0ir/C1K ¼ð d3rδ(r)(ξ+(i∇r)þB/C1M+(i∇r))e/C0ir/C1k1þe 2/C22hr/C1(B/C2∇k)þO(B2)/C16/C17 ¼ξ+(k)þB/C1M+(k)þð d3rδ(r)ξ+(i∇r)e/C0ir/C1ke 2/C22hr/C1(B/C2∇k)þO(B2) ¼ξ+(k)þB/C1M+(k)þe 2/C22hð d3rδ(r)e/C0ir/C1kξ+(kþi∇r)r/C1(B/C2∇k)þO(B2) ¼ξ+(k)þB/C1M+(k)þe 2/C22hð d3rδ(r)e/C0ir/C1k[i∇kξ+(k)/C1∇r]r/C1(B/C2∇k)þO(B2) ¼ξ+(k)þB/C1M+(k)þie 2/C22hv+/C1(B/C2∇k)þO(B2), (C7) where v+¼∇kξ+(k) and the third equality uses the fact that r/C1k commutes with r/C1(B/C2∇k). With the expression shown above, we get the first order correction to the Green function which is eG+ 0(k,ωn)(B/C1M+(k)þie 2/C22hv+/C1(B/C2∇k))eG+ 0(k,ωn): (C8) Note that v+/C1(B/C2∇k)eG+ 0(k,ωn)¼v+/C1(B/C2v+) (iωn/C0ξ+(k))2¼0, (C9) we finally get solution to Eq. (C4) to the first order of B, eG+(k,ωn)¼1 iωn/C0ξ+(k)þB/C1M+(k) (iωn/C0ξ+(k))2: (C10) APPENDIX D: DERIVATION OF (17) In this part, we derive Eq. (17) following Ref. 46. 1. General expression of the superconducting free energy FSC According to Eq. (4), the interacting part of the action reads SI¼ðβ 0dτ1 2VX qX a¼0,1VaPa(q,τ)Py a(q,τ)/C2/C3 , (D1)where τis the imaginary time, /C22ck,τis the Grassman field, P0(q,τ)¼X k/C22ckþq 2,τ(Γ0γ=2)(/C22c/C0kþq 2,τ)T(D2) and P1(q,τ)¼X k/C22ckþq 2,τ(a2gk/C1Γγ=2)(/C22c/C0kþq 2,τ)T: (D3) Using Hubbard –Stratonovich transformation, we have exp(/C0SI)¼ð DΔDΔ*exp"ðβ 0dτX q,a /C2 /C01 2Pa(q,τ)Δa(q,τ)/C01 2Py a(q,τ)Δ* a(q,τ) þVjΔa(q,τ)j2 2Va!# , (D4) where ð DΔDΔ*¼Y τ,q,aVdτ /C02πVað dΔa(q,τ)dΔ* a(q,τ): (D5) Assume that Δi(q,τ) is uniform in τ, and thereby it can be re-labeled as Δi(q). Then, the partition function becomesJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063904 (2020); doi: 10.1063/5.0013596 128, 063904-12 Published under license by AIP Publishing.Z¼ð DcD /C22cDΔDΔ*exp/C0S0þX q,ωn,a/C01 2Pa(q,ωn)Δa(q)/C01 2Py a(q,ωn)Δ* a(q)/C18/C19 þX q,aβV 2VajΔa(q)j2"# , (D6) where S0is the non-interacting action, P0(q,ωn)¼X k/C22ckþq 2,ωn(Γ0γ=2)(/C22c/C0kþq 2,/C0ωn)T, (D7) P1(q,ωn)¼X k/C22ckþq 2,ωn(a2gk/C1Γγ=2)(/C22c/C0kþq 2,/C0ωn)T, (D8) and the Fourier transformation relation Eq. (A2) is used. We can express Piin eigen-wavefunctions of ξ+bands, Pa(q,ωn)¼X k/C22ψkþq 2,ωnna(k,q)/C22ψ/C0kþq 2,/C0ωn/C16/C17T , (D9) where n0(k,q)¼Uykþq 2/C16/C17 (Γ0γ=2)U*/C0kþq 2/C16/C17 (D10) and n1(k,q)¼Uykþq 2/C16/C17 (a2gk/C1Γγ=2)U*/C0kþq 2/C16/C17 : (D11) To simplify na(k,q), we would first neglect the qdependence, i.e.,na(k,q)/C25na(k, 0) re-labeled as na(k). The reason is given below. Typically, the order parameter with large qis not the minimum of the Free energy, and thus jqjis small compared with the Fermi momen- tumkF. Then, we can expand U(kþq 2)i nt e r m so f jqj=kF, U(kþq 2)¼U(k)þq 2/C1∇kU(k)þ/C1/C1/C1 : (D12) Since U(k) only depends on the direction of k,i . e . , U(k)¼U(^k), we thereby have Ukþq 2/C16/C17 ¼U(^k)þq 2k/C1∇^kU(^k)þ/C1/C1/C1 ,( D 1 3 ) where1 k∇^kstands for the angular part of ∇koperator. Then, we can conclude that the qnterm brought by the expression of U(kþq 2)o n the Fermi surface is of order (jqj kF)ncompared with the original term. Theqnterm in the free energy can also be given by the Green func- tion. To estimate that contribution, let us assume an isotropic form of the Green function [ iω/C0(kþq=2)2=(2m*)þμ]/C01.I nt h i sc a s e , jqjn term of the Green function on the Fermi surface is of order (jqj kFjμj kBT)n compared with the original term. Here, we replace 1 =(iω)nby (kBT)n because the other part of 1 =(iω)nwill just contribute to a convergent d i m e n s i o n l e s se x p r e s s i o na f t e rs u m m i n go v e r ω. Since we assume jμj kBT/C291, the qdependence in the U(kþq 2) can be neglected.We further simplify ni(k) by making the approximation na(k)/C25na þ(k) na /C0(k)/C18/C19 , (D14) where na +(k) are shown in Eqs. (B7) and(B8). This approximation is legitimate since the inter-band contribution is of order ϵc=(2Qck2 F)/C281, where ϵcis the energy cut-off of the attractive interaction. Therefore, we have Pa(q,ωn)¼X kX λ¼+/C22ψkþq 2,ωn,λna λ(k)(/C22ψ/C0kþq 2,/C0ωn,λ)T: (D15) Since the Green function does not have translational invari- ance, it is better to deal with the problem in the position space.After the Fourier transformation, we have X q,ωnPa(q,ωn)Δa(q)¼1 Vð d3rPa(r)Δa(r) (D16) with Pa(r1)¼X ωn,λð d3r2/C22ψr1þr2 2,ωn,λna λ(r2)/C22ψr1/C0r2 2,/C0ωn,λ/C16/C17T (D17) and VX qjΔa(q)j2¼ð d3rjΔa(r)j2, (D18) where na λ(r)¼P kna λ(k)eik/C1randΔa(r)¼P qeiq/C1rΔa(q). Then, we have Z¼ð DcD /C22cDΔDΔ*exp" /C0S0þX að d3r /C2 /C01 2VPa(r)Δa(r)/C01 2VPy a(r)Δ* a(r)þβ 2VajΔa(r)j2!# ,(D19) where /C0S0¼X ωn,λð d3r/C22ψr,ωn,λ(iωn/C0Eλ(Kr,B))ψr,ωn,λ: (D20) Using the expression of the Green function Eqs. (15) and(16), we can integrate out the fermionic field and get the effective actionS eff[Δ] with the partition function being Z¼ð DΔDΔ*exp(/C0Seff[Δ]): (D21)Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063904 (2020); doi: 10.1063/5.0013596 128, 063904-13 Published under license by AIP Publishing.Under the mean-field approximation, we have Z/C25exp(/C0Seff[Δ]) (D22) withΔsatisfying δSeff[Δ] δΔ* a(r)¼0: (D23) Then, the mean-field free energy (Ginzburg –Landau free energy) reads F¼/C01 βln (Z)¼1 βSeff, (D24)which gives the superconducting free energy, FSC¼F/C0FN¼1 β(Seff[Δ]/C0Seff[0]), (D25) where FNmeans the mean-field free energy with zero Δ. In order to get the critical temperature of this second-order phase transition, we only need to derive FSCto the second order of Δ, which is FSC¼/C0X að d3r1 2VajΔa(r)j2/C01 2V2 /C2X a1,a2ð d3r1ð d3r2Δ* a1(r1)Sa1a2(r1,r2)Δa2(r2)þO(jΔj4):(D26) Here, Sa1a2(r1,r2)¼X ωn,λð d3ρ1d3ρ21 βTrGλr1þρ1 2,r2þρ2 2,ωn/C16/C17 na2 λ(ρ2)GT λr1/C0ρ1 2,r2/C0ρ2 2,/C0ωn/C16/C17 [na1 λ(ρ1)]yhi ¼X ωn,λð d3ρ1d3ρ21 βTreGλr1þρ1 2/C0r2/C0ρ2 2,ωn/C16/C17 na2 λ(ρ2)eGT λr1/C0ρ1 2/C0r2þρ2 2,/C0ωn/C16/C17 [na1 λ(ρ1)]yhi e/C0ie /C22h[2r1/C1A(r2)þρ1/C1A(ρ2 2)] ¼e/C0i2e /C22hr1/C1A(r2)X ωn,λ1 βX k,qeiq/C1(r1/C0r2)eGλkþq 2,ωn/C16/C17hi β2α1eGT λ/C0kþq 2,/C0ωn/C16/C17hi α2β1[Λ(k)]a1a2 α1α2β1β2, (D27) with the summation over α1α2β1β2implied, and [Λ(k)]a1a2 α1α2β1β2¼1 V2ð d3ρ1d3ρ2e/C0ie /C22hρ1/C1A(ρ2 2)eik/C1(ρ1/C0ρ2)[na2 λ(ρ2)]α1α2[na1 λ(ρ1)]y β1β2 ¼1 V2ð d3ρ1d3ρ2X k0,k00e/C0ie /C22hρ1/C1Aρ2 2ðÞei(k/C0k0)/C1ρ1ei(k00/C0k)/C1ρ2[na2 λ(k00)]α1α2[na1 λ(k0)]y β1β2 ¼e/C0ie 4/C22h∇k0/C1(B/C2∇k00)[na2 λ(k00)]α1α2[na1 λ(k0)]y β1β2/C12/C12/C12 k0,k00!k¼eie 4/C22hB/C1(∇k0/C2∇k00)[na2 λ(k00)]α1α2[na1 λ(k0)]y β1β2/C12/C12/C12 k0,k00!k: (D28) Clearly, as long as a1¼0o r a2¼0, [Λ(k)]a1a2 α1α2β1β2has no magnetic field dependence since n0 λ(k)i skindependent as shown in Eq. (B7). According to Eq. (B8), the contribution to [ Λ(k)]11 α1α2β1β2of first order of Bvanishes since it is proportional to the cross product of two same gradients. Therefore, we have [Λ(k)]a1a2 α1α2β1β2¼[ni λ(k)]y β1β2[nj λ(k)]α1α2(D29) to the first order of B. Using Eqs. (B7),(B8), and (D29) and to the first order of B,w eh a v e eGλkþq 2,ωn/C16/C17hi β2α1eGT λ/C0kþq 2,/C0ωn/C16/C17hi α2β1[Λ(k)]a1a2 α1α2β1β2 ¼TreGλkþq 2,ωn/C16/C17 na2 λ(k)eGT λ/C0kþq 2,/C0ωn/C16/C17 [na1 λ(k)]yhi ¼Tr[na2 λ(k)[na1 λ(k)]y] (iωn/C0ξλ(kþq 2))(/C0iωn/C0ξλ(/C0kþq 2))þTr[B/C1Mλ(kþq 2)na2 λ(k)[na1 λ(k)]y] (iωn/C0ξλ(kþq 2))2(/C0iωn/C0ξλ(/C0kþq 2)) þTr[na2 λ(k)[B/C1Mλ(/C0kþq 2)]T[na1 λ(k)]y] (iωn/C0ξλ(kþq 2))(/C0iωn/C0ξλ(/C0kþq 2))2¼Tr[na2 λ(k)[na1 λ(k)]y] (iωn/C0ξλ(kþq 2))(/C0iωn/C0ξλ(/C0kþq 2)), (D30)Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063904 (2020); doi: 10.1063/5.0013596 128, 063904-14 Published under license by AIP Publishing.where the summation over α1α2β1β2is implied in the first expres- sion and the last equality uses the fact that B/C1M+(k) are traceless. Therefore, the Zeeman coupling does not contribute to the firstorder term of magnetic fields in the free energy. Eventually, we have S a1a2(r1,r2)¼e/C0i2e /C22hr1/C1A(r2)X ωn,λ1 βX k,qeiq/C1(r1/C0r2) /C2Tr[na2 λ(k)[na1 λ(k)]y] (iωn/C0ξλ(kþq 2))(/C0iωn/C0ξλ(/C0kþq 2)):(D31) 2. Simplification of Sa1a2(r1,r2) In this part, we further simplify Sa1a2(r1,r2) following Ref. 46. The Einstein summation notation for repeated indexes is used inthis part. First, consider the expansion of the following expression to the second order of jqj: X ωn1 (iωn/C0ξλ(kþq 2))(/C0iωn/C0ξλ(/C0kþq 2)) ¼S0(ξλ)þqiqj[S2(ξλ)vλ ivλ jþS1(ξλ)wλ ij], (D32) where ξλis short for ξλ(k),ξλ(/C0k)¼ξλ(k) is used, vλ i¼@kiξλ(k), wλ ij¼@ki@kjξλ(k), S0(ξλ)¼X ωn1 ω2 nþξ2 λ¼βtanhβξλ 2/C16/C17 2ξλ, (D33) S1(ξλ)¼X ωn1 4(iωn/C0ξλ)2(/C0iωn/C0ξλ)¼1 8S0 0(ξλ), (D34) and S2(ξλ)¼X ωn1 2(iωn/C0ξλ)3(/C0iωn/C0ξλ) /C0X ωn1 4(iωn/C0ξλ)2(/C0iωn/C0ξλ)2 ¼/C0β3cosh/C03βξλ 2/C16/C17 sinhβξλ 2/C16/C17 32ξλ: (D35) Then, we have 1 VβX ωn,λ,kTr[na2 λ(k)[na1 λ(k)]y] (iωn/C0ξλ(kþq 2))(/C0iωn/C0ξλ(/C0kþq 2)) ¼Ka1a2 0þqiqjKa1a2 1,ijþO(jqj4), (D36)where Ka1a2 0¼1 VβX λ,kTr[na2 λ(k)[na1 λ(k)]y]S0(ξλ) (D37) and Ka1a2 1,ij¼1 VβX λ,kTr[na2 λ(k)[na1 λ(k)]y] /C2[S2(ξλ)vλ ivλ jþS1(ξλ)wλ ij]: (D38) Ka1a2 0has been carried out in Ref. 1, which has the following expression: K0¼xN0 2uy1y2 y2y3/C18/C19 u, (D39) where x¼ln (2e/C22γβϵc=π),/C22γis Euler ’s constant, u¼diag(sgn( c1), 2mμa2), and expressions of y1,2,3are in Appendix A . Now, we simplify Ka1a2 1,ij. First, we show that Ka1a2 1,ijis propor- tional to δij. Since ξλ(R/C01k)¼ξλ(k) for any operation RinOh group with ( R/C01k)i¼R/C01 ii0ki0,w eh a v e vλ i(k)¼@ξλ(k) @ki¼@k0 i0 @ki@ξλ(R/C01k0) @k0 i0¼vλ i0(k0)Ri0i (D40) and wλ ij(k)¼wλ i0j0(k0)Ri0iRj0j, (D41) where k0¼Rk. Due to na1 λ(R/C01k)¼na1 λ(k), we can derive that Ka1a2 1,ij¼Ka1a2 1,i0j0Ri0iRj0j (D42) holds for any operation RinOhgroup, which leads to Ka1a2 1,ij¼Ka1a2 1,zzδij¼Ka1a2 1δij: (D43) Among Ka1a2 1, the term including S1reads Ia1a2 1¼1 VβX λ,kTr[na2 λ(k)[na1 λ(k)]y]wλ zzS1(ξλ), (D44) the term including S2reads Ia1a2 2¼1 VβX λ,kTr[na2 λ(k)[na1 λ(k)]y]vλ zvλ zS2(ξλ), (D45) andKa1a2 1¼Ia1a2 1þIa1a2 2.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063904 (2020); doi: 10.1063/5.0013596 128, 063904-15 Published under license by AIP Publishing.ForIa1a2 1andIa1a2 2,w eh a v e Ia1a2 1¼1 8βX λðdΩ 4πθ(emλ)Nλ(0)ðϵc /C0ϵcdξλTr[na2 λ(k)[na1 λ(k)]y]wλ zzS00(ξλ)ffiffiffiffiffiffiffiffiffiffiffiffiffi ξλ μþ1s /C25/C01 8βX λðdΩ 4πθ(emλ)Nλ(0)ðϵc /C0ϵcdξλd dξλTr[na2 λ(k)[na1 λ(k)]y]wλ zzffiffiffiffiffiffiffiffiffiffiffiffiffi ξλ μþ1s ! S0(ξλ) /C25/C0x 8X λðdΩ 4πθ(emλ)Nλ(0)d dξλTr[na2 λ(k)[na1 λ(k)]y]wλ zzffiffiffiffiffiffiffiffiffiffiffiffiffi ξλ μþ1s ! /C12/C12/C12/C12/C12 ξλ!0(D46) and Ia1a2 2¼1 βX λðdΩ 4πθ(emλ)Nλ(0)ðϵc /C0ϵcdξλTr[na2 λ(k)[na1 λ(k)]y](vλ z)2S2(ξλ)ffiffiffiffiffiffiffiffiffiffiffiffiffi ξλ μþ1s /C25X λðdΩ 4πθ(emλ)Nλ(0)Tr[ na2 λ(kF,λ)[na1 λ(kF,λ)]y](vλ z(kF,λ))2/C07β2 16π2ζ(3), (D47) where jkF,λj¼ffiffiffiffiffiffiffiffiffiffiffi2mλμp,Nλ(0)¼N0em3=2 λ,ζ(/C1/C1/C1) is the Riemann ζ function, and the result is to the leading order of 1 =(βϵc)/C281 and ϵc=jμj/C281. Before further derivation, let us first estimate the order of those two terms in the isotropic case. In that case, mλ,ξλ,na1 λ, andNλ(0) are independent of the angle. Then, we can get the mag- nitude dependence for the following quantities: Wλ zz/(mλ)/C01, (vλ z(kF,λ))2/μ(mλ)/C01, and Tr[ na2 λ(kF,λ)[na1 λ(kF,λ)]y]/(mλμa2)a1þa2. Finally, we have Ia1a2 1/x μX λθ(emλ)Nλ(0)(mλμa2)a1þa2 mλ(D48) compared with Ia1a2 2/β2μX λθ(emλ)Nλ(0)(mλμa2)a1þa2 mλ: (D49) Since ( βμ)2/C29(βϵc)2/C29βϵc/C29ln (βϵc)/differencex,w eh a v e Ia1a2 2/C29Ia1a2 1 and can neglect Ia1a2 1to get Ka1a2 1/C25Ia1a2 2. Then, by defining ( evλ z)2 andz1,2,3as in Appendix A ,w eh a v e K1¼/C0N0 2β2μ 2muz1z2 z2z3/C18/C19 u, (D50) where the expressions of z1,2,3are shown in Appendix A . As a result, we have Sa1a2(r1,r2)¼V2[Ka1a2 0δ(r1/C0r2) þe/C0i2e /C22hr1/C1A(r2)Ka1a2 1(i∇r2)2δ(r1/C0r2)]:(D51) Substituting the expression shown above into Eq. (D26) , we can get Eq.(17).APPENDIX E: DERIVATION OF (20)AND (21) At first, we derive the eigenvalues of D2. Suppose B¼B^e3. Assume that ^e1and ^e2are the two orthogonal directions perpendic- ular to ^e3and satisfy ^e1/C2^e2¼^e3. Then, D2¼D2 1þD2 2þD2 3.I ti s the similar to the Landau level problem. Since D¼/C0i∇rþe /C22h(B/C2r), we have D3¼/C0i@r3, [D1,D2]¼/C0i2e /C22hB (E1) as well as [ D1,D3]¼[D2,D3]¼0. Define ^a¼ffiffiffiffiffi /C22h 4eBq (D1/C0iD2), we have FIG. 6. This shows the αfactor1 /C0α2e/C22γ π/C0/C1 2of the slope /C0dBc,2=B0 dT=T0as a function of SSOC j2mc 1j. The large change of the αfactor in the intermediate region between regimes III and II does not have much effects on the slope due to thesmall T cfactor, as shown in Fig. 3(a) .Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063904 (2020); doi: 10.1063/5.0013596 128, 063904-16 Published under license by AIP Publishing.[^a,^ay]¼/C22h 4eB2i[D1,D2]¼1: (E2) In this case, D2can be re-written as D2¼4eB /C22h^ay^aþ1 2/C18/C19 þ(/C0i@r3)2, (E3) of which the eigenvalue is l2¼4eB /C22hnþ1 2/C18/C19 þ(k3)2, (E4) with n/C210 and k3being the component of the momentum along the magnetic field direction. Next, we solve for the upper critical field. The linearized gap equation directly given by Eq. (17) reads eΔa1(r)¼/C0X a2eVa1(eKa1a2 0þeKa1a2 1D2)eΔa2(r): (E5) Since Eq. (E5) is linear and D2is Hermitian, eigenfunctions of D2 with different eigenvalues cannot be coupled. Suppose eΔa(r)’sa r e the eigenfunctions of D2with eigenvalue l2, then the linearized gap equation becomes eΔa1(r)¼/C0X a2eVa1(eKa1a2 0þeKa1a2 1l2)eΔa2(r): (E6) Equation (20) is just the matrix version of Eq. (E6). Assume the l2is of the same order as eB=/C22hmeaning that the order of k3is no larger than the order of eB=/C22handnis not large. The resulted expression of the transition temperature Tto the first order of Breads T Tc¼1þβ2 cμ2xc 8mμαl2: (E7) Typically, we have α,0, meaning that the highest Tis given by smallest l2that is 2 eB=/C22h. Replacing BbyBc,2,w eh a v eE q . (21). APPENDIX F: DISORDER AVERAGE AND REPLICA TRICK In this part, we follow Ref. 55to introduce the Replica trick based on our model. Note that, in this part, we temporarily abandon the previous defined x¼2e/C22γβϵc πand define x¼(τ,r) instead. We start from discussing the disorder average of a certain observable for the disorder term Eq. (25) and the probability measure Eq. (26). Given a non-interacting partition function with the random disorder Z0[V]¼ð D/C22cDcexp(/C0S[/C22c,c,V]), (F1)where S[/C22c,c,V]¼S0[/C22c,c]þð dxV(r)/C22cxcx, (F2) /C22c,care Grassmann fields if appearing in the action, /C0S0¼X k,ωn/C22ck,ωn(iωn/C0h(k))ck,ωn, (F3) x¼(τ,r), and τis the imaginary time. Suppose we want to compute thermal average of certain observable Oi(cy,c) in the pres- ence of the random disorder, Oi(/C22c,c) hi ¼Ð DcyDcO i(/C22c,c)exp(/C0S[/C22c,c,V]) Z0[V] ¼δ δJiln (Z[V,J])/C12/C12/C12/C12 J!0, (F4) where Z[V,J]¼ð D/C22cDcexp/C0S[/C22c,c,V]þð dXX iJiOi ! (F5) andXdenotes the imaginary time and position dependence of Oi. Now, one may take the disorder average of Oi(/C22c,c) hi . However, due toZ0[V] in denominator of Eq. (F4), the disorder average is hard to carry out directly. One way to overcome it is the replica trick. Since ln ( Z[V,J])¼limR!0(Z[V,J]R/C01)=R,w eh a v e δ δJiln (Z[V,J])/C12/C12/C12/C12 J!0¼lim R!01 Rδ δJiZ[V,J]R/C12/C12/C12/C12 J!0: (F6) IfRis integer, Z[V,J]R¼ð D/C22ΨDΨexp(/C0S[/C22Ψ,Ψ,V] þð dXX iJiOi(/C22Ψ,Ψ)), (F7) where Ψ¼(c1,...,cR)T, /C22Ψ¼(/C22c1,...,/C22cR),Oi(/C22Ψ,Ψ)¼P a Oi(/C22ca,ca),S[/C22Ψ,Ψ,V]¼PR a¼1S[/C22ca,ca,V], and a¼1,...,Ris the replica index. Then, we have δ δJiZ[V,J]R/C12/C12/C12/C12 J!0¼ð D/C22ΨDΨOi(/C22Ψ,Ψ)exp(/C0S[/C22Ψ,Ψ,V]):(F8) Then, the disorder average becomes Oi(/C22c,c) hihidis ¼lim R!01 Rð D/C22ΨDΨOi(/C22Ψ,Ψ)hexp(/C0S[/C22Ψ,Ψ,V])idis, (F9)Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063904 (2020); doi: 10.1063/5.0013596 128, 063904-17 Published under license by AIP Publishing.where hexp(/C0S[/C22Ψ,Ψ,V])idis¼exp(/C0S0[/C22Ψ,Ψ]) ¼exp/C0ð dxV(r)/C22Ψ(x)Ψ(x)/C18/C19/C28/C29 dis(F10) with exp/C0ð dxV(r)/C22Ψ(x)Ψ(x)/C18/C19/C28/C29 dis ¼ÐDVP [V]exp(/C0Ðd3rV(r)Ðdτ/C22Ψ(x)Ψ(x))ÐDVP [V] ¼Ð DVexp/C01 2γ2 dÐ d3r[V2(r)þ2γ2 dV(r)Ð dτ/C22Ψ(x)Ψ(x)]/C16/C17 ÐDVP [V] ¼expγ2 d 2ð d3rð dτ/C22Ψ(x)Ψ(x)/C18/C192"# ¼expγ2 d 2ð dxdx0δ(r/C0r0)/C22Ψ(x)Ψ(x)/C22Ψ(x0)Ψ(x0)/C20/C21 : (F11) Here, the probability measure P[V] is defined in Eq. (26), and the limitation of Rshould be taken as the limitation of the analytic continuity of the function of integer R’s. Although the failure of this trick is possible since the limit of the analytic continuity maynot be the real limit, the trick works well for most of the times. Next, we discuss the Feymann rule. Recall the non-interacting action /C0S 0[/C22Ψ,Ψ]¼X aX k,ωncy a,k,ωnG/C01 0(k,ωn)ca,k,ωn, (F12) where G0(k,ωn)¼(iωn/C0h(k))/C01: (F13) Based on the expression, if using δ/C22ca,k,ωn,α1 δ/C22ca0,k0,ω0n,α2¼δk,k0δωn,ω0nδa,a0δα1,α2, (F14) the fermion line corresponds to /C0[G0(k,ωn)]α1α2δa1a2, which con- serves the replica index and momentum ( k,ωn). The Fourier transform of the four fermionic field interaction generated by integrating out the disorder potential in Eq. (F10) reads γ2 d 2ð dxdx0δ(r/C0r0)/C22Ψ(x)Ψ(x)/C22Ψ(x0)Ψ(x0) ¼γ2 d 2VX ω,ω0X k1,k2,k3,k4X a1,a2X α1,α2δk1þk3,k2þk4 /C2/C22ca1,k1,ω,α1ca1,k2,ω,α1/C22ca2,k3,ω0,α2ca2,k4,ω0,α2: (F15) This is clear that this effective ( /C22cc)2vertex corresponds toγ2 d V.T h e vertex can be noted as a dashed line (disorder line) connected withtwo fermionic lines at either end. It conserves replica index a,s p i n index α, and frequency ωnonly for the fermionic lines and conserves spacial momentum kfor the entire vertex. For any disorder average of fermionic operators [must contain equal number of /C22candcdue to global U(1) symmetry], say npairs of/C22cand c, the graph must contain nfermionic lines without forming fermionic loops. Each of the nlines would give δaa¼1, which eventually leads to a factor of Rafter the summation of replica index in the definition of the observable with replica index.That Rcancels the one in the denominator of Eq. (F9). If a graph contains at least one fermionic loop, each loop would give a factor ofR. After the limitation R!0, it is clear that all graphs with fer- mionic loops would give zero. Therefore, we only need to considergraphs without fermion loops. In this case, we can simplify the dis-order average of fermionic fields O i(/C22c,c) by neglecting the replica index and considering the all graphs without fermionic loops of the following expression: Oi(/C22c,c) hihidis¼ð D/C22cDcO i(/C22c,c) exp/C26 /C0S0[/C22c,c]: þγ2 d 2ð dxdx0δ(r/C0r0)/C22cxcx/C22cx0cx0/C27 : (F16) APPENDIX G: DERIVATION OF LINEARIZED GAP EQUATION WITH DISORDER (27) In this section, we derive Eq. (27). We first derive the disorder- averaged normal state green function, then derive the superconduct-ing free energy with disorders, and finally get Eq. (27). 1. Disorder-averaged normal state Green function The disorder-averaged Green function is defined as /C22G αβ(k,ωn)¼/C0 ck,ωn,α/C22ck,ωn,β/C10/C11/C10/C11 dis: (G1) It can be expressed as [/C22G(k,ω)]/C01¼[G0(k,ω)]/C01/C0Σ(k,ω), (G2) where Σ(k,ω) is called the self-energy. Since we adopted the Born approximation, Σ(k,ω) only depends on ωand satisfies the self-consistent Born approximation (SCBA) equation, Σ(ω)¼γ2 d VX k0G0(k0,ω)þγ2 d VX k0G0(k0,ω)Σ(ω)G0(k0,ω):(G3) Define P+(k)¼1 2+h(k)/C0ξk 2k2Qc(G4) to be the projection operators to ξ+bands, respectively. The normal state Green function without disorder is given in Eq. (F13) , which can be expressed in terms of projection operatorsJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063904 (2020); doi: 10.1063/5.0013596 128, 063904-18 Published under license by AIP Publishing.G0(k,ω)¼X λ¼+1 iω/C0ξλ(k)Pλ(k): (G5) Using the expression of Pλ(k), we have X kG0(k,ωn)¼X k1 21 iωn/C0ξþ(k)þ1 iωn/C0ξ/C0(k)/C18/C19 þX k1 iωn/C0ξþ(k)/C01 iωn/C0ξ/C0(k)/C18/C19h(k)/C0ξk 2k2Qc: (G6) Since the second term can be re-written asP kP5 i¼1fi(k)gk,iΓi with fi(k) being Ohinvariant, the second term should be zero. Then, we have X kG0(k,ωn)¼X k1 21 iωn/C0ξþ(k)þ1 iωn/C0ξ/C0(k)/C18/C19 , (G7) which is proportional to the identity matrix. Similarly, we have X kG0(k,ωn)G0(k,ωn) ¼X k1 (iωn/C0ξþ(k))2Pþ(k)þ1 (iωn/C0ξ/C0(k))2P/C0(k)"# ¼X k1 21 (iωn/C0ξþ(k))2þ1 (iωn/C0ξ/C0(k))2"# : (G8) Then, by induction, we get that Σ(ω) is proportional to the identity matrix. Thereby, Eq. (G3) can be re-written as Σ(ω)¼γ2 d VX k0G0(k0,ω)1/C0γ2 d VX k00G2 0(k00,ω)"#/C01 : (G9) Now estimate the order of G2 0(ω,k0) term. The term can be re-written as γ2 d VX k0G2 0(k0,ω)¼(γ2 d)ð dεN(ε) 21 (iω/C0ε)2, (G10) where N(ε)¼hNþ(ϵ)iΩþhN/C0(ϵ)iΩ (G11) andhN+(ε)iΩ¼1 VP k0δ(ε/C0ξ+(k0)) are density of states of ξ+ bands at εwithout spin index and h/C1 /C1 /C1iΩis the average over the solid angle. Then, we have γ2 dð dεN(ε) 2(iω/C0ε)2/differenceγ2 dN0(0)/differenceγ2 dN(0) μ, (G12) where we assume that jωjis no larger than the energy cut-off ϵc which is small compared with chemical potential μ.T h i sm e a n s ,when dealing with the disorder problem, we will integrate the energy band first and then sum up the frequency, which is the same as the other way to the leading order of 1 =βϵc/C281. Since we assume γ2 dNF jμj/C281w i t h NF¼N(0)¼N0y1, we can neglect the G2 0(ω,k0)t e r m since we only keep the leading order ofγ2 dNF jμj. Then, we have Σ(ω)¼γ2 d VX k0G0(k0,ω): (G13) Since1 VP k0G0(k0,ω) is equal to ð dεN(ε) 21 iω/C0ε/C25/C0iπsgn(ω)NF 2/C0ϵ0 γ2 d, (G14) with ϵ0¼γ2 dPÐ dεN(ϵ) 2ε/C2/C3 ,w eh a v e /C22G(k,ω)¼1 iω/C0h(k)þi1 2τdsgn(ω)þϵ0, (G15) with 1 =τd¼γ2 dπNF. Moreover, if choosing the isotropic limit, we can estimate the order of ϵ0by choosing the range of integration to be (/C0jμj,jμj), which is ϵ0τd/difference1. In terms of the projection opera- tors, the disorder-averaged Green function reads /C22G(k,ω)¼X λ¼+/C22Gλ(k,ω)Pλ(k), (G16) with /C22Gλ(k,ω)¼1 iω/C0ξλ(k)þi1 2τdsgn(ω)þϵ0: (G17) 2. Disorder-averaged superconducting free energy The mean-field free energy with disorder reads F¼/C0kBTlnð DcyDcexp[/C0S/C0SΔþβfΔ]/C26/C27 , (G18) where Sis shown in Eq. (F2), Δ(k,q)¼Δ0(q)Γ0γ 2þΔ1(q)a2gk/C1Γγ 2, (G19) SΔ¼1 2X ω,k,q/C22cω,kþq 2Δ(k,q)(/C22c/C0ω,/C0kþq 2)T" þX ω,k,qcT ω,/C0kþq 2Δy(k,q)c/C0ω,kþq 2# , (G20) fΔ¼X q,aV 2VajΔa(q)j2, (G21) and the order parameter is assumed to be uniform with respect to the imaginary time. It is clear that /C0ΔT(/C0k,q)¼Δ(k,q).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063904 (2020); doi: 10.1063/5.0013596 128, 063904-19 Published under license by AIP Publishing.Then, the mean-field superconducting Free energy reads FSC¼/C0fΔ/C0kBTln { exp[ /C0SΔ] hi }, (G22) where exp[/C0SΔ] hi ¼Ð D/C22cDcexp[/C0S/C0SΔ] Z0[V](G23)andZ0[V]i ss h o w ni nE q . (F1). Since exp[ /C0SΔ] hi ¼eWwith W contains all the connected gr aphs, we have the disorder- averaged FSC FSChidis¼/C0fΔ/C0kBTWhidis: (G24) According to Eq. (F16) ,Whidisto the second order of Δreads W(2)/C10/C11 dis¼(/C0SΔ)2 2!/C28/C29/C28/C29 dis¼ð D/C22cDc(/C0SΔ)2 2!exp[/C0Sdis] ¼1 4X ω,k,qX ω0,k0,q0Δα1α2(k,q)Δ* α4,α3(k0,q0)ð D/C22cDc/C22cω,kþq 2,α1/C22c/C0ω,/C0kþq 2,α2cω0,/C0k0þq0 2,α3c/C0ω0,k0þq0 2,α4exp[/C0Sdis], (G25) where the summation of α1,2,3,4 is implied, /C0Sdis¼/C0S0þγ2 d 2ð dxdx0δ(r/C0r0)/C22cxcx/C22cx0cx0, (G26) the internal fermionic loops are abandoned, and it is not necessary to specifically rule out the disconnected graphs to this order sinceall non-zero contribution is given by connected graphs. We adopt the Born approximation to abandon all graphs with crossed disorder lines and only include the Cooperon modes. 55In this case, we have W(2)/C10/C11 dis ¼1 2X ω,k,qTr /C22Gkþq 2,ω/C16/C17 D(k,q,ω)/C22GT/C0kþq 2,/C0ω/C16/C17 Δy(k,q)hi , (G27) where D(k,q,ω)¼Δ(k,q) þγ2 d VX k0/C22Gk0þq 2,ω/C16/C17 D(k0,q,ω)/C22GT/C0k0þq 2,/C0ω/C16/C17 : (G28) If the order parameter is uniform, then we can choose Δ(k,q)¼Δ(k)δq,0. In this case, combining Eq. (G23) and the two equations shown above, we can get FSC¼/C0VX ajΔaj2 2Va /C01 2βX ωn,kTr[/C22G(k,ωn)D(k,ωn)/C22GT(/C0k,/C0ωn)Δy(k)], (G29)where D(k,ω)¼Δ(k)þγ2 d VX k0/C22G(k0,ω)D(k0,ω)/C22GT(/C0k0,/C0ω):(G30) 3. Further simplification of FSCwith disorder First, recall the property of the projection operator in our case, Pλ(k)¼P2 λ(k), Pλ(k)¼Py λ(k), Pλ(/C0k)¼Pλ(k), Pλ(k)ΓiγPT λ(k)/Pλ(k)γPT λ(k),(G31) where the first two are general, the third one is due to the inversion symmetry of our model, and the last one is for i¼0,..., 5. In the following, we will use the four relations again and again, and wewill not refer to them for convenience. The trace term in Eq. (G28) can be expressed by the projection operator using Eq. (G16) to the leading order of ϵ c=(2Qck2 F)/C281, X ωn,kTr[/C22G(k,ωn)D(k,ωn)/C22GT(/C0k,/C0ωn)Δy(k)] ¼X ωn,k,λ/C22Gλ(k,ωn)/C22Gλ(/C0k,/C0ωn)d* λ(k)Tr[Dλ(k,ωn)γy λ(k)], (G32) where γλ(k)¼Pλ(k)γPT λ(/C0k),Pλ(k)Δ(k)PT λ(/C0k)¼dλ(k)γλ(k), dλ(k)¼Δ0 2þλΔ1 2a2k2sgn(c1)fQ, and Dλ(k,ω)¼Pλ(k)D(k,ω)PT λ(/C0k). Using Eq. (G29) , the equation satisfied by Dλ(k,ω) reads Dλ(k,ωn)¼dλ(k)γλ(k)þγ2 d VX k0,λ0/C22Gλ0(k0,ωn) /C2/C22Gλ0(/C0k0,/C0ωn)Pλ(k)Dλ0(k0,ωn)PT λ(/C0k), (G33)Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063904 (2020); doi: 10.1063/5.0013596 128, 063904-20 Published under license by AIP Publishing.where we also neglect the inter-band contribution as before, i.e., only keep terms to the leading order of ϵc=(2Qck2 F)/C281. Since γT λ(k)¼γT λ(/C0k)¼/C0γλ(k), we can show that /C0DT λ(k,ωn) satisfies the same equation as Dλ(k,ωn), meaning that /C0DT λ(k,ωn)¼Dλ(k,ωn). Thereby, Dλ(k,ωn) can be expressed in terms of Γiγwith i¼0,..., 5. Furthermore, we have Dλ(k,ωn)¼/C22Dλ(k,ωn)γλ(k), (G34) where /C22Dλ(k,ωn) is a scalar function. Using Tr( γλ(k)γy λ(k))¼2, Eqs. (G32) and(G33) ,w eh a v e Tr[Dλ(k,ωn)γy λ(k)]¼2/C22Dλ(k,ωn) (G35) and /C22Dλ(k,ωn)¼dλ(k)þγ2 d 2VX k0,λ0/C22Gλ0(k0,ωn) /C2/C22Gλ0(/C0k0,/C0ωn)/C22Dλ0(k0,ωn)Tr[γλ0(k0)γy λ(k)]:(G36) Since the non-interacting Hamiltonian is Ohinvariant, we have URh(R/C01k)Uy R¼h(k) for any R[Oh, where URis the unitary representation of Rforj¼3 2fermions. Using Eq. (G4) ,w eh a v e URPλ(R/C01k)Uy R¼Pλ(k) and thereby URγλ(R/C01k)UT R¼γλ(k). Using this relation and the fact that dλ(k)a n d /C22Gλ(k,ωn)a r e Ohinvariant, we can get that /C22Dλ(R/C01k,ωn) satisfies the same equation as /C22Dλ(k,ωn), meaning that /C22Dλ(k,ωn)i s Oh invariant: /C22Dλ(R/C01k,ωn)¼/C22Dλ(k,ωn). Moreover, according to Schur ’s lemmas,52we haveP kf(k)gk,i¼0a n dP kf(k)gk,igk,j ¼P kf(k)g2 k,iδijiff(k)i s Ohinvariant and i,j¼1,...,5 . Combining the previous facts, we can get X k0/C22Gλ0(k0,ωn)/C22Gλ0(/C0k0,/C0ωn)/C22Dλ0(k0,ωn)γλ0(k0) ¼X k0/C22Gλ0(k0,ωn)/C22Gλ0(/C0k0,/C0ωn)/C22Dλ0(k0,ωn)γ 2,( G 3 7 ) where ΓiγΓT i¼γfori¼1,...,5 i s u s e d . T h e n , E q . (G35) becomes /C22Dλ(k,ωn)¼dλ(k)þγ2 d 2VX k0,λ0 /C2/C22Gλ0(k0,ωn)/C22Gλ0(/C0k0,/C0ωn)/C22Dλ0(k0,ωn): (G38) Define /C22L0(ωn)¼ffiffiffiffiffiffi γ2 d 2Vs X k,λ/C22Gλ(k,ωn)/C22Gλ(/C0k,/C0ωn)dλ(k)( G 3 9 )and /C22L(ωn)¼ffiffiffiffiffiffi γ2 d 2Vs X k,λ/C22Gλ(k,ωn)/C22Gλ(/C0k,/C0ωn)/C22Dλ(k,ωn):(G40) Then, Eq. (G37) relates /C22Land /C22L0, /C22L(ωn)¼/C22L0(ωn)þ/C22b(ωn)/C22L(ωn), (G41) which gives /C22L(ωn)¼/C22L0(ωn) 1/C0/C22b(ωn),( G 4 2 ) with /C22b(ωn)¼γ2 d 2VX k,λ/C22Gλ(k,ωn)/C22Gλ(/C0k,/C0ωn): (G43) Eventually, Eq. (G31) becomes X ωn,kTr[/C22G(k,ωn)D(k,ωn)/C22GT(/C0k,/C0ωn)Δy(k)] ¼X ωn,k,λ/C22Gλ(k,ωn)/C22Gλ(/C0k,/C0ωn)d* λ(k)2/C22Dλ(k,ωn) ¼2X ωn,k,λ/C22Gλ(k,ωn)/C22Gλ(/C0k,/C0ωn)d* λ(k)dλ(k) þ2X ωn/C22L* 0(ωn)/C22L(ωn) ¼2X ωn,k,λ/C22Gλ(k,ωn)/C22Gλ(/C0k,/C0ωn)d* λ(k)dλ(k) þ2X ωnj/C22L0(ωn)j2 1/C0/C22b(ωn), (G44) where the second equality uses Eqs. (G31) and (G34) , the third equality uses Eq. (G37) ,/C22Gλ(k,ωn)/C22Gλ(/C0k,/C0ωn) is real and defini- tions of /C22L(ωn) and /C22L0(ωn), and the last equality uses Eq. (G40) . Combined with Eq. (G28) , we eventually have FSC¼/C01 βX ωn,k,λ/C22Gλ(k,ωn)/C22Gλ(/C0k,/C0ωn)d* λ(k)dλ(k) /C01 βX ωnj/C22L0(ωn)j2 1/C0/C22b(ωn)/C0VX ajeΔaj2 2eVa: (G45) 4. Derivation of (27) First, we derive a general expression that will be used repeat- edly later,Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063904 (2020); doi: 10.1063/5.0013596 128, 063904-21 Published under license by AIP Publishing.1 VX λ,k/C22Gλ(k,ωn)/C22Gλ(/C0k,/C0ωn)fλ(k) /C25X λðdΩ 4πN0em3=2 λθ(emλ)fλ(kF,λ)ð dξλ1 (jωnjþ1 2τd)2þ(ξλ)2 ¼π jωnjþ1 2τdX λhN0em3=2 λθ(emλ)fλ(kF,λ)iΩ, (G46) where the first equality uses two things: (i) ϵ0/difference1=τdand we neglect terms of order 1 =(τdμ) and (ii) 1 =(ξ2þϵ2)h a sap e a ka t ξ¼0a n d drops fast away from the peak when ϵis small. The range of the inte- gration of ξλis from /C01to1since the energy cut-off ϵcis included in the limit of the summation of ωnasjωnjþ1 2τd/C20ϵc. Using the formula derived above, we have X k,λ/C22Gλ(k,ωn)/C22Gλ(/C0k,/C0ωn)d* λ(k)dλ(k) ¼N0V 4π jωnjjωnj jωnjþ1 2τdeΔyy1y2 y2y3/C18/C19 eΔ, (G47) witheΔ¼(eΔ0,eΔ1)T, 1/C0/C22b(ωn)¼jωnj jωnjþ1 2τd, (G48) /C22L0(ωn)¼Vffiffiffiffiffiffi γ2 d 2Vs π jωnjþ1 2τdN0sgn(c1)eΔ0 2y1þeΔ1 2y2 ! , (G49) and thereby j/C22L0(ωn)j2 1/C0/C22b(ωn)¼N0V 4π jωnj1 2τd (jωnjþ1 2τd)1 y1jeΔ0y1þeΔ1y2j2 ¼N0V 4π jωnj1 2τd (jωnjþ1 2τd)eΔyy1 y2 y2y2 2=y1/C18/C19 eΔ: (G50) Here, 1 =τd¼γ2 dπNFis used. Substituting Eqs. (G46) and(G49) into Eq. (G44) ,w eh a v e FSC¼/C0VX ajeΔaj2 2eVa /C0N0V 4βX ωnπ jωnjeΔyy1 y2 y2jωnj jωnjþ1 2τdy3þ1 2τd (jωnjþ1 2τd)y2 2=y10 @1 AeΔ:(G51) Assuming ϵcτd/C291 and βϵc/C291, we have π βX ωn1 jωnj¼ln2e/C22γβϵc π/C18/C19 þO1 βϵc,1 ϵcτd/C18/C19 , (G52)π βX ωn1 jωnjþ1 2τd¼ln2e/C22γβϵc π/C18/C19 /C0Fβ 4πτd/C18/C19 þO1 βϵc/C18/C19 , (G53) and π βX ωn1 2τd jωnjjωnjþ1 2τd/C16/C17 ¼Fβ 4πτd/C18/C19 þO1 βϵc,1 ϵcτd/C18/C19 , (G54) where F(β 4πτd)¼Ψ(0)(β 4πτdþ1 2)/C0Ψ(0)(1 2),Ψ(0)(x) is the digamma function, and the range of the sum is jωnjþ1 2τd/C20ϵc. Then, the free energy becomes FSC¼/C0V 2eΔy1 eV0 1 eV10 @1 AeΔ /C0N0V 4xeΔyy1 y2 y2y3þF x(y2 2=y1/C0y3) ! eΔ, (G55) whereeΔ¼(eΔ0,eΔ1)T. Then, the linearized gap equation reads ~Δ0 ~Δ1/C18/C19 ¼xλ0 2y1λ0 2y2 λ1 2y2λ1 2y3b1 ! ~Δ0 ~Δ1/C18/C19 , (G56) where b1¼1þF(β 4πτd) xy2 2 y1y3/C01/C18/C19 : (G57) Next, we will show 0 ,b1/C201. Let us define tλ¼emλθ(emλ), and thereby tλ/C210. In this case, y1¼h(t3=2 þþt3=2 /C0)iΩ, y2¼hfQ(t5=2 þ/C0t5=2 /C0)iΩ, and y3¼hf2 Q(t7=2 þþt7=2 /C0)iΩ. According to Cauchy –Schwarz inequality, t+/C210 and fQ/C210, we have y1y3¼hf2 Q(t7=2 þþt7=2 /C0)iΩh(t3=2 þþt3=2 /C0)iΩ /C21hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f2 Q(t7=2 þþt7=2 /C0)(t3=2 þþt3=2 /C0)q i2 Ω ¼hfQffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (t5 þþt5 /C0þt3=2 þt7=2 /C0þt7=2 þt3=2 /C0)q i2 Ω /C21hfQffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (t5 þþt5 /C0þ2t5=2 þt5=2 /C0)q i2 Ω /C21hfQffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (t5 þþt5 /C0/C02t5=2 þt5=2 /C0)q i2 Ω ¼hfQjt5=2 þ/C0t5=2 /C0ji2 Ω /C21j hfQ(t5=2 þ/C0t5=2 /C0)iΩj2¼jy2j2¼y2 2: (G58) It gives 0 /C20y2 2=(y1y3)/C201 since y1,3.0. Combined with F(β 4πτd),x,w eh a v e0 ,b1/C201. To have b1¼1, we either needJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063904 (2020); doi: 10.1063/5.0013596 128, 063904-22 Published under license by AIP Publishing.1 τd¼0 meaning that there is no disorder or need the system to be in regime II ( tþt/C0¼0 in any direction) and isotropic ( c1¼c2). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1J. Yu and C.-X. Liu, Phys. Rev. B 98, 104514 (2018). 2G. Goll, M. Marz, A. Hamann, T. Tomanic, K. Grube, T. Yoshino, and T. Takabatake, Phys. B Condens. Matter 403, 1065 (2008). 3N. P. Butch, P. Syers, K. Kirshenbaum, A. P. Hope, and J. Paglione, Phys. Rev. B 84, 220504 (2011). 4T. V. Bay, T. Naka, Y. K. Huang, and A. de Visser, Phys. Rev. B 86, 064515 (2012). 5F. F. Tafti, T. Fujii, A. Juneau-Fecteau, S. René de Cotret, N. Doiron-Leyraud, A. Asamitsu, and L. 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5.0019985.pdf

Appl. Phys. Lett. 117, 092405 (2020); https://doi.org/10.1063/5.0019985 117, 092405 © 2020 Author(s).Magnetic and magnetocaloric properties of layered van der Waals CrCl3 Cite as: Appl. Phys. Lett. 117, 092405 (2020); https://doi.org/10.1063/5.0019985 Submitted: 30 June 2020 . Accepted: 17 August 2020 . Published Online: 01 September 2020 Suchanda Mondal , A. Midya , Manju Mishra Patidar , V. Ganesan , and Prabhat Mandal Magnetic and magnetocaloric properties of layered van der Waals CrCl 3 Cite as: Appl. Phys. Lett. 117, 092405 (2020); doi: 10.1063/5.0019985 Submitted: 30 June 2020 .Accepted: 17 August 2020 . Published Online: 1 September 2020 Suchanda Mondal,1A.Midya,2Manju Mishra Patidar,3 V.Ganesan,3and Prabhat Mandal1,a) AFFILIATIONS 1Saha Institute of Nuclear Physics, HBNI, 1/AF Bidhannagar, Calcutta 700 064, India 2Department of Physics, City College, 102/1, Raja Rammohan Sarani, Calcutta 700 009, India 3UGC-DAE Consortium for Scientific Research, University Campus, Khandwa Road, Indore 452 001, India a)Author to whom correspondence should be addressed: prabhat.mandal@saha.ac.in ABSTRACT We have investigated the magnetic and magnetocaloric properties of van der Waals (vdW) layered CrCl 3from magnetization and heat capacity measurements. CrCl 3exhibits complicated magnetic properties due to the strong competition between the ferromagnetic and antiferromagnetic interactions: a ferromagnetic ordering around 17 K followed by an antiferromagnetic ordering at 14.3 K. A large magnetic entropy change ( /C0DSM)o f1 9 Jk g/C01K/C01, an adiabatic temperature change ( DTad) of 6.2 K, and a relative cooling power of 600 J kg/C01are observed for a field change of 7 T near the transition temperature, and the mechanical efficiency ( gm) at 18 K and 0–3 T is 1.17. These values of magnetocaloric parameters are significantly larger than those for CrI 3and other layered vdW systems. The scaling analysis shows that all the rescaled DSM(T,H) data collapse into a single curve, which indicates the second order nature of magnetic phase transition. The above results suggest that environmentally friendly CrCl 3can be a phenomenal alternative to very expensive rare-earth material for the magnetic refrigeration for liquefaction of hydrogen. Published under license by AIP Publishing. https://doi.org/10.1063/5.0019985 Refrigeration is the process of artificially cooling down a system below its ambient temperature, which has a great advantage not only for domestic purposes but also largely for commercial use such as industrial freezers, cryogenics, and the fuel industry. But the ever increasing demand of refrigeration causes a chronic effect on our envi- ronment as synthetic refrigerants mostly based on chlorofluorocarbon, hydrofluorocarbon, and hydrochlorofluorocarbon deplete the ozone layer, the shield against ultraviolet radiation. Magnetic refrigeration, an environmentally friendly cooling alternative to gaseous refrigera- tion, can achieve low temperature based on the magnetocaloric effect.In this process, the change in magnetic entropy determines the adia- batic temperature change of the material under the variation of the magnetic field. Using adiabatic demagnetization, one can reach ultra- low temperature, which has implicit usage in spacecraft, but it faces major challenges due to thermal and magnetic hysteresis. Layered van der Waals (vdW) materials have attracted immense interest due to their remarkable physical properties and potential applications in microelectronics, spintronics, and optoelectronic devices. 1,2Graphene and transition metal dichalcogenides are widely studied layered materials with adjacent layers bonded via weak vdW force having enormous applications in nanoelectronics and energydevices.3,4However, the absence of intrinsic magnetism in the 2D limit restricts their usage in spintronic devices. Recently, some layered vdW materials such as Cr X3(X¼C l ,B r ,I ) ,C r 2X2Te6(X¼Ge, Si), Fe3GeTe 2,a n dV I 3have attracted tremendous interest because they retain intrinsic magnetism down to the monolayer or bilayer limit, which facilitates them in spin/valley electronics, vdW heterostructures, magnetoelectronics, and magneto-optics.5–10CrI3, a member of the chromium trihalide family, shows long range ferromagnetism even at the monolayer limit and further shows bilayer antiferromagnetism, which is a promising candidate for low-dimensional spintronics aswell as in fundamental physics 5,11and CrBr 3; another ferromagnetic member of this family11retains ferromagnetism down to the monolayer. In contrast, bulk CrCl 3shows intralayer ferromagnetism with antiferromagnetic interaction between consecutive layers having aNeel temperature of T N¼14 K with comparatively weak anisotropy.12 In typical antiferromagnets, the magnetic moments are ordered, but the neighboring moments align along the opposite direction, resulting net zero magnetization ( M). So in antiferromagnets, the stray current is zero. They have fascinating applications in spintronics, in which the spin–orbit torque can be controlled by the application of the electric Appl. Phys. Lett. 117, 092405 (2020); doi: 10.1063/5.0019985 117, 092405-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplfield.13CrI3has strong anisotropy and exhibits small anisotropic mag- netic entropy changes ( /C0DSM) as compared to conventional rare earth magnetocaloric materials.14 From extensive magnetic and thermal measurements, we report magnetic entropy changes along with the adiabatic temperaturechange ( DT ad)f o rC r C l 3single crystals. The coefficient of performance (COP), g, which determines the energy efficiency of a refrigerator is also estimated to be very high, 1.17 for a field change of 0–3 T at 18 K. These evaluations of the magnetocaloric effect (MCE) refer to moder-ately large values of /C0DS M;DTad,a n d g, which suggest that CrCl 3can be a magnificent alternative to rare earth material as a magnetic refrig-erant near liquid hydrogen temperature in the commercial sector due to the high cost and less availability of the later corresponding to their comparable efficiencies. CrCl 3single crystals were grown by the chemical vapor transport method by recrystallizing CrCl 3(99.995%, Alfa Aesar). Anhydrous CrCl 3was sealed in an evacuated quartz tube and placed in a gradient furnace with a hot end at 700/C14C and a cold end at 550/C14Cf o rap e r i o d of seven days. Thin violet colored plate-like crystals were collectedfrom the cold end of the tube. The crystals were freshly cleaved beforecharacterization and measurements. The structural analysis is carriedout using a high-resolution Rigaku, TTRAX III (Cu-K aradiation) x-ray diffractometer. The x-ray diffraction pattern from the surface of a cleaved thin single crystal of CrCl 3corresponds to (0 0 l) peaks, con- firming that the flat surface is perpendicular to the caxis with an inter- layer spacing of 5.8086 A ˚(see the supplementary material ). Magnetization and heat capacity measurements were performed using a physical property measurement system (PPMS, Quantum Design). The heat capacity was measured down to 2 K by the relaxationtechnique. Figure 1(a) shows the temperature dependence of magnetization below 75 K with the Hjjabplane and the Hjjcaxis at 500 Oe and 1 kOe. Magnetization is comparatively smaller for the field along the c axis. Measurements are performed for both ZFC and FC conditions,but the data for the ZFC condition are shown for clarity. No significantdifference between ZFC and FC data is observed. Mincreases sharply below 20 K, indicating the onset of ferromagnetic ordering. At low fields, Mshows a cusp at 14 K due to antiferromagnetic ordering asso- ciated with antiparallel alignment of moments between two consecu-tive layers. 12,15For further understanding of the magnetic groundstate, inverse susceptibility ( v/C01) as a function of temperature is plot- ted in Fig. 1(a) .v/C01shows a nearly linear dependence on Tboth above and below the structural transition TS/C25240 K. In the temperature range of 250–380 K, susceptibility ( v) follows the Curie–Weiss law, v¼C=ðT/C0hÞ, with an effective paramagnetic moment of leff ¼3:72lB/Cr3þand a Weiss temperature of h¼57:65 K and a linear fit to the plot in the temperature range of 100–185 K gives leff¼4:06 lB/Cr3þandh¼35:43 K. The observed value of leffis close to the theoretical spin-only moment ( S¼3/2). leffis close to the value reported earlier, whereas hderived from v/C01(T) above TSis higher than the value in the previous report.12The positive value of Weiss temperature suggests that ferromagnetic interaction is of dominating nature in the paramagnetic state. In order to understand the effect ofthe magnetic field on the ground state, the isothermal magnetizationfor the Hjjabplane at 2 K is shown in the inset of Fig. 1(a) .M(H) shows negligible hysteresis. As the field increases, Mincreases sharply and tends to saturate at a field less than 3 kOe. This field is sufficient tocompletely orient the magnetic moments from the antiferromagneticstate to a field-induced spin-flip ferromagnetic state. 12,16At 2 K and 5 T, the observed value of M¼2:9lB/Cr3þcorresponds to spin only moment of Cr. For insight into the magnetic phase transition inCrCl 3,M2as a function of H/M(Hjjab)i sp l o t t e di n Fig. 1(b) to con- struct the Arrott plot around the onset of the magnetic ordering. Thenature of the Arrott plot for the present system differs significantlyfrom the critical behavior of the conventional ferromagnet. The curva-tures in the M 2vsH/Misotherms in the low-field region are strongly temperature dependent. For T>17 K, the isothermal curves mimic the Arrott plot of conventional ferromagnetic phase transition with magnetic fluctuations above TC. The magnetic fluctuation above TCin CrCl 3is also reflected from the large Weiss temperature, which is about two times of TC.B e l o w T¼17 K, the nature of the curvature of M2vsH/Misotherms at low fields changes rapidly with decreasing temperature. The squared magnetization falls off rapidly as the field isdecreased to zero for isotherms below 14 K. The deviation of theArrott plot from that observed in a conventional ferromagnetic phase transition is consistent with the fact that the long-range magnetic order in CrCl 3sets in at low temperatures in a two-step process.12 Below the Weiss temperature, ferromagnetic correlation developsbetween the Cr spins lying in the ab-plane and becomes the strongest at around 17 K. However, the onset of the antiferromagnetic FIG. 1. (a) Magnetic moment vs tempera- ture measured for the applied field both par- allel and perpendicular to the plane. Temperature-dependent inverse susceptibil-ityv /C01ðTÞfor CrCl 3single crystals mea- sured at a field of 1 kOe with the Hjjab plane; solid lines are the Curie–Weiss fit. Isothermal magnetization with the Hjjab plane at T ¼2 Ki ss h o w ni nt h ei n s e t .( b ) T h eA r r o t tp l o tf o rC r C l 3.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 092405 (2020); doi: 10.1063/5.0019985 117, 092405-2 Published under license by AIP Publishingcorrelations between the ferromagnetic ab-layers at about 14 K rein- forces the system to enter a long-ranged antiferromagnetic ground state at low temperatures and breaks the time reversal symmetry. The universality class of the system is, thus, governed by the antiferromag-netic phase transition at T N¼14 K, where the correlation length diverges rather than the transition to short-range ferromagnetic order- ing at T¼17 K. The evolution of the true order parameter of the phase transition, i.e., the spontaneous magnetization with tempera-tures and T C, cannot be determined from the extrapolation of the Arrott plot or modified Arrott plot isotherms at high fields to zero andfrom quadratic extrapolation of the Arrott plot to zero field as have been done in several systems. 17However, the sub-lattice magnetization can be deduced from the neutron scattering experiments at differenttemperatures around 14 K to determine the true nature of the mag-netic phase transition in CrCl 3. M(H) curves for the Hjjabplane and the Hjjcaxis are depicted inFigs. 2(a) and2(b)after demagnetization correction as discussed in the earlier report.12At low temperature, M(H) for the Hjjcaxis also saturates at fields less than 3 kOe. To test whether this material is suit-able for magnetic refrigeration, magnetic entropy changes have been calculated as DS M¼P iMiþ1/C0Mi Tiþ1/C0Ti/C16/C17 DHi,w h e r e Miþ1andMiare the magnetic moments at temperatures Tiþ1andTi, respectively, with the change in magnetic field DHi.DSM(T)f o rt h e Hjjabplane andthe Hjjcaxis is shown in Figs. 3(a) and 3(b), respectively. DSMincreases with increasing Tand reaches maximum at TCfor low fields. As the applied field increases, the maximum value of DSM (DSmax M) increases and DSmax Mbecomes 19.8 J kg/C01K/C01forHjjaband 19.5 J kg/C01K/C01forHjjcat 7 T. Thus, DSMis almost isotropic in CrCl 3. The negative sign of DSMindicates the reduction of tempera- ture when the magnetic field changes adiabatically. For understanding the nature of the magnetic ground state and intrinsic magnetocaloric properties, heat capacity is measured as a function of temperature and field ( Fig. 4 ). At zero field, CPexhibits a sharp k-like anomaly at 14 K, which diminishes with the field and dis- appears above H¼2k O e . T h e k-like anomaly is followed by a broad facet centered at TC. With the application of the field, this facet further broadens and shifts toward higher temperature. The CP(T)c u r v efi t s well with the combined Debye plus Einstein model of lattice heat capacity ( CL) over a wide temperature range as shown in Fig. 4(a) (see thesupplementary material ). By subtracting the lattice contribution from total heat capacity, the magnetic contribution ( Cm) is determined as depicted in Fig. 4(b) . The entropy estimated by integrating Cm=Tis about 10.2 J mol/C01K/C01at zero field, which is close to the theoretical value of Rlnð2Sþ1Þ¼11.5 J mol/C01K/C01as shown in the inset of Fig. 4(a) . To check whether the magnetic entropy change from the M(H) plot is consistent or not, /C0DSMis calculated from heat capacity 0255075100 02460255075100H| |a bMoment (emu g-1)2K 56 K ΔT(2-38K)=2 K , ΔT(41-56K)=3 K(a) (b) H| |c 56 K2K ΔT(2-38K)=2 K , ΔT(41-56K)=3 KMoment (emu g-1) Field (T) FIG. 2. Isothermal magnetization curves for the field applied along (a) Hjjabplane and (b) Hjjcaxis.048121620 20 40 60 8005101520(b) -ΔSM(Jk g-1K-1)0-1T 0-2T 0-3T 0-4T 0-5T 0-6T 0-7TH| |a b (a) H| |c-ΔSM(Jk g-1K-1) T( K )0-1 T 0-2 T 0-3 T 0-4 T 0-5 T 0-6 T 0-7 T FIG. 3. (a) Temperature variation of /C0DSMcalculated from magnetization data for theHjjabplane. (b) Temperature variation of /C0DSMcalculated from magnetization data for the Hjjcaxis.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 092405 (2020); doi: 10.1063/5.0019985 117, 092405-3 Published under license by AIP Publishingdata as shown in Fig. 5(a) . This plot reveals that both the values of DSMare close to each other. A small difference between these two may be due to the overestimation of lattice contribution. The adiabatic tem- perature change DTad¼Ti/C0Tfis defined as the temperature change of the system from initial temperature Ti(H¼0) to final temperature Tf(H6¼0) in an isentropic process caused by the intrinsic magneto- caloric effect. We have calculated DTadfrom zero field heat capacity andDSMdata.DTad(T) at different fields is shown in Fig. 5(b) .W i t h the increase in the field, the maximum value of DTadincreases and becomes 6.8 K for 0–7 T. Both DTadandDSMfor bulk CrCl 3single crystals are quite large compared to those reported for other vdW sys-tems ( Table I ). A phenomenological curve for DS Mfor different applied mag- netic fields has been proposed by Franco et al. as a way to determine the order of magnetic phase transition.19In this method, the parame- ters related to DSM(T) curves follow a series of power laws dependent on the field: jDSmax Mj/Hn;dTFWHM /Hb,a n dR C P /Hc,w h e r e dTFWHM is the full-width at half maximum and RCP is the relative cooling power.19,20For temperature well above TC,t h ev a l u eo f nfor all applied fields tends to 2, which points toward the second ordernature of phase transition 21(for plots and analysis, see the supplemen- tary material ). From scaling analysis, all the DSMðT;HÞplots collapse into a single curve for the Hjjabplane, which reveals that thecorresponding phase transition is second order in nature24(for details, see the supplementary material ). For understanding the nature of magnetic interaction in CrCl 3, a detailed neutron scattering study of this layered material can be very interesting. In conclusion, the calculated values of DSM;DTad,a n dR C Pa r e large compared to those of CrI 3and other layered vdW systems. /C0DSMvalues for two directions of the applied field with the Hjjab plane and the Hjjcaxis are almost the same, which is attributed to the very weak anisotropy in CrCl 3. Our result suggests that single crystal CrCl 3can have potential application as a magnetic refrigerant at liquid hydrogen temperature. Note that after submitting this manuscript we came to know about the investigation of the magnetocaloric effect in CrCl 3single crystals by Liu et al.,25which does not explore the isotropic nature of the MCE, considering the demagnetization correction. Also, they have not calculated magnetic entropy changes and adiabatic temperature changes from the heat capacity measurement.0 100 200 300050100 2468101202 0 4 0 2468101202 0 4 0 01 5 3 0 4 50510Cp CLCp(J mol-1K-1) T(K)T(K) Sm CmCm,Sm(J/mol K)R*ln4 (b)(a) 0T 0.5T 1T 2T 3T 5T 7TCm(J mol-1K-1) T(K) FIG. 4. (a) The zero field heat capacity data and combined Debye–Einstein fit (solid line). Estimated magnetic entropy and magnetic heat capacity are shown in the inset. (b) Temperature dependence of magnetic heat capacity for different fields.05101520 02 5 5 0 7 5 1 0 00246Open Symbol- M(T, H) Solid line-Cp(T, H) 0.5T 1T 2T 3T 5T 7T-ΔSM(J kg-1K-1) (b)0-0.5T 0-1T 0-2T 0-3T 0-4T 0-5T 0-6T 0-7TΔTad(K) T(K)(a) FIG. 5. (a) Temperature variation of /C0DSMat different fields calculated from the heat capacity data. (b) Temperature dependence of adiabatic temperature changeDT adat different fields. TABLE I. Parameters for layered vdW refrigerant materials at liquid-hydrogen tem- perature for a field change of 0–2 T. vdW /C0DSmax M TmaxdTFWHM RCP systems (J kg/C01K–1) (K) (K) (J kg/C01) References CrCl 3 8.97 18 16.09 144.35 This work Crl3 2.4 60 18.8 45 14 Fe3/C0xGeTe 2 0.8 155 50 40 18 Cr2Ge2Te6 26 8 2 1 3 8 22 Vl3 1.9 50 8.9 17 23Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 092405 (2020); doi: 10.1063/5.0019985 117, 092405-4 Published under license by AIP PublishingSee the supplementary material f o rd e t a i l e da n a l y s i sa n dp l o t sf o r determination of order of transition and efficiency. The authors are very much thankful to Dr. Nazir Khan for his valuable assistance in the revised manuscript. The authors also would like to thank A. Paul for his help during the measurements. V.G. contributed to this work while in service, acknowledges UGC- DAE CSR for LTHM facilities, and would like to thank Senior Cryogenic Engineer Er. P. Saravanan. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1J. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. Seyler, W. Yao, and X. Xu,Nat. Rev. Mater. 1, 16055 (2016). 2E. Pomerantseva and Y. Gogotsi, Nat. 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Conde, Appl. Phys. Lett. 89, 222512 (2006). 21J. Y. Law, V. Franco, L. M. Moreno-Ram /C19ırez, A. Conde, D. Y. Karpenkov, I. Radulov, K. P. Skokov, and O. Gutfleisch, Nat. Mater. 9(1), 2680 (2018). 22W. Liu, Y. Dai, Y.-E. Yang, J. Fan, L. Pi, L. Zhang, and Y. Zhang, Phys. Rev. B 98, 214420 (2018). 23J. Yan, X. Luo, F. C. Chen, J. J. Gao, Z. Z. Jiang, G. C. Zhao, Y. Sun, H. Y. Lv, S. J. Tian, Q. W. Yin, H. C. Lei, W. J. Lu, P. Tong, W. H. Song, X. B. Zhu, and Y.P. Sun, Phys. Rev. B 100, 094402 (2019). 24C. Romero-Mu ~niz, R. Tamura, S. Tanaka, and V. Franco, Phys. Rev. B 94, 134401 (2016). 25Y. Liu and C. Petrovic, Phys. Rev. B 102, 014424 (2020).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 092405 (2020); doi: 10.1063/5.0019985 117, 092405-5 Published under license by AIP Publishing

cjcp2005061.pdf

Chin. J. Chem. Phys. 33, 443 (2020); https://doi.org/10.1063/1674-0068/cjcp2005061 33, 443 © 2020 Chinese Physical Society.Topological transition in monolayer blue phosphorene with transition-metal adatom under strain Cite as: Chin. J. Chem. Phys. 33, 443 (2020); https://doi.org/10.1063/1674-0068/cjcp2005061 Submitted: 05 May 2020 . Accepted: 24 May 2020 . Published Online: 18 September 2020 Ge Hu , and Jun Hu ARTICLES YOU MAY BE INTERESTED IN Effect of passivation on piezoelectricity of ZnO nanowire Chinese Journal of Chemical Physics 33, 434 (2020); https://doi.org/10.1063/1674-0068/ cjcp1911208 Geometric and electronic structures of pyrazine molecule chemisorbed on Si(100) surface by XPS and NEXAFS spectroscopy Chinese Journal of Chemical Physics 33, 417 (2020); https://doi.org/10.1063/1674-0068/ cjcp1910180 Doping copper ions in a metal-organic framework (UiO-66-NH 2): Location effect examined by ultrafast spectroscopy Chinese Journal of Chemical Physics 33, 394 (2020); https://doi.org/10.1063/1674-0068/ cjcp2005070CHINESE JOURNAL OF CHEMICAL PHYSICS VOLUME 33, NUMBER 4 AUGUST 27, 2020 ARTICLE Topological Transition in Monolayer Blue Phosphorene with Transition-Metal Adatom under Strain Ge Hu ;Jun Hu School of Physical Science and Technology & Jiangsu Key Laboratory of Thin Films, Soochow University, Suzhou 215006, China. (Dated: Received on May 5, 2020; Accepted on May 24, 2020) We carried out rst-principles calculations to investigate the electronic properties of the monolayer blue phosphorene (BlueP) decorated by the group-IVB transition-metal adatoms (Cr, Mo and W), and found that the Cr-decorated BlueP is a magnetic half metal, while the Mo- and W-decorated BlueP are semiconductors with band gaps smaller than 0.2 eV. Compressive biaxial strains make the band gaps close and reopen, and band inversions occur during this process, which induces topological transitions in the Mo-decorated BlueP (with strain of 5:75%) and W-decorated BlueP (with strain of 4:25%) from normal insulators to topological insulators (TIs). The TI gap is 94 meV for the Mo-decorated BlueP and 218 meV for the W-decorated BlueP. Such large TI gaps demonstrate the possibility to engineer topological phases in the monolayer BlueP with transition-metal adatoms at high temperature. Key words: Topological transition, Monolayer Blue phosphorene, Biaxial strain, Transition-metal adatom I. INTRODUCTION Advances in modern technologies accelerate minia- turization of electronic devices, which drives the en- deavors for exploring exotic materials in nano scale. Ever since the discovery of graphene which was later predicted to be a topological insulator (TI) [1{4], two- dimensional (2D) crystals have received extensive re- search interest because their unique physical and chem- ical properties that are not found in their bulk coun- terparts are promising for applications in the future electronics and spintronics devices. So far, various types of 2D materials have been fabricated such as graphene, hexagonal boron nitride, transition metal dichalcogenide monolayers, and so on, with the elec- tronic properties varying from semiconductors, half metals, metals, TIs and so on [5{7]. Among them, the 2D TIs which are a new state of condensed matters in- spire great interest in recent years, because they exhibit intriguing quantum spin hall (QSH) states. Usually, a QSH state is characterized by the combination of insu- lating bulk state and quantized helical conducting edge state which provides intrinsic spin lock and is robust against elastic backscattering and localization, so that the ITs are ideal for various applications that require dissipationless spin transport [3, 4, 8{11]. Although QSH state was rstly predicted in graphene, it is still Author to whom correspondence should be addressed. E-mail: jhu@suda.edu.cndicult to obtain practical 2D TIs with sizable TI gaps to frustrate thermal uctuations at high temperature. Recently, the family of 2D elemental phosphorus (termed as phosphorene) has attracted great attention [12], since semiconducting few-layer Black Phospho- rene (BlackP) has been fabricated [13, 14]. The eld- eect transistors made of few-layer BlackP exhibit high charge-carrier mobilities up to 1000 cm2
V1
S1[14{ 16], which demonstrates the potential for applications in nanoelectronics devices. Apart from the well-known BlackP, other forms of 2D phosphorene such as the monolayer Blue Phosphorene (BlueP), -phosphorene, and -phosphorene have also been reported [17, 18]. In particular, the monolayer BlueP which crystallizes buckled graphene-like hexagonal structure has been grown on the Au(111) surface [19{22]. The band gap (Eg) of the free-standing monolayer BlueP was pre- dicted to be 2 eV at the generalized gradient ap- proximation (GGA) of the density functional theory (DFT) level [17], while the measurement in experiment for the monolayer BlueP grown on the Au(111) surface yields a smaller band gap of 1.1 eV, partially due to the interaction between the BlueP and metallic substrate [20]. Nevertheless, the monolayer BlueP maintains its semiconducting character, even though it is grown on a metallic substrate, unlike the case of silicene on Ag(111) [23]. Accordingly, the monolayer BlueP is expected to be a promising candidate as a 2D channel material for electronic and optoelectronic devices [24]. It is remarkable that a rich diversity of electronic and magnetic properties may be achieved by decorating 2D DOI:10.1063/1674-0068/cjcp2005061 443 c⃝2020 Chinese Physical Society444 Chin. J. Chem. Phys., Vol. 33, No. 4 Ge Hu et al. phosphorene with metal adatoms, especially for the BlackP and BlueP [25{31]. For instance, metallic 2D phosphorene may be obtained with alkali and alkaline- earth adatoms, some transition-metal adatoms such as Cr may result in half-metallicity, and most 3d transition-metal adatoms are magnetic on 2D phospho- rene. On the other hand, it was found that a variety of ways could be used to turn monolayer 2D phosphorene from normal insulator into TIs. A monolayer BlackP undergoes topological transition via electric eld, dop- ing, hydrostatic pressure, uniaxial strain, or even polar- ized laser [32{37]. A monolayer BlueP may become TI, even through its original electronic band gap is much larger than that of the monolayer BlackP, through oxi- dization, hydrogenation, uorination, or strain [38{40]. These investigations thus open a route toward the re- alization of topological phases in a monolayer 2D phos- phorene. Clearly, it is interesting whether transition- metal adatoms on a monolayer 2D phosphorene pro- duce topological phases, as that was demonstrated in graphene [41{43]. In this work, we chose the monolayer BlueP as the prototype to explore the possibility of produc- ing TI states in 2D phosphorene with transition- metal adatoms. Based on rst-principles calculations, we found that the monolayer BlueP with either Mo adatoms or W adatoms may become TI under certain compressive strain. The TI gap of the Mo-adsorbed BlueP is about 94 meV and that of W-adsorbed BlueP is about 218 meV. Analysis of the electronic structures reveals that the TI states originate from the band in- version between dierent components of the d orbital of the adatoms. II. COMPUTATIONAL METHODS The structural and electronic properties were calcu- lated with DFT as implemented in the Vienna ab initio simulation package [44, 45]. The interaction between valence electrons and ionic cores was described within the framework of the projector augmented wave (PAW) method [46, 47]. The spin-polarized GGA was used for the exchange-correlation potentials and the spin-orbit coupling (SOC) eect was invoked self-consistently [48]. The energy cuto for the plane wave basis expansion was set to 400 eV. The monolayer BlueP has a buckled graphene-like hexagonal primitive cell with two phos- phorous atoms per unit cell, as shown in FIG. 1(a). The optimized lattice constant is 3.275 A, and the buckling height is 1.24 A, in agreement with previous experimen- tal measurement [20]. In this geometry, the P P bond length and P PP bond angle are 2.26 A and 92.92, respectively. To explore the electronic properties of transition-metal adatoms on the monolayer BlueP, we used a 2 2 supercell and the 2D Brillouin zone was sampled by a 15 15k-grid mesh. There are four high symmetric adsorption sites: the top (T) and valley (V) FIG. 1 (a) Top (upper panel) and side (lower panel) views of a monolayer BlueP. The coral and plum spheres stand for the two sublayers of P atoms. The dark spheres stand for adatoms at dierent adsorption sites: hollow (H), cen- ter (C), valley (V), and top (T). Note that the hollow site is above the BlueP plane, while the center site is at the middle point between the two sublayers. The parallelogram indicates a 2 2 supercell. (b) PBE (black solid curves) and HSE06 (blue dashed curves) band structure without SOC of pure Blue phosphorene. The VBM is set to 0 eV. sites over the top and bottom P atoms, respectively, the hollow site (H) above the two P layers, the center site (C) between the two P layers, as notated in FIG. 1(a). With an adatom at each of these sites, a vacuum of 15 A is added to avoid fake interaction between neighboring monolayers which is produced by the periodic boundary condition. Both the lattice constant and atomic posi- tions were fully relaxed with a criterion that requires the forces on each atom smaller than 0.01 eV/ A. The band topology is characterized by the topological in- variant Z2, with Z2= 1 for TIs and Z2= 0 for normal insulators [49]. We adopted the so-called n-eld scheme to calculate Z2[50{52]. III. RESULTS AND DISCUSSION We rst calculated the band structure of the mono- layer BlueP, as shown in FIG. 1(b). It can be seen that the monolayer BlueP is an indirect-band-gap semicon- dutor with the valence band maximum (VBM) and the conduction band minimum (CBM) near the midpoint along the path KandM, respectively. The band gap at the GGA level is 2 eV, in accordance with the previous calculations [17]. It is known that the regular GGA calculations usually underestimate band gaps of semiconductors, so we carried out further calculation with the hybrid functional HSE06 [53] to obtain more accurate band gap. As seen from FIG. 1(b), the dis- persions of the bands from the HSE06 calculation are similar to those from the GGA calculations, but the DOI:10.1063/1674-0068/cjcp2005061 c⃝2020 Chinese Physical SocietyChin. J. Chem. Phys., Vol. 33, No. 4 Topological Transition in Monolayer Blue Phosphorene 445 TABLE I Binding energy for an adatom at dierent ad- sorption sites: center (C), hollow (H), valley (V), and top (T). Aadatom Binding energy/eV C H V T Cr 0.09 1.24 1.48 0.75 Mo 1.84 1.71 2.61 0.94 W 3.10 2.84 3.38 1.41 band gap increases signicantly to 2.8 eV, which in- dicates that the monolayer BlueP is a wide-band-gap semiconductor. The group-VI B transition-metal elements (Cr, Mo and W) were chosen to be the adatoms. To nd the most stable adsorption congure, we calculated the binding energy ( Eb) with one adatom at each of the four dierent adsorption sites as indicated in FIG. 1(a) in a 2 2 supercell. Ebis calculated as Eb=EBlueP +EAdatom EAdatom/BlueP where EBlueP ,EAdatom and EAdatom/BlueP are the to- tal energies of the pure monolayer BlueP, isolated transition-metal atom and the BlueP with adatom, re- spectively. As listed in Table I, all the adatoms pre- fer the V site, in line with the previous calculations [25, 27, 29]. In this conguration, the adatom mainly bonds to the P atom underneath and the bonding be- tween the adatom and the three neighboring P atoms which form the upper bound of the valley is weaker. For all adatoms, the amplitudes of Ebat the V site are large, which implies strong hybridization between the BlueP and adatoms. In addition, Ebincreases from 3d to 5d adatom, because of the increasing chemical activ- ity for large adatom. Moreover, the Cr adatom at the V site is magnetic, with magnetic moment of 4.1 B, while both Mo and W adatoms are nonmagnetic. FIG. 2 shows the element-resolved band structure for the monolayer BlueP with a Cr adatom at the V site. Attractively, the system is half-metallic, the same as the previous reports [25, 27, 29]. The bands in the en- ergy range of 2.5 to 2.0 eV are comprised of pure P states. In the majority spin channel FIG. 2(a), a few bands (from 1.7 eV to 0.7 eV) appear in the band gap of the BlueP ( 2.2 eV to 0 eV), and they are con- tributed almost completely from the 3d orbitals of the Cr adatom. In the minority spin channel (FIG. 2(a)), two bands cross the Fermi level, and they are from the hybridization between the P and Cr atoms as indicated by the color bar. Clearly, there is a pseudo band gap in the energy range from 2.1 to 0.3 eV, corresponding to the original band gap of the BlueP. Therefore, the Cr adatom donates electron charge to the BlueP, which pushes the Fermi level to the conduction band of the BlueP. Along with the hybridization between the BlueP and Cr adatom, the whole system becomes a magnetic FIG. 2 (a) Spin-up and (b) spin-down band structures of the monolayer BlueP with a Cr adatom adsorbed at a valley site. The color bar indicates the weights from the host P atoms and Cr adatom. The Fermi energy is set to 0 eV. half metal which may be used in nano-spintronics de- vices. The monolayer BlueP with either Mo or W adatom is nonmagnetic and semiconducting. However, the band gap decreases strikingly down to 0.16 eV for the Mo- adsorbed BlueP and 0.11 for the W-adsorbed BlueP, as shown in FIG. 3 (a) and (d). In both cases, the bands near 2:0 eV are contributed from the P atoms, similar to the case with the Cr adatom. The bands around the Fermi level ( 1.0 to 1.0 eV) mainly origi- nate from the adatoms, slightly hybridized with the P atom, as indicated by the color bar. To further iden- tify the atomic characteristics of these bands in details, we calculated the weights of each atomic orbital for all the bands, as plotted in FIG. 3. Due to the lo- cal symmetry, the d orbitals of the adatoms split into three groups: in-plane orbitals with degenerate d xyand dx2y2, cross orbitals with degenerate d xzand d yz, and perpendicular orbital with d z2. From FIG. 3, it can be seen that the hybridization in the energy range from 1.0 eV to 1.0 eV mainly occurs between d xz=yz of the adatom and p x=yof the P atom underneath, while the other orbitals do not hybridize notably. Moreover, the three neighboring P atoms surrounding the adatom do not contribute to these bands. Interestingly, there are obvious intra-atomic hybridizations between the d or- bitals of the adatoms, which produce three local band gaps around the Fermi level. The two gaps near the midpoints along the paths M andK(denoted as gap-1 and gap-2) are derived from the hybridization be- tween d xz=yz and d z2, while the gap at Kpoint (denoted as gap-3) is mainly from d xy=x2y2and d xz=yz . This feature oers the opportunity to engineer topological phases, because band inversion could be induced by bi- axial strain for small-band-gap semiconductors [54, 55]. By applying tensile or compressive strain, the small gap may close at certain strength of strain and then reopen as the strain further increases. This closing-reopening DOI:10.1063/1674-0068/cjcp2005061 c⃝2020 Chinese Physical Society446 Chin. J. Chem. Phys., Vol. 33, No. 4 Ge Hu et al. FIG. 3 Band structures of the monolayer BlueP with (a{c) Mo or (d{f) W adatom adsorbed at a valley site. The Fermi energy is set to 0 eV. (a) and (d) The complete band structures with the weights of the host P atoms and Mo or W adatom indicated by the color bar. (b) and (e) Bands between 1:0 eV and 1.0 eV with the weights of the d orbitals of the Mo or W adatom. (c) and (f) Bands between 1.0 eV and 1.0 eV with the weights of the p orbitals of the host P atom underneath the adatom. FIG. 4 (a) Charge states and (b) nonrelativistic band gaps at Kpoint (gap-3) for the Mo- and W-deposited BlueP (adatom at the valley site) under dierent biaxial strains. The insets in (a) show the electron charge redistribution caused by the interaction between the host P atoms and adatoms. The yellow and blue regions indicate electron charge gain and loss, respectively. process results in band inversion beside the gap, which is a typical characteristic for TIs. To investigate the eect of biaxial strain on the elec- tronic property, we applied biaxial strain ( ") from 7% to 5% where negative and positive values denote com- pressive and tensile strains, respectively. Intuitively, the strain might alter the hybridization between the adatomand BlueP, which ought to cause change of charge state of the adatoms. As shown in FIG. 4(a), compressive strain makes the charge state of the Mo adatom increase from +1.05e at zero strain to maximum value of +1.17e at strain of 6%. On the contrary, tensile strain does not lead visible change of the charge states of the Mo adatom up to strain of 5%. For the W adatom, the over- DOI:10.1063/1674-0068/cjcp2005061 c⃝2020 Chinese Physical SocietyChin. J. Chem. Phys., Vol. 33, No. 4 Topological Transition in Monolayer Blue Phosphorene 447 all charge states (+0.57e to +0.61e) are much smaller than those of the Mo adatom (+1.04e to +1.17e). In addition, both compressive and tensile strains reduce the charge state of the W adatom slightly. The inset in FIG. 4(a) shows the details of the charge redistribu- tion caused by the hybridization between the adatom and BlueP. It can be seen that the charge transfer hap- pens not only between the adatom and its neighboring P atoms but also between dierent orbitals of the adatom. Nevertheless, the charge transfer from the adatoms to the BlueP dominates, which results in positive charge state of the adatoms. Then we calculated the band structures of BlueP with Mo and W adatoms under strains. For both Mo and W adatoms, gap-3 undergoes closing-reopening process under compressive strain, while this happens for gap- 1 and gap-2 under tensile strain. Since gap-3 locates at highly symmetric Kpoint, it may be more interest- ing than the other two gaps. Hence, we focus on gap-3 in the following. FIG. 4(b) shows the nonrelativistic values of gap-3 as a function of strains. It can be seen that as the strain varies from 7% to 5%, gap-3 of both Mo- and W-adsorbed BlueP decreases linearly rst and then increases linearly. The amplitude of gap-3 reaches zero at 5:75% and 4:25% for Mo- and W-adsorbed BlueP, respectively, which indicates the critical point of the closing-reopening process. Obviously, normal insu- lator to topological insulator transition is expected at these critical strains. So it is worth investigating the corresponding band structures in detail, as plotted in FIG. 5 and FIG. 6. From FIG. 5(a), it can be seen that two bands cross the Fermi level at Kpoint and they are linearly disper- sive near the Fermi level. This attribute implies that the low-energy electrons behave as massless Dirac Fermions as in graphene [3, 4]. Moreover, these bands are mainly from the d xy=x2y2and d xz=yz orbitals of Mo, mixed slightly with the p x=yorbital of the P atom underneath, as presented in FIG. 5 (b) and (c). Including the SOC eect, the degeneracy of the linear bands is lifted due to the Rashba eect and a gap of 32.5 meV opens at K point (FIG. 5(d)). The atomic-orbital resolved bands in FIG. 5(d) shows that the highest occupied and low- est unoccupied levels at Kpoint are contributed com- pletely from d xz=yz and d xy=x2y2, respectively, which indicates band inversion induced by the SOC eect. To conrm that the band inversion indeed results in topo- logical phase in the Mo-adsorbed BlueP, we calculated Z2with the n-eld method [50{52]. By summing the positive and negative n-eld numbers over half of the torus as indicated in the inset in FIG. 5(d), we obtained Z2=1 for the Mo-adsorbed BlueP, clearly demonstrat- ing the nontrivial band topology. For the W-adsorbed BlueP, the linear dispersion also appears for the two bands crossing the Fermi level at Kpoint if the SOC eect is ignored, as shown in FIG. 6(a). When the SOC eect is considered, a gap of 76.1 meV is induced around the crossing point. However, the SOC eect makes the FIG. 5 Band structures of the monolayer BlueP with a Mo adatom adsorbed at a valley site under compressive strain of5.75% (a, b) without and (c, d) with the spin-orbit coupling. The Fermi energy is set to 0 eV. The blue dashed curves in (a) and (c) are bands calculated by the HSE06 method. The insets in (a) and (c) show low-energy bands near the Fermi energy. (b) and (d) Bands with the weights of the d orbitals of the Mo adatom. The inset in (d) shows the n-eld conguration. The nonzero points are denoted by red ( n=1) and blue ( n=1) dots, respectively. The Z2 invariant is obtained by summing all the nin half Brillouin zone marked by the shadow. upper bands of gap-1 and gap-2 decrease down to the Fermi level, so it seems that the W-adsorbed BlueP is a semimetal. However, the W-adsorbed BlueP is actually a TI because the topological invariant Z2is 1 as calcu- lated with the n-eld method (see the inset in FIG. 6). It should be pointed out that the topological tran- sition is size-dependent. In fact, we calculated the band structures of BlueP with one adatom in a 3 3 or 44 supercell. However, the systems are only nor- mal insulators. This is because the adatoms interact with each other mediated by the Bloch wavefunctions of BlueP. In a 2 2 supercell, the distance between neigh- boring adatoms is small, so the interaction between the adatoms is strong. Therefore the impurity bands cross each other in the original band gap of BlueP, which results in the nontrivial topological phase induced by the spin-orbit coupling. However, if the supercell size is large (3 3 and 4 4), the interaction between the adatoms is weak. Then, the impurity bands keep sep- arate from each other, which makes the system as a normal insulator. Fortunately, the high-density deposi- DOI:10.1063/1674-0068/cjcp2005061 c⃝2020 Chinese Physical Society448 Chin. J. Chem. Phys., Vol. 33, No. 4 Ge Hu et al. FIG. 6 Band structures of the monolayer BlueP with a W adatom adsorbed at valley site under compressive strain of 4.25% (a) without and (b) with the spin-orbit coupling. The blue dashed curves are bands calculated by the HSE06 method. The Fermi energy is set to 0 eV. The inset shows the n-eld conguration. tion of adatoms could be achieved under delicate growth conditions, which means that the topological phase pre- dicted in this work may be realizable. As mentioned above, the GGA calculations usually underestimate band gaps of semiconductors. Therefore, it is necessary to verify the Dirac states in the Mo- adsorbed BlueP and W-adsorbed BlueP at the HSE06 level. To save computational resources, we only cal- culated the band structure of the path -M-Kwhich is good enough to capture the feature of the bands near the Fermi level. Interestingly, the linearly dis- persive Dirac bands maintain in the HSE06 calcula- tions for both cases, and the SOC eect induces gaps of 94.3 and 217.5 meV near Kpoint respectively for the Mo-adsorbed BlueP and W-adsorbed BlueP, as seen in FIG. 5 and FIG. 6. Most importantly, the topolog- ical invariant Z2is still 1, which demonstrates that the Mo-adsorbed BlueP and W-adsorbed BlueP is a TI at the HSE06 level. Furthermore, it can be seen that W-adsorbed BlueP has an indirect global band gap of 146.1 meV (see FIG. 6(b)), which implies that the W-adsorbed BlueP is a semiconductor rather than a semimetal one which is predicted by the GGA calcu- lations. IV. CONCLUSION Based on rst-principles calculations, we studied the electronic properties of the monolayer BlueP with group-IV B transition-metal adatoms (Cr, Mo and W). We found that the Cr-adsorbed BlueP is magnetic half metal, while the Mo- and W-adsorbed BlueP are semiconductors with band gap smaller than 0.2 eV. The hybridization between the host P atoms and theadatom generates a few bands in the original band gap of the monolayer BlueP, leading the reduction of the band gaps of the Mo- and W-adsorbed BlueP. Intrigu- ingly, compressive strains turn the Mo- and W-adsorbed BlueP into TIs with large TI gaps. The TI gap of the Mo-adsorbed BlueP is 94.3 meV and that of the W- adsorbed BlueP can be as large as 217.5 meV. Investi- gation of the band structures reveals that the band in- version between dierent components of the d orbital of the adatoms is responsible for the topological transition. Our ndings manifest that the Mo- and W-adsorbed BlueP may be good platforms to investigate the QSH eect in 2D TIs at high temperature and hence promis- ing for practical applications in spintronics devices. V. ACKNOWLEDGMENTS This work is supported by the National Natural Sci- ence Foundation of China (No.11574223), the Natural Science Foundation of Jiangsu Province (BK20150303) and the Six Talent Peaks Project of Jiangsu Province (2019-XCL-081). [1]K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, A. A. Firsov, Science, 306, 666 (2004). [2]K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotkevich, S. V. Morozov, and A. K. Geim, Proc. Natl. Acad. Sci. 102, 10451 (2005). [3]C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005). [4]C. L. 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5.0015679.pdf

AIP Conference Proceedings 2251 , 040040 (2020); https://doi.org/10.1063/5.0015679 2251 , 040040 © 2020 Author(s).The effect of Zn doping on thermal properties and antimicrobial of ZnxFe2-xO3 nanoparticles Cite as: AIP Conference Proceedings 2251 , 040040 (2020); https://doi.org/10.1063/5.0015679 Published Online: 18 August 2020 Kormil Saputra , Sunaryono Sunaryono , Nizar Velayati Difa , Samsul Hidayat , and Ahmad Taufiq ARTICLES YOU MAY BE INTERESTED IN Magnetic driven electrical conductivity and band gap energy of SrTi 1-XCrxO3 AIP Conference Proceedings 2251 , 040003 (2020); https://doi.org/10.1063/5.0015818 Photoelectric conductivity of SrTi 1-xNixO3 driven by orbital and magnetic field AIP Conference Proceedings 2251 , 040013 (2020); https://doi.org/10.1063/5.0015819 Green synthesis of Fe 3O4 nanoparticles based on biosurfactant Saccharum officinarum extract AIP Conference Proceedings 2251 , 040035 (2020); https://doi.org/10.1063/5.0015631The Effect of Zn Doping on Thermal P roperties a nd Antimicrobial of Zn x Fe 2 - x O 3 Nanoparticles Kormil Saputra 1 , Sunaryono Sunaryono 1, 2, * ) , Nizar Velayati Difa 1 , Samsul Hidayat 1 , and Ahmad Taufiq 1, 2 1 Department of Physics, Faculty of Mathematics and Natural Sciences, Universitas Negeri Malang, Jl. Semarang No. 5, Malang 65145 , Indonesia 2 Center of Advanced Materials for Renewable Energy (CAMRY) , Universitas Negeri Malang, Jl. Semarang No. 5 Malang 65145 , Indonesia * ) Corresponding a uthor: sunaryono .fmipa@um.ac.id Abstract . As a natural material, iron oxides can be applied and advanced across discip lines, such as industrial, medical, and technological fields. Among various kinds of iron oxides material, there is hematite. In this research, Zn doping on hematite has been successfully executed through a co - precipitation method. Mass variation (0.1 and 0.3 g) was also completed to construe the hematite characteristics through sample characterization using X - Ray Diffraction (XRD) , Fourier - transform infrared (FTIR) spectroscopy, Magneto - thermal instrument, and antimicrobial bacteria testing. Those tests were conducted to 1) see the structural pattern, particle size, and the effect of Zn doping on the crystal structure of the sample , 2) know the functional group of the sample, 3) k now the thermal pattern, and 4) see the thermal - activity of the sample. By using XRD data analysis, 5 and 60 nm grains sized of Zn x Fe 2 - x O 3 nanoparticles w ere obtained for Zn mass of 0.1 and 0.3 g, respectively. The result of the magneto - thermal test exhibi ted a specific absorption rate (SAR) of 1.017 and 1.075 W/g for 0.1 and 0.3 g Zn, respectively. Besides, there was a stronger zone of inhibition of Zn x Fe 2 - x O 3 particle with 0.1 and 0.3 g Zn doping applied over E. Coli (~ 13 and 10 mm) rather than applied t o S. Aureus (~ 7 and 6 mm). Thus, hematite with Zn doping has a potential as the candidate for anti - bacterial agents, particularly for the 0.1 g Zn doping composition. INTRODUCTION Recently, studies on natural material based magnetic nanoparticles have been intriguing among scholars. Hematite is one of the successfully developed magnetic nanoparticles [1]. Hematite ( α - Fe 2 O 3 ) is a highly stable iron oxide [2]. Additionally, hematite can be applied across disciplines as ingredients in pi gment [3], catalyst [4], s ensor [5], photocatalytic [6], magnetic material [7], anticorrosion coating [8], and an anti - bacterial agent [9]. Also, hematite is a kind of spintronic materials with biocompatible [10], low toxic [11], and ferromagnetic characte ristics [12] . Hematite is frequently doped with a specific compound to strengthen its chemical and physical characteristics. On the other hand, Zn is one of the atoms that is generally used for magnetite doping material. Zn material was chosen as the dopin g atom due to its higher radius than Fe. Thus, it regulates hematite [13]. Ayed et al . reported that Zn doping (α - Fe 2 O 3 ) with cubical phase and hexagonal polyhedron pyramided with Zn doping has high index bias due to the radius shifting in the hematite sam ple [14]. Furthermore, Nikolic et al . reported that the result of Zn doping on hematite samples shows its better conductive properties than the natural hematite as well as semiconductor properties with the narrow band (1.9 eV). Thus, it can be potentially applied as the catalyst for photocatalytic [15]. From the results of studies explained above, several attempts have been carried out to synthesize hematite compound with Zn doping (Zn x Fe 2 - x O 3 ) , which are sol - gel method [13], microemulsion [16], combustion [17], solvothermal [18] , hydrothermal [19] , solvent evaporation [20] and so forth. However, those methods require special tools, high temperatures, and a sterilized synthesis environment. As a consequence, they generally involve a long period and high cost. Therefore, this research was developed to synthesize Zn x Fe 2 - x O 3 through co - precipitation, which has been rarely employed. The Zn x Fe 2 - x O 3 nanoparticles w ere synthesized with Zn doping of 0.1 and 0.3 g. During International Conference on Electromagnetism, Rock Magnetism and Magnetic Material (ICE-R3M) 2019 AIP Conf. Proc. 2251, 040040-1–040040-7; https://doi.org/10.1063/5.0015679 Published by AIP Publishing. 978-0-7354-2010-6/$30.00040040-1the synthesis process, for the transition of γ hematite to α - Fe 2 O 3 , the calcination process with + 400 ° C temperature was employed. The examination of the Zn x Fe 2 - x O 3 nanoparticles’ structure and the functional group was characterized using XRD and FITR instruments. In addition, the examination of thermal properties and anti - microbial activities in Zn x Fe 2 - x O 3 nanoparticles were characterized by using magneto - thermal and antimicrobial bacteria testing instruments. EXPERIMENTAL METHOD Sample synthesis was started by the separation process of the iron sand, resulted in 20 g iron sand that was reacted to Hydrochloric acid ( HCl ) . The solution was reacted on a hot plate magnetic stirrer for a while and filtered through filter paper. The filtered solution of ferrous chloride ( FeCl 2 ) and Zinc chloride ( ZnCl ) with a variation of 0.1 and 0 .3 g were mixed and spun on the hot plate magnetic stirrer for 15 minutes. Subsequently, a drop of Ammonium hydroxide ( NH 4 OH ) was added on the room temperature and turned on the a ngular velocity of 750 rpm. The sample, later, was calcinated on 450 ° C and washed by using distilled water until pH = 7 was earned. The grain from the washing was characterized by using XRD to identify the phase structure formed and the crystallite size, FITR to see the functional group formed by Zn x Fe 2 - x O 3 , antimicrobial activity t o know various sample activities as an anti - bacterial agent, and magneto - thermal instrument to identify the SAR rate of the sample. RESULTS AND DISCUSSION The result of Zn x Fe 2 - x O 3 nanoparticles XRD characterization for Zn mass variation of 0.1 and 0.3 is shown in Fig . 1. The Zn x Fe 2 - x O 3 nanoparticles with 0.1 g Zn mass diffraction profile show eight main peaks with the highest intensity is in 2 θ ( ° ) angle located at 2 3 , 33, 35. 5 , 41.1, 4 7 .6, 5 3 , 5 8 , 6 3 , 64. 9 , and 6 8 .8°. On the other hand, the Zn x Fe 2 - x O 3 nan oparticles with 0.3 g Zn mass diffraction profile are presented in Table 1. The representation of diffraction pattern expresses crystal hkl plane of Fe 2 O 3 at (012), (104), (110), (113), (024), (116), (112), (214), (300) , and (208) . This finding is in line with research reported by Rufus et al . [21]. They have successfully identified hematite particle in 2 θ (°) located at 24.4, 33.2, 35.8, 41.1, 49.6, 54.2, 57.8, 62.6, 64.1, and 69.8° [21] . TABLE 1 . Analysis of Zn x Fe 2 − x O 3 nanoparticles diffraction peaks for Zn mass of 0.1 and 0.3 g Sam ple Po sition of peak hkl plane [ 012 ] [ 014 ] [ 110 ] [ 113 ] [ 024 ] [ 116 ] [ 112 ] [ 214 ] [ 300 ] [ 208 ] Zn x Fe 2 - x O 3 (Zn = 0.1 g) 23 .0 ° 33 .0 ° 35.5 ° 41.1 ° 47.6 ° 53 . 0 ° 58 .0 ° 63 .0 ° 64.9 ° 68.8 ° Zn x Fe 2 - x O 3 (Zn = 0.3 g) 23.7 ° 33.4 ° 35.1 ° 39.7 ° 46.7 ° 52 .0 ° 56 .0 ° 63 .0 ° 64.9 ° 68.2 ° FIGURE 1 . X - ray diffraction pattern on Zn x Fe 2 - x O 3 for Zn m ass of 0.1 and 0.3 g 040040-2The XRD pattern of Zn x Fe 2 - x O 3 nanoparticles presented in Fig . 1 demonstrates the absence of a Zn atom peak of 2 θ angle between 20 and 30° that becomes Zn Atom properties. The result of XRD pattern analysis is dominated by the hematite sample. This finding corresponds to research condu cted by Chu et al . [22]. They successfully synthesized Zn x Fe 2 - x O 3 without impurity as well as reported the absence of Zn peaks detected on Zn x Fe 2 - x O 3 nanoparticles diffraction pattern [22]. The absence of Zn atom on the Zn x Fe 2 - x O 3 nanoparticles diffraction pattern can be caused by a bigger radius of Fe than Zn, along with the relatively small mass composition of Zn doping. Thus, the causes of magnetite particle domination are essential for this doping process. Through a crystal structure visualized with Vista software, Zn x Fe 2 - x O 3 nanoparticles are manifested with AMCSD 0000143 and presented in Fig . 2. In determining the average size of Zn x Fe 2 - x O 3 nanopa rticles , the XRD sample data were analyzed by using the Debye - Scherrer equation, as presented in Equation (1) [23]. cos Κλ D β θ (1) i n which D is the average of crystallite size, λ is the length of X - ray waves, β represents the FWHM on the maximum peak, θ represents the Bragg angle and K is the constant with 0 . 9 value. The analysis of peak (104) using Gauss fitting to obtain the FWHM which will be used in Debye - Scherrer equation cal culation is shown in Fig . 3. FIGURE 2. Crystal structure model of Zn x Fe 2 - x O 3 nanoparticles FIGURE 3. Result of Gauss fitting on peaks (104) The result of the Debye - Scherrer equation displays Zn x Fe 2 - x O 3 nanoparticles with Zn doping mass of 1.0 and 0.3 have crystal lite sizes average of 5 and 6 nm, respectively. The vast difference between those crystal lite sizes average is caused by the highly controlled aggregation of Zn x Fe 2 - x O 3 nanoparticles with Zn dop ing mass of 0.3 g. The high aggregation on Zn x Fe 2 - x O 3 nanoparticles, particularly for the Zn doping mass of 0.3 g, corresponds with research conducted by Lemine et al . [24]. They reported that Fe 2 O 3 nanoparticles tend to have high aggregation due to the shifted of phases during the formation of Fe 2 O 3 . Thus, Fe 2 O 3 nanoparticles generally have a bigger average size of grain [24]. Every functional group of Zn x Fe 2 - x O 3 nanoparticles w as correlated with FTIR referential data to see the assembled atomic bonding. The result of FTIR data analysis and characterization is presented in Fig . 4, and the detailed result of data analysis is shown in Table 2. Atomic bonding vibration of Zn x Fe 2 - x O 3 nanoparticles w as found to have different characteristics, known from data analysis. One of the vibrations is Metal - O vibration with wave number absorption elevation of 4 08 - 50 6 cm - 1 . According to previous research reported by Aisida et al . [25], they reported that vibrati on on the wave elevation of 4 80 - 580 cm - 1 characterized the bonding for Fe - O and An - O octahedral vibration. This further emphasizes that the vibration on Fe 2+ ion in this research is replaced by Zn 2+ ion. In addition, the Fe - O tetrahedral was identified on the wavenumber of 611 cm - 1 that corresponds with the research conducted by Rathod et al . [26]. They reported that Fe - O tetrahedral atom would vibrate on wave number of 580 - 670 cm - 1 . Furthermore, FTIR data analysis shows O - H atom vibration on wave number of 1640 and 331 4 – 35 00 cm - 1 . O - H atom vibration detected in this research coincides with the functional group characteristic that vibrates on wave number of 1636 dan 331 7 - 35 14 cm - 1 [ 27]. Besides, the atom functional group characteristics were also detected on the wavenumber of 1434 cm - 1 that represents CO 2 atom. The detection of CO 2 in this research comes from CO 2 evaporation from the Zn x Fe 2 - x O 3 nanoparticles synthesis residual. This result corresponds with research conducted by Luo et al . [28]. They recorded that the detection of CO 2 material in the wavenumber of 1367 cm - 1 is the effect of the ionic solution synthesis residual caused by the reaction on the cation - anion sample [28] . 040040-3 FIGURE 4. Functional groups of Zn x Fe 2 - x O 3 nanoparticles TABLE 2 . FTIR spectrum of the samples Researched Wavenumber (cm - 1 ) Referential Wavenumber (cm - 1 ) Description 3314 - 3500 3317 - 3514 O - H [27] 1640 1636 O - H [27] 1 434 1367 C - O [28] 4 08 - 506 480 - 580 Fe - O , Zn - O octahedral [25] 611 583 - 670 Fe - O tetrahedral [26] The antimicrobial activity of Zn x Fe 2 - x O 3 nanoparticles w as completed through the diffusion method. In relation to the antimicrobial activity, this research employed two different types of bacteria, namely Staphylococcus aureus , and E. Coli , as positive bacteria. The examination of antimicrobial activity was obtain ed from the inhibition zone diameter of the sample toward the bacteria growth. The illustration of the inhibition zone toward the bacteria growth is presented in Fig . 5. From the result of the examination on the antimicrobial test, the sample’s inhibitio n zone diameter toward the microbe’s growth shows a favorable result. The Zn x Fe 2 - x O 3 nanoparticles with Zn doping mass of 0.1 g acquired a wider inhibition zone diameter than the ones with a Zn doping mass of 0.3 g. Thus, the amount of Zn doping mass does not significantly improve the microbial inhibition zone and Zn x Fe 2 - x O 3 nanoparticles interaction, while increasing the amount of Zn doping mass decreases the microbial membrane. Besides, the Zn atom impact attached to Zn x Fe 2 - x O 3 nanoparticles has an electrostatic attribute materialized from higher affinity energy. The effect of this affinity energy tends to grow and generate an opposing interaction between Zn x Fe 2 - x O 3 nanoparticles and the used bacteria. Therefore, the increasing amount of Zn doping does not effectively obstruct the growth of the bacteria. This is also approved by a finding reported by Zhao et al . [29] . They stated that the cause of the increasing Zn x Fe 2 - x O 3 antimicrobial activity due to the interaction between Zn x Fe 2 - x O 3 and the microbial membrane decelerating the bacteria’s growth [29]. Thus, from this explanation, microbes require a relatively long period to perform cell division. The essential part of this activity relies on the reactive oxygen ability to disent angle microbial cell’s DNA and protein. Figure 6 also displays a more robust inhibition zone from the sample toward the microbe’s growth on E. Coli (~13 mm for Zn doping mass of 0.1 g and 10 mm for Zn doping mass of 0 . 3 g) compared to S. Aureus (~7 mm f or Zn doping mass of 0 . 1 g and 6 mm for Zn doping mass of 0 . 3 g). This phenomenon is induced by E. Coli bacteria, and Zn x Fe 2 - x O 3 nanoparticles can easily permeate membranes. This result is in line with research conducted by Bhushan et al . [30] . They stated that the increasing inhibition zone diameter of E. Coli is caused by the membranes’ layer vulnerability produced by the growing affinity of Fe radius. 040040-4 FIGURE 5. Illustration of antimicrobial activity FIGURE 6. Histogram of Zn x Fe 2 - x O 3 inhibition zone diameter The magneto - thermal test on Zn x Fe 2 - x O 3 nanoparticles was completed by using 965 Hz frequency. The effect of temperature changes on Zn x Fe 2 - x O 3 nanoparticles in a particular time frame is presented in Fig . 7. The result of observation shows Zn x Fe 2 - x O 3 nanoparticles experience an increasing temperature up to the 25 th minute for every variation of Zn doping. There are no significant differences between the increasing temperatures of each doping mass variation . This is caused by the diamagnetic property of the Zn atom. The diamagnetic material within the Zn x Fe 2 - x O 3 nanoparticles will affect the spin and orbital magnetic moment motion of the magnetite. The increasing temperature on the two samples is not optima lly detected due to the total magnetic moment do not experience substantial changes. However, the semi - conductor property of Zn atom that heated up at a particular temperature creates significant improvement on Zn x Fe 2 - x O 3 nanoparticles. This result corresponds to the research and assumption stated by Mandal et al . [31] . They mentioned that Fe 2 O 3 nanoparticles with Zn doping has high content polarization that accelerates the sample’s temperature up to >177 ° C on 100 kHz fre quency [31]. FIGURE 7. Result of magneto thermal test on Zn x Fe 2 - x O 3 nanoparticles In addition, the small temperature changes in the sample are also affected by the frequency used. In this research, the 965 Hz frequency was used. That frequency is relatively low compared to other researches that use above 50 kHz frequency. As the research conducted by Ramanujan and L ao [32] . They used up to 3 77 kHz frequency resulted in the nano magnetite temperature changes up to 40 ° C [32]. The usage of frequency affects the SAR rate of the material. The analysis of the SAR rate of a material is determined by the temperature changes ( ∆ T ) on a specific period ( ∆ t ). The SAR rate is established by Equation (2). s p np m c T SAR m t (2) 040040-5in which m s is the sample’s mass, m np is the sample’s particle, and c p is Zn x Fe 2 - x O 3 nanoparticles’ thermal capacity. Through Equation (2), the SAR rate of Zn x Fe 2 - x O 3 nanoparticles was successfully identified. The ∆ T / ∆ t can be obtained from the linear fitting of the temperature curve toward time [33]. By using the frequency of 965 Hz, the temperature changes graphic toward the time functi on of the sample (Fig . 8) has 1.017 and 1.075 W/g SAR rate, respectively, for the Zn doping mass of 0.1 and 0.3 g). This result is substantially different from a finding reported by Ramanujan and Lao [32] that successfully obtained a magnetite SAR rate of 8.4 W/g with ~375 kHz frequency. On the other hand, Miaskowski et al . successfully obtain a magnetite SAR rate of 20 W/g with 400 kHz frequency [33]. The low SAR rate of Zn x Fe 2 - x O 3 nanoparticles in this research is caused by two main reasons, which is the low frequency used to create the temperature changes and the Zn doping usage that reduce the magnetization value of the sample due to its diamagnetic characteristics. The decreasing magnetization value of Zn x Fe 2 - x O 3 will affect its frail spin and orbital m agnetic motion that results in the low - temperature changes of the sample. (a) (b) FIGURE 8. Data of magneto thermal linear fitting for Zn x Fe 2 - x O 3 nanoparticles with Zn doping mass of (a) 0.1 g and (b) 0.3 g CONCLUSION Synthesis on Zn x Fe 2 - x O 3 nanoparticles has been successfully completed through the co - precipitation method for atom Zn mass of 0.1 and 0.3 g. The structure pattern of Zn x Fe 2 - x O 3 nanopa rticles has been analyzed and in line with the AMCSD 0000143 data model. By using the D ebye - S cherrer equation, the crystal lite size average of Zn x Fe 2 - x O 3 nanoparticles obtained is 5 and 6 mm, respectively, for Zn mass of 0.1 and 0.3 g. The result of antiba cterial activity demonstrates that hematite with Zn doping is the potential candidate for antibacterial agents. This result is derived from the sample’s zone of inhibition after the antibacterial activity test. The Zn x Fe 2 - x O 3 nanoparticles with Zn mass of 0.1 and 0.3 g have a different zone of inhibition. The microbes’ growth using E. Coli is generally stronger than using S. Aureus . Through SAR data analysis, the Zn doping choice is ineffective in accelerating the magnetization value of Zn x Fe 2 - x O 3 nanoparticles. The ineffectiveness of this material is caused by the low spin and orbital magnetic motion that provokes the minimum temperature changes on the sample. Therefore, the SAR rate of Zn x Fe 2 - x O 3 nanoparticles should be improv ed by the doping material with high spin and orbital magnetic patterns. ACKNOWLEDGMENT S This research was financially supported by the UM PNBP on behalf of SN. The authors wish to thank the Rector of UM for this research grant. 040040-6R EFERENCES 1. B . Saiphaneendra, T. Saxena, S. A. Singh, G. Madras, and C. Srivastava, J. Environ. Chem. Eng. , vol. 5, no. 1, pp. 26–37, Feb. 2017. 2. K. K. Kefeni, T. A. M. Msagati, T. T. I. Nkambule, and B. B. Mamba, J. Environ. Chem. Eng. , vol. 6, no. 2, pp. 1865–1874, Apr. 2018. 3. J. Carneiro et al., Ceram. Int. , vol. 44, no. 4, pp. 4211–4219, Mar. 2018. 4. J. Zhang, R. García-Rodríguez, P. Cameron, and S. Eslava, Energy Environ. Sci. , vol. 11, no. 10, pp. 2972–2984, 2018. 5. D. Li et al. , Nanotechnology , vol. 27, no. 18, p. 185702, Mar. 2016. 6. T. C. Araújo, H. dos S. Oliveira, J. J. S. Teles, J. D. Fabris, L. C. A. Oliveira, and J. P. de Mesquita, Appl. Catal. 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5.0009445.pdf

J. Appl. Phys. 128, 040904 (2020); https://doi.org/10.1063/5.0009445 128, 040904 © 2020 Author(s).Metallic antiferromagnets Cite as: J. Appl. Phys. 128, 040904 (2020); https://doi.org/10.1063/5.0009445 Submitted: 31 March 2020 . Accepted: 14 July 2020 . Published Online: 30 July 2020 Saima A. Siddiqui , Joseph Sklenar , Kisung Kang , Matthew J. Gilbert , André Schleife , Nadya Mason , and Axel Hoffmann COLLECTIONS Paper published as part of the special topic on Antiferromagnetic Spintronics Note: This paper is part of the special topic on Antiferromagnetic Spintronics. This paper was selected as Featured Metallic antiferromagnets Cite as: J. Appl. Phys. 128, 040904 (2020); doi: 10.1063/5.0009445 View Online Export Citation CrossMar k Submitted: 31 March 2020 · Accepted: 14 July 2020 · Published Online: 30 July 2020 Saima A. Siddiqui,1,2Joseph Sklenar,3Kisung Kang,1Matthew J. Gilbert,4André Schleife,1,2,5 Nadya Mason,2,6 and Axel Hoffmann1,2,4,6 ,a) AFFILIATIONS 1Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA 2Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA 3Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA 4Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA 5National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA 6Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA Note: This paper is part of the special topic on Antiferromagnetic Spintronics. a)Author to whom correspondence should be addressed: axelh@illinois.edu ABSTRACT Antiferromagnetic materials have recently gained renewed interest due to their possible use in spintronics technologies, where spin transport is the foundation of their functionalities. In that respect, metallic antiferromagnets are of particular interest since they enable complex inter-plays between electronic charge transport, spin, optical, and magnetization dynamics. Here, we review the phenomena where the metallic conductivity provides a unique perspective for the practical use and fundamental properties of antiferromagnetic materials. The future direc- tion is outlined with respect to the current advances of the field. Published under license by AIP Publishing. https://doi.org/10.1063/5.0009445 I. INTRODUCTION As conventional ferromagnetic digital storage devices reach the end of scaling, 1interest has burgeoned in exploring antiferro- magnetic materials for information storage and manipulation. This interest is largely motivated by the robustness of antiferromagneticorder to moderate external magnetic fields, zero net magnetizationthat does not produce stray fields, and switching timescales thatcorrespond to switching rates in the THz regime. In particular, the precession frequency of antiferromagnetic order is set by the geo- metric mean of the anisotropy and exchange energies, 2,3leading to antiferromagnetic switching that is up to two orders of magnitudefaster than ferromagnetic switching. While research on antiferro-magnets has been ongoing for decades, antiferromagnets had proven difficult to manipulate and read. However, the field has now been newly motivated by recent experiments and theory thatseem to show that antiferromagnetic order can be manipulated,possibly by spin –orbit-torques generated by charge currents, 4stag- gered local relativistic fields induced by electrical currents,5domain wall motion,6and optical excitation by circularly polarized light.7 Yet, many aspects regarding manipulation of antiferromagneticorder are still unknown, including the influence of thermal effects,8 the timescale of antiferromagnetic switching and manipulation, and even the mechanism and robustness of the switching itself.8–10 These unknowns motivate further experimental and theoretical study of antiferromagnetic materials. A large number of antiferromagnetic materials are available in nature. Insulating antiferromagnets, which are mostly oxides such as NiO and halides such as MnF 2, have been well-studied recently because of their potential to carry chargeless spin waves (magnons). Conducting antiferromagnets, while regularly used as sources of exchange bias in magnetic spin-valve and tunnel junction-based devices,11have been less considered as spintronic devices. Conducting antiferromagnets also have great potential for funda- mental studies and applications due to their high electrical andthermal conductivities, and the strong interactions of electrons, spin, phonons, and photons. In this perspective, we focus on conducting antiferromagnets, which include materials that are of high current research focus such as CuMnAs, Mn 3Sn, Mn 2Au, and FeRh. Detailed reviews on antiferromagnetic spintronics have been published previously.3,12–15In this perspective, we discuss recentJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 040904 (2020); doi: 10.1063/5.0009445 128, 040904-1 Published under license by AIP Publishing.progress and understanding of key properties of antiferromagnetic metals: charge transport, dynamics, and optical effects ( Fig. 1 ). Numerical values of different effects in antiferromagnetic metalscompared to other relevant materials are shown in Table I . We first discuss the appearance of anisotropic magnetoresistance (AMR) effects in antiferromagnetic metals where the electrical resistancedepends on the relative orientation of the current and the Néelvector; we also mention the related spin-Hall magnetoresistance(SMR) effect. We then discuss how antiferromagnetic metals can be used to generate spin currents via spin Hall or anomalous Hall effects, as well as absorb spin currents, possibly leading to tunablesub-THz frequency oscillators. The mechanisms behind all of these“charge-related ”effects are active areas of investigation. Thus, we discuss how understanding these phenomena may depend on aninterplay between antiferromagnetism and topology, and further show how tunable magnetism may be derived from first-principlescalculations. Beyond charge transport, we discuss progress inlayered antiferromagnetic materials and coupling of light to antifer- romagnetic materials, including studies of the antiferromagnetic structure, phases, and dynamics using linear, quadratic, and non-linear magneto-optical techniques. II. CHARGE TRANSPORT IN METALLIC ANTIFERROMAGNETS A. Magnetoresistance Anisotropic magnetoresistance (AMR) is a long-studied electrical property of ferromagnetic metals where resistivity FIG. 1. Illustration of the recent concepts of antiferromagnets. (a) Anisotropic magnetoresistance. The red and the gray arrows show the magnetic moments. (b) Spin dynamics in antiferromagnets. Magnetization precession of the two sublattices of the two magnon modes. The red and the blue arrows show the magnetic m oments of two sublattices. H! 0is the applied field along the easy axis of the antiferromagnet. (c) Weyl semimetal state with two single Weyl nodes with the opposite (embedded image) chiral charges (left); Fermi surface map at the k-space corresponding to the terminations of the crescent Fermi arcs and bulk Fermi surface map of the k-space region corresponding to the Weyl nodes (right). (d) Schematics of polar magneto-optical Kerr microscopy on a non-collinear antiferromagnet. Repro duced figures in (b) with permission from Rezende et al., J. Appl. Phys. 126, 151101 (2019). Copyright 2019 AIP Publishing LLC. Reproduced figures in (c) with permission from Xu et al., Science 349, 613 (2015). Copyright 2015 The American Association for the Advancement of Science. TABLE I. Overview of physical properties in antiferromagnetic metals in comparison with other materials. Properties Reference materials Antiferromagnetic metals Anomalous Hall conductivity 500 −2×1 05Ω−1cm−1(ferromagnets) 10 –50Ω−1cm−1 Spin Hall conductivity 800 /C08000( /C22h=e)Ω/C01cm−1(heavy metals) 20 –1100 ( /C22h=e)Ω/C01cm−1(Refs. 16–19) Seebeck coefficient 160 −170μVK−1(Bi2O3) (Ref. 20)7 5 μVK−1(Ref. 21) Resonant frequency 5 −20 GHz (ferromagnets) 20 –1000 GHz Magneto-optical Kerr rotation angle 0.003 −3 deg (ferromagnets) 0.02 deg (zero field) (Ref. 22)Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 040904 (2020); doi: 10.1063/5.0009445 128, 040904-2 Published under license by AIP Publishing.depends upon the relative orientation between the magnetization and current or crystalline orientation.23In spintronics research involving ferromagnetic materials, AMR is used in a variety ofcontexts. The mixing of AMR with microwave currents in spin-tronic devices leads to a rectification effect 24,25that can be used to detect high-frequency magnetization dynamics excited by spin torques;26,27this effect is often used to quantify spin torque sym- metries and torque magnitudes.28,29AMR also enables the emission of microwave radiation in spin-torque oscillators that are driven byelectric currents. 30–33AMR is not exclusive to ferromagnets; in antiferromagnet metals, AMR refers to the dependence of the elec- trical resistance upon the relative orientation between the current and the magnetic order vector (e.g., Néel vector) or sometimes onthe direction relative to the crystal axes. As we will discuss, AMR isa useful way to read out a magnetic memory state stored within anantiferromagnetic metal. 4,34,35As interest in these materials evolves, it remains to be seen if the above noted applications of AMR in fer- romagnets will have analogs in antiferromagnets. In ferromagnetic materials, a straightforward way to characterize the AMR of a sample is to apply a large enough magnetic field thatovercomes internal demagnetization fields and anisotropies of the system. 36By saturating the magnetization and rotating the applied magnetic field, the resistivity can be measured as a function of theangle between the current and the field/magnetization of a givensample. Similar measurements can be made in antiferromagnets, provided the magnetocrystalline anisotropies and exchange energy of the material do not restrict rotation of the Néel vector. This hasrecently been shown for Mn 2Au, where large resistance changes were observed after exceeding the spin-flop field of 30 T, which is given bythe geometric mean of anisotropy and exchange fields. 37A good example of characterizing AMR with rotating fields can be found in a synthetic antiferromagnet consisting of different alloys of CoGd.38 In bilayers of CoGd films with different compositions, the tempera-ture can be adjusted to a “compensation ”point where the bilayer behaves as two antiferromagnetically coupled macrospins. At the compensation temperature, a phase shift of 90 /C14in the AMR signal occurs relative to the AMR traces at higher (or lower) temperatureswhen there is a net moment. The phase shift arises from competitionbetween the Zeeman and antiferromagnetic exchange interaction,leading to a 90 /C14angular offset between the Néel vector and the exter- nal field. Examples of rotating magnetic fields leading to an AMR signal have been reported in antiferromagnetic materials, includingSr 2IrO 4,39,40MnTe,41and EuTiO 3.42In Sr 2IrO 4and EuTiO 3,af i e l d - dependent AMR signal was reported where both the angular period and the AMR amplitude were shown to depend on the magnitude of the rotating field. The origin of the field-dependent AMR in thesematerials is still under active investigation. In antiferromagnetic spintronics, AMR and related effects can be used to “read ”out memory states, e.g., of metallic antiferromag- nets or heterostructures incorporating antiferromagnetic layers. An early example was the usage of AMR in FeRh. 34,43FeRh possesses a first order phase transition from ferro- to antiferromagnetismwhich occurs near room temperature. 44By applying a field in the ferromagnetic phase, and cooling to the antiferromagnetic phase, the Néel vector can be initialized perpendicular to the field-cooling orientation. The field-cooling process “writes ”information into the FeRh, and AMR is used to “read ”out the memory state. Usingx-ray linear magnetic dichroism, combined with the AMR measure- ment, a higher (lower) resistance state was found when the Néel order was parallel (perpendicular) to the current.34In Sec. II C,w e discuss how antiferromagnetic materials such as CuMnAs andMn 2Au can have memory states “written ”by electrical means.4,35 Although the writing process differs, the read-out mechanism in these in CuMnAs and Mn 2Au was attributed to their intrinsic AMR. Phenomenologically similar to AMR is the SMR effect. SMR was first discovered in bilayers of a ferromagnetic insulator adjacentto a spin Hall metal. 45Here, an anisotropic resistance is endowed into the spin Hall metal that depends upon the relative orientation between the spin polarization in the spin Hall metal and the magnetic order. SMR also exists in all-metallic bilayers.46,47More recently, SMR has been discovered in bilayers consisting of an anti-ferromagnetic insulator adjacent to a spin Hall metal. 48–51The resistance depends on the orientation of the Néel order relative to the spin Hall effect induced spin polarization. Using spin torque effects from the spin Hall metal, switching experiments that usedSMR to read out the memory state were reported in NiO/Pt. 52The switching experiments in NiO/Pt were qualitatively quite similar tothe experiments in CuMnAs and Mn 2Au. However, recently, there have been a series of experiments in NiO/Pt8,53and Fe 2O3/Pt,54,55suggesting that the AMR/SMR signal, i.e., the read-out in electrical switching experiments, can be a thermalor electromigration artifact arising from the high current densities needed to switch the magnetic state (see Fig. 2 ). In the Fe 2O3/Pt bilayer system, intentional thermal annealing of the sample was usedto distinguish two distinct types of switching in the magnetoresis-tance. A saw-tooth magnetoresistance shape was attributed to thethermal artifact, while a smaller amplitude step-like change in the resistance was identified as antiferromagnetic switching. The implica- tions of these new SMR switching experiments have not been fullyreconciled yet with the earlier AMR switching experiments. Clearly,a future goal in the field of antiferromagnetic spintronics will be toidentify and separate magnetoresistance effects from the resistance modulation arising from thermal or electromigration artifacts in both SMR and AMR based systems and devices. A combination of trans-port and magnetic imaging of the antiferromagnetic states may beeffective in this aspect. 56 B. Spin current generation In Sec. II A, we discussed general features of the charge trans- port, and how the charge transport interacts with the spin structurewithin an antiferromagnet. Another important question is howmetallic antiferromagnets can be used to generate spin currents that can be injected into other adjacent materials. These spin cur- rents may originate from charge currents, temperature gradients, ormagnetization dynamics. The generation of spin currents from charge currents in the bulk of conducting materials is known as spin Hall effects. 57They exist in any conducting materials and are a consequence of spin – orbit coupling. Therefore, one can naively expect that spin Hall effects are more pronounced for materials with heavier elements, and, in fact, a systematic study of different CuAu-I-type Mn-based metallic antifer- romagnets demonstrated such a dependence, both experimentallyJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 040904 (2020); doi: 10.1063/5.0009445 128, 040904-3 Published under license by AIP Publishing.and theoretically.16Using spin pumping and inverse spin Hall effect measurements it was shown that the spin Hall angles, which are material-specific parameters describing the efficiency ofthe charge- to spin-current conversion, follow the relationshipPtMn.IrMn.PdMn.FeMn. In fact, the experimentally observed spin Hall conductivities are reasonably well explained by first-principles calculations of the intrinsic spin Hall effects in these alloys. Similar measurements were subsequently performedby spin-torque ferromagnetic resonance, 58which has become one of the standard approaches for quantifying spin Hall effects.When an rfcurrent is passed through a bilayer of the spin Hall material and a ferromagnetic metallic layer [such as Ni 80Fe20,p e r - malloy (Py)], it results in different torques acting on the magneti-zation of the ferromagnet, as is shown schematically in Fig. 3(a) . An optical image of a sample integrated into a terminated coplanar waveguide is shown in Fig. 3(b) . Here, the current passing through the antiferromagnetic layer may generate an Oersted fieldh rf, which results in a torque τ?/M/C2hrf. It also generates a damping-like torque τk, perpendicular to the torque from the Oersted field, via the spin Hall effect. Note that for investigating spinHall effects in metallic antiferromagnets, one commonly inserts anon-magnetic layer (typically copper) in between the ferromagnetand the antiferromagnet to avoid additional spurious magnetic inter- actions. Since the two torques are perpendicular to each other, they drive the magnetization dynamics in the ferromagnet with differentphases. Consequently, the magnetization dynamics in the ferromag-net results, through its anisotropic magnetoresistance, in a resistanceat radio frequency. The time-varying change in resistance mixes with the original rfcharge current resulting in a phase sensitive detection of the ferromagnetic magnetization dynamics. 24,25Therefore, ana- lyzing the resonance line shape allows the determination of themagnitude of the spin Hall angle. 26,59The results in spin-torque ferromagnetic resonance experiments are shown in Figs. 3(c) and3(d), which show voltage spectra as a function of the applied FIG. 2. (a) Illustration of a multi- terminal device with electrical leads for both the “writing ”and “reading ”of infor- mation into a heterostructure containingPt with or without NiO. (b) Switching behavior in both the longitudinal and transverse “read ”configuration is observed both with and without NiO.Reproduced with permission from Chiang et al., Phys. Rev. Lett. 123, 227203 (2019). Copyright 2019American Physical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 040904 (2020); doi: 10.1063/5.0009445 128, 040904-4 Published under license by AIP Publishing.magnetic field measured for fixed rffrequency for PtMn and IrMn, respectively.17In both cases, the line shape is a combination of an antisymmetric and a symmetric Lorentzian, which correspond tothe Oersted and spin Hall torques, respectively. Thus, the larger sym- metric component in Fig. 3(c) compared to Fig. 3(d) indicates a larger spin Hall angle for PtMn compared to IrMn. The initial experimental results 16were obtained for polycrys- talline films of the metallic antiferromagnets. At the same time, first-principles calculations16suggested pronounced anisotropies of the spin Hall conductivities, which reflect the different broken sym-metries due to the antiferromagnetic order. Subsequent measure-ments for epitaxial films grown along different directions confirmedpronounced anisotropies, e.g., PtMn films grown with an a-axis ori- entation have a spin Hall conductivity about twice as large as PtMn films grown with a c-axis orientation. 17Similarly, large anisotropies for different crystalline orientations have also been observed forother antiferromagnetic systems, such as IrMn 3.60 Generally, these strong anisotropies directly reflect the possi- bility of additional spin Hall contributions, once the antiferromag- netic order reduces the symmetry.29,61Therefore, the question arises, in general, what the role of the antiferromagnetic spin struc-ture is with respect to the spin Hall effects in metallic antiferro-magnets. 16,62A first attempt to investigate this question was pursued by assuming that different exchange bias configurations also reflect different microscopic spin configurations in the antifer-romagnets. The experiment shows that the antidamping-like torquein the NiFe/IrMn system is related to the exchange-bias field. 59 However, other experiments with IrMn show that the spin Halleffects are mostly independent of exchange bias. 63,64Nevertheless,these experiments were performed for polycrystalline films and more conclusive investigations may require measurements forepitaxial systems. In fact, recent measurements with epitaxialMn 3GaN films showed that spin –orbit torques with new symme- tries appear below the Néel ordering temperature.29Another open question is how spin Hall effects in antiferromagnetic metallicalloys depend on composition and doping. Interestingly, the spin Hall effects in antiferromagnetic metals that incorporate heavier elements, such as Ir or Pt, are comparable in efficiency to the spin Hall effects in other commonly used non-magnetic metals, or may even exceed them, i.e., for Mn 2Au.65 Thus, antiferromagnetic metals can be used for switching magneti-zation via spin –orbit torques. At the same time, they may provide an effective field via exchange bias on the magnetization in an adja- cent ferromagnetic layer. This turns out to be useful for switchingmagnetization in ferromagnetic layers with perpendicular anisotro-pies. For many magnetic memory devices, it is often beneficial tohave information stored in perpendicular magnetized layers, since dipolar fields can assist with the switching and, therefore, result in better thermal stability compared to in-plane magnetized layer forsimilar switching fields or currents. However, in order to havedeterministic electric switching of perpendicular magnetizations viaspin –orbit torques, an additional symmetry breaking in-plane mag- netic field is required. Thus, if the symmetry breaking magnetic field is provided via exchange bias, switching of perpendicularmagnetizations can be achieved even without any additional exter-nally applied magnetic field. 66,67Furthermore, inhomogeneities of exchange bias in polycrystalline metallic antiferromagnets may result in magnetization switching via complex intermediate FIG. 3. (a) Schematic of torques gener- ated due to charge current flow in aNi 80Fe20(Py)/Cu/antiferromagnet (AF) multilayer sample. The torques gener- ated from Oersted fields ( τ?) are per- pendicular to the damping-like spin –orbit torques ( τk). (b) Optical image of a sample integrated into a terminated coplanar waveguide together with aschematic of a spin-torque ferromagneticresonance measurement (ST-FMR). (c) and (d) ST-FMR measurements at differ- ent resonance frequencies for PtMn andIrMn. Reproduced with permission fromZhang et al., P h y s .R e v .B 92, 144405 (2015). Copyright 2015 American Physical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 040904 (2020); doi: 10.1063/5.0009445 128, 040904-5 Published under license by AIP Publishing.magnetization states, which can be exploited for memristive behav- ior66that has already been used for implementing associative memory devices.68 In addition to the ordinary spi n Hall effects discussed so far, the magnetic structure in metallic antiferromagnets may alsogive rise to spin current generation with unusual symmetries. In particular, antiferromagnets with chiral non-collinear spin structures are expected to show anomalous Hall effects. This wasfirst discussed theoretically for strained γ-FeMn, 69which has a 3Qspin-structure where spins are arranged on the corner of a tet- rahedron and either point toward or away from the center of the tetrahedron. More recently, similar effects have been theoretically predicted for metallic antiferromagnets with spins arranged on aKagome-lattice, 70,71and indeed corresponding anomalous Hall effects have been observed experimentally.18,19Note that in ferro- magnets, it has already been de monstrated that the anomalous Hall effect is accompanied by a transverse spin current, which can give rise to spin accumulations and spin –orbit torques.72–74 Thus, an open question is whether the anomalous Hall effects in antiferromagnets can similarly give rise to concomitant transversespin currents. Toward this end, it has already been demonstrated that the response of the non-collinear spin structure to an applied field can give rise to a magnetic spin Hall effect, which is odd inmagnetic fields. This has been observed for Mn 3Sn75and Mn 3Ir.76,77Furthermore, it was shown that for [001] oriented Mn 3Ir films, the generated spin cu rrent can have a significant polarization in the out-of-plane direction,78which provides inter- esting new perspectives for manipulating magnetization of ferro-magnets with perpendicular anis otropies. In addition, the close relationship between the broken symmetries due to antiferromag- netic spin structure and the resultant spin currents 79opens up entirely new perspectives for reconfigurable spin –orbit torques. Aside from using charge currents in antiferromagnets for gen- erating spin currents, it is also known that heat current due tothermal gradients can inject spin current from antiferromagnets into ferromagnets, 80,81a phenomenon known as the spin Seebeck effect. Note that due to the compensated nature of the spin struc-ture there can be degenerate magnon modes with opposite spins inantiferromagnets. 82Therefore, a magnetic field is required to lift this degeneracy in order to generate a net spin Seebeck signal. It is known that magnons can contribute to spin transport in metallic antiferromagnets.83,84However, there has been so far one report of the spin Seebeck effect in metallic antiferromagnets.21For metallic ferromagnets, spin Seebeck effects are hard to distinguish from anomalous Nernst effects.85This should not be an issue with col- linear antiferromagnets, where anomalous Hall effects are assumedto be absent. However, recent theoretical and experimental worksshow that anomalous Hall effects can be present in collinearantiferromagnets with specific crystal symmetries that allow to dis- tinguish the two magnetic sublattices. 86,87At the same time, non- collinear, chiral antiferromagnets may give rise to anomalousNernst effects, 88and therefore, just as with the above discussed anomalous Hall effects, the question arises, whether these anoma-lous Nernst effects also give rise to concomitant spin currents. Another possibility for generating spin currents from antifer- romagnets is via spin pumping from antiferromagnetic magnetiza-tion dynamics, 89in analogy to the well-established spin pumpingfrom ferromagnetic resonance.90,91As already mentioned, magnons in antiferromagnets may carry two opposite directions of angular momentum, and thus in principle it is possible to have two differentspin polarizations pumped from an antiferromagnet. Very recently, ithas been shown that such spin pumping is indeed possible frominsulating antiferromagnets, 92–94but similar measurements with metallic antiferromagnets are still missing. C. Spin torques Just as metallic antiferromagnets can be utilized for generating spin currents, they may also absorb spin currents. The absorptionof spin currents becomes of particular interest when the angular momentum associated with the spin currents gets absorbed into the antiferromagnetic spin structure and results in changes ordynamic excitations of the spin structures through spin transfertorques. For ferromagnetic systems with a net magnetization M, one generally distinguishes between field-like torques τ fl/M/C2σ and damping-like torques τdl/M/C2(σ/C2M), where σis the polarization direction of the injected spin current. If one adopts thesame torques due to spin current injections to antiferromagneticsystems, then it is easy to see 3that the field-like torques cancel each other, due to the opposite direction of the two antiferromagnetic spin sublattices. However, since the damping-like torque is even inthe magnetization direction, it creates identical torques for bothsublattices. Thus, if a spin current is injected into the antiferromag-net with a polarization perpendicular to the Néel vector, then the spin-torque induced canting of the two sublattices should lead to a rotation of the Néel vector via the torques from the antiferromag-netic exchange interactions [see Fig. 4(a) ]. 95–97As shown by theo- retical calculations, see Fig. 4(b) , this may then give rise to tunable oscillators in the sub-THz frequency range. So far, an experimental demonstration of such dccurrent driven oscillations is still missing. One issue might be that the spin diffusion lengths in metallic anti-ferromagnets are very short, 17,98,99and so far are reported to be mostly below 2 nm.12Nevertheless, measurements in ultrathin 1-nm thick IrMn films may suggest some spin-torque related mag- netization changes in the metallic antiferromagnet.100 Another possible way to electrically manipulate the antiferro- magnetic spin structure is via Néel spin –orbit torques, as will be discussed further in Sec. III C . The basic idea is that if the crystal structure of the antiferromagnet is such that each crystal site for the two antiferromagnetic spin sublattices has locally oppositeinversion asymmetry, then this may result in local staggered (Néel)spin accumulation with opposite signs for each antiferromagneticspin, and, therefore, result in identical field-like torques for both antiferromagnetic sublattices. This idea was first theoretically pro- posed for Mn 2Au,101 –103but experimental observation of sublattice switching via Néel spin –orbit torques was first demonstrated for CuMnAs.4Subsequently, similar experimental results were obtained for Mn 2Au films.35,104Since there are specific symmetry require- ments for the crystal structure of the metallic antiferromagnet in order to observe Néel spin orbit torques, this effect has only beenreported for CuMnAs and Mn 2Au. Another experimental compli- cation is that all-electrical measurements of the current induced switching of antiferromagnetic spin structures often require mea- surements with currents applied in two different directions withJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 040904 (2020); doi: 10.1063/5.0009445 128, 040904-6 Published under license by AIP Publishing.respect to the crystalline orientation, and with current densities that are close to the breakdown threshold of the devices. Thus,extrinsic effects due to electromigration may very often be mistakenfor changes of the antiferromagnetic spin-structure. 8,53,54Therefore, further exploration of Néel torque related effects will benefit from detailed direct experimental detection (e.g., x-ray magnetic lineardichroism with photoemission electron microscopy 105) of the antifer- romagnetic order in these devices. III. NEW MATERIALS Recently, there have many novel antiferromagnetic materials been predicted from first principle calculations and discovered experimentally. They possess interesting magnetic properties forscientific exploration and technological advances. In this section,we will discuss the most prominent novel antiferromagnetic materi- als in the field.A. Layered materials Layered ferromagnetic and antiferromagnetic materials have been studied since the early 1960s. Only recently, due to the advances in exfoliation processes, has it been possible to separate van der Waals materials into layers with thicknesses corresponding to asingle unit cell. One example is Cr X 3(X¼C l ,B r ,a n dI ) ,w h i c hh a s magnetic ordering down to monolayers at low temperature. Among the trihalides, CrI 3is the most studied antiferromagnetic insulator and has an out-of-plane anisotropy. The intralayer Cr3þions in CrI 3 are coupled ferromagnetically, while the interlayers are coupled antiferromagnetically. On the other hand, CrCl 3has similar mag- netic coupling but with an in-plane anisotropy and CrBr 3is iden- tified as a Heisenberg ferromagnetic insulator.106Other layered materials with antiferromagnetic ordering include Cr 2Ge2Te6,107 XPS3(X¼Mn and Fe),108,109VY2(Y¼Sa n dS e ) ,110and RuCl 3.111 –113In MnPS 3, all nearest-neighbor interactions within a layer are antiferromagnetic114[see Fig. 5(b) ], whereas in FePS 3, FIG. 4. (a) Schematic of antiferromag- netic magnetization dynamics inducedvia spin –orbit torques in a bi-axial antifer- romagnet. (b) Numerically calculated rotation frequency of the Néel vector as a function of driving current. The orangedashed line corresponds to an analyticalapproximation, while the blue and green dashed lines indicate the minimum current densities required to initiate andmaintain the dynamics, respectively.R. Khymyn, I. Lisenkov, V . Tiberkevich, B. A. Ivanov, and A. Slavin, Sci. Rep. 7, 43705 (2017). Copyright 2017 Author(s),licensed under a Creative CommonsAttribution (CC BY) license. FIG. 5. (a) Crystal structure of XPS3(X¼Mn and Fe). Spin orientation in (b) MnPS 3and (c) FePS 3. Reproduced (c) with permission from Lançon et al., Phys. Rev. B 94, 214407 (2016). Copyright 2016 American Physical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 040904 (2020); doi: 10.1063/5.0009445 128, 040904-7 Published under license by AIP Publishing.Fe2þis coupled ferromagnetically to two of the nearest neighbors and antiferromagnetically to the third, so that within the layer the Fe2þmoments appear as ferromagnetic chains coupled antiferro- magnetically to each other [see Fig. 5(c) ].115For both compounds, the magnetic moments are perpendicular to the layers plane. Tunable magnetism has also been identified in many other novel materials from first-principles calculations.116 –118Only recently, the antiferromagnetic van der Waals metal GdTe 3has been identified.119Leiet al. have experimentally shown for low temperatures that the antiferromagnetic order of GdTe 3persists down to three monolayers. Other layered antiferromagnetic metals have been predicted from first-principles calculations to have high spin –orbit torque and Néel temperatures well above room tempera- ture.120However, the experimental demonstration of such materials yet has to be explored. One limitation is the air sensitivity oflayered antiferromagnetic materials, which adds to the challenges for their technological applications. B. Antiferromagnetic Weyl metals In addition to layered materials, the advent of topological materials has cast a new light on many different aspects of theproperties of materials that were, heretofore, considered to be well-understood. More specifically, there is growing experimental and theoretical evidence that there is a strong connectionbetween magnetism and topology within condensed matter andmaterials, 121though the connection is not understood. The lack of understanding represents a unique opportunity to explore antiferromagnetic semimetals for signatures of the presence of topology. In this context, we discuss the current understanding ofboth the theoretical and experimental search for antiferromag-netic Weyl and Dirac semimetals in an effort to not only uncover t h eo r i g i no ft h ec o e x i s t e n c eo ft h e s et w os e e m i n g l yd i s p a r a t e orders, but to assess their poten tial usefulness for future spin- tronic technologies. Before descending to the current state of research in topologi- cal antiferromagnetic semimetals, we briefly review some of the important topological concepts that are needed to understand the developments. To date, the vast majority of observed topologicalphases in non-magnetic metals are stabilized as a result of the pres-ence of time-reversal ( T) and inversion ( P) symmetry. 122,123When either time-reversal symmetry or inversion symmetry is broken, the resultant non-degenerate conduction and valence bands may touch at discrete points or lines within the Brillouin zone. The low-energyquasiparticle excitations around the non-degenerate band touchingpoints are twofold degenerate and described by Weyl fermionswhose Hamiltonian is H(k)¼X i,j¼x,y,zvijkiσj, (1) where σi¼x,y,zare the Pauli matrices and vijis the Fermi velocity assuming that det[ vij]=0. Materials that possess such band struc- tures, with the Weyl points close to the Fermi level, are referred to as Weyl semimetals.124The band touching points, or Weyl nodes, in Weyl semimetals may not be removed by perturbations due tothe fact that there are no remaining Pauli matrices that may be added to the Hamiltonian. Weyl nodes come in pairs and act as monopoles of the Berry curvature with one Weyl node acting as a source of the Berry cur-vature and the other as the sink. The locations of the Weyl nodeswithin the Brillouin zone is determined by the preserved symmetry present in the material. Considering the two aforementioned canonical symmetries individually, we note that when Tis present, then a Weyl node located at kmust have a time-reversed partner located at /C0kthat carries the same topological charge. Therefore, to avoid having a non-zero topological charge within a given mate- rial, there must be two additional Weyl nodes present that both carry oppositely compensating topological charge to ensure thatthe total topological charge remains zero. On the other hand, whenPis present, a Weyl node located at kmust have a partner of oppo- site topological charge located at /C0k. The theory of charge transport in Weyl semimetals is well established and many of the predictions revolve around manifesta-tions of axion electrodynamics. To be precise, axion electrodynamicsrefers to the addition of an axion term to the traditional electromag-netic Lagrangian, where the action is S θ¼e2 2πhÐdtdrθ(r,t)E/C1B.I n the preceding equation, θis the axion background, or magnetoelec- tric polarization, and EandBare the electric and magnetic fields, respectively. The presence of this magnetoelectric term in the crystalproduces prominent charge transport responses such as the chiral anomaly and the anomalous Hall effect (AHE). 125,126 The presence of the AHE arises from contributions that are both intrinsic to the crystal, such as broken Tand spin –orbit cou- pling, and extrinsic, such as defect scattering.127Recent theoretical work has shown that antiferromagnetic metals that have broken time-reversal symmetry and non-collinear spin order will have a non-zero Berry curvature and, consequently, possess an AHE.128 To illustrate the AHE in an antiferromagnetically ordered Weylsemimetal, consider the Heusler compound Mn 3Sn.Figure 6(a) shows the crystal structure of Mn 3Sn. While it is known that Mn 3Sn129may crystallize in both tetragonal and hexagonal struc- tures, Fig. 6(a) depicts the more common hexagonal form of the Mn 3Sn crystal structure with magnetic ordering temperature well above room temperature.18,88The crystal consists of a Kagome lattice formed by the Mn atoms within the ab-plane that are subse- quently stacked vertically along the c-axis to form a tube of face- sharing octahedra. Figure 6(b) shows the ab initio calculated band structure of Mn 3Sn to illustrate the existence of Weyl nodes near the Fermi energy, EF¼0 eV, giving rise to a large Berry curvature at the Fermi surface, where the adiabatic motion of the quasiparti- cles in the Berry curvature leads to the AHE.130InFig. 6(c) , the AHE is plotted for Mn 3Sn at room temperature along two distinct crystal directions producing an anomalous Hall conductivity18of 20Ω/C01cm/C01. In the closely related material, i.e., Mn 3Ge,131that value is 50 Ω/C01cm/C01, and the predicted spin Hall conductivity19is 1100 ( /C22h=e)Ω/C01cm/C01, which is comparable to that of platinum.132 In close association to the Fermi surface properties of the AHE, measurements on the anomalous Nernst effect, shown inFig. 6(d) , similarly demonstrate that Mn 3Sn possesses a large Nernst effect resulting in a Seebeck coefficient of /C250:35μVK/C01 without an externally applied magnetic field at room temperature.88 More exciting results have been reported recently on the electricalJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 040904 (2020); doi: 10.1063/5.0009445 128, 040904-8 Published under license by AIP Publishing.switching of a topological antiferromagnetic state in the antiferro- magnetic Weyl metal Mn 3Sn at room temperature using AHE as the detection method.133Last, we would like to point out that anti- ferromagnetic Weyl semimetals may also have sizeable spin Halleffects. 134 C. Antiferromagnetic Dirac metals In addition to the existence of Weyl fermions in antiferromag- netic materials, it is possible to find additional fermionic excita-tions. To begin to see how one may find Dirac fermions, consider that in order for the band structure of a Weyl semimetal to remain twofold degenerate, the material may not possess both Pand T.The resulting energy spectrum places Weyl nodes of opposite topo- logical charge at the same point in momentum space, resulting in a fourfold degenerate band crossing that is not topologically stable. However, if there is an additional symmetry present in the semime-tallic crystal structure, then Weyl nodes of opposite topologicalcharge may be stabilized at the same point in momentum space.Fortunately, the presence of additional crystalline point group or space group symmetries is capable of constraining the Hamiltonian such that the mixing of Weyl nodes is forbidden, leading to a stablefourfold degenerate band crossing. The stable merger of differentWeyl nodes realizes a (3 þ1)-D Dirac vacuum and materials con- taining such fourfold degenerate nodes are referred to as Dirac semimetals. 124 FIG. 6. (a) Schematic representation of the ab-plane of the Mn 3Sn crystal lattice. The connections between the atoms consisting of alternating large and small triangles illustrate the breathing modes of the Kagome lattice. Reproduced with permission from Ikhlas et al., Nat. Phys. 13, 1085 (2017). Copyright 2017 Springer Nature. (b) Electronic band structure of Mn 3Sn from ab initio calculations using the spin-density functional approximation. Reproduced with permission from Chen et al., Europhys. Lett. 120, 47002 (2017). Copyright 2017 IOP Publishing. (c) Measured AHE in Mn 3Sn at room temperature along two different crystal directions. Reproduced with permis- sion from Nakatsuji et al., Nature 527, 212 (2015). Copyright 2015 Springer Nature. (d) Measured Nernst signal in Mn 3Sn at room temperature showing clear hysteresis as the in-plane magnetic field is varied. Reproduced with permission from Ikhlas et al., Nat. Phys. 1 3, 1085 (2017). Copyright 2017 Springer Nature.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 040904 (2020); doi: 10.1063/5.0009445 128, 040904-9 Published under license by AIP Publishing.In antiferromagnetic metals, the inherent magnetism breaks either one or both PandT. Therefore, in order for a Dirac semi- metal to preserve the fourfold band degeneracy in the presence ofantiferromagnetism, it must possess an emergent antiunitary sym-metry that serves to stabilize relativistic band crossings within themetal. 135Such conditions are satisfied in CuMnAs in which Pand Tare individually broken but the combined PTis preserved in the presence of an additional nonsymmorphic space group D2h.136,137 The resulting band structure of this material, shown in Fig. 7(a) , shows several degenerate band crossings within the Brillouin zonethat are protected by the expansion of the little group resulting from the presence of the nonsymmorphic crystalline symmetries. Moreover, in groundbreaking experimental work, charge transportmeasurements have demonstrated that in CuMnAs one maymanipulate the antiferromagnetic order using the Néel spin –orbit torque. 4 The work on utilizing the Néel spin –orbit torque clearly points to the potential for both reading and writing states in anti-ferromagnets via manipulation of the position of the Néel vectorbetween gapped and gapless phases of the topological antiferro-magnet. Clearly, there are additional materials that possess a similar nonsymmorphic crystal structure, collinear antiferromag- netism, and augmented antiunitary symmetries that may becapable of operating at higher temperatures than for CuMnAs. Figures 7(b) and 7(c) show the band structure calculated via ab initio methods for MnPd 2for two different orientations of the Néel vector. In Fig. 7(b) , the Néel vector is aligned along the [001]-direction where several topologically protected degenerateband crossings are observed, denoted by the arrows. 138Using the Néel spin –orbit torque, the Néel vector may be reoriented to the [010]-direction that results in the magnetic orientation breakingthe underlying crystal symmetries that serve to protect the gapless nature of the topological phase, creating gaps in spectrum. Figures 7(b) and7(c)show biaxial anisotropy energy corresponding to the gapped and gapless phases, respectively. The principle behind the Néel spin –orbit torque is that charge transport reorients the Néel vector and underlying antiferromag- netic order. While it is certain that the Néel spin –orbit torque provides sufficient torque to reorient the magnetism within thetopological Dirac semimetal when the phase is initially gapless, it isunclear if sufficient torque is produced when the semimetal is inthe gapped phase. Another methodology to reorient the antiferromagnetic order within a topological Dirac semimetal is to manipulate the locationof the chemical potential within the material using electrostaticgating. 139Figure 7(d) shows the calculated anisotropy energy for semimetallic CuMnAs as a function of the location of the chemical potential. Note that CuMnAs is semimetallic only in the ortho- rhombic phase. The anisotropy energy in CuMnAs is defined asthe energetic difference between the gapless phase when the Néelvector is aligned along the [100]-direction and the gapped phasewhen the Néel vector is aligned along the [001]-direction with pos- itive values indicating the system prefers the gapless phase. We see clearly that by simply manipulating the chemical potential we areable to change the preferred state of the system without the needfor a charge current. Therefore, one is able to move between the two bistable phases with less energy. IV. DYNAMICS Ferromagnetic resonance (FMR) and the associated magnon mode spectrum are the fundamental dynamic excitations of FIG. 7. (a) Calculated band structure for orthorhombic CuMnAs showing theclear Dirac bandcrossings within theBrillouin zone. Reproduced with per- mission from Šmejkal et al., Stat. Sol.: Rap. Res. Lett. 11, 201700044 (2017).Copyright 2017 John Wiley and Sons.Calculated electronic band structure for MnPd 2with the Néel vector aligned along the (b) [001]-direction and (c)[010]-direction. Reproduced with per-mission from Shao et al., Phys. Rev. Lett. 122, 077203 (2019). Copyright 2019 American Physical Society. (d)Calculated anisotropy energy forCuMnAs as a function of the chemical potential where a positive value indi- cates that the system prefers a gaplessphase and a negative value a gappedphase. Reproduced with permission from Kim et al. , Phys. Rev. B 97, 134415 (2018). Copyright 2018American Physical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 040904 (2020); doi: 10.1063/5.0009445 128, 040904-10 Published under license by AIP Publishing.magnetization, and are ubiquitous across many areas of magnetism such as spintronics140and magnonics.141FMR can be thought of as an infinite wavelength magnon mode. For long wavelength excita-tions, precession frequencies are in the GHz range and are set bythe external magnetic field, magnetic anisotropies, and the dipolarinteraction. 142,143Short wavelength magnons in ferromagnets can have much higher frequencies that are set by the exchange interac- tion.144In antiferromagnets, there are both acoustic and optical antiferromagnetic resonance (AFMR) dynamic modes.145The two modes are distinguished by a phase difference in the precession ofthe antiferromagnetically coupled magnetic sub-lattices. The energy scale of optical magnon modes is set by the exchange interaction across all length scales , and optical AFMR can have frequencies in the THz range. 3,146,147Because spatially uniform modes are easier to access experimentally, long wavelength THz modes in antiferromag-nets represent a unique difference compared with ferromagnets. These modes are actively being considered for their technological potential in terms of ultrafast switching of magnetic memories or aspotential sources for THz electromagnetic radiation. Basic research into the dynamics of antiferromagnets can be more readily enabled if the antiferromagnetic exchange interaction is reduced, since this lowers the resonance frequencies into a more accessible range of values. Synthetic antiferromagnets are an artificialmaterial system typically comprised of multiple magnetic layers thatare weakly coupled. 148A simple example is the insertion of a non- magnetic spacer layer between two ferromagnets which facilitates an antiferromagnetic interaction via the Ruderman –Kittel –Kasuya – Yosida interaction. For typical synthetic antiferromagnets, consistingof two magnetic layers spaced out with a Ru interlayer, opticalAFMR can be observed between 10 and 20 GHz. 149 –152Recently, these types of synthetic magnets have been theoretically and experi- mentally used as a platform to study parity-time symmetry breakingeffects which can lead to exceptional points in the optical and acoustic AFMR eigenvalue spectra.153 –156Exceptional points can be reached by modifying either the interlayer exchange interactionof the relative magnetic damping between layers, and in the vicin-ity of an exceptional point the AFMR frequencies are sensitive toboth parameters. Also recently, magnetic garnet materials known to have exceptionally low magnetic damping have been used to create synthetic antiferromagnets. 157Low damping ferromagnetic insulators, such as yttrium iron garnet (YIG), have previouslyenabled unique coherent magnon phenomena like the Bose –Einstein condensation of magnons. 158There has been previous interest exploring Bose –Einstein condensation of magnons within antiferro- magnets as well.159,160It will be intriguing to see if synthetic antifer- romagnets, employing low-damping insulators, can bridge these twoareas of interest. From the perspective of magnetization dynamics, two dimen- sional magnets based upon van der Waals materials 107,161(see also Sec.III A ) are quite similar to synthetic antiferromagnets. In insu- lating CrI 3and CrCl 3individual atomic layers are ferromagnetic, but there is also a weaker antiferromagnetic interlayer coupling.162 Thick platelets of CrCl 3have been used to study both optical and acoustic AFMR at GHz frequencies [see Fig. 8(a) ].163After the observation of GHz-AFMR in CrCl 3it was reported that thinner samples near the monolayer limit have an increased interlayerexchange coupling. 164This observation may help explain very recent experiments where magnons are optically detected in CrI 3 with reported frequencies varying from tens of GHz165to the THz regime.166In the thick platelet limit, out-of-plane magnetic fields have been used to hybridize optical and acoustic magnon modes inCrCl 3. This is appealing because parallel efforts involving ferro- magnetic materials have identified magnon –photon167 –170and magnon –magnon171 –173hybridized modes as being promising FIG. 8. (a) CrCl 3platelets are fixed onto a co-planar wave guide andabsorption of microwaves in the waveguide, at a fixed frequency, is mea- sured as a function of external field. Both an optical and acoustic AFMRmode are observed at less than10 GHz. Reproduced with permission from MacNeill et al. , Phys. Rev. Lett. 123, 047204 (2019). Copyright 2019 American Physical Society. (b) Anexperimental setup for measuring AFMR in the frequency domain up to 1.2 THz is illustrated. The transmissionof THz radiation passing through theNiO sample is measured as a function of frequency, and an AFMR is identified near 1.0 THz. Reproduced with permis-sion from Moriyama et al. , Phys. Rev. Mater. 3, 051402 (2019). Copyright 2019 American Physical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 040904 (2020); doi: 10.1063/5.0009445 128, 040904-11 Published under license by AIP Publishing.platforms for quantum information processing. Antiferromagnetic materials may have unique potential in these hybrid quantum systems simply because of the separate optical and acoustic modeswhich can be independently targeted for hybridization with eachother or with microwave photons. The discovery of intrinsic Néel spin –orbit torques and the associated current-induced switching of memory states in antiferro- magnets possessing these torques 4,35raises the question of whether current-induced switching at THz speeds is possible.102,174In mate- rials with Néel spin –orbit torques, like CuMnAs and Mn 2Au, the state that is switched is comprised of many magnetic domains.10,105 The switching process itself involves a redistribution of magnetic domains through domain wall motion.6,175A current induced switching process which exploits THz dynamics in an antiferro-magnetic metal has thus far not been demonstrated. Other promis-ing work indicates that a pulse train of THz pulses can lead to a switching process in CuMnAs similar to how switching occurs after applying electrical current pulses. 176,176 Looking ahead, research into the magnetization dynamics of antiferromagnets will benefit from experimental techniques, i.e.,measuring antiferromagnetic resonance in the frequency domain. Recently, terahertz spectroscopy techniques have been used to elec- trically detect AFMR via spin pumping and the inverse spin Halleffect in Cr 2O3.92In addition, broadband techniques working within the frequency domain have helped to study the origin of magnetic damping in both polycrystalline and single crystalline NiO [see Fig. 8(b) ].177By measuring the antiferromagnetic reso- nance in the frequency domain, and quantifying the linewidth ofAFMR as a function of field and temperature, damping mecha-nisms may be partially elucidated. Future devices, such as anti- ferromagnetic spin-torque oscillators, will greatly benefit from identifying materials that have low damping; theoretical progressis being made in this area. 178From this standpoint, FeRh becomes an intriguing material. Magnetic damping in the ferromagnetic phase ofFeRh has been reported, 179and it is a relatively low damping material similar to permalloy. It remains to be seen if the low damping observed in the ferromagnetic phase has any implications fordamping within the antiferromagnetic phase, and this would appearto be an interesting direction to pursue. V. OPTICAL AND MAGNETO-OPTICAL PROPERTIES Interaction of materials with light provides rich information both statically, and also dynamically with femto-second time-resolution. For magnetically ordered materials, this includesvisualization of details of the magnetic structure and magnetic domains, i.e., fundamental material properties that are also essen- tial for applications. For the specific example of antiferromagneticmaterials, the Néel vector orientation of domains can be deter-mined optically, which turns out difficult to probe otherwise dueto the lack of net magnetization. 13,180 For device applications, magneto-optical effects are discussed in the literature, e.g., for reading and writing information throughmanipulation of the magnetic ordering. This constitutes a challengeespecially for antiferromagnetic metals : While neutron diffrac- tion 103,181and synchrotron x-ray techniques182,183analyze the mag- netic structure with high resolution, such measurements requirelarge-scale experimental facilities. Transmission electron diffraction through Lorentz microscopy was successfully implemented for antiferromagnetic NiO,184but suffers from the same problem, com- pared to much more easily accessible optical or magneto-opticalprobes. We now discuss experimental and theoretical results foraccessing fundamental optical and magneto-optical effects in metallic antiferromagnets, 185many of which rely on relatively simple, laboratory-scale experimental setups.186,187Our discussion will be divided into linear optical and non-linear optical effects (see Fig. 9 ), where the former refer to optical processes that merely affect lightpolarization, but do not change the frequency of the light, and the latter allow for such frequency changes. 180 In particular, linear (see Sec. VA) and quadratic magneto- optical effects (see Sec. VB), both of which are linear optical, are suitable for reading magnetic configurations of different domains.Nonlinear optical effects (see Sec. VC), such as second-harmonic generation, require strong electromagnetic fields but can provide direct information of the antiferromagnetic order. 180Manipulation of the magnetic order on time scales of about 100 fs has beenachieved by excitation with short laser pulses (see Sec. VD ). In addition, throughout we will point out magneto-optical effects that so far were only observed in semiconducting or insulating antifer- romagnets, but that also have high potential for yielding importantinsight into metallic antiferromagnets. For a more detailed intro-duction and more comprehensive overview, we refer to the excel- lent reviews in Refs. 13and180. A. Linear magneto-optical effects Magneto-optical effects describe the change of polarization of light upon interaction with the magnetic configuration of a mate-rial. Depending on the magnetic symmetry of the specific material, linear and quadratic magneto-optical effects can occur (see Fig. 9 ). In particular, for two-sublattice collinear antiferromagnets the Néelvector is a good magnetic order parameter and magneto-opticaleffects can be expressed through an expansion of the complex,frequency-dependent dielectric tensor, 147,188 –190 ϵij¼ϵ(0) ijþKijkMkþGMM ijklMkMlþGLL ijklLkLlþGML ijklMkLlþ/C1/C1/C1 :(2) Here, Mis the net magnetization ( M¼M1þM2),Lis the Néel vector ( L¼M1/C0M2),ϵ(0) ijis the magnetization-independent dielectric tensor, Kijkis the linear magneto-optic tensor, and Gijklis the quadratic magneto-optic tensor. Linear magneto-optical effects, as the first-order term ( KijkMk) in the expansion in Eq. (2), are proportional to the net magnetiza- tion M. Examples for these include the magneto-optical Kerr effect (MOKE)191that is measured in reflected light and the Faraday effect,192measured in transmitted light. Due to the zero net magne- tization of collinear antiferromagnets the off-diagonal components of the dielectric tensor typically vanish, precluding observation of linear magneto-optical effects in these materials. When the chemi-cal structures of the sublattices are distinguishable, there is thepossibility to generate off-diagonal components of the dielectric tensor even for collinear antiferromagnets, as recently proposed by Smejkal et al. 86Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 040904 (2020); doi: 10.1063/5.0009445 128, 040904-12 Published under license by AIP Publishing.However, contrary to ferromagnets, antiferromagnetic materi- als also comprise of systems with non-collinear or non-coplanar magnetic ordering (see Fig. 9 ), for which the Néel vector cannot be defined and Eq. (2)is not applicable. For these, symmetry analysis and first-principles simulations recently lead to the prediction ofanomalous Hall conductivity 128and magneto-optical Kerr effect (MOKE).193These seminal works illustrate that magneto-optical effects are not simply linked to the net magnetization, but instead to the underlying magnetic and crystalline symmetries as repre-sented in the off-diagonal elements of the dielectric tensor ϵ ij.194 This insight, along with potential applications for visualizing anti- ferromagnetic order, triggered large interest in magneto-optical effects also for antiferromagnetic metals and in particular, materialswith (i) non-collinear/non-coplanar orderings and (ii) canted col- linear orderings, or combinations thereof (see Fig. 9 ). In 2015, Feng et al. were the first to conclude from their first- principles simulations that three non-spinpolarized, non-magneticmetals Mn 3X (with X ¼Rh, Ir, Pt) show large MOKE.193They attributed this to strong spin –orbit interaction and degeneracy- breaking band splitting, arising from non-collinear antiferromag- netic ordering. The first experimental observation of MOKE in an antiferromagnetic metal was reported shortly after for Mn 3Sn, which shows large zero-field Kerr rotation, comparable in its mag-nitude to ferromagnets. 22Mn 3Sn also shows non-collinear order- ing, with an inverse triangular spin structure and uniform negative vector chirality of the in-plane Mn magnetic moments. While the FIG. 9. Schematic categorization of optical and magneto-optical effects in antiferromagnets (AFMs). Each color label represents a magnetic ordering,studied by different optical or magneto-optical methods. i,s,m-AFM correspond to insulating, semiconducting, and metallic antiferromagnets, respectively.Note that an exception is possible evenfor collinear antiferromagnet if the chemical structures of the sublattices are distinguishable. 86Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 040904 (2020); doi: 10.1063/5.0009445 128, 040904-13 Published under license by AIP Publishing.authors also note that the magnetic moments are slightly canted, causing a small net ferromagnetic moment, they discuss that the large Hall resistivity and field-dependent MOKE measurementsindicate that this ferromagnetic moment is not responsible for thelarge MOKE signal they observed. 22 This claim is supported by symmetry analyses and cluster multipole moments that are suggested as an order parameter to measure symmetry breaking for commensurate non-collinear mag-netic order. 195Cluster multipole moments work similar to ferro- magnets and can generate large linear MOKE and anomalous Halleffect. Higo et al. specifically invoked magnetic octupole domains 22 in their work to explain their large MOKE signals. In addition, Ref. 22reports first-principles calculations that discuss the fully compensated antiferromagnetic state of Mn 3Sn, also confirming large MOKE signals in the absence of any f erromagnetic contributions. More recently, other works followed up on these results and investigated MOKE in non-collinear as well as non-coplanar anti- ferromagnetic metals: Wimmer et al.196used symmetry arguments to discuss magneto-optical phenomena and non-zero off-diagonalelements of the frequency-dependent conductivity tensor of copla-nar, non-collinear Mn 3Ir and Mn 3Ge. Zhou et al. investigated dif- ferent non-collinear antiferromagnetic orderings of Mn 3XN (X¼Ga, Zn, Ag, Ni) and reported strong MOKE signals as well as their dependence on the specific magnetic ordering. In addition tothese coplanar antiferromagnetic metals, Feng et al. recently identi- fied compensated non-coplanar orderings as candidates for strong MOKE signals and illustrated this using first-principles results forKerr rotation angles of γ-Fe 0:5Mn 0:5.197 Most of the above results focus on polar MOKE, i.e., MOKE for surfaces perpendicular to the direction characterizing mag- netic ordering, e.g., that of weak magnetization. In addition, Higo et al. also reported longitudinal MOKE for Mn 3Sn, where weak ferromagnetism lies within the surface plane.22Also, Balk et al. recently measured longitudinal MOKE for non-collinear antiferromagnetic Mn 3Sn with an extremely small in-plane mag- netic moment.198They studied three different antiferromagnetic orderings by increasing the temperature above the Néel tempera-ture of about 420 K. Their results further point to a difference ofsurface and bulk magnetism that requires a more detailedinvestigation. Spontaneously canted collinear antiferromagnets with a weak ferromagnetic contribution show linear magneto-optical effects.Such spontaneous canting of magnetic moments is typically on theorder of 1 /C14and can arise, for instance, due to the Dzyaloshinskii – Moriya interaction.13,199,200This was observed early on for the spontaneously canted antiferromagnetic insulator α-Fe2O3using the Faraday effect.201While orthoferrites are also not metallic, they show canting and are another example that illustrates observa-tion of Faraday rotation without any external stimulation. 202,203 Providing direct evidence of magneto-optical effects also in canted collinear antiferromagnetic metals, and understanding its magni-tude quantitatively, is a promising but outstanding goal. Finally, collinear antiferromagnetic materials can also be studied using MOKE if they possess special crystal symmetry. 86,87 Developing in-depth understanding of these collinear antiferromag- nets using optical analysis is one of the promising future directionsin the spintronics field.In addition to using optical and magneto-optical measurements for visualizing domains, they can also be used to identify magnetic ordering and magnetic phase transitions of a material. Saidl et al. illustrated this for the transition from antiferromagnetic to ferromag-netic FeRh using reflectivity and transmittance measurements. 204 While the non-collinear antiferromagnetic metal Mn 3Sn has three different magnetic phase transitions (see Fig. 10 ), their optical detec- tion was studied near the magnetic transition temperature Tm:198 Balk et al. reported that longitudinal MOKE and anomalous Hall effect are almost zero below Tm, while finite signals arise when the material is heated right above Tm.198Motivated by earlier studies, the combination of accurate first-principles simulations and experiments has successfully identified signatures in optical spectra that correlatedirectly with the crystal structure. 205,206This was also explored for themagnetic structure, e.g., of Mn 3Sn. To this end, first-principles density functional theory calculations were performed within the Vienna Ab Initio Simulation Package (VASP),207,208using the projector-augmented wave method to describe electron –ion interac- tion.209Kohn –Sham states were expanded into plane waves with a cutoff energy of 600 eV. Relaxed atomic geometries and electronicand optical properties 210were computed using a 13 /C213/C213 Monkhorst –Pack211k-point grid to sample the Brillouin zone. Exchange and correlation were described using the generalized-gradient approximation by Perdew et al. 212 From the first-principles calculations of the frequency- dependent complex dielectric tensor across the visible spectral range, similar results were found for polar MOKE around Tmas discussed from experiment,198allowing the distinction of the B2gand the E1g phases (see Fig. 10 ). It was also noted that the result for E1gagreed to within 0.2 eV with that of Higo et al.22The polar MOKE for a total of seven non-collinear antiferromagnetic configurations (E1g-Tx,E1g-Ty,B2g-Tx,B1g-Ty,A2g-Tz,E2g-Txyz,a n d E2g-Tz)a n d three collinear antiferromagnetic configurations (Anti- x,A n t i - y, Anti- z) was calculated. Of these, only E1g-TxandE1g-Tyshow non- vanishing MOKE signals that are very similar in their magnitude, but exhibit different directionality: E1g-TxandE1g-Tycan be detected from different surface orientations, i.e., along the xaxis and yaxis, respectively. Further increasing the temperature from Tmeventually turns Mn 3Sn paramagnetic above the Néel temperature TN(see Fig. 10 ). No polar MOKE signal is expected for paramagnetic Mn 3Sn, indicating that both magnetic phase transitions can be dis- tinguished by magneto-optical detection. However, magneticmoments of the paramagnetic state can align in the presence of astrong enough external field, leading to a ferromagnetic configura- tion. For three different orientations of this phase our results show sizable polar MOKE with different spectral behavior, enabling opticaldistinction of these orientations. B. Quadratic magneto-optical effects While linear magneto-optical effects occur only in antiferro- magnets with certain magnetic orderings, most antiferromagnetic metals exhibit collinear ordering. Even though quadratic magneto-optical effects are typically weaker than their linear counterparts,they enable studying typical collinear antiferromagnets for which linear magneto-optical properties vanish. For collinear ordering with a Néel vector larger than the magnetization the second-orderJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 040904 (2020); doi: 10.1063/5.0009445 128, 040904-14 Published under license by AIP Publishing.term ( GLL ijklLkLl) of the expansion in Eq. (2)dominates and is proportional to the square of the Néel vector. Quadratic magneto-optical effects include magnetic linear dichroism213,214and magnetic linear birefringence of reflected190,215 (also called quadratic MOKE or Hubert –Schäfer effect) or trans- mitted light187(also called Voigt effect or Cotton –Mouton effect). Magnetic linear birefringence arises from a contribution to the dielectric tensor that is separate from crystal-structure driven terms. Itcan be measured indirectly through large changes of the birefringencenear the Néel or Curie temperature of magnetic phase transitions. Inthis case, changes of the birefringence are typically attributed exclu- sively to magnetic contributions. The magnetic contribution can also be measured directly, e.g., in cubic systems with vanishing structure-driven birefringence, in which magnetism reduces the symmetryfrom cubic to uniaxial, leading to magnetic birefringence. 216 In the context of collinear antiferromagnetic metals, Saidl et al. used the Voigt effect to optically determine the orientation ofthe Néel vector in thin films of CuMnAs on a GaP substrate.187 They used a pump –probe setup to accomplish separating the small polarization rotation due to the Voigt effect from all other changesof polarization in the experiment, such as strain. Interestingly, mea-surements with a laser pump –probe system also allow studying the connection of the temperature dependence of the magnetic heat capacity and magnetic linear birefringence: 215For metallic antifer- romagnetic Fe 2As, it was shown that this connection is mediated by the exchange interaction.190We also note that magnetic linear birefringence was studied in the canted collinear antiferromagnets DyFeO 3217and α-Fe2O3;218however, these two materials are not metallic. Finally, Pisarev et al. reported linear dichroism for antiferro- magnetic KNiF 3, attributed it to a purely magneto-optical origin, and disentangled this contribution from strain effects.213 Kharchenko et al. reported the observation of magnetic linear dichroism in MnF 2with the magnitude of the effect being large FIG. 10. Detecting magnetic phase transitions through switching of magneto-optical effects for Mn 3Sn. Left: Polar magneto-optical Kerr effect signals for different magnetic phases. The blue curve shows a computational result by Higo et al. , Ref. 22. Right: Magnetic phase transitions and corresponding temperatures. Above the Néel tempera- ture, Mn 3Sn becomes paramagnetic, but can exhibit ferromagnetic ordering under strong external magnetic fields.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 040904 (2020); doi: 10.1063/5.0009445 128, 040904-15 Published under license by AIP Publishing.enough to visually observe antiferromagnetic domains in the material.214While both of these materials are non-metallic, the potential of linear magnetic dichroism for visualizing antiferro-magnetic domains renders this effect of interest also for metallicantiferromagnets. C. Non-linear optical effects While the previous two sections discussed linear-optical effects, i.e., processes that do not change the frequency of the light,also non-linear optical processes couple to magnetic properties andmagnetic ordering of materials. Due to their low efficiency, theserequire high electromagnetic field strengths, making their experi- mental realization more involved and, thus, less common. To the best of our knowledge, non-linear optical effects were not studiedin metallic antiferromagnets so far; instead, we now highlightimportant examples of semiconductors and insulators to illustrate the potential of non-linear optics for antiferromagnetic metals. Nonlinear magneto-optical effects have also been discussed, espe- cially for ferromagnetic materials, in the literature. 219,220 The most common non-linear optical technique in the present context is second-harmonic generation. Similar to the above discus- sion of magnetic linear birefringence, there are also crystal-structure and magnetic-structure driven contributions to second-harmonicgeneration 221that manifest themselves in the nonlinear susceptibility tensor of a given material. It is reported that second-harmonic gener-ation is particularly well-suited for studying magnetic ordering with broken inversion symmetry 180and it is sensitive to the direction of the Néel vector or net magnetization. This can be measured usingthe difference between left and right circularly polarized light, non-linear rotation, and ellipticity of linearly polarized light, or studyingthe temperature dependence of the second-harmonic signal near the Néel temperature. 222Furthermore, second-harmonic spectra can dis- tinguish 180/C14Néel vector domains, which cannot be achieved using linear-optical methods.221The effect was used to distinguish the sign change under the time-reversal operation, which allows to investigate different domains, e.g., in antiferromagnetic Cr 2O3.222Theoretical predictions exist that second-harmonic generation can be used toprobe antiferromagnetism at surfaces and in thin films of NiO 219,223 and experimental results were presented for the model systems CoO and NiO.221Second-harmonic generation was also studied from first principles for NiO.224Finally, this effect was used to study non- collinear antiferromagnets such as RMnO 3(R= Sc, Y, In, Ho, Er, Tm, By, Lu).221,225 –229ForRMnO 3, Fiebig et al. reported that they can distinguish the different magnetic phases corresponding to dif-ferent non-collinear antiferromagnetic configurations. 221Manz et al. identified antiferromagnetic spin cycloids in TbMnO 3by the helicity of the structure.230 Higher order non-linear optical effects are even more rare; however, one example is the use of the inverse Faraday effect, as athird-order nonlinear optical effect, to induce a magnetization in antiferromagnetic NiO that was subsequently probed by means of the Faraday effect. 231 The above examples represent studies of insulating or semi- conducting systems, i.e., materials with a spectral region of optical transparency. While a comprehensive discussion of the experimen- tal feasibility of non-linear optical and magneto-optical effects inantiferromagnetic metals is beyond the scope of this paper, we note that second-harmonic generation has been accomplished in ferro- magnetic metals232and third-order nonlinear optics was studied for metallic thin films233and, thus, we envision that it can also be a powerful tool to study antiferromagnetic metals. D. Laser-induced dynamics One exciting potential application of antiferromagnetic mate- rials in general and metals in particular, is information storage,since antiferromagnets are expected to show orders of magnitude faster spin dynamics, compared to ferromagnets. 234Reading and writing of information is a prerequisite for such applications andhas, for instance, been achieved in non-collinear, non-metallic anti-ferromagnets. 235Electrical switching of metallic antiferromagnets has indeed been reported, e.g., in CuMnAs4and Mn 2Au.104In addition, magneto-optical effects are effective in reading magnetic information from metallic antiferromagnets, and they may also beutilized to write magnetic information optically by reorientingspins. This triggered interest in the question of whether ultrafastswitching can be achieved optically in metallic antiferromagnets. 180 So far, this question was investigated experimentally and computa- tionally only for semiconducting or insulating antiferromagnets(see Fig. 11 ). The prospect of applications and the fundamental interest in ultrafast phenomena 13,180are the reason why laser- induced dynamics remains an interesting, rapidly evolving research direction and below we provide a current overview. A review oflaser-induced phenomena can be found in Ref. 7. FIG. 11. Schematic categorization of laser-induced dynamics in antiferromag- nets. Color labels represent magnetic structures studied with different magneto- optical methods. i,s,m-AFM correspond to insulating, semiconducting, andmetallic antiferromagnets, respectively.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 040904 (2020); doi: 10.1063/5.0009445 128, 040904-16 Published under license by AIP Publishing.Laser-induced dynamical phenomena include demagnetization,236 –239Néel vector reorientation,234,240and spin oscillations.147,188,241 –243Most of these optical techniques to manipulate magnetic order were applied to insulating and semi-conducting antiferromagnets (see Fig. 11 )a n db e l o ww eb r i e f l y discuss key insights, since these are promising research directions also for metallic antiferromagnets. To the best of our knowledge, only optical-pump induced demagn etization was realized experi- mentally for a metallic antiferromagnet: 190Fe2As was studied using a pump –probe technique and the observed change of the magnetic birefringence signal near the Néel temperature was attributed to laser-induced demagnetization.190 In 2001, Trzeciecki et al. developed a theoretical description of ultrafast spin dynamics in antiferromagnetic NiO and reportedfemto-second time scales for dephasing –rephasing dynamics. 237 Shortly thereafter, laser-induced demagnetization was shown exper- imentally via the optically induced phase transition from antiferro- magnetic to paramagnetic FeBO 3236and a time scale of 700 ps was reported. The work on NiO was followed up later to investigatedemagnetization and magneto-optical switching for bulk and the(001) surface of antiferromagnetic NiO. 238,239 Spin reorientation was triggered by Kimel et al. in antiferro- magnetic TmFeO 3using a short, 100 fs full width at half maximum laser pulse and detected using time-resolved linearmagnetic birefringence. 234They report a reorientation of spins by several tens of degrees within a few picoseconds and explain the underlying mechanism by an o ptical excitation that subse- quently causes a change of the magnetocrystalline anisotropyaxis via electron –phonon and phonon –phonon coupling. Their measurement also relies on an optical approach, making the entire process of influencing and detecting the spin orientation an all-optical technique. More recently, Kimel et al. used a magnetic field pulse, gener- ated from a 100 fs circularly polarized optical pump pulse by meansof the inverse Faraday effect, to demonstrate an inertia-mediated spin switching mechanism in antiferromagnetic HoFeO 3.240They report switching between the Γ12magnetic state, where the Néel vector is in the zyplane, and the Γ24state with the Néel vector in thexzplane. This spin dynamics was monitored via the Faraday effect in the probe pulse. The inertia-driven mechanism allows spin switching with extremely short laser pulses, since it circumvents the use of very strong fields that are typically needed for ultrashortpulses, but that are detrimental because they destroy the magneticorder. It also decouples addressing a given bit from actually switch- ing it, which has large potential for writing large amounts of data. Optically induced spin and Néel vector oscillations in anti- ferromagnets were reported early on using the antiferromagneticresonance 244,245or magnon generation.246The first report of opti- cally induced coherent spin oscillations leading to a net magneti- zation is for MnF 2.247Later, building on a study by Satoh et al. ,241 Tzschaschel et al. used optical pumping (90 fs, 0.98 eV) and probing (50 fs, 1.55 eV) with linearly and circularly polarized lightto study the excitation of two optical magnon modes in antiferro-magnetic NiO. 147They disentangle in-plane and out-of-plane magnon modes by observing the Faraday effect and magnetic linear birefringence, in agreement with their theoretical predictions, includ-ing selection rules for both modes. Essenberger et al. further studythe magnon dispersion of the antiferromagnetic transition-metal oxides NiO, FeO, MnO, and CoO from first principles. 248 Earlier experimental studies of optically induced spin oscilla- tions showed that linearly polarized 150 fs light pulses excite coher-ent spin precession in antiferromagnetic FeBO 3242and DyFeO 3.188 These were performed in the transparent regime of the antiferro- magnetic materials, which prevented heating of the sample and allowed non-thermal excitation; this may constitute a difficultywhen applied to antiferromagnetic metals. In an earlier work onTmFeO 3, Kimel et al. showed that thermal excitations can still excite antiferromagnetic resonances, however, at a different fre- quency than the resonance that is excited non-thermally.243 Recently, first-principles techniques are increasingly applied for simulating real-time dynamics of magnetic order. Differenttechniques are available to study (de-)magnetization dynamicsof ferromagnetic metals, 249 –252but also Heusler compounds253and nanoclusters.251The time scales found in these studies generally agree with experiment. In addition, the simulations provide valu-able insight into the underlying mechanisms: This is illustrated,for instance, by the phase diagram of all-optical spin switching inRef. 254, by attributing demagnetization of Ni and Co to spin flips, 255and by distinguishing mechanisms for demagnetization in bulk from those at surfaces of Ni.256Spin selective charge transfer between magnetic sublattices was identified as the underlyingmechanism for ultrafast switching of magnetic order in Fe-Mn and Co-Mn multilayers and antiferromagnetically ordered NiO 257and was also identified as an important mechanism near the Co/Cuinterface of a ferromagnetic heterostructure. 258This mechanism was also shown to be important in FeNi alloys.259Real-time propa- gation was also applied to study magnons in Fe, Co, and Ni.260 VI. OUTLOOK AND CONCLUSIONS Over the past five years, the interest in metallic antiferromag- nets has significantly increased due to the realization that chargetransport and magnetic spin structures can have very complex interactions. While these interactions were often inspired by phe- nomena that have already been well studied with respect to spin-tronics based on ferromagnets, it turns out that the differentsymmetries of antiferromagnetic materials enable new types of phe-nomena. In particular, antiferromagnets with non-collinear chiral or non-coplanar spin structures do not have easy corresponding systems in typical ferromagnets. Therefore, exploring further therole of symmetry and topology will remain a very fruitful researchfield in the foreseeable future. Clearly, many open questions remain regarding the interplay of charge currents and magnetic structure for metallic antiferro- magnets. In particular, a better understanding of the correlation ofthe magnetic structure with charge transport is required as is indi-cated by recent results showing that electromigration can mimictransport signatures commonly associated with magnetic structure changes. Related to this is the challenge to identify materials where smaller current densities may result in sufficient spin-torques tomanipulate antiferromagnetic spin order. This will help to identifyand overcome thermal artifacts. Another challenge is to obtain a clear understanding of the magnetization dynamics, especially in non-collinear and non-coplanar structures.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 040904 (2020); doi: 10.1063/5.0009445 128, 040904-17 Published under license by AIP Publishing.Furthermore, an important open question is whether dccur- rents can efficiently manipulate the dynamics in antiferromagnets. Can we electrically change the damping in antiferromagnets to thepoint where they spontaneously start to oscillate? If so, then thismay provide completely new perspectives for THz devices andtechnologies. Beyond the connection with THz radiation, it will also be important to better understand the interaction of antiferro- magnetic spin structures with optical photons, which ultimatelymay more readily enable characterizations of the domain structures.Thus, progress with new optical experimental approaches may becrucial for understanding the microscopic physics. Last, in terms of exploring material systems, the investigation of metallic antiferromagnets is really just in its infancy. The researchcommunity is just starting to explore large areas of unusual materialplatforms, such as two-dimensional layered systems and topologicalsemimetals. Therefore, one can expect many new interesting phe- nomena to emerge, which will enrich our fundamental under- standing of antiferromagnets, and also provide new technologicalsolutions that are both robust and energy efficient. ACKNOWLEDGMENTS The preparation of this manuscript was primarily supported by the National Science Foundation (NSF) through the University of Illinois at Urbana-Champaign Materials Research Science and Engineering Center No. DMR-1720633 and was carried out in partin the Materials Research Laboratory Central Research Facilities,University of Illinois. This work also made use of the IllinoisCampus Cluster, a computing resource that is operated by the Illinois Campus Cluster Program (ICCP) in conjunction with the National Center for Supercomputing Applications (NCSA) andwhich is supported by funds from the University of Illinois atUrbana-Champaign. The authors thank Eric Huang for his helppreparing the manuscript. 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5.0018924.pdf

Appl. Phys. Lett. 117, 082403 (2020); https://doi.org/10.1063/5.0018924 117, 082403 © 2020 Author(s).Evolution of strong second-order magnetic anisotropy in Pt/Co/MgO trilayers by post- annealing Cite as: Appl. Phys. Lett. 117, 082403 (2020); https://doi.org/10.1063/5.0018924 Submitted: 19 June 2020 . Accepted: 08 August 2020 . Published Online: 25 August 2020 Hyung Keun Gweon , and Sang Ho Lim ARTICLES YOU MAY BE INTERESTED IN Low Gilbert damping and high thermal stability of Ru-seeded L1 0-phase FePd perpendicular magnetic thin films at elevated temperatures Applied Physics Letters 117, 082405 (2020); https://doi.org/10.1063/5.0016100 Skyrmion Brownian circuit implemented in continuous ferromagnetic thin film Applied Physics Letters 117, 082402 (2020); https://doi.org/10.1063/5.0011105 Realization of mutual synchronization of spin torque nano-oscillators under room temperature by noise reduction technique Applied Physics Letters 117, 082404 (2020); https://doi.org/10.1063/5.0016400Evolution of strong second-order magnetic anisotropy in Pt/Co/MgO trilayers by post-annealing Cite as: Appl. Phys. Lett. 117, 082403 (2020); doi: 10.1063/5.0018924 Submitted: 19 June 2020 .Accepted: 8 August 2020 . Published Online: 25 August 2020 Hyung Keun Gweon1and Sang Ho Lim2,a) AFFILIATIONS 1Center for Spintronics, Korea Institute of Science and Technology, Seoul 02792, South Korea 2Department of Materials Science and Engineering, Korea University, Seoul 02841, South Korea a)Author to whom correspondence should be addressed: sangholim@korea.ac.kr .Tel.:þ82-2-3290-3285; Fax:þ82-2-928-3584 ABSTRACT In this study, the first- ( K1) and second-order ( K2) magnetic anisotropies are investigated as a function of post-annealing temperature ( Ta)i n Pt/Co/MgO heterostructures. We find that both extrinsic and intrinsic mechanisms contribute to K2with their relative contributions signifi- cantly depending on the quality of the Co/MgO interface, which is sensitively affected by the Tavalue. In contrast with previous studies that mainly considered the extrinsic effects on K2, we obtain high K2values of up to 2.04 /C2106erg/cm3for the stack annealed at Ta¼400/C14Cb y maximally utilizing the intrinsic effect at the Co/MgO interface, thus facilitating robust easy-cone anisotropy. We also demonstrate that thecanted magnetization can be efficiently manipulated by the spin–orbit torques generated from the Pt layer, which is an important step toward the application of easy-cone states in various spintronic devices. Published under license by AIP Publishing. https://doi.org/10.1063/5.0018924 Magnetic anisotropy, which governs the orientational stability of a magnetic system, has been extensively utilized in the design of mag-netic devices. For example, owing to their good scalability and improved functionality over their in-plane counterparts, 1magnetic structures with perpendicular magnetic anisotropy (PMA) have beenof particular interest in recent spintronic applications, such as mag-n e t i cr a n d o ma c c e s sm e m o r i e s( M R A M s ) , 2,3magnetic sensors,4,5and spin-torque oscillators.6It was recently demonstrated that the intro- duction of an easy-cone (EC) state, in which the equilibrium magneti- zation direction is canted from the film normal and forms a cone, can improve the switching characteristics of a spin-transfer-torque (STT)-MRAM 7–12and enable field-free operation of STT oscillators.13 Furthermore, the EC state is able to host spin superfluids associatedwith the spontaneous breaking of the U(1) spin-rotational symme- try 14,15and induce unique EC domain-wall dynamics.16 The EC state can be described by considering magnetic anisotro- pies up to the second order, EðhÞ¼Keff 1sin2hþK2sin4h: (1) Here, Keff 1¼K1/C02pM2 sis the effective first-order anisotropy energy density, which includes the demagnetization energy (with K1 and Msrepresenting the first-order anisotropy energy density andsaturation magnetization, respectively); K2is the second-order anisot- ropy energy density; and his the polar angle of the magnetization. Using Eq. (1), it is straightforward to construct a magnetic phase diagram as functions of Keff 1andK2,a ss h o w ni n Fig. 1(a) .17,18For the sine series expansion as in Eq. (1), the EC state arises when Keff 1<0 and K2>/C01=2Keff 1. Observation of EC anisotropy was recently reported in several systems that include Co/Pt multilayers,19Ta/ CoFeB/MgO,20–24TaN/CoFeB/MgO,12Pt/CoFeB/MgO,25and dual CoFeB/MgO.26,27However, the window of the ferromagnetic metal (FM) layer thickness displaying the EC anisotropy is extremely narrow.19–21,23Furthermore, the physical origin of K2remains elusive. Recently, we reported large values of K2being sufficient to achieve the EC state, using the interfacial anisotropy arising from the strong inver-sion asymmetry at the FM/oxide interface. 28Herein, we report a study on a stack of Pt/Co/MgO to demonstrate that K2is closely related to the quality of the Co/MgO interface, such as the degree of oxidation and formation of Co–O–Mg bonds. To effectively control the quality of the interface, the Pt/Co/MgO stacks were annealed at variouspost-annealing temperatures ( T a).29,30In contrast to previous reports,21,23,26,27which mainly considered the extrinsic effects on K2,31,32we focus on the intrinsic effect of the symmetry breaking at the Co/MgO interface28to promote this anisotropy and achieve an even Appl. Phys. Lett. 117, 082403 (2020); doi: 10.1063/5.0018924 117, 082403-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplhigher K2value. We then experimentally realize a robust EC state in the Pt/Co/MgO structure at an optimized Taand Co layer thickness (tCo). We further demonstrate that the canted magnetization can be manipulated in a controllable way by the spin–orbit torques (SOTs)generated from the Pt layer, 33thereby demonstrating the applicability of an EC state in various spintronic devices. The stack structure examined in this study consisted of a Si sub- strate (wet-oxidized)/Ta (5 nm)/Pt (5 nm)/Co (1.2 nm)/MgO (2 nm)/ Ta (3 nm). For a comparative study, an MgO-free model stack of Pt(5 nm)/Co (1.2 nm)/Cu (2 nm) with the same buffer and capping layerswas also prepared, with Cu being chosen to replace MgO owing to a weak PMA at the Co/Cu interface 34and its immiscibility with Co.35 The stacks were fabricated using an ultrahigh-vacuum DC/RF magne- tron sputtering system consisting of two chambers with base pressuresof 5/C210 /C08Torr and 1 /C210/C08Torr. The metallic layers were deposited by DC sputtering under an Ar pressure of 2 /C210/C03Torr, whereas the MgO layer was grown by RF sputtering under an Ar pressure of 1/C210/C03Torr (and at a fixed sputtering power of 4.39 W/cm2). The as-deposited samples were annealed in a vacuum of 9 /C210/C06Torr for 30 min at four different Taof 250, 300, 350, and 400/C14C. The Msof the continuous films was measured using a vibrating sample magnetome- ter. The continuous films were patterned into a Hall bar structure using photolithography and inductively coupled plasma etching. A 50-nmthick Pt electrode was deposited on top of the patterned structures. TheHall measurements involved injecting an in-plane current ( I x¼5m A ) along the xdirection and sensing the Hall voltage induced along the y direction [ Fig. 1(b) ]. The generalized Sucksmith–Thompson (GST)method28,36was used to accurately determine the magnetic anisotropy constants ( K1and K2), and in-plane harmonic response measure- ments37were carried out to characterize the SOT properties. Current- induced switching experiments were conducted by applying a pulse current ( Ipulse)w i t haw i d t ho f1 0 ls and a sensing current of 100 lA. Figure 1(c) shows the mz–Hextplots for the Pt/Co/MgO stack in the as-deposited state and also after annealing at Ta¼250–400/C14C (where mzandHextdenote the zcomponent of normalized magnetiza- tion and the external magnetic field, respectively). These curves wereobtained by normalizing the anomalous Hall effect (AHE) results with respect to the anomalous Hall voltages. H extwas applied at a polar angle ( hH)o f8 0/C14to rotate the magnetization coherently. These plots were then reconstructed into aHextvs 1/C0m2 z[Fig. 1(d) ][ w i t h adefined as mzsinhH/C0ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C0m2 zp coshH/C16/C17 . mzffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1/C0m2 zp ]28to extract the effective first-order ( Heff K1) and second-order ( HK2) anisotropy fields from their intercept and slope, respectively. The accuracy of the anisotropy fields was confirmed by comparing the measured mz–Hextresults (symbols) with those from macrospin simulations (dashed lines) using the extracted Heff K1andHK2values as inputs. As shown in Fig. 1(c) , the agreement between the two sets of results is excellent over the entire Hext, except for the as-deposited sample for which a large difference is observed at Hext<1.5 kOe. According to our previous studies, this deviation is associated with the formation of magnetic multidomains due to strong interpenetration of the oxy- gen atoms into the underlying Co layer during the deposition of theMgO layer. 38,39For the as-deposited Pt/Co/MgO sample, therefore, the anisotropy fields were obtained using the results at Hext >1.5 kOe. The anisotropy constants were obtained from the rela- tionships K1¼Ms;effHeff K1=2þ2pM2 s;effandK2¼Ms,effHK2/4. Here, Ms,effdenotes the effective Msvalue taking into account the mag- netic dead layer ( tdead) [see Fig. 2(d) and the supplementary material Sec. S1 for details], which was taken from Ref. 30. Figure 2 shows the K1(a),K2(b),K2/K1(c), and tdead(d) as a function of Tafor the Pt/Co/MgO (MgO sample) and Pt/Co/Cu (Cu sample) structures. In the as-deposited state, the MgO sample exhibits an even lower value of K1than the Cu sample [ Fig. 2(a) ], whereas the former has a much higher tdeadvalue than the latter [Fig. 2(d) ]. These K1andtdeadresults in the as-deposited MgO sample were found to be due to strong interpenetration of oxygen atoms intothe Co layer. 30,34The K1of the MgO sample, however, improves greatly by post-annealing up to the highest Taof 400/C14C. In contrast, the annealing effects are small for the Cu sample. Considering the pri-mary role of the Pt/Co interface in K 1for the Cu sample,34,39,40the observed results strongly suggest that the effect of intermixing between Pt and Co (promoted by annealing) on K1is small for both the Cu and MgO samples. The large variation of K1observed for the MgO sample, therefore, can be explained by the change in the Co/MgO interface caused by the diffusion of the interpenetrated oxygen atoms toward the interface upon annealing.29,30,34,39,41 For the MgO sample, the Tadependence is even stronger for K2, indicating an important role played by the oxygen in forming K2. Furthermore, unlike for K1, the variation is non-monotonic with respect to Ta; it initially decreases as the sample is annealed at Ta ¼250/C14C and then increases sharply with increasing Tato form a para- bolic curve. This parabolic shape of K2implies the presence of more than two underlying mechanisms contributing to this anisotropy. FIG. 1. (a) Magnetic phase diagram as functions of Keff 1and K2, showing four dis- tinct magnetic states28[reproduced with permission from Gweon et al. , NPG Asia Mater. 12, 23 (2020). Copyright 2020 Springer Nature]. (b) Schematic of the Hall bar device used for magnetotransport measurements, together with the definition ofvarious angles for mand H ext. (c) mz–Hextplots for the as-deposited and annealed (250–400/C14C) Pt/Co/MgO trilayers. The symbols and dashed lines represent the results of AHE measurements and macrospin simulations, respectively. (d) aHextvs 1/C0m2 zplots reconstructed from the results shown in (c). The solid lines are the lin- ear fittings to the data, which were used to extract Heff K1and HK2values.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 082403 (2020); doi: 10.1063/5.0018924 117, 082403-2 Published under license by AIP PublishingPrevious studies have shown that there are both extrinsic and intrinsic mechanisms of K2; the former is related to K2arising from either the spatial fluctuation of K131or interface-concentrated K1,32whereas the latter accounts for the K2formed from either the bulk magnetocrystal- line cubic anisotropy42or the inversion asymmetry at the interface.28 For the as-deposited MgO sample, the strong oxygen contamination at the Co/MgO interface30,34is likely to result in large local variations ofK1and/or exchange energy, hence producing a sizable value of K2 by the extrinsic effects.31,32Note that the strong oxygen contamination of the as-deposited MgO sample is further evidenced by its incoherentmagnetization behavior [ Fig. 1(c) ]a n dah i g h t deadvalue [ Fig. 2(d) ]. Upon annealing, the as-deposited MgO sample exhibits two major structural changes: (1) the intermixing of Pt and Co at the Pt/Co interface and (2) the de-mixing of interpenetrated oxygen atomsfrom the Co layer. 30,34Similarly to K1, the effect of intermixing between Pt and Co on K2can be estimated from the results of the Cu sample. For this sample, the sign of K2is negative in the as-deposited state,28but becomes positive at Ta¼300/C14Co rh i g h e r[ Fig. 2(b) ]. This implies that the sign of K2is related to the intermixing of Pt and Co, such that it becomes more positive with a greater extent of intermixing (via extrinsic mechanisms).21,26However, the effect of intermixing on K2is small because the variation of K2with Tais small for the Cu sam- ple. For the MgO sample, intermixing at the Pt/Co interface will alsooccur (thus causing a similar change in K 2), although it is clearly not sufficient to account for the strong variation of K2with Taseen for this sample [ Fig. 2(b) ]. Instead, the strong Tadependence can be explained by a structural change at the other interface of Co/MgO. In terms of the extrinsic origin of K2, the de-mixing of the inter- penetrated oxygen atoms upon annealing is likely to result in a reduc-tion in the spatial distribution of K 1and/or exchange energy, thus causing a decrease in K2. This expectation is met for the MgO sampleannealed at Ta¼250/C14C, where a slight dip occurs in the K2–Tacurve. At higher Ta,h o w e v e r ,t h e K2increases gradually up to Ta¼350/C14C and then shows a strong upturn with the increase in Tafrom 350 to 400/C14C. It should then be necessary to invoke intrinsic effects to explain such a strong increase in K2. Our recent study showed that a large positive K2can be obtained for the Pt/Co/MgO structure annealed at Ta ¼400/C14C, which originates from the interfacial effect associated with the strong symmetry breaking at the Co/MgO interface.28In this regard, one can possibly conclude that the K2of the MgO sample is directly related to the quality of the Co/MgO interface, such as its roughnessand the number of Co–Mg–O bonds formed at the interface. 29,41 As the amount of interpenetrated oxygen atoms in the Co layer is the major source of tdeadfor the MgO sample, this parameter can be a good indicator for the quality of the Co/MgO interface at various Ta. At the highest Taof 400/C14C, the tdeadvalues of the MgO and Cu sam- ples are of comparable magnitudes (within the error limit), implying that most of the interpenetrated oxygen atoms are diffused out from the Co layer at this temperature, thus forming a well-defined Co/MgOinterface. Accordingly, at this T a, the number of Co–Mg–O bonds at the Co/MgO interface is highest,29allowing for a significant increase inK2. The formation of a sharp Co/MgO interface, as verified by our previous transmission electron microscopy results,30may also contrib- ute to this increase. In this study, a K2value of 2.04 /C2106erg/cm3 is obtained for the Pt/Co (1.2 nm)/MgO structure annealed at Ta¼400/C14C, which is higher than those reported previously in heavy metal/FM/oxide systems. The results shown in Fig. 2(b) clearly demonstrate that K2can be effectively controlled by Ta. It is reminded that the conditions of Keff 1<0a n d K2>/C01=2Keff 1must be satisfied for the EC state to be achieved.18This means that K2/K1,r a t h e rt h a n K2, is a more relevant parameter in forming the EC state. As shown in Fig. 2(c) ,t h e Ta dependence of K2/K1is similar to that of K2, indicating that the increase in K2is greater than that in K1upon annealing. Owing to the sharp upturn, it is possible to achieve the EC state for the MgO sam- ples annealed at Ta¼400/C14C. For the experimental realization of a robust EC state, tCowas tuned from 1.75 to 1.77 nm (in nominal thick- ness), the range of which corresponds to the spin reorientation transition.34 InFig. 3(a) ,mz–Hextplots are shown for two Pt/Co ( tCo)/MgO stacks showing PMA ( tCo¼1.75 nm) and EC anisotropy (tCo¼1.77 nm), both of which were annealed at Ta¼400/C14C. At tCo ¼1.75 nm, the mzis 1 at Hext¼0, confirming the PMA. At tCo ¼1.77 nm, EC anisotropy is formed with mz¼0.76. This canted mag- netic state is expected to break the collinearity between the magnetiza- tions of the reference and storage layers in the STT-MRAM, enabling an instant-on spin torque that could further improve the reversalspeed, power consumption, and stochastic switching behavior. 8,11,12A previous study also indicated that the characteristic cone angle [hc¼sin/C01ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C0Keff 1=2K2p/C16/C17 ] is an important design parameter deter- mining the performance of the memory device employing the EC state.8In our Pt/Co/MgO trilayer system, hccan be modified by fine- tuning the tCo.I nFig. 3(b) ,t h e Keff 1andK2values for the Pt/Co ( tCo)/ MgO stacks with tCo¼1.2, 1.75, 1.76, and 1.77 nm are overlaid on the magnetic phase diagram. Figure 3(c) depicts the hcfor several tCoval- ues. These results clearly indicate that the TaandtCoare important parameters in controlling the hcin the Pt/Co/MgO structure. FIG. 2. (a)K1, (b) K2, (c) K2/K1, and (d) tdead as a function of Tafor samples of Pt/ Co/MgO (circles) and Pt/Co/Cu (squares). The tdead results shown in (d) are repro- duced with permission from Gweon et al. , Sci. Rep. 8, 1266 (2018). Copyright 2018 Springer Nature.30Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 082403 (2020); doi: 10.1063/5.0018924 117, 082403-3 Published under license by AIP PublishingNext, we demonstrate that the canted magnetization can be effi- ciently switched by the SOT arising from the Pt layer. Our carefulanalysis (see the supplementary material Sec. S2) indicates that the Pt/ Co (1.76 nm)/MgO stack annealed at T a¼400/C14C exhibits a damping- like torque efficiency ( nDL) of 0.17 60.005, which is sufficient to induce a magnetization reversal via SOT. Even for the EC state, withits equilibrium magnetization direction tilted away from the z-axis, an in-plane H extis required to ensure deterministic switching.33,43 Therefore, the switching experiments were performed under an in- plane Hextof/C0500,/C0100, 0, 100, and 500 Oe, and the results are shown in Fig. 3(d) . The switching polarity and the reversal of the switching direction upon changing the sign of Hextprovide evidence that the magnetization switching of the EC device is facilitated by theSOT effects of the Pt layer with a positive n DL.33,44From the switching experiment performed under an in-plane Hextof6500 Oe, a critical switching current (density) of 16.3 mA (3.56 /C2107A/cm2) is obtained. The observed bipolar switching in an EC magnet suggests that heavy metal/FM/oxide heterostructures with high positive K2values can be suitable choices for spin-based logic applications, in which fasterswitching speed and low-power operation are preferred over datastability. 12,45,46 See the supplementary material for the effective saturation mag- netization of the MgO and Cu samples and the evaluation ofdamping-like torque efficiency in the EC sample. This research was supported by the Creative Materials Discovery Program through the National Research Foundation ofKorea (No. 2015M3D1A1070465) and by the Samsung Electronics’ University R&D program. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1B. Dieny and M. Chshiev, Rev. Mod. Phys. 89, 025008 (2017). 2S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey, B. D. Terris, and E. E. Fullerton, Nat. Mater. 5, 210 (2006). 3S. Ikeda, K. 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5.0010981.pdf

Appl. Phys. Lett. 117, 094001 (2020); https://doi.org/10.1063/5.0010981 117, 094001 © 2020 Author(s).Temperature dependence of hole transport properties through physically defined silicon quantum dots Cite as: Appl. Phys. Lett. 117, 094001 (2020); https://doi.org/10.1063/5.0010981 Submitted: 27 April 2020 . Accepted: 19 August 2020 . Published Online: 31 August 2020 N. Shimatani , Y. Yamaoka , R. Ishihara , A. Andreev , D. A. Williams , S. Oda , and T. Kodera COLLECTIONS This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN Perspective on the pressure-driven evolution of the lattice and electronic structure in perovskite and double perovskite Applied Physics Letters 117, 080502 (2020); https://doi.org/10.1063/5.0014947 Impact of the resistive switching effects in ZnMgO electron transport layer on the aging characteristics of quantum dot light-emitting diodes Applied Physics Letters 117, 093501 (2020); https://doi.org/10.1063/5.0019140 Microwave engineering for semiconductor quantum dots in a cQED architecture Applied Physics Letters 117, 083502 (2020); https://doi.org/10.1063/5.0016248Temperature dependence of hole transport properties through physically defined silicon quantum dots Cite as: Appl. Phys. Lett. 117, 094001 (2020); doi: 10.1063/5.0010981 Submitted: 27 April 2020 .Accepted: 19 August 2020 . Published Online: 31 August 2020 N.Shimatani,1Y.Yamaoka,1R.Ishihara,2,3 A.Andreev,4,5D. A. Williams,6S.Oda,1 and T. Kodera1,3,a) AFFILIATIONS 1Department of Electrical and Electronic Engineering, Tokyo Institute of Technology, Meguro, Tokyo 152-8552, Japan 2QuTech, Kavli Institute of Nanoscience and Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, Delft 2628CD, The Netherlands 3Quantum Computing Unit and Tokyo Tech World Research Hub Initiative, Institute of Innovative Research,Tokyo Institute of Technology, Meguro, Tokyo 152-8552, Japan 4A-Modelling Solutions Ltd, 11 Forster Road, Guildford GU2 9AE, United Kingdom 5Hitachi Cambridge Laboratory, J.J.Thomson Ave., Cambridge CB3 0HE, United Kingdom 6Cavendish Laboratory, University of Cambridge, J.J.Thomson Ave., Cambridge CB3 0HE, United Kingdom a)Author to whom correspondence should be addressed: kodera.t.ac@m.titech.ac.jp .Tel.:þ81–3-5734-3421. Fax:þ81-3-5734-3421 ABSTRACT For future integration of a large number of qubits and complementary metal-oxide-semiconductor (CMOS) controllers, higher operation temperature of qubits is strongly desired. In this work, we fabricate p-channel silicon quantum dot (Si QD) devices on silicon-on-insulatorfor strong confinement of holes and investigate the temperature dependence of Coulomb oscillations and Coulomb diamonds. The physically defined Si QDs show clear Coulomb diamonds at temperatures up to 25 K, much higher than for gate defined QDs. To verify the temperature dependence of Coulomb diamonds, we carry out simulations and find good agreement with the experiment. The results suggest a possibilityfor realizing quantum computing chips with qubits integrated with CMOS electronics operating at higher temperature in the future. Published under license by AIP Publishing. https://doi.org/10.1063/5.0010981 Single electron transistors (SETs) 1–4are expected to have wide- spread application in information processing devices. Semiconductorquantum dots (QDs) have been studied because of their potential for quantum bit (qubit) applications, 5,6such as spin qubits and charge qubits. Specifically, silicon (Si) QDs have been studied extensively7–12 due to the absence of nuclear spins in28Si and possibility of the on- chip integration with complementary metal-oxide-semiconductor (CMOS) technology. P-channel Si QD devices have attracted signifi-cant attention recently 13–15because in such devices, the spin manipu- lation can be done just by applying oscillating electric fields using the fact that holes in Si QDs have moderate spin–orbit coupling. Therefore, with p-channel Si QDs, a simple structure without a stripline or a micromagnet used for spin resonance 16can be realized. A simple structure of the device will allow more device integration in the future. Future quantum computer chips may contain millions of QDs in 2D grids, which all need to be addressed.17The challenge here is thelimited cooling power of the refrigerator and heat flux from wires con- necting between QDs and controllers. Many authors report the opera-tion temperature of Si QDs at around 20 mK–1.2 K, 18–20where currently available cooling power is only on the order of tens of lWt o tens of mW. Compatibility of classical CMOS electronics and Si QDsallows their fabrication on the same device. The multiplexing strategy by the on-chip CMOS controllers will mitigate the heat flux; however, the limited cooling power will still restrict the power dissipation forCMOS electronics. Since cooling power significantly increases withtemperature, for future quantum computing systems, higher operation temperature of QD SETs is strongly desired, also allowing the reduc- t i o ni nt h ep h y s i c a ls i z ea n dc o s to ft h ee q u i p m e n t . In this work, we fabricated physically defined p-channel Si QDs on a silicon-on-insulator (SOI) substrate by electron beam lithography (EBL) and SF 6dry etching6,21–23and investigated the temperature dependence of Coulomb oscillations and Coulomb diamonds both in the many-hole regime where tens of holes are stored in the QD and Appl. Phys. Lett. 117, 094001 (2020); doi: 10.1063/5.0010981 117, 094001-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplthe few-hole regime. To verify the temperature dependence of Coulomb diamonds, we have carried out simulations and found goodagreement with the experiment. In comparison with the gate-defined QDs, the physically defined Si QDs are expected to have stronger con- finement. We demonstrate that the physically defined QDs can oper-ate at temperatures up to 25 K in the few-hole region, which isattractive for future on-chip CMOS electronics integration in quantum computing chips. Note that in this work, we have not demonstrated spin qubit operation but the large addition energy of single hole to theSi QD. The large addition energy is not the only limiting energy scalefor the whole process of spin quantum information processing in sili- con. For the energy selective readout of spins using Zeeman splitting, the Zeeman splitting energy needs to be large compared to the temper-ature in order to keep the fidelity high enough. On the other hand, forthe scheme of dispersive readout of spins using Pauli spin blockade, we think that the Zeeman splitting energy does not necessarily have to be greater than the energy of temperature. In this scheme, the additionenergy in one QD of double QDs must be greater than the energy oftemperature. In order to keep the fidelity high enough, the large addi- tion energy of single hole to the Si QD is desirable. Figures 1(a) and 1(b) show a schematic cross section of the p-channel Si QD device and a scanning electron microscope (SEM) image, respectively. The device consists of QDs and side gates, whichwere patterned by electron beam lithography and SF 6dry etching on a 40-nm-thick SOI layer. Here, we used lightly Boron-doped UNIBOND SOI formed by smart cut technology. The doping level isabout 10 15=cm3. We used natural Si without isotope purification. By applying voltage to the side gates, the electric potential inside QDs can be tuned. The structure is thermally oxidized at 1000/C14Cf o r5 m i nt o passivate the surface states of the SOI layer. The top gate, which indu-ces holes in the QDs, was formed by depositing gate oxide (65-nm-thick SiO 2) and polycrystalline silicon by low-pressure chemical vapor deposition. Then, Boron ions (Bþ) were implanted in the source and drain with the dose of 1 :0/C21020cm/C03. Activation annealing was per- formed at 1000/C14C for 30 min. Holes are induced in the QDs by apply- ing negative voltages to the top gate like in a metal-oxide- semiconductor field effect transistor (MOSFET). The device presented inFig. 1(b) consists of a single QD (lower part—marked as SQD) and a double QD (upper part—marked as DQD). In this paper, we onlypresent the results related to the single QD. To model the current through the device in the many-hole regime, we have used a detailed 3D modeling of transport through Si QDs [see Fig. 1(c) ]. The model 22includes several stages: 3Ddigitization of the real QD device; electron wavefunction calculations in the QDs through the solution of coupled Poisson’s andSchr€odinger-like equations (we used the kp model); determination of the effective capacitance matrix of the device; and transport calcula- tions in the Coulomb blockade regime using the solution of the masterequations. It is essential that the QDs are considered to be dielectricmaterials when solving the Poisson equation, which means that the electric field induced by the gates is varying across the volume of the QD, in contrast to the widely used metallic model when the QD isconsidered to be a metal and the boundary conditions in the form ofthe constant electric potential across the whole QD are set. This effec- tive capacitance matrix is then used as an input for master equation solution 24for the calculation of the current through the QD (the mas- ter equation solution is temperature dependent). It is important tonote that the model does not contain any adjustable or fitting parame- ters and by using only the device structure and geometry, we were able to predict qualitatively and quantitatively the relative variation of thecurrent through devices in the many-hole regime. The details of themodeling are available in the supplementary material . For the few-hole regime, the hole localization details may be very much affected by the potential from possible surface charges on the silicon and silicon oxide interfaces, possible strain near the source-QD and QD-drain junction [see Fig. 1(b) —it is the narrowing that forms the QD] and the presence of defects. All these effects lead to the reduc- tion in the effective QD size and the increase in the charging energy, thus allowing high temperature operation of the QD. Without know-ing the physical details of these important factors, it is not possible topredict the energy levels and extent of the wavefunctions of the local- ized holes and, hence, the effective capacitances and current stability diagrams without using fitting parameters. We, therefore, followed adifferent route compared to modeling the many-hole regime andemployed the effective circuit model for fitting the experiment. In this model, the capacitances and the effective energy levels (the addition energies) of the localized holes are used as fitting parameters whensolving the master equations for the calculation of the transport prop-erties of the device. The measurements of the SET current with single QDs were per- formed at various temperatures from 4.2 K to 60 K as shown in Fig. 2 . We observed Coulomb oscillations by sweeping side gate voltage ( V sg) with a fixed top gate voltage of VTG¼/C02.5 V and a source-drain voltage of Vds¼1.0 mV. According to a basic theory, to observe the Coulomb oscillations, the charging energy ECof the dot should be much larger than the thermal energy, EC¼e2=C/C29kBT.H e r e , Cis FIG. 1. (a) Schematic cross section of the p-channel silicon QD device. (b) SEM image of the p-channel silicon QD device. All the transport data in this paper ar e obtained using the SQD channel. (c) 3D model of the device used for capacitance calculations in the many-hole regime, showing the results of the 3D electrical po tential variation for the case of uniform charge distribution in the QD.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 094001 (2020); doi: 10.1063/5.0010981 117, 094001-2 Published under license by AIP Publishingthe total capacitance of the QD in the SET, kBis the Boltzmann con- stant, and Tis the temperature. Current peaks correspond to single hole tunneling, which is clearly observed at low temperature. There isa fall in the peak-valley ratio as the temperature increases due to ther-mally activated tunneling. Current peaks could not be observed in the many-hole region ( V sg¼0–4 V) at 30 K. On the other hand, a change in the derivative of the current at the Coulomb peak position(V sg¼4.8 V) was observed at up to 60 K in the few-hole region. The less the dot stores holes, the smaller the size of dot becomes, resultingin larger charging energy E C. Figures 3(a)–3(c) show measured QD current in the many-hole regime at various temperatures of 4.2 K, 10 K, and 15 K.Figures 3(d)–3(f) show the results of the simulation at the same tem- perature as Figures 3(a)–3(c) , respectively. To check that the measure- ment temperature of the device is controlled at these specified values, we have fitted the measured current peaks of the QD (current depen- dence on the side gate voltage, I dsðVsgÞ, at fixed Vds¼1m e V )w i t ht h e current calculated using master equation solution described above, butconsidering temperature as a varying parameter. This procedure con- firmed that the temperatures of the device are indeed 10 K and 15 K with the accuracy better than 0.5 K. Figure 3(a) shows that the Coulomb diamonds are clearly visible in the QD stability diagram, which is the plot of I dsas a function of Vsgand Vdsat 4.2 K. The current flow is suppressed inside the dia- m o n d sd u et oC o u l o m bb l o c k a d ea ss h o w ni n Fig. 3(a) . Using the modeling technique described above, we have calculated the effectivecapacitances to be as follows: 12.5 aF, 12.0 aF, and 4.0 aF for the QD-source, QD-drain, and QD-top gate, respectively; the total QD capaci-tance is calculated to be C¼31.3 aF, which corresponds to a charging energy of E C¼5:1 meV. This is the size of the Coulomb diamonds we obtain if we calculate the stability diagram at low temperatures(T¼0.1 K). However, at T¼4.2 K, the calculated Coulomb diamond size is about 3.5 meV only [black area in Fig. 3(d) ]a n dt h es a m es i z e of the Coulomb diamond is observed in experiment, Fig. 3(a) .T h i sclearly shows that the area of the Coulomb blockade region (the Coulomb diamond) shrinks with temperature and this effect is already visible even at T¼4.2 K. When the temperature increases to 10 K, the Coulomb diamonds are still visible in the stability diagrams and they almost disappear at T¼15 K. Single hole transport through the QDs can be observed when the charging energy is larger than thermal energy. To achieve the Coulomb blockade region, we put the quantitative condition that min- ima of the charging energy are 4 times larger than thermal energy (E C>4kBT).25,26Figure 3(b) shows that the Coulomb blockade occurs even at the temperature of 10 K because the charging energy EC (/C255 . 1 m e V )i sl a r g e rt h a n4 kBTat 10 K. However, the shape of dia- monds becomes blurred and the size of the Coulomb diamond seems to be smaller ( /C252 meV). The smaller peak-valley ratio at 10 K [see Figs. 3(b) and3(e)] leads to such a blurred Coulomb diamond shape. At 15 K, as shown in Fig. 3(c) , the Coulomb diamonds could not be observed since Coulomb blockade is suppressed due to thermally acti- vated tunneling at high temperature. This is because the charging energy ( /C255.1 meV) is about 4 kBTat 15 K in the many-hole regime. The calculated Coulomb diamonds also could not be observed at 15 K. The calculated stability diagrams at various temperatures show good agreement with the experimental data. This means that the modeling confirms that in the many-hole regime, the effective capacitance of the QD (which determines the Coulomb diamond size) is determined only by the geometry and size of the QD, and hence, the holes are localized in the whole QD. Figure 4(a) shows the Coulomb diamonds in the few-hole regime measured at 4.2 K. It should be noted that the current level of pA for t h ef e w - h o l er e g i m ei n Fig. 4 is three orders of magnitude smaller than the current level of nA in the many-hole regime. In the few-hole regime, the localization is stronger (only in some part of the QD), and hence, the effective energy barriers are wider in real space and higher in energy; the tunneling rates are, therefore, much lower than those in the many-hole regime when the holes are localized in the whole FIG. 2. Temperature dependence of Coulomb oscillations as functions of side gate voltage ( Vsg) with Vds¼1.0 mV and VTG¼/C0 2.5 V. FIG. 3. (a) Coulomb diamonds obtained by mapping Idsas a function of Vdsand Vsgwith VTG¼/C0 2.5 V at 4.2 K. The number of holes in the QD is estimated to be more than 10 in the many-hole regime. (b) Coulomb diamonds measured at 10 Kunder the same conditions as in (a). Coulomb blockade is still observable; (c) Coulomb diamonds measured at 15 K. (d)–(f) Calculated stability diagrams at the same temperatures as in (a) /C0(c).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 094001 (2020); doi: 10.1063/5.0010981 117, 094001-3 Published under license by AIP Publishingv o l u m eo ft h eQ D .T h ea r e ao fz e r oc u r r e n ta tl a r g ev a l u e so fs i d eg a t e voltages Vsg>6.3 V corresponds to zero number of holes in the QD [seeFig. 4(a) ]. Various sizes of Coulomb diamonds are observed in the few-hole regime, which indicates that the effect of quantisation of holes is essential; this correlates with strong localization of holes and large Coulomb diamond size, as shown in Fig. 4(a) . To calculate the current stability diagrams in the few-hole regime, we used the following parameters for the hole addition energy EN: E1¼33 meV, E2¼5:2m e V , E3¼12:1m e V ,a n d EN¼5:2m e V forN>3 and the capacitances of 1 aF, 28 aF, and 1.8 aF for the QD- source, QD-drain, and QD-gate, respectively. In this model, the size of the Coulomb diamond corresponding to the occupation of Nholes (N>0) in the QD is given by the addition energy. The largest Coulomb diamond corresponds to N¼1 [see Figs. 4(a) and 4(g)]. This Coulomb diamond is very asymmetric, which reflects the fact that the QD in the few-hole regime is formed closer to drain. The addition energy for N¼1i sm u c hl a r g e rt h a nc h a r g i n ge n e r g y ECin the many-hole regime. The Coulomb diamond corresponding to N¼2 is small, and the third Coulomb diamond for N¼3h a st h es i z e ofE3¼12:1 meV; all other Coulomb diamonds with N>3h a v et h e size of the charging energy ECin the many-hole regime, see Fig. 4(g) . This calculation describes well the experimentally observed features in the stability diagram as it is clear from comparison of Figs. 4(a) and 4(g). Note that the reason of the large difference in the size of the Coulomb diamonds in the few-hole regime is not clear. It may be explained by the filling in several disordered QDs or in dopants in the QDs or by shell filling in the QDs.27,28 The large addition energy satisfies the condition of Eadd/C29kBT sufficiently. Therefore, the few-hole regime is robust even at high temperatures. To prove this, we measured Coulomb diamonds at higher temperatures (10 K, 15 K, 20 K, 25 K, and 30 K) as shown in Figs. 4(b)–4(f) . The shape of Coulomb diamonds becomes blurred with the increasing temperature due to the reduction of the peak- valley ratio. However, we have observed Coulomb diamonds at 25 K as shown in Fig. 4(e) . It clearly shows that the few-hole region is robust against thermal energy because the addition energy Eaddislarge enough in the few-hole region. Single hole tunneling was observed above Vsg¼5.7 V. Current always flows and no Coulomb blockade regions were observed when Vsgis smaller than 5.7 V since the charging energy is small. Figure 4(f) shows the measurement of Coulomb diamonds at 30 K. The regular shape of Coulomb diamond could not be observed anymore even in the few-hole region. We demonstrated that our QD devices can work in the few-hole regionuntil 25 K. Using the described model, we then tried to reproduce the tem- perature dependence of the stability diagram. We found that in experi- ment, the Coulomb diamonds shrink about two times faster withtemperature compared to the modeled temperature dependence of the stability diagrams. The largest Coulomb diamond almost disappears at T/C2460 K, while in experiment, it occurs at T/C2430 K. This indicates that the simple model we used for the few-hole regime does not take into account the effect of variation of the addition energy with temper-ature due to hole redistribution in the physically defined QD and partial screening of the potential when the holes are localized in the few-hole regime. In conclusion, we fabricated physically defined p-channel silicon QDs on the silicon-on-insulator substrate by electron beam lithogra- phy and SF 6dry etching and investigated the temperature dependence of Coulomb oscillations and Coulomb diamonds. We demonstratedhigher temperature operation of single hole transport in the few-hole regime. We conclude that the physically defined QDs can operate at 25 K, which is attractive for future on-chip CMOS electronics integra-tion for quantum computing chips. See the supplementary material for the details of the theoretical model and effective capacitance calculations. Part of this work was financially supported by KAKENHI Grants-in-Aid (Grant No. 20H00237), JST-CREST (No. JPMJCR1675), and Quantum Leap Flagship Program (Q-LEAPGrant No. JPMXS0118069228) of the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT). FIG. 4. Upper row—(a) Coulomb diamonds obtained by mapping Idsas a function of Vdsand Vsgwith VTG¼/C0 2.5 V at 4.2 K. (b)–(e) Coulomb diamonds measured under con- ditions as (a) but at temperatures of 10 K, 15 K, 20 K, and 25 K. (f) Stability diagram measured at 30 K; Coulomb diamonds are not visible. Lower row—(g)–(l ) Calculated tem- perature dependence of the stability diagrams in a few-hole regime at 4.2 K, 20 K, 30 K, 40 K, 50 K, and 60 K, respectively.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 094001 (2020); doi: 10.1063/5.0010981 117, 094001-4 Published under license by AIP PublishingDATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1M. A. Kastner, Rev. Mod. Phys. 64, 849 (1992). 2M. Manoharan, B. Pruvost, H. Mizuta, and S. Oda, IEEE Trans. Nanotechnol. 7, 266 (2008). 3Y. Ono, Y. Takahashi, K. Yamazaki, M. Nagase, H. Namatsu, K. Kurihara, and K. Murase, IEEE Trans. Electron Devices 47, 147 (2000). 4N. D. Akhavan, A. Afzalian, C. W. Lee, R. Yan, I. Ferain, P. Razavi, R. Yu, G. Fagas, and J. P. Colinge, IEEE Trans. Electron Devices 58, 26 (2011). 5H. W. Liu, T. Fujisawa, Y. Ono, H. Inokawa, A. Fujisawa, K. Takashina, and Y. Hirayama, Phys. Rev. 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Coppersmith, M. A. Eriksson, and L. M. K. Vandersypen, Nature 555, 633–637 (2018).12D. M. Zajac, A. J. Sigillito, M. Russ, F. Borjans, J. M. Taylor, G. Burkard, and J. R. Petta, Science 359(6374), 439–442 (2018). 13K. Yamada, T. Kodera, T. Kambara, and S. Oda, Appl. Phys. Lett. 105, 113110 (2014). 14R .L i ,F .E .H u d s o n ,A .S .D z u r a k ,a n dA .R .H a m i l t o n , Nano Lett. 15, 7314 (2015). 15R. Maurand, X. Jehl, D. Kotekar-Patil, A. Corna, H. Bohuslavskyi, R. Lavi /C19eville, L. Hutin, S. Barraud, M. Vinet, M. Sanquer, and S. De Franceschi, Nat. Commun. 7, 13575 (2016). 16M. Veldhorst, J. C. C. Hwang, C. H. Yang, A. W. Leenstra, B. de Ronde, J. P. Dehollain, J. T. Muhonen, F. E. Hudson, K. M. Itoh, A. Morello, and A. S. Dzurak, Nat. Nanotechnol. 9, 981 (2014). 17L. M. K. Vandersypen, H. Bluhm, J. S. Clarke, A. S. Dzurak, R. Ishihara, A. Morello, D. J. Reilly, L. R. Schreiber, and M. Veldhorst, npj Quantum Inf. 3,3 4 (2017). 18J. T. Muhonen, J. P. Dehollain, A. Laucht, F. E. Hudson, T. Sekiguchi, K. M. Itoh, D. N. Jamieson, J. C. McCallum, and A. S. Dzurak, Nat. Nanotechnol. 9, 986 (2014). 19R. E. George, W. Witzel, H. Riemann, N. V. Abrosimov, N. N €otzel, M. L. W. Thewalt, and J. J. L. Morton, Phys. Rev. Lett. 105, 067601 (2010). 20J. J. Pla, K. Y. Tan, J. P. Dehollain, W. H. Lim, J. J. L. Morton, D. N. Jamieson, A. S. Dzurak, and A. Morello, Nature 489, 541 (2012). 21T. Kambara, T. Kodera, Y. Arakawa, and S. Oda, Jpn. J. Appl. Phys., Part 1 52, 04CJ01 (2013). 22S. Ihara, A. Andreev, D. A. Williams, T. Kodera, and S. Oda, Appl. Phys. Lett. 107, 013102 (2015). 23K. Horibe, T. Kodera, and S. Oda, Appl. Phys. Lett. 106, 083111 (2015). 24M. Kirihara, K. Nakazato, and M. Wagner, Jpn. J. Appl. Phys., Part 1 38, 2028 (1999). 25Y. Takahashi, A. Fujiwara, M. Nagase, H. Namatsu, K. Kurihara, K. Iwadate, and K. Murase, Int. J. Electron. 86, 605 (1999). 26U. Meirav, M. A. Kastner, and S. J. Wind, Phys. Rev. Lett. 65, 771 (1990). 27L. P. Kouwenhoven, D. G. Austing, and S. Tarucha, Rep. Prog. Phys. 64, 701 (2001). 28D. H. Cobden and J. Nyga ˚rd,Phys. Rev. Lett. 89, 046803 (2002).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 094001 (2020); doi: 10.1063/5.0010981 117, 094001-5 Published under license by AIP Publishing

5.0013634.pdf

J. Chem. Phys. 153, 064108 (2020); https://doi.org/10.1063/5.0013634 153, 064108 © 2020 Author(s).A computational exploration of aggregation-induced excitonic quenching mechanisms for perylene diimide chromophores Cite as: J. Chem. Phys. 153, 064108 (2020); https://doi.org/10.1063/5.0013634 Submitted: 13 May 2020 . Accepted: 28 July 2020 . Published Online: 13 August 2020 Nastaran Meftahi , Anjay Manian , Andrew J. Christofferson , Igor Lyskov , and Salvy P. Russo COLLECTIONS Paper published as part of the special topic on 65 Years of Electron Transfer Note: This paper is part of the JCP Special Topic on 65 Years of Electron Transfer. ARTICLES YOU MAY BE INTERESTED IN Charge transfer via spin flip configuration interaction: Benchmarks and application to singlet fission The Journal of Chemical Physics 153, 064109 (2020); https://doi.org/10.1063/5.0018267 PSI4 1.4: Open-source software for high-throughput quantum chemistry The Journal of Chemical Physics 152, 184108 (2020); https://doi.org/10.1063/5.0006002 Hole–hole Tamm–Dancoff-approximated density functional theory: A highly efficient electronic structure method incorporating dynamic and static correlation The Journal of Chemical Physics 153, 024110 (2020); https://doi.org/10.1063/5.0003985The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp A computational exploration of aggregation-induced excitonic quenching mechanisms for perylene diimide chromophores Cite as: J. Chem. Phys. 153, 064108 (2020); doi: 10.1063/5.0013634 Submitted: 13 May 2020 •Accepted: 28 July 2020 • Published Online: 13 August 2020 Nastaran Meftahi,1,a) Anjay Manian,1 Andrew J. Christofferson,2 Igor Lyskov,1 and Salvy P. Russo1,b) AFFILIATIONS 1ARC Centre of Excellence in Exciton Science, School of Science, RMIT University, Victoria 3001, Australia 2School of Science, College of Science, Engineering and Health, RMIT University, Victoria 3001, Australia Note: This paper is part of the JCP Special Topic on 65 Years of Electron Transfer. a)Author to whom correspondence should be addressed: nastaran.meftahi@rmit.edu.au b)Electronic mail: salvy.russo@rmit.edu.au ABSTRACT Perylene diimide (PDI) derivatives are widely used materials for luminescent solar concentrator (LSC) applications due to their attractive opti- cal and electronic properties. In this work, we study aggregation-induced exciton quenching pathways in four PDI derivatives with increasing steric bulk, which were previously synthesized. We combine molecular dynamics and quantum chemical methods to simulate the aggregation behavior of chromophores at low concentration and compute their excited state properties. We found that PDIs with small steric bulk are prone to aggregate in a solid state matrix, while those with large steric volume displayed greater tendencies to isolate themselves. We find that for the aggregation class of PDI dimers, the optically accessible excitations are in close energetic proximity to triplet charge transfer (CT) states, thus facilitating inter-system crossing and reducing overall LSC performance. While direct singlet fission pathways appear endother- mic, evidence is found for the facilitation of a singlet fission pathway via intermediate CT states. Conversely, the insulation class of PDI does not suffer from aggregation-induced photoluminescence quenching at the concentrations studied here and therefore display high photon out- put. These findings should aid in the choice of PDI derivatives for various solar applications and suggest further avenues for functionalization and study. Published under license by AIP Publishing. https://doi.org/10.1063/5.0013634 .,s I. INTRODUCTION Organic chromophores are a class of materials with current and potential application in the dye industry, fluorescence labeling, semiconductors, and light harvesting.1Materials that can convert near-ultraviolet and ultraviolet light into white light are consid- ered a promising material in producing white light-emitting diodes (WLEDs).2–5Among organic dyes, polycyclic aromatic hydrocar- bons (PAHs) have been of interest for the past few decades because of their physiochemical applications in organic electronics and advanced materials.6–15Perylene diimides (PDIs) and their deriva- tives are polycyclic aromatic chromophores and are among the most important members of the PAH family16due to their favor- able charge transfer (CT), luminescence, and thermal stabilitiesin addition to their specific self-assembly properties. The interest- ing behaviors of PDIs are a result of their near-planar backbone and extended conjugated perylene core with a strong propensity to exhibit intermolecular π–πinteractions that cause molecular aggre- gates.17–19These aggregates can change the photophysical proper- ties of PDI compounds drastically,20–24but the π–πstacking can also cause a dramatic decrease in photoluminescence quantum yield (PLQY).23 Preventing unfavorable π–πintermolecular interactions between PDIs in elevated concentrations by adding covalent substituents to the PDI core can potentially yield a higher PLQY.20There are three possible regions available on PDIs for substitution: bay (1, 6, 7, 12- positions), shoulder (2, 5, 8, 11-positions), and imide. Bay substi- tution25was found to change the near-planar conformation of the J. Chem. Phys. 153, 064108 (2020); doi: 10.1063/5.0013634 153, 064108-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp perylene core in order to prevent the π–πinteractions, which also altered the photophysical characteristics of the PDI core. While the π–πstacking was found to be reduced but not fully removed, the effects could be positive or negative depending on the desired appli- cation.26–28Substitution at the shoulder and imide positions should decrease the π–πinteractions between PDI molecules via molecular isolation.20In this manner, the π–πstacking can be eliminated with- out changing the near-planar conformation of the perylene core by positioning derivatives around the core to sterically stop π–πinter- actions.29,30By this method, the core will stay near-planar, and con- sequently, the original photophysical properties of PDI are expected to remain the same.31,32 In 2017, Zhang and co-workers investigated the PLQY and energy migration behavior, as mediated by Forster Resonance Energy Transfer (FRET)33,34[Fig. 1(a)], of PDI chromophores from dilute solutions to the solid state.35As substitution through the shoulder position of PDI was found to change the optical and elec- tronic properties of the PDI,32,36they focused on substitution at the amide position of the chromophore. Four derivatives of PDI, labeled bPDI-1, -2, -3, and -4 [Fig. 1(c)], were synthesized by substitution on the amide position in order to identify the correlation between the size of amide substituent and PLQY of PDI molecules whendispersed in a poly(methyl methacrylate) (PMMA) matrix. Zhang et al. found that bPDI-1 exhibited low (i.e., ∼20%) PLQY at 10 mM concentration in PMMA, while for bPDI-2, PLQY was found to be above 90% and close to 100% for bPDI-3 and -4 at the same concentration. While these results suggested the presence of both emissive excited state aggregates and aggregation-induced quench- ing, a molecular-level understanding of the effects of PDI orientation and proximity on emission and potential quenching mechanisms is highly desirable. In the quantum chemical (QC) picture, aggregate geometries and their classifications are derived from transition charge distri- butions and the angle between the inter-chromophore separation vector and transition dipole. Known as Kasha aggregates,37–39these aggregates can be broadly sorted into two groups: H-aggregates, which exhibit side-by-side configurations resulting in blue-shifted absorption energies and low PLQYs, and J-aggregates, which exhibit head-to-tail configurations yielding red-shifted energies and greater PLQYs. In Kasha’s model, classification of geometries is derived through Coulomb coupling terms only. However, recent stud- ies40,41illustrate how strong π-stacking systems such as perylene and 7,8,15,16-tetraazaterrylene (TAT) enable charge transfer (CT) states to compete and mix with Coulomb coupling, generating FIG. 1 . (a) Schematic pathway describing fluorescent quenching in the solid state. Energy is transferred between molecules until it becomes trapped within an aggregate cluster. (b) Possible de-excitation quenching pathways for a delocalized singlet excited state. (c) The structures of the four bPDI compounds synthesized by Zhang et al.35 J. Chem. Phys. 153, 064108 (2020); doi: 10.1063/5.0013634 153, 064108-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp non-standard behavior from Kasha-type coupled systems. This mix- ing can alter the band structure and therefore the photophysical properties of the system. To describe this, a new notation was devel- oped,42the double index (DI) notation, which builds on the descrip- tion of the Kasha model. In this DI notation framework, dimers are given a two-letter H- or J-label to describe both the nature of the Coulomb (first letter) and the CT (second letter) state, where the case of the letter describes the relative coupling strength. In such a paradigm, standard Kasha geometries can be labeled Jh-, Jj-, or Hh-, Hj-, and so on. The new DI CT type geometries were shown to dis- play interesting photophysical properties, dependent on the nature of each coupling term. This tunability of aggregate properties there- fore becomes a very useful tool in terms of physical application of the theory. It was hypothesized by Zhang et al.35that exciton quench- ing is caused by aggregation-induced pathways. Such pathways could include non-radiative relaxation (NR), inter-system crossing (ISC),43and singlet fission (SF)44–49[Fig. 1(b)]. These mechanisms propagate via the dissipation of energy via heat, a singlet-to-triplet spin conversion, and a process whereby a singlet exciton is irre- versibly split into a pair of uncorrelated triplets, respectively. Since NR has been shown to be mediated by electronic and nuclear vibra- tional (vibronic) mode interactions,43the number of vibronic con- figurations that need to be considered to accurately calculate the rate of internal conversion is ∼15N, where N is the number of vibra- tional modes. For the bPDI chromophores in this study, vibra- tional analysis indicated that the number of vibrational modes N was≥40. As such, the calculation was beyond our resource capac- ity to compute. Therefore, we concentrated here on ISC and SF as possible mechanisms for fluorescence quenching. We applied molecular dynamics (MD) simulations and quantum chemical (QC) calculations to investigate the aggregation behaviors and result- ing optical and electronic properties of the four bPDI compounds [Fig. 1(c)]. While we would expect all of the bPDI to aggregate in some way at higher concentrations, our goal here was to explore whether we could spontaneously form aggregates at the relatively lower concentrations where the greatest PLQY is found experimen- tally and whether those aggregates would be emissive or quench- ing. Since the optical properties of the chromophore are defined by the PDI core,35any and all quenching of fluorescence is facilitated by aggregation. We examined the energies and correlated quantum chemical properties of various low-lying electronic excited states of both singlet and triplet multiplicity to gain insight into terms known to affect ISC and SF. We discuss the feasibility of both of these photoluminescent quenching pathways in aggregated bPDI dimers. II. METHODS A. Structure preparation and MD simulation protocol Experimentally, the bPDI and PMMA are dissolved sepa- rately in chloroform, then mixed to form a casting solution that is deposited as a thin film via spin coating, followed by evapora- tion of the chloroform at 100○C.35Here, we employed an anal- ogous simulation protocol by first performing MD simulations of bPDI dimers and tetramers in chloroform, separately relaxing PMMA in chloroform, followed by simulations of bPDI tetramersin chloroform and PMMA, where chloroform was sequentially removed from the system to model the experimental evaporation (Fig. 2). To examine the behavior of bPDIs in chloroform, we started with three sets of MD simulations. First, dimer configurations of bPDIs from the x-ray crystal structures35were placed in 53.819 ×54.118 ×53.032 Å3pre-equilibrated boxes of chloroform, and any chloroform within 2 Å of bPDI was removed, resulting in 970, 936, 932, and 942 chloroform molecules for bPDI-1, -2, - 3, and -4, respectively. Second, the distances between the cen- ters of mass of x-ray crystal dimers were extended to 20 Å and when placed in the same initial chloroform box resulted in 967, 956, 934, and 914 chloroform molecules for bPDI-1, -2, - 3, and -4, respectively, after removal of chloroform within 2 Å of bPDI. Third, tetramers with 20 Å separation between centers of mass were placed in the same initial chloroform box, with 934, 921, 968, and 940 chloroform for bPDI-1, -2, -3, and -4, respectively, after removal of chloroform within 2 Å of bPDI. The resulting concen- tration of bPDI in the dimer and tetramer systems was ∼20 mM and∼40 mM, respectively, compared to an experimental concentra- tion of 5 mM.35All simulations were run for 100 ns at 298 K in the isothermal–isobaric (NPT) ensemble and repeated three times with different random initial atomic velocities. For the simulations of bPDI in the PMMA matrix, initial seg- ments of atactic PMMA with 25 monomer units were constructed, and 35 segments were randomly placed in a 50 ×50×50 Å3box using PACKMOL.50The PMMA was solvated with chloroform and relaxed via 100 ns of simulation in the canonical (NVT) ensem- ble, and then, the box dimensions were extended in all directions to 100 Å. The bPDI tetramers with 20 Å separation were added and fully solvated with chloroform to produce simulation boxes of 35 PMMA segments and 4535, 4460, 4405, and 4337 chloroform for bPDI-1, -2, -3, and -4, respectively. Each system was equilibrated first by linearly heating the chloroform from 10 K to 375 K over 1 ns with a 1 fs time step and then linearly heating chloroform, bPDIs, and PMMA from 10 K to 375 K over 1 ns to relax any close contacts between the solvent and PMMA and PDIs. Then, MD simulations in the NPT ensemble were run for 100 ns at 375 K. In order to sim- ulate the evaporation of chloroform, 10% of the initial amount of chloroform was deleted from the simulation box at 10 ns intervals, resulting in zero chloroform after 100 ns of simulation and a final bPDI concentration of ∼20 mM, which corresponded to the opti- mal experimental concentration of 10 mM–20 mM.35This proce- dure was repeated three times, with different random initial atomic velocities, for each bPDI. For all simulations, the CGenFF version 3.0.1 program51,52and CGenFF force field53version 1.0.0 were used to generate compatible parameters of bPDI-1, -2, -3, and -4 and PMMA for the CHARMM force field.54MD simulations were performed using the GROMACS 2018 MD code.55In each case, an energy minimization using the steepest descent algorithm with a tolerance of 500 kJ mol−1nm−1 was performed prior to MD simulation. Pairwise electrostatic inter- actions were cut off at 1.1 Å, with a particle mesh Ewald (PME) treat- ment of long–range interactions. A switching function was used to reduce the van der Waals interactions to zero between 9 Å and 10 Å. Covalent bonds involving hydrogen were constrained using the LINCS algorithm. The Nosé–Hoover thermostat was used to main- tain the temperature for all simulations. Simulations in the NPT J. Chem. Phys. 153, 064108 (2020); doi: 10.1063/5.0013634 153, 064108-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Schematic depiction of the computational protocol followed in this work. ensemble utilized the Parrinello–Rahman barostat to maintain pres- sure at 1 bar. All production simulations were run with a 2 fs time step and 10 ps output frequency, with analysis carried out using VMD 1.9.3.56 B. Quantum mechanics calculations Geometries obtained from the MD simulations of bPDI in the PMMA matrix were extracted and paired into dimer configura- tions. Using the VMD software package, the geometry-optimized electronic ground state of the PDI core was root-mean-square devi- ation (RMSD) fit to each dimer subunit. The electronic ground state geometry optimization was performed at the density functional the- ory (DFT) level, using the B3LYP exchange-correlation functional57 and the triple-zeta valence polarization (TZVP) basis set,58using the TURBOMOLE software package.59Normal coordinate analysis confirmed that the structure was in a local minimum. Analysis of the various PDI dimers was performed with the DFT-based multireference configuration interaction (MRCI), known as the DFT/MRCI method.60For each chromophore pair, the one-particle basis was computed using the Becke’s half-and-half (BHLYP) exchange-correlation functional61as implemented in the TUBROMOLE software package using the split valence Gaussian- type orbital basis set def-SV(P).62The resolution-of-the-identity approach was employed using a def-SV(P) auxiliary basis set63for calculating four-index two-electron integrals over molecular orbitals (MO). The reference space for DFT/MRCI was generated iteratively by including all electron configurations with expansion coefficients greater than 10−3in a probe DFT/MRCI run. Here, any configu- rations with energy greater than 0.8 E hcompared to the highest reference energy were discarded, yielding the truncated DFT/MRCI expansion.60 The resulting singlet DFT/MRCI wavefunctions of excited PDI dimers were then used to generate one-particle transition densitymatrices.64–66We used this to gain insight into the extent of charge delocalization upon excitation to a particular state by generating a CT matrix, Ωn=[ωn AAωn AB ωn BAωn BB]. (1) Here, ndenotes the electronically excited state. Components ωAA andωBBrepresent the probability of an exciton residing entirely on a single respective monomer (A or B), analogous to a Frenkel exci- ton. Conversely, ωABandωBArepresent the probability of an electron being donated from one monomer to the other or a CT exciton. Elements of a CT matrix shown in Eq. (1) are obtained from the one- particle transition density matrix in the atomic orbital (AO) basis ρ′[ΦΨn]given as64 ρ′[ΦΨn] μv=∑ pqcμp⟨Φ∣ˆEpq|Ψn⟩cvq, (2) where indices pqandμνdenote the MO and AO, respectively. Ele- ments of cμpare the MOLCAO amplitudes of AO μin the MO p.Φand Ψndenote the electronic ground and nth excited state respectively, while Êpqrepresents the one-particle excitation opera- tor. Application of the AO overlap matrix Sto the transition density matrix in the AO basis can then be used to compute elements of the CT matrix,65 ωn A→B=1 2∑ μ∈A∑ ν∈B(ρ′[ΦΨn]S) μv(Sρ′[ΦΨn]) μv. (3) Here, AOs μare localized on molecule A, while AOs νare localized on molecule B in a PDI dimer. Components of the CT matrixΩnelude to the characterization of a particular electronic state; either a localized excitation ( ωAA,ωBB) or a CT state ( ωAB, ωBA). In the case where the two-electron contribution is larger than the one-electron contribution, the state is classified as an excimeric J. Chem. Phys. 153, 064108 (2020); doi: 10.1063/5.0013634 153, 064108-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp excitation. Whether the state is classified as a SF state depends on the structure of the DFT/MRCI wavefunction. When the lat- ter shows a dominant contribution from a four open-shell con- figuration, it is considered to be a SF-mediated state. The same methodology for CT analysis was used recently for other PDI derivatives.67 III. RESULTS A. Simulations of bPDI in chloroform For MD simulations of bPDI dimers in chloroform in the x- ray crystal structure configuration, we found that for bPDI-1, the core π–πstacking of the crystal structure was maintained through- out the simulation. Similarly, for bPDI-2, the interlocking of the PDI cores was maintained throughout the simulation. For bPDI- 3 and bPDI-4, we did not observe any aggregation behavior after 100 ns (see the supplementary material). When the initial dimer sep- aration was extended to 20 Å between the bPDI centers of mass, during the 100 ns of MD simulation, we observed that for bPDI- 1, the two monomers spontaneously aggregated to form the crystal structure π–πstacking. However, for bPDI-2, bPDI-3, and bPDI- 4, the two monomers had only transient contacts, and no aggre- gation was observed within 100 ns (see the supplementary mate- rial). For bPDI tetramers with center of mass separation of 20 Å, we observed that with bPDI-1 the four monomers spontaneously aggregated in a manner very similar to the crystal structure within the 100 ns timeframe, as shown in Fig. 3(a). For bPDI-2, two of the monomers were observed to aggregate and form the inter- locked configuration of the x-ray crystal structure, while the other two monomers did not aggregate, shown in Fig. 3(b). This sug- gests a concentration-dependent interlocking behavior for bPDI-2. For bPDI-3 and bPDI-4, frequent transient contacts were observed, but no persistent aggregation was found within 100 ns, even at this relatively high concentration of bPDI (see the supplementary material). In general, we found in our simulations that for bPDI-1, dimer- ization via π–πstacking can occur in chloroform, even at mM con- centration, and higher-order π–πstacking aggregates can sponta- neously form in chloroform as well. For bPDI-2, dimerization via interlocking appeared qualitatively concentration dependent. For bPDI-3 and bPDI-4, no aggregation was observed in chloroform at this concentration due to the bulky substituents around the core of perylene.B. Simulations of bPDI in chloroform and PMMA Based on our simulations of bPDI in chloroform, we chose the initial tetramers with 20 Å center of mass separation as the starting configurations for simulations of bPDI with PMMA. We found that the bPDI could move freely through the polymer matrix until the chloroform reached 10% of its initial value, after which the bPDI became locked into place by the PMMA matrix, with no further migration. Inter-chromophore separation distributions and example snapshots are presented in Fig. 4. Following our chloro- form evaporation protocol, MD simulations revealed that in the case of bPDI-1, in all three simulations, the four monomers showed a definite aggregation via π–πstacking, with an average inter- chromophore separation of 10 ±7 Å. For bPDI-2, a broader range of inter-chromophore separations was observed, with an average of 24 ±10 Å. Similar to the simulations of tetramers in chloro- form, in one case, the spontaneous interlocking to form the x- ray crystal dimer configuration was observed, while in other cases, aggregation occurred via interactions between the side groups of the monomers or between the side group of one monomer with the PDI core of another monomer. For bPDI-3 and -4, no inter- chromophore separations below 12 Å were found, with average values of 25 ±6 Å and 27 ±7 Å, respectively, and the tight pack- ing of the x-ray crystal structures was not observed. In both cases, aggregates interacted via the side groups, as the PDI cores were sufficiently insulated. In general, there was a correlation between the inter-chromophore separation and the size of the amide sub- stituent at this concentration for bPDI aggregates in a PMMA matrix. While MD simulations on the ns timescale are unlikely to gen- erate all possible PDI configurations that occur experimentally dur- ing the evaporation of chloroform over a timescale of 10 min, these results are in qualitative agreement with experiment, where small inter-chromophore separations, such as those found in MD simula- tions of PDI-1, are correlated with low PLQY.35Importantly, these simulations have generated an ensemble of viable configurations for further study in QC calculations. C. QC calculations Prior to analysis of dimer configurations, the PDI core was opti- mized at the B3LYP/TZVP level of theory. Hydrogen in the bay positions of the PDI compound generates a natural twist through the near-planar chromophore, stretching the aromatic bond lengths of the inner-most long-axis C–C bonds from 1.42 Å to 1.47 Å. FIG. 3 . Final configuration of (a) bPDI-1 and (b) bPDI-2 tetramers in chloroform after 100 ns of MD simulation. J. Chem. Phys. 153, 064108 (2020); doi: 10.1063/5.0013634 153, 064108-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . Histogram distributions of inter-chromophore separations across all bPDI chromophores and representative configurations of (a) bPDI-1, (b) bPDI-2, (c) bPDI-3, and (d) bPDI-4 tetramers in PMMA after simulated evaporation of chloroform. The nitrogen atoms at either end of the compound cause simi- lar stretching along the C–C bonds while also generating C–N– C bond angles of 128○, in addition to carbonyl bond lengths of 1.22 Å. Hydrogen bond lengths along the bay and shoulder positions were measured at 1.08 Å, while imide positioned hydrogens were observed to be slightly shorter at 1.01 Å. The rest of the molecule displays typical atomic structural properties, with near-planar aro- matic bond angles of 119○–121○. The computed geometry compares well with results obtained by Mete et al. ,68Zhang et al. ,69and Li and Wang,70all using time-dependent (TD) DFT methods. Coordi- nates of the optimized structure can be found in the supplementary material. The vertical excitation energies for the PDI monomer at the S0 geometry are shown in the supplementary material. Here, we see that theS1state possesses a large oscillator strength of f= 0.861. The underlying wavefunction is due to the one-electron excitation from the highest occupied MO to the lowest unoccupied MO, i.e., HOMO →LUMO. Frontier monomer orbitals are shown in Fig. S7 of the supplementary material. The vertical DFT/MRCI energy of 2.430 eV compares well with experimental absorption spectra showing max- imum intensity at 2.353 eV.35Similar studies on PDI monomers conducted by Mete et al.68found an excitation energy of 2.45 eV using TDDFT methods. Further studies by Zhang et al.69and Li andWang70reported energies of 2.47 eV and 2.50 eV, respectively, with both using TDDFT methods. In contrast to the S1, the second sin- glet state S2with an energy of 3.172 eV displays multiconfigurational character with a dominant contribution due to two-electron excita- tion having HOMO2→LUMO2type. Due to the rules of symmetry, this state has an oscillator strength of f= 0.0 and is therefore optically dark. A strong exchange interaction between HOMO and LUMO orbitals significantly stabilizes the T1state compared to its singlet S1counterpart. The DFT/MRCI computation places the T1state at 1.401 eV, which agrees very well with the TDDFT results pub- lished by Yu et al.16who reported an excitation energy of 1.382 eV. Other triplets are energetically more than 1.3 eV greater than theT1. Overall, the MD simulations of bPDI in PMMA yielded 72 relevant snapshots for possible PDI dimer configurations. Global trends showed that dimers with small intermolecular separations displayed standard H-type geometries. As inter-chromophore sep- arations increased to sizes larger than 15 Å, aggregation classifica- tions shift toward J-type aggregates. Dimers with separations less than 15 Å displayed pronounced energy splitting between the S1 andS2states resulting from excitonic coupling, shown in Fig. 5(a). Since the molecules forming the dimer are identical, each molecule should have the same excitation energies, observed typically as J. Chem. Phys. 153, 064108 (2020); doi: 10.1063/5.0013634 153, 064108-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . (a) Energies of the first two opti- cally active electronic singlet states in relation to each other. (b) Energies of the first two optically active electronic singlet states with respect to the first two singlet charge transfer levels for the eight dimers with the smallest inter- chromophore separation. completely degenerate Frenkel-like states. This is seen for dimers with separations larger than 15 Å; here, dimers are well insulated with respect to excitonic coupling between each subunit. A conver- gence in energies begins from around 5 Å and concludes at ∼15 Å. In this regime, excitonic coupling strengths weaken as separations increase, resulting in less mixing of wavefunction components. In the large inter-chromophore separation limit, wavefunctions are completely isolated upon either respective subunit. Clark et al.71noted similar behavior in their study. At inter-chromophore sepa- rations smaller than 5 Å, the S1–S2energy splitting becomes more pronounced and its energetic centroid is shifted to the red. This is commonly understood through the realization that low-energy CT states emerge in the excitation spectrum of a dimer and interact with theS1andS2states. As discussed, any dimer systems with inter-chromophore sepa- rations larger than around 5 Å inferred very strong CT coupling with J. Chem. Phys. 153, 064108 (2020); doi: 10.1063/5.0013634 153, 064108-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp respect to Coulomb coupling, resulting in deviations from the stan- dard behavior of bPDI dimers. As such, we now choose to narrow our scope of investigation to dimers with inter-chromophore sep- arations smaller than 6 Å. Screening of dimers is not limited to the species of bPDI, only to their relative center of mass separations. Fig- ures 5(b) and 6 show the energies and respective geometries of eight dimer configurations, which meet our selection criteria. We label them ascending numerically, i.e., dimers 1–8, based on their inter- chromophore separations. In this strongly coupled regime, energy levels are closely packed and interact with each other. However, once inter-chromophore separations increase to 5 Å and larger, exci- tonic components were found to be less mixed. Frenkel states for dimer 8, for example, displayed CT contributions of 3% and 9.5%for the S1andS2states, respectively. In fact, computation of prop- erties of dimers just outside of the selection criteria displayed little to no CT contributions in any low-lying states. Dimers with inter- chromophore separations larger than 5 Å show CT states well above the Frenkel states such that the energy separations at such distances are large enough to minimize mixing between excitonic components (see the supplementary material). Among these eight dimers, dimer 1 is composed of bPDI-2 molecules, while the other seven are bPDI-1 dimers. Small inter-chromophore separations less than 5 Å charac- terize a high degree of local MO overlap, resulting in the DFT/MRCI states exhibiting a large extent of mixing between Frenkel and CT components. Analysis of the CT matrix shows that the lowest S1 andS2states experience significant contributions from CT excitons. FIG. 6 . Geometries and inter-chromophore separations of each of the eight studied dimers. The center of mass of each chromophore is shown as a pink sphere, and the inter-chromophore separation in angstroms is given in parentheses below the dimer label. J. Chem. Phys. 153, 064108 (2020); doi: 10.1063/5.0013634 153, 064108-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp For example, examination of the excited state wavefunctions shows that while the monomer displays a clear localized HOMO →LUMO contribution to the S1state, dimer 1 exhibits clear contributions from CT excitons. As per Table S3 of the supplementary material, it displays CT components of 10.6% and 10.2% for the S1andS2 states, respectively. The largest CT contribution to a local state can be observed in dimer 5, where the S2state displays a CT contri- bution of 37.4%. Hence, the CT components energetically stabilize the lowest singlet states at small dimer separations, as discussed previously. We note that dimer 1 does not display similar photophysical properties with respect to the other seven dimers. Rather, energy splitting between the S 1and S 2states is minimized for the inter- locked twisted cofacial geometry. Furthermore, it is the only dimer of the eight to display a bright S 1state, with almost even distribution of oscillator strengths between the first two singlet states. As such, the emission properties of this dimer are superior to the others. This is likely caused by strong CT coupling destructively reinforcing the Frenkel state, lowering the energies of all low lying excited states. These results agree with a previous study67in which the geometric configuration and excited state properties were very similar. While the inter-chromophore separation in the previous study of 4.21 Å is greater than the 3.663 Å in the present study, both values fall within the range of inter-chromophore separations found in the MD sim- ulations of interlocked bPDI-2, which had an average value of 3.5 Å and a maximum of 4.8 Å. While the difference is relatively small, CT states are well known to be highly distance sensitive.41Consequently, even a sub-angstrom difference between interlocking geometries can alter the photophysics of the system. As per the stronger opti- cal properties, dimer 1 should not be classified as a hH-aggregate,rather as a hJ-aggregate. However, the DI geometry is difficult to determine with clarity without direct computation of transfer integrals.40 It is further expected that CT excitons can alter optical prop- erties of low-lying excited states. Partial distribution of oscilla- tor strengths between localized excitations and CT configurations occurs within all eight studied dimers due, in no small part, to strong exciton coupling. Notably, the S4state of Dimer 5 illus- trates a pronounced oscillator strength of f= 0.411 while clearly skewing more toward the CT state classification ( ωAB= 0.640 and ωBA= 0.025). Similar effects were observed in dimer 4, where the S2state has a large CT contributing term ( ωBA= 0.512) yet still dis- plays a large oscillator strength of f= 0.607. Oleson et al.40showed that H-aggregate perylene dimers separated by 3.5 Å experienced strong CT contributions to their wavefunctions, and the photophys- ical properties would change based on the nature of the CT coupling; either constructively (hH-) or destructively (hJ-). The geometries presented in this work therefore appear to be hH-type aggregates. The low-lying CT states clearly stabilize the lowest singlet state as per H-aggregate CT states, while the partial distribution of oscilla- tor strengths is typical of Kasha type H-aggregates. However, the DI geometry is difficult to determine with clarity without direct computation of transfer integrals.40 For all the dimers, DFT/MRCI calculations characterize the S5state by double excitations from the ground state. As an exam- ple, we consider the S5wavefunction in dimer 1. The DFT/MRCI expansion yields three leading electronic configurations, namely, HOMO −1, HOMO →LUMO, LUMO + 1 and HOMO −1, HOMO →LUMO2, and HOMO −1, HOMO →LUMO + 12(see Table I and Fig. 7). The former four-open-shell configuration is expected TABLE I . Vertical excitation energies of a dimer perylene diimide from dimer 1 geometry computed with DFT/MRCI. State Primary configurations Weight ∆E(eV) ∆E(nm) f(L) S0 GS 0.893 . . . . . . . . . S1 HOMO →LUMO 0.558 2.264 548 0.475 HOMO →LUMO + 1 0.184 S2 HOMO −1→LUMO 0.618 2.312 536 0.582 HOMO −1→LUMO + 1 0.142 S3 HOMO →LUMO + 1 0.627 2.338 530 0.080 HOMO →LUMO 0.213 S4 HOMO −1→LUMO + 1 0.634 2.404 516 0.021 HOMO −1→LUMO 0.189 S5 HOMO −1, HOMO →LUMO, LUMO + 1 0.283 2.936 422 0.000 HOMO −1, HOMO →LUMO20.218 HOMO −1, HOMO →(LUMO + 1)20.208 T1 HOMO →LUMO + 1 0.424 1.347 920 . . . HOMO →LUMO 0.382 T2 HOMO −1→LUMO 0.491 1.374 902 . . . HOMO −1→LUMO + 1 0.314 T3 HUMO −1→LUMO + 1 0.425 2.269 546 . . . HOMO →LUMO + 1 0.319 T4 HUMO −1→LUMO + 1 0.418 2.347 528 . . . HOMO −1→LUMO 0.322 J. Chem. Phys. 153, 064108 (2020); doi: 10.1063/5.0013634 153, 064108-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 7 . Frontier molecular orbitals for a perylene diimide dimer. Orbitals gener- ated from dimer 1 geometries. as it is commonly viewed as an intermediate in SF theory.72The latter two describe two-open-shell configurations and emerge nat- urally to compensate for delocalization of virtual orbitals. Thus, in the basis of localized excitons, the S5state is seen as two triplets, each of which reside on an individual molecule, coupled overall to a sin- glet spin1TT. The1TTstate energy is expected to be nearly equal to a sum of two independent triplets, which is slightly off as com- puted by DFT/MRCI by ∼0.2 eV. An important criterion driving the SF process is such an energy condition: the1TTstate is lower than the brightest singlet state with the largest oscillator strength. Oth- erwise, fluorescence becomes a competing process. Our calculations predict that1TTstates appear on average 0.449 eV above the opti- cally accessible singlet for bPDI-1 dimers and 0.624 eV for bPDI-2 dimers. As these SF states are fundamentally dark and inaccessible by experiment methods, it is difficult to compare our energies with any solid reference. However, DFT/MRCI methods have been shown to statistically reproduce experimental energies with errors not exceed- ing 0.2 eV.60Therefore, the energy criterion for a direct SF pro- cess is endothermic and is unlikely to influence the PLQY of bPDI dimers. Similar studies concerning the SF pathway within PDI sys- tems can be argued to agree with the results presented thus far. Renaud et al. found that SF rates computed for dimer PDI sys- tems using Marcus theory48led to low SF quantum yields (SFQY). Indeed, it was shown that SF rate constants are hyper-sensitive to dimer geometries, increasing the SFQY for only a small series of sub- Angstrom slip displacements along a set of 2-dimensional coordi- nates. The authors went as far as to suggest an optimum H-aggregate geometry that could maximize the SFQY. While several dimers in this study fall within the suggested bounds of optimum SF capa- bilities, the energy condition required for a direct pathway is not met, and any SF occurring within these systems would require anintermediate state. When the quantum effects on SF were investi- gated on the SF rates in a follow-up study,49it was shown that a CT state within energetic proximity to an optically active state could act as the virtual intermediate required for fast SF. Direct compar- isons of semiclassical and quantum mechanical methods highlighted the importance of excitonic coupling to the SFQY, with further suggestions of constructive and destructive reinforcement of rate constants brought on by dimer geometries. Quantum interference effects can clearly be observed within the dimers of the present study, with excitonic coupling stabilizing energies of low lying electronic excited states. While this in turn rendered the possibility of direct SF highly improbable, it also spawned low-lying CT states in energetic proximity of both the optically active and1TTstates within both spin multiplicities, allowing for the strong possibility of a CT medi- ated SF (CT-SF) pathway. It should be noted that in these studies, dimer energies were approximated from the monomer properties. While this may be a widely used approximation, when molecules aggregate, energy levels delocalize and shift from their monomer positions, as noted in other studies exploring aggregate PDI sys- tems40and as shown in Fig. 5(a). Additionally, TDDFT methods are well known for being unable to compute two-electron inter- molecular excitations accurately. Conversely, DFT/MRCI methods have been shown to account for these terms quiet well. It there- fore becomes evident that a conclusion concerning a CT-SF pathway requires further quantum chemical analysis and computation of the various coupling terms47using a methodology that can accurately account for all important interference effects. This knowledge is vital to the understanding of the true effect of quantum interference on SF. Conversely, very small energy differences between the optically brightest localized excitation and triplet CT state (L-CT) infer the possibility of a singlet-to-triplet pathway via ISC. A similar pathway J. Chem. Phys. 153, 064108 (2020); doi: 10.1063/5.0013634 153, 064108-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 8 . (a) Energy of the brightest localized excitation compared to the energies of the first triplet CT exciton and correlated TT pair computed using DFT/MRCI. (b) Different energy differ- ences between the brightest localized excitation and both the correlated TT pair (1TTBarrier) and triplet CT exci- ton (L-CT Barrier). Sign of difference dic- tated whether the process is endother- mic (positive) or exothermic (negative). has been shown experimentally in anthracene complexes,73where donor–acceptor bridged chromophores exhibited large triplet yields caused by a low lying CT triplet energy level. As shown in Fig. 8(a), we see that at inter-chromophore separations up to 5 Å, energy dif- ferences are less than 0.07 eV at most. In fact, such small energy gaps are indicative of electronic states in near-resonance with one another. ISC is mediated primarily by two quantum chemical prop- erties: the spin–orbit coupling between the transitioning states and the Franck–Condon (FC) overlap between both states.43Highly res- onant excitations produce strong FC overlaps, facilitating fast decay mechanisms. As such, the larger the overlap between states, the larger the rate of transition. Some dimer configurations even suggest the opportunity for exothermic pathways, as per Fig. 8(b), further increasing the feasibility of ISC. These data are listed in detail in the supplementary material. As all QC calculations in this work were performed on equi- librium geometries obtained from MD snapshots, analysis does not account for intramolecular distortions resulting from electronic excitation. Noted here and by others,47,49frontier MOs are strongly localized on the core of the PDI molecule, resulting in minimal fluctuations of electronic excitation energies. However, it has been observed that intramolecular vibrational modes relax the excitonic states. As such, the approach here neglects the effects of the solvent and strong excitonic coupling on the proximity of the CT states to the optically accessible state and even how the bulky side groups may interplay with intermolecular coupling effects; assuming both to be negligible. While the latter presents a reasonable assumption, the former may likely present a significant effect. Therefore, it is crucial that further analysis on intramolecular perturbations to the overall wavefunction be addressed. IV. CONCLUSIONS In this work, we used MD simulations to investigate the aggre- gation behavior of the four bPDI chromophores by performing an in silico evaporation protocol similar to the experimental procedure for the preparation of films of bPDI in a polymer matrix. MD simula- tions showed that for bPDI-1, dimerization and higher-order aggre- gation can occur spontaneously in chloroform at relatively low con- centration, while for bPDI-2, the interlocking dimerization found in the x-ray crystal structure can occur spontaneously in chloroform but may depend on concentration. For bPDI-3 and bPDI-4, we didnot observe any consistent aggregation in chloroform at the concen- trations studied due to the bulkier substituents around the perylene core. Configurations of bPDI-3 and -4 in PMMA were varied, gener- ally J-type, and did not present inter-chromophore separations less than 12 Å. Broadly speaking, these results are in line with previ- ous work that classified bPDI-1 and -2 as aggregating and bPDI-3 and -4 as insulating,67where aggregation in this case refers to inter- chromophore separation of ∼5 Å or less, where quenching pathways can reduce PLQY. Geometries of bPDI aggregates with inter-chromophore sep- arations less than 5 Å were obtained from MD simulations and had an optimized gas-phase monomer RMSD fit to each subunit to use in subsequent QC calculations. The bPDI-1 molecule was clearly shown to form several dimers with inter-chromophore sep- arations much smaller than 5 Å, allowing localized excitations and CT states to mix. For bPDI-2, there was a broader distribution of inter-chromophore separations due to the potential for interlock- ing not found in bPDI-3 and -4. For bPDIs 3–4, coupling occurred at very similar distances, and while each molecule exhibited its own unique behavior and tendencies, the average distance remained similar. Of the available geometries, eight were found to display inter- chromophore separations that displayed heavy splitting of energies between the first two singlet states and partial distributions of oscil- lator strengths. The archetypal Kasha H-aggregates were all shown to display qualities ascribed to low-lying CT states. Based on the observed photophysical effects, we determined that the geometries could be reclassified as hH-aggregates under the DI paradigm, with the exception of dimer 1, which was classified as a hJ-aggregate. Upon measurement of the energies of the T1and1TTstates, an energy difference of ∼0.2 eV was noted. While this is within the tolerable uncertainty of the DFT/MRCI methodology, the corre- lated state was shown to be in an endothermic position with respect to the optically active state, appearing ∼0.5 eV higher in energy. This suggests that a direct SF transition is unlikely. However, were we to factor in the previously noted quantum interference effects, it is likely a CT-SF pathway would become available due to the low-lying CT states across both the singlet and triplet multiplic- ities. With the relevant coupling terms all being heavily tied to the geometry of each dimer, the parameters underpinning exci- tonic coupling here are of the utmost importance. Furthermore, it was found that across all strongly coupled dimers, a CT state of J. Chem. Phys. 153, 064108 (2020); doi: 10.1063/5.0013634 153, 064108-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp triplet multiplicity could be found in near-resonance to the opti- cally brightest state, suggesting the option for an additional fast ISC pathway. In conclusion, aggregation-class bPDI chromophores were shown to display large CT contributions to the low-lying excited states of dimers. Consequently, excitations are relaxed non- radiatively via a fast ISC mechanism due to low lying CT triplet states or a CT-SF pathway, quenching the fluorescence. However, the interlocking dimer geometry found in bPDI-2 was shown to mini- mize these CT terms, resulting in an optically bright configuration and an unquenched aggregating emitter. SUPPLEMENTARY MATERIAL See the supplementary material for plots of the inter- chromophore separation from MD simulations, frontier orbitals of the PDI core, a breakdown of electronic structure of the PDI monomer at the electronic ground state geometry in the gas phase, a summary of the vertical excitation energies for each closely stud- ied dimer of localized excitations, low-lying singlet and triplet CT states, and correlated triplet–triplet pairs, a review of localized and CT contributions to the first 5 singlet and first 4 triplet excited state wavefunctions for each dimer, the energies of each electronic state, and their oscillator strengths where appropriate, and the geometry- optimized coordinates of the PDI monomer in the gas phase at the electronic ground state. AUTHORS’ CONTRIBUTIONS N.M. and A.M. contributed equally to this work. ACKNOWLEDGMENTS This work was supported by the Australian Government through the Australian Research Council (ARC) under the Center of Excellence scheme (Project No. CD170100026). This work was also supported by computational resources provided by the Australian Government through the National Computational Infrastructure (NCI) National Facility and the Pawsey Supercomputing Center. Professor Chris F. McConville is gratefully acknowledged for help- ful discussions. We thank the anonymous reviewers for constructive comments. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding authors upon reasonable request. REFERENCES 1M. K. Bera, P. Pal, and S. Malik, J. Mater. Chem. C 8, 788 (2020). 2T. Guner, E. Aksoy, M. M. Demir, and C. Varlikli, Dyes Pigm. 160, 501 (2019). 3Y. Kajiwara, A. Nagai, K. Tanaka, and Y. Chujo, J. Mater. Chem. C 1, 4437 (2013). 4G.-H. Pan, H. 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5.0007489.pdf

J. Appl. Phys. 128, 125706 (2020); https://doi.org/10.1063/5.0007489 128, 125706 © 2020 Author(s).Interference and electro-optical effects in cubic GaN/GaAs heterostructures prepared by molecular beam epitaxy Cite as: J. Appl. Phys. 128, 125706 (2020); https://doi.org/10.1063/5.0007489 Submitted: 17 March 2020 . Accepted: 06 September 2020 . Published Online: 23 September 2020 B. E. Zendejas-Leal , Y. L. Casallas-Moreno , C. M. Yee-Rendon , G. I. González-Pedreros , J. Santoyo-Salazar , J. R. Aguilar-Hernández , C. Vázquez-López , S. Gallardo-Hernández , J. Huerta-Ruelas , and M. López-López ARTICLES YOU MAY BE INTERESTED IN In-situ spectroscopic analysis of the recombination kinetics in UVB LEDs during their operation Applied Physics Letters 117, 121104 (2020); https://doi.org/10.1063/5.0018751 Point defects in two-dimensional hexagonal boron nitride: A perspective Journal of Applied Physics 128, 100902 (2020); https://doi.org/10.1063/5.0021093 Effects of deposition conditions on the ferroelectric properties of (Al 1−xScx)N thin films Journal of Applied Physics 128, 114103 (2020); https://doi.org/10.1063/5.0015281Interference and electro-optical effects in cubic GaN/GaAs heterostructures prepared by molecular beam epitaxy Cite as: J. Appl. Phys. 128, 125706 (2020); doi: 10.1063/5.0007489 View Online Export Citation CrossMar k Submitted: 17 March 2020 · Accepted: 6 September 2020 · Published Online: 23 September 2020 B. E. Zendejas-Leal,1 Y. L. Casallas-Moreno,2 C. M. Yee-Rendon,3 G. I. González-Pedreros,4 J. Santoyo-Salazar,1 J. R. Aguilar-Hernández,5 C. Vázquez-López,1,a) S. Gallardo-Hernández,1 J. Huerta-Ruelas,6 and M. López-López1 AFFILIATIONS 1Departamento de Física, Centro de Investigación y de Estudios Avanzados del IPN, Av. IPN 2508, Gustavo A. Madero, 07360 Ciudad de México, Mexico 2CONACYT-Unidad Profesional Interdisciplinaria en Ingeniería y Tecnologías Avanzadas, Instituto Politécnico Nacional, Av. IPN 2580, Gustavo A. Madero, 07340 Ciudad de México, Mexico 3Facultad de Ciencias Físico-Matemáticas, Universidad Autónoma de Sinaloa, Av. de las Américas y Blvd. Universitarios, Culiacán 80000, Sinaloa, Mexico 4Universidad Distrital Francisco José de Caldas, Facultad de Ciencias y Educación, Bogotá 11021-110231588, DC, Colombia 5Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, San Pedro Zacatenco, Gustavo A. Madero, 07360 Ciudad de México, Mexico 6Centro de Investigación en Ciencia Aplicada y Tecnología Avanzada, Instituto Politécnico Nacional, Cerro Blanco 141, Querétaro 76090, Mexico a)Author to whom correspondence should be addressed: cvlopez@fis.cinvestav.mx ABSTRACT Cubic GaN (c-GaN) samples on GaAs (0 0 1) substrates were grown by RF-plasma-assisted molecular beam epitaxy, in which an As 4 overpressure was employed for the nucleating layer. Photoreflectance spectra were obtained in the temperature range from 14 to 300 K. Two independent phenomena were noticed. The first one consisted in optical interference features below the c-GaN bandgap, whose origin is a thermo-optical effect: the ultraviolet perturbation beam changes the refractive index of the c-GaN. The second one represents electro-optical phenomena in which two classical band-to-band transitions occur: the first transition for c-GaN layer in which the temperature dependencereveals defects in the film attributed to a hexagonal fraction estimated previously between 3% and 10%, and a second transition for theGaAs substrate that shows Franz –Keldysh oscillations. Published under license by AIP Publishing. https://doi.org/10.1063/5.0007489 I. INTRODUCTION Metastable cubic GaN (c-GaN) presents attractive advantages in comparison to hexagonal GaN (h-GaN) because of its high degreeof crystallographic symmetry, with the absence of spontaneouspolarization and strong piezoelectric fields. 1–3Besides, the cubic phase may present a high mobility of carriers, the ease to get p-type doping, and more suitable cleaving planes.4One of the major difficulties for obtaining high quality c-GaN is the absence oflattice-matched substrates.2,5This problem results in the inclusion of a hexagonal GaN component. In this work, we studied samples with optimized exposure time to As 4overpressure during the nucleating layer process, influencing the relaxation process of c-GaN. These samples were characterized by photoreflectance (PR) spectroscopy,which is a standard method to determine the energy gap, the pres- ence of defects, and in some conditions the static electric field at the surface and at the interface between layers in heterostructures.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 125706 (2020); doi: 10.1063/5.0007489 128, 125706-1 Published under license by AIP Publishing.II. MATERIALS AND METHODS A. Materials The growth process of the materials was reported by Casallas-Moreno et al .6On the buffer layer, a c-GaN nucleation layer was grown at 700 °C during 18 s with a constant As 4pressure. The pressure of As 4and Ga were 8.3 × 10−6and 2.7 × 10−7Torr, respectively. The predominance of the cubic phase was demon- strated through x-ray diffraction reciprocal space mapping. Hall effect measurements at 300 K resulted in an n-type carrier concen-tration N = 10 18cm−3, originated from Ga vacancies, substitution of nitrogen by oxygen, and planar defects. B. Methods A conventional experimental arrangement was used to obtain the PR spectra. A dual phase SR530 lock-in amplifier was used. Forlow temperature measurements, the sample was cooled using aclosed cycle He refrigerator. Two experimental setups were used tomeasure different spectral ranges: in the range of 300 –700 nm, a 10 mW UV LED with λ= 285 nm was used as perturbation beam, while the probe beam was obtained by dispersing light from a1000 W, Xe lamp, using a 125 mm monochromator. An enhancedUV Si photodetector was used with a long pass filter at 325 nm. Inthe spectral range of 700 –1035 nm, a 2 mW AlGaAs LED with λ= 635 nm was used as the perturbation beam, while the probe beam was obtained by dispersing light from a 250 W halogen tung-sten lamp using a 500 mm monochromator. A long pass filter at700 nm was used at the photodetector entrance. III. EXPERIMENTAL RESULTS AND DISCUSSION A. The reflectance spectra The three media [ambient/film (GaN)/substrate (GaAs)] model is illustrated in Fig. 1 . The total reflected amplitude of anoptical plane wave is given by 7 R¼(RkþR?)/2, (1) where the contribution of the parallel and perpendicular polariza- tion are indicated. In the Appendix , the reflection amplitude is written in terms of the interface Fresnel reflection coefficients. In Fig. 2 , a typical experimental reflectance spectrum of the studied samples is shown (dashed curve). The corresponding simulationusing Eq. (1)in terms of the optical parameters of Ref. 8and the formalism written in the Appendix is shown (continuous curve). The sample thickness was used as a fitting parameter, resulting in d s= 324 nm, which agrees well with the thickness measured by scanning electron microscopy. B. The modulated reflectance spectra in the range between 1.8 and 4.0 eV Measured photoreflectance spectra at different temperatures are shown in Fig. 3 . 1. Band-to-band transition in c-GaN Above 3 eV, the Aspnes third-derivative line shape9was used for obtaining the energy gap and the broadening parameter. The structure corresponds only to a one band-to-band transition with abroadening parameter Γof 100 meV, approximately. The large line- width prevents us from resolving the main transition E 0from the spin–orbit split-off one (E 0+Δ0).2In the PR spectra reported by Ramírez-Flores et al.,10the values of Γvary from 10 meV at 5 K to 20 meV at 300 K. In our spectra, this parameter is roughly indepen-dent of temperature. This could be understood in terms of thedefects consisting in Ga vacancies, planar and stacking defects, and FIG. 1. Oblique reflection and transmission of a plane wave by an ambient(0) – film(1) –substrate(2) system with parallel-plane boundaries. d sis the film thick- ness, θ0= 45° is the angle of incidence in the ambient, and θ1andθ2are the angles of refraction in the film and the substrate, respectively.7 FIG. 2. Typical reflectance spectrum at 300 K (dashed curve) at photon energies smaller than the bandgap of c-GaN. The continuous curve is thecorresponding simulation using the interface optical constants.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 125706 (2020); doi: 10.1063/5.0007489 128, 125706-2 Published under license by AIP Publishing.also the presence of a hexagonal component between 3% and 10%, determined by mapping the reciprocal space.6Thus, the PR line- width does not depend on the lifetime broadening of the initial and final electronic states involved in the optical transition. Some oscil-latory features above the bandgap were observed in one of thesamples (not shown). But these oscillations cannot be consideredFranz –Keldysh oscillations (FKOs) due to (a) the above-mentioned large amount of defects, (b) the large broadening parameter(Γ/C2925 meV) as a consequence, and (c) the high doping level (N should be in the range of 10 15–1016cm−3).11 2. Modulated interference effect: Thermo-optic effect At energies below 3.2 eV, oscillations in the optical spectra are observed. These oscillations are present even at low temperatures, as shown in Fig. 3 . Considering the absorption coefficient of c-GaN,8αs= 2.55 × 107m−1,a tλ= 285 nm, the modulating beam has a penetration depth of 36 nm in the semiconductor. The spectra for probe energies below 3.2 eV must arise because of a change of the optical refractive index of the film, which occurs in phase with the periodic perturbation, the choppedperturbation beam in PR. Here, we propose a thermo-optical effectas the origin of these oscillations. The change in temperature mayinfluence the refractive index, as we will show below. In our experiment, the perturbation UV beam is chopped at f = 527 Hz, and the thermal diffusivity of c-GaN is α th= 0.43 cm2/s.12 Thus, the thermal diffusion length can be calculated as13 Λ¼ffiffiffiffiffiffiffiffiffiffiffiffi αth/πfp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:43/527 πp ¼161μm: (2) This length is much higher than the c-GaN film thickness. Therefore, the c-GaN layer can be considered a thermally thin film. Thus, the temperature change does not have any local depth depen-dence, and we expect a uniform temperature rise throughout thefilm. In order to estimate the temperature change, a classical model of the photothermal effect is considered. 13The model consists of a sample s of thickness d scorresponding to c-GaN, on a backing b, (GaAs substrate) placed at the ambient medium l, (air) on which the UV perturbation beam is incident. This model allows the calcu-lation of the complex change in temperature ΔT at the surface, 13,14 whose amplitude is jΔTj¼A α2 s/C0σ2 s(r/C01)(hþ1)eσsds/C0(rþ1)(h/C01)e/C0σsdsþ2(h/C0r)e/C0αsds (gþ1)(hþ1)eσsds/C0(g/C01)(h/C01)e/C0σsds/C20/C21/C26/C27 exp(/C0σlxþiωt)/C12/C12/C12/C12/C12/C12/C12/C12, (3) where A¼ αsI0 2κs,I0¼6:4/C2103W/m2, (4) κsis the thermal conductivity of c-GaN, Dj¼κj ρjcj,aj¼ffiffiffiffiffi πf Djs (5) [ j = s (sample: c-GaN), b (backing: GaAs substrate), l(fluid: air)]. For the j medium, D jis the thermal diffusivity, ρjis the density, c jis the specific heat, and a jis the reciprocal of the diffusion length.The other parameters are σj¼(1þi)aj,r¼(1/C0i)αs 2as,h¼abκb asκs, (6) g¼alκl asκs: (7) The material parameters employed are summarized in Table I . The result calculated using Eq. (3)isΔT = 0.003 K. With this change in the temperature amplitude, we can obtain thethermo-optic effect. The exciting UV beam modulates the sample temperature, which in turn modulates the GaN refractive index. The change in FIG. 3. Photoreflectance spectra at different temperatures in the spectral range from 1.75 to 4.0 eV .Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 125706 (2020); doi: 10.1063/5.0007489 128, 125706-3 Published under license by AIP Publishing.the real part of the refractive index due to ΔTi s Δn1¼@n1 @TΔT¼@n1 @T/C18/C19 (0:003), (8) where@n1 @Tis the thermo-optic coefficient of c-GaN, which is an energy-dependent function.10,12 InFig. 4 , an experimental modulated reflectance spectrum (noisy curve) is shown. The change in the reflectance induced by the thermo-optical effect is ΔR¼@R @n1@n1 @TΔT, (9) where R is obtained from Eq. (1)and the formalism of the Appendix . The adjusting parameter is the thermo-optic coefficient. We propose as the fitting function the thermo-optic coeffi- cient with a quadratic dependence in energy E, @n1 @T¼A0þA1EþA2E2, (10)where the calculated fitting parameters are A0¼/C01:35/C210/C03,A1¼1:02/C210/C03,A2¼/C01:039/C210/C04: (11) The resulted fitting function is shown in Fig. 5 . The experi- mental change of the reflectance is properly simulated, as shown bythe continuous curve in Fig. 4 . In a comparison of Fig. 5 with two reported cases of thermo-optical coefficients for hexagonal GaN prepared byMOCVD, 15,16we observed similar order of magnitude and increas- ing value behaviors of the parameter with energy. 3. Other possible mechanisms for the modulated interference Since GaN films were grown on (001)-oriented GaAs sub- strates, an alternative explanation for PR oscillations is based onthe modulation of the GaN refractive index by a linear electro-opticeffect. In this case, the dependence of the amplitude of the oscilla-tions on the laser intensity would be logarithmic instead of linear. 17In order to identify different features in the spectra, the phase settings of the lock-in amplifier were used. The phase wasselected in order to diminish the c-GaN band-to-band transition inone channel of the lock-in (output channel X). This channel willcorrespond mainly to the oscillatory low energy features, while the output channel Y will correspond to the band-to-band transition. 14 Different modulation powers of the UV perturbation beam were applied. In Fig. 6(a) , the amplitude of the band-to-band transition (channel Y) as a function of the logarithm of the normalized mod- ulating beam is shown. In Fig. 6(b) , the amplitude of the oscillatory PR signal (channel X of the lock-in amplifier) as a function of the FIG. 4. Experimental change of the reflectance (noisy curve) and calculated fitting (continuous curve).TABLE I. Material parameters employed in Eq. (3). Index j MaterialThermal conductivity (W m−1K−1)Heat capacity (J kg−1K−1)Density (kg m−3) S (sample) c-GaN 130 490 6150 b (backup) GaAs 55 330 5320 l(fluid) Air 0.026 1040 1.29 FIG. 5. Thermo-optic coefficient calculated to achieve the fitting to experimental ΔR.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 125706 (2020); doi: 10.1063/5.0007489 128, 125706-4 Published under license by AIP Publishing.modulation beam intensity is shown. In this way, we demonstrate that the electro-optical effect (above 3 eV) and the thermo-opticeffect (oscillations below 3 eV) give different behaviors in the PRspectrum. We also verified that the phase of the oscillations is not polarization-dependent of the probe light. Another possible mechanism of modulation of the refractive index could be electrons drifting from the GaN surface to theGaN/GaAs interface. 18However, this is not the case for our samples since the interference effect is present even when the tem- perature is reduced, as observed in Fig. 3 . In Ref. 18, the drifting mechanism disappears due to the photovoltage, which decreasesthe electric field as the temperature decreases. 19 For all of these reasons, we consider that the thermo-optical effect in the refractive index is the origin of the observed modulated reflectance at energies below the bandgap. 4. Electro-optical effect in the substrate In the spectral region around 1.5 eV, the light is not absorbed by the c-GaN film; hence, the probing beam reaching thec-GaN/GaAs interface is not attenuated. This is evident in the spec- trum shown in Fig. 7 , in which another type of oscillation shows up, FKOs, whose asymptotic behavior is described in Ref. 20, ΔR R/C25(/C22hω/C0Eg)/C01exp/C0Γ(/C22hω/C0Eg)1/2 (/C22hΩ)3/2 ! /C2cosθþ4 3/C22hω/C0Eg /C22hΩ/C18/C193/2"# , (12) FIG. 6. (a) Logarithmic dependence on the beam intensity of the maximum of the spectrum above 3 eV (band-to-band transition). (b) Linear dependence with the beam intensity of the amplitude oscillations below 3 eV . FIG. 7. PR spectrum of GaAs. FKO extrema are indicated by arrows. FIG. 8. FKO analysis of the GaAs in contact with c-GaN.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 125706 (2020); doi: 10.1063/5.0007489 128, 125706-5 Published under license by AIP Publishing.where E gis the bandgap, /C22hΩis the electro-optic energy, Γis the phenomenological broadening parameter, θis an arbitrary phase factor, and /C22hωis the probe beam energy. The extremum values of Eq. (12) are determined by the cosine argument, nπ¼θþ(4/3)[( E(n)/C0Eg)//C22hΩ]3/2, (13) where the energy E(n) corresponds to the extremum labeled by index n. This equation may be written in the following form: 4 3π[E(n)/C0Eg]3/2¼(/C22hΩ)3/2n/C01 π(/C22hΩ)3/2θ: (14) A typical analysis of the FKOs PR spectra is shown in Fig. 8 . The slope obtained by least squares fit to a linear function is identi-fied with Eq. (14) as Slope ¼(/C22hΩ) 3/2. Then, the electro-optic energy is given by /C22hΩ¼(Slope)2/3, (15) and the slope is related with the electric field (F) by20 /C22hΩ¼(e2F2/C22h2/2μk)1 3, (16) where μkis the reduced effective mass in the direction of the electric field [001]. From the analysis of the extrema shown in Fig. 8 , an electric field of 7.9 kV/cm and an electro-optic energy of 7.54 meV wereobtained. IV. CONCLUSIONS We have performed an analysis of the photoreflectance spectra of GaN/GaAs heterostructures. For probe energies between1.8 and 3 eV, the photoreflectance oscillations originate from athermo-optical effect. The perturbation UV beam induces a tem- perature modulation whose amplitude is 0.003 K, resulting in refractive index changes, whose influence is revealed as reflectancechanges. This is the most likely modulation mechanism giving riseto oscillatory features in PR. For probe energies higher than 3.1 eV, a feature corresponding to band-to-band optical transition appears, which is influenced by crystallographic defects. The c-GaN/GaAsinterface presents a PR signal with Franz –Keldysh oscillations. The electric field and the electro-optic energy at the surface of theGaAs substrate were calculated to be 7.9 kV/cm and 7.54 meV, respectively. ACKNOWLEDGMENTS This work was partially supported by Prodep Fortalecimiento de CA, Clave: CINVESTAV-CA-15. The authors would like to thank A. Guillén and M. Guerrero for their technical assistance. Fruitful discussions with A. Cruz-Orea are acknowledged.APPENDIX: OPTICAL REFLECTANCE AND PHASE CHANGE In terms of the interface Fresnel coefficients, the reflection coefficients are defined as 7 Rk¼r01kþr12ke/C0i2β 1þr01kþr12ke/C0i2βandR?¼r01?þr12?e/C0i2β 1þr01?þr12?e/C0i2β, (A1) where the subscripts identify the ambient –film (0 –1) and film –sub- strate (1 –2) interfaces. In terms of the refractive indices, the Fresnel coefficients are r01k¼n1cosθ0/C0n0cosθ1 n1cosθ0þn0cosθ1, (A2) r01?¼n0cosθ0/C0n1cosθ1 n0cosθ0þn1cosθ1, (A3) r12k¼n2cosθ1/C0n1cosθ2 n2cosθ1þn1cosθ2, (A4) r12?¼n1cosθ1/C0n2cosθ2 n1cosθ1þn2cosθ2: (A5) The phase change βin Eq. (A1) results from multiple reflec- tions of the wave inside the film as it travels through the film once from one boundary to the other. βis given by β¼2πds λ/C18/C19 (~n2 1/C0n2 0sin2θ0)1/2, (A6) where d sis the film thickness, λis the free-space wavelength, and θ0is the angle of incidence. The complex index of refraction of GaN film is given by ~n1¼n1þiκ1, (A7) where n 1is the refractive index and κ1the extinction coefficient of c-GaN, respectively. The values were obtained from Ref. 8. Typical reflectivity spectra show an oscillatory behavior below the bandgap of the thin film and a damped oscillatory behavior above and around the bandgap. The period of these oscillations depends on the refractive index and thickness of the films under consideration. DATA AVAILABILITY Data sharing is not applicable to this article as no new data were created or analyzed in this study. REFERENCES 1M. Feneberg, M. Röppischer, C. Cobet, N. Esser, J. Schörmann, T. Schupp, D. J. As, F. Hörich, J. Bläsing, A. Krost, and R. Goldhahn, Phys. Rev. B 85, 155207 (2012). 2O. C. Noriega, A. Tabata, J. A. N. T. Soares, S. C. P. Rodrigues, J. R. Leite, E. Ribeiro, J. R. L. Fernandez, E. A. Meneses, F. Cerdeira, D. J. As, D. Schikora, and K. Lischk, J. Cryst. Growth 252, 208 (2003).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 125706 (2020); doi: 10.1063/5.0007489 128, 125706-6 Published under license by AIP Publishing.3D. Moss, A. V. Akimov, S. V. Novikov, R. P. Campion, C. R. Staddon, N. Zainal, C. T. Foxon, and A. J. Ken, J. Phys. D Appl. Phys. 42, 115412 (2009). 4Y. L. Casallas-Moreno, D. Cardona, E. Ortega, C. A. Hernández-Gutiérrez, S. Gallardo Hernández, L. A. Hernández-Hernández, H. Gómez-Pozos, A. Ponce, G. Contreras-Puente, and M. López-López, Thin Solid Films 647, 64 (2018). 5A. K. Jain, S. Yadav, M. Mehra, S. Sapra, and M. Singh, MRS Adv. 4, 567 (2019). 6Y. L. Casallas-Moreno, S. Gallardo-Hernández, F. Ruiz-Zepeda, B. M. Monroy, A. Hernández-Hernández, A. Herrera-Gómez, A. Escobosa-Echavarría, G. Santana, A. Ponce, and M. López-López, Appl. Surf. Sci. 353, 588 (2015). 7M. Born and E. Wolf, Principles of Optics , 6th ed. (Pergamon, New York, 1989). 8M. Muñoz, Y. S. Huang, F. H. Pollak, and H. Yang, J. Appl. Phys. 93, 2549 (2003). 9D. E. Aspnes, Surf. Sci. 37, 418 (1973). 10G. Ramírez-Flores, H. Navarro-Contreras, A. Lastras-Martínez, R. C. Powel, and J. E. Green, Phys. Rev. B 50, 8433 (1994).11H. Shen and F. H. Pollak, Phys. Rev. B 42, 7097 (1990). 12V. Bougrov, M. E. Levinshtein, S. L. Rumyantsev, and A. Zubrilov, in Properties of Advanced Semiconductor Materials GaN, AlN, InN, BN, SiC, SiGe , edited by M. E. Levinshtein, S. L. Rumyantsev, and M. S. Shur (John Wiley & Sons, Inc., New York, 2001), pp. 1 –30. 13A. Mandelis, J. Appl. Phys. 54, 3404 (1983). 14S. Ghosh and B. M. Arora, J. Appl. Phys. 81, 6968 (1997). 15N. Watanabe, T. Kimoto, and J. Suda, J. Appl. Phys. 104, 106101 (2008). 16G. Y. Zhao, H. Ishikawa, G. Yu, T. Egawa, J. Watanabe, T. Soga, T. Jimbo, and M. Umeno, Appl. Phys. Lett. 73, 22 (1998). 17R. E. Wagner and A. Mandelis, Phys. Rev. B 50, 14228 (1994). 18H. K. Lipsanen and V. M. Airaksinen, App. Phys. Lett. 63, 2863 (1993). 19V. M. Airaksinen and H. K. Lipsanen, Appl. Phys. Lett. 60, 2110 (1992). 20H. Shen and M. Dutta, J. Appl. Phys. 78, 2151 (1995).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 125706 (2020); doi: 10.1063/5.0007489 128, 125706-7 Published under license by AIP Publishing.

5.0005940.pdf

J. Chem. Phys. 153, 044304 (2020); https://doi.org/10.1063/5.0005940 153, 044304 © 2020 Author(s).Effect of chemical structure on the ultrafast spin dynamics in core-excited states Cite as: J. Chem. Phys. 153, 044304 (2020); https://doi.org/10.1063/5.0005940 Submitted: 27 February 2020 . Accepted: 02 July 2020 . Published Online: 28 July 2020 Vladislav Kochetov , Huihui Wang , and Sergey I. Bokarev ARTICLES YOU MAY BE INTERESTED IN Massively Parallel Quantum Chemistry: A high-performance research platform for electronic structure The Journal of Chemical Physics 153, 044120 (2020); https://doi.org/10.1063/5.0005889 Modern quantum chemistry with [Open]Molcas The Journal of Chemical Physics 152, 214117 (2020); https://doi.org/10.1063/5.0004835 Dyson-orbital concepts for description of electrons in molecules The Journal of Chemical Physics 153, 070902 (2020); https://doi.org/10.1063/5.0016472The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Effect of chemical structure on the ultrafast spin dynamics in core-excited states Cite as: J. Chem. Phys. 153, 044304 (2020); doi: 10.1063/5.0005940 Submitted: 27 February 2020 •Accepted: 2 July 2020 • Published Online: 28 July 2020 Vladislav Kochetov,1 Huihui Wang,2and Sergey I. Bokarev1,a) AFFILIATIONS 1Institut für Physik, Universität Rostock, A.-Einstein-Strasse 23-24, 18059 Rostock, Germany 2State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University, 030006 Taiyuan, China a)Author to whom correspondence should be addressed: sergey.bokarev@uni-rostock.de ABSTRACT Recent developments of the sources of intense and ultrashort x-ray pulses stimulate theoretical studies of phenomena occurring on ultrafast timescales. In the present study, spin–flip dynamics in transition metal complexes triggered by sub-femtosecond x-ray pulses are addressed theoretically using a density matrix-based time-dependent configuration interaction approach. The influence of different central metal ions and ligands on the character and efficiency of spin–flip dynamics is put in focus. According to our results, slight variations in the coordination sphere do not lead to qualitative differences in dynamics, whereas the nature of the central ion is more critical. However, the behavior in a row of transition metals demonstrates trends that are not consistent with general expectations. Thus, the peculiarities of spin dynamics have to be analyzed on a case-by-case basis. Published under license by AIP Publishing. https://doi.org/10.1063/5.0005940 .,s I. INTRODUCTION Novel light sources such as High Harmonic Generation (HHG) and X-ray Free Electron Laser (XFEL) are steadily improving in terms of increasing intensity, energy, and shortening pulse duration and temporal resolution down to attoseconds.1–5Such an advance allows one to study electron dynamics on a few femtosecond and subfemtosecond timescales.6,7The key point is the preparation of a superposition of quantum states by pulses, which have a broad linewidth in the frequency domain. This non-stationary superpo- sition then coherently evolves in time. Examples of such behav- ior were demonstrated experimentally and reinforced theoretically for the different cases of charge migration.8Due to their ultra- fast character, the early electron dynamics appear to be almost iso- lated from nuclear motion and other effects taking place at longer times. Another kind of coherent dynamics reported recently and con- sidered in this paper is the spin dynamics initiated by x-ray light.9,10 The principle of this process is briefly illustrated in Fig. 1(a) show- ing the many-body state patterns without [Spin–Free (SF)] and with [Spin–Orbit (SO)] strong Spin–Orbit Coupling (SOC), which arecharacteristic for core-excited states. Initially, only the spin-allowed transitions with ΔS= 0, i.e., between the green and red SF states, are occurring upon light absorption. The creation of a core-hole in 2porbitals, i.e., the L 2,3-edge absorption, in transition metal com- plexes is followed by the mixing of states with different spins evolv- ing in time.10,11For certain pulse characteristics, this process leads to a spin–flip, taking place within about a femtosecond, which is extremely fast compared to the conventional spin-crossover times, taking, as a rule, more than 50 fs.12–16However, in exceptional cases, it may take notably less time.17Since SOC for the deeper holes with non-zero angular momentum is, in general, much larger than in the valence band, simulating dynamics initiated by high-energy photons is of interest. From the general viewpoint, the ultrafast spin–flip should occur when a superposition of states with 2 p3/2and 2 p1/2core holes is effectively prepared by excitation with a broadband pulse. Thus, one can expect this process to be purely dictated by the proper- ties of these core holes, with the chemical environment and the details of the pulse characteristics being less relevant. However, previous works9,10concluded that the carrier frequency and the width of the pulse are essential to trigger the efficient spin–flip J. Chem. Phys. 153, 044304 (2020); doi: 10.1063/5.0005940 153, 044304-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . (a) Scheme of many-electron energy levels in the system without (left) and with (right) SOC. States of different spins are marked with red and blue colors; we additionally distinguish the “ground” states (green), which can be pop- ulated due to the finite temperature. The light pulse with carrier frequency Ωand bandwidth ˜σis shown in gray; it prepares the superposition of the SOC states. (b) Population pattern of the core (C) and valence (V) states with spins Sgand Sfenabled by the light absorption (Abs.), stimulated emission (SE), and SOC (VSOC). The style of arrows corresponds to the primary, secondary, and tertiary processes. transition. In particular, the creation of the superposition of the 2p3/2and 2 p1/2core holes was not always a prerequisite for such a spin transition to occur. Moreover, the peculiarities of the dynamics were discussed only for one particular system—hexaaqua iron (II) complex [Fe(H2O)6]2+, where a sub-femtosecond transition from quintet to triplet states has been observed. Therefore, the question of the conditions for the efficient spin transition calls for additional study. The central question of this article is how the nature of the excited metal atom and its chemical environment (coordination sphere) influences the dynamics and the spin–flip yield. Focusing on the transition metal (Ti, Cr, Fe, and Ni) complexes with dif- ferent weak- and strong-field ligands, we deduce the compounds, from which one can expect significant changes in the populations of states with different spins, and discuss conditions, at which one can observe them. This article is organized as follows: First, we present the theo- retical method used in this work in Sec. II. Furthermore, the logic behind the choice of the objects under investigation is explained in Sec. III, and the essential parameters of the computation are presented in Sec. IV. The influence of ligands and the nature of the central metal on the dynamics are presented in Sec. V and are further analyzed in Sec. VI. Finally, the conclusions are given in Sec. VII. II. METHOD The approach we use for the study of the populations of spin states is the quite general density matrix-based Time-Dependent Restricted Active Space Configuration Interaction (TD-RASCI) described elsewhere.11,18Since the whole process of interest lasts no more than few fs, explicit nuclear motion is neglected in the calculation. Its effect is taken into account implicitly via cou- pling of the electronic subsystem, having Hamiltonian ˆH, to a vibrational heat bath. The dynamics of an open system and itsreduced density operator ˆρare described by the Liouville–von Neu- mann equation,19 ∂ ∂tˆρ=−i[ˆH,ˆρ]−Rˆρ, (1) whereRis a dissipation superoperator. Note that here and below atomic units are used unless stated otherwise. If one writes the Hamilton operator in the basis of configuration state functions,20it has the form H(t)=HCI+VSOC+Uext(t). (2) Here, HCIandVSOCare the Configuration Interaction (CI) Hamil- tonian, responsible for electron correlation effects, and the SO interaction part, respectively. The light field contributes to the Hamiltonian with the time-dependent light–matter interaction term Uext(t)=−d⋅⃗E(t)in the dipole approximation, where dis a transition dipole matrix and ⃗E(t)is an external electric field. However, in practical applications to the spin–flip dynam- ics, the so-called SF basis {Φ(Si,MSi) i}is more convenient. These {Φ(Si,MSi) i}are the eigenfunctions of HCIand correspond to a par- ticular spin Sand its projection MSonto the quantization axis and, thus, are the eigenfunctions of the ˆS2and ˆSzoperators. The term VSOCcouples different SF functions such that the eigenfunctions of HCI+VSOCare the linear combinations of SFs with different spins. The density matrix in the SF basis is given by ρ(t)=∑ i,jρ(Si,MSi),(Sj,MSj) ij (t)∣Φ(Si,MSi) i⟩⟨Φ(Sj,MSj) j∣. (3) Diagonal elements ρiiin such representation are the pop- ulations of the corresponding SF states ∣Φ(Si,MSi) i⟩. For the sim- plicity of the analysis, ρiiwith the same total spin Shave been summed, P(S)=∑ iδSSiρ(Si,MSi),(Si,MSi) ii . (4) We use the following notation for the different groups of states: P(GS) is the population of a single or several ground states with the spin Sgsplit by SOC and found in thermal equilibrium at finite tem- perature. P(Sg) is the population of the excited states with the same spin Sg, which is also called the “main” spin below. P(Sf) is the pop- ulation of the excited states with the “flipped” spin Sfdifferent from that of the ground one. The mean value of the spin squared operator has been calculated in the SF basis as ⟨ˆS2⟩=tr[ˆρˆS2]=∑ iρiiS2 ii=∑ SP(S)⋅S(S+ 1) (5) and is further used as an integral characteristic of the spin–flip process. For simplicity, the incoming electric field was chosen to be a single linearly polarized pulse with a temporal Gaussian envelope, ⃗E(t)=A⃗eexp(−t2/(2σ2))sin(Ωt), (6) J. Chem. Phys. 153, 044304 (2020); doi: 10.1063/5.0005940 153, 044304-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp although the pulse trains are more efficient to induce spin–flip tran- sitions.10Here, A,⃗e, and Ω are the amplitude, polarization, and carrier frequency. The pulse width σwas chosen such as to cover a wide range of valence–core excitations; thus, it corresponds to the ultrashort pulse in the time domain. We apply the following strategy to estimate the influence of phase and energy relaxation due to the vibrational degrees of free- dom. We employ a system-bath approach, where we assume that the coupling to the reservoir can be treated in the second order of per- turbation theory and the Markov approximation. Furthermore, we distinguish the “intracomplex” (high-frequency) and “outer” (low- frequency) vibrational baths. The high-frequency part is specific to the first solvation shell, coupled to a secondary bath given by the second and further solvation shells. We assume that it repre- sents a collection of harmonic oscillators in thermal equilibrium coupled to the electronic transitions in a Huang–Rhys-like fash- ion.19The modes corresponding to the low-frequency part are usually more delocalized, and the distinction between solvation shells is not meaningful. Hence, we employ the multi-mode Brow- nian oscillator model21for the “intracomplex” bath and utilize the Ohmic spectral density for the low-frequency one. In total, the effect of the reservoir is described by the spectral density of the form Jij(ω)=∑ ξω2 ξg2 ij,ξωωξγ (ω2−ω2 ξ)2+ω2γ2+θ(ω)j0ωe−ω/ωc, (7) where iandjlabel coupled electronic states. With this, we assume no correlation between the two baths; moreover, the low-frequency part is state-independent. In the first term of Eq. (7), ξis a normal mode with ground-state frequency ωξ, parameter γaccounts for the influence of the secondary bath, and gij,ξis the dimensionless shift of the i’s state harmonic potential energy surface with respect to the potential of state j. The couplings g2 ij,ξ=1 2ωξ(ΔQgi,ξ−ΔQgj,ξ)2δSiSjδMSiMSj(8) have been obtained in the SF basis from the Cartesian gradients in the excited states iandjat the geometry of the ground state gby pro- jecting them onto a normal mode ξ. These gradients in normal mode coordinates give mass-weighted shifts ΔQgi,ξ. In the summation in the first term of Eq. (7), only the normal modes with frequencies above 200 cm−1are taken into account since lower-energy vibrations are highly anharmonic and cannot be mapped to a harmonic model. The second Ohmic term compensates for this fact. In this term, θ(ω) is the Heaviside step function, j0is a coupling strength, and ωcis a cutoff frequency. For the details of spectral density parameterization, see Sec. IV. Finally, we employ the Bloch model, which decouples pop- ulation relaxation and coherence dephasing. In this case, the only non-zero elements of the relaxation matrix Rab,cdare given by Rii,jj=δij∑ lki→l−kj→i (9) for population relaxation andRij,ij=1 2(∑ lki→l+∑ lkj→l) (10) for coherence dephasing. The relaxation rates ki→jhave been obtained from the spectral density, Eq. (7), as ki→j=2[1 +n(ω)][Jij(ω)−Jij(−ω)], (11) where n(ω)=(exp(ω/kBT)−1)−1is the Bose–Einstein distribution function. In the initial density matrix, SO states were populated accord- ing to the Boltzmann distribution: ρij(0) =δijexp(−Ei/(kBT)), which is then transformed to the SF basis for propagation. Pure dephas- ing due to the Auger decay is neglected for the clarity of discus- sion; its effect has been estimated elsewhere10,11and does not qual- itatively change the picture beyond the relatively uniform decay of diagonal elements ρii. Furthermore, for the simplicity of dis- cussion, most of the calculations are performed without dissipa- tion to the environment (nuclear degrees of freedom), i.e., R=0 in Eq. (1). Propagation of the density matrix according to Eq. (1) has been performed in the SF basis with the Runge–Kutta–Cash–Karp method22,23of the 4(5) order of accuracy. All the computations for the density matrix propagation are carried out employing the locally modified version of the OpenMOLCAS package.24 III. INVESTIGATED SPECIES In this study, we have focused on transition metal complexes as convenient objects to study the effect of the chemical structure on the spin dynamics. These complexes exhibit states of different multi- plicities that can be close in energy, as shown in Fig. 1. Their relative energies are governed by the interplay of the ligand-field splitting and pairing (exchange) energy and depend on the position of the lig- and in the spectrochemical series. In turn, SOC increases from left to right in the row of transition metals. Both effects influence relative stability and spin crossover properties.12,13 When talking about the soft x-ray excitation of metal atoms, the electronic states relevant for such dynamics are of the 2 p→3dtype and are strongly dipole allowed, in contrast to the weaker 2 p→4s ones. Depending on the number of electrons in the d-shell, the num- ber of these states also varies because of the difference in the available d-holes for an excited 2 pelectron. Variation in the total number of accessible states may also strongly influence the spin dynamics, in addition to differences in SOC strength, and its impact has been studied here. Furthermore, the 3 p→3dtransitions could also be relevant for observing spin dynamics and seem to be more attrac- tive as M-edge absorption requires less energetic radiation, but at the same time, SOC for the 3 pholes is notably smaller than that for the 2 pones. This type of transitions has also been considered in this paper. Two main sets of compounds have been studied; see Table I. The weak-field d6iron hexaaqua complex [Fe(H2O)6]2+with a quintet ground state is used as a reference as it has been recently studied and demonstrated an efficient spin–flip transition.10Set 1 includes this hexaaqua iron (II) complex and its derivatives with the J. Chem. Phys. 153, 044304 (2020); doi: 10.1063/5.0005940 153, 044304-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Details of the geometric and electronic structure of studied complexes: the metal–ligand distance, the number of 3 delectrons in the ground state, the magnitude of L3/L2energy splitting, the total number of SOC electronic states with different multiplicities considered in the dynamics, and the global XAS shift for the comparison with the experiment. Compound R(M–L)a(Å) 3 delectrons L 3/L2SOC splittingb(eV) States (2 Sg+ 1) States (2 Sf+ 1) XAS shift (eV) Reference [Fe(H2O)6]2+2.04, 2.27 [2.095]256 12.8 175 (5) 585 (3) 1.65 Set 1 [Fe(H2O)5(NH 3)]2+2.16, 2.20, 2.26c12.6 1.60 [Fe(NH 3)6]2+2.30 6 12.6 175 (5) 585 (3) 1.55 [Fe(H2O)5(CN)]+2.05, 2.11, 1.92c12.5 1.68 [Fe(CO)5]01.68, 1.77 [1.810, 1.842]268 11.0 751 (1) 3015 (3) 11.0 Set 2 [TiO 6]8−[1.95, 1.98]280 5.3 16 (1) 45 (3) 8.1 [Cr(H2O)6]3+2.00 [1.966]253 7.2 640 (4) 650 (2), 90 (6) 5.9 [Ni(H2O)6]2+2.09 [2.044–2.064]278 17.9 75 (3) 30 (1) 13.5 aDue to the Jahn–Teller effect or the presence of axial/equatorial ligands, distances may vary. All different distances are given in this case. Values in square brackets correspond to the experiment (where available). Data: Ref. 25 aqueous solution EXAFS, Ref. 26 gas-phase electron diffraction, and Refs. 27 and 28 x-ray diffraction. bThe energy splitting between the L 3and L 2highest peaks. cThe order of distances: equatorial H 2O, axial H 2O, axial NH 3or CN−ligand. general formula [FeX n(H2O)6−n]2+, where water molecules are par- tially or completely replaced by stronger ligands X = NH 3(n= 1 or 6), or even stronger CN−(n= 1). This set is intended to test the influence of ligand strength; complexes of set 1 are listed in Table I in the ascending order of the spectrochemical strength of ligands. The second set comprises six-coordinated complexes [TiO 6]8− and[M(H2O)6]n+with M = Cr, Fe, Ni ordered by the SOC value or equivalently by their nuclear charge. The perovskite building block, [TiO 6]8−cluster, has been chosen because of its high relevance to many functional materials; besides, it resembles the reference [Fe(H2O)6]2+complex, also having a nearly octahedral coordina- tion sphere of oxygen atoms. It is also interesting from the viewpoint of the number of possible singly excited 2 p−13d1configurations as this number is quite small (Table I). The nickel complex, possess- ing an almost filled d-shell, also features the small number of rel- evant electronic configurations similar to Ti, but its SOC constant is larger by about a factor of three. The chromium complex has a more intricate electronic structure with the d3ground state, result- ing in lots of excited states similar to the reference iron compound. A standalone compound, in some sense, is [Fe(CO)5]0. Due to the strong-field ligands, it has a singlet ground state, and the spin-state energetic pattern is substantially different from the other high-spin complexes. IV. COMPUTATIONAL DETAILS All structures and vibrational frequencies ωξwere obtained at the Density Functional Theory (DFT) level with the B3LYP func- tional and aug-cc-pVTZ basis set in the Gaussian 09 program pack- age.29For[TiO 6]8−, the rutile experimental geometry has been used.28 The calculation of SF states and interstate couplings has been performed at the Restricted Active Space Self-Consistent Field(RASSCF) level of theory. Scalar relativistic effects were introduced via the Douglas–Kroll–Hess transformation30up to the second order in perturbation theory in conjunction with the all-electron ANO- RCC basis set31of VTZP quality. The results up to the fourth order are benchmarked in the supplementary material. The active space of eight orbitals (three 2 pand five 3 d) was found to give a good approximation32and is used for all species except for [Fe(CO)5]0. Full CI has been done for the 3 dsubspace (RAS2), while for the 2 p subspace (RAS1), only one hole has been allowed; RAS3 subspace has been left empty. For [Fe(CO)5]0, the 3 dσ(a′ 1), four 3 d(e′and e′′), and 3 dσ∗(a′∗ 1) orbitals were put to the RAS2 as well as four π∗orbitals to the RAS3 with only one electron allowed, resulting in 13 orbitals in the active space. Note that all the experimental X-ray Absorption Spectra (XAS) were shifted to be aligned with calcula- tions for the computational consistency as opposed to the conven- tional way of doing vice versa. The respective shifts can be found in Table I. The particular construction of basis functions and respec- tive matrices in Eq. (2) has been done as follows: First, molecular orbitals were optimized in a state-averaged RASSCF procedure,33 where averaging over all possible electron configurations has been performed. These orbitals were kept frozen during the propaga- tion. Thus, the polarizability of ligands is accounted for only stati- cally, and orbital relaxation in the course of dynamics is neglected. The SOC matrix VSOC is computed by means of the state interac- tion approach,34,35implementing the atomic mean field integral36,37 method. It has proven itself to be a versatile tool for computing the L2,3-edge absorption spectra of transition metal complexes.32,38–40 The respective calculations have been done with the OpenMOLCAS program package.24 The polarization vector ⃗e, see Eq. (6), has been selected to point along one of the metal–ligand bonds. Different polarizations have relatively little effect on the dynamics, as has been shown in Ref. 11, and thus have not been addressed here. The width of the light pulse J. Chem. Phys. 153, 044304 (2020); doi: 10.1063/5.0005940 153, 044304-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II . Summary of the pulse characteristics, see Eq. (6). Compound σ(fs)̵hΩ (eV) A (a.u.) [TiO 6]8−0.2 470.0 1.5 [Cr(H2O)6]3+0.2 588.0 2.5 [Fe(H2O)6]2+0.2 716.0 6.0 0.2 713.2 2.5 [Fe(CO)5]00.2 728.0 6.0 [Ni(H2O)6]2+0.2 875.0 9.0 σwas set to 0.2 fs in the time domain for all simulations. The carrier frequency Ω was chosen to correspond to the center between the L 3 and L 2bands, see Table II. The amplitude Aof the pulse was adjusted to ensure approximately the same depletion of the ground state in all simulations [ P(GS)<0.2]. The initial population of near-degenerate ground states in the high-spin complexes corresponded to the tem- perature of T= 300 K. Additional calculations for T= 0 K are shown in the supplementary material. To estimate the effect of dephasing due to the width of the vibrational wave function in the initial state on the dynamics, 250 geometries were sampled from the vibrational Wigner function41,42 atT= 300 K using the tool wigner.py provided in the SHARC 2.1 suite.43For comparison, results for T= 0 K are given in the supplementary material. The couplings to the intracomplex high-frequency heat bath g2 ij,ξin Eq. (7) were computed for SF states from the RASSCF gradi- ents of the excited states at the ground state optimized geometry. The parameters γandj0were chosen to correspond to the rate of vibra- tional relaxation to the second solvation shell of about (100 fs)−1, i.e.,γ= 300 cm−1andj0= 5.92 ⋅10−4. The cutoff frequency ωcof the Ohmic counterpart of the spectral density was set to 80 cm−1as it corresponds to the characteristic changes in the second solvation shell caused by solute in liquid water.44Relaxation rates, Eq. (11), were computed at T= 300 K. V. RESULTS In Subsections V A and V B, we analyze the influence of purely electronic factors such as the strength of the ligand field and SOC that are decoupled from the influence of vibrations for simplicity. Nuclear effects are discussed in Subsection V C. A. Influence of ligand strength The influence of the surrounding ligand on the dynamics has been studied on the example of Fe2+complexes with H 2O, NH 3, and CN−ligands. Despite different positions in the spectrochem- ical series (especially that of H 2O and CN−), all these complexes have a quintet ground state, see Table I. XAS for all members of the set represents dipole-allowed transitions from the 2 p3/2and 2 p1/2 orbitals to the non-bonding 3 d(t2g) and anti-bonding 3 dσ∗(eg) lev- els. Although one sees clear differences in the nature and energy of individual transitions between complexes, e.g., in the extent of spin- mixing, these differences are washed out upon lifetime broadening.Therefore, the XAS spectra of different species are fairly similar, showing only minor variations in the L 3/L2energy splitting as well as in the structure of the L 3edge, see Fig. 2(a). The spin–flip dynamics occurring in these complexes upon light excitation is illustrated in Figs. 2(b) and 2(c), where the time- dependent values of ⟨ˆS2⟩are presented. One can see that upon excitation, the expectation value ⟨ˆS2⟩first quickly drops from the value of 6̵h2, corresponding to the quintet state manifold, and then slowly evolves after the pulse is over, exhibiting some oscil- lations. However, within the considered time interval, ⟨ˆS2⟩does not reach the triplet value of 2̵h2, evidencing a notable contri- bution from quintet states in the superposition. In other words, the closed system reaches a quasi-equilibrium with respect to the influence of VSOC, and the populations do not change strongly anymore. Concerning the influence of ligands, the same statement as for XAS can also be made for the dynamics. It can be seen from a similar form of the respective ⟨ˆS2⟩curves as a function of time in Figs. 2(b) and 2(c) for two different pulses. These pulses have different car- rier frequencies and amplitudes and thus involve different groups of states in the dynamics. For instance, the gray pulse [panel (c)], centered between L 3and L 2, overlaps with the latter edge in energy, whereas the orange pulse [panel (b)] barely touches it. This fact explains the larger yield of triplet states in the case of the gray pulse. Comparing the curves for different ligands, one can conclude that at least for short pulses (broad in energy), the smearing of the fine FIG. 2 . (a) Calculated XAS of the reference complex [Fe(H2O)6]2+and com- plexes from set 1 , see Table I. Spectra are shifted vertically for visual clarity. Dashed lines show the shape of the pulses in the frequency domain. [(b) and (c)] Time evolution of ⟨ˆS2⟩for these complexes for two different pulses; their characteristics are given in the respective panels, and filled curves depict the time envelopes. The centers of the corresponding excitation bands in the fre- quency domain are also depicted in panel (a) with two vertical lines. The ampli- tude has been selected to give a comparable depletion of the ground state [P(GS)≈0.1]. J. Chem. Phys. 153, 044304 (2020); doi: 10.1063/5.0005940 153, 044304-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp details of the electronic structure occurs, leveling the differences due to ligands. To consider a qualitatively different case, we now address the results for [Fe(CO)5]0. For this complex, all ligands have a strong field, and in contrast to [Fe(H2O)5(CN)]+, this results in the low- spin singlet ground state. The [Fe(CO)5]0spectrum is less consis- tent with the experiment45than that of the [Fe(H2O)6]2+species [see Fig. 3(a)] as the SOC splitting is underestimated by 2 eV– 3 eV, but main spectral features can be clearly recognized. The lower intensity pre-peak at about 720 eV is due to transitions to the 3 dσ∗ (a′∗ 1) orbital and is, thus, somewhat similar to the transitions dis- cussed before for set 1 . In contrast, the pronounced second peak of the L 3edge at 722 eV is a fingerprint of a strong π-backdonation45 as it mainly corresponds to transitions from the 2 p3/2to the lig- andπ∗orbitals, which are notably mixed with the iron 3 dorbitals. Thus, excitation with the light pulse occurs from the ground sin- glet state to predominantly charge-transfer ones because of the larger transition strengths of the latter. The partial contributions of differ- ent spin states to the SOC-coupled ones are illustrated in Fig. 3(a). The total intensity is partitioned according to the fraction of the singlet (red curve) and triplet (blue curve) SF state contributions to the respective SOC-state. Essential for the current discussion is that the L 3states have approximately equal contributions from singlet and triplets SF states, whereas for the L 2, triplets distinctly dominate. Dynamics in [Fe(CO)5]0is shown in Fig. 3(b) for the pulse centered between the L 3and L 2edges and overlapping with all dipole-allowed transitions. One sees the spin–flip from singlet to triplet happening much faster than the pulse duration, i.e., shortly after the initial singlet–singlet excitation (red curve). Remarkably, in contrast to other iron complexes from set 1 , the efficient spin transition occurs independent of the pulse characteristics. In this case, the final ⟨ˆS2⟩is closer to the target triplet value of 2̵h2. Note- worthily, in the case of [Fe(CO)5]0, pronounced oscillations in the state populations and the ⟨ˆS2⟩are observed. The time period of these oscillations (0.35 fs) corresponds to the SOC-splitting of 11.0 eV. FIG. 3 . Results of modeling for the [Fe(CO)5]0complex: (a) Experimental (black dotted line) and calculated (green line) XAS. The total intensity is decomposed according to the fraction of the singlet (red curve) and triplet (blue curve) character of the respective states, see text. (b) Evolution of the population of singlet Sg= 0 (red) and triplet Sf= 1 (blue) SF states initiated by the pulse with characteristics given in Table II. The dashed line shows the expectation value of the ˆS2operator. The value of 2̵h2, marked with a horizontal line, corresponds to the pure triplet Sf= 1.Naturally, when the symmetry is changed and the electronic struc- ture is altered by the strong-field ligands and the dominant contri- butions from the charge-transfer states, the time-evolution changes qualitatively. The reasons for this fact will be further analyzed in Sec. VI. B. Transition metal series The influence of the central atom is studied on the exam- ple of complexes from set 2 , see Table I. In the upper row of Fig. 4, the calculated L 2,3absorption spectra are presented. Over- all, reasonably good agreement with the experiment is reached at the RASSCF level of theory. The SOC splitting between the L 3 and L 2bands, as well as the ligand-field splitting within the L 3 band, is well reproduced for all systems under study. The [TiO 6]8− spectrum is compared to the Ti L 2,3-edge XAS in SrTiO 346as a reference for the d0system in the octahedral field of oxygen atoms. It shows four clear peaks originating from the 2 p−1 3/2t2g, 2p−1 3/2eg, 2p−1 1/2t2g, and 2 p−1 1/2egstates. A somewhat similar multi- plet configuration can be roughly recognized for other species of the row, but one sees an intensity redistribution and appear- ance of additional peaks due to the stronger multiconfigurational character. The first row in Fig. 4 depicts the decomposition of spectra in the spin multiplicity of final states (red and blue curves). Note that almost everywhere, the contributions from the states with the main or ground spin Sg(red) are higher than from the ones with the flipped spin Sf(blue). The only exceptions are the L 2band of [TiO 6]8−and the 712 eV–727 eV region of [Fe(H2O)6]2+shown in Figs. 4(a1) and 4(c1), respectively. The spin dynamics of set 2 are shown in the lower row of Fig. 4. The [TiO 6]8−compound features strong oscillations between the main-spin and flipped-spin states with a period of about 0.6 fs, which corresponds to 6.9 eV energy [Fig. 4(a2)], correlating with the L 3/L2SOC splitting. Conceivably, it is caused by the rela- tively small number of states involved in the dynamics giving rise to the Rabi-like oscillations. Other complexes demonstrate the same trend, i.e., oscillations have the characteristic period inversely pro- portional to the value of SOC splitting, see Table I. However, it is difficult to rationalize why they are very pronounced in some cases and, in others, notably washed out. These oscillations result from the joint behavior of hundreds and even thousands of states and depend on their interference. They should also depend on the interplay with some other energetic differences (and their charac- teristic timescales), e.g., ligand-field splittings. For instance, in the cases of chromium [Fig. 4(b2)] and iron [Fig. 4(c2)], the oscil- lations are dumped slowly, which is related to a huge number of involved states. Similar behavior has also been observed for [Fe(CO)5]0(Fig. 3). Note that these oscillations also persist if the inhomogeneous distribution of geometries (Wigner sampling) and relaxation to the environment are taken into account (see Sec. V C). The prevailing population of states with the flipped spin (blue line in the lower row of panels in Fig. 4) is observed only in [TiO 6]8−and[Fe(H2O)6]2+complexes, whereas for [Cr(H2O)6]3+ and[Ni(H2O)6]2+, the spin of the ground state (red line) stays dominant. This behavior is also observed for the ⟨ˆS2⟩curves J. Chem. Phys. 153, 044304 (2020); doi: 10.1063/5.0005940 153, 044304-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . Results for species from set 2 (Table I): panels (a) [TiO 6]8−, (b)[Cr(H2O)6]3+, (c)[Fe(H2O)6]2+, and (d) [Ni(H2O)6]2+. Upper row: the comparison of the experimental XAS spectra38–40,46with the calculated ones. Lower row: SF-state population dynamics in the corresponding complexes initiated by pulses with characteristics given in Table II. All pulse amplitudes (light-gray) are normalized to the same height for the sake of clarity. Partial populations [Eq. (4)] of the ground states GS, excited states with the same spin Sg, and spin distinct by ±1 (flipped spin) Sfare depicted in green, red, and blue, respectively. Thick dashed lines give the actual ⟨ˆS2⟩(t) for a particular complex. Horizontal dashed lines indicate the expectation values S(S+ 1)̵h2of the ˆS2operator for half-integer and integer spins Srelevant for each complex. (dashed line). For the former two cases, the ⟨ˆS2⟩significantly devi- ates from the initial value and tends to the flipped value, namely, the singlet (0̵h2) to triplet (2̵h2) transition for [TiO 6]8−and quintet (6̵h2) to triplet (2̵h2) for[Fe(H2O)6]2+. In turn, only a moderate spin tran- sition can be seen for [Cr(H2O)6]3+(b2) and [Ni(H2O)6]2+(d2). To summarize, there is no correlation between the efficiency of spin transition and the value of SOC, which is opposite to what can be anticipated from general considerations. To strengthen this conclusion, one has to exclude the influence of the light pulse because the amplitude has been selected differ- ently for different complexes. It seems natural that the pulse strength notably influences the ⟨ˆS2⟩for the same system, and one might argue that by substantially increasing the amplitude, one could achieve a more efficient spin conversion. This fact is illustrated in Fig. 5, where the dependence of ⟨ˆS2⟩(t) on the amplitude of the pulse ranging from 1 a.u. to 7 a.u. for iron (a) and from 4 a.u. to 10 a.u. for nickel (b) is shown. One can see that already at 6 a.u. for iron and 7 a.u. for nickel, there is no further increase in the yield of spin transition. Observed saturation takes place when the ground state population is almost completely depleted. These values of amplitude, at which the total population of the ground states GSdrops below 0.1, were chosen for all the complexes to exclude the influence of the pulse strength possibly. In the transition metal row, the characteristic A value increases as we excite in the center of L 2,3-edge, meaning that a stronger pulse is needed to overlap with more energetically distanttransitions separated by SOC efficiently. As an example, note in Fig. 5(a) the growing prominence of oscillations with the increasing amplitude, witnessing an involvement of more distant L 3/L2groups of states. Finally, the 3 p→3dexcitations have been considered. The respective XAS spectra are given in the supplementary material. Although SOC is also notable in this case [SOC splittings are up to 6 eV in[Ni(H2O)6]2+], the Sgstates are by far prevailing among the FIG. 5 . The dependence of ⟨ˆS2⟩(t) on the amplitude Aof the incoming pulse, which is given in atomic units in the legend, exemplified for (a)̵hΩ=716eV andσ= 0.2 fs for[Fe(H2O)6]2+and (b)̵hΩ=875eV andσ= 0.2 fs for [Ni(H2O)6]2+. J. Chem. Phys. 153, 044304 (2020); doi: 10.1063/5.0005940 153, 044304-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp bright states, and no spin–flip dynamics is observed. That is why this case will be not further discussed here. C. Nuclear effects The effect of nuclear dynamics on the spin–flip processes can be roughly considered being twofold: the initial nuclear distribu- tion may cause electronic dephasing, and the vibrational degrees of freedom may act as a heat bath causing phase and population relaxation between participating states. The first one refers to the inhomogeneous and the second to the homogeneous broadening of the respective spectroscopic lines. Both effects may be considered as related to the chemical structure discussed herein since the nature of ligands determines their possible contribution. Figure 6 presents the effect of nuclear distribution in the system’s initial state on three examples. It shows the time dependence of the population account- ing for the width of the initial nuclear wave packet as compared to the results for a single “equilibrium” geometry, corresponding to the center of the distribution. In[Fe(H2O)6]2+, three lowest SF quintet states lie close to each other in energy, which corresponds to 15 SO states slightly split due to SOC. Different geometries affect the relative energies of these states and notably change equilibrium populations of the lowest five “ground” microstates at T= 300 K, i.e., the diagonal elements of the initial density matrix ρ(0). Hence, before the pulse comes, one sees a broad distribution of the populations of these “ground” states and other quintet states, including the other ten closely lying microstates. Since all these states have the same spin ( S= 2), the ⟨ˆS2⟩ value is sharp before excitation, as indicated in panel (a2). However, the spin–flip dynamics itself is less sensitive to the initial popula- tions and changes in the electronic structure due to different geome- tries. The scope of population curves (color-filled ranges) is rela- tively narrow, and most importantly, the curves averaged over 250 geometries closely resemble those for a single optimized geometry, cf. respective solid and dotted lines in Fig. 6(a1). Therefore, Wigner sampling leads to only minor changes. This fact may be rational- ized by noting that the potential energy surfaces of the core-excited states in this complex are almost parallel to each other, as has been illustrated for the symmetric Fe–O stretching mode,10leading to small dephasing. Interestingly, this dephasing does not destroy fastoscillations due to the most distant states split by SOC, which is discussed in Sec. V B. The situation seems to be qualitatively different for [Fe(H2O)5(CN)]+, see panels (b1) and (b2). In this case, due to the strong-field CN−ligand, a triplet state comes close to the ground quintet one. Geometries from the flanks of the Wigner function induce a flipped order of the lowest quintet and triplet states. It leads to a very broad distribution of the population curves as the dynamics comprise both quintet →triplet and reverse transitions. However, because of the small statistical weight of these points, the averaged dynamics stay almost the same as for a single optimized geometry. For the third example of [Ni(H2O)6]2+, the initial density matrix, ρ(0), is not influenced that much by geometrical variations. Nevertheless, the averaged dynamics are substantially different from the single-point one. In this case, one might expect a stronger dependence of the slopes of core-excited states’ potential surfaces on nuclear displacements, leading to a significant dephasing. One should also note that Wigner sampling leads to a higher Sfyield for the nickel complex. The influence of ligands due to the dissipation to the vibra- tional heat bath has been estimated for three Fe2+complexes, having the same ground state and featuring similar non-dissipative dynam- ics:[Fe(H2O)6]2+,[Fe(H2O)5(CN)]+, and[Fe(NH 3)6]2+. For all these cases, one can barely see the changes in the total popula- tions and ⟨ˆS2⟩due to relaxation. An example of [Fe(H2O)5(CN)]+, showing the most pronounced differences, is presented in Fig. 7; the other two examples can be found in the supplementary mate- rial. This finding is in accord with the previous study10and can be explained by the following two observations: the considered elec- tronic timescale is very short for nuclear dynamics to come into effect, and the number of the kij’s that are large enough to cause notable relaxation within 5 fs is quite small. The latter statement is illustrated in the panel (a) of Fig. 7 where only few ijpairs out of 52 670 have rate values larger than 0.1 eV. Note that the dissipa- tion results should not be overinterpreted because these largest rates exceed the energetic distances ( <0.1 eV–0.2 eV), which are typical for neighboring core states. Therefore, they approach the applica- bility limits of the second-order perturbation theory in system-bath coupling. FIG. 6 . Spin–flip dynamics accounting for the width of the initial vibrational wave packet in the nuclear phase space (only the distribution of geometries is con- sidered) for [Fe(H2O)6]2+[panels (a)], [Fe(H2O)5(CN)]+[panels (b)], and [Ni(H2O)6]2+[panels (c)]. Solid lines denote the population dynamics aver- aged with the Wigner distribution func- tion; color-filled intervals show the over- all range of Wigner-sampled trajectories; dotted lines correspond to a single “equi- librium” geometry calculation. J. Chem. Phys. 153, 044304 (2020); doi: 10.1063/5.0005940 153, 044304-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 7 . (a) The distribution of the relaxation rates kijbetween the valence- (red bars) and core-excited (green bars) SF states of [Fe(H2O)5(CN)]+. Note that bars for the range 0.00–0.01 eV are cut not to show all 52 670 rates. (b) Respective spin-dynamics with (solid lines) and without (dashed lines) dissipation. Of course, different ligands have different fingerprints in relax- ation dynamics. For instance, in [Fe(H2O)6]2+, the largest contribu- tions to the rates correspond to the Fe–O symmetric and asymmetric stretching modes. A similar situation is observed in [Fe(NH 3)6]2+ where, in addition to Fe–N stretching vibrations, a symmetric NH 3 rocking mode is also contributing. For [Fe(H2O)5(CN)]+, the prominent modes are (in order of decreasing importance) axial H2O torsion, axial and equatorial Fe–ligand stretching, and equa- torial H 2O wagging modes. However, these individual features are averaged out when the total populations of 175 quintet and 585 triplet states or the integral ⟨ˆS2⟩are considered on a short timescale. Nevertheless, for longer times, one can expect the relaxation in [Fe(H2O)5(CN)]+to be most and for [Fe(NH 3)6]2+least efficient out of these three examples. VI. DISCUSSION From the discussion in Secs. V A–V C, one can make four gen- eral observations. First, the qualitative character of the dynamics is only barely dependent on the chemical nature of ligands unless the electronic structure is altered completely. The examples are high- spin complexes of set 1 , demonstrating a very similar behavior, and [Fe(CO)5]0, possessing the low-spin ground state and exhibiting a completely different energetic pattern of spin-states. Second, the related differences between the complexes in the dephasing and relaxation rates to the vibrational bath also do not cause substan- tial changes in the dynamics. The respective structural reorganiza- tion responsible for energy dissipation occurs on a notably longer timescale of hundreds of femtoseconds to few picoseconds,15,16 whereas the spin–flip is a sub-femtosecond process. Third, in con- trast to expectations, the value of SOC splitting does not play a deci- sive role in the character of dynamics, as seen from the comparisonof different metals. For instance, [Ni(H2O)6]2+has the largest SOC constant in the considered series but does not show prominent spin– flip dynamics. On the other hand, [TiO 6]8−, with its SOC constant being a factor of three smaller, demonstrates intricate dynamics. Fourth, the critical point is the ratio between the SF states with dif- ferent spins constituting the SOC-eigenstates. It can be seen from the XAS decomposition into SgandSfcontributions in Figs. 3 and 4. Indeed, a significant spin–flip was observed for titanium and iron compounds, where bright states with the prevailing amount of the Sf spin contributions are dominating in XAS for the high-energy flank of the L 3and the whole L 2edges. It means that at specific energy ranges, more SF flipped states can be accessed by the excitation and SOC mediated population transfer. For the 3 pexcitation, there are no such ranges, and spin–flip is not observed. However, a profound analysis going beyond these simple observations is complicated due to the vast amount of the electronic states, which are coupled in a complex way. To shed light on the reasons for such behavior and attain a more mechanistic understanding, let us consider a somewhat sim- plified model, see Fig. 1(b). Assuming that we are working in the saturated regime (Fig. 5) and, thus, can neglect the details of the incoming light pulse for simplicity, the dynamics are governed by two factors—strengths of the dipole transition and that of the SOC. Let us follow the density matrix evolution and, namely, its diago- nal elements in the basis of SF states. Initially, an entire population resides in the ground state and the lowest excited states, which are populated according to the respective Boltzmann factors, see Sec. IV. The light pulse couples these initial states with the core ones through the respective transition dipole dmatrix elements. Note that the spin quantum number is conserved GS→Sgdue to the spin selection rules. Reflecting this fact, in Fig. 4, the red line, P(Sg), rises simul- taneously with the arriving pulse. In the SOC picture, states with strictly defined spin do not exist as the spin quantum number is not conserved; thus, the predominant population of the SgSF states cor- responds to a non-stationary superposition of SOC eigenstates. After the initial population of the core-excited Sgstates, all SF states get mixed regardless of their spin through VSOC. That is why the blue Sf line goes up parallel to the red one but after a short delay (see Fig. 4). Once the pulse is switched off, the dynamics are governed solely by the elements of VSOC. This free dynamics is then determined by the populations accumulated during the pulse in the bright core-excited states with spin Sgand their SO coupling to other states with both Sg andSf. Provided a large number of coupled states, we employ a con- cept widely used in the data analysis to illustrate some trends. Let us start the discussion from the example of the [Fe(CO)5]0com- plex; see Fig. 8. This figure is obtained with the NetworkX pack- age47implementing the force-directed graph drawing algorithm by Fruchterman and Reingold.48Here, each node corresponds to one of the 3766 SF basis states Φ(S,MS) i , and the color encodes their nature, e.g., ground as well as excited states with spins Sgand Sf. The size of the nodes, in turn, denotes whether the state is involved in dynamics (we call it participating) or stays mainly unpopulated (spectator). The distances between nodes are optimized to minimize spring-like forces between them. If the pulse characteristics are left besides the discussion, the force ( Fij=−κijΔxij) between nodes iand jcorresponds to the spring constant, J. Chem. Phys. 153, 044304 (2020); doi: 10.1063/5.0005940 153, 044304-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 8 . Force-directed graph showing the clustering of the states of [Fe(CO)5]0 according to the transition-dipole and SOC; see Eq. (12). Each node corresponds to one of the 3766 SF basis states Φ(S,MS) i: green— GSsinglet states, red— excited singlet states, and blue—excited triplet states. Large circles indicate states participating in the dynamics, i.e., having a notable maximal population; small circles correspond to the “spectator” states acquiring no population. κij=c∣(VSOC)ij∣+∣dij∣, (12) where cis a factor governing the relative importance of the two couplings. It has been adjusted for visual clarity to illustrate the clus- tering of states. These two quantities in the sum are correlated with the degree of spin conversion. The dipole matrix identifies states that can be directly populated by the light absorption from the initial state manifold (which is denoted as green nodes in Fig. 8). The sub- sequent dynamics is governed mainly by the strength of SOC. The combination of these two quantities allows considering both effects together. Looking at Fig. 8, one can note the following peculiarities. According to the above criteria, the states group in two mainclusters, which are separated from each other and, thus, are con- nected neither by transition dipole nor by SOC. Inside both clusters, one can distinguish four subgroups. The red one corresponds to the singlet ( Sg) excited states ( MS= 0); the three blue subgroups are triplet ( Sf) states grouped by their MSquantum number. Inside of the smaller subgroups, both dandVSOCcontributions keep nodes together, whereas between them, only the SO interaction is non- zero. This is due to the spin selection rules ( ΔS= 0,ΔMS= 0) for the dipole transitions, causing the blocked structure of d. Remark- ably, the participating states (big nodes) are found only in the cluster, where the single ground state (green node) is entering. Thus, the sec- ond big cluster is completely excluded from the dynamics. Even in the former cluster, a relatively small amount of states (about 200 out of 1800) are populated during dynamics. The last important notice is that the amount of the spin-flipped states in this participating cluster is larger than that of the spin-conserved ones. The graphs for the other compounds of set 2 and the refer- ence[Fe(H2O)6]2+complex are presented in Fig. 9. For most of the species {apart from [Ni(H2O)6]2+}, the states also group in two major clusters. [TiO 6]8−[panel (a)], however, does not show subdi- vision according to the MSquantum number for the triplet states. For this complex, the overall number of states is the lowest among all systems. The singlet ground state enters only one cluster simi- lar to[Fe(CO)5]0. Analogously to the latter, the amount of triplet Sfstates is dominating over the Sgsinglet states. [Cr(H2O)6]3+ and[Fe(H2O)6]2+systems shown in panels (b) and (c), respec- tively, demonstrate similar clustering. In these two cases, the MS- components of the ground state are distributed between two major clusters, thus leading to the involvement of both groups of states into the dynamics. Therefore, almost all considered states are popu- lated within the first femtosecond. However, this behavior depends on the temperature: for low temperatures, only one component of the ground state may be initially populated, and thus, only one clus- ter is participating. For instance, compare with Figs. S2 and S3 in the supplementary material for the case of dynamics in [Fe(H2O)6]2+ FIG. 9 . Clustering of states for the complexes of set 2 : (a)[TiO 6]8−, (b) [Cr(H2O)6]3+, (c)[Fe(H2O)6]2+, and (d)[Ni(H2O)6]2+; see caption of Fig. 8. J. Chem. Phys. 153, 044304 (2020); doi: 10.1063/5.0005940 153, 044304-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp atT= 0 K. Both clusters show a distinct splitting according to the MSquantum number. The difference between the two systems is the ratio between numbers of Sfand Sgstates. The flipped states are prevailing in the case of [Fe(H2O)6]2+and represent a minor- ity in the case of [Cr(H2O)6]3+.[Ni(H2O)6]2+is somewhat simi- lar to[TiO 6]8−since the total number of states is quite small. The three components of the ground state are also uniformly distributed throughout the cluster. The overall MSgrouping is less pronounced but still can be seen in the central part of the panel (d). In con- trast to other cases, the separation of states into two clusters is not present. One should also note the dominating number of the Sg states. Although the graphs given in Figs. 8 and 9 provide a conve- nient visualization of the connections between different states, they do not allow making an unambiguous conclusion about the deci- sive factors, which could be used for the a priori assertion on the efficiency of spin–flip for an arbitrary system. The only factor that seems to favor the efficient transition is the dominating number of spin-flipped states over the states with the ground state spin. Such a situation is observed for [Fe(CO)5]0,[Fe(H2O)6]2+, and[TiO 6]8−, being efficient systems, and is not observed for [Cr(H2O)6]3+and [Ni(H2O)6]2+, showing no prominent spin transition. As described for the case of set 2 , this domination can be present in some energy ranges and be absent for the other. This fact explains the dependence of the efficiency on the particular pulse characteristics used for the excitation, see Refs. 9 and 10. In this respect, the proper pulse lead- ing to spin transition should necessarily overlap with the spectral regions, where Sfstates dominate. VII. CONCLUSIONS This article represents an extension of the previous study of the ultrafast spin–flip dynamics in the core-excited states, which has been performed for a prototypical Fe2+complex.9,10There, the occurrence of the spin transition within the time window of hun- dreds of attoseconds has been observed, being also dependent on the characteristics of the exciting x-ray light pulse. Here, we address the main question: what is the crucial factor influencing the spin dynam- ics in terms of the yield of the spin-flipped states? For example, how the central metal ion and surrounding ligands influence the extent of the transition. An intuitive answer can be suggested based on the two-level model, where the probability of the transition between states is pro- portional to the square of the coupling matrix element.19In par- ticular, the efficiency of spin–flip should be proportional to the SOC constant. Therefore, one expects the population transfer from states with the spin of the ground state Sgto ones with a different spin Sfto increase from left to right in the periodic table. How- ever, the situation appears to be more complicated. Although the values of the SOC matrix VSOC responsible for L 2,3-splitting are indeed important, the number of the relevant states plays a deci- sive role. For instance, the SOC strength in [Ni(H2O)6]2+is three times larger than in [TiO 6]8−, but the small number of the acces- sible spin-flipped Sfstates makes the whole process inefficient in the former case, while in the latter, the spin dynamics is much more prominent. Importantly, the exciting pulse should overlap with the spec- tral regions where Sfstates dominate. Here, the decomposition ofXAS provides a hint about how many states of different multiplic- ities are presented and what is the chance to have enough relevant states in order to observe a target effect. Relevant states are those that are coupled to the ground states by the dipole matrix elements either directly or indirectly through SOC. The effect seems to be stable to moderate changes in the coor- dination sphere. For instance, the exchange of ligands situated close to each other in the spectrochemical series (e.g., H 2O and NH 3) does not lead to the qualitative changes in the rate and completeness of the spin dynamics. These ligands strongly affect neither the electronic structure nor the dephasing and relaxation rate to the environment. In general, nuclear effects were found to have little influence on the dynamics as their characteristic timescale is longer than the process considered in this article. However, strong-field ligands can substan- tially change the electronic structure of the outer valence shell, alter- ing the relative energetic stability of the spin states. This is observed for[Fe(CO)5]0, where the ground state spin is changed to a sin- glet in contrast to [Fe(H2O)6]2+with its quintet ground state. In conclusion, the character and efficiency of the dynamics have to be analyzed on a case-by-case basis as no general trends have been observed. SUPPLEMENTARY MATERIAL The supplementary material contains the M 2,3absorption spec- tra of the complexes of set 2 and their decomposition in terms of the involved spin states, the results of dynamics and the graph of partic- ipating states at T= 0 K, the comparison of energies calculated with the second- and fourth-order Douglas–Kroll–Hess transformation, Wigner sampling at T= 0 K, and dynamics including relaxation for [Fe(H2O)6]2+and[Fe(NH 3)6]2+. ACKNOWLEDGMENTS The authors would like to thank Professor Dr. Oliver Kühn and Dr. Olga S. Bokareva for fruitful discussion. Financial support from the Deutsche Forschungsgemeinschaft (Grant No. BO 4915/1- 1) (V.K. and S.I.B.) and the National Natural Science Foundation of China (Grant No. 11904215) (H.W.) is gratefully acknowledged. DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material. REFERENCES 1M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. 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5.0020145.pdf

Appl. Phys. Lett. 117, 122411 (2020); https://doi.org/10.1063/5.0020145 117, 122411 © 2020 Author(s).Paramagnetism and martensite stabilization of tensile strained NiTi shape memory alloy Cite as: Appl. Phys. Lett. 117, 122411 (2020); https://doi.org/10.1063/5.0020145 Submitted: 28 June 2020 . Accepted: 14 September 2020 . Published Online: 24 September 2020 A. Kyianytsia , E. Gaudry , M. Ponçot , P. Boulet , B. Kierren , and T. Hauet ARTICLES YOU MAY BE INTERESTED IN Magnon-mediated spin currents in Tm 3Fe5O12/Pt with perpendicular magnetic anisotropy Applied Physics Letters 117, 122412 (2020); https://doi.org/10.1063/5.0023242 Strong spin-lattice coupling in tetragonal-like BiFeO 3 films with thermal expansion anomalies Applied Physics Letters 117, 122901 (2020); https://doi.org/10.1063/5.0014767 Crossover from normal to relaxor ferroelectric in Sr 0.25 Ba0.75 (Nb1−xTax)2O6 ceramics with tungsten bronze structure Applied Physics Letters 117, 122902 (2020); https://doi.org/10.1063/5.0020853Paramagnetism and martensite stabilization of tensile strained NiTi shape memory alloy Cite as: Appl. Phys. Lett. 117, 122411 (2020); doi: 10.1063/5.0020145 Submitted: 28 June 2020 .Accepted: 14 September 2020 . Published Online: 24 September 2020 A.Kyianytsia, E.Gaudry, M.Ponc ¸ot,P.Boulet, B.Kierren, and T. Haueta) AFFILIATIONS Institut Jean Lamour, Universit /C19e de Lorraine-CNRS, Nancy 54000, France a)Author to whom correspondence should be addressed: thomas.hauet@univ-lorraine.fr ABSTRACT We present an experimental and theoretical study of Pauli paramagnetism and martensite stabilization in a near equiatomic NiTi shape memory alloy. We demonstrate a direct correlation between strain-induced shear of the B190NiTi lattice and its electronic and thermody- namical features. An increase in the monoclinic angle bfrom 97.4 to 98/C14induces a 7% decrease in the magnetic susceptibility because of a shift and deepening of a dip in B190density of states at the Fermi level. It also produces a decrease in the B190enthalpy, which translates into an increase in the martensite-to-austenite transition temperature by 60 K. Published under license by AIP Publishing. https://doi.org/10.1063/5.0020145 Near equiatomic NiTi alloys carry remarkable properties such as shape memory effect and pseudoelasticity. Both effects are born of first-order reversible phase transformation between the low-symmetry monoclinic B190martensite phase and the high-symmetry cubic B2 austenite phase. Tuning the transition temperatures and martensite transformation hysteresis has been a constant objective. It has yet been achieved by adjusting the Ni xTi1-xconcentration1or by substituting the element in ternary NiTiX alloys (with X ¼Hf, Pd, Cu,…).2,3 Geometrical compatibility of B190and B2 phases,4change of the valence electron number,5and thermodynamics1,6have been investi- gated to explain the influence of these chemical changes on NiTi- based alloy characteristics. Interestingly, no direct correlation between electronic features, transition temperatures, and NiTi lat- tice distortion for a fixed concentration has yet been demonstrated, whereas the complex microstructure of NiTi under strain has been heavily studied.7–9Moreover, magnetism, which is known as a good probe of the electronic properties, has been rarely studied in NiTi alloys10–13although its impact on the magnetic resonance imaging of NiTi-based stents and biomedical implants has often been highlighted.14,15 Here, we perform systematic x-ray diffraction (XRD) and magne- tization measurements in order to characterize the influence of the tensile strain on both the martensite-to-austenite transition tempera- ture and magnetic susceptibility of a NiTi sheet. Ab initio calculations correlate the measured strain-induced shear of the martensite lattice with particular features of the B190energy and density of states at the Fermi level.We use NiTi (50.6 at. % Ti; 49.4 at. % Ni 60.5%) free-standing polycrystalline 20 lm thick sheets, grown by DC magnetron sputtering as described in detail in Ref. 16.Figure 1 shows the DC magnetic sus- ceptibility vmeasured by a SQUID magnetometer as a function of temperature under a constant magnetic field of 2T. The magnetic field is applied in the plane of the NiTi sheet. Note that vdoes not depend on the measurement direction in the film plane because of the poly- crystalline nature of the NiTi sheet. The as-grown NiTi sample over- comes a hysteretic transition between a value vM¼23 mJ T/C02kg/C01in the martensite phase and vA¼32.5 mJ T/C02kg/C01in the austenite phase, in agreement with Ref. 10. A magnetic transition temperature of around 360 K (330 K) matches the martensite-to-austenite (respec- tively, austenite-to-martensite) transition in XRD curves, in agreement with Refs. 8and17for similar concentrations. Tensile tests are per- formed using a Deformation Device System commercialized by Kammrath and Weiss at 2 lm/s and room temperature, i.e., in the martensite phase. During stretching, elongation of the specimen is measured by a linear displacement gauge while the corresponding force is recorded. 3D strain fields, obtained by the ARAMIS 6M DIC system, confirm the good homogeneity of strain magnitudes over the useful zone of specimens. All details can be found in our previous report.18 InFig. 1 ,r e da n db l u ec u r v e ss h o w vas a function of temperature during the martensite-to-austenite transformation of the 5% (red) and 16.5% (blue) strained NiTi samples. They are compared to the v(T) cycle of the non-deformed NiTi sample (in black). Stretching modifies the magnetic response of NiTi in two ways. First, the martensite-to- Appl. Phys. Lett. 117, 122411 (2020); doi: 10.1063/5.0020145 117, 122411-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplaustenite transition temperature (so-called Austenite start temperature As) is pushed toward higher temperature when strained, up to 415 K in the case of 16.5% strain. The positive shift increases linearly withincreasing strain [see Fig. 2(b) ]. This effect, known as stress-induced martensite stabilization and whose origin is still under debate, 4–6,19is one of the two phenomena, which we focus on here. In Fig. 1 ,t h es e c - ond noticeable difference between the as-grown and strained samplesis the difference between magnetic susceptibility values at 300 K in theB19 0phase ( vM), whereas austenite susceptibility ( vA) is preserved at high temperature. Figure 2(a) summarizes all our magnetic susceptibil- ity measurements as a function of strain. vMis found to decrease monotonically when strain amplitude increases. In order to access average lattice parameters as a function of tem- perature and strain and to later use them in ab initio calculations, we perform x-ray diffraction (XRD) measurements with a Co K asource.Heating is accomplished in situ by means of a DHS 1100 oven at atmospheric pressure [ Fig. 3(a) ]. The measured lattice parameters of the unstrained B190phase at 300 K are a¼2.894 A ˚,b¼4.115 A ˚, c¼4.637 A ˚,a n d b¼97.41/C14, which are close to previous theoretical reports.20,21T h eB 2a u s t e n i t ep h a s ep a r a m e t e ri s aA¼3.016 60.002 A ˚. To analyze the strain-induced change of martensite lattice, XRD is performed for five representative samples with various strain values ranging from 0 to 16% [ Fig. 3(a) ]. The monoclinic angle bof FIG. 2. (a) Magnetic susceptibility vMof NiTi in the martensite phase at 300 K and (b) austenite start temperature As as a function of strain amplitude. FIG. 3. (a) Room temperature XRD measurements for 0% strained (black), 5% strained (red), and 16% strained (blue) martensite NiTi sheets. Dashed lines indi-cate the family of crystallographic planes corresponding to each intensity peak of the unstrained sample. (b) Monoclinic angle bas a function of tensile strain ampli- tude deduced from XRD measurements. (c)–(e) Lattice parameters of the martens-ite phase as a function of monoclinic angle bextracted from XRD experiments and compared with theoretical values (dashed lines) from Ref. 20. FIG. 1. Magnetic susceptibility vof NiTi vs temperature for the as-grown sample (black curve), 5% strained sample (red curve), and 16.5% strained sample (blue curve).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 122411 (2020); doi: 10.1063/5.0020145 117, 122411-2 Published under license by AIP Publishingthe B190phase increases with tensile strain and tends to saturate at high strain values. The saturation is due to the increasing amount of energy relaxation via plastic deformation [see Fig. 3(a) ]. The same ten- dency as that of bis observed for both aand cparameters, while it is inversed for b.InFig. 3(b) , a relative change of a,b,andcparameters as a function bmatches quantitatively well earlier theoretical predic- tions obtained by ab initio calculations for relaxed B190lattices with fixed b.20,21Significant deviation only occurs for the 16% strained sam- ple, most probably due to large level of plastic deformation. Interestingly, the austenite lattice parameter measured by XRD afterheating is found to be insensitive to the level of strain applied to the martensite state and a A¼3.01660.002 A ˚.T h em a i no b s e r v e df e a t u r e in the B2 XRD data is a broadening of the diffraction peak for larger strains, which we associate with microstresses induced by dislocation networks and/or twin domains reminiscent from B190deformations.7,9 Based on the XRD data, Density Functional Theory (DFT) calcu- lations of Ni 50Ti50are performed with the Vienna Ab initio Simulation Package (VASP).22The interactions between the valence electrons and the ionic core are described using the projector-augmented wave (PAW) method and the calculations are performed with the general- ized gradient approximation (GGA-PBE).23,24Spin polarization is considered. A plane wave basis set for electron wave functions with acutoff energy of 500 eV is used. Integrations in the Brillouin zone are performed using a k-grid generated according to the Monkhorst–Pack scheme: 33 /C223/C223 for the martensite NiTi. Density of states (DOS) calculations are performed with the tetrahedron method for Brillouin-zone integrations. 25Atomic positions used for B190in all cal- culations are for Ni: x¼0.0372, y¼1/4, and z¼0.1752 and for Ti: x¼0.4176, y¼1/4, and z¼0.7164. The total DOS of B2 and B190as a function of energy is presented in the inset in Fig. 4(a) . Our DFT cal- culations show no net atomic magnetic moment at rest for both crystal structures. The DOS at the Fermi level D(EF) is 0.72 states atom/C01 eV/C01for the martensite phase and 1.38 states atom/C01eV/C01for the austenite phase. It is interesting to note that both values agree wellwith two previous DFT calculations, 26,27but the martensite one is 25% lower than another value calculated by Bihlmayer et al.28This discrep- ancy originates from the appearance of a DOS dip at E F[Fig. 4(a) ] when considering a monoclinic angle around 97.8/C14as in Refs. 26 and27, while this dip is much reduced for an angle of 96.8/C14as in Ref. 28. We calculate that this dip at EForiginates mostly from Ni and Ti d orbitals, which contribute equally to Pauli paramagnetism. Therefore,tuning of bmust allow us to tune the density of states at E Fand finally magnetic susceptibility. Using our DFT results in the equation of Pauli susceptibility in the nearly free electron model, vPauli¼lB2D(E F), and subtracting the Landau diamagnetic term (1/3 of vPauli), we obtain theoretical values: vMT¼2.9 mJ T/C02kg/C01in the martensite phase and vAT ¼5.5 mJ T/C02kg/C01in the austenite phase. Experimental values are 7.6 and 5.8 times larger, respectively. A similar discrepancy is reported for many Pauli paramagnetic metals such as Pd (factor of 10)29or Ni50Zr50(factor of 6).30The electronic exchange interaction in the nearly free electron model29–31accounts for such an enhancement of magnetic susceptibility. In NiTi, we find an interaction energy of 1.1 eV in B190and 0.55 eV in B2, in line with an earlier analytical pre- diction by Mitchell et al.11InFig. 4(a) ,D O Sa saf u n c t i o no fe n e r g yi s zoomed-in around EFand plotted for the different B190unstrained and strained B190lattices found by XRD in Fig. 3 .W ee x t e n d e do u rset of theoretical data with one more point at lower bangle ( b ¼97.3/C14) considering the lattice parameters calculated in Ref. 20.A sb increases, D(EF) is found to decrease because the DOS dip at EFshifts toward larger energy and gets deeper and wider. An overall decrease in D(EF) by 0.04 atom/C01eV/C01as observed in Fig. 4(a) would translate to a reduction of vMby 1.2 mJ T/C02kg/C01, in good agreement with the experimental data in Fig. 2(a) . Therefore, we conclude that NiTi is an exchange-enhanced Pauli paramagnet whose magnetic susceptibility issensitive to strain because of its dependence to B19 0lattice shear. The correlation between Asandbis revealed when comparing Figs. 2(b) and3(a). Between 0 and 10% strain, blinearly increases by 0.5/C14, while As rises linearly by 40 K. Although here only the shear is affected and no chemical modification is operated, the correlation ratio80 K/deg is quite similar to that obtained when tuning the Hf concen- t r a t i o ni nN i T i H f 2or when tuning the Ni concentration in NiTi.1In order to enlighten the origin of the influence of B190shear on the As increase with tensile strain, the difference of energy (enthalpy) betweenB19 0and B2 states, EB2-EB190, calculated for 0 K, is plotted in Fig. 4(b) as a function of the bangle. It rises when bis enhanced. In other words, the calculated B2 energy being constant with strain (since the FIG. 4. (a) Total DOS of the martensite phase as a function of energy around the Fermi level for various values of monoclinic angle b. Inset: total DOS of the mar- tensite phase (black curve) with b¼97.4/C14and austenite phase (pink curve) as a function of energy. (b) The energy difference between austenite and martensitestates, as a function of the calculated measured martensite bangle. The inset shows the scheme of free energy as a function of bfor 0 K, 300 K and As,a s reported in Ref. 6. The black open square corresponds to the region of interest in our study.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 122411 (2020); doi: 10.1063/5.0020145 117, 122411-3 Published under license by AIP Publishingmeasured lattice parameter by XRD is constant), only B190energy is found to decrease by 0.6 meV atom/C01when bincreases by 0.5/C14. Such a decrease in energy for the NiTi B190state with increasing bagrees with the many ab initio calculations at 0 K reported in the litera- ture.6,20The link between the variation of enthalpy at 0 K and the mar- tensite-to-austenite transition temperature above room temperature can be understood by considering the difference of free energy G B2- GB190¼(HB2-HB190)- T/C3(SB2-SB190) ,w h e r eGi st h ef r e ee n e r g y ,Hi s the enthalpy at 0 K, T is the temperature, and S is the entropy. Due tothe significant difference of phonon-based entropy of B2 and B19 0, GB2-GB190almost linearly decreases with increasing temperature and reaches zero for a value just below As,1,6as schemed in the inset of Fig.4(b) . Therefore, any variation of enthalpy at 0 K leads to a shift of As. We conclude that martensite stabilization can be produced in NiTi, without changing its composition, by engineering strain-inducedshear of the B19 0lattice. In summary, we present x-ray diffraction and magnetometry studies of a Ni 49,4Ti50,6alloy free standing film as a function of tensile strain amplitude. We characterize a strain-dependent shear-like trendin the B19 0lattice with the monoclinic angle bincreasing from 97.4/C14 to 98/C14, which we correlate with a reduction of magnetic susceptibility (vM) and an increase in martensite-to-austenite transition temperature (As). Using DFT calculations, we show that the dependence of NiTi exchange-enhanced Pauli paramagnetism originates from the shiftand the deepening of a density of state dip at the Fermi level, whilestrain-induced martensite stabilization is coherent with the lowering of B19 0state enthalpy with shear. The authors thank L. Calmels, P. Scheid, and J. Gorchon for fruitful discussions, as well as S. Migot and J. Ghanbaja for TEM analysis. This work was supported partly by CRYOSCAN SAS, bythe French PIA project “Lorraine Universit /C19e d’Excellence,” Reference No. ANR-15-IDEX-04-LUE, and by the SONOMAproject co-funded by FEDER-FSE Lorraine et Massif des Vosges 2014–2020, a European Union Program. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1J. Frenzel, A. Wieczorek, I. Opahle, B. Maa b, R. Drautz, and G. Eggeler, Acta Mater. 90, 213 (2015).2H. Sehitoglu, Y. Wu, L. Patriarca, G. Li, A. Ojha, S. 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5.0021031.pdf

Appl. Phys. Lett. 117, 072402 (2020); https://doi.org/10.1063/5.0021031 117, 072402 © 2020 Author(s).Engineering the magnetocaloric properties of PrVO3 epitaxial oxide thin films by strain effects Cite as: Appl. Phys. Lett. 117, 072402 (2020); https://doi.org/10.1063/5.0021031 Submitted: 06 July 2020 . Accepted: 31 July 2020 . Published Online: 18 August 2020 H. Bouhani , A. Endichi , D. Kumar , O. Copie , H. Zaari , A. David , A. Fouchet , W. Prellier , O. Mounkachi , M. Balli , A. Benyoussef , A. El Kenz , and S. Mangin COLLECTIONS This paper was selected as Featured Engineering the magnetocaloric properties of PrVO 3epitaxial oxide thin films by strain effects Cite as: Appl. Phys. Lett. 117, 072402 (2020); doi: 10.1063/5.0021031 Submitted: 6 July 2020 .Accepted: 31 July 2020 . Published Online: 18 August 2020 H.Bouhani,1,2,a) A.Endichi,1,2 D.Kumar,3O.Copie,1H.Zaari,2A.David,3A.Fouchet,3 W.Prellier,3 O.Mounkachi,2 M.Balli,4A.Benyoussef,2A.El Kenz,2and S. Mangin1 AFFILIATIONS 1Universit /C19e de Lorraine, CNRS, Institut Jean Lamour, F-54000 Nancy, France 2Laboratory of Condensed Matter and Interdisciplinary Sciences (LaMCScI), Faculty of Science, Mohammed V University, 1014 Rabat, Morocco 3Normandie University ENSICAEN UNICAEN CNRS, CRISMAT 6 Boulevard Mar /C19echal Juin, F-14050 Caen, Cedex 4, France 4AMEEC Team, LERMA, ESIE, International University of Rabat, Parc Technopolis, Rocade de Rabat-Sal /C19e, 11100, Morocco a)Author to whom correspondence should be addressed: hamza.bouhani@univ-lorraine.fr ABSTRACT Combining multiple degrees of freedom in strongly correlated materials such as transition-metal oxides would lead to fascinating magnetic and magnetocaloric features. Herein, the strain effects are used to markedly tailor the magnetic and magnetocaloric properties of PrVO 3thin films. The selection of an appropriate thickness and substrate enables us to dramatically decrease the coercive magnetic field from 2.4 T pre- viously observed in sintered PVO 3bulk to 0.05 T for compressive thin films making from the PrVO 3compound a nearly soft magnet. This is associated with a marked enhancement of the magnetic moment and the magnetocaloric effect that reaches unusual maximum values ofroughly 4.86 l Band 56.8 J/kg K with the magnetic field change of 6 T applied in the sample plane in the cryogenic temperature range (3 K), respectively. This work strongly suggests that taking advantage of different degrees of freedom and the exploitation of multiple instabilities in a nanoscale regime is a promising strategy for unveiling unexpected phases accompanied by a large magnetocaloric effect in oxides. Published under license by AIP Publishing. https://doi.org/10.1063/5.0021031 The interplay between several degrees of freedom in complex functional materials gained a lot of interest due to its potential to enhance the caloric effects in new alternative cooling technologies such as magnetic cooling.1–11The latter, which is based on the magne- tocaloric effect (MCE), is an emergent, innovating, and potentially low carbon technology. The MCE, which is an intrinsic property of certain magnetic materials, results in a change of their thermal state whensubjected to an external magnetic field. Currently, the gadolinium metal (Gd) is the magnetic material used in the vast majority of mag- netic cooling prototypes, mainly due to its magnetic phase transition taking place at 294 K, leading to excellent magnetocaloric properties close to room temperature. 9However, this metal presents multiple dis- advantages such as its easy oxidation as well as the limitation of its working temperature range close to room temperature. In addition, gadolinium cannot be used in large scale applications because of itshigh cost. These issues have motivated the scientists to search for cheaper, safe, and performant magnetocaloric materials under moder- ate magnetic fields including intermetallic and oxides. 1–11The magne- tocaloric effect in manganite-based perovskites exhibiting multiferroicbehaviors has become an interesting topic because of the potential application of these oxides in some specific applications such as the liquefaction of hydrogen and space industry.10,11This sort of com- pounds fulfills the necessary conditions for practical applications asthey unveil a strong chemical stability, high electrical resistivity, low hysteresis, and mechanical stability. 1In contrast, the magnetocaloric potential of the RVO 3vanadates (R ¼rare earth) has not yet been explored except for the bulk HoVO 3.12However, perovskite-type vanadium oxides RVO 3display a great variety of phase transitions associated with a series of charge and spin and orbital ordering phe-nomena, making them interesting candidates from a magnetocaloric point of view. 12 Today’s research activities on magnetocaloric thin films attract widespread interest due to their potential integration in miniaturized electronic devices.13,14This particularly motivated us to investigate the MCE in RVO 3thin films since their behavior strongly depends on the cooperative nature of the Jahn–Teller distortion, making them sensi- tive to strain effects including the induced biaxial strain due to the lat-tice mismatch between the substrate and the film. Such structural Appl. Phys. Lett. 117, 072402 (2020); doi: 10.1063/5.0021031 117, 072402-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apleffects tend to play an important role in tuning the film properties.15,16 In this work, we mainly focus on exploring and tuning the magnetic and magnetocaloric properties of high quality epitaxial PrVO 3(PVO) thin films by applying a compressive strain via a proper choice of substrate. The epitaxial PVO films were grown by pulsed-laser deposition (PLD) on two different cubic substrates, namely, (001)-oriented (La,Sr)(Al,Ta) O3(LSAT) and (001)-oriented SrTiO 3(STO). Their in- and out-of-plane lattice parameters are found to be 3.868 A ˚and 3.95 A ˚ with a thickness of 41.7 nm for the LSAT film, while those of the SrTiO 3film are 3.905 A ˚and 3.92 A ˚with a thickness of 100 nm. The lattice parameters have been determined by performing x-ray diffrac- tion measurements; see Refs. 21and23for details. The film deposition was carried out with a KrF excimer laser ( k¼248 nm) with a repeti- tion rate of 2 Hz and a laser fluence of /C242J / c m2focusing on stoichio- metric ceramic targets. The thin films were deposited at an optimumgrowth temperature ( T GROWTH )o f6 5 0/C14C and under an oxygen partial pressure ( PGROWTH )o f1 0/C06bar. In ZFC measurements, our sample was cooled to the desired temperature under no magnetic field, and then data were collected while heating under the magnetic field. For the FC process, the sample was cooled in the presence of an external magnetic field to the desired temperature. We used two procedures, field cooled cooling (FCC) and field cooled warming (FCW) where data are collected during the cooling and heating processes, respec- tively. Magnetization was performed by using a Quantum Design SQUID-VSM. Each hysteresis loop was measured after a 150 K excur- sion above the N /C19eel temperature and corrected by subtracting the dia- magnetic contribution arising from the substrate. To investigate the strain effect on the magnetic properties of PrVO 3thin films, the magnetization dependence on temperature was measured for PVO/SrTiO 3under an applied magnetic field of 50 Oe as shown in Fig. 1 . The hysteretic loop is also obtained at 10 K in mag- netic fields changing between –6 and 6 T (see the inset of Fig. 1 ). Magnetic measurements indicate hard-ferromagnetic behavior below 80 K, which is similar to that reported previously for bulk PVO.17–19 In fact, the S-shape of magnetization depicts a metamagnetic transi- tion, which is defined as the transition between antiferromagnetic (AF) and ferromagnetic (F) configurations of spins under the effect of magnetic fields or temperature change.20The intrinsic coercivity is /C242.8 T, while the remanence magnetization is /C2448 emu/cm3.T h e magnetization saturation is found to be /C2454 emu/cm3being equiva- lent to only 0.291 lB/f.u at 10 K. The presence of a soft component can be seen at a magnetic field of/C240.2 T indicated by the shape of the M vs H loop caused by the magnetic field-induced transition, which is absent at higher tempera- tures as observed earlier.21The high coercivity may arise from the pin- ning mechanism due to the microstructure as well as the film unit cell orientation compared to the in-plane and out-of plane magnetic field directions. On the other hand, we observed a reduction of TN(see Fig. S2) compared to bulk PVO ( TN/C25140 K), which could be explained by the oxygen vacancy-induced film lattice distortion.22XRD reveals that the pseudo cubic volume of PVO unit cell when deposited on SrTiO 3(’60.91 ˚A3) is larger than its equivalent of the bulk (’58.86 ˚A3).23As a result, the volume expansion decreases the transfer integral, which tends to reduce the neighbor exchange interactions as the magnetic interactions in this system are governed by superex- change mechanisms. It is worth mentioning that no significantmagnetic anisotropy is observed when comparing the performed mag- netic measurements under magnetic fields applied in and out of the sample plane. This can be related to the crystallographic orientation as a strong perpendicular magnetic anisotropy is revealed when the sub-strate orientation is changed from (001)- to (111)- or (110)-orientedSrTiO 3.24 Figures 2(c) and2(d) display some selected isothermal magneti- zation curves for two different orientations measured up to 6 T of aFIG. 1. Magnetization dependence of temperature for the PrVO 3film on the SrTi O3substrate performed under an in-plane applied magnetic field of 50 Oe. The inset displays the magnetic hysteresis loops measured at 10 K after subtracting thediamagnetic contribution of the substrate and the holder. FIG. 2. (a) Temperature dependence of magnetization of PrVO 3on the LSAT thin film under an in-plane applied magnetic field of 50 Oe. Inset: differentiation of thetemperature-dependent magnetization. (b) Temperature dependence of magnetiza- tion in zero field-cooling (ZFC), field cooled cooling (FCC), and field cooled warm- ing (FCW) conditions. (c) and (d) Some selected magnetic isotherms in thetemperature range of 3–13 K for PVO/LSAT collected in-plane (c) and out-of-plane(d) applied magnetic fields.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 072402 (2020); doi: 10.1063/5.0021031 117, 072402-2 Published under license by AIP PublishingPVO film deposited on a (001)-oriented LSAT substrate. As shown, the magnetization saturation is markedly enhanced when compared to PVO/SrTiO 3, reaching about ’900 emu/cm3and’785 emu/cm3at 3K f o r H ?[Fig. 2(d) ]a n dH / / [ Fig. 2(c) ], respectively. At 10 K, the corresponding magnetization saturations are about ’402 emu/cm3 and’305 emu/cm3, respectively. These values are much larger than those of PVO =SrTiO 3as can be clearly seen from the inset of Fig. 1 .I n addition, the coercive field is largely reduced to attain about 0.3 T and1.1 T for hysteretic loops performed in plane and out of plane ( Fig. 2 ), respectively. More surprisingly, the coercive field decreases dramati- cally at 3 K reaching only 0.05 T for magnetic fields applied within thefilm plane as shown in Fig. 2(c) . This markedly contrasts with the con- ventional magnets in which usually the thermal excitations lead to the reduction of coercivity. This contrast could be attributed to FM and AFM couplings and/or the spin and orbital transitions usually leading to stair-like hysteresis observed in bulk PrVO 3.18The enhancement of coercivity when heating may also be attributed to the stress-induced magnetic anisotropy due to the relaxation of the surface stress reported for various magnetic thin films.24The temperature dependence of magnetization at an in-plane applied magnetic field of 50 Oe is dis- played in Fig. 2(a) . As can be clearly observed, a sharp decrease in magnetization at low temperature and a magnetic transition from paramagnetic (PM) to an antiferromagnetic (AFM) phase transition occur at TN¼125 K. Such a transition is attributed to the beginning of a G-type spin ordering (G-SO).17The differentiation of the temperature-dependent magnetization is displayed in the inset of Fig. 2(a) where two additional magnetic transitions take place at T2’20 K and T3’80 K. These transitions were absent in bulk PVO,25but they were recently reported in strained PVO films17and in doped Pr 1/C0xCaxVO 3compounds.26Upon cooling down to 3 K, the plot of the first derivative of the magnetization temperature depen- dence exhibits a minimum at very low temperature, which could be explained by the polarization of the praseodymium magnetic moments. The newly established order is due to the fact that the anti-ferromagnetic vanadium sublattice produces an exchange field that results in a ferrimagnetic structure of Pr sublattice under cooling as already observed in other vanadates. 26This is supported by the pres- ence of a soft component at temperatures below 20 K [ Figs. 2(c) and 2(d)] and also by the fact that the magnetization saturation reaches 4.86 and 5.54 lBat 3 K when a magnetic field is applied in and out- of-plane, respectively. On the other hand, these findings inform us on the contribution of both Pr3þand V3þions to the whole magnetiza- tion since the theoretical saturated moments of free Pr3þand V3þ i o n sa r e3 . 2 2a n d2 . 1 2 lB, suggesting that all the praseodymium and vanadium moments are fully aligned parallel to the magnetic field. The ZFC, FCW, and FCC curves were measured in the 50 Oe field applied in the sample plane from 3 to 300 K for PVO =LSAT as shown in Fig. 2(b) . The bifurcation between FC and ZFC magnetiza- tions indicates an intrinsic disorder and irreversibility, which is the characteristic of a complex system. This difference reflects the impactof the anisotropy on the shapes of ZFC and FC curves below the ordering temperature since the coercivity is related to the magnetic anisotropy. The latter plays an important role in determining the mag-netization at a given field strength during both the ZFC and FC pro- cesses since it aligns the spins in a preferred direction. During the ZFC process, the spins are locked in random directions since no magnetic field is applied while cooling the thin films to the desired temperature.When a small magnetic field is applied at temperatures far below T N and as the system is anisotropic,27the magnetization decreases to reach negative values, indicating a possible competition between anti- ferromagnetic interactions, a characteristic that is observed in orthova- nadate RVO 3compounds.28A small negative trapped field in the sample space as well as the coercivity could be responsible for the neg-ative magnetization. 29During the FC process, the PVO film is cooled under the application of a magnetic field. Therefore, the spins will bealigned in a specific direction depending on the strength of the appliedmagnetic field. Consequently, M FCcontinuously increases below TN as the temperature decreases. Magnetic isotherms collected under magnetic fields going from 0 up to 6 T at different temperatures are shown in Figs. 3(a) and3(b)for the PVO =LSAT films. Except the isothermal magnetization at 3 K, which shows a typical behavior of a ferromagnetic material, all theother isotherms follow a sharp increase when the magnetic field isbelow 30 kOe, indicating a field induced first order metamagnetic tran-sition from the AFM to FM state as a result of the strong competitionbetween Pr 4f and V 3d spins. 21Such a competition often leads to a giant MCE in strongly correlated materials.30–32A similar behavior is found in the corresponding Arrott plots (M2vs H/M), confirming the first order nature of the transition according to the Banerjee33criterion as the curves show negative slopes at some points (not shown here). The large field-induced metamagnetic transition in PVO =LSAT films and the soft component in M vs H below 30 K are a clear indica-tion of a possible giant magnetic entropy change. In order to explorethe magnetocaloric effect in PVO films, the magnetic field-inducedentropy change /C0DS Mwas calculted from magnetic isotherms by using the well-known Maxwell relation (MR) given as follows:1 DSM¼ðH 0@M @T/C18/C19 HdH: We are aware that the utilization of the Maxell relation to evaluate entropy changes in materials showing large hysteretic effects could FIG. 3. Magnetic and MCE properties of PrVO 3on LSAT. (a) Magnetic isotherms in the temperature range of 3–31 K with a step of 2 K under in-plane (a) and out-of- plane (b) magnetic fields. (c) and (d) Temperature dependence of the magneticentropy change of PrV O 3/LSAT under some selected magnetic fields applied within (c) and out-of-plane (d).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 072402 (2020); doi: 10.1063/5.0021031 117, 072402-3 Published under license by AIP Publishinglead to spurious values as already demonstrated by one of the present paper authors.34However, the Maxwell relation can also be used to reasonably evaluate the MCE in terms of the entropy change even in first-order phase transition materials, provided that the remaining magnetization from previous isotherms is suppressed via a thermal loop.35On the other hand, it has been demonstrated recently that the magnetocaloric effect in multiferroics could be evaluated perfectly via the Maxwell relation.36In this way, it has been particularly found that the deduced entropy change from magnetization measurements of EuTiO 3fits perfectly with that obtained from specific heat data.36For more information about the impact of hysteretic phenomena on the MCE, we refer the interested reader to Refs. 34and35. In our case, magnetic isotherms of Figs. 3(a) and3(b)are used to calculate the entropy change exhibited by the PVO films on LSAT. However, since the entropy change is proportional to the area between two successive isotherms, DSMwas directly calculated without sub- tracting the magnetic contribution arising from the substrate as already done in the case of La 2NiMnO 6thin films.37For PVO =STO, it was difficult to calculate the MCE because of the overlap between M vs H curves as well as the very low (negligible) magnetization (see the supplementary material ) at low temperatures. The temperature dependence of the magnetic entropy change unveils larger values at very low temperature for PVO =LSAT films. /C0DSMreaches roughly a maximum value of 56.7 J kg/C01K/C01for a magnetic field changing from 0 to 6 T applied in the sample plane [Fig. 3(a) ], being about 63% of its theoretical limit given by R/C3Ln(2Jþ1). In a similar field change applied out of plane [ Fig. 3(d) ], —DSMis slightly lower and found to be about 52.7 J kg/C01K/C01.A l s o , the magnetic entropy change shows a large magnetocaloric effectunder relatively low magnetic fields, which can be easily reached via permanent magnets. In the magnetic field change of 2 T applied within and out of thin film plane,— DS Mreaches 19.5 and 16.3 J kg/C01K/C01, respectively. As shown in Figs. 3(c) and3(d), a large magnetocaloric effect can be induced below the AFM transition temperature. This sug- gests that a major part of the contribution to the MCE comes from the praseodymium 4f spins. A list of some relevant magnetocaloric materi- als working in the cryogenic temperature range are given in Table I for comparison. As shown, the exhibited DSMby the strained PVO film is significantly larger than its equivalent reported for several rare-earth metal transition oxides, making the PVO films potential candidatesfor low temperature magnetic refrigeration. To sum up, we have investigated the magnetic and magneto- caloric properties of PVO films grown by pulsed laser deposition, in view of their potential application in cryogenic magnetic cooling. The obtained results reveal that the magnetic and magnetocaloric proper- ties of PVO compounds can be easily tailored by using the thin filmapproach. Particularly, the coercive magnetic field was dramatically decreased making from the PVO compound a nearly soft magnet.Accordingly, a giant MCE is exhibited by PVO thin films grown onLSAT substrates at low temperatures, pointing out the great impactof strain effects and the competition between AFM and FMexchange interactions. These finding would open the way for theimplementation of PVO thin films in some specific applicationssuch as on-chip magnetic micro-refrigeration and sensor technol-ogy. Our result not only suggests that epitaxial PVO thin films pre-sent non-negligible potential for refrigeration at cryogenictemperatures but may also pave the way for new applications taking,for example, advantage of the possibility to tailor their magneticcoercivity. However, we are aware that the reported entropy changevalues in PVO thin films are too large when compared to the bestmagnetocaloric materials working in a similar working temperaturerange. The n ecessary was done to reasonably evaluate the MCE in terms of the entropy change by considering the impact of hystereticphenomena. In order to accurately estimate DS, the measurements of specific heat in equilibrium conditions are highly required. This pointwill be certainly addressed in the future. See the supplementary material for in-plane MH loops for the PVO/STO film as well as its differentiation of the temperature-dependent magnetization and higher temperature MH loops for thePVO/LSAT film with the magnetic field applied in- and out-of-plane. This work was supported by the PHC Toubkal 17/49 Project, the French PIA project “Lorraine Universit /C19e d’Excellence” Reference No. ANR-15-IDEX-04-LUE, and the Institut CarnotICEEL. M. Balli highly appreciates the financial support from the International University of Rabat. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1M. Balli, S. Jandl, P. Fournier, and A. Kedous-Lebouc, Appl. Phys. Rev. 4, 021305 (2017). 2S. Murakami and N. Nagaosa, Phys. Rev. Lett. 90, 197201 (2003). 3Y. Tokura, Rep. Prog. Phys. 69, 797 (2006). 4Y. Taguchi, H. Sakai, and D. Choudhury, Adv. Mater. 29, 1606144 (2017). 5D. Choudhury, T. Suzuki, D. Okuyama, D. 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E. Vickers, E. Defay, S. S. Dhesi, and N. D. Mathur, Nat. Mater. 12, 52 (2013). 14Y. Liu, C. Phillips, R. Mattana, M. Bibes, A. Barth /C19el/C19emy, and B. Dkhil, Nat. Commun. 7, 11614 (2016).TABLE I. Maximum magnetic entropy change DSMshown by PrVO 3deposited on the LSAT substrate compared to some relevant cryomagnetocaloric compounds. SC means single crystal. Materials T(K) DH (T) DS(J/kg K) References PVO/LSAT (//- ?) 3 6 T 56.7–52.7 Present work GdFeO 3(SC) 2.5 6 T 43.1 38 EuTiO 3 5.6 5 T 42.4 36 HoVO 3(SC) 15 7 T 17.2 12Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 072402 (2020); doi: 10.1063/5.0021031 117, 072402-4 Published under license by AIP Publishing15R. A. Rao, D. Lavric, T. K. Nath, C. B. Eom, L. Wu, and F. Tsui, Appl. Phys. Lett. 73, 3294 (1998). 16M. Chen, S. Bao, Y. Zhang, Y. Wang, Y. Liang, J. Wu, T. Huang, L. Wu, P. Yu, J. Zhu, Y. Lin, J. Ma, C. W. Nan, A. J. Jacobson, and C. Chen, Appl. Phys. Lett. 115, 081903 (2019). 17O. 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5.0018229.pdf

Appl. Phys. Lett. 117, 052409 (2020); https://doi.org/10.1063/5.0018229 117, 052409 © 2020 Author(s).Large topological Hall effect in an easy-cone ferromagnet (Cr0.9B0.1)Te Cite as: Appl. Phys. Lett. 117, 052409 (2020); https://doi.org/10.1063/5.0018229 Submitted: 12 June 2020 . Accepted: 24 July 2020 . Published Online: 07 August 2020 Yangkun He , Johannes Kroder , Jacob Gayles , Chenguang Fu , Yu Pan , Walter Schnelle , Claudia Felser , and Gerhard H. Fecher ARTICLES YOU MAY BE INTERESTED IN Magnetic transition behavior and large topological Hall effect in hexagonal Mn 2−xFe1+xSn (x = 0.1) magnet Applied Physics Letters 117, 052407 (2020); https://doi.org/10.1063/5.0011570 Size-dependent anomalous Hall effect in noncollinear antiferromagnetic Mn 3Sn films Applied Physics Letters 117, 052404 (2020); https://doi.org/10.1063/5.0011566 Current-induced bulk magnetization of a chiral crystal CrNb 3S6 Applied Physics Letters 117, 052408 (2020); https://doi.org/10.1063/5.0017882Large topological Hall effect in an easy-cone ferromagnet (Cr 0.9B0.1)Te Cite as: Appl. Phys. Lett. 117, 052409 (2020); doi: 10.1063/5.0018229 Submitted: 12 June 2020 .Accepted: 24 July 2020 . Published Online: 7 August 2020 Yangkun He,a) Johannes Kroder, Jacob Gayles, Chenguang Fu,YuPan, Walter Schnelle, Claudia Felser, and Gerhard H. Fecher AFFILIATIONS Max Planck Institute for Chemical Physics of Solids, D-01187 Dresden, Germany a)Author to whom correspondence should be addressed: yangkun.he@cpfs.mpg.de ABSTRACT The Berry phase understanding of electronic properties has attracted special interest in condensed matter physics, leading to phenomena such as the anomalous Hall effect and the topological Hall effect. A non-vanishing Berry phase, induced in momentum space by the bandstructure or in real space by a non-coplanar spin structure, is the origin of both effects. Here, we report a sign conversion of the anomalousHall effect and a large topological Hall effect in (Cr 0.9B0.1)Te single crystals. The spin reorientation from an easy-axis structure at high tem- perature to an easy-cone structure below 140 K leads to conversion of the Berry curvature, which influences both, anomalous and topological, Hall effects in the presence of an applied magnetic field and current. We compare and summarize the topological Hall effect in four catego-ries with different mechanisms and have a discussion into the possible artificial fake effect of the topological Hall effect in polycrystallinesamples, which provides a deep understanding of the relation between the spin structure and Hall properties. VC2020 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/10.1063/5.0018229 In condensed matter physics, Berry phases have enabled a wider understanding of many physical concepts and phenomena, such as chiral anomalies,1,2magnetic monopoles,3and the anomalous Nernst effect.4Among them, the intrinsic anomalous Hall effect (AHE) requires the absence of time reversal symmetry and the orbital degen- eracy to be lifted. The former not only is usually seen in ferromagneticsystems but also can be found in specific antiferromagnetic systems. The latter not only is due to relativistic effects such as the spin–orbit interaction but can also be induced by a non-collinear magnetic spin texture. 5,6The combination of these phenomena leads to the momentum-space Berry curvature as a linear response to an appliedelectric field. 7,8However, a real-space Berry phase originating from non-coplanar spin texture or magnetic topological excitations like sky- rmions9with non-zero scalar spin chirality can also play the role of the magnetic field and contribute to the Hall signal, referred to as the topological Hall effect (THE).10 A topological Hall effect was first observed in skyrmions in non- centrosymmetric materials, such as the B20 compounds MnSi11,12and FeGe,13,14in which it is stabilized by the Dzyaloshinskii–Moriya inter- action. In these cubic systems, the topological Hall resistivity is usually as small as 10/C03–10/C02lXcm. In centrosymmetric materials with uni- axial magnetic anisotropy, such as MnNiGa,,15the biskyrmionic phaseshows a large topological Hall effect of 0.15 lXcm. A topological Hall effect was also observed in systems with a non-coplanar anti- ferromagnetic spin structure, such as Mn 5Si3,16MnP,17and YMn 6Sn6.18Under an applied magnetic field strong enough for a metamagnetic or spin-flop transition, the (partially) antiferromag- netic coupled or canted spins align to the field direction due to theZeeman energy, a process during which a large topological Hall resistivity of up to 10 /C01lXcm has been observed. More recently, a topological Hall effect was also observed during the magnetization process along the hard axis of ferromagnets, such as in strong uniaxial Cr 5Te819and Fe 3GeTe 220with a magnetic field applied in-plane. None of these materials, however, shows a topological Hall effect with a field along the easy axis ( c- a x i s ) .T h i ss u g g e s t sa complex behavior during magnetization along the hard axis in fer-romagnets. Many Mn-based Heusler compounds crystallize in an inverse structure 21–25and have a non-collinear spin structure at low temperature. They exhibit a topological Hall effect that belongsto a mixed type of the above two cases. Recently, the THE was also reported in frustrated magnets. 26,27 The topological Hall effect is one of the characteristics of sky- rmions, and electrical transport is easy to measure. Therefore, it can beused to select materials for potential skyrmion applications. In Appl. Phys. Lett. 117, 052409 (2020); doi: 10.1063/5.0018229 117, 052409-1 VCAuthor(s) 2020Applied Physics Letters ARTICLE scitation.org/journal/apladdition, the topological Hall effect can be used to confirm some non- coplanar spin structures without the need for expensive neutron studies. Ferromagnetism exists in a large range of compositions in Cr 1/C0xTe (0<x<0.4) with different Curie temperatures Tcand saturation magne- tization Ms.28These compounds share a similar hexagonal structure, with Cr vacancies in every second Cr layer, while the Te layer is fully occupied. The vacancies induce small deviations from the hexagonal symmetry, leading to monoclinic Cr 3Te4,t r i g o n a lC r 2Te3, and trigonal or monoclinic Cr5Te8.T r i g o n a lC r 5Te8is a strong uniaxial ferromagnet with a magneto- crystalline anisotropy constant K1of 0.8 MJ m/C03.19,29However, for mate- rials with a higher Cr concentration and smaller anisotropy ( K1<0.5 MJ m/C03),30,31the magnetic structure is much more complicated. A canted ferromagnetic structure at low temperature was observed by neutron dif- fraction32and magnetization measurements,28which could lead to a pos- sible real-space Berry phase and a topological Hall effect, providing a candidate material for skyrmion bubbles. In a previous study, we reported the magnetic structure of (Cr 0.9B0.1)Te.33O w i n gt ot h ed i f fi c u l t yi ns y n - thesizing stoichiometric CrTe, the chromium vacancies are filled by boron, stabilizing the hexagonal structure as well as shifting the Fermi energy to modify the magnetism. The magnetic moment changes fromcollinear along cat high temperature to an easy-cone structure below the spin-reorientation transition temperature T SR¼140 K. The tilt angle varies with temperature. Here, we report the magneto-electronic transport properties of (Cr 0.9B0.1)Te single crystals. The spin reorientation leads to a change in the Berry curvature, which significantly influences both, anomalousand topological, Hall effects depending on the applied magnetic field and current direction. Single crystals of (Cr 0.9B0.1)Te were grown by an annealing process followed by water quenching. The details of the crystal growth, composition, crystal structure, magnetic properties, and electronic structure are published in Ref. 33. The longitudinal and Hall resistivi- ties were measured using a Quantum Design PPMS 9 by a standard four- or five-probe method. (Cr 0.9B0.1)Te crystallizes in a B81structure (prototype: NiAs, hP4, P63mmc , 194) with alternating Cr and Te layers. The lattice constants area¼4.0184(6) A ˚andc¼6.2684(7) A ˚. It is assumed that B replaces only Cr atoms of every second Cr layer. A collinear spin structure withan easy axis along cis observed at high temperature, whereas the mag- netic moments localized at the Cr atoms become gradually tilted away from the c-axis at temperatures below 140 K. T h ee l e c t r i ct r a n s p o r tp r o p e r t i e so f( C r 0.9B0.1)Te single crystals are shown in Fig. 1 with H//c[0001], I//ab plane [01-10] in Figs. 1(a)–1(c) and H//abplane [2-1-10], I//c[0001] in Figs. 1(d)–1(f) . Along both the c-axis and the abplane, the longitudinal resistance shows a metallic behavior with a kink at Tc¼336 K, although the value in the abplane is almost twice as large as that along the c-axis. The small residual-resistance ratio (RRR) is about 1.7 in-plane and 2.1 along the c-axis, indicating a large number of dislocations (vacancies or B atoms) inside the crystals. For magnetization along the c-axis, the magnetoresistance is almost zero, and as the field increases further, its value gradually decreases during heating to /C01.5% at 300 K under 3 T due to spin-disorder scattering. However, the magnetoresistance is positive (1.5%) in the abplane at 2 K during magnetization. With this additional effect, the negative magnetoresistance region at 3 T rises toabove 200 K.(Cr 0.9B0.1)Te also exhibits a large, anisotropic anomalous Hall effect that depends strongly on temperature. When the applied fielddirection is along the c-axis, the anomalous Hall resistivity q AHE is positive at high temperature with a collinear spin structure. However, it decreases during cooling and then changes its sign at TSR, finally reaching /C02.3lXcm at 2 K, as shown in Fig. 1(c) . When the field is in-plane, as shown in Fig. 1(f) , the anomalous Hall effect is always neg- ative and changes during cooling from /C03.5lXcm at 300 K to about /C00.7lXcm at 2 K. Skew scattering is the dominant mechanism of the anomalous Hall effect for both I//abandI//c,19a ss h o w ni nt h el i n e a r fitting of qAHEvs longitudinal resistivity in the supplementary mate- rial.qAHEoffsets the trend when the temperature approaches Tc.T h e skew-scattering mechanism confirms (Cr 0.9B0.1)Te as a bad metal with a large number of defects. The total Hall effect can be regarded as the sum of the ordinary Hall effect due to the Lorenz force, the anomalous, and the topologicalHall effect using the following formula for the Hall resistivity: q H¼R0BþRsl0MþqTHE; (1) where R0and Rsare the ordinary and anomalous Hall coefficients, respectively. The fitted ordinary Hall resistivity is negligible, indicatinga high charge carrier density of more than 10 22cm/C03; therefore, it is not shown here. A low mobility of <1c m2V/C01s/C01further demon- strates that it is a bad metal, which is also confirmed by the low resid- ual resistivity ratio (RRR) and the low thermal conductivity of 3.8 WK /C01m/C01at 300 K. When the field is applied along the c-axis, Rs changes its sign during cooling, whereas it remains negative with the field along the a-axis, as shown in the supplementary material . Moreover, the Hall signal during magnetization causes an addi- tional effect, namely, the topological Hall effect, when H//[2-1-10] and I//[0001] with the easy-cone structure at low temperature, as shown in Fig. 2 . At 250 K, with the collinear spin structure, there is no topologi- cal Hall effect. However, it appears below TSRof 140 K. At 2 K, the value is as large as 0.21 lXcm. A similar result is observed when both the current and the field are in two in-plane perpendicular directions.A large topological Hall effect appears near saturation, when the easy-cone structure has already been destroyed by the applied magnetic field. Note that the in-plane magnetization curve below 140 K is not linear before saturation, with a kink at around 0.2 T ( supplementary material ), which is also the field at which the topological Hall effect starts to show a large value. This indicates a non-coplanar spin struc- ture before saturation, with a solid angle Xshowing spin chirality as sketched in Fig. 3 . However, when the field is parallel to the c-axis, the topological Hall effect is too small to be distinguished from the noise. This can beexplained by the lack of a non-coplanar intermediate phase duringmagnetization along the c-axis, as indicated by the absence of a kink in the magnetization curve (see the supplementary material ). The domain wall motion and spin reorientation occur simultaneously dur-ing magnetization instead. This collinear spin structure does not giverise to an additional contribution to the Hall signal from the Berry cur- vature with X¼0. Similar phenomena have been observed in Cr 5Te819and Fe 3GeTe 2,20which also show a topological Hall effect w h e nt h efi e l di sa l o n gt h eh a r da x i sr a t h e rt h a nt h ee a s y c-axis due to the non-coplanar spin structure. Figure 4 shows the phase diagram according to the above data with the field in-plane. Here, Tcand TSRare collected from the M(T)Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 052409 (2020); doi: 10.1063/5.0018229 117, 052409-2 VCAuthor(s) 2020curves, whereas the saturation field l0Hsand the maximum field of the easy cone are collected from the M(H) curves. It is clearly shown that the topological Hall effect appears at low temperature with a non-coplanar spin structure near saturation, when the spin is alreadyshifted away from the initial cone. We compare materials exhibiting a topological Hall effect in Table I . Generally, a large topological Hall effect requires a largemagnetic field (see the supplementary material ). The first category consists of the skyrmion materials. However, in general, the topologi-cal Hall effect induced by skyrmions is small. The cubic B20 com-pounds, such as MnSi 11,12or FeGe,13,14only exhibit a topological Hall resistivity of 10/C03–10/C02lXcm. In Mn 1.4PtSn above the spin- reorientation temperature, no topological Hall effect was found,23 although skyrmions still existed.34The second category covers the FIG. 1. Transport properties of (Cr 0.9B0.1)Te. (a) and (d) Longitudinal resistance. (b) and (e) Magnetoresistivity at 2–300 K. (c) and (f) Hall signal at 2–300 K. The demagnetizing factors here are approximately 0.40 and 0.65, respectively. The crystal structure is shown in the inset in (a). FIG. 2. Topological Hall effect of (Cr 0.9B0.1)Te with H//[2-1-10] and I//[0001]. The demagnetizing effect has already been corrected here to remove the possibility of an artificial effect. (a)–(f) correspond to different temperatures from 2 K to 250 K, respectively.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 052409 (2020); doi: 10.1063/5.0018229 117, 052409-3 VCAuthor(s) 2020possibility to realize metamagnetic or spin-flop transitions under applied magnetic fields in antiferromagnetic materials with a non-coplanar spin structure, as in the cases of Mn 5Si3,16MnP,17and YMn 6Sn6.18The third category covers magnetization from the hard axis, including Cr 5Te8,19 Fe3GeTe 2,20and (Cr 0.9B0.1)Te. Many Mn-based Heusler com- pounds21–25,35have combined materials from all three categories. The topological Hall effects for the second and third categories are larger, with a large magnetic field, depending on the exchange coupling strength or magnetocrystalline anisotropy. (Cr 0.9B0.1)Te belongs to the third group and is one of the materi- als that can achieve a large anomalous Hall effect with a mild field owing to its small magnetocrystalline anisotropy. Note that thet o p o l o g i c a lH a l lr e s i s t i v i t yo f0 . 2 1 lXcm is already comparable to the anomalous Hall resistivity of 0.75 lXcm. Larger values of the topolog- ical Hall effect are also observed in thin films,36which disappear when the thickness increases, indicating that the topological Hall effect issensitive to the lattice constant, which is easily affected by strain. Thesmall applied field is due to vanishing small single-ion anisotropy ofCr 3þions (3 d3) with a negligible orbital moment.33Note that B signifi- cantly increases the Cr moment from 2.7 lBin the binary compound28 to 3.1 lBhere and decreases the magnetocrystalline anisotropy K1 from 500 kJ m/C0331to/C0100 kJ m/C03at 2 K. This reduced field is impor- tant for potential applications. The findings also allow us to interpret transport data from previ- ously reported polycrystalline materials. We propose a topological-like anomalous Hall effect (topological AHE) that originates in polycrystals from two magnetic sublattices with different anisotropy constants, thevalues of which are approximately the same magnitude, but of oppo-site sign. We simulated a textured structure (80% cþ20%a)a sa n example using the data at 300 K in Fig. 1 ,a ss h o w ni nt h es u p p l e m e n - tary information. Note that there is no topological Hall effect at this temperature . The magnetization along the c-axis saturates fast, domi- nating the initial magnetization curve and the anomalous Hall effect(positive). However, after saturation in the “ c”-texture in a higher field, an additional anomalous Hall effect only emerges from the “ a”-tex- ture, giving a negative contribution. As a result, a bump, namely, atopological-like anomalous Hall effect, appears near saturation. Thetopological-like anomalous Hall resistivity is 1.3 lXcm after fitting and, thus, much larger than the real topological Hall resistivity observed at low temperature. Dijkstra et al. 28also reported Hall measurements on polycrystalline Cr 0.9Te and Cr 0.8Te. Both samples, especially Cr 0.8Te, showed similar behavior of two kinks before satura- tion, which can now be explained by topological and topological-likeanomalous Hall effects. The first kink might come from the competi- tion between anomalous Hall effects with different signs, whereas the second kink should come from the topological Hall effect of the non-coplanar spin structure. Interestingly, this topological-like anomalousHall effect has also been realized in films in which two materials withopposite signs of the anomalous Hall effect were selected. 37Our study points out that this can be realized in one material as well. Owing to the topological-like anomalous Hall effect, great care should be takenwith the transport properties of polycrystalline materials with ananisotropic crystal structure to exclude the possibility of an ‘artificialtopological Hall effect’. For this reason, some polycrystalline materials without texture, especially those with high magnetocrystalline anisot- ropy, are not included in Table I . In conclusion, the spin reorientation transition at 140 K has a sig- nificant effect on the transport properties of (Cr 0.9B0.1)Te. The non- FIG. 3. Schematic magnetic and crystal structures of (Cr 0.9B0.1)Te with (a) H¼0, (b)H//abplane, and (c) H//c. The moment is tilted from cand collinear in the ground state. The in-plane field leads to a non-coplanar spin structure with different tilted angles before saturation. The moment is collinear with the c-axis magnetic field. FIG. 4. Phase diagram of (Cr 0.9B0.1)Te with the in-plane field. TABLE I. Comparison of different categories of materials showing a topological Hall effect. Materials MnSi, FeGe Mn 5Si3, MnP, YMn 6Sn6 Cr5Te8,F e 3GeTe 2,C r 0.9B0.1Te Mn 1.4PtSn, Mn 2RhSn Ground state Helical (AFM) Non-collinear AFM FM Non-collinear FM Process Skyrmion Metamagnetic transition Hard-axis magnetization MixtureField direction All All Hard axis AllMax THE ( lXcm) 10 -3–10/C0210-1–10010-2–10110-2–10/C01 Magnetic field Small Large Anisotropy dependent Anisotropy dependentApplied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 052409 (2020); doi: 10.1063/5.0018229 117, 052409-4 VCAuthor(s) 2020coplanar spin structure at low temperature leads to a non-vanishing Berry phase, further causing a highly anisotropic anomalous Hall effectand a topological Hall effect that strongly depends on the field direc-tion. Consequently, a sign change of the anomalous Hall effect and alarge topological Hall resistivity of 0.21 lXcm are observed. Our study provides a deep understanding of the non-coplanar magnetic structureand topological Hall effect. See the supplementary material for the mechanism of the anoma- lous Hall effect, topological-like anomalous Hall effect, magnetizationcurves, and comparison of the topological Hall effect with othermaterials. This work was financially supported by the European Research Council Advanced Grant (No. 742068) “TOPMAT,” theEuropean Union’s Horizon 2020 Research and InnovationProgramme (No. 824123) “SKYTOP,” the European Union’sHorizon 2020 Research and Innovation Programme (No. 766566)“ASPIN,” the Deutsche Forschungsgemeinschaft (Project-ID No.258499086) “SFB 1143,” the Deutsche Forschungsgemeinschaft(Project-ID No. FE 633/30-1) “SPP Skyrmions,” and the DFGthrough the W €urzburg–Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter ct.qmat (EXC 2147,Project-ID No. 39085490). DATA AVAILABILITY Data are available on request from the authors. REFERENCES 1E. Liu, Y. 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5.0010822.pdf

APL Mater. 8, 071105 (2020); https://doi.org/10.1063/5.0010822 8, 071105 © 2020 Author(s).Rational design of 2D organic magnets with giant magnetic anisotropy based on two- coordinate 5d transition metals Cite as: APL Mater. 8, 071105 (2020); https://doi.org/10.1063/5.0010822 Submitted: 16 April 2020 . Accepted: 17 June 2020 . Published Online: 07 July 2020 Jianpei Xing , Peng Wang , Zhou Jiang , Xue Jiang , Yi Wang , and Jijun Zhao ARTICLES YOU MAY BE INTERESTED IN First-principles investigation of a new 2D magnetic crystal: Ferromagnetic ordering and intrinsic half-metallicity The Journal of Chemical Physics 152, 244704 (2020); https://doi.org/10.1063/5.0013393 Observation of near-infrared sub-Poissonian photon emission in hexagonal boron nitride at room temperature APL Photonics 5, 076103 (2020); https://doi.org/10.1063/5.0008242 Applied Physics Reviews publishes original research Applied Physics Reviews 6, 030401 (2019); https://doi.org/10.1063/1.5117150APL Materials ARTICLE scitation.org/journal/apm Rational design of 2D organic magnets with giant magnetic anisotropy based on two-coordinate 5d transition metals Cite as: APL Mater. 8, 071105 (2020); doi: 10.1063/5.0010822 Submitted: 16 April 2020 •Accepted: 17 June 2020 • Published Online: 7 July 2020 Jianpei Xing,1 Peng Wang,2 Zhou Jiang,1Xue Jiang,1,a) Yi Wang,1,a)and Jijun Zhao1 AFFILIATIONS 1Key Laboratory of Materials Modification by Laser, Ion and Electron Beams (Dalian University of Technology), Ministry of Education, Dalian 116024, China 2School of Physical Science and Technology, Southwest University, Chongqing 400715, China a)Authors to whom correspondence should be addressed: jiangx@dlut.edu.cn and yiwang@dlut.edu.cn ABSTRACT As a new class of single-molecule magnets, two-coordinate complexes of open-shell transition metals are comparatively rare and have attracted interest due to their high degree of coordinative unsaturation. However, the dynamic distortion associated with the low coordination number of the metal center hinders the applications of high-density information storage, quantum computing, and spintronics. Here, we propose a series of stable 2D metal–organic frameworks constructed by ideal (1, 3, 5)-benzenetricarbonitrile (TCB) molecules and 5d transition met- als (Hf, Ta, W, Re, Os, and Ir) with a highly symmetrical ligand field and rigid πconjugated framework. Among them, TCB-Re exhibits intrinsic ferromagnetic ordering with a considerably large magnetic anisotropic energy (MAE) of 19 meV/atom and high Curie tempera- ture ( TC) of 613 K. Under biaxial strain, diverse magnetic states (such as ferromagnetic, paramagnetic, and antiferromagnetic states) can be achieved in TCB-Re by the complicated competition between the in-plane d–px/y–dand out-of-plane d–pz–dsuperexchange interac- tions. At a small compressive strain of 0.5%, the MAE for perpendicular magnetization increases substantially to 120 meV/atom; meanwhile, the magnetization and TCabove room temperature are well retained. Our results not only extend two-coordinate transition metal com- plexes to continuous 2D organic magnets but also demonstrate an effective method of strain engineering for manipulating the spin state and MAE. ©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/10.1063/5.0010822 .,s I. INTRODUCTION In the past century, magnetic materials have been widely devel- oped for information storage, mainly motivated by the commer- cial desire to increase the magnetic storage density.1,2The dimen- sion of the magnetic materials has to decrease from traditional three-dimensional (3D) to 2D, 1D, or even a single atom.3–5To this end, the magnetic anisotropic energy (MAE) of these mag- nets, namely, the energy difference between different magnetized orientations, must be large enough. This is the most critical require- ment for magnetic storage, since a large MAE can overcome the ambient thermal fluctuations, and thus enables long-term stable magnetic storage over 10 years.6The theoretical required value of MAE is about 26 meV (kT at room temperature).7With regard toinformation storage applications, a high Curie temperature ( TC) of the low-dimensional ferromagnets is also necessary. More- over, reversible manipulation of the magnetic behavior of low- dimensional ferromagnets is crucial in spin-based information pro- cessing technologies. Numerous studies have demonstrated that the materials hav- ing low coordination, low dimensionality, 4 d- or 5 d-transition metal (TM) with strong spin–orbit coupling (SOC), and suitable ligand field8–12are likely to induce large MAEs. The complexes with three- coordinate,13,14tetra-coordinate,15,16and hexa-coordinate17transi- tion metals have been investigated widely in the past 10 years. Transition metals can be embedded in the framework of planar organic molecules. Thanks to the robust coordinate environment, dynamic distortion can be suppressed in many complexes, such as APL Mater. 8, 071105 (2020); doi: 10.1063/5.0010822 8, 071105-1 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm porphyrin, phthalocyanine, and so on. However, the orbital angular momentum will be quenched owing to the high coordinate envi- ronment. Therefore, the ideal complexes with large spin magnetism moment and strong ferromagnetic ordering generally contain two- coordinate, open-shell transition metals.18–20The linear coordinate plays an important role in obtaining high spin moments due to the high degree of coordinative unsaturation.21–23For example, a linear cobalt(II) complex with maximal orbital angular momen- tum (L = 3) and a large energy barrier about 450 cm−1has been reported by Bunting et al. in the year 2018.24In linear dyspro- sium metallocene single-molecule magnets, the highest magnetiza- tion reversal energy barrier up to 1541 cm–1is observed.25Besides, the effective magnetic moment of Fe(N(SiMe3)Dipp)2 can reach up to 5.89 μBat 300 K.22Even so, the low coordinate transition metal (TM) complexes especially for 2D heavy TM materials are rarely reported because of the significant formation of aggregates. Thus, designing 2D two-coordinate transition metal complexes with high stability is a meaningful and urgent need for spintronic device applications. In addition, reducing the dimensionality is of course another effective way to gain giant intrinsic MAEs. For instance, Ruiz-Díaz’s group found that the MAE reaches up to 5.2 meV/atom for the sys- tem of the Fe single layer capped with the Pt layer, much larger than that of conventional bulk Fe crystals with only 10−3meV/atom.26 Another method for MAE enhancement is to properly introduce ele- ments with strong SOC, especially 4 d- or 5 d-transition metals. van Vleck27has proven that the magnetic anisotropy mainly comes from the SOC effect. Meanwhile, low coordination numbers of the metal atoms could preserve the first-order orbital angular momentum and also significantly improve the magnetic anisotropy.28For example, Odkhuu et al. found a large perpendicular MAE of 10 meV per Ir atom in the Ir-capped Fe (001) system.9A series of two-coordinate transition metal complexes as outstanding single molecule mag- nets have been widely investigated.28For example, two-coordinate cobalt imido complexes have a highest record relaxation barrier (413 cm−1) in the transition metal based single-molecule mag- nets.29The overlaying effect of the adjacent crystal field environ- ment might be another way to tailor the MAE. Combining with the structural stability, a series of 2D environments of crystal fields have been designed, such as 2D metal–organic frameworks (MOFs), defective graphene, ultrathin Fe/XTiO 3(X = Sr, Ba) films, and the metal-free graphitic g-C 4N3sheet.30–33For example, the MAE reaches up to 100 meV/atom in individual Os and Ru atoms at a S vacancy in MoS 2.33In the systems of Ru and Os adatoms on the MgO (001) surface, an enormously large MAE as high as 110 meV and 208 meV as a consequence of combined effects of the ligand field, orbital multiplet, and large SOC constants has been found, respectively.8 Besides searching for magnetic materials with an intrinsic large MAE, several approaches have been exploited to enhance the MAE of low-dimensional magnetic structures, such as via functionaliza- tion, electric field, charge injection, and strain.7,26,33–38Among them, strain engineering has been shown to be a precise and flexible way to tune the MAE of 2D materials by taking advantage of substrate lattice mismatching.38Webster and Yan39revealed that the MAE of CrI 3can be increased by 47% under a compressive strain of ε= 5%. In addition, it was also shown that CrCl 3, CrBr 3, and CrI 3 exhibit ferromagnetic ordering at the ground state and will undergophase transition to the antiferromagnetic state by applying a com- pressive strain. Zhuang et al.40explored the strain effect on the MAE of 3D Stoner ferromagnets Fe 3GeTe 2and demonstrated that tensile strains of 2% would enhance their MAEs by 50%. When a com- pressive strain is applied for Fe/MoS 2, the MAE gradually increases almost linearly from 0.02 meV to 2.0 meV. Meanwhile, the inversion of the easy-axis MAE is found by applying tensile strain due to the decrease in dxyorbital contribution in the Fe layer.36Jena’s group found that, by applying an external electric field or a biaxial tensile strain, a MAE value as high as 140 meV can be achieved on phthalo- cyanine sheets decorated by 5 dtransition metal atoms such as Os and Ir.10According to Zheng’s study, compared with the unstrained system, the MAE of the 1T-FeCl 2monolayer has been enhanced by about 36.77% under a compressive strain of 4%.41 In this paper, we propose that the above-mentioned advanta- geous characteristics for data storage can be manifested in a class of 2D MOFs, which is composed by (1, 3, 5)-benzenetricarbonitrile and 5 dtransition metal (TCB-TM) (TM = Hf, Ta, W, Re, Os, and Ir), namely, C 18H6N6TM 3. C 18H6N6TM 3has only one-atom layer thickness with near minimal and regular ligation. The TM atoms in TCB-TM locate in the linear coordination environment. Thus, the requirements of the two-coordinate principle and stability are simultaneously fulfilled in the designed materials. The crystal bond- ing, stability, and electronic and magnetic properties of these new 2D MOFs have been investigated from density functional theory (DFT) calculations. We find that the TCB-Re has the largest value of MAE and TC. Therefore, we focus on this novel system, TCB-Re. It is found that the MAE of TCB-Re can be significantly enlarged by more than four times only under a tiny tensile/compressive strain of 0.5%. Moreover, the strain dependence of other mag- netic properties (magnetization, magnetic ground phase, and TC) of TCB-Re has been discussed. Interestingly, we found that the control- lable transition between ferromagnetic (FM), paramagnetic (PM), and antiferromagnetic (AFM) states can be realized in TCB-Re by applying relatively small strains (1%). Our present results not only indicate that strain is an effective way to improve the MAE of 2D MOFs but also rise a novel opportunity for magnetic state controlling. II. COMPUTATIONAL METHODS Our first-principles calculations on 2D periodic structures were performed based on density functional theory (DFT), as imple- mented in the Vienna Ab initio Simulation Package (VASP).42 The ion–electron interaction was described by the projector aug- mented wave (PAW) approach.43,44Generalized gradient approxi- mation (GGA) in the scheme of the Perdew–Burke–Ernzerhof (PBE) functional was used to describe the exchange–correlation interac- tion.45,46The cutoff energy of the plane-wave basis was set to 500 eV. The Brillouin-zone was sampled by a Γ-centered Monkhorst–Pack k-point mesh of 5 ×5×1 for the structural relaxation and 9 ×9×1 for electronic and magnetic calculations. A slab model was built with a vacuum separation of 15 Å to avoid the interac- tions between two periodic images. Lattice parameters and atomic position were fully relaxed until the force on each atom was less than 0.02 eV/Å and the total energy change was less than 10−5eV. TCof ferromagnetic structures were estimated by the mean-field APL Mater. 8, 071105 (2020); doi: 10.1063/5.0010822 8, 071105-2 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm FIG. 1 . (a) The top view and (b) side view of the TCB-TM unit-cell (C 18H6N6TM3). (c) The charge density difference of TCB-Re plotted with the isovalue of 0.012 a.u. (d) The partial charge den- sity of TCB-Re with the value of 0.012 a.u. in the energy range of E F−4.0 eV to E F. Here, green, gray, blue, and pink spheres represent Re, C, N, and H atoms, respectively. approximation (MFA) based on an Ising model. It was known that a strong correlation of the localized dorbitals of transition metals may not be properly described by conventional DFT calculations.47 To consider this strong correlation effect on dorbitals of Re, test- ing calculations were performed by including a Hubbard U term, namely, the GGA + U method introduced by Dudarev et al.48Our results show that the large MAE of TCB-Re is still retained and even increased when the effective on-site Coulomb interaction and exchange interaction are implemented by an effective U. The mag- netic moment and MAE under different U effare shown in Table S2. When U effis equal to 1, the MAE increases from 19.05 meV/atom to 77 meV/atom. For our 5 dtransition metal compounds, the effects of spin– orbit coupling are described by first principles calculations with second-order perturbation Hamilton, which is quite different with the multiconfiguration ab initio method. The former one demon- strated their good performance in 5 dtransition metal com- pounds,49–51and the latter one is also suitable for more heavier lan- thanide compounds.52Based on this method, the orbital moment and MAE were calculated by spin–orbit coupling together with the magnetic non-collinearity.53Moreover, the projector augmented wave method is successful in calculating the non-collinear mag- netic structures, which allows the atomic and magnetic structures to relax simultaneously and self-consistently. In the self-consistent calculations, the MAE is defined as MAE =Etot[∥]−Etot[/⊙◇⊞]. (1) In this equation, E tot[∥] refers to the total energy of the state when the magnetization direction is parallel to the XY plane (in-plane). In order to verify the minimum in-plane energy, according the geomet- ric symmetry, two in-plane magnetization directions were consid- ered, i.e., along and perpendicular to the N–TM–N axis [Fig. 1(a)]. For device applications, a positive MAE is more valuable because materials have a preference for out-of-plane magnetization, which enables a magnetic domain to remain at bistable states.32To clarify the origin of the large MAEs, the torque method proposed by Wang and co-workers was adopted.54,55In this regime, the MAE can be defined in terms of the angular momentum operators Lx(orLy) andLzas MAE ≈ξ2∑o,u,α,β(2δαβ−1)[⟨o,α∣Lz∣u,β⟩2 Eu,α−Eo,β−⟨o,α∣Lx∣u,β⟩2 Eu,α−Eo,β]. (2) Here, o and u represent the occupied and unoccupied states and α andβrepresent the two spin channels. When αand βare the same spin channel, δαβis equal to 1; otherwise, it is equal to 0. We can analyze the origin of MAE by examining the contributions of differ- ent spin channels, including majority spin states (uu), minority spin states (dd), and cross spin states (ud + du). In this system, minority spin states dominate the MAE according to our torque calculations, soδαβin Eq. (2) should be equal to 1 and the MAE can be expressed as MAE ≈ΔEdd=(ξ)2∑o−u−∣⟨o−∣Lz∣u−⟩∣2−∣⟨o−∣Lx∣u−⟩∣2 εu−−εo−. (3) From Eq. (3), we know that ΔEddis not only determined by the orbital character of the occupied states but also depends on the cou- pling with the empty states and the splitting between them through the energy denominator. Obviously, the MAE is sensitive to the change in the minority spin states. The TCvalues of some fer- romagnetic structures are estimated by MFA based on an Ising model. III. RESULT AND DISCUSSION Inspired by the recently proposed stable two-coordinate, open-shell transition metal complexes,28here, we constructed a series of 2D two-coordinate MOFs with a hexagonal symmetry of the P6/mmm space group. The relaxed structure of TCB-TM (C18H6N6TM 3) is illustrated in Figs. 1(a) and 1(b). In particular, each TM atom connects with two TCB molecules forming a lin- ear ligand. Taking advantage of both the strong SOC effect and partially occupied dorbitals of 5 dtransition elements, the physi- cal properties of Hf, Ta, W, Re, Os, and Ir combined with TCB molecules are calculated and their parameters are listed in Table I. For TCB-TM monolayers, the lattice parameters and TM–N bond lengths have the same tendency, that is, they decrease gradually with APL Mater. 8, 071105 (2020); doi: 10.1063/5.0010822 8, 071105-3 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm TABLE I . The equilibrium lattice parameters (a 0), TM–N bond length ( dTM–N ), bind- ing energy ( EB), bandgap from DFT–PBE calculations, magnetic phase, magnetic moment of the unit cell ( Mtot) and TM atom ( MTM), Curie temperature ( TC) by MFA based on the Ising model, and magnetic anisotropy energy (MAE) of each metal atom of TCB-TM. TM Hf Ta W Re Os Ir a0(Å) 21.16 20.98 20.77 20.65 20.51 20.43 dTM–N (Å) 2.10 2.05 1.99 1.95 1.92 1.90 EB(eV/atom) 5.17 5.12 5.78 5.74 5.30 5.87 Bandgap (eV) Metal Metal 0.156 0.063 0.49 0.57 Magnetic state AFM FM FM FM AFM AFM Mtot(μB) 6.0 12.8 12.0 9.0 6.0 3.0 MTM(μB) 0.60 2.77 3.19 2.58 2.0 1.0 TC(K) . . . 153 15 613 . . . . . . MAE (meV/atom) −9.52−8.36 15.77 19.05 1.00 −25.48 the increase in the atomic number. The length of the TM–N bond is in the range of 1.88 Å–2.10 Å. Owing to the strong bonding between C- porbitals, N- porbitals, and TM- dorbitals, the average binding energy of those TCB-TM systems is 5.49 eV/atom. Such a value is comparable to some common 2D monolayer materials, such as 5.14 eV/atom for MoS 2and 4.52 eV/atom for graphene. The electronic bandgap, magnetic moment, and MAE of these TCB-TM frameworks are also listed in Table I. In these systems, TCB-Ta is metallic, while TCB-W, TCB-Re, TCB-Os, and TCB-Ir are semiconductors with a narrow bandgap less than 1 eV. Inter- estingly, TCB-Hf shows a half-metallic behavior, which is metal- lic in the spin-up channel and semiconducting with a bandgap of 0.97 eV in the spin-down channel, respectively, as shown in Fig. S1. Each metal atom in TCB-Hf, TCB-Ta, TCB-W, TCB-Re, TCB-Os, and TCB-Ir 2D MOFs possesses an on-site magnetic moment of 0.6μB, 2.77 μB, 3.19 μB, 2.58 μB, 2μB, and 1 μB, respectively. The MAE of each system is determined by considering the SOC effect through non-collinear calculations. It should be noted that TCB-W and TCB-Re have larger MAEs of 15.77 meV/atom and 19.04 meV/atom, respectively, which are more suitable for magnetic data storage. To further examine the magnetic behavior, we optimized a 2×1 supercell and considered three possible magnetic configura- tions (Fig. S2): one ferromagnetic (FM) configuration and two anti- ferromagnetic (AFM) ones. The results show that the FM ground state is more stable in TCB-Ta, TCB-W, and TCB-Re. With MFA based on the Ising model, the ferromagnetic state in TCB-Re can be preserved up to 613 K (i.e., TC= 613 K). Among the six systems, 2D TCB-Re shows the largest MAE, highest TC, and large magnetic moment. Moreover, its thermal stability was further assessed by ab initio molecular dynamics (AIMD) simulation at 300 K (Fig. S3). Compared to the initial equilibrium structure, the final structure is well maintained and shows a small buckling of 0.5 Å. The vari- ance of total energy and a snapshot structure prove that TCB-Re has superior thermal stability. Therefore, it is necessary to uncover the origins of those good properties and explore new approaches to further improve them.We first investigated the electronic behaviors of TCB-Re. The charge density difference and the partial charge density of the TCB- Re framework are calculated and shown in Figs. 1(c) and 1(d), respectively. One can clearly see the localized in-plane σbonds between C atoms and the delocalized out-plane πbonds. Specifi- cally, the p(pxandpy) electrons of the C6 ring form σbonds and hybrid with the p(py) electrons of the bridge C atom. The pyorbital of the bridge C atom forms a σbond with the surrounding pyorbital of the N atom. Meanwhile, the d(dxyanddyz) states of Re hybrid with the neighboring N- p(pxand pz) states. Bader charge analy- sis indicates that each Re atom gives nearly 0.43 electrons to the neighboring C and N atoms. In a word, the coexistence of σbonds and πbonds results in a high structural stability of such 2D organic frameworks. The spin-polarized band structures in Fig. S4(a) show that TCB-Re is a narrow-bandgap semiconductor with a spin-up gap of 1.34 eV and a spin-down gap of 0.082 eV, respectively. The elec- tronic states near the Fermi level arise mainly from the spin-down orbitals, i.e., dxyanddyzorbitals of Re, pxandpzorbitals of bridge C, and the pzorbital of C6 ring, as shown in Fig. 2. Comparison of different magnetic couplings further shows that the ground state of 2D TCB-Re is FM with an exchange energy of 544 meV. As listed in Table I, a magnetic moment of 2.58 μBon each Re site is obtained, while the contribution from organic molecules is nearly negligible, totally 0.07 μBon C and N atoms. This is also revealed by the spin charge density in Fig. S4(b). The origin of the magnetic moment of 2.58 μBcan be understood from the following. Due to the twofold symmetry around Re atoms, the dorbitals are split into the higher- energy dxzorbitals, dz2/dx2-y2orbitals, and lower-energy dxyanddyz orbitals. As is known, the valence electronic configuration of Re is 5d56s2. From the Re-5 dorbital projected density of states (PDOS) (Fig. 2), one can clearly see that the dxyorbital is fully occupied in both spin-up and spin-down states, while the magnetic moments from two higher-energy dorbitals of dz2and dxz(about 1.6 μB) are nearly unquenched due to the absence of ligands that interact directly with these orbitals. Moreover, dyz,dx2-y2, and sorbitals are FIG. 2 . Projected density of states (PDOS) of the C6 ring, bridge C, bridge N, and Re. In the PDOS of the C6 ring, the pxorbitals are degenerate with pyorbitals. Here, the Re atom refers specifically to the atom on the OA axis, as marked in Fig. 1(a), and the C ring, bridge C, and bridge N are the atoms connected with that Re atom. APL Mater. 8, 071105 (2020); doi: 10.1063/5.0010822 8, 071105-4 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm half filled in the spin-down channel with occupancy numbers of 0.85, 0.27, and 0.34, contributing to the total on-site spin moment of ∼2.58 μBper Re atom. Next, the TCof TCB-Re was evaluated by MFA calculation based on the Ising model. From the estimated exchange parameters (J1=−4.9 meV and J2= 9.8 meV), a high TC(613 K) was predicted, and the details of calculations are given in Scheme S1 of the supple- mentary material. As displayed in Fig. 1, the distance between the nearest Re ions is about 10.32 Å, which is too long to induce any direct magnetic exchange interaction. The magnetic exchange mech- anism could be understood by active dyz/dx2-y2orbital coupling between two neighboring Re ions through superexchange interac- tions bridged by the πconjugated framework. From the partial den- sity of state (Fig. 2), one can see apparent hybridization between the dyzorbitals of the Re atom and the pzorbital of the C atom around the Fermi level. In addition, there is also an obvious overlap between thedxyorbitals of the Re atom and the pxorbital of C atoms. Thus, formation of cyanido-bridged Re–N–C leads to strong d–px–dand d–pz–dexchange interactions between the nearest neighboring Re atoms, which can also be proved by the partial charge density of TCB-Re in the energy range from EF−4.0 eV to EF, as shown in Fig. 1(d). A similar image is also applicable to describe the other Re atoms in the 2D TCB-Re systems. The physical origin of MAE was discussed using the torque method.56,57In Fig. 3, we plot the total and decomposed MAEs of 2D TCB-Re with respect to the Fermi level of the system. The coupling between occupied and unoccupied spin-down dstates of Re atoms (ddterm) contributes to the major part of the positive MAE com- pared with other couplings near the Fermi level. In contrast, the cou- pling between majority spin channels ( uuterm) and crossover spin channels ( ud+duterm) generates small MAE values. By decompos- ing the contribution from each pair of states involved in the SOC Hamiltonian, we found that the strong SOC interactions between occupied dxyand unoccupied dx2-y2of Re atoms in the minority spin channel near the fermi level contribute to the large positive FIG. 3 . Fermi level dependent total and decomposed MAEs of TCB-Re. Here, majority (red line), minority (blue line), cross (green line) represent the cou- pling between spin-up channels, spin-down channels, and spin-up with spin-down channels, respectively. The Fermi level (E F) is set to zero.MAE. However, the other coupling channels including dx2-y2/dyz anddz2/dyzpairs result in a negative MAE. According to the above analysis, TCand MAE are closely related to the alignments of a few specific electronic orbitals, which can be precisely manipulated by mechanical deformation. Hence, we propose to use biaxial strain as an effective strategy to tailor the spin coupling strength of 2D TCB-Re; meanwhile, one can retain the large on-site moment of the Re atom. A tiny biaxial strain between −1% and 1.5% is applied to TCB-Re, which is defined as ε=(a−a0)/a0, (4) where a and a 0are the lattice constants of the strained and unstrained states, respectively. As expected, the spin moments of TCB-Re remain nearly unchanged (Table II), including those of Re atoms ( MRe), bridge N atoms ( MN), bridge C atoms ( MCbridge ), and each C6 ring atom ( MC1andMC2represent the C atoms that connect with H and C bridge atoms, respectively). The magnetic ground states for 2D TCB-Re under given strains (−1%,−0.5%, 0.5%, 0.75%, 1%, and 1.5%) were determined by total energy calculations. The strain-induced magnetic phase transition is shown in Fig. 4. Under compression, a paramagnetic (PM) state is observed for strain less than −0.5%. Then, the TCB-Re changes gradually from the PM to FM state with strain increasing to 0.75%. As the tensile strain exceeds 0.75%, it transits from the FM to AFM state. The competition of the exchange coupling of the first nearest- neighbor ( J1) and second nearest-neighbor ( J2) in TCB-Re deter- mines such variations (Fig. 4). For example, the coupling parameter J1switches from negative to positive under the tensile strain, while an opposite trend is observed for coupling parameter J2. That is to say, AFM coupling from the second nearest-neighbor is much stronger than FM coupling from the first nearest-neighbor when the tensile strain is greater than the crossover point of 0.75%. However, when the compressive strain is bigger than −0.5%, TCB-Re behaves as a paramagnetic system. To get insight into the magnetic transition, the dominant superexchange interaction between in-plane σbonding and out- plane πbonding has been investigated in detail. According to the Goodenough–Kanamori–Anderson (GKA) mechanism,28the in- plane superexchange interactions overlapping with the same ligand porbital favor antiferromagnetic ordering. However, the exchange interaction between occupied and empty dorbitals on neighboring TM ions overlapping with the same porbital leads to a ferromag- netic state. The occupation numbers of dyzand dx2-y2orbitals of TABLE II . The spin moments on C6 ring atoms ( MC1andMC2represent the C atoms that connect with the H and C bridge, respectively), and the spin moment of the bridge C atom ( MCbridge ), bridge N atom ( MN), and Re atom ( MRe). MAE on each Re atom under the strain range from −0.5% to 0.5%. MC1MC2MCbridge MN MRe MAE Strain (%) ( μB) (μB) ( μB) ( μB) ( μB) (meV/atom) −0.5 0.04 0.006 0.03 0.024 2.576 120.21 0.25 0.04 0.006 0.03 0.024 2.583 76.12 0 0.04 0.006 0.03 0.024 2.584 19.05 0.25 0.04 0.006 0.03 0.024 2.585 35.06 0.5 0.04 0.006 0.03 0.024 2.590 91.13 APL Mater. 8, 071105 (2020); doi: 10.1063/5.0010822 8, 071105-5 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm FIG. 4 . The left scale is the orbital energy level difference line of TCB-Re under the strain range from −1% to 1.5%. The black line represents the difference value of energy between unoccupied dx2-y2and occupied dxy(ΔE1), and the red line represents unoccupied dz2/dx2-y2and occupied dyz(ΔE2). The right scale is the exchange parameter of the first and second nearest-neighbor in the strain range from−1% to 1.5%. The green and blue lines represent the first ( J1) and second (J2) nearest-neighbor exchange parameters, respectively. the Re atom, pyorbital of the N atom, and pxand pzorbitals of the C atom were calculated under zero strain, 1% tensile strain, and 1% compressive strain, respectively. At the strain-free state, the coupling strength of d–pz–dis higher than d–px–d; thus, FM coupling is much stronger than the AFM one. Under 1% tensile strain, electrons are transferred from the Re atom and C atoms and the electron occupancy on spin-down dyz(pz) orbitals decreases from 0.85 (0.77) to 0.64 (0.68), while the electron occupancy on pxorbitals of C–C6 rings increases from 0.83 to almost fully occu- pied of 0.96. At the same time, the occupations on the other orbitals remain unchanged. Therefore, we infer that the in-plane superex- change interactions of the intermediary electron acceptor by the C6 ring and C–N bridge have been strengthened through the px orbital and the out-of-plane superexchange interactions through pz orbitals have been reduced. In the 2D TCB-Re system, under the tensile strain, AFM exchange interaction is stronger than the FM exchange coupling. Not surprisingly, the magnetic state of TCB- Re behaves as AFM when the tensile strain is bigger than 0.75%. Under 1% compressive strain, only dx2-y2of the Re atom in the spin- down channel has been changed and the corresponding electron occupation number increases from 0.27 to 0.37. Compressive strain induced stronger FM coupling is expected, and the on-site spin moments orient randomly, leading to a FM to PM transition. The change in coupling strength originated from electron occupation numbers is also responsible for the variation of the TC, as given in Table S1. More importantly, the values of MAE can be effectively engi- neered via external strains. Here, we only discuss the change in MAE under the strain range from −0.5% to 0.5%, under which the mag- netic state favors FM. As listed in Table II, the value of the MAE abruptly increases under both compressive and tensile strains. The maximum MAE values of 91.13 meV and 120.21 meV are obtained under a tensile strain of 0.5% and a compressive strain of −0.5%, respectively. In other words, the MAE is enhanced by more than fourtimes compared with the original MAE value of 19.05 meV with- out strain. To reveal the origin of such remarkable MAE enhance- ment, we decomposed the contribution from the dx2-y2and dxy orbital pair, dx2-y2and dyzpair, and dz2and dyzpair from torque calculations using Eq. (3). First, according to nonvanishing angu- lar momentum matrix elements between dstates,55the coupling between the dx2-y2anddxyorbital pair, dx2-y2anddyzpair, and dz2 and dyzpair is 2, 1, and√3, which are strain invariant. Second, with increasing tensile and compressive strains, the energy differ- ences between the three pairs of energy levels are shown in Fig. S5. The negative contributions of the dx2-y2anddyzpair, and dz2and dyzpair are reduced greatly, while the positive contributions of the dx2-y2anddxypair are decreased slightly, which result in significant enhancement of MAE. For example, the orbital energy level differ- ence between dx2-y2anddxyunder zero strain and 0.5% tensile strain is 0.32 eV and 0.40 eV, respectively, while the differences between dx2-y2/dz2anddyzare 0.46 eV and 0.59 eV for 0% and 0.5% tensile strains. In summary, we have carried out in-depth DFT calculations on the magnetic properties of 2D TCB-TM with two-coordinate and open-shell 5d transition atoms. The transition metal atoms are stabilized by ideal (1, 3, 5)-benzenetricarbonitrile (TCB) molecules under a rigid hexagonal lattice. We found that the TCB-Re mono- layer can be a robust ferromagnetism semiconductor with 63 meV bandgap, a large magnetic anisotropy energy of 19 meV/atom, and aTCof 613 K. Moreover, strain driven transitions between PM, FM, and AFM states have been observed. The competition between in-plane d–px/y–dand out-of-plane d–pz–dsuperexchange inter- actions is mainly responsible for such phenomena. The MAE of the 2D TCB-Re monolayer also exhibits an evident strain depen- dence, i.e., it increases from 19 meV/atom to 91 meV/atom and 120 meV/atom under 0.5% tensile strain and 0.5% compressive strain, respectively. The proposed 2D MOFs provide an ideal ligand field to fulfill the requirements of preventing aggregates and main- taining strong magnetic anisotropy of 5d transition metal atoms with twofold coordination. Although the realization of the free-standing 2D organometal- lic framework is still a very challenging work, many groups have performed pioneering attempt to synthesize such 2D metal–organic frameworks in recent works. For example, Pei et al. found that the two-dimensional metal organic matrix structure nanosheet can be obtained by mixing the organic solution, metal ion aqueous solution, and organic solvent through the solvothermal technology.58For our designed TCB-Re, one-unit cell of TCB-Re is consisted by two (1, 3, 5)-benzenetricarbonitrile molecules and three Re ions. Among them, (1, 3, 5)-benzenetricarbonitrile can be obtained by a variety of methods such as via chlorination, ammoniation, and dehydra- tion with (1, 3, 5)-benzenetricarboxylic acid as raw materials59or catalytic production of aromatic nitrile.60As we know, a cyanide ion (CN-) is classified as soft bases, so they can form rigid bonds with low coordinated heavy metal ions of soft acids.61Re is a kind of heavy metal with a density of 21.8 g/cm3. One can design a chemical reaction by mixing the (1, 3, 5)-benzenetricarbonitrile solution and Re ion aqueous by using an organic solvent with suitable proportion and conditions. Above all, our predicted 2D organic materials are highly anticipated and the present results are also valuable for the coordination chemistry of two-coordinate or quasi-two-coordinate transition metal complexes. APL Mater. 8, 071105 (2020); doi: 10.1063/5.0010822 8, 071105-6 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm SUPPLEMENTARY MATERIAL See the supplementary material for the total DOS of TCB-TM (TM = Hf, Ta, W, Os, and Ir); magnetic configuration in the 2 ∗1 supercell; Curie temperature calculation details; AIMD simulation; the spin-polarized band structure; spin density; PDOS of dorbitals of the Re atom for TCB-Re under different strains; and the GGA + U test for the magnetism of TCB-Re. ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Grant No. 11874097) and the Fundamen- tal Research Funds for the Central Universities of China (Grant No. DUT19LK12). We acknowledge the Xinghai Scholar project of Dalian University of Technology and the project of Dalian Youth Science and Technology Star (Grant No. 2017RQ012). 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APL Mater. 8, 071105 (2020); doi: 10.1063/5.0010822 8, 071105-7 © Author(s) 2020

5.0015542.pdf

Appl. Phys. Lett. 117, 060501 (2020); https://doi.org/10.1063/5.0015542 117, 060501 © 2020 Author(s).Quantum size effect in nanocorrals: From fundamental to potential applications Cite as: Appl. Phys. Lett. 117, 060501 (2020); https://doi.org/10.1063/5.0015542 Submitted: 29 May 2020 . Accepted: 01 July 2020 . Published Online: 10 August 2020 Qili Li , Rongxing Cao , and Haifeng Ding COLLECTIONS This paper was selected as Featured Quantum size effect in nanocorrals: From fundamental to potential applications Cite as: Appl. Phys. Lett. 117, 060501 (2020); doi: 10.1063/5.0015542 Submitted: 29 May 2020 .Accepted: 1 July 2020 . Published Online: 10 August 2020 QiliLi,1 Rongxing Cao,1,2 and Haifeng Ding1,3,a) AFFILIATIONS 1National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China 2College of Physics Science and Technology, Yangzhou University, Yangzhou 225002, China 3Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China a)Author to whom correspondence should be addressed: hfding@nju.edu.cn ABSTRACT Conventional silicon-based devices are approaching the scaling limits toward super miniaturization, where the quantum size effect naturally emerges with increasing importance. Exploring the quantum size effect may provide additional functionality and alternative architectures forinformation processing and computation. Scanning tunneling microscopy/spectroscopy is an ideal tool to explore such an opportunity as it can construct the devices in an atom-by-atom fashion and investigate their morphologies and properties down to the atomic level. Utilizing nanocorrals as examples, the quantum size effect is demonstrated to possess the great capability in guiding the adatom diffusion and the self-assembly, controlling the statistical fluctuation, tuning the Kondo temperature, etc. Besides these fundamentals, it also shows strong potentialin logic operations as the basic logic gates are constructed. Published under license by AIP Publishing. https://doi.org/10.1063/5.0015542 In the past several decades, the semiconductor industry had a fast development along the guideline of Moore’s law. Generally, the num-ber of transistors on a microprocessor chip doubled every two years. Nowadays, conventional silicon-based devices are approaching the scaling limits toward super miniaturization. 1With decreasing the device size, one of the well-known phenomena, the quantum size effect (QSE) naturally evolves with increasing importance. The QSE describes a system whose size is comparable with the effective de Broglie wavelength of the carriers, resulting in the size quantization ofthe energy of the carriers. 2Exploring the QSE may provide additional functionality and alternative architectures for information processing and computation. Generally, the QSE can be classified into three categories according to the number of dimensions in size quantization: (i) one- dimensionally (1D) confined ultrathin films; (ii) 2D confined nanocor- rals and islands; and (iii) 3D confined quantum dots. The experimental exploration of the 1D-QSE in thin films can be dated to 1966, where the thickness dependences of the resistivity, Hall coefficient, magneto- resistance, and tunnel spectroscopy were observed in thin bismuthfilms. 3,4From then on, the QSE in thin films was also extended to the investigation of optical phenomena, phase transitions, superconductiv- ity, magnetism, etc.2,5Toward application, the QSE in ultrathin films is able to tune the coupling between magnetic layers and control giantmagnetoresistance.6,7For the 3D-QSE in quantum dots, it arose in the late 1980s8and has been booming fast since then. Quantum dots have been widely used for medical and display applications9and also exhibit great potential in multiple applications such as solar harvesting,10flexi- ble nonvolatile memory,11and energetic material.12Moreover, quan- tum dots are also an alternative architecture toward the quantum computation.13With regard to the 2D-QSE, it is the mostly related QSE to the super miniaturization where the lateral confinement is dominant. Here, we focus our discussion on the 2D-QSE with nanocorrals (Fig. 1 ) created by atom manipulation14since the investigation can be made in a designed manner. Besides the pioneering works showingthe quantum confinement of the electronic states 15and projecting the Kondo resonance of one focal point of the ellipses to the other focal point (quantum mirage effect),16the QSE is also used in modulating the atom diffusion, guiding the self-assembly,17controlling the statisti- cal fluctuation,18and tuning the Kondo temperature.19Al a t e s tw o r k demonstrates that non-Kondo effect based quantum mirages can exist in a wide energy range beyond the Fermi level.20What is more, the signal of the Kondo-free mirage can be even stronger than that of the object. With these merits, the manipulation of Kondo-free mirages is exploited to realize basic logic operations, such as NOT, FANOUT, and OR gates.20 Appl. Phys. Lett. 117, 060501 (2020); doi: 10.1063/5.0015542 117, 060501-1 Published under license by AIP PublishingApplied Physics Letters PERSPECTIVE scitation.org/journal/aplLateral confined electronic surface states introduced by the molecular network21,22or parallel atomic/molecular chains23,24have been shown to substantially affect the atomic motion behavior on thesurface. Recently, inspired by the theoretical prediction, 25quantum guided atomic diffusion and self-assembly have also been experimen- tally demonstrated inside the nanocorrals.17Utilizing the Fe corrals constructed by atom manipulation and subsequently deposited Gdatoms on the Ag(111) surface, the statistics on the Gd adatom diffu-sion inside the corral has been harvested from hundreds of consecutivescanning tunneling microscopy (STM) images. Three concentric pref- erential orbits of adatom motion plus one preferred location at the center can be observed in the cumulative image [ Fig. 2(a) ]. We note that the brighter spots with much higher occupation probability onthe outermost orbit can be attributed to the geometric deviation of the constructed nanocorral from the perfect circular shape. 17The orbital separation of /C243.8 nm is consistent with half of the Fermi wavelength of the Ag(111) surface. This probability distribution is closely related to the characteristic electronic standing-wave patterns induced by the QSE,15,17while in sharp contrast to the random walk on a flat terrace free of nanocorrals.23The same conclusion was also confirmed in other different sized nanocorrals.26These experimental findings dem- onstrate that lateral quantum confinement can be used to engineer atom diffusion. Based on the observed higher occupancy probability of atom dif- fusion in the outmost orbit inside the corral, one can expect that upon increasing the Gd coverage, most of the adatoms will be located near the quantum corral, forming a ring-like structure. Indeed, the experi- mental observation highlights this effect unambiguously [ Fig. 2(b) ]. The deposition of more adatoms will make the inner orbits occupied and form an onion-like structure at optimal coverage as predicted the- oretically.25These atomic structures are different from the self- assembled hexagonal superlattice.17Besides the circular nanocorrals, the QSE was also studied in different shaped corrals such as triangular ones and exhibited the ability to control the orientation of the atomic structures. Therefore, the diffusion behavior and self-assembled structures of adatoms can be significantly modified by introducing the QSE. Since these corrals can also be built by advanced lithography, further combining them with quantum engineering will open possibilities for local functionality design down to the atomic scale. The statistical fluctuation phenomenon is fundamental and usu- ally unavoidable and can deteriorate the uniformity/reproducibility of the desired atomic structures. As a rule of thumb, the fluctuation scales with the square root of the number of atoms. The smaller the number, the stronger the fluctuation as compared to the total number. For instance, the number of deposited adatoms inside the prepared nanocorrals strongly fluctuates, resulting in a broad distribution of the occupancy histogram.18,22In contrast to the conventional closed corrals, Cao et al. found that the QSE in open nanocorrals can be used to control the statistical fluctuation of the atomic structures.18 Open nanocorrals were built by missing one Fe adatom while keeping the other Fe adatoms on Ag(111) with a separation between 2.0 and 2.5 nm the same as that for the closed corrals. The opening forms a gate to regulate the flow of Gd adatoms traveling in and out, resulting in quantized atom trapping. Depending on the diameters of the open corrals (5–12 nm), one to seven Gd adatoms end up being trapped inside the nanocorrals, preferring to self-arrange in regular polygons [ Fig. 2(c) ]. A systematic measurement on the number of trapped Gd adatoms as a function of the open-corral diameter hasbeen performed further, for one through seven trapped adatoms except for six. The absence of 6-atom trapping can be understood as follows. As shown in Fig. 2(c) , the trapped atoms prefer to form equi- lateral polygon structures. If an equilateral hexagon is formed by 6 Gd atoms, an additional Gd atom can also be stabilized at the center posi- tion of the hexagon since its distance to the surrounding Gd atom is the same as the separation between the surrounding Gd atoms. In such a case, the total energy of forming a 7-atom structure is lower than that of a 6-atom structure plus one atom located outside the corral. The mechanism of this quantized trapping was revealed by STM imaging and energy landscape calculations, which can be FIG. 1. Sketch of the quantum size effect in nanocorrals and their fields of exploration. FIG. 2. (a) Statistics on Gd adatom’s diffusion inside the Fe corral on Ag(111). (b) Ring-like structure formed upon deposition of more Gd adatoms. Cao et al. , Phys. Rev. B 87, 085415 (2013). Copyright 2013 American Physical Society. (c) Topographies of the quantized Gd-atom trapping in open Fe corrals with differentdiameters. Cao et al. , Phys. Rev. B 90, 045433 (2014). Copyright 2014 American Physical Society.Applied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 117, 060501 (2020); doi: 10.1063/5.0015542 117, 060501-2 Published under license by AIP Publishingattributed to the QSE induced self-regulating process: when the num- ber of adatoms inside the corral is insufficient, trapping of additional atoms is automatically triggered; whereas if there are too many atoms inside the corral, an efficient overall repulsion process sets in and the extra atoms go out. Remarkably, the wide plateaus of the staircase in the curve of the number of trapped Gd adatoms vs the open-corral diameter illustrate the robustness and stability of this open-corral atom trapping, giving sufficient tolerance for reliable nanostructure design and fabrication via this method. The effect of controlling the statistical fluctuation utilizing the QSE in open nanocorrals was further illustrated with the direct com- parison between open and closed corrals in the context of their trap- ping capabilities. The statistical histogram for the number of trapped Gd adatoms in an array of open 8.5-nm corrals shows a single value, while a broad distribution in the histogram was found in an array of closed corrals of the same diameter. These experimental findings demonstrate that the QSE in open nanocorrals can be utilized to enforce a number-selection rule and control the statistical fluctuation of the atomic structures over a wide range of experimentally relevant conditions, whereas closed corrals do not. This method opens a potential way to improve the uniformity/ reproducibility of the desired device structures and processes. The Kondo effect describes the scattering of conduction electrons by the local spin of a magnetic impurity and has inspired many advan- ces of both theories and experiments. One of the continuous motiva- tions to explore that the Kondo effect is the spin control toward spintronic devices. When the system temperature is much lower than the characteristic temperature, i.e., Kondo temperature, the spin of a magnetic impurity will be screened, while the spin remains, when the system temperature is higher than Kondo temperature. In addition, one can also manipulate the spin state of magnetic impurities at ambi- ent temperature by controlling the Kondo temperature. Utilizing the QSE in thin Pb films, the Kondo temperature of the MnPc molecule shows oscillation behavior with the film thickness and can reach a value up to 419 K.27For quantum dots, the Kondo temperature can be tuned by means of a gate voltage as a single-particle energy state near the Fermi energy.28 Given those studies in both 1D-confined thin films and 3D- confined quantum dots, a natural question of whether the in-plane QSE can modulate the Kondo effect or not arises. The in-plane quan- tum size effect requires the wave vector to be parallel to the surface. Thus, the surface state comes into the consideration, e.g., the surface state of noble metal with (111) orientation. In fact, whether and how t h es u r f a c es t a t ei n fl u e n c e st h eK o n d ot e m p e r a t u r eh a v eb e e ni nh o t debate for a certain time.29–35This discrepancy is recently solved by the extended experiments, which showed that Kondo temperature ofCo adatoms placed at the center of quantum corrals atop Ag(111) [Fig. 3(a) ] oscillates strongly as a function of the diameter of the cor- ral. 19The authors compared the dI/dV measurements of the empty corral by removing the Co atom at the center [ Fig. 3(b) ]. The summa- rized corral radius r-dependent Kondo resonance width win the occu- pied corral [ Fig. 3(c) ] and the dI/dV signal at the Fermi level measured in the center of the empty corrals [ Fig. 3(d) ] exhibit similar oscilla- tions. The maximum value of wreaches 25.1 meV (corresponding to a Kondo temperature of /C24291 K), which is about three times the value obtained for a single Co monomer on a wide terrace. The oscillation period is /C243.8 nm, which is consistent with the half Fermi wavelengthof the Ag(111) surface state. It is generally accepted that the Kondo temperature can be expressed as kBTK¼Dexpð/C01 JqðEFÞÞ,27,36where Dis the band cutoff, Jis the exchange constant, qðEFÞis the local den- sity of state (LDOS) at the Fermi level, and kBis the Boltzmann con- stant. The authors extended the expression to accommodate both thebulk and surface states, i.e., two-band contribution with the form of k BTK¼~Dexpð/C01 JbqbðEFÞþJsqsðEFÞÞ,w h e r e Jband Jsare the exchange constants of the adatom with the bulk and surface states, respectively, and ~Dis the effective band cutoff. Fitting the measured Kondo reso- nance [red curve in Fig. 3(c) ] and the extracted LDOS from the dI/dV spectrum [red curve in Fig. 3(d) ], the exchange values of single adatom with the bulk and surface states, namely, JbandJswere obtained. Moreover, the subsequent works also find site-dependent Kondo temperature for single Co adatom placed at different positions within nanocorrals37or on one monolayer Ag-covered Cu(111) surface.38 Note that the latter surface has surface reconstruction, which confines the surface states to spatially modulated LDOS. Based on these, one may envision an atomic-scale magnetic memory patterned with arrays of paired low-and-high LDOS units via STM lithography or nanofabri-cation for Co/Ag(111) or Co/1 ML Ag/Cu(111), where the Co adatoma tt h eh i g hL D O Ss i t ei sn o n m a g n e t i cm e a n i n g“ 0 ”a n dt h eo n ea tt h e low LDOS site is magnetic meaning “1.” The writing procedure can be realized through atom manipulation by moving the Co atom from thehigh LDOS site to the low LDOS site and vice versa . The readout can be made via the scanning tunneling microscopy/spectroscopy (STS). Together with the findings in thin films and quantum dots, theseobservations complement the viable ways to modulate the Kondo effect via the QSE. The QSE in nanocorrals can also be used to build atomic logic gates. As demonstrated by Manoharan et al. in 2000, the Kondo effect- based quantum mirages show great potential in information transportat the nanometer scale. 16The observed quantum mirages are Kondo FIG. 3. (a) and (b) show the topographic view of Co-centered and empty quantum corrals with the radius of the nanocorrals, respectively. (c) r-dependent Kondo reso- nance width wfor the occupied corral. (d) r-dependent dI/dV at the Fermi level. Red curves in (c) and (d) are fittings. Li et al. , Phys. Rev. B 97, 035417 (2018). Copyright 2018 American Physical Society.Applied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 117, 060501 (2020); doi: 10.1063/5.0015542 117, 060501-3 Published under license by AIP Publishingeffect based and thus are limited to near the Fermi level only. Recently. Liet al.20found that a Kondo-free mirage rooted in the inversion effect,39which has high signal transport efficiency and can operate in a wide energy range.20Taking advantage of these merits, they demon- strate the atomic logic gates of NOT, FANOUT, and OR. In their con-ceptual design, the input “1”/“0” is the presence/absence of theadatom at one focal point and the output “1”/“0” is the triggered quan-tum mirages with a high/low difference of dI/dV intensity at the other focus. The NOT gate is a two-terminal device, which just corresponds to the two foci of an elliptical quantum corral by utilizing inverted mirage or anti-mirage. To realize FANOUT and OR gates, a specialgeometry by combining two elliptical quantum corrals with one jointfocus was devised to form the three-terminal device [ Fig. 4 ]. The pres- ence/absence of an adatom at joint focus A is as the input “1”/“0” anddI/dV intensities at B and C are as the outputs. The dI/dV map obtained from “0” input [ Fig. 4(a) ] at a bias voltage of 34 mV shows low contrast at both foci B and C [ Fig. 4(b) ], corresponding to “0” out- puts. When shifting the input to “1” by placing an Fe adatom at jointfocus A [ Fig. 4(c) ], the corresponding dI/dV map [ Fig. 4(d) ]s h o w s high intensity (“1”) at both outputs. The relation between the input and outputs satisfies the function of a FANOUT gate. Besides, an OR gate was also demonstrated when swapping the input and output. Wenote that the output is sensitive to the chosen bias voltage due to thequantum interference. When a destructive condition is used, input 1can result in an output 0. This was applied to construct a NOT gate in an elliptical quantum corral. 20 Although the atomic basic logic gates have been demonstrated, there are still many further explorations need to be done before theirapplications. First, as discussed in the original work of Li et al. , 20the logic threshold of the on-off signal needs to be improved. As the pres- ence of the bulk state significantly reduces the on-off ratio by contrib- uting as a constant background, seeking a system with a reduced oreven vanishing bulk state can achieve a higher on-off ratio. Besides,seeking a system with a strong confinement effect can also enhance the on-off ratio. Furthermore, devices with more than three terminals can also be designed by combining three or four elliptical quantumcorrals with one joint focus to form four or five terminals. 40Finally, one can even envision a structure with cascaded FANOUT or ORgates. We have discussed the QSE in nanocorrals on noble metal (111) surfaces, from quantum-guided diffusion and self-assembly, to con- trolling statistical fluctuation, tuning Kondo temperature, and building atomic logic gates. On one hand, noble metal (111) surfaces providean excellent playground to explore the QSE in nanocorrals and thereremains much room to be explored. On the other hand, seeking other potential materials is also highly desired. This effort on noble metal (111) surfaces should be transferrable to the potential materials. Here,we list a few potential candidates that may produce fascinating physi-cal phenomena in the QSE. For example, the topological insulators,which have surface states only near the Fermi level, are good candi- dates to investigate. A recent work showed that nanocorrals with Rb atoms can be built on the Bi 2Se3(111) surface and demonstrated the ability to create tailored electronic potential landscapes on topologicalsurfaces with atomic-scale control. 41As topological insulators are gen- erally topological protected from nonmagnetic impurities, the quan- tum confinement for nonmagnetic impurities on topological insulators is hard to observe. A recent theoretical prediction, however,showed that magnetic nanocorrals can yield strong spin-polarizedquantum well states. 42This could be probed via the state-of-the-art spin-polarized STM, which shows great capability in resolving com- plex magnetic structures like magnetic skyrmions.43,44Another candi- date is the superconductor substrate, which is also considered asthe potential candidate for quantum computation and quantuminformation. 45,46As aforementioned, the QSE in nanocorrals on the noble metal surface demonstrates the ability of information and logic gates. When taking advantage of the QSE in nanocorrals, it might pro-mote the explorations of quantum computation and quantum infor-mation. Actually, a theoretical work has showed the quantum mirage effect for a magnetic adatom on the superconductor substrate confined in a nanocorral. 47In addition, two-dimensional materials are boom- ing, while the study of the lateral quantum confinement in them is stillat an early stage. We believe that those two-dimensional materialssuch as transitional metal dichalcogenides 48,49are worth investigating. Finally, we would like to point out that the QSE in nanocorrals is observed under ultrahigh vacuum, which might be too expensive touse commercially. For practical application, one could consider theencapsulation to protect devices from the ambient environment, simi- lar to what has been done in quantum dots. 50 AUTHORS’ CONTRIBUTIONS Q.L. and R.C. contributed equally to this work. FIG. 4. (a) Topography of a confocal elliptical quantum corral (EQC). Both the EQCs used to build the confocal EQC have the same size of major axislength¼6.6 nm and eccentricity ¼0.7. Joint focus A is used as the input, and the other two foci B and C are the outputs. (b) The corresponding dI/dV map of panel (a) at the bias voltage of 34 mV. The dI/dV values of outputs B and C are 30.2 nS and 31.7 nS, respectively. (c) Topography of the same confocal EQC in panel (a)but with an additional Fe adatom placed at joint focus A. (d) The corresponding dI/ dVmap of panel (c) at the same bias voltage of 34 mV. The dI/dV values of outputs B and C are 35.6 nS and 35.2 nS, respectively. Dashed circles mark the focal posi-tions, for which black means 1 and white means 0. Li et al. , Nat. Commun. 11, 1400 (2020). Copyright 2020 Springer Nature.Applied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 117, 060501 (2020); doi: 10.1063/5.0015542 117, 060501-4 Published under license by AIP PublishingThis work was supported by the National Key R&D Program of China (Grant Nos. 2017YFA0303202 and 2018YFA0306004), theNational Natural Science Foundation of China (Grant Nos.11974165, 51971110, and 11734006), the China PostdoctoralScience Foundation (Grant No. 2019M651766), and the NaturalScience Foundation of Jiangsu Province (Grant Nos. BK20190057 and BK20180889). 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5.0012748.pdf

Appl. Phys. Lett. 117, 022411 (2020); https://doi.org/10.1063/5.0012748 117, 022411 © 2020 Author(s).Iodine orbital moment and chromium anisotropy contributions to CrI3 magnetism Cite as: Appl. Phys. Lett. 117, 022411 (2020); https://doi.org/10.1063/5.0012748 Submitted: 04 May 2020 . Accepted: 01 July 2020 . Published Online: 17 July 2020 Y. Choi , P. J. Ryan , D. Haskel , J. L. McChesney , G. Fabbris , M. A. McGuire , and J.-W. Kim ARTICLES YOU MAY BE INTERESTED IN Ultrahigh tunneling magnetoresistance in van der Waals and lateral magnetic tunnel junctions formed by intrinsic ferromagnets Li 0.5CrI3 and CrI 3 Applied Physics Letters 117, 022412 (2020); https://doi.org/10.1063/5.0013951 Magnetic skyrmions in atomic thin CrI 3 monolayer Applied Physics Letters 114, 232402 (2019); https://doi.org/10.1063/1.5096782 Oxide 2D electron gases as a reservoir of defects for resistive switching Applied Physics Letters 116, 223503 (2020); https://doi.org/10.1063/5.0003590Iodine orbital moment and chromium anisotropy contributions to CrI 3magnetism Cite as: Appl. Phys. Lett. 117, 022411 (2020); doi: 10.1063/5.0012748 Submitted: 4 May 2020 .Accepted: 1 July 2020 . Published Online: 17 July 2020 Y.Choi,1,a) P. J.Ryan,1D.Haskel,1J. L.McChesney,1 G.Fabbris,1M. A. McGuire,2 and J.-W. Kim1 AFFILIATIONS 1Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA 2Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA a)Author to whom correspondence should be addressed: ychoi@anl.gov ABSTRACT The recent discovery of two-dimensional (2D) magnets, with a number of interesting magnetic properties, has drawn much interest due to their potential for future 2D spintronic device applications. CrI 3, a van der Waals magnet, exhibits two-dimensional ferromagnetism even in monolayer form, stabilized by strong magnetic anisotropy. Its interlayer magnetic ordering is coupled to structural layer stacking, implyingthat the charge density distribution mediating van der Waals interactions plays a key role in the magnetic interaction between the layers.Using polarization-dependent x-ray spectroscopy, we investigated the response of the electronic environment around Cr and I sites to struc-tural changes of layer stacking order. The highly anisotropic nature of the Cr local environment is significantly enhanced and is accompanied by changes in the valence band, in the rhombohedral phase. Magnetic spectroscopy measurements reveal a sizable iodine orbital moment, indicating the iodine contribution to magnetic anisotropy. Our results uncover an important role for the extended nature of anisotropic Crorbital states in dictating interlayer magnetic interactions and the iodine contribution to magnetic anisotropy. Published under license by AIP Publishing. https://doi.org/10.1063/5.0012748 The realization of two-dimensional magnetism in van der Waals material CrI 3, with readily cleavable atomically flat surfaces, makes it a potential spintronic material for magnetic tunneling junctions.1–4 Although magnetic ordering is unstable in an isotropic Heisenberg two-dimensional system due to thermal fluctuations, a strong mag- netic anisotropy can stabilize long-range magnetic order.5With a lay- ered structure, CrI 3exhibits ferromagnetism with a Curie temperature of 60 K due to sizable magnetic anisotropy along the out-of-plane direction. This anisotropy stabilizes long-range ferromagnetic order even in a single CrI 3layer.6In bulk form, the crystal structure under- goes a structural transition near 210 K from rhombohedral to mono-clinic on warming, which involves displacement of layer stacking. 7 While ferromagnetism in CrI 3is readily observable, the mecha- nisms behind ferromagnetism and magnetic anisotropy still remain an active research topic.8–16An experimentally observed value of /C243lB/Cr suggests that the Cr3þions in CrI 3should have S ¼3/2 with a quenched orbital moment, indicating insignificant single ion anisot-ropy contribution from the Cr atoms. Indeed, x-ray magnetic circular dichroism (XMCD) at the Cr L 2,3absorption edges, probing the Cr 3d states, has not shown any measurable orbital moment contribu-tion, 15,17and thus, this naturally leads to a postulation that the iodine ion is likely the main contributor to magnetic anisotropy. Fromprevious studies, it is found that sizable spin–orbit coupling from I atoms and strong hybridization between the Cr 3d and I 5p states arelinked to the stabilization of long-range ferromagnetic ordering andstrong magnetic anisotropy. 8–15Another interesting aspect of the CrI 3 material is an intricate coupling between layer stacking and exchangeinteractions. Modifying the layer stacking can tune the interlayer inter-action between antiferromagnetic and ferromagnetic, with the mono-clinic phase favoring the antiferromagnetic interlayer coupling, whereas the rhombohedral favors ferromagnetic coupling, as observed in experimental studies 18–20and supported by theoretical calcula- tions.10,12,13,21In this study, we investigate the contributions of Cr and I to the interlayer magnetic coupling and the magnetic anisotropy using x-ray spectroscopy measurements on bulk CrI 3crystals. The above recent studies have shown a strong tie between inter- layer magnetic interaction and structural transition, and thus, we probe the change in the local electronic environment across the transi-tion. x-ray linear dichroism (XLD), the difference between x-rayabsorption near edge structure (XANES) spectra obtained with two polarizations, parallel and perpendicular to a crystalline axis, reflects the anisotropy of the electronic states probed in the absorption pro-cess. Figure 1(a) shows temperature-dependent XANES at the Cr K edge. The spectra taken with incident polarization parallel to the ab Appl. Phys. Lett. 117, 022411 (2020); doi: 10.1063/5.0012748 117, 022411-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplplane (E jjab) and the c-axis (E jjc) show noticeable differences. The derived XLD spectra as a function of temperature are shown in Fig. 1(b). The Cr K XANES and XLD exhibit a strong temperature depen- dence, revealing changes in electronic anisotropy around the Cr atoms. The changes are significant across the known structural transition(210–230 K), whereas minor changes were observable below 210 K.The increased Cr XLD indicates that the electronic environmentaround the Cr site becomes more anisotropic in the low temperaturerhombohedral phase. To understand the changes in linear dichroism across the mono- clinic to rhombohedral structural transition, we carried out simula- tions of polarization-dependent Cr K-edge XANES using the FEFF8code. 22The ab initio , self-consistent approach used real space full mul- tiple scattering calculations within an 8 A ˚cluster centered on the Cr absorbing atom. The cluster was constructed using crystallographicdata reported in Ref. 7for both monoclinic (250 K) and rhombohedral (90 K) phases. The x-ray polarization vector was oriented within or perpendicular to the CrI 3layers, the former using an average over the in-plane directions. Calculations with linear polarization along the aand b directions show negligible in-plane linear dichroism, in eitherstructure. A Debye model was used to approximate thermal effects onthe fine structure although these are, in general, small near absorptionthreshold. The simulations in Fig. 1(e) show a significant increase in XLD in the low temperature rhombohedral phase, in agreement withexperiment. The simulations indicate that changes in c-axis XANES are the main source behind the XLD changes, as expected from achange in layer stacking, which preserves the ab-plane local structure around Cr ions. Despite this agreement, the simulated fine structuredoes not fully capture the experimental fine structure, suggesting that more complex structural models including potential local structure distortions may be at play. Despite this shortcoming, we note that thepredominant structural change across the transition is the shearing ofthe 2D layers, as illustrated in the bottom schematic in Fig. 1(c) ,a n d that this layer stacking change leads to a more anisotropic local envi-ronment around the Cr ions. While the XLD data show an anisotropic electronic environment around Cr, deviation from centrosymmetry in the Cr octahedral envi- ronment is insignificant and invariant across the structural transition.The relevant feature in the current Cr XANES data is the pre-edgeregion, as indicated by a vertical arrow and replotted in Fig. 1(d) .T h e pre-edge structures in transition metal K-edge XANES are linked tothe 3d character, via quadrupolar transitions (1s !3d) or dipole tran- sitions to mixed 3d/4p orbitals. While insignificant in centrosymmetric symmetry, a non-centrosymmetric distortion can enhance the pre- edge features via dipole-allowed transitions. 23–25The vanishingly small pre-edge intensity in Fig. 1(d) , invariant with temperature, indicates the absence of non-centrosymmetric distortion of the CrI 6octahedra across the monoclinic-to-rhombohedral transition. This result is con-sistent with x-ray diffraction data. 7In few-monolayer-thick CrI 3sam- ples, probing magnetism is nontrivial, and in recent studies,photoluminescence (PL) 26and second-harmonic generation (SHG)19 techniques have been used to observe emerging antiferromagnetic FIG. 1. (a) Temperature dependence of XAS at the Cr K edge. Vertical arrow marks the pre-edge region in the Cr XANES spectra. (b) XLD as a function of temperature . (c) Schematic diagrams of the CrI 3structure, highlighting edge-sharing octahedra. Purple spheres represent iodine atoms, and Cr (in blue and red) atoms are located at the center of each octahedron. Top schematics show the iodine ions only. Bottom schematics highlight relative Cr orientation between neighboring layers. Note that the layers are shifted fractional lattice coordinates from an ABC stacking, and thus, the Cr ions between the layers are not aligned vertically, whereas it appears so in the s ide view. (d) Temperature dependence of the pre-edge feature, denoted by the vertical arrow, in (a) is replotted. (e) Simulated Cr K XAS spectra for the high temperature monocli nic and low temperature rhombohedral phases. (f) Temperature-dependent XPS results with the incident photon energy of 750 eV. Below the structural transition, the valence band maximum (VBM) shows an extra spectral weight indicated by the arrow, leading to a center of mass (COM) shift of 0.28 eV. The energy level was calibrated by using the I 5d core levels as ref- erence points. To compensate the temperature-dependent charging effects, the low temperature spectrum was shifted to match the 5d core levels. The e ffective energy resolu- tion was better than 0.2 eV.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 022411 (2020); doi: 10.1063/5.0012748 117, 022411-2 Published under license by AIP Publishinginterlayer ordering. These techniques are sensitive to inversion and centrosymmetry breaking, respectively. Our result here shows that the centrosymmetry breaking remains insignificant in both monoclinic and rhombohedral phases, supporting that the origin of the enhanced PL and SHG signals is local spin configuration changes rather than structural ones. Next, we explored a possible link between the iodine electronic structure across the structural transition. Temperature-dependent changes in I L 1,2,3XANES and XLD are shown in Fig. 2 .U n l i k et h e strong temperature dependence in the Cr XANES ( Fig. 1 ), the I L XANES spectra show negligible changes between 40 and 220 K. Moreover, while the XLD reveals anisotropies of the probed 5p (I L 1) and 4d, 5s (I L 2,3) states, there is no clear temperature-dependent changes across the magneto-structural transition. This suggests that changes in the iodine 4d, 5s, and 5p orbital anisotropy across the struc- tural transition are subtle. The schematics in Fig. 1(c) , highlighting the iodine ions, provide clues to the negligible change in XLD. The local environment around the iodine ions remains nearly unchanged between the monoclinic and rhombohedral structures, consistent with the insignificant changes in the iodine L-edge XLD. To further investigate the electronic change across the structural transition, we performed x-ray photoemission spectroscopy (XPS) on CrI3crystals. While XANES is sensitive to the unoccupied states, XPS provides valuable insight into the occupied (valence) states. Temperature-dependent XPS spectra below and above the structural transition are shown in Fig. 1(f) , revealing a noticeable change in the Cr 3d and I 5p features near the valence band maximum (VBM). Upon cooling from 230 to 180 K, the spectral weight shifts toward lower binding energies, indicating changes in the Cr 3d and I 5p states triggered by the structural transition. Band structure calculations have shown a strong iodine character in the valence band9,11,27,28with a strong Cr–I covalency.17Combined with the absence of the tempera- ture dependence in the I 5p states in L1 XLD [ Fig. 2(c) ], the observed weight shift in the XPS spectra suggests potential changes in the Cr 3d features across the structural transition. There is a strong link between the electronic structure near the Fermi level and magnetism, as thep–d covalency and the Cr 3d–I 5p–Cr 3d bond have significant impacts on the anisotropic exchange interactions and consequently on the magnetic anisotropy. 9,15,16Electronic structure calculations have shown that applied strain or magnetic alignment can have effects on the valence band structure.27,29,30Furthermore, recent studies demon- strate the tunability of ferromagnetic and antiferromagnetic ordering in CrI 3via electrostatic doping.31–33 Stemming from strong hybridization between I 5p and Cr 3d, iodine is expected to contribute to magnetism in CrI 3,a ss h o w ni nt h e - oretical,9,11–14,21,34hyperfine field,35and iodine M 4,5XMCD results.15 The XMCD data at the Cr L 2,3edges reveal no discernable Cr 3d orbital moment contribution,15,17and thus, the role of iodine becomes more relevant. In particular, the spin–orbit coupling of the heavy iodine atoms is expected to play a major role in emerging magnetic anisotropy via the anisotropic superexchange9and the induced iodine moment is predicted to be /C00.12–/C00.14lB/I.11–13Along the chro- mium trihalide series CrCl 3, CrBr 3,a n dC r I 3, magnetic anisotropy energy increases, hinting an important role of spin–orbit coupling of the ligands.36,37Utilizing the element and orbital selectivity of XMCD technique, we carried out XMCD measurements at the I L 1,2,3edges, in order to verify the presence of a significant iodine orbital moment.The XMCD measurements were conducted on bulk CrI 3single crys- tals that are about 8 lm thick along the out-of-plane direction (deter- mined from the x-ray transmission measurement across the I L 1 absorption edge). The XMCD spectra were measured at 10 K, with an applied field of 1 Tesla along the easy axis (c-axis) of the CrI 3structure [shown in Fig. 1(c) ]. Considering the strong hybridization between I 5p and Cr 3d states, we first investigated the iodine orbital moment contribution using I L 1XMCD since the iodine 5p state is more directly accessibleFIG. 2. Temperature and polarization dependence of XAS at the I L 3(a), L 2(b), and L 1(c) edges. The inset in each plot shows that the temperature dependence with the incident polarization parallel to in-plane (E jjab).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 022411 (2020); doi: 10.1063/5.0012748 117, 022411-3 Published under license by AIP Publishingat the L 1edge (2s !5p). While XMCD sum rules38–40allow separat- ing orbital (m l¼/C0 h Lzi)a n ds p i n( m s¼/C02hSzi) magnetic moments, at the K and L 1absorption edges, due to the absence of spin–orbit split core levels, XMCD sum rules provide orbital moment contribu- tion only.38,40At the K and L 1absorption edges, the integrated XMCD area is proportional to the p state orbital moment, as demonstrated in As K XMCD studies.41–43In these studies, a positive XMCD peak corresponds to the p-state orbital moment that is antiparallel to the applied field direction. The XMCD sum rule analysis of I L 1in Fig. 3(d) gives an iodine 5p orbital moment of /C00.018 (60.004) lB/I. In calculating the orbital moment, we used one hole in the 5p state. This value is an overestimation, considering that the 5p state is full in the nominal electronic configuration of the I-oxidation state. However, as revealed in the density of state calculation studies, the 5p states in CrI 3remain less than full.9–13,33,44The orbital moment from the XMCD sum rules is simply proportional to the number of 5p holes, and the result can be considered as an upper limit estimatefor the iodine 5p orbital moment. The negative sign of the orbital moment here indicates that the iodine 5p orbital moment is antiparal- lel to the applied field direction and, thus, antiparallel to the dominant Cr 3d spin moment. This resembles the antiparallel alignment between the Mn 3d spin and As 5p orbital moments in MnAs- and Ba 2Mn 2As2-based alloys.41,43 Figures 3(a) and3(b) show sizable XMCD signals at the I L 2,3 edge, corresponding to the 2p !4d,5s transitions. The field depen- dence of the I L 2XMCD in the inset in Fig. 3(b) shows that the signal is saturated well below an applied field of 1 Tesla and is overall consis- tent with the magnetometry result from bulk CrI 3.7The iodine 4d states are nominally full, as evidenced by a negligible XAS peak (whiteline) near the I L 2,3absorption threshold energy. Thus, transitions into states with pure 4d character are unlikely, and it is plausible that the probed state includes the iodine 5s state, which is hybridized with the iodine 5p state. This scenario is supported by the observed iodine mag- netic hyperfine signature, originating from the hybridized iodine 5s-5porbitals, in Mossbauer measurements, 35and in the presence of Cr(3d)- I(5s5p) hybridization in a theoretical study.44Due to uncertainties in the number of I 5s and 4d holes and negligible white-line features in XAS, isolating orbital and spin moment via XMCD sum rules38–40is not straightforward. Instead, we focus on the ratio between orbital and spin moments, which is independent of these factors. The correspond- ing orbital to spin ratio from the I L 2,3XMCD data is estimated to be 6.5 (60.3)%, and the result reveals that the iodine orbital moment is small relative to its spin counterpart. The overall I L XMCD results show clear evidence for induced iodine magnetic moment and, in par- ticular, for iodine 5p orbital moment. Given the hybridized and extended nature of the iodine 5p ligands in CrI 3,t h ei o d i n eo r b i t a l moment may be involved in mediating anisotropic exchange for incip- ient magnetic anisotropy. In summary, element-specific x-ray measurements revealed a measurable orbital moment at the iodine sites, as well as changes in the anisotropy of the Cr local environment across the structural transi-tion. Moreover, below the structural transition temperature, the observed anisotropy increase is accompanied by a XPS spectral weight shift, alluding to potential changes in the Cr 3d states. As shown in the doped tetradymite chalcogenides 45and dilute magnetic semiconduc- tors,46modification in the 3d density of states near the Fermi level can have a significant impact on the p-d exchange magnetic interactions.Our results highlight the electronic environment changes across the structural transition, which may be linked to the incipient magneticordering and anisotropy in the low temperature phase. Our iodineXMCD result provides direct evidence for small iodine 5p orbital moment, alternatively suggesting the prominence of the anisotropicFIG. 3. XMCD at the I L 3,2,1 edges at 10 K, with an external field of 1 Tesla along the c-axis of CrI 3. (a) Iodine L 3XMCD, (b) L 2XMCD, and (c) L 1XMCD. The shaded areas in the XMCD spectra indicate the ranges used for XMCD sum rules. The inset in (b) shows the field dependence of the L 2XMCD at the incident energy (4.857 keV) indicated by the vertical line. (d) Simulated edge jump and peaks usedin the I L 1XMCD sum rule analysis.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 022411 (2020); doi: 10.1063/5.0012748 117, 022411-4 Published under license by AIP Publishingsuperexchange interaction as the main source of the magnetic anisot- ropy in CrI 3, as predicted from the theoretical studies.9,15,16 X-ray absorption and photoemission experiments were carried out at beamlines 4-ID-D, 6-ID-B, and 29-ID of the Advanced Photon Source, Argonne National Laboratory. The work performed at the Advanced Photon Source was supported by the U.S.Department of Energy, Office of Science, and Office of Basic EnergySciences under Contract No. DE-AC02-06CH11357. Crystal growthand characterization at ORNL were supported by the U. S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division under Contract No.DE-AC05-00OR22725. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1T. Song, X. Cai, M. W.-Y. Tu, X. Zhang, B. Huang, N. P. Wilson, K. L. Seyler, L. Zhu, T. Taniguchi, K. Watanabe, M. A. McGuire, D. H. Cobden, D. Xiao, W. Yao, and X. Xu, Science 360, 1214 (2018). 2D. R. Klein, D. MacNeill, J. L. Lado, D. Soriano, E. Navarro-Moratalla, K. Watanabe, T. Taniguchi, S. Manni, P. Canfield, J. Fern /C19andez-Rossier, and P. Jarillo-Herrero, Science 360, 1218 (2018). 3Z. Wang, I. Guti /C19errez-Lezama, N. Ubrig, M. Kroner, M. Gibertini, T. Taniguchi, K. Watanabe, A. Imamo /C21glu, E. Giannini, and A. F. Morpurgo, Nat. Commun. 9, 2516 (2018). 4H. H. Kim, B. Yang, T. Patel, F. Sfigakis, C. Li, S. Tian, H. Lei, and A. W. Tsen, Nano Lett. 18, 4885 (2018). 5N. D. Mermin and H. 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5.0016289.pdf

J. Chem. Phys. 153, 104101 (2020); https://doi.org/10.1063/5.0016289 153, 104101 © 2020 Author(s).The effect of descriptor choice in machine learning models for ionic liquid melting point prediction Cite as: J. Chem. Phys. 153, 104101 (2020); https://doi.org/10.1063/5.0016289 Submitted: 04 June 2020 . Accepted: 13 August 2020 . Published Online: 08 September 2020 Kaycee Low , Rika Kobayashi , and Ekaterina I. Izgorodina COLLECTIONS Paper published as part of the special topic on Machine Learning Meets Chemical Physics ARTICLES YOU MAY BE INTERESTED IN When machine learning meets multiscale modeling in chemical reactions The Journal of Chemical Physics 153, 094117 (2020); https://doi.org/10.1063/5.0015779 Committee neural network potentials control generalization errors and enable active learning The Journal of Chemical Physics 153, 104105 (2020); https://doi.org/10.1063/5.0016004 Machine learning approaches for structural and thermodynamic properties of a Lennard- Jones fluid The Journal of Chemical Physics 153, 104502 (2020); https://doi.org/10.1063/5.0017894The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The effect of descriptor choice in machine learning models for ionic liquid melting point prediction Cite as: J. Chem. Phys. 153, 104101 (2020); doi: 10.1063/5.0016289 Submitted: 4 June 2020 •Accepted: 13 August 2020 • Published Online: 8 September 2020 Kaycee Low,1 Rika Kobayashi,2 and Ekaterina I. Izgorodina1,a) AFFILIATIONS 1Monash Computational Chemistry Group, Monash University, 17 Rainforest Walk, Clayton, VIC 3800, Australia 2ANU Supercomputer Facility, Leonard Huxley Building 56, Mills Road, Canberra, ACT 2601, Australia Note: This paper is part of the JCP Special Topic on Machine Learning Meets Chemical Physics. a)Author to whom correspondence should be addressed: katya.pas@monash.edu. URL: http://www.https://mccg.erc.monash.edu ABSTRACT The characterization of an ionic liquid’s properties based on structural information is a longstanding goal of computational chemistry, which has received much focus from ab initio and molecular dynamics calculations. This work examines kernel ridge regression models built from an experimental dataset of 2212 ionic liquid melting points consisting of diverse ion types. Structural descriptors, which have been shown to predict quantum mechanical properties of small neutral molecules within chemical accuracy, benefit from the addition of first-principles data related to the target property (molecular orbital energy, charge density profile, and interaction energy based on the geometry of a single ion pair) when predicting the melting point of ionic liquids. Out of the two chosen structural descriptors, ECFP4 circular fingerprints and the Coulomb matrix, the addition of molecular orbital energies and all quantum mechanical data to each descriptor, respectively, increases the accuracy of surrogate models for melting point prediction compared to using the structural descriptors alone. The best model, based on ECFP4 and molecular orbital energies, predicts ionic liquid melting points with an average mean absolute error of 29 K and, unlike group contribution methods, which have achieved similar results, is applicable to any type of ionic liquid. Published under license by AIP Publishing. https://doi.org/10.1063/5.0016289 .,s I. INTRODUCTION Ionic liquids, which display a wide range of physico-chemical properties due to their unique structure, have incited much research into their characterization for use as new-generation solvents and catalysts. One of the main focuses is on their potential as high- conductivity, low-volatility electrolytes, which requires properties such as low viscosity and low melting temperature for practical applications.1,2Considering the number of possible ionic liquids, estimated to be in the trillions, synthesis of every possible cation and anion combination would be a near endless task.3The prediction of thermochemical and physical properties of an ionic liquid based on only its cation and anion structure represents one of the great challenges in computational chemistry, with progress being made in the fields of molecular dynamics, ab initio MD, and quantumchemical calculations.4–6However, the timescale required for these rigorous methods still exceeds what is desirable for high-throughput screening. As an alternative, machine learning algorithms have been making strides in predicting materials properties using a small set of reference calculations only, requiring inputs such as chemical formula descriptors, or, in the case of graph networks, molecular connectivity information.7,8 Recent research into chemical machine learning has resulted in significant successes in predicting materials properties, includ- ing but not limited to crystal lattice energies,9thermal conductiv- ities of perovskites,10and melting temperature of solids.11,12Ionic liquids present a particularly unique challenge, being a cocktail of interactions including electrostatic repulsion and attraction between like-charged and oppositely charged ions, hydrogen bonding, and London dispersion forces between cations and anions, and alkyl J. Chem. Phys. 153, 104101 (2020); doi: 10.1063/5.0016289 153, 104101-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp groups on the cation.13Experimentally, the range of properties such as density and conductivity exhibited by ionic liquids is wider when compared with that of standard solvents and electrolytes. Thus, the relationship between the individual cation and anion structure and bulk ionic liquid properties may not be straightfor- ward to elucidate via machine learning. Previous work in this field has focused on relating experimental or calculated properties of the ions, such as molecular volume or HOMO–LUMO energies,14 to observed properties such as viscosity or melting temperature via quantitative structure–property relationship (QSPR) studies.15,16 Throughout these studies, there has been little focus on the ratio- nale behind the selection of descriptors and effect of the descriptor on the machine learning model. Descriptor choice varies accord- ing to the study at hand, and considerations such as time and computational resources. Numerical vectors built using molecular properties have long been utilized in the quantitative structure– activity and structure–property relationship (QSAR and QSPR) fields,17though these descriptors can miss often crucial informa- tion about the substructure and connectivity of molecules.18While incorporating quantum mechanical (QM) information is fairly com- mon, many studies have kept to lower levels of theory and semi- empirical optimizations using methods such as PM6 or PM7.19,20 To reduce computational cost, the dataset size is frequently lim- ited to several hundred ionic liquids or smaller.16,21Other than numerical vector descriptors built from molecular and electronic properties, descriptors can be based on the one-dimensional, two- dimensional, or three-dimensional structure of a molecule: atomic formula; bonding and connectivity; or 3D geometry, respectively. One-dimensional descriptors constructed from the chemical for- mula, such as some molecular fingerprints, require little effort when inputting these easily computed features such as the presence or absence of a functional group into a bit vector. Including molecular structure information such as size, shape, cyclicity, and symmetry provides a two-dimensional descriptor. Two-dimensional molecu- lar fingerprints encode graph structural information, including a larger radius over two or more bonds, compared to one-dimensional fingerprints.17Since one-dimensional and two-dimensional descrip- tors do not include stereochemical information, recently developed three-dimensional descriptors encode the structure and bonding, e.g., the Coulomb matrix22(CM) or the bag of bonds (BoB) model,23 though these require optimized geometries of the molecules of interest. Three-dimensional descriptors eliminate the need to cre- ate hand-engineered vectors and have achieved impressive accuracy when applied to electronic molecular properties such as atomization energy,24,25though they have not yet been applied to ionic liquid systems. The application of 3D descriptors in machine learning has mostly been limited to small, neutral molecules, in learning prob- lems where the exact form of the property is known and the aim is to reduce computational expense via machine learning, for example, in predicting the atomization energy of molecules.22In this work, we apply such structural descriptors to assess whether or not they are viable for use in surrogate models where the relationship between the input (structure) and the target property (melting point) is not known. Beyond the choice of descriptor, there still remains the choice of an appropriate machine learning algorithm. As stated by the no free lunch theorem,26each method performs differently given a different application. As the purpose of this work is not tocompare the performance of different models but rather the effect of descriptor choice on such models, we choose kernel ridge regression (KRR), a regression algorithm that has been widely used for pre- dictions of materials properties, including molecular orbital27and atomization energies.28The relatively few hyperparameters (essen- tially the choice of kernel function, the kernel width, and a reg- ularization parameter) make KRR ideal for a study attempting to deconvolute the effect of descriptor choice from the machine learn- ing algorithm. While end-to-end learning using deep neural net- works, where the representation is learned from the input data, has exploded in popularity in fields where millions of data points are available—such as image search and text-to-speech—in the scien- tific literature, such volumes of data are rare. A comparison of KRR with a deep neural network would be ideal to evaluate descrip- tor performance but is not possible at this stage as deep learning suffers in performance with smaller datasets.29Hence, this work focuses on the effect of descriptor choice applied to a small dataset of∼2000 experimental ionic liquid melting points using the KRR algorithm. With a plethora of descriptors and algorithms available for use, this paper rationalizes the effect of each descriptor type on the performance of regression models applied to predicting the melt- ing temperature of ionic liquids from a relatively small (in machine learning terms) experimental dataset. We evaluate the “accuracy” of given descriptors (here, 1D vector-based descriptors are considered to be lower in accuracy than quantum-chemical ones) with respect to their performance in several machine learning algorithms. The outputs of quantum mechanical calculations are used in addition to the structural descriptors, with the aim of overcoming the errors associated with a limited size dataset in machine learning. Although several studies have shown the accuracy of structure-encoding descriptors for large datasets typically with tens of thousands of sam- ples,18,27this work utilizes a small dataset, representative of what can be measured experimentally rather than using theoretical data. By adding molecular information via ab initio calculations such as fron- tier orbital energies and interaction energies to simple 1D-chemical and 2D-chemical formula-based descriptors, we consider whether machine learning models applied to a dataset of limited size can make increasingly accurate predictions aided by the information included in high-level calculations. Few have looked at the combi- nation of 3D descriptors with ab initio data, though in the same vein Tchagang and Valdés recently showed that the combination of the Coulomb matrix and atomic composition improved prediction of atomization energy by an average of 0.5 kcal/mol for molecules in the QM7 database.30Additionally, we investigate whether 3D descriptors encoding structural features that have achieved near chemical accuracy for predicting quantum mechanical properties—a relatively noise-less property—can perform well for the experimen- tal property of melting temperature, which is likely to have noise in the measured value. II. DATASET AND MOLECULAR DESCRIPTORS The dataset used in this work is sourced from a study of Venka- traman et al. , which used PM6-calculated descriptors for predicting ionic liquid melting points using machine learning.312212 experi- mental values were extracted from over 300 literature sources, with J. Chem. Phys. 153, 104101 (2020); doi: 10.1063/5.0016289 153, 104101-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp ionic liquids containing various types of ions and a melting point range from 177 K to 632 K. Any experimental measurement such as melting point characterization may have its result influenced by impurities and the measurement technique, and discrepancies in the experimental data are common. Though steps were taken to reduce discrepancies in the dataset, such as taking the most frequent melt- ing point for ionic liquids with multiple reported values, there is still likely some amount of error present (further details on dataset clean- ing and refinement can be found within the source paper, Ref. 31). The difference in experimental melting point temperatures mea- sured for a single ionic liquid can vary greatly: by around 10 K–20 K in a good example, or by up to 70 K for a temperamental ionic liq- uid type.32This is obviously an issue for machine learning models as errors impart noise into the data. Hence, choosing an appropriate descriptor that works even when applied to noisy data is essential for a successful ionic liquid predictive model. Distribution of the target property, melting point, in this dataset is shown in Fig. 1. The mean melting temperature is 361 K, with a standard deviation of 78 K. Further separation of the dataset by cation type reveals that the majority of ionic liquids are imida- zolium or ammonium cation based, which is expected as these are the most widely studied cations in the field of ionic liquids. This may cause problems for the ML model, as the dataset is not so chem- ically diverse. A model trained on mostly ammonium-based and imidazolium-based ionic liquids is likely to make errors when pre- dicting values for the underrepresented ions such as sulfonium and phosphonium, which we investigate further in Sec. IV C. The simplest types of descriptors utilize only atom types in a molecule (1D), or atom and bonding information (2D) stored FIG. 1 . Distribution of the ionic liquid melting point in the entire dataset (top) and separated by cation type (bottom).either in a vector indicating the presence/absence of a group or as a fingerprint array. The appeal of these descriptors is that the three-dimensional structure of the molecule is not needed; a line drawing or SMILES string is sufficient to extract relevant infor- mation. These fingerprints are popular in cheminformatics as they facilitate simple substructure searches. One of the most prominent examples is the extended connectivity fingerprint (ECFP),33which has well-established performance in virtual screening, for example, when identifying compounds with similar bioactivity. In this study, we use ECFP4, which encodes functional groups up to four bonds away from the central atom, as this has performed well in benchmark virtual screening studies.34The ECFP4 fingerprint was encoded as a 2048-bit vector as implemented in rdkit ,35as this length was found to give better performance compared to all available shorter lengths. A plot of bit vector length vs cross-validated mean average error showing that the 2048-bit vector length produces the lowest error is included in the supplementary material (Fig. S1). 3D geometrical descriptors have been shown to lead to good performance for the prediction of thermodynamic and electronic properties.18,25The Coulomb matrix36(CM) encodes the three- dimensional geometry of the molecule in a square matrix Mwhere the off-diagonal elements correspond to the nuclear repulsion between atoms, whereas diagonal elements represent the electronic potential energy of the free atom, MIJ=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩0.5Z2.4 I, I = J ZIZJ ∣RI−RJ∣, I≠J.(1) The CM implementation in the qmmlpack package36was used, with rows of the matrix sorted by their L1-norm. To obtain the geometry required for 3D descriptors as well as ab initio values, structures of the cations and anions and their corresponding neu- tral single ion pairs were optimized at the M06-2X/cc-pVDZ level of theory in G AUSSIAN1637using an ethanol solvent, using the conductor- like polarizable continuum (CPCM) solvation model.38,39All indi- vidual ions were screened for the lowest possible energy conforma- tion. Where more than one interaction site between the cation and the ion was possible, multiple configurations were optimized and the configuration with lowest energy was chosen for further anal- ysis. The supplementary material contains exact details of geome- try optimizations. Single point energy calculations to extract inter- action energies for use as ab initio descriptors were conducted at the SRS-MP2/cc-pVTZ level of theory using counterpoise-corrected HF/cc-pVTZ for the calculation of Hartree–Fock interaction energy (for more details, see Ref. 40). The quantum mechanical descriptors including the interaction energy, density functional theory (DFT)- computed HOMO–LUMO energies and bandgap, and charge den- sity distribution ( σ-profile) were calculated for both the single ions and the ion pair. As the melting point of an ionic liquid is largely the product of inter-molecular and intra-molecular interactions, the quantum mechanical descriptors were chosen to best represent these interactions in numerical form, based on our chemical intuition. The σ-profile, originally developed by Klamt,41is calculated as the distribution pi(σ)=Ai(σ) Ai, where Ai(σ)is the area on the surface of the molecule with charge density σand Aiis the total cavity surface area. Sigma profiles have previously been used in a number of ionic J. Chem. Phys. 153, 104101 (2020); doi: 10.1063/5.0016289 153, 104101-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp liquid machine learning studies as a descriptor and are assumed to have some correlation with properties such as melting point and viscosity.16,42 The diversity of the dataset as represented by the CM descrip- tor for the ionic liquid ion pair is shown using the t-distributed Stochastic Neighbor Embedding (t-SNE) technique in two dimen- sions. t-SNE is a method of dimensionality reduction that seeks to preserve the distances between neighboring points during projection of a high-dimensional data space to a lower dimensional one.43As shown in Fig. 2(a), there are several obvious clusters of ionic liquid, which have been grouped by anion type according to the t-SNE anal- ysis. Interestingly, the t-SNE dimensions seem to correspond mostly to the anion contribution, and the contribution from the cation is not clear (t-SNE plot colored by cation type shows no clear clus- ters; this plot is included in the supplementary material, Fig. S2). The most common anion is bromide, and as can be seen in Fig. 2(b), these ionic liquids tend to be associated with a higher melting tem- perature and are grouped around the outer edges of the t-SNE plot. The average melting point for the bromide-based ionic liquids is 423 K, compared to the average melting point of 361 K for the whole dataset. Outside of the clusters belonging to each anion, the inter- cluster distance is relatively large by comparison, indicating that there are structural discontinuities in the dataset as represented by the CM. This is expected for a varied dataset of this size where the majority of cations and anions belong to one or two chemical groups. This may cause problems when predicting the melting point of less frequently found ions as fewer training examples will be present. For multi-component systems such as ionic liquids, the ques- tion of how to best represent the ion pair via descriptor choice arises. Many of the descriptors used in materials science do not include information about which atom belongs to which molecule. One solution is to treat the cation–anion ion pair as a single entity, for example, by constructing a single Coulomb matrix or a single fingerprint based on the single ion pair geometry, and use this as the input to the ML model. Alternatively, separate descriptors can be constructed for the cation and anion individually, resulting in two vectors that are concatenated together to provide a single vector input to the ML model. Both of these options have been investigated in the following.III. MACHINE LEARNING MODEL Due to its computational efficiency and good performance in previous materials prediction studies, we base all machine learn- ing models on the kernel ridge regression (KRR) algorithm.7,44In essence, KRR establishes a mapping between the input features and the target property by projecting the inputs and outputs into a high- dimensional space where a relationship is learned. It is assumed that similar compounds, defined by a distance measure in the high- dimensional space, should exhibit similar properties. An estimate for the target property pof a compound xtestnot in the training set is obtained as the weighted sum of the Nkernel functions placed on each training compound xi, p(xtest)=N ∑ i=1αiK(xtest,xi). (2) Solutions for the coefficients aiare obtained from the training process that minimizes the expression N ∑ i=1(ppred(xi)−ptrue(xi))2+λN ∑ i=1α2 i, (3) where λis a regularization parameter that penalizes larger regression coefficients and complex models.45We use KRR as implemented in the scikit -learn package46with a Laplacian kernel, which is defined by KLaplacian (xi,xj)=exp−∥xi−xj∥1 2σ2, (4) where the distance measure in the exponential term is the L1-norm. σis the kernel width, and the second optimizable hyperparameter in KRR. The hyperparameters λand σwere optimized over a square grid from 10−10to 10−1using fivefold cross-validation. The available data were split into a training set (80%) with the remaining 20% as a test set. The hyperparameters with best performance as determined via cross-validation were then applied to a model, which was trained on the entire training set, and then the test set was applied to make predictions. The final test error values as presented in Sec. IV cor- respond to these test set predictions. No test data were used in the generation or tuning of the models. To examine the robustness of FIG. 2 . t-SNE projection of the ionic liquid database into two dimensions as represented by the Coulomb matrix descriptor. Points are colored by anion type [(a) “other” corresponds to infrequent anions such as Al−and As−] and melting point value (b). J. Chem. Phys. 153, 104101 (2020); doi: 10.1063/5.0016289 153, 104101-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp each model and account for variations in prediction associated with randomly splitting the data into training and test sets, the results for each model have been averaged over 100 different randomly selected training/test splits. Hyperparameters for each model can be found in the supplementary material. IV. RESULTS AND DISCUSSION This section presents the results of each KRR model beginning with the structural descriptors (ECFP4 and CM) alone, followed by the results of the model when quantum mechanical information is added. The standard metrics presented are the test mean absolute error (MAE), test root mean square error (RMSE) as defined in Eq. (5), and test R2score. The standard deviation associated with the mean of each metric over 100 random training/test splits is also provided. RMSE=⌟roo⟪⟪op ⌟roo⟪mo⟨⌟roo⟪mo⟨⌟roo⟪mo⟨⌟roo⟪⟨o⟪1 NN ∑ p=1(ptrue−ppred). (5) The notation in the following tables and figures is as follows: the CAsuffix added to the descriptor indicates that the cation and anion of the ionic liquid have been treated separately (e.g., optimized as separate ions), and the ILsuffix indicates that the ionic liquid was treated as a single ion pair. Of the structural descriptors, the descriptor with the best per- formance for predicting ionic liquid melting points was the ECFP4 fingerprint. As shown in Table I, by comparison, the performance of the KRR model decreases by up to 6 K on average when using the CM descriptor. The CM is based on interatomic distances and charge and was originally developed for predicting atomization energies, a property based on exactly those values.22Hence, it is not so surprising that there is a weak relationship between the CM and the melting point. By comparison, the ECFP4 fingerprint performs considerably better. Fingerprints, which were developed for sub- structure searching, indicate the presence or absence of functional groups and bonding information in molecules. For ionic liquids, the relationship between the melting point and functionality is evident, as shown by the R2value of 0.74 ±0.02 for ECFP4- CA. This is likely one of the reasons for the failure of the CM: since all interatomic distances in the molecule are treated equally [as seen in Eq. (1)], the KRR model is not able to learn relationships based on simi- lar functional groups, whereas functionality is a key feature of the ECFP4 descriptor that focuses more on the local environment and TABLE I . Performance of structural descriptors for out-of-sample ionic liquid melt- ing point predictions. Errors and R2value are reported as the average over 100 repetitions, accompanied by the standard deviation of the mean. Descriptor MAE (K) RMSE (K) R2 ECFP4- IL 33.20 ±1.40 43.77 ±2.00 0.68 ±0.02 ECFP4- CA 29.78 ±1.20 39.84 ±1.81 0.74 ±0.02 CM-IL 39.92 ±1.29 52.11 ±2.01 0.55 ±0.03 CM-CA 35.92 ±1.44 47.30 ±2.22 0.63 ±0.03bonding. Interestingly, for both descriptors, a greater predictive accuracy is associated with the use of the CA formulation com- pared to ILresults. For the ECFP4 fingerprint, this is probably due to the fingerprinting algorithm that removes duplicate substructures present in both the cation and the anion; the overall result is the loss of important structural information for one of the ions. When the cation and anion are described with separate fingerprints, sub- structures present in both the cation and the anion would not be considered duplicates and would thus be present in both vectors. The CM has a similar result, showing an improvement in the predic- tion accuracy when individual ion geometries are used. This could be due to the inflexibility of an attempt to represent all possible ionic liquid interactions through the geometry of a single ion pair, when multiple ion configurations are possible due to the importance of induction and dispersion forces.47The use of individual ion geome- tries does not constrain the ionic liquid to being described through a single ion pair configuration of the multiple possible, resulting in improved accuracy. Perhaps, the biggest issue with the CM descriptor is in the use of standard atomic charges to describe ionic liquids, a topic that has been widely considered both experimentally and computationally in the field.48,49It is well known that atomic charges in ionic liq- uids fluctuate significantly, leading to fractional atomic charges.50,51 Hence, the use of standard whole charges as in the CM formula- tion would not be representative of the true atomic charges in the ionic liquid. The fluctuation of atomic charges in ionic liquids occurs on both an inter-molecular and an intra-molecular scale. To include a charge-based descriptor, we have considered the σ-profile for the individual ions and ion pair. These, and the other chosen quantum mechanical descriptors, are discussed in Sec. IV A. A. Quantum mechanical features In theory, the σ-profile of an ionic liquid should be the sum of theσ-profiles of the individual ions.42However, as significant charge transfer may occur between the single ion pair to a greater extent than present in bulk ionic liquids, we have decided to compare the sigma profile of the individual ions with that of the ion pair, analo- gous to the ILand the CAformulation of the structural descriptors. 51 numeric values describing the ionic liquid or ion’s σ-profile were included, within a range from −0.025 to 0.025 e/Å2, which is the commonly used range as based on the σ-profile of water.52Molec- ular orbital data refer to the energy of the HOMO, LUMO, and band gap in eV for the single ion pair, calculated using M06-2X/cc-pVDZ. Interaction energy data contain a value representing the strength and nature of interaction between the cation and the anion: the ratio of the correlation (i.e., dispersion) interaction energy to the total interaction energyIEtotal IECorr, where the total interaction energy and correlation interaction energy are defined as follows: IEtotal=Eion pair −(Eanion + E cation), (6) IEtotal=IEHartreeâ ˘A¸ SFock + IE Correlation . (7) We have previously shown that the ratioIEtotal IECorris correlated with the melting point for clusters of ionic liquids containing two ion pairs.53Results of the structural descriptors combined with quan- tum mechanical data are shown in Table II. As the CAdescriptors J. Chem. Phys. 153, 104101 (2020); doi: 10.1063/5.0016289 153, 104101-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II . Performance of structural descriptors with additional quantum mechani- cal features for out-of-sample ionic liquid melting point predictions. Errors and R2 value are reported as the average over 100 repetitions, accompanied by the standard deviation of the mean. QM features MAE (K) RMSE (K) R2 ECFP4- CAσ-profile- IL 31.64 ±0.95 41.79 ±1.75 0.71 ±0.02 σ-profile- CA 30.72 ±1.11 41.02 ±1.77 0.72 ±0.03 MO energy 29.15 ±1.06 39.50 ±1.82 0.74 ±0.02 Int. energy 29.62 ±0.99 39.88 ±1.72 0.74 ±0.02 All QM 30.73 ±1.08 41.07 ±1.72 0.74 ±0.02 CM-CAσ-profile- IL 35.03 ±1.28 47.96 ±2.61 0.62 ±0.04 σ-profile- CA 35.00 ±1.30 47.84 ±2.64 0.62 ±0.04 MO energy 36.05 ±1.39 49.02 ±2.60 0.60 ±0.04 Int. energy 36.09 ±1.39 49.07 ±2.60 0.60 ±0.04 All QM 34.82 ±1.33 45.85 ±2.19 0.65 ±0.03 showed better prediction accuracy than ILdescriptors based on the results in Table I, and this pattern was also observed upon addi- tion of QM descriptors, only results for ECFP4- CAand CM- CAare shown in Table II. Results from ECFP4- ILand CM- ILmodels with QM descriptors can be found in the supplementary material. In this section, each QM descriptor has been examined individually, and the combination of all QM descriptors ( σ-profile- CA, molecular orbital energies, and interaction energy ratio) is also included. Parity plots for the predicted vs true values corresponding to the first run out of 100 random splits are shown for the best performing ECFP4 and CM models in Fig. 3. Although the σ-profile is a popular descriptor in the ionic liquid QSAR community for predicting physicochemical properties,16,42in ECFP4 models, there is no appreciable increase in prediction accu- racy when adding the σ-profile to structural descriptors: the test MAE increases slightly, by an average of 0.94 K. Using σ-profiles alone as inputs to KRR results in MAE values between 34 K and 52 K (see the supplementary material for KRR models using only ab initio data), with poorer results for the σ-profile- ILdescriptor. The use of σ-profile- CA however gives the best QM-only model, reinforcing the notion that an apt description of ion charge distri- bution is necessary to predict the ionic liquid melting temperature. From a chemical point of view, the charge density of each moleculeis likely to play an important role in determining the melting point of a liquid. In the case of ionic liquids however, which have regions of inhomogeneity throughout the bulk structure,54different nanos- tructural organization within the ionic liquid would produce slightly different σ-profiles for each ion pair depending on its surround- ings and configuration, a complex effect that cannot be captured through a σ-profile calculation of a single ion pair geometry. Hence, we observed that the σ-profile for the individual ions gave better results than that of the single ion pair. The CM descriptor in combi- nation with the σ-profile produces a slightly better model compared to using the CM alone, with the test MAE decreasing by an average of 0.92 K. This indicates that incorporating information about charge distribution within an ionic liquid is important for this particular descriptor, which otherwise contains an overly simplistic illustration of the atomic charges for ionic liquid systems. A small increase in prediction accuracy is afforded to the ECFP4 descriptor by addition of molecular orbital energies. On average, the ECFP4- CA test MAE decreased by 0.63 K. The CM showed an insignificant increase in accuracy with the addition of molecular orbitals, by 0.13 K on average. At first glance, there is no clear correlation between the melting points in the dataset and the energy of the frontier orbitals of the ion pair, and a KRR model based on only these three values produces weak test scores (MAE = 45.26 ±1.65 K, see the supplementary material for full met- rics). The energy of the bandgap is related to the stability of the ionic liquid, and HOMO/LUMO energies are related to the overall molecular structure. While there is no linear correlation between the bandgap and the melting point as shown in Fig. 4, there may be some non-linear correlation between the combination of frontier orbital energies and the ECFP4 descriptor, which is uncovered through the kernel regression. The use of the ECFP4 descriptor, which encodes functionality and local bonding, plus the frontier orbital energies providing some information about the overall molecu- lar structure, suggests that the combination of this information increases predictive power in comparison with using each descriptor alone. Including the correlation interaction energy ratio has a surpris- ingly small effect on performance metrics. Examining the relation- ship between the melting point andIEtot IEcorr, as shown in Fig. 5, provides some insight as to why. There does not appear to be a strong cor- relation between the two values, linear or otherwise. The majority of ionic liquids are clustered around aIEtot IEcorrratio of 10 but display a wide range of melting points from 177 K to 632 K. This type FIG. 3 . Parity plots comparing the predicted melting point values with the corresponding experimental data for the best performing ECFP4 (a) and CM (b) models. Plots correspond to predictions for the first run out of 100 repetitions. J. Chem. Phys. 153, 104101 (2020); doi: 10.1063/5.0016289 153, 104101-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . Plots of the frontier orbital energy [(a) highest occupied molecular orbital (HOMO) and (b) lowest unoccupied molecular orbital (LUMO)] and (c) bandgap energy (eV) vs melting point (K) for the ion pairs in the dataset. of relationship is unlikely to be a successful input for kernel ridge regression, which is reinforced by the metrics of the models built from the interaction energy ratio alone: MAE = 57.00 ±1.91 K, R2 = 0.14 ±0.02. Searching over several types of kernels (linear, Gaus- sian, and Laplacian) fails to improve these results. The plots of molecular orbital energies show a similar trend with weak corre- lation, but the addition of MO data improves model performance. Of course, it is given that—similar to the results of the σ-profile— a description of interaction energy using a single ion pair confor- mation is an overly simplistic approximation. In contrast, frontier orbital energies and bandgap remain the same regardless of the number of ions present. Therefore, when calculating interaction energies in the future work for use in these models, it would be worthwhile to consider the possible range of interaction energies and furthermore use a two ion pair cluster rather than a single ion pair, as this would likely display a stronger relationship with the melting point. The difference in the approach of the two structural descrip- tors ECFP4 and CM is highlighted by prediction metrics when all quantum mechanical data are included in the models. In the case of the ECFP4 descriptor, accuracy decreases slightly, whereas for the CM, the addition of the QM data results in the best perform- ing model of all: the MAE decreases by 1.20 K on average and the RMSE by 2.55 K compared to those in the CM model alone. This FIG. 5 . Plot of the SRS-MP2/cc-pVTZ dispersion interaction energy ratio vs melting temperature (K) as calculated for single ion pairs in the dataset.indicates that newer 3D structural descriptors may benefit predic- tive accuracy by adding simple but high-quality quantum mechan- ical data that are related to the property of interest. Of course, the biggest problem lies in filtering through data that are relevant and discarding those that add unhelpful noise to the model. As we have shown, some first-principles data that intuitively correspond to the target property from a chemical point of view—for example, inter- action energy and melting point—may prove to not have shown any correlation when incorporated in a machine learning model. The performance of the two structural descriptors reflects their his- tory and intended use, with one (ECFP4) having been developed for use in the QSAR/QSPR field to correlate the molecular substruc- ture with an experimental value such as drug potency, which as with any experiment measurement would have a high degree of noise in the measurement. On the other hand, the CM was developed for predicting results of quantum mechanical calculations, which are essentially noiseless. For this ionic liquid melting point dataset, where there is likely some amount of noise due to variation in the melting point characterization method as well as the presence of impurities, ECFP4 proves its worth as a structural descriptor on its own with excellent performance without any QM data added to this dataset. There is a certainty merit in the addition of QM data to the CM however, as this provides the best performing CM model. B. Error analysis of best performing models As the range of ions in the investigated dataset is dispropor- tionate in its representation of certain ions, we investigate in this section whether the above models perform better on certain subsec- tions of the dataset. For example, 822 of the 2212 ionic liquids have a bromine anion; hence, it is possible that the model would have a lower test error for bromine-based ionic liquids. To test this hypoth- esis, we have plotted the histogram of errors for the best performing ECFP4 and CM models for the first model from the 100 training/test split runs. In Fig. 6, the orange ticks on each histogram represent the error for a bromide-based ionic liquid in the test set. While the range reduces somewhat, the errors are still for the most part distributed equally within the possible range of errors, which is [ −113, 172] for the ECFP4 model and [ −92, 152] for the CM model. Further nar- rowing down the ionic liquid type to be a bromide anion with an imidazolium cation (one of the most common cation types in the dataset) is shown by the red ticks on the right-hand side of Fig. 6. As can be seen in the two histograms, the range of residual errors nar- rows down slightly, from [ −113, 117] to [ −84, 117], for the ECFP4 model but does not change for the CM model. As both ranges still encompass most of the entire residual error range, we can assume that the models are not biased toward a certain type of ionic liquid, despite the prevalence of both bromide and imidazolium ions in the dataset. This lack of predictive accuracy is likely due to descriptor choice. For further analysis, we examine the identity of the worst- performing ionic liquids for each model, as this can provide some insight into the shortfalls of the model in terms of what features are not being captured by the chosen descriptors. The structure of these four ionic liquids is shown in Fig. 7. For the ECFP4 model, the ionic liquid with the highest residual error (+172 K) is structurally not complex: an ammonium cation with a pyridinylmethanolate anion J. Chem. Phys. 153, 104101 (2020); doi: 10.1063/5.0016289 153, 104101-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 6 . Histogram of residual errors (y true −ypred) for the best performing ECFP4 model [ECFP4 + MO] (top) and the best performing CM model [CM + all QM] (bot- tom). Errors corresponding to bromide- based ionic liquids and imidazolium– bromide based ionic liquids are high- lighted in the plots using orange and red ticks, respectively. [Fig. 7(a)]. Incidentally, this is also the worst ionic liquid for the CM model, with a slightly higher residual error of +177 K. The worst negative residual error for ECFP4 is associated with a pyridinium bromide ionic liquid [Fig. 7(b), −106 K]. For the CM model, the largest negative residual error −106 K occurs for triazolium nitrite [Fig. 7(d)]. There appear to be some similarities and differences in the types of molecules that each model predicts poorly. For the ECFP4 model, a pyridine ring appears in both ionic liquids, whereas the CM appears to perform worse with systems with diffuse charges: both the formate and nitrate anions as well as the triazolium cation are all highly charge-delocalized structures. Figure 6 shows that the CM model has several outliers grouped on the outer right of the histogram with residual errors greater than 170 K: two are ammo- nium formate ionic liquids—we describe issues with representing ammonium systems in the following paragraph—and one is sulfo- nium tetrafluoroborate, another highly charge-diffused system. This could reflect the shortfalls of the CM in describing these ionic sys- tems, which as discussed previously require a non-static descrip- tion of charges. The future work will investigate the use of newer 3D descriptors, which include the atomic charge but lack the fitted exponent term present in the CM [Eq. (1)], which allows for the FIG. 7 . Ionic liquids with the largest positive and negative residual errors for the ECFP4 model: (a) ammonium cation and formate anion (+172 K) and (b) pyri- dinium bromide ( −106 K), and the CM model: (c) the same ammonium formate ionic liquid as for ECFP4 (+177 K) and (d) triazolium cation with nitrate anion (−106 K). Atom coloring: carbon = gray, hydrogen = white, oxygen = red, nitrogen = blue, and bromine = burgundy.substitution of fixed atomic charges with quantum-mechanically determined ones. For ionic liquids, we have previously shown that the Geodesic charge scheme can best capture the extent of charge transfer and delocalization present in ionic liquids, out of sev- eral available partial charge schemes.55Substituting a fixed, stan- dard charge description with Geodesic charges is likely to have a positive effect on prediction accuracy based on assessment of the worst-performing ionic liquids for the CM model. Considering the worst-performing ionic liquids for the ECFP4 model, the presence of elements and functional groups (pyridine, carboxylate, bromide) in both ionic liquids that are relatively com- mon throughout the entire dataset indicates that something in the descriptor, in either the cation–anion ECFP4 fingerprint or the added frontier orbital energies, lacks the detailed structural encod- ing required to learn the relationship between the descriptor and the melting point. Examining plots of the residual error vs frontier orbital energy (HOMO, LUMO, and bandgap) for each ionic liq- uid in the test set, Fig. 8, it can be seen that different ionic liquid types with similar frontier orbital energy values to the ionic liquids in Figs. 7(a) and 7(b) have much smaller residual errors. Narrowing these down to focus only on ionic liquids similar to those in Fig. 7(a), with ammonium formate ions, colored blue in Fig. 8, shows that these types of ions have a wide distribution of residual errors: sev- eral are scattered about the y= 0 residual error line, but equally three ammonium formate ionic liquids have some of the highest residual errors of all test set examples. Similarly, the errors associated with pyridinium bromide ionic (orange dots in Fig. 8) are randomly scat- tered around the y= 0 line, though the range of errors is not as large compared to that of ammonium formate ionic liquids. As sim- ilar ionic liquid types with similar frontier orbital energies can be associated with lower residual errors, we speculate that it is either the description provided by the ECFP4 fingerprint or the use of the single cation–anion representation—or, likely, a combination of the two—which lacks key structural information needed for prediction of the target property, melting point. In terms of the cation–anion representation, molecular dynamics simulations have shown that J. Chem. Phys. 153, 104101 (2020); doi: 10.1063/5.0016289 153, 104101-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 8 . Residual errors (K) for all test set ionic liquids and their frontier orbital energies (eV). Frontier orbital values corresponding to the ionic liquid with the largest positive error [Fig. 7(a)] and largest negative residual error [Fig. 7(a)] for the ECFP4-MO model are highlighted in blue and orange, respectively. Similar ionic liquid types are shown by the same colored markers, and other ionic liquid types are shown in gray. in ammonium ionic liquids such as that in Fig. 7(a), alkyl–alkyl chain aggregation interactions can overtake cation–anion interac- tions, especially for chain lengths with n= 14 or larger.54,56Equally, ionic liquids with a pyridinium cation such as that in Fig. 7(b) are known to have a large dispersion component in the interaction ener- gies of bulk clusters.47For these systems, many-body effects are par- ticularly important in their contribution to the melting point. Hence, the representation of the bulk structure using a single ion results in significant errors, which is emphasized through the fact that pyri- dinium ionic liquids delete have high residual errors in both ECFP4 and CM models. In terms of the ECFP4 descriptor itself, it is possible that not including information about molecular 3D geometry contributes to these errors, as the ECFP4 descriptor is only based on bonding infor- mation. This would result in erroneous predictions, for example, for chiral ionic liquids, which would have the same ECFP4 repre- sentation but display different properties.57Given this information based on analysis of each model’s errors, we can suppose that an ideal structural descriptor for predicting ionic liquid melting points should contain information including 3D geometry, as in the CM, in addition to 2D bonding information over four or more bonds, as in ECFP4. For interest’s sake, the results of a model combin- ing the ECFP4-CA and CM-CA descriptors were trained, but the performance of this model was slightly worse than that of ECFP4 models alone (though better than that of CM only models), with a test MAE of 34.19 ±1.37, RMSE of 45.1 ±2.19, and R2equal to 0.66 ±0.02. Realistically, the solution is not so simple as com- bining the two structural descriptors chosen in this study: based on our results, we hypothesize that the inclusion of all“bonds” within a molecule, regardless of distance between atoms, in the CM imparts a fair amount of irrelevant information into the model. An ideal descriptor for these ionic liquid systems should strike a bal- ance between incorporating the overall three-dimensional structure and shape of the ions while including local bonding information— encapsulating bond order and functional group patterns—within some cutoff for each atom. While such a descriptor may not exist yet, much progress is being made in the field of developing new representations to describe local environments. Several examples from the past few years include the many-body tensor represen- tation58and symmetry functions,59,60which were not used in the present work but will be considered in the future studies. The Sherrill group very recently extended the application of weighted symmetry functions to intermolecular systems to predict thesymmetry-adapted perturbation theory (SAPT) interaction energy between neutral dimers;61investigating the performance of these descriptors with charged systems such as ionic liquids would be enlightening. Finally, it would be worthwhile to consider for certain types of ionic liquids where aggregation between cations is known to occur—e.g., pyridinium and long-chain ammonium cations—a cluster approach involving two or more ion pairs when computing ab initio descriptors, as single ion interactions clearly do not suffice for these cases. C. Model generalizability To evaluate generalizability and performance of the model when applied to unseen ion types, we evaluate our best performing ECFP4-MO model in several cases where ion types in the test set are not present in the training set. The results are shown in Table III for four anions: NTf− 2, PF− 6, Cl−, and BF− 4, and three cation types: pyri- dinium, pyrrolidinium, and triazolium. These anions and cations are the second-most, third-most, and fourth-most prevalent ion types in the whole dataset. The most common ions, bromide anion and imi- dazolium/ammonium cations, were not considered—since each of these ions makes up over one-third of the original dataset, studying these types would have resulted in an insufficiently small training set. The anion with lowest errors from this study is the bistriflim- ide anion NTf− 2. Both the test MAE (30.51 K) and the test RMSE (38.58 K) are close to the original test metrics reported in Sec. IV, suggesting that the model is able to extrapolate to anion types that were not present in the training dataset. Parity plots for the NTf− 2 TABLE III . Performance of the ECFP4-MO model when various ion types in the test set are excluded from the training set. NtrainandNtestrefer to the number of ionic liquids in the training set and test set, respectively. Anion/cation type Ntrain Ntest MAE (K) RMSE (K) NTf− 2 1975 237 30.51 38.58 PF− 6 2097 115 36.50 49.11 BF− 4 2100 112 38.25 50.28 Cl−2132 80 34.47 44.89 Pyridinium 1919 293 41.53 51.42 Pyrrolidinium 2112 100 47.59 61.83 Triazolium 2114 98 40.73 54.51 J. Chem. Phys. 153, 104101 (2020); doi: 10.1063/5.0016289 153, 104101-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp model (included in the supplementary material, Fig. S3) show that although many of the predicted melting points lie on or close to the y=xline, there are also several outliers that have predicted melt- ing points within ±100 K of the true value. For all anions studied, the presented results are similar to or worse than the metrics from the original (randomly split) dataset. However, as this study used a much smaller test set in comparison with the original test set, out- liers have a larger effect on test metrics and particularly the R2value. This was also seen in the BF− 4results; the anion with the largest test MAE and RMSE had the highest R2value since points were clustered between an equal positive and negative distance about the y=xline. Hence, we consider test MAE and RMSE values only to evaluate model performance in this section. Table III shows that the ECFP4-MO model has acceptable predictive accuracy when applied to ion types outside of the training set. This is particularly true of the results for the Cl−anion: considering only 70 cations and 8 anions in the training set that contain a chlorine atom after all chloride- containing ionic liquids were removed, the test MAE and test RMSE of 34.47 K and 44.80 K for chloride-based ionic liquids are indeed promising. Errors for the three studied cations are slightly higher in com- parison with those for the anions, which is expected as pyridinium, pyrrolidinium, and triazolium cation groups encompass multiple molecules with a shared structural backbone. Prediction accuracy is best for the pyridinium and triazolium cations (MAE = 41.53 K and 40.73 K, respectively) and worst for the pyrrolidinium cation (MAE = 47.59 K). This is likely due to the structure of the training set: pyrrolidinium and triazolium cations both share a conjugated backbone, similar to the imidazolium cation that makes up over 35% of the dataset. Hence, these two ion types can inherit learned fea- tures from the imidazolium cations. In contrast, fewer cations in the dataset share similar features to pyrrolidinium, a cation type that encompasses a wide range of structures due to numerous possible variations in the two ammonium R-groups including alkyl chains or ring motifs. Naturally, errors are higher for predictions made for ionic liquids containing this cation. By examining models trained on different subsets of the dataset that exclude various anion and cation types, we have shown that the applicability domain of the ECFP4-MO model extends to ion types outside of which the model was trained on. It is possible to achieve good prediction accuracy for previously unseen anion types, ranging from small halogen anions to a larger anion such as NTf− 2. Further- more, examining applicability of the model to various unseen cation structures shows that while the model performs better if similar ion types to the test samples are present in the training set (e.g., imida- zolium ions in the training set extend well to other types of planar, conjugated cations such as triazolium), the prediction accuracy is still reasonable for cation types that differ in the backbone struc- ture. Ultimately, the most important factor in achieving a robust model with reliable predictions for any new type of ion is to ensure the diversity of the training dataset, dataset quality being one of the key factors in successful application of machine learning models to predict experimental properties. D. Comparison with literature models This section compares the results of our best performing model, ECFP4-MO, with those of selected other models in the literature.However, as there are still relatively few studies on the application of machine learning for ionic liquids (most literature studies have targeted a specific property for an intended purpose, such as CO 2 solubility21), results in the literature are mostly based on QSAR or group contribution methods. The previous work in the literature has reported similar or higher MAE values for melting point prediction, such as in the source paper for the database: the random forest-based models used by Venkatraman et al.31achieved a test MAE of 33 K and test RMSE of 45 K, compared to our values of 29 K and 40 K, respectively. The test R2score and test RMSE as reported by several other authors are shown in Table IV, along with the type of model, number of features used as the input (we consider each ECFP4 descriptor as a single feature in our case, though argument could be made against this as it is a 2048-length bit vector), model type, and size of the dataset. As machine learning for ionic liquid property predictions is still a relatively new field, the majority of existing papers using QSAR or group contribution approaches were limited to a small sub- set of ionic liquids such as only imidazolium-based ones.62To keep comparisons on an equal footing, we have included only literature sources that studied a varied ionic liquid dataset containing several hundred ionic liquids or more. Compared to the models in Table IV, our model uses a fewer number of features yet achieves comparable accuracy to the other models. The best performing model is based on a group contribu- tion method, which involves enumerating each possible substruc- ture in the ionic liquid dataset and assigning it a value. In order to make predictions on new data with chemical substructures outside of the original dataset, the group contribution scheme would need to be re-devised. By comparison, our model makes use of readily available features: the ECFP4 fingerprint that is generated from the SMILES string and the molecular orbital energies that can be com- puted from DFT-optimized geometries. Hence, any type of ionic liq- uid can theoretically be included, though performance will depend on the composition of the training set, as explored in Sec. IV C. This reflects the larger size of our dataset in comparison with that in Ref. 63. In terms of other computational methods for calculating ionic liquid melting points, molecular dynamics has been a popular choice in the field. However, it is accepted that ionic liquids require specialized force fields and additional treatment of polarization effects for the most accurate description of intermolecular interac- tions.66,67Simulations using these force fields have shown a simi- lar range of errors to the models in Table IV,68though some MD TABLE IV . Test metrics for predictions of the ionic liquid melting temperature as reported in the literature. Nrefers to the size of the entire dataset. GC = group contribution; ANN = artificial neural network; RF = random forest. References N Features Model R2 test RMSE (K) 63 799 80 GC 0.89 24.86 64 799 40 ANN 0.54 33.33 65 808 12 QSPR 0.72 26.85 19 2212 226 RF 0.66 45.00 This work 2212 5 KRR 0.76 38.54 J. Chem. Phys. 153, 104101 (2020); doi: 10.1063/5.0016289 153, 104101-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp simulations have come within ±5 K of the experimental value depending on the type of ionic liquid and methodology used.69For example, imidazolium-based ionic liquids (especially [C 1−4mim]+) have been widely studied, and the use of specialized force fields allows molecular dynamics simulations to accurately reproduce some transport properties for these cation types.70,71Molecular dynamics simulations of other cation types—such as morpholinium and sulfonium, which are both present in our dataset—are unlikely to give accurate results, since parameterized force fields for these cations do not yet exist. Considering the time and computational resources required for MD simulations, by comparison, the machine learning models such as those presented herein offer a suitable alter- native that provides comparable accuracy for ionic liquid property prediction. Touching on the time requirements for each descriptor examined in this work, ECFP4 fingerprints are naturally much faster to compute than the CM and all QM descriptors, which require an optimized geometry. Geometry optimizations of the 141 anions required an average of 6.8 central processing unit (CPU) hours per anion, roughly doubling in time for the cations, requiring 12.5 CPU hours on average for each of the 1369 cations. However, based on these results presented when adding all QM descriptors, it is likely that a QM descriptor outside of those considered in this work could have a more significant effect on performance, such that the gain in accuracy outweighs the increased computational demand. The great advantage of machine learning is that it requires far less compu- tational time than simulation, as once training is completed, pre- dictions for new compounds can be made effectively in an instant. Machine learning models are therefore ideal for high-throughput screening of chemical databases and can efficiently filter out pos- sible candidates, which fall outside of the desired property range. The addition of QM properties to machine learning models, while increasing computational costs, will ultimately save experimental hours by further filtering the pool of candidates once an initial screen using semi-empirical methods has taken place. V. CONCLUSION In the ionic liquid field, the QSAR model with hundreds of hand-engineered features has long been the best and accepted approach for predicting physicochemical properties such as viscos- ity and melting point.32While impressive results can be achieved using these methods, this study strives to move away from this “shot- gun” approach and toward structural descriptors plus a focus on a small number of quantum mechanical descriptors, which are related to the target property. Our approach reduces the need to calculate large feature vectors followed by the process of downward feature selection, in addition to being able to apply such models to any type of ionic liquid and ion types outside of the training dataset with acceptable accuracy. The application of newer structural descrip- tors such as the Coulomb matrix (which have so far only been applied to small, organic, neutral molecules) to ionic liquid ions has shown that they can feasibly be used in such surrogate mod- els where the exact form of the relationship between the structure and the property is unknown. However, in the case of ionic liquid melting points, the correlation between the CM alone and the melt- ing point is weak, achieving an R2score of 0.63 ±0.03. Nonetheless, the three-dimensional geometric information contained within the CM proved to be important for the prediction of the ionic liquidmelting point, and better predictive performance was achieved upon combining the CM with all QM descriptors (R2= 0.65 ±0.03). The ab initio information and particularly the ion σ-profiles likely com- pensate for the use of fixed charges in the CM formulation by pro- viding a more flexible description of charge. Overall, the best results were achieved using the ECFP4 fingerprint (R2= 0.74 ±0.02), which offers a faster computation in comparison with the CM as it does not need 3D geometries and contains only 2D bonding information. ECFP4 has been a favorite among the many possible QSAR descrip- tors for years. With the addition of DFT molecular orbital energies, a very small increase in accuracy was afforded compared to using the ECFP4 descriptor alone (the MAE decreases by 0.63 K on average). Although the time requirements for computing molecular orbitals in comparison with computing the ECFP4 alone are not likely to jus- tify such a small increase in predictive accuracy, it is likely that there are other quantum mechanical features outside of those explored in this work, which could lower the error more significantly, as was observed for the CM. Given the number of possible ionic liquids that could be con- sidered for an application, the models developed in this manuscript present a viable alternative to molecular simulation for rapid screen- ing of ionic liquids with a desired melting point range. The use of a relatively expensive DFT method, M06-2X/cc-pVDZ for optimiza- tions, was chosen based on our previous studies.72,73This combina- tion contrasts with that in the literature where semi-empirical meth- ods are commonly employed for rapid exploratory searches of large datasets.19,20However, we believe that these two approaches would work best in tandem: the use of higher levels of theory is more suited for medium-sized datasets due to increased accuracy of the predicted geometries and should follow an initial screening of a large dataset using a semi-empirical method. Using a higher level of theory could further refine semi-empirical predictions and narrow the potential pool of suitable ionic liquids to be synthesized, potentially elim- inating several ionic liquids that were incorrectly predicted using semi-empirical methods or adding ones where effects could not be captured properly using a lower level of theory. Several questions remain to be answered in future work: what role the cation plays as these models seem to favor contribution from the anion, in addition to finding a descriptor for these systems that can achieve better pre- diction accuracy. Based on the few descriptors studied herein and their errors, the ideal descriptor for predicting ionic liquid melting points should combine bonding information with some 3D geome- try, an accurate description of the charge environment. Determining the exact form of the structural descriptor for the chosen target prop- erty remains the challenge at hand. With the right combination of the structural descriptor and appropriate high-level ab initio data, we are confident that a level of accuracy is possible such that machine learning becomes the new norm for property prediction in the ionic liquid field, as it has become so in many other areas of materials science. SUPPLEMENTARY MATERIAL The supplementary material includes the following: SMILES strings and experimental melting point values, details of geometry optimizations, evaluation of the ECFP4 vector length, t-SNE anal- ysis colored by cation type, KRR models using ab initio data only, J. Chem. Phys. 153, 104101 (2020); doi: 10.1063/5.0016289 153, 104101-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp ECFP4-IL and CM- ILmodels with QM descriptors’ results, all KRR model parameters, model generalizability parity plots, and ab ini- tiodescriptor data. 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Matter Radiat. Extremes 5, 064201 (2020); https://doi.org/10.1063/5.0014158 5, 064201 © 2020 Author(s).Dielectronic recombination in non-LTE plasmas Cite as: Matter Radiat. Extremes 5, 064201 (2020); https://doi.org/10.1063/5.0014158 Submitted: 18 May 2020 . Accepted: 02 September 2020 . Published Online: 05 October 2020 F. B. Rosmej , V. A. Astapenko , V. S. Lisitsa , and L. A. Vainshtein COLLECTIONS Paper published as part of the special topic on Atomic and Molecular Physics for Controlled Fusion and Astrophysics ARTICLES YOU MAY BE INTERESTED IN Region-of-interest micro-focus computed tomography based on an all-optical inverse Compton scattering source Matter and Radiation at Extremes 5, 064401 (2020); https://doi.org/10.1063/5.0016034 Challenges of x-ray spectroscopy in investigations of matter under extreme conditions Matter and Radiation at Extremes 4, 024201 (2019); https://doi.org/10.1063/1.5086344 Non-resonant inelastic X-ray scattering spectroscopy: A momentum probe to detect the electronic structures of atoms and molecules Matter and Radiation at Extremes 5, 054201 (2020); https://doi.org/10.1063/5.0011416Dielectronic recombination in non-LTE plasmas Cite as: Matter Radiat. Extremes 5,064201 (2020); doi: 10.1063/5.0014158 Submitted: 18 May 2020 •Accepted: 2 September 2020 • Published Online: 5 October 2020 F. B. Rosmej,1,2,3,4,a) V. A. Astapenko,3 V. S. Lisitsa,3,4,5and L. A. Vainshtein6 AFFILIATIONS 1Sorbonne University, Faculty of Science and Engineering, UMR 7605, Case 128, 4 Place Jussieu, F-75252 Paris Cedex 05, France 2LULI, Ecole Polytechnique, CNRS-CEA, Physique Atomique dans les Plasmas Denses (PAPD), Route de Saclay, F-91128 Palaiseau Cedex, France 3Moscow Institute of Physics and Technology MIPT (National Research University), Dolgoprudnyi 141700, Russia 4National Research Nuclear University —MEPhI, Department of Plasma Physics, Moscow 115409, Russia 5National Research Center “Kurchatov Institute ”, Moscow, Russia 6P. N. Lebedev Physical Institute of the Russian Academy of Sciences, Moscow 119991, Russia Note: This paper is part of the Special Issue on Atomic and Molecular Physics for Controlled Fusion and Astrophysics. a)Author to whom correspondence should be addressed: frank.rosmej@sorbonne-universite.fr ABSTRACT Novel phenomena and methods related to dielectro nic capture and dielectronic recombination are st udied for non-local thermodynamic equilibrium (LTE) plasmas and for applications to non-LTE ionization balance. It is de monstrated that multichannel autoion ization and radiative decay strongl y suppress higher-order contributions to the tota l dielectronic recombination rates, which are ov erestimated by standard approaches by orders of magnitude. Excited-state coupling of dielectronic capture is shown to b e much more important than ground-state contributions, and electron collisional excitation is also identi fied as a mechanism driving effective dielectronic recombina tion. A theoretical description of the effect of angular- momentum-changing collisions on die lectronic recombination is developed from an atomic k inetic point of view and is visualized with a simple analytical model. The perturbation of the autoionizing states due to electric fields is discussed with respect to ionization potential depression and perturbation of symmetry properties of a utoionization matrix elements. The first steps in the development of statistical methods are presented and are realized in the framework of a local plasma freque ncy approach. Finally, the impact of collisional –radiative processes and atomic population kinetics on dielectronic recombination is critically discu ssed, and simple analytical formulas are presented. © 2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creat ivecommons.org/ licenses/by/4.0/). https://doi.org/10.1063/5.0014158 I. INTRODUCTION Atomic populations are of fundamental importance in a variety of areas in both pure and applied science. Examples include the equation of state in thermodynamics; absorption, emission, andscattering processes in matter; lasing; radiation transport; radiativecooling and energy loss; diagnostic and spectroscopic methods thatemploy the radiative properties of matter; astrophysics and planetaryscience; the physics of radiation sources; and fusion science andtechnology. 1,2 In a plasma, several charge states usually exist simulta- neously, and therefore the total radiation emission arises from excited states of different ionic c harges. This indicates that not only are the populations of excite d states and their excitation mechanisms relevant but so too is the ionization balance. In hotplasmas, while the excited-state p opulations are essentially driven by electron collisional excitation, the ionization balance dependsstrongly on ionization and recom bination processes. As shown by Burgess 3in the context of an analysis of solar emission, electron collisional ionization and radiative recombination alone could not account for observations, and d ielectronic recombination was proposed to explain a low level of ionization. In considering ra- diation balance, it is therefore important to take account of die-lectronic recombination. Similarly, the concept of localthermodynamic equilibrium (LTE ) depends not only on radiative and collisional rates (as in the t raditional description), but on autoionization rates too. Dielectronic recombination (DR) is the capture of an electron with simultaneous excitation of an atomic core and subsequent radiative stabilization of th e core. Within the framework of as i m p l i fied model, DR can be viewed as a sequence involving dielectronic capture of a continuum electron into the state nland subsequent radiative stabilization. The first step is the capture of Matter Radiat. Extremes 5,064201 (2020); doi: 10.1063/5.0014158 5,064201-1 © Author(s) 2020Matter and Radiation at ExtremesREVIEW scitation.org/journal/mrean electron into the state nlwith simultaneous excitation of the atomic core from state α0to state α: dielectronic capture :A+Z(α0)+e→A+(Z−1)**(αnl). (1.1) After dielectronic capture, autoionization or radiative stabilization can take place: autoionization :A+(Z−1)**(αnl)→A+Z(α0)+eAuger. (1.2) The radiative stabilization involves the core and the captured electron (spectator electron): core stabilization :A+(Z−1)**(αnl)→A+(Z−1)*(α0nl)+Zωcore, (1.3) spectator electron stabilization :A+(Z−1)*(α0nl)→A+(Z−1)(γ)+Zωnl. (1.4) Here, A+(Z−1)**denotes a doubly excited state, A+(Z−1)*a singly excited state, and γthe ground state of the ion in charge state Z−1, i.e., A+(Z−1)(γ)/equalsA+(Z−1)(ground). The radiation emission of the core [the relation (1.3) ] is known as dielectronic satellite emission and has properties that are of great importance for plasma temperature di- agnostics, as revealed in the pioneering work by Gabriel.4The study of dielectronic satellite emission has become a major aspect of the characterization of a variety of complex phenomena in non-equilibrium plasmas: determination of electron density by angular- momentum-changing collisions 2,5–7and by the Stark effect;8,9 characterization and measurement of hot electrons and classi fication of related instabilities;10–12determination of Auger electron heating phenomena in X-ray free-electron laser (XFEL) interaction with solid matter;13radiation field analysis via hollow ion X-ray emission;14,15 collisional phenomena induced by laser-produced plasma jets;16,17 impurity transport and charge-exchange phenomena with the neutral background in magnetic fusion plasmas;18,19relaxation phenomena influctuating plasmas;20disappearance of resonance line emission followed by accumulation of dielectronic satellite emission;21–24 ionization potential depression analysis via two-dimensional maps of hollow-ion X-ray emission in XFEL –solid matter interaction.25It should be noted here that for diagnostic purposes, radiation transport of radiative transitions originating from autoionizing states should be avoided, because re-emission of the photons is considerably reduced owing to the high autoionization rates.16 Comparison of the relations (1.1) and(1.4) shows that effective recombination occurs because an ion A+Zisfinally transformed into an ion A+(Z−1). As was demonstrated in Refs. 3and26, DR can even be the most important recombination process. As a rule, the DR rate is high for ions with a complex core that exhibits transitions between levels without any change in principal quantum number, i.e., transitions with Δn/equals0, such as 2 s→2ptransitions in lithium-like and more complex ions. It is obvious that, particularly for ions with a complex core, the number of angular momentum coupling possibilities for the whole series of captured electrons is enormous. Therefore, the corresponding atomic structure calculations that combine all the single contributions (1.1)–(1.4) to give the total DR of each ion are numerically prohibitive, and this has manifested itself in a continuing controversy regarding the calculation of the ionic fractions.27–29 Moreover, the total DR rate is not an issue of atomic structure calculations alone, but also requires consideration of populations thatare strongly out of LTE and consideration of the plasma electric micro field. Therefore, the theoretical determination of the total DR in a nonequilibrium plasma cannot be done within the framework of atomic structure calculations alone. The present paper is therefore devoted to a critical analysis of methods for the determination of the total DR rate and to develop a new framework to address the challenges faced. In this respect, a rather surprising element is discovered that can be illustrated via the above-mentioned Δn/equals0 transitions, e.g., 2 s→2ptransitions in lithium-like and more complex ions. The transition energy ΔE /equalsZωcoreforΔn/equals0 and Z≫1 is of the order of ZRy, while the ionization energy is of the order of Z2Ry≫ΔE. The 2 s→2ptransition is the main channel for both the autoionization and radiative decay. Since the energy Eof the incident recombining electron is in all cases smaller than the excitation energy, this implies the following in- equality [using Ry/equals1 2me(cα)2andE/equals1 2mev2 e]: Z2Ry E/equalsZαc ve/parenleftBigg/parenrightBigg2 ≡η2≫1, (1.5) where ηis the Coulomb parameter (note that the standard Coulomb parameter determines the possibility of a quasiclassical consider- ation). The condition (1.5) indicates the validity of the quasiclassical regime in which, for example, the spectrum from the quantum theory of bremsstrahlung30coincides with the classical spectrum.31Thus, an important part of the overall DR processes is described well by a quasiclassical approach. This opens up new ways to treat important phenomena occurring in plasmas other than purely quantum me- chanical atomic structure calculations (e.g., the multicon figuration Dirac –Fock method). On the other hand, DR related to Δn/equals1 transitions may require quantum mechanical approaches. A typical example is provided by the core transitions 1 s→2pin H-like, He-like, etc. ions. Here, the 1s→2ptransition is the main radiation channel, while the auto- ionization channel depends strongly on the state nlof the captured electron: for large quantum numbers n, we encounter autoionization according to 2 pnl→2s+e, while for small quantum numbers, the autoionization channel 2 pnl→1s+eis dominant. For the case 1s→2p, the transition energy ΔE/equalsZωcoreis of the order of Z2Ryand thus of the same order as the ionization energy. Therefore, the Coulomb parameter is about η≈1 and quantum mechanical cal- culations might be required. The paper considers the most recent developments that allow a unique description of the relevant phenomena in DR in non- equilibrium plasmas. For a review of standard methods in the theory of DR, the reader is referred to Refs. 26and32–34. In Sec. II, general DR formulas are derived in terms of atomic structure and collisio- nal–radiative decay probabilities. In Sec. III, quantum mechanical multichannel methods are developed and then in Sec. IV, excited- state coupling driven by electron collisional excitation is considered. In Sec. V, the impact of angular-momentum-changing collisions on DR is considered and visualized with a simple analytical approach. Plasma electric field effects are discussed in Sec. VI, where the impacts of ionization potential depression and of symmetry perturbations of autoionizing matrix elements are described. Finally, Sec. VIIdevelops a local plasma frequency statistical approach to calculate DR rates for very complex ions. Matter Radiat. Extremes 5,064201 (2020); doi: 10.1063/5.0014158 5,064201-2 © Author(s) 2020Matter and Radiation at ExtremesREVIEW scitation.org/journal/mreII. AUTOIONIZATION, DIELECTRONIC CAPTURE, AND DIELECTRONIC RECOMBINATION Let us recall the essence of the DR process. An incident electron with energy Eexcites an ion core with excitation energy ΔE/equalsZωcore. In this case, if the energy Eis smaller than ΔE, the electron is finally captured by the ion into a state with energy Ef≈−RyZ2 eff/n2 f(Zeffis the effective charge of the excited ion core and nfis the principal quantum number of the captured electron) obeying the condition E−Ef/equalsΔE/equalsZωcore≈E+Ry Z2 eff/slashBign2 f. (2.1) This capture results in a doubly excited state of the ion; i.e., the ion core electron is excited with energy ΔE, while the captured electron occupies a highly excited level of the ion. This state of the ion can decay in two possible ways: (i) by relaxation of the ion core electron into the initial ground state with simultaneous ejection of the captured electron from the ion:this process is known as autoionization [compare with the relation (1.2) ]; (ii) by radiative decay of the ion core electron, resulting in its return to the initial state after the emission of a photon of energy Zω≈Zω core /equalsΔE, whereas the captured electron remains bound to the ion [compare with the relation (1.3) ]. For illustration, Fig. 1 shows the relevant energy –level diagram for the He-like 2 l2l′autoionizing levels from which the so-called Ly α satellites originate. The energy of the 2 l2l′levels is given approxi- mately by E2l2l′≈2Z2 effRy/4/equalsZ2 effRy/2 (in the H-like approxima- tion), which is about half of the ionization energy of the H-like ground state Z2 nRy(where Znis the nuclear charge). The series limit of the autoionizing levels 2lnl ′is the first excited state 2 l. Radiative decay (dielectronic satellite emission) from the 2 l2l′levels populates the singly excited levels 1 s2l1,3L, from which further radiative decays proceed (e.g., the resonance line W /equals1s2p1P1−1s21S0and the intercombination line Y /equals1s2p3P1−1s21S0) that finally populate the ground state 1 s21S0. The chain of processes of dielectronic capture (1 s+e→2l2l′), radiative decay to singly excited levels (2 l2l′−1s2l+h]), and radiative decay to the ground state (1 s2l1,3L→1s21S0+h]′) is called die- lectronic recombination (the DR channel) because an effective re- combination has taken place from the H-like ground state 1 s2S1/2to the He-like ground state 1 s21S0. Thus, the DR process as well as the photorecombination (PR) process result in the capture of an incident electron and its simul- taneous photon emission. The difference is that the photon is emitted by the ion core electron in the DR process rather than by the incident electron as in the PR process. Note that the relationship between the PR and DR processes is analogous to that between standard bremsstrahlung and polarization bremsstrahlung.35 The DR rate can be calculated from the autoionization rate of a given atomic state with the help of the principle of detailed balance. Thefirst step is the application of the principle of detailed balance to dielectronic capture, i.e., nZ jΓZ,Z+1 jk/equalsnZ+1 kne〈DC〉kj, (2.2) where nZ jis the atomic population o f the autoionizing state, ΓZ,Z+1 jk is the autoionization rate from state jto a state kwith populationnZ+1 k,a n d 〈DC〉kjis the dielectronic capture rate from state kto the upper state j.T os o l v eE q . (2.2) for the dielectronic capture rate, we need to specify the populations . For this purpose, we consider a system in thermodynamic equi librium. In this case, the pop- ulations nZ jandnZ+1 kare linked via the Saha –Boltzmann equation, because states jandkbelong to different ionic states ZandZ+1 , respectively, i.e., nZ j nZ+1 k/equalsnegZ j 2gZ+1 k2πZ2 mekTe/parenleftBigg/parenrightBigg3/2 expΔEZ+1,Z k,j kTe⎛⎝⎞⎠, (2.3) where gZ jandgZ+1 kare the statistical weights of states jandk,neis the electron density, meis the electron mass, and Teis the electron temperature. The energy difference ΔEZ+1,Z kjis related to the so-called dielectronic capture energy EDC kjby (see also Fig. 1 ) ΔEZ+1,Z k,j/equals−EDC kj, (2.4) where EDC kjis the energy of the Auger electron if the autoionizing state jdecays via autoionization to state k. Combining Eqs. (2.2) –(2.4) ,w e find the general expression for the dielectronic capture rate: FIG. 1. Energy-level diagram of the He-like autoionizing levels 2 l2l′and their associated radiative decays, so-called Ly αsatellites. After radiative decay, the singly excited states 1 s2l1,3Lare formed, from which further radiative decay proceeds (e.g., the resonance and intercombination lines W and Y , respectively). Alsoindicated are the Li-like autoionizing levels 1 s2l2l′. Matter Radiat. Extremes 5,064201 (2020); doi: 10.1063/5.0014158 5,064201-3 © Author(s) 2020Matter and Radiation at ExtremesREVIEW scitation.org/journal/mre〈DC〉kj/equalsgZ j 2gZ+1 k2πZ2 me/parenleftBigg/parenrightBigg3/2 ΓZ,Z+1 jkexp(−EDC kj/slashbiggkTe) (kTe)3/2, (2.5) or, in convenient units (with ΓZ,Z+1 jkin s−1, and EDC kjandTein eV), 〈DC〉kj/equals1.656310−22gZ j gZ+1 kΓZ,Z+1 jkexp(−EDC kj/slashbiggTe) T3/2 e(cm3/slashbigs)(2.6) [note that Eq. (2.6) assumes a Maxwellian electron energy distribution function with temperature Te]. IfPZ j,gris the probability that the autoionizing state jof charge state Zdecays to the ground state of the same charge state, then the quantity PZ j,gr〈DC〉kjis called the die- lectronic recombination rate coef ficient (with units cm3s−1) into state kvia the intermediate state j: 〈DR〉Z+1,Z kj/equalsPZ j,gr〈DC〉Z+1,Z kj. (2.7) Note that it is important to underline that the assumption of thermodynamic equilibrium to eliminate the populations of Eq. (2.2) with the help of Eq. (2.3) is only a convenient method to apply the principle of detailed balance, but does notmean that relations derived with the help of the detailed balance relations are only valid under the assumption of thermodynamic equilibrium. In fact, the dielectroniccapture rate [Eq. (2.6) ] can also be derived from purely quantum mechanical relations (micro-reversibility), providing a dielectronic capture rate that is also valid for an arbitrary electron energy dis- tribution function F(E): 20 〈DC〉kj/equalsπ2Z3 /radicaltpext 2√ m3/2 egZ j gZ+1 kΓZ,Z+1 jkF(EDC kj)/radicaltpext/radicaltpext/radicaltpext/radicaltpext EDC kj/radicalBig , (2.8) or, in convenient units (with ΓZ,Z+1 jkin s−1, and F(E),EDC kj, and Te in eV), 〈DC〉kj/equals2.9360310−40ΓZ,Z+1 jkgZ j gZ+1 kΓZ,Z+1 jkF(EDC kj)/radicaltpext/radicaltpext/radicaltpext/radicaltpext EDC kj/radicalBig (cm3/slashbigs).(2.9) III. TOTAL RATES OF DIELECTRONIC RECOMBINATION: LOW-DENSITY APPROXIMATION A. General considerations In general, the probability PZ j,grfrom Eq. (2.7) is a function of density and temperature, i.e., PZ j,gr/equalsPZ j,gr(ne,Te). (3.1) The probability function (3.1) has to be determined from nu- merical calculations of a multilevel multi-charge-state atomic population kinetics that explici tly involves all necessary auto- ionizing states as “active levels ”(“active ”here means that the populations of the autoionizing l evels are calculated in a similar way to the ground and singly excited states in the population kinetics). If collisions are neglig ible compared with spontaneous r a d i a t i v ed e c a yr a t e sa sw e l la sw i t h autoionization rates, then the probability PZ j,grcan be approximated by t he so-called satellite branching factors BZ ji:3PZ j,gr→/C229 iBZ ji/equals/C229 iAZ ji /C229 lAZjl+/C229 kΓZ,Z+1 jk⎧⎪⎪⎪⎨ ⎪⎪⎪⎩⎫⎪⎪⎪⎬ ⎪⎪⎪⎭. (3.2) That is to say, Eq. (3.2) is the low-density approximation of the probability P Z j,gr. In this case, the DR rate is given by the following approximate expression: 〈DR〉Z+1,Z kj≈/C229 iBZ ji.〈DC〉Z+1,Z kj /equals/C229 iAZ ji /C229 lAZjl+/C229 kΓZ,Z+1 jk〈DC〉Z+1,Z kj⎧⎪⎪⎪⎨ ⎪⎪⎪⎩⎫⎪⎪⎪⎬ ⎪⎪⎪⎭. (3.3) With the help of Eq. (2.5) , Eq. (3.3) can be written as follows: 〈DR〉 Z+1,Z kj≈1 2gZ+1 k2πZ2 me/parenleftBigg/parenrightBigg3/2 3exp(−EDC kj/slashbiggkTe) (kTe)3/2/C229 igZ jΓZ,Z+1 jkAZ ji /C229 lAZ jl+/C229 kΓZ,Z+1 jk⎧⎪⎪⎪⎨ ⎪⎪⎪⎩⎫⎪⎪⎪⎬ ⎪⎪⎪⎭.(3.4) The term in braces { ···} is the so-called dielectronic satellite intensity factor Q Z+1,Z k,ji/equalsgZ jΓZ,Z+1 jkAZ ji /C229 lAZ jl+/C229 kΓZ,Z+1 jk,(3.5) which has been calculated by the pioneering work in Refs. 36and 37with unprecedented precision via the multicon figuration Z-expansion method. T herefore, under the assumptions made in Eq.(3.2) , the DR rate is given by the sum of the dielectronic satellite intensity factors. One can see that the essential quantities that appear in Eqs. (3.4) and(3.5) to calculate the DR rate in the low- density approximation are the dielectronic capture energy, sta- tistical weights, and radiative an d autoionizing decay rates. These quantities can nowadays routinely be generated from atomic structure calculations. However, as one can see, even for the simplest con figura- tions 2 lnl′, the numerical calculations are rather cumbersome because very large quantum numbers nl′have to be involved to achieve convergence for the DR rates. For large quantum numbers, however, convergence is dif ficult to achieve in purely quantum numerical atomic structure calculations. Moreover, to obtain the total DR rate from H-like to He-like ions, one needs to invoke all possible intermediate states j/equals3lnl′, 4lnl′,5lnl′,.... One can easily understand that for more complex con fig- urations, the number of autoionizing states that need to be involved rapidly becomes numerically prohibitive for purely quantum mechanical numerical calculations. For practical ap- plications, it is therefore mandatory to invoke additional methods, such as the Burgess approach,3,32Vainshtein ’s Matter Radiat. Extremes 5,064201 (2020); doi: 10.1063/5.0014158 5,064201-4 © Author(s) 2020Matter and Radiation at ExtremesREVIEW scitation.org/journal/mresimpli fied quantum mechanical dielectronic recombination model,26,38a quasiclassical approach,31,39or a statistical approach.40,41 B. The Burgess approximation To illuminate the essence of the various approximations cur- rently employed, we consider first the most general expression for the total dielectronic recombination rate 〈DR〉Z+1,Z tot . For the total rate, all DR rates 〈DR〉Z+1,Z kj/equalsPZ j,gr〈DC〉Z+1,Z kjhave to be summed with re- spect to the initial state kand also with respect to the intermediate states j, i.e., 〈DR〉Z+1,Z tot /equals/C229 k/C229 j〈DR〉Z+1,Z kj/equals/C229 k/C229 jPZ j,gr〈DC〉Z+1,Z kj.(3.6) Because the probability PZ j,gris a function of density and temperature [see Eq. (3.1) ], it is very dif ficult to obtain general and closed formulas for the DR rate coef ficient. Only in the low-density approximation, where Eq. (3.3) holds, can general formulas for the DR rate co- efficients be obtained. One of the most widely used general approximate empirical formulas in the framework of the approximation (3.2) is the so- called Burgess formula,3i nw h i c hi ti sa s s u m e dt h a tt h e nl spectator electron is not interacting with the core and can be treated in the hydrogenic approximation and that the capture cross section averaged over the resonances can be obtained with the aid of the correspondence principle by extrapolating below threshold the partial cross section for the core excitation α0→α: 〈DR〉Z+1,Z kj:/equalsDZ+1,Z(α0→α,n l). (3.7) For the total DR rate, we have 〈DR〉Z+1,Z tot :/equalsDZ+1,Z/equals/C229 α0/C229 α/C229 n/C229n−1 l/equals0DZ+1,Z(α0→α,n l).(3.8) For the simplest example of autoionizing states 2 l2l′outlined in Fig. 1 ,α0/equals1sandα1/equals2p, i.e., the transition α0→αcorresponds to the Ly αtransition in H-like ions. For these con figurations, DR into the ground state is the most important transition; i.e., there exists a single state k/equalsα0/equals1s. Therefore, α0coincides with the atomic ground state, and the sum over α0can be suppressed: DZ+1,Z≈/C229 α/C229 n/C229n−1 l/equals0DZ+1,Z(α0→α,n l). (3.9) The DR rate coef ficient DZ+1,Z(α0→α,nl) can then be expressed via the following analytical empirical expression:3 DZ+1,Z(α0→α,n l)/equals4.8310−11fα0αBdβ3/2e−βχd(cm3/s),(3.10) with β/equals(z+1)2Ry kTe, (3.11)χd/equalsχ 1+0.015z3 (z+1)2, (3.12) χ/equalsΔE(α0→α) (z+1)2Ry. (3.13) Here, zis the so-called spectroscopic symbol of the doubly excited ion after recombination, z/equalsZn−Nbound + 1, where Nbound is the number of bound electrons and Zneis the nuclear charge. If the first resonance transition is a Δn/equals0 transition, then the branching factor Bdis given by the following fitting formula (the so-called Burgess –Mertz formula):32 Bd/equalszχ z2+13.4/parenleftbigg/parenrightbigg1/2 1 1+0.105(z+1)χ+0.015(z+1)2χ2.(3.14) ForΔn≠0 the fitting function for the branching factor Bdis dif- ferent:32 Bd/equalszχ z2+13.4/parenleftbigg/parenrightbigg1/2 0.5 1+0.210(z+1)χ+0.030(z+1)2χ2.(3.15) Let us recall the meaning of the branching factor Bd: after dielectronic capture, a doubly excited state is formed that can decay via auto- ionization or radiative decay. For DR, only the radiative decays contribute finally to recombination, since autoionization returns the autoionizing state to the original state. According to Eq. (3.10) ,α0is the ground state, and therefore fα0αis the electric dipole absorption oscillator strength for the resonance transition α0→αwith transition energy ΔE(α0→α)i n eV. As the absorption oscillator strength decreases rapidly with increasing principal quantum number of the upper level, it is usually suf ficient to consider only the first two α-terms in the sum in Eq.(3.9) ,a n dw e finally obtain the desired expression for the total DR coef ficient: DZ+1,Z≈DZ+1,Z(α0→α1)+DZ+1,Z(α0→α2), (3.16) with DZ+1,Z(α0→α):/equals/C229 n/C229n−1 l/equals0DZ+1,Z(α0→α,n l). (3.17) Let us consider DR into neutral helium as an example (note that a single * indicates a singly excited state, while ** corresponds to a multiply excited state): He1+(1s)+e→He0+**(nln′l′)→He0+(1s2). (3.18) For this example, α0/equals1s,α1/equals2p,α2/equals3p,.... Therefore, fα0α1 corresponds to the electric dipole absorption oscillator strength of the resonance line, namely, the H-like Ly αline of singly ionized helium, while fα0α2corresponds to the Ly βline. The oscillator strengths are f1s→2p/equals0.4164 and f1s→3p/equals0.079 14, and their transition energies are ΔE(1s→2p)/equals40.81 eV and ΔE(1s→3p)/equals48.37 eV. The spec- troscopic symbol is z/equals1 and Δn≠0 [therefore, Eq. (3.15) applies]. As one can see, higher- noscillator strengths make almost negligible contributions to the total DR rate (note that this has to be Matter Radiat. Extremes 5,064201 (2020); doi: 10.1063/5.0014158 5,064201-5 © Author(s) 2020Matter and Radiation at ExtremesREVIEW scitation.org/journal/mredistinguished from the fact that high- nspectator electrons can make quite important contributions). From Eqs. (3.11) –(3.13) , we obtain β/equals4Ry/ kTe,Bd(1s→2p)/equals0.0825, Bd(1s→3p)/equals0.0846, χd(1s→2p) /equals0.747, and χd(1s→3p)/equals0.886. For the rate coef ficients at kTe/equalsRy (β/equals4), we obtain DHe1+,He0+(1s→2p)/equals1.65310−12cm3/s, and DHe1+,He0+(1s→3p)/equals3.2310−13cm3/s. This con firms that the leading terms for DR are indeed given by Eq. (3.16) . C. Quantum mechanical multichannel approach Comparison of the results from Eqs. (3.10) –(3.15) with more precise quantum mechanical calculations carried out with Vain- shtein ’s ATOM code42,43show that the χdvalues are in quite good agreement, whereas the Bdvalues differ strongly. For the resonance transitions 1 s→2pand 1 s→3p, the Bdvalues obtained from Vainshtein ’s simpli fied quantum mechanical multi-channel (QMMC) approach38,44for the above example of helium are sig- nificantly different, namely, Bd,ref(1s→2p)/equals0.155, Bd,ref(1s→3p) /equals0.0144, χd,ref(1s→2p)/equals0.744, and χd,ref(1s→3p) /equals0.888, giving DHe1+,He0+ ref(1s→2p)/equals3.10310−12cm3/s and DHe1+,He0+ ref(1s→3p)/equals5.46310−14cm3/s, i.e., the Burgess formula underestimates the Bdvalue for α0→α/equals1s→2pby a factor of 2 and overestimates the Bdvalue for α0→α/equals1s→3pby a factor of 6. These are general observations: the precision of Eqs. (3.10) –(3.15) is very difficult to estimate: it might be abou t a factor of 2 for the strongest resonance transition, but it might also deviate by orders of magnitude. Understanding the large overestimate of the Bdvalue for the transition 1 s→3pis of particular importance and is related to the fact that the Burgess formulas take into account only one autoionizing channel. For example, the 3 lnl′-configurations (which are related to the transition α0→α2/equals1s→3pin the above example) autoionize not only to the ground state but to excited states too: 3lnl′→1s+eAuger 2l+eAuger/braceleftBigg/bracerightBigg . (3.19) Complex numerical multicon figuration Hartree –Fock calcula- tions show23,45that the autoionization rates to the excited states “2l”are even more important than to the ground state “1s.”This reduces considerably the branching factor for DR [the Bdfactor in Eq.(3.10) ]. In fact, as one can see from Eq. (3.15) ,v e r ys i m i l a r branching factors are given for the transitions α0→α1/equals1s→2p andα0→α2/equals1s→3pbecause only one autoionizing channel is taken into consideration. It is very important for the practical application of DR rates in the modeling of ionization balance to explore in more detail the influence of the various multiple channels for Auger and radiative decay. Below, we perform QMMC calculations for DR in the simpli fied Vainshtein approach,38,42 –44and we fitt h en u - merical results to an analytical ex pression in order to facilitate the application and allow direct comparison with the Burgess formulas: DZ+1,Z(α0→α,n l)/equals10−83m 2l0+1Bdβ3/2e−βχd(cm3/s),(3.20)β/equalsZ2Ry kTe, (3.21) where Ry/equals13.6057 eV, kTeis the electron temperature in eV, mis the number of equivalent electrons of state α0,Zis the charge of the ion where the core transition α0→αtakes place (e.g., for the 2 lnl′ autoionizing states of He-like argon, the core transition is the 1 s→2p transition in H-like argon, Z/equals18), and l0is the corresponding orbital momentum of state α0. The physical meaning of the parameter χdis related to the fact that all contributions from the con figurations with different spectator electrons nlhave to be summed up to give the total DR rate with different energies [see Eq. (2.1) ]. The parameter χdprovides a fit to the numerical results to replace the sum of different energies in the best manner through an average energy parameter χdβ. Finally, the total sum is replaced by an average amplitude Bdto provide a simple analytical expression without the need for summation. As a demonstration, let us consider the various mechanisms via a study of the DR related to the core transition 2 s–4p. For example, the numerical calculations in the single-channel approximation for Be atoms give B(1−channel ) d(2s−3p)/equals3.4310−5, whereas B(1−channel ) d(2s−4p)/equals1.6310−5; i.e., 2 s–4ptransitions are only reduced by a factor of about two compared with 2 s–3ptransitions. Numerical calculations including multichannel decay provide an entirely different picture, with B(6−channel ) d(2s−3p)/equals2.0310−6, but B(6−channel ) d(2s−4p)/equals3.5310−7; i.e., the QMMC numerical calcu- lations indicate that higher-order DR rates are strongly suppressed.This is a general observation that multichannel decay can reduce DR considerably and can even lead to a quite different interpretation of its importance. Table I shows the numerical calculation for the DR B dfactors for single- and multichannel decay into Li-like ions for different orders and elements in comparison with the standard Burgess formula. One observes that the Burgess formula is in reasonable agreement with the numerical results for single-channel decay, although it differs by up to a factor of 3 in some cases. However, comparison with the numerical calculations using the QMMC approach reveals extremely large overestimates of DR obtained by the Burgess formula. In particular for light elements, the overestimation can be by as much as one or two orders of magnitude: for example, fo r the DR related to the autoionizing states 1 s24lnl′of Be, we have B(multichannel ) d(2s−4p)/equals3.47310−7and B(Burgess ) d(2s−4p)/equals1.10310−5; i.e., the Burgess formula over- estimates Bdby more than a factor of 30. It is therefore notrecommended to calculate higher-order contributions to DR via the Burgess approach. InTables II –IV, we present the numerical results of the QMMC approach for H-, He-, and Li-like ions, which are the most important forK-shell spectroscopy. For ease of application, we have fitted all results to the simple analytical formula (3.20) . Table II presents the results of a numerical calculation of the total DR rate into H-like ions for the core transitions 1 s–2pand 1 s–3pfor all elements from He ( Z/equals2) up to Mo ( Z/equals42) and the corresponding fitting parameters according to Eq. (3.20) . It can be seen that for low- Z elements, the DR related to the core transition 1 s–2pis dominant, whereas for large Z, the relative contribution of the DR with the core transition 1 s–3pincreases. The Burgess formula provides amplitudes Bdthat are about a factor of 3 smaller than the present numerical calculations. Matter Radiat. Extremes 5,064201 (2020); doi: 10.1063/5.0014158 5,064201-6 © Author(s) 2020Matter and Radiation at ExtremesREVIEW scitation.org/journal/mreFor the 3 lnl′states, the Burgess formula considerably over- estimates the DR rate because it does not take into account multi- channel radiative and Auger decays. This is of particular importance for low- Zelements. For example, for C, the single-channel ap- proximation gives Bd/equals6.75310−5, whereas the four-channel ap- proximation gives Bd/equals6.32310−6, i.e., a reduction by a factor of 10. Multichannel decay is much less important for higher Z: for example, for Fe, Bd/equals5.13310−6, whereas the four-channel approximation gives Bd/equals2.60310−6. Table III presents numerical results for the total DR rate into He- like ions for the core transitions 1 s–2pand 1 s–3pfor all elements from He (Z/equals2) up to Mo ( Z/equals42) and the corresponding fitting parameters according to Eq. (3.20) . It can be seen that for low- Zelements, the DR related to the core transition 1 s–2pis dominant, whereas for large Z, the relative contribution of the DR with the core transition 1 s–3p increases. The Burgess formula gives amplitudes Bdthat are about a factor of 3 smaller than the present numerical calculations. For the 1s3lnl′states, the Burgess formula considerably overestimates the DR rate because it does not take into account multichannel radiative and Auger decays. This is of particular importance for low- Zelements. For example, for C, the single-channel approximation gives Bd/equals6.76 310−5, whereas the four-channel approximation gives Bd/equals2.98 310−6, i.e., a reduction by a factor of 20. Multichannel decay is muchless important for higher Z: for example, for Fe, Bd/equals5.34310−6, whereas the four-channel approximation gives Bd/equals2.60310−6. Table IV presents numerical results for DR into Li-like ions related to the core transition 2 s–2p, i.e., a Δn/equals0 transition. Therefore, thefitting parameter χdis rather small and the associated exponential factor for DR does not vary much. In addition, the con figurations 1s22lnl′are only autoionizing for rather high principal quantumTABLE II. Fitting coef ficients according to Eqs. (3.20) and(3.21) for DR into H-like ions originating from the 2lnl ′and 3lnl ′autoionizing levels, with Z/equalsZn,m/equals1, and l0/equals0. The numerical data include corrections for multiple decay channels (two channels for 2 lnl′ and four channels for 3lnl ′). 2lnl′:α0/equals1s→α/equals2p 3lnl′:α0/equals1s→α/equals3p Element Bd χd Bd χd He 3.12 310−40.744 5.48 310−60.888 Li 3.72 310−40.736 6.41 310−60.887 Be 3.67 310−40.727 6.53 310−60.885 B 3.42 310−40.718 6.47 310−60.883 C 3.13 310−40.709 6.32 310−60.881 N 2.85 310−40.700 6.31 310−60.879 O 2.58 310−40.691 5.92 310−60.877 F 2.33 310−40.682 5.70 310−60.874 Ne 2.11 310−40.673 5.48 310−60.872 Na 1.90 310−40.665 5.26 310−60.870 Mg 1.72 310−40.657 5.04 310−60.868 Al 1.56 310−40.649 4.84 310−60.866 Si 1.41 310−40.642 4.63 310−60.863 P 1.27 310−40.636 4.43 310−60.861 S 1.15 310−40.630 4.24 310−60.859 Cl 1.05 310−40.624 4.05 310−60.857 Ar 9.50 310−50.620 3.87 310−60.856 K 8.61 310−50.616 3.69 310−60.854 C 7.82 310−50.612 3.52 310−60.852 Sc 7.09 310−50.609 3.35 310−60.851 Ti 6.45 310−50.606 3.19 310−60.849 V 5.85 310−50.604 3.04 310−60.848 Cr 5.33 310−50.602 2.89 310−60.847 Mn 4.85 310−50.601 2.74 310−60.846 Fe 4.42 310−50.599 2.60 310−60.845 Co 4.03 310−50.598 2.47 310−60.844 Ni 3.68 310−50.598 2.34 310−60.843 Cu 3.37 310−50.597 2.22 310−60.842 Zn 3.08 310−50.597 2.10 310−60.842 Ga 2.83 310−50.596 1.99 310−60.842 Ge 2.60 310−50.596 1.88 310−60.841 As 2.39 310−50.596 1.78 310−60.841 Se 2.20 310−50.596 1.68 310−60.841 Br 2.03 310−50.596 1.59 310−60.841 Kr 1.88 310−50.596 1.50 310−60.841 Rb 1.74 310−50.597 1.42 310−60.841 Sr 1.61 310−50.597 1.34 310−60.842 Y 1.50 310−50.597 1.27 310−60.842 Zr 1.39 310−50.598 1.20 310−60.842 Nb 1.30 310−50.599 1.13 310−60.843 Mo 1.21 310−50.599 1.07 310−60.843TABLE I. Bdfactors according to Eqs. (3.20) and (3.21) for DR into Li-like ions originating from the 1 s2nln′l′autoionizing levels, with Z/equalsZn−2,m/equals1, and l0/equals0. The numerical data show single- and multichannel approximations as well as thecorresponding factors according to the Burgess approach (note that the differentnumerical coef ficients and the oscillator strength in the original Burgess formula [Eq. (3.10) ] compared with Eq. (3.20) have been included in the value for B (Burgess ) dto facilitate comparison of the different methods). Element1s22lnl′:α0/equals1s22s→α/equals1s22p B(1−channel ) dB(multichannel ) dB(Burgess ) d Be 8.09 310−5... 1.34310−4 C 5.18 310−5... 7.99310−5 Mg 1.34 310−5... 1.94310−5 Ar 6.87 310−6... 8.65310−6 Fe 4.02 310−6... 4.88310−6 Mo 3.11 310−6... 3.87310−6 1s23lnl ′:α0/equals1s22s→α/equals1s23p Be 3.44 310−51.97310−62.88310−5 C 6.45 310−56.61310−66.98310−5 Mg 6.43 310−52.57310−56.96310−5 Ar 4.55 310−52.42310−55.15310−5 Fe 2.61 310−51.57310−53.54310−5 Mo 8.61 310−66.48310−61.89310−5 1s24lnl ′:α0/equals1s22s→α/equals1s24p Be 1.60 310−53.47310−71.10310−5 C 2.52 310−53.39310−72.23310−5 Mg 2.06 310−51.30310−61.87310−5 Ar 1.29 310−52.05310−61.27310−5 Fe 6.54 310−62.00310−68.01310−6 Mo 1.87 310−61.17310−63.82310−6 Matter Radiat. Extremes 5,064201 (2020); doi: 10.1063/5.0014158 5,064201-7 © Author(s) 2020Matter and Radiation at ExtremesREVIEW scitation.org/journal/mrenumbers. This is quite different to the DR related to the core transition 2s–3p, where the states are autoionizing for rather low quantum numbers nland the temperature dependence is very different owing to an order-of-magnitude difference in the χdfactor. Unlike DR into H- and He-like ions ( Tables II andIII), the DR related to the n/equals3 core transition is very important compared with the 2 s–2prelated recombination. For this reason, the temperature dependence of the total recombination rate (which is the sum of theDR rates related to the 2 s–2p,2s–3p,..., core transitions) is complex and differs qualitatively from the rates of DR into H- and He-like ions, which are dominated by a single exponential factor. IV. EXCITED-STATE COUPLING OF DIELECTRONIC RECOMBINATION IN DENSE PLASMAS Table V shows the DR rates related to the excited states 1 s22pof Li-like ions. It can be seen from a comparison of the numerical data inTABLE III. Fitting coef ficients according to Eqs. (3.20) and(3.21) for DR into He-like ions originating from the 1 s2lnl′and 1 s3lnl′autoionizing levels, with Z/equalsZn−1,m/equals2, andl0/equals0. The numerical data include corrections for multiple decay channels (two channels for 1 s2lnl′and four channels for 1 s3lnl′). 1s2lnl′: α0/equals1s2→α/equals1s2p1s3lnl′: α0/equals1s2→α/equals1s3p Element Bd χd Bd χd Li 3.39 310−51.11 1.57 310−61.27 Be 9.94 310−50.961 2.12 310−61.14 B 1.53 310−40.891 2.51 310−61.07 C 1.93 310−40.848 2.98 310−61.03 N 2.17 310−40.818 3.40 310−61.00 O 2.34 310−40.795 3.92 310−60.983 F 2.17 310−40.775 4.23 310−60.967 Ne 2.05 310−40.757 4.50 310−60.956 Na 1.88 310−40.740 4.56 310−60.945 Mg 1.72 310−40.726 4.54 310−60.937 Al 1.57 310−40.713 4.47 310−60.929 Si 1.43 310−40.701 4.36 310−60.922 P 1.30 310−40.690 4.22 310−60.916 S 1.18 310−40.681 4.07 310−60.910 Cl 1.07 310−40.672 3.92 310−60.905 Ar 9.72 310−50.664 3.76 310−60.901 K 8.83 310−50.658 3.61 310−60.897 C 8.02 310−50.652 3.45 310−60.893 Sc 7.28 310−50.647 3.30 310−60.889 Ti 6.62 310−50.642 3.15 310−60.886 V 6.02 310−50.638 3.01 310−60.883 Cr 5.47 310−50.635 2.87 310−60.880 Mn 4.98 310−50.632 2.73 310−60.877 Fe 4.54 310−50.629 2.60 310−60.875 Co 4.14 310−50.627 2.47 310−60.873 Ni 3.78 310−50.625 2.35 310−60.871 Cu 3.46 310−50.623 2.23 310−60.869 Zn 3.16 310−50.622 2.11 310−60.868 Ga 2.90 310−50.620 2.00 310−60.867 Ge 2.67 310−50.619 1.90 310−60.865 As 2.45 310−50.619 1.80 310−60.864 Se 2.26 310−50.618 1.70 310−60.864 Br 2.08 310−50.617 1.61 310−60.863 Kr 1.93 310−50.617 1.52 310−60.862 Rb 1.78 310−50.616 1.44 310−60.862 Sr 1.65 310−50.616 1.36 310−60.861 Y 1.53 310−50.616 1.29 310−60.861 Zr 1.43 310−50.616 1.22 310−60.861 Nb 1.33 310−50.616 1.15 310−60.861 Mo 1.24 310−50.616 1.09 310−60.861TABLE IV. Fitting coef ficients according to Eqs. (3.20) and(3.21) for DR into Li-like ions originating from the 1 s22lnl′and 1 s23lnl′autoionizing levels, with Z/equalsZn−2,m/equals1, and l0/equals0. The numerical data include corrections for multiple decay channels (one channel for 1s22lnl′and four channels for 1 s23lnl′). 1s22lnl ′: α0/equals1s22s→α/equals1s22p1s23lnl ′: α0/equals1s22s→α/equals1s23p Element Bd χd Bd χd Be 8.09 310−50.057 1 1.97 310−60.197 B 6.86 310−50.040 0 2.85 310−60.173 C 5.18 310−50.030 6 6.61 310−60.161 N 3.95 310−50.024 8 1.06 310−50.153 O 3.09 310−50.020 7 1.47 310−50.149 F 2.47 310−50.017 9 1.85 310−50.145 Ne 2.02 310−50.015 6 2.17 310−50.142 Na 1.69 310−50.013 9 2.41 310−50.140 Mg 1.43 310−50.012 6 2.57 310−50.138 Al 1.23 310−50.011 5 2.67 310−50.136 Si 1.07 310−50.010 5 2.71 310−50.135 P 9.43 310−60.009 81 2.69 310−50.133 S 8.41 310−60.009 14 2.60 310−50.131 Cl 7.57 310−60.008 58 2.53 310−50.130 Ar 6.87 310−60.008 09 2.42 310−50.128 K 6.25 310−60.007 72 2.31 310−50.127 C 5.76 310−60.007 36 2.19 310−50.126 Sc 5.35 310−60.007 04 2.09 310−50.124 Ti 5.00 310−60.006 77 1.97 310−50.123 V 4.67 310−60.006 58 1.86 310−50.122 Cr 4.42 310−60.006 37 1.76 310−50.120 Mn 4.20 310−60.006 20 1.66 310−50.119 Fe 4.02 310−60.006 05 1.57 310−50.118 Co 3.86 310−60.005 92 1.48 310−50.117 Ni 3.72 310−60.005 81 1.40 310−50.116 Cu 3.61 310−60.005 71 1.32 310−50.115 Zn 3.51 310−60.005 64 1.25 310−50.114 Ga 3.42 310−60.005 58 1.18 310−50.113 Ge 3.35 310−60.005 53 1.11 310−50.112 As 3.25 310−60.005 56 1.05 310−50.111 Se 3.20 310−60.005 54 9.96 310−60.110 Br 3.20 310−60.005 46 9.43 310−60.109 Kr 3.17 310−60.005 46 8.92 310−60.108 Rb 3.15 310−60.005 47 8.45 310−60.107 Sr 3.13 310−60.005 48 8.01 310−60.106 Y 3.12 310−60.005 51 7.59 310−60.105 Zr 3.11 310−60.005 54 7.20 310−60.105 Nb 3.11 310−60.005 58 6.83 310−60.104 Mo 3.11 310−60.005 63 6.48 310−60.103 Matter Radiat. Extremes 5,064201 (2020); doi: 10.1063/5.0014158 5,064201-8 © Author(s) 2020Matter and Radiation at ExtremesREVIEW scitation.org/journal/mreTables IV andVthat the contribution from the excited states is even more important than that from the ground state. For example, for Be, Bd(2s−3p)/equals1.97310−6, whereas Bd(2p−3d)/equals1.78310−4and Bd(2p−4d)/equals1.88310−5. This means that the excited-state con- tribution is up to two orders of magnitude larger than the ground- state contribution. Therefore, even for rather moderate densities with small populations of the excited states, their contribution to DR can be important.Particular important cases are encountered if the first excited states are related to Δn/equals0 radiative transitions. Because these transition probabilities are orders of magnitude lower than those for Δn>0 transitions, Boltzmann populations with respect to the ground state are already achieved for rather low electron densities. For ex- ample, for Be, at densities of about 1015cm−3, the population of the excited state 1 s22pis more important than that of the ground state 1s22s.46Therefore, all excited states of beryllium (e.g., for tokamaks at typical divertor densities) make larger contributions than the ground state. The excited-state contribution could even be important at very low densities if the excited states are metastable states. Therefore, for heavy ions, where we encounter excited states that are close to ground states, related either by a dipole-allowed radiative transition or by multipole transitions, DR is extremely complex even at rather low densities. This is the main reason why, to date, ionic balance cal- culations of heavy elements differ strongly from one method to another and why DR remains an active field of research and of considerable interest for a number of applications (nuclear fusion, astrophysics, radiation sources, spectroscopic diagnostics, etc.). In conclusion, the excited-state contribution is driven by atomic kinetics that can have a much greater impact than any other com- plicated effects related to ground-state contributions. This points again to the great practical importance of approximate methods, including the quasiclassical approach, that provide the possibility of obtaining numerical data even for large quantum numbers (which can be quite important for DR). We underline that the excited-state contributions to DR up to high quantum numbers for the corresponding core transitions may exceed the ground-state contribution by many orders of magnitude. It is for this reason that it is essential to include excited-state contri- butions as much as possible, even if these are based on atomic structure calculations of limited precision, rather than attempting to improve via sophisticated atomic structure calculations the simplest core-transition-related DR rates while ignoring higher-order and excited-state contributions. V. ANGULAR-MOMENTUM-CHANGING COLLISIONS A strict consideration of angular-momentum-changing colli- sions requires a very extended atomic-level system that includes all details of the autoionizing states in order to treat properly the col- lisional redistribution of populations. We restrict ourselves here to a discussion of principles with the help of the most frequently employed formula for DR and proceed from dielectronic capture from channel kand with radiative transition j→i[see also Eqs. (3.3) –(3.5) ]: 〈DR〉Z+1,Z k,ji≈1 2gZ+1 k2πZ2 me/parenleftBigg/parenrightBigg3/2gZ jΓZ,Z+1 jkAZ ji /C229 lAZjl+/C229 kΓZ,Z+1 jkexp(−EDC kj/slashbiggkTe) (kTe)3/2. (5.1) Let us now consider a simple illustrative example, namely the Ly α dielectronic 2 l2l′satellites of He-like ions and depict two levels, one that has very large autoionization rate and one that has a negligible one. For the first case, we consider the level j′/equals2p21D2,k/equals1s2S1/2 and the radiative transition j′/equals2p21D2→i′/equals1s2p1P1. Atomic structure calculations for carbon ( Zn/equals6) deliver37TABLE V. Fitting coef ficients according to Eqs. (3.20) and (3.21) for DR into excited states of Li-like ions originating from the 1 s23lnl′and 1 s24lnl′autoionizing levels, with Z/equalsZn−2,m/equals1, and l0/equals1. The numerical data include corrections for multiple decay channels (three channels for 1 s23lnl′and six channels for 1 s24lnl′). 1s23lnl′: α0/equals1s22p→α/equals1s23d1s24lnl′: α0/equals1s22p→α/equals1s24d Element Bd χd Bd χd Be 1.78 310−40.140 1.88 310−50.190 B 2.99 310−40.137 2.01 310−50.189 C 3.74 310−40.135 2.04 310−50.188 N 4.44 310−40.133 2.18 310−50.187 O 5.15 310−40.131 2.35 310−50.187 F 5.52 310−40.130 2.53 310−50.186 Ne 5.65 310−40.128 2.67 310−50.185 Na 5.76 310−40.127 2.88 310−50.181 Mg 5.73 310−40.125 3.28 310−50.174 Al 5.61 310−40.124 3.32 310−50.172 Si 5.39 310−40.122 3.33 310−50.171 P 5.19 310−40.120 3.48 310−50.167 S 4.96 310−40.119 3.46 310−50.165 Cl 4.71 310−40.117 3.44 310−50.164 Ar 4.48 310−40.115 3.41 310−50.163 K 4.25 310−40.114 3.38 310−50.161 C 4.04 310−40.112 3.34 310−50.160 Sc 3.83 310−40.110 3.30 310−50.159 Ti 3.64 310−40.109 3.25 310−50.158 V 3.45 310−40.107 3.20 310−50.157 Cr 3.27 310−40.105 3.14 310−50.156 Mn 3.11 310−40.104 3.08 310−50.156 Fe 2.95 310−40.102 3.02 310−50.155 Co 2.80 310−40.101 2.95 310−50.154 Ni 2.66 310−40.0992 2.88 310−50.154 Cu 2.53 310−40.0978 2.80 310−50.153 Zn 2.40 310−40.0964 2.72 310−50.153 Ga 2.28 310−40.0951 2.64 310−50.153 Ge 2.17 310−40.0939 2.56 310−50.152 As 2.06 310−40.0927 2.47 310−50.152 Se 1.96 310−40.0916 2.39 310−50.152 Br 1.86 310−40.0905 2.30 310−50.152 Kr 1.77 310−40.0895 2.22 310−50.152 Rb 1.68 310−40.0885 2.14 310−50.152 Sr 1.60 310−40.0876 2.05 310−50.152 Y 1.52 310−40.0867 1.97 310−50.152 Zr 1.45 310−40.0859 1.89 310−50.152 Nb 1.38 310−40.0851 1.82 310−50.152 Mo 1.31 310−40.0844 1.74 310−50.152 Matter Radiat. Extremes 5,064201 (2020); doi: 10.1063/5.0014158 5,064201-9 © Author(s) 2020Matter and Radiation at ExtremesREVIEW scitation.org/journal/mreΓZ,Z+1 j′k/equals2.531014s−1,AZ j′i′/equals1.431012s−1, /C229 lAZ j′l/equals1.431013s−1,/C229 kΓZ,Z+1 j′k/equals2.531014s−1. For the second case, we consider the autoionizing con figura- tion j/equals2p23P1,k/equals1s2S1/2and the radiative transition j/equals2p23P1→i/equals1s2p3P2. Atomic structure calculations (again for Zn/equals6) deliver ΓZ,Z+1 jk/equals0,AZ ji/equals6.031011s−1, /C229 lAZ jl/equals1.431012s−1,/C229 kΓZ,Z+1 jk/equals0, from which it follows that QZ+1,Z k,ji/equals0. Assuming a two-level system where only dielectronic capture and angular momentum changing collisions (characterized by the rate coef ficient Cj′j) contribute, the atomic populations nZ jandnZ j′are given by nZ j′/C229 lAZj ′l+/C229 kΓZ,Z+1 j′k+neCj′j /parenleftBigg/parenrightBigg /equalsnZ+1 kne〈DC〉Z+1,Z k,j′+nenZ jCjj′, (5.2) nZ j/C229 lAZjl+/C229 kΓZ,Z+1 jk+neCjj′ /parenleftBigg/parenrightBigg /equalsnZ+1 kne〈DC〉Z+1,Z k,j+nenZ j′Cj′j, (5.3) where 〈DC〉Z+1,Z k,q/equals1 2gZ+1 k2πZ2 me/parenleftBigg/parenrightBigg3/2 gZ qΓZ,Z+1 qkexp(−EDC kq/slashbiggkTe) (kTe)3/2,(5.4) with q/equalsj,j′. In the absence of collisions, Eqs. (5.1) –(5.4) become n(0),Z q /equalsnZ+1 kne1 2gZ+1 k2πZ2 me/parenleftBigg/parenrightBigg3/2gZ qΓZ,Z+1 qk /C229 lAZ ql+/C229 kΓZ,Z+1 qkexp(−EDC kq/slashbiggkTe) (kTe)3/2, (5.5) where the superscript “(0)”indicates the low-density case. As can be seen from Eq. (5.5) , the low-density dielectronic intensity satellite factor [Eq. (3.5) ] is reproduced from Eq. (5.2) if the angular-momentum-changing collisions are negligible, i.e., if /C229 lAZ j′l+/C229 kΓZ,Z+1 j′k≫neCj′j. Note that for very closely spaced levels, ion –ion collisions might also be of importance. To understand the effect of angular-momentum-changing collisions on the total DR rate, we need to consider the sum for the two levels, i.e., 〈DR coll〉Z+1,Z tot /equals〈DR coll〉Z+1,Z k,ji+〈DR coll〉Z+1,Z k,j′i′, (5.6)where the subscript “coll”for the single DR rates 〈DR coll〉Z+1,Z k,jiand 〈DR coll〉Z+1,Z k,j′i′indicates that these rates include collisional processes. This has to be distinguished from Eq. (5.1) , which is a low-density approximation. It is of principal interest to understand the change inDR due to collisions with reference to the low-density case, i.e., we consider the ratio 〈DR coll〉Z+1,Z tot 〈DR〉Z+1,Z tot/equals〈DR coll〉Z+1,Z k,ji+〈DR coll〉Z+1,Z k,j′i′ 〈DR〉Z+1,Z k,ji+〈DR〉Z+1,Z k,j′i′. (5.7) The collisional DR rates cannot be determined from relations like Eq. (5.5) but need to be determined directly from the populations, i.e., 〈DR coll〉Z+1,Z k,ji}nZ jAZji, (5.8) because the product of the level population with the radiative decay is the rate at which the excited state decays to the ground state, which is equivalent to DR [note that the usual branching ratios that appear in formulas like Eq. (5.1) are already included via the equilibrium population] if the right-hand sides of Eqs. (5.2) and(5.3) are driven by dielectronic capture and angular-momentum-changing collisions between the autoionizing levels under consideration. Combining Eqs. (5.7) and(5.8) , we obtain 〈DR coll〉Z+1,Z tot 〈DR〉Z+1,Z tot/equalsnZ jAZ ji+nZ j′AZ j′i′ n(0),Z jAZ ji+n(0),Z j′AZ j′i′, (5.9) i.e., 〈DR coll〉Z+1,Z tot 〈DR〉Z+1,Z tot/equalsnZ j′ n(0),Z j′+nZ jAZ ji n(0),Z j′AZ j′i′ n(0),Z jAZ ji n(0),Z j′AZ j′i′+1. (5.10) Because EDC kj≈EDC kj′, we have for the population ratio in the low- density case (for the example given above) n(0),Z j n(0),Z j′≈gZ jΓZ,Z+1 jk gZ j′ΓZ,Z+1 j′k/C229 lAZ j′l+/C229 kΓZ,Z+1 j′k /C229 lAZ jl+/C229 kΓZ,Z+1 jk≈0, (5.11) since ΓZ,Z+1 jk≪ΓZ,Z+1 j′kandΓZ,Z+1 jk≪/C229lAZ jl(see the above example). Therefore, population is essentially transferred by angular-mo-mentum-changing collisions from level j′to level j, but not vice versa. Let us now consider the above example with the populations given by Eqs. (5.2) and(5.3) : n Z j/C229 lAZjl+neCjj′ /parenleftBigg/parenrightBigg ≈nenZ j′Cj′j (5.12) and nZ j′/C229 lAZj ′l+/C229 kΓZ,Z+1 j′k/parenleftBigg/parenrightBigg ≈nZ+1 kne〈DC〉Z+1,Z k,j′. (5.13) Equations (5.2) ,(5.3) , and (5.8) indicate that for autoionizing levels with very large autoionization rates, the populations are close to the Matter Radiat. Extremes 5,064201 (2020); doi: 10.1063/5.0014158 5,064201-10 © Author(s) 2020Matter and Radiation at ExtremesREVIEW scitation.org/journal/mrelow-density case. Equation (5.4) therefore corresponds to the low- density case [Eq. (5.5) ], i.e., nZ j′≈n(0),Z j′. (5.14) Substituting Eqs. (5.11) ,(5.12) , and . (5.14) into Eq. (5.9) , we obtain 〈DR coll〉Z+1,Z tot 〈DR〉Z+1,Z tot≈1+neCj′j /C229 lAZ jl+neCjj′AZ ji AZ j′i′.(5.15) Because gj′Cj′j≈gjCjj′for closely spaced levels, Eq. (5.15) takes the form 〈DR coll〉Z+1,Z tot 〈DR〉Z+1,Z tot≈1+gZ j gZ j′AZ ji AZ j′i′1 1+/C229 lAZ jl/slashBiggneCjj′⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (5.16) If/C229lAZ jl≈neCjj′, then the term in parentheses in Eq. (5.16) is about one-half, and the relation indicates that the total DR rate is enhanced (i.e., 〈DR coll〉Z+1,Z tot/〈DR〉Z+1,Z tot>1) owing to angular-momentum- changing collisions. This can be understood in a transparent quali- tative picture: for the level j′with high autoionization rate, the dielectronic capture is high and, owing to the large autoionization rate, the branching factor for radiative deexcitation is small. If, however, a certain fraction of population is collisionally transferred to another level before autoionization and radiative decay disintegrate the upper level j′, then level jis effectively populated by collisions from j′→j(because the population of level jis small since dielectronic capture is insigni ficant owing to a small autoionization rate). The transferred population, however, has a very favorable branching factor for level jcompared with level j′. In the above example, AZ ji/slashBigg/C229 lAZjl+/C229 kΓZ,Z+1 jk /parenleftBigg/parenrightBigg /equals6.031011/slashBigg1.431012/equals0.43 for level j, whereas AZ j′i′/slashBigg/C229 lAZj ′l+/C229 kΓZ,Z+1 j′k/parenleftBigg/parenrightBigg /equals1.431012/slashBigg2.631014/equals0.0088 for level j′. Therefore, the transferred population is more effectively transferred to the ground state to finally contribute to DR. The impact of angular-momentum-changing collisions on the total DR rate is dif ficult to observe in dense plasmas, because the impact of such collisions is only indirect, namely, via the change inionization balance, where other recombination processes (e.g., three- body recombination and radiative recombination) also come into play. However, angular-momentum-changing collisions can be di- rectly observed via the collisional induced change in the spectral distribution of the corresponding dielectronic satellite transitions. Figure 2 demonstrates this effect via He-like Ly αtransitions and their associated satellite transitions 2 lnl′→1s2l+h]. Numerical calculations of the spectral distribution have been carried out using the MARIA suite of codes,10,47 –49taking into account an extended level structure, with LSJ-split levels of different ionization stages for ground, single, and multiple excited states being simultaneously included. Strong density effects are indicated by red arrows in Fig. 2 .Not only do the 2 l2l′satellites show strong density effects near λ≈0.853 nm, but so do the 2 l3l′satellites near λ≈0.847 nm. The density sensitivity of the 2 l3l′satellites starts at lower densities, be- cause the collisional rates between the 2 l3l′configurations are in general larger than those for the 2 l2l′configurations [the collisional rates C(2lnl′–2lnl″) increase with principal quantum number n, the corresponding radiative rates A(2lnl″–1snl″) are almost independent ofn, and the autoionization rates Γ(2lnl ″–1s) decrease with n]. Also indicated in Fig. 2 is the so-called “blue satellite ”emission located on the blue wing of the Ly αresonance line. These satellite transitions exhibit negative screening7that is due to the strong effect of angular- momentum coupling ( Fstates). [Note that the term “negative screening ”arises from use of the Bohr formula E/equalsZ2 effRy n2/equals(Zn−σ)2Ry n2 to match the actual energy Evia a screening constant σ: in cases where the match can only be obtained for effective charges Zeff/equalsZn−σ, the FIG. 2. MARIA simulations of dielectronic satellite emission near Ly αof H-like Mg ions for different values of the electron density at kTe/equals100 eV. The red arrows indicate the rises in intensity of particular satellite transitions with increasing density. Satellites indicated in blue have effective negative screening due to strong angular- momentum coupling effects. Matter Radiat. Extremes 5,064201 (2020); doi: 10.1063/5.0014158 5,064201-11 © Author(s) 2020Matter and Radiation at ExtremesREVIEW scitation.org/journal/mrescreening constant is negative; see also Ref. 12.] As can be seen from Fig. 2 , angular-momentum-changing collisions have little effect on blue satellites (because their autoionization rates are quite large). In dense-plasma spectroscopy, the perturbation of the spectral dielectronic satellite distribution due to angular-momentum- changing collisions is employed for density diagnostics.5–7Note that although the satellite intensity shows variations with temperature too, density determination via dielectronic satellite transitions is essen- tially nota two-parameter problem: the temperature determination relies on the intensities of satellite transitions (with high auto- ionization rates) relative to the resonance line, while the density determination is based on the relative intensities of dielectronic satellites among the transitions themselves (with low and high autoionization rates). The density diagnostic is usually rather in- dependent of the electron temperature, because all cross sections of angular-momentum-changing collisions are typically in the Born limit (the energy differences are typically much smaller than the electron temperature). VI. ELECTRIC FIELD EFFECTS A. Ionization potential depression of spectator electron orbitals Atomic population kinetics of gases and plasmas has been applied very successfully to the study of low- density environments, where atoms and ions are essentially field-free. As the density increases, however, the free-atom model breaks down, resul ting in a perturbation of the atomic energy levels. This perturbation manifests itself essentially in a broad- ening and a shift. Such perturbations can be observed in high-resolution spectroscopic experiments via analysis of the line broadening, the line shift, and the disappearance of the line emission corresponding to the ionization potential depression (IPD) of the upper level. The IPD is of great interest for applications in thermodynamics and also for the understanding of various radiative properties, such as emission, ab- sorption, and scattering (for further reading on current developments in IPD research with spectroscopic precision, the interested reader is referred to Refs. 50and51and references therein). The fundamental origin of the perturbation of energy levels is the plasma electric micro field, which in turn also limits the number of bound states. Electric field ionization starts at a critical field strength Fcritgiven by39,52 Fcrit≈6.83108Z3 eff n4 F[V/cm], (6.1) where Zeffis the effective ion charge and nFis the principal quantum number at which field ionization starts. To estimate the limited number of quantum states that effectively take part in the re- combination process, we identify the critical field strength Fcritwith the mean field strength, which can be expressed in terms of the Holtsmark normal mean field strength F0(ZPis the perturber charge andNPis the perturber density),52a Fmean≈3.41F0/equals3.434π4 15/parenleftbigg/parenrightbigg2/3Ry e/parenleftbigg/parenrightbigg a0ZPN2/3 P ≈1.3310−6ZPN2/3 P[cm−3][V/cm], (6.2)and we identify the principal quantum number nFwith the maximum principal quantum number of the spectator electron nlof the autoionizing con figuration, i.e., nF≈nspectator max ≈4.83103 Z3/4 eff Z1/4 PN1/6 P(cm−3). (6.3) nspectator max limits the contribution of high- nspectator electrons according to Eq. (3.17) [and therefore effectively limits the Bdvalues in Eqs. (3.14) ,(3.15) , and . (3.20) ]. Equation (6.3) is useful to un- derstand the various effects that limit DR (apart from the electric field effect, kinetic effects, for example, may likewise limit the effective rate coefficient; see below). Note, however, that Eq. (6.3) itself provides only a rough estimate of the principal quantum number of the spectator electrons due to the electric field effect, since (a) it neglects the fact that usually up to five perturber charge states are present in a dense plasma, (b) the field strength distribution is only taken into account via the mean field value, and (c) charged-particle correlations that in fluence the field strength distribution itself are ignored. Effective DR is not only limited by the maximum principal quantum number nspectator max from Eq. (6.3) , but also by collisional disintegration of the autoionizing level: in fact, in a dense plasma, not only does the high density of charged particles result in a high micro field, but the high density also implies a signi ficant rate of collisional processes. For autoionizing states, we need to consider two stages: a first stage involving collisional processes associated with core relaxation and a second stage following core relaxation. Concerning thefirst stage, we need to compare the radiative stabilization rate of the autoionizing con figuration (which is the origin of effective re- combination) with the collisional ionization rate of the spectator electron (which is the origin of collisional disintegration of the autoionizing level). Therefore, effective DR requires Arad(α→α0)≫neIn, (6.4) where Arad(α→α0) is the radiative stabilization rate from Eqs. (3.10) and (3.20) (note that this rate is distinctly different from the radiative transition rate of the spectator electron itself), neis the electron density, and Inis the electron collisional ionization rate coef ficient of the spectator electron with principal quantum number n.N o t et h a tf o r autoionizing states, the stabilization rate is almost independent of the high- nspectator electron, which is distinctly different from the scaling law for singly excited s tates (approximately A}1/n3). If the radiative stabilization is associated with a c hange in principal quantum number, then the H-like approximati on can be used to estimate Arad(α→α0): Aradα→α0,Δn>0 () ≈1.5731010Z4 eff nα0n3 α(n2 α−n2 α0)[s−1], (6.5) where Zeffis the effective ion charge. If Arad(α→α0) is associated with a transition without change in principal quantum number, i.e., Δn/equals0, then numerical Hartree –Fock calculations including angular-mo- mentum coupling are required to obtain reasonable estimates of the transition probability. Numerical calculations indicate that for Li-like and Be-like ions, transition probabilities and oscillator strengths can be roughly estimated from the expressions Arad(1s22p2P→1s22s2S)≈1.23108(Zn−3)(s−1), Matter Radiat. Extremes 5,064201 (2020); doi: 10.1063/5.0014158 5,064201-12 © Author(s) 2020Matter and Radiation at ExtremesREVIEW scitation.org/journal/mref(1s22s2S→1s22p2P)≈1.2/slashbig(Zn−2), Arad(1s22s2p1P→1s22s21S)≈83108(Zn−3)[s−1], f(1s22s21S→1s22s2p1P)≈4.5/slashbig(Zn−2). The ionization rate of the spectator electron can be estimated from the following expression (note that there are many competing formulas available in the literature,26but for the present estimates, the simple Lotz-like formula appears to provide a good balance between com- plexity and precision): In≈6310−8Ry En/parenleftBigg/parenrightBigg3/2/radicaltpext/radicaltpext βn/radicalBig e−βnln 1 +0.562+1.4βn βn(1+1.4βn)/bracketleftBigg/bracketrightBigg [cm3/s], (6.6) where Ry/equals13.6056 eV and βn/equalsEn kTe. (6.7) Enis the ionization potential of the spectator electron, which can be approximated by En≈(Zeff−1)2Ry n2. (6.8) I ti si m p o r t a n tt op o i n to u tt h a tt h ec o n d i t i o n (6.4) is necessary for DR to be effective, but it is notsufficient. After radiative sta- bilization α→α0of the core, we need to radiatively stabilize the spectator electron nlto the ground state “gr”(if it disappears in the continuum, no recombination is encountered), i.e., we must have Arad(nl→gr)≫neIn. (6.9) For spectator electrons, Δn/equals0 transitions are not relevant and an expression similar to Eq. (6.5) describes all cases of practical interest: Arad(nl→gr,Δn>0)≈1.5731010(Zeff−1)4 ngrn3(n2−n2 gr)[s−1]. (6.10) Whether condition (6.4) or(6.9) is more stringent depends on the type of core transition. Let us consider as an example a magnetically confined deuterium fusion plasma containing Li-like iron impurities. Assuming ZP/equals2,NP/equals1013cm−3,ne/equals1014cm−3, and kTe/equals10 keV, we obtain from Eq. (6.3) with Zeff/equals24 a maximum principal quantum number of spectator electrons of about nspectator max ≈300. For Li-like iron, DR is associated with the core transition α0→α/equals1s22s2S →1s22p2Pwith Arad(1s22p2P→1s22s2S)≈3.53109s−1and f(1s22s2S→1s22p2P)≈0.066. As the relevant principal quantum numbers are of the order of 100, we have βn/equalsEn kTe/equals(24−1)2313.6 104n2≈10−4, and we can approximate the βn-dependent terms in Eq. (6.6) roughly by 0.1 and solve Eq. (6.4) for the critical principal quantum number of the spectator electron nspectator crit [note that the critical principalquantum number is obtained from Eq. (6.4) with equality sign]: nspectator crit ≈(731021/ne)1/3≈400. Let us now investigate the condition (6.9) . For Li-like iron, ngr/equals2 and Eq. (6.10) provides roughly Arad(nl→gr,Δn>0)≈2 31015/n5[s−1]. Taking into account the ndependence of Arad(nl→ gr,Δn>0), we can solve Eq. (6.9) for the critical principal quantum number nspectator crit [note that the critical principal quantum number is obtained from Eq. (6.10) with equality sign] of the spectator electron: nspectator crit ≈(431023/ne)1/8≈16. Therefore, in the above example, the condition (6.10) is much more stringent than conditions (6.3) and(6.4) . Let us now consider the above example for parameters typical of inertial fusion plasmas. We assume that iron is employed as a - diagnostic tracer element in a compressed dense plasma. Assuming ZP/equals2,NP/equals1023cm−3,ne/equals1025cm−3, and kTe/equals10 keV, we obtain from Eq. (6.3) with Zeff/equals24 a maximum principal quantum number of the spectator electron of about nspectator max ≈6. As before, α0→α/equals 1s22s2S→1s22p2Pwith Arad(1s22p2P→1s22s2S)≈3.53109s−1and f(1s22s2S→1s22p2P)≈0.066. As the relevant principal quantum numbers are of the order of 10, we have βn/equalsEn kTe/equals(24−1)2313.6 104n2≈10−2, and we can approximate the βn-dependent terms in Eq. (6.6) roughly by 0.4 and solve Eq. (6.4) for the critical principal quantum number of the spectator electron nspectator crit [note that the critical principal quantum number is obtained from Eq. (6.4) with equality sign]: nspectator crit ≈(231021/ne)1/3<1. Therefore, DR associated with the Li- like core transition α0→α/equals1s22s2S→1s22p2Pis ineffective. The essential physical reason is related to the low transition probability for theΔn/equals0 core transition. Therefore, total recombination rates are effectively related to core transitions involving the Kshell because, in this case, the radiative decay rates are very large owing to the strong Z scaling [see Eq. (6.5) ]: assuming Zeff/equals24 as before, but taking nα0/equals1 andnα/equals2 in Eq. (6.5) , we have Arad(α→α0,Δn>0)≈1.131014s−1, i.e., a value five orders of magnitude higher compared with the Δn/equals0 core transition discussed above. In this case, K-electron involvement in a core transition allows a few principal quantum numbers of the order of 1 to survive and to contribute effectively to DR. What is the conclusion from the above estimates? They indicate that it might not be appropriate to adopt a purely atomic structure point of view to obtain convergence of the sum of DR rates over a large range of principal quantum numbers of the spectator electron nlin Eq.(3.17) : owing to micro fields and collisional –radiative competi- tion, large principal quantum numbers might not effectively con- tribute to the total DR rate. Therefore, total DR rates calculated with rather approximate methods (quasiclassical methods or Vainshtein ’s simpli fied QMMC method) but taking into account the plasma micro field and collisional processes might be more accurate than purely adopting sophisticated atomic structure calculations. In view of these results, we now address the in fluence of the plasma micro field itself on the autoionization rates. B. Perturbed autoionization rates The in fluence of the electric field on the autoionization and corresponding DR rates was first studied in Refs. 53–55in the context of the simplest atomic system of He-like autoionizing states 2 l2l′.I t Matter Radiat. Extremes 5,064201 (2020); doi: 10.1063/5.0014158 5,064201-13 © Author(s) 2020Matter and Radiation at ExtremesREVIEW scitation.org/journal/mrewas realized that forbidden autoionizing processes (i.e., forbidden in theLS-coupling scheme) become allowed by electric field mixing of autoionizing bound-state wavefunctions. The allowed autoionization width is given by the first-order transition rate Γ(d→c)/equals2π Z∣〈d∣V∣c〉∣2δ(Ed−Ec), (6.11) where Vis the electrostatic interaction. Because Vis a scalar op- erator, the autoionization vanishes unless there are available ad- jacent continuum states cwith the same angular momentum and parity as those of the discrete levels d.32B e c a u s eo ft h ea b s e n c eo f even-parity Pstates below the second ionization threshold, the 2p23P-state of He-like ions is metastable against autoionization decay. In the presence of perturbing electric fields, however, autoionization of the state a/equals2p23Pmay occur by a second-order process involving a field-induced transition to the nearby auto- ionizing state d/equals2s2p3P.I naq u a s i s t a t i ci o n field, the field-induced autoionization rate is given by Γ(a→c)/equals2π Z/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/C229 d〈a∣Q→·E→∣d〉〈d∣V∣c〉 (Ea−Ed)+iZ(Γd+Ad)/slashbig2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 δ(Ea−Ec),(6.12) where Q→is the electric dipole moment operator, and ΓdandAdare respectively the autoionization and radiative widths of the state d. Therefore, the first-order contribution from the field-induced tran- sition decays directly into the nonresonant continuum c/equals1sεp3P. It should be noted that for practical applications, not only field- induced transitions have to be considered, but intermediate coupling, configuration, and magnetic interactions too. In particular for highly charged ions, these “non-electric- field effects ”may make a consid- erable contribution to the forbidden autoionization width, as isdemonstrated by the results in Table VI , which have been calculated using the FAC code. 56In addition, the Breit interaction56ainduces an autoionization rate for the 2 p23P1state. Table VI also illustrates the general effect that if the nuclear charge increases, then the autoionization widths are more evenly distributed over the levels. Therefore, electric field effects are best studied for low- Zelements. The table also demonstrates that auto- ionization rates are strongly dependent on LSJquantum numbers: therefore, simple summations over lquantum numbers might be a quite inappropriate way to simplify complex atomic structures in kinetics.48 From the relationship between the corresponding capture and autoionization rates, it follows that the electric field can induce DR through normally inaccessible high-angular-momentum states thathave large statistical weights.54,55In fact, in a plasma, the angular momentum lis no longer a good quantum number, because the presence of an electric field destroys the spherical symmetry. How- ever, the projection LZofL⃗ , which generates the magnetic quantum number m,d efined with respect to the direction of the electric field, remains a good quantum number. For nonzero quantum numbers m, this results in a twofold degeneracy of the outer electrons in addition to the twofold degeneracy due to spin. The appropriate trans- formation of the field-free substates lhas the form ∣nλm〉/equals/C229n−1 l/equals∣m∣∣nlm〉〈nlm,∣nλm〉〉, (6.13) where the electric quantum number λ, which replaces lin the presence of the electric field, can take integer values in the range λ/equals0,...,n− |m|−1. Calculations54,55,57,58demonstrate that the dependence of the autoionization rates on the quantum number λis rather smooth, in contrast to the field-free case, where the autoionization rates decrease rapidly with quantum number l. For this reason, dielectronic capture increases in the presence of an electric field, because it is proportional to the autoionization rate and the statistical weight: 〈DC〉Z+1,Z k,j}gZ jΓZ,Z+1 jk. Because the DR rate is proportional to die- lectronic capture rate [see Eq. (2.7) ], this results in a considerable increase in the total DR rate. For example, for the autoionizing states 1s22pnlin Be-like Fe22+, an approximately threefold increase in the DR rate was found even for densities as low as 1014cm−3.54This dramatic increase at rather low densities is connected in particular with the fact that for the 1 s22pnlconfiguration, the resonance spontaneous transition probability 2 s–2pis not very large and high- n states have autoionization rates larger than radiative decay rates for n quantum numbers up to about 100. Consequently, high- nstates contribute considerably to the DR rate. As high- nstates are likewise strongly affected by rather small electric fields, a considerable impact on the total recombination rate is encountered even for rather low plasma densities (being of importance for typical densities of the solar corona or magnetic fusion plasmas). Interaction with an electric field makes atomic structure cal- culations extremely complex, and it is dif ficult to derive general conclusions. However, it has been demonstrated57,58that the qua- siclassical approach combined with a transformation to parabolic quantum numbers [Eq. (6.13) ] gives results that are in surprisingly good agreement with those of extremely complex numerical calcu- lations.59Moreover, the quasiclassical approach combined with the transformation to parabolic quantum numbers59a,59benables the derivation of a closed-form expression for the autoionization rate in an electric field: TABLE VI. Field-free autoionization decay rates (s−1) including intermediate coupling, con figuration, and magnetic interaction. State Zn/equals3 Zn/equals6 Zn/equals13 Zn/equals18 Zn/equals26 Zn/equals42 2p21S0 8.4310105.1310121.3310131.9310133.4310137.031013 2p21D2 1.5310142.5310143.1310143.1310142.3310142.131014 2p23P0 2.931072.331092.3310111.2310123.7310122.831012 2p23P1 000000 2p23P1with Breit interaction 2.6 31076.831081.9310107.2310103.2310112.231012 2p23P2 1.131093.1310103.0310122.1310131.1310141.531014 Matter Radiat. Extremes 5,064201 (2020); doi: 10.1063/5.0014158 5,064201-14 © Author(s) 2020Matter and Radiation at ExtremesREVIEW scitation.org/journal/mreΓ(n,λ,m)/equals/integraldisplaylmax lminP(nl;λm)Γ(nl)dl, (6.14) with l2 min/equals1 2/braceleftbigg[(n−1)2+m2−λ2] −/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext [(n−1)2+m2−λ2]2−4(n−1)2m2/radicalBig /bracerightbigg (6.15) and l2 max/equals1 2/braceleftbigg[(n−1)2+m2−λ2] +/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext [(n−1)2+m2−λ2]2−4(n−1)2m2/radicalBig /bracerightbigg, (6.16) where Γ(nl) is the standard autoionization rate in spherical polar coordinates (which is independent of mowing to spherical symmetry) andP(nl;λm) is a joint probability (with normalization equal to unity) for the appearance of spherical ( nl) and parabolic ( nλm) quantum numbers that can be expressed in terms of Clebsch –Gordan co- efficients. For large quantum numbers and the condition m<l«n(the quasiclassical limit of Clebsch –Gordan coef ficients that is of practical interest), the joint probability can be approximated by58 P(nl;λm)≈1 π2l/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext (l2−l2 min)(l2 max−l2)/radicalBig .(6.17) Substituting quasiclassical values for the autoionization rate31,32,36,38,42Γ(nl) into Eq. (6.14) and using Eq. (6.13) , we obtain an autoionization rate in parabolic quantum numbers expressed in terms of universal functions ( t/equalsl/leff,leff/equals(3Z2/ω)1/3): Γ(n,λ,m)/equalsfij πn3I(tmin,tmax), (6.18) I(tmin,tmax)≈2 lmax3Z2 ω/parenleftBigg/parenrightBigg2/3 Yl min(ω/slashbig3Z2)1/3/parenleftBig/parenrightBig , (6.19) Y(x)≈0.284 exp (−2x3), (6.20) where fijis the oscillator strength of the core transition with charge Z (e.g., the oscillator strength corresponding to the transition 1 s–2pin H-like Al for the He-like 2lnl ′satellites, Z/equals13). The expressions (6.14) –(6.20) demonstrate similarly a broad distribution over the electric quantum number λthatfinally results in an increase of the DR rate. VII. THE LOCAL PLASMA FREQUENCY APPROACH TO DIELECTRONIC RECOMBINATION As discussed already, DR is the most effective recombination channel in electron –heavy-ion collisions. Owing to the complex electronic structure of multielectron ions, providing a proper account of all necessary channels is a very dif ficult task, in particular for open- shell con figurations. In addition, in dense plasmas, dielectroniccapture might effectively proceed from excited states (see also Sec. IV), thus considerably increasing the number of quantum channels for dielectronic capture. Moreover, in heavy ions, numerous metastable states may play the role of excited states even in rather low-density plasmas, thereby increasing the numerical complexity of fully quantum calculations considerably. At present, DR of heavy ions is still a matter of controversy and is one of the main sources of dis- crepancy between different methods of calculation for radiation loss and ionic charge state distributions. It is therefore of great interest to develop different methods for the calculation of the DR rate in heavy ions that permit more general studies, including analysis of scaling laws. Below, we develop a twofold statistical approach that is realized by a combination of the statistical theory of atoms 60–65with the local plasma frequency approximation.40,41,66 Let us start from Eq. (3.4) and rewrite the formula for the total DR rate as 〈DR(Te)〉/equals4πRy kTe/parenleftBigg/parenrightBigg3/2 3a3 0gf giA/C229 n,lΓ(n, l) A+Γ(n, l)exp−Zω kTe+Z2 iRy n2kTe/parenleftBigg/parenrightBigg /bracketleftBigg/bracketrightBigg , (7.1) where kTeis the electronic temperature in eV, giandgfare the sta- tistical weights of the initial and final states of the atomic core, Ais the radiative transition probability inside the core, Γis the autoionization decay rate of an excited atomic energy level, ℏωis the transition energy inside the core, Ziis the ion charge, a0is the Bohr radius, and nandl are the principal and orbital quantum numbers, respectively, of the captured electron. The radiative decay rate is expressed simply in terms of the oscillator strength fijfor the transition inside the core: A/equals2ω2fif c3, (7.2) where cis the speed of light. To obtain an expression for the auto- ionization decay rate Γ(n,l), we use a relationship between the decay rateΓ(n,l) and the partial electron excitation cross section σex(n,l)i n the semiclassical representation. The quantities Γ(n,l) and σex(n,l) describe mutually inverse processes, so the relationship between them can be obtained from the detailed balance between ions XZi+1and XZi. Thus, we obtain (2l+1)gfΓ(n, l)/equalsZ2 n3ωgiσex(n, l) π2a2 0. (7.3) The electron excitation cross section in the semiclassical approxi- mation takes the form σex(n, l)/equals8π 3Z mV e/parenleftBigg/parenrightBigg2gf gififZ−2 i(l+1 2)2Gω(l+1 2)3 3Z2 i/parenleftBigg/parenrightBigg , (7.4) where the function G(u) is given by G(u)/equalsu[K2 1/3(u)+K2 2/3(u)], (7.5) where K1/2andK3/2are the Macdonald functions (modi fied Bessel functions of the second kind). Taking into account that the essential values of the argument of the function G(u) are never close to zero, it is possible to replace G(u) by its asymptotic expansion: Matter Radiat. Extremes 5,064201 (2020); doi: 10.1063/5.0014158 5,064201-15 © Author(s) 2020Matter and Radiation at ExtremesREVIEW scitation.org/journal/mreG(u)≈3.4 exp (−2u). (7.6) With these approximations, the autoionization decay rate takes the form Γ(n, l)≈0.72ω/parenleftbiggl+1 2/parenrightbiggfij n3exp−2ω(l+1 2)3 3Z2 i/bracketleftBigg/bracketrightBigg .(7.7) The sum of the absorption oscillator strengths satis fies the Tho- mas–Reiche –Kuhn sum rule, i.e., Ne/equals/C229 ffif (7.8) (note that Neis the number of electrons, while neis the electron density and nthe principal quantum number). In the statistical model, the oscillator strengths are expressed in terms of the atomic electron density ne(r,q,Zn), and the statistical sum rule is given by Ne/equals/integraldisplayne(r, q, Z n)dV. (7.9) The application of the semiclassical statistical model to the general formula (7.1) for the total DR is achieved by using the relationships /C229 ffif→/integraldisplayr0 0dr4πr2ne(r, q, Z n) (7.10) and Eif→ω/equals/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext /radicaltpext 4πne(r, q, Z n)/radicalBig . (7.11) After all the substitutions, we obtain for the DR rates 〈DR[cm3/slashbigs]〉/equals0.61310−8〈DR(a.u.)〉, (7.12) 〈DR(a.u.)〉/equals54.5 T3/2 eZn Zi/parenleftBigg/parenrightBigg2 /integraldisplayx0 0dx x2φ(x, q) x/bracketleftBigg/bracketrightBigg9/4 3/integraldisplay∞ 1dtexp−ω(x) Te1−1 t2/parenleftbigg/parenrightbigg/bracketleftBigg/bracketrightBigg 3/integraldisplaylmax/equalstn1−1 0dl(l+1 2)exp[−2ω(x)(l+1 2)3/slashbig3Z2 i] t3+A(x, l),(7.13) A(x, l)/equals5.23106/parenleftbiggl+1 2/parenrightbiggexp/bracketleftbigg−2ω(x)(l+1 2)3/slashbig3Z2 i/bracketrightbigg Z3 i/radicaltpext/radicaltpext/radicaltpext/radicaltpext/radicaltpext ω(x)/radicalbig ,(7.14) ω(x)/equals1.2Znφ(x, q) x/bracketleftBigg/bracketrightBigg3/4 , (7.15) with Te[a.u.] /equalsTe[eV]/27.21 and t/equalsn/n1, where n1is the minimum possible quantum number. n1is the lowest level at which electron capture is possible and corresponds to an energy of an incident electron Ei, given byEi/equalsω−Z2 i 2n2, (7.16) that is equal to zero, i.e., 0/equalsω−Z2 i 2n2 1, (7.17) from which it follows that n1/equalsZi/radicaltpext/radicaltpext/radicaltpext 2ω√. (7.18) In the framework of the Thomas –Fermi model,60–65the electron density distribution of a particular element and charge state is given by ne(x, q, Z n)/equals32 9π3Z2 nφ(x, q) x/bracketleftBigg/bracketrightBigg3/2 , (7.19) with x/equalsr rTF, (7.20) rTF/equals9π3 128/parenleftBigg/parenrightBigg1/31 Z1/3 n/equals0.8853Z−1/3 n, (7.21) q/equalsZ Zn, (7.22) where Znis the nuclear charge, Zis the ion charge, qcharacterizes the degree of ionization, and rTFis the Thomas –Fermi radius. The Thomas –Fermi function φ(x,q) can be approximated by the Som- merfeld method61,62,67as an exact particular solution of the Tho- mas–Fermi differential equation: φ(x, q)/equalsφ0(x)1−1+z(x) 1+z0(x)/bracketleftBigg/bracketrightBiggλ1/λ2⎧⎨ ⎩⎫⎬ ⎭, (7.23) with z(x)/equalsx 1443/2/parenleftbigg/parenrightbiggλ2 , (7.24) z0(x)/equalsx0(q) 1443/2/bracketleftBigg/bracketrightBiggλ2 , (7.25) φ0(x)/equals1 1+z(x) []λ1/2, (7.26) λ1/equals0.57+/radicaltpext/radicaltpext 73√ /parenleftbig/parenrightbig /equals7.77200 , (7.27) λ2/equals0.5−7+/radicaltpext/radicaltpext 73√ /parenleftbig/parenrightbig /equals0.77200 . (7.28) The reduced radius x0(q) is determined from the boundary condition x0dφ(x0) dx/equals−q. (7.29) Matter Radiat. Extremes 5,064201 (2020); doi: 10.1063/5.0014158 5,064201-16 © Author(s) 2020Matter and Radiation at ExtremesREVIEW scitation.org/journal/mreIn a high-temperature plasma, i.e., when the degree of ionization q/equalsZ/Znis not too low, the reduced radius can be approximated by x0(q)/equals2.961−q q/parenleftBigg/parenrightBigg2/3 if 0.2<q≤1, 6.841 q3ifq<0.05.⎧⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎩(7.30) Note that the use of the Thomas –Fermi model described above, which ignores exchange corrections, is quite appropriate within the framework of the approximations of the statistical DR model itself (see also the further discussion below). The ionization energy of an atom or ion is then given by I Z/equalsZ2 nRy128 9π2/parenleftbigg/parenrightbigg1/32Z Z5/3 nx0(q, Z n)/braceleftBigg/bracerightBigg . (7.31) As can be seen from Eq. (7.31) , the hydrogenic approximation Z2 nRyof the ionization potential of an ion with charge Znis corrected via the Thomas –Fermi electron density distribution, which depends on the nuclear and ionic charges [the factor in braces { ···}i nE q . (7.31) ]. A comparison of the ionization energies obtained from Eq. (7.31) with the results of detailed Hartree –Fock calculations shows reasonable agree- ment for heavy elements over a wide range of degrees of ionization.66 Note that more accurate descriptions of the ionization potentials cancertainly be obtained from a direct fit to the vast number of ionization p o t e n t i a l st h a ta r ek n o w na sf u n c t i o n so f ZandZ n:68 IZ≈0.221Ry(1+Z)4/3 1−0.961+Z Zn/parenleftBigg/parenrightBigg0.257.(7.32) Many modi fications of the Thomas –Fermi model have been proposed with the aim of including shell structure, obtaining improved ionization energies, and , in particular, approaching the Hartree –Fock results for the electron density distribution. In further developments to improve the statistical approach, how- ever, one must not lose sight of the requirement that the funda- mental equations of the statistical model of atoms, including the various corrections terms, should not be too complicated, in particular no more complicated than the basic equations of the quantum mechanical many-body a pproximation (e.g., the mul- ticon figuration Hartree –Fock methods). One must always bear in mind that the statistical theory of atoms is only a rough ap- proximation of the quantum atom and that its advantage is its extreme simplicity both in structure and application to determine the electron and potential distributions of atoms, to derive ele- mentary processes in collisional –radiative regimes, to shed light on detailed atomic structure calcu lations (in particular for heavy atoms), and, in particular, to derive general scaling laws that could hardly be obtained otherwise. It is this practical philosophy that we adopt when we consider Eqs. (7.19) –(7.31) for the statistical framework of the atom/ion and its realization via the local plasma frequency. In the simplest version of the statistical model, the atomic density, excitation energies, and oscillator strengths do not depend on the orbital momentum quantum number l. If we average the branching factor over orbital momentum l, i.e.,/C229 n,lΓ(n, l) A+Γ(n, l), then we obtain for the total DR rate 〈DR(a.u.)〉/equals39.2 T3/2 eZn Zi/parenleftBigg/parenrightBigg2Zn Z2 i/integraldisplayx0 0dx x2φ(x) x/bracketleftBigg/bracketrightBigg3 3/integraldisplay∞ 1dt t2exp−ω(x) T1−1 t2/parenleftbigg/parenrightbigg/bracketleftBigg/bracketrightBigg 3/integraldisplaytn1 0dl(l+1 2)2exp/bracketleftbigg−2ω(x)(l+1 2)3/slashbigg3Z2 i/bracketrightbigg t3+A(x, l), (7.33) where the function A(x,l) is given by Eqs. (7.14) and(7.15) . Instead of averaging over the branching factor, we may investigate averaging the autoionization decay rate Γ(n,l) from Eq. (7.7) over the orbital quantum number, i.e., 〈WA(n, l)〉/equals1.7fifZ2 i πn5ω. (7.34) For the corresponding total DR rate, we then obtain 〈DR(a.u.)〉/equals0.863102 T3/2 eZn Zi/parenleftBigg/parenrightBigg2 /integraldisplayx0 0dx x2φ(x, q) x/bracketleftBigg/bracketrightBigg9/4 3/integraldisplay∞ 1dtexp−1.2Z Teφ(x, q) x/bracketleftBigg/bracketrightBigg3/4 1−1 t2/parenleftbigg/parenrightbigg⎧⎨ ⎩⎫⎬ ⎭ t5+A(x), (7.35) A(x)/equals4.563106 Z3 i/radicaltpext/radicaltpext/radicaltpextZn√ φ(x, q) x/bracketleftBigg/bracketrightBigg3/8. For heavy ions, the quantum mechanical level-by-level calcu- lations are very complex and have so far been carried out mainly for closed-shell con figurations. Only recently have open-shell con figu- rations also been considered.69,70In open-shell con figurations (e.g., the open 4 p,4d, and 4 fshells, or even higher ones such as the 5 p,5d,5f, and 5 gshells), excitation –autoionization channels are very complex, and the overall completeness of quantum mechanical level-by-level calculations should still be considered with care. Analysis shows that order-of-magnitude disagreements can be expected at low temper- atures, while at high temperatures, different level-by-level quantum mechanical models differ by about a factor of 2, and the Bur- gess–Mertz approach32may deviate by many orders of magnitude and also gives an entirely inadequate temperature dependence, as demonstrated by more detailed calculations.71 Below, we compare the different approaches with detailed quantum mechanical level-by-level calculations of the DR rates. Figure 3 shows the total DR rates of xenon Xe26+a n dg o l dA u51+(the Ni-like 3s23p63d10configuration into which dielectronic capture proceeds) calculated with the l-averaged statistical model from Eqs. (7.33) –(7.35) that employs the Thomas –Fermi model of Eqs. (7.19) –(7.31) ,t h e Matter Radiat. Extremes 5,064201 (2020); doi: 10.1063/5.0014158 5,064201-17 © Author(s) 2020Matter and Radiation at ExtremesREVIEW scitation.org/journal/mreBurgess –Mertz formula from Eqs. (3.10) –(3.15) , and the quantum level- by-level calculations from Ref. 71. The statistical model compares quite well (within a factor of two) over a very large temperature interval until very low temperatures, while the Burgess approach entirely fails to describe the total DR rate of heavy ions. Similar observations are made for other isoelectronic sequences. Figure 4 shows a comparison of the results from the different approaches for the DR rates of Sr-like (4 s24p64d2) and Zn- like (4 s2) tungsten W36+and W44+, respectively.70 It is particularly impressive that the statistical model provides a rather good approximation of the total DR rate in the low-tem- perature region that is numerically exceedingly dif ficult to treat by fully quantum mechanical level-by-level calculations. Thus, the statistical model in its simplest version seems to provide even the possibility of estimating the order-of-magnitude correctness of very complex quantum level-by-level calculations. Moreover, it should be remembered that currently even the most sophisticated quantum level-by-level calculations70have been performed only in the low- density limit (the coronal model: a low-density limit in which three- body recombination may be entirely neglected), where the branching factors are entirely determined by radiative and autoionization decayrates while dielectronic capture proceeds from the respective ground states of the various charge states only. In high-density plasmas, however, as discussed above, collisional depopulation is due to electron collisional ionization or collisional transfer to other levels. In addition, excited states are highly populated, and very ef ficient channels of DR may proceed from these. This may entirely change the properties of the total DR rate, because dielectronic capture into excited states can be even more important than the corresponding capture to the ground state. This effect has been explicitly con firmed by high-resolution X-ray spectroscopy of dense laser-produced plasmas, where it has been shown that DR into excited states can exceed by many orders of magnitude the corresponding DR into ground states.23,45For high- Zelements and open M,N, and Oshells, excited states might be highly populated even at rather moderate electron densities. Therefore, all current detailed quantum level-by- level calculations to determine the DR rate have to be considered with care for each particular application. In this respect, the properties and the innovation potential of the statistical model look very advanta- geous for the determination of total DR rates for heavy elements. Finally, it should be noted that the inclusion of more levels in the detailed quantum mechanical level-by-level calculations may not FIG. 4. Comparison of the l-averaged statistical approach with the Burgess and quantum level-by-level calculations for the Sr-like sequence 4 s24p64d2of W36+and the Zn-like sequence 4 s2of tungsten W44+. FIG. 3. Comparison of the l-averaged statistical approach with the Burgess and quantum level-by-level calculations for the Ni-like sequence 3 s23p63d10of Xe26+and Au51+. Matter Radiat. Extremes 5,064201 (2020); doi: 10.1063/5.0014158 5,064201-18 © Author(s) 2020Matter and Radiation at ExtremesREVIEW scitation.org/journal/mrenecessarily result only in an increase in the DR rate, but can also lead to a decrease, as discussed in Sec. III. Therefore, at present, the simple statistical method as presented above compares quite well with other available much more complex methods of calculation and has theadvantages of generality and ease of application. In addition, there is much room for improvement to the statistical model via improve- ments to the Thomas –Fermi model (ionization energies, l-quantum- number dependence, adopting the Vlasov approach instead of the local plasma frequency etc.). VIII. CONCLUSION Dielectronic recombination (DR) can be cast as a product of dielectronic capture and a probability for radiative stabilization of the excited core followed by radiative decay of the spectator electron to the ground state. In the limiting case of negligible collisions compared with radiative decay, the total DR rate is a function of atomic structure constants only ;i . e . ,t h eD Rr a t ei saf u n c t i o no f radiative decays, Auger rates, energ y levels, statistical weights, and temperature. The quantum mech anical multichannel coupling (QMMC) approach demonstrates that the Burgess theory (in- cluding the Burgess –Mertz formulas) of DR may overestimate higher-order contributions by orders of magnitude. Collisional processes are identi fied to have multiple impacts on DR: (a) electron collisional excitation drives excited-state couplings that are often even more important than ground-st ate contributions; (b) angular- momentum-changing collisions between autoionizing states change the effective core relaxation and induce DR rates that are dependent on density —an effect that manifests itself in a perturbation of the spectral distribution of dielectroni c satellite spectra; (c) collisional ionization of the spectator electron reduces high- ncontributions to DR if ionizations are more frequent than radiative stabilizations of t h ec o r ea n dt h es p e c t a t o re l e c t r o n .T h ep l a s m am i c r o field strongly influences high- nDR contributions via ionization potential de- pression and perturbations of the spherical symmetry of auto- ionization matrix elements. Approximate calculations of DR rates, such as the quasiclassical method and Vainshtein ’s approximate QMMC approach (which allows treatment of very large quantum numbers without convergence p roblems) combined with atomic population kinetic effects (excite d-state-driven DR, angular-mo- mentum-changing collisions, ionization potential depression, and collisional ionization of the spec tator electron before radiative re- laxation) are identi fied as providing effectively higher precision for the total DR rate than pure atomic structure calculations (even though the latter are more sophisticated). Finally, the first steps have been undertaken in a statistical approach to DR that is based on the local plasma frequency approximation rather than on standard atomic structure calculations. Quite good agreement with the most advanced quantum mechanical calculations so far available have been obtained, opening up a new field of activity for the plasma atom approach. 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Rev. A 54, 3070 (1996). Matter Radiat. Extremes 5,064201 (2020); doi: 10.1063/5.0014158 5,064201-20 © Author(s) 2020Matter and Radiation at ExtremesREVIEW scitation.org/journal/mre

5.0023205.pdf

J. Chem. Phys. 153, 134305 (2020); https://doi.org/10.1063/5.0023205 153, 134305 © 2020 Author(s).Infrared spectroscopy and anharmonic theory of H3+Ar2,3 complexes: The role of symmetry in solvation Cite as: J. Chem. Phys. 153, 134305 (2020); https://doi.org/10.1063/5.0023205 Submitted: 27 July 2020 . Accepted: 16 September 2020 . Published Online: 06 October 2020 D. C. McDonald , B. M. Rittgers , R. A. Theis , R. C. Fortenberry , J. H. Marks , D. Leicht , and M. A. Duncan ARTICLES YOU MAY BE INTERESTED IN Nonlinear fluorescence spectroscopy of layered perovskite quantum wells The Journal of Chemical Physics 153, 134202 (2020); https://doi.org/10.1063/5.0021759 Enhanced Jarzynski free energy calculations using weighted ensemble The Journal of Chemical Physics 153, 134116 (2020); https://doi.org/10.1063/5.0020600 Infrared spectroscopy of H+(CO) 2 in the gas phase and in para-hydrogen matrices The Journal of Chemical Physics 153, 084305 (2020); https://doi.org/10.1063/5.0019731The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Infrared spectroscopy and anharmonic theory of H 3+Ar2,3complexes: The role of symmetry in solvation Cite as: J. Chem. Phys. 153, 134305 (2020); doi: 10.1063/5.0023205 Submitted: 27 July 2020 •Accepted: 16 September 2020 • Published Online: 6 October 2020 D. C. McDonald II,1 B. M. Rittgers,1R. A. Theis,2R. C. Fortenberry,3 J. H. Marks,1D. Leicht,1 and M. A. Duncan1,a) AFFILIATIONS 1Department of Chemistry, University of Georgia, Athens, Georgia 30602, USA 2Department of Chemistry and Biochemistry, Georgia Southern University, Statesboro, Georgia 30460, USA 3Department of Chemistry and Biochemistry, University of Mississippi, University, Mississippi 38677, USA a)Author to whom correspondence should be addressed: maduncan@uga.edu ABSTRACT The vibrational spectra of H 3+Ar2,3and D 3+Ar2,3are investigated in the 2000 cm−1to 4500 cm−1region through a combination of mass- selected infrared laser photodissociation spectroscopy and computational work including the effects of anharmonicity. In the reduced symmetry of the di-argon complex, vibrational activity is detected in the regions of both the symmetric and antisymmetric hydrogen stretch- ing modes of H 3+. The tri-argon complex restores the D3hsymmetry of the H 3+ion, with a concomitant reduction in the vibrational activity that is limited to the region of the antisymmetric stretch. Throughout these spectra, additional bands are detected beyond those predicted with harmonic vibrational theory. Anharmonic theory is able to reproduce some of the additional bands, with varying degrees of success. Published under license by AIP Publishing. https://doi.org/10.1063/5.0023205 .,s I. INTRODUCTION The H 3+molecular ion was first detected in mass spectrometry by Thompson1and has been studied extensively throughout the his- tory of gas phase ion chemistry.2–17This ion is produced efficiently in any ionization or plasma environment containing hydrogen. H 2 has a low proton affinity (PA = 100.9 kcal/mol),18and collisions of H3+with other neutral molecules usually lead to proton transfer, producing the corresponding protonated molecule. H 3+has been recognized for many years as a pivotal species in interstellar chem- istry, acting as a protonating agent for many neutrals.19–27Some reaction schemes even make this molecule the progenitor of all subsequent gas phase astrochemistry.19–27Cosmic ray ionization of ubiquitous H 2creates the relatively unstable H 2+cation that quickly reacts with more H 2to produce H 3+, eliminating a hydrogen atom. Because of the abundance of H 2, this simple reaction may be one of (if not the) most common chemical reactions in the universe.However, the abundant H 3+cation is difficult to detect in space. Because it has D3hsymmetry and no dipole moment, H 3+cannot be detected with pure rotational spectroscopy in the microwave or millimeter-wave regions, the mainstays of astrochemical observa- tion. It was, therefore, not until its infrared spectrum was measured in the lab by Oka that H 3+could be searched for and detected in space.28It is now recognized to be present in a variety of astro- nomical environments, providing a convenient indicator of the local chemistry.29–33Because of its widespread abundance and rich chem- istry, complexes of H 3+with abundant interstellar neutrals are also of significant interest. If the partner in such complexes has a lower proton affinity than H 2, the H 3+moiety can retain its structural iden- tity in ion–neutral complexes. Here, we investigate the spectroscopy of such complexes with argon (PA = 88.7 kcal/mol).18 After the initial laboratory spectrum was measured for H 3+, this ion and its isotopologues became the subjects of intense exper- imental and computational investigations over a period of many J. Chem. Phys. 153, 134305 (2020); doi: 10.1063/5.0023205 153, 134305-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp years.34–52The spectroscopy of all its isotopic variants is now well-known in both the fundamental and overtone regions. The infrared active e′ν2mode of H 3+has a frequency of 2521.4 cm−1, while the IR-forbidden a 1′symmetric stretch has a frequency of 3178.2 cm−1.53The latter frequency was measured first via an autoionizing Rydberg series in the photoionization of H 343and later as an extremely weak infrared spectrum made possible by vibration– rotation coupling interaction and intensity borrowing from an acci- dental degeneracy with other allowed transitions.46This system exhibits a strong vibration–rotation coupling and has become a benchmark for computational quantum chemistry studies of a com- plex rovibrational motion. In high pressure environments, espe- cially in supersonic molecular beams, hydrogen cluster ions with H3+at their core can be formed, and these clusters have also been studied both experimentally and computationally.4,7,14–16,54–72Of these clusters, the H 5+cation has attracted much recent attention because of its highly fluxional behavior and extremely anharmonic vibrations.57–72 The formation of hydrogen–noble gas ions has also been known in mass spectrometry for many years,73–79and the infrared spectroscopy of several RG-H+diatomics has been performed.80–83 Of these, the ArH+argonium ion has been detected in various astronomical sources, making it the first naturally occurring noble gas molecule.84Likewise, the long-sought HeH+ion was recently detected in space.85In larger complexes, proton-bound dimers of the noble gases argon and helium have recently been characterized in the infrared.86,87Larger H 3+Arncomplexes have been detected in mass spectrometry and studied computationally, demonstrating consid- erable bonding stability.73–79Because of this, it has been suggested that H 3+may be a sink of noble gas atoms in the outer solar sys- tem.88–91However, until now, the only spectroscopy on H 3+–noble gas complexes is that for H 3+Ar. This cation has been characterized with pure rotational spectroscopy and studied computationally.92–97 In unpublished work, Dopfer has measured an infrared spectrum in the region of the ν2fundamental with a complex rotational structure that could not yet be assigned.98 The so-called “tagging” with noble gas atoms is a common methodology used to enhance photodissociation yields in ion vibra- tional spectroscopy.99–106In many cases, tagging enables the spec- troscopy without significant perturbation of the vibrational levels. However, the attachment of noble gas atoms such as argon to light hydrogen atoms in a molecular structure is known to induce sig- nificant shifts in those vibrations, and the tag is no longer “inert.” This latter scenario is expected for H 3+Arncomplexes. Recent high- level computational work on the H 3+Ar complex shows that the inclusion of the Ar atom splits the H 3+degenerate ν2frequency from 2521 cm−1into 1765.6 cm−1and 2009.8 cm−1frequencies.97 The symmetry breaking from the inclusion of argon also allows the IR-inactive totally symmetric ν1mode to become IR-active for H3+Ar, and this is predicted to occur at 3284.7 cm−1. Similar sym- metry breaking should occur from the addition of a second argon, but a third argon is expected to produce a D3hcomplex with IR activity more like that of H 3+.75,79Similar symmetry breaking and restoring effects were observed previously in the spectra of NH 4+Arn complexes.107,108In each of these H 3+Arnsystems, the bonding to argon may produce significant shifts in the H 3+vibrations with the possibility of strong anharmonic interactions resulting from the large amplitude motion. These systems, therefore, provide rareexamples of light molecules in weakly bonded complexes with heavy atoms. We investigate these issues in the infrared spectroscopy and theoretical treatments of H 3+Ar2and H 3+Ar3. II. METHODOLOGY A. Experimental setup H3+Arn(n = 1–50) ions are generated in a pulsed dis- charge/supersonic expansion of Ar containing 10% H 2using meth- ods described previously.106The ions are entrained in the expansion gas and skimmed into a differentially pumped chamber downstream where they are pulse-extracted into a reflectron time-of-flight mass spectrometer.109The desired H 3+Arnspecies is mass-selected by pulsed deflection plates in the first flight tube of the instrument. It is then intersected with an infrared optical parametric oscilla- tor/optical parametric amplifier (OPO/OPA) laser system at the turning point of the reflectron field. Photodissociation results in the loss of one or two Ar atoms from the parent ion, depending on the excitation energy. The fragment ion signal is recorded with a digital oscilloscope as a function of photon frequency. There is no correc- tion for the wavelength variation of the laser pulse energy because the spatial beam shape and overlap with the ion beam change in an unpredictable way as the laser scans. B. Computational details In a method we abbreviate as “CC,” structures were pre- optimized at the MP2 level of theory using the Gaussian09110pro- gram package and then re-optimized at the coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] level of theory with the ANO1 basis set111using the CFOUR112,113program package. Anhar- monic frequency calculations using second-order perturbation the- ory (VPT2) were performed at the CCSD(T)/ANO1 level. This same level of theory is used to compute the anharmonic CC intensities from finite differences of analytic Hessians as available in CFOUR. This method has been demonstrated in recent studies to be useful for anharmonic problems114,115similar to the present system and is available in standard software packages accessible to experimental groups.112,113 In a second approach, CCSD(T)-F12b/cc-pVTZ-F12116,117 anharmonic vibrational frequencies were computed from quartic force fields (QFFs), fourth-order Taylor series expansions of the internuclear Hamiltonian.118This level of theory, abbreviated as F12-TZ hereafter, combined with QFFs, has been shown to produce accurate frequencies when compared with higher-level theory and even experimental values in some cases.118–121After initial F12-TZ geometry optimizations, the symmetry-internal coordinates are dis- placed with step sizes of 0.005 Å and 0.005 rad. The resulting ener- gies are then fit and refit via a least-squares method, and the result- ing force constants are then transformed into Cartesian coordinates via the INTDER program.122Second-order vibrational perturba- tion theory (VPT2)123,124within the SPECTRO program125is then utilized via the Watson Hamiltonian to produce the anharmonic vibrational frequencies. The weak association of the argon atoms with the H 3+core makes numerical differentiation of the cubic and quartic terms dif- ficult for those coordinates that involve motion of the argon atoms. J. Chem. Phys. 153, 134305 (2020); doi: 10.1063/5.0023205 153, 134305-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Similar behavior within QFFs has been shown in other weakly bound systems,126,127in contrast to the case of a single-argon atom bonded to the trihydrogen cation in H 3+Ar, which performed well when compared with the experiment.97Consequently, the harmonic terms for H 3+Ar2and H 3+Ar3are utilized for all modes reported herein from the F12-TZ QFF that do not involve the motions of the H 3+ core ( ν4and higher for both complexes). The anharmonic funda- mental, overtone, and combination frequencies solely involving the hydrogen atoms of these two complexes are computed explicitly via the F12-TZ QFF. Approximations to the H 3+Ar2and H 3+Ar3over- tones and combination bands that involve motions of the argon atoms are determined from the harmonic fundamental of the argon motion ( ν4and higher) added to the anharmonic trihydrogen core frequency ( ν1–ν3). All CCSD(T) electronic structure computations for the QFFs utilize the MOLPRO 2015.1 quantum chemical program,128except for the MP2/6-31+G(d) double-harmonic intensities129,130com- puted within Gaussian09, which have been shown to give at least semi-quantitative agreement with higher-level computations.127,131 The F12-TZ QFF anharmonic frequencies for isolated H 3+are also computed in the same scheme as that done for c-N3+.132The F12-TZ frequencies of 3170.7 cm−1forν1and 2510.7 cm−1for ν2compare well to the corresponding experimental frequencies of 3178 cm−1and 2521 cm−1, implying that the anharmonic frequen- cies of the trihydrogen cation core of H 3+Ar2and H 3+Ar3should be well-represented. The two anharmonic overtones and one com- bination band of H 3+are also determined here at 6241.8 cm−1, 4704.9 cm−1, and 5516.2 cm−1, respectively. III. RESULTS AND DISCUSSION A. Mass spectra and photodissociation A pulsed discharge in pure hydrogen gas produces strong sig- nals for hydrogen clusters of the form H(H 2)n+, for nvalues of up to and beyond seven, and much weaker signals for H 4+and H 6+. A sample mass spectrum is shown in the supplementary material as Fig. S1. The most intense clusters are the H 3+, H 5+, H 7+, and H 9+ species. These pure hydrogen clusters have been the subject of pre- vious infrared spectroscopy studies from our group and others.54–60 When an argon–hydrogen gas mixture is employed, a variety of clus- ters of the forms H+Arnand H 3+Arn(n = 1–7) are produced. This mass spectrum is shown as Fig. S2. The H+Arnspecies were the sub- ject of a previous study from our group,86which showed that the core ion is the proton-bound dimer Ar–H+–Ar that is solvated by argon in the larger clusters. In the present study, we examine the H3+Arncomplexes. We find experimentally that we are not able to dissociate the H3+Ar complex with excitation in the 2000 cm−1–3500 cm−1region. This is presumably because the dissociation energy of this com- plex is greater than the infrared photon energy in this region. Several ion chemistry studies have derived experimental dissocia- tion energies for this ion in the range of 6 kcal/mol–7 kcal/mol (2100 cm−1–2450 cm−1),75–79,89implying that infrared dissociation could have been detected. However, higher-level computational studies with zero point corrections find higher dissociation ener- gies (e.g., 8.5 kcal/mol = 2980 cm−1).97Thermal effects are known to cause lower collisional dissociation energies in conventional massspectrometry measurements, whereas our ions are cooled to very low temperature (10 K–20 K), perhaps explaining these discrepan- cies. A result inconsistent with ours is that Dopfer has previously detected a photodissociation spectrum of this ion with a much nar- rower linewidth OPO laser.98The spectrum had a complex rota- tional structure that could not be assigned. It is not clear what kind of dissociation dynamics or instrument sensitivity issue made that experiment possible where ours failed. However, as shown below, we find efficient photodissociation for larger H 3+Arncomplexes, consis- tent with the significantly lower dissociation energies that have been reported for these species (e.g., 3.7 kcal/mol = 1295 cm−1for H 3+Ar2 and 3.6 kcal/mol = 1260 cm−1for H 3+Ar3).79 B. Spectroscopy measurements Figure 1 shows the infrared photodissociation spectra mea- sured for H 3+Ar2and D 3+Ar2by mass-selecting the respective par- ent ions, exciting them with tunable infrared radiation, and measur- ing the yield of the H 3+Ar or D 3+Ar fragment ions. In the region above 4000 cm−1, dissociation occurs via the loss of two argon atoms, and this is the fragment channel recorded. The elimination of only one argon via photodissociation in the energy region below 4000 cm−1is consistent with the dissociation energies discussed above. In the previously measured absorption spectra of isolated H3+, an extensive rovibrational structure was detected spanning FIG. 1 . Infrared photodissociation spectra of H 3+Ar2and D 3+Ar2measured in the mass channel corresponding to the loss of one argon atom, except in the region above 4000 cm−1, where the fragmentation is measured in the loss of two argons. J. Chem. Phys. 153, 134305 (2020); doi: 10.1063/5.0023205 153, 134305-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp several hundred cm−1.28–52However, these noble gas complexes have much smaller rotational constants, and at the low tempera- ture of the experiment ( ∼20 K), the structure seen is likely that of individual vibrational bands with widths determined by a combi- nation of rotational contours and predissociation. Because we have no rotational resolution, we cannot address any issues regarding the ortho –para species of H 3+, which are of course highly interesting for astrochemistry.9,19–33 The spectrum of H 3+Ar2exhibits a complex pattern of three main bands at 2142 cm−1, 2281 cm−1, and 2715 cm−1with a weaker broad structure in the 2400 −3300 cm−1region and two distinct weaker bands at 4176 cm−1and 4357 cm−1. The dashed vertical red lines in the figure show vibrational frequencies for the corre- sponding isolated H 3+(2521.4 cm−1and 3178.2 cm−1) and D 3+ (1834.7 cm−1and 2300.8 cm−1) ions.53The three most intense bands are, therefore, all at frequencies lower than those of the iso- lated H 3+ion. Shifts to lower frequencies are generally expected for vibrations involving hydrogen interacting with argon. The high- frequency bands are at roughly twice the frequency of the 2142 cm−1 and 2281 cm−1features, consistent with overtones of these fre- quencies. D 3+Ar2also exhibits a pattern of three sharp features at frequencies to the red of those for isolated D 3+at 1604 cm−1, 1724 cm−1, and 1975 cm−1.53The spectrum is dominated by an intense sharp band at 3110 cm−1, which is at roughly twice the frequency of the 1604 cm−1band, again consistent with an over- tone. Although overtone bands usually have much weaker intensities than fundamentals, here we measure the photodissociation rather than absorption, and higher energy bands can appear more intense because of their enhanced dissociation yield. Additionally, the laser pulse energy is much higher above 2000 cm−1than it is below this value, and the spectra here are not corrected for the laser output vari- ations. Both of these factors likely contribute to the greater intensity of the 3110 cm−1band. In their computational study of H 3+Ar, Theis and Fortenberry found a C 2vstructure which caused the antisymmetric stretch of H 3+ to split into two bands predicted at 1765.6 cm−1and 2009.8 cm−1, and the symmetric stretch became infrared active as a single feature at 3284.7 cm−1.97The three main bands seen here for each of these ions, therefore, make sense if they represent a similar vibrational pat- tern for these di-argon complexes, but with smaller shifts relative to the isolated ions. Previous theoretical studies found a C 2vstructure for H 3+Ar2with one argon attached to each of two hydrogens,75,79 which should produce similar vibrational activity to that seen for H3+Ar. The weaker structure seen across this spectrum is consis- tent with combinations involving argon bending and/or stretch- ing modes, which are often seen in the vibrational spectroscopy of noble gas complexes. However, these tentative assignments require confirmation from theory. Figure 2 shows the infrared photodissocation spectra measured for H 3+Ar3and D 3+Ar3, again measured in the mass channel corre- sponding to the elimination of one argon. The vibrational frequen- cies for the corresponding isolated H 3+and D 3+ions are indicated with dashed vertical lines. These spectra are simpler than those in Fig. 1, with fewer bands and a less underlying structure. The main feature in the H 3+Ar3spectrum is a doublet at 2172 cm−1/2216 cm−1, with weaker signals at 2486 cm−1, 2798 cm−1, and 4260 cm−1. The 4260 cm−1band is at roughly twice the frequency of the 2172/2216 doublet, again consistent with an overtone. The D 3+Ar3has a FIG. 2 . Infrared photodissociation spectra of H 3+Ar3and D 3+Ar3measured in the mass channel corresponding to the loss of one argon atom, except in the region above 4000 cm-1, where the fragmentation is measured in the loss of two argons. doublet at 1622 cm−1/1654 cm−1, much weaker bands at 1860 cm−1, 3106 cm−1, and 3297 cm−1, and a strong sharp band at 3163 cm−1. Again, the strong band at 3163 cm−1appears at roughly twice the frequency of the 1622 cm−1/1654 cm−1doublet, consistent with an overtone, and its intensity is likely enhanced by the greater dissocia- tion yield and laser pulse energy here. The main features for both of these ions are shifted to frequencies lower than those of the isolated H3+and D 3+ions, respectively, consistent with the argon complexa- tion interaction. These spectra for tri-argon ions are more similar to those of the isolated ions, which have only a single IR-active vibra- tion each, than those of the di-argon ions. According to the existing theory for these ions, the tri-argon species have D3hstructures,75,79 which would lead to reduced vibrational activity. These spectra are, therefore, consistent with lower symmetry and greater IR activity for the di-argon ions and higher symmetry with reduced IR activity for the tri-argon ions. C. Computational studies To investigate these spectra in more detail, we have employed computational chemistry to explore the possible isomers and their spectra. Figure 3 shows the isomeric structures determined here for the H 3+Ar2and H 3+Ar3species and their relative energies at the CCSD(T)/ANO1 level. As shown, each ion has three isomers, including the most stable C 2vform identified previously for H 3+Ar2 and the D3hstructure determined for H 3+Ar3. Each ion also has two different Ar–H+–Ar proton-bound dimer structures with a more loosely associated H 2molecule. The proton-bound dimer structure was studied previously by our group,86and it has a distinct spectrum J. Chem. Phys. 153, 134305 (2020); doi: 10.1063/5.0023205 153, 134305-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Isomeric structures and relative energies (kcal/mol; ZPE corrected) for the isomers of H 3+Ar2and H 3+Ar3. with intense bands at 900 cm−1–1000 cm−1and a series of weaker features extending out to 2100 cm−1, with nothing at higher fre- quency. We can, therefore, eliminate these kinds of structures from further consideration here. Apparently, the slightly stronger binding energy and greater concentration of argon in the present experi- ment preclude the formation of these species with weakly bonded H2units. Table I shows the relative energies and argon binding ener- gies computed here for the most stable isomers of these complexes at the CCSD(T)/ANO1 level of theory. The binding energies are much TABLE I . Absolute and zero point corrected energies for H 3+Arnand D 3+Arn(n = 2, 3) along with argon tag binding energies computed at the CCSD(T)/ANO1 level of theory. Energy ZPVE corrected Argon binding Complex (hartree) (hartree) energy (kcal/mol) Ar −527.007 37 H3+−1.341 9 −1.3220 H3+Ar −528.364 3 −528.3432 8.63 H3+Ar2−1055.379 8 −1055.3562 3.60 H3+Ar3−1582.394 2 −1582.3686 3.12 D3+−1.341 9 −1.3277 D3+Ar −528.364 3 −528.3491 8.78 D3+Ar2−1055.379 8 −1055.3629 4.03 D3+Ar3−1582.394 2 −1582.3765 3.95 FIG. 4 . Infrared spectrum of H 3+Ar2compared to those predicted by anhar- monic theory at the CCSD(T)/ANO1/VPT2 (CC; middle trace; blue) and CCSD(T)- F12b/cc-pVTZ-F12 (F12-TZ; lower trace; red) levels. higher for the single-argon species than for the di- and tri-argon complexes, consistent with previous computational work and with the dissociation behavior observed experimentally. To investigate the vibrational patterns in these complexes, we employ two different methods to account for anharmonicity, as described in the experimental section. For convenience, we indicate these as the CC and F12-TZ methods, respectively. Figures 4–7 show the comparison of the spectra measured for the di- and tri-argon complexes of H 3+and D 3+to those predicted by anharmonic theory for these ions. Tables II and III provide a list of the vibrational mode numbering for these complexes, and Tables IV and V present the observed band positions compared to the bands predicted by theory with the two different methods. D. Comparison between experiment and theory Figure 4 shows the comparison of experiment and theory for the H 3+Ar2complex. In this case, the spectra predicted by anhar- monic theory are quite different for the CC (middle trace; blue) vs F12-TZ (lower trace; red) methods. We first consider the H 3+core ion fundamentals. Both theory methods predict a strong doublet at lower frequency corresponding to the two components of the anti- symmetric stretch ( ν3andν2), but neither the absolute frequencies nor the splitting between these bands is described particularly well by either method. Both methods agree that there should be two J. Chem. Phys. 153, 134305 (2020); doi: 10.1063/5.0023205 153, 134305-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . Infrared spectrum of D 3+Ar2compared to those predicted by anhar- monic theory at the CCSD(T)/ANO1/VPT2 (CC; middle trace; blue) and CCSD(T)- F12b/cc-pVTZ-F12 (F12-TZ; lower trace; red) levels. relatively strong bands in this general region, but they do not agree on their positions. Compared to the strongest experimental bands at 2142 cm−1and 2281 cm−1(splitting of 139 cm−1), the CC method predicts bands at 1923 cm−1and 2047 cm−1(splitting of 124 cm−1) and the F12-TZ method predicts bands at 2039 cm−1and 2216 cm−1 (splitting of 177 cm−1). The ν1symmetric stretch of H 3+is pre- dicted at 3062 cm−1by the F12-TZ method, and the CC method predicts this at 3098 cm−1; both methods agree reasonably well on the frequency of this vibration and suggest that it has a relatively weak intensity. It is not obvious from the experiment which band should be assigned to this, as there are no bands in the predicted region. The other peaks in this congested spectrum of the H 3+Ar2com- plex must represent overtone and combination bands. The only high-frequency bands from the H 3+core are expected to be the hydrogen stretch overtones and combinations. Both theory meth- ods predict a ν2+ν3combination and a 2 ν2overtone to have rea- sonable intensity, although the MP2/6-31+G(d) double-harmonic intensities conjoined to the F12-TZ frequencies are much higher. Most notably, the positions of the bands predicted are quite dif- ferent, with the CC method predicting much lower frequencies: 3829 cm−1and 3914 cm−1vs 4208 cm−1and 4259 cm−1for the F12-TZ method. The higher frequency bands from the F12-TZ pre- diction agree much better with the experimental bands at 4176 cm−1 and 4357 cm−1, although the predicted spacing is smaller than that observed. Nevertheless, the combined theory results are enough to FIG. 6 . Infrared spectrum of H 3+Ar3compared to those predicted by anhar- monic theory at the CCSD(T)/ANO1/VPT2 (CC; middle trace; blue) and CCSD(T)- F12b/cc-pVTZ-F12 (F12-TZ; lower trace; red) levels. establish that the high-frequency experimental bands come from the ν2+ν3combination and the 2 ν2overtone. Because the only other vibrations are those involving argon motions, the frequency intervals from such combinations are small. Both theory methods predict strong bands for each of the two anti- symmetric stretches in combination with the ν9argon bending mode (harmonic frequency of 43.9 cm−1). The predicted intervals vary slightly between the two theory methods, but both agree that these combinations should be relatively intense. The experimental bands are broad, and somewhat asymmetric, so these combination bands may be obscured within their widths. There is no agreement between the two theory methods on any other combination bands involving argon vibrations. The F12-TZ method predicts a number of strong transitions in the 2300 cm−1–3500 cm−1region, consistent with the strong signals here in the experiment, but the CC method predicts virtually no bands here with any significant intensities. The F12-TZ method predicts a pattern of six bands here with reasonable MP2/6- 31+G(d) intensities, but none of these matches well with the exact positions of the experimental bands. In Table IV, we have assigned strong features in the experiment to the best matches in the theory, but these assignments are by no means certain. Figure 5 shows the spectrum of D 3+Ar2vs the predictions of anharmonic theory. The CC spectrum is in the center trace (blue), and the F12-TZ spectrum is in the lower trace (red). The spectrum has fewer bands than that of the H 3+Ar2complex, and the bands observed are, expectedly, shifted to lower frequencies. The spectrum J. Chem. Phys. 153, 134305 (2020); doi: 10.1063/5.0023205 153, 134305-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 7 . Infrared spectrum of D 3+Ar3compared to those predicted by anhar- monic theory at the CCSD(T)/ANO1/VPT2 (CC; middle trace; blue) and CCSD(T)- F12b/cc-pVTZ-F12 (F12-TZ; lower trace; red) levels. has two bands at lower frequencies (1604 cm−1and 1724 cm−1), a weaker one at the intermediate frequency (1975 cm−1), a strong sharp band at the high frequency (3110 cm−1), and a few weaker fea- tures. The 3110 cm−1band occurs at roughly twice the frequency of the 1604 cm−1band, consistent with an overtone. Both theory methods agree that there should be relatively intense signals for the ν2andν3antisymmetric hydrogen stretch fundamentals, as seen TABLE II . Description of the vibrational modes for H 3+Ar2and D 3+Ar2. Harmonic frequencies were determined at the CCSD(T)/ANO1 level of theory. Harmonic Mode Symmetry Approx. description freq. H (D) ν1 a1 H3+breathing 3298.3 (2333.1) ν2 b2 H–H antisym. stretch 2488.4 (1760.9) ν3 a1 H3+scissors bend 2260.1 (1598.9) ν4 a2 H3+out of plane bend 783.2 (554.3) ν5 b2 ArHH antisym. bend 685.4 (485.4) ν6 b1 H3+out of plane bend 542.9 (384.5) ν7 a1 Ar sym. stretch 366.6 (270.8) ν8 b2 Ar antisym. stretch 303.8 (220.2) ν9 a1 Ar in-plane scissors bend 43.9 (42.8)TABLE III . Description of the vibrational modes for H 3+Ar3and D 3+Ar3. Harmonic frequencies were determined at the CCSD(T)/ANO1 level of theory. Harmonic Mode Symmetry Approx. description freq. H (D) ν1 a1′H3+breathing 3238.7 (2291.0) ν2 e′H–H antisym. stretch 2456.7 (1738.1) ν3 e′′H3+out of plane bend 703.9 (498.1) ν4 a2′H3+twist 692.6 (490.2) ν5 e′Ar antisym. stretch 350.1 (252.5) ν6 a2′′H3+drumbeat 129.4 (105.7) ν7 a1′Ar sym. stretch 105.7 (92.6) ν8 e′Ar in plane bend 25.4 (25.2) for the corresponding H 3+Ar2complex. Such as the predictions for the H 3+Ar2complex, both methods predict the frequencies of these bands to be lower than those in the experiment, but the spacings and intensity ratios match the experiment reasonably well. Both methods predict the ν1symmetric stretch to have a much weaker intensity and to occur in a region near 2200 cm−1, where there is no signal in the experiment. In the hydrogen stretch overtone/combination region, both methods predict bands with non-negligible intensities for tran- sitions involving 2 ν3,ν2+ν3, and 2 ν2(in the order of increasing frequency), with the ν2+ν3band having the greatest intensity. The CC method finds much weaker bands at lower frequencies than the experiment, whereas the F12-TZ method matches the experiment quite nicely if we assume that the higher frequency 2 ν2transition matches a higher frequency shoulder on the experimental band at 3110 cm−1. The D 3+Ar2spectrum is also predicted to have bands from combinations involving the argon motions. As in the H 3+Ar2spec- trum, both the theory methods predict combinations of the ν2andν3 fundamentals with the ν9argon bending mode. The 1604 cm−1band is broad and has a hint of a high-frequency shoulder that might be consistent with this, but the 1724 cm−1band is rather sharp and iso- lated. As for the H 3+Ar2complex, the CC method does not predict many argon-based combination bands with strong intensity, but the F12-TZ method predicts much activity in the 1700 cm−1–2500 cm−1 range. The ν2+ν8andν2+ν7combinations are predicted to have reasonably strong intensities near the middle of the spectrum, where the 1975 cm−1experimental feature is found. These same combi- nations were also predicted to be strong for the H 3+Ar2complex and likely explain the experimental feature at 2715 cm−1in its spec- trum. Several other argon-based combinations are predicted, but there are few experimental bands corresponding to these. Again, we make our best assignments in Table IV, but these have considerable uncertainty. Figure 6 shows the experimental spectrum for the H 3+Ar3com- plex compared to the predictions of anharmonic theory. The CC predicted spectrum is in the center trace (blue), and the F12-TZ predicted spectrum is in the lower trace (red). The experimen- tal spectrum has fewer bands than those of the H 3+Ar2spectrum, consistent with the higher symmetry D3hstructure predicted by theory. The most prominent experimental feature is a doublet at J. Chem. Phys. 153, 134305 (2020); doi: 10.1063/5.0023205 153, 134305-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE IV . Band positions (in cm−1) for H 3+Ar2and D 3+Ar2compared with anharmonic theory at the CCSD(T)-F12b/cc- pVTZ-F12 (F12-TZ) and CCSD(T)/ANO1 levels of theory. Band intensities (km/mol) are given in parentheses. Harmonic frequencies for the fundamentals at the CCSD(T) level are also given for comparison. Expt. F12-TZ Modes CCSD(T) Modes Harm. freq. H3+Ar2 2142 2039 (443) ν3 1923 (254) ν3 2260 2078 (232) ν3 +ν9 1965 (125) ν3 +ν9 1986 (37) ν2 +ν8 2281 2216 (1672) ν2 2047 (901) ν2 2488 2255 (1365) ν2 +ν9 2089 (521) ν2 +ν9 2394 (456) ν3 +ν8 2150 (17) ν3 +ν8 2406 (305) ν3 +ν7 2247 (0.5) ν3 +ν7 2544 (232) ν3 +ν6 2408 (4) ν3 +ν6 2398 (2) ν2 +ν7 2715 2571 (956) ν2 +ν8 2686 (5) ν2 +ν5 2584 (1263) ν2 +ν7 2667 (235) ν3 +ν5 2692 (16) ν3 +ν5 2721 (891) ν2 +ν6 2687 (5) ν2 +ν6 2766 (230) ν3 +ν4 2959 2844 (1045) ν2 +ν5 2943 (840) ν2 +ν4 3062 (35) ν1 3098 (58) ν1 3298 3100 (22) ν1 +ν9 3137 (0.2) ν1 +ν9 3291 3417 (470) ν1 +ν8 3482 (25) ν1 +ν8 3429 (167) ν1 +ν7 3421 (0.2) ν1 +ν7 3566 (20) ν1 +ν6 3573 (0.6) ν1 +ν6 3689 (27) ν1 +ν5 3782 (111) 2 ν3 3512 (46) 2 ν3 3789 (18) ν1 +ν4 3786 (4) ν1 +ν5 4176 4208 (1058) ν2 +ν3 3829 (161) ν2 +ν3 4357 4259 (418) 2 ν2 3914 (15) 2 ν2 D3+Ar2 1604 1527 (221) ν3 1431 (161) ν3 1599 1565 (114) ν3+ν9 1473 (37) ν3+ν9 1724 1655 (834) ν2 1542 (603) ν2 1761 1692 (421) ν2+ν9 1583 (142) ν2+ν9 1576 (15) ν2+ν8 1784 (336) ν3+ν8 1608 (7) ν3+ν8 1799 (182) ν3+ν7 1680 (0.2) ν3+ν7 1885 (112) ν3+ν6 1786 (2) ν3+ν6 1975 1912 (643) ν2+ν8 1927(488) ν2+ν7 1803 (0.8) ν2+ν7 1972 (115) ν3+ν5 1945 (6) ν3+ν5 2012 (419) ν2+ν6 1990 (2) ν2+ν5 2042 (111) ν3+ν4 2029 (2) ν2+ν4 2099 (150) ν2+ν5 2169 (150) ν2+ν4 2211 (17) ν1 2236 (26) ν1 2333 2248 (12) ν1+ν9 2276 (0.2) ν1+ν9 2468 (150) ν1+ν8 2498 (9) ν1+ν8 2482 (80) ν1+ν7 2484 (0.1) ν1+ν7 2568 (10) ν1+ν6 2586 (0.2) ν1+ν6 2655 (13) ν1+ν5 2725 (9) ν1+ν4 J. Chem. Phys. 153, 134305 (2020); doi: 10.1063/5.0023205 153, 134305-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE IV .(Continued.) Expt. F12-TZ Modes CCSD(T) Modes Harm. freq. D3+Ar2 2861 2896 (55) 2 ν3 2694 (18) 2 ν3 2709 (1) ν1+ν5 3110 3098 (528) ν2+ν3 2904 (61) ν2+ν3 3152 (209) 2 ν2 2994 (6) 2 ν2 3655 (119) ν1+ν3 3595 (1) ν1+ν3 3783 (150) ν1+ν2 3656 (16) ν1+ν2 4372 (4) 2 ν1 4412 (0.01) 2 ν1 TABLE V . Band positions (in cm−1) for H 3+Ar3and D 3+Ar3compared with anharmonic theory at the CCSD(T)-F12b/cc- pVTZ-F12 (F12-TZ) and CCSD(T)/ANO1 levels of theory. Band intensities (km/mol) are given in parentheses. Harmonic frequencies for the fundamentals at the CCSD(T) level are also given for comparison. Expt. F12-TZ Mode CCSD(T) Mode Harm. freq. H3+Ar3 2172 2229 (1159) ν2 2128 (412) ν2 2457 2216 2252 (580) ν2+ν8 2149 (287) ν2+ν8 2373 2225 (9) ν2+ν7 2486 2349 (642) ν2+ν6 2323 (13) ν2+ν5 2324 (13) ν2+ν5 2405 (13) ν2+ν5 2406 (13) ν2+ν5 2772 (4) ν2+ν3 2798 2613 (905) ν2+ν5 2799 (19) ν2+ν4 . . . ν1 . . . ν1 3239 3012 (1) ν1+ν8 3044 (1) ν1+ν8 3109 (63) ν1+ν6 3139 (5) ν1+ν6 4053 (94) 2 ν2 4055 (94) 2 ν2 4260 4141 (290) 2 ν2 4151 (192) 2 ν2 1622 1631 (579) ν2 1573 (316) ν2 1738 1654 1654 (290) ν2+ν8 1596 (76) ν2+ν8 1717 (10) ν2+ν7 1739 (319) ν2+ν6 1860 1909 (542) ν2+ν5 1743 (23) ν2+ν5 2087 (10) ν2+ν4 1745 (23) ν2+ν5 2100 (10) ν2+ν3 1786 (23) ν2+ν5 . . . ν1 . . . ν1 2291 2192 (1) ν1+ν8 1787 (23) ν2+ν5 2051 (14) ν2+ν4 2277 (30) ν1+ν6 2447 (10) ν1+ν5 3106 3044 (35) 2 ν2 3047 (35) 2 ν2 3163 3104 (145) 2 ν2 3094 (72) 2 ν2 3297 3718 (10) ν1+ν2 J. Chem. Phys. 153, 134305 (2020); doi: 10.1063/5.0023205 153, 134305-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 2172 cm−1/2216 cm−1, which is matched well in both theory meth- ods. It is assigned to the ν2antisymmetric stretch fundamental and a combination with the ν8argon in-plane bend. A similar combi- nation was predicted for the H 3+Ar2complex, but here, the doublet structure is clearly resolved in the experiment. A broad experimen- tal band centered at 4260 cm−1is reproduced reasonably well by both theory methods and assigned to the ν2overtone. As we saw for the H 3+Ar2complex, the F12-TZ method predicts more bands with greater intensities for argon-based combinations lying at frequencies above the ν2fundamental. Two main bands are predicted, and two are observed in the experiment, but their frequencies do not match. The CC method produces much weaker combination bands, but the feature predicted at 2799 cm−1for a ν2+ν4combination matches the position of the experimental band at 2798 cm−1. The exact com- bination band assignments are, therefore, again uncertain, but we present our best attempts at this in Table V. Figure 7 shows the spectrum of the D 3+Ar3complex com- pared to the predictions of anharmonic theory. The CC predicted spectrum is in the center trace (blue), and the F12-TZ predicted spectrum is in the lower trace (red). The experimental spectrum again has fewer bands than those of the D 3+Ar2spectrum, con- sistent with the higher symmetry D3hstructure predicted by the- ory. The most prominent experimental features are a doublet at 1622 cm−1/1654 cm−1and an intense single band at 3163 cm−1. Both theory methods again predict a fundamental band from the ν2 vibration, accompanied by a combination of this with the ν8argon bend. Both methods again predict an overtone of the ν2vibration, but both undershoot the experimental frequency of the 3163 cm−1 band. As in the case of the D 3+Ar2complex, the intensity of the ν2overtone is exaggerated in the experiment, most likely because of the higher laser power in this region and the higher dissociation yield here. Argon-based combination bands are again predicted to be stronger in the F12-TZ theory than for the CC method, but nei- ther predicts the position of the single significant band at 1860 cm−1. The closest predicted band is the 1909 cm−1feature predicted to be quite intense by the F12-TZ method and assigned to the ν2+ν9 combination. This same combination is predicted to be the most intense by the CC method, although the frequency predicted is lower. Although the comparison between experiment and theory throughout these data is less than ideal, there is enough consensus between the two theory methods to make confident assignments for the main spectral features. The bands at 2142 cm−1and 2281 cm−1 for H 3+Ar2and those at 1604 cm−1and 1724 cm−1for D 3+Ar2are, therefore, the ν3andν2antisymmetric stretch fundamentals of the core H 3+(D3+) ion, respectively. These modes are derived from theν2vibration in the isolated H 3+(D3+) ion, whose degeneracy is split in the reduced symmetry of the di-argon complexes. In the tri- argon species, the band at 2172 cm−1for the H 3+Ar3complex and that at 1622 cm−1for the D 3+Ar3complex represent the single ν2 vibration whose degeneracy has been “restored” in the D3hsymme- try of these complexes. The ν1symmetric hydrogen stretch is IR- forbidden for H 3+, D3+, and their tri-argon analogs, but is predicted to become weakly allowed in the di-argon complexes. Nevertheless, no experimental band corresponding to this can be identified. This is either because of the weak intensity or the fact that such a band is obscured by other features from argon-based combinations in the same region. The high-frequency bands for the di-argon complexescan be assigned to the ν2+ν3combination bands, which are pre- dicted to be more intense than either of the pure-mode overtones. Those for the tri-argon complexes can be assigned to the ν2overtone. These bands appear to be more intense for the respective deuterated complexes, but this is likely a result of the more favorable laser power in the region of 3000 cm−1. There is less consensus for the assignments of the other bands, which are mostly attributed to combinations of the ν2andν3hydro- gen stretches indicated above with various argon-based vibrations. Both theory methods find strong combinations for the antisymmet- ric hydrogen stretch with the in-plane argon bends ( ν9for H 3+Ar2 andν8for H 3+Ar3), and there is clear evidence for this in the tri- argon complexes. Other argon bends and stretches are predicted by both theory methods; while there is some consensus on certain fre- quencies and relative intensities, it is not generally possible to make specific assignments of these to specific features in the experiment. Virtually, all of the argon-based combination bands are predicted to be much more intense in the MP2/6-31+G(d) double-harmonic intensities used with F12-TZ theory than in the CC data. This fascinating complex between a light molecular cation and heavy argon atoms presents a challenging problem for anharmonic vibrational theory. Both the CC and F12-TZ methods struggle to predict vibrational frequencies, with most of the predicted bands falling at lower frequencies than those in the experiment. The CC method seems to perform worse in this respect than the F12-TZ method. The F12-TZ method and its MP2/6-31+G(d) intensities also seem to perform better with regard to both absolute and rel- ative intensities of the vibrations, especially those involving the argon-based combination bands. The explicitly correlated F12-TZ approach includes a better description of the electronic wavefunction due to the F12 terms even though it is nominally a CCSD(T)-based method such as that in the CC computations utilized here. Additionally, the modern cc- pVTZ-F12 basis set is finely tuned to interact with the explicitly correlated wavefunctions further increasing the accuracy of F12-TZ over CC by as much as an entire zeta level. The SPECTRO treat- ment of the F12-TZ QFF also allows for the inclusion of Fermi resonances and even Fermi resonance polyads which augment the more complete description of the F12-TZ vibrational modes.119–121 Granted, the exceedingly large anharmonicities from the H 3+core for the di- and tri-argon cations are difficult to treat for either method since both utilize VPT2. Possible complexities in these spectra caused by tunneling of the H 3+core would also not be described by these theory methods. This could contribute to the simpler appearance of the spectra for the deuterated species. Still, the additional factors present in the F12-TZ computations are likely responsible for better predictions of the fundamental vibrational frequencies. Additionally, double-zeta basis MP2 computations are well- known to provide an exceedingly fortuitous cancellation of errors for energies,133and such has also been documented for infrared inten- sities.127,131The analytic harmonic nature of the MP2/6-31+G(d) intensities also gives better predictions since the weak interactions in the anharmonic CC intensities are likely corrupted in the numerical dipolar derivatives beyond the harmonic level. Hence, MP2 intensi- ties coupled with more complete quantum chemical predictions of the frequencies is likely why the F12-TZ fundamental frequencies give somewhat better correlation to the experiment. J. Chem. Phys. 153, 134305 (2020); doi: 10.1063/5.0023205 153, 134305-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp IV. CONCLUSIONS Cationic complexes of H 3+and D 3+with two or three attached argon atoms were produced in a pulsed supersonic beam and stud- ied with infrared photodissociation spectroscopy in the 2000 cm−1– 4500 cm−1region. Structured spectra were obtained for each of the four complexes studied. The spectra can be interpreted qualitatively as perturbed H 3+or D 3+ions with a reduced symmetry in the di- argon complexes and argon solvation in both the di- and tri-argon complexes. The spectra reveal the antisymmetric stretch vibrations, whose degeneracy is split into two components in the di-argon com- plex but is restored to a single feature in the tri-argon complexes. Argon solvation shifts these frequencies substantially to the red from the free-ion vibrations. Additionally, overtones of the antisymmetric stretch appear with significant intensities for each of these com- plexes. Additional vibrational bands are observed for each of the four complexes that are attributed to combinations of the antisymmetric stretching vibration of the core ion with various argon stretching and bending modes. These spectra are investigated with two forms of anharmonic vibrational theory. Each method provides qualitative insights into the spectroscopy, but neither is able to provide consis- tently reliable quantitative assignments of the complex vibrational structure due to weak atomic interactions with the argon atoms and notable anharmonicities in the H 3+core. SUPPLEMENTARY MATERIAL See the supplementary material for mass spectra of the cluster ions and for optimized geometries and harmonic frequencies of all computed isomers. ACKNOWLEDGMENTS This work was sponsored by the National Science Founda- tion through Grant No. CHE-1764111 (M.A.D.). R.C.F. acknowl- edges funding from the NSF (Grant No. OIA-1757220), the NASA (Grant No. NNX17AH15G), the Mississippi Center for Supercom- puting Research, and startup funds provided by the University of Mississippi. DATA AVAILABILITY The data that support the findings of this study are available within this article and its supplementary material. REFERENCES 1J. J. Thomson, “XXVI. Rays of positive electricity,” Philos. Mag. 21, 225 (1911). 2P. H. Dawson and A. W. Tickner, “Detection of H 5+in hydrogen glow dis- charge,” J. Chem. Phys. 37, 672 (1962). 3K. Buchheit and W. Henkes, “Investigations of mass spectra of hydrogen cluster ions with a momentum analyzer of the magnetic sector type,” Z. Angew. Phys. 24, 191 (1968). 4R. Clampitt and L. Gowland, “Clustering of cold hydrogen gas on protons,” Nature 223, 815 (1969). 5S. L. Bennett and F. H. Field, “Reversible reactions of gaseous ions. 7. Hydrogen systems,” J. Am. Chem. Soc. 94, 8669 (1972). 6R. Johnsen, C. M. Huang, and M. A. 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5.0017637.pdf

Appl. Phys. Lett. 117, 042408 (2020); https://doi.org/10.1063/5.0017637 117, 042408 © 2020 Author(s).Perpendicular magnetic tunnel junctions based on half-metallic NiCo2O4 Cite as: Appl. Phys. Lett. 117, 042408 (2020); https://doi.org/10.1063/5.0017637 Submitted: 09 June 2020 . Accepted: 19 July 2020 . Published Online: 30 July 2020 Yufan Shen , Daisuke Kan , I-Ching Lin , Ming-Wen Chu , Ikumi Suzuki , and Yuichi Shimakawa ARTICLES YOU MAY BE INTERESTED IN Sizable spin-transfer torque in the Bi/Ni 80Fe20 bilayer film Applied Physics Letters 117, 042407 (2020); https://doi.org/10.1063/5.0009798 High quality epitaxial thin films and exchange bias of antiferromagnetic Dirac semimetal FeSn Applied Physics Letters 117, 032403 (2020); https://doi.org/10.1063/5.0011497 Spintronics with compensated ferrimagnets Applied Physics Letters 116, 110501 (2020); https://doi.org/10.1063/1.5144076Perpendicular magnetic tunnel junctions based on half-metallic NiCo 2O4 Cite as: Appl. Phys. Lett. 117, 042408 (2020); doi: 10.1063/5.0017637 Submitted: 9 June 2020 .Accepted: 19 July 2020 . Published Online: 30 July 2020 Yufan Shen,1Daisuke Kan,1,a) I-Ching Lin,2Ming-Wen Chu,2Ikumi Suzuki,1and Yuichi Shimakawa1 AFFILIATIONS 1Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan 2Center for Condensed Matter Sciences, National Taiwan University, Taipei 10617, Taiwan a)Author to whom correspondence should be addressed: dkan@scl.kyoto-u.ac.jp ABSTRACT Spin polarization and magnetic anisotropy are key properties that determine the performance of magnetic tunnel junctions (MTJs), which are utilized in various spintronic devices. Although materials that have both high spin polarization and sufficient perpendicular magnetic anisotropy are desirable for further developments of MTJs, such materials are rare because electronic structures necessary for achieving thesetwo properties are distinct. Here, we experimentally show the inverse spinel ferrimagnet NiCo 2O4(NCO), whose band structure is theoretically predicted to be half-metallic, has both high spin polarization and perpendicular magnetic anisotropy. Perpendicular MTJs withNCO magnetic electrodes exhibit magnetoresistance up to 230%, indicating that the spin polarization of perpendicularly magnetized NCO is as high as /C073%. Our experimental results demonstrate the potential of NCO as a half-metal with perpendicular magnetic anisotropy, which will lead to new paradigms for designing and developing all-oxide spintronic devices. Published under license by AIP Publishing. https://doi.org/10.1063/5.0017637 Magnetic tunnel junctions (MTJs) with perpendicular mag- netic anisotropy (PMA) 1–4have been regarded as an imperative component for next-generation high-density non-volatile memorydevices. 5,6The tunnel magnetoresistance (TMR) ratio in MTJs, defined as (R AP-RP)/RP, where R Pand R AP, respectively, are the resistance in parallel (P) and anti-parallel (AP) magnetization con- figurations of magnetic electrodes, is closely tied with spin polari- zation of conduction carriers in magnetic electrodes.7Actually, MTJs composed of half-metals8,9whose conduction carriers are fully spin-polarized (100% spin polarization) have been reported to show large TMR ratios.10–13Therefore, exploring magnetic materials that have high spin polarization as well as sufficient perpendicular magnetic anisotropy (PMA) is crucial for further developing perpendicular MTJs. However, stabilizing PMA in half-metals is rather challenging. Spin–orbit interactions for inducing PMA could somehow modify the half-metallic bandstructures, in which one of the spin subbands (either the majority- or minority-spin subband) is partially occupied while another one has an energy gap (the half-metallic energy gap) at the E F.8 Fabricating artificial structures has been shown as an approach for stabilizing the PMA in half-metals for example in Heusler alloys.14,15However, for this approach, the half-metals have to be thin enough (less than a few nanometers thick), and consequently,their characteristic band structures are more or less affected. Hence, finding new materials with high spin polarization and suffi- cient PMA is still in need. Spinel-structured oxides accommodate various transition metals in crystallographically distinct tetrahedral (T d) and octahedral (O h) sites, and magnetic interactions between these cations lead to magnetic properties that are useful for spintronic devices such as spin-filter tun- nel junctions.16,17The inverse-spinel NiCo 2O4(NCO),18–22in which the T d-site is populated by Co, and the O h-site is occupied by Ni and Co evenly [ Fig. 1(a) ], is a ferrimagnetic metal with a transition temper- ature above 400 K and saturated magnetization of 1.5–2 lBper for- mula unit (f.u.). Compressively strained epitaxial films of NCO (grown on MgAl 2O4substrates) have also been revealed to possess PMA, whose anisotropy energy is as large as 0.2 MJ/m3at room tem- perature.23,24Theoretical calculations indicated that majority- and minority-spin subbands dominantly originate from Co and Ni, respec- tively, and that the density of states at the E Fconsists of only the minority-spin subband, leading to the half-metallic band structurewith the spin polarization of /C0100% 25–27[Fig. 1(b) ]. In addition, recent investigations including x-ray magnetic circular dichroism characterizations have shown that the PMA in NCO results from the orbital magnetic moment dominantly arising from the d x2-y2orbital in the T d-site Co and that the O h-site cations have negligibly small Appl. Phys. Lett. 117, 042408 (2020); doi: 10.1063/5.0017637 117, 042408-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplcontributions to the PMA.28,29These observations imply that NCO is a candidate material that has both high spin polarization and sufficientperpendicular magnetic anisotropy, highlighting its potential applica- tion for spintronic devices. Here, we show that all-oxide perpendicular MTJs, consisting of NCO as magnetic electrodes and the spinel oxide MgAl 2O4(MAO) as a tunnel barrier layer, have TMR ratios as large as 230% under out-of- plane magnetic fields. This result indicates that the spin polarization inperpendicularly magnetized NCO is as high as /C073%. These results clarify the potential of NCO as a half-metal with PMA. The trilayer NCO(40)/MAO(3)/NCO(25) stack for MTJs, in which the MAO layer (bandgap /C247.8 eV in bulk) serves as a tunnel barrier, was deposited on (100) MAO substrates (5 mm /C25m m i n size) by pulsed laser deposition. The numbers in parentheses denote the layer’s thickness in nanometers. The NCO and MAO layers were,respectively, deposited by ablating NiCo 2Oxand MgAl 2Oyceramic tar- gets with a KrF excimer laser ( k¼248 nm) with a laser spot energy density of 1.2 J/cm2. While the trilayer stack was deposited at the fixed temperature of 315/C14C, the oxygen pressure during the deposition was changed to tune the properties of each constituent layer. The bottom NCO layer was grown under 100 mTorr oxygen pressure. Then theMAO layer was deposited on top of it under 50 mTorr oxygen pres- sure, which was followed by in situ annealing for 15 min without changing the pressure. The top NCO layer was deposited on the MAOlayer under 200 mTorr oxygen pressure, and the grown trilayer stackwas cooled to room temperature without changing the pressure. Figure 1(c) shows the x-ray 2 h/hdiffraction profile for the grown trilayer stack. For references, the diffraction patterns for the single-layer NCO(25) film and the bilayer MAO(3)/NCO(25) stack are also shown in the figure. The (004) NCO reflection from the trilayer stackis found to split. The observed reflection profiles are attributed to the interference between x-rays diffracted from the top and bottom NCO layers and, as shown in the figure, can be well reproduced by our struc- tural model calculation. Figure 1(d) shows the reciprocal space map around the (408) MAO reflections for the trilayer stack. The (408) NCO reflection from the stack appears in the same position along the horizontal axis (the in-plane direction) as that of the MAO substrate. These observations indicate that the trilayer stack is epitaxially grown on the substrate, and that both top and bottom NCO layers are underthe substrate-induced compressive strain ( /C240.4%). We also note that the MAO layer has a smooth surface with the root mean square roughness below 0.5 nm as confirmed from surface morphology observed by atomic force microscopy [inset of Fig. 1(c) ]. Figure 2 shows the magnetic field dependence of the magnetiza- tions for the single-layer film and the trilayer stack. The diamagnetic signal from the substrates was subtracted by linearly fitting the data in larger magnetic fields. The single-layer film displays a saturation mag- netization (M s)o f1 lB/f.u. at room temperature, which is in close agreement with previous studies.23,24The out-of-plane magnetization [Fig. 2(a) ] exhibits a square-shape hysteresis with coercive field Hc of 17 mT. On the other hand, as shown in Fig. 2(b) , the in-plane magne- tization displays a negligibly small hysteresis against the field sweep direction and is saturated in the magnetic field region larger than 3 T.These behaviors of the magnetization confirm that the NCO layer has perpendicular magnetic anisotropy. The room-temperature hysteresis loop of the out-of-plane magnetization for the trilayer stack [ Fig. 2(c) ] displays two distinct steps, at around 17 and 260 mT, which are associ- ated with the magnetization switching in each NCO layer. Given thatthe magnetic field at which the first magnetization switching occurs is about the same as the Hc for the single-layer NCO(25) film, the mag- netization switching in the bottom NCO layer leads to the formation FIG. 1. (a) Crystal structure of the inverse spinel oxide NiCo 2O4(NCO). (b) Schematic band diagram around the E Fin NCO. The diagram is based on the cal- culated spin-resolved densities of states (Refs. 25–27 ). (c) x-ray 2 h/hdiffraction profiles around the (004) MgAl 2O4(MAO) reflection for the NCO(40)/MAO(3)/ NCO(25) trilayer stack epitaxially grown on (001) MAO substrates, together with thesimulated profile. As references, the diffraction profiles of the single-layer NCO(25)film and the bilayer MAO(3)/NCO(25) stack are also shown. In the figure 5lm/C25lm AFM surface morphology of the bilayer MAO/NCO stack is included. (d) Reciprocal space mapping around the (408) MAO reflection for the NCO(40)/MAO(3)/NCO(25) trilayer stack. The numbers in parenthesis denote the layer thick-ness in nanometers. FIG. 2. (a) and (b) Room-temperature magnetic-field dependence of (a) out-of- plane and (b) in-plane magnetization of the single-layer NCO(25) film. (c) Room- temperature magnetic-field dependence of the out-of-plane magnetization for the NCO(40)/MAO(3)/NCO(25) trilayer stack. The arrows in the figure indicate the fieldsweep direction. (d) Major (red) and minor (blue) loops of the out-of-plane magneti-zation at 35 K for the trilayer stack.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 042408 (2020); doi: 10.1063/5.0017637 117, 042408-2 Published under license by AIP Publishingof the anti-parallel configuration between the magnetizations in the top and bottom NCO layers in the magnetic field region between 17 and 260 mT at room temperature. These observations indicate that the Hc of the top NCO(40) layer is larger than that of the bottom NCOlayer. In addition, the M svalue of the trilayer stack [ Fig. 2(c) ]i m p l i e s that the magnetization of the top NCO layer is largely reduced (/C240.3lB/f.u. at room temperature, and also see Fig. S1). Cross- sectional scanning transmission electron microscope (STEM) observa-tions of the trilayer stack (Fig. S2) shows that the upper NCO/MAOinterface is much rougher than the lower NCO/MAO interface. Thisobservation suggests that larger amounts of defects and cation disor- ders, which deteriorate NCO’s magnetic properties, 23,24are introduced in the top NCO layer, which explains why the top NCO layer has thereduced M sand the increased Hc. As shown in Fig. 2(d) , the two-step magnetization switching in the trilayer stack is also seen at low tem-peratures, such as 35 K. In addition, the minor hysteresis loop exhibits the sharp single-step magnetization switching in the bottom NCO layer in the returning field sweep (from the negative field to the posi-tive one), and the switching fields in the forward and returning sweepsof the minor loop are almost the same. These observations indicatethat the exchange coupling between the magnetizations in the top and bottom NCO layers through the MAO barrier is very weak. To evaluate TMR in the trilayer stack, we employed microfabri- cation processes based on conventional photolithography and Ar ionmilling and patterned the stack into 8 lm/C28lm pillars. A schematic of the fabricated devices was provided in the inset of Fig. 3(a) .T h e resistance of the fabricated TMJs was measured by a standard two- probe method and in a voltage-controlled mode. Magneto-transportmeasurements were carried out under vacuum in a closed-cycle cryo- stat system. Figure 3(a) shows the I–V curves for the fabricated junc- tions at various temperatures without magnetic fields. We defined thecurrent flowing from the top NCO layer to the bottom one as positive. At high temperatures, I–V curves are linear due to large leakage cur- rents in the MAO barrier. With decreasing temperatures, the leakage currents are suppressed and non-linear behavior begins to appear. In addition, as shown in the inset of the figure, the derivative of the I–Vcurve (dI/dV) at 35 K exhibits the parabolic dependence on the bias voltage, signifying that the electron tunneling process through the bar- rier becomes dominant at low temperatures. To minimize the leakagec u r r e n ti s s u e sa sm u c ha sp o s s i b l e ,w ef o c u so nd a t at a k e na t3 5 K , which is the lowest temperature that could be reached by our measure- ment setup. Figure 3(b) shows the out-of-plane and in-plane magnetic field dependence of the junction resistance measured when applying the bias voltage of 0.5 mV. When magnetic fields are swept along theout-of-plane direction, the junction resistance exhibits a clear hystere- sis with high and low resistance states in the anti-parallel (AP) and parallel (P) magnetization configurations, respectively. The observed TMR is positive, implying that the spin-down electrons (in the minority-spin subband) in both NCO layers dominantly contribute tothe electron tunneling process and that the NCO layer has the negative spin polarization expected from the calculated band structure. 25–27We also note that when in-plane magnetic fields are applied, no TMR isdetected and the junction resistance is almost constant [the bottom panel in Fig. 3(b) ], ensuring that our TMR junctions have the perpen- dicular anisotropy. The TMR ratio calculated from the resistance at 0.2 T is about 230% [the upper panel in Fig. 3(b) ]. By simply assuming that the spin polarization ( P) of the top and bottom NCO layers is the same, the Julliere Model 7can give the following relation between the TMR and spin polarization: TMR ratio ¼2jPj2/(1-jPj2)A c c o r d i n g l y , the observed TMR ratio indicates that the NCO’s geometrical mean value of spin polarization is jPj¼73%, and considering the band structure shown in Fig. 1(b) , we can conclude that NCO has the spin polarization of /C073%. It should be pointed out that as discussed above, the magnetic properties of the top and bottom NCO layers are in fact different and that the Pof the upper layer pr obably deteriorates due to some defects associated with the roughness of the upper NCO/MAO interface of the trilayer stack (Fig. S2). This implies that the actual spin polarization of the bottom NCO layer is larger than that estimated by the Julliere model. For MTJs having MAO barriers, spin-dependent coherent tunneling processes have been shown to be possible for elec-trons in the D 1band, leading to high TMR ratios.30This band symme- try is compatible with the sand d22 3z-rorbitals,31,32suggesting possible occurrence of coherent tunneling processes in NCO/MAO/NCO junc- tions. However, local density of states of the NCO layers near the MAO barrier strongly depend on strained states and interface struc-tures like chemical compositions and bonding at the NCO/MAO interfaces. Further investigations such as first-principle calculations will be necessary to evaluate the mechanism of spin-dependent tunnel-ing processes in NCO/MAO/NCO junctions. It should be pointed out that the resistance of the AP magnetiza- tion configuration strongly depends on the magnitude of the appliedbias. Figure 3(c) shows the out-of-plane magnetic field dependence of the junction resistance measured with the applied bias of 1, 5, and 10 mV, and the inset of the figure shows the bias dependence of the junction resistances in the P and AP magnetization configurations. FIG. 3. (a) I–V characteristics for the fabricated NCO/MAO/NCO junctions at differ- ent temperatures without magnetic fields. The insets show the schematic of ourMTJ devices and the derivative of the I–V curve at 35 K. (b) Out-of-plane and in-plane magnetic field dependence of the junction resistance at 35 K. The red arrows indicate the field sweep direction. (c) Out-of-plane magnetic field dependence of the junctions’ resistance measured under different bias voltages. The inset showsthe bias dependence of junctions’ resistance in the parallel (P) and anti-parallel(AP) magnetization configurations.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 042408 (2020); doi: 10.1063/5.0017637 117, 042408-3 Published under license by AIP PublishingWhile the resistance of the P magnetization configuration (the low-resistance state) remains almost unchanged against the bias vol- tages, that of the AP configuration (the high-resistance state) is largely reduced with increasing the bias from 0.5 mV to 10 mV, resulting inthe strong bias-dependence of the TMR ratio. While bias-dependentreductions in the TMR ratio are often ascribed to a zero-bias conduc- tion anomaly due to the spin-wave excitation at interfaces between the magnetic electrodes and the barrier, 33–35the observed reduction in the TMR ratio is too large to be explained by the spin-wave excitation sce-nario. Instead, we point out that large bias-dependent reductions ofTMR ratio in the low-bias region were observed for MTJs embedding the half-metallic Heusler alloy Co 2MnSi as magnetic electrodes and were regarded as a manifestation of the half-metallic energy gap.11,36 In the AP magnetization configuration, the half-metallic gap prohibitselectrons in the E Ffrom tunneling. Applying the bias relaxes this pro- hibition of the electron tunneling, reducing the resistance in the AP configuration. Therefore, the bias-dependence of the resistance in the AP configuration for the NCO/MAO/NCO junction is probably thesign of the existence of the half-metallic energy gap in NCO. In summary, we fabricated perpendicular MTJs consisting of NiCo 2O4magnetic electrodes, whose band structure is theoretically shown to be half-metallic. The fabricated junctions exhibit TMR ratios as large as 230% under out-of-plane magnetic fields, while no TMR is observedunder in-plane fields. The observed TMR ratio indicates that the spinpolarization of the perpendicularly magnetized NCO film is as high as/C073%. In addition, the junction resistance in the antiparallel magnetiza- tion configuration is strongly bias-dependent even in the relatively low b i a sr e g i o n ,i n d i c a t i v eo ft h ee x i s t e n c eo ft h eh a l f - m e t a l l i ce n e r g yg a pi nNCO. These observations reveal the potential of NCO as a half-metal withperpendicular magnetic anisotropy. The unique properties of NCO will lead to new paradigms for designing and developing spintronic devices. See the supplementary material for (1) M–H curve of NCO films grown under 200 mTorr oxygen pressure, and (2) cross-sectionalSTEM observations of the NCO/MAO/NCO trilayer stack. This work was partially supported by a grant for the Integrated Research Consortium on Chemical Sciences, by Grants-in-Aid forScientific Research (Grant Nos. JP16H02266, JP17H04813,JP19H05816, and JP19H05823), by a JSPS Core-to-Core program (A), and by a grant for the Joint Project of Chemical Synthesis Core Research Institutions from the Ministry of Education, Culture,Sports, Science, and Technology (MEXT) of Japan and by ISHIZUE2020 of Kyoto University Research Development Program. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno, Nat. Mater. 9, 721 (2010). 2N. Nishimura, T. Hirai, A. Koganei, T. Ikeda, K. Okano, Y. Sekiguchi, and Y. Osada, J. Appl. Phys. 91, 5246 (2002). 3H. Ohmori, T. Hatori, and S. Nakagawa, J. Appl. Phys. 103, 07A911 (2008).4G. Kim, Y. Sakuraba, M. Oogane, Y. Ando, and T. Miyazaki, Appl. Phys. Lett. 92, 172502 (2008). 5J.-G. Zhu and C. Park, Mater. Today 9, 36 (2006). 6A. Hirohata, K. Yamada, Y. 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5.0016230.pdf

J. Chem. Phys. 153, 074309 (2020); https://doi.org/10.1063/5.0016230 153, 074309 © 2020 Author(s).Energetic degeneracy and electronic structures of germanium trimers doped with titanium Cite as: J. Chem. Phys. 153, 074309 (2020); https://doi.org/10.1063/5.0016230 Submitted: 03 June 2020 . Accepted: 31 July 2020 . Published Online: 19 August 2020 Le Nhan Pham , and Salvy P. Russo ARTICLES YOU MAY BE INTERESTED IN Bond dissociation energies of transition metal oxides: CrO, MoO, RuO, and RhO The Journal of Chemical Physics 153, 074303 (2020); https://doi.org/10.1063/5.0021052 Electronic structure software The Journal of Chemical Physics 153, 070401 (2020); https://doi.org/10.1063/5.0023185 Quasi-symmetry effects in the threshold photoelectron spectrum of methyl isocyanate The Journal of Chemical Physics 153, 074308 (2020); https://doi.org/10.1063/5.0017753The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Energetic degeneracy and electronic structures of germanium trimers doped with titanium Cite as: J. Chem. Phys. 153, 074309 (2020); doi: 10.1063/5.0016230 Submitted: 3 June 2020 •Accepted: 31 July 2020 • Published Online: 19 August 2020 Le Nhan Pham1,2,a) and Salvy P. Russo1,a) AFFILIATIONS 1ARC Centre of Excellence in Exciton Science, School of Science, RMIT University, Melbourne 3000, Australia 2Department of Chemistry, The University of Dalat, 670000 Dalat, Vietnam a)Authors to whom correspondence should be addressed: le.nhan.pham@rmit.edu.au and salvy.russo@rmit.edu.au ABSTRACT Geometries and electronic structures of germanium trimer clusters doped with titanium TiGe 3−/0were studied making use of the com- plete active space self-consistent field followed by second-order perturbation theory, explicitly correlated coupled cluster singles and doubles method with perturbative triples corrections CCSD(T)-F12, and Tao-Perdew-Staroverov-Scuseria methods. Two electronic states (2A′and 2A′′) of the anion (pyramid shape) were determined to be nearly degenerate and energetically competing for the anionic ground state of TiGe 3−. These two anionic states are believed to be concurrently populated in the experiment and induce six observed anion photoelectron bands. Total 14 electronic transitions starting from the2A′and2A′′states were assigned to five out of six visible bands in the experimental anion photoelectron spectrum of TiGe 3−. Each band was proven to be caused by multiple one-electron detachments from two populated anionic states. The last experimental band with the highest detachment energy is believed to be the result of various inner one-electron removals. Published under license by AIP Publishing. https://doi.org/10.1063/5.0016230 .,s I. INTRODUCTION In the search for new materials used in semiconductors, ger- manium arises as a potential candidate for high performance materials taking part in future non-silicon transistors.1Several works were conducted to identify new forms and properties of pure germanium materials.2–7In order to discover new germanium-based counterparts and improve their expected properties, pure germa- nium materials are doped with a second element, usually transition or noble metals. In doing so, dozens of metal doped germanium materials were studied either experimentally or theoretically.8–19 Once the new materials are synthesized, several experimen- tal techniques can be used to characterize these new materials. Anion photoelectron (PE) spectroscopy is one of the frequently used techniques for materials in the form of clusters. In princi- ple, the synthesized clusters after being mass selected are irradiated with a laser beam of photons, leading to removals of valence elec- trons. The signals of removed electrons are then recorded in the anion PE spectra, and detachment energies of electrons are also determined. On the basis of detachment energies and cluster sizes, quantum chemical computations are performed subsequently toidentify geometries, electronic structures, and related properties of obtained clusters.11–14,16,18,19 Multiconfigurational or strongly correlated characters are inherent in systems containing transition metals and their excited states and therefore have strong effects on energetic properties of systems.20For pure germanium clusters and their metal doped ones, energetic degeneracy and multiconfigurational features were found to significantly contribute to the potential energy surfaces of corre- sponding clusters.21–24As a result, single reference methods density functional theory (DFT), coupled cluster singles and doubles with perturbative triples corrections [CCSD(T)] may not give reliable energetic values.25–29To accurately describe such systems and their excited states, the use of multireference quantum chemical methods is inevitable.30Such use also shed light on the dominant electronic configurations of the studied systems that single reference calcula- tions can be based on subsequently. Hence, simultaneous utilization of both single reference and multireference methods to support and confirm results of each other is expected to be more reliable in the description of non-single reference systems.23,24,31–36 Among the germanium clusters doped with titanium, TiGe 3−is one of the smallest ones synthesized and spectroscopically probed.19 J. Chem. Phys. 153, 074309 (2020); doi: 10.1063/5.0016230 153, 074309-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . Experimental anion photoelectron spectrum of TiGe 3−. Reproduced with permission from Deng et al. , RSC Adv. 4, 25963 (2014). Copyright 2014 Royal Society of Chemistry. In the anion photoelectron spectrum of TiGe 3−, six distinguished ionization levels were reported. The lowest one, usually correspond- ing to a ground–ground electronic transition, has an adiabatic detachment energy (ADE) of 1.43 eV, while its vertical detachment energy (VDE) is 1.67 eV. Five higher ionization levels have VDEs of 1.95 eV, 2.38 eV, 2.68 eV, 3.20 eV, and 3.43 eV in ascending order. For the sake of convenience, all six experimental bands are denoted by X to E in the experimental spectrum of TiGe 3−, as can be seen in Fig. 1. Basic explanation for the obtained experimental spectrum of TiGe 3−was made on the basis of DFT calculations.19A C stetra- hedral isomer [see Fig. 2(c)] was found to be the most stable isomer for both the neutral (3A) and anion (2A). Therefore, the first band X was assigned to the electronic transition2A→3A. This assign- ment is not clear enough because the most stable isomer of TiGe 3− belongs to the C spoint-group symmetry, and hence, the involved electronic states need to be more specific. Five more experimen- tal bands corresponding to higher levels of ionization observed in the spectrum of TiGe 3−are still unexplained. Because the TiGe 3−/0 clusters are among the smallest metal doped clusters of germanium, accurate description of such small systems will contribute to a sys- tematic understanding of whole series doped with titanium and will partially shed light on the selection of possible DFT function- als for future study of larger clusters. Hence, we decided to study TiGe 3−/0by using rather high-level theory of wave function methods in combination with appropriate density functionals. FIG. 2 . Three isomers of TiGe 3−/0and the coordinate systems used in this work.II. COMPUTATIONAL METHODS The computational process was performed through two steps including geometrical optimization and additional single point cal- culations. Because the difference in energy of important states is negligible, a few more calculations were conducted to assess the effects of zero-point energy (ZPE) on the nearly degenerate states. To make things easier to follow, three quantum chemical programs used in this work are mentioned here first. Specifically, multiconfigura- tional calculations were conducted making use of the OpenMolcas code;37single point energies were obtained from calculations done with the Molpro program 2019;38and the Turbomole 7.2 package39 was employed for DFT vibrational frequencies and ZPEs. A general geometrical pattern of atomic clusters is often unknown, and therefore, the first step is to identify the most stable geometrical structure of TiGe 3−/0. Since electronic states of TiGe 3−/0 are expected to be multiconfigurational, geometries of TiGe 3−/0were optimized at the complete active space self-consistent field followed by second-order perturbation theory level of theory.40From the pre- vious report19and our preliminary DFT calculations, three geomet- rical shapes of TiGe 3−/0should be considered at this level of theory. These three geometrical structures and their pre-defined coordinate systems are given in Fig. 2. The complete active space self-consistent field calculations need to be fed with an active space for generation of all possible configuration state functions. For TiGe 3−/0, an active space of 15 orbitals consisting of all six valence orbitals (4s and 3d) of Ti and nine 4p ones of the Ge 3moiety. Note that the number of electrons in the anion is higher than that in the neutral by one. Hence, with the same active space size, the numbers of electrons in the active space of the anion and the neutral have a difference of 1, being 11 and 10, respectively. For each CASSCF/CASPT2 calcu- lation, dynamic correlation energy is computed on the top of the CASSCF wave function by implementation of the second order per- turbation CASPT2. All 3p and inner core orbitals of both titanium and germanium atoms were not taken into account (kept frozen) in the perturbative calculations. All the CASSCF/CASPT2 calcula- tions were done in combination with the ANO-RCC basis sets con- tracted to [7s6p4d3f2g]41and [6s5p3d1f]42for titanium and germa- nium, respectively. Such large ANO basis-set contraction was used to study several small systems containing titanium and proven to be relevant.23,43,44Additionally, the above contraction of ANO basis sets was used with the intention to significantly reduce computing resources for CASPT2 optimization in this work. Various single point energies of optimized electronic states were computed using the CASPT2 geometries. Single point calcu- lations at two levels of theory [explicitly correlated restricted-spin coupled-cluster single-double plus perturbative triple excitation45 and Tao-Perdew-Staroverov-Scuseria46] were carried out to syner- gistically support the CASPT2 results. The TPSS functional was used because it is quite good for treatment of systems containing transition metals with multiconfigurational characters.32,49,50The explicitly correlated RCCSD(T)-F12 method can recover dynamic correlation energy quite well with a relatively small-size basis set in comparison to the restricted-spin coupled-cluster single-double plus perturbative triple excitation. Therefore, two triple- ζbasis sets aug-cc-pVTZ47and cc-PVTZ48were used for titanium and germa- nium at this level. For TPSS calculations, a larger quadruple- ζbasis set aug-cc-pVQZ-DK47,48was used for both Ti and Ge. Due to the J. Chem. Phys. 153, 074309 (2020); doi: 10.1063/5.0016230 153, 074309-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp importance of relativistic effects in transition metals and heavy ele- ments, all electron scalar relativistic effects were taken into account by treatment of the second-order Douglas–Kroll–Hess Hamiltonian (DKH2).51Calculations of correlation energy in the RCCSD(T)-F12 implementation exclude dynamic recovery from frozen core orbitals as in the CASPT2 calculations. Unless otherwise stated, relative energies of listed states were deduced from bare electronic energies without ZPE correction because ZPEs seem not to significantly affect potential energy surfaces of small systems. In order to ensures neg- ligible contribution of ZPEs to nearly degenerate electronic states of TiGe 3−/0, harmonic vibrational frequencies of two nearly degener- ate states were analytically calculated at the TPSS level employing the triple- ζdef2-TZVP basis set.52 III. RESULTS AND DISCUSSION A. Most stable isomer and electronic ground states The most stable geometrical and electronic structures of the anionic and neutral ground states are paramount important in elu- cidation of the observed TiGe 3−spectrum. This is because most of the visible bands appearing in the spectrum originate from the most stable anionic isomer and its ground state. Therefore, to ensure that findings of the most stable isomer and the ground states of TiGe 3−/0are reliable, energies of various electronic states were computed at three levels of theory [CASSCF/CASPT2, RCCSD(T)- F12, and TPSS] simultaneously. The relative state energies of three probed isomers (Fig. 2) are tabulated in Table I disclosing the most stable isomer and lowest electronic states of the anion and neu- tral TiGe 3−/0. Three used methods determined that the tetrahedral η3-(Ge) 3Ti (isomer A in Fig. 2) was found to be the most stable geometrical structure of the anionic and neutral TiGe 3−/0clusters, which is in agreement with a previous report.19To be more detailed, at the CASPT2 level, the2A′′state of the tetrahedral η3-(Ge) 3Ti− was identified as the anionic ground state, which is 0.30 eV and 0.47 eV more stable than the lowest low-lying states of the cyclic η3-(Ge) 3Ti−(2B2, isomer B) and rhombic η2-(Ge) 3Ti−(4B2, isomer C), respectively. For the neutral cluster TiGe 3, the3A′′state of iso- mer A is clearly the global ground state of the neutral. Two next low-lying states (3B2and3A2) of isomer C are ∼0.6 eV higher than the3A′′state of isomer A. The RCCSD(T)-F12 and TPSS results also reinforce geometrical findings from CASPT2 optimizations. Note that for some low-lying electronic states represented by hyper open-shell electronic configurations, single-reference methods can- not treat these state wave functions, and therefore, their relative energies at RCCSD(T)-F12 and TPSS are not available in Table I. From Table I, one can see that two anionic states2A′and2A′′ of isomer A are nearly degenerate and compete with each other for the ground state of the anion TiGe 3–. The difference in energy of these two states is ∼0.01 eV, in which the2A′′state is slightly more stable than the2A′one. Because the difference is insignificant, we need to examine the effect of ZPEs on energies of these two com- petitive states. Relative energies corrected with the TPSS ZPEs of the two states2A′and2A′′are given in Table II. Obviously, ZPE does not substantially affect the energy of the two nearly degener- ate states. Especially, at the CASPT2 level of theory, the difference is almost close to zero (0.002 eV, and 0.001 eV with larger contraction of [10s9p8d6f4g2h] for Ti, and [7s6p4d2f1g] for Ge). Therefore, it isstrongly believed that these two nearly degenerate states were con- currently populated in the experiment and underwent one-electron removals simultaneously. Let us recapitulate a few main points related to geometrical structures and electronic ground states of TiGe 3−/0identified thus far. The most stable geometrical structures of TiGe 3−/0have a pyra- mid form (see Fig. 2) with a low symmetry group C s. However, the Cssymmetry is only true for the geometries of two lowest states (2A′and2A′′) of the anionic cluster TiGe 3−. The actual symme- try point group of the neutral cluster belongs to a higher symmetry point group C 3v. Therefore, the correct electronic ground state of the neutral should be3E instead of3A′′. It is worth noting that while the anion has nearly degenerate electronic states (2A′and2A′′) vying for the ground state, the neutral has only one state3E (3A′′in C s) being the ground state. Explanations for difference between symmetry of the anionic and neutral ground states and for the degeneracy in the most stable anionic cluster will be provided later in detail once the electronic structures of these states are analyzed. B. Electronic structures and possible electron detachments Electronic structures play an import role in prediction of pos- sible electronic transitions causing visible bands in the experimen- tal spectrum of TiGe 3−. Understanding of electronic structures will unveil insights into geometrical properties and energetic degeneracy in the studied clusters TiGe 3−/0. For the sake of electronic analy- sis, the dominant electronic configurations of the anionic and neu- tral ground states and of several low-lying ones extracted from the CASSCF wave functions are provided in Table III. Leading electronic configurations of the two lowest doublet states of the anion TiGe 3- are given in Table III. Depending on irre- ducible representations of the sole singly occupied active molecular orbital (MO) in the active space of these states, the total symmetry feature of each state is determined correspondingly. A close look at the active orbitals reveals that these MOs are clearly constructed by hybridization of germanium (4p) and titanium (3d and 4s) valence orbitals. In other words, chemical bonds between the metal tita- nium atom and germanium ones are formed and such chemical bonds make the clusters TiGe 3−/0chemically stable. In detail, two hybridization types between metallic valence atomic orbitals (AOs) and germanium 4p ones can be recognized: (i) hybridized MOs formed by Ti 4s and Ge 4p AOs and (ii) by Ti 3d and Ge 4p AOs. The former can be seen in MOs 35a′in the leading configurations of the two lowest anionic states. The remaining occupied active MOs (34a′, 36a′, 37a′, 22a′′, and 23a′′) of the two lowest anionic electronic states are categorized as the latter. A clear understanding of MO components will aid us in predic- tion of one-electron removal ordering. Among the above-mentioned MOs, two active MOs 37a′and 23a′′are dominantly contributed by the metallic AOs of titanium, whereas the active MOs 34a′, 36a′, and 22a′′have more characteristics of Ge 4p AOs. To be more specific, the metallic AOs 3d x2−y2and 3d yzare found to be primary in the MOs 37a′, while a major part of 3d xzis present in the MOs 23a′′. With a bit difference, the 35a′MOs have equally large components of the Ti 4s and Ge 4p AOs. The other active MOs (34a′, 36a′, and 22a′′) are composed of smaller parts from the metallic AOs of Ti, which means these MOs are expected to have more impacts from the Ge 4p AOs. To have an intuitive view on the characteristics of all J. Chem. Phys. 153, 074309 (2020); doi: 10.1063/5.0016230 153, 074309-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Relative energy of three isomers (A, B and C) and their electronic states at the three used levels of theory. CASPT2 geometryRelative energy (eV) Isomer Sym. State (´Å) (r 1, r2, r3, r4) TPSS RCCSD(T)-F12 CASPT2 c-anion C s2A′2.49, 2.43, 2.63, 2.85 0.01 0.01 0.01 2A′′2.42, 2.47, 2.78, 2.56 0.00 0.00 0.00 4A′2.47, 2.45, 2.63, 2.84 0.45 0.79 0.60 4A′′2.53, 2.51, 2.60, 2.60 0.22 0.48 0.22 b-anion C 2v2A1 2.69, 2.51, 2.39, 4.09 0.60 0.36 0.45 2B1 2.71, 2.47, 2.50, 4.17 1.06 2B2 2.70, 2.44, 2.48, 4.11 0.43 0.20 0.30 2A2 2.93, 2.49, 2.42, 3.94 0.56 0.29 0.37 4A1 3.27, 2.53, 2.42, 3.73 0.83 1.10 0.81 4B1 2.89, 2.49, 2.45, 4.01 0.56 0.65 0.48 4B2 2.64, 2.45, 2.42, 4.09 0.68 0.53 0.76 4A2 2.75, 2.52, 2.40, 4.07 0.60 0.52 0.77 a-anion C 2v2A1 4.23, 2.52, 2.45, 2.61 1.05 0.95 1.05 2B1 4.45, 2.74, 2.42, 2.58 1.32 0.94 1.07 2B2 4.39, 2.64, 2.43, 2.53 0.82 2A2 4.39, 2.66, 2.42, 2.55 2.24 2.04 0.70 4A1 3.26, 2.53, 2.42, 3.73 0.83 1.10 0.81 4B1 4.47, 2.73, 2.42, 2.60 0.57 0.96 0.70 4B2 4.41, 2.65, 2.43, 2.53 0.38 0.81 0.47 4A2 4.42, 2.70, 2.42, 2.58 0.86 1.15 0.69 c-neutral C s1A′2.44, 2.46, 2.65, 2.69 2.22 1.63 1.64 1A′′2.45, 2.45, 2.69, 2.61 1.34 3A′2.51, 2.50, 2.83, 2.44 2.14 2.38 2.00 3A′′2.46, 2.46, 2.64, 2.64 1.39 1.63 1.27 b-neutral C 2v1A1 2.52, 2.39, 2.59, 4.28 2.49 1.86 3.11 1B1 2.73, 2.49, 2.49, 4.15 2.55 1B2 2.60, 2.51, 2.43, 4.21 2.50 1A2 2.72, 2.55, 2.39, 4.12 2.57 3A1 2.86, 2.63, 2.40, 4.12 2.74 2.84 2.58 3B1 2.87, 2.48, 2.45, 4.01 2.40 2.46 2.26 3B2 2.54, 2.50, 2.44, 4.23 2.32 2.36 2.14 3A2 2.70, 2.57, 2.38, 4.13 2.36 2.68 2.23 a-neutral C 2v1A1 4.03, 2.46, 2.44, 2.78 2.49 2.30 2.28 1B1 4.45, 2.61, 2.47, 2.43 2.97 1B2 4.20, 2.49, 2.45, 2.62 2.47 1A2 4.39, 2.56, 2.47, 2.46 2.52 3A1 4.24, 2.63, 2.40, 2.71 2.21 2.60 2.12 3B1 4.22, 2.66, 2.38, 2.75 1.89 2.64 2.18 3B2 4.21, 2.58, 2.41, 2.66 1.99 2.43 1.89 3A2 4.23, 2.56, 2.41, 2.62 1.86 2.38 1.89 active MOs in the two nearly degenerate anionic states, the visualiza- tion of these MOs is pictorially presented in Fig. 3. For the purpose of prediction on deeper electron removals, three more MOs (32a′, 33a′, and 21a′′) of each anionic state are also plotted in Fig. 3. These threeMOs mainly originate from three 4s AOs of germanium atoms. By checking leading configurations of the remaining states in Table III and orbital plots in Fig. 3, one can derive electronic properties of corresponding states qualitatively. J. Chem. Phys. 153, 074309 (2020); doi: 10.1063/5.0016230 153, 074309-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II . Effects of zero-point energy (ZPE) on the energetic minima of two lowest anionic states2A′and2A′′. CASPT2 RCCSD(T)-F12 TPSS State Without ZPE With ZPE Without ZPE With ZPE Without ZPE With ZPE 12A′′0.00 0.00 0.00 0.00 0.00 0.00 12A′0.01 0.00a0.01 0.01 0.01 0.01 aThe actual value is 0.002. This value is determined to be 0.001 with a larger contraction of [10s9p8d6f4g2h] and [7s6p4d2f1g] for Ti and Ge, respectively. From electronic analysis of MOs above, the energetic order- ing of one-electron removals occurring in the experiment can be roughly estimated. Fundamentally, MOs dominantly composed of metallic AOs tend to undergo one-electron detachments first. With that in mind, we can deduce that for one of the two anionic states(2A′and2A′′), one-electron removals are expected to happen to two MOs 37a′and 23a′′with dominant metallic features first. Alterna- tively stated, electron binding energies of single electrons located in these two MOs are estimated to be lowest. Signals of these removed electrons are often recorded as X bands in the experiment. In total, TABLE III . Leading electronic configurations obtained from the CASSCF wave functions and predicted electronic transitions. Dominant AO components of ionized orbitals are noted in brackets. State Leading electronic configuration Weight (%)aIonization Ionized orbital 12A′32a′233a′234a′235a′236a′237a′↑38a′021a′′222a′′223a′′224a′′072 12A′′32a′233a′234a′235a′236a′237a′238a′021a′′222a′′223a′′↑24a′′071 11A′32a′233a′234a′235a′236a′237a′038a′021a′′222a′′223a′′224a′′055 12A′→11A′37a′[Ti 3d x2-y2, Ti 3d yz] 21A′32a′233a′234a′235a′236a′237a′238a′021a′′222a′′223a′′024a′′055 12A′′→21A′23a′′[Ti 3d xz] 31A′32a′233a′234a′235a′236a′237a′238a′021a′′222a′′↑23a′′↓24a′′033 41A′32a′233a′234a′235a′236a′↑37a′↓38a′021a′′222a′′223a′′224a′′057 11A′′32a′233a′234a′235a′236a′237a′↑38a′021a′′222a′′223a′′↓24a′′071 12A′→11A′′23a′′[Ti 3d xz] 21A′′32a′233a′234a′235a′236a′237a′↑38a′021a′′222a′′↓23a′′224a′′037 12A′→21A′′22a′′[Ge 4p] 31A′′32a′233a′234a′235a′236a′↑37a′238a′021a′′222a′′↓23a′′224a′′037 41A′′32a′233a′234a′235a′236a′237a′038a′↑21a′′222a′′223a′′↓24a′′060 13A′32a′233a′234a′235a′236a′237a′238a′021a′′222a′′↑23a′′↑24a′′069 12A′′→13A′22a′′[Ge 4p] 23A′32a′233a′234a′235a′236a′↑37a′↑38a′021a′′222a′′223a′′224a′′067 12A′→23A′36a′[Ge 4p] 33A′32a′233a′234a′235a′↑36a′237a′↑38a′021a′′222a′′223a′′224a′′051 12A′→33A′35a′[Ti 4s, Ge 4p] 43A′32a′233a′234a′235a′236a′237a′038a′021a′′222a′′223a′′↑24a′′↑53 53A′32a′233a′234a′235a′236a′237a′↑38a′↑21a′′222a′′223a′′024a′′052 63A′32a′233a′234a′235a′236a′237a′038a′021a′′222a′′223a′′↑24a′′↑24 73A′32a′233a′234a′↑35a′236a′237a′↑38a′021a′′222a′′223a′′224a′′039 12A′→73A′34a′[Ge 4p] 13A′′32a′233a′234a′235a′236a′237a′↑38a′021a′′222a′′223a′′↑24a′′069 12A′′→13A′′37a′[Ti 3d x2-y2, Ti 3d yz] 12A′→13A′′23a′′[Ti 3d xz] 23A′′32a′233a′234a′235a′236a′↑37a′238a′021a′′222a′′223a′′↑24a′′056 12A′′→23A′′36a′[Ge 4p] 33A′′32a′233a′234a′235a′236a′237a′↑38a′021a′′222a′′↑23a′′224a′′039 12A′′→33A′′22a′′[Ge 4p] 43A′′32a′233a′234a′235a′↑36a′237a′238a′021a′′222a′′223a′′↑24a′′065 12A′′→43A′′35a′[Ti 4s, Ge 4p] 53A′′32a′233a′234a′235a′236a′037a′238a′↑21a′′222a′′223a′′↑24a′′043 63A′′32a′233a′234a′↑35a′236a′237a′238a′021a′′222a′′223a′′↑24a′′029 12A′′→63A′′34a′[Ge 4p] 73A′′32a′233a′234a′235a′236a′↑37a′238a′021a′′222a′′↑23a′′224a′′066 83A′32a′233a′↑34a′235a′236a′237a′↑38a′021a′′222a′′223a′′224a′′012A′→83A′33a′[Ge 4s] 93A′32a′↑33a′234a′235a′236a′237a′↑38a′021a′′222a′′223a′′224a′′012A′→93A′32a′[Ge 4s] 103A′32a′233a′234a′235a′236a′237a′238a′021a′′↑22a′′223a′′↑24a′′012A′′→103A′21a′′[Ge 4s] 83A′′32a′233a′↑34a′235a′236a′237a′238a′021a′′222a′′223a′′↑24a′′012A′′→83A′′33a′[Ge 4s] 93A′′32a′↑33a′234a′235a′236a′237a′238a′021a′′222a′′223a′′↑24a′′012A′′→93A′′32a′[Ge 4s] 103A′′32a′233a′234a′235a′236a′237a′↑38a′021a′′↑22a′′223a′′224a′′012A′→103A′′21a′′[Ge 4s] aSome higher excited states are predicted by removing one electron from inner MOs of the two lowest anionic states2A′and2A′′, and hence, the coefficient weights are not available. J. Chem. Phys. 153, 074309 (2020); doi: 10.1063/5.0016230 153, 074309-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Occupied orbitals in the active space and three additional 4s(Ge) hybrid ones of the two lowest anionic states 2A′and2A′′. The orbitals are obtained from CASSCF wave functions. Occupa- tion numbers are noted in parentheses. The titanium atom is on the top of the tetrahedral cluster. five electronic transitions starting from the two nearly degenerate anionic states2A′and2A′′were computationally predicted. These electronic transitions are presented as 12A′→11A′, 12A′′→21A′, 12A′→11A′′, 12A′′→13A′′, and 12A′→13A′′in Table III. Note that two separate one-electron detachments can occur in the MO 23a′′of the2A′state obtained from analysis of the CASSCF state wave functions. Conversely, electrons from the MOs characterized with larger contributions of the Ge 4p AOs have a propensity to be removed with higher detachment energies. Therefore, three occupied MOs in the active space 34a′, 36a′, and 22a′′of each anionic state need higher energy of photon beams to be detached. As a result, the elec- tron detachments from these MOs are believed to be responsible for higher-energy bands in the experimental spectrum of TiGe 3−. At least seven electronic transitions were identified to have involvement with one-electron removals from these three orbitals (see Table III). In addition, one other MO (35a′) has a large part of the Ge 4p AOs and hence is expected to have high detachment energy val- ues. Because this MO is occupied in the leading electronic config- uration of the states2A′and2A′′, two more electronic transitions 12A′→33A′and 12A′′→43A′′were predicted from our CASSCF calculations. Apart from aforementioned MOs in the active space, three more MOs beyond the active space of each anionic state were further used for prediction of more six inner one-electron removals responsible for the experimental band with very high elec- tron binding energy that outer one-electron detachments cannot be attributed to. As discussed above, symmetry point groups of the clusters are transformed from C sto C 3vunder the removals of one electron in the two electronic transitions 12A′→13A′′and 12A′′→13A′′. The highest symmetry point group that both the anion and neutral can possess is C 3v. Considering the electronic structure of the anion in its highest symmetry point group, one can deduce that there will be two twofold degenerate MOs in electronic configurations of any dou- blet state (see Fig. 4). Depending on electron occupation schemes of the HOMO pair, two degenerate electronic states are expected to be simultaneously generated. In this instance, Jahn–Teller effects are expected to occur and distort the C 3vgeometry of the anion. Two geometrical distortion directions corresponding to two electronoccupation schemes lead to two lower-symmetry states within the C s point group. This is what we obtained from CASPT2 optimization of the anionic cluster TiGe 3−. For the neutral, energetic degeneracy of states cannot happen in the triple ground state3E (3A′′in C s). Two unpaired electrons occupy two twofold degenerate HOMOs, and therefore, there will be no Jahn–Teller distortion. As a result, symmetry of the triplet neutral ground state is unchanged, being C 3v. The MO diagram of this neutral ground state is provided in Fig. 5. In addition, the MO diagrams of several3A′′excited states are given as well. Each of these excited states, one by one, theoretically has its correspond- ing degenerate state once permutations of electron occupation are FIG. 4 . MO diagrams of the twofold degenerate anionic states2A′and2A′′in the C3vgeometry and the Jahn–Teller effect causing geometrical distortion. The black (red) orbital labels are noted with respect to the C s(C3v) spatial symmetry. Only occupied MOs in the active space are presented. J. Chem. Phys. 153, 074309 (2020); doi: 10.1063/5.0016230 153, 074309-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . MO diagrams of the neutral ground state and its low-lying states in the C3vgeometry. The black (red) orbital labels are noted with respect to the C s(C3v) spatial symmetry. Only occupied MOs in the active space are presented. applied to twofold degenerate MOs. As a result, Jahn–Teller effects are expected to happen to these states, and their relaxation geome- tries will be constrained within the C sspatial symmetry. A further study of Jahn–Teller effects on excited states is of future interest. C. Anion photoelectron band assignments On the basis of electronic structures, energetic orderings of sev- eral predicted one-electron detachments can be roughly estimated and subsequently used for assignments of visible bands in the exper- imental anion PE spectrum of TiGe 3−. The general idea is that detachments of one electron from MOs resulting from hybridiza- tion of major metal AOs will be taking place at lower photon energy and therefore will be responsible for PE bands with lower binding energies. Those MOs comprised of large parts of Ge 4p AOs will have higher ionization energies and will be causing higher-energy PE bands. In conjunction with calculated ADEs and VDEs, PE bands in the spectrum of TiGe 3−will be explained and assigned. Because two lowest anionic states (2A′and2A′′) were determined to be energet- ically degenerate and simultaneously populated in the experiment, all experimental bands in the anion PE spectrum of TiGe 3−are considered based on these two states. In the anion PE spectrum of TiGe 3−, the lowest band, labeled with X in Fig. 1, ranges from ∼1.40 eV to ∼1.8 eV. Conventionally,the X band is attributed to one-electron transitions from the anionic ground state to the neutral one. Interestingly, two low-lying degen- erate states (2A′and2A′′) of the anion are known to be competing for the anionic ground state, and hence, the removals of one-electron from these two anionic states to form the neutral ground state3A′′ are apparently responsible for the X band. Our calculations show that the CASPT2 ADEs of two electronic transitions2A′′→3A′′and 2A′→3A′′are∼1.27 eV, which is ∼0.16 eV lower than the experi- mental ADE of the X band, being 1.43 eV. The experimental VDE of this band was determined to be 1.67 eV, and as usual, the CASPT val- ues of the ground–ground transitions are seriously underestimated for the lowest band. In this situation, single reference methods often give better theoretical values. Indeed, the RCCSD(T)-F12 VDEs are ∼1.70 eV, which is 0.03 eV deviating from the experimental VDE of 1.67 eV. The TPSS results for this band are also acceptedly reason- able with a deviation of ∼0.20 eV from the experimental values. Two ground–ground electronic transitions, their detachment energies, and experimental ones of the X band are presented in Table IV. Note that two one-electron removals causing the X band involve MOs with large hybridization compositions of Ti 3d AOs (see Table III). There are three more one-electron removals originating from MOs characterized with dominant metal AOs as well, and therefore, these electron removals are believed to take part in producing signals of the X band. The VDEs of these three transitions (12A′→11A′, 12A′′→21A′, and 12A′→11A′′) identified at the CASPT2 level are 1.78 eV, 1.70 eV, and 1.74 eV, which all point to the experimental value (1.67 eV). The good agreement between the experimental and calculated VDEs strongly supports the involvement of these three electronic transitions in the X band. In total, five electronic transi- tions were proven to concurrently arise under the irradiation of the experimental photon beam and give rise to the X band. The next band in the anion PE spectrum of TiGe 3−was recorded at 1.95 eV, noted as A in Fig. 1. Two electronic transi- tions 12A′′→13A′and 12A′→23A′were found to be energetically befitting of this band at the CASPT2 level. To be more detailed, the CASPT2 VDEs of these two transitions are determined to be 2.08 eV, being in good correlation with the experimental VDE of 1.95 eV. Therefore, these two electronic transitions are believed to underlie the A band. In contrast to the CASPT2 values, the VDEs obtained from single reference methods [TPSS and RCCSD(T)-F12] are quite overestimated (see Table IV). Such inaccuracy produced at single configuration levels of theory in treatment of excited sates and systems containing transition metals are not unexpected due to com- plicated wave functions of treated systems and correlated electron effects. An inner ionization level of TiGe 3−was experimentally deter- mined to be 2.38 eV, being ∼0.40 eV vertically higher than the A band. In Table IV, three electronic transitions starting from two competitive anionic ground states2A′and2A′′to both singlet and triplet states (21A′′, 23A′′, and 33A′′) have the CASPT2 VDE val- ues of∼2.35 eV lying within the error bar (0.08 eV) of experi- mental VDE. Without a doubt, these three low-lying states of the neutral were assigned to band B. From this level of excitation, ener- gies of low-lying states cannot be accessed from the single config- uration formulations due to wave function convergence issues and controlling. Two more visible bands in the experimental spectrum of TiGe 3−can be energetically assigned within the scope of our used J. Chem. Phys. 153, 074309 (2020); doi: 10.1063/5.0016230 153, 074309-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE IV . Theoretical and experimental detachment energies of five bands from X to D in the experimental PE spectrum of TiGe 3−. Relative energy (eV)a State TPSS RCCSD(T)-F12 CASPT2 Transition Expt.aBand 12A′′0.00 0.00 0.00 12A′0.01 0.01 0.01 11A′2.07 1.47 1.78 12A′→11A′1.67 X (2.13) (1.62) (1.64) (1.43) 21A′1.70 12A′′→21A′1.67 X 31A′2.34 41A′2.59 11A′′1.74 12A′→11A′′1.67 X (1.65) (1.43) 21A′′2.42 12A′→21A′′2.38 B 31A′′2.73 41A′′2.92 13A′2.23 2.49 2.08 12A′′→13A′1.95 A (2.14) (2.38) (2.00) 23A′2.08 12A′→23A′1.95 A 33A′2.59 12A′→33A′2.68 C 43A′2.67 53A′2.75 63A′3.01 73A′3.16 12A′→73A′3.20 D 13A′′1.48 1.71 1.34 12A′′→13A′′1.67 X (1.39) (1.63) (1.27) (1.43) 1.46 1.69 1.33 12A′→13A′′1.67 X (1.38) (1.62) (1.26) (1.43) 23A′′2.37 12A′′→23A′′2.38 B 33A′′2.33 12A′→33A′′2.38 B 43A′′2.72 12A′′→43A′′2.68 C 53A′′2.74 63A′′3.09 12A′′→63A′′3.20 D 73A′′3.17 aADE values in parentheses. active space orbital. The lower one has a VDE of 2.68 eV, and the higher one has a VDE of 3.20 eV. The initial states are clearly two lowest anionic states as mentioned above. The final neutral states should be formed by one-electron removals from deeper anionic MOs. Intuitively, from the MO diagrams of two anionic states2A′ and2A′′in Fig. 4, one can predict that electron removals can hap- pen to two inner MOs 35a′and 34a′of the2A′and2A′′states. Our CASPT2 calculations found four electronic transitions starting from these MOs whose calculated VDEs are in good correlation with the experiment. Specifically, for the C band, the CASPT2 VDEs of two transitions 12A′→33A′and 12A′′→43A′′are 2.59 eV and 2.72 eV, respectively. In comparison to the experimental VDE of the C band (2.68 eV), the errors of two calculated energies are in a range from 0.04 eV to 0.09 eV, which is quite convincing for assignments of the C band to two mentioned electronic transitions. Similarly, twotransitions 12A′→73A′and 12A′′→63A′′with the calculated VDEs of 3.16 eV and 3.09 eV were ascribed to the appearance of the D band centered at 3.20 eV in the spectrum. Apparently, an additional visible band, represented by E in Fig. 1, was recorded at a higher electron binding energy of 3.43 eV in the experimental spectrum of TiGe 3−. To fully theoretically probe this band, a larger active space of >18 MOs must be employed in the CASSCF/CASPT2 calculations. The inclusion of >18 MOs would be super computationally expensive. Thus, the CASPT2 VDEs of inner electronic transitions involving deeper one-electron removals are not computed. Only possible one-electron transitions, which do not violate the selection rules in anion photoelectron spectroscopy and originate from inner MOs of the two lowest anionic2A′and 2A′′states, are predicted. These one-electron transitions, electronic features of initial and final states, and ionized orbitals can be found J. Chem. Phys. 153, 074309 (2020); doi: 10.1063/5.0016230 153, 074309-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp in Table III. All ionized MOs causing these transitions are mainly consisted of the Ge 4s AOs (see Fig. 3), leading us to an estima- tion that their detachment energies will be quite high. As a result, multiple electronic transitions with involvement of these MOs are believed to generate spectroscopic signals of the E band and higher ones if possibly observed. IV. CONCLUSION Geometries, ground and excited states, and electronic struc- tures of the TiGe 3−/0clusters were studied by using relatively high levels of quantum chemical methods including TPSS, RCCSD(T)- F12, and CASSCF/CASPT2. The most stable geometrical structure of both the anion and the neutral were identified to have a tetrahe- dral arrangement of atoms. Due to a Jahn–Teller effect, the stable anionic cluster TiGe 3−cannot keep the highest possible symmetry (C3v) of a tetrahedral η3-(Ge) 3Ti isomer but the C sone. Conversely, the Jahn–Teller effect does not occur in the neutral ground state, and therefore, its atomic tetrahedral cluster belongs to the spatial point group of C 3v. The ground states of the anion and neutral play an important role in explanation of all visible bands in the experimental PE spec- trum of TiGe 3−. From the calculation results, two states of the anion (2A′and2A′′) were identified to be nearly degenerate and competing for the anionic ground states on the potential energy hypersurface, and a triplet state (3A′′) was concluded to be the ground state of the neutral. As a result, these two states are expected to be pop- ulated in the experimental measurement process. On the basis of the two experimentally populated states of the anion, multiple one- electron detachments were proven to underlie all visible bands in the anion photoelectron spectrum of TiGe 3−. Particularly, five simulta- neous one-electron removals from two anionic ground states were found to be responsible for the X band. The appearance of the A band in the spectrum was caused by two electronic transitions from the anionic states to two low-lying triplet ones. Furthermore, band B was ascribed to three deeper one-electron detachments. For the two higher bands C and D, each was attributed to two one-electron detachments originating from two competitive ground states of the anion. The highest visible band E could not be definitively assigned to specific electronic transitions due to highly demanding compu- tation, but six detachments of one electron were predicted and are believed to give rise to this band and possible higher ones. ACKNOWLEDGMENTS This work was supported by the Australian government through the Australian Research Council (ARC) under the Cen- tre of Excellence scheme (Project No. CE170100026). It was also supported by computational resources provided by the Australian government through the National Computational Infrastructure National Facility (NCI_NF) and the Pawsey Supercomputer Centre. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request.REFERENCES 1R. Pillarisetty, Nature 479, 324 (2011). 2S. Bals, S. Van Aert, C. P. Romero, K. Lauwaet, M. J. Van Bael, B. Schoeters, B. Partoens, E. Yücelen, P. Lievens, and G. Van Tendeloo, Nat. Commun. 3, 897 (2012). 3T. P. Martin and H. Schaber, J. Chem. Phys. 83, 855 (1985). 4Y. Liu, Q. L. Zhang, F. K. Tittel, R. F. Curl, and R. E. Smalley, J. Chem. Phys. 85, 7434 (1986). 5G. Pacchioni and J. Koutecký, J. Chem. Phys. 84, 3301 (1986). 6J. M. Hunter, J. L. Fye, M. F. Jarrold, and J. E. Bower, Phys. Rev. Lett. 73, 2063 (1994). 7G. R. Burton, C. Xu, and D. M. Neumark, Surf. Rev. Lett. 03, 383 (1996). 8V. Kumar and Y. Kawazoe, Phys. Rev. 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5.0007517.pdf

Appl. Phys. Lett. 117, 072403 (2020); https://doi.org/10.1063/5.0007517 117, 072403 © 2020 Author(s).Quantum oscillations with magnetic hysteresis observed in CeTe3 thin films Cite as: Appl. Phys. Lett. 117, 072403 (2020); https://doi.org/10.1063/5.0007517 Submitted: 13 March 2020 . Accepted: 03 August 2020 . Published Online: 19 August 2020 Mori Watanabe , Sanghyun Lee , Takuya Asano , Takashi Ibe , Masashi Tokuda , Hiroki Taniguchi , Daichi Ueta , Yoshinori Okada , Kensuke Kobayashi , and Yasuhiro Niimi ARTICLES YOU MAY BE INTERESTED IN A four-state magnetic tunnel junction switchable with spin–orbit torques Applied Physics Letters 117, 072404 (2020); https://doi.org/10.1063/5.0014771 Engineering the magnetocaloric properties of PrVO 3 epitaxial oxide thin films by strain effects Applied Physics Letters 117, 072402 (2020); https://doi.org/10.1063/5.0021031 Reconfigurable spin orbit logic device using asymmetric Dzyaloshinskii–Moriya interaction Applied Physics Letters 117, 072401 (2020); https://doi.org/10.1063/5.0020953Quantum oscillations with magnetic hysteresis observed in CeTe 3thin films Cite as: Appl. Phys. Lett. 117, 072403 (2020); doi: 10.1063/5.0007517 Submitted: 13 March 2020 .Accepted: 3 August 2020 . Published Online: 19 August 2020 Mori Watanabe,1Sanghyun Lee,1Takuya Asano,1Takashi Ibe,1Masashi Tokuda,1Hiroki Taniguchi,1Daichi Ueta,2 Yoshinori Okada,2Kensuke Kobayashi,1,3 and Yasuhiro Niimi1,4,a) AFFILIATIONS 1Department of Physics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan 2Okinawa Institute of Science and Technology Graduate University, Okinawa 904-0495, Japan 3Institute for Physics of Intelligence and Department of Physics, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan 4Center for Spintronics Research Network, Osaka University, Toyonaka, Osaka 560-8531, Japan a)Author to whom correspondence should be addressed: niimi@phys.sci.osaka-u.ac.jp ABSTRACT We have performed magnetotransport measurements in CeTe 3thin films down to 0.2 K. It is known that CeTe 3has two magnetic transitions atTN1/C253 K and TN2/C251 K. A clear Shubnikov–de-Haas (SdH) oscillation was observed at 4 K, demonstrating the strong two-dimensional nature in this material. Below TN2, the SdH oscillation has two frequencies, indicating that the Fermi surface could be slightly modulated due to the second magnetic transition. We also observed a magnetic hysteresis in the SdH oscillation below TN1. Specifically, there is a unique spike in the magnetoresistance at B/C250:6 T only when the magnetic field is swept from a high enough field (more than 2 T) to zero field. Published under license by AIP Publishing. https://doi.org/10.1063/5.0007517 Research studies on layered materials have attracted much attention over the last decade.1,2This interest was triggered by the dis- covery of graphene, where not only the polarity but also the density of carriers can be controlled by the electric field.3–5A wide range of mate- rials not only limited to semiconductors,6–8such as insulators,9,10 superconductors,11–14and ferromagnetic materials,15–19have been actively studied with the aim of controlling the physical properties orthe phase transition temperature by applying the electric field to suchthin film devices. More recently, the research field, so-called van derWaals engineering, has become an important stream. 1,2The most striking discovery is the superconductivity in twisted bilayer graphene,which is an originally zero gapped semiconductor. 20By stacking a strong spin–orbit transition metal dichalcogenide on a ferromagnetic thin layer, a magnetic skyrmion phase can be induced.21Thus, it is an urgent task to investigate a variety of materials, which can be fabri-cated into atomically thin films and explore atomically stacked deviceswith new physical properties. Specifically, magnetic materials could beuseful for future spintronic applications. 22,23 CeTe 3is a layered material in the family of rare earth ( R) tritellur- ides, i.e., RTe3. It is known as a heavy fermion system with a localized 4f1orbital at the Ce3þsite. Its crystal structure consists of a NaCl-type CeTe layer, which is responsible for its magnetic properties, separated by two Te sheets, which are responsible for the highly two-dimensional(2D) electric transport,24as shown in Fig. 1(a) .D u et ot h eh i g h l y2 D electrical transport, the material forms an incommensurate charge density wave (CDW) from well above room temperature, which has been studied extensively.25–31 This material is also known to show two magnetic phase transi- tions at low temperatures.24,32–34The first magnetic transition at TN1¼3:1 K is understood to be from a paramagnetic state to an anti- ferromagnetic (possibly short range ordering) state. In this phase, themagnetic moment at the Ce site is antiferromagnetically coupled andaligned to an easy axis perpendicular to the layer stacking direction. T h es e c o n dm a g n e t i ct r a n s i t i o ni sk n o w nt ob ea n o t h e ra n t i f e r r o m a g - netic (possibly long range ordering) transition 24,32atTN2¼1:3K . Unlike the case of the first transition, a clear peak in the heat capacityhas been observed below T N2and the magnetic moment is still aligned t ot h ei n - p l a n ed i r e c t i o n( c a l l e dt h e non-parallel easy axis), but differ- ent from the easy axis in the first transition.33Nevertheless, the details of these magnetic ordering states are poorly understood. Furthermore,there are no reports on thin film transport measurements. In the present work, we have performed magnetotransport mea- surements with 30–40 nm thick CeTe 3devices. We have observed the coexistence of quantum oscillation with unique magnetotransportphenomena. Specifically, a clear Shubnikov–de-Haas (SdH) oscillation was observed in the magnetic field Brange of 3–8 T even above T N1, Appl. Phys. Lett. 117, 072403 (2020); doi: 10.1063/5.0007517 117, 072403-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apldemonstrating the existence of a small Fermi surface pocket. Since such a SdH oscillation has never been reported in bulk CeTe 3,t h e result reveals its strong 2D nature, which is possibly enhanced by the thin film fabrication. Furthermore, the SdH oscillation has two fre- quencies below TN2. This could originate from the modification of the Fermi surface due to the second magnetic transition at TN2.W ea l s o observed a magnetic hysteresis in the SdH oscillation below TN1.I n particular, a sharp resistance peak appears at B/C250:6 T only when the magnetic field is swept from a high enough magnetic field (more than2 T) to zero field (see Fig. 3 ). Single crystals of CeTe 3were synthesized in an evacuated quartz tube. The tube was heated up to 900/C14C and slowly cooled to 550/C14C over a period of 7 days. To fabricate thin film devices from the bulkCeTe 3, we used the mechanical exfoliation technique using scotch tapes.3–6It is noted that all the following fabrication processes should be carried out inside a glovebox with an Ar purity of 99.9999% since CeTe 3is extremely sensitive to ambient air. After the mechanical exfoliation process, many CeTe 3thin flakes on the scotch tape were transferred onto a thermally oxidized silicon substrate. We then spin- coated polymethyl-methacrylate (PMMA) resist onto the substrate.The substrate was taken out from the glovebox, and electrode patterns were printed using electron beam lithography. After lithography, the substrate was put back into the glovebox again for the development ofthe resist. The Au electrodes were deposited using electron beam depo-sition in a vacuum chamber next to the glovebox. Before the deposi- tion of Au, Ar milling was performed to remove the residual resist and any possibly oxidized layers of CeTe 3. It should be noted that contact resistance varies greatly with electrode material. Au electrodes werefound to have a minimum contact resistance compared to Ti/Au or Cu. In order to avoid further damage in ambient conditions after the fabrication, the device was capped with PMMA shortly after the elec- trode deposition and the lift-off process. Figure 1(b) shows a scanning electron microscopy (SEM) image of one of the thin film devices. Electrical transport measurements were performed by the con- ventional four-probe method using a lock-in amplifier. The device wascooled with a 3He/4He dilution refrigerator down to 0.2 K, and the external magnetic field was applied using a superconducting magnet. The thicknesses of all measured CeTe 3thin films were confirmed by using a commercially available atomic force microscope after finishing all the electrical measurements. The resistivity qðTÞmeasured with the CeTe 3thin film device in Fig. 1(b) is plotted as a function of temperature in Fig. 1(c) . Although the CeTe structure is half-metallic on its own, the high conductivity between the two Te sheets gives rise to a highly metallic temperature dependence in CeTe 3.32There are two resistivity changes in the low temperature region: the first resistivity drop at T¼2.7 K and the second resistivity kink structure at T¼1.3 K. These behaviors are con- sistent with the anomalies observed in bulk CeTe 3resistivity measure- ments, which correspond to the two magnetic phase transitions at TN1¼3:1K a n d TN2¼1:3K .24,32The resistivity at room tempera- ture of this device is 81.3 lX/C1cm, resulting in the residual resistivity ratio (RRR) of 59.2 with respect to qðT¼1:5KÞ. This is 1.3 times higher than the previously reported value of 44.9.24At the lowest tem- perature ( T¼0.2 K), RRR reaches a value of 140, indicating that the device is a high-quality single crystal CeTe 3thin film. We next performed magnetoresistance measurements up to B¼8Tf o rt h et e m p e r a t u r er a n g eo f0 . 4 – 4 K ,a ss h o w ni n Fig. 2(a) . The external magnetic field Bwas applied perpendicular to the plane, i.e., along the b-axis of CeTe 3and swept from zero to 8 T. A large posi- tive magnetoresistance qxxðBÞwas observed in the low field regime (B<1 T) along with a clear SdH oscillation, which develops from magnetic fields as low as B¼2 T in the case of T¼2 K. As far as we know, such a quantum oscillation (including the de Haas-van Alphen effect) has never been reported so far even for bulk CeTe 3although Lei et al. have recently reported a SdH oscillation in GdTe 3thin films.35 This fact is possibly related to much stronger 2D nature in thin films, compared to bulk crystals. Furthermore, the magnetoresistance behav- ior drastically changes below T¼0.8 K, which is below the second magnetic transition temperature TN2. In order to extract the oscillatory part of the magnetoresistance, the derivative of qxxðBÞwith respect to 1 =Bwas obtained numerically. Fast Fourier transform (FFT) was then performed in order to obtain the frequency fof the quantum oscillation. The results are presented in Fig. 2(b) . The main oscillation observed at all temperatures corre- sponds to a frequency of f0¼32:4 T, which is about two times smaller than that of GdTe 3.35Below T¼0.8 K, there is an additional oscilla- tion with f2¼11:0 T, which can be seen as the secondary peak in the FFT spectrum in Fig. 2(b) . The frequencies of these oscillations are proportional to the extremal cross-sectional areas of the Fermi surface perpendicular to the applied magnetic field: S¼2pef=/C22h,w h e r e Sis the Fermi surface area, eis the elementary charge, fis the frequency in the unit of T, and /C22his the reduced Planck constant.36Using this equa- tion, we obtained the Fermi surface areas of S0¼3:09/C21013cm/C02 forf0¼32:4Ta n d S2¼1:05/C21013cm/C02forf2¼11:0T .T h em a i n oscillation is present even above the two magnetic transition FIG. 1. (a) Crystal structure of CeTe 3. The red and blue spheres represent Ce and Te atoms, respectively. The black lined box indicates the unit cell, with unit latticevectors of a/C24c/C244.4 A˚andb/C2426 A˚. (b) SEM image of a typical thin film device. CeTe 3is shown in yellow (false color). (c) Temperature dependence of the resistiv- ity in the CeTe 3thin film device. There are two resistivity drops at 2.7 K and 1.3 K, which correspond to the two magnetic transition temperatures TN1and TN2, respectively.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 072403 (2020); doi: 10.1063/5.0007517 117, 072403-2 Published under license by AIP Publishingtemperatures, but well below its CDW transition temperature (TCDW>500 K).26–29The origin of this Fermi surface pocket can be attributed to the reconstruction of the Fermi surface due to the incom- mensurate CDW, which has been observed through photoemissionspectroscopy experiments 26,28as well as through quantum oscillations of other RTe3materials.35,37,38 Although the angular dependence of the SdH oscillation has not been measured in the present study, it was already demonstrated in a similar tritelluride thin film device, i.e., GdTe 335where the SdH oscilla- tion follows 1 =cosh(his the angle between the applied magnetic field and the layer stacking direction). This indicates a highly 2D geometryof the Fermi pockets. Since the crystal structure and the high conduc- t i o no ft h eT el a y e r sa r et h es a m ef o rG d T e 3and CeTe 3,w eb e l i e v e that the SdH oscillation observed in CeTe 3also originates from a highly 2D Fermi surface pocket. Specifically, S0in the CeTe 3device is two times smaller than that in a thin film GdTe 3device, where the effective mass is known to be as small as /C250:1m0(m0is the bare elec- tron mass).35This fact suggests that the conduction electrons betweenthe Te sheets in the CeTe 3device have a similarly small effective mass. In addition, we detected the secondary oscillation with a frequency of f2, which does not exist in GdTe 3and develops only after the second magnetic transition temperature TN2.A c c o r d i n gt oR e f . 32,t h es e c o n d magnetic transition at TN2is related to a spin density wave transition with formation of heavy quasiparticles. Thus, such reconstruction ofthe Fermi surface along with this transition could be a possible cause of this new oscillation, but further investigation is required to confirm the hypothesis. We note that these SdH frequencies have also beenobserved in multiple different CeTe 3devices, confirming their reproducibility. In a typical SdH oscillation, the FFT amplitude for a given fre- quency decreases with increasing temperature. However, FFT ampli- tudes of the main oscillation remained mostly constant, with the exception at T¼2 K. A similar result has been reported in GdTe 3, where the FFT amplitude plateaus below its antiferromagnetic transi- tion temperature, while showing a typical temperature dependence well above the transition temperature.35Our scenario is consistent with this report; in other words, the unconventional temperature dependence of the FFT amplitude depends strongly on the interaction between the magnetic order at the CeTe site and the conduction electrons. In addition to the SdH oscillations discussed above, we observed a magnetic hysteresis behavior, superimposed onto the SdH oscilla-tions, below the first antiferromagnetic ordering temperature T N1. This is highlighted in Fig. 3 . At 4 K above TN1, there is no hysteresis in the SdH oscillation. Below TN1, however, a clear magnetic hysteresis is observed with a sharp peak structure at B/C250:6 T. The magnetic hys- teresis in the SdH oscillation vanishes above B/C254T , a n d t h e s h a r p peak appears only when Bis swept from a high enough field (in this case, Bmax¼8 T) to zero field, which we call the negative field sweep- ing. Note that quantum oscillations with magnetic hysteresis arereproducible for other CeTe 3thin film devices, even below TN2,a n d also when the applied magnetic field is negative (see Fig. 4 ).FIG. 2. (a) Magnetoresistance measured with a 40 nm thick CeTe 3thin film device at several different temperatures. The external magnetic field was applied perpen-dicular to the plane and swept from zero to 8 T. (b) FFT for the derivative of q xxðBÞ vs 1=B. The main oscillation peak has been observed at f0¼32:4 T below 4 K. The secondary oscillation peak has been observed at f2¼11:0 T only below 0.8 K. The peak at 0 T is due to the DC offset of the frequency.FIG. 3. Hysteresis in magnetoresistance measured at T¼2 and 0.4 K. There is a clear hysteresis between the blue and red curves where the magnetic field is sweptfrom 0 to 8 T and from 8 to 0 T, respectively. A clear jump in resistance wasobserved only for the red curve at B/C250:6 T. Such a hysteresis was not observed atT¼4 K above T N1.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 072403 (2020); doi: 10.1063/5.0007517 117, 072403-3 Published under license by AIP PublishingFurthermore, the peak amplitude depends on the absolute value of Bmax.A t T¼0.4 K, when jBmaxjis lower than 1.5 T, the peak at B/C250:6 T vanishes, while still preserving the hysteresis behavior, as shown in Fig. 4(b) . Such a hysteresis has never been reported previ- ously in bulk CeTe 3as well as in thin film GdTe 3devices.35On the other hand, a hysteresis closely resembling our measurements has been observed for the magnetoresistance measurements in CeTe 2,39a variation of CeTe 3.C e T e 2has the same crystal structure as CeTe 3,b u t has a single Te layer instead of the double Te layers, and is known to order ferrimagnetically with an easy axis along the layer stacking direc- tion.40This is different from the magnetic order of bulk CeTe 3,w h e r e t h em a g n e t i cm o m e n t sa tt h eC es i t e si nt h et w om a g n e t i cp h a s e sa r ebelieved to be aligned to the basal plane. 32,33One possible scenario to explain the magnetoresistance hysteresis and peak structure observed in CeTe 3devices is that by thin film fabrication, we have induced a canting of the magnetic moments along the layer stacking direction,resulting in a similar magnetoresistance effect to CeTe 2. There is a sup- portive result where the perpendicular anisotropy of a van der Waals ferromagnet Fe 5GeTe 2is enhanced by making it atomically thinner.41 In order to elucidate more details, it would be desirable to perform fur- ther experiments in the future on the thickness dependence of thepeak structure.In conclusion, we have performed magnetotransport measure- ments of 30–40 nm thick CeTe 3thin film devices. A clear SdH oscilla- tion was observed from T¼4 K, indicating a highly two-dimensional character of the conduction electrons, possibly enhanced due to thin film fabrication. Below the second magnetic transition temperature TN2¼1:3 K, on the other hand, SdH oscillations with two different frequencies were obtained. The FFT analysis revealed the existence oftwo small Fermi pockets whose sizes are 3 :09/C210 13cm/C02and 1:05/C21013cm/C02. In addition, a magnetic hysteresis superimposed to the SdH oscillation was detected below the first antiferromagnetic tem- perature TN1¼2:7 K. Specifically, a sharp peak at B/C250:6T w a s clearly observed when the magnetic field was swept from the high enough field to zero field. Materials where quantum oscillations and magnetic hysteresis coexist are extremely scarce. Along with the ease of thin film fabrication through mechanical exfoliation, further research on CeTe 3could pave the way for f-orbital spintronics and could provide an ideal stage for understanding the interplay between electronic quantum conduction and localized heavy fermion spins. W et h a n kH .S a k a i ,K .K u r o k i ,M .O c h i ,a n dK .D e g u c h if o r fruitful discussions. The cell structure of CeTe 3was visualized using VESTA.42This work was supported by JSPS KAKENHI (Grant Nos. JP16H05964, JP17K18756, JP19K21850, JP20H02557, JP26103002,JP19H00656, and JP19H05826), the Mazda Foundation, the Shimadzu Science Foundation, the Yazaki Memorial Foundation for Science and Technology, the SCAT Foundation, the Murata Science Foundation, the Toyota Riken Scholar, and the Kato Foundation for Promotion of Science. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1A. K. Geim and I. V. Grigorieva, Nature 499, 419 (2013). 2K. S. Novoselov, A. Mishchenko, A. Carvalho, and A. H. Castro Neto, Science 353, aac9439 (2016). 3K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004). 4K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005). 5Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, Nature 438, 201 (2005). 6K. 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5.0021099.pdf

Appl. Phys. Lett. 117, 090501 (2020); https://doi.org/10.1063/5.0021099 117, 090501 © 2020 Author(s).Magnon-squeezing as a niche of quantum magnonics Cite as: Appl. Phys. Lett. 117, 090501 (2020); https://doi.org/10.1063/5.0021099 Submitted: 07 July 2020 . Accepted: 10 August 2020 . Published Online: 31 August 2020 Akashdeep Kamra , Wolfgang Belzig , and Arne Brataas Magnon-squeezing as a niche of quantum magnonics Cite as: Appl. Phys. Lett. 117, 090501 (2020); doi: 10.1063/5.0021099 Submitted: 7 July 2020 .Accepted: 10 August 2020 . Published Online: 31 August 2020 Akashdeep Kamra,1,2,a) Wolfgang Belzig,3 and Arne Brataas1 AFFILIATIONS 1Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway 2Kavli Institute for Theoretical Physics, University of California Santa Barbara, Santa Barbara 93106, USA 3Department of Physics, University of Konstanz, D-78457 Konstanz, Germany a)Author to whom correspondence should be addressed: akashdeep.kamra@ntnu.no ABSTRACT Spin excitations of ordered magnets – magnons – mediate transport in magnetic insulators. Their bosonic nature makes them qualitatively distinct from electrons. These features include quantum properties traditionally realized with photons. In this perspective, we present an intuitive discussion of one such phenomenon. Equilibrium magnon-squeezing manifests unique advantages with magnons as compared tophotons, including properties such as entanglement. Building upon the recent progress in the fields of spintronics and quantum optics, weoutline challenges and opportunities in this emerging field of quantum magnonics. Published under license by AIP Publishing. https://doi.org/10.1063/5.0021099 Spin excitations of ordered magnets, broadly called “magnons,” carry spin information 1–8and offer a viable path toward low- dissipation, unconventional computing paradigms. Their bosonicnature enables realizing and exploiting phenomena not admitted byelectrons. 9–15The field of “magnonics” has made rapid progress toward fundamental physics as well as potential applications in recent years.1,4,5Several studies have also emphasized the quantum nature of magnon quasiparticles resulting in the spin-off entitled “quantummagnonics.” In this perspective, we outline some recent insights andemerged opportunities focusing on the specific topic of equilibriummagnon squeezing. 16–19There are many other exciting advancements in the field,20–24which we will not discuss further here. An overview of these can be found in recent review articles.25,26Since the terminology – quantum vs classical – sometimes depends on the criterion chosen,we briefly mention the latter as a footnote at appropriate places whileemploying the term “quantum” in our discussion. A brief comparison between the fields of quantum 27optics28,29 and magnonics is in order since the ideas to be discussed here take inspiration from the former field. While photons and magnons are bosonic excitations described by similar theoretical toolboxes, crucial differences in their physical properties make them complementary interms of experimental platforms and parameter regimes. We limit thediscussion here to only two of these distinctions. First, long opticalwavelengths make photons suitable for large systems, while magnonsfit in on-chip nanodevices. Second, photons need external matter to mediate interactions between them, while magnons are intrinsically interacting. A corollary is that photons have much longer coherencelengths, while magnons provide a compact platform for quantum 30 effects and manipulation via interactions. The latter point partly allowsthe unique niche of magnon squeezing to be discussed here. Inessence, the two fields are complementary and can gain from each other. We find it convenient to introduce the equilibrium magnon- squeezing physics first and later place it in the context of the more mature and widely known nonequilibrium squeezing phenome- non. 28,29,31,32Magnons in a ferromagnet admit single-mode squeezing mediated by the relatively weak spin-nonconserving interactions,16,17 thereby providing an apt start of the discussion. Antiferromagneticmodes manifest large two-mode squeezing, mediated by the strongexchange interaction, 17,18and are discussed next. Altogether, these understandings open avenues toward exploiting quantum phenomena in “classically ordered” magnets. Consider a uniformly ordered ferromagnetic ground state with all the spins pointing along the z direction. A spin flip at one of the lat-tice sites may be seen as a spin /C0/C22hquasiparticle – magnon – superim- posed on the perfectly ordered ground state. In the absence of ananisotropy in the x-y plane, such quasiparticles delocalized in the formof plane waves with wavevectors labeled kconstitute the eigen Appl. Phys. Lett. 117, 090501 (2020); doi: 10.1063/5.0021099 117, 090501-1 Published under license by AIP PublishingApplied Physics Letters PERSPECTIVE scitation.org/journal/aplexcitations. We focus on the spatially uniform mode corresponding to k¼0, which describes the sum over all spins in the ferromagnet. As per the Heisenberg uncertainty relation, the total spin in the ferromag- net may not point exactly along the z-direction since that would entail a vanishing uncertainty in both the transverse (x and y) spin compo-nents. The latter is not allowed by the Heisenberg principle as the operators for S xand Sydo not commute. Thus, even in the ground state, the total spin manifests quantum33fluctuations and a corre- sponding uncertainty region schematically depicted in Fig. 1(a) ,r i g h t panel. The corresponding wavefunctions for the ground state and eigenmode – magnon – are depicted in Figs. 1(b) and 1(c) on the right. Now, let us include an anisotropy that levies a larger energy cost on the y, as compared to the x, component of the total spin.34As a result, the system adapts its ground-state quantum fluctuations into an ellipse [ Fig. 1(a) , left panel]. In this way, it minimizes its energy while obeying the Heisenberg uncertainty principle, which only constraints the area of the uncertainty region and not the shape. The correspond- ing ground state wavefunction [ Fig. 1(b) , left] is constituted by a super- position of the even magnon-number Fock states28,29and is related to the magnon vacuum via the so-called single mode squeeze operator S(r).16Here, ris the so-called squeeze parameter determined for the case at hand by the transverse (x-y) anisotropy. Naively, one canexpect the corresponding excitation [ Fig. 1(c) , left] to be obtained by superimposing an additional spin flip (magnon) on the vacuum 35,36 [compare Figs. 1(b) and1(c)]. The resulting eigenexcitation is corre- spondingly related to the magnon wavefunction via the squeeze opera- tor [Fig. 1(c) ] and is, therefore, termed the squeezed magnon.16 The uncertainty region ellipticity, depicted in Fig. 1(a) ,r e p r e s e n t s a phenomenon distinct from the spin precession ellipticity in the Landau–Lifshitz phenomenology. The former pertains to the shape ofthe quantum fluctuations around the average spin direction, while the latter describes the trajectory of the average spin in a coherent excited state. 9,37,38While determined by the same anisotropies for the case at hand, their manifestations and dependencies differ. The squeeze parameter rð>0Þcaptures the degree of squeezing and theconcomitant quantum effects, such as superposition and entangle- ment. For excitations with high frequencies, the relative energy contri- bution of the anisotropies becomes small, resulting in a diminishing r. Furthermore, in contrast to the above single-mode case, modes with k6¼0 manifest two-mode squeezing,16which will be discussed in the context of antiferromagnets below. To sum up, anisotropies in the transverse plane modify the quan- tum fluctuations in a ferromagnet [ Fig. 1(a) ]. This results in squeezed vacuum and squeezed magnon as the corresponding ground state [Fig. 1(b) ] and eigenexcitation [ Fig. 1(c) ].16The anisotropies arise from dipolar or spin–orbit interaction, thereby mediating an effective coupling between magnons required for squeezing.17The effect and importance of anisotropies diminish with increasing eigenmode fre- quency and can become relatively weak. Specifically, the noninteger average spin /C22h/C3of the squeezed magnon16,17corresponding to the Kittel mode is /H11407/C22hsince the x-y plane anisotropy contribution is typi- cally important. This increase in spin arises from the quantum39super- position of odd magnon number states, which describes the excitation depicted in Fig. 1(c) . The average spin approaches /C22has the eigen excita- tion frequency significantly exceeds the anisotropy contribution.40 There are several key differences in antiferromagnets, one being that the strong exchange interaction, and not anisotropy, causes squeezing.18Consider a bipartite antiferromagnet in its N /C19eel ordered state such that all spins at the A (B) sublattice point along the (against) z-direction. We disregard anisotropies17and “turn off” the antiferro- magnetic exchange for the moment. The two sublattices are then equivalent to two isotropic ferromagnets with spins oriented antiparal- lel to each other. The eigenmodes are spin-down and spin-up mag- nons residing on sublattices A (red in Fig. 2 )a n dB( b l u ei n Fig. 2 ), respectively. As per the Heisenberg uncertainty relation for the total spin ( k¼0 mode) on each sublattice, the quantum fluctuations in the ground state are now circular in both phase spaces [ Fig. 2(a) ]. The total spins on the two sublattices SAandSBfluctuate independently, FIG. 1. Schematic depiction of the spatially uniform ferromagnetic vacuum and magnon mode in the presence (left) and absence (right) of anisotropy in the trans- verse (x-y) plane. The ensuing relation between squeezed (left) and unsqueezed (right) modes is also depicted. (a) Heisenberg uncertainty region in the ferromag-netic ground state saturated along the z-axis. The anisotropy in the x-y plane (leftpanel) causes ellipticity to minimize energy. Schematic depiction of the (b) ground state and (c) magnon-mode wavefunctions for squeezed (left) and unsqueezed (right) ferromagnets related by the squeeze operator S(r). An empty ket and a dou- ble arrow denote a fully saturated ferromagnet and a spin- /C22hmagnon, respectively, which become the ground state and eigenexcitation in the isotropic case (right).FIG. 2. Schematic depiction of the spatially uniform antiferromagnetic ground state and eigenmodes. (a) Heisenberg uncertainty regions in the phase spaces of totalsublattice spins. The quantum fluctuations in the two sublattices become correlatedin order to keep their spins antiparallel to minimize the exchange energy cost. (b) The uncertainty region of quantum fluctuations in a combined phase space con- structed out of both sublattices. The gray circular region corresponds to the N /C19eel state, and the green squeezed one represents the actual antiferromagnetic orderedground state. Schematic depiction of the (c) ground state and (d) spin-up magnon- mode wavefunctions for actual, two-mode squeezed (left) and N /C19eel, unsqueezed (right) antiferromagnets related by the two-mode squeeze operator S 2ðrÞ. An empty ket denotes the perfectly ordered N /C19eel state devoid of any red and blue magnons, which reside on sublattices A and B, respectively.Applied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 117, 090501 (2020); doi: 10.1063/5.0021099 117, 090501-2 Published under license by AIP Publishingwhich corresponds to no magnons on either of the sublattices in the ground state. Now, let us “turn on” the antiferromagnetic exchange, which forces the spins to remain antiparallel. The uncorrelated quantum fluc- tuations of SAandSBwould cost high energy now as fluctuating inde- pendently, and SAis not always antiparallel to SB. Thus, mediated by the strong exchange interaction, SAnow fluctuates while maintaining its antiparallel direction with respect to SB. The system minimizes its energy, while obeying the Heisenberg rule, by bestowing quantum41 correlated noise to SAand SB, which individually maintain circular uncertainty regions [ Fig. 2(a) ]. The squeezing now takes place in the phase space constituted by SAxþSBxandSAy/C0SBy,18as depicted in Fig. 2(b) . Distinct from the ferromagnet k¼0 case, the ground state here is two-mode squeezed where the participating modes are the spin- down and spin-up magnons residing on sublattices A and B,18hence- forth simply called “red” and “blue” magnons. The ensuing ground state is formed by a superposition of states with an equal number of red and blue magnons and is related to the N /C19eel state via the two-mode squeeze operator S2ðrÞ[Fig. 2(c) ]. The corresponding spin-up eigenex- citation [ Fig. 2(d) ] may be understood as a result of adding a blue mag- non to the ground state18,35,36[compare Figs. 2(c) and2(d)]. We considered the above k¼0 modes since they admit relatively simple physical pictures. However, the treatment and interpretation fork6¼0 modes are mathematically analogous42and are implicit in the above two-mode squeezing interpretation differing only in the par- ticipating modes and the squeeze parameter r, which is wavevector dependent. In antiferromagnets, the squeeze parameter ris large (theo- retically divergent for isotropic magnets) for k¼0 eigenmodes. It decreases with an increasing k and vanishes as kapproaches the Brillouin zone boundary.18The squeezing being mediated by the exchange interaction in antiferromagnets bestows them with their unique strong quantum43fluctuations and nature. The notion of squeezing has been developed and exploited in the field of quantum optics.28,29,32For light or photons, the two noncom- muting variables, often called quadratures, which embody the Heisenberg uncertainty region, are the associated electric and magnetic fields. In this case, the uncertainty region in equilibrium is circular. A squeezing of the fluctuations is achieved by generating pairs of quan- tum correlated photons via four-wave mixing44or parametric down- conversion,45for example. The ensuing squeezed state is a transient, nonequilibrium state that decays as the drive is turned off. While such squeezed states are perhaps best known for enabling a beyond-quan-tum-limit sensitivity of Laser Interferometer Gravitational-wave Observatory (LIGO) 46,47that detected gravitational waves,48several other quantum properties such as entanglement are inherent to these states and have been studied in great detail.49–52 Similar nonequilibrium squeezed states have also been realized in antiferromagnets.53–55By generating correlated pairs of red and blue magnons via Raman scattering with light, experiments observed spin dynamics, which could only be explained in terms of a nonequilibrium two-mode squeezed state.55The participating modes here are the red and blue magnons with wavevectors at the Brillouin zone boundary, which are the antiferromagnetic eigenmodes on account of a vanishing equilibrium squeezing at these wavevectors. The situation is, thus, dis- tinct from our discussion of equilibrium squeezing above. The squeezing perspective and picture presented here capitalize on insights developed in the field of quantum optics to shed fresh lighton magnons, which were investigated56four decades before the notion of squeezing was developed.31This perspective is largely based on the direct mathematical relation between the Bogoliubov transformation56 and the squeeze operator identified initially in the context of magnon spin current shot noise theory.16The latter could be understood in terms of the noninteger average spin of squeezed magnons in ferro-magnets 16,17consistent with the schematic in Fig. 1(c) .T h i s squeezing-based picture of magnons further allows us to predict and exploit various effects, such as entanglement18,19,57,58and exponen- tially enhanced coupling,18,59,60already established for light49–52,61,62 but now in magnetic systems manifesting certain advantages. The two key strengths of this magnon squeezing are (i) its equi- librium nature, i.e., squeezing here results from energy minimization, and (ii) large squeeze parameter, relative to what is achieved with light,32due to strong interactions in magnets. These unique features open new avenues. For example, on account of attribute (i), the entan-glement inherent to these squeezed states is stabilized against decay bythe system’s tendency to minimize its energy. Can we design architec-tures harnessing this entanglement stability for phenomena such as quantum computing and teleportation? This protection from decay is quantified in terms of strong squeezing. How is this stability affectedand limited by the dephasing and decoherence processes? The equilib-rium nature also makes these phenomena somewhat different fromthe nonequilibrium squeezing physics encountered in the field ofquantum optics. Understanding these differences requires further investigation and will be crucial for exploiting these phenomena toward applications. The experimental demonstration and exploitation of these effects may capitalize on a large body of knowledge from quantum opticstogether with a multitude of tools available for solid state systems. Several approaches, such as spin current noise, 16,40,63–69magnon- photon interaction,20–24,57,58,70and nitrogen-vacancy (NV)-center magnetometry,71–76which offer access and control over magnons, have emerged in recent years. These techniques can be broadly classi-fied into (i) detecting the average effect of squeezing-mediated quan-tum fluctuations, such as the already demonstrated coupling enhancement 59or spin dynamics,53–55and (ii) those probing the fluc- tuations themselves. The latter class of methods is expected to offerdirect insights into and pathways to exploit the quantum correlationsand constitute an active area witnessing rapid developments. We conclude this perspective with a geometrical argument underpinning the robustness of phenomena discussed here. Many quantum effects vanish in the limit of /C22h!0, and the smallness of /C22h underlies their fragile nature. In contrast, the squeezing and relatedquantum effects result from geometrically deforming the Heisenberguncertainty region irrespective of its area. Thus, while these squeezingeffects fundamentally rely on the Heisenberg uncertainty principle and the corresponding quantum fluctuations, they continue to persist in the limit of /C22h!0. The robust geometrical nature of equilibrium mag- non squeezing, therefore, offers unique possibilities toward realizingquantum devices. We acknowledge financial support from the Research Council of Norway through its Centers of Excellence funding scheme,Project No. 262633, “QuSpin,” and the DFG through SFB 767. Thiswork was also supported in part by the National ScienceFoundation under Grant No. NSF PHY-1748958.Applied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 117, 090501 (2020); doi: 10.1063/5.0021099 117, 090501-3 Published under license by AIP PublishingDATA AVAILABILITY Data sharing is not applicable to this article as no new data were created or analyzed in this study. REFERENCES 1V. V. Kruglyak, S. O. Demokritov, and D. Grundler, “Magnonics,” J. Phys. D 43, 264001 (2010). 2K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh, “Spin see-beck insulator,” Nat. Mater. 9, 894–897 (2010). 3H. Adachi, K. Uchida, E. Saitoh, and S. Maekawa, “Theory of the spin seebeck effect,” Rep. Prog. Phys. 76, 036501 (2013). 4G. E. W. Bauer, E. Saitoh, and B. J. van Wees, “Spin caloritronics,” Nat. Mater. 11, 391 (2012). 5A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. 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5.0023286.pdf

Appl. Phys. Lett. 117, 132406 (2020); https://doi.org/10.1063/5.0023286 117, 132406 © 2020 Author(s).Observation of surface dominated topological transport in strained semimetallic ErPdBi thin films Cite as: Appl. Phys. Lett. 117, 132406 (2020); https://doi.org/10.1063/5.0023286 Submitted: 28 July 2020 . Accepted: 11 September 2020 . Published Online: 29 September 2020 Vishal Bhardwaj , Anupam Bhattacharya , A. K. Nigam , Saroj P. Dash , and Ratnamala Chatterjee ARTICLES YOU MAY BE INTERESTED IN Electrostatic-doping-controlled phase separation in electron-doped manganites Applied Physics Letters 117, 132405 (2020); https://doi.org/10.1063/5.0024431 A half-metallic ferrimagnet of CeCu 3Cr4O12 with 4f itinerant electron Applied Physics Letters 117, 132404 (2020); https://doi.org/10.1063/5.0020199 Metal-insulator transition of ultrathin films embedded in superlattices Applied Physics Letters 117, 133105 (2020); https://doi.org/10.1063/5.0020615Observation of surface dominated topological transport in strained semimetallic ErPdBi thin films Cite as: Appl. Phys. Lett. 117, 132406 (2020); doi: 10.1063/5.0023286 Submitted: 28 July 2020 .Accepted: 11 September 2020 . Published Online: 29 September 2020 Vishal Bhardwaj,1Anupam Bhattacharya,1,2A. K. Nigam,3Saroj P. Dash,4 and Ratnamala Chatterjee1,a) AFFILIATIONS 1Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi-110016, India 2Department of Mechanical Engineering, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India 3Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400005, India 4Department of Microtechnology and Nanoscience, Chalmers University of Technology, SE 41296 G €oteborg, Sweden a)Author to whom correspondence should be addressed: ratnamalac@gmail.com andrmala@physics.iitd.ac.in ABSTRACT In this Letter, we present experimental observation of surface-dominated transport properties in [110]-oriented strained ( /C241.6%) ErPdBi thin films. The resistivity data show typical semi-metallic behavior in the temperature range of 3 K /C20T/C20350 K with a transition from semi- conductor- to metal-like behavior below 3 K. The metallic behavior at low temperature disappears entirely in the presence of an externalmagnetic field >1 T. The weak-antilocalization (WAL) effect is observed in magneto-conductance data in the low magnetic field region and follows the Hikami–Larkin–Nagaoka (HLN) model. HLN fitting estimated single coherent channel, i.e., a/C24/C00.51 at 1.9 K, and the phase coherence length ( L /) shows the L//C24T/C00.52power law dependence on temperature in the range of 1.9 K–10 K, indicating the observation of 2D WAL. Shubnikov–de Haas (SdH) oscillations are observed in magneto-resistance data below 10 K and are fitted to standard LifhsitzKosevich theory. Fitting reveals the effective mass of charge carriers /C240.15m eand a finite Berry phase of 0.86 p60.16. The sheet carrier con- centration and mobility of carriers estimated using SdH data are ns/C241.35/C21012cm/C02andls¼1210 cm2V/C01s/C01, respectively, and match well with the data measured using the Hall measurement at 1.9 K to be n/C241.22/C21012cm/C02,l¼1035 cm2V/C01s/C01. These findings indicate the non-trivial nature and surface-dominated transport properties of strained (110) ErPdBi thin films at low temperatures. Published under license by AIP Publishing. https://doi.org/10.1063/5.0023286 Materials classification based on the topology of material band structures1–4has opened up a new arena of research in contemporary condensed matter physics.5–7In this regard, half Heusler alloy-based topological semi-metals are currently the hot topic of research due totheir topologically protected transport properties. 7–9These new classes of materials have the coexistence of Weyl/Dirac fermionic electron states, magnetism, and beyond10–14and, thus, provide the researchers with a material-science platform to test predictions of the laws of topo- logical physics. The carrier transport in topological semimetals is robust against crystalline disorder and imperfections, which results inhigh carrier mobility, chiral anomaly, and large magneto-resistance. 15 The bulk boundary correspondence in these semimetals gives birth to the topologically protected Fermi arcs and drumhead surface states.14 Therefore, the experimental and theoretical realization of these materi- als opens a new dimension in spintronic,6,16thermoelectric,17and quantum computing18,19-based applications.Rare-earth (R)-based RPdBi ternary half-Heusler (HH) alloys are predicted as the potential candidates to show non-trivial topological properties based on density functional theory (DFT) calculations.20–23 The rare-earth chalcogenide HH compounds are cubic ternary inter-metallic alloys with composition XYZ (X and Y can be transition metal, noble metal, or rare-earth elements, and Z is a main group ele- ment) that crystallize in the MgAgAs-type (F43(m space group) struc- ture that lacks inversion symmetry.24Many of the RPdBi-based half Heusler (e.g., YPdBi, DyPdBi, LuPdBi, and ScPdBi)25–29topological semimetals have shown the observation of the weak-antilocalization(WAL), Shubnikov–de Haas (SdH) effect, and unconventional super-conductivity in magneto-transport measurements. Recently, Nakajima et al. 9have theoretically predicted ErPdBi (EPB) as a topologically non-trivial semimetal that shows negative band inversion strength(DE¼E U8/C0EU6)>0 at the equilibrium lattice constant. It is interest- ing to note that in an earlier work, Pan et al.30reported the coexistence Appl. Phys. Lett. 117, 132406 (2020); doi: 10.1063/5.0023286 117, 132406-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplof magnetic ordering ( TN/C241.06 K) and superconductivity ( TC /C241.22 K) in EPB single crystals. Later, Pavlosiuk et al.31observed a magnetic order ( TN¼1.2 K) and SdH oscillations in EPB single crys- tals, but they could not find any distinct evidence of the superconduct- ing state down to 0.4 K. The magneto-transport measurements on RPdBi half Heusler crystals are a powerful tool to study the non-trivial topology of their bulk bands and related topologically protected surface or edge states; the time-reversal symmetry protection of these topological states can be broken in the presence of an external magnetic field.10,11,14 However, it should be noted that in the transport method, it is veryhard to avoid the contribution of bulk states. 25,26One method to sup- press the bulk channel contribution in transport experiments is to grow single crystalline thin films of topological materials.32The thin film growth increases the surface to volume area ratio and decreases the overall bulk contribution.33There are two important signatures in magneto-transport experiments while probing the topological surface states: (i) observation of 2D WAL and (ii) observation of quantum oscillations in the resistivity, i.e., SdH oscillations.10,14The extraction of the topological order parameter like pBerry’s Phase from these experimental signatures provides us with an indirect method to verify the topological non-triviality of these materials.5,10,14 For designing future spintronics and quantum computing-based devices, thin films are the desirable geometry and the aim of this work is to grow non-trivial single crystals like EPB thin films. When a single crystal like/oriented crystal is grown on a substrate, there will be strain effects visible in the structure. Chadov et al. in 2010 had demonstrated the strain induced band structure changes in RPdBi systems;21evi- dently, it is of utmost importance to study the transport behavior in non-trivial EPB thin films (strained crystal). In this work, we investi- gate the [110]-oriented thin films ( /C2425 nm) of EPB with a tensile strain of /C241.6%. The magneto-transport properties of these films are studied in detail. This work reports the observation of the 2D-WAL effect in the temperature range of 1.9 K /C20T/C2010 K with /C00.51/C20a /C20/C00.56 and shows the variation of the phase coherence length ( L/) with the temperature as L//C24T/C00.52, confirming the surface- dominated transport. The robustness of topologically protected surface states is investigated by extracting Berry’s phase /C240.86p60.16 from the Landau level fan diagram of SdH data. The mobility and concen- tration of carriers are extracted from two complementary methods, i.e., SdH data ( ns/C241.35/C21012cm/C02andls¼1210 cm2V/C01s/C01)a n d Hall measurements ( n/C241.22/C21012cm/C02,l¼1035 cm2V/C01s/C01), which can be useful for future spintronics device fabrication. The polycrystalline EPB bulk sample is prepared in an especially designed RF induction melting furnace under a high-purity Argon atmosphere.28,33High-purity Er ingot (99.99%), Pd ingot (99.99%), and Bi chunks (99.99%) are used as the starting materials in a molar ratio of 1:1:3 and subsequently placed in a water-cooled copper hearth. Thin films of EPB are prepared from the same bulk EPB target, using pulsed laser deposition (PLD) on MgO(100) substrates of dimensions /C243m m /C210 mm with Ta ( /C245 nm) as a seed layer. For this study, thin films of thickness /C2425 nm are grown at an optimized substrate tem- perature of /C24270/C14C using the PLD technique with a base pressure of chamber /C245/C210/C07Torr. The crystal structure of the bulk sample and the thin film is determined using a PANalytical X’Pert Highscore diffractometer equipped with a Cu K a(1.54A ˚)s o u r c e .T h et h i nfi l m growth rate is optimized from x-ray reflectivity (XRR) plots measuredusing a D8 Bruker multi-function high resolution x-ray diffractometer. Atomic force microscopy (AFM) is performed using a BrukerDimension Icon scanning probe micro-scope. Electrical and magneto-transport studies are performed on thin films of dimensions3m m/C210 mm /C225 nm using the four-point probe method, and cop- per wire contacts are prepared using silver paste, which is cured at room temperature. Transport measurements are performed using aQuantum Design Physical Properties Measurement System (PPMS-9T) in the four-probe method using a DC of 50 lAi nt h et e m p e r a t u r e range of 1.9 K–350 K and a maximum magnetic field of 69T . H a l l resistivity data are also recorded using a Quantum design 9 T PPMSusing 0.1 mA AC of frequency 133 Hz. Magnetization vs temperature measurements are performed using a Quantum Design SQUID 7 T magnetometer. Figure 1(a) shows the powder XRD spectra and Rietveld refine- ment of the bulk EPB sample measured at room temperature. TheRietveld refinement confirms the C1 bcrystal structure of EPB with the F4 3(m (216) space group and the refined lattice constant of the fcc unit cell ab/C246.597 A ˚, which is in good agreement with the previous reports.30,31,34The gonio-mode XRD spectra of the thin film EPB sam- ple shown in Fig. 1(b) indicate the [110]-oriented growth with a lattice constant of at/C246.70 A ˚.T h ei n s e to f Fig. 1(b) shows the x-ray rocking curve scan ( x-2h) of (220) reflection and the (110) unit cell of EPB. A very small FWHM /C240.081/C14is calculated from the Gauss fit of rocking curve data, which indicates a highly oriented growth along [110]. Both(100) MgO (B 1,a/C244.21 A ˚) and EPB (C1 b,a/C246.59 A ˚) have a cubic structure; however, there is a significant lattice mismatch ( /C2456%) between them along [100]. The [110]-oriented growth of the EPB thin film can be explained using Fig. 1(c) ,w h i c hs h o w st h ef o u ra t o m i c FIG. 1. (a) Powder XRD pattern and Rietveld refinement of the bulk EPB polycrys- talline sample. (b) The XRD pattern of the EPB thin film recorded in Gonio mode,showing [110]-oriented growth. The inset shows the EPB (110) unit cell and x-ray rocking curve x-2hscan of the (220) peak. (c) Atomic arrangement of four layers of (110) EPB atoms on top of two layers of (100) MgO atoms. (d) XRR pattern ofMgO/Ta(5 nm)/EPB(25 nm); the inset shows the tilted 3D-AFM topographic imageof the same sample.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 132406 (2020); doi: 10.1063/5.0023286 117, 132406-2 Published under license by AIP Publishinglayers of (110) EPB atoms on top of two (100) MgO atomic layers, simulated using the coincident site lattice method.33This atomic arrangement with /C2427/C14rotation between the surfaces of (110) EPB and (100) MgO layers results in a mean absolute strain of /C241.9%, which is of the same order as the tensile strain of /C241.6% calculated from XRD data between bulk and thin film samples. Figure 1(d) shows the raw XRR data (black curve) fitted (simulated red curve) using seg-mented fit and generic algorithms to estimate the EPB and Ta filmthicknesses to be /C2425 nm and 5 nm, respectively. The inset of Fig. 1(d) shows the tilted 3D AFM image of the EPB thin film. The interfacialroughness between Ta and EPB layers estimated using XRR is R XRR /C241.5 nm and surface roughness measured using AFM is RAFM /C241.3 nm. The temperature-dependent (1.9 K /C20T/C20350 K) sheet resistance (Rsh) data of the EPB thin film are shown in Fig. 2(a) . In the tempera- ture range of 3 K /C20T/C20350 K, a typical semi-metallic curve is observed, whereas at lower temperatures 1.9 K /C20T/C203K , Rsh increases with the increase in temperature (metallic like) [see the inset inFig. 2(a) ]. This results in a cusp in Rshvs T data at /C243K .A ne a r l i e r report on EPB single crystals by Pan et al.30shows the coexistence of antiferromagnetic ( TN¼1.06 K) and superconducting states ( TC /C241.22 K), whereas Pavlosiuk et al.31reported only the antiferromag- netic state with TN¼1.2 K. We performed zero-field cooling (ZFC) and field cooled cooling (FC) magnetization vs temperature measure-ments on the EPB thin film in the temperature range of 1.9 K /C20T /C2050 K [see the lower inset in Fig. 2(a) ]. Our experimental conditions permit the lowest temperature of 1.9 K, and no discernible anomaly isobserved in magnetization vs temperature data around 3 K; instead, anoverall paramagnetic state can be seen down to 1.9 K. 9,35This indicatesthat any magnetic ordering is not the reason for this transition in Rsh below 3 K. This downturn is further explored by cooling the sample in the magnetic field and measuring Rshin the presence of the same mag- netic field in heating cycles in the temperature range of 1.9 K– 50 K[seeFig. 2(b) ]. Interestingly, as R shis measured in the presence of the magnetic field (0 <B/C207 T ) ,t h eo b s e r v e dc u s pi n Rshvs T starts to dis- appear with the increase in the magnitude of the magnetic field. Anoverall semiconducting behavior of R shis observed above 1 T down to 1.9 K, and we can see approximately no change in the slope of RSh with respect to temperature ( dRSh/dT) at 1 T, as shown in the inset of Fig. 2(b) . Since ErPdBi is a semimetal, one cannot rule out the contri- bution of bulk bands to the transport properties using resistivity mea- surements only. The lowest temperature available in our system is1.9 K; therefore, we also cannot perform the magnetic measurementsto investigate the region below 1.9 K, in thin films. Thus, the reasonbehind the sharp fall observed at 1.22 K (by Pan et al.) 30and at 1.2 K (by Pavlosiuk et al. ),31which is designated as an onset of unconven- tional/topological superconductivity (by Pan et al. ),30is beyond the scope of our investigations. Therefore, the observation of metallic-likeresistivity behavior below 3 K possibly can be attributed either to thedominance of surface states and/or bulk bands below this temperatureor an onset of unconventional/topological superconductivity. 30,31 Figure 2(c) shows the normalized magnetoresistance (MR) vs magnetic-field data at different temperatures in /C09T/C20B/C20þ 9T, applied perpendicular to the plane of the thin film. The MR% isexpressed as [R(B)-R(0)]/ q(0)]/C2100%, where R(B) and R(0) are the longitudinal resistances measured at magnetic fields B and zero,respectively. A positive saturating MR /C242% is obtained at 1.9 K at 9 T, and MR% decreases with the increase in temperature to almost0% at 50 K. The MR data below 10 K show a prominent cusp aroundthe low magnetic field region and SdH oscillations in the high fieldregion. This cusp around 0 T is ascribed to the quantum interferencephenomenon called the weak anti-localization (WAL) effect emanat-ing from nontrivial pBerry’s phase associated with the topologically protected surface states. pBerry’s Phase associated with these helical surface/edge states prevents backscattering from impurities and givesquantum correction to coherent time reversed closed paths of car-riers. 36,37A sharp rise in resistance due to destructive interference of these closed paths is observed in an external perpendicular low mag-netic field due to the breaking of time reversal symmetry of thesestates. 38,39Figure 2(d) shows the variation of magneto-conductance, DGxx¼Gxx(B)- G xx(B¼0), measured in the perpendicular magnetic field (/C00.5 T to þ0.5 T) in the temperature range of 1.9 K /C20T/C2010 K described in a strong spin–orbit interaction limit using the following HLN equation:40 DGxx¼ae2 2p2/C22hlnBu B/C0W1 2þBu B/C18/C19 /C20/C21 ; (1) whereWis the digamma function, Bu¼/C22h 4el2u,L/is the phase coher- ence length, /C22his Dirac’s constant, and eis the electron charge. The temperature dependence of fitting parameters aandL/is shown in Fig. 3(a) ,a n d aincreases from /C00.51 at 1.9 K to /C00.56 at 10 K with the increase in temperature. L/d e c r e a s e s f r o m 4 8n m t o 2 0n m w i t h the increase in temperature from 1.9 K to 10 K and follows the L//C24 T/C00.52temperature dependence. It should be noted that in topological materials for 2D surface states, ais expected to be /C0(1/2) and L// T/C0(1/2)for a single coherent topological surface channel, whereas in a FIG. 2. (a) Longitudinal sheet resistance vs temperature measurement; the upper inset shows the zoomed view of the transition. The lower inset shows the magneti-zation vs temperature plot. (b) The longitudinal sheet resistance vs temperature measured in the presence of external magnetic field B ¼0 T–7 T. (c) MR curves as a function of magnetic fields /C09T –þ9 T measured in the temperature range of 1.9 K–50 K. (d) HLN fitting (solid red line) of magneto-conductance data around thelow field region ( /C00.5 T–þ0.5 T) in the temperature range of 1.9 K–10 K.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 132406 (2020); doi: 10.1063/5.0023286 117, 132406-3 Published under license by AIP Publishing3D system, for two independent coherent transport channels with a similar coherence length, atakes the value of /C01a n d L//T/C0(3/4).28,32,33,40The observation of metallic-like conductivity behavior at low temperature ( <3K ) a n d a¼/C00.50 at 1.9 and 2 K indicates the conductance through surface states only at these lowesttemperatures. However, the slightly increased values of ain the tem- perature range of 3 K <T/C2010 K could indicate a partial bulk-surface coupling. Therefore, overall behavior of above results implies theobservation of the 2D WAL effect ( a/C24/C00.50 and L //C24T/C01/2)a n d topologically non-trivial nature of EPB thin films.10,28 In addition to the 2D WAL effect, SdH oscillations are also observed in MR data at high magnetic fields and low temperatures.The Landau level quantization of the density of states in the presence of the high magnetic field leads to the observation of SdH oscillations in resistivity. 41Thorough investigation of SdH oscillations can provide information about the non-triviality of material, the presence of Diracfermions, and various band structure parameters like the effective mass and mobility of carriers. 8,28,29,42Figure 3(b) shows the SdH oscil- lations after subtracting the smooth background from longitudinalresistance ( DR xx) data as a function of 1/H. The SdH oscillation ampli- tude decreases with the increase in temperature and survives up to T/C2010 K. A single frequency of SdH oscillations fSdH/C2456 T is esti- mated using the Fast Fourier Transform (FFT) technique as shown inFig. 3(c) . Assuming a spherical Fermi surface, f SdHis related to the cross section (A F) of the Fermi surface in momentum space, using the Onsager relation as fSdH¼ðh 4p2eÞAF,w h e r e AF¼pk2 Ffor the 2DFermi surface system, his Planck’s constant, and kFis the Fermi vector. The sheet carrier concentration ns/C241.35/C21012cm/C02is calculated from kF/C240.0568A ˚/C01using the relation ns¼k2 F 4p:43The SdH oscillations shown in Fig. 3(b) can be described using the following standard Lifshitz–Kosevich expression:41,44 DRxx/exp/C0kDTD DEnBðÞ/C18/C19 /C2kDT=DEnBðÞ sinhðkDT=DEnBðÞ /C2cos 2 pfSdH Hn/C01 2þb/C18/C19/C20/C21 ; (2) where TDandDEnðBÞare the fitting parameters, kD¼2p2jB,i n which jBis the Boltzmann constant. The maxima and minima in DRxxcorrespond to (n þ1/2)th and nth landau levels, respectively. The transport lifetime ( s) of the charge carriers can be deduced fromthe Dingle temperature expression as TD¼/C22h/2psj B.Here, TDis a term accounting for the reduced oscillation amplitude due to landaulevel broadening due to electron scattering. 41The energy difference between two consecutive Landau levels is given by parameter DEnðHÞ¼/C22heH/m/C3,w h e r e /C22his Dirac’s constant and eis the electron charge. In topologically non-trivial semi-metals, Berry’s phase factorb¼1/2 and Berry phase 2 pb¼p. 10,37Experimentally, this phase factor bcan be estimated using the Landau Level (LL) Fan diagram as shown in Fig. 4(a) . For this analysis, minima (n) and maxima (nþ1/2) in the DRxxvs 1/Hplot are assigned the Landau level index (n) as shown in the inset of Fig. 3(a) .45The assigned values of the n-index are plotted against 1/ Hn, and linear fit to data points yields the intercept on the n-axis as b¼0.4360.1 as per Eq. (2).T h i s bvalue corresponds to Berry’s phase ¼0.86p60.16, which is very close to p FIG. 3. (a) Power law dependence of L/with temperature (1.9 K /C20T/C2010K) as L//C24T/C00.52. (b) SdH oscillations at different temperatures. (c) FFT of SdH oscillations, with single frequency fSdH/C2456 T. FIG. 4. (a) Landau-level fan diagram of minima and maxima landau level’s n index vs 1/H at 1.9 K. The inset shows indexing criteria of landau levels. (b) Temperature-dependent amplitude (red data points) of SdH oscillation fitting (black solid line) tothe thermal damping expression of Lifshitz–Kosevich theory at 8.4 T. (c) Dingle tem- perature T D/C247 K estimated from the slope of the Semilog Dingle plot at 1.9 K–4 K. (d) Variation of the carrier concentration (n) and carrier mobility ( l) mea- sured using the Hall-effect measurement with temperature. Inset: magnetic fielddependence of Hall resistivity of EPB measured at 1.9 K.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 132406 (2020); doi: 10.1063/5.0023286 117, 132406-4 Published under license by AIP Publishingand provides further manifestation of non-trivial nature of the EPB thin film.10,14The cyclotron effective mass( m/C3) of carriers can be deduced from fitting of temperature-dependent amplitude of SdH oscillations in DRxxto the thermal damping expressionkDT=DEnðBÞ sinhðkDT=DEnðBÞ of Eq. (2).41,44Figure 4(b) shows the fitting of experimental data to this expression; extracted parameters are DEnðBÞ¼7m e V a n d m/C3¼0.15 m e(me/C249.1/C210/C031kg). Therefore, the Fermi velocity (VF),Fermi vector ( kF), and Fermi level from linear band crossing ( Es F) parameters are calculated to be 0.04 A ˚/C01,3 . 1 5 /C2105ms/C01,a n d /C2486 meV, respectively. Figure 4(c) shows the semi-log Dingle plot25,28 for oscillations measured at 1.9 K, 2 K, 3 K, and 4 K; the slope of fittings gives TD/C247 K, and the carrier lifetime is s/C242.21/C210/C013s. The mean free path of carriers ( l¼VFs) and mobility ( ls¼es/m/C3)a r ee s t i - mated to be /C24110 nm and 1035 cm2V/C01s/C01, respectively. All esti- mated parameters extracted from SdH data are listed in Table I and compared with the values reported in Ref. 31. The band structure parameters like m/C3,kF,EF, etc., estimated in this work, are different from those reported for EPB single crystals in Ref. 31. The band inver- sion strength of topologically non-trivial materials is known toincrease with the increase in tensile strain in the lattice; resulting in themodified band structure of the crystal. 21Thus, the values of band structure parameters of EPB thin films ( /C241.6% strained EPB lattice) are expected to be different from those of the EPB single crystals [seeTable I ]. Figure 4(d) shows the variation of the surface carrier concentra- tion (n) and carrier mobility ( l) with temperature extracted from Hall-effect measurements on the EPB thin film assuming a one-bandmodel. 46The inset of Fig. 4(d) shows the magnetic field dependence of Hall resistivity qxyat 1.9 K. The negative and linear slope up to 9To b s e r v e di n qxyimplies electrons as the dominant charge carriers. n/C241.22/C21012cm/C02andl¼1035 cm2V/C01s/C01are estimated at 1.9 K, which match well with the ns/C241.35/C21012cm/C02and ls¼1210 cm2V/C01s/C01data extracted from SdH oscillations. In summary, /C241.6% strained [110]-oriented EPB thin films are grown using the PLD technique. The MR data of these films show thepresence of sharp cusp (WAL) around the low magnetic field regionwith clear SdH oscillations at high magnetic fields. The WAL data arewell described using the HLN model, and SdH data are fitted to thestandard Lifshitz–Kosevich equation. The HLN fitting results in /C00.51 /C20a/C20/C00.56 and L //C24T/C00.52in the temperature range of 1.9 K–10 K, which indicates the observation of the 2D-WAL effect. Berry’s phase/C240.86pis extracted using the LL fan diagram, which indicates the non-trivial band structure of the strained EPB lattice. 9,21Av e r yl o w m/C3¼0.15 m eand large ls¼1035 cm2V/C01s/C01of carriers are esti- mated from SdH data with s/C242.21/C2105ms/C01. The application of strain in the lattice constant plays an important role in tuning theband structure between topologically trivial and non-trivial states. 21 Therefore, complementary DFT calculations would help in under-standing the magneto-transport experimental results of the strainedEPB lattice. However, our experimental results clearly demonstrated the non-triviality and surface-dominated transport in strained EPBthin films at low temperature /C2010 K. The authors would like to thank the NRF, CRF, PPMS, and SQUID (Department of Physics) IIT Delhi for providing characterization facilities. Mr. Vishal Bhardwaj would like to thankProfessor Jenh-Yih Juang for PLD facility and TEEP exchange program of MOE Taiwan for providing funding to visit NCTU Taiwan as an exchange student. R.C and S.P.D acknowledge the DST-Sweden VR Programme 2019. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982). 2M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). 3B. A. Bernevig and T. L. Hughes, Topological Insulators and Topological Superconductors (Princeton University Press, 2013). 4M. Nakahara, Geometry, Topology and Physics (CRC Press, 2003). 5V. Bhardwaj and R. Chatterjee, Resonance 25, 431 (2020). 6F. Casper, T. Graf, S. Chadov, B. Balke, and C. 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5.0013799.pdf

J. Chem. Phys. 153, 054105 (2020); https://doi.org/10.1063/5.0013799 153, 054105 © 2020 Author(s).Double-hybrid density functional theory for g-tensor calculations using gauge including atomic orbitals Cite as: J. Chem. Phys. 153, 054105 (2020); https://doi.org/10.1063/5.0013799 Submitted: 14 May 2020 . Accepted: 13 July 2020 . Published Online: 03 August 2020 V. A. Tran , and F. Neese ARTICLES YOU MAY BE INTERESTED IN The ORCA quantum chemistry program package The Journal of Chemical Physics 152, 224108 (2020); https://doi.org/10.1063/5.0004608 Recent developments in the PySCF program package The Journal of Chemical Physics 153, 024109 (2020); https://doi.org/10.1063/5.0006074 Implementation of the iterative triples model CC3 for excitation energies using pair natural orbitals and Laplace transformation techniques The Journal of Chemical Physics 153, 034109 (2020); https://doi.org/10.1063/5.0012597The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Double-hybrid density functional theory for g-tensor calculations using gauge including atomic orbitals Cite as: J. Chem. Phys. 153, 054105 (2020); doi: 10.1063/5.0013799 Submitted: 14 May 2020 •Accepted: 13 July 2020 • Published Online: 3 August 2020 V. A. Tran and F. Neesea) AFFILIATIONS Max-Planck-Institut für Kohlenforschung, Mülheim an der Ruhr, Germany a)Author to whom correspondence should be addressed: frank.neese@kofo.mpg.de ABSTRACT An efficient implementation for calculations of the electronic g-tensor at the level of second-order Møller–Plesset perturbation the- ory (MP2) is presented. The resolution of identity (RI) approximation is applied for the efficient treatment of two-electron integrals, and gauge including atomic orbitals are used to circumvent the gauge problem present in all magnetic property calculations. Further- more, given that MP2 is an ingredient in double-hybrid density functional theory (DHDFT), the latter is also featured in the imple- mentation. Calculated g-shifts with RI-MP2 and DHDFT using the double-hybrid density functionals B2PLYP and DSD-PBEP86 are compared to experimental data and published data from other methods including coupled cluster singles doubles. Additionally, the computational performance for medium to large size molecular systems was studied using the RIJK and RIJCOSX approximations for the two-electron integral treatment in the formation of Fock and Fock-like matrices necessary for the calculation of analytic second derivatives. Published under license by AIP Publishing. https://doi.org/10.1063/5.0013799 .,s I. INTRODUCTION Electron paramagnetic resonance (EPR) spectroscopy is a pow- erful tool for the analysis of open shell systems, starting from small radicals up to large biologically active systems. Advances in EPR technology allow for high resolution data to be obtained. In addition, the high sophistication reached in the interpreta- tion of the spectra based on the Spin-Hamiltonian (SH) formal- ism provides a great stimulus for quantum chemistry to develop more accurate methods for the prediction of SH parameters.1–3In fact, quantum chemistry can be of great use for the interpreta- tion of high resolution EPR spectra as thoroughly discussed in the literature.4 Comprehensive introductions into the theory of EPR param- eters can be found in the books of McWeeny5–7and Harriman.8 One of the central SH parameters is the electronic g-tensor that describes the interaction of the fictitious electron spin with an exter- nal magnetic field. The g-tensor (more properly referred to as g- matrix as g itself does not have tensorial properties) is a globalproperty of a paramagnetic system. As such, it contains information about the geometric structure as well as the spin distribution of the investigated systems. Some early work has investigated g-tensors in a wave func- tion based ab initio context.9–13A substantial amount of effort has gone into the implementation of accurate g-tensor calculations starting from an unrestricted Slater determinant as in Hartree– Fock (HF) theory. The required quantities are typically computed from a non-relativistic wave function using linear response the- ory.14–16It is clear from these studies that dynamic electron corre- lation is essential in order to obtain accurate results.16Lushington and Grein were the first to use elaborate multireference configu- ration interaction (MR-CI) methods in g-tensor calculations using a sum-over-states (SOS) formulation.17,18Along the same lines, we have previously introduced the so-called spectroscopy oriented mul- tireference configuration interaction (SORCI) variant and applied it, among other systems, to some smaller transition metal com- plexes.19,20However, given the high-computational effort and the lack of size consistency, such methods are impractical for routine J. Chem. Phys. 153, 054105 (2020); doi: 10.1063/5.0013799 153, 054105-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp use in computational chemistry on larger systems. Hence, size con- sistent single-reference methods are particularly promising candi- dates for accurate g-tensor calculations. In fact, Gauss et al. imple- mented the g-tensor at the coupled cluster singles doubles (CCSD) level, making it the most accurate method available up to now.21 Extended work along these lines has recently been reported by Perera et al.22These studies all have demonstrated the substantial impact that correlation has on the accuracy of the computed g- tensor, whereas the effects of orbital relaxation were found to be very limited. Given the explosive costs of the post-HF methods with increas- ing system size, the focus of method development has subsequently shifted toward density functional theory (DFT), which has shown very good price/performance ratios in many areas of computational chemistry. Schreckenbach and Ziegler implemented an algorithm for the g-tensor calculation using gauge including atomic orbitals (GIAOs)23to circumvent the “gauge dependency problem” of mag- netic properties, albeit its effect on the g-tensor is not as pronounced as for nuclear magnetic resonance (NMR).24–27This method was then applied to large molecules.28,29We have previously presented an elaborate study on the electronic g-tensor based on the cou- pled perturbed Hartree–Fock and Kohn–Sham theory (CPSCF), making the use of unrestricted HF (UHF) and hybrid functional calculations possible.16,30More recently, Glasbrenner et al. stud- ied the gauge-origin dependence of g-tensor calculations in detail and implemented a low-order scaling method at the DFT level.31,32 Distinct effort was also made in the treatment of the spin–orbit (SO) coupling by which the g-tensor is dominantly affected. The importance of high-order SO effects for heavy-element compounds amplified the development within the two- and four-component approaches.33Important contributions to relativistic DFT were made by van Lenthe et al. Applying the zeroth-order regular approx- imation (ZORA) to the Dirac equation, SO coupling effects are taken into account variationally.34An alternative DFT formalism was presented by Malkina et al. They reported the usage of both the all-electron atomic mean-field approximation to the complete Breit–Pauli SO operators and the combination of quasi-relativistic effective core potentials (ECPs) with SO-ECPs for g-tensor calcu- lations, allowing an inexpensive yet accurate treatment of the SO coupling.35Mean field spin–orbit operators and their multi-center generalization36were also used in our previous work.16In this study, we have reported calculations on a series of small organic radicals comparing HF to different density functionals covering a range of the “Jacob’s ladder”:37local density approximation (LDA), gener- alized gradient approximation (GGA), and hybrid functionals (e.g., B3LYP).16The results were compared not only to experimental val- ues but also to other approaches of Schreckenbach and Ziegler, MR-CI results by Lushington and Grein, and the values resulting from the ZORA based treatment of the SO coupling.25,26,34,35The best performance was found with the hybrid functional B3LYP. The inclusion of exact Fock exchange seems to compensate the missing explicit treatment of electron correlation effects to a certain extent. Nonetheless, DFT is still limited within the accurate prediction of g- tensors for transition metal complexes as reported by Kaupp and co- workers in extensive studies on the performance of different density functional methods over the years, including global and local hybrid functionals.38–41The highest level (fifth rung) DFT methodology according to “Jacob’s ladder” includes not only exact Fock exchange but also non-local correlation contribution by taking into account virtual molecular orbitals (MOs). Such functionals may be realized in the double-hybrid DFT (DHDFT) context as proposed by Grimme.42 In these methods, a fraction of wave function based correlation is mixed into a DFT functional and computed using the second- order Møller–Plesset perturbation (MP2) theory energy expression. Double-hybrid density functionals (DHDFs) have shown excellent performance for total energies and other properties in a large body of benchmark calculations.43Given the more elaborate form of the energy expression and its non-variational nature, the calcula- tion of properties is much more involved in DHDFT compared to standard DFT. Early on, the geometry gradient has been for- mulated by us,44,45while analytic second derivatives were reported by Johnson et al.46and Stanton et al.47More recently, Stoychev et al. have shown that the DHDFT approach allows for the effi- cient and accurate prediction of nuclear magnetic resonance shield- ing tensors.48In that work, the first implementation of analytic second derivatives including GIAOs for closed shell systems was reported. Motivated by the success of DHDFT, we report, in this work, the first implementation of analytic second derivatives for property calculations in the unrestricted formalism at the level of MP2 and DHDFT, including the resolution of identity (RI) approximation and the usage of GIAOs. This method is then applied to a set of small molecules for the computation of the g-tensor. This paper is organized as follows: We first present an overview of the theoretical background, covering the foundation of the g- tensor and the theory of DHDFT. The important working equations for the implementation of the perturbed (response) electron and spin-densities are then derived. Finally, numerical calculations are reported, which demonstrate both the accuracy and the computa- tional cost of the new methodology. II. THEORY A. The g-tensor in linear response theory In the one-component scheme, the g-tensor is described as a second order property by linear response theory, i.e., being the sec- ond energy derivative with respect to the magnetic field Band the total electron spin S, g=1 μB∂2E ∂B∂S∣ B,S=0, (1) with μBbeing the Bohr magneton, which is1 2in atomic units. The g-tensor can be separated into four main contributions,7,8,49,50 g=ge1+ΔgRMC+ΔgGC+ΔgOZ/SOC, (2) of which the latter three correction terms basically describe the deviations from the g-value of a free-electron,51 ge=2.002319304386 (20). (3) The second and third terms refer to the relativistic mass cor- rection and the diamagnetic correction to the g-tensor, which can be expressed by means of the spin density Pα−β μν, J. Chem. Phys. 153, 054105 (2020); doi: 10.1063/5.0013799 153, 054105-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp ΔgRMC=−α2 S∑ μνPα−β μν⟨μ∣−1 2⃗∇2∣ν⟩, (4) ΔgGC pq=α2 4S∑ A,iZA eff∑ μνPα−β μν⟨μ⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪δpq⃗rA⃗rO−⃗rA,p⃗rO,q ∣⃗ri−⃗RA∣3⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪ν⟩ p,q∈{x,y,z}, (5) where ZA effis the semiempirically chosen effective nuclear charge from atom Aat position ⃗RA.52⃗rOis the vector relative to the chosen origin O, and analogously, ⃗rAis the electronic position with respect to nucleus A.α,μ/νare the fine structure constant and the atomic basis functions, respectively. The last term is the actual second order orbital Zeemann/spin- orbit coupling (SOC) term that contains the perturbed (response) spin density,4 ΔgOZ/SOC pq=−1 S∑ μν∂Pα−β μν ∂Bp⟨μ∣ˆhSOC q∣ν⟩ p,q∈{x,y,z}. (6) Within this framework, the g-tensor can be calculated at the chosen level of theory, provided that the spin density and per- turbed spin density are available for the particular method. The per- turbed spin density is obtained by the difference of the perturbed α- and perturbed β-density, which can be computed by the unre- stricted CPSCF.16,30In this work, the spin–orbit operator ˆhSOC q is chosen to be the spin–orbit mean field operator53applied with the 1X-approximation [SOMF(1X)].36 As it is known for magnetic properties, the g-tensor is origin dependent for finite basis sets54even though its dependency is not as pronounced as for nuclear magnetic shieldings because it is a molec- ular property, i.e., averaged over the whole molecule. As already mentioned, this “gauge dependency problem” can be circumvented by using GIAOs23for which the latter two correction terms of the g-tensor change to ΔgGC pq=α2 4S∑ A,iZA eff∑ μνPα−β μν⟨μ⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪δpq⃗rA⃗rN−⃗rA,p⃗rN,q ∣⃗ri−⃗RA∣3⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪ν⟩ p,q∈{x,y,z}, (7) ΔgOZ/SOC pq=−1 S∑ μν⎡⎢⎢⎢⎢⎣∂Pα−β μν ∂Bp⟨μ∣ˆhSOC q∣ν⟩ +Pα−β μν∂ ∂Bp⟨μ∣ˆhSOC q∣ν⟩] p,q∈{x,y,z}, (8) with the perturbed SOMF(1X) integrals being ∂ ∂Bp⟨μ∣ˆhSOC q∣ν⟩=QMN⟨μ∣rNˆhSOC∣ν⟩, (9) where⃗rNrefers to the electronic position with respect to nucleus N. QMNis the antisymmetric matrix,QMN=⎡⎢⎢⎢⎢⎢⎢⎣0−ZMN YMN ZMN 0−XMN −YMN XMN 0⎤⎥⎥⎥⎥⎥⎥⎦, (10) which can be used to rewrite a cross product as a matrix multiplica- tion⃗RMN×⃗r=QMN⃗r. B. Double-hybrid density functional theory The general expression for spin-component-scaled double- hybrid DFT (DSD-DFT)55–57is EDSD−DFT=TS+J+Ene+cXEHF X+(1−cX)EDFT X+cCEDFT C ⌟⟨⟨⟪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⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫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⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫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⟩⟩⟩⟪ ESCF +cOEMP2 O+cSEMP2 S⌟⟨⟨⟪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⟫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⟫r⟩⟩⟩⟪ EMP2+s6ED, (11) with the terms being the kinetic energy, the electron–electron and electron–nuclear Coulomb energy, the exact (HF) exchange energy, the exchange and correlation contribution of the DF, and the oppo- site and the same spin energy contribution at the MP2 level, respec- tively. The last term takes into account the dispersion correction. Depending on the coefficients, different cases may be considered as follows: ●cX=cO=cS= 0: “pure” DFT. ●cX= 1,cO=cS=cC=s6= 0: pure HF. ●cX=cO=cS= 1,cC=s6= 0: pure MP2. ●cO=cS= 1−cC,s6= 0: simpler double-hybrid functionals, e.g., B2PLYP. Analogous to the split of energy contributions in Eq. (11), the g-tensor can also be separated into an SCF- and MP2-part, g=gSCF+gMP2. (12) Derivations for the SCF-part including the GIAO framework are already well established and widely present throughout the lit- erature.16,24,26,48In the following, we will first give an overview of the SCF response density and focus on the MP2-part of the theory later on. C. SCF response density and CPSCF equations The spin density is simply the difference between the α- and β-density, Pα−β=Pα−Pβ, (13) with the density matrix elements being Pσ μν=i∈Nσ ∑ icσ∗ μicσ νi, σ∈{α,β}. (14) The response density then is obtained by taking the derivative of the spin density with respect to the magnetic field, d dBPα−β=Pα−β,B=Pα,B−Pβ,B. (15) J. Chem. Phys. 153, 054105 (2020); doi: 10.1063/5.0013799 153, 054105-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp In the following, we will look at the spin- αcase only and there- fore will omit the superscript σfor clarity. The response density then becomes PB μν=∑ i(c∗,B μicνi+c∗ μicB νi) =∑ iaUB ai(cμicνa−cμacνi)−∑ ijS(B) ijcμjcνi, (16) where we have expanded the perturbed MO coefficients as cB μp=∑ qcμqUB qp (17) and made use of the orthonormality condition U∗,B qp+S(B) pq+UB pq=0 (18) as well as the antisymmetry of the perturbed overlap matrix S(B)so that UB ij=−UB ji=−U∗,B ij=U∗,B ji=−1 2S(B) ij. (19) Throughout this work, i,j, and kdenote the occupied MOs, a,b, and cdenote the virtual MOs, and p,q, and rdenote any MOs. The superscript Brefers to the total derivative with respect to the magnetic field, whereas the superscript ( B) refers to solely the basis function and operator derivatives if dependent on the mag- netic field, not the MO coefficients. Note that the field derivative has to be taken for each component ( x,y,z) of the magnetic field. The occupied-virtual/virtual-occupied part of the U-coefficients as needed in Eq. (16) are obtained by taking the derivative of the Brillouin condition, 0=d dB∑ μνc∗ μaFμνcνi=∑ μν(c∗,B μaFμνcνi+c∗ μaFB μνcνi+c∗ μaFμνcB νi), (20) where Fμν=(μ∣ˆh∣ν)+∑ κλPκλ(μν∥κλ)+∑ κλPκλ(μν∣κλ) (21) in the unrestricted case. The (antisymmetrized) two-electron inte- grals are given in Mulliken notation, i.e., (1∗1∣2∗2), and the over- bar denotes the opposite spin (here, β) contribution. The field derivatives of the individual terms are ∑ μνc∗,B μaFμνcνi=∑ μν∑ qc∗ μqU∗,B qaFμνcνi=(−UB ai−S(B) ai)εi, (22) ∑ μνc∗ μaFμνcB νi=∑ μνc∗ μFμν∑ qcνqU∗,B qi=εaUB ai, (23) ∑ μνc∗ μa(μ∣ˆh∣ν)Bcνi=∑ μνcμacνih(B) μν, (24)∑ μνκλc∗ μa(Pκλ(μν∥κλ))Bcνi =∑ μνκλcμaPB κλ(μν∥κλ)cνi+∑ μνκλcμaPκλ(μν∥κλ)(B)cνi =∑ jbUB bj[(aj∣bi)−(ab∣ji)]−∑ jkS(B) jk(ai∥kj)+∑ j(ai∥jj)(B), (25) ∑ μνκλc∗ μa(Pκλ(μν∣κλ))Bcνi =∑ μνκλcμaPB κλ(μν∥κλ)cνi+∑ μνκλcμaPκλ(μν∥κλ)(B)cνi =−∑ kjS(B) jk(ai∣kj)+∑ j(ai∣jj)(B). (26) Collecting all terms containing the U-coefficients on one side and combining the indices aito a single index gives the final CPSCF equations in matrix notation, XUB=b, (27) with Xai,bj=(εa−εi)δai,bj+(aj∣bi)−(ab∣ji), (28) bai=S(B) aiεi−h(B) ai+∑ jkS(B) jk(ai∥kj)+∑ jkS(B) jk(ai∥kj) −∑ j(ai∥jj)(B)−∑ j(ai∣jj)(B). (29) In the case of DFT, an additional term due to the XC- functional58VXChas to be considered on the right-hand side (RHS), bai←∑ μνcμacνid⟨μ∣VXC∣ν⟩ dB, (30) and for hybrid DFT, the HF exchange contributions have to be scaled accordingly in the CPSCF equations. In the unrestricted framework, the same set of CPSCF equa- tions has to be formulated analogously for the β-spin case. Hence, a linear equation system of the following form needs to be solved iteratively: (Xα0 0Xβ)(Uα Uβ)=(bα bβ). (31) D. RI-MP2 response density and perturbed Z-vector equations The general formulation of perturbation theory does not pro- vide variational energies. Instead, using the Lagrange method of undetermined multipliers to reformulate the MP2 energy expres- sion can make it variational.59,60The starting point is, therefore, J. Chem. Phys. 153, 054105 (2020); doi: 10.1063/5.0013799 153, 054105-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the Hylleraas MP2 functional that has the following form for the unrestricted case using the RI formulation: JUHF 2[t]=∑ ijab,P{1 4[BP iaBP jb−BP ibBP ja]tij ab+1 4[BP aiBP bj−BP ajBP bi]tij∗ ab +1 4[BP iaBP jb−BP ibBP ja]tij ab+1 4[BP aiBP bj−BP ajBP bi]tij∗ ab +BP iaBP jbtij ab+BP aiBP bjtij∗ ab}+⟨D′†F⟩+⟨D′†F⟩, (32) with tbeing the MP2 amplitudes, Fthe being the Fock matrix, and D′being the unrelaxed density. The overbar denotes again the oppo- site spin case. The unrestricted formulation distinguishes between three cases: αα,ββ, and αβ=βα. The pointy brackets imply the trace of the resulting matrix. The three-index RI integrals are61–63 ∑ PBP prBP qs=∑ PQ(pr∣Q)(V−1)PQ(Q∣qs), (33) VPQ=(P∣Q), (34) BP pq=∑ Q(pq∣Q)(V−1 2) QP, (35) where P,Qdenote the auxiliary basis functions. The amplitudes and the occupied/virtual part of the unrelaxed density are defined as same spin: tij ab=BP aiBP bj−BP ajBP bi εi+εj−εa−εb, (36) opposite spin: tij ab=BP aiBP bj εi+εj−εa−εb, (37) D′ ij=−∑ k[1 2⟨Tjk∗Tki⟩+⟨Tjk∗Tki⟩], (38) D′ ab=∑ ij[1 2Tji∗Tij+Tji∗Tij] ab, (39) with Tbeing the amplitude matrices for the given occupied MO indices. The relaxed (perturbed) MP2 density is needed for the g-tensor, i.e., allowing the mixing between occupied and virtual orbitals. This is achieved by adding the Brillouin condition to formulate the Lagrangian64,65 L=J2+∑ iaZaiFai+∑ iaZaiFai. (40) Solving the resulting Z-vector equation that is given here for the α-case, ∑ iaZai[(εb−εj)δabδij+(ai∥jb)+(ai∥bj)] =−2{∑ aPBP ba(ss ΓP ja+os ΓP ja)−∑ iPBP ij(ss ΓP ib+os ΓP ib)+1 2R[D′]bj}, (41)by making the Lagrangian stationary with respect to orbital rotations provides the relaxed density by filling the occupied-virtual/virtual- occupied part of the density matrix according to66–68 Dai=−1 2Zia=Dia. (42) The three-index density matrix and the Fock response operator are defined as68,69 same spin:ss ΓP ia=∑ jbtij abBP jb, (43) opposite spin:os ΓP ia=∑ jbtij abBP jb, (44) R[D]rs=2⎡⎢⎢⎢⎢⎣∑ pqDpq(pq∥rs)+∑ pq(pq∣rs)+RXC⎤⎥⎥⎥⎥⎦. (45) The (antisymmetrized) two-electron integrals can be approx- imated using RIJK70or RIJCOSX.71The contribution RXCarising from the XC functional is given and discussed in Ref. 72. The β-case is defined analogously. The relaxed perturbed MP2 density is obtained by taking the derivative of the stationary conditions of the Lagrangian with respect to the magnetic field. The necessary quantities for the calculation of the relaxed response density are given in the following. For the unrelaxed perturbed density, we need the perturbed amplitudes, tij,B ab={∑ P[BP,B aiBP bj+BP aiBP,B bj−BP,B ajBP bi−BP ajBP,B bi] −∑ k[tik abFB kj+tkj abFB ki]+∑ c[tij acFB cb+tij cbFB ca]}/Δijab=−tji,B ba, (46) tij,B ab={∑ P[BP,B aiBP bj+BP aiBP,B bj]−∑ k[tik abFB kj+tkj abFB ki] +∑ c[tij acFB cb+tij cbFB ca]}/Δijab=−tji,B ba, (47) D′B ij=∑ k[1 2(⟨Tjk,BTki⟩−⟨TjkTki,B⟩)+⟨Tjk,BTki⟩ −⟨TjkTki,B⟩]=−D′B ji, (48) D′B ab=∑ ij[1 2(Tji,BTij−TjiTij,B)+Tji,BTij−TjiTij,B] ab =−D′B ba. (49) The relaxed perturbed density then is defined as the field derivative of Eq. 42, J. Chem. Phys. 153, 054105 (2020); doi: 10.1063/5.0013799 153, 054105-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp DB ai=1 2ZB ai=−DB ia. (50) Therefore, the perturbed Z-vector equation needs to be solved according to ∑ iaZB ai[(εb−εj)δabδij+(aj∣bi)−(ab∣ji)] =2{∑ aP[(ss ΓP,B ja+os ΓP,B ja)BP ba+(ss ΓP ja+os ΓP ja)BP,B ba] −∑ iP[(ss ΓP,B ib+os ΓP,B ib)BP ij+(ss ΓP ib+os ΓP ib)BP,B ij]−∑ a1 2ZajFB ab+∑ i1 2ZbiFB ji −∑ pq[Dpq(pq∥jb)(B)+Dpq(pq∥jb)(B)]−∑ r1 2UB rbR[D]jr +∑ r1 2UB rjR[D]rb−∑ r1 2UB rbR[D]jr+∑ r1 2UB rjR[D]rb +1 2R[D′B]bj+1 2RXC,(B)[D]bj−R[UBD]bj}, (51) with R[DB]rs=2∑ pqDB pq(pr∣sq)=−R[DB]sr. (52)The derivatives of the RI three-index integrals and three-index density matrices are BP,B pq=BP,(B) pq +∑ k[UB kqBP pk−UB kpBP kq] +∑ k[UB cqBP pc−UB cpBP cq]=−BB qp, (53) BP,(B) pq=∑ Q(pq∣Q)(B)(V−1 2) QP=−BP,(B) qp , (54) ss ΓP,B ia=∑ jb[tij,B abBP jb+tij abBP,B jb], (55) os ΓP,B ia=∑ jb[tij,B abBP jb+tij abBP,B jb]. (56) The final relaxed perturbed MP2 density is then completed as DB=D′B+ZB. (57) ALGORITHM 1 . Pseudocode of the unrestricted RI-MP2 response density implementation in ORCA. begin preparation: ∙make and store all BPandBP,B ∙readDandUB ai ∙complete UBandFB forop1∈{α,β}do forbatch∈{number of batches }do fori∈batch do forop2∈{α,β}do forj∈{occupied }do ∙make amplitudes tij ∙make Γintegrals ∙make derivative amplitudes tij,B ∙make derivative ΓBintegrals ∙make contribution to virtual block of response density DB ab end forj∈{occupied }do ∙make contribution to occupied block of perturbed density D′B ij end end ∙make three-external ΓP,BBPandΓPBP,Bto RHS of perturbed Z-vector end end end ∙Fock response contribution to RHS of perturbed Z-vector ∙solve perturbed Z-vector equations ∙complete relaxed response density DB end J. Chem. Phys. 153, 054105 (2020); doi: 10.1063/5.0013799 153, 054105-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp E. Implementation details The implementation of the RI-MP2 relaxed perturbed density in ORCA as shown in Algorithm 1 using the unrestricted formalism is very similar to the closed shell case, which is thoroughly described in Ref. 72. Care has to be taken of the different spin cases, i.e., there exist an αdensity and a βdensity that are coupled via mixed spin amplitudes [Eqs. (37) and (47)] or Coulomb integral contributions of the Fock response [Eq. (45)]. The loops over operators ( op1 and op2) take care of this spin case distinction. An important point to consider is the treatment of the two- electron integrals occurring on the left-hand side (LHS) and the RHS of the Z-vector and perturbed Z-vector equations [Eqs. (41) and (51), respectively], as well as in the perturbed Fock matrix FBcontributions to the perturbed amplitudes [Eqs. (46) and (47), respectively]. Even though the same treatment of the two-electron integrals as in the SCF procedure would be most consistent, the terms on the LHS are treated with the so-called RIJ-DX approxi- mation since the RI approximation for the exchange integrals does not improve the computational performance when they have to be contracted with the density matrix defined for the entire MO space. In the RIJ-DX method, the RI approximation is only applied to the Coulomb integrals, whereas the exchange integrals are com- puted in an integral-direct fashion employing traditional four center repulsion integrals. III. COMPUTATIONAL DETAILS All calculations were done using the current development ver- sion of ORCA 4.2 with the following tasks in mind:73(1) A bench- mark study is conducted based on sets of small radicals since, to our knowledge, no data have been provided so far in the lit- erature concerning g-tensor calculations at the MP2 level with GIAOs. This benchmark study, therefore, includes basis set conver- gence and comparison to experimental values quoted from Refs. 24 and 74 and highly accurate data at the CC level.21(2) The tim- ing and computational scaling with the system size is investi- gated in order to determine the limits and the efficiency of the methodology. The geometries of the small radical test sets were used from Refs. 16 and 21, respectively, whereas the geometries of the medium to large size molecules are provided in the supplementary material. Two double-hybrid density functionals (DHDFs) were chosen for this study: B2PLYP as a representative of the “simple” DHDFs andDSD-PBEP86 as a representative of the dispersion-corrected spin- component-scaled DHDFs.43Very tight SCF convergence criteria were applied, as well as a threshold of 10−8for the convergence of the Z-vector solution. All calculations were done using GIAOs for the gauge origin treatment. Basis sets of the def2-family by Weigend and Ahlrichs75and aug-cc-family by Dunning et al.76,77were used for the calculations. IV. RESULTS AND DISCUSSION A. Benchmark 1. Basis set convergence The basis set convergence was tested using the NH radical. Only thegxx-component is investigated since the gzz-component shows almost no effect as, for symmetry reasons, its main contribution is the isotropic relative mass correction term that is not very chal- lenging from a computational point of view. The basis set conver- gence for the gisovalue shows the same behavior as for the gxxvalue since it is the main contribution to the isotropic value, as shown in Fig. 1. It can be seen that both tested DHDFs converge to almost the same value for a sextuple- ζbasis set. Their convergence toward the complete basis set (CBS) limit is rather slow, which is characteris- tic for wave function based correlation theories. B2PLYP converges faster than DSD-PBEP86 up to the quintuple- ζbasis, but the next step to the sextuple- ζbasis leads to a decrease in the g-tensor compo- nents. This behavior is presently not understood. The convergence data for the DSD-PBEP86 functional show a more regular trend. However, for both functionals, the improvement by changing from double- ζto triple- ζis the most pronounced. Thus, the use of at least a triple- ζbasis set for the g-tensor calculation with double-hybrid functionals appears to be the minimum requirement. At this level, basis set incompleteness errors on the order of 50 ppm would need to be tolerated. 2. Comparison with experiment In the next step, a set of small radicals was chosen for the comparison of the calculated with experimental principle g-shifts: H2O+, CO+, HCO, C 3H5, NO 2, NF 2, and MgF.24,74This was done for two different triple- ζbasis sets, namely, def2-tzvpp and aug-cc-pvtz. Figure 2 shows the plot of the experimental vs calculated g-shifts for both DHDFs including the corresponding linear regression plots (colored solid lines), as well as for the pure RI-MP2 method. The FIG. 1 . Basis set convergence for B2PLYP and DSD-PBEP86 DHDFs using the aug-cc-pvXz (X = d, t, q, 5, 6) basis set family on the NH radical. J. Chem. Phys. 153, 054105 (2020); doi: 10.1063/5.0013799 153, 054105-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Plots of calculated vs experimental Δg-values including linear regression fit (depicted by the colored solid lines) for pure RI-MP2, the B2PLYP, and DSD- PBEP86 DHDF using different triple- ζbasis sets. black solid line corresponds to the ideal match between experimen- tal and computational data. It is evident that the difference between both basis sets is very small. The computed values of both DHDFs are in good agreement with the experimental ones except for theΔgyyvalue of H 2O+, which was excluded from this dataset, since its deviation of about 3000 ppm would have distorted not only the visualization but also the statistical analysis. The data of the sta- tistical analysis including the linear regression parameters are pre- sented in Table I. They confirm the good agreement of the cal- culated values with the experimental ones with a slope of almost 1 for all tested cases. As expected for molecular properties, pure RI-MP2 performs less well than both tested DHDFs, which simi- larly show good performance. According to the statistical evalua- tion, the def2-tzvpp basis set gives slightly better results than the aug-cc-pvtz. 3. Comparison to CCSD In this section, we compare the performance of the DHDFs to other methods with CCSD as the most accurate reference currently available. For consistency, the same set of small radicals as in the Ref. 21 that is used for comparison was tested. Note that the values taken from the reference were all computed without using GIAOs. It was shown that the use of GIAOs does not lead to a very pro- nounced change for g-tensor calculations (unlike NMR chemical shifts). This may be related to the fact that, on the one hand, in con- trast to nuclear chemical shieldings, the g-tensor is a global property and, on the other hand, that g-shifts tend to be much larger than chemical shieldings, while gauge non-invariance errors tend to be more comparable.21 Again, the Δgyycomponent of H 2O+was removed from the dataset for the statistical evaluation for the same reasons as before since all methods show difficulties in treating the H 2O+radical. Figure 3 shows the plots of the pure RI-MP2 and DHDF g-shifts vs the CCSD g-shifts with the solid colored lines corresponding to a lin- ear fit of the datasets. For comparison, values calculated with HF are also provided. A clear improvement to HF g-shifts with DHDFT is observed. This was expected as correlation effects are non-negligible for g-tensor calculations. However, pure RI-MP2 performs distinctly worse than both DHDFs and interestingly even worse than HF. This observation is quantified through a statistical analysis presented in Table II. The detailed data for each molecule in the test set are given in Table III. One source for poor performances of unrestricted wave func- tion approaches may be spin contamination. Its measurement is the difference between the expectation value ⟨ˆS2⟩and the ideal value computed by S(S+ 1), where Sis the total spin of the TABLE I . Statistical evaluation of the calculated Δg-values for the set of seven small radicals (H 2O+, CO+, HCO, C 3H5, NO 2, NF 2, and MgF) and the linear regression parameters of the fit depicted in Fig. 2. RI-MP2 B2PLYP DSD-PBEP86 def2-tzvpp aug-cc-pvtz def2-tzvpp aug-cc-pvtz def2-tzvpp aug-cc-pvtz R20.983 0.983 0.988 0.988 0.988 0.988 Slope 0.934 0.913 0.966 0.943 0.957 0.936 Standard deviation [ppm] 551 579 445 469 443 484 Maximum error [ppm] 1053 1128 962 948 861 970 Mean unsigned error [ppm] 475 474 386 399 385 423 Mean signed error [ppm] 158 157 12 56 12 75 J. Chem. Phys. 153, 054105 (2020); doi: 10.1063/5.0013799 153, 054105-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Plot of calculated Δg-values using different methods/functionals (RI- MP2, B2PLYP, DSD-PBEP86, and HF) vs CCSD-based values including linear regression fit (depicted by the colored solid lines). system. This difference is an indicator for the quality of the underly- ing wave function. The mean spin contamination values are 0.0617, 0.0617, 0.0146, and 0.0274 for pure RI-MP2, HF, B2PLYP, and DSD-PBEP86, respectively. The values for RI-MP2 and HF are two to four times larger than those for the DHDFs, which sup- ports the observation given in Fig. 3 and Table III. However, somewhat surprisingly, looking at the dependency of the g-shift component error with respect to the CCSD values on the cor- responding spin contamination, no correlation can be observed (Fig. 4). In addition, the spin contamination for DSD-PBEP86 is intrinsically larger than that for B2PLYP as by construction, DSD-PBEP86 contains a higher fraction of HF exchange. Nonethe- less, it shows a slightly better performance than B2PLYP. Hence, the good performance most likely relies on error cancellation. In any case, no clear correlation can be established between the amount of spin contamination of the corresponding error in g-shift calculations. In addition to the methods so far discussed, some data for the popular hybrid functional B3LYP were taken into account for the statistical analysis, but not plotted in Fig. 3 for clarity. It is evident from the comparison that despite the improvement of the calculated g-shifts by using DHDFT over HF and pure RI-MP2, both DHDFs are still being outperformed by B3LYP with respect to both accuracy, at least for the herein chosen test set, and efficiency. There are some isolated exceptions where one of the DHDFs performs better thanB3LYP (B2PLYP—NF 2,Δgyy, and NF3+Δgzzand DSD-PBEP86— NH,Δgzz, OH+Δgzz, and H 2CO+Δgzz). For pure RI-MP2 and the DSD-PBEP86 functional, a sign error is, however, observed for the Δgyyvalue of H 2CO+. Thus, overall, the hope that wave function cor- relation could “repair” some of the shortcomings of DFT for this specific magnetic property has not materialized. B. Computational costs In Secs. IV A 2 and IV A 3, the accuracy of using DHDFT for g-tensor calculations was discussed with regard to experimen- tal and other available computational methods. In this section, the computational costs of the implemented method are analyzed and discussed. Four medium to large size molecules were studied using the DSD-PBEP86 DHDF with the def2-tzvpp basis set. The detailed data are given in Table IV and visualized in Fig. 5. The total time is split into its main contributions: the SCF part, the CPSCF part that includes both the assembling of the RHS and the solution of the CPSCF equations, the computation of the SOC integrals, the formation of the relaxed MP2-density ( D), and the relaxed MP2 response density ( DB). It is clearly visible that the computational effort is dominated by the calculation of the MP2 response density. It formally scales as O(N5)and, therefore, can be 20–30 times more expensive than the evaluation of D.72 Two different approximations for the two-electron integrals contributions to the Fock matrix are used here. In this section, the integrals ( ia|jb) and ( ij|ab) (also the corresponding other spin integrals) are generated by an RI transformation and stored on disk, denoted as RIJK.70The results then depend on the speed of disk I/O operations. Here, a RAID 0 hard disk configuration was used. The RIJCOSX notation refers to the RI approximation for the Coulomb integrals and the chain of spheres (COS) approxima- tion for the exchange integrals.71In this case, the RI transformed integrals are not stored on disk but generated in an AO-direct fashion. A quick overview is provided by comparing the final row in Table IV, which denotes the total time of each calculation. For the two smaller systems, phenyl and tyrosyl, no distinct differ- ences are observable between both approximations, but for the two larger examples, α-tocopheryl and α-chlorophyll, a clear difference between RIJK and RIJCOSX can be seen where the latter requires less time for the computation. The larger the molecule (with respect to the number of electrons and therefore the number of basis func- tions), the more distinct the time difference between the two tested TABLE II . Statistical evaluation of the calculated Δg-values at different levels of theory for the set of 14 small radicals given in Table III and the linear regression parameters of the fit depicted in Fig. 3. RI-MP2 B2PLYP DSD-PBEP86 HF21B3LYP21 R20.927 0.998 0.998 0.960 0.998 Slope 1.223 0.886 0.927 1.204 1.017 Standard deviation [ppm] 1971 572 414 1539 236 Maximum error [ppm] 8295 1376 1397 5708 736 Mean unsigned error [ppm] 871 342 291 822 146 Mean signed error [ppm] 643 −120 −25 140 22 J. Chem. Phys. 153, 054105 (2020); doi: 10.1063/5.0013799 153, 054105-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE III . Comparison of RI-MP2, the DHDFs B2PLYP, and DSD-PBEP86 with CCSD, HF, and B3LYP Δg-values given in ppm and taken from Ref. 21. Component CCSD21RI-MP2 B2PLYP DSD-PBEP86 HF21B3LYP21 CN Δgxx −2 151 −1 930 −1 980 −1 997 −2 237 −2 193 Δgzz −124 −161 −141 −151 −81 −134 CO+Δgxx −2 598 −1 524 −2 299 −2 097 −3 225 −2 656 Δgzz −125 −216 −142 −158 −63 −133 BO Δgxx −1 870 −1 244 −1 622 −1 546 −2 113 −1 857 Δgzz −60 −103 −70 −75 −27 −68 NH Δgxx −105 −105 −106 −105 −109 −106 Δgzz 1 465 1 294 1 278 1 240 1 133 1 363 OH+Δgxx −173 −173 −174 −172 −178 −174 Δgzz 4 119 3 742 3 586 3 497 3 405 3 704 H2O+Δgxx −188 −210 −189 −192 −155 −188 Δgyy 16 667 14 863 13 469 13 249 13 123 13 574 Δgzz 4 940 4 477 4 312 4 247 4 052 4 681 CH 3 Δgxx −84 −78 −76 −71 −84 −89 Δgzz 646 546 545 534 506 649 O2 Δgxx −199 −187 −197 −196 −232 −199 Δgzz 2 669 2 066 2 339 2 401 3 498 2 677 O3−Δgxx −706 810 −491 −371 −1 502 −555 Δgyy 18 062 26 357 16 016 17 331 23 770 18 429 Δgzz 10 668 18 585 9 031 10 232 16 103 11 032 CO 2−Δgxx 840 584 738 716 1 048 932 Δgyy −5 104 −4 436 −4 400 −4 432 −5 709 −5 122 Δgzz −779 −694 −687 −684 −927 −716 H2CO+Δgxx 6 172 6 100 5 613 5 724 5 806 5 910 Δgyy 144 −168 22 −62 662 24 Δgzz 721 1 573 417 910 3 039 91 NO 2 Δgxx 3 596 2 883 3 138 3 108 4 278 3 628 Δgyy−11 728 −10 172 −10 352 −10 331 −12 588 −11 837 Δgzz −762 −234 −647 −578 −1 195 −695 NF 2 Δgxx −699 −504 −620 −612 −1 038 −667 Δgyy 6 704 6 370 6 270 6 155 5 757 6 988 Δgzz 3 766 3 605 3 631 3 537 2 889 4 126 NF 3+Δgxx −2 010 −1 209 −1 707 −1 666 −3 667 −1 806 Δgzz 5 178 5 816 5 221 5 258 4 020 5 914 FIG. 4 . Study of the relation between spin contamination and g-shift errors of different methods (RI-MP2, B2PLYP, DSD-PBEP86, and HF) with respect to CCSD. On the left column, the princi- ple g-shift values are plotted vs the spin contamination, whereas the right column shows two bar plots of the isotropic g- shift error (top) and the spin contamina- tion (bottom) for the respective methods for each molecule of the test set. J. Chem. Phys. 153, 054105 (2020); doi: 10.1063/5.0013799 153, 054105-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . Computational performance of DSD-PBEP86 given in minutes for the phenyl and tyrosyl radical in hours for theα-tocopheryl and α-chlorophyll rad- ical. The timings are shown by stacked bar plots grouped into sets of two for each compound. The total height of each bar refers to the total time, whereas each stack refers to the corresponding con- tribution as decoded in the legend. For each compound, the left bar (no pattern) refers to the RIJK treatment of the two- electron integrals, whereas the right bar (dotted pattern) refers to the RIJCOSX treatment. TABLE IV . Computational timing data of DSD-PBEP86 given in minutes for four medium to large size radicals with Natoms , Nel,Nbasis,Nauxbasis , and grid size being the number of atoms, number of electrons, basis set dimension, auxiliary basis set dimension, and grid size for the COSX approximation. The calculations were performed on 8 Intel ®Xeon®CPU E5-2640 v3 2.60 GHz cores with 16 GB RAM per core. Phenyl Tyrosyl α-tocopheryl α-chlorophyll Natoms 11 23 80 73 Nel 41 95 239 293 Nbasis 256 543 1 647 1 720 Nauxbasis 531 1 138 3 091 3 612 Grid size 11 434 23 521 68 641 136 669 RIJK RIJCOSX RIJK RIJCOSX RIJK RIJCOSX RIJK RIJCOSX SCF 0.1 0.1 0.8 0.5 10.6 4.0 16.4 11.6 CPSCF 0.3 0.5 1.6 3.2 149.4 36.9 503.3 87.3 SOC 0.5 0.5 2.4 2.4 26.1 26.0 32.2 32.5 D 0.2 0.4 3.6 4.0 654.8 177.4 1 675.4 351.0 DB2.1 2.0 27.3 24.3 3498.6 3135.4 7 845.7 7154.6 Total time 3.2 3.6 35.7 34.4 4339.5 3379.8 10 073.1 7637.0 approximations.45Therefore, it can be stated that for smaller sys- tems, the choice of approximation is non-relevant, whereas for larger systems with more than 100 electrons, the RIJCOSX approximation is clearly more efficient. For instance, the α-chlorophyll g-tensor cal- culation with RIJCOSX took about 5 days, whereas the RIJK option took 7 days. V. CONCLUSION AND OUTLOOK In this work, we presented an efficient implementation of the electronic g-tensor at the MP2 level using GIAOs. The working equations were derived from an unrestricted Ansatz. With this scheme, the usage of DHDFT was enabled for g-tensor calcula- tions. Our computed g-shift values with pure RI-MP2, B2PLYP, and DSD-PBEP86 are in overall reasonable to good agreement with the experimental data, whereas pure RI-MP2 performs slightlyless well than the DHDFs, as is expected for molecular proper- ties. The comparison to other available computational methods shows a clear improvement of DHDFT to pure RI-MP2. Both tested DHDFs, B2PLYP and DSD-PBEP86, give g-shifts close to the CCSD method, which was taken as a reference being the up-to-now most accurate computational method available for g- tensor calculations. RI-MP2 performs even worse than HF, but the DHDFs are still outperformed by the hybrid DF B3LYP for the used test set of small radicals. Comparing both tested DHDFs to one another, DSD-PBEP86 shows a better performance. The study of the computational costs shows that for molecular sys- tems with more than 100 electrons, the choice of two-electron inte- gral approximations becomes relevant. Indeed, for small systems, RIJK and RIJCOSX marginally differ, but for large systems, e.g., theα-chlorophyll radical, RIJCOSX reduces the computational costs significantly. J. Chem. Phys. 153, 054105 (2020); doi: 10.1063/5.0013799 153, 054105-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp We can conclude that DHDFT clearly improves the com- putational performance for the calculation of the electronic g-tensor compared to pure RI-MP2. Nonetheless, somewhat frus- tratingly, for the investigated set of small radicals, the B3LYP func- tional still shows slightly better agreement with CCSD. However, it is well known that DFT, up to the hybrid functionals, is still failing in computing accurate g-shifts for transition metal com- pounds.16Therefore, it is of interest to study the performance of DHDFT for the latter group of molecules and make efficient higher level theoretical methods, e.g., CC, available for g-tensor calcula- tions of large molecular systems by using linear scaling correlation approaches. SUPPLEMENTARY MATERIAL See the supplementary material for the detailed data underlying the studies in Sec. IV A 2 and the geometries of the medium to large size molecules used for the study of Sec. IV B. ACKNOWLEDGMENTS The authors gratefully acknowledge generous financial sup- port of this work by the Max Planck Society and the IMPRS- RECHARGE. APPENDIX: ORIGIN OF ΔgGCAND ΔgOZ/SOC The origin of the last two correction terms for the g-tensor in Eq. (2) is rooted in the treatment of the SO interaction. It is a commonly used approximation to treat the two electron term resulting from the Breit–Pauli SOC operator as “screening” of the nuclear charges. Therefore, the SOC term can be treated as a one- electron term, and the effective SOC operator then has the following form: ˆhSOC=∑ A∑ iξ(riA)(⃗rA i×⃗pi)⃗si with ξ(riA)=α2 2ZA eff r3 iA. (A1) In the case of a magnetic field, the particle momentum operator is replaced by π=−i⃗∇+A(⃗r) with A(⃗r)=1 2⃗B×⃗rO, (A2) with A(⃗r)as the vector potential introducing the dependency on the magnetic field. Plugging this into Eq. (A1) and expanding the terms gives the operator for the gauge correction term of the SOC for ΔgGC in the no-GIAO case. In the case of using GIAOs, field dependent atomic orbitals of the following form are applied: ˜χ(⃗rM,AM)=exp(−iAM⋅⃗r)χ(⃗rM) with AM=1 2⃗B×RMO. (A3) Inserting these into the molecular integrals and reformulating them in the regular AO basis give⟨˜μ(⃗rM,AM)∣ˆhSOC∣˜ν(⃗rN,AN)⟩, (A4) =⟨μ∣exp(iAM⋅⃗r)ˆhSOCexp(−iAN⋅⃗r)∣ν⟩, (A5) =⟨μ∣exp(iAMN⋅⃗r)∑ i,Aξ(riA) ×[(⃗si×⃗riA)(−i⃗∇)+1 2(⃗si×⃗riA)(⃗B×⃗rN)]∣ν⟩ (A6) due to the non-zero commutator, [−i⃗∇, exp(−iAN⋅⃗r)]=−exp(−iAN⋅⃗r)AN. (A7) Taking the partial derivatives∂2 ∂B∂S∣ B,S=0and∂ ∂B∣B,S=0then results in the final GIAO correction terms of the g-tensor given in Eqs. (7) and (8). DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material. REFERENCES 1J. A. Weil, J. R. Bolton, and J. E. Wertz, Electron Paramagnetic Resonance: Elementary Theory and Application (Wiley-Interscience, 1994). 2N. M. Atherton, Principles of Electron Spin Resonance (Ellis Horwood Limited, 1993). 3A. Schweiger and G. Jeschke, Principles of Pulse Electron Paramagnetic Resonance (Oxford University Press, 2005). 4F. 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5.0023131.pdf

J. Appl. Phys. 128, 144501 (2020); https://doi.org/10.1063/5.0023131 128, 144501 © 2020 Author(s).Origin of irreversible to reversible transition in acetone detection for Y-doped BiFeO3 perovskite Cite as: J. Appl. Phys. 128, 144501 (2020); https://doi.org/10.1063/5.0023131 Submitted: 27 July 2020 . Accepted: 19 September 2020 . Published Online: 09 October 2020 S. Neogi , and R. Ghosh ARTICLES YOU MAY BE INTERESTED IN Strain induced Co/Mn ionization and magnetic properties in double-perovskite Nd 2CoMnO 6 thin films Journal of Applied Physics 128, 145305 (2020); https://doi.org/10.1063/1.5143222 Introducing an extremely high output power and high temperature piezoelectric bimorph energy harvester technology based on the ferroelectric system Bi(Me)O 3-PbTiO 3 Journal of Applied Physics 128, 144102 (2020); https://doi.org/10.1063/5.0005789 Specific absorption rate of magnetic nanoparticles: Nonlinear AC susceptibility Journal of Applied Physics 128, 143901 (2020); https://doi.org/10.1063/5.0018685Origin of irreversible to reversible transition in acetone detection for Y-doped BiFeO 3perovskite Cite as: J. Appl. Phys. 128, 144501 (2020); doi: 10.1063/5.0023131 View Online Export Citation CrossMar k Submitted: 27 July 2020 · Accepted: 19 September 2020 · Published Online: 9 October 2020 S. Neogi1,2and R. Ghosh1,2,a) AFFILIATIONS 1CSIR-Central Mechanical Engineering Research Institute, Durgapur 713209, India 2Academy of Scientific and Innovative Research (AcSIR), CSIR-CMERI Campus, Durgapur 713209, India a)Author to whom correspondence should be addressed: ghosh.ranajit@gmail.com ABSTRACT To eliminate the demerits of irreversibility associated with a gas-sensing material, it is always indispensable to know the exact origin of the same. In this view, the present study associates a perovskite to investigate the origin of irreversible sensing by a non-conventional bismuth ferrite (BiFeO 3) nanomaterial prepared by a simple solgel technique. Yttrium (Y) doping in BiFeO 3significantly enhances the response performance and eliminates the irreversible nature, showing a reversible-type sensing behavior for selective detection of acetone (crossselectivity, R cross=RAcetone /Rxylene : 20) along with the virtue of very low sensing ability (1 ppm), long-term stability with a negligible deviation in response value ( R= 3.5 ± 0.25) toward 5 ppm acetone tested repeatedly for 300 days, and excellent repeatable over nine loops character desirable for practical application of the perovskite material. A remarkably highest response ( R=ΔG/Ga) of 52 was achieved toward acetone utilizing Y-doped BiFeO 3perovskite. A possibility of the formation of closed pores and incomplete desorption of the reducing species (volatile organic compounds) from the active sites of the sensor has been considered the prime origin of unfinishedrecovery. Finally, irreversible to reversible transition in sensing has correlated with the structural and morphological change resulting fromthe substitution of the Bi 3+ion by the Y3+ion, particularly at the A-site of the perovskite BiFeO 3due to Y-doping. Published under license by AIP Publishing. https://doi.org/10.1063/5.0023131 I. INTRODUCTION In a state when the total signal change (change in current/ resistance/conductance) of a sensor upon exposure to a particular gas/vapor (response) will be exactly or nearly equal to the samewhen switched from a gas to air environment (recovery), thesensing element is said to be a reversible one, otherwise irrevers-ible. 1,2For commercial purposes, metal –oxide –semiconductor (MOS) type resistive sensors are widely used due to a bunch of advantages associated with the materials.3–7For initial screening of the gas/volatile organic compound (VOC) sensing characteristicsof this type of resistive sensors, the materials are periodicallyexposed to a test gas/VOC (response) and air (recovery) to record the dynamic response. During this process, it is often observed that there will be a significant drift in the baseline originating from thepartial recovery process. 1Incomplete and slow recovery indicated by the dynamic response curve confirms the irreversible natureassociated with any particular gas sensor. In view of practical appli- cation, irreversibility is one of the prime bottlenecks for a typical sensing element. It has been observed that, for an irreversiblegas/VOC sensor, the response time is found to be a function of the exposed gas/VOC concentration, which is independent of a reversible-type sensor. 1It is worth mentioning that for an efficient and commercially viable gas sensor, reversible-type sensing ele-ments are always expedient. 8–10However, according to literature studies, a variety of sensing elements are available that showirreversible-type sensing along with a reversible-type subject to a wide range of sensed gas species. For better understanding of irreversible-type sensing, Table I illustrates some important irreversible as well as reversible gas-sensing phenomena for resis-tive type sensors along with the possible reasons for the same.The irreversible-type sensing character mainly for MOS sensorsbelongs to the pathways of interaction between the target gas/va-por and oxygen (chemisorbed on the surface of the sensor ini-tially) to finally form the reaction product. So, realization of theadsorption (in response) and desorption (in recovery) process ofthe analytes on the sensor surface is the prime objective foranswering the entire “why ”and “how ”related to the irreversibility in sensors. 11Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 144501 (2020); doi: 10.1063/5.0023131 128, 144501-1 Published under license by AIP Publishing.Although the immense field of gas/VOC sensing is crowded with conventional binary MOS materials, such as ZnO,17,18SnO 2,19 or Fe 2O3,20but as humid atmosphere for practical application is concerned, the long-term stability is still uncertain for these materi-als. 21In this context, it is important to mention that, due to stabil- ity in structure and induced oxygen vacancies resulting from non-stoichiometric ratio, ternary perovskite oxides with the general formula ABO 3can be a promising candidate for gas-sensing.22 Perovskite oxides are mostly attractive for high-temperature appli-cations providing high melting and decomposition temperatures along with microstructural and morphological stability to improve reliability and long-term performance for a sensor. 23Ferrites, par- ticularly belonging to the perovskite category, such as BiFeO 3, LuFeO 3, LaFeO 3, and many others, exhibit different degrees of gas sensing24,25apart from their multiferroic properties. Among them, bismuth ferrite (BFO) shows excellent gas/VOC sensing character- istics irrespective of the possible application in the field of spin-tronics and multiferroics. 22,26–28BFO shows a p-type conduction behavior when exposed to a reducing gas (VOC) environment.Moreover, it has been always desired to enhance or improve the sensing properties of a material concerning its commercial applica- bility. Doping with different elements particularly at the A-site ofBFO can enhance as well as improve various properties of thematerial. 22 Rare earth elements due to the availability of the 4f shell are the operative choices as a dopant to improve different sensing char- acteristics of a MOS type sensor.29As per the available previous reports, the introduction of rare earth elements in MOS improvestheir sensing performance by modifying the surface morphology and grain size. 30,31The smaller grains of rare earth doped materials resulting from the impediment of the growth of grains usually leadto an increase in active sites for sensing and enhancement of the performance.32The rare earth yttrium (Y) doping results in a struc- tural transition of BFO from rhombohedral ( R3c) toward ortho- rhombic ( Pna21 /Pnma ) depending on the dopant concentration (usually >10 wt. %).33Graf et al.34observed a similar phase transi- tion induced by Y-doping in BiFeO 3. On the other hand, according toSheng et al .,35a structural transition from rhombohedral ( R3c) to orthorhombic ( Pna21 ) was observed with increasing x above 0.10 in Bi 1−xYxFeO 3films. Table II summarizes the effect of Y-doping on the structural and morphological properties of BFO. This structural change is followed by an enhancement of various functional properties of the material. A report by Mukherjeeet al. 32shows an enhancement of electrical and optical properties due to the introduction of the Y3+ion at the A-site of BFO. Gautam et al .38observed significant improvements of magnetic properties of BFO with Y-doping. So, among the possible pathways to eliminate the irreversible sensing character associated with a pro-totype, doping can be considered a wise choice to fulfill the saidpurpose. A doped material can perform better as compared to apure one due to its change in structural and functional properties resulting from proper doping. To the best of our knowledge, most of the previous studies related to Y-doped BFO were only con-cerned about the effect of yttrium doping on the ferroelectric andmagnetic properties, i.e., multiferroic properties of BFO correlatedwith the structural change of the material. Here, we have investi- gated the effect of Y-doping on the gas/VOC sensing characteristics using the non-conventional MOS materials such as BiFeO 3and sys- tematically explore the mechanism behind the observations. In this study, we have observed initially an irreversible-type acetone sensing characteristics by a non-conventional MOS type bismuth ferrite chemiresistive sensor and most importantly aTABLE I. Reversible and irreversible-type gas sensing for various chemiresistive type sensors. Sensing element Target gas/vapor Nature of sensing Possible reason Reference ZnO and Mg 0.5Zn0.5Fe2O4 H2and CO Irreversible using ZnO and reversible for Mg 0.5Zn0.5Fe2O4ZnO and Mg 0.5Zn0.5Fe2O4shows irreversible and reversible sensing due to different surface morphology and porous nature1 Peptide –hydrogel-based composite materialp-xylene Irreversible sensing and long recovery without carbon dot and reversible with carbon dotsIncorporation of carbon dots increases charge carrier concentration and responsible for fast and reversible sensing12 CuO-thin films H 2S Reversible at low and irreversible at high H 2S concentrationFormation of CuS on the CuO surface responsible for irreversibility13 m-carbon nanotube(CNT)/NaClO 4/ polypyrrole (composite)C2H5OH Irreversible (without m-CNT) and reversible (with m-CNT)Irreversible due to strong chemisorption. CNT increases the electron concentration and weakens the chemisorption process and shows reversible-type sensing14 Polypyrrole film NH 3 Irreversible sensing Irreversible response is due to the strong chemical bond formation between the tested analyte and film surface15 Carbon nanotube NH 3 Both irreversible as well as reversible-type sensingReversible for physisorption of NH 3, whereas in the case of chemisorption of NH 3 irreversible-type sensing occurs (prolonged exposure)16Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 144501 (2020); doi: 10.1063/5.0023131 128, 144501-2 Published under license by AIP Publishing.typical irreversible to reversible sensing transition, unique of its kind, achieved by doping (10 wt. %) the pure material with rare earth yttrium (Y3+ion) at the A-site of BiFeO 3(Y-BFO) prepared by a simple, low-cost solgel protocol. We have doped BFO with dif-ferent concentrations of Y (1 –15 wt. %), and as a fully reversible character along with highest response value was achieved at 10 wt. %, this sample was taken for further study. Apart from theextremely reversible character for simple BFO, the doped materialalso shows highly sensitive and selective detection of acetone vapor.The difference in ionic radius of Bi 3+and Y3+ions results in a large lattice distortion followed by a structural phase transformation and morphological change considered to be the possible reason for thisirreversible to reversible sensing transition along with an enhance-ment of sensor response toward acetone vapor. II. EXPERIMENTAL PROCEDURE A. Materials All analytical grade chemical reagents were purchased and used for this experiment without any further purification. Bismuthnitrate pentahydrate [Bi(NO 3)35H2O] (≥98%), iron nitrate nano- hydrate [Fe(NO 3)39H2O] (≥99%), yttrium nitrate hexahydrate [Y (NO 3)36H2O] (≥99%), and tartaric acid [C 4H6O6](≥98%) were procured from MERCK, India. Ultrapure water (Milli Q, resistivity >18.2 M Ωcm) was used for materials synthesis. B. Synthesis of BiFeO 3and Y-doped BiFeO 3 nanomaterial In a typical solgel synthesis method, bismuth nitrate pentahy- drate [Bi(NO 3)35H2O], iron nitrate nanohydrate [Fe(NO 3)39H2O], and yttrium nitrate hexahydrate [Y(NO 3)36H2O] were taken as the starting materials. For synthesizing pure BiFeO 3, equimolar amount (0.03M) of Bi(NO 3)35H2Oa n dF e ( N O 3)39H2O were mixed with 10 ml ultrapure water in two separate beakers and stirred for 30 min.Now to maintain the pH value of t he final solution to 2, 70% HNO 3 was added drop wise separately in two beakers. After that, the two sol- utions were mixed together and stirred for another 1 h. A tartaric acid solution was prepared by dissolving 0.06M in 30 ml water, and it wasused as a complexing agent. The as prepared tartaric acid solution wasthen added to the nitrate solution for better solubility of bismuth salt in the solution and to get more porous precursor powder. After adding the tartaric acid, the resulta nt solution was stirred for another1 h and then dried in a hot air oven at a constant temperature of 100 ° Cf o r6 –7 h. At the final stage, the dried precursor powder was grinded and then calcined at a temperature of 600 °C (873 K) for 2 hwith a heating rate of 400 K/h. Finally, a brown color BiFeO 3powder was obtained for further characterization. For the synthesis of yttrium the (Y) doped sample (Bi1−xYxFeO 3with x = 0.10), the calculated amount [0.65 g of Bi(NO 3)35H2O and 0.06 g of Y(NO 3)36H2O] of Y(NO 3)36H2O and Bi(NO 3)35H2O were taken for the total solution concentration of 0.03M and mixed in 10 ml ultrapure water and followed the same protocols as already discussed above for preparing the pure BFO sample to finally get Y-doped BiFeO 3(Y-BFO). Finally, gas-sensing measurements were carried out using BFO and Y-BFO pallets.A uniaxial pressing instrument (CARVER, Indiana, USA) was usedto compact the synthesized powder at a compaction pressure of 520 MPa for fabricating circular pallet specimens of 10 mm diameter and 7 mm thickness. C. Characterizations The phase formation of the synthesized BFO and Y-BFO was studied by analyzing the x-ray diffraction (XRD) pattern (X 0Pert PRO, PANalytical, UK) recorded using a Cu K α1 monochromatic radiation of 1.5406 Å wavelength. All the powder diffraction datawere collected applying a tube voltage of 40 kV with a tube currentof 30 mA. A field emission scanning electron microscope (FESEM)(Evo 60, Carl Zeiss, Germany) was used to investigate the surface morphology of the prepared samples. The identification of micro- structure was done by a transmission electron microscope (TEM)(GSM-2010, JOL, Tokyo, Japan). X-ray photoelectron spectroscopy(XPS) analysis was carried out utilizing a PHI 5000 Versa probe IIscanning XPS microprobe (ULVAC-PHI, USA). The gas-sensing measurements were carried out using an indigenously developed gas-sensing measurement system shown in Fig. S1 in thesupplementary material . A precession electrometer unit (2612 A, Keithley Instruments, USA) furnished with a data assembly systemwas used for I–V(current –voltage) measurement. In primary stage, the transient current was measured for a fixed bias voltage (5 V) at different operating temperatures through periodically exposing thesensor element in air and target VOC vapor of various concentra-tions (parts per million, ppm). A precession temperature controller connected to the heating element has been utilized to control the sensing temperature. The required concentration of VOCs in theTABLE II. Effect of Y-doping on structure and morphology of BiFeO 3. Sample Structural results Morphological results Reference Bi1−xYxFeO 3(x = 0, 0.18) Reduction of rhombohedral (R3c) phase and increase of orthorhombic (Pbnm) phaseReduction in particle size 36 Bi1−xYxFeO 3(x = 0, 0.05 –0.2) Rhombohedrally distorted perovskite (R3c); additional phase Y3Fe5O12appears at x ≥0.05Grain size gradually decreases 37 Bi1−xYxFeO 3(x = 0.1 –0.3) Structural distortion and decrease in tolerance factor Particle size reduces 38 BY10FO thin films Only structural distortion within the R3c phase Particle size ranges from 100 to 140 nm39 Y-doped BiFeO 3(x = 0 –0.2) Decrease in lattice parameters Grain size reduced 40Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 144501 (2020); doi: 10.1063/5.0023131 128, 144501-3 Published under license by AIP Publishing.form of liquids was injected by a microliter syringe with a volume precision of <2.5% into the sensing chamber. The details for calcu- lating VOC concentration in ppm is shown in the supplementary material . About 26% of relative humidity was maintained through- out the sensing process. Finally, the response to a particular VOCexpressed in terms of the change in conductance of the sensing element by the equation R¼ΔG/G a, where ΔG(¼Gg/differenceGa), implies the change in conductance of the sensing material when itis exposed to the VOC from air. For the p-type sensing behavior,ΔG¼(G a/C0Gg). Again, Ggis the conductance measured in the presence of the gas/vapor being sensed and Gais the conductance measured in air (i.e., in the absence of the vapor being tested).III. RESULTS AND DISCUSSION A. Structural and morphological study The x-ray diffraction (XRD) results for BiFeO 3(BFO) and Y-doped BiFeO 3(Y-BFO) as shown in Fig. 1 implies a single phase formation of the respective perovskites. The observed diffractionpeaks reveal a rhombohedral structure with the R3cspace group for both the BFO and Y-BFO samples. The inset in Fig. 1(b) compares the XRD pattern of BFO and Y-BFO considering (104) and (110) peaks. It can be clearly identified that in the Y-doped sample split-ting of (104) and (110) peaks (very close to 2 θ= 32°) almost disap- pears indicating a reduction of the rhombohedral phase and FIG. 1. Rietveld refinement analysis spectra with the obtained crystallographic structure of unit cells for (a) BFO and (b) Y-BFO perovskite. Inset in (b) sh ows the compari- son of (104) and (110) peaks of BFO and Y-BFO.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 144501 (2020); doi: 10.1063/5.0023131 128, 144501-4 Published under license by AIP Publishing.increase of the orthorhombic phase followed by a possibility of structural phase transformation of BFO from rhombohedral towardorthorhombic. A similar phenomenon of phase transformation dueto the introduction of rare earth elements in BFO had beenobserved in different studies. 33–35Furthermore, the peaks (104) and (110) shift toward higher 2 θvalues in Y-BFO implying a proper substitution of Bi3+with Y3+of smaller ionic radius.32,40For an in detailed crystallographic analysis of the as prepared materials,Rietveld refinements 41of XRD data were carried out by utilizing theR3cspace group. Figures 1(a) and1(b) show the Rietveld analy- sis spectrum along with the obtained crystallographic structure of the unit cell for BFO and Y-BFO, respectively. In the perovskite(ABO 3) unit cell, the corner positions (A-site) are occupied by Bi3+ (or Y3+in Y-BFO), Fe3+is in the body centered position (B-site), and all the face centered positions are being occupied by O2−ions. The as obtained crystallographic information for BFO and Y-BFO is shown in Table III . By reviewing Table III , it can be clearly seen that the unit cell volume of the Y-doped sample is smaller com-pared to the pure one (BFO) as indicated by the contracted R3c unit cell of Y-BFO. The fact can be justified by a smaller ionicradius of Y 3+ion (1.04 Å) as compared to the Bi3+ion (1.17 Å) considering the substitution ensued at the A-site of the perovskite. On the other hand, as per the crystallite size is concerned, Y-BFOshows a lower crystallite size of 20.6 nm as compared to BFO(36.2 nm) due to the incorporation of rare earth yttrium, whichimpedes the grain growth. The Y-doped BFO also shows higher lattice strain (0.00058 for Y-BFO and 0.00027 for BFO) due to the lattice distortion resulting from the proper doping by yttrium andchemical pressure induced by the dopants. Now, to quantify thestructural stability of a perovskite material, the tolerance factor ( t) can be calculated as 38,42 t¼(RAþRB)ffiffiffi 2p (RBþRO), (1) where RAis the mean ionic radius of the Bi3+and Y3+ions, ROis the ionic radius of the O2−ion, and RBis the mean ionic radius of the Fe3+and Fe4+ions. The calculated values of the tolerance factor (t) for both the samples are shown in Table III . The decrease in the tolerance factor ( t) for the Y-doped sample (0.88 for BFO and 0.85 in Y-BFO) implies an increase in the driving force responsible for octahedral rotation and a strong possibility of structural transi- tion.38,42The rare earth Y-doping can change the density of the charge carrier and modulate the electrical properties by substitutionof Bi 3+by Y3+ion. Thus, the inclusion of yttrium in BFO results a structural change of the material, and a rhombohedral –orthorhom- bic phase boundary may exist in the Y-doped sample (Y-BFO). The morphological characteristics of BFO and Y-BFO have been observed from the SEM image as shown in Figs. 2(a) and 2(b), respectively. It can be clearly seen from Fig. 2 that the grain size reduces significantly to 28 ± 3 nm for Y-doped BFO com- pared to undoped BFO to 57 ± 5 nm as already indicated by the XRD analysis. Suppression in grain growth is found due to theTABLE III. Structural and refinement parameters and tolerance factor ( t) for BFO and Y-BFO sensor. Sensor BFO Y-BFO a/Å 5.5769 5.5641 c/Å 13.8695 13.8239V/Å 3373.5736 370.6406 Crystallite size/nm 36.2 20.6 Lattice strain/10−42.7 5.8 GOF 1.5 1.2Tolerance factor ( t) 0.88 0.85 FIG. 2. FESEM images for (a) BFO and (b) Y-BFO perovskite.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 144501 (2020); doi: 10.1063/5.0023131 128, 144501-5 Published under license by AIP Publishing.inclusion of yttrium in BFO and also results in the densification of the material. The smaller grains provide more effective surface area which accelerates the sensing process enhancing the responsetoward VOCs. Moreover, the highly agglomerated morphology ofBFO may result in the formation of capsulated zones (closed pores)in the sample. On the other hand, a comparatively less agglomer- ated structure of Y-BFO lowers the possibility of the formation of closed pores and results in an open porous system, which is advan-tageous for gas-sensing applications. B. XPS study To identify the formation of pure and Y-doped BFO nanopar- ticles, x-ray photoelectron spectroscopy measurements were carried out as shown in Fig. 3 . To study the valence states of Bi, Fe, O, and Y,Fig. 3(a) outlines the full survey of BFO as well as Y-BFO nano- particles. Figures 3(b) –3(d) show a detailed study for the recorded XPS spectra. First, two Gaussian peaks at 156 eV and 161 eV were observed corresponding to the Bi 4f 7/2and Bi 4f 5/2states,43respec- tively, as shown in Fig. 3(b) . The peaks at 707.8 eV and 720.8 eV related to Fe 2p 3/2and Fe 2p 1/2, as shown in the inset in Fig. 3(b) , respectively, confirm the 3+ oxidation states resulting from spin – orbit coupling.44,45Apart from the characteristic XPS peaks for Bi (4f7/2and 4f 5/2), O 1s and Fe (2p 3/2and 2p 1/2), characteristic peaks corresponding to Y 3d 5/2and Y 3d 3/2can also be observed in the survey spectra of Y-BFO [ Fig. 3(c) ], confirming the desired doping. To understand the oxygen states of the prepared samples, the O 1s XPS spectra for both BFO and Y-BFO are displayed in Figs. 3(d) and 3(e), respectively. It can be seen from O 1s XPS spectra that an asymmetric peak appears very close to 528 eV,while an additional peak near 530 eV can be evidently observed inboth samples. These two observed peaks are allocated as low binding energy (LBE) corresponding to the O 1s binding energy of the BFO phase and higher binding energy (HBE) peak related tothe loss of oxygen in the samples. 22For the Y-BFO sample, the asymmetric nature of the O 1s curve increases as indicted by theincrease of the area corresponding to the HBE peak, which in turn confirms the enhanced oxygen vacancies in the sample. The forma- tion of oxygen vacancy in Y-BFO may be due to the substitution ofthe Bi 3+ion by Y3+at the A-site of the perovskite.22 C. Gas-sensing performance All the gas/VOC sensing characteristics of the as prepared sensors (BFO and Y-BFO) were recorded using circular pellets employing the DC two probes measurement technique. A fixed bias voltage of 5 V was applied to activate the sensing. Figure 4(a) shows the variation of irreversibility with the increasing Y-dopingconcentration. It can be clearly seen from Fig. 4(a) that, at 10 wt. %, the material exhibits a fully reversible character (for 200 ppm acetone) with the highest response value [ Fig. 4(b) ]. We have uti- lized the 10 wt. % Y-doped sample (Y-BFO) for further study. Toachieve the optimum operating temperature (highest response),sensing was performed at different temperature ranges (250 –450 °C). Figure 5(a) shows the temperature dependent response for both the sensing prototypes of BFO and Y-BFO. It can be seen that thehighest response ( R max) was achieved at 350 °C for 50 ppm acetone and considered as the optimum operating temperature ( Topt)f o r sensing. After that particular temperature of 350 °C, the response again starts to decrease. The fact of reducing the response beyond a par-ticular temperature can be enlightened by the adsorption/ FIG. 3. The XPS spectra for BFO and Y-BFO perovskite: (a) full survey spectra, (b) Bi 4f (inset shows Fe 2p), (c) Y 3d, (d) O 1s (for BFO), and (e) O 1s (for Y-BFO).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 144501 (2020); doi: 10.1063/5.0023131 128, 144501-6 Published under license by AIP Publishing.desorption phenomena when the desorption rate becomes domi- nant, which leads to the decrease in response of the sensor. Forhighest response, the adsorption rate becomes equal to the desorp-tion rate. On the other hand, this fact can also be explained by using the following equations in connection to the depletion region concept of semiconducting materials: 46 LD¼ffiffiffiffiffiffiffiffiffiffi ε0kT n0e2s , (2) R¼Δn n0/C2LD, (3) where LDis the width of depletion region, ε0is the static dielectric constant, kis the Boltzmann constant, Tis the absolute tempera- ture, n0is the total concentration of carriers, eis the charge of each carrier, and Δnis the change in carrier concentration. In a semi- conducting material, the charge carrier concentration ( n0) and hence the conductance increases with the rise in temperature. Fromthe above equations, it is obvious that beyond a particular tempera-ture, the response of a MOS sensor is reduced due to the reduction in the depletion region width. The ability to operate at a higher temperature region ( T> 300 °C) comes with the advantage of reducing the influence of air humidity on the gas-sensing proper-ties of the material. 47No response was observed at room tempera- ture for both the sensors. The minimum operating temperature was 200 °C. The prime outcome of Y-doping in BFO is the elimination of irreversibility in connection to the sensing process. In view of this,Figs. 5(b) and5(c) compare the dynamic response curves for BFO and Y-BFO originating from the exposure of acetone vapor in a concentration range of 1 –200 ppm. Both the response curves (forBFO and Y-BFO) initially reflect the common nature of enhancing the response with the enhancement of the respective vaporconcentration. But, all the response characteristics related to BFO[Fig. 5(b) ] undoubtedly show an irreversible-type sensing behavior irrespective of the vapor concentration. Furthermore, it is obvious that with the increase in acetone concentration, the degree of irre-versibility increases for the BFO sensor. On the other hand, con-cerning the acetone sensing, performed by Y-BFO, it can be clearlyvisible that all the response characteristics (from low to high con- centration) reveals a reversible-type detection without any remark- able baseline drift as shown by Fig. 5(c) . To provide supportive evidence regarding irreversible as well as reversible-type sensing, the variation of response time ( τ res) (time taken to reach 90% of the total signal change after exposing to respective vapor) with acetone concentration ( C) has been pre- sented in Fig. 5(d) . It is well known that, for reversible gas sensing, the response time ( τres) can be expressed as1,2 τres¼(1/k)/C2[K/(1þCK)], (4) where kis the forward rate constant and Kis the reversible rate constant for the reversible reaction where the incoming VOCs(reducing gases) react with adsorbed oxygen ions on the sensorsurface to finally form the reaction products [Eq. (12)]. In the lim- iting case, i.e., in the presence of low concentrations of test gases, the response time ( τ res) can be expressed as lim C!0τres¼limC!0(1/k)/C2[K/(1þCK)]¼K/k: (5) Based on this consideration, it can be quantified that for rever- sible sensing, the response time ( τres) should be independent of the gas being used for a constant operating temperature. FIG. 4. (a) Variation of irreversibility with Y-doping concentration. (b) Variation of sensor response with Y-doping concentration tested for 200 ppm acet one at 350 °C.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 144501 (2020); doi: 10.1063/5.0023131 128, 144501-7 Published under license by AIP Publishing.In the present study, the response time ( τres) was found to depend on the acetone concentration ( C) in the case of BFO oper- ating at a constant temperature of 350 °C, and it follows an expo- nential relation as τres¼τ0þAeBC, (6) where τresis the response time (in seconds); τ0,A, and Bare the constants; and Cis the acetone concentration (in ppm). The inset inFig. 5(d) shows a continuous decrease in the response time ( τres) with the increasing acetone concentration ( C) for BFO. It confirms the irreversible nature of the BFO sensor otherwise. On the otherhand, an independent nature of response time ( τres) confirms rever- sibility of Y-BFO, a necessary feature for a viable gas/VOC sensor. Moreover, the mean response time ( τres) decreases in the case of the Y-doped sample, which is desirable for the practical applica- tion of the material. A lowest response time ( τres) of about 25 s was achieved for 50 ppm acetone vapor (which is about 40 s for BFO)using a Y-BFO sensor. Figure 6(a) shows the selectivity of the reversible-type Y-BFO sensor, where a bunch of VOCs (acetone, ethanol, methanol,hexane, toluene, and xylene) were initially tested to check the selec-tive nature of the present sensors. The sensor Y-BFO was found tobe selective toward acetone vapor among other tested VOCs. Moreover, the cross response ( R cross=RAcetone /RVOC) values [inset FIG. 5. (a) T emperature dependent response for BFO and Y-BFO sensor toward 50 ppm acetone. Dynamic response curves for acetone (1 –200 ppm): (b) BFO sensor and (c) Y-BFO sensor. (d) Variation of response time ( τres) with acetone concentration (C) for BFO and Y-BFO sensor (inset shows the exponential decay of response time with acetone concentration for BFO sensor).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 144501 (2020); doi: 10.1063/5.0023131 128, 144501-8 Published under license by AIP Publishing.inFig. 6(a) ] in the range 2 < Rcross≤20 exceptionally confirms the selectivity of the prototype (Y-BFO) toward acetone vapor. Figure 6(b) compares the response toward acetone (1 –200 ppm) for BFO and Y-BFO sensors. Except the prime advantage of reversibledetection, the Y-doped sample (Y-BFO) that additionally providesan enhancement of response (1.5-fold for 200 ppm acetone) inacetone sensing may be due to the decrease in the crystallite size and the increase in the effective surface area of the sensing proto- type as already confirmed by XRD and morphological study. The increment in response is also in accordance with the results obtained from the XPS study where an enhancement of oxygen vacancy has been observed, which has a positive impact on the sensing properties. 22Figures 6(c) and6(d) show the fitting ofsensing response ( R) for BFO and Y-BFO toward different concen- trations (C) of acetone (1 –200 ppm) to the Freundlich isotherm equation with linear regression,2,27 logR¼logαþβlogC, (7) where Ris the response of the sensor, αis proportionality factor, β is the exponent, and Cis the concentration of acetone. It can be clearly observed from the linear fitted curves that an excellent line-arity has been obtained in the case of the Y-BFO sensor (Adj.R 2= 0.98) compared to BFO (Adj. R2= 0.90). Most interestingly, the BFO sensor follows a non-linear nature [the inset image of Fig. 6(c) ] more precisely (Adj. R2= 0.97) rather than a linear one FIG. 6. (a) Selectivity of Y-BFO sensor (inset shows the cross response values). (b) Variation of response with acetone concentration for BFO and Y-BFO senso r. (c) Linear fitting of sensor response with acetone concentration (1 –200 ppm) for BFO (inset shows nonlinear fit) and (d) Y-BFO sensor using the Freundlich isotherm equation.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 144501 (2020); doi: 10.1063/5.0023131 128, 144501-9 Published under license by AIP Publishing.confirming the irreversible-type sensing associated with it. It is worth mentioning that, a high degree of linearity associated with the fitted curve of Y-BFO once again confirms the ability of fullyreversible sensing of acetone vapor. As mentioned in Figs. 6(c) and6(d),βvalues for both BFO and Y-BFO sensors are 0.63 and 0.76, respectively, which are similar and consistent with the values reported in literature studies. 27 To describe the irreversible to reversible transition for the present study, it is necessary to investigate the recovery characteris-tics separately for the BFO/Y-BFO sensor. To fulfill the need,Figs. 7(a) and 7(b) show only the recovery part of the dynamic response curve subject to 200 ppm exposure of acetone for BFO and Y-BFO sensors, respectively. To understand the recovery kinet-ics, the respective curves have been fitted by using the Langmuiradsorption equation as 28 R(t)¼R0þR1[1/C0exp(/C0t/τrecov)]: (8) Calculating the recovery rate of the sensors, the rate of desorp- tion of the reaction product from the sensor surface duringrecovery process can be obtained by the time derivative of theequation, i.e., dR(t) dt¼R1 τrecovexp(/C0t/τrecov): (9) It was found that for Y-BFO sensors, the rate of desorption increases significantly (0.05 for BFO and 0.1 for Y-BFO), which ultimately results in a reversible-type sensing of acetone. The enhancement of the desorption rate may be attributed to thestrained structure of Y-BFO (contracted R3c), which accelerates therate of desorption of the reaction products from the active sites ofthe sensor and helps the signal to reach the baseline (reversible sensing). Along with reversible sensing, reproducibility/repeatability and long-term stability are the two unavoidable requirements for a par-ticular sensing prototype concerning its practical applicability. Tofulfill the mentioned requirements, Fig. 8(a) represents more pre- cisely the dynamic response curves showing the irreversible to reversible transition and the reversible-type sensing character of theY-BFO sensor. On the other hand, Fig. 8(b) delineates the long- term stability of the reversible Y-BFO sensor with negligible devia-tion in response values, i.e., R= 3.5 ± 0.25, 17 ± 0.16, and 52 ± 0.2 toward 5, 50, and 200 ppm acetone, respectively, tested repeatedly for 300 days. The inset in Fig. 8(b) shows nine consecutive loops originating from the periodic exposure of 5, 50, and 200 ppmacetone and air, and confirms the excellent repeatable character ofthe sensing prototype. D. Gas-sensing mechanism The exact mechanism of sensing a gas/vapor by a sensing element is still controversial, complex, and not completely under-stood. Although, in the present study, we are trying to demonstratethe plausible sensing mechanism behind the irreversible to reversi-ble transition by utilizing the well accepted adsorption –desorption process of VOCs by a p-type MOS followed by specific change in the structure and morphology occurring from Y-doping. Initially,in the presence of air, the oxygen molecules are absorbed on thesensing surface and converted to O −ions (at the operating temper- ature of >200 °C) by the electron capture mechanism in the con- duction band (CB). So, in the presence of air, a moderate conductivity is achieved by the sensing element (the baseline). Ahole accumulation layer (HAL) is formed in this process. 48The near-surface HAL is responsible for the conduction process in the FIG. 7. Exponential fitting of the recovery characteristics for (a) BFO and (b) Y-BFO sensor exposed to 200 ppm acetone vapor.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 144501 (2020); doi: 10.1063/5.0023131 128, 144501-10 Published under license by AIP Publishing.p-type material. Now, when the reducing gases/VOCs ( R) are intro- duced in the sensing chamber, they will react with the surface adsorbed O−ions and remove them from the sensing surface and hence unrestraint one electron associated with each O−ion. Thesefree electrons from the conduction band (CB) will recombine with the holes in the valance band (VB) and thus lowers the charge carrier concentration and hence the conductivity of a p-type mate-rial, which in turn decrease the HAL. 49The reaction products FIG. 8. (a) Irreversible to reversible transition using BFO and Y-BFO sensor. (b) Long-term stability and repeatability (inset) for Y-BFO sensor. FIG. 9. Possible sensing mechanism behind irreversible to reversible transition for (a) BFO and (b) Y-BFO sensor.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 144501 (2020); doi: 10.1063/5.0023131 128, 144501-11 Published under license by AIP Publishing.(mainly CO/CO 2and H 2O) are desorbed from the sensing grains when air is introduced again and considered the most important step of the sequential surface reactions occurring in the process ofsensing. 27This step determines the irreversible/reversible nature of the sensing prototype. The incomplete desorption of the reactionproducts leads to an irreversible-type sensing, whereas complete and fast desorption results in reversible and quick recovery, which depends on the structural and morphological properties of thematerial. So, after the re-introduction of air, a reversible sensor canreach the baseline (initial conductance) but the signal lags behindthe baseline value in the case of irreversible-type sensing. The sequential surface reactions of the sensing process are given as 50 O2(gas)!O2(ads), (10) O/C0 2(ads)þe/C0!2O/C0(ads), (11) Rþ2O/C0$RO 2þe/C0[Rate and type determining step], (12) e/C0þhþ!Null : (13) In the present study, we observe an irreversible-type sensing by BFO and reversible sensing by the Y-BFO sensor. The probablesensing mechanisms behind the observation are represented in Figs. 9(a) and9(b). As already indicated by the structural and mor- phological study, BFO exhibits a highly agglomerated morphologi-cal character with comparatively large crystallite size whereasY-BFO shows remarkably lower agglomeration with small crystal-lites. The highly agglomerated structure imparts a negative impact to the sensing properties of the material. A higher degree of agglomeration may form capsulated zones (closed porosity) inBFO, which is undesired for sensing applications. As shown byFig. 8(a) , the reaction products (mainly CO 2and H 2O) cannot completely desorb from the sensing grains as they are entrapped inside the closed pores of the material. As a result, although the air is flashed inside the chamber, the hole accumulation layer (HAL) isnot able to restore its initial condition, which finally gives theirreversible-type sensing. On the other side, for the Y-BFO sensoras the degree of agglomeration is lower with the small particle size, the formation of closed pores is hard to occur. The reaction prod- ucts can easily channelize through the open pores and completelydesorb from the sensing surface leaving the HAL in its initial state.In this way, the Y-BFO sensor due to the morphological advantagesprovides a reversible-type sensing as shown by Fig. 9(b) . In continuation to this, it is important to mention that the structural change and induced strain due to Y-doping acceleratethe desorption rate of the reaction products from the sensing grainsand are considered to be an effective reason of reversible-type sensing. So, the origin of irreversible to reversible transition in the detection process of acetone lies in the pathway of structural andmorphological changes of the material due to Y-doping. IV. CONCLUSIONS In summary, we have investigated the possible reason behind the irreversible to reversible transition while detecting acetonevapor by utilizing a non-conventional MOS type pure BFO and Y-doped BiFeO 3perovskite materials. While doped with yttrium (Y), the material (Y-BFO) exhibits an excellent reversible-type,highly selective acetone sensing with enhanced sensor performance.The irreversible nature of the pure BiFeO 3sensor was confirmed by the dependency of the response time with the tested vapor concen- tration and also by the non-linear preference in the variation of sensor response with increasing vapor concentration fitted usingthe Freundlich isotherm equation. The structural change occurringdue to the substitution of the Bi 3+ion by Y3+at the A-site of the perovskite is found to be favorable to accelerate the recovery process, which ultimately results in a reversible sensing. Along with the reversible character, the Y-doped BiFeO 3sensor reflects excel- lent repeatability and long-term stability, highly desirable for com-mercial applications of the material. Finally, the probable sensingmechanism associated with the sensors was described by correlat- ing the desorption of reaction products from the sensor surface with a structural/morphological change resulting from the properdoping of yttrium in BiFeO 3and, most importantly, the formation of closed and open pores inside the pure and Y-doped sensingmaterial, respectively. SUPPLEMENTARY MATERIAL See the supplementary material for calculations of ppm and Fig. S1 for the gas/VOC sensing setup for the detection of acetone. ACKNOWLEDGMENTS The authors acknowledge the funding received from SERB (DST), New Delhi, via Grant No. EMR/2017/000058. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1K. Mukherjee, A. P. S. Gaur, and S. B. Majumder, J. Phys. D Appl. Phys. 45, 505306 (2012). 2M. Sinha, R. Mahapatra, B. Mondal, T. Maruyama, and R. Ghosh, J. Phys. Chem. C 120, 3019 (2016). 3J. D. P. 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5.0017921.pdf

J. Appl. Phys. 128, 063907 (2020); https://doi.org/10.1063/5.0017921 128, 063907 © 2020 Author(s).Strong band-filling-dependence of the scattering lifetime in gated nanolayers induced by the opening of intervalley scattering channels Cite as: J. Appl. Phys. 128, 063907 (2020); https://doi.org/10.1063/5.0017921 Submitted: 10 June 2020 . Accepted: 25 July 2020 . Published Online: 13 August 2020 Davide Romanin , Thomas Brumme , Dario Daghero , Renato S. Gonnelli , and Erik Piatti COLLECTIONS Paper published as part of the special topic on 2D Quantum Materials: Magnetism and Superconductivity ARTICLES YOU MAY BE INTERESTED IN Magnetic-field modeling with surface currents. Part I. Physical and computational principles of bfieldtools Journal of Applied Physics 128, 063906 (2020); https://doi.org/10.1063/5.0016090 Engineering hematite/plasmonic nanoparticle interfaces for efficient photoelectrochemical water splitting Journal of Applied Physics 128, 063103 (2020); https://doi.org/10.1063/5.0015519 Magnetic field modeling with surface currents. Part II. Implementation and usage of bfieldtools Journal of Applied Physics 128, 063905 (2020); https://doi.org/10.1063/5.0016087Strong band-filling-dependence of the scattering lifetime in gated MoS 2nanolayers induced by the opening of intervalley scattering channels Cite as: J. Appl. Phys. 128, 063907 (2020); doi: 10.1063/5.0017921 View Online Export Citation CrossMar k Submitted: 10 June 2020 · Accepted: 25 July 2020 · Published Online: 13 August 2020 Davide Romanin,1 Thomas Brumme,2,3Dario Daghero,1 Renato S. Gonnelli,1,a) and Erik Piatti1,b) AFFILIATIONS 1Department of Applied Science and Technology, Politecnico di Torino, I-10129 Torino, Italy 2Wilhelm-Ostwald-Institut für Physikalische und Theoretische Chemie, Linnéstraße 2, 04103 Leipzig, Germany 3Theoretische Chemie, Technische Universität Dresden, Bergstraße 66c, 01062 Dresden, Germany Note: This paper is part of the Special Topic on 2D Quantum Materials: Magnetism and Superconductivity a)Author to whom correspondence should be addressed: renato.gonnelli@polito.it b)Electronic mail: erik.piatti@polito.it ABSTRACT Gated molybdenum disulfide (MoS 2) exhibits a rich phase diagram upon increasing electron doping, including a superconducting phase, a polaronic reconstruction of the band structure, and structural transitions away from the 2H polytype. The average time between twocharge-carrier scattering events —the scattering lifetime —is a key parameter to describe charge transport and obtain physical insight into the behavior of such a complex system. In this paper, we combine the solution of the Boltzmann transport equation (based on ab initio density- functional theory calculations of the electronic band structure) with the experimental results concerning the charge-carrier mobility in order to determine the scattering lifetime in gated MoS 2nanolayers as a function of electron doping and temperature. From these dependencies, we assess the major sources of charge-carrier scattering upon increasing band filling and discover two narrow ranges of electron dopingwhere the scattering lifetime is strongly suppressed. We identify the opening of additional intervalley scattering channels connecting thesimultaneously filled K =K 0and Q =Q0valleys in the Brillouin zone as the source of these reductions, which are triggered by the two Lifshitz transitions induced by the filling of the high-energy Q =Q0valleys upon increasing electron doping. Published under license by AIP Publishing. https://doi.org/10.1063/5.0017921 I. INTRODUCTION In the last decade, the ionic gating technique has become a fundamental tool for probing the ground-state properties of low- dimensional systems as a function of doping. Indeed, thanks to thefield-effect transistor (FET) architecture, it is possible to investigatethe rich phase diagrams of (quasi) two-dimensional (2D) materialsand surfaces in an almost continuous way. 1–20The transition metal dichalcogenides (TMDs) represent a notably tunable class of mate- rials thanks to the occurrence of both superconducting (SC) andcharge-density-wave (CDW) phases. 21,22Among them, molybde- num disulfide (2 H-MoS 2) has been the most studied both theoreti- cally and experimentally, owing to its stability at ambient pressure and temperature, the ease by which it can be exfoliated, its sizeable bandgap,23and the indirect-to-direct gap transition that it undergoeswhen thinned from the bulk to the single-layer,23–25which make it eminently suitable for electronic and optoelectronic applica- tions.23,26,27This layered semiconductor develops a SC phase with a maximum transition temperature Tc/difference11 K either via ion intercala- tion28,29or by electrostatic ion accumulation at the interface between the material and an electrolyte.2,30,31 When 2 H-MoS 2is electrostatically electron-doped in the FET configuration [ Fig. 1(a) ], the presence of the electric field along the direction orthogonal to the surface breaks inversion symmetry and leads to a Zeeman-like spin –orbit splitting of the conduction bands32,33in the Brillouin Zone (BZ). The conduction band minima lie at the inequivalent K =K0points (located at the corner of the hexagonal BZ) and Q =Q0points (which lie more or less halfway between K =K0and the center of the BZ Γ), as depicted in the insetJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063907 (2020); doi: 10.1063/5.0017921 128, 063907-1 Published under license by AIP Publishing.ofFig. 1(b) . The corresponding spin-split electron pockets are the so-called valleys common to all TMDs in the 2 Hcrystal structure34–38that become filled upon electron doping. As a conse- quence, the geometry of the Fermi surface (FS) of gated MoS 2 strongly depends on their occupation. Such valley filling is in turn strongly dependent on the number of layers, on the strength of the electric field, and on the tensile strain of the sample.34,35The Zeeman-like spin –orbit splitting is crucial in determining the proper- ties of the gate-induced SC state,33as it leads to the spin-valley locking of the Copper pairs39,40and the so-called 2D Ising SC and its ultrahigh out-of-plane critical magnetic field.39,40 The filling of the K =K0and Q =Q0valleys can be probed exper- imentally by means of electric transport measurements. Whenhigher-energy subbands are crossed by the Fermi level, characteris-tic kinks appear in the doping-dependence of the conductivity of ion-gated TMD nanolayers. 41,42This also allows directly probing the change in the topology of the Fermi surface, i.e., the occurrenceof Lifshitz transitions: At low doping, only the K =K0valleys are filled, giving rise to two electron pockets only, whereas as the elec- tron doping increases, the Q =Q0valleys become filled as well, gen- erating six new electron pockets.34,35,41Recent developments in density functional theory (DFT) allow computing the electronic34,35,43and vibrational44properties of materials in the FET configuration from first principles by fullytaking into account the presence of an orthogonal electric field in aself-consistent way. In such a way, it has been possible to obtainthe electronic structure of many gated TMDs, 34,35to study the flex- ural phonons in graphene,44to explore the anomalous screening of an electric field at the surface of niobium nitride,45and to predict a possible high- TcSC phase transition in diamond thin films.46,47 More specifically, in Ref. 56, we showed that DFT calculations can reliably reproduce the experimental doping-dependence both of the conductivity and of the valley filling in ion-gated MoS 2 nanolayers, when the presence of the transverse electric field, the number of layers, and the level of strain in the experimentalsamples are taken into account. However, our analysis provided noinformation on the charge-carrier scattering lifetime, which is a crucial physical quantity necessary to describe charge transport in the system. The scattering lifetime τis the average time between two successive scattering events experienced by a given chargecarrier, and it directly determines key parameters for both thephysics of the system and the device operation, such as, for example, the charge-carrier mobility, the mean free path, and the degree of metallicity of the system. In this work, we tackle this issue directly byfollowing the approach introduced in Ref. 35: We start by computing theab initio band structure of gated 4L /C0MoS 2and subsequently combine the Hall mobility-to-lifetime ratio, obtained by solving the Boltzmann transport equation,54with the Hall mobility calculated from the doping-dependence of the conductivity reported inRefs. 41and56. We find that when the Q =Q0valleys are filled by the increasing field-induced electron doping, i.e., when the Lifshitz transitions occur, the scattering lifetime undergoes a strong reduction. We show that this observation can in turn belinked to the opening of new intervalley scattering channelsbetween the simultaneously filled K =K 0and Q =Q0electron pockets. We discuss how this phenomenon can strongly affect key properties of gated MoS 2reported in the literature, such as the electron –phonon coupling, the gate-induced SC state, the polar- onic reconstruction of the K =K0Fermi sea, and the low- temperature incipient localization often observed in real devices. II. METHODS A. Computational details In order to precisely match the experimental conditions of Ref. 41, we considered a four-layer MoS 2crystal (4L /C0MoS 2), set the in-plane lattice parameter to the experimental bulk value, and added 0 :13% tensile strain.56We then performed the DFT calculations using the plane-wave pseudopotential method asimplemented in Quantum ESPRESSO. 51,52We made use of fully relativistic projector-augmented pseudopotentials57and of the Perdew –Burke –Ernzerhof exchange-correlation functional58includ- ing van der Waals dispersion corrections.59We set the energy cutoff for the wave functions to 50 Ry and that for the chargedensity to 410 Ry. We performed the Brillouin zone integrationusing a Monkhorst –Pack grid 60of 32/C232/C21kpoints with a Gaussian broadening of 2 mRy and set the self-consistency con- ditions for the solution of the Kohn –Sham equations to 10/C09Ry FIG. 1. (a) Schematic view of the four-layer molybdenum disulphide (4L-MoS 2 crystal in the FET configuration. Yellow spheres are S atoms and purple –gray spheres are Mo atoms. The positive ions accumulated at the electrolyte –sample interface are represented by red spheres. The negative (positive) inducedcharge at the surface of the 4L /C0MoS 2crystal is depicted with blue (red) clouds around atoms using isosurfaces at 1 =15th of the maximum charge density at an electron doping level n2D¼7/C21013cm/C02. The gate electric field ~Eis depicted as violet arrows. (b) Electronic band structure and density of states (DOS) of gated 4L /C0MoS 2at an electron doping level n2D¼7/C21013cm/C02. The gray line represents the Fermi energy EF. The inset shows the first Brillouin zone of 2 H-MoS 2where the Γ,K=K0, and M high- symmetry points are highlighted. Band edges Q 1and Q 2of the spin –orbit split subbands at the Q =Q0points are highlighted by red-dashed lines.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063907 (2020); doi: 10.1063/5.0017921 128, 063907-2 Published under license by AIP Publishing.for the total energy and to 10/C04Ry=Bohr for the total force acting on the atoms during the structure relaxation. After con- vergence of the ground-state density, we then performed anadditional non-self-consistent calculation on a denser grid of64/C264/C21kpoints that will be used later for an accurate solu- tion of the Boltzmann equation. We modeled the FET setup using the method described in Refs. 34,43, and 45, where a dipole correction is employed in order to get rid of spurious Coulomb interactions along the non-periodicdirection due to repeated images of the system under study. Weplaced the dipole for the dipole correction at z dip¼ddip=2 with ddip¼0:01L,Lbeing the size of the unit cell in the z-direction, and the charged plane mimicking the gate electrode slightly closerto the MoS 2crystal at zmono¼0:011L. A potential barrier of height V0¼2 Ry and width db¼0:1Lis placed between the gate and the MoS 2crystal in order to prevent charge spilling. To avoid unphysi- cal interactions between repeated images of the system due to the periodic boundary conditions, /difference30/differenceA/C14of vacuum were added to the supercell along the zdirection. The Boltzmann transport equation is solved in the constant- relaxation-time approximation, i.e., τi,k¼τ(EF), as implemented in the BoltzTraP54code starting from the eigenvalues of the Kohn – Sham hamiltonian obtained after the non-self-consistent computa-tion. The ratio between the number of plane waves and the numberof band energies is set to 5. In order to solve the integrals for the computation of transport tensors, we took into account bands that fall into an energy window of 0 :04 Ry around the Fermi energy. B. Solution of the Boltzmann equation For 2D systems, the conductivity tensors σ αβand σαβγat the temperature Tand the chemical potential EFare34,35 σαβ(T,EF)¼e2 4π2X ið τi,kvi,k αvi,k β/C0@fEF(T,εi,k) @ε/C20/C21 dk, (1) σαβγ(T,EF)¼e3 4π2X ið τ2 i,kϵγδρvi,k αvi,k ρ(Mi,k βδ)/C01/C0@fEF(T,εi,k) @ε/C20/C21 dk, (2) where eis the elementary charge, /C22his the reduced Planck constant, εi,kis the energy of the ith band at the momentum k¼(kx,ky), ϵαβγis the Levi –Cività symbol, vi,k α¼/C22h/C01@εi,k=@kαis the group velocity along the αth k-component, ( Mi,k αβ)/C01¼/C22h/C02@2εi,k=@kα@kβ is the inverse mass tensor for the αth and βth k-components, and fEF(T,ε) is the Fermi distribution function. Notice that for a general 3D system, all of the indices { α,β,γ,δ,ρ} are run over by all of the Cartesian coordinates { x,y,z}; however, in a 2D system, γ¼zand { α,β,δ,ρ} are limited to the in-plane coordinates { x,y}. Thanks to the conductivity tensors, it is possible to compute the Hall tensor as Rijk¼(σαj)/C01σαβk(σiβ)/C01: (3) While the relaxation time τi,kcan be both band- and momentum-dependent, in the often-used constant -relaxation-timeapproximation, one sets τi,k¼τ(EF)¼τ(where EFis the Fermi level). In this approximation, both σαβ=τand the Hall tensor are independent of τand can be directly computed with BoltzTraP54 from the ab initio band structure obtained on a fine kmesh. Thus, the theoretical value of the in-plane Hall mobility-to-lifetime ratiocan be computed simply as 35 μth H τ(T,EF)¼σxx τ(T,EF)/C0Rxyz(T,EF)/C2/C3 , (4) where σxx=τis the in-plane conductivity-to-lifetime ratio and Rxyz is the Hall coefficient, i.e., the component of the Hall tensor with the induced electric field along y, the current flowing along x, and the magnetic field applied along z[see Fig. 1(a) ]. Here, we have made use of the fact that the conductivity tensor σαβof crystals with hexagonal symmetry, such as MoS 2, has only two independent components (in-plane σxxand out-of-plane σzz).34,35,55To allow for a more reliable comparison with the experimental results, from Rxyz, we also directly determine the Hall carrier density nH¼/C01=eRxyzsince in principle in TMDs, nHis known to some- times strongly deviate from the actual doping charge n2D.34,35 Finally, we also compute the thermally smeared 2D density of states (DOS) as DOS( T,E)¼e2 4π2X ið /C0@fEF(T,εi,k) @ε/C20/C21 dk: (5) C. Determination of the scattering lifetime We determine the scattering lifetime by means of the approach originally developed in Ref. 35, where it was applied to gated WS 2, and that has later been successfully applied to other gated materials such as few-layer graphene48,49and epitaxial diamond films.20Specifically, once the dependence of μth H=τas a function of nHis known, the scattering lifetime τin a gated device can be easily obtained from the experimental values of the Hall mobility. Here, we directly calculate it as μexp H¼σxx enH(6) from the values of σxxand nHwe experimentally measured in Ref.41and summarized in Ref. 56. The scattering lifetime can then be recovered by τ¼μexp H μth H=τ(7) for any value of nHand Tfor which both the experimental Hall mobility and the theoretical mobility-to-lifetime ratio have been determined.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063907 (2020); doi: 10.1063/5.0017921 128, 063907-3 Published under license by AIP Publishing.III. RESULTS AND DISCUSSION A. Density of states and transport coefficients We first consider the effect of the band filling upon increasing electron doping on the electronic structure and transport coeffi- cients in 4L /C0MoS 2. The most profound impact can be observed in the doping-dependence of the density of states at the Fermi levelDOS( E F), which we plot in Fig. 2(a) for different values of T.A t T¼0 K, DOS( EF) exhibits the typical staircase behavior of a multi- band 2D system, increasing in a nearly step-like fashion wheneverthe Fermi level crosses the bottom of a subband and remainingnearly constant otherwise. By inspecting the electronic dispersionrelations for increasing values of n 2Dshown in Fig. 1(b) and in Ref.56, we can attribute the two sudden jumps in DOS( EF) around n2D/C251:5 and 7 /C21013cm/C02to the filling of the Q 1and Q 2 spin-split subbands, respectively. At finite T, the 2D-like behavior of DOS( EF) is quickly lost due to thermal smearing. Already at T¼10 K, only small “humps ”can be observed in the doping- dependence of DOS( EF) in correspondence with the crossing of the Q1and Q 2subbands. These humps disappear almost completely at T¼50 K, and at T¼100 and 200 K, the doping-dependence of DOS( EF) is fully smooth. This strong influence of a finite Ton DOS( EF) can be directly attributed to the small spin –orbit splitting Δsoof a few meV between the subbands: In the K =K0valleys, Δsois doping-independent and equal to about 3 meV, whereas in theQ=Q0valleys, it slowly increases with doping. In particular, in the doping range before the crossing of Q 2,Δso&10 meV in the Q=Q0valleys.56 The doping dependencies of the transport coefficients σxx=τ and Rxyzare much less affected by both the band filling and T. Upon increasing T,σxx=τslightly decreases in the entire doping range. For any value of T,σxx=τsmoothly increases with increasing n2D[Fig. 2(b) ], and the only effect of band filling is to progressively reduce the power-law exponent of the increase (from σxx=τ/n0:99 2D for n2D&1:5/C21013cm/C02to σxx=τ/n0:76 2D for n2D*7/C21013cm/C02).Rxyz, on the other hand, is found to be almost T-independent and smoothly decreases as n/C01 2Din the entire doping range [ Fig. 2(c) ]. Furthermore, the Hall carrier density nH¼ /C01=eRxyzis always almost identical to the doping charge n2D,a s s h o w ni nt h ei n s e to f Fig. 2(c) . This is consistent with what was r e p o r t e di nt h ec a s eo fg a t e d1 L - ,2 L - ,a n d3 L /C0MoS 234,35and is due to the good parabolicity of all subbands in both the K =K0and Q =Q0 valleys and their comparable effective masses at any doping n2D&2/C21014cm/C02. B. Mobility and scattering lifetime We now turn to the determination of the doping-dependent scattering lifetime. In Fig. 3(a) , we show the theoretical mobility-to-scattering lifetime ratio as a function of the Hall carrierdensity, determined with Eq. (4)from the data shown in Fig. 2 . While the dependencies of σ xx=τandRxyzonnHchange little upon increasing T, the dependencies of μth H=τare significantly affected instead. At low T/C2050 K, μth H=τmonotonically decreases at the increase of nHand the effects of band filling are negligible. Conversely, at intermediate and high T/C21100 K, the nH-dependence ofμth H=τbecomes non-monotonic and dependent on band-filling. At lownH&1:7/C21013cm/C02, where only the K =K0valleys are filled, μth H=τincreases with increasing nHand is strongly suppressed by increasing T.A tl a r g e r nH*2/C21013cm/C02,w h e r ea l s ot h eQ =Q0 valleys become filled, μth H=τdecreases with increasing nHand is much less sensitive to the Tincrease. On the experimental side, in Fig. 3(b) , we show the doping- dependence of the Hall mobility of ion-gated 4L /C0MoS 2directly calculated from the transport data we measured in Ref. 41.μexp His starkly dependent on both band filling and temperature. Before dis-cussing them, we note that these dependencies are much stronger than those exhibited by μ th H=τ. As a direct consequence, the behav- ior of the scattering lifetime τdetermined using Eq. (7)as a FIG. 2. (a) Density of states at the Fermi level DOS( EF) as a function of the doping charge density n2Dfor different values of T. Curves at finite Tare rigidly shifted by 0 :5e V/C01spin/C01cell/C01for clarity. (b) In-plane conductivity-to-lifetime ratioσxx=τand (c) the Hall coefficient Rxyzas a function of n2Dcomputed with BoltzTraP54for different values of T. The inset shows the n2D-dependence of the ratio of the Hall carrier density nH¼/C0 1=eRxyzandn2D. Solid lines are guides to the eye. Black dashed lines in (b) highlight the power-law scaling atlow and high n 2D.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063907 (2020); doi: 10.1063/5.0017921 128, 063907-4 Published under license by AIP Publishing.function of nHandT[shown in Fig. 3(c) ] is completely dominated by that of μexp H. Since μexp Hand τshare the same dependencies, in the following, we focus on discussing the behavior of τ. This behav- ior is non-trivial and can be separated in three main doping ranges.The first range occurs at low doping before the crossing of Q 1 (nH&1/C21013cm/C02), where τincreases with increasing nHat any T. The second range occurs at intermediate doping between the crossings of Q 1and Q 2(2&nH&6/C21013cm/C02), where the behavior of τstrongly depends on T: It increases with nHat T¼10 K, is nearly independent of nHatT¼50 K, and decreases with nHatT¼100 and 200 K. The third range occurs at very large doping after the crossing of Q 2(nH*8/C21013cm/C02), where τ decreases with increasing nHat any T. Additionally, in the narrowdoping ranges corresponding to the Q 1and Q 2band crossings, τis starkly non-monotonic below 50 K and becomes smooth at higher T, mirroring the “kinks ”observed in the doping-dependence of the conductivity.35,41,56 C. Scattering mechanisms Let us first consider the three main doping ranges away from the band crossings, where the number of bands crossing the Fermilevel is constant and the electronic DOS is almost constant as well. In gated MoS 2, the mobility and scattering lifetime are dominated by four main sources of scattering:62(i) acoustic-phonon scattering, (ii) charged-impurity scattering, (iii) substrate-optical phonon scat-tering, and (iv) charged traps. In our case, the first two mechanismsare certainly the most important, if not the only ones. Indeed, in all three doping ranges, τdecreases with increasing T(except at the lowest measured value of n H≃7/C21012cm/C02), ruling out charged traps.62Substrate-optical phonon scattering can also be ruled out since it is weak for T&200 K62and is further suppressed in liquid- gated devices even close to room T.63 The acoustic-phonon scattering rate is expected to be doping-independent in each of the aforementioned doping ranges.Moreover, this scattering mechanism is negligible at very low T and increases with T. 62,63The scattering rate due to charged impu- rities, instead, is strongly doping-dependent at any Tsince it is strongly suppressed by the improved electrostatic screening upon increasing the carrier density.62In ion-gated devices, however, this scattering rate can also increase upon increasing doping due to the extrinsic scattering centers introduced by the ions in the electric double layer (EDL), leading to a competition.6,11,18,20,29,48–50,64–67 Furthermore, in MoS 2, the charged-impurity scattering rate can in general lead to a T-dependence of the scattering rate very similar to that due to acoustic-phonon scattering.62 AtT¼10 K, where the acoustic-phonon scattering is negligi- ble, the doping dependence of τcan be entirely ascribed to charged-impurity scattering. Its increase is thus due to the improvedelectrostatic screening; its decrease in the high-doping range beyondthe Q 2band crossing [see the last two blue points in Fig. 3(c) ]i s very likely to be due to the disorder introduced by the ions in the EDL.6,11,18,20,29,48–50,64–67These two mechanisms are certainly acting at any T, but at higher temperatures, the scattering from acoustic phonons suppresses τ, more and more effectively as Tincreases. The idea that phonon scattering (rather than charged-impurity scattering) is the main factor that determines the Tevolution of the curves is suggested by the fact that the suppression of τis approximately uniform in each doping range. At high T, when the thermal smear- ing makes the DOS be smoothly doping-dependent [see Fig. 2(a) ], the suppression is practically uniform for any nH*1/C21013cm/C02. Another proof that phonon scattering dominates at high Tis the fact that, in the intermediate doping range, τdecreases as a function of doping, while it should increase (as it does at low T) if the scattering was mainly due to charged impurities. The interplay of the differentscattering mechanisms is depicted schematically in Fig. 4 . D. Intervalley scattering and Lifshitz transitions We now consider the two narrow doping ranges corresponding to the Q 1and Q 2band crossings where the kinks in the doping- FIG. 3. (a) Theoretical Hall mobility-to-lifetime ratio μth H=τ, (b) experimental Hall mobility μexp H, and (c) scattering lifetime τas a function of the Hall carrier density nHat different temperatures T. Data in (b) are directly computed from the doping-dependent conductivity values we reported in Refs. 41and 56. Solid lines are guides to the eye. Vertical dashed lines highlight the different doping ranges as indicated in panel (a) and discussed in the main text.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063907 (2020); doi: 10.1063/5.0017921 128, 063907-5 Published under license by AIP Publishing.dependent conductivity and mobility are experimentally observed in Ref. 41. At a first approximation, the presence of the kinks in the conductivity and the mobility can be attributed to the strong reduc-tion in the average Fermi velocity, which occurs when the bottom ofa high-energy subband becomes filled. 35,56However, the reduction in the conductivity is entirely accounted for by the reduction in the Fermi velocity only when the scattering lifetime is exactly inverselyproportional to the density of states, τ/DOS( E F)/C01, for all values of doping.35,56Since at low Tin 2D systems DOS( EF) follows a stair- case behavior, similar kinks can be expected also in the doping- dependence of τ. Indeed, as we show in Fig. 3(c) , these kinks do appear in the doping-dependence of τin gated MoS 2devices and are similarly smeared out by increasing temperature. We now investigatewhether the kinks in τcan be simply explained in terms of the doping-dependence of DOS( E F). To do so, we focus our attention on the data at T¼10 K, where the kinks are most evident, and normalize the scatteringlifetime by its value at the lowest Hall density, τ(n H)=τ(nH,min) (blue squares in Fig. 5 ). As highlighted by the black dashed lines, theτat 10 K does indeed exhibit a “canted ”staircase dependence onnH, which is somewhat reminiscent of the DOS( EF) computed atT¼0 and shown in Fig. 2(a) . However, when τ/DOS( EF)/C01 is computed from the DOS( EF)a tT¼10 K [red circles in Fig. 5(a) ], it becomes apparent that this simple approximation fails to repro- duce most of the features of the scattering lifetime determined from the experimental mobility. τ/DOS( EF)/C01is obviously unable toreproduce any increase inτas a function of nHsince this stems from the doping-dependent charged-impurity scattering and not from theintrinsic DOS of gated MoS 2. The sudden drops in τassociated with the subband crossings (highlighted by the black arrows in Fig. 5 ) also cannot be reproduced satisfactorily by τ/DOS( EF)/C01.I nt h e case of the Q 1crossing at T¼10 K, the disagreement is limited: The τdetermined from the experimental mobility drops by a factor /difference4 upon this first Lifshitz transition, whereas τ/DOS( EF)/C01estimates a smaller drop of only a factor /difference2 at the same T.T h e r e f o r e ,t h e simple approximation correctly gauges the order-of-magnitude ofthe lifetime reduction but fails in accounting for nearly half the effectobserved experimentally. Most importantly, τ/DOS( E F)/C01predicts that almost no drop inτshould be observed upon crossing Q 2at T¼10 K, in clear contrast with the τdetermined from the experi- mental mobility. This finding is consistent with our results inRef.56, where the intensity of the kink in the conductivity at Q 2was severely underestimated in a model based on τ/DOS( EF)/C01. Therefore, another mechanism must be responsible for the large drop in τobserved upon the second Lifshitz transition occurring due to the crossing of Q 2. On top of increasing the DOS, filling high-energy bands can strongly alter the scattering lifetime by opening previously forbid- den interband scattering channels, thereby strongly increasing the scattering rate.68Indeed, the kinks in the doping-dependence of the conductivity of ion-gated few-layer graphene were explicitly attrib-uted to the activation of interband scattering by the filling of high-energy bands. 48,49,69,70In gated MoS 2, the evolution of the Fermi surface upon electron doping leads to the simultaneous filling of the low-energy K =K0valleys and the high-energy Q =Q0valleys.41,56 Therefore, we attribute the strong reductions in the scattering life- time to the opening of those intervalley scattering channels that are forbidden when only the low-energy K =K0valleys are populated. These include scattering channels connecting the electron pockets FIG. 4. Schematic explanation of the effects of different scattering mechanisms on the behavior of τvsnHandT. The doping ranges corresponding to the Q 1 and Q 2Lifshitz transitions have been excluded from the analysis so as to focus the attention on the regions where the number of bands crossing the Fermi level is constant. At low T, two competing mechanisms (improved screening and induced disorder), both ascribed to charged impurities, determine the trendofτ, the latter being dominant at high doping. At higher T, acoustic-phonon scattering comes into play and determines a decrease in τ, with a different mag- nitude in each of the three doping regions separated by the Lifshitz transitions.The DOS smearing further changes the shape of the τ(n H) curve. The same arguments apply also to the trends of μexp H. FIG. 5. Scattering lifetime normalized by its value at the lowest doping, τ(nH)=τ(nH,min), as a function of the Hall density nHatT¼10 K. Blue squares and red circles are obtained via Eq. (7) and the simple approximation τ/DOS( EF)/C01, respectively. Black arrows highlight the drops in the scattering lifetime associated with the Q 1and Q 2band crossings. Dashed black lines are guides to the eye.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063907 (2020); doi: 10.1063/5.0017921 128, 063907-6 Published under license by AIP Publishing.at Q$Q0,Q$K, Q0$K0,Q$K0, and Q0$K. It is very important to note that the opening of these intervalley scattering channels has a profound influence not only on the low- Tscattering lifetime and mobility, but on several other key properties ofgated MoS 2. Specifically, the availability of these intervalley scattering chan- nels is paramount in optimizing the nesting efficiency of the Fermi surface,61,71thereby allowing to strongly enhance the electron – phonon coupling (EPC) in the system.41,53,72–76This in turn leads to significant changes in the vibrational spectrum,53,72,74such as the pronounced doping-dependent phonon softenings, which have been observed in ion-gated MoS 2and other semiconducting TMDs by means of Raman spectroscopy.74,78In this context, the large suppres- sion of the scattering lifetime at the crossing of the Q 2subband points to a dominant role of this second Lifshitz transition in theopening of intervalley scattering channels and associated strong boost to the EPC, with respect to the milder effect of the first Lifshitz transition induced by the crossing of Q 1.T h i si sc o n s i s t e n t with the stronger Fermi surface nesting associated with the simulta-neous filling of all the available subbands in both the K =K0and Q=Q0valleys.41,75Moreover, both the sharp increasing part of the SC dome of gated MoS 241,72,73—which develops as a function of doping from a quantum-critical point in the same doping range where theQ 2Lifshitz transition is observed2,39,41,73,79—and the polaronic reconstruction of the Fermi sea in the K =K0valleys revealed by high- resolution angle-resolved photoemission spectroscopy77,80have been explicitly attributed to the strong increase in the EPC induced by theLifshitz transition, which allows the opening of additional intervalleyscattering channels. Upon further increasing the electron doping, therelated Fermi surface nesting has also been predicted to become so efficient as to destabilize the 2 Hcrystal structure of pristine MoS 2,75 thus potentially triggering the onset of a charge-density wave67,81 and/or a structural transition toward the 1 T=1T0polytypes67,82–85 and thus suppressing the SC state. E. Intervalley scattering and carrier localization Finally, we show that the strong suppressions of the scattering lifetime in correspondence with the Q 1and Q 2Lifshitz transitions may help in explaining another puzzling feature often observed inthe two associated doping ranges in gated MoS 2. Specifically, when the kinks due to the subband crossings are observed in the doping- dependence of the conductivity, for the same doping levels, theT-dependence of the resistivity often exhibits a slight upturn at a very low T. 39,41,86Reference 86attributed this behavior purely to the carrier localization effect due to trap states introduced by the ions in the EDL. Our results here paint a more complex picture. While the gate-induced extra scattering centers do play a significantrole in determining the scattering rate and the mobility, the largestsuppressions of τat a low Tarise from the opening of the interval- ley scattering channels [see Figs. 3(c) and5]. These suppressions in τcould indeed lead to carrier localization by bringing the system closer to the insulator-to-metal transition (IMT). Following our approach in Ref. 20, we address this issue quanti- tatively by calculating the Ioffe –Regel parameter x¼(E F/C0Ec)τ=/C22h as a function of nHfrom our band structure calculations for 4L/C0MoS 2(Fig. 6 ). Here, EF/C0Ecis the chemical potentialmeasured from the bottom of the conduction band Ec. According to the Mott –Ioffe–Regel criterion,87the Ioffe –Regel parameter charac- terizes the IMT in disordered systems in terms of how close themean free path is to the lattice periodicity. When x/C291, the mean free path is much larger than the lattice periodicity, leading to good metallic behavior. The opposite limit x/C281 suggests that the system is approaching the strong localization regime. The condition x/difference1 plays the role of a conventional crossover between the two regimes.AtT¼100 K —where the kinks are smeared out and no resistance upturn is experimentally observed —the Ioffe-Regel parameter increases smoothly in the whole doping range. The increase is veryfast at low doping, as the gate-induced 2D electron gas (2DEG)rapidly becomes more metallic due to the filling of the K =K 0valleys, while it is almost constant at intermediate and high doping, likely due to the scattering lifetime being limited by electron –phonon scat- tering. While the 2DEG never becomes a “good ”metallic conductor (x/C2110) at high temperatures, it is nevertheless firmly in the metallic side of the IMT as attested by its conductivity and mobility increas- ing with decreasing T.A tT¼10 K, on the other hand, the doping- dependence of the Ioffe –Regel parameter becomes non-monotonic: In the two doping ranges associated with the Q 1and Q 2band cross- ings, the sudden increase in the intervalley scattering rate reduces τ and brings the 2DEG back closer to the IMT. In the doping range corresponding to the crossing of Q 1,t h e2 D E Gi s lessmetallic at 10 K than it is at 100 K, whereas this inversion is not observed in thedoping range corresponding to the crossing of Q 2. However, in the latter case, the reduction in metallicity is comparatively muchstronger and brings the 2DEG away from the “good metal ”regime reached immediately before the Lifshitz transition and back to a more localized regime. Both behaviors are consistent with apicture of incipient localization at low Tbut are not strong enough to trigger a re-entrant IMT as in the case of gated ReS 211 and WS 2,66thus allowing for superconductivity to develop unim- peded in the system. FIG. 6. Ioffe–Regel parameter ( EF/C0Ec)τ=/C22hvsnH,a t T¼10 and 100 K. Solid lines are guides to the eye. Doping ranges where the crossings of the Q 1 and Q 2subbands occur are highlighted.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 063907 (2020); doi: 10.1063/5.0017921 128, 063907-7 Published under license by AIP Publishing.IV. CONCLUSIONS In summary, we have performed ab initio density-functional theory calculations of the band structure of gated and strainedMoS 2nanolayers upon electron doping. We have employed the Boltzmann transport equation in the constant-relaxation-time approximation to calculate the theoretical mobility-to-scatteringlifetime ratio as a function of the Hall carrier density. By combin-ing it with the experimental data of the Hall mobility, we havedetermined the scattering lifetime in the system as a function of temperature and electron doping and have discussed its behavior in terms of the major sources of charge-carrier scattering uponincreasing band filling. We have shown that the scattering lifetimeis strongly reduced in correspondence with the two Lifshitz transi-tions induced by the filling of the high-energy Q =Q 0valleys upon electron doping owing to the opening of additional intervalley scat- tering channels, which become available only when both the K =K0 and Q =Q0valleys are simultaneously occupied. We have also dis- cussed how the opening of these intervalley scattering channels canstrongly increase the electron –phonon coupling, potentially trigger- ing the onset of the gate-induced superconducting state and of the polaronic reconstruction of the Fermi sea, as well as leading to alow-temperature incipient localization as reported in the literature. ACKNOWLEDGMENTS We acknowledge funding from the MIUR PRIN-2017 program (Grant No. 2017Z8TS5B —“Tuning and understanding Quantum phases in 2D materials –Quantum2D ”). 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5.0010773.pdf

J. Appl. Phys. 128, 053903 (2020); https://doi.org/10.1063/5.0010773 128, 053903 © 2020 Author(s).Magnetic impurities in thin films and 2D Ising superconductors Cite as: J. Appl. Phys. 128, 053903 (2020); https://doi.org/10.1063/5.0010773 Submitted: 14 April 2020 . Accepted: 16 July 2020 . Published Online: 04 August 2020 David Möckli , Menashe Haim , and Maxim Khodas COLLECTIONS Paper published as part of the special topic on 2D Quantum Materials: Magnetism and Superconductivity ARTICLES YOU MAY BE INTERESTED IN Electrical readout of the antiferromagnetic state of IrMn through anomalous Hall effect Journal of Applied Physics 128, 053904 (2020); https://doi.org/10.1063/5.0009553 Current driven chiral domain wall motions in synthetic antiferromagnets with Co/Rh/Co Journal of Applied Physics 128, 053902 (2020); https://doi.org/10.1063/5.0012453 A practical guide for crystal growth of van der Waals layered materials Journal of Applied Physics 128, 051101 (2020); https://doi.org/10.1063/5.0015971Magnetic impurities in thin films and 2D Ising superconductors Cite as: J. Appl. Phys. 128, 053903 (2020); doi: 10.1063/5.0010773 View Online Export Citation CrossMar k Submitted: 14 April 2020 · Accepted: 16 July 2020 · Published Online: 4 August 2020 David Möckli,a) Menashe Haim, and Maxim Khodas AFFILIATIONS The Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel Note: This paper is part of the Special Topic on 2D Quantum Materials: Magnetism and Superconductivity a)Author to whom correspondence should be addressed: d.mockli@gmail.com ABSTRACT In the theory of dilute magnetic impurities in superconductors, the effect of all impurity spin-components is expressed via a single magnetic scattering rate Γm. In a more realistic setting, magnetic impurities are anisotropic. In this case, the spatial randomness of three spin-compo- nents of impurities gives rise to generally different scattering rates Γi(i¼1, 2, 3). We explore the effects of anisotropic magnetic impurities on the in-plane critical field in 2D superconductors. We discuss singlet, triplet, and parity-mixed order parameters allowed in systemswithout the inversion center. Also, the addition of a small amount of magnetic impurities may cause singlet to triplet crossovers. In allcases, different components of impurity spin affect the magnetic field —temperature phase diagram differently. We show that anisotropy of the magnetic impurities can serve as a probe of unconventional triplet or parity-mixed superconductivity. Published under license by AIP Publishing. https://doi.org/10.1063/5.0010773 I. INTRODUCTION The study of the effect of non-magnetic and magnetic impuri- ties on superconductors has been of great importance, as it made the distinction between isotropic and anisotropic superconductivity possible. 1,2Non-magnetic (scalar) impurities do not affect the iso- tropic singlet s-wave order parameter. In stark contrast, a magnetic scattering is very efficient in destroying s-wave superconductivity. The non-magnetic and magnetic impurities are equally detrimental to the critical temperature Tcof anisotropic superconducting states3–5as summarized in Table I . The studies of magnetic impurities in superconductors date back to Bardeen –Cooper –Schrieffer (BCS) theory.6The traditional theories consider the magnetic exchange interaction σ/C1Sbetween the spin of the conduction electrons σand impurity spins S.1,7–10 When the impurities are dilute, their spins are randomly oriented and uncorrelated.11An averaging procedure over all magnetic impurity sites yields a single magnetic scattering rate Γm/differenceS2¼S(Sþ1), where Sis the total impurity spin.12In this case, all Si(i¼1, 2, 3) components contribute equally to Γm, S2 i¼S2=3. Within this model of magnetic impurities, the gapless superconductivity has been discussed.13,14 The above models ignore the orientation anisotropy of the moments of magnetic impurities. We argue that such an anisotropyis potentially relevant to the 2D superconductors that are of a current research interest.15–22T h em a g n e t i ci m p u r i t i e sl o c a t e da t or near the interface between a 2D superconductor and adjacent monolayers are often anisotropic due to the reduced spatialsymmetry. 23If the spin of such an impurity tends to point, e.g., out-of-plane, the contribution of the out-of-plane spin- component to the scattering domi nates the contribution of the in-plane components. Recently, superconducting devices were fabricated by exfoli- ating one- to few-layers of transition metal dichalcogenides.24 These systems often lack an inversion center. This implies that the electronic bands are split by spin –orbit coupling (SOC) which is an odd function of the electron momentum and hence, aniso-tropic. Besides, the lack of an inversion symmetry leads to aparity-mixed superconductivity with coexisting parity-even singletand parity-odd triplet order parameters. 25–27These 2D supercon- ductors are essentially spin anisot ropic, and it is in this situation, the out-of-plane and in-plane spin-components of impurities playa distinct role. The superconductivity in 2D materials based on transition metal dichalcogenides with horizontal mirror symmetry, referred to as Ising superconductors withstand in-plane magnetic fields farbeyond the Pauli limit. 28–30Such a large field introduces aJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 053903 (2020); doi: 10.1063/5.0010773 128, 053903-1 Published under license by AIP Publishing.significant in-plane spin anisotropy. We demonstrate that this makes the two in-plane spin-components of impurities to beinequivalent in the way they affect the superconductivity. To study the effect of the anisotropy of impurity spins, we assume for clarity that different impurity spin-components are stat-istically independent. In result, the total scattering rate due to mag- netic impurities is Γ m¼Γ1þΓ2þΓ3, where Γi/S2 ioriginates from the spatial randomness of ith spin-component of impurities. We study the effect of the magnetic impurity anisotropy by consid-ering the scattering rates Γ ias independent parameters. In materi- als where σ/C1Sis the only spin-dependent interaction, the magnetic anisotropy of impurity spins is inconsequential. In this case, thetotal scattering rate Γ mcharacterizes the effectiveness of the mag- netic impurities in suppressing the superconductivity, see Table I . We demonstrate that the presence of SOC and in-plane magnetic field make the effect of all three spin-components of magnetic impurities distinct. Recently, the authors demonstrated that the combined action of an in-plane magnetic field and Ising-type SOC converts isotro-pic singlet s-wave Cooper pairs to equal-spin-triplet pairs. 31,32It is then natural to ask, if time-reversal symmetry breaking by a magnetic field can cause singlet to triplet conversion, could mag-netic impurities also cause conversion? We demonstrate thatalthough each impurity contribute a local exchange field, on average the effect of such a local field cancels out in the absence of a total net spin polarization of magnetic impurities, see Sec. VII for further details. The paper is organized as follows. In Sec. II, we introduce our model Hamiltonian. In Sec. III A , we assume that the Fermi energy is the largest energy scale and develop the quasi-classical theory of the superconducting state. We obtain the Eilenberger equation thatincludes the effects of arbitrary spin-fields and impurities. Next, inSec.III B , we specialize to the case with Ising-SOC and an in-plane Zeeman field and obtain the corresponding Eilenberger equation that describes the superconducting transition. In Sec. IV,w e address the case when there is pairing only in the singlet channel.We obtain the transition lines in the magnetic field –temperature phase diagram for both singlet ( s-wave, d-wave, etc.) and triplet (p-wave, f-wave, etc.) order parameters. In Sec. V, we address the case when there is pairing in the triplet channels only and discuss the role of the Cooper pair spin polarization. In Sec. VI, we con- sider the case when the singlet s-wave pairing channel is dominant, but a sub-dominant triplet channel exists. Then, the increase of magnetic impurities can drive a crossover from a pure singlet to a pure triplet state. In Sec. VII, we analyze the case when singlet andtriplet order parameters coexist. The joint presence of SOC and Zeeman fields selects a specific triplet component to couple to the s-wave singlet. We discuss how the impurities affect this coupling. In Sec. VIII, we explain our results in the context of the current lit- erature and conclude. In the Appendix , we derive the impurity self- energy in the self-consistent Born approximation. II. THE MODEL We employ a coordinate system such that the 2D supercon- ductor lies in the xy-plane. The Hamiltonian of a generic disor- dered non-centrosymmetric superconductor in the presence of a magnetic Zeeman field is 33 H¼X k,sξ(k)cy kscksþX k,ss0γ(k)/C0B ðÞ /C1 σss0cy kscks0 þ1 2X k,k0X {si}Vs1s2 s0 1s02k,k0ðÞ cy ks1cy /C0ks2c/C0k0s0 2ck0s01 þ1 2X k,k0X ss0Uss0(k/C0k0)cy ksck0s0: (1) Here, ξ(k)¼ξ(/C0k) is the symmetric part of the normal state dis- persion counted from the chemical potential. The vector of Paulimatrices is denoted by σ¼(σ 1,σ2,σ3) with subscripts denoting the spatial directions ( x,y,z). The anti-symmetric part γ(k)¼ /C0γ(/C0k) is the spin –orbit coupling (SOC) that arises in crystals lacking an inversion center. We indicate the Fermi surface average of the SOC as hjγ(k)j2iFS¼Δ2. Here, the magnetic field Babsorbs the prefactor gμB=2 with the g-factor and the Bohr magneton and has dimension of energy. The Zeeman field breaks the time-reversal symmetry of the Hamiltonian. The superconducting pairing interaction includes the singlet and triplet pairing channels allowed by crystal symmetry, Vs1s2 s0 1s02k,k0ðÞ ¼X Γ,jvs,Γ^ψΓj(k)iσ2hi s1s2^ψΓj(k0)iσ2hi* s0 1s02 þX Γ,jvt,Γ^dΓj(k)/C1σiσ2hi s1s2^dΓj(k0)/C1σiσ2hi* s0 1s02:(2) Γlabels a particular point group irreducible representation. ^ψΓj(k) is an even basis function of Γand ^dΓj(k) are the odd basis func- tions. The vs(t),Γ,0 denote the attractive pairing interactions for each pairing channel. Later, we will associate a superconducting critical temperature to each channel. We consider scalar and magnetic impurities distributed ran- domly and independently in the system. The potential produced bya scalar and magnetic impurities are given by U 0(k/C0k0)δss0and J(k/C0k0)S/C1σss0, respectively. We assume that the spins of magnetic impurities are static (see Refs. 2and34) and uncorrelated at different sites. This assumption neglects the quantum effects that can arisefrom magnetic impurities such as Kondo screening. Furthermore, weconsider the distribution of impurities ’spins and the distribution of their spatial location as statistically independent. This allows us to reduce the averaging over the magnetic impurities to averaging overTABLE I. Summary of the effects of non-magnetic and magnetic impurities on iso- tropic (singlet) and anisotropic (singlet or triplet) superconducting states. αis the pair-breaking parameter that informs how strongly Tcis suppressed. The scalar (magnetic) scattering rate is denoted by Γ0(Γm). A generic order parameter is indi- cated by χ(k). Order parameter Condition Non-magnetic Magnetic Isotropic (singlet)P kχ(k)¼const : α=0 α=2Γm AnisotropicP kχ(k)¼0 α=Γ0+Γm α=Γ0+ΓmJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 053903 (2020); doi: 10.1063/5.0010773 128, 053903-2 Published under license by AIP Publishing.the spin orientations and averaging over the spatial locations done independently. III. THE EILENBERGER EQUATION A. General theory We use the Pauli matrices { ρi} (plus identity ρ0) to generate Nambu-space and { σi} for spin-space. After a mean-field decou- pling of the superconducting term in Hamiltonian (1), we write the Hamiltonian in 4 /C24 matrix form in the basis defined by the Kronecker product ρi/C10σj(ρiσjfor short).8The matrix of the impurity part is5,8,11 ^U(k/C0k0)¼U0(k/C0k0)ρ3σ0þJ(k/C0k0)S/C1α, (3) where, for brevity, we have combined the contributions of the stat- istically independent scalar and magnetic impurity potentials andα¼ρ 3σ1,ρ0σ2,ρ3σ3 ðÞ . For simplicity, we consider a short range scattering impurity potential such that the scattering amplitudes inEq. (3) are momentum independent, U 0(k/C0k0)¼U0and J(k/C0k0)¼J. In what follows, we keep track of the impurity spin- components S¼(S1,S2,S3) explicitly, which will allow us to iden- tify special effects associated to specific components Si. We derive the quasi-classical theory in the 4 /C24 Nambu-spin basis.32,35–37We assume that the Fermi energy EFis the largest energy scale compared to all other quantities. The procedure of deriving the quasi-classical Eilenberger equations is described indetail in Refs. 32and 37. The first step is to obtain the Gor ’kov equations from Eq. (1)that determine the 4 /C24 Gor ’kov Green ’s function ^Gk,ω n ðÞ , where ωn¼(2nþ1)πTare the fermionic Matsubara frequencies. Next, instead of solving the Gor ’kov equa- tions for ^Gk,ωn ðÞ , one introduces the simpler quasi-classical prop- agator that depends on the Fermi momentum kF(the assumption of large EFenters here), ^gkF,ωn ðÞ ¼ð1 /C01dξk πiρ3σ0^Gk,ωn ðÞ ¼g(kF,ωn) /C0if(kF,ωn) /C0if*(/C0kF,ωn)/C0g*(/C0kF,ωn),/C20/C21 , (4) where g(k;ωn)¼g0(k;ωn)σ0þg(k;ωn)/C1σ and f(k;ωn)¼ f0(k;ωn)σ0þf(k;ωn)/C1σ) ½/C138 iσ2. The new quasi-classical propagator ^gk,ωn ðÞ that substitutes the Gor ’kov Green ’s function ^Gk,ωn ðÞ is the central quantity of the quasi-classical theory. The quasi-classical propagator is found by solving the com- mutator Eilenberger equation, iωnρ0σ0/C0^Σ(ωn)/C0^S(k)/C0^V(k)/C0/C1 ρ3σ0,^g(k;ωn)/C2/C3 ¼0, (5) together with the normalization condition ^g2(k;ωn)¼ρ0σ0. The impurity self-energy ^Σ(ωn) is obtained within the self-consistent Born approximation (see the Appendix for details) and has the components ^Σ¼^Σ0þ^Σ1þ^Σ2þ^Σ3defined by ^Σ0(ωn)¼/C0 iΓ0ρ0σ0h^g(k;ωn)iFSρ3σ0; (6)^Σ1(ωn)¼/C0 iΓ1ρ0σ1h^g(k;ωn)iFSρ3σ1; (7) ^Σ2(ωn)¼/C0 iΓ2ρ3σ2h^g(k;ωn)iFSρ0σ2; (8) ^Σ3(ωn)¼/C0 iΓ3ρ0σ3h^g(k;ωn)iFSρ3σ3: (9) Here, h...iFSstands for the angular averaging over the directions of ^k,Ð2π 0dwk 2π(...). The scalar impurity scattering rate is Γ0¼πn0N0U2 0, where n0is the number of scalar impurities and N0 is the density of states per spin at the Fermi level. The magnetic scattering rate components are Γi¼πnmN0J2S2 i, where nmis the number of magnetic impurities. For an isotropic distribution of impurities ’spins, we have Γ1¼Γ2¼Γ3¼Γm=3 with Γm¼πnmN0J2S(Sþ1), where Sis the total impurity spin. In the more generic situation considered here, the scattering rates Γican be different. The spin-fields enter Eq. (5)through the matrix ^S(k)¼(γ(k)/C0B)/C1σ 02/C22 02/C22 (γ(k)þB)/C1σT/C20/C21 (10) and the superconducting order parameters enters Eq. (5)through the matrix ^V(k)¼02/C22Ψ(k) Ψy(k)0 2/C22/C20/C21 , (11) where Ψ(k)¼ψ(k)σ0þd(k)/C1σ ½/C138 iσ2.38–41The Pauli principle enforces the function ψ(k) parametrizing Cooper spin-singlets to be even, and the d-vector d(k) parametrizing Cooper spin-triplets to be odd. The diagonal elements of Ψ(k) describes the S¼1 trip- lets, and the off-diagonal elements contain the S¼0 singlets and triplets. These order parameters are related to the anomalous prop- agators { f0,f} through the self-consistency condition given by32 di(k) logT Tc þπTX1 n¼/C01di(k) jωnj/C0^di(k)^di(k0)fi(k0;ωn)DE FS/C20/C21 ¼0, (12) where ^di(k) is a basis function of the corresponding order parame- terdi(k) that belongs to a specific irreducible representation of the crystal point group, and Tcis the superconducting transition tem- perature of the pairing channel. The same self-consistency condi-tion holds for ψ(k) with basis function ^ψ(k). In Eq. (12) the basis functions are normalized, h^d 2 i(k)iFS¼h ^ψ2(k)iFS¼1. To obtain the pair-breaking equation that describes the super- conducting transition T(B), we solve the linearized Eilenberger equation for the superconducting propagators { f0,f}, substitute the solutions into the self-consistency conditions (12), and perform the summation over the Matsubara frequencies. The result of this sum-mation contains all the information on how the superconductingorder parameters are affected by the spin-fields and impurities.Equation (12) can then be written in the form log ( T=T c)þS¼0, whereSis a Matsubara sum. In simple special cases, such as whenJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 053903 (2020); doi: 10.1063/5.0010773 128, 053903-3 Published under license by AIP Publishing.the order parameter is purely singlet/triplet and in the absence of one of the spin-fields, the Matsubara sum can be performed analyt- ically and the pair-breaking equation adopts the form42 logT TcþReψ1 2þα 2πT/C18/C19 /C0ψ1 2/C18/C19 ¼0, (13) where αis a generic pair-breaking parameter that can be a combi- nation of scattering rates and spin-fields, ψ(z) is the digamma function, and Tis the new α-affected critical temperature. We plot the contour of Eq. (13) inFig. 1 . In the absence of spin-fields, only magnetic impurities are pair-breakers in isotropic singlet supercon-ductors with α¼2Γ m. In anisotropic singlet superconductors, both scalar and magnetic impurities are equally pair-breaking with α¼Γ0þΓm. Since isotropic singlets suffer twice as much from magnetic impurities than anisotropic singlets, the effect of magneticimpurities can give indications of isotropy/anisotropy of the super-conducting state. In triplet superconductors, α¼Γ 0þΓm, which is the same as in anisotropic singlet superconductors. This changes in the presence of SOC and the Zeeman field. B. The case of Ising superconductivity Up to now, the discussion applies generically to non- centrosymmetric superconductors with an arbitrary direction of the spin-fields. A case of special interest is the one when the magnetic field is in-plane such that orbital depairing effects can be neglected.Moreover, if the SOC is out-of-plane, its action on the Cooper pairspins makes them more robust against paramagnetic depairing byan in-plane magnetic field. A similar situation with γ(k)?Bin a 2D superconductor can also be realized when γ(k) is restricted to the plane, such as helical Rashba SOC, and Bapplied perpendicu- larly to the plane. In this case, the orbital depairing mechanism isexpected to play a dominant role. In what follows, we specialize to the Ising-SOC with an in-plane Zeeman field. In fact, an Ising superconductor is the simplest non-centrosymmetricsuperconductor because it only has one SOC component [0, 0, γ 3(k)] as opposed to Rashba (or more complicated SOCs) which has two components [ γ1(k),γ2(k), 0]. Therefore, without loss of generality for the Ising case, we henceforth set B¼(B,0 ,0 ) andγ(k)¼[0, 0, γ(k)]. To study the superconducting transition, we linearize Eilenberger equation (5)and write it in matrix form as ~ωn /C0iB 00 /C0iB ~ωn γ(k)0 0 /C0γ(k) ~ωn 0 000 ~ωn2 6643 775f 0(k;ωn) f1(k;ωn) f2(k;ωn) f3(k;ωn)2 6643 775¼s~ψ(k;ω n) ~d1(k;ωn) ~d2(k;ωn) ~d3(k;ωn)2 6643 775, (14) where s¼sgn(ω n) and the impurity rescaled quantities are defined by ~ωn¼ωnþsΓ0þΓm ðÞ ; (15) ~ψ(k;ωn)¼ψ(k)þΓ0/C0Γm ðÞ h f0(k;ωn)iFS; (16) ~di(k;ωn)¼di(k)þΓ0 ihfi(k;ωn)iFS, (17) with Γ0 0¼Γ0/C0Γ1/C0Γ2/C0Γ3; (18) Γ0 1¼Γ0/C0Γ1þΓ2þΓ3; (19) Γ0 2¼Γ0þΓ1/C0Γ2þΓ3; (20) Γ0 3¼Γ0þΓ1þΓ2/C0Γ3: (21) Here, Γm¼Γ1þΓ2þΓ3while we are not making an assumption of the components Γibeing equal. We also introduce the notation Γ¼Γ0þΓm. In Secs. IV–VII, we present the results for solving special cases of our master equation (14). We consider the purely singlet, purely triplet, and the general singlet –triplet mixed cases. We also discuss the possibility of ma gnetic impurity induced cross- overs from leading isotropic singlet states to sub-leading aniso- tropic states. IV. SINGLET SUPERCONDUCTORS We examine two cases for the momentum structure ψ(k)o f the singlet order parameters. Case (A): the order parameter is iso-tropic ( s-wave) ψ(k)¼ψ 0, where ψ0is a constant (the basis func- tion ^ψ(k)¼1). Case (B): the order parameter is anisotropic with hψ(k)iFS¼0 and the basis function ^ψ(k) is even. Below we analyze these cases. A. Isotropic order parameter We set di(k)¼0 in Eq. (14) and using γ(k)¼Δsgn[γ(k)] for simplicity, the pair-breaking equation for ψ0reads log ( T=Tc)þ FIG. 1. Contour plot of pair-breaking equation (13) showing the suppression of the superconducting critical temperature T=Tcfor different pair-breaking param- eters α. The blue curve shows the case for a purely imaginary pair-breaking parameter, which is typically the case for spin-fields (magnetic field and SOC). The green and the red curves show the case for a purely real parameter, whichis typically the case for impurities.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 053903 (2020); doi: 10.1063/5.0010773 128, 053903-4 Published under license by AIP Publishing.Ss¼0 with Ss¼πTX1 n¼/C011 jωnj/C20 /C0j~ωnj(jωnjþ2Γ1)þΔ2 j~ωnjB2þ(jωnjþ2Γm)j~ωnj(jωnjþ2Γ1)þΔ2/C2/C3# : (22) In the absence of the Zeeman field B, the pair-breaking equation reduces to Eq. (13) with α¼2Γm(see the red curve in Fig. 1 ). In this case, all magnetic components of Γi(i¼1, 2, 3) are equally detrimental to superconductivity. In the low temperature limit,superconductivity is obliterated for α=T c¼π=(2eγ)/C250:88, where γ/C250:58 is the Euler constant. The constant γalways appears in the exponent and should not cause any confusion with SOC. 1. Inversion symmetric case, Δ=0 InFig. 2(a) , we show the case with inversion symmetry, for which Δ¼0. The clean (gray) curve is described by the imaginary pair-breaking parameter α¼iB. By setting Δ¼0 in Eq. (22), one can see that superconductivity remains indifferent to scalar impuri-ties. All the magnetic components act as pair-breakers but in differ-ent ways. The magnetic impurities Γ 1that are parallel to the Zeeman field have a weaker effect than the magnetic impurities with perpendicular components Γ2(3). To show this, we first plot the case when the magnetic impurity directions are randomly ori-ented such that Γ 1¼Γ2¼Γ3¼Γm=3 [see the green curve in Fig. 2(a) ]. If we now consider anisotropic magnetic impurities with all impurity spins Γ1parallel to the Zeeman field B, the critical field at low temperatures is higher (see the blue curve). Similarly, for spin impurities, Γ2(3)aligned only perpendicularly to the Zeeman field, suppression is maximized (see the red curve). Theseperpendicular spin impurities add a new direction of depairing,making them more detrimental. We briefly comment on the nature of the normal to supercon- ducting phase transition. In the clean case, it is known that fortemperatures T=T c&0:56, the transition is of first-order.43–45 However, both SOC and impurities increase the spin susceptibility in the superconducting state, which turns the first-order into a con- tinuous transition.2,20,46–48Due to the linearization of the Eilenberger equations, our pair-breaking equations capture the con-tinuous phase transitions.2. The case of no inversion symmetry, Δ=0 InFig. 2(b) , we show the case without inversion symmetry by adding Δ¼Tc. The SOC enhances the critical field because it counteracts the Zeeman field. In contrast to the case with inversion, non-magnetic impurities Γ0now suppress the critical field (black curve) because they undo the enhancement caused by SOC.32The larger the SOC, the harder it becomes to distinguish the differenteffects of the spin impurity components Γ i. B. Anisotropic order parameter The Matsubara sum for the case with ψ(k)i s Ss¼πTX1 n¼/C011 jωnj/C0~ω2 nþΔ2 j~ωnj(~ω2 nþB2þΔ2)/C20/C21 : (23) The sum can be performed exactly but we choose to maintain it in this simple form for discussion purposes. In contrast to the case of isotropic singlet order parameter, now any type of impu-rity is equally detrimental to superconductivity. In the absence ofthe Zeeman field, the pair-breaking equation reduces to Eq. (13) with α¼Γ 0þΓm. Anisotropic order parameters have an intrinsic phase structure, which makes the depairing by magnetic impuri-ties less effective as compared to isotropic order parameters. InFigs. 3(a) and 3(b), we show the effect of arbitrary impurities Γ on the transition line with inversion symmetry ( Δ¼0) and without inversion symmetry ( Δ¼T c), respectively. Throughout the paper, we use Γwithout a subscript if the nature of the impu- rities is unimportant. In Fig. 3(c) , we compare the transition lines of an isotropic singlet order parameter ψ0(orange) to an aniso- tropic order parameter ψ(k) for an equal amount of magnetic impurities Γm. Isotropic superconductivity is obliterated for Γm=Tc/C250:44 and anisotropic for Γm=Tc/C250:88. V. TRIPLET SUPERCONDUCTORS In triplet superconductors, the order parameter is the three component vector d(k)¼(d1(k),d2(k),d3(k)). We set ψ(k)¼0 in the linearized Eilenberger equation (14), which together with the self-consistency condition (12) yields the pair-breaking equa- tions for the d-vector components di(k)(i¼1, 2, 3), which read FIG. 2. The effect of magnetic and non-magnetic impuri- ties on purely isotropic ( s-wave) singlet superconductors. The magnetic impurity component Γ1that is parallel to the magnetic field has a weaker effect than the perpendic-ular components Γ 2(3). The different effects of Γi (i¼1,2,3) become less relevant for larger values of SOC. For the green curve, we used Γ1¼Γ2¼Γ3¼Γm=3.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 053903 (2020); doi: 10.1063/5.0010773 128, 053903-5 Published under license by AIP Publishing.log (T=Tc)þSti¼0, with the sums given by St1¼πTX1 n¼/C011 jωnj/C20 /C0j~ωnjðjωnjþ2Γ2Þ ðjωnjþ2Γ2ÞB2þj~ωnjj~ωnjðjωnjþ2Γ2ÞþΔ2/C2/C3# ;(24) St2¼πTX1 n¼/C011 jωnj/C20 /C0ðjωnjþ2Γ1Þðjωnjþ2ΓmÞþB2 j~ωnjB2þð jωnjþ2ΓmÞj~ωnjðjωnjþ2Γ1ÞþΔ2/C2/C3# ;(25) St3¼πTX1 n¼/C011 jωnj/C01 j~ωnj/C20/C21 ;ðα¼Γ0þΓmÞ: (26) For simplicity, we assume that the Tc’s for all triplet components are the same. A. Analysis of the results Let us analyze these results in detail. First, note that in the absence of spin-fields ( B¼Δ¼0), the d-vector components areindistinguishable ( St1¼St2¼St3), and the pair-breaking parame- ter is α¼Γ0þΓm, which is the same for the anisotropic singlet case. For finite spin-fields, the d3component is parallel to SOC and perpendicular to the Zeeman field such that it remains unaffectedby them. Only impurities suppress the d 3component. It is then reasonable to expect that in the presence of spin-fields, the d3may become a dominant superconducting channel. To discuss how the spin-fields affect the in-plane d-vector components, let us consider the effect of SOC and the Zeemanfield separately. In Fig. 4(a) , we show the situation with time- reversal symmetry ( B¼0) but finite SOC. Equations (24) and(25) assume the form S B!0 t1(2)¼πTX1 n¼/C011 jωnj/C0jωnjþ2Γ2(1) j~ωnj(jωnjþ2Γ2(1))þΔ2/C20/C21 : (27) The presence of SOC gives a preferred spin-structure to Cooper pairs and makes the dicomponents inequivalent in how they respond to the impurity components Γi. In the clean case, d1(2)is suppressed by SOC with α¼iΔ[see black curve in Fig. 4(a) ]. With magnetic impurities, the Γ1(2)component has a weaker effect on d2(1)thanΓ1(2)ond1(2)[compare red and green curves in Fig. 4(a) ]. InFig. 4(b) , we consider only the Zeeman field ( Δ¼0), and we obtain the pair-breaking parameters α¼{iBþΓ0þΓm,Γ0þ Γm,Γ0þΓm} for { d1,d2,d3}, respectively, whereas the Zeeman field suppresses the parallel d1component, it has no effect on the perpendicular components d2,3. We show the case with both SOC and Zeeman field in Fig. 4(c) . Thed1component that points along the Zeeman field is the only one suppressed by it. It is also suppressed by SOC such that even in theclean case the transition temperature for d 1channel T=Tc,1. We draw special attention to the behavior of d2. The joint presence of SOC and the Zeeman field favors the d2component. An increasing magnetic field minimizes the suppression caused by SOC [see howthe green curves asymptotically approach the vertical red lines in Fig. 4(c) ]. B. Polarization of the triplets A more intuitive understanding can be gained by interpreting the effects of the impurity components in terms of the Cooper pair spin polarization. The momentum dependent spin polarization of aCooper pair is defined as the expectation value 38 P(k)¼1 2trΨy(k)σΨ(k)hi ¼ψ(k)d*(k)þψ*(k)d(k)þid(k)/C2d*(k): (28) Since ψ(k) is even and d(k) is odd, only q¼id(k)/C2d*(k) poten- tially contributes to the total polarization averaged over the Fermisurface hP(k)i FS. For the present pure triplet situation, ψ(k)¼0. We now provide a heuristic motivation for the phases of (d1,d2,d3) to derive the polarization P¼(P1,P2,P3), which can also be obtained more rigorously.31Since SOC respects time- reversal, and the Zeeman field is in-plane, it is reasonable to expect P3¼0. This imposes d1d* 2¼d2d* 1such that both d1and d2are either purely real or purely imaginary. We know that d2is FIG. 3. The effect of impurities on purely anisotropic singlet ( d-wave, h-wave, etc.) superconductors. Magnetic and non-magnetic impurities have the same effect, which is indicated by a generic scattering rate Γ. In (c), we compare the effect of a magnetic scattering rate Γmon an isotropic singlet (orange) and anisotropic singlet (purple) state assuming that both of their clean Tcis the same.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 053903 (2020); doi: 10.1063/5.0010773 128, 053903-6 Published under license by AIP Publishing.promoted by B, see numerator in (25), and is expected to break time-reversal. Therefore, we choose d1(2)(k)¼iη1(2)^d(k)t ob e imaginary, where η1(2)are real coefficients and ^d(k) are basis func- tions as in Eq. (12). We write d3(k)¼η3^d(k), where η3is complex.With this, we obtain the total polarization, hP(k)iFS¼2Re(η3)(/C0η2,η1, 0), (29) where we used h^d2(k)iFS¼1. This tells us that d1(2)is responsible for a Cooper pair spin polarization along y(x), provided that η3 has a real component. Now we present an intuitive picture of these results. We argue that in a triplet superconductor, a finite Cooper pair spin polariza- tion can explain why different spin-components of impurities have a distinct effect on superconductivity. Recall that in the case of iso-tropic singlet order parameter, Eq. (22), the impurity spin- component parallel to the applied magnetic field has a weaker effect on the critical field, compared to the perpendicular compo- nents. In a similar way, in the triplet superconductor with theCooper pairs with a net polarization along a specific directionsuffer less from the magnetic impurity component Γ ithat is paral- lel to that direction. From Eq. (29), we see that d2is responsible for a Cooper pair polarization along xsuch that, according to Eq. (27), theΓ1(2)component has a weaker (stronger) effect. Similarly, Γ1(2) has a stronger (weaker) effect on d1that is responsible for polariza- tion along y. This can be summarized as follows: the Γicomponent that is parallel to a net-field (either magnetic field or Cooper pair polarization) has a weaker effect. VI. CROSSOVER FROM SINGLET TO TRIPLET SUPERCONDUCTIVITY BY MAGNETIC IMPURITIES In this section, we consider a leading isotropic singlet ( s-wave) channel with superconducting critical temperature Tcsand a sub- leading triplet channel with corresponding critical temperatureT ct,Tcs. The discussion applies to both the cases with and without inversion. For the arguments of this section, the effects of specific Γicomponents is less relevant such that we set Γ1¼Γ2¼Γ3¼Γm=3. Let us say that both singlet and triplet channels belong to the same crystal symmetry irreducible represen-tation, such as ψ 0and d3of Secs. IVandV. We also learned from Sec. Vthat d3dominates over d1and d2in the presence of spin- fields, so it is reasonable to concentrate on d3. The singlet transition temperature Tcssuffers more from mag- netic impurities Γmthan the triplet transition temperature Tct(see Fig. 1 ). With Tcs.Tct, it is then possible to observe a magnetic impurity induced crossover from a singlet to a triplet state. The precise value of Tct=Tcsis an issue of pairing mechanism physics. Sub-leading triplet instabilities are present even in conventionalpairing mechanisms. 49It is then natural to ask the guiding question of this section: what is the minimum ratio of Tct=Tcsto observe such a singlet to triplet crossover? A. No magnetic field AtB¼0, a crossover occurs if Tct=Tcs.1=2. This result can be analytically obtained by combining two pair-breaking equations of the form in Eq. (13), one for the singlets and another for the triplets. This is illustrated in Fig. 5(a) . In the clean case, the singlet (blue) curve starts from a critical temperature T=Tcs¼1 and is obliterated for Γm=Tcs/C250:44. The triplet (red) curve starts from T=Tcs¼0:75 and is obliterated for Γm=Tcs/C250:66. The blue region FIG. 4. The effect of impurities on purely triplet ( p-wave, f-wave, etc.) supercon- ductors. (a) The suppression of the critical temperature T=Tcas a function of x, where xis a combination of an impurity scattering rate with SOC. The black curve shows how SOC is detrimental to the in-plane d-vector components and isdescribed by α¼iΔ. The blue curve shows how any impurities suppress the d-vector components in the absence of spin-fields. The green and red curves show how the in-plane Γ iaffects the in-plane d-vectors components. Suppression is maximum (red curve) when Γiis perpendicular to the Cooper pair polarization and weaker (green) when Γiis parallel to the pair polarization. (b) The effect of impurities in the presence of the magnetic field without SOC. The components d2(3)that are perpendicular to the Zeeman field remain unaffected by it and only suffer from impurities. The d1component that is parallel to the field suffers from both paramagnetic limiting and the impurities. (c) The effect of magnetic impuritieswhen both the SOC and Zeeman fields are present. Here, the value of SOC is fixed to Δ=T c¼0:77 such that the components d1(2)that are perpendicular to SOC are suppressed by SOC even at B¼0 [see (a)]. The order of increasing robustness to the impurities in the purely triplet case is: d1,d2,d3.I np a r t i c u l a r , d2(green) displays a peculiar behavior at high magnetic fields. The green curves asymptotically approach the red curves for high magnetic fields. The Zeeman field undoes the suppression caused by SOC.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 053903 (2020); doi: 10.1063/5.0010773 128, 053903-7 Published under license by AIP Publishing.shows a purely singlet ψ0state, and the red region shows a purely triplet d3(k) state. The precise crossover boundaries inside the phase diagram would depend on a treatment beyond linearization. The red-dashed curve shows the case with Tct=Tcs¼1=2. B. With magnetic field InFig. 5(b) , we show the case at finite magnetic fields for a fixed value of magnetic impurities Γm=Tcs¼0:1. The value of the ratio is set to Tct=Tcs¼0:3 such that no crossover at zero field is possible. Because of the magnetic impurities, the effective singlet (triplet) transition temperatures T* cs(T* ct) start at lower values. Since thed3triplets remain unaffected by spin-fields, it will always be the leading instability at high magnetic fields below T* ct. VII. SINGLET –TRIPLET CONVERSION BY SPIN-FIELDS In Secs. IV–VI, the superconducting order parameter was either ap u r es i n g l e to rap u r et r i p l e t .I nt h i ss e c t i o n ,w ea d d r e s st h et h e general situation with coexisting order parameters { ψ(k),d(k)}. To do this, we study the general solution of Eilenberger equation (14). A coupling between singlet and triplet order parameters can originate from mainly two reasons: (i) The densities of states ofthe spin-split bands are different. 27,29,50This would lead to a cou- pling of ψ0and d3(k), which belongs to the same irreducible rep- resentation. Here, we neglect the possible difference of thedensities of states such that d 3(k) remains decoupled. This infor- mation is already built into the Eilenberger equation (14).( i i )T h e joint action of SOC and magnetic field component that are per- pendicular to each other. This can be seen from the matrix struc- ture of the linearized Eilenberger equation (14). The presence of both spin-fields will select the d2(k) triplets to couple to the ψ0 singlets. We have pointed this out in our previous works, and we refer to Refs. 31and 32for more details. Here, we focus on the effect of magnetic impurities. The structure of the matrix in Eq. (14) reveals how the spin- fields couple the propagator components { f0,f} and, consequently, the order parameters { ψ(k),d(k)}. The role of the Zeeman field B is to couple the singlet pairing correlations (propagators) to the triplet pairing correlations. Yet, the role of Ising-SOC is to couplethe in-plane triplet correlations f1and f2. The f3triplet correlations remain unaffected by the spin-fields because they are parallel toIsing-SOC and perpendicular to the Zeeman field. We now assume an isotropic singlet state ψ(k)¼ψ 0, solve Eq.(14) for { ψ0,d(k)}, which together with the self-consistency condition for the order parameters (12) results in three sub- systems: {{ ψ0,d2(k)}, {d1(k)}, {d3(k)}}. The solutions for d1(3)are the same as in Eqs. (24) and (26), respectively. The sub-system {ψ0,d2(k)} is coupled by the joint presence of SOC and the Zeeman field, and the pair-breaking equation is given by the char-acteristic equation detln T TcsþSsSs,t2 Ss,t2 lnT TctþSt2"# ¼0, (30) withSsdefined in Eq. (22),St2in Eq. (25), and Ss,t2¼X1 n¼/C01(πT)BΔ j~ωnjB2þ(jωnjþ2Γm)j~ωnj(jωnjþ2Γ1)þΔ2/C2/C3 :(31) Equation (31) is what couples d2(k)t oψ0. It is only possible for BΔ=0. The singlet to triplet coupling is impossible between an aniso- tropic singlet ψ(k) and d2(k). The reason is that the product of basis function involving an anisotropic singlet state h^ψ(k)^d2(k)iFS vanishes. Another way to interpret this is that while a finite BΔcan convert isotropic singlets ψ0into triplets, it necessarily depairs anisotropic singlet Cooper pairs. This can also be understood in terms of a spin-rotation argument, which is discussed in Ref. 31. Since time-reversal symmetry breaking by the magnetic field (together with SOC) couples the ψ0singlet to the d2triplet order parameter, it is natural to ask whether time-reversal symmetry breaking by magnetic impurities could also lead to such a coupling? From the structure of the matrix in Eqs. (14) and(31), the answer is clearly: no. All impurities rescale the diagonal quantities in thematrix such that no coupling is possible. It is also clear that since the scattering rates only appear in the denominator of Eq. (31), the impurities suppress the coupling. While a magnetic field can FIG. 5. Magnetic impurity induced crossover from a leading isotropic singlet ψ0(s-wave) to a sub-leading anisotropic triplet state d3(k)(p-wave, f-wave, etc.). (a) Possible crossovers at a zero magnetic field. At a zero magnetic field, a crossover happens for Tct=Tcs.1=2 (see the red-dashed curve). (b) At finite magnetic field, the d3triplets are always favorable below T/C3 ct, for field above the blue curve. The value Δ=Tcs¼0:5 is only used for illustration purposes.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 053903 (2020); doi: 10.1063/5.0010773 128, 053903-8 Published under license by AIP Publishing.convert singlet Cooper pairs to triplet, randomly oriented impuri- ties lead to depairing. A singlet –triplet coupling might be possible in a situation where the impurity spins are aligned such that thereis an average impurity net magnetization. 51 InFig. 6 , we plot some contours of the pair-breaking equation (30) for the parity-mixed state { ψ0,d2(k)}. For illustration purposes, we used the fixed value Tct=Tcs¼0:2. The divergent behavior of the clean (gray) curve reflects this value. For any scatter-ing rate Γ=T cs.π=(2eγ)(Tct=Tcs)/C250:18, the d1(3)triplets would already be obliterated and are not shown in Fig. 6 . For illustration, we use a relatively small value for Δ=Tcs¼0:27 such that the differ- ences in the effects of Γiare still visible. The important messages of Fig. 6 are that now the d2triplets acquire the same robustness to impurities as the ψ0singlets and that magnetic impurities are destructive to the singlet –triplet coupling. VIII. DISCUSSION AND CONCLUSION We now discuss the results in the context of the literature. As examples, we comment on monolayer transition metal dichalcoge-nides (TMDs), superconducting thin films, AND heavy-fermiontriplet superconductors and then conclude. In this work, we explored ways in which the magnetic impuri- ties can serve as a tool for probing unconventional superconductiv- ity. The 2D superconductors which are of great technological andtheoretical significance are anisotropic in the response to the in-and out-of-plane magnetic field. Also, in systems without inversionsymmetry, the SOC is anisotropic in momentum. We studied how superconductivity in the presence of the Zeeman field and SOC can be probed by magnetic impurities that are randomly orientedand distributed across the system. We demonstrate that the anisot-ropies in the distribution of the impurity spin orientation can beused as a tool to study the momentum texture of specific properties of the order parameter and its evolution with the Zeeman field. In the absence of spin-fields such as the SOC and the Zeeman field, the effect of the scalar and magnetic impurities on the singletsuperconductivity is well known. The effect of a dilute concentration of magnetic impurities in triplet and parity-mixed superconductors remains largely unexplored. To our knowledge, the few efforts arelimited to Refs. 52–54. Here, we take into account the SOC, the Zeeman field, and the possible anisotropy of the superconducting order parameter. As a specific example, we focused on the 2D Ising supercon- ductors. We addressed several relevant aspects of these systems inRefs. 31,32, and 55with special emphasis on monolayer transition metal dichalcogenides (TMDs). For completeness, we now comment on specifics for TMDs. The crystal point group of thehexagonal family of monolayer TMDs is D 3hlacking the inversion element. In the present context, the group has three relevantpairing channels: 31an even s-wave A0 1channel with basis function ^ψ(k)¼1, an odd f-wave A0 1channel with basis function ^γ(k)^z, and an if-wave E00channel with basis functions { ^γ(k)^x,^γ(k)^y}, where ^γ(k) is the same basis function used for Ising-SOC. Therefore, the superconducting state in monolayer TMDs is generically denotedby an sþfþifstate, where the srefers to ψ 0,ftod3,a n d ifto {d1,d2}. According to the discussion in Sec. VII, a small amount of impurities of any kind Γf¼π=(2eγ)Tctobliterates the f-wave such that we are left with the sþif(ψ0þd2(k)) state. The sþifstate is obliterated by magnetic impurities Γm¼π=(2eγ)Tcs.Γf. Nonetheless, the sþifstate cannot be obliterated by scalar impu- ritiesΓ0. In monolayer TMDs, the SOC energy scale is larger than Tcsuch that the different effects of the magnetic components Γi are most likely insignificant. In conventional BCS superconducting thin films, the in-plane paramagnetic critical field is of the order of a few teslas.56,57 Figure 2(a) shows that at low temperatures, the difference of the effects of Γ1andΓ2(3)should be of easy access to magnetometers. Perhaps a greater challenge is to prepare/find thin films that havemagnetic impurities with preferred orientations. Ordered magnetic impurities have been reported in superfluid 3He aerogels (see Ref. 58and references therein). It is less clear if such situations could be produced (artificially or naturally) in singlet superconduc-tors. The situation in Fig. 2(b) could occur in Ising systems such as thin Pb films grown on a silicon substrate. 59The difference in the ordered magnetic impurity effects is better seen for smaller values of SOC. In anisotropic singlet superconductors ( Fig. 3 ), these effects are expected to be irrelevant since the transition lines areaffected equally by any kind of impurities. To see these effects in triplet superconductors might be more challenging. To this date, to our knowledge, the only consensual triplet superconductors (besides superfluid 3He) are the Uranium based ferromagnets.60–62Their critical temperature is usually below a Kelvin (and sometimes high pressures are needed), which makes impurity effects significantly harder to observe. Using the self-consistent Born approximation, we addressed the following cases: singlet, triplet, impurity induced crossoversfrom pure singlet to pure triplet, and the parity-mixed case. Itmight be worth extending the treatment beyond the Born approxi- mation and investigate additional effects that could arise due to Shiba bands. 2In singlet superconductors, the magnetic impurities that are parallel to the in-plane magnetic field (the direction ofpolarization) have a weaker effect than the perpendicular impuri-ties. In the triplet case, the in-plane components d 1(2)are the ones that SOC suppresses. For these in-plane components, Γ1(2)has a weaker effect on d2(1). The d1(2)components are responsible for a Cooper pair polarization in the y(x) directions. Thus, a similar FIG. 6. Effect of impurity components on the { ψ0,d2(k)}-coupled superconduct- ing state. The triplet components d1(3)are already obliterated for Γ=Tcs¼0:2 and are thus not shown in the figure. The clean (gray) curve diverges at T¼0:2Tcs¼Tct.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 053903 (2020); doi: 10.1063/5.0010773 128, 053903-9 Published under license by AIP Publishing.rule to the singlet situation applies: the impurity component Γi that is parallel to the direction of polarization has a weaker effect . In the parity-mixed case, the joint action of SOC and in-plane magnetic field selects the triplet component that is polarized along B,n a m e l y , d2, to couple to ψ0. We discussed the effect of different magnetic impurity com ponents on the superconducting transition curves for systems with and without inversion. For thecase without inversion, we specialized to the Ising-SOC type. However, these effects are general and can be generalized to other types of SOC. We argue that the anisotropy of the spin ori-entation of magnetic impurities can serve as a tool for manipula-tion, control, and characterization of superconducting states ofboth definite and mixed parity. ACKNOWLEDGMENTS D.M. acknowledges the support from the Swiss National Science Foundation, Project No. 184050, and the authors acknowl- edge the support from the Israel Science Foundation, Grant No. 1287/15. APPENDIX: DERIVATION OF THE EXPRESSIONS FOR THE IMPURITY SELF-ENERGY In this Appendix , we derive the expressions for the self- energies, (6)–(9). We consider the electrons scattered off the ran- domly distributed scalar and magnetic impurities. In theNambu-spin bases, the magnetic exchange interaction S/C1σgives rise to the term S/C1αpresent in the matrix of the impurity part of the Hamiltonian presented in Eq. (3). The positions of impurity sites and the orientation of the impurity spins are statisticallyuncorrelated. Therefore, we perform the averaging over the impu-rity position and spin orientations independently. This procedure leads to the the self-energy, ^Σ¼n 0X k1U0k/C0k1 ðÞ ρ3σ0 ½/C138 ^Gk1,ωn ðÞ U0k1/C0k ðÞ ρ3σ0 ½/C138 þnmX k1X3 j¼1S2 jJk/C0k1 ðÞ αj/C2/C3^Gk1,ωn ðÞ Jk1/C0k ðÞ αj/C2/C3 , (A1) where /C1/C1/C1denotes averaging over of all the magnetic impurities. In writing Eq. (A1), we assumed that the covariance matrix of spin- components is diagonal, SiSj¼0 for i=j. The resulting self- energy, Eq. (A1), is presented in Fig. 7 .Using the relationship X k/C25N0ð1 /C01dξkð2π 0dwk 2π(A2) and the definition of the matrices αwhich follows Eq. (3),w e rewrite (A1) as the sum of four contributions, ^Σ¼^Σ0þ^Σ1þ^Σ2þ^Σ3. The contribution of the short range scalar impurities given by the first term of Eq. (A1), ^Σ0¼/C0 iπn0N0U2 0ρ0σ0ðdwk 2πðdξk πiρ3σ0^Gk,ωn ðÞ ρ3σ0 ¼/C0 iΓ0ρ0σ0^gk,ωn ðÞhiFSρ3σ0, (A3) reproduces Eq. (6)with the scattering rate Γ0defined in the main text. The contribution of the ximpurity spin-component can be similarly obtained from the j¼1 term in Eq. (A1), ^Σ1¼/C0 iπnmN0J2S2 1ρ0σ1ðdwk 2πðdξk πiρ3σ0^Gk,ωn ðÞ ρ3σ1 ¼/C0 iΓ1ρ0σ1^gk,ωn ðÞhiFSρ3σ1: (A4) Equation (A4) coincides with (7)with the scattering rate Γ1 defined in the main text. The two remaining contribution to the self-energy, (A1) are similarly shown to reproduce Eqs. (8)and(9). For isotropic distribution of impurity spin orientations, S2 1¼S22¼S23, which leads to separate scattering rates to be equal, Γ1¼Γ2¼Γ3¼1 3Γm, where Γmis defined following Eq. (9). In the general case of anisotropic spin distribution, S2 1þS22þS23¼ S(Sþ1) which leads to the relations, Γ1þΓ2þΓ3¼Γm. The latter serves as a constraint on a separate scattering rates imposed by the magnitude of impurity spin being constant. DATA AVAILABILITY The data that support the findings of this study are available within the article. REFERENCES 1L. P. Gor ’kov, “Theory of superconducting alloys, ”inSuperconductivity (Springer, Berlin, 2008), pp. 201 –224. 2A. V. Balatsky, I. Vekhter, and J. X. 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5.0011316.pdf

AVS Quantum Sci. 2, 031701 (2020); https://doi.org/10.1116/5.0011316 2, 031701 © 2020 Author(s).Integrated single photon emitters Cite as: AVS Quantum Sci. 2, 031701 (2020); https://doi.org/10.1116/5.0011316 Submitted: 20 April 2020 . Accepted: 28 July 2020 . Published Online: 20 August 2020 Junyi Lee , Victor Leong , Dmitry Kalashnikov , Jibo Dai , Alagappan Gandhi , and Leonid A. Krivitsky COLLECTIONS Paper published as part of the special topic on Special Topic: Quantum Photonics Note: This paper is part of the special topic on Quantum Photonics. This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN Photonic quantum metrology AVS Quantum Science 2, 024703 (2020); https://doi.org/10.1116/5.0007577 Observation of near-infrared sub-Poissonian photon emission in hexagonal boron nitride at room temperature APL Photonics 5, 076103 (2020); https://doi.org/10.1063/5.0008242 Perspective of self-assembled InGaAs quantum-dots for multi-source quantum implementations Applied Physics Letters 117, 030501 (2020); https://doi.org/10.1063/5.0010782Integrated single photon emitters Cite as: AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 Submitted: 20 April 2020 .Accepted: 28 July 2020 . Published Online: 20 August 2020 Junyi Lee,1 Victor Leong,1 Dmitry Kalashnikov,1 Jibo Dai,1Alagappan Gandhi,2and Leonid A. Krivitsky1,a) AFFILIATIONS 1Institute of Materials Research and Engineering, Agency for Science, Technology and Research (A/C3STAR), 2 Fusionopolis Way, #08-03 Innovis, 138634 Singapore 2Institute of High Performance Computing, Agency for Science, Technology and Research (A/C3STAR), 1 Fusionopolis Way, #16-16 Connexis North, 138632 Singapore Note: This paper is part of the special topic on Quantum Photonics. a)Electronic mail: Leonid_Krivitskiy@imre.a-star.edu.sg ABSTRACT The realization of scalable systems for quantum information processing and networking is of utmost importance to the quantum information community. However, building such systems is difficult because of challenges in achieving all the necessary functionalities on aunified platform while maintaining stringent performance requirements of the individual elements. A promising approach that addressesthis challenge is based on the consolidation of experimental and theoretical capabilities in quantum physics and integrated photonics.Integrated quantum photonic devices allow efficient control and read-out of quantum information while being scalable and cost effective. Here, the authors review recent developments in solid-state single photon emitters coupled with various integrated photonic structures, which form a critical component of future scalable quantum devices. Their work contributes to the further development and realization ofquantum networking protocols and quantum logic on a scalable and fabrication-friendly platform. VC2020 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/10.1116/5.0011316 TABLE OF CONTENTS I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 II. THEORETICAL BACKGROUND . . . . . . . . . . . . . . . . . . . 2 A. NV/C0center as an illustrative SPE . . . . . . . . . . . . . . 3 1. Photostability, saturated count rates, and emission into ZPL . . . . . . . . . . . . . . . . . . . . . . . . 4 2. Indistinguishability . . . . . . . . . . . . . . . . . . . . . . . . 4 3. Single photon purity . . . . . . . . . . . . . . . . . . . . . . 6 B. Enhancement of ZPL emission using resonant cavities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 III. INTEGRATED NV/C0CENTERS IN BULK DIAMOND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 A. Integration with waveguides . . . . . . . . . . . . . . . . . . . 9 B. Integration with resonators . . . . . . . . . . . . . . . . . . . . 10 1. Diamond ring resonators . . . . . . . . . . . . . . . . . . 102. Diamond 2D photonic crystal cavities . . . . . . . 113. Diamond 1-D photonic crystal cavities . . . . . . 12 C. Larger scale integration . . . . . . . . . . . . . . . . . . . . . . . 12 D. Deterministic integration. . . . . . . . . . . . . . . . . . . . . . 13E. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 IV. COLOR CENTERS IN NANODIAMONDS . . . . . . . . . . 15A. Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 B. Integration with photonic structures . . . . . . . . . . . . 15 V. QUANTUM DOTS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 A. Introduction to quantum dots . . . . . . . . . . . . . . . . . 17 B. As a single-photon emitter . . . . . . . . . . . . . . . . . . . . 18 C. Spin–photon Interfaces . . . . . . . . . . . . . . . . . . . . . . . 18D. Interfacing multiple QDs. . . . . . . . . . . . . . . . . . . . . . 19 1. Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2. Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3. Electric field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194. Frequency conversion . . . . . . . . . . . . . . . . . . . . . 20 VI. 2D MATERIALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 A. Introduction into 2D materials. . . . . . . . . . . . . . . . . 21 B. Single photon emitters in TMDCs . . . . . . . . . . . . . . 21C. Single photon emitters in hBN. . . . . . . . . . . . . . . . . 21D. Deterministic Creation and Control of Single Photon Emitters in 2D Materials. . . . . . . . . . . . . . . 22 E. Enhancement of emission from 2D materials by coupling into resonant modes. . . . . . . . . . . . . . . . . . 23 F. Coupling and transfer of emission from 2D materials into photonic structures . . . . . . . . . . . . . . 23 AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-1 VCAuthor(s) 2020. AVS Quantum Science REVIEW scitation.org/journal/aqsVII. INTEGRATION APPROACHES . . . . . . . . . . . . . . . . . . . 24 A. Random dispersion. . . . . . . . . . . . . . . . . . . . . . . . . . . 24 B. Targeted creation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1. Irradiation and annealing in diamond . . . . . . . 252. Implantation and annealing in diamond . . . . . 263. Laser writing and annealing in diamond . . . . . 264. In situ lithography . . . . . . . . . . . . . . . . . . . . . . . . 27 C. Wafer bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 D. Pick-and-place. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1. Transfer printing . . . . . . . . . . . . . . . . . . . . . . . . . 292. Microprobe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 VIII. CONCLUSION AND OUTLOOK . . . . . . . . . . . . . . . . . 30 I. INTRODUCTION The control and manipulation of physical objects at the quantum level have progressed considerably in the past decade. This quantum control promises fascinating advances to both technology and funda- mental science. For example, the use of quantum phenomena in data systems allows one to speed up computation and database search algo- rithms and to develop highly secure communication networks. 1,2A new class of devices is now in active development to fundamentally exploit the paradigm of quantum information and to make it accessi- ble in practical applications. A variety of physical systems have been identified as candidates for emerging quantum technologies, such as quantum dots (QDs), atomic defects in solids, and atoms. Solid-state systems possess out- standing quantum optical properties that can be used to build practi- cal quantum devices. Quantum information can be stored, for example, in the electron spin of a defect and the nuclear spin of nearby atoms with relatively long coherence times (a few millisec- onds) even at room temperature. During this time, it is feasible to record, manipulate, and read-out quantum information. Quantum logic can be implemented with incident microwaves and RF fields, driving transitions between electron and nuclear sublevels. These blocks can interact with each other via photon-mediated interaction. Optical links connect the nodes as they enable reliable and fast trans- fer of quantum information. Building the quantum data system outlined above requires, among other things, efficient interfaces between solid-state single pho- ton emitters (SPEs) and optical networks. A scalable and cost efficient approach toward implementing such interfaces relies on the use of integrated photonic technologies. The stationary nodes that encode the quantum information (for example, a spin state of an atomiclike defect) can be interconnected via optical waveguides made of low-loss materials. Optically resonant micro- and nanostructures can enhance the coupling of photons emitted from the stationary nodes into wave- guides. Moreover, enhancement can also be obtained in waveguide quantum electrodynamics (QED) due to slow light effects. Photons can then be routed to different nodes on the same chip or between dif- ferent chips to create quantum entanglement between the nodes. Furthermore, one can use compact and highly sensitive photodetectorsfabricated on the same chip for the read-out of quantum information. Generation, transport, manipulation, and detection of quantum infor- mation can all be accomplished on this scalable, intrinsically stable, and fabrication-friendly platform. Besides applications in quantum information processing, the same physics and engineering concepts can be further applied inquantum metrology and sensing. 3,4Combining solid-state quantum systems with compact photonic devices will lead to the development of a new family of highly sensitive and compact temperature, stress,inertia, and electric and magnetic field sensors with high spatial resolu- tion. These sensors will find applications in microelectronics, bio- chemical, and healthcare industries. In this paper, we review recent experimental efforts in developing integrated solid-state single photon sources, which serve as a keyenabling component for scalable quantum devices. Broader reviews of other necessary components in a quantum photonic chip are available elsewhere. 5,6Due to the multidisciplinary nature of integrating solid state SPEs onto photonic chips, we have striven to make this review accessible to a broad audience with varying backgrounds by giving atheoretical overview of important SPE metrics and providing details related to the fabrication and integration of SPEs with resonant pho- tonic structures. A shorter review covering solid state SPEs of slightly different systems can be found in Ref. 7. We start off with a review of the nitrogen vacancy (NV) defect center in diamond as an illustrative example of a solid state SPE in Sec.II. 8After describing its basic photo-physical properties, we then provide a generic theoretical framework for the interaction of resonantphotonic structures with quantum emitters. We then proceed with a review of experiments that have integrated NV centers in bulk dia- monds with optical waveguides and resonant structures before discus- sing their prospects of larger scale integration with other photonic components and different material platforms (Sec. III). In Sec. IV,w e describe the integration of color centers in nanodiamonds. The limita- tions and benefits of color centers in nanodiamonds vs those in bulkdiamond are also discussed. Section Vis focused on integrated SPEs in QDs. Following a brief introduction into the photo-physics of QDs and their interaction withoptical cavities, we describe methods for manipulating the QDs’ spin states. We then discuss experiments on interfacing multiple QDs coherently with the goal of generating entanglement between spatially separated QDs on the same chip. In Sec. VI, we discuss SPEs in 2D materials, namely, in transi- tional metal dichalcogenides (TMDCs) and hexagonal boron nitride (hBN). Following a brief overview of different types of 2D SPEs, we discuss their interfaces with resonant photonic structures. In Sec. VII, we discuss various techniques for the integration of solid state SPEs with nanophotonic structures. This section describes dedicated nanofabrication and mechanical nanomanipulationmethods. Finally, we present two benchmarking tables in Sec. VIIIfor com- paring various experimentally integrated SPEs and resonators beforeconcluding with a general outlook for the field. II. THEORETICAL BACKGROUND Two-level quantum mechanical systems are natural candidates for true SPEs. At first sight, discrete energy levels within a solid-state system normally described by valence and conduction bands mightseem at odds with intuition, but they can exist under special circum- stances near a lattice defect or in quantum dots, where electrons are physically confined to such small spatial volumes that their eigen- energies are discrete. For many applications, however, it is not merely sufficient to have a two-level system since there are many other met- rics to consider. In Sec. II A, we use a nitrogen vacancy defect inAVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-2 VCAuthor(s) 2020diamond as an illustrative solid-state SPE to discuss these other con- siderations and to motivate the benefits of integrating solid-state SPEswith resonant photonic structures. Although we use the nitrogenvacancy defect in diamond as a concrete example, many of the chal-lenges we point out are broadly applicable to quantum dots and 2Dmaterials, where these challenges can be similarly mitigated by theirintegration with photonic structures. A. NV /C0center as an illustrative SPE A NV center in diamond consists of a nitrogen atom (a substitu- tional defect) that is paired together with a neighboring vacant site (seeFig. 1 ). The substitution-vacancy pair can be aligned along any of the equivalent h111idirections in the crystal, and typically, all four possi- ble orientations of the NV centers are found in equal proportionsalthough relatively recent work has successfully created preferentiallyoriented NV centers. 9–11Although NV centers are known to exist in two distinct states (traditionally labeled as NV0and NV/C0),12it is the NV/C0state that has, of late, received the most attention due to its attractive optical and spin properties that have made it amenable to avariety of technological applications including quantum computa-tion, 13quantum information14and microscopic magnetic,15electric,16 stress,17inertia,18and even thermal19sensing. Figure 2 shows the energy levels of the ground ( a2 1e2) and first excited ( a1 1e3)m o l e c u l a r orbital (MO) configurations20of the NV/C0.S i n c et h eN Vc e n t e rh a sa C3vpoint symmetry with 2 one-dimensional irreducible representa- tions ( A1and A2) and a two-dimensional irreducible representation (E),21,22the nomenclature of the states and orbitals are typically given by their transformation properties under C3v.T h e3A2orbital singlet ground state has been amply confirmed by electron paramagnetic resonance (EPR) in the dark,23optical hole burning,24optically detected magnetic resonance25,26(ODMR), and Raman heterodyne measurements27to be a spin triplet that is split, at zero magnetic field, by/C252.88 GHz24,28into a spin singlet A 1(with ms¼0) and a spin dou- blet E x,Ey(with ms¼61) state due to spin–spin interactions.21,29,30 Similarly, the orbital-doublet excited3E state is also known to be a spin triplet via ODMR measurements.31,32The degeneracy of these states is likewise lifted by spin-spin and spin–orbit interactions21into Ex,Eystates with ms¼0a n d A1,A2,Ex,a n d Eystates with ms¼61 (seeFig. 2 ). Compared to the ground state, however, the excited state is considerably more sensitive to shifts induced by lattice strains,31–33 and the exact ordering of its states is more variable. Nevertheless, it is clear from EPR measurements that the zero-phonon line (ZPL) at 637 nm is associated with a spin-triplet excited state and that the triplet state consists of a spin-singlet (i.e., the Ex,Eystates with ms¼0) and a spin-doublet (consisting of A1,A2,Ex,a n d Eystates with ms¼61) state, which are separated by a zero-field splitting Desof /C251.42 GHz.31,32Early uniaxial stress studies,34together with the mea- surements described above, indicate that the prominent ZPL observed FIG.2 .Energy levels of the ground and first excited MO configuration of the NV/C0. States labeled here with a jn;msinotation are spin–orbit states that transform according to a particular row of an irreducible representation of C3v(labeled by n; see Refs. 21and20), which are the convenient basis states to use in the presence of spin–orbit/spin–spin interactions. Note that they are linear combinations of states with definite azimuthal spin quantum numbers msand can hence have ms¼61. PSB denotes the phonon side band. Solid lines with single arrow heads denoteoptical transitions, while solid lines with double arrow heads denote microwave tran-sitions. Nonradiative inter-system crossings (ISCs) are denoted by dashed lines. A darker ISC line between the m s¼61 states of3Et o1A1is used to illustrate the faster ISC rate for that transition. The spacing of the energy levels is not drawn toscale. FIG.1 . Structure of the NV/C0center. Gray circles denote carbon atoms. The vacancy is denoted with a dashed vacant circle, while the nitrogen substitutional defect is shown as an orange circle. sp3bonds are illustrated by gray rods, while dangling bonds to the vacancy are drawn as purple ellipses. As shown in the figure,the NV center can have a C 3vsymmetry about any of the equivalent h111idirections in a diamond that it is aligned with. A NV/C0state is negatively charged because it acquires and traps an additional electron from another donor.Reprinted from Gali, Nanophotonics 8, 1907 (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution 4.0 License.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-3 VCAuthor(s) 2020at 1.945 eV ( /C25637 nm) is due to a3A2!3E transition (see Fig. 3 ). Moreover, for a NV in a low strain environment, the ZPL emission is mostly linearly polarized with the plane of polarization depending on the NV’s axial orientation, indicating that the transitions are mostlyspin-conserving. 34,35 1. Photostability, saturated count rates, and emission into ZPL Despite receiving less attention than the NV/C0,t h eN V0state can affect the dynamics of NV/C0in important ways such as its photostabil- ity to which we now turn. An important characteristic of a SPE is itsphotostability. Although organic fluorescent dyes are an importantclass of single photon emitters, such sources have a significant disad-vantage because of their susceptibility to photobleaching in which pho- tochemical changes induced by the excitation light cause the emitter to degrade and permanently lose its fluorescence. 36This is, for obvious reasons, undesirable for many quantum information and communica-tion applications. Fortunately, SPEs like defect centers in diamond, 2-Dmaterials, and quantum dots are considerably more robust. However,despite the excellent photostability of the NV /C0center, it is known that the fluorescence of NV/C0can, nevertheless, be significantly (but mostly reversibly) quenched37–40(sometimes called blinking )w h e np r o b e da t high (typically pulsed) laser intensity and that this has been partiallyattributed to a spin-dependent ionization of NV /C0to NV0.41Moreover, oscillations between NV/C0and NV0have been observed as a function of the excitation wavelength,42and it is estimated that the NV center can be in the NV0state for /C2430% of the time under usual operating conditions.43Nevertheless, these effects in NV may be mitigated by fur- ther annealing at 1200/C14C,44,45and more generally, the equilibrium concentration46and stability of the NV/C0state depend on the Fermi level, which may be altered, among other things, by doping (N is itselfa deep donor), irradiation, heating, photo-excitation, surface-termina- tion, and annealing conditions.12,47–51Similarly, although the exact mechanisms may differ, quantum dots in nanocrystals are also suscep- tible to blinking.52,53 A related metric is the source’s saturated count rate at which point further increasing the excitation power no longer induces signifi- cantly more fluorescence. Generally, most applications would benefit from a brighter source since there are always losses in any real-world application, and in particular, repeat-until-succeed quantum informa- tion schemes55–57benefit from a higher rate of success with brighter sources. Table I lists some experimentally measured count rates that give an idea of the brightness of various sources. We caution, however, that the numbers do not enable a fair comparison between different references since the measured count rates are highly dependent on experimental conditions such as the collection optics/position, detec- tion efficiency, and excitation intensity, which are highly variable from one experiment to another. Physically, the count rate from a single emitter is inversely pro- portional to its excited state lifetime that can, as we further discuss in Sec.II B, be decreased by integrating the SPE with a resonator so that its rate of spontaneous emission into a resonant mode of the cavity is enhanced. This is useful for several reasons. Firstly, as Fig. 3 illustrates, emission from the NV/C0’s ZPL constitutes only a small fraction of its entire emission, with the majority of it coming from red-shifted transi- tions to the ground state phonon sidebands. The Debye-Waller factor, which quantifies this fraction, is particularly small for the NV/C0and is approximately 0.04.58,59This is undesirable for many quantum infor- mation applications since coherent information may be lost to the phonon reservoir when transitions to the phonon sideband occur. A small Debye-Waller factor is also undesirable for many nanosensingapplications involving the NV /C0since many of them rely on the NV/C0’s spin dependent ZPL emission to read out the NV/C0’s spin state (see Fig. 4 ). Fortunately, by integrating the SPE with a resonator that has been engineered to be resonant at the SPE’s ZPL, high count rates into the desired ZPL transition can be achieved. 2. Indistinguishability Indistinguishability of photons is another important metric of SPEs designed for on-chip quantum information applications such as linear optical quantum computing,64quantum teleportation,65–67and entanglement swapping68,69that uses two-photon interference. In gen- eral, photons can be distinguished by their spectral/temporal shape, as well as their polarization and time-of-arrival at a particular location. Although two identical but spatially separated two-level systems should, in theory, emit photons with the same spectral content, this is typically spoilt by the emitters’ coupling to two different local environ- ments. At short (compared to the radiative lifetime) time scales, inter- actions with the solid-state environment through, for example,phonons, charge, or spin noise, 70,71perturb the energies of the two- level system and induce dephasing of the optical transitions that homogeneously broaden the linewidth and decrease the indistinguish- ability of emitted photons. On the other hand, slower interactions (relative to the radiative lifetime) will induce spectral diffusion of the emission wavelength and inhomogeneously broaden the linewidth [seeFigs. 5 and17(a) ]. Moreover, the excitation wavelength can also affect the distinguishability of emitted photons. FIG.3 .Typical photoluminescence spectra of a NV center (on a sapphire substrate). Red-shifted emission to the ground state phonon sidebands constitutes a large part of the spectrum, and the NV ZPLs constitute only a small fraction of the emissionspectra. Reprinted with permission from Brenneis et al. , Nat. Nanotechnol. 10, 135 (2015). Copyright 2015, Springer Nature Customer Service Center GmbH.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-4 VCAuthor(s) 2020In general, using a resonant excitation (637 nm for NV/C0)i sm o r e advantageous to the creation of indistinguishable photons since iteliminates timing jitters (which decreases indistinguishability) associ-ated with relaxation through phonons. 72Furthermore, higher fre- quency nonresonant excitation has greater potential of ionizing defectsaround the SPE, leading to larger charge fluctuations 73that will, in turn, induce spectral diffusion. Consequently, resonant excitation isgenerally preferred for generating indistinguishable photons.However, we note that for the NV /C0, resonant excitation cannot by itself generate ZPL photons continuously due to a permanent photo- ionization into the dark NV0state.74However, this can be alleviated by using a weak ( /C24100 nW) repump laser that is resonant with the NV0ZPL (575 nm). Although a nonresonant 532 nm repump is also a popular choice, for reasons noted above, a weak resonant 575 nmrepump is crucial to reducing longer term spectral diffusion bydecreasing the probability of ionizing defects around the NV center 74 (seeFig. 5 ). At short timescales, the photon’s indistinguishability can be esti- mated by the metric75 n/C17T2 2T1¼1 CT1; (1) where T1is the emitter’s radiative lifetime and T2is the coherence time of the optical transition that is defined as75 1 T2¼1 2T1þ1 T/C3 2¼C 2: (2) Here, T/C3 2is the reciprocal of the dephasing rate C/C3¼2=T/C3 2that causes additional (on top of the natural linewidth) homogeneous broadening, while Cis the (angular) FWHM of the emission’s homogeneouslyTABLE I.Comparison of selected integrated SPEs. kis the SPE’s emission wavelength, Rcount gives the count rate, and Vis the HOM visibility. gð2Þð0ÞandT1are as defined in Eqs. (4)and(1), respectively. C†is the experimentally measured FWHM of the optical transition that may or may not be homogeneously broadened. n†is computed using Eq. (1)withC!C†. We note that although n†2½0;1/C138no longer predicts the size of a HOM dip (since C†6¼Cfor an inhomogeneously broadened line), it still serves as a mea- sure of indistinguishability, with n†¼1 indicating perfect indistinguishability. We have made the following abbreviations for the sake of brevity: WG—waveguide, RR—ring reso- nator, NW—nanowire, NB—nanobeam, PCW—photonic crystal waveguide, PCC—photonic crystal cavity, GC—grating coupler, and BS—beamsplitter. Measu rement uncertainties are given in brackets. SPE Platform k(nm) gð2Þð0Þ Rcount T1(ns) C†/2p(GHz) n†V References InAs QD GaAs WG on SiN WG 1130 0.0(1) — 1.014(4) — — — 256 InAs QD GaAs RR on SiN WG 1110 0.52(8) — 0.263(7) — — — 256 InAsP QD InP NW on SiN WG 988 0.03 22.4 MHz 1.3 14.5 0.008 — 233 InAs QD GaAs WG on SiN WG 916 0.11(4) — 1.39(4) 2.20(19) 0.052(6) 0 :89þ0:11 /C00:29 266 InAs QD InP NB on Si WG/GC 1300 0.33 2.1 MHz 1.25 — — — 234 InAs QD GaAs NB/GC 921 — — 0.182(1) 0.96(7) 0.91(7) — 416 InAs QD GaAs NB/GC 927 0.05 2.9 MHz 1.4 — — — 246 InGaAs QD GaAs NB 921 0.006 — 0.185 1.12(3) 0.77(2) 0.94(1) 264 InAs QD GaAs PCW/GC 944-950 — — 0.346(2) 0.54 0.852(4) — 228 InAs QD GaAs PCW/GC 895 0.02(2) — 0.70(3) 0.22(3) 1.0(2) 0.80(3) 263 InAs QD GaAs PCW to fiber 931 0.20(8) 8.2(1.7) MHz 0.885 — — — 417 InGaAs QD GaAs WG 941 0.009(2) — 0.50(1) — — 0.975(5) 265 NV/C0Diamond RR 637 — — 8.3 40 0.000 5 — 151 NV/C0Diamond RR/WG/GC 638 — 325 Hz 8 51 0.000 4 — 155 NV/C0Diamond RR/WG/GC 637 0.24 15 kHz — — — — 154 NV/C0Diamond L3 PCC 637 0.38 13.2 kHz 4 10 0.004 — 109 NV/C0Diamond NB 638 0.2 — — 491 — — 123 NV/C0Diamond NB 637 0.28 60 kHz — 15 — — 137 NV/C0Diamond WG 637 0.07 1.45 MHz — 0.323 — — 129 NV/C0in ND GaP PCC 643 <0.5 1 MHz 12.7 — — — 199 NV/C0in ND HSQ WG/GC 650-700 <0.5 — 6 — — — 202 NV/C0in ND HSQ Cavity/WG/GC 700–750 <0.5 — 3 — — — 203 NV/C0in ND Ag Nanopatch antennas 650 0.41 56.3 MHz 0.36 — — — 205 SiV in ND Optical microcavity 737–759 <0.5 0.106 0.46–1.97 21 — — 210 to fiber –1.78 MHz GeV in ND HSQ WG/GC 602 <0.5 — 3.8 — — — 184 2D nanoflake WSe 2on silver NW 736.74 — 30 kHz 2.4 44.2 0.001 5 — 361 2D nanoflake WSe 2on MIM WG 737.1 — 300 Hz 3.2(1.1) 55.2 0.000 9 — 362 2D nanoflake WSe 2on LiNbO 3BS 720–760 — 4 kHz — — — — 364 2D nanoflake WSe 2on Si 3N4WG 730–750 0.47 100 kHz 7.99 725 27.5 /C210/C06— 363AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-5 VCAuthor(s) 2020broadened spectrum. We note that in general T2/C202T1, and therefore, n2½0;1/C138. Experimentally, a common way to measure the indistin- guishability of photons from a SPE is to measure the two-photon interference in a Hong–Ou–Mandel (HOM) type experiment76in which two indistinguishable photons arriving at a 50/50 beam-splitter at the same time should always end up in the same output port. In such a setup, coincidence counts of photons at both output ports should decrease to zero for two indistinguishable photons. It can be shown that Eq. (1)gives the efficiency (or the normalized size) of a HOM dip with n¼1 corresponding to perfect distinguishability of the photons.75HOM interference between spatially separated defect cen- ters has been experimentally demonstrated,77–79and as Table I dem- onstrates, multiple integrated QDs have also demonstrated near ideal indistinguishability. An intuitive way of understanding Eq. (1)is to see a transition with coherence time T2as emitting a photon wavepacket of temporal width T2=2, which sets the width of a HOM interference dip since that is the maximum temporal overlap between two distinct photons. Moreover, there is a time jitter of order T1f o rt h es p o n t a n e o u se m i s - sion to occur, and therefore, the probability of having two such distinct photon wavepacket interfere successfully is /C24T2=ð2T1Þ. This suggests that one way of increasing the indistinguishability of photons from SPEs is to reduce their radiative lifetime T1by placing them into a res- onant cavity. For example, this has been successfully done for quan-tum dots in micropillar cavities where a HOM dip was successfully measured. 80,81Moreover, such resonant cavities can be potentially used to implement other schemes for generating indistinguishable photons including cavity-assisted spin flip Raman transitions.72,82 Furthermore, cavities can be used to select a particular polarization,which is also important for indistinguishability, and a particular spatial mode, which can make outcoupling to an in-plane waveguide easier.Table I summarizes some of the experimentally realized T 1values of i n t e g r a t e dS P E s .W eh a v ea l s op r o v i d e d C†=ð2pÞvalues, which are the experimentally measured FWHM values that are not necessarily from homogeneously broadened lines. n†, which is calculated using Eq. (1) withC!C†, is also tabulated as a measure of indistinguishability. Despite the utility of cavities described above, we acknowledge that they can only help with short term dephasing processes that homogeneously broaden the linewidth. For longer term fluctuations due, for example, to ionization of nearby defects that lead to local charge fluctuations78,83or drifting strains84that shift the energies of the excited states, a different strategy is required. Passive solutions include carefully fabricating the material with tailored annealing and surface treatments,83while active solutions have also been investigated whereby the energies of the excited states are actively shifted via the Stark effect to stabilize the ZPL frequency.84Using these strategies, near life-time limited linewidths ( /C2413 MHz) have been obtained for NV/C0centers in bulk diamond at long time scales. 3. Single photon purity Although a “single photon” can, in principle, be obtained by suf- ficiently attenuating a classical light source like a laser, there is a subtle but important difference between such attenuated sources and true SPEs: whereas a true SPE will never emit two photons at the same time, an attenuated source can occasionally deliver two photons in a single pulse. This is highly undesirable for some applications like FIG.4 .Fluorescence from a NV/C0center after initialization to either a ms¼0o r ms¼1 state. The spin dependent fluorescence is due to a much faster nonradiative inter-system crossing (ISC) from the ms¼61 excited states to the1A singlet state (see Fig. 2 ), and consequently, fluorescence at the ZPL is due mostly to radiative decays from the ms¼0 excited state (Ref. 60). This spin-selective depopulation of 3Et o1A1also enables an /C2480% polarization of the ms¼0 ground state by optical pumping (Refs. 61and 62) that can then (if desired) be transferred to the ms¼61 states by applying a microwave ppulse. Data in this trace were obtained after averaging over 3 /C2107measurements. The optimal duration for photon counting can be obtained by using a maximum likelihood analysis to optimize the discrimination between ms¼0v s ms¼1 states. Reprinted with permission from Gupta et al. , J. Opt. Soc. Am., B 33, B28 (2016). Copyright 2016, The Optical Society (Ref. 63). FIG.5 .(a) and (b) Photoluminescence as a function of excitation frequency for a NV/C0in an appropriately processed diamond. The NV/C0is repumped at 532 nm for (a) and 575 nm for (b). Notice the significant decrease in spectral wandering for aresonant (of NV0) repump. Reprinted with permission from Chu et al. , Nano Lett. 14, 1982 (2014). Copyright 2014, American Chemical Society. (c) and (d): Photoluminescence as a function of excitation frequency for a NV without (c) and with (d) active feedback of the ZPL transition. Reprinted with permission fromAcosta et al. , Phys. Rev. Lett. 108, 206401 (2012). Copyright 2012, Author(s), licensed under a Creative Commons Attribution 3.0 Unported License.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-6 VCAuthor(s) 2020quantum key distribution since security of the distributed key will be compromised in such cases and some of the key exchanged betweenthe two parties will have to be discarded to ensure the security of theprotocol. 85,86An important metric that is typically used to measure a source’s single photon purity (in this sense) is the second order corre-lation function, 87 gð2Þðx1;x2Þ¼TrqE/C0ðx1ÞE/C0ðx2ÞEþðx1ÞEþðx2Þ/C2/C3 TrqE/C0ðx1ÞEþðx1Þ ½/C138 TrqE/C0ðx2ÞEþðx2Þ ½/C138; (3) where xiare space-time coordinates, qis the density matrix of the pho- tons, and E6ðxiÞare positive/negative frequency components of the electric field operator. Typically, the field is assumed to be stationary,and we are only interested in the time difference s¼x 0 2/C0x0 1so that Eq.(3)reduces to gð2ÞðsÞ¼hIðtÞIðtþsÞi hIðtÞi2; (4) where I(t) denotes the intensity (or count rate) and the brackets h…i can be interpreted as a time average. Intuitively, Eq. (4)can be under- stood as the number of photons detected after a delay sfrom the detection of a preceding photon at time t, normalized by the average count rate. Since the field is assumed to be stationary, tdrops out of the argument of gð2Þ. Evidently, given that a true SPE can only emit a single photon at any particular instance of time, gð2Þð0Þshould equal zero for a true SPE since the number of photons detected immediately after the detection of one photon should be exactly zero. In reality, additional background photons from other sources and finite time res-olution and timing jitter in photodetectors and time-to-digital convert-ers result in a nonzero g ð2Þð0Þ. For SPEs like NV centers, quantum dots, and defects in 2D materials, it is typically not possible to opticallyresolve two closely separated emitters, and a g ð2Þvalue of less than 0.5 is, therefore, typically used to discern whether emission is being collected from more than one emitter.88Table I lists some of the mea- sured gð2Þð0Þvalues of various SPEs from the experiment. In cases where corrected gð2Þvalues are available, we give those that have been corrected for the timing response of the equipment used. B. Enhancement of ZPL emission using resonant cavities We motivated in Secs. II A 1 andII A 2 how an enhanced sponta- neous emission rate for SPEs is beneficial for numerous applications. In this section, we review how such an enhancement can be achievedby integrating SPEs with a resonant cavity. Classical electromagnetism shows that the time-averaged power radiated by a dipole emitter can be written as P¼x 2Imp/C3/C1Eðr0Þ ½/C138 ; (5) where p,x,a n d Eðr0Þare the dipole moment, angular frequency, and electric field at position r0, respectively. The Purcell factor Fof a dipole emitter gives the enhanced emission rate of an emitter in an opticalcavity, normalized with respect to its emission rate in free space (i.e.,in the absence of the cavity). Using the well-known expression forE 0ðr0Þ, the electric field of a dipole in a homogeneous dielectric medium, the power radiated by a dipole in a homogeneous dielectricmedium may be, from Eq. (5), succinctly written asP 0¼l0jpj2nx4=ð12pcÞ,w h e r e nis the refractive index of the dielec- tric medium and cis the speed of light.89With these expressions for P andP0, the Purcell factor is given by F¼P=P0. One can also derive similar expressions for PandP0using Fermi’s golden rule. The quan- tum derivations will have an additional factor of 4, and this is related to fields from vacuum fluctuations.90Nevertheless, the factor cancels out in the ratio of Fso that both classical and quantum derivations yield the same result. When a dipole is placed in a structured dielectric medium like photonic crystal cavities (PCCS), the corresponding electric field can be expressed as a sum of the dipole’s own field E0ðrÞand the scattered electric field Esðr0Þ.U s i n g F¼P=P0and the expression for P0,i tc a n be shown that F¼1þ6pc l0njpj2x3Imp/C3/C1EsðrcÞ ½/C138 : (6) Equation (6)may be evaluated by numerically simulating a dipole source in a finite-difference time-domain91(FDTD) simulation of Maxwell’s equations where, in general, the field created by the dipole consists of its own field and the scattered field, but the scattered field EsðrÞmay be obtained by Fourier transforming the electric fields after the dipole excitation is switched off.92 The fields EðrÞandHðrÞare the total fields in the presence of the dipole emitter with current density, JðrÞ¼/C0 ixpdðr/C0r0Þ; (7) where pis the dipole moment and dðrÞis the Dirac delta function. EðrÞandHðrÞobey Maxwell’s equations for time harmonic fields, r/C2 EðrÞ¼ixl0HðrÞ; (8) r/C2 HðrÞ¼JðrÞ/C0ix/C15ðrÞEðrÞ: (9) Let us assume that the fields may be expanded using the quasi-normal modes of the optical cavity.93–95Under single-mode conditions, the expansions can be written as E¼X nanEn¼a0E0; (10) H¼X nanHn¼a0H0; (11) where a0is the expansion coefficient for the single mode.96,97E0and H0are the electric and magnetic fields of the quasinormal single mode, and they obey (in the absence of free currents) r/C2 E0ðrÞ¼i~x0l0H0ðrÞ; (12) r/C2 H0ðrÞ¼/C0 i~x0/C15ðrÞE0ðrÞ; (13) where ~x0is the complex frequency of the quasinormal mode. Applying the Lorentz reciprocity theorem [96] to the set of fields (E;H)a n d( E0;H0), we haveÐ d3rr/C1ð E/C2H0/C0E0/C2HÞ¼0. Subsequently, Eqs. (8),(9),(12),a n d (13) can be combined as iðx/C0~x0ÞÐd3rðE/C1/C15ðrÞE0/C0l0H/C1H0Þ¼Ðd3rJ/C1E0.U s i n gt h i s result and Eqs. (7),(10),a n d (11), it can be shown that the complex expansion coefficient a0is a0¼/C0xp/C1E0ðr0Þ ðx/C0~x0ÞI; (14)AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-7 VCAuthor(s) 2020where Iis given by I¼ð d3rðE0/C1/C15ðrÞE0/C0l0H0/C1H0Þ: (15) Using this integral, a mode volume Vfor the single mode can be defined as V/C17I 2/C150n2E0ðr0Þ/C1p ½/C1382; (16) where /C150is the vacuum permittivity and n2i st h es q u a r eo ft h er e f r a c - tive index. For cavities with a high quality factor, the quasinormal modes can be approximated to be normal modes where the integral Iand the mode volume Vare real-valued. If we assume that E02Re, we see from Eqs. (12) and (13) that H02Im, and therefore, I¼Ð d3r ð/C15jE0j2þl0jH0j2Þ¼2Ð d3r/C15jE0j2, where we have used the fact thatÐ d3r/C15jE0j2¼Ð d3rl0jH0j2, which follows from the identity r/C1ð ~E/C2~HÞ¼ ~H/C1r/C2 ~E/C0~E/C1r/C2 ~H: (17) And the requirement thatÐd3rr/C1ð E0/C2H/C3 0Þ¼0 for normal modes in which there is no out flow of energy from the cavity. By assuming arealIand using Eqs. (10)and(5),i tm a yb es h o w nt h a tt h em a x i m u m Purcell factor on resonance (i.e., x¼x 0¼Re½~x0/C138)i sg i v e nb y Fc¼P P0¼3 4p2k0 n/C18/C193Q V; (18) where Q¼Im½~x0/C138=ð2Re½~x0/C138Þandk0¼c=ð2px0Þ.I nt h ec a s eo fa slight deviation off resonance, it is straightforward to show thatF¼F cLsðxÞ,w h e r e LsðxÞis the Lorentzian line shape function, LsðxÞ¼x2 0 x2x20 x2 0þ4Q2ðx/C0x0Þ2: (19)Sauvan et al.96showed that for cavities with small quality factors, it is important to include the leakage part of the fields in the integral of I. Consequently, this leads to a complex volume, and in this case, onecan obtain a generalized F, F¼F cLsðxÞ1þ2Qx/C0x0 x0ImV½/C138 2ReV½/C138 ! ; (20) where Fcis now Fc¼3 4p2k0 n/C18/C193Q ReV½/C138: (21) Equations (18)and(21)indicate that to maximize the spontane- ous emission at the ZPL, it is necessary to achieve a high Q/Vratio at the ZPL wavelength. Typical cavity structures include microdisks,99,100 micropillars,101nanopockets,102–104and photonic crystals.105The whispering gallery modes of microdisk cavities can have very highquality factors of Q/C2410 5but relatively large mode volumes.99For micropillar resonators with integrated Bragg mirrors, very highQ>250,000 and small V<ðk=nÞ 3can be achieved, where kis the wavelength and nis the refractive index.101Nanowires106and nanopil- lars98(seeFig. 6 for an example), like micropillars, are also efficient vertical-emitting photon sources when coupled with a quantum emit- t e rs u c ha saq u a n t u md o to ra nN V/C0center. Yet, although they are useful for applications requiring outcoupling of light from the plane ofthe device, they are not well suited for planar routing of light. PCCspresent an appealing compromise between high Q,l o w V, and efficient in-plane coupling. PCCs commonly take the form of a membrane ofmaterial with a periodic lattice of air holes, where selected holes havebeen displaced to form a defect in the photonic crystal bandgap. Aprominent example is the L3 cavity with three missing holes in a line(seeFig. 10 for an example), which supports polarized single modes with a relatively wide spectral margin. 105For fabricated L3 cavities FIG.6 .(a) Scanning electron microscopy (SEM) image of diamond nanopillars. (b) and (c) FDTD simulations of the E-field’s magnitude for orthogonal orientat ions of the optical dipole (shown with white arrows). Reprinted with permission from Neu et al. , Appl. Phys. Lett. 104, 153108 (2014). Copyright 2014, AIP Publishing.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-8 VCAuthor(s) 2020with embedded QDs on a GaAs platform, Q/C24104andV/C24ðk=nÞ3 have been demonstrated,107,108while an NV/C0center integrated with an all-diamond L3 cavity achieved a Purcell factor of 70.109AH 1 PCC, which consists of a single central hole in a triangular 2D lattice,w a sa l s or e c e n t l yu s e dt oa c h i e v ea4 3f o l ds p o n t a n e o u se m i s s i o nenhancement from an InGaAs quantum dot in a GaAs cavity. 110 For optimal coupling to the cavity, it is important for the quan- tum emitters, which may be modeled as dipole emitters, to be correctlyoriented and positioned to match the cavity’s mode and polarization.For example, light emission from a strain-free NV /C0center has a single ZPL transition that may be modeled using a pair of orthogonal dipoles111(with equal strength of dipole moment) perpendicular to the NV axis, while SiV centers, which have four ZPL transitions at cryo-genic temperatures, 112,113may be modeled as single dipole emissions. A misalignment of the optical dipole’s orientation with respect to the cavity can significantly affect the modes it excites as Fig. 6 shows. Cavities are not only useful for enhancing spontaneous emis- sion but also useful in enhancing absorption, which can be beneficialunder certain spin-to-charge, 114,115spin-to-photocurrent,54,116and magnetometry117,118read-out schemes. Besides enabling enhanced emission/absorption and out-of-plane waveguiding of quantum emitters, integrated photonic structures can also of course providefor in-plane waveguiding through conventional waveguides or line-defect PCC waveguides. In Secs. III–VII , we review various examples of integrated SPEs and different integration techniques that have been employed. III. INTEGRATED NV /C0CENTERS IN BULK DIAMOND NV/C0centers have been successfully integrated with a variety of photonic structures in bulk diamond, which we here define as dia- mond substrates that are larger than nanodiamonds. Although nano-diamonds are in some ways easier to integrate with dissimilarphotonic structures, they tend to suffer from poorer photostability and shorter coherence times due to their larger surface area to volume ratio that makes them particularly susceptible to surface effects. It is, there-fore, desirable to integrate NV /C0centers in bulk diamond to other pho- tonic structures. These structures can be fabricated from the same bulkdiamond substrate or they could be made of a dissimilar material and coupled to NV /C0centers in a bulk diamond substrate evanescently. Moreover, with the advent of pick-and-place techniques, it is possibleto envision NV /C0in diamond photonic structures that are, in turn, coupled to other dissimilar material systems that could offer additional functionalities. For fabricating all diamond photonic structures that contain only a single mode around the NV/C0center’s ZPL wavelength (637 nm), it is necessary to use thin membranes of diamond ( n/C252:4 at 637 nm), which are /C24200 nm thick. Although such membranes may be obtained from nanocrystalline diamond films grown on a substrate,119,120their optical quality is typically worse compared to bulk single-crystal diamonds due to increased absorption and scat-tering. 119It is, therefore, preferable to obtain such thin diamond membranes from (typically oxygen plasma) reactive ion etching (RIE) of bulk single-crystal diamonds,121which is a fabrication pro- cess that has been demonstrated to be compatible with a moderatelylong NV /C0spin coherence lifetime of /H11407100ls (Ref. 122) while also being consistent with low optical losses.106,109Moreover, RIE (with oxygen plasma) can be used to create surface-termination of thediamond membrane that encourages the conversion of NV0to NV/C0 states.51To obtain good mode confinement, it is also typical to undercut the structures so as to achieve a large refractive index con-trast between diamond and air. This may be achieved in severalways. For example, the diamond membrane can be first mounted on a sacrificial substrate, processed, and then made into a freestanding structure by a final isotropic etch step that removes the sacrificialsubstrate under the area of interest. 109,123Alternatively, angular124,125 and quasi-isotropic100,126,127RIE etching can also be employed to create such freestanding structures. Instead of creating a freestanding structure, another typical variation is to further etch the substrate to create a pedestal, which would reduce the leaking of fields into thesubstrate. 128 Such all diamond photonic structures can then, in principle, be transferred to other dissimilar systems by using a pick-and-place tech-nique to create a hybrid material platform. For example, this was dem-onstrated in Ref. 129where NV /C0containing diamond waveguides were transferred to a silicon platform containing SiN waveguides. GaP-diamond is another popular hybrid platform due to both thehigh refractive index of GaP ( /C253:3 at 637 nm) (compared to diamond /C252:4 at 637 nm) and its relative ease of fabrication using standard semiconductor processing technology. In addition, unlike an all (bulk)diamond platform, which by inversion symmetry has a zero second- order nonlinear susceptibility ( v ð2Þ) (we note, however, that diamond has a nonzero third-order nonlinear susceptibility vð3Þand that it has a relatively high nonlinear refractive index that allows it to be harnessedfor nonlinear four-wave mixing processes 130) GaP possesses a rela- tively large vð2Þthat allows it to be used in nonlinear processes such as second harmonic generation.131Moreover, unlike diamond that is for- bidden by symmetry to have a bulk linear electro-optic coefficient,132 GaP has a nonzero linear electro-optic coefficient ( r41/C25/C00:97 pm/V at 633 nm (Ref. 133) that enables it to be used for active electro-optic switching applications (as has been demonstrated in AlN material sys-tems 134,135) and as a III-VI semiconductor, GaP can potentially host on-chip integrated single-photon detectors as demonstrated on GaAs waveguides.136In Subsections III A–III E , we review some examples of integrated NV/C0. A. Integration with waveguides NV/C0centers have been integrated with waveguides in a variety of ways. One direct approach is to fabricate an all-diamond waveguide on a thin diamond membrane using a mask and RIE etch. Since the diamond membrane will have randomly dispersed native NV/C0cen- ters, some of these diamond waveguides will, by chance, have NV/C0 centers in the approximately correct location within the waveguides.These NV /C0integrated waveguides can then be postselected and used to form more complex photonic circuits. This approach was taken in Ref. 129where tapered diamond microwaveguides were fabricated from a 200 nm thick single crystal diamond membrane with a Si maskand RIE etch. In this case, the Si mask was separately fabricated usingwell developed silicon fabrication processes and then transferred ontoa diamond substrate using a mask transfer technique. Due to the mature silicon technology, such masks can be fabricated with stringent tolerances, and their patterns can then be transferred to the diamondsubstrate after a RIE etch. The resulting diamond waveguides are,then, characterized by photoluminescence measurements and thosethat are found to have a single NV /C0in the center of the waveguide, asAVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-9 VCAuthor(s) 2020verified by gð2Þð0Þmeasurements, are selected and placed on top of an air gap in between SiN waveguides by a probe (see Fig. 7 ). Due to the air gap and taper of the diamond waveguides, up to 86% of the NV/C0’s ZPL emission can be coupled to the SiN waveguides.129The back- ground corrected saturated count rate from one end of the SiN wave-guides was estimated to be 1 :45/C210 6photons/s, and a gð2Þð0Þvalue as low as 0.07 was obtained. Photoluminescence excitation measure-ments of the NV /C0revealed a FWHM of 393 MHz, and ODMR Hahn- echo measurements revealed a relatively long spin coherence time of T2/C25120ls, which is, as in Ref. 137, close to the spin coherence time of NV/C0centers in high quality bulk diamond crystals. This can likely be extended to the ms range if isotopically purified12Cc a r b o n(13C has a nuclear spin that decoheres the NV/C0spin) is used instead.138 It is also possible to integrate NV/C0with diamond waveguides using fs-laser writing. As discussed in Sec. VII B 3 , fs laser pulses are capable of creating NV/C0centers in diamond. Moreover, as in the case of crystals like LiNbO 3139,140and sapphire,141it is possible to inscribe waveguides in diamond with fs laser writing. This may be accom-plished by writing two parallel lines in diamond, which results ingraphitization of material within the focus, leading to a decreasedrefractive index that, in turn, enables the confinement of an opticalmode between the two laser written lines. In addition, the graphitized material, which has lower density, expands and causes stress-induced modification to the refractive index of the surrounding diamond,which leads to vertical confinement of the optical mode. 142 Importantly, the laser inscribed waveguides in diamond surviveannealing at 1000 /C14C,143which is commonly required for the forma- tion of NV/C0centers, but are not necessarily guaranteed as in the case of laser inscribed waveguides in sapphire.141Since the same fs laser system can be used to both create NV/C0centers and write waveguides within bulk diamond, submicrometer relative positioning accuracy ispossible between the NV /C0center and waveguide. Using this technique, single NV/C0centers, with a 31 69% probability of creation per 28 nJ pulse, were successfully incorporated into the midst of laser writtendiamond waveguides, and waveguiding of their spontaneous emission was confirmed 144(seeFig. 8 ). The measured gð2Þð0Þvalues of such NV/C0emission were as low as 0.07, confirming that single NV/C0centers were indeed deterministically created in the waveguides.144 NV/C0centers in bulk diamond have also been integrated with GaP waveguides, although there has not, to our knowledge, yet been a4demonstration of deterministic single NV /C0integration with GaP wave- guides. However, NV/C0centers created by ion implantation in high- pressure high-temperature (HPHT) type Ib diamonds (at a depth of/C25100 nm) have been evanescently coupled to a 120 nm thick GaP rib waveguide that was transferred onto diamond via epitaxial liftoff145after removal from its underlying Al 0.8Ga0.2P sacrificial layer atop a GaP sub- strate.146Evanescent coupling between the NV/C0and GaP waveguide was successfully observed when NV/C0emission was detected after send- ing in a 532 nm excitation beam through the GaP waveguide.146The evanescent coupling requires that NV/C0centers are created close to the diamond’s surface and for gaps between GaP and the diamond substrate to be minimized. Indeed, a significant disadvantage of a (bulk) hybrid platform compared to an all-diamond one is that the NV/C0centers can- not generally be placed in a maxima of the optical mode that would oth- erwise enable good optical coupling and an enhancement of spontaneous emission rates. To mitigate this, NV/C0centers may be cou- p l e dt oo p t i c a lr e s o n a t o r ss u c ha sm i c r o d i s k sw i t has u f fi c i e n t l yh i g h quality factor,147and light within these resonators may then be out- coupled via coupling with another waveguide148,149(see Sec. III C). Despite the high refractive index of GaP, waveguiding in a GaP wave- guide atop a diamond substrate can still be significantly lossy due to the reduced effective index of the guided mode and the moderately high refractive index of diamond. To enable waveguiding and to reduce losses due to mode leakage into the substrate, it is common to decrease the effective index of the substrate (from its bulk value) by etching it so as to create a diamond pedestal beneath the resonator.128,147–150 B. Integration with resonators 1. Diamond ring resonators One of the earliest demonstration of NV/C0integration with a res- onator came in 2011 with the successful coupling of a NV/C0with a FIG.7 .Top: Schematic of single-mode diamond waveguides containing single NV/C0 centers suspended over single-mode SiN waveguides assembled using a tungsten probe. Bottom: Electric field profiles from FDTD simulations showing the transfer of mode from the suspended diamond waveguide (gray) into the underlying SiN wave- guide (brown). Reprinted with permission from Mouradian et al. , Phys. Rev. X 5, 031009 (2015). Copyright 2015, Author(s), licensed under a Creative CommonsAttribution 3.0 Unported License. FIG.8 .Raman and NV fluorescence for various excitation (E)/collection (C) modes of an integrated laser written diamond waveguide and NV/C0center. (a) Raman scat- tered light collected above the waveguide after butt-coupled excitation of the wave- guide, demonstrating waveguiding within the diamond. (b) Fluorescence of NV/C0 above the waveguide after sending in excitation light from the end of the wave- guide, demonstrating that the NV/C0is sufficiently close to the optical mode of the waveguide to interact with it. Fluorescence was measured using a home-built con- focal microscope. (c) Same as in (b), but the fluorescence is measured using an electron multiplying charged coupled detector (EMCCD). (d) Fluorescence of theNV/C0measured at the end of the waveguide using an EMCCD after the NV/C0center was excited from above. Reprinted with permission from Hadden et al. , Opt. Lett. 43, 3586 (2018). Copyright 2018, The Optical Society.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-10 VCAuthor(s) 20204.8lm outer diameter and a 700 /C2280 (width /C2height) nm diamond microring resonator on top of a 300 nm high SiO 2pedestal with mode volumes in the range of /C2517/C032ðk=nÞ3(Ref. 151)( s e e Fig. 9 ). Photoluminescence measurements resolved roughly ten native NV/C0 lines within the resonator. Characterization of the resonator was doneat cryogenic temperatures ( <10 K), and xenon was flowed through the cryostat, which allowed tuning of the cavity’s resonance as the xenon condensed on the cavity and altered its resonance wavelength. 152A spontaneous emission enhancement of /C2512 was obtained on resonance with a FWHM of /C2540 GHz and a radiative lifetime of 8.3 ns. The broad linewidth has been attributed to strain within the diamond. 2. Diamond 2D photonic crystal cavities As discussed in Sec. II B, although ring resonators are capable of achieving high Qfactors, yet they tend to have large mode volumesand are therefore not as ideal in achieving Purcell factors. On the other hand, photonic crystal cavities provide a good compromise between having high quality factors and small mode volumes, which makes them particularly useful for enhancing a SPE’s spontaneous emission rate. Accordingly, there have been various attempts at integrating NV/C0centers with PCCs. In Ref. 109, an all-diamond suspended L3 cavity with a theoretical mode volume of /C250.88ðkmode=nÞ3and Q¼6000 was designed and fabricated to have a resonance close to the NV/C0’s ZPL (see Fig. 10 ). Confocal characterization of a native NV/C0 center that was successfully coupled to the cavity gave a measured gð2Þð0Þvalue of 0.38 and an on-resonant radiative lifetime of 4 ns. As in Ref. 151, the cavity’s resonance wavelength was tuned by flowing xenon in a cryogenic environment. High resolution photolumines-cence excitation measurements revealed two distinct peaks from the coupled NV /C0center, with the FWHM of the main peak at /C258G H z . Given that gð2Þð0Þis 0.38, it is likely that the double peaks are due to FIG.9 .A microring resonator fabricated out of a single diamond crystal using EBL followed by RIE. Insets show the side wall roughness and how the SiO 2substrate has been etched by 300 nm during the fabrication process. A NV/C0center coupled to the resonator was observed to have a spontaneous emission enhancement of /C2412. Although theo- retical Qvalues exceeded 106, the achieved quality factors were only /C245000 due to the surface roughness of the ring. Reprinted with permission from Faraon et al. , Nat. Photonics 5, 301 (2011). Copyright 2011, Springer Nature Customer Service Center GmbH. FIG. 10. (a) Simulated electric field energy density of an L3 cavity. (b) SEM image of an L3 cavity fabricated on a single crystal diamond membrane using EBL and RI Ea s described in the text. Reprinted with permission from Faraon et al. , Phys. Rev. Lett. 109, 033604 (2012). Copyright 2012, the American Physical Society.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-11 VCAuthor(s) 2020strain-split branches of the same NV/C0and not to two spatially sepa- rated NV/C0centers. 3. Diamond 1-D photonic crystal cavities Besides 2-D PCCs, NV/C0centers have also been successfully inte- grated with 1-D PCCs. A relatively common 1-D PCC is a nanobeamthat consists of a suspended diamond waveguide that contains peri-odic holes in it with some (optical) defects introduced near its center.Typically, the defect consists of either missing holes or holes with slightly different periodicity near the center. However, it is also possi- ble to introduce a defect by increasing or decreasing the width of thewaveguide in the middle. 153In Ref. 123, 1D PCCs were created out of suspended 500 nm wide diamond waveguides that had 130 nm diame- ter air holes in it with a periodicity of 165 /C0175 nm and a 400 nm tapered width in the middle (see Fig. 11 ). Photoluminescence excita- tion and white light transmission spectra give a spectrometer limitedQof above 6000 (simulated Qwas/C255/C210 5), and the mode volume is estimated to be 1.8 ðk=nÞ3. In this case, a two pronged strategy was employed to tune the cavity’s resonance to the ZPL of native NV/C0 centers within the nanobeams. First, a coarse tuning by means of con- trolled oxygen plasma etching was employed to blue shift the reso-nance, and then, the device is later placed in a cryogenic environment (4 K) and xenon gas was introduced, as above, to red shift the reso- nance more precisely. A spontaneous emission enhancement of 7 wasobserved on resonance, and the g ð2Þvalue of 0.2 can be obtained. However, we note that the FWHM of the plotted photoluminescenceis rather large at /C24490 GHz. In Ref. 137, 1-D nanobeams were also fabricated from a diamond membrane, albeit with a new Si mask transfer and RIE etch technique. NV /C0centers were then created by15N implantation and annealing. The cavities had a theoretical mode volume of 1.05 ðk=nÞ3and quality factors ranging in the 1000s. For a nanobeam with Q¼17006300, a Purcell factor of 8 and 15 was achieved, for the ExandEybranches of a single NV/C0’s ZPL, respectively. The measured gð2Þvalue was 0.28. As before, the enhancement was lower than expected from the Q/Vratio, but this is here attributed to a poor alignment of the NV axis andnonideal spatial position in the cavity. In another nanobeam with a Q factor of 3300 650, a larger Purcell factor of 62 was achieved, but in this case, there were multiple NV/C0centers present (as judged by mul- tiple spectrally distinct ZPL transitions). Interestingly, however, the single NV/C0center was observed to retain a long spin coherence time of/C24230ls, which is similar to the spin coherence time of NV/C0cen- ters in the parent unprocessed diamond. This indicates that the fabri- cation process using a Si mask did not adversely degrade the properties of the NV/C0center and is promising for future applications requiring long spin coherence times in cavity coupled SPEs. C. Larger scale integration Larger scale integration has also been achieved on an all- diamond platform consisting of microring resonators coupled with waveguides and grating couplers.154,155The first demonstration154was characterized at room temperature with a confocal microscope that had two independent collection arms, with one of the collection arms also being used to excite the cavity. This allowed the structure to beexcited at one location, while emitted photons were collected at a dif- ferent location. The microring resonator had an outer diameter of 40lm and a 1000 /C2410 nm cross-section. Fluorescence collected from the output of both gratings gave a saturated count rate of ð1560:1Þ/C210 3Hz with a saturated pump power of ð10064ÞlW. FDTD modeling suggests that the total collection efficiency is /C2415%, and therefore, it appears that there remains room for significant improvement. In addition, coincidence counts of photons collected from the ring and each grating under simultaneous excitation ofthe ring gave a g ð2Þvalue of /C240.24, indicating that a single native NV/C0center in the ring had successfully outcoupled to the gratings (seeFig. 12 ). Moreover, the fluorescence spectrum suggests a loaded Q ofð3:260:4Þ/C2103at 665.9 nm. However, no attempt was made to determine if the ring had successfully enhanced the spontaneous emis-sion rate of the coupled NV /C0center. This was accomplished in a slightly later demonstration of a very similar system consisting of a 4.5 lm outer diameter microring resona- tor, a ridge waveguide of about /C25100 nm away, and grating FIG.1 1 . (a) Left: SEM top view of the outcoupling region of a suspended diamond 1D nanobeam. Right: SEM top view of a suspended diamond 1D nanobeam. Note the tape r- ing of the width in the middle. (b) Transmission measurement with a supercontinuum source coupled in from the right. Notice the localization of the mod e in the center of the nanobeam. (c) Transmission spectrum indicating Q/C246000. Reprinted with permission from Hausmann et al. , Nano Lett. 13, 5791 (2013). Copyright 2013, American Chemical Society.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-12 VCAuthor(s) 2020couplers.155In this experiment, a spontaneous emission enhancement of/C2512 was reported after applying the same technique as in Ref. 151 to tune the cavity’s resonance to native NV/C0centers in the resonator. Transmission measurements of the mode used to enhance the NV/C0 center(s) ZPL line gave a coupled Qfactor of 5500, and the mode vol- ume is estimated to be /C2515ðk=nÞ3. Excitation of the native NV/C0cen- ter(s) in the bulk material showed that when the cavity was onresonance, approximately 25 times more photons were collected fromthe grating than from the bulk. However, given that no g ð2Þmeasure- ments were performed, it is not clear if only one NV/C0center was excited in each case, and it is, therefore, difficult to make an unambig- uous comparison. Larger scale integration has also been achieved on a GaP-on-dia- mond hybrid architecture.148,149In Ref. 149,a1 2 5 n ml a y e rt h i c ko f GaP was transferred onto a diamond substrate, which had previouslybeen implanted and annealed to produce NV /C0centers approximately 15 nm below the surface. The GaP was, then, patterned using electron-beam lithography and etched through using a Cl 2/N2/Ar RIE. To obtain better mode confinement, the diamond substrate was further etched using O 2RIE to get an /C25600 nm high diamond pedestal. GaP disk resonators, waveguides, directional couplers, and gratingcouplers on a diamond pedestal were fabricated using this approach 149 (seeFig. 13 ). The coupled disk resonators were measured via transmis- sion measurements to have loaded Qfactors in the range of 2500/C010 000, and a full range of coupling ratios were obtained by varying the directional couplers’ coupling region’s length, whichconsisted of two 160 nm ridge waveguides spaced 80 nm apart [seeFig. 13(b) ]. Emission from NV /C0below the ridge waveguide was suc- cessfully outcoupled by the grating couplers, but unfortunately, noNV/C0ZPL line was observed at the grating couplers when the coupled disk resonators were excited by 532 nm light due to a mismatch of thecavity’s resonance (no attempt was made to tune the resonance here). D. Deterministic integration Most of the examples we have cited thus far relied on native NV /C0centers that were randomly dispersed in the diamond mem- brane. Given that optimal placement of the NV/C0center within the cavity is crucial to obtaining ideal spontaneous emission enhancement,deterministic integration of NV /C0centers is an important technological milestone. A 1-D deterministic integration of NV/C0centers in the verti- cal dimension was first attempted by delta-doping156a high purity chemical vapor deposition (CVD) grown diamond membrane with athin (/C246 nm) layer of nitrogen impurities. 157This produces a layer of NV/C0centers, which is well localized in the vertical dimension. Following this, 1-D suspended nanobeams are fabricated on the mem-brane with a theoretical Qvalue of /C24270,000 and a mode volume of /C240.47ðk=nÞ 3. However, measured photoluminescence spectra indi- cated that the highest Qobtained experimentally was /C2424,000, which is generally attributed to imperfections in fabrication. A spontaneousemission enhancement of 27 was obtained by cooling the diamond tocryogenic temperatures (4.5 K) and flowing nitrogen into the cryostatto tune a cavity with the mode of Q/C247000 to the NV /C0’s ZPL. An on- resonance lifetime of 10.43 60.5 ns was also measured, which when combined with the off resonance life-time of 22.34 61.1 ns, and a Debye-Waller factor of 0.03, gives a similar Purcell factor of 22. It isw o r t hn o t i n gt h a tt h ee n h a n c e m e n ti ss o m e w h a ts m a l l e rt h a nexpected based on the structure’s Q/Vratio. Compared to Ref. 123,t h e Q/Vratio is larger by a factor of 30, but the enhancement is only FIG. 12. (a) and (b) Confocal images of the microring resonator, waveguide, and grating coupler. (c)–(e) gð2Þmeasurements made using correlation from light collected at C11, C21,C22, and C23in (a), (b). Reprinted with permission from Hausmann et al. , Nano Lett. 12, 1578 (2012). Copyright 2012, American Chemical Society.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-13 VCAuthor(s) 2020/C244t i m e sl a r g e r .S i m i l a r l y ,t h e Q/Vratio of this nanobeam is /C245t i m e s larger than the L-3 cavity in Ref. 109, but the enhancement is actually /C243t i m e s smaller . The surprisingly low enhancement in this case is attributed to the fact that there is likely to be several NV/C0centers in the nanobeam, and their linewidths has been estimated to be/C24150–260 GHz, which is, for comparison, /C244t i m e sl a r g e rt h a ni n Ref.109. A consequence of this large linewidth is that the NVs’ ZPL is poorly coupled to the cavity’s resonance since the cavity mode is con- siderably narrower. The low enhancement could also plausibly be due to poor alignment of the NV axis (which affects the polarization of itsemission) to the cavity. Deterministic placement of NV /C0in 2D photonic crystals has also been attempted using ion implantation through a hole in an atomicforce microscopy (AFM) tip. 158,159In Ref. 158,N V/C0centers were cre- ated at the center of 2D PCCs with a measured Qin the range of 150/C01200 and estimated Vmode/C251ðk=nÞ3. The created NV/C0sh a da relatively large spectral diffusion limited FWHM of /C24250 GHz at 10 K and were expected to have a lateral spatial resolution of <15 nm and a vertical spatial resolution of /C253 nm. Unfortunately, the spatial resolution of the NV/C0centers was only measured to be less than 1 lm, and it is not obvious if the expected resolution was actually obtained.Moreover, the NV /C0creation yield was quite low at 0.8 60.2%. Higher deterministic single NV/C0creation yield can be obtained by using hard Si masks for both implantation and diamond patterning where a single NV-cavity system yield of 26 61% was obtained.160By having a high Si mask aspect ratio for the “implantation” holes, theetching rate for the underlying diamond substrate is negligible, but Nions during implantation are still able to implant into the diamonddue to different conditions for etching and implantation. 160This allows for the use of a single mask for both diamond patterning andNV /C0positioning, thereby eliminating any loss of accuracy due to realignment of separate masks. Unfortunately, there was no definite measurement of the overall spatial resolution obtained although a 1Dnanobeam with Q¼577 and coupled single NV /C0center was reported. Nevertheless, considering that spatial positioning of NV/C0to about /C2410 nm has been obtained using masks,161,162this scalable approach to deterministically position NV/C0centers within photonic structures is promising. E. Outlook Moving forward, we believe that there is still much room for improvement in deterministically integrating high quality single NV/C0 centers to complex photonic circuits using diverse strategies that havebeen developed over the years. Although most of the work discussedabove, which demonstrated coupling of NV /C0centers to all-diamond photonic structures, relied on randomly positioned native NV/C0cen- ters, we note that spatial positioning of NV/C0centers to about /C2410 nm in all three dimensions161,162has already been separately demon- strated. In these demonstrations, delta doping of CVD grown diamond(see Sec. VII B 1 ) is typically used together with a mask for accurately creating vacancies via irradiation, which then lead to NV /C0formation in the thin nitrogen doped layer after annealing. Previously, such amask consisted of spin-coated resist 161and other additional layers162 that can be difficult to coat with even thickness over a large area.However, the recent development of mask transfer techniques (see Sec.VII D 2 ) opens up the possibility of using high quality Si masks FIG. 13. Examples of integrated hybrid GaP-diamond systems where a thin 125 nm film of GaP was epitaxially grown and transferred onto a diamond substrate before being patterned with EBL and RIE. Top: Schematic view overlaid with FDTD simulations. Bottom: SEM images of integrated devices. (a) Waveguide-coupled dis k resonators, (b) directional coupler, and (c) grating coupler. Reprinted with permission from Gould et al. , J. Opt. Soc. Am., B 33, B35 (2016). Copyright 2016, The Optical Society.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-14 VCAuthor(s) 2020that can be positioned with submicrometer or even nm scale accuracy on the diamond substrate and then later removed mechanically. It is, therefore, possible to imagine using a single mask to create both NV/C0s and photonic structures at deterministic positions.160Successful NV/C0 integrated photonic structures, as characterized by optical measure-ments, can then be picked-and-placed (as in Ref. 129) by a microprobe to integrate with other photonic structures of a potentially dissimilar material that will further unlock other functionalities. This allows NV /C0centers to be created in high quality single crystal diamond (potentially with isotopically enriched12C) where they can have long optical and spin coherence times, while still being able to be efficiently routed and processed by other photonic elements on a chip. IV. COLOR CENTERS IN NANODIAMONDS As discussed in Sec. III, NV centers are not the only defects in diamond, which exhibit discrete energy levels with optical transitions although they are arguably the most studied defect. Recently, defect centers consisting of group-IV elements such as silicon vacancy (SiV),163,164germanium-vacancy (GeV),165and tin-vacancy (SiV)166,167centers in diamond have been of particular interest due to symmetries in their configuration, which leads to a higher Debye- Waller factor and narrower spectral lines that increase the indistin- guishability of their emitted photons. Moreover, although there has, as Sec.IIIshows, been a great deal of work in bulk diamond, there has also been considerable work in integrating color centers in nanodia- monds to hybrid photonic structures. We note, however, that NV cen- ters in nanodiamonds are less photo-stable and tend to have significantly larger inhomogeneously broadened ZPL linewidths as compared to their bulk counterparts. Although this makes them lesssuitable for many quantum computing/processing applications, nano- diamonds are more suited for bio-sensing/labeling applications, 168–170 and interestingly enough, the spontaneous emission rates of NV cen-ters in nanodiamonds can also be enhanced by encasing them in phenol-ionic complexes. 171Moreover, nanodiamonds with a high con- centration of SiV centers can also be used as temperature sensors.172 In this section, we give examples of other color centers in nanodia-monds that have been coupled to photonic structures. A. Fabrication Nanodiamonds are synthesized by various techniques such as detonation, laser assisted synthesis, HPHT high energy ball milling of microcrystalline diamond, hydrothermal synthesis, CVD growth, ion bombardment on graphite, chlorination of carbides, and ultrasonic cavitation. 173In the laboratory, the detonation method and HPHT growth are commonly employed to synthesize NV containing nano- diamonds on a large scale.174CVD growth is another promising technique that has successfully synthesized single NV centers in nano- diamonds.175More recently, a new metal-catalyst free method to syn- thesize nanodiamonds with varying contents of NV and SiV centersproduced high-quality color centers with almost lifetime-limited linewidths. 176,177 In Ref. 178, the authors reported the first direct observation of NV centers in discrete 5 nm nanodiamonds at room temperature. Although the luminescence of those NV centers was intermittent (i.e., they undergo blinking), the authors were able to modify the surface of the nanodiamonds to mitigate the undesirable blinking. In another work, the authors showed the size reduction of nanodiamonds by airoxidation and its effect on the nitrogen-vacancy centers that they host.179The smallest nanodiamond in their samples that still hosted a NV center was about 8 nm in size. SiV centers in nanodiamonds have subsequently been investi- gated.180–182Reference 180described the first ultrabright single photon emission from SiV centers grown in nanodiamonds on iridium. TheSiV centers were grown using microwave-plasma-assisted CVD, andthose single SiV /C0defects achieved a photon count rate of about 4.8 Mcounts/s (at saturation). Bright luminescence in the 730–750 nm spectral range was observed using confocal microscopy. No blinking was observed, but photobleaching occurred at high laser power.Enhanced stability might be gained by controlling the surface termina-tion of the nanodiamonds, as was shown for the case of NV centers. 50 Residual silicon in CVD chambers often results in the formation of SiV/C0centers in most CVD-grown nanodiamonds.177,180Likewise, due to silicon-containing precursors, many HPHT-synthesized nano-diamonds also include SiV /C0centers.183In Ref. 182, the authors dem- onstrated optical coupling of single SiV/C0centers in nanodiamonds and were able to manipulate the nanodiamonds both translationallyand rotationally with an AFM cantilever. Fabrication of other color centers such as GeV centers in nano- diamond were also recently demonstrated. For example, single GeVcenters in nanodiamonds were successfully fabricated by the authors in Ref. 184after they introduced Ge during HPHT growth of the nanodiamonds. More generally, in Ref. 185, the authors studied a larger variety of group IV color centers in diamond, including SiV,GeV, SnV, and PbV centers. We note that it is possible to control the size and purity of the HPHT nanodiamonds down to 1 nm. 186In other works, the size of nanodiamonds is typically tens of nanometers,187–189which makes nanomanipulation of them feasible. For example, emission from singleNV centers hosted in uniformly sized single-crystal nanodiamondswith a size of 30.0 65.4 nm has been reported. 189 Although high count rates are, in general, achievable for NV and SiV color centers in nanodiamonds,190these high count rates were sometimes reported to be correlated with blinking.191Compared to SiV centers in the bulk, SiV centers in nanodiamonds have signifi- cantly less reproducible spectral features and can feature a broad rangeof ZPL emission wavelengths and linewidths. 192More generally, the linewidths of SiV centers in nanodiamonds have been shown todepend on the strain of the diamond lattice. 192Nevertheless, it is sometimes possible to obtain nearly lifetime-broadened optical emis-sion in SiV centers in nanodiamonds at cryogenic temperatures, 176,193 and indeed, nearly lifetime limited zero-phonon linewidths have been obtained in both NV and SiV centers in nanodiamonds. For example, despite spectral diffusion and spin-nonconserving transitions, zero-phonon linewidths as small as 16 MHz have been reported for NVcenters in type Ib nanodiamond at low temperature. 194 For GeV centers in HPHT nanodiamond, the stability of its ZPL emission wavelength and linewidth has been attributed to the symme-try of its molecular configuration, although a large variation of life-times was also reported. 195T h ea u t h o r st h e r ee s t i m a t eaq u a n t u m efficiency of about 20% for GeV centers in HPHT nanodiamonds. B. Integration with photonic structures As mentioned in Sec. III, a hybrid GaP-diamond platform is attractive for multiple reasons, and there has been work involving notAVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-15 VCAuthor(s) 2020just bulk GaP-diamond systems but also hybrid GaP-nanodiamond systems. For an extensive review, see Ref. 174. Purcell enhancement of the ZPL emission by a factor of 12.1 has been reported in a hybridnanodiamond-GaP platform where the ZPL of an NV center is cou-p l e dt oas i n g l em o d eo faP C C . 196In that work, both the nanodia- mond and cavity are first preselected, and the resonance of the cavityis then tuned to the ZPL of the NV center by locally oxidizing the GaPwith a focused blue laser. 196Finally, the preselected nanodiamond is, then, transferred to the GaP cavity using a pick-and-place tech-nique 197,198(seeFig. 14 ). Alternatively, a GaP PCC may be transferred using a micropolydimethylsiloxane (PDMS) adhesive on a tungstenprobe (briefly discussed and illustrated in Sec. VII D 2 andFig. 29 )t oa preselected nanodiamond containing a NV /C0center of desirable properties.199 Nanodiamonds were also integrated with silica microresonators to achieve cavity QED (cQED) effects. In one early attempt, diamondnanocrystals were attached to silica microresonators by dipping silica microdisks with a diameter of 20 lm into an isopropanol solution con- taining suspended nanodiamonds with a mean diameter of 70 nm. 188 Initially, the microdisks had a quality factor of Q¼40000 at room temperature, but after deposition of the nanodiamonds, the low tem-perature measurement showed that the quality factor decreased signifi-cantly to around 2000–3000. By condensing nitrogen gas to tune thecavity modes, the authors observed that a single NV center couldcouple to two cavity modes simultaneously. However, there was no significant change in the spontaneous emission rate, which was proba- bly due to, in addition to the emitters’ large linewidth, the resonator’s large mode volume and limited quality factor. In Ref. 200, a tapered fiber is used to both pick up and position NV containing nanodia- monds onto a high-Q SiO 2microdisk cavity. The same tapered fiber could, then, also be used to characterize light transmission through the system. Coupling in the strong cQED regime has also been achieved between NV centers in nanodiamonds and silica microsphere resonators.201 Besides silica resonators, there has also been work on polystyrene microsphere resonators. Nanodiamonds with a mean diameter of25 nm can be attached to polystyrene microspheres with a diameter of /C245lm by both first dispersing on a cover slip and then using near- field scanning optical microscopy tips to bring the microspheres close to a preselected nanodiamond containing a single NV /C0center.187 Touching a nanodiamond with a microsphere, then, attaches the for-mer to the latter. Using this technique, the authors demonstrated cou- pling of two single NV centers found in two different nanodiamonds to the same microsphere resonator. 187 Silicon carbide is another material that can be integrated with diamond due to its similarity with diamond. For example, the authors in Ref. 174developed a scalable hybrid photonic platform, which inte- grates nanodiamonds with 3C-SiC microdisk resonators fabricated on a silicon wafer. By condensing argon gas on the structure, the authors were able to continuously red shift the resonator’s resonance and tune it to the color center’s emission to observe an enhancement of the cen-ter’s spontaneous emission. It is also possible to enhance the spontaneous emission rate of a quantum emitter coupled to waveguiding structures such as dielectric- loaded surface plasmon polariton waveguides (DLSPPWs) where the significantly confined mode volume of the surface plasmon polariton (SPP) can enable Purcell factors above unity (see Fig. 15 ). Experiments involving embedded nanodiamonds with NV centers in a DLSPPW consisting of a hydrogen silsesquioxane (HSQ) waveguide on top of a silver film demonstrated a spontaneous emission enhancement of up to 42 times. 202,203In a similar vein, a GeV center embedded within a similar DLSPPW was successfully excited by 532 nm light propagating within the waveguide and achieved a three-fold enhancement in its spontaneous emission rate due to the small mode volume within the waveguide.184Although likely to be less useful than coupling to DLSPPWs due to higher losses, coupling of single NV centers in nano-diamonds to silver nanowires can enable interesting studies of SPPs as in Ref. 204where a wave-particle duality was demonstrated for SPPs excited by single photons from a nanodiamond. The spontaneous emission rate of a quantum emitter can be sig- nificantly enhanced when coupled to a plasmonic nanoantenna. For example, enhancement factors of up to 90 times were observed for aNV center within a nanodiamond that was coupled to a nanopatch antenna. 205Even higher enhancement of up to 300 times has been the- oretically proposed by coupling SiV centers in a nanodiamond to a specific geometry of gold dimers.206 B e s i d e ss t a t i cr e s o n a t o r ss u c ha sm i c r o d i s k sa n dm i c r o s p h e r e s ,a fiber-based microcavity technique, where a tunable cavity is typically formed by the combination of a fiber and macroscopic mirror, can also be applied to NV and SiV centers in diamond to enhance theirefficiency, brightness, and single photon purity. 207–210Finally, it is also FIG. 14. (a) AFM image of the core of a photonic crystal fiber. (b) A preselected nanodiamond is then transferred to the center of the photonic crystal fiber using a pick-and-place technique. (c) A nanodiamond in the center of a gallium phosphidephotonic crystal membrane cavity. Reprinted with permission from Schell et al. , Rev. Sci. Instrum. 82, 073709 (2011). Copyright 2011, AIP Publishing LLC.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-16 VCAuthor(s) 2020possible to directly couple the color centers in a nanodiamond to an optical fiber. For example, in Ref. 211, preselected NV containing nanodiamonds were placed directly on a fiber facet to create an align-ment free single photon source. High coupling efficiency was also reported in a nanodiamond-tapered fiber system 212–214(seeFig. 16 ), and in Ref. 215, NV containing nanodiamonds were successfully embedded in tellurite soft glass. V. QUANTUM DOTS A. Introduction to quantum dots A QD is a small nanometer-sized three-dimensional inclusion of a narrower bandgap material within a wider bandgap matrix. The 3Dconfinement potential of the QD leads to a discretization of energy lev- els and gives it localized, atom-like properties. By controlling and manipulating these properties, QDs can be utilized in many aspects ofquantum technologies, such as SPEs or as qubit systems. QDs havealso been studied extensively and developed for numerous optoelec-tronic applications, including light-emitting diodes, 216photovoltaic devices,217and flexible displays.218 Compared to other atomic systems (e.g., trapped ions) used in early experimental realizations of quantum logic,219QDs are embed- ded within a solid-state medium and thus do not require bulky andcomplicated vacuum systems and optical trapping setups. Moreover, QD-based devices can take advantage of well-established growthtechniques, e.g., molecular beam epitaxy (MBE) 220–222or metalorganic vapor phase epitaxy,223–225which allow for monolithic growth with monolayer precision. Coupled with the ability to electrically controlthese devices, 226–228QDs have attracted extensive research efforts in developing and realizing QD photonic devices. QDs used in integrated photonic applications are typically based on III-V materials, especially In(Ga)As in (Al, Ga)As matrices. The most common QD growth approach uses the Stranski-Krastanovmechanism: 229as the QD material is successively deposited and reaches a critical thickness, strain energy from mismatched lattice con-stants drive the formation of 3D nanoislands through a self-assemblyprocess, which allows for more efficient strain relaxation. Thedownside of self-assembly is that the QDs are randomly positioned,but site-controlled growth techniques have been developed to gaindeterministic control over the QD positioning 230,231and their coupling to nanophotonic structures.232 QD devices have numerous applications in quantum integrated photonics. They can serve as tunable, high-quality single-photon sour-ces that can be integrated into nanophotonic structures such as wave-guides 233and beamsplitters.234To complement this, photonic device components for photon manipulation, such as modulators,235fre- quency sorters,236and frequency converters,237have been developed. High-speed near-infrared detectors238,239based on QDs have also been demonstrated in recent years. By controlling the QD spin, spin- FIG. 15. (a) DLSPPW circuits are built around nanodiamonds containing a singlephoton emitter with an enhanced emission rate. (b) Scanning electron micrograph of a HSQ waveguide fabricated on a silver-coated silicon substrate. Reprinted withpermission from Siampour et al. , ACS Photonics 4, 1879 (2017). Copyright 2017, American Chemical Society. FIG. 16. (a) Nanodiamonds were attached to the fiber taper, and single photons emitted by the color centers were collected either through the objective or the fib er ends. (b) The tapered fiber was dipped into a small droplet of a nanodiamond solution on the facet of a thin glass rod. The nanodiamonds were attached to the fiber tap er as it is moved through the droplet. Reprinted with permission from Schr €oder et al. , Opt. Express 20, 10490 (2012). Copyright 2012, The Optical Society.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-17 VCAuthor(s) 2020photon interfaces can also be realized, allowing the QD to be used as a quantum memory, as well as a range of additional applications such as single-photon switching.240 In this work, we will focus more on examples and applications of QDs integrated on photonic platforms; a broad-spectrum overviewcan be found in another recent review. B. As a single-photon emitter SPEs can be realized from QDs by utilizing the radiative recombi- nation from an excitonic state of a single QD. 244,245The first demon- strations of single-photon emission from QDs were performed under optical pumping,244and then by electrical pumping.226Beyond single- photon generation, multiphoton generation via demultiplexing ofhigh-brightness integrated QDs has also achieved a four-photon coinci-d e n c er a t eo f >1H z . 246QD-based SPEs have been extensively studied, and in-depth discussions can be found in other review articles.247,248 The photon statistics of QD single-photon sources can be degraded by imperfections such as multiphoton emission from multi-exciton states or if light is also collected from nearby QDs. As dis-cussed in Sec. II A 3 , we will use the g ð2Þð0Þvalue as a measure of the source’s single-photon purity. For nonresonant excitation, multiphoton emission can result from the QD capturing additional carriers after the first photon emis-sion, which can subsequently recombine. Therefore, to obtain a lowg ð2Þð0Þvalue, relaxation into the QD and the radiative cascade causing recombination should occur on a longer timescale than the decay of the initial carriers.249,250With resonant, pulsed excitation, gð2Þð0Þval- ues close to zero have been demonstrated,251–253while the lowest reported gð2Þð0Þvalues of below 10/C04have been achieved with two- photon excitation.254,255However, we note that these lowest values were not obtained from QDs integrated with on-chip planar wave- guides; demonstrations with integrated QDs have reported more mod- estgð2Þð0Þvalues due to factors such as increased background emission from cavity modes256(see also Table I ). The radiative cascade of high-energy carriers also results in a temporal uncertainty (i.e., jitter) of photon emission,249which leads to decreased indistinguishability for higher excitation powers.81 However, this can be overcome with strictly resonant pumping schemes.251,257Moreover, resonant pumping and adding a weak auxil- iary continuous wave reference beam to the excitation beam of the QDcan help to suppress charge fluctuations 258that would otherwise lead to spectral diffusion. To suppress the effects of phonon interactions, one can operate at cryogenic temperatures although we acknowledge that for InGaAs QDsat 4 K, PSB emissions can still represent /C2410% of emission [see Fig. 17]. 241–243Also, spectral filtering of the QD ZPL can yield high indistin- guishability close to unity,259albeit at the expense of photon rates. Since the first HOM two-photon interference experiment with QDs reported an indistinguishability of /C2470%,80near-unity values have been consistently achieved257,260–262(e.g., 0.995 60.007, Ref. 257). Recent reports have also reproduced high HOM visibility value QDs integrated with nanophotonic waveguides.263–266 C. Spin–photon Interfaces By accessing and manipulating their spin, QDs can provide not only photonic qubits but also spin qubits. Various level structures can beexploited for qubit encoding, and rapid spin initialization, manipulation, and read-out can be achieved with short optical pulses (in the nanosec-ond range). 267Such spin-photon interfaces can enable many quantum information processing tasks, such as deterministic spin-photon entan- glement and mediating strong photon-photon interactions. The strong nonlinearity at a single-photon level has led to dem- onstrations of the photon blockade268and tunable photon statistics via the Fano effect.269Single-photon switches and transistors have been realized via a QD spin108,270,271[seeFig. 18(a) ]. The coherent control of the QD spin has also been achieved, with Ref. 272demon- strating Ramsey interference with a dephasing time T/C3 2¼2.260.1 ns [seeFig. 18(b) ]. The strong light confinement in nanophotonic waveguides also opens up the possibilities of chiral, or propagation-direction-dependent, FIG. 17. (a) The bottom part shows a typical spectrum of multiple InAs/GaAs QDs. The top part shows the temporal evolution of the spectrum, showing spectral wan-dering of the QD transitions. Reprinted with permission from Rodt et al. , Phys. Rev. B71, 155325 (2005). Copyright 2005, the American Physical Society. (b) Calculated absorption spectra of an InAs QD. At lower temperatures, the broad- band phonon interactions are suppressed, while the ZPL is asymmetrically broad-ened. The inset compares the calculated broadening with experimental data(circles). Reprinted with permission from Muljarov and Zimmermann, Phys. Rev. Lett. 93, 237401 (2004). Copyright 2004, the American Physical Society.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-18 VCAuthor(s) 2020quantum optics.273This can be used to deterministically induce unidirec- tional photon emission from quantum dot spin states, i.e., r6transitions emit in different directions274,275[seeFig. 18(c) ]. This can help to realize complex on-chip nonreciprocal devices such as single-photon opticalcirculators. 276Although chirality has only been demonstrated to date using waveguides, recent theory papers have shown that chirality withsignificant Purcell enhancement should be possible using a ring resona- tor geometry.276,277 D. Interfacing multiple QDs Hybrid quantum photonic platforms aim to integrate multiple quantum sources, including dissimilar quantum systems, onto the same device. Two-photon interference has been demonstrated between QDs and other quantum emitters, including atomicvapors, 282Poissonian lasers,283,284parametric down-conversion sour- ces,285,286and frequency combs.287The rest of this section will focus on the interfacing of multiple QDs on the same photonic circuit. To obtain high intereference visibility, the emitted photons have to be identical, but it is experimentally challenging to find two QDs withalmost identical emission energies, linewidths, and polarization. Whilemuch effort has been invested in fabricating highly reproducible QDs, 288it is often necessary to employ tuning mechanisms, both for the QDs and the cavity structures they are embedded in, to match the emis-sion properties and ensure a high indistinguishability of the photons. The photonics community has been actively developing multiple techniques for tuning on-chip resonators. 289–291However, certain methods such as wet-chemical etching292and gas condensation152 (unless used in conjunction with local heating) are not suited for tun-ing individual devices on a chip or an array. We emphasize here that ascalable solution for fully integrated quantum photonics would requirethat the tuning can be applied locally and independently to individual emitter devices. 1. Temperature Temperature tuning can affect the bandgap structure, which strongly tunes the QD energy; it can also alter the refractive index and cause physi- cal expansion, which would shift the resonances of a cavity device coupledt ot h eQ D .T h es i m p l e s tw a yt oa c h i e v et h i si st oc h a n g et h es a m p l et e m -perature in the cryostat, but this does not allow for the tuning of individ- ual devices. 293,294Instead, local temperature changes can be applied via electrical heaters or laser irradiation278,295,296[seeFig. 19(a) ]. A recent report has also employed temperature tuning of two QDs in a nanopho-tonic waveguide to achieve superradiant emission. 297 2. Strain QD energy is sensitive to strain tuning, and strain sensors have been demonstrated by detecting energy shifts at the leV level for InGaAs QDs embedded in a photonic crystal membrane.298However, to achieve larger tuning ranges, strain can be induced via piezoelectriccrystals, and a tuning rate of /C241p m / V ( R e f . 279) has been achieved [seeFig. 19(b) ]. A difficulty with this approach is the relatively large fabrication overhead of integrating piezoelectric materials. However, another recent work has shown that strain tuning can be achieved vialaser-induced local phase transitions of the crystal structure, which cir-cumvents this issue. 299 3. Electric field The application of electric fields across the QD can be used to control the energy of the QD excitonic lines via the quantum-confinedStark effect. 300–302The application of a forward bias voltage leads to a FIG. 18. (a) Schematic of the single-photon switch and transistor. A gate photon controls the state of the spin, and then, the spin determines the polarization of thesignal field. Reprinted with permission from Sun et al. , Science 361, 57 (2018). Copyright 2018, AAAS. (b) Ramsey interference of a QD spin embedded in a nano- beam waveguide, with a dephasing time T 2/C3¼ 2.260.1 ns and a contrast of 0.04 for the first period. Reprinted with permission from Ding et al. , Phys. Rev. Appl. 11, 031002 (2019). Copyright 2019, the American Physical Society. (c) (Left) QD level scheme featuring two circularly polarized exciton transitions. (Right) Calculated directional emission patterns of rþandr/C0transitions. Reprinted with permission from S €oollner et al. , Nat. Nanotechnol. 10, 775 (2015). Copyright 2015, Springer Nature Customer Service Center GmbH.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-19 VCAuthor(s) 2020blueshift of the QD emission wavelength of several nanometers. This can be complemented by independently tuning the cavity mode of thephotonic crystal structure, e.g., via electromechanical actuation 280,303 [seeFig. 1(c) ]. Beyond wavelength tuning, recent work has also shown that the electrical control of QDs is crucial to obtain the best opticalproperties for integrated QDs. 2284. Frequency conversion An alternative strategy is to tune the emitted photons via on-chip quantum frequency conversion (QFC).237Reference 281performed QFC separately on the output of two QDs to convert them from 904 nm to the telecom C-band, achieving a two-photon interferencevisibility of 29 63% [see Fig. 19(d) ]. FIG. 19. (a) (Top) Schematic of a QD device temperature tuned by laser irradiation on a heating pad. (Bottom) Dependence of the QD detuning on the heating laser p ower. Reprinted with permission from Katsumi et al. , Appl. Phys. Lett. 116, 041103 (2020). Copyright 2020, AIP Publishing LLC. (b) (Top) Artistic representation of a waveguide- coupled QD photon source fabricated on a strain-tunable substrate. (Bottom) Emission spectra of the nanowire QD as a function of the applied voltage t o the piezoelectric sub- strate. Reprinted with permission from Elshaari et al. , Nano Lett. 18, 7969 (2018). Copyright 2018, American Chemical Society. (c) (Top) Annotated SEM picture of a device with two integrated QD devices. The QDs and cavities can be electrically tuned individually. (Bottom) Spectra showing the electromechanical tuning of the cavities (left) and the Stark tuning of the QDs (right). Reprinted with permission from Petruzzella et al. , APL Photonics 3, 106103 (2018). Copyright 2018, Author(s) licensed under a Creative Commons Attribution 4.0 License. (d) (Left) Schematic of a device where the emission of two QDs is individually tuned via QFC. (Right) Energy diagram d epicting the QFC scheme. Reprinted with permission from Weber et al. , Nat. Nanotechnol. 14, 23 (2019). Copyright 2019, Springer Nature Customer Service Center GmbH.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-20 VCAuthor(s) 2020VI. 2D MATERIALS A. Introduction into 2D materials Single photon sources in 2D materials have unique advantages compared to other quantum emitters in the 3D bulk material. Confined in an atomically thin material, they can potentially have high photon extraction efficiencies, and their emission properties canbe controlled by a variety of effects including strain, temperature, pres- sure, and applied electric and magnetic fields. Indeed, single photon sources in a monolayered 2D material can have almost unity out- coupling efficiency as none of the emitters are surrounded by a high refractive index material and their emitted light is consequently notaffected by Fresnel or total internal reflection. 304,305In addition, 2D materials can be easily transferred and integrated with photonic struc- tures or other 2D materials to form synergistic heterostructures that combine the advantages of various materials together in one unified structure.306In this review of single photon sources in 2D materials, we restrict ourselves to TMDCs and hBN although we acknowledge that there are other important examples of 2D materials including graphene, anisotropic black phosphorus, and borophene.307–309An important feature of single photon emitters in these 2D materials is that similar to NV/C0centers in diamonds and QDs, they can be used for efficient spin–photon quantum interfaces by tailoring the light- matter interactions due to the broken inversion symmetry.310,311The zero field splitting in TMDC materials can be up to /C240.7 meV, which is about 50 times higher than InAs/GaAs self-assembled quantum dots, and it has a surprisingly large anomalous g-factor of /C248–10 that can potentially allow for extremely fast coherent spin coupling.312On the other hand, hBN has a considerably smaller zero-field splitting of0.00145 meV and a more modest g-factor of 2. 313 B. Single photon emitters in TMDCs A monolayer of TMDC can be described as a MX 2sandwich structure with M being a transition metal atom (e.g., Mo and W)enclosed between two lattices of chalcogen atom X (e.g., S, Se, and Te). 315,316Depending on the choice of elements and the number of layers present, TMDC materials can have widely varying electrical, optical, chemical, thermal, and mechanical properties.317–322Although TMDCs have strong in-plane covalent bonds, they are only weaklybonded in between the layers by van der Waals forces, which allows them to be easily exfoliated to form monolayer flakes. Alternatively, single layer TMDCs can be fabricated using CVD or MBE. 323,324 Despite the fact that multilayer TMDCs have indirect bandgaps,monolayer TMDCs are actually direct bandgap semiconductors, whichenables them to have enhanced interactions with light. 317,325–327There are two distinctive properties that are associated with monolayer TMDCs: strong excitonic effects and valley/spin-dependent properties. The latter can be attributed to the fact that there is no inversion center for a monolayer structure, which opens up a new degree of freedom for charge carriers, i.e., the k-valley index, which brings new valley- dependent optical and electrical properties into play.310,321,328,329In contrast, TMDC’s strong excitonic effects are due to strong Coulomb interactions between charged particles (electrons and holes) and reduced dielectric screening, which result in the formation of excitons with large binding energies (0.2 to 0.8 eV), charged excitons (trions), and biexcitons.330–334These manifest themselves by broadband photo- luminescence (tens of nanometers) in the visible and near-IR ranges atroom and cryogenic temperatures. At the same time, TMDCs can pos- sess quantum-dot like defects, which exhibit themselves in the photo-luminescence spectrum as a series of sharp peaks with peak intensitiesup to several hundred times larger than the excitonic photolumines-cence with a linewidth around 100 leV and an excited state lifetime range of 1 to several nanoseconds 312,334–336(Fig. 20 ). For exfoliated samples, these defects are usually associated with local strain and typi- cally appear at cracks or edges of the flake, while for grown samples, they are mostly due to impurities and can appear everywhere. A num-ber of works have shown that these defects emit in the single photonregime and can be controlled by induced strain, applied temperature,and electric and magnetic fields. 312,335–338However, a significant draw- back is that the single photon emission quenches at temperaturesabove 20 K although some recent research has shown that specialtreatment of TMDC flakes can lead to a redistribution of the energy levels and enable emission at room temperature. 339,340 C. Single photon emitters in hBN hBN monolayers are structurally similar to TMDC, whereas SPEs in TMDCs are associated with localized excitons, SPEs in insulat-ing hBN are, similar to color centers in diamond, attributed to atomic-like defects of the crystal structure. 341–343These defects in hBN are some of the brightest single photon sources in the visible spectrumand have large Debye-Waller factors with good polarization contrasts.Like NV /C0centers, their electronic levels are within the bandgap (/C246 eV), resulting in stable and extremely robust emitters at room temperature over a wide spectral bandwidth ranging from green to FIG. 20. Single photon emitters in TMDC materials: (a) contour plot of the photolu- minescence scanning experiment for the WSe 2flake with emission spots at the edges of the flake; (b) these spots possess narrow line emission spectra; and (c) measurement of g2(0) for one of the narrow emission lines; the value below 0.5 clearly indicates the single photon properties of the emitter. Reprinted with permis-sion from Koperski et al. , Nat. Nanotechnol. 10, 503 (2015). Copyright 2015, Springer Nature Customer Service Center GmbH.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-21 VCAuthor(s) 2020near infrared with most emitters emitting around the yellow-red region344(Fig. 21 ). These SPEs in hBN are generally characterized by short excited state lifetime (several ns), absolute photon stability, and high quantum efficiency.341–344Close to Fourier transform limited linewidths below 100 MHz have been recorded with resonant excita- tion at cryo and room temperatures.345,346Recent research indicates that various types of defects are responsible for the multiplicity of observed ZPL emissions, including NV defects, carbon substitutional (of a nitrogen atom) defects, and oxygen related defects.343,347 Interestingly, the asymmetric linewidths of some of these ZPLs havebeen attributed to the existence of two independent electronic transitions. 348 D. Deterministic Creation and Control of Single Photon Emitters in 2D Materials The random distribution of SPEs in 2D materials is a signifi- cant obstacle that prevents their integration with photonic struc- tures. Deterministically creating SPEs in 2D materials is, therefore, an important technological goal. One way to do so is to induceSPEs by introducing strain to the material. This concept was suc- cessfully realized by several groups that transferred 2D materialsonto the tops of metallic or dielectric nanopillar arrays 337,338,349 (Fig. 22 ). The SPE [as confirmed by measured g2(0)<0.1] yield of this technique exceeds 90%, and the quality of these engineeredSPEs was reported to be even higher than naturally occurringdefects with 10 times less spectral diffusion ( /C240.1 meV). 338An even simpler (but less scalable) way to create SPEs with strain wassuggested by Rosenberger et al. : place a deformable polymer film below the 2D material of interest and apply mechanical force to thefilm using an AFM tip. 350Although this approach is less scalable, it can enable one to tune the SPE’s optical properties through carefulstrain engineering with the AFM tip. Remarkably, emission from SPE in 2D material defects can be controlled through the application of a voltage. 336For example, in Ref. 336, photoluminescence and electroluminescence were observed from point defects in a 2D material (WSe 2) sandwiched between hBN layers that served as tunneling barriers between WSe 2and its graphene elec- trodes. Moreover, Schwarz et al. showed that it was possible to tune the emission wavelength of the SPE ( /C240.4 meV/V) by changing the FIG. 21. Single photon emitters in hBN nanoflakes: (a) scanning confocal map of the hBN nanoflake with bright luminescent spots corresponding to single defects ; (b) room temperature emission spectrum for single defects in mono- and few-layer hBN nanoflakes; and (c) measurements of g2(0) for single defects in mono- and few-layer hBN nano- flakes. The curves are shifted for clarity demonstrating g2(0)<0.5. Reprinted with permission from Tran et al. , Nat. Nanotechnol. 11, 37 (2016). Copyright 2016, Springer Nature Customer Service Center GmbH.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-22 VCAuthor(s) 2020applied bias voltage, which makes the platform amenable to a host of technological applications. E. Enhancement of emission from 2D materials by coupling into resonant modes Although the tunability and large oscillator strengths of SPEs in 2D materials make them attractive in photonic applications, their sub- nanometer thickness results in a small light-matter interaction length that limits their efficiency. However, as noted in Sec. II B,t h i sd i s a d - vantage may be mitigated by coupling them with resonant photonicstructures where both their absorption and emission can potentially be enhanced. 352Fortunately, the atomic thickness of 2D materials makes them especially amenable to integration with photonic structures suchas planar photonic crystal cavities, ring resonators, and opticalmicrocavities.The first demonstration of this came from the coupling of photo- luminescence from SPEs in TMDC materials to dielectric and plas-monic nanocavities 351,354–356[Figs. 23(a)–23(c) ]. In Ref. 354,a coupling efficiency of over 80% was demonstrated, and a spontaneous emission enhancement of over 70 times was reported. Intriguingly, the photoluminescence enhancement can be controlled via an optical spin–orbit coupling, which depends on both the resonant nanopar-ticles’ geometry and the incident laser’s polarization and power 351 [Figs. 23(a) and23(b) ]. Subsequently, single photon emission from a single emitter in a hBN nanoflake was successfully coupled to bothone and two resonant gold nanospheres 353[Figs. 23(d)–23(g) ]. These nanospheres were brought into close proximity with a precharacter- ized SPE [verified by measuring gð2Þð0Þ<0:5] by means of an AFM tip, and the emitter was observed to have a photon flux of about 6 MHz that corresponded to a 3-fold Purcell enhancement.353 Natural structures for SPEs in 2D materials to couple to are noble metal nanopillars since they can kill two birds with one stone by first facilitating the deterministic creation of SPEs (as discussed in Sec.VI D) and then enhancing the created SPEs’ spontaneous emission through the SPEs’ coupling with surface plasmon resonances of the metallic nanopillars. This has been successfully implemented for both TMDCs and hBN at cryogenic and room temperatures where increased brightness, shorter lifetimes, and enhanced spontaneousemission of the SPEs were all reported. 357–360Coupling of the emitter to plasmonic modes results in linearly polarized emission that depends on the geometry of the nanopillars and the orientation of the opticaldipole. 358A record high Purcell enhancement of 551 times was achieved with metallic nanocubes.359 F. Coupling and transfer of emission from 2D materials into photonic structures To fully integrate SPEs in 2D materials onto an on-chip photonic platform, it is also necessary to couple emission from SPEs into wave- guiding photonic structures. To this end, a few groups have success-fully coupled emission from SPEs in 2D materials into the surface plasmon polariton modes of silver based waveguides. For example, localized SPEs formed from the intrinsic strain gradient formed alongaW S e 2monolayer when it was deposited on top of a silver nanowire were efficiently coupled to the guided surface plasmon modes of the nanowire.361A coupling efficiency of 39% was measured for a single SPE by comparing the intensity of the laser excited SPE and the emis- sion intensities at both ends of the silver nanowire.361Separately, S. Dutta et al. demonstrated the coupling of single emitters in WSe 2 to propagating surface plasmon polaritons in silver-air-silver and metal–insulator–metal (MIM) waveguides362[Figs. 24(a)–24(d) ]. The waveguides were fabricated using EBL, followed by metal deposition of Cr and Ag, and then a liftoff in acetone with subsequent protection by a 4 nm buffer layer of oxide. As before, strain gradients on themonolayer due to the waveguide generated sharp localized SPEs that were intrinsically close to the plasmonic mode. Due to the subwave- length confinement of the surface plasmon polariton modes, a 1.89times enhancement of the SPE’s radiative lifetime was observed under illumination at 532 nm at cryogenic (3.2 K) temperatures, and bright narrow lines associated with the SPE’s emission were measured. Although surface plasmon polariton modes on silver waveguides may help to enhance the spontaneous emission of a SPE, the metal interface results in significantly lossy propagation that is undesirable. FIG. 22. Deterministic creation of single photon emitters by strain: (a) SEM image of the nanopillar substrate, fabricated using electron beam lithography; the black scalebar corresponds to 2 lm; (b) illustration of the fabrication method; and (c) dark-field optical image of the monolayer WSe 2flake on top of the nanopillar array. The green rectangle indicates the six adjacent nanopillars where single photon emitters weremeasured; (d) examples of measured spectra at different pillars demonstrating nar-row emission lines and (e) measurements of g 2(0) for the emission lines in (d). Reprinted with permission from Palacios-Berraquero et al. , Nat. Commun. 8, 15093 (2017). Copyright 2017, Author(s) licensed under a Creative Commons Attribution4.0 License.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-23 VCAuthor(s) 2020Such high propagation losses can be circumvented by using dielectric waveguides instead. In Ref. 363, integration of a monolayer WSe 2flake with a 700 nm wide Si 3N4waveguide that was patterned using stan- dard EBL techniques was achieved by carefully picking up and releas-ing a bulk exfoliated flake using a GelPak stamp [ Figs. 24(e)–24(g) ]. Confocal scans of the WSe 2monolayer with a 532 nm excitation laser at cryogenic temperatures indicated that several SPEs were sufficientlyclose to the Si 3N4waveguide to enable coupling to it. Although some luminescence was measured at the end of the waveguide, which pro-vided proof of a nonzero coupling, the SPE’s coupling to the wave-guide is strongly dependent on the orientation of its optical dipole, and consequently, the photoluminescence spectra for a confocal scan can be significantly different from those obtained at the end of thewaveguide. Brighter emission and saturation counts of up to 100 kHzcan be obtained by exciting the SPE at close to the free exciton wave-length ( /C25702 nm) with a tunable Ti:Sapphire laser. Besides enabling brighter emission, excitation at 702 nm also produces less backgroundfluorescence. Measurements of the background subtracted g ð2Þcorrela- tion function in confocal geometry revealed an antibunching dip withg ð2Þð0Þ¼0:47, which suggested the existence of a SPE. Finally, we note that there has been successful integration of SPEs in 2D materials to an on-chip beamsplitter in the form of a lithiumniobate directional coupler 364[Figs. 24(h)–24(i) ]. Indeed, emission from an excited SPE in a strain engineered WSe 2monolayer wascoupled into one input port of the directional coupler and its photolu- minescence, which consisted of strong emission lines corresponding to emission from the WSe 2flake, was successfully measured at the other output port of the directional coupler [ Fig. 24(e) ], demonstrating the desired operation of the beam splitter and showing that an on-chip Hanbury Brown and Twiss measurement is possible. VII. INTEGRATION APPROACHES Numerous techniques have been developed to directly integrate solid-state quantum emitters with on-chip nanophotonic structures. Doing so would allow for dense integration of these emitters on a large scale and also provide potential benefits in coupling efficiencies, device stability, and ease of control. In this section, we will provide an overview of hybrid integration approaches and discuss their applicability to the solid-state emitters presented in this review. A detailed reference on integration methods for hybrid quantum photonics can be found in Ref. 6 A. Random dispersion A simple integration method is to forego deterministic position- ing and rely on the random placement of the quantum emitters. For QDs229,239and 2D materials,305,312,314,335the emitters may already be randomly distributed in their as-grown state. FIG. 23. Enhancement of spontaneous emission with nanocavities: (a) Schematic of the spiral ring structure with circularly polarized light excitation. The photoluminescence intensity is enhanced when the MoS 2monolayer couples with plasmonic spiral structures. (b) Spectra of the MoS 2monolayer with and without spiral structures, under the exci- tation of different circularly polarized light; the inset shows the cross-sectional view of a 2-turn spiral ring. The scale bar is 400 nm. (a) and (b) Re printed with permission from Liet al. , ACS Nano 11, 1165 (2017). Copyright 2017, American Chemical Society. (c) Spectra of the MoS 2monolayer on a SiO 2/Si substrate (blue) and in the nanocavity (red) obtained using a diffraction-limited excitation spot. The inset presents normalized spectra for the two cases along with the scattering spectrum fo r a typical nanocavity (gray). Reprinted with permission from Akselrod et al. , Nano Lett. 15, 3578 (2015). Copyright 2015, American Chemical Society. (d) Deterministic positioning of gold nanospheres with the hBN nanoflake by the AFM tip; the white scale bar is 250 nm. (e) A comparison between time-resolved measurements of pristine and double particle arra ngements. (f) A comparison of fluorescence saturation curves between the pristine, single particle, and double particle arrangements. (g) Measurements of g2(0) for pristine, single particle, and double particle arrangements. The curves are shifted for clarity. (d)–(g) Republished with permission from Nguyen et al. , Nanoscale 10, 2267 (2018). Copyright 2018, the Royal Society of Chemistry.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-24 VCAuthor(s) 2020In the case of NV/C0centers, type Ib diamonds, which, by defini- tion, have significant singly dispersed nitrogen impurities,365–367natu- rally host an ensemble of randomly positioned NV/C0centers. Such diamonds can be found naturally or manufactured using a High- Pressure High-Temperature process.365Similarly, type IIa diamonds, which, by definition, have much lower concentrations of nitrogen impurities compared to type Ib diamonds, also host a sparse ensemble of randomly positioned NV/C0centers. Although rare in nature, type IIa diamonds can be grown using chemical vapor deposition.368 Photonic structures fabricated on diamond membranes will, therefore,have randomly positioned NV centers, and indeed, although the yield for such structures is poor with suboptimal coupling between the pho- tonic structure and NV center, many early experiments relied on sucha random dispersal technique. Other color centers in diamond do not form naturally and have to be integrated using more deterministic techniques. Colloidal QDs and nanodiamonds containing color centers can also be randomly dispersed onto photonic structures via drop casting or spin coating. 369Nanodiamonds have also been dip coated directly onto single mode optical fibers.212In this technique, the tapered fiber is dipped into a droplet of nanodiamond-containing solution on the tip of a glass rod. The tapered fiber is then pulled by a linear stage along the axis of the fiber. However, SPEs in the form of nanoparticles often have a large surface area, which may lead to optical instabilities such as blinking or bleaching. This is due to the stronger influence of surface states and enhanced Auger recombination.52Moreover, ran- dom dispersion is not suitably scalable for quantum photonic applica- tions where efficient, deterministic coupling between emitters and the photonic circuit is crucial.To improve the positioning precision, and thus the coupling efficiency, a lithography-based masking method can be used to selec- tively deposit dispersed emitters on top of the photonic structures370 [Fig. 28(a) ]. Despite its limitations, randomly positioning of emitters can still be a useful method to rapidly prototype hybrid quantum photonic platforms. B. Targeted creation 1. Irradiation and annealing in diamond Deterministic positioning of NV centers can be achieved by deliberately creating vacancies in type Ib diamonds with irradiation of either a focused ion,371–373proton,374or electron371,375,376beam. Since vacancies in diamond can, with an activation energy of /C252.3 eV, migrate during annealing12,377,378at/H11407600/C14C, the diamond is then typically annealed after irradiation to allow the vacancies to diffuse and be “captured” by an existing nitrogen impurity.12,34,379The spatial resolution of this technique is, therefore, dependent on not only the resolution of the focused ion or electron beam but also the concentra- tion of nitrogen impurities in the diamond, which determines how far a vacancy has to diffuse before being captured by a nitrogen impurity. In fact, despite having a significantly larger concentration of nitrogen compared to type II diamonds, the spatial resolution of this technique in type Ib diamonds can still be limited by the concentration of nitro- gen impurities.371The diffusion length lDof the vacancies may be esti- mated using lD/C24ffiffiffiffiffiffiffiffiffi DDtp ,w h e r e Dis the diffusion coefficient and Dtis the anneal time. Dmay be obtained via the Arrhenius equation as follows: FIG. 24. Coupling and transport of single photon emission from 2D materials in photonic structures: (a) Scheme of an MIM waveguide covered by a WSe 2monolayer. The yel- low dipole is a quantum emitter, and the blue arrows denote the excitation and collection points. (b) Spectra of the defect with the excitation spot fixe d at the location of the defect at the waveguide and collection spot moved to the far end of the waveguide. (c) and (d) Lifetime measurements for emitters located on the MIM wave guide and out of it. (a)–(d) Reprinted with permission from Dutta et al. , Appl. Phys. Lett. 113, 191105 (2018). Copyright 2018, AIP Publishing. (e) Illustration of the Si 3N4photonic device with a WSe 2flake integrated on top of a 220 nm thick single mode Si 3N4waveguide, separated by two air trenches from the bulk Si 3N4. (f) Confocal and waveguide-coupled spectrum of the emitter excited by a 702 nm pump. The waveguide spectrum is multiplied by 10 and offset by 2000 counts/s for improved visualization. The inset sho ws confocal spectra obtained by either green excitation at 532 nm (green curve) or excitation at 702 nm (blue curve). (g) Background-corrected measurements of g2(0) for the coupled emitter made in confocal geometry. (e)–(g) Reprinted with permission from Peyskens et al. , Nat. Commun. 10, 4435 (2019). Copyright 2019, Author(s) licensed under a Creative Commons Attribution 4.0 License. (h) Scheme of Ti in a diffused lithium niobate directional coupler with a WSe 2flake at the input facet. Emitters are excited in confocal geome- try. Their emission is detected using a confocal microscope and through the two output ports (port 1 and port 2). (i) Spectra measured through the waveg uide output port and confocally when the emitter is excited at the facet. (h) and (i) Reprinted with permission from White et al. , Opt. Mater. Express 9, 441 (2019). Copyright 2019, Author(s) licensed under a Creative Commons Attribution 4.0 License.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-25 VCAuthor(s) 2020D¼D0exp /C0Ea kBT/C18/C19 ; (22) where Eais the activation energy, kBis the Boltzmann constant, Tis the temperature, and D0is the maximal diffusion constant calculated to be around 3 :7/C210/C06cm2/s for vacancies near the diamond sur- face.380For typical conditions, the diffusion length is likely to be /C24100 nm, and indeed, vacancies have been observed to diffuse by a few hundred nm in the vertical (along the irradiation axis) direc- tion,381and a vacancy diffusion limited transverse spot size of less than 180 nm was obtained using a focused Gaþion beam.371Although there have been reports of vacancy transverse diffusion lengths that are in the tens of lm range,382,383which seem to defy the simple ld/C24ffiffiffiffiffiffiffiffiffi DDtp estimate, we note that this could potentially be explained by the scattering of ions/electrons on masks384–387if they were used on the diamond’s surface.388Nevertheless, we note that if the mask is carefully designed, transverse spatial resolution in the tens of nano- meters can be achieved using an implantation and annealing approach162(seeFig. 26 and Sec. VII B 2 ) .F u r t h e r m o r e ,w en o t et h a t the choice of radiation used can significantly affect the vertical distri- bution of vacancies. In general, the heavier ions deposit most of their energy within a narrow band and create vacancies at a more well- defined depth, whereas the lighter electrons tend to create a more uniform depth profile of vacancies.389 An alternate pathway to creating NV/C0centers is to grow an iso- topically pure12C (which has no nuclear spin) diamond layer on an existing substrate using plasma assisted CVD and then introducing nitrogen gas during the last stages of the growth156,390,391(seeFig. 25 ). It is possible using this procedure to make single NV/C0centers with a long spin coherence time of T2/C251:7m s .390To achieve 3-D localiza- tion of NV/C0centers using such an approach, the depth of the nitrogen doped layer, which determines the depth of the NV centers, can first be carefully controlled by slowing the growth rate down to/C240.1 nm/min,156which allows a depth precision of a few nanometers (delta doping). Transverse localization ( /H11351450 nm) of long coherence NV/C0centers with T2/C251 ms can then be achieved by using a trans- mission electron microscope (TEM) to create vacancies within thenitrogen doped layer followed by annealing. 392The long coherence time of those NV centers are not limited by either13C nuclear spins or lattice damage induced by the electrons (which is thought to be small compared to ion implantation) but rather the presence of other nitro-gen impurities that were not converted into NV centers. 392A variation to using electron irradiation via a TEM is to use irradiation of12C (Ref. 161)o rH eþ(Ref. 393) ions to create vacancies. Compared to electron irradiation, using ions allows for a more localized layer ofvacancies, which will reduce the unwanted creation of NV centers inthe substrate of the CVD grown diamond. However, we note that using such an approach can possibly decrease the T 2of the NV centers due to increased lattice damage caused by the ions. 2. Implantation and annealing in diamond A slightly different approach is to start with a type IIa diamond, which does not contain significant amounts of nitrogen impurities, and to introduce both the vacancy and nitrogen impurity at the same time by implanting Nþ(or Nþ 2) ions with a focused ion beam and then annealing at /H11407600/C14C. Using this approach, it is possible to fab- ricate single NV/C0centers with transverse spatial resolution of tens of n ma n day i e l do f /C2450% using 2 MeV Nþions with a beam diameter of 300 nm.394The NV/C0yield, which is defined as the ratio of active NV/C0centers to the number of implanted Nþions, is proportional to the ion beam’s energy with a particularly strong slope in the keV region.395This is most likely due to the fact that the number of vacan- cies an ion generates is also proportional to its energy, and SRIM (Ref.396) calculations show that the NV /C0yield shows a very similar energy dependence.395A less energetic beam should, therefore, be used to decrease the NV/C0yield (for single NV/C0creation), but a less energetic beam also results in shallower NV/C0centers within the diamond, which can be undesirable for some applications. If a focused ion beam is not available, high spatial resolution can also be achieved by the appropriate use of a mask and (unfocused) Nþ ion beam implantation. In Ref. 162,a nN V/C0spatial resolution of /C2410 nm in all three directions was accomplished using Nþion implantation with the mask shown in Fig. 26 . 3. Laser writing and annealing in diamond While traditional irradiation or implantation of particles typically creates a trail of vacancies following the implanted particle’s path (with increased straggling for lighter particles), irradiation of the dia- mond by focused femtosecond (fs) laser pulses can create vacancies ata more localized depth. Moreover, their transverse spatial resolutioncan be better than the diffraction limit due to the nonlinear processesinvolved. In addition, fs laser pulses from the same optical system canalso be used to fabricate other photonic structures on the same dia-mond, opening a convenient avenue of integrating photonic structureswith NV centers on the same diamond. Implementing fs laser machining in diamond is complicated by the fact that there is a significant mismatch between the refractive indi-ces of diamond ( /C252:4 at 790 nm) and air or (more commonly) immersion oil ( /C251.5). This can result in considerable aberration of the FIG. 25. Top: NV/C0centers created by CVD growth and nitrogen delta doping of a bulk diamond substrate. Vacancies were created by electron irradiation at 2 MeV.Bottom: NV/C0centers created in this way at a depth of 52 nm exhibited a Hahn- echo T2coherence time of 765 ls at room temperature. Using a similar procedure, Ref. 392achieved a T2of/C251.7 ms in isotopically enriched12C diamond. Reprinted with permission from Ohno et al. , Appl. Phys. Lett. 101, 082413 (2012). Copyright 2012, AIP Publishing LLC.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-26 VCAuthor(s) 2020focal point within diamond and cause a significant elongation of the focal volume (which reduces the peak field intensity and localization of vacancies) in the beam’s direction of propagation. Indeed, an earlyexperiment attempting to create NV centers using fs lasers focused the beam above the diamond’s surface instead of within it and relied on the ionization of O 2and N 2molecules in air to generate free electrons and ions that are subsequently accelerated by the light’s electric field into the diamond.397 Aberrations caused by refractive index mismatch between dia- mond and air (or immersion oil) may be corrected by using adaptiveoptics such as membrane deformable mirror and/or spatial light mod- ulators (SLM) that modify the light’s wavefront to compensate for the refractive index mismatch. Indeed, NV centers have been successfullycreated using focused fs laser pulses with a SLM that had a transverse spatial resolution of /C25200 nm that was limited by the diamond’s nitro- gen concentration. 401More recently, it was demonstrated that the same fs laser system may be used to both create a vacancy andanneal the diamond (locally) by careful control of the laser pulse energy.398Coupled together with real-time monitoring of the fluorescence, NV centers at a single site could be generated with near-unity yield, and statistically selective generation of NV centers with a particular orien- tation is even possible by monitoring the polarization pattern of the fluorescence (which is correlated with the NV center’s orientation) and keeping the annealing pulse on until a desired polarization pattern is generated (NV centers with the “wrong orientation” can be destroyed after creation by keeping the annealing pulses on)398(see Fig. 27 ). 4. In situ lithography Photonic structures can be fabricated on adjacent layers to a QD sheet. However, in the absence of site-controlled growth,232the self- assembled QDs would be randomly located402and thus would not be optimally placed with respect to the photonic structures for efficient coupling. To circumvent this problem, the QDs can first be located and preselected, e.g., via cathode luminescence, and the waveguide FIG. 27. Top: Optical setup enabling both vacancy creation and annealing using the same fs laser with a SLM to correct for aberrations. Vacancy creation uses a single 27 nJ pulse, while annealing occurs with a 1 kHz pulse train consisting of 19 nJ pulses. Real time monitoring of photo-luminescence is also possible by incorporating a sepa-rate optical arm consisting of both a 532 nm excitation laser and a single photon ava-lanche detector with a long-pass filter. Bottom: Fluorescence counts measured during the annealing pulses, showing first the creation and destruction of a NV /C0center ( /C2528 to 30 ns) followed by the creation of a second NV/C0center ( /C2552 ns). Reprinted with permission from Chen et al. ,O p t i c a 6, 662 (2019). Copyright 2019, Author(s) licensed under a Creative Commons Attribution 4.0 License. FIG. 26. Fabrication of NV/C0centers by implantation of15N ions on a masked dia- mond substrate. (a) Electron beam lithography (EBL) is used to pattern a poly- methyl methacrylate (PMMA) mask spin coated on top of an Au layer that sits on an 8 nm thick Cr adhesion layer. (b) After development of the PMMA resist, a hardTi mask is deposited at 30 /C14to reduce the aperture size and to protect the resist’s top surface from an O 2plasma reactive ion etch (RIE) that removes any resist resi- due from the bottom of the aperture. (c) The pattern is then transferred to the Au mask by Ar plasma etching. Note that the Cr layer is not etched. (d) 10 keV15N ions were implanted to introduce nitrogen atoms at /C257.5 nm below the surface. e Mask is removed by wet chemical treatment. (f) Sample was annealed at 1000/C14C to form NV/C0centers with a spatial resolution of /C2413 nm. Reprinted with permission from Scarabelli et al. , Nano Lett. 16, 4982 (2016). Copyright 2016, American Chemical Society.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-27 VCAuthor(s) 2020structures can, then, be patterned and etched via an in situ lithography technique266[seeFig. 28(b) ]. With this technique, very small systematic misalignments (<10 nm), as well as minimal fabrication-induced spectral shifts (/H113511 nm), have been achieved, which can be compensated via tuning of the QD.403 This approach has also been used for nanodiamonds, where DLSPPWs were fabricated with deterministic positioning to include precharacterized nanodiamonds with NV-centers.202,203The authors were able to control the in-plane position of the nanodiamond with the desired NV-center within about 30 nm. C. Wafer bonding Quantum emitters embedded in a high-quality bulk crystalline material are able to produce stable single-photon emission with highpurity and indistinguishability. Ideally, hybrid heterostructures of QDs and photonic components can be grown directly on a single wafer. However, growing such heterostructures directly often results in poor crystal quality due to the formation of antiphase boundaries and largemismatches in material lattice constants, thermal coefficients, and charge polarity. 6 Wafer-to-wafer bonding is a method for integrating dissimilar material platforms.404Consider the transfer of a III-V material onto a silicon nitride photonic circuit: Each material is grown separately with optimized substrates and conditions, thus maintaining high crystal quality for both compounds. The III-V wafer is flipped and bondedonto the top surface of the photonic wafer; subsequent removal of the substrate of the transferred wafer leaves a thin membrane structure on top of the photonic circuit. Photonic structures are then patterned using lithographic techniques; for example, the coupling of the emis- sion from InAs QDs in a GaAs waveguide cavity structure into an FIG. 28. Integration strategies for QDs on photonic circuits. (a) Schematic of QDs deposited via spin-coating. A lithographic mask allows for deterministic positioning on top of photonic cavity devices. Reprinted with permission from Chen et al. , Nano Lett. 18, 6404 (2018). Copyright 2018, American Chemical Society. (b) Schematic of a photonic waveguide patterned via in situ electron beam lithography (EBL) around a preselected QD, ensuring optimal alignment between the QD and waveguide. Reprinted with permis- sion from Schnauber et al. , Nano Lett. 19, 7164 (2019). Copyright 2019, American Chemical Society. (c) (Left) A GaAs nanobeam on a Si 3N4waveguide fabricated via EBL from a wafer-bonded GaAs/Si 3N4heterostructure. (Right) Simulation results showing the efficient coupling between the GaAs and Si 3N4waveguides. Reprinted with permission from Davanco et al. , Nat. Commun. 8, 889 (2017), Author(s) licensed under a Creative Commons Attribution 4.0 License. (d) Optical image (left) and schematic (right) of InAs QDs integrated with a photonic chip via orthogonal wafer bonding. Reprinted with permission from Murray et al. , Appl. Phys. Lett. 107, 171108 (2015). Copyright 2015, AIP Publishing. (e) (Top) Schematic of the transfer printing technique using a transparent rubber stamp. (Bottom)Optical image of a transfer-printed Q D device on top of a silicon waveguide. Reprinted with permission from Katsumi et al. , APL Photonics 4, 036105 (2019). Copyright 2019, Authors(s) licensed under a Creative Commons Attribution 4.0 License. (f) (Left) Schematic of pick-and-place positioning of QD devices using a sharp microprobe. (Right) Electron microscopy image of a nanowire QD attached to a micro- probe. Reprinted with permission from Elshaari et al. , Nat. Commun. 8, 379 (2017). Copyright 2019, Authors(s) licensed under a Creative Commons Attribution 4.0 License.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-28 VCAuthor(s) 2020underlying silicon nitride waveguide has been observed, with an over- all coupling efficiency of /C240.2 (Ref. 256)[Fig. 28(c) ]. Similar techni- ques can be employed to integrate diamond with other materialplatforms. For example, numerous GaP-on-diamond plat-forms 146,148,149have been realized by bonding a thin epitaxial film of GaP to a diamond substrate via Van der Waals bonding. However, random positioning of SPEs with respect to the pho- tonic structures in such wafers will result in nonoptimal coupling,leading to a low yield of efficiently coupled devices across the wafer.This can be improved via in situ lithography (see Sec. VII B 4 )a r o u n d preidentified SPEs after the wafer bonding step. 266 The wafer bonding can also be performed orthogonally [ Fig. 28(d) ] for optimized SPE out-of-plane emission (e.g., with distributed Braggreflector cavities 399,405) to be efficiently coupled into photonic wave- guides. However, since only devices at the wafer edge can be integrated( a so p p o s e dt oa c r o s st h ee n t i r ew a f e rf o rn o n o r t h o g o n a lb o n d i n g ) ,t h i sapproach appears to be less scalable. D. Pick-and-place Another approach that allows for precise positioning of emitters on the photonic circuit is to pick-and-place individual emitters (or thenanostructures they are embedded in) instead of having a singlewafer-scale integration step. Emitters can be precharacterized and pre-selected and then selectively integrated at desired positions on the pho-tonic circuit. For example, they can be either placed on top of existingwaveguides, 129,233,236or at specific points relative to a marker for sub- sequent waveguide encapsulation, i.e., the waveguide material is depos-ited and patterned over the emitter. 406Besides ensuring optimal coupling of the emitters to the photonic circuit, this method alsoallows for a greater flexibility in the choice of emitter host material anddevice geometry. There are two common techniques for performing the pick-and- place transfer: transfer printing via an adhesive stamp and using asharp microprobe. 1. Transfer printing The transfer printing method typically uses a stamp made of an adhesive and transparent material, such as PDMS or Gelfilm fromGelpak, which allows for precise alignment of the structures under anoptical microscope during the transfer. This has been successfully dem-onstrated for exfoliable layered crystals 407and QDs278,400,408[Fig. 28(e) ]. In the case of 2D flakes, using a dry viscoelastic stamp is advantageouscompared to wet processes, 409–411since there are no capillary forces involved, which could potentially collapse the suspended material. First, the emitter (e.g., QD nanowire or exfoliated 2D flake) is attached to the stamp. Next, the emitter is brought to the sample sur-face using XYZ micromanipulators. To release the emitter, the stampis pressed against the surface and then peeled off slowly. Due to thestamp’s viscoelasticity, it behaves as an elastic solid at short timescaleswhile slowly flowing at longer timescales. Consequently, the viscoelas-tic material can detach from the emitter by slowly peeling the stampoff the surface. Different strategies to control the adhesion of the stampare detailed in a separate review paper. 412 There are several challenges in using the transfer printing tech- nique. The stamping process induces a force over a large sample area and may damage parts of the fragile photonic circuit, although thismay be mitigated by using a sufficiently small stamp not much larger than the transferred material.413M o r e o v e r ,i ti sd i f fi c u l tt or e p o s i t i o n the emitters as the adhesion between the integrated structures is typi- cally much stronger than their adhesion to the stamp. Nevertheless, this method provides close to 100% success rate for transfer onto atomically flat materials, though for rougher surfaces theyield is lower due to reduced adhesion forces. With the aid of addi- tional alignment markers, positioning accuracies better than 100 nm have been achieved. 408 2. Microprobe An alternative is to perform the pick-up and transfer using a sharp microprobe [ Fig. 28(f) ]. A small amount of adhesive (e.g., PDMS) can be added to the probe tip to aid the transfer, analogous to am i c r o s t a m p .414Although this technique requires precise control of the microprobe, it is able to transfer small, fragile structures such assingle nanowires with high accuracy and controllability when aided by an optical microscope, 236,279,406and especially so when using an elec- tron microscope.234Currently, the manual transfer of individual devi- ces one by one can be very time-consuming, but there is great potential in automating the process and allowing for scalable fabrica- tion of many integrated devices. Besides picking up single nanowires, a microprobe enabled pick- and-place technique allows for the mechanical transfer and removal ofSi masks onto diamond substrates (see Fig. 29 ). This allows for the fab- rication of high quality Si masks due to existing mature Si processing technology, which, when combined with a negligible Si etch rate dur-ing O 2RIE etching of the diamond membrane, allows for tight fabrica- tion tolerances of the diamond membrane. Moreover, such a mask can be reused multiple times due to the negligible Si etch rate. One FIG. 29. Schematic of the mask transfer technique. (I) A micro-PDMS adhesive on a tungsten probe tip is used to transfer a Si mask (orange) onto a diamond mem- brane (gray) on top of a Si substrate. (II) The Si mask serves as a etch mask forO 2plasma RIE. (III) Pattern on the Si mask is transferred to the diamond mem- brane after etching and removal of the Si mask. (IV) An isotropic SF 6dry etch can then be used to undercut the Si substrate to create a suspended diamond photonic structure if desired. Reprinted with permission from Li et al. , Sci. Rep. 5, 7802 (2015). Copyright 2015, Authors(s) licensed under a Creative Commons Attribution4.0 license.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-29 VCAuthor(s) 2020significant advantage of this mask transfer technique is that the dia- mond membrane is never exposed to damaging irradiation that can adversely affect the NV/C0centers’ properties. Indeed, using such a fab- rication procedure, a spin coherence lifetime of /C24200ls was measured for NV/C0centers coupled to suspended 1-D photonic crystal defects, which are similar to lifetimes measured in their parent CVD crys- tals.137Masks positioned in this way can be placed with submicrome- ter or even nm scale accuracy if integrated with an electron microscope.414,415 Another commonly used microprobe is an AFM tip. Combined with a confocal microscope, nanoparticles such as nanodiamonds containing desired color centers can be picked up and integrated with the optical devices such as photonic crystal cavities.197Scanning the AFM tip in intermittent contact mode over the focus of the confocal microscope allows identification of the precharacterized nanodiamonds. The tip is then pressed on the center of the nanodiamond. A force is applied, and surface adhe- sion allows the nanodiamond to be attached to the tip. After suc- cessful picking up, which could take over 50 trials, the tip is pressed over the new desired place to allow the nanodiamond to be integrated with the photonic structure. However, there is only lim- ited success rate for the placing stage.197 VIII. CONCLUSION AND OUTLOOK In this paper, we have reviewed a variety of SPEs that are conve- niently embedded within a solid. These emitters have promising prop- erties and can be used to form the building blocks of future quantum networks. As discussed in Sec. II A, there are various important met- rics to evaluate a SPE in terms of its suitability for various applications. For the sake of easy comparison, Table I lists a selected range of inte- grated SPEs, along with some of the metrics introduced in Sec. II A. Similarly, Table II tabulates various resonators that have been inte- grated with SPEs and characterizes them based on a few relevant properties.AsTables I andIIshow, there are a few ways of integrating SPEs with on-chip optical structures and thus realizing functional quantumdevices. Indeed, these interfaces enhance the light-matter interactionand allow efficient interaction and entanglement between the distantemitters. However, translation from proof-of-concept laboratory dem-onstrations of individual components to full-scale quantum devices isstill quite immature. Considerable efforts are required to overcomeissues associated with the material incompatibility of quantum emit-ters, photonic circuits, and other required components on the samechip. Furthermore, there are still challenges in developing high-throughput and reliable integration techniques. We propose the fol-lowing four critical steps to address these challenges: First, quantum photonic devices should be thoroughly designed, fabricated, and tested since quantum information processing imposesstringent demands on loss and fabrication accuracy. These demands,which are at the limits of conventional silicon photonic technology,might require fabrication for quantum applications to be achieved atthe expense of scalable fabrication by, for example, using time-consuming electron beam lithography instead of photolithography. Second, the coupling between quantum systems and resonant photonic cavities should be optimized through the accurate position-ing of the emitter in the cavity. This calls for further improvements inthe reliability and throughput of methods such as AFM manipulation,nanopatterning, and various transfer techniques that have beenemployed for accurate positioning of these quantum emitters. Third, full scalability implies integration with on-chip single-pho- ton sensitive detectors and lasers on-chip. The development of on-chip active devices would eliminate the need for using bulk optics andallow a significantly smaller footprint for the photonic platform.Design, fabrication, and characterization of on-chip photodetectorsand lasers operating at desired performance levels are challenging taskssince they involve multiple fabrication steps involving various materi-als that require state-of-the-art clean room facilities and a comprehen-sive strategy for heat management and integrating associatedoptoelectronics. TABLE II.Comparison of various integrated resonators. QtheandQexpare the theoretical and experimental quality factors, respectively, and Vmode,k, and Fexpare the mode vol- ume, resonance wavelength, and experimental Purcell factor, respectively. Type Material Qthe Qexp Vmodeðk=nÞ3k(nm) Fexp References Ring resonator Diamond — 5500 15 638 12 155 L3 PCC Diamond 6000 3000 0.88 637 70 109 Ring resonator Diamond >106/C244000 17–32 637 12 151 1D nanobeam Diamond 1041635 3.7 638 7 123 Ring resonator þWG Diamond — ð3:260:4Þ/C2103— 665.9 — 154 1D nanobeam Diamond — 7000 0.47 637 22 157 1D nanobeam Diamond — 3300 — 637 62 137 Disk resonator GaP — 2500 /C010 000 — 637 — 149 L3 PCC ND þGaP 1000 603 0.75 639.5 12.1 196 L3 PCC ND þGaP 6000 610 0.74 643 7.0 199 Microdisk cavity ND þSi O 2 34 0000 170 000 82 637 — 200 Optical microcavity ND þplanar mirror — 19 000 3.4 737 /C0759 9.2 210 þSM optical fiber Microdisk resonator ND þ3C-SiC 150 000 2700 5.5 737 2–5 174AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031701 (2020); doi: 10.1116/5.0011316 2, 031701-30 VCAuthor(s) 2020Fourth, a crucial building block for quantum networks is the real- ization of quantum-mechanical interaction and entanglement betweentwo separate quantum nodes on the same optical chip. Photons emit-ted by two independent nodes should be able to coherently interfereon a beamsplitter and produce an interference signal. This taskrequires demanding control of quantum emitters and photonic ele-ments, but demonstration of such an interaction would lead to more complicated schemes of quantum networking, including the interac- tion of a large number of quantum emitters on the same chip. Clearly, a consolidation of new technologies is required to address these challenges and to demonstrate a platform for quantum networking that is scalable and amenable to mass manufacturing. Thisprogram should leverage on a broad collaboration between experts inquantum physics, integrated photonics, materials science, andelectronics. ACKNOWLEDGMENTS This work was supported by the National Research Foundation Grant No. NRF-CRP14-2014-04, “Engineering of aScalable Photonics Platform for Quantum Enabled Technologies.”The authors also acknowledge support from the QuantumTechnologies for Engineering (QTE) program of A /C3STAR Project No. A1685b0005. DATA AVAILABILITY Data sharing is not applicable to this article as no new data were created or analyzed in this study. REFERENCES 1H. J. Kimble, Nature 453, 1023 (2008). 2N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, Rev. Mod. Phys. 83, 33 (2011). 3J. R. Maze et al. ,Nature 455, 644 (2008). 4G. Balasubramanian et al. ,Nature 455, 648 (2008). 5A. W. Elshaari, W. Pernice, K. Srinivasan, O. Benson, and V. Zwiller, Nat. Photonics 14, 285 (2020). 6J.-H. Kim, S. Aghaeimeibodi, J. Carolan, D. Englund, and E. Waks, Optica 7, 291 (2020). 7I. Aharonovich, D. Englund, and M. Toth, Nat. Photonics 10, 631 (2016). 8/C19A. Gali, Nanophotonics 8, 1907 (2019). 9A. M. Edmonds, U. 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5.0009677.pdf

J. Chem. Phys. 153, 054110 (2020); https://doi.org/10.1063/5.0009677 153, 054110 © 2020 Author(s).Simulated field-modulated x-ray absorption in titania Cite as: J. Chem. Phys. 153, 054110 (2020); https://doi.org/10.1063/5.0009677 Submitted: 01 April 2020 . Accepted: 03 July 2020 . Published Online: 04 August 2020 Pragathi Darapaneni , Alexander M. Meyer , Mykola Sereda , Adam Bruner , James A. Dorman , and Kenneth Lopata ARTICLES YOU MAY BE INTERESTED IN PSI4 1.4: Open-source software for high-throughput quantum chemistry The Journal of Chemical Physics 152, 184108 (2020); https://doi.org/10.1063/5.0006002The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Simulated field-modulated x-ray absorption in titania Cite as: J. Chem. Phys. 153, 054110 (2020); doi: 10.1063/5.0009677 Submitted: 1 April 2020 •Accepted: 3 July 2020 • Published Online: 4 August 2020 Pragathi Darapaneni,1 Alexander M. Meyer,2 Mykola Sereda,2Adam Bruner,2,a) James A. Dorman,1,b) and Kenneth Lopata2,3,b) AFFILIATIONS 1Cain Department of Chemical Engineering, Louisiana State University, Baton Rouge, Louisiana 70803, USA 2Department of Chemistry, Louisiana State University, Baton Rouge, Louisiana 70803, USA 3Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA a)Current address: Department of Chemistry and Physics University of Tennessee at Martin, Martin, Tennessee 38238, USA. b)Authors to whom correspondence should be addressed: jamesdorman@lsu.edu and klopata@lsu.edu ABSTRACT In this paper, we present a method to compute the x-ray absorption near-edge structure (XANES) spectra of solid-state transition metal oxides using real-time time-dependent density functional theory, including spin–orbit coupling effects. This was performed on bulk-mimicking anatase titania (TiO 2) clusters, which allows for the use of hybrid functionals and atom-centered all electron basis sets. Furthermore, this method was employed to calculate the shifts in the XANES spectra of the Ti L-edge in the presence of applied electric fields to understand how external fields can modify the electronic structure, and how this can be probed using x-ray absorption spectroscopy. Specifically, the onset of t 2gpeaks in the Ti L-edge was observed to red shift and the e gpeaks were observed to blue shift with increasing fields, attributed to changes in the hybridization of the conduction band (3 d) orbitals. Published under license by AIP Publishing. https://doi.org/10.1063/5.0009677 .,s I. INTRODUCTION Transition metal oxides possess a wide range of optical,1elec- trical,2and magnetic properties3–5that stem from the character and occupations of the d-orbitals.6–9These properties can be tailored using external stimuli including electric,10optical,11and chemi- cal fields12due to the changes in the energy/hybridization of the transition metal d-orbitals.13,14Subsequently, many modern tech- nologies such as photovoltaics,15,16solid-state display panels,17and non-volatile memory devices18,19that demand charge mobility and reversibility4are currently based on transition metal oxides. Recent experiments, for example, have shown that the electronic struc- ture of transition metal oxides can be modulated using electric fields induced by surface ligands, which can be used to tune the optoelectronic properties.12,20For all of these applications, under- standing the relationship between applied fields and the electronic structure is critical to elucidate the physical mechanisms and ulti- mately design new functional inorganic materials. Experimental approaches to this include UV–Vis absorption spectroscopy,21,22x-ray photoemission/absorption spectroscopy,20,23,24and magnetic measurements using the superconducting quantum interference device (SQUID).22,25 Among these, x-ray absorption near-edge structure (XANES) spectroscopy has emerged as a powerful tool for probing the elec- tronic structure of transition metal oxides26–28due to its atomic specificity and ability to capture subtle changes in the unoccupied electronic states (conduction band)29that result from the changes in the lattice geometry,30oxidation state,31band spacing,32and band populations.33XANES has been applied to measure changes due to weak fields, such as the effect of surface ligands on the d-orbitals of TiO 2,12as well as strong field processes such as band-tunneling and transient metallization.34Interpreting XANES spectra, however, can be quite challenging, which necessitates first-principles simula- tions for relating the observed spectra to the underlying electronic structure and/or dynamics. There are numerous methods for computing XANES spec- tra for molecules and solids.35–37Semi-empirical methods such as crystal-field multiplet (CFM)38and charge-transfer multiplet J. Chem. Phys. 153, 054110 (2020); doi: 10.1063/5.0009677 153, 054110-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp (CTM)39,40can give transition metal spectra that match with exper- iments quite well12,41but require choosing empirical crystal field parameters. Alternatively, one can use ground-state-based first- principles ( ΔSCF) methods to compute a spectrum directly from the transition between the core and valence states. These ΔSCF approaches require some description of the core hole, which is typ- ically done by constraining the occupancy.42–47For the electronic structure method, density functional theory (DFT)43–45,48–51is often used due to a good tradeoff between accuracy and efficiency. Mul- ticonfigurational methods are also applied to XANES, such as com- plete/restricted active space SCF (CAS/RASSCF),46,52multireference configuration interaction (MRCI),53and multireference coupled- cluster (MRCC).54These are especially successful for partially occu- pied degenerate ground states and partially filled orbitals, but care must be taken in choosing the active space. While SCF-based meth- ods naturally capture relaxation, they may suffer from variational collapse.55Additionally, they may require modifications to explicitly enforce orthogonality.56 Excited-state methods, which do not suffer from these limita- tions, are also used to compute XANES spectra. Single-determinant excited-state methods such as static exchange (STEX),57,58lin- ear response (LR),59–61and real-time (RT)59,62–70time-dependent density functional theory (TDDFT) have been quite success- ful but may fail for double excitations and multiplet effects and often give inaccurate absolute energies due to incomplete core-hole relaxation.59These problems can often be remedied somewhat by using post Hartree–Fock (HF) methods, such as equation-of-motion coupled-cluster (EOM-CC)54,71and real-time EOM-CC,72,73which are systematically improvable but with sig- nificantly increased computational cost. Green’s function (GW) approaches, such as multiple-scattering with SCF potentials,74,75 Bethe–Salpeter-Equation (BSE typically with the GW approxima- tion),76–78and algebraic diagrammatic construction (ADC),79cap- ture relaxation well but typically require transitions to be calcu- lated separately, which can make them inconvenient for broadband spectroscopy.80 In particular, XANES calculations of transition metal oxides typically use some form of periodic boundary conditions with either grids or a planewave basis, primarily using DFT,48multiple- scattering,74,75and GW/BSE.76–78These give a good description of the band structure and reliable spectra. For DFT, however, these basis sets are usually limited to local density (LDA) and general- ized gradient (GGA) approximations or Hubbard-correct versions of these functionals (LDA/GGA+U).81–83On the other hand, low concentration of dopants and defects can be challenging, and for practical reasons, these basis sets preclude the wavefunction-based methods such as hybrid DFT or post-HF techniques. As an alter- native, finite simulations can be used, where a cluster is embedded chemically/electrostatically to emulate the bulk.62,67,84This enables the use of all-electron atom-centered basis sets, which is conve- nient for inner-shell spectra and allows for efficient evaluation of exchange integrals. However, they can struggle to properly repre- sent long range interactions and can suffer from unphysical finite size effects. Some examples of previous XANES simulations using cluster models include TiO 2using DFT85and multiple-scattering,86 CaF 2using ROCIS,85and iron oxides using MRCI.53Finally, rel- ativistic effects can be significant in transition metal oxides as the spin–orbit (SO) coupling is often on the order of the peaksplitting in a XANES spectrum.87In TiO 2, for example, the SO splitting in the 2 porbitals is roughly 6 eV,12,13meaning one can- not simply do a separate L IIIand L IIsimulation as for larger Z elements.59 TDDFT-based methods are convenient for valence proper- ties of condensed-phase systems88–92due to the good tradeoff between reasonable computational cost, predictability, and ability to be extended to include SO coupling.64,67,87,93Although less com- mon than the valence, TDDFT has also been applied to solid-state XANES including silica,62alkaline-earth oxides,94and titania.95In this context, atom-centered basis sets and cluster models are espe- cially useful as they obviate the preparation of core holes or tran- sition potentials and allow for efficient evaluation of hybrid func- tionals, which have been shown to give improved bandgaps.96,97 Additionally, hybrids have been shown to give better XANES spec- tra in molecules vs LDA/GGAs, with 50% exact exchange giving the most accurate absolute energies.98LR-TDDFT can be challeng- ing due to the convergence issues of the large number of roots required for the calculation of the XANES spectra.99Real-time methods, where the density matrix or orbitals are propagated in time, are well-suited to solid-state XANES simulations as they yield the entire spectrum from valence-to-core transitions without hav- ing root convergence issues.100,101Another advantage of using a real-time method is the ability to simulate non-linear63,66and time- resolved processes, such as transient absorption spectroscopy.102,103 The methods developed in this paper were done with this in mind. This paper demonstrates the use of RT-TDDFT simulations with the inclusion of SO coupling to elucidate the effect of external fields on the electronic structure (XANES spectra) of a prototypi- cal transition metal oxide system, i.e., anatase TiO 2.104,105First, we develop bulk-mimicking clusters for anatase and validate SO-RT- TDDFT for computing the Ti L-edge XANES spectra. Next, we apply a range of static fields and calculate the field-modified XANES spec- tra to better understand how observed changes in spectra can be related to the changes in the Ti d-orbital energy landscape. Ulti- mately, this work has implications in solar cells,15(bio)sensing,106,107 tunable displays,108and capacitors,109etc., where application of external fields can be used to modify the optoelectronic properties. II. METHODS All calculations were performed using a customized develop- ment version of NWChem110with Gaussian-type orbital (GTO) basis sets and relativistic effects described using zeroth order regular approximation (ZORA).111The B3LYP exchange-correlation func- tional was used for the calculations in this study as it has performed well in previous studies for ground and excited-state properties of TiO 2.96,112All the geometry optimizations and similar calculations, along with the XANES calculations, were performed with the fol- lowing basis sets:113Ti atoms were given a Def2-TZVP basis set, O atoms were given a Def2-SVP basis sets, and the pseudo-H atoms were given a 6-31G basis set. In this work, we use bulk-mimicking clusters, which is a well-developed approach for weak-field ver- tical valence and core-level excited states in non-metallic materi- als where the excitations are localized in space.114Herein, Ti xOy J. Chem. Phys. 153, 054110 (2020); doi: 10.1063/5.0009677 153, 054110-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp clusters were developed using a covalent embedding procedure.84,89 The experimental bulk anatase TiO 2structure was cut to yield a Ti centered finite cluster, which was then “chemically passivated” with pseudo-hydrogen atoms at the boundaries (Fig. S1). This is done by replacing the outermost atoms in the cluster by appropriately charged H atoms. In anatase TiO 2, Ti forms a distorted octahedron with the six surrounding O atoms, while O atoms form a trigonal planar structure with the three surrounding Ti atoms. Based on the formal oxidation states of Ti (+4) and O ( −2), each Ti atom will share an effective charge of +2/3 and each O atom will possess an effective charge of −2/3 to their neighboring atoms. The boundary pseudo-hydrogen atoms, which replace the outermost Ti atoms, will therefore have an effective charge of +2/3. These charges are inde- pendent of the applied electric field. Additionally, by varying the O–H bond lengths, one can control how much electron density is donated or withdrawn from the cluster. The O–H bond distance was chosen as 1.0 Åbased on previous work.89The clusters were then geometry optimized with the interior atoms allowed to move, while pinning the boundary O atoms and fixing the O–H bond lengths and H–O–H angles. Next, the XANES spectra were computed using the NWChem real-time TDDFT module65using a two-component SO approach, similar to previous relativistic RT-TDDFT implementations.61,67,93 A manuscript detailing the validation and technical aspects of this two-component implementation is in preparation.115In this approach, the single particle density matrix is propagated in time after broadband pulse excitation. The equation of motion in the von Neumann representation is given by i∂P′(t) ∂t=[F′(t),P′(t)], (1) where F′(t) andP′(t) are the Fock and density matrices in the canon- ical basis.65The details regarding the integration of the equation of motion and calculation of the time-advanced Fock matrix are given in Ref. 116. To save computation time, our propagator is con- structed via the exponential midpoint of the extrapolated (future) Fock matrix without self-consistent iteration, P′(t+Δt)=eΩP′(t)e−Ω, (2) Ω≡−iF′(t+Δt 2)Δt. (3) This approximation, which was tested for a few spectra, is valid due to the relatively short time steps, which are required to capture x- ray frequency spectra. The time step used for these calculations is ∆t= 7.3 ×10−4fs (0.03 a.u.), and the total time of propagation is 9.7 fs (400 a.u.). The time step was chosen to be short enough to adequately capture the frequency according to the energy range of the XANES spectra. To compute the spectra, the system was excited using a delta-function (broadband) electric field E(t)=κexp[−(t−to)2 2w2]ˆd. (4) For every simulation, the field amplitude was taken to be κ= 0.0001 a.u, the center of the pulse was to= 20 a.u = 0.48 fs, and the width wasw= 0.024 fs. ˆd=x,y,zdenotes the polarization. This electric field is coupled into the Fock matrix in the atomic orbital basis through an external potential ( V) by its product with the transition dipole matrix ( D), under the assumption of the uniform electric field across the system, V(t)=−D⋅E(t). (5) This approximation is valid for relatively low x-ray frequencies stud- ied here. For higher energies, however, quadrupole and higher terms may need to be considered. A Padé accelerated method of Fourier transform analysis116was employed to convert the spectra from time-domain to frequency- domain using only the contributions from the Ti 2 pcore spin- orbitals. This converges much more rapidly with simulation time than conventional Fourier transform of the dipole moment and also allows one to only include contributions from the L-edge in the spec- trum. We checked convergence with simulation time and observed that the spectra were converged for signals longer than 8.5 fs (350 a.u.). An alternative acceleration method is to use a time-correlation function with the energy of the core-level factored out to allow for a larger time step.117 To compute the spectra, the time-dependent dipole moment is first written as a sum of occupied-virtual pair dipoles, μ(t)=μo+∑mocc i=1∑m a=mocc+1μia(t), (6) μia(t)=DMO iaPMO ai(t)+DMO aiPMO ia(t), (7) where i= 1,. . .,moccare the virtual orbitals and a=mocc+1,. . .,mare the occupied orbitals. These are computed by projecting the density matrix onto the ground state molecular orbitals, DMO=C′†(0)D′C′(0), (8) PMO(t)=C′†(0)P′(t)C′(0), (9) where C′(0) is the eigenvector matrix of the ground state (in the canonical basis). Now the dipole polarizability for each dipole contribution ( αia dd) is obtained by taking Padé approxi- mant to the Fourier transform for each dipole contribution signal separately, αia dd=μia dd(ω)E∗ d(ω) ∣Ed(ω)∣2. (10) Here, ˆd=x,y,zand the ddsubscripts denote the on-diagonal part of the polarizability tensor, e.g., xxmeans xdipole resulting from x polarized kick. The dipole strength function is then computed from the polarizability as S(ω)=4πω 3cIm[αxx(ω)+αyy(ω)+αzz(ω)]. (11) Finally, to better match experimental spectra that have lifetimes that generally decrease with increasing energy above the edge, we apply energy-dependent broadening to our spectrum separately for both J. Chem. Phys. 153, 054110 (2020); doi: 10.1063/5.0009677 153, 054110-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp LIIIand L IIedges. This is achieved by fitting the spectrum to a num- ber of Lorentzian curves and then broadening those curves accord- ing to τ(E), which takes the form of an exponentially decreasing core-hole lifetime given by τ(E)=τoe−α(E−Eo), (12) where αhas dimensions of inverse energy, E ostands for edge energy and is equal to 458.1 eV for the L IIIedge, and τvaries from 70 s−1to 60 s−1. For the L IIedge, Eo= 463.4 eV and τvaries from 47 s−1to 25 s−1. Another option for energy-dependent broad- ening is to add a small imaginary potential to the Fock matrix with the values chosen phenomenologically, e.g., exponential in the eigenvalues in the basis of the Kohn Sham orbitals.118This results in peaks with increasing widths as you go higher above the edge. However, since the operator is non-Hermitian, strong applied fields will cause significant ionization, which can give unphysical spectra. III. RESULTS Before computing the effect of electric fields on the XANES spectra, we first validate our approach for the case of anatase TiO 2 without an applied field. To confirm convergence of results with the cluster size, we checked the optical gaps and orbital char- acter for a Ti 9O38H60(107 atom) and a smaller Ti 3O14H24(41 atom) cluster (Fig. 1) carved from the large one. For convenience, we computed the optical gaps using LR-TDDFT with 10 roots. In principle, RT-TDDFT, could also be used but would require longer simulation times. The optical gap of the 107 atom clus- ter was computed to be 3.1 eV with the “valence band” domi- nated by O 2 pand the “conduction band” dominated by Ti 3 d orbitals. The smaller 41 atom cluster had molecular orbitals simi- lar to that of the large cluster with an optical gap of 3.7 eV. This is an overestimate of the experimental value of 3.2 eV,119likely due to the quantum confinement effects observed in smaller clusters. Although the L-edge spectra should not necessarily depend on the FIG. 1 . Ti3O14H24bulk-mimicking anatase cluster.specific value of the optical gap, the value and character of the gap serve as indicators that the cluster is semiconductor-like. Based on these results, the smaller 41 atom cluster adequately mimics bulk anatase and is thus employed for all subsequent calculations. For the XANES calculations, we use our spin–orbit (SO)-RT-TDDFT ver- sion of NWChem since the LR code in NWChem does not have SO coupling. Figure 2 shows the resulting Ti L-edge XANES spectrum of the bulk-mimicking Ti 3O14H24anatase cluster. This cluster has 370 basis functions and took approximately two days on using 80 pro- cessors to complete the time propagation. To match the experi- ment, we include energy-dependent broadening and then shift our XANES spectrum by +9.33 eV to account for core-hole relaxation effects that are inadequately captured by TDDFT.120,121Overall, it is observed that there is good agreement with the experimental spec- trum, including the value of crystal field splitting energy (10 Dq). The two-component SO-RT-TDDFT captures both the L III(Ti 2 p3/2 →Ti 3d) and L II(Ti 2 p1/2→Ti 3d) edges, as well as the energy splitting between them ( ∼6 eV). The peak splitting of the e gpeak in the L IIIedge is attributed to dz2and dx2−y2orbitals due to the deviation of the Ti from O hsymmetry in anatase.122However, in the simulated spectra, the peak intensities of these e gpeaks are reversed, likely due to finite size effects. While RT-TDDFT would be tractable for the 9-Ti (107) atom cluster, we found that the 3-Ti (41) atom cluster already gave adequate agreement with exper- iment, and thus, we did not perform the larger calculation due to computational cost. It would require 3000 processors for the 9-Ti atom cluster, which has 1057 basis functions, to compute the spec- tra in the same amount of time. This choice of 3-Ti atom cluster is consistent with the previously reported restricted open-shell cal- culations of TiO 2using the B3LYP functional by Maganas and co- workers, who showed that a 3-Ti atom cluster gave nearly converged spectra.85 FIG. 2 . Comparison of experimental (dashed) vs calculated (solid) XANES of the Ti L-edge for the Ti 3O14H24bulk-mimicking cluster. Experimental spectrum repro- duced with permission from Darapaneni et al. J. Phys. Chem. C 122, 22699 (2018). Copyright 2018 American Chemical Society. J. Chem. Phys. 153, 054110 (2020); doi: 10.1063/5.0009677 153, 054110-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . (a) Ti L III/IIXANES spectra of anatase TiO 2and (b) expansion of the LIIIedge, showing an increase in the value of 10 Dq with increasing applied electric field. To elucidate the effects of applied fields on the Ti L-edge XANES spectra and the resulting d-orbitals, the system was con- verged in the ground state in the presence of static electric fields ranging from 0 V/nm to 0.07 V/nm applied in the x-axis direction. The largest magnitude of field roughly corresponds to half of mate- rial’s breakdown voltage ( ∼0.14 V/nm) for this cluster, i.e., field at which the bandgap disappears. In order to create a quasi-uniform electric field on the Ti 3O14H24cluster, two point charges, with mag- nitudes ranging from ±43eto 300 e, were separated at a distance of ±100Åin the x-axis of the cluster. Similar calculations with fields applied in yandz-axes were performed but did not show any sig- nificant differences. To better resolve the spectral features, the Ti L-edge XANES spectra shown in Fig. 3(a) is uniformly broadened instead of using energy-dependent broadening. Looking at the L III edge [Fig. 3(b)], first, we notice a subtle red shift of the onset t 2g peak. Second, we observe a splitting of the dz2(#) peak at higher fields (0.07 V/nm), and third, an increase in the intensity of the dx2−y2 (∗) peak with increasing field is observed. The red shift of the t 2g peak at higher fields suggests non-degeneracy of the dxy,dyz, and ordzxorbitals123upon the application of the fields. Analogous to this, the dz2(#) peak was also observed to split at higher fields. This phenomenon of d-orbital Stark splitting in the presence of applied static fields is similar to that of rare earth 4 forbitals, which split in the presence of electromagnetic radiation.124,125An increase in the intensity of the dx2−y2(∗) features with electric field amplitude is indicative of less hybridization with the p-states and, thus, higher oscillator dipole strengths.126This is attributed to the overlap of egwith the ligand p-states that are along the axes, resulting in a variation in hybridization between the Ti ion and the surrounding ligand. Additionally, the value of crystal field splitting energy (10 Dq), which is calculated as the energy difference between the t 2g and dz2(#) peaks in the L IIIedge, is observed to slightly increase when applied external fields are above a critical strength, in this case between 0.03 V/nm and 0.07 V/nm (see the supplementary material). Qualitatively, our calculations are consistent with surface- functionalization experiments on Ni2+-doped TiO 2films using polarized ligands ( μ=±5 D).12Due to the shallow penetration(<5 nm) of the ligand-induced fields127and soft x rays,128as well as the surface segregation of the Ni2+ions, these experimental observations are essentially surface selective. In these experiments, the e gpeak in the Ni L II-edge was observed to increase in intensity with the increasing ligand-induced electric field. In this regard, our calculated Ti L III-edge XANES spectra show a similar effect, albeit at a ten times stronger field strength compared to the experiments. This is likely due to the partially filled d-orbitals in Ni2+ions, which are strongly influenced by the field than the unoccupied d-orbitals in Ti4+.129While we cannot directly compare the values of 10 Dq from our calculations (TiO 2) and experiments (Ni2+-doped TiO 2), the 10 Dq calculated from the Ti L III-edge spectra shows a similar trend as the experiments,12,23,130i.e., an increase in its value for fields above 0.03 V/nm. In these ligand-bonding experiments, the Ni2+L-edge spec- tra were fit to an empirical model using ligand field multiplet the- ory.39,131Based on the fitting, the changes in spectra upon ligand- bonding were interpreted as a slight field-induced elongation of the axial bonds around the transition metal ions at the inorganic– organic interface.12To differentiate between the roles of electronic effects vs geometry on the spectra, we use a single-Ti atom cluster as an extreme case of possible distortions in the presence of fields. While there was an axial bond distortion upon relaxation of this cluster, these geometry relaxation effects counteracted the electronic effects on the spectra. In other words, the shifts in the spectra due to geometry relaxations are opposite to that of the static field-induced changes (see the supplementary material). Since the single-Ti atom cluster grossly overestimates the distortions, it is reasonable to con- clude that the geometry would be less perturbed in the bulk, thus having less effect on the spectra. Therefore, the observed changes in the spectra can be attributed to primarily electronic (hybridization) effects. IV. CONCLUSIONS In summary, we have developed a spin–orbit real-time TDDFT method to compute the XANES spectra of TiO 2under the pres- ence of external fields, which captures both the L IIIand L IIedges J. Chem. Phys. 153, 054110 (2020); doi: 10.1063/5.0009677 153, 054110-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp of Ti. Spin–orbit coupling is crucial for the calculation of L-edge spectra in first-row transition metal oxide systems as the coupling is of the order of a few eVs. For this purpose, bulk-mimicking anatase clusters were developed, and the bandgap was evaluated to verify the accuracy of the cluster. This finite cluster method offers some advantages over other ab initio methods as it uses all electron basis sets and allows for the use of hybrid functionals to improve the quality of the XANES spectra. This technique was used to eluci- date the field-induced changes in the electronic structure of TiO 2for static electric fields varying from 0 V/nm to 0.07 V/nm. Although it is experimentally challenging to disentangle electronic effects from geometric field-induced effects, our calculations indicate that fields can modify the electronic structure without geometry distortions. Critically, in the limit that geometry relaxation effects are negligi- ble, these changes in the d-orbital hybridization can be probed via XANES. In particular, the onset of the t 2gpeaks is red shifted and the egpeaks are blue shifted with increasing fields, along with an increase in the intensity of the dx2−y2peak. While these spectral changes are specific to the anatase TiO 2system, which has a distorted octahedral Ti4+site (D 2d), similar effects are likely to be observed for other tran- sition metal oxides for which first-principles calculations may assist in interpretation. SUPPLEMENTARY MATERIAL The supplementary material for this article contains the geome- tries of 107 atom and 41 atom clusters and the effect of geometry optimization on the single-Ti cluster. AUTHORS’ CONTRIBUTIONS A.M.M. and P.D. developed the TiO 2cluster models. A.M.M. performed the RT-TDDFT calculations. P.D. and A.M.M. ana- lyzed the computed XANES spectra. A.B. and M.S. assisted in the Padé analysis and energy-dependent broadening. P.D. and A.M.M. contributed equally to this work. ACKNOWLEDGMENTS K.L. acknowledges support by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Atomic, Molec- ular and Optical Sciences, under Contract No. DE-SC0017868. J.A.D. acknowledges the National Science Foundation (NSF) under Grant No. CHE-1709902 for financial support. P.D. is thank- ful to the U.S. Department of Energy (DOE) under EPSCOR Grant No. DE-SC0012432 for financial support and the Gradu- ate School of Louisiana State University for the Dissertation Year Fellowship. This research was conducted with high performance computational resources provided by Louisiana State University (http://www.hpc.lsu.edu). 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5.0012709.pdf

J. Chem. Phys. 153, 074701 (2020); https://doi.org/10.1063/5.0012709 153, 074701 © 2020 Author(s).Hole-punching for enhancing electrocatalytic activities of 2D graphene electrodes: Less is more Cite as: J. Chem. Phys. 153, 074701 (2020); https://doi.org/10.1063/5.0012709 Submitted: 04 May 2020 . Accepted: 27 July 2020 . Published Online: 17 August 2020 Yunxiang Gao , Lipeng Zhang , Zhenhai Xia , Chang Ming Li , and Liming Dai COLLECTIONS Paper published as part of the special topic on 2D Materials Note: This paper is part of the JCP Special Topic on 2D Materials. 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Phys. 153, 074701 (2020); doi: 10.1063/5.0012709 Submitted: 4 May 2020 •Accepted: 27 July 2020 • Published Online: 17 August 2020 Yunxiang Gao,1,2Lipeng Zhang,3Zhenhai Xia,4Chang Ming Li,5 and Liming Dai1,6,a) AFFILIATIONS 1Department of Macromolecular Science and Engineering, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 44106, USA 2Department of Chemistry, Prairie View A&M University, Prairie View, Texas 77446, USA 3State Key Laboratory of Organic–Inorganic Composites, Beijing Advanced Innovation Center for Soft Matter Science and Engineering, College of Chemical Engineering, Beijing University of Chemical Technology, Beijing, China 4Department of Materials Science and Engineering, Department of Chemistry, University of North Texas, Denton, Texas 76203, USA 5Institute for Advanced Cross-field Sciences, Qingdao University, 308 Ningxia Road, Qingdao, Shandong, China 6UNSW-CWRU International Joint Laboratory, School of Chemical Engineering, University of New South Wales, Sydney, NSW 2052, Australia Note: This paper is part of the JCP Special Topic on 2D Materials. a)Author to whom correspondence should be addressed: l.dai@unsw.edu.au ABSTRACT Using a polymer-masking approach, we have developed metal-free 2D carbon electrocatalysts based on single-layer graphene with and without punched holes and/or N-doping. A combined experimental and theoretical study on the resultant 2D graphene electrodes revealed that a single-layer graphene sheet exhibited a significantly higher electrocatalytic activity at its edge than that over the surface of its basal plane. Furthermore, the electrocatalytic activity of a single-layer 2D graphene sheet was significantly enhanced by simply punching microholes through the graphene electrode due to the increased edge population for the hole-punched graphene electrode. In a good consistency with the experimental observations, our density function theory calculations confirmed that the introduction of holes into a graphene sheet generated additional positive charge along the edge of the punched holes and hence the creation of more highly active sites for the oxygen reduction reaction. The demonstrated concept for less graphene material to be more electrocatalytically active shed light on the rational design of low-cost, but efficient electrocatalysts from 2D graphene for various potential applications ranging from electrochemical sensing to energy conversion and storage. Published under license by AIP Publishing. https://doi.org/10.1063/5.0012709 .,s I. INTRODUCTION Graphene with an atomic layer of carbon atoms that are densely packed in a 2D honeycomb crystal lattice has received tremen- dous interest for various potential applications.1,2Electrochem- istry plays a major role to make graphene useful in biosensors3–5 and electrochemical energy conversion and storage, including fuelcells,6supercapacitors,7and batteries.8,9Although many studies on electrochemistry of graphene have been reported,10,11most of them focused on the collective performance of numerous individual graphene sheets on an electrode. While the electrochemical activ- ity of the graphitic electrode was demonstrated to rely on their edge planes,12–16the edge vs basal plane electrochemistry has been less discussed for a single-layer graphene sheet.17,18 J. Chem. Phys. 153, 074701 (2020); doi: 10.1063/5.0012709 153, 074701-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp We have previously studied the electrochemical difference of the tip and sidewall of carbon nanotubes (CNTs) with polymer- masking techniques in which the tip of carbon nanotubes was selectively exposed by sealing a super-long vertically aligned carbon nanotube (SVA-CNT) rope with polymers, followed by scissor-cutting of the circular cross section of the CNT rope.19 With the development of chemical vapor deposition (CVD) based graphene growth, large-dimensional graphene sheets have been easily obtained,20making the fabrication of edge-only graphene electrode possible with similar polymer-masking techniques to investigate the electrochemistry of the edge and basal plane for a single-laye graphene.18 In order to gain a better understanding of the edge vs basal plane electrochemistry for a single-layer graphene sheet, we report a systematic study on the electrochemical properties of a single-layer CVD-grown graphene sheet by polymer masking. The graphene electrode was prepared by polymer-masking to selectively expose their surface or edge for comparison. Potassium ferricyanide [K3Fe(CN) 3], ascorbic acid (AA), cysteine, nicotinamide adenine dinucleotide (NADH), and oxygen reduction reaction (ORR) were used as electrochemical probes for recording the electrochemical differences on the graphene surface and edge, respectively. For a single-layer graphene sheet, we found that the current density on graphene edge is at least four orders of magnitude higher than that on the planar surface, with the electron transfer rate also being much faster on the edge. Following this observation, we further found that the electrochemical performance of a single piece of CVD- grown graphene can easily be improved by simply punching the graphene sheet with microholes to expose more edges. Compared with the wet chemical approach used in our previous study18that could readily introduce heteroatoms (e.g., O), leading to function- alized graphene edges to cause the edge or bulk structure changes, the mechanical punching that we used in the present study can retain the pristine structure of the graphene lattice and keep the graphene edge free from any possible functionalization for a more reliable comparison between the graphene edge and basal plane. In a good consistency with the experimental observations, our den- sity function theory (DFT) calculations confirm that the introduc- tion of holes into a graphene sheet generated additional positive charge along the edge of the punched hole and hence the creation of more highly active sites for ORR and beyond. This work repre- sents a conceptually novel, but facile and scalable, approach to highly efficient single-layer graphene electrodes for potential applications ranging from electrochemical sensing to energy conversion and storage. II. RESULTS AND DISCUSSION A. Experiments 1. Sample preparation and characterization Scheme 1 shows the polymer-masking method for preparing the edge-only and surface-only single-layer graphene electrodes. The as-grown graphene on a copper film was covered with 10% PMMA solution in toluene and dried to form a hard PMMA stamp. If there are any minor cracks on the graphene sheet, the crack gap can be sealed with the PMMA coating. Thus, there should be no crack SCHEME 1 . Strategy used for fabrication of the surface- and edge-exposed single- layer graphene electrodes. edges exposed for subsequent testing. The Si-substrate-supported graphene and PMMA layers were then immersed in a saturated FeCl 3solution to etch off the copper catalyst film and release the PMMA stamp with the transferred graphene film on it. A copper extension electrode was then attached on the graphene side with silver paste. To fabricate the surface-exposed graphene electrode, the edge of the PMMA stamp with transferred graphene, as well as most of the peripheral area of the graphene surface, was brush- coated with PMMA solution and dried, leaving only a small area about 4 mm2in the center of the graphene sheet exposed to the air (Fig. S1). The electrode thus prepared was then used as the surface- exposed electrode for electrochemical testing. Upon completion of studies on the surface graphene electrode, the whole electrode was then rapidly brush-coated with PMMA solution again and dried for a complete electrode-sealing. We then cut the graphene electrode embedded in PMMA to get the edge-exposed graphene electrode (Fig. S1). In a typical experiment, graphene film was grown on a SiO 2/Si substrate with a 300-nm thick Cu film as the catalyst in a CVD FIG. 1 . (a) Raman spectrum of the CVD-grown graphene with the Cu catalyst. (b) AFM topography image of a CVD-grown graphene film that was transferred onto a silicon wafer. J. Chem. Phys. 153, 074701 (2020); doi: 10.1063/5.0012709 153, 074701-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp chamber. The Cu catalyst was activated at 900○C under a H 2gas flow (100 sccm). The sample was then heated up to 1000○C inside a quartz tube under a mixture flow of Ar/CH 4/H2(Ar/CH 4/H2 200/50/65) and reacted for 5 min–6 min.18Finally, the growth sys- tem was subjected to rapidly cooling down to room temperature by taking out the quartz tube from the furnace under the protection of Ar gas flow, leading to the production of an ultrathin graphene layer. The Raman spectrum in Fig. 1(a) shows that the 2D/G ratio of the graphene film is close to 4:1, indicating that the graphene film is a single layer. Atomic force microscopy (AFM) image [Fig. 1(b)] shows that the graphene film is continuous and has typical wrinkles and folds. B. Electrochemical performance 1. The pristine graphene Using the surface- and edge-graphene electrodes prepared according to the strategy shown in Scheme 1, we tested the electro- chemical response of the graphene surface and edge, respectively, to 5 mM K 3[Fe(CN) 6] in a 0.1M phosphate buffer saline (BPS, PH = 6.5). As shown in Fig. 2, the electrode with only the edge exposed exhibited well-identified redox peaks [Fig. 2(a)], while the surface- only electrode showed almost no noticeable peak, but just tail-like shapes with small shoulders. The electron transfer rate can be stud- ied by the peak-to-peak potential separation;21the smaller the peak- to-peak separation, the faster the electron transfer rate. Thus, well- defined redox peaks for the edge-only electrode indicate that the electron transfer is much faster on graphene edge than that on the graphene surface. At all the scan rates investigated in this study, the onset potential of the electroxidation of [Fe(CN) 6]+4was observed around +0.1 V on the edge-only electrode, while the surface-only electrode appeared around +0.3 V, indicating a lower activation energy for ferrocyanide oxidation at the graphene edge. Our edge- only electrode has a graphene edge of 1 cm in length and 0.8 nm in height, while the surface-only electrode exposed an area of about 2×2 mm2. Thus, the apparent geometrical area ratio between the edge and the surface ( Aedge:Asurface ) is about 1:200 000. Based on this electrode area ratio, the current density ratio ( jedge vsjsurface ) is∼15 000:1. Moreover, due to the existence of folds and wrinkleson the graphene surface, the actual surface area for the surface-only electrode could be even larger than its geometrical area as we esti- mated above, making the ratio of jedgetojsurface even higher than the aforementioned value of 15 000:1. Thus, the graphene edge could generate a current density, which is at least four orders of magni- tude higher than that of the graphene surface. Clearly, therefore, the edge of a single-layer graphene sheet is much more electrochemically active than the surface of the same electrode. In order to confirm that the electrolyte did not diffuse into the PMMA coating, we made electrochemical measurements on the completely PMMA-sealed graphene sheet before cutting it for the edge exposure. No electrical signal was detected (see Fig. S2 of the supplementary material). As carbon materials are attractive for sens- ing biologically important molecules,3–5we continued to investi- gate the edge and surface effects of single-layer graphene electrodes on biologically important molecules, including ascorbic acid (AA), cysteine, and NADH. Figures 3(a) and 3(b) shows typical cyclic voltammograms of 2.0 mM ascorbic acid (AA) in 0.1M PBS elec- trolyte (pH 6.5) at the edge-only and surface-only graphene elec- trodes, respectively. The CV curves corresponding to the edge-only graphene electrode [Fig. 3(a)] show a distorted peak shape with some sigmoidal characters due to its ultra-small size.22In contrast, the CV curves corresponding to the surface-only graphene electrode [Fig. 3(b)] reveal tail-like waves associated with the AA oxidation.10 A simple comparison between Fig. 3(a) with 3B suggests that the electron transfer rate on the graphene surface is much slower than that at the edge. Furthermore, the onset potential for AA oxida- tion is about 0.15 V at the edge [Fig. 3(a)], a value that is 550 mV lower than that of the surface electrode [Fig. 3(b)], indicat- ing that a much lower energy is required for the graphene edge to catalyze AA oxidation. Similar electrochemical behavior was also observed for the oxidation of cysteine on the surface and edge of graphene, respectively. CV curves collected on the edge-only elec- trode show a small oxidation peak at 0.35 V [Fig. 3(c)] while just tail-like waves with a higher onset potential for the surface-only elec- trode [Fig. 3(d)]. For the oxidation of NADH, the oxidation peaks are not so strong on both the edge- and surface-only electrodes, which is consistent with the CNT electrode previously reported for NADH.19However, if the NADH oxidation signal is compared with the CV curve collected in the pure buffer solution without FIG. 2 . Cyclic voltammograms of 5.0 mM K3[Fe(CN) 6] recorded in a pure buffer solution (PBS) solution (pH 6.5) at the edge-exposed (a) and surface-exposed (b) graphene electrode sealed in PMMA. Scan rate: 200 mV s−1, 150 mV s−1, 100 mV s−1, 50 mV s−1, 20 mV s−1, and 10 mV s−1. J. Chem. Phys. 153, 074701 (2020); doi: 10.1063/5.0012709 153, 074701-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Cyclic voltammograms of (a) recorded in 2.0 mM ascorbic acid (AA) in 0.1M PBS solution (pH 6.5) at the edge-exposed (a) and surface-exposed (b) graphene electrode sealed in PMMA, 2.0 mM cysteine in 0.1M PBS solution (pH 6.5) at the edge-exposed (c) and surface-exposed (d) graphene electrode sealed in PMMA, and 2.0 mM NADH in 0.1M PBS solution (pH 6.5) at the edge-exposed (e) and surface-exposed (f) graphene electrode sealed in PMMA. Scan rate: 100 mV s−1, 50 mV s−1, 20 mV s−1, 10 mV s−1, and 5 mV s−1. NADH, we can clearly distinguish a small NADH oxidation peak at 0.7 V on the edge-only electrode with an onset potential of 0.55 V [Fig. 3(e)]. However, the corresponding NADH oxidation peak on the surface-only graphene electrode appears 100 mV higher at 0.8 V with a higher onset potential of 0.65 V [Fig. 3(f)]. These results indicate that both the oxidation of cysteine and NADH occurred easier and faster at the edge of the graphene with respect to its surface.2. The pristine graphene with punched holes Based on the above observation that the edge of a single-layer graphene sheet is much more active than its surface for electrocatal- ysis, we further introduced more edges to a single-layer graphene sheet by punching microholes through a transferred graphene sheet on the PMMA substrate (Fig. 4) with a micro needle array (see Fig. S3 of the supplementary material). Figure 4(a) schematically J. Chem. Phys. 153, 074701 (2020); doi: 10.1063/5.0012709 153, 074701-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . Scheme (a) and microscope image (b) of the punched micro-hole arrays on a graphene electrode. (c) Cyclic voltammograms of 5 mM K 3[Fe(CN) 6] recorded in a phosphate buffer solution (pH 6.5) on the punched and unpunched graphene electrode, respectively; scan rate: 20 mV s−1. (d) Cyclic voltammograms of 5 mM K 3[Fe(CN) 6] recorded in a phosphate buffer solution (pH 6.5) at the hole-punched graphene electrode recorded at different scan rates. shows a single-layer graphene sheet with punched holes, while Fig. 4(b) shows a microscopic image of the hole-punched graphene electrode. Compared to the bare graphene surface that did not show any well-defined redox peak for 5 mM K 3[Fe(CN) 6] in PH = 6.5 phosphate buffer, the CV curves recorded with the hole-punched graphene electrode show well-defined redox peaks reflecting a faster reaction of the analyte [Fig. 4(c)]. Furthermore, the oxidation peak potential decreased by 300 mV by introducing more edges for the graphene electrode through hole-punching due to the better electrocatalytic activity at the edge. Owing to the extremely small area of the edge for a single graphene sheet, the edge-only graphene electrode can be regarded as a nanoelectrode. Ultrasmall microelectrodes or nanoelectrodes have attracted long attention for various electrochemical applica- tions with multiple advantages, including a high mass transport, high signal-to-noise ratio, and good capability for measuring the steady-state current.23However, the current generated from a single ultrasmall microelectrode is usually very low due to its low electro- chemical active surface area. In order to increase the current signal of ultrasmall microelectrodes, ultramicroelectrode arrays (UMEAs) have been developed based on pillar-like electrode arrays generated, for instance by complex lithography techniques.24Instead of the pillar arrays as conventional UMEAs, the observed superior electro- chemical performance of the graphene edge to its surface prompted us to punch holes (pore arrays) over a graphene sheet as the UMEAs [Figs. 4(a) and 4(b)]. An ultramicroelectrode is often characterized by a sigmoidal shaped CV curve at steady states. As can be seen in Fig. 4(d), sigmoidal shaped CV curves are evident at slow scanning rates. 3. N-doped graphene with and without punched holes In view of the recent development of heteroatom-doped carbon nanomaterials (e.g., N-doped vertically aligned carbon nanotubes25and CVD-grown graphene sheets26) as efficient metal-free electro- catalysts for the oxygen reduction reaction (ORR) in fuel cells,6,27 ORR and oxygen evolution reaction (OER) in metal-air batteries,28,29 OER and hydrogen evolution reaction (HER) for water-splitting to generate hydrogen energy,30,31and many other energy-related reac- tions,32–34we further tested nitrogen-doped graphene UMEAs based on the pore arrays for ORR. To keep all the tests on the same graphene layer, we used ammonia plasma for nitrogen dopping35 of the pre-tested pristine graphene electrode, followed by the FIG. 5 . Linear scan voltammetry (LSV) for ORR catalyzed by the as-grown pristine graphene, ammonia-plasma treated (N-doped) graphene, and hole-punched N- doped graphene in 0.1M KOH. J. Chem. Phys. 153, 074701 (2020); doi: 10.1063/5.0012709 153, 074701-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp hole-punching. After the NH 3plasma treatment, Raman spectrum shows the appearance of a new D-band, indicating the occurrence of N-doping in the graphene lattice (see Fig. S4 of the supplementary material). As shown in Fig. 5, the pristine graphene electrode showed a low current and onset potential. Upon the NH 3-plasma treatment for 2 min,35however, both the onset potential and the current inten- sity significantly increased (Fig. 5). The two oxidation steps at −0.6 V and−0.8 V for the LSV curve of the pristine graphene electrode merged into one step after the NH 3-plasma treatment, indicating an transition from a 2-electron to 4-electron pathway for the ORR.25As the N-doped graphene electrode was punched, the mass of the effec- tive graphene layer participated in ORR was reduced, but the cath- ode current intensity from ORR increased significantly due to the good ORR activity associated with the N-doped graphene edge. This “less is more” phenomenon sheds light on the rational design of newlow-cost, but effective, electrodes and electrocatalysts. As expected, the N-doped hole-punched graphene electrode showed almost zero current in the N 2-saturated 0.1M KOH. C. DFT calculations The role of graphene edge in electrocatalytic activity was also studied theoretically via the density function theory (DFT) calcula- tions through Gaussian 03 (Revision E.01; Gaussian, Inc., Walling- ford, CT, 2004). The details of the calculation can be found in the supplementary material and also in Ref. 36. To be consistent with the experimental samples, we built four types of graphene structures: the pristine (pure) graphene (C 100H26), graphene of the same size but with a hole, and these graphene sheets doped with nitrogen, as shown in Fig. 6. FIG. 6 . The distributions of charge density on (a) the pristine (pure) graphene, C 100H26, and (b) the same graphene with a hole and spin density on (c) graphene with a N dopant in the center (C 100H26N) and (d) graphene with a hole and a N atom on the hole edge. The small and large balls stand for hydrogen and carbon atoms, respectively. The nitrogen atoms are highlighted by dotted circles. J. Chem. Phys. 153, 074701 (2020); doi: 10.1063/5.0012709 153, 074701-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The atomic spin and charge densities determine the catalytic capability of carbon materials for ORR.32,36Our previous work on graphene shows that the carbon atoms with positive spin or charge density larger than ∼0.15 are most likely to serve as catalytic active sites.36,37Thus, the number of atoms with large positive spin or charge density is a measure of the catalytic capability of carbon materials. The atomic spin density and charge density were calcu- lated for these graphene structures. It was found that all the atoms with large positive spin or charge densities were located near the edge (Fig. 6). The charge or spin density is almost zero in the mid- dle of the graphene sheet, as shown in Figs. 6(a) and 6(c). When a hole is made on graphene, additional positive charges are gener- ated along the edge of the punched holes, suggesting that punching a hole will create more active sites for ORR. This could explain why the graphene edge has better catalytic performance for ORR, and hole-punching can enhance the catalytic activity of a single-layer graphene electrode. It is known that nitrogen-doping introduces positive-charge and spin densities on carbon atoms near nitrogen atoms in graphene, which facilitates ORR.36In this calculation, we found that when graphene was doped by nitrogen at its center, the atoms with large positive spin were located at its edge, instead of surrounding the dopant, as shown in Fig. 6(c). Thus, even when a dopant is located around the center of a graphene sheet, it enhances the catalytic capability of the edge of the graphene. On the other hand, when a nitrogen atom exists at the edge of a punched hole on a graphene sheet, it generates more active sites along the edge with even larger spin density [ ∼3 folds, Fig. 6(d)] than those on a per- fect graphene sheet with a nitrogen dopant at its center. As a result, the catalytic capability of graphene edge could be further enhanced by the combination of the doping and edge effect. The higher ORR activity of the edge than the basal plane of graphene is also confirmed by the ORR overpotential calculations.38 Another simple indicator of catalytic activity is the bandgap energy [i.e., energy separation of the highest occupied molec- ular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO)]. A smaller HOMO–LUMO gap implies lower kinetic sta- bility and higher chemical reactivity because smaller energy gap means that the state of the graphene is energetically favorable to add electrons to a high-lying LUMO and to extract electrons from a low-lying HOMO and so to readily form the activated complex for any potential reactions.39We have calculated the HOMO–LUMO energy gap of the pristine perfect and punched graphene structures [Figs. 6(a) and 6(b)]. The graphene with a hole has energy gaps of 0.31 eV–0.64 eV, with an average value over two times lower than the perfect graphene (0.8 eV). Thus, the punched graphene is catalyt- ically more active for ORR, which is consistent with the experimental results. III. CONCLUSIONS In summary, we have developed 2D carbon electrocatalysts based on graphene with and without punched holes and/or N- doping and performed a comprehensive combined experimental and theoretical study on the edge and surface electrocatalytic activi- ties of single-layer 2D graphene sheets. It was found that a single- layer graphene sheet exhibited a higher electrocatalytic activity at its edge than that over the surface of its basal plane. The currentdensities generated from the K 3[Fe(CN) 6] electrochemical probe and certain biologically important molecules (e.g., ascorbic acid, cys- teine, and NADH) at the edge-only graphene electrode can be up to several orders of magnitude higher than that of its surface-only counterpart. More interestingly, it was, for the first time, demon- strated that punching microholes through the graphene electrode could significantly increase its electrocatalytic activity due to the increased edge population for the hole-punched graphene elec- trode. N-doping can further enhance the electrocatalytic activity of the single-layer graphene electrode with and without holes for ORR. DFT calculations confirmed that the introduction of holes into a graphene sheet generated additional positive charge along the edge of the punched hole and hence the creation of more highly active sites for ORR and beyond. These results indicate that the less (graphene material) is more (electrocatalytically active) and shed light on the rational design of new low-cost, but efficient, electro- catalysts from graphene and other 2D materials for various poten- tial applications, ranging from electrochemical sensing to energy conversion and storage. SUPPLEMENTARY MATERIAL See the supplementary material for photographs of the microneedles, the edge-only and surface-only graphene electrodes, Raman spectrum for of ammonia-plasma treated graphene elec- trode, and cyclic voltammograms of 5.0 mM K 3[Fe(CN) 6] recorded in phosphate buffer solution (PBS, pH 6.5) at the edge-exposed and surface-exposed graphene electrode. ACKNOWLEDGMENTS The authors are grateful for financial support of this research from the Australian Research Council (Grant Nos. ARC, DP 190103881, FL 190100126, and IH180100020), the National Key Research and Development Program of China (Grant No. 2017YFA0206500), the National Natural Science Foundation of China (Grant No. 51732002), the Distinguished Scientist Program at BUCT (buctylkxj02), Fundamental Research Funds For the Central Universities (buctrc202008) and the U.S. National Science Founda- tion (Grant Nos. 1561886 and 1662288). 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5.0012967.pdf

Appl. Phys. Lett. 117, 052101 (2020); https://doi.org/10.1063/5.0012967 117, 052101 © 2020 Author(s).On the origin of red luminescence from iron- doped β-Ga2O3 bulk crystals Cite as: Appl. Phys. Lett. 117, 052101 (2020); https://doi.org/10.1063/5.0012967 Submitted: 07 May 2020 . Accepted: 20 July 2020 . Published Online: 03 August 2020 Rujun Sun , Yu Kee Ooi , Peter T. Dickens , Kelvin G. Lynn , and Michael A. Scarpulla On the origin of red luminescence from iron-doped b-Ga 2O3bulk crystals Cite as: Appl. Phys. Lett. 117, 052101 (2020); doi: 10.1063/5.0012967 Submitted: 7 May 2020 .Accepted: 20 July 2020 . Published Online: 3 August 2020 Rujun Sun,1 Yu Kee Ooi,1Peter T. Dickens,2Kelvin G. Lynn,2and Michael A. Scarpulla1,3,a) AFFILIATIONS 1Electrical and Computer Engineering, University of Utah, Salt Lake City, Utah 84112, USA 2Materials Science and Engineering Program, Washington State University, Pullman, Washington 99164, USA 3Materials Science and Engineering, University of Utah, Salt Lake City, Utah 84112, USA Note: This paper is part of the Special Topic on Ultrawide Bandgap Semiconductors. a)Author to whom correspondence should be addressed: mike.scarpulla@utah.edu ABSTRACT Currently, Fe doping in the /C241018cm/C03range is the most widely available method for producing semi-insulating single crystalline b-Ga 2O3 substrates. Red luminescence features have been reported from multiple types of Ga 2O3samples, including Fe-doped b-Ga 2O3, and attributed to Fe or N O. Herein, however, we demonstrate that the high-intensity red luminescence from Fe-doped b-Ga 2O3commercial substrates con- sisting of two sharp peaks at 689 nm and 697 nm superimposed on a broader peak centered at 710 nm originates from Cr impurities presentat a concentration near 2 ppm. The red emission exhibiting a twofold symmetry, peaks in intensity for excitation near the absorption edge,seems to compete with the Ga 2O3emission at a higher excitation energy and appears to be intensified in the presence of Fe. Based on the polarized absorption, luminescence observations, and the Tanabe–Sugano diagram analysis, we propose a resonant energy transfer of photo- generated carriers in the b-Ga 2O3matrix to octahedrally coordinated Cr3þto give red luminescence, possibly also sensitized by Fe3þ. Published under license by AIP Publishing. https://doi.org/10.1063/5.0012967 b-Ga 2O3exhibits an ultrawide bandgap from 4.5 to 4.8 eV (with the optical axis dependent), suggesting its application for electronic devi-ces requiring high breakdown field. 1Both widely variable n-type con- ducting and semi-insulating layers can be achieved with extrinsicdoping. Specifically, the controllable n-type doping can be achieved inmultiple growth techniques using Si dopant, as well as Ge, Sn, Zr, andHf. Dopants of Fe, Mg, and N as deep acceptors are used to constructb-Ga 2O3FET lateral devices, current blocking layers in vertical devices, and to provide highly resistive substrates/buffer layer for other applica- tions.2,3Computations show that oxygen vacancy is a deep donor, while gallium vacancies and their various complexes, especially with hydro-gen, are the most probable dominant compensating native defects. 4–8 Fe-doped b-Ga 2O3semi-insulating crystals are commercially available and widely used as substrates for epitaxial b-Ga 2O3layers. In these crystals, the Fe dopant concentration is around 1018cm/C03to compensate background donors and pin the Fermi level.9–11The bandgap of Fe-doped b-Ga 2O3has been reported as /C244.5 eV12and is reduced from /C244.6 eV to /C242.9 eV with [Fe]/([Ga] þ[Fe]) from 0.0 to 0.4.13The deep donor-like level (E2) of Fe is measured as E c-0.78 eV using deep level transient spectroscopy.14The Fe2þ/3þcharge transi- tion level has been determined as E c-0.8460.05 eV using noncontactspectroscopy methods (DLTS).15However, other work has reported the optically induced change from Fe3þas 1.360.2 eV using steady- state photo-induced electron paramagnetic resonance (EPR) measure-ments. 16For luminescence, Polyakov et al. reported two sharp emission lines near 1.78 eV and 1.80 eV at low temperature,17sugges- ting an origin from highly localized atomic states (e.g., d orbitals). They ascribed the 1.78 eV peak to the4T1!6A1intracenter transition of Fe3þ. This assignment was based on a logical but circumstantial argument that since Fe is the highest-concentration intentional impu-rity, this emission is probably related to Fe. Hany et al. reported that a red to near-infrared band (R-NIR) emerged after annealing in the air,and two extremely sharp R 1and R 2peaks appeared below 140 K.12 The R-NIR sharp peaks were ascribed to nitrogen incorporated duringair annealing. However, this is also circumstantial and at odds with the general finding that transitions arising from states mixed with b-Ga 2O3matrix states are broadened by the strong carrier-lattice cou- pling.18Finally, similar red peaks are observed in thermally stimulated luminescence of UID,19Mg-doped,20and Fe-doped21b-Ga 2O3 crystals and electroluminescence of Si and Cr co-doped b-Ga 2O3 crystals,22and origins from Fe and Cr are claimed but also lack detailed investigation. Appl. Phys. Lett. 117, 052101 (2020); doi: 10.1063/5.0012967 117, 052101-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplHere, we seek to clarify some details of the red and near-infrared luminescence from Fe-doped b-Ga 2O3crystals. Reduced optical bandgap and increased subbandgap absorption are observed. In PL, b- Ga2O3emissions are strongly quenched, and an additional structure emerges of a broad red peak around 710 nm with two sharp peaks at688 nm and 696 nm. The sharp and broad red peaks are assigned to the emission of 2E!4A2and4T2!4A2of Cr internal transitions, respectively. Finally, we discuss the possible origins of the Cr co- doping and possible mechanisms within the luminescence pathway. Three types of Fe-doped crystals [Syn (100) and Syn (010) grown by the Czochralski (CZ) method by Synoptics, and NCT (010) grown by the edge-fed growth method by Novel Crystal Technology] were studied. Additionally, we measured a (100) unintentionally doped (UID) crystal grown at Washington State University (WSU) by verti- cal gradient freeze (VGF). We use polarization-dependent transmis-sion measurements to determine the bandgaps and absorption coefficient. Photoluminescence was collected using a fiber coupled spectrometer in an integrating sphere as a function of polarization of the incident laser. For temperature-dependent PL, the laser beam was normal to the sample surface, while the luminescence was collected with an optical fiber located perpendicular to the laser beam. We quantified the concentrations of Fe and Cr using inductively coupledplasma mass spectrometry (ICPMS). The fundamental absorption of b-Ga 2O3occurs from the valence band maxima mostly composed of O-2p orbitals to the conductionband minima composed of Ga-4s orbitals. 23The absorption edges for different axes result from selection rules and splitting related to the O- 2p states. Figures 1(a) and1(b)show the transmittance vs polarization angle of the Syn (100) and Syn (010) Fe-doped b-Ga 2O3,r e s p e c t i v e l y . For the Syn (100) sample, the absorption onset rises sharply with the photon energy. The optical transition thresholds deduced from Taucplots of ðahvÞ2vshvfor Ejjc and E jjb are 4.54 eV and 4.81 eV, respec- tively, in good agreement with the prior reported values.23For the Syn (010) sample, the absorption slowly increases with photon energy for each incident direction, and the bandgaps of E jjc and E jja/C3are 4.25 eV and 4.34 eV, respectively. The ordering and energies of these thresh- olds are also consistent with the literature: E g, Ejjc<Eg,Ejja/C3<Eg,Ejjb.24 The reduction of the apparent threshold for E jjc and E jja/C3of the Syn (010) sample could be explained by the larger concentration of Fe and associated disorder.13A peculiar feature of the transmittance data is that the data for all incident angles cross at one particular energy near4.30–4.40 eV for both samples. This crossing of the data may result from changes in reflectivity near the band edge associated with the birefringence and changes in the refractive index for the two directions near the optical absorption transition. Moreover, in Fig. 1(c) , we observe anisotropy in the subbandgap absorption. Both samples’ data illustrate regions that show the expo- nential energy deference, i.e., a¼a 0expE/C0Eg EU/C16/C17 ,w h e r e a0is a con- stant; and E,Eg,a n d EUare the photon energy, optical bandgap, and Urbach energy, respectively. Fitting using the above expression yields EU¼76 meV and 52 meV for Ejjba n d Ejjc, respectively, for the Syn (100) Fe-doped sample. These are considerably larger values as com- pared, for example, to crystalline Si and GaAs (11–18 meV), but are in the range for hydrogenated amorphous Si (50–100 meV).25 Furthermore, EUfor the Syn (010) Fe-doped sample are 600 meV and 360 meV for Ejja/C3and Ejjc, respectively [ Fig. 1(c) ]. These values are similar to the levels found for ion-implanted GaAs (300–520 meV).25 The typical physics of positional disorder induced Urbach energies would be expected to produce isotropic sub-gap absorption, whereas the fact that we observe anisotropy seems to suggest that the transi-tions are still tied to specific anisotropic or selection rule features in the band structure or anisotropic defect absorption. Therefore, we con- sider it unlikely that the true Urbach parameter is so large given that the samples are crystalline. It is possible that the slopes of the data in this region reflect complications of the shape of the absorption edge bythe dipole-forbidden minimum bandgap transition, which is close in energy. 26Finally, we note that no additional absorption bands are detectable across the wider visible range for both the Syn (010) andSyn (100) samples [ Fig. 1(d) ]. This shows that these samples contain low enough concentrations of transition metals that their intracenter absorption bands are below detection limits. Figure 2(a) shows the typical PL spectra for NCT (010), Syn (100), and Syn (010) Fe-doped b-Ga 2O3. The UID sample shows the typical UV, UV0, blue, and green emissions.27The Fe-doped samples show UV0, blue, and green peaks of /C24100 times lower in intensity as compared to those from the UID samples. Additionally, the dominant red emission from these Fe doped samples consists of a broad peak centered at 710 nm with two sharp peaks at 689 nm and 697 nm (R 1 and R 2) superimposed. Low-temperature PL shows that the two sharp peaks intensify while the broad peak decreases and eventually dimin-i s h e sa st h et e m p e r a t u r ed e c r e a s e s[ Fig. 2(b) ]. The quenching of the broad peak at low temperatures indicates that its emission requires an excitation over a thermal barrier. The two sharp peaks exhibit verynarrow bandwidth (FWHM <3 nm for all measured temperatures). The energy difference between the two sharp peaks is 18.1 meV at around 100 K. The red emission shows a twofold symmetry, which coincides with the incident polarization parallel to the a /C3,b ,a n dc FIG. 1. Transmittance spectra for different light polarization angles for the (a) Syn (100) and (b) Syn (010)-oriented Fe-doped b-Ga 2O3crystals. (c) Estimated absorp- tion coefficient corrected by the refractive index and its Urbach tail fitting. (d) Un-polarized transmittance over a wider sub-gap energy range for the Syn (100) andSyn (010) samples.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 052101 (2020); doi: 10.1063/5.0012967 117, 052101-2 Published under license by AIP Publishingaxes. This implies that the optical absorption mainly occurs in the b-Ga 2O3matrix. The photoluminescence excitation (PLE) spectra for both the UID WSU (100) and the Syn (100) Fe-doped samples showsimilar trends in the E//b direction [ Figs. 2(c) and2(d)] but differ in E//c. The intensity of the UV emission from the UID sample continu-ously drops with an increasing excitation energy ( E ex). The intensity of a 689 nm emission of the Syn (100) sample fades above 4.9 eV andthen the emission intensity integrated from 300 to 600 nm from b- Ga 2O3emerges. The red emission in E//c is inefficient above a 4.9 eV excitation and the b-Ga 2O3emission occurs simultaneously, sugges- ting a competition for photocarriers between these two emissionpathways. The red PL emission originating from nitrogen incorporation 28 differs in three aspects as compared to those from the Fe-doped b- Ga2O3bulk crystal: (1) the centroid shifts from 1.71 eV to 1.65 eV with temperature decreasing to 10 K; (2) the nitrogen related peak exhibitsonly one broad peak from 300 to 10 K (with FWHM /C240.4 eV); and (3) this broad peak intensifies with a decreasing temperature rather thandisappearing.Hybrid functional calculations show that self-trapped holes and many extrinsic defects can give rise to very broad luminescencebands 18due to the strong electron–phonon coupling. This applies for acceptors such as Mg, Ca, and N; shallow donors of Si and Sn; and o t h e ri m p u r i t i e sl i k eB ,N a ,A l ,S ,C l ,P b ,a n dB ii ft h e ya r ei n v o l v e di nPL. We stress here that it is the expectation for emission arising fromstates that mix significantly with the states of the b-Ga 2O3matrix, as opposed to those arising from the atomic-like d and f shell transitions.The narrow bandwidths of the R 1and R 2peaks in the Fe-doped b-Ga 2O3suggest that it arises from the internal transitions of ions, specifically, spin-forbidden transitions with long lifetime. The spin-allowed transitions exhibit fast decay leading to lifetime-broadenedpeaks. 29We are well aware that many extrinsic impurities, including some transition metals, are found in the b-Ga 2O3melt-grown crystals, some originating in the feedstock materials and some introduced dur- ing growth (especially from crucibles). Transition metal impurities inb-Ga 2O3crystals include Ir, Zr, Ti, and Ni.30These elements are not believed to contribute to the sharp red emission structure we discusshere in this paper. Transitions related to all mentioned but the Fe andCr are summarized in Table I . Ir has not been observed as a lumines- cent center in minerals and semiconducting or insulating compounds,although some of its complexes do luminesce. Now, we discuss internal transitions from Cr 3þand Fe3þin both b-Ga 2O3and Al 2O3(corundum/sapphire). Tanabe–Sugano diagrams predict the dependence of the transition energy on the perturbing crystalfield imposed on transition metals by the nearest-neighbor atomstreated as point charges. This allows the determination of the crystalfield strength and predicting unobserved transitions using knownabsorption bands/lines. Both the experiment and theory calculationsindicate that Fe and Cr in b-Ga 2O3show a preference for incorporation on the octahedrally coordinated Ga IIsite.14,36The Tanabe–Sugano dia- grams for d3and d5electron configurations of Cr3þand Fe3þin the octahedral coordination are presented in Fig. 3 .37 Two absorption bands at 428 nm and 600 nm in the Cr-doped b-Ga 2O3crystals ( >0.1 at. %)38,39correspond to the4A2(F)!4T1(F) and4A2(F)!4T2(F) absorption transitions, which imply that theD0 B parameter is near 24.8. This value is close to 23.1–23.540,41reported for other Cr-doped Ga 2O3crystals. Interestingly, this value of B pre- dicts that another absorption transition for4A2(F)!4T1(P) should exist near 271 nm (4.58 eV), which is nearly resonant with the funda-mental absorption of b-Ga 2O3for E//a/C3and c. This value ofD0 Bdeter- mined from the absorption implies that the2E(G)!4A2(F) emission band should be near 693 nm, which is very close to the sharp red emis- sion we observe (at 689 and 696 nm in Fig. 2 ). For comparison, Cr in Al2O3exhibits absorption bands at 400 nm and 555 nm,42,43indicating D0 Bof 28.0 because of the slightly smaller bond lengths. The well- characterized ruby R 1and R 2emission lines at 692 nm and 694 nm in FIG. 2. (a) PL spectra in the arbitrary scale for the Fe-doped crystals under an exci- tation of 266 nm. (b) Representative temperature-dependent PL spectra for the Fe-doped b-Ga 2O3crystal. (c) PLE for the UID WSU (100) crystal observed at 367 nm. (d) PLE spectra for the Syn (100) Fe-doped b-Ga 2O3observed at 689 nm; the blue triangle represents the integrated intensity from 300 nm to 600 nm at E//c. TABLE I. Ir, Zr, Ti, and Ni related emission properties in the octahedral Ga sites. Elements Electron configuration Transitions Note Ir4þ[Xe]4f145d52T2(I) split by spin-orbital coupling: expected sharp peak at 0.64 eV31,32Spin-forbidden Zr4þ[Kr] … … Ti3þ[Ar]3d12E!2T2: broad peak at 1.64 eV33,34Spin-allowed Ni2þ[Ar]3d83T2!3A2: broad peaked at 0.86 eV35Spin-allowedApplied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 052101 (2020); doi: 10.1063/5.0012967 117, 052101-3 Published under license by AIP PublishingAl2O3result from the splitting of2E due to a combination of the crys- tal field and spin–orbit interaction.44The Tanabe–Sugano diagrams in Fig. 3 show that the4A2!4T1and4A2!4T2transition energies significantly change with the crystal field strength, but that of the 2E!4A2transition does not. Thus, the experimental observation that the absorption bands in the Cr doped Al 2O3andb-Ga 2O3differ while the emission transitions are very close in energy is consistent withthe assignments above. The broad red luminescence comes from 4T2!4A2due to a partial thermal population from2Et o4T2.45With decreasing temperature, the thermal population of4T2states is sup- pressed, and thus the broad peak disappears as we observe. We now examine the possibility that the red luminescence might arise from Fe in b-Ga 2O3.F e3þabsorption lines are confirmed by the yellowish color of crystals and distinct absorption around 455 nm for 6A1(S)!4A1(G)/4E(G), which is rather independent of the crystal field for the high-spin Fe3þregime.13,46TheD0 Bof octahedral Fe in the b-Ga 2O3single crystals is calculated as 21 using the two absorption bands at 460 nm and 689 nm13corresponding to6A1(S)!4A1(G) and6A1(S)!4T2(G) transitions, respectively. This restricts its emis- sion from4T1!6A1to be at 980 nm (1.265 eV), which is very far away from the red emissions near 700 nm (1.75 eV) that we (and others) observed from the Fe-doped b-Ga 2O3. Similarly, this value of B predicts that another possible absorption transition for6A1! 4A2(F) near 278 nm (4.46 eV). This is nearly resonant with b-Ga 2O3 matrix and with the Cr absorption discussed early. The luminescence of the Fe3þdoped a-Ga 2O3powder is observed at 950 nm (1.305 eV),47 where a-Ga 2O3, just like corundum/ruby a-Al2O3, contains only octa- hedrally coordinated cation sites. Finally,D0 Bfor octahedral Fe3þin Al2O3is 22.0–22.9,46,48predicting an emission band for the4T1!6A1 transition to be close to 1050–1130 nm (1.181–1.097 eV). Therefore, we can conclude that red luminescence originating from Fe3þin the Fe-doped b-Ga 2O3is highly unlikely. Note that Fe3þ,C r3þ,a n dI r3þcan form Fe2þ,C r4þ, and Ir4þ, respectively, by capturing the photogenerated electrons and holes.Since the Fermi energy is fixed at the Fe 3þ/Fe2þlevel ( /C24Ec/C00.8 eV) for the Fe-doped b-Ga 2O3semi-insulating samples (for the n-type doped sample, EFis near Ecdue to shallow doping), most of Cr and Ir (deep donors) are in neutral states, namely, Cr3þand Ir3þ.T h u s ,o n l ya small part of these ions changes charges.15Besides, none of them has observed to give a sharp red emission near 1.8 eV. Hence, the red emission we observe from the Fe-doped b-Ga 2O3 likely originates from Cr3þinstead of from Fe3þitself. The NCT (010) Fe-doped crystal was analyzed by ICPMS. We measured 12 ppm [Fe] (/C241.1/C21018cm/C03), which is in agreement with the reported /C248/C21017cm/C03[Fe] by glow discharge mass spectrometry (GDMS) in samples from the same vendor.9Our ICPMS did detect 2 ppm [Cr] corresponding to 1.8 /C21017cm/C03. Note that 0.3 ppm [Cr] is reported in UID crystals30and 0.4–0.6 ppm in our Zr-doped crystals. Approximately, /C246/C2higher [Cr] is observed in the Fe-doped crystal. The effective segregation coefficient of Cr is larger than 1.0 (reported to be 1.45–3.1149)s u c ht h a tC rf r o mt h eG a 2O3feedstock tends to spontaneously concentrate into the growing crystal. Cr (along with Si and Fe) would also be expected to leach from the Ir crucible (99.99% purity is common) into melts and ultimately into growing crystals. We propose an energy transfer process between b-Ga 2O3and Cr, probably sensitized by Fe in the Fe-doped b-Ga 2O3; namely, the elec- tron–hole pairs generated in b-Ga 2O3recombine by transferring energy to excite Fe3þand Cr3þ.E x c i t e dF e3þreturns to its ground state by transferring the energy to excite Cr3þ. The red luminescence occurs from the excited Cr3þ. The lines of evidence we use to reach this conclu- sion are as follows: (1) the red emission intensifies with Eex/C25Egfor PLE and shows a twofold symmetry, which coincides with E//a/C3,b ,a n d c axes for polarized excitation; (2) the red emission competes with the b-Ga 2O3matrix emission (the red emission fades and the b-Ga 2O3 emission emerges with Eex>4.90 eV in E//c) [ Fig. 2(d) ]; (3) near- resonant energy levels exist, predicted by the Tanabe–Sugano diagram; and (4) the red emission is clearly observed by naked eyes, with /C2430 mW power in the Fe-doped b-Ga 2O3crystals but not measurable in UID or other intentionally doped b-Ga 2O3crystals. Such difference can- not be explained by /C246/C2higher Cr in the Fe-doped Ga 2O3compared to the UID one (roughly 0.5 ppm of the Fe content in UID30). The fol- lowing phenomena also support our hypothesis. A resonant energy transfer has been proposed in the Cr-doped b-Ga 2O3crystals and films, evidenced by the following: (1) the blue emission is more reduced thanUV and green emissions in the higher Cr-doped b-Ga 2O3crystal39and (2) the red luminescence peaks with Eex/C25Egand is less efficient with Eex>5.06 eV.50In in a-Ga 2O3powder, Cr luminescence is observed to be nonlinearly enhanced by the coexistence of Fe.47In terms of the non- radiative process, deep state recombination centers at Fe or Cr, or some- thing else unknown might also exist. More work is needed to illustrate the details of the energy transfer process in the Fe-doped b-Ga 2O3. However, our findings clearly demonstrate that the red luminescence we observe originates from Cr centers, not Fe ones. In conclusion, the polarized transmittance observed a reduced optical bandgap and increased sub-bandgap absorption. A red emis- sion consisting of two sharp peaks at 689 nm and 697 nm superimpos- i n gab r o a dp e a ka t7 1 0n mw a so b s e r v e df r o ma l lF e - d o p e d b-Ga 2O3 samples under deep UV excitation, while the typical b-Ga 2O3PL spec- trum from 300 to 600 nm was quenched. The red luminescence proba- bly originated from Cr3þ, instead of Fe and N, based on the low temperature PL and Tanabe–Sugano analyses. The red emission in E//c was inefficient above a 4.9 eV excitation, and the b-Ga 2O3 emission occurred simultaneously. We proposed that an energy trans- fer process exists between Ga 2O3and Cr, probably sensitized by Fe in the Fe-doped b-Ga 2O3for red luminescence. FIG. 3. Tanabe–Sugano diagrams for the octahedral site of (a) d3and (b) d5elec- tron configurations. The x-axis isD0 B, where D0and B denote the crystal splitting energy and Rach parameter related to electron repulsion, respectively, and the y- axis is the transition energy E normalized to B. The solid and dash lines mean thespin-allowed and spin-forbidden transitions, respectively.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 052101 (2020); doi: 10.1063/5.0012967 117, 052101-4 Published under license by AIP PublishingAUTHORS’ CONTRIBUTIONS R.S. and Y.K.O. contributed equally to this work. This material is based upon the work supported by the Air Force Office of Scientific Research under Award No. FA9550-18-1-0507 (Program Manager: Dr. Ali Sayir). Any opinions, findings, conclusions, or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of theUnited States Air Force. The authors thank Mr. John Blevenisat AFRL for supplying us with the Synoptics Fe-doped Ga 2O3 substrates and Professor Steve Blair at the University of Utah forproviding polarized optical measurement facilities. This work isdedicated to the memory of the late Professor Kelvin G. Lynn. 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Krebs and W. G. Maisch, Phys. Rev. B 4(3), 757 (1971). 49Z. Galazka, K. Irmscher, R. Schewski, I. M. Hanke, M. Pietsch, S. Ganschow, D. Klimm, A. Dittmar, A. Fiedler, T. Schroeder, and M. Bickermann, J. Cryst. Growth 529, 125297 (2020). 50S. Fujihara and Y. Shibata, J. Lumin. 121(2), 470 (2006).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 052101 (2020); doi: 10.1063/5.0012967 117, 052101-5 Published under license by AIP Publishing

5.0020019.pdf

Appl. Phys. Lett. 117, 131104 (2020); https://doi.org/10.1063/5.0020019 117, 131104 © 2020 Author(s).High frequency lithium niobate film- thickness-mode optomechanical resonator Cite as: Appl. Phys. Lett. 117, 131104 (2020); https://doi.org/10.1063/5.0020019 Submitted: 28 June 2020 . Accepted: 18 September 2020 . Published Online: 29 September 2020 Mohan Shen , Jiacheng Xie , Chang-Ling Zou , Yuntao Xu , Wei Fu , and Hong X. Tang ARTICLES YOU MAY BE INTERESTED IN Incorporation of erbium ions into thin-film lithium niobate integrated photonics Applied Physics Letters 116, 151103 (2020); https://doi.org/10.1063/1.5142631 Temporal acoustic wave computational metamaterials Applied Physics Letters 117, 131902 (2020); https://doi.org/10.1063/5.0018758 Acousto-optic modulation in lithium niobate on sapphire APL Photonics 5, 086104 (2020); https://doi.org/10.1063/5.0012288High frequency lithium niobate film-thickness- mode optomechanical resonator Cite as: Appl. Phys. Lett. 117, 131104 (2020); doi: 10.1063/5.0020019 Submitted: 28 June 2020 .Accepted: 18 September 2020 . Published Online: 29 September 2020 Mohan Shen,1Jiacheng Xie,1 Chang-Ling Zou,2 Yuntao Xu,1Wei Fu,1and Hong X. Tang1,a) AFFILIATIONS 1Department of Electrical Engineering, Yale University, New Haven, Connecticut 06511, USA 2Key Laboratory of Quantum Information, Chinese Academy of Sciences, University of Science and Technology of China, Hefei, 230026 Anhui, People’s Republic of China a)Author to whom correspondence should be addressed: hong.tang@yale.edu ABSTRACT High-frequency optomechanical resonators are in demand as transduction devices to bridge microwave and optical fields. Thin-film lithium niobate is a promising platform for implementing high-frequency optomechanics for its low optical loss and strong piezoelectric coefficients.However, its strong piezoelectricity is also known to introduce excess phonon loss. Here, we present lithium niobate optomechanical resona-tors with film-thickness-mode mechanical resonances up to 5.2 GHz, reaching the operating frequency regime of superconducting qubits. By engineering the mechanical anchor to minimize the phonon loss, we achieve a high quality factor up to 12 500 at cryogenic temperatures and, hence, a frequency-quality factor product of 6.6 /C210 13. Our system also features interference between piezo-optomechanical and electro-optic modulation. A theoretical model is derived to analyze these two effects and their interference. Published under license by AIP Publishing. https://doi.org/10.1063/5.0020019 Phonons piezoelectrically coupled to microwaves have enabled active control and manipulation of light1–5and coherent signal con- version between microwave and optical domains.6–10To effectively couple microwaves and phonons, significant effort has been focusedon piezoelectric materials such as aluminum nitride, 11,12gallium nitride,13and gallium arsenide14,15in a variety of device structures. However, to realize both high optical and mechanical quality factors,strong electromechanical and optomechanical coupling (MO) on asingle platform is still challenging. The thin-film lithium niobate-on-insulator (LNOI) platform with large electro-optic (EO), acousto-optic, and piezoelectric coefficients 16has recently drawn tremendous attention. High quality factor optical17,18and mechanical resonators19 and efficient microwave-to-mechanical coupling efficiency18have been demonstrated on the LNOI platform recently. On the otherhand, pushing for higher frequency mechanical mode is also of greatimportance such as in quantum optomechanics in order to reducethermal occupation. Another critical parameter for optomechanicalsystems is the mechanical frequency-quality factor ( fQ) product, which plays an important role in demonstrating quantum optomechanicsand entanglement creation at elevated bath temperatures. 20–22 Therefore, in optomechanics experiments, it is essential to achievehigh mechanical modal frequency fandfQproducts simultaneously. Piezoelectricity enables efficient coupling of microwave to pho- non modes. However, the same coupling is also responsible for excesspiezoelectric losses. 23,24P r e v i o u s l y ,w eh a v es h o w nt h a tA l N - b a s e d piezo-optomechanical resonators can operate at the microwaveX-band 25and its use for microwave-to-optics coherent conversions.9 LiNbO 3, being a stronger piezoelectric material than AlN, promises to deliver even higher microwave-to-optics coupling efficiency if similarmechanical performance can be realized. Recently, high fQproduct (on the order of 10 12/C01013) MEMS resonators based on LiNbO 3thin films have been reported.26,27Here, in this paper, we demonstrate piezo- optomechanical racetrack resonators on thin film lithium niobate withmechanical resonance as high as 5.2 GHz, realized by exploiting the filmthickness mode of suspended waveguides. The mechanical modes aredriven by two electrodes placed across a gap from the suspended wave-guide and readout optomechanically. By engineering the mechanicalanchors, we achieve a high quality factor of 12 500 at 4 K, leading to anfQproduct of 6 :6/C210 13, a value approaching that of the state-of-the- art cryogenic AlN devices.9The mechanical anchor design also pre- serves the high optical quality factor of around 2 /C2106, allowing opera- tions in the resolved-sideband regime. Due to the coexistence of theacousto-optic and electro-optic effect in lithium niobate, our systemexhibits interference between opto-mechanical and electro-optic modu- lation for which we present a detailed analysis for effective extraction of the mechanical quality factor. On the LNOI platform, partially etched waveguide resonators show the highest optical quality factor compared to photonic crystal Appl. Phys. Lett. 117, 131104 (2020); doi: 10.1063/5.0020019 117, 131104-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apland microdisk resonators.17The cross section of this type of wave- guide normally has a width of a few micrometers and a thickness of several hundreds of nanometers. With the thickness being the smallerdimension, thickness mode-like mechanical modes will possess higherfrequency with its wavevector along the thickness direction. 25On the other hand, to drive mechanical modes using on-chip electrodes, co- planar electrodes can be more easily realized from the fabrication pointof view, which will produce electric fields mainly in the in-plane direc- tion. Given the vanishing d 31andd32piezoelectric tensor elements of lithium niobate,16X-cut LNOI would allow us to excite thickness mode-like mechanical modes using the in-plane electric field, whilethis cannot be done with Z-cut LNOI. Therefore, our devices are fabri- cated on a commercial X-cut LNOI wafer with a 600 nm-thick lithium niobate film. Figure 1(a) shows a fully suspended racetrack resonator. In the fabrication process, the lithium niobate film is etched twice using Argon milling with patterned hydrogen silsesquioxane (HSQ) as resist. The lithium niobate film is first etched by 350 nm to fabricate half-etched optical waveguides. The second etch stops at the silicon dioxide layer and defines the slab. The slab under the optical waveguide is 3.5lm wide and is connected with large slab pads through narrow l i n k sa ss h o w ni n Figs. 1(b) and1(c). These links provide support for t h eo p t i c a lw a v e g u i d ea f t e rb e i n gr e l e a s e d .T h en a r r o wl i n k sa l s o unavoidably introduce anchor loss for the mechanical modes, and sothe spacing between the anchors along the optical waveguide should be not only as long as possible to reduce anchor loss but also not toofar apart; otherwise, the optical waveguide may break without suffi-cient support. In our devices, this spacing is chosen to be 50 lm. After etching lithium niobate, 300 nm thick gold electrodes are deposited through liftoff using polymethyl methacrylate (PMMA)resist. Electrodes are placed along the arms of the racetrack resonatorat both sides. The electrode-to-electrode gap is 8.1 lm. This narrow gap can provide a strong electric field but still large enough to preventmetal-induced photon absorption. Finally, the optical racetrack is fullyreleased by removing the underneath 2 lm-thick silicon dioxide in buffered oxide etchant (BOE), while the slabs across the air gap fromthe released waveguide still have silicon dioxide underneath and pro-vide side-anchors for mechanical suspension. In this design, the opticalmode is well confined in the half-etched waveguide, does not see thediscontinuity along the edges of the slab, and, therefore, prevents scat-tering loss and maintains the high quality factor of the optical resona-tor.Figure 1(e) shows an optical TE mode resonance around 1550 nm of a racetrack resonator fabricated from this fabrication process, which has a fitted quality factor of 2 :1/C210 6. By applying a RF drive on the electrodes, a horizontal electric field is formed, which excites the mechanical motion of the suspendedarm and, thus, in turn, modulates light in an optical cavity, via movingthe boundary effect 28a sw e l la st h ee l a s t o - o p t i ce f f e c t .W ed e n o t et h i s optical modulation induced by electrically excited mechanical motionelectrical–mechanical–optical (EMO) modulation. Besides, the appliedelectric field also directly modulates the optical field through theelectro-optic (EO) effect. To characterize the modulation efficiencythrough the two different mechanisms, we define two parameters asfollows: g EO¼/C0@x @V0/C12/C12/C12 EO; (1) gEMO¼/C0@x @V0/C12/C12/C12 EMO; (2) where xis the optical resonance frequency and V0is the voltage amplitude applied between the electrodes. These two parameters mea-sure the modulation efficiency by the optical resonance frequency shiftunder a unit microwave voltage drive. For the electro-optic interaction, g EO¼/C0@x @V0/C12/C12/C12 EO¼x0 2ðX ijEi/C1ðdeij=dV0Þ/C1EjdV ð ~E/C1~DdV; (3) with deij=dV0¼/C0P ke2 ijrijkEmw k,w h e r e Emwis the microwave electric field when the voltage between the electrodes is 1 V and rijkis the third rank electro-optic tensor. Compared to the EO effect, EMO modula-tion features a strong frequency dependence due to mechanical reso-nance. This dependence can be separated into two cascaded parts, the microwave to mechanical coupling and optomechanical coupling, or g EMO¼/C0@x @V0jEMO¼/C0@x @a@a @V0,w h e r e ais the maximum mechanical displacement within the waveguide. The response of a driven har- monic oscillator gives a¼gEMV0=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðX2/C0X2 bÞ2þC2 mX2q ;um ¼arctan ð/C0CmX=ðX2 b/C0X2ÞÞ,w h e r e gEMis a constant, Xbis the FIG. 1. (a) Optical image of a racetrack piezo-optomechanical resonator. The yel- low part is deposited gold electrodes. (b) SEM image of the coupling part of the racetrack resonator. (c) SEM image showing the suspended optical waveguide sup- ported by the slab. (d) Cross section of the released optomechanical waveguidestructure and simulated electric field (shown by arrows) (e) Optical transmissionspectrum of the resonator showing a loaded quality factor of 2 :1/C210 6.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 131104 (2020); doi: 10.1063/5.0020019 117, 131104-2 Published under license by AIP Publishingmechanical resonant angular frequency, Cmis the mechanical mode loss rate, and Xis the driven frequency. The optomechanical coupling (MO) can be described by1 gMO¼/C0@x @a ¼x0 2ð De~Es2/C0D1 e~Dn2/C18/C19 ð~qð~rÞ/C1^nÞdSþðX ijEideij daEjdV ð ~E/C1~DdV; (4) with deij=da¼/C0e2 ijðP klpijkluklþP krijkEpkÞ,w h e r e x0is the optical resonant frequency, ~qð~rÞis the normalized mechanical displacement field, ~Esis the tangential electric field at the boundary, ~Dnis the nor- mal electric displacement field at the boundary, pijklis the fourth rank elasto-optic tensor, rijkis the third rank electro-optic tensor, uklis the normalized strain field, and Epis the normalized piezoelectric field. Therefore, the overall electrical–mechanical–optical (EMO) modula-tion has the following expression: g EMO¼/C0@x @V0/C12/C12/C12 EMO¼gMOgEMffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðX2/C0X2 bÞ2þC2 mX2q : (5)Considering both EMO and EO modulation, the cavity dynamics can be written as _a¼/C0j 2a/C0ix0/C0gEMOV0sinXtþum ðÞ ð /C0gEOV0sinXtÞaþffiffiffiffiffiffijexpse/C0ixt; ¼/C0j 2a/C0ix0/C0AsinXtþ/ ðÞ ðÞ aþffiffiffiffiffiffijexpse/C0ixt; (6) under the existence of an exerted voltage between the electrodes V¼V0sinXt,w h e r e ais the annihilation operator of the optical mode, xis the pump laser angular frequency, sis the optical pump, j is the total loss rate, jexis the external loss rate, and Aand/are the a m p l i t u d ea n dp h a s er e s p o n s eo fc o m b i n e dE M Oa n dE Oe f f e c t s . A ss h o w ni nE q . (5), the EMO modulation has resonant behav- ior, while for the EO modulation, its amplitude and phase responseshould remain a constant while sweeping the microwave frequency,thus, resulting in a flat background in the S 21spectrum. By solving for the solution apðtÞof Eq. (6)and utilizing cavity input–output theory jaoutj2¼j/C0 se/C0ixtþffiffiffiffiffiffijexpapðtÞj2, we could obtain the output optical signal oscillating at X. Under the resolved sideband limit X/C29j,a n d with a pump detuning D¼x/C0x0/C256j=2, the optical beating signal at Xreceived by the photodetector (PD) in Fig. 2(a) is reduced tojaoutj2=jsj2/C257cosðXtþ/Þ/C1ð2jex=jÞ/C1ðA=XÞ.S i n c et h e FIG. 2. (a) Measurement setup. (b) S21spectrum when pumping at the slope of an optical resonance. (c) Simulation of mechanical eigenmodes. The bottom mechanical mode is the film thickness mode. (d) Zoom in of the 4.2 GHz resonance with a Qvalue of 2200, fitted using the model described above. The inset demonstrates the S21phase dia- gram, with a red vector representing the EO modulation strength and a purple vector representing the EMO modulation strength.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 131104 (2020); doi: 10.1063/5.0020019 117, 131104-3 Published under license by AIP Publishingelectromechanical link is very under-coupled, the terminal impedance of the microwave transmission line does not change much when thefrequency of the vector network analyzer (VNA) sweeps across the mechanical resonance. Therefore, the S 21signal measured by the setup depicted in Fig. 2(a) can be written as jS21j¼k/C1ð2jex=jÞ/C1ðA=X), /S21¼/þ/0,w h e r e kand/0are assumed to be constants in the interested frequency range. Figure 2(b) shows the measured S21when the probe IR laser fre- quency is set at the slope of an optical resonance. The mechanical modes can be simulated using COMSOL Multiphysics, as seen in Fig. 2(c) . One can find that the measured resonances are of Fano shape, resulting from the interference between EMO and EO modula-tion. Using the model we described above, we can extract the Q factor of the mechanical mode and the g EMO=gEOr a t i ob yfi t t i n gt h em e a - sured S21response. An example of a 4.2 GHz mechanical resonance is shown in Fig. 2(d) , with a fitted Q factor of 2200 and jgEMO=gEOjX¼Xb¼0:7 on the mechanical resonance. To understand the EMO and EO interference more intuitively, one can resort to the phase diagramshown in the inset of Fig. 2(d) , where the resonant behavior of the mechanical mode shown in the phase diagram is circle like, while the non-resonant behavior of the EO modulation can be viewed as a vec- tor that offsets the resonant circle. Here, the length of the red vectorand purple vector in Fig. 2(d) represents the EO and EMO modulation strength, respectively. The ratio between the length of the purple vec- tor and that of the red vector is the same as the absolute value of the ratio between g EMO andgEO, which is around 0.7 for this mechanical mode. From the COMSOL simulation, we obtain jgMOj¼2p/C17:8 /C2106Hz/pm for the mechanical resonance around 4.2 GHz as shown inFig. 2(d) andjgEOj¼2p/C18:8/C2107Hz/V (around 0.7 pm/V for the optical telecom band). The vacuum coupling rate defined byg 0;MO¼gMOffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C22h=2meffXbp gives an absolute value of 2 p/C1288 Hz, with an effective mass meffof 1.46 pg. Using the fitted ratio between gEMO andgEO, we further obtain gEMjj¼2:5/C2106m/(V s2), which relates the voltage and the maximum mechanical displacement. Thus, for avoltage of 1 V, the maximum displacement of such a mechanical mode is 7.83 pm. The corresponding microwave-to-phonon efficiency g¼C e=Cm¼Cm 4/C11 2meffX2ba2=ð0:5V0Þ2 2Z0¼3:8/C210/C05,w h e r e Ceis the mechanical resonator coupling rate to the microwave channel, Cmis the total mechanical loss rate, V0is the voltage between the electrodes, andZ0is the characteristic impedance of the transmission line. The mechanical mode of particular interest is the film thickness mode at 5.2 GHz as shown in Fig. 2(c) . For the thickness mode, the simulated jgMOj¼2p/C13:6/C2107Hz/pm and jg0;MOj¼2p/C11:7k H z , with an effective mass of 0.69 pg. To obtain the Q factor and micro- wave coupling efficiency of the thickness mode, fitting of the measured spectrum is required. However, around this thickness mode, there areseveral other mechanical modes that are only a few MHz away fromthe thickness mode as shown in Fig. 3 at 200 K. At room temperature, when the quality factor is not high enough, the optomechanical responses of those modes are similar in amplitude, and so the mea-sured optomechanical response peak of the thickness mode is actuallythe envelope of mechanical modes around the thickness mode and the thickness mode cannot be resolved. To resolve the thickness mode, we perform measurements at cryogenic temperatures because the phononloss is reduced at lower temperatures. 29 The temperature dependence of the quality factor and frequency of thickness mode is shown in Fig. 3 . As the temperature decreasesfrom 200 to 4 K, the quality factor of the thickness mode increases from 5500 to 12 500. With the increasing quality factor and, therefore, stronger resonance enhancement, the thickness mode has a strongerresponse as the temperature decreases. So the thickness mode standsout from the surrounding modes with decreasing temperature as shown in Fig. 3 .A t4K ,a n fQproduct of 6 :6/C210 13is achieved, which approaches the state-of-the-art value of that on AlN platforms30and GaAs platforms31at similar temperatures. From the trend of the qual- ity factor’s dependence on temperature shown in the inset of Fig. 3 , thefQproduct could be higher at lower temperatures. Resulting from the improved mechanical Q factor and, therefore, stronger microwave to mechanical coupling efficiency, the fittedjg EMO=gEOjratio at 4 K is approximately 13, with estimated jgEMj ¼2:7/C2106m/(V s2)a n d g¼1/C210/C04. The microwave to phonon coupling efficiency could be further improved with the higher mechanical Q factor and by better impedance matching the micro-wave drive. A recent work on piezo-optomechanical transducer hasdemonstrated microwave to phonon coupling efficiency up to 17%. 18 In summary, we demonstrate a high frequency piezo- optomechanical system up to 5.2 GHz by employing the film thickness mode on the thin film lithium niobate platform. The mechanicalmode exhibits a high quality factor around 12 500 at 4 K, yielding astate-of-the-art fQproduct on this important material platform. Together with a high optical quality factor, our system shows great potential for strong light modulation and efficient microwave- to-photon conversion. AUTHORS’ CONTRIBUTIONS M.S. and J.X. contributed equally to this work. This work was supported by ARO Grant No. W911NF-18–1- 0020. H.X.T. acknowledges partial funding support from the NSFFIG. 3. Optomechanical response of the thickness mode at different temperatures from 4 to 200 K. The measurement setup is identical to that in Fig. 2 , but with the device mounted in a cryostat. The temperature dependence of the quality factor and resonance frequency of the thickness mode is shown in the inset.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 131104 (2020); doi: 10.1063/5.0020019 117, 131104-4 Published under license by AIP PublishingEFRI grant (No. EFMA-1640959) and Packard foundation. Funding for substrate materials used in this research was provided by theDOE/BES grant under Award No. DE-SC0019406. The authorsthank Yong Sun, Mike Rooks, Sean Rinehart, and Kelly Woods for their assistance provided in the device fabrication. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1S. A. Tadesse and M. Li, “Sub-optical wavelength acoustic wave modulation of integrated photonic resonators at microwave frequencies,” Nat. Commun. 5, 5402 (2014). 2L. Fan, C.-L. Zou, M. Poot, R. Cheng, X. Guo, X. Han, and H. X. 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Lett. 103, 241112 (2013).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 131104 (2020); doi: 10.1063/5.0020019 117, 131104-5 Published under license by AIP Publishing

1.5144537.pdf

J. Appl. Phys. 127, 153904 (2020); https://doi.org/10.1063/1.5144537 127, 153904 © 2020 Author(s).Size dependent chaotic spin–orbit torque induced magnetization switching of a ferromagnetic layer with in-plane anisotropy Cite as: J. Appl. Phys. 127, 153904 (2020); https://doi.org/10.1063/1.5144537 Submitted: 08 January 2020 . Accepted: 02 April 2020 . Published Online: 17 April 2020 BingJin Chen , Hong Jing Chung , and Sze Ter Lim ARTICLES YOU MAY BE INTERESTED IN Robust magnetic domain of Pt/Co/Au/Cr 2O3/Pt stacked films with a perpendicular exchange bias Journal of Applied Physics 127, 153902 (2020); https://doi.org/10.1063/5.0002240 Spin–orbit parameters derivation using single-frequency analysis of InGaAs multiple quantum wells in transient spin dynamics regime Journal of Applied Physics 127, 153901 (2020); https://doi.org/10.1063/5.0002821 Frequency enhancement and power tunability in tilted polarizer spin-torque nano-oscillator Journal of Applied Physics 127, 153903 (2020); https://doi.org/10.1063/1.5143195Size dependent chaotic spin –orbit torque induced magnetization switching of a ferromagnetic layer with in-plane anisotropy Cite as: J. Appl. Phys. 127, 153904 (2020); doi: 10.1063/1.5144537 View Online Export Citation CrossMar k Submitted: 8 January 2020 · Accepted: 2 April 2020 · Published Online: 17 April 2020 BingJin Chen,1,a) Hong Jing Chung,2 and Sze Ter Lim2 AFFILIATIONS 1Institute of High Performance Computing (IHPC), Agency for Science, Technology and Research (A*STAR), 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632 2Institute of Materials Research and Engineering (IMRE), Agency for Science, Technology and Research(A*STAR), 2 Fusionopolis Way, #08-03, Innovis, Singapore 138634 a)Author to whom correspondence should be addressed: chen_bingJin@ihpc.a-star.edu.sg ABSTRACT Understanding the magnetization switching dynamics induced by the spin –orbit torque (SOT) in a ferromagnetic layer is crucial to the design of the ultrafast and energy-saving spin –orbit torque magnetic random access memory. Here, we investigate the SOT switching dynamics of a ferromagnetic layer with in-plane anisotropy with various elliptic sizes in different easy-axis orientations using micro- magnetic simulations. The reliable and ultrafast magnetization switching can be realized by tilting the easy axis to an optimum angle with respect to the current injecting direction. The switching time, in general, decreases smoothly with an increasing current density, and theoptimum tilting angle is determined for small device sizes with width smaller than 100 nm. This optimum angle is a small angle deviatingfrom a case when the in-plane easy axis is orthogonal to the current direction. It depends on the size, the current density, and also thedamping constant. However, with the device increasing to a certain size (e.g., 250 nm), especially at small tilting angles, we observe chaotic switching behavior where the switching times fluctuate locally with the current density. We attribute this size dependent chaotic switching phenomenon to the nucleation and formulation of complex multi-domains during switching. This chaotic phenomenon can be alleviatedby enhancing the field-like torque in the device and thus decreasing the switching times. Consequently, the shape and size of the devicesshould be carefully taken into account while designing a practical fast switching and low power SOT device with in-plane anisotropy. Published under license by AIP Publishing. https://doi.org/10.1063/1.5144537 I. INTRODUCTION The in-plane charge current passing through a tri-layered structure [heavy metal (HM)/ferromagnet (FM)/oxide] generates an out of plane spin current due to the spin –orbit interaction from the HM bulk or HM/FM interface. 1–7The spin polarized current transfers torque (spin –orbit torque, SOT) to the ferromagnet and induced magnetization switching. Magnetization switching using SOT can be realized through a three-terminal magnetic tunnelingjunction (3T-MTJ), which separates the write and read paths. The3T-MTJ structure solves the issues faced by using the spin-transfertorque switching method in a two-terminal MTJ, 8–10where the write/read current paths are coupled and result in a contradictorydesign in terms of write/read performance. 11–20SOT magnetization switching has attracted tremendous inter- ests in the research community. Both in-plane anisotropy (IPA) and perpendicular anisotropy (PMA) MTJ have been addressed and explored extensively.5,11,13–41PMA MTJ has the advantage of improving the storage density by decreasing the bit cell size sincethe thermal stability is maintained by the interfacial anisotropyinstead of the shape anisotropy as in IPA MTJ. However, the need of an external magnetic field for the deterministic switching com- plicates the structural design and also compromises the thermalstability of the MTJs. 21–29,42,43 In contrast, deterministic switching can be achieved without the assistance of an external magnetic field by using IPA MTJ with a much simpler structural design.11,44–47In particular, IPA MTJ utilizes the shape anisotropy to maintain the thermal stability, thusJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 153904 (2020); doi: 10.1063/1.5144537 127, 153904-1 Published under license by AIP Publishing.the cross section of the in-plane bit cell is usually elliptic. Ultrafast SOT switching of the in-plane magnetization has been demon- strated recently using the configuration that the easy axis of thedevice is perpendicular to the charge current direction. 45The easy- axis direction is found to have significant effect on the switchingperformance for a device of 100 × 400 nm 2experimentally.47A micro-magnetic study on a device size of 80 × 180 nm2is also carried out to show that ultrafast switching of the in-plane magne-tization can be achieved by tilting the easy axis. 46In order for a thorough understanding of the switching mechanisms, a moredetailed study on exploring the different switching performance for various device sizes and easy-axis directions is imperative, but cur- rently still lacking. II. SIMULATION METHODS Here, we address this issue by using micro-magnetic study and provide a guideline for optimizing the switching performance forultrafast and energy-saving SOT switching of IPA MTJ. Figure 1(a) shows the studied in-plane SOT-MTJ device and the coordinate systems. The electric charge current (J e) flows along the x0direction in the heavy-metal layer (HM), while the MTJ stack, including thefree layer (FL), the oxide layer (Ox), and the reference layer (RL), isattached along the z 0direction. The geometrically determined spin polarization of the spin current is along the y0direction. We only consider FL in the simulation. The magnetization ~m satisfies the Landau –Lifshitz –Gilbert –Slonczewski (LLGS) equation (see the supplementary material ) and can be numerically solved by using the Object Oriented Micro-Magnetic Framework (OOMMF) public code.48The parameters used for the simulations are the same as in the literature30–41(see the supplementary material ). III. RESULTS AND DISCUSSION To understand the switching performance of different device sizes with various easy axis, three device sizes are studied in thesimulations, namely, W × L = 50 × 100 nm2, 100 × 200 nm2, and 250 × 500 nm2. Here, W, L, and the easy-axis direction angle ware defined accordingly as the width, length of the device, and theangle between the easy-axis direction ( y) and the charge current (J e) flowing direction ( x0) [as shown in Fig. 1(b) ]. The results are depicted in Fig. 2 in terms of the switching times as a function of w at current densities of 2 × 1012A/m2and 3 × 1012A/m2, respec- tively. Here, the switching time at a current density is defined asthe time required for the current density to change the sign of theeasy-axis component of the magnetization and its value reaches90% of its saturated magnetization. Figure 2 reveals that for a specific current density, the fastest switching occurs at an easy axis tilting at a small angle deviatingfrom w= 90° (this has been experimentally demonstrated 47); this optimum switching angle depends on the size, the current density,and also the damping constant. For small device, such as 50 × 100 nm 2, there is a single optimum angle for the fastest switching; however, such definition of optimum angle becomescomplicated for large device (e.g., for device >100 × 200 nm 2). Referring to Fig. 1(b) , the accumulated spins are along the y0 direction, tilting with an angle wfrom the initial magnetization direction y. In the case of w= 90°, the accumulated spins are collin- ear with the magnetization. Switching is similar to the conventionalspin-transfer torque switching, and it is a field-free switching. Themagnetization needs to go through precessions to switch its direc- tion. The threshold current density highly depends on the pulse width and proportional to the damping constant, while the switch-ing time is inversely proportional to the current density. However,in the w= 0° case, where the accumulated spins are orthogonal with the magnetization, switching occurs without precessions, and an external field is needed to achieve a deterministic switching. The threshold current density is insensitive either to the pulse width orto the damping constant. Switching is fast and less dependent onthe current density. Hence, it is evident that the collinear andorthogonal components of the accumulated spins play different roles during the magnetization switching. For any wbetween 0° and 90°, the ratio of the two components varies, resulting in anoptimum angle for the fastest switching. On the other hand, thisratio becomes dynamic for large devices due to domain nucleationand formulations during switching, and thus the optimum angle become complicated for large devices. To further explore the dependence of switching performance on the device size, Fig. 3 plots the switching times as a function of J eat various wfor different device sizes with α= 0.005. Figure 3 shows that (i) for small device of 30 × 50 nm2, the switching times decrease smoothly with increasing the currentdensity in all easy-axis tilting angles [ Fig. 3(a) ]; (ii) for devices of 50 ×100 nm 2and 100 × 200 nm2, the switching times decrease smoothly with increasing the current density only in larger axis tilting angles ( w> 60°). Interestingly, there are local fluctuations in the curves for small tilting angles ( w< 60°) [ Figs. 3(b) and 3(c)]; and (iii) for large device of 250 × 500 nm2, such a chaotic switching phenomenon is observed at all tilting angles [ Fig. 3(d) ]. For example, in Fig. 3(c) , the switching times decreases monotonously with the current density for w= 90° but fluctuates for w= 15°; in Fig. 3(d) , the switching times fluctuate with the current densities for all angles (only w= 45° and w= 90° are shown). FIG. 1. (a) Schematic structure of the studied SOT device and (b) geometry and easy-axis orientations of the free layer. Here, W and L are the width and length of the free layer, respectively, and wis the angle between the charge current density J e(along x0axis) and free layer easy axis (along yaxis).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 153904 (2020); doi: 10.1063/1.5144537 127, 153904-2 Published under license by AIP Publishing.FIG. 2. Switching times as a function of easy-axis orientation angles at current densities of 2 × 1012A/m2and 3 × 1012A/m2for various device sizes and damping con- stants: (a) and (b) for 50 × 100 nm2; (c) and (d) for 100 × 200 nm2; (e) and (f) for 250 × 500 nm2. Damping constant for (a), (c), and (e) is 0.003 and that for (b), (d), and (f) is 0.01.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 153904 (2020); doi: 10.1063/1.5144537 127, 153904-3 Published under license by AIP Publishing.To explain the above size dependence distinctive switching phenomena, we study the time evolution of the average magnetiza-tion component along the easy-axis direction y at specific current densities for device 100 × 200 nm 2atw= 15° in Fig. 4(a) and w= 90° in Fig. 4(c) , respectively. The initial magnetization is assumed to be along the positive y direction and relaxed 1 nsbefore the charge current with a pulse width of 10 ns is applied.Forw= 15°, the successful switching is achievable for both Je = 2.4 × 10 12A/m2and 2.8 × 1012A/m2, yet failed to switch at Je = 2.6 × 1012A/m2. However, in the latter case ( w= 90°), the suc- cessful switching is always achievable so long as the current densityexceeds the critical value, i.e., 0.6 × 10 12A/m2for 10 ns pulse width. The corresponding magnetization configurations during switching shown in Figs. 4(b) and 4(d) indicate that for both tilting angles,the magnetization switching is always triggered by domain nucle- ation. These domains are more likely to be nucleated near the endsof the easy axis. This may be due to the fact that the magneto-static fields are larger at the two ends. For w= 90°, the collinear spins dominate switching. The magnetization switches to the oppositedirection after undergoing many precessions, resulting in the lowerswitching current. In contrast, the orthogonal spins dominateswitching for w= 15°. The magnetization switches to the opposite direction without precessions, resulting in higher switching current. The switching mechanism becomes complicated for the large device of 250 × 500 nm 2. We plot the time evolution of the average magnetization component along easy-axis direction y at some specific current densities for device 250 × 500 nm2forw= 45° inFig. 4(e) and for w= 90° in Fig. 4(g) , respectively. FIG. 3. Switching times as a function of current densities at various easy-axis angles for different device sizes: (a) 30 × 50 nm2, (b) 50 × 100 nm2, (c) 100 × 200 nm2, and (d) 250 × 500 nm2. Damping constant is fixed at 0.005.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 153904 (2020); doi: 10.1063/1.5144537 127, 153904-4 Published under license by AIP Publishing.The corresponding magnetization configurations during switching are shown in Figs. 4(f) and 4(h). It is shown that for a larger device with any easy-axis tilting angles, switching is also triggeredby domain nucleation. However, the position where the initial mag-netic domain nucleated is much more complicated and shows chaotic behaviors during switching. It may nucleate at any position whether at edges or inside the ferromagnet, which may depend onvarious factors, such as device size, device shape, current density, damping constant, etc. We also study a very small device (30 × 50 nm 2). The magneti- zation switching is clearly shown to be uniform and coherentdespite the tilting angle (see the supplementary material ). The charge current passing through the heavy metal may gen- erate an Oersted field. This Oersted field acts as an in-plane FIG. 4. (a) and (c) Time evolution of the average magnetization component alongeasy axis y at some specific current den- sities for w= 15° and w= 90°, respec- tively, for device 100 × 200 nm2. (b) and (d) Snapshots of magnetization configu-rations during switching at various current densities and time steps related to (a) and (c), respectively. (e) and (g)Time evolution of the average magneti-zation component along easy axis y at some specific current densities for w= 15° and w= 90°, respectively, for device 250 × 500 nm 2. (f ) and (h) Snapshots of magnetization configu- rations during switching at various current densities and time steps relatedto (e) and (g), respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 153904 (2020); doi: 10.1063/1.5144537 127, 153904-5 Published under license by AIP Publishing.magnetic field on the ferromagnetic free layer and alleviates the non-uniform magnetization configurations to assist the magnetiza- tion switching.45We do not include this Oersted field in our simu- lations, since its effect is similar to the field-like torque, which wediscuss next. Field-like torque plays a similar role as a magnetic field along the spin polarization direction and may affect the switching dynamics. Figure 5 plots the switching times as a function of current density at various r FL(rFLis a factor reflecting the ratio of the field-like torque to damping-like torque, see the supplementary material ) for (a) small device size 50 × 100 nm2and (b) large device size 250 × 500 nm2. It demonstrates that the field-like torque does not affect small devices much unless the ratio is larger thanone. In fact, switching is more coherent and field-like torque onlychanges the energy landscape. In contract, field-like torque pro- motes coherent switching in large devices. Figure 5(c) plots the time evolution of the average magnetiza- tion component along easy axis y at J e= 2.0 × 1012A/m2and w= 45° for the small device size and Fig. 5(e) for the large device. Figures 5(d) and 5(f) are snapshots of magnetization configura- tions during switching at various field-like torque ratios and time steps related to (c) and (e), respectively. It is shown thatthe field-like torque also affects the switching mechanisms for thelarge device size. Without the field-like torque, switching is viathe nucleation and formulation of complex multi-domains. Hence, results in a chaotic switching phenomenon. With the field- like torque, switching becomes more coherent and is moredeterministic. FIG. 5. (a) Switching times as a func- tion of current densities at various field-like torque ratios for device size 50 × 100 nm2a n d( b )l a r g ed e v i c e size 250 × 500 nm2. (c) and (e) Time evolution of the average magnetiza- tion component along easy axis y at Je=2×1 012A/m2andw= 45° for small device size 50 × 100 nm2and 250 × 500 nm2, respectively. (d) and (f) Snapshots of magnetization con- figurations during switching at variousfield-like torque ratios and time stepsrelated to (c) and (e), respectively. Damping constant is fixed at 0.005.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 153904 (2020); doi: 10.1063/1.5144537 127, 153904-6 Published under license by AIP Publishing.IV. CONCLUSION In summary, we investigated the switching dynamics of a fer- romagnetic layer with in-plane anisotropy with different devicesizes at easy-axis tilting for various angles. We demonstrated that the nanosecond scale of field-free spin –orbit torque switching can be achieved by using in-plane anisotropy. The fast switching canbe obtained by tilting the device easy-axis to some degree awayfrom the accumulated spin current direction. For small device(<100 nm) with coherently or nearly coherently switching, this tilting angle is single value and determined. However, the optimum tilting angle for large device becomes complicated. The switchingprocess depends on various parameters, such as device ’s geometry, free layer ’s damping constant, the Dzyaloshinskii –Moriya interac- tion (DMI) constant, and the applied current. Chaotic switching phenomenon is observed for larger devices where the switching times fluctuate with the current densities. This chaotic phenome-non can be alleviated by enhancing the field-like torque in thedevice and thus decreasing the switching times. Consequently, theshape and size of the devices should be carefully taken into account while designing a practical fast switching and low power spin –orbit torque device with in-plane anisotropy. SUPPLEMENTARY MATERIAL See the supplementary material for the LLGS equation, simu- lation parameters, switching dynamics of a very small device,effect of device shape, effect of damping constant, and effect ofDzyaloshinskii –Moriya interaction (DMI) on the switching dynamics. ACKNOWLEDGMENTS This work was supported by A*STAR RIE2020 AME Programmatic Grant (Grant No. A18A6b0057). REFERENCES 1L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 109, 096602 (2012). 2S. Emori, U. Bauer, S. M. Ahn, E. 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5.0002590.pdf

J. Appl. Phys. 128, 093105 (2020); https://doi.org/10.1063/5.0002590 128, 093105 © 2020 Author(s).Concentrating, diverging, shifting, and splitting electromagnetic beams using a single conical structure Cite as: J. Appl. Phys. 128, 093105 (2020); https://doi.org/10.1063/5.0002590 Submitted: 26 January 2020 . Accepted: 16 August 2020 . Published Online: 03 September 2020 Wallysson Barros , Antônio de Pádua Santos , and Erms Pereira ARTICLES YOU MAY BE INTERESTED IN Ultra-broadband terahertz absorber based on a multilayer graphene metamaterial Journal of Applied Physics 128, 093104 (2020); https://doi.org/10.1063/5.0019902 Progress on and challenges of p-type formation for GaN power devices Journal of Applied Physics 128, 090901 (2020); https://doi.org/10.1063/5.0022198 Axially controllable multiple orbital angular momentum beam generator Applied Physics Letters 117, 021101 (2020); https://doi.org/10.1063/5.0011445Concentrating, diverging, shifting, and splitting electromagnetic beams using a single conical structure Cite as: J. Appl. Phys. 128, 093105 (2020); doi: 10.1063/5.0002590 View Online Export Citation CrossMar k Submitted: 26 January 2020 · Accepted: 16 August 2020 · Published Online: 3 September 2020 Wallysson Barros,1,a) Antônio de Pádua Santos,2 and Erms Pereira1,2 AFFILIATIONS 1Polytechnic School of Pernambuco, Universidade de Pernambuco, Rua Benfica, 455, 50720-001 Recife, PE, Brazil 2Department of Physics, Universidade Federal Rural de Pernambuco, 52171-900 Recife, PE, Brazil a)Author to whom correspondence should be addressed: wallyssonklaus1@gmail.com ABSTRACT Bending, shifting, and splitting light rays are some of the basic operations in optics. A change of operation generally requires the device associated with a particular operation to be replaced by another one, resulting in delays. Here, we propose a structure that switches amongbidirectional bending, shifting, and splitting of a light beam when rotating it. It is an anisotropic dielectric structure that makes light feel aneffective asymmetric conical space. Such a system arises spontaneously in nematic liquid crystals, living liquid crystals, and active nematicsand, in any case, can be realized with optical metamaterials. We numerically solve the wave equation to demonstrate bending, shifting, and splitting as noted above. When fabricated with liquid crystals, its functionality can vary with temperature. Published under license by AIP Publishing. https://doi.org/10.1063/5.0002590 I. INTRODUCTION In optics and electrical engineering, it is important to control a light beam ’s path. Simple ways for changing the direction and splitting beams use mirrors and prisms, while shifting requires aplane-parallel slab of glass. 1Recently, we have advanced methods that increase the control of light paths. For example, heating a polymer using a second light beam,2 recurring to graded photonic crystals,3or using hyperbolic meta- materials4can control the bending of a light trajectory. Acrylic working as a waveguide for transmitted laser deflects light due toan inhomogeneous thermo-optic effect by a heating laser. 2It is also possible to use SiO 2rods with position-dependent radii to deflect light in exotic ways.3Finally, core –shell nanoparticle dispersion in nematic liquid crystals produces hyperbolic effective space for light,allowing for the creation of perfect absorbers. 4Different optical effects require distinct devices, even using simple materials ormetamaterials. If it is possible to incorporate different optical out- comes in a single device, it will promote the safety of the material in optical experiments and fast interchange between desired results. We solve these problems by showing how a single asymmetric conical structure deflects, shifts, and splits light rays only by rotatingit around its axis. The proposed asymmetric structures spontane- ously arise in nematic liquid crystals, 5–7solid materials,8–10living liquid crystals,11and active nematics.12For the nematic phase of liquid crystal-like systems, the components can be elongated ones (asin this article), and they are randomly positioned in space and orien- tated on average along a directional field represented by the versor ^n named director . When the liquid crystal transits from the isotropic phase to the nematic phase, topological defects can arise in thelatter, and the topological charge k¼...,/C00:5,þ0:5,þ1, þ1:5,...indicates the defect ’s strength. 5 Transformation optics13–16when using metamaterials17can lead to some interesting predictions and realizations. The casesrange from the classical cloaking devices 18and hyperlenses19to particle-and-field-inspired systems.20–24For instance, the adoption of symmetric slices of metamaterials or rotational metamaterials can simulate an effective conical space of cosmic strings.25,26 Here, we apply the transformation optics13,27to describe the dielectric properties of the proposed asymmetric structure. Afterthat, we solve the electromagnetic wave equation using the finiteelement software COMSOL, and we show the raising of different optical effects —bidirectional bending, splitting, and shifting —by rotating the structure around its axis. The structure has aJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 093105 (2020); doi: 10.1063/5.0002590 128, 093105-1 Published under license by AIP Publishing.cylindrical symmetry, and we assume that its perpendicular plane contains all electromagnetic wave ’s polarizations. II. METHODS Transformation optics13,27answers what the dielectric proper- ties of the material must be for an electromagnetic beam to feel aneffective curvature in its propagation, even though the real space isflat. This solution raises from a coordinate transformationrepresented by a matrix with components 28,29Λi i0¼@xi=@xi0, where coordinate irefers to the flat space and i0refers to the curved space. The coordinate transformation appears in the determination of the new electric permittivity with Cartesian components28,30 εi0j0¼detΛi i0/C0/C1/C12/C12/C12/C12/C01Λi i0Λj j0ε0δijand in the effective metric tensor gi0j0¼detΛi i0/C0/C1/C12/C12/C12/C12/C01Λi i0Λj j0δij, allowing one to write4,28,31,32 εi0j0¼ε0gi0j0: (1) FIG. 1. Perpendicular plane of the k¼þ 1=2,c¼π=4, and k¼/C0 1=2, c¼0 disclinations (asymmetric conical structures) in a nematic liquidcrystal showing their director fields ^n. FIG. 2. Optical effects on the 1 V/m-plane wave with wavelength ¼2μm emerging from a 3 μm-aperture and central incidence for different incident angles. The red arrows represent the Poynting vectors, and the white bar marks the fixed angular position of the circle. The outer part of the 15 μm-circle is air, and the inner part is the asymmetric structure with k¼/C0 0:5,c¼0, and the ratio between refractive indexes α¼no=ne¼0:8. Splitting occurs for rotation 0 rad [case (a)] and 1 :5708 rad [case (e)]; bending to the right occurs for rotation 0 :15708 rad [case (b)]; bending to the left occurs for rotation 0 :34907 rad [case (c)]; and shifting occurs for rotation 0:95993 rad [case (d)].Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 093105 (2020); doi: 10.1063/5.0002590 128, 093105-2 Published under license by AIP Publishing.Previous work studied light propagation around symmetric conical structures30with an effective metric identical to that of cosmic strings.33,34They presented the same optical effect, diverg- ing or converging, for different incident directions, where a particu- lar optical action demands the creation of a distinct material. Here,we use an asymmetric conical structure with the following lineelement in polar coordinates: 35 ds2¼X ijgijdxidxk ¼cos2ξþα2sin2ξ/C0/C1 dρ2þα2cos2ξþsin2ξ/C0/C1 ρ2df2 /C02 sin ξcosξ(α2/C01)ρdρdf,(2)with ξ¼(k/C01)fþc.36This equation is valid when light ’sp o l a r i z a - tion is in the perpendicular plane of the conical structure. Such asystem rises spontaneously in confined nematic liquid crystals 5and in active nematics12as wedge disclinations with strength k,l a t e r a lm o l e c u - lar angular orientation c, and the ratio between refractive indexes α¼no=ne(Fig. 1 ), where noand neare the molecular ordinary and extraordinary refractive indexes, respectively. A previous work35has found that, for k=1a n d c¼0 asymmetric conical structures, the nonzero Riemann curvature tensor components are Rρfρf¼/C0 R0,t h e Ricci tensor components are Rij¼R0gij(αρ)/C02, and the Ricci scalar is R¼2R0 α2ρ2þ2kδ(ρ) αρ1/C01 2αþ1 α/C18/C19 /C20/C21 , (3) FIG. 4. T emperature influence on the results of Fig. 2 . FIG. 3. Relative permittivity ’s components εijfor the conical structure k¼/C0 0:5,c¼0, and α¼0:8.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 093105 (2020); doi: 10.1063/5.0002590 128, 093105-3 Published under license by AIP Publishing.where R0¼k(k/C01)(1/C0α2)cos(2 ξ), and the δ-function δ(ρ)i s responsible for the conical singularity. III. RESULTS AND DISCUSSION Using the finite element software COMSOL, we solved Maxwell ’s equations in an electric permittivity replacing Eq. (2)in Eq.(1). To achieve different optical effects through rotation and to avoid additional refractive effects from flat boundaries, our device is a right circular cylinder with the inner region filled by theasymmetric conical permittivity from Eq. (2)in Eq. (1). Running simulation in device ’s perpendicular plane and using k¼/C00:5, c¼0, and α¼n o=ne¼0:8, we found the occurrence of four optical effects only rotating in the incidence direction and independently on the incident wave ’s frequency: concentrating, diverging, shifting, and splitting (see Fig. 2 and animations feixe_k-05c0ddi0rott0a90.webm for central incidence and feixe_k05cPiover4ddi2rott0a90.webm for impact factor 2 μm in the supplementary material ). These roles arise because the asymmetric conical structure has anisotropic electric permittivity ( Fig. 3 ). Explaining the results using the molecular director, each rotationmakes the wave to feel a different molecular director orientation and a different local refractive index, with Fermat ’s principle ruling the path that the beam travels. 36Since we used asymmetric cosmic string ’s line element Eq. (2), an inspired-gravity explanation of the optical effects is that such cosmic objects have an alternating distri-bution of positive and negative masses around it, 35sometimes pushing and others pulling the light beam. For lateral incidence with a non null impact parameter, we noticed that the opticaleffects raise at different angles (see animationfeixe_k05cPiover4ddi2rott0a90.mp4 in the supplementary mate- rial), obeying the anisotropic relativity permittivity ( Fig. 3 ). Other results for k¼þ1, 5, c¼0,α¼0:8, and k¼/C01, 5, c¼0, α¼0:8 are in the animations feixe_k15c0alpha08.mp4 and feixe_k-15c0alpha08.mp4 in the supplementary material . The temperature is an important aspect to influence optical results in the laboratory. Figure 4 shows that the angle for the right bending deflection and the splitting, and the lateral shift decreasewith an increase in temperature. This implies that the temperaturecan finely tune the intensity of the optical results. Some liquid crys-tals have their birefringence dependent on the temperature, where increasing the temperature moves α!1. 37Our results are opposite to that of an acrylic slab serving as a waveguide and heated by asecond light spot; 2however, Ref. 2also shows that the temperature can modify the magnitude of the effects. Using the liquid crystal, one is limited to the possible configu- ration that such materials can sustain, as k¼þ0:5,c¼π=4, and α¼0:8 disclination. However, using optical metamaterials,13one can achieve different effects. See the supplementary material to find out videos of the optical effects for k¼þ0:5,c¼π=4 and α¼0:8, k¼/C00:5,c¼0, and α¼0:5 (a metamaterial simulating a liquid crystal with an extremely high birefringence), and k¼þ2,c¼0 and α¼0:8 (an arbitrary metamaterial). We highlight the latter case where a relevant back-scattering occurs at 1 :0297 rad. Our results are generalizations of the ones in a previous work that studied the influence of the symmetric conical structure on light.30In such work and agreement with our presented study fork¼þ1, the optical effects are the same for all incident angles and depend on different values of α, with αbeing the topological charge according to the Gauss –Bonnet theorem.38In our work, we have anisotropic optical effects when k=1 for different incident angles, and kand αshare the role of the topological charge.35 FIG. 5. Ricci scalar Rvsαforρ¼1 (arbitrary unit) and different values of the angular position f. (T op panel) k¼/C0 0:5. (Bottom panel) k¼þ 0:5.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 093105 (2020); doi: 10.1063/5.0002590 128, 093105-4 Published under license by AIP Publishing.Thus, asymmetric conical structures have increased utilization because they comprise a unique structure with different optical actions on light waves. Another advantage of asymmetric conicalstructures is that, beyond can be tailored using metamaterials, theyare spontaneously generated by confined liquid crystals, as the defects with k¼/C00:5 and k¼þ0:5. 5 When light propagates in a conical space, a positive Ricci scalar attracts it radially, while a negative Ricci scalar repels it.39Thus, FIG. 7. Ricci scalar Ras a function of kfor different values of αandρ¼1 (arbitrary unit). (T op panel) f¼0. (Bottom panel) f¼π=4. FIG. 6. Ricci scalar Rvsρ(arbitrary unit) for α¼0:8 and different values of the angular position f. (T op panel) k¼/C0 0:5. (Bottom panel) k¼þ 0:5.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 093105 (2020); doi: 10.1063/5.0002590 128, 093105-5 Published under license by AIP Publishing.another relevant analysis is associated with the behavior of the Ricci scalar as a function of parameters α,k,a n d ρas given by Eq. (3)and using c¼0. From Figs. 5 and6, one notes that Ricci scalar ’ss i g n changes for different f, indicating positive and negative effectivemasses around the asymmetric conical structure.35The behavior of the Ricci scalar as a function of the parameter αis shown in Fig. 5 . While αincreases from negative values, Rjjdecreases, and it changes its sign at α¼1 and converges to 2 k(1/C0k)ρ/C02cos 2 ξðÞ forα/C291. A consequence is that such a device is sensible for variations in therefractive index, for example, using temperature dependent refractive index liquid crystals, 37when α,1 and it is insensible for α/C291. The cancellation of the Ricci scalar when α¼1 is also according to Ref. 40. On the other hand, the behavior of the Ricci scalar as a function of ρfor different angles fis shown in Fig. 6 .O n e notes that, once the values of α,k,a n dfare set, the Ricci scalar decreases with an increase in ρ, and its sign does not change. Such results, in accordance to Ref. 36, imply that the optical effects are smaller and farther away from the center of the device, in conformitywith Ref. 2. The behavior of Ricci scalar as a function of kis presented in Figs. 7 and8,f o r ρ¼1 (arbitrary units) and different values of α and for f. We note the occurrence of the zeros at k¼0,k¼1, the emergence of new ones when f!π, high oscillations for low values of α, and an asymmetric behavior around k¼0. Despite the effective electric permittivity can vary smoothly (see Fig. 3 ), these results indicate that the Ricci scalar ’s variation on k,α,a n d f explains the variety of light paths for different incident angles and its sensibility on αand k. Finally, to summarize some previous results, we plotted the Ricci scalar as a function of kandfforρ¼1a n d α¼0:8i n Fig. 9 . The surface describes regions where the Ricci scalar is positive, null, and negative. We highlight our results that deal light as rays and the topo- logical charge k, with candα, rules the dielectric properties, chang- ing trajectories. However, the real-space topology of the conical FIG. 8. Ricci scalar Ras a function of k, for different values of αandρ¼1 (arbitrary unit). (T op panel) f¼π=2. (Bottom panel) f¼π. FIG. 9. Ricci scalar Ras function of kandfforρ¼1 considering 0/C20f/C20π=2.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 093105 (2020); doi: 10.1063/5.0002590 128, 093105-6 Published under license by AIP Publishing.structures is more evident when using wave optics. For example, it is possible to manipulate the wave ’s orbital angular momentum using liquid-crystalline topological defects with different topologi-cal charges. 41 Finally, when using liquid crystals, one expects that the pro- posed device would work for other classical waves, for example, acoustic ones. The reason for such waves is that the local molecular director determines the material properties that rules the wavepropagation, as shown in Ref. 42. An approach diverse from the metric one for acoustic waves generated the study on acoustic rectifiers 43. IV. SUMMARY AND CONCLUSION In conclusion, we have demonstrated that asymmetric conical dielectrics can bidirectionally deflect, shift, and split an electromag-netic beam only when rotating them relative to the incident beam. Rising spontaneously in nematic liquid crystals and active nematics, such devices produce an effective metric tensor similar to theasymmetric cosmic strings, and we used transformation optics tocalculate their relative permittivities. We also showed new optical effects when optical metamaterials are used for creating such asym- metric conical structures, broadening the application of asymmetricconical devices. This article provides a new beam controller using asingle passive optical device, promoting, for example, saving mate-rial in optical experiments. We aim to study in future works the influence of the turbulence of active nematics with such asymmet- ric conical structures ( Fig. 10 ) on electromagnetic beams. SUPPLEMENTARY MATERIAL See the supplementary material for animations feixe-k- 05c0ddi0rott0a90.webm for central incidence and feixe- k05cPiover4ddi2rott0a90.webm for impact factor 2 μm, and opticaleffects for conical structures with k¼þ0:5,c¼π=4a n d α¼0:8, k¼/C00:5,c¼0, and α¼0:5 (a metamaterial simulating a liquid crystal with an extremely high birefringence). ACKNOWLEDGMENTS The authors thank the National Council for Scientific and Technological Development (CNPq —No. 465259/2014-6), the Coordination for the Improvement of Higher Education Personnel (CAPES), the National Institute of Science and TechnologyComplex Fluids (INCT-FCx), the São Paulo Research Foundation(FAPESP —No. 2014/50983-3), and the Pernambuco Research Foundation (FACEPE). REFERENCES 1D. C. O ’Shea, Elements of Modern Optical Design (Wiley, New York, 1985), Vol. 51. 2C. Sheng, H. Liu, S. Zhu, and D. A. Genov, “Active control of electromagnetic radiation through an enhanced thermo-optic effect, ”Sci. Rep. 5, 8835 (2015). 3B. Vasi ć, G. Isi ć, R. Gaji ć, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime, ”Opt. Express. 18, 20321 –20333 (2010). 4S. Fumeron, B. Berche, F. Santos, E. Pereira, and F. Moraes, “Optics near a hyperbolic defect, ”Phys. Rev. A 92, 063806 (2015). 5M. Kleman and O. D. Laverntovich, Soft Matter Physics: An Introduction (Springer Science & Business Media, 2007). 6S. Fumeron, E. Pereira, and F. Moraes, “Principles of thermal design with nematic liquid crystals, ”Phys. Rev. E 89, 020501 (2014). 7S. Fumeron, F. Moraes, and E. Pereira, “Retrieving the saddle-splay elastic cons- tant K 24of nematic liquid crystals from an algebraic approach, ”Eur. Phys. J. E 39, 83 (2016). 8M. Katanaev and I. Volovich, “Theory of defects in solids and three- dimensional gravity, ”Ann. Phys. 216,1–28 (1992). 9S. Fumeron, E. Pereira, and F. Moraes, “Generation of optical vorticity from topological defects, ”Phys. B: Condens. Matter 476,1 9–23 (2015). FIG. 10. Director field ^naround the k¼þ 1=2 and k¼/C0 1=2 defects in active nematics. Image extracted from A. Doostmohammadi, J. Ignés-Mullol, J. M. Yeomans, and F . Sagués, Nat. Commun., 9,1–13 (2018). Copyright 2018 Author(s), licensed under a Creative Commons Attribution (CC BY 4.0) license.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 093105 (2020); doi: 10.1063/5.0002590 128, 093105-7 Published under license by AIP Publishing.10S. Fumeron, B. Berche, F. Moraes, F. A. Santos, and E. Pereira, “Geometrical optics limit of phonon transport in a channel of disclinations, ”Eur. Phys. J. B 90, 95 (2017). 11S. Zhou, A. Sokolov, O. D. Lavrentovich, and I. S. Aranson, “Living liquid crystals, ”Proc. Natl. Acad. Sci. U.S.A. 111, 1265 –1270 (2014). 12A. Doostmohammadi, J. Ignés-Mullol, J. M. Yeomans, and F. Sagués, “Active nematics, ”Nat. Commun. 9,1–13 (2018). 13J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields, ” Science 312, 1780 –1782 (2006). 14H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamateri- als,”Nat. Mater. 9, 387 –396 (2010). 15V. M. 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Pereira, “High rectification in a broad- band subwavelength acoustic device using liquid crystals, ”J. Appl. Phys. 125, 204503 (2019).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 093105 (2020); doi: 10.1063/5.0002590 128, 093105-8 Published under license by AIP Publishing.

5.0011576.pdf

J. Appl. Phys. 128, 134102 (2020); https://doi.org/10.1063/5.0011576 128, 134102 © 2020 Author(s).Partial cation ordering, relaxor ferroelectricity, and ferrimagnetism in Pb(Fe1 −xYbx)2/3W1/3O3 solid solutions Cite as: J. Appl. Phys. 128, 134102 (2020); https://doi.org/10.1063/5.0011576 Submitted: 22 April 2020 . Accepted: 18 September 2020 . Published Online: 06 October 2020 S. A. Ivanov , D. C. Joshi , A. A. Bush , D. Wang , B. Sanyal , O. Eriksson , P. Nordblad , and R. Mathieu Partial cation ordering, relaxor ferroelectricity, and ferrimagnetism in Pb(Fe 1−xYbx)2/3W1/3O3 solid solutions Cite as: J. Appl. Phys. 128, 134102 (2020); doi: 10.1063/5.0011576 View Online Export Citation CrossMar k Submitted: 22 April 2020 · Accepted: 18 September 2020 · Published Online: 6 October 2020 S. A. Ivanov,1,2D. C. Joshi,2A. A. Bush,3 D. Wang,4B. Sanyal,4 O. Eriksson,4P. Nordblad,2 and R. Mathieu2,a) AFFILIATIONS 1Department of Chemistry, M.V. Lomonosov Moscow State University, Leninskie Gory 1/3, Moscow 119991, Russia 2Department of Materials Science and Engineering, Uppsala University, Box 35, SE-751 03 Uppsala, Sweden 3MIREA—Russian Technological University (RTU MIREA), Moscow 119454, Russia 4Department of Physics and Astronomy, Uppsala University, Box 516, 751 20 Uppsala, Sweden a)Author to whom correspondence should be addressed: roland.mathieu@angstrom.uu.se ABSTRACT The structural, magnetic, and dielectric properties of ceramic samples of Yb-doped PbFe 2/3W1/3O3have been investigated by a variety of methods including x-ray powder diffraction, magnetometry, and dielectric spectroscopy. In addition, theoretical investigations were madeusing first-principles density functional calculations. All the doped samples Pb(Fe 1−xYbx)2/3W1/3O3(PFYWO) (0.1 ≤x≤0.5) were found to crystallize in an ordered cubic ( Fm/C223m) structure with partial ordering in the B-perovskite sites. Observed changes in the cationic order were accompanied by differences in the dielectric and magnetic responses of the system. While pure PbFe 2/3W1/3O3is antiferromagnetic, the doped Pb(Fe 1−xYbx)2/3W1/3O3PFYWO samples display excess moments and ferrimagnetic-like behavior, associated with differences in B0 and B00site occupancies of the magnetic Fe3+cations. The magnetic transition temperature of the ferrimagnetic phase is found to decrease with increasing Yb content, from TN∼350 K of the undoped sample down to 137 K for x= 0.5. All PFYWO compounds display a ferroelec- tric relaxor behavior akin to that of PbFe 2/3W1/3O3, albeit our results show significant changes of the frequency and temperature dependence of the dielectric properties. The changes of the properties of PFYWO with increasing Yb substitution can be explained by the changes in the cation size/charge mismatch and the size difference of the two ordered positions. © 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/10.1063/5.0011576 INTRODUCTION Studies of dielectric and magnetic properties of Pb-based perovskites have revealed several new multiferroic (MF)materials. 1–7The consequences and cause of cationic order on the B-site in such perovskites have been discussed in the literature.8–10 Ionization potentials, cation coordination geometry, and the A-cation/ B-cation size ratio are factors that influence the degree of ordering.11PbFe 2/3W1/3O3(PFWO) was the first reported MF material in the double perovskite family Pb B3+ 2/3B6+ 1/3O3.12The initial idea behind the preparation of PFWO was based on the search for ferrimagnetic perovskites with some kind of order of theB-site cations creating two sublattices corresponding to theformula Pb[Fe] 0.5[Fe1/3W2/3]0.5O3. In this case, the magnetic moments of two sublattices directed oppositely to each other are not compensated, and the ferroelectric material also becomes fer- rimagnetic. Some experimental results indicate that partial ordermay occur and that the structural formula for PFWO could bewritten Pb[Fe 1−yWy]0.5[Fe1/3+yW2/3−y]0.5O3with 0 ≤y≤1/3.12 However, it has been found impossible to realize substantial cat- ionic order in undoped PFWO. PFWO is a Pb-based 2:1 perovskite that combines magneti- cally active Fe3+cations and ferroelectrically active W6+cations.13,14 PFWO orders antiferromagnetically at about 350 K and shows relaxor ferroelectric behavior between 150 and 200 K. In the lastJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 134102 (2020); doi: 10.1063/5.0011576 128, 134102-1 ©A u t h o r ( s )2 0 2 0few years, several attempts have been made to modify the properties of PFWO by substitutions at the Aor (and) Bcrystallographic sites. Results from studies using various solid solutions have been reported,for instance, by A-site doping 15–17and B-site substitution.18–23These studies show that the magnetic and dielectric properties of PFWOcan be modified and controlled by cation substitution. Although major advances have been made in optimizing PFWO for a variety of applications, several fundamental aspectsof their behavior, in particular, the nature of the compositionaland spin ordering and its relation to the relaxor properties remainunresolved. 24–27 The aim of the present work is to clarify the effect of substitu- tion of Yb3+for Fe3+on the structural, magnetic, and dielectric properties of PFWO. The selection of Yb3+substitution is related to its large cation size and small electronegativity providingoptimal conditions to create cation ordering and polar bonds. We have prepared phase-pure, stoichiometric Pb(Fe 1−xYbx)2/3W1/3O3 (PFYWO) samples of composition x= 0, 0.1, 0.2, 0.3, 0.4, and 0.5 and investigate these by x-ray powder diffraction, magnetometry,dielectric spectroscopy, and electron microscopy. The Yb doping isfound to promote cationic ordering of the Fe(Yb) and W cations on the Bsite. The experiments are accompanied by theoretical studies of magnetic interactions for several key Yb concentrationsusing first-principles calculations. METHODS Pb(Fe 1−xYbx) 2/3W1/3O3(0≤x≤0.5) samples were synthe- sized as ceramics by a conventional multiple-step solid state reac- tion method in the air;18see the supplementary material for more details about the synthesis and the characterization methods.Chemical composition was checked using scanning electronmicroscopy (SEM) images, x-ray energy dispersive spectroscopy(EDS), and inductively coupled plasma (ICP) analyses. X-ray powder diffraction (XRPD) was used for both phase analysis and advanced Rietveld analyses using the FULLPROF program. 28The (magneto)dielectric measurements were performed using a LCRmeter and a customized PPMS probe, while magnetic measure-ments were collected using a superconducting quantum interfer- ence device (SQUID) magnetometer from Quantum Design Inc. We have performed first-principles density functional calcula- tions to study the energetics and structural and magnetic propertiesof PFYWO, employing the VASP package. 29,30Perdew –Burke – Ernzerhof31generalized gradient approximation (GGA) has been considered for the treatment of exchange-correlation functional. Strong electron correlations in the d-orbitals have been included ina static mean-field approach (GGA + U). 32,33We have considered the U eff(U–J where U is the Coulomb parameter and J is the Hund parameter) value of 4 eV for the Fe-d electrons;34see the supplementary material for more information. RESULTS XRPD studies The SEM images indicate that as in, e.g., the doping of PFWO by Mn or Sc,18,22the average grain size ( ∼2–6μm for undoped PFWO) is slightly decreasing as the doping of Yb increases (seeFig. SM1 in the supplementary material ). The PFYWO ceramics have 90% –93% of the theoretical density values. EDS analyses evi- dence close to nominal stoichiometry of Pb, Fe, Yb, and W cations(see Table SM1 in the supplementary material ). Room temperature XRPD patterns of the PFYWO (0 ≤x≤0.5) samples are shown in Fig. 1 . The position and intensities of the reflections in the XRPD patterns of the samples were consistent with those expected for PFWO-based solid solutions with the cubic perovskite structure. 18 Reflections from secondary phases were not detected. Structuralrefinements of the XRPD powder diffraction patterns of PFYWOwith x< 0.1 confirm that these samples stabilize in the cubic Pm/C223m symmetry at room temperature. For samples with 0.1 ≤x≤0.5, a set of additional superstructure reflections were found [e.g., (111)at around 19° in addition to the Bragg (200) at around 22°, thex-dependences of which are illustrated in Fig. 2 ] and the crystal structure of PFYWO ( x≥0.1) stabilizes in the cubic Fm/C223msymme- try. Thus, the addition of few percent of Yb to PFWO leads to B-site cation ordering (introduction of two independent B-sites). The evolution of the lattice parameter ( a) for the different samples is plotted in Fig. 3(a) , and there is a continuous increase of awith increasing Yb content. This increase is in accord with the size dif- ference between Yb 3+(0.868 Å) and Fe3+(0.645 Å) cations for coordination number 6.36 The structural refinements of XRPD patterns measured on the x= 0.1, 0.2, 0.3, 0.4, and 0.5 samples [ Figs. 1(b) –1(f)] were per- formed in the Fm/C223mspace group (No. 225). The Fe, Yb, and W atoms are located at the 4a(0,0,0) and 4b(0.5,0.5,0.5) crystallo- graphic Wyckoff sites [which we refer to as B(4a)a n d B(4b)i n the following]. The O atoms are found at the 24esite ( x,0,0) for an ideal primitive cubic cell and its deviation from 0.25 deter- mines the difference between B(4a)–Oa n d B(4b)–O distances. Finally, the Pb atoms occupy the 32f(y,y,y) sites, presenting an off-center displacement along the [111] di rection. The Fm/C223m space group does not allow any tilt nor rotation of the BO6coor- dination octahedra nor polyhedral distortions; however, different B–O bond distances are permitted owing to the presence of partial cation ordering. The refined B–O bond lengths are indeed different and their x-dependence plotted in Fig. 3(b) . The distribution of the three different transition metal cations on the two available Bsites and the Pb and O stoichiometry have been refined assuming the nominal Fe:Yb:W ratio and using the determined ratio of (Fe + Yb) /W on the two sites from the refine- ment of XRPD data. Due to their different scattering form, factorsFe 3+and Yb3+are well distinguishable in x-ray scattering. The refine- ments indicate an increase of the degree of cation site order as the Yb content increases. Normally,18,37determination of the distribu- tion of three different kinds of atoms (Fe, Yb, and W) on two avail-able crystallographic sites ( 4aand 4binFm/C223m) requires at least two sets of diffraction data with differing atomic cross sections. However, using only XRPD data and the nominal composition in the refine- ments, conclusive results about the cation distribution were achieved.The resulting cation distributions on the 4aand 4bsites are pre- sented in Fig. 4 . Polyhedral analysis was made by the IVTON soft- ware, and the results are presented in Table SM3 in the supplementary material . The oxidation states of the cations, deter- mined by bond valence calculations, are consistent with Pb 2+,F e3+, Yb3+,a n dW6+(see Table SM1 in the supplementary material ).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 134102 (2020); doi: 10.1063/5.0011576 128, 134102-2 ©A u t h o r ( s )2 0 2 0Theoretical results Our experimental results indicate that Yb replaces Fe in the B sublattice with W sites remaining almost unaffected. For thisreason, Fe atoms have been substituted solely by Yb atoms in our calculations. Two concentrations of Yb were considered initially with two different configurations to check whether the Yb atomsare more likely to be periodically distributed in the structure for high Yb concentrations. We built structural models with the peri- odic and clustered arrangement of Yb atoms for two different con-centrations: x= 0.25 and x= 0.375. The supercells were built up with six octahedra along the c axis andffiffi ffi 2p a/C2ffiffi ffi 2p ain-plane geometry. In Table I , we show the relative energies of FM and FIG. 1. X-ray powder diffraction patterns for polycrystalline Pb(Fe 1−xYbx)2/3W1/3O3for 0≤x≤0.5 together with their Rietveld refinements. x= (a) 0, (b) 0.1, (c) 0.2, (d) 0.3, (e) 0.4, and (f ) 0.5; see Table SM2 in the supplementary material for quality R-factors and goodness of fit values.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 134102 (2020); doi: 10.1063/5.0011576 128, 134102-3 ©A u t h o r ( s )2 0 2 0AFM alignments of Fe moments. First, we note that structures with periodic arrangement always have lower energy than the clustered structure. The energy difference between periodic andnon-periodic distribution increases as the concentration increases(from 12.33 meV/atom to 16.50 meV/atom). Second, AFM config-urations are always energetically more favorable than the FM con- figurations. In addition, if we only focus on the ground state configuration, i.e., periodic arrangement of Yb, as the Yb concen-tration increases, the energy difference between AFM and FMdecreases (from 11.17 meV/atom to 2.17 meV/atom). This isbecause Fe atoms on the average are situated more distantly, and hence the strength of exchange interaction decreases. For simulating a lower concentration, we built another supercell with six octahedra along the c axis and 2ffiffi ffi 2p a/C22ffiffi ffi 2p ain-plane geometry. In this case, we performed calculations for the lowest con-centration, x= 0.125, in the supercell containing 240 atoms. We have also simulated the highest concentration ( x= 0.375) with the super- cell containing 240 atoms to check the consistency of results obtainedwith the supercell containing 60 atoms. Figure 5 is an example of the structure we used for the simulation. The results are shown inTable II . At high Yb concentration, the lowest energy state occurs for the periodic structure with the AFM configuration, and the second-lowest energy state is the random structure with the AFMconfiguration; the energy difference between these two states being9.75 meV/atom. On the contrary, at low Yb concentration, the ground state turns out to be a randomly distributed structure where the energies of the periodic and clustered structures are2.38 meV/atom and 3.71 meV/atom higher than the ground state,respectively. According to our simulation, as the Yb concentrationincreases, we expect a transition from a randomly distributed phase to a periodically distributed phase. We also compared the average magnetic moment of Fe for different Yb doping and the averagevolume of the octahedra before and after doping. The results areshown in Tables III andIV. It has been observed that the volume of PFWO increases after Yb doping. It is also seen that the FeO 6octa- hedra are modified more significantly than the WO 6octahedra. Besides that, according to the histograms of the optimized bondlengths shown in Fig. 6 , we can conclude that a significant variation FIG. 2. (a) (200) Bragg reflections and (b) (111) superlattice reflection for Pb (Fe 1−xYbx)2/3W1/3O3(0.1≤x≤0.5). FIG. 3. Compositional dependence of (a) lattice parameter and (b) bond length B–O for 4aand4bsites; the theoretical average ionic radius is added in panel (b) for com- parison. In order to compare these data with the lattice parameter of ordered Fm/C223msamples, we have doubled the lattice parameter of the compound with x=0 .Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 134102 (2020); doi: 10.1063/5.0011576 128, 134102-4 ©A u t h o r ( s )2 0 2 0exists among the species. It is found that most of the Pb –Fe bond lengths are smaller than the average Pb –B bond length, while Pb –W bond lengths are bigger. The corresponding Fe –O–Fe bond angle dis- tribution is shown in Fig. SM2 in the supplementary material , angles ranging from 158° to 175°. According to the Goodenough –Kanamori rule, when the bond angle is close to 180°, the Fe –O–Fe bond with d5-d5interacting cations shows AFM superexchange interaction. One can clearly see that for x= 0.125, there are more Fe –O–Fe networks, and more importantly most of the bond angles are close to 180°giving rise to stronger AFM superexchange. Magnetic and dielectric analyses A sharp kink of the M vs T curve (H = 100 Oe) is observed across the Néel temperature T N∼350 K for undoped PFWO as s h o w ni nt h ei n s e to f Fig. 7 . In the main frame of the figure, the tem- perature dependence of the magnetization M(T) measured under ZFC and FC conditions at a constant magnetic field H = 100 Oe forvarious Yb doping concentrations (0.1 ≤x≤0.5) is displayed. All these curves show a sharp upturn across the transition temperature. Below the ordering temperature, all the M(T) curves exhibit irrevers- ibility in ZFC and FC curves suggesting uncompensated magneticmoments at low temperatures. As the Yb doping concentrationincreases from x= 0.1 to 0.5, the transition temperature T mag decreases from 292 K to 137 K (see Table SM4 in the supplementary material ). The incorporation of Yb inside the PFWO matrix hence alters the magnetic ordering from antiferro to ferrimagnetic leadingto the excess/uncompensated moments. For higher doping concentra-tions x≥0.3, this sharp upturn becomes less prominent, reflecting the decrease of the magnitude of the excess moment at large x(see below), as well as dilution effects; x= 0.5 implies the chemical compo- sition PbFe 1/3Yb1/3W1/3O3,w i t hF e3+accounting for only one- of the B-site cations. FIG. 4. Compositional dependence of site occupancies for Fe, Yb, and W for (a)4asite and (b) 4bsite determined from the XRPD data. TABLE I. The result from calculations with 60 atoms structures, relative energies, unit in meV/atom. Periodic Cluster x FM AFM FM AFM 0.250 11.17 0.00 22.00 12.33 0.375 2.17 0.00 27.67 16.50 FIG. 5. Structural models with 240 atoms used in the first-principles calcula- tions: (a) clustered, (b) periodic, and (c) random structures of Pb(Fe 1−xYbx)2/ 3W1/3O3forx= 0.375; structural representations drawn using VESTA.35 TABLE II. The result from calculations with 240 atoms structures, energy difference, unit in meV/atom. Cluster Periodic Random x AFM FM AFM FM AFM FM 0.125 3.71 … 2.38 … 0.00 16.63 0.375 13.33 … 0.00 12.46 9.75 …TABLE III. Average magnetic moments of Fe (Fe and Yb), unit in μB. x Cluster Periodic random 0.125 4.17 (3.66) 4.18 (3.67) 4.18 (3.66) 0.250 4.19 (3.15) 4.21 (3.18) … 0.375 4.18 (2.63) 4.19 (2.64) 4.20 (2.65)Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 134102 (2020); doi: 10.1063/5.0011576 128, 134102-5 ©A u t h o r ( s )2 0 2 0Figure 8(a) shows the field dependence of magnetization M(H) represented in μB/f.u. recorded at constant temperature T = 5 K for different Yb doping concentrations 0 ≤x≤0.5. The M(H) curves represented in emu/g are shown in Fig. SM3 in the supplementary material . The observed magnetic moments M(μB/f.u.) in all the Yb-doped samples are higher as compared to the undoped PFWO, which is associated with the presence ofexcess moment caused by unbalanced magnetic sub-lattice. The coercivity of the doped samples, ∼200 Oe, may be observed in the inset of Fig. 8(a) .Figure 9(a) shows the temperature dependence of dielectric permittivity ε r(T) at two selected frequencies f= 2 kHz and 0.2 MHz, recorded under zero magnetic field for 0 ≤x≤0.50. The corresponding dielectric loss tan δ(T) for one selected fre- quency (2 kHz) is shown in Fig. 9(b) . As the temperature increases, εrincreases and attains a maximum across T m. This peak tempera- ture T mshifts toward higher temperature (188 K →230 K) with increasing Yb concentration x(0→0.50) ,while the magnitude of dielectric permittivity decreases (2911 →497). The parameters evaluated from the dielectric measurements are listed inTable SM5 in the supplementary material . At higher temperatures (T > 250 K), the much higher dielectric permittivity of undopedPFWO is likely associated with the presence of oxygen vacancies or Pb vacancies. 21,22The frequency dispersion of the high temper- ature dielectric permittivity was fitted to the Debye relaxationmodel (Arrhenius law). 22The effect of oxygen vacancies on εr(T) and tan δ(T) for x= 0.1 is discussed in Ref. 21. The contribution of oxygen vacancies seems to be less effective with increasing Yb concentration. Another important characteristic observed from Fig. 9(a) is the frequency dispersion of the dielectric maximum εr(Tm). In order to observe the frequency dispersion, the εr(T) and tan δ(T) data for x= 0.2 are shown for various frequencies 200 Hz ≤f≤2 MHz in Figs. 10(a) and10(b) , respectively. The frequency dependence of T mfollows the Vogel –Fulcher (VF) law [the inset of Fig. 10(a) ]a sg i v e n byτ¼τ0expEa kB(Tm/C0TVF)hi , where E a(=0.12 eV) is the activation energy, T VF( = 159 K) is the freezing temperature of the polariza- tion function, kBis the Boltzmann constant, and the pre- exponential factor τ0(=2 . 2×1 0−13s) is known as the characteris- tic relaxation time (Ref. 21and references therein). For other compositions, these parameters are listed in Table SM5 in the supplementary material . The observed value of τ0decreases with increasing x, while the activation energy E aincreases. For compo- sitions x= 0, 0.1, and 0.2, these values are in good agreement with lead magnesium niobate ( τ0=1 . 5 4×1 0−13sa n dE a=0 . 0 4e V ) , which is an example of a classical relaxor ferroelectric.19It is also reported in Ref. 21that the dielectric properties of the x=0 . 1 sample shows an unusual re-entrant ferroelectric relaxor behaviorwith a ferroelectric like transition at 280 K and a re-entrantrelaxor behavior around 190 K.TABLE IV . Average octahedral volume change after doping, unit in Å3. x WO 6 FeO 6 0.125 0.05 0.25 0.250 0.04 0.200.375 0.10 0.20 FIG. 6. Pb–B bond length distribution of the ground state structure in 240 atoms structural model: (a) x= 0.125 and (b) x= 0.375. The vertical line is the average Pb –B bond length. FIG. 7. T emperature dependence of magnetization M(T) measured under ZFC (open symbols) and FC (filled symbols) conditions at a constant magnetic fieldH = 100 Oe for various Yb doping concentrations (0.1 ≤x≤0.5). The inset shows the corresponding curve for x=0 .Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 134102 (2020); doi: 10.1063/5.0011576 128, 134102-6 ©A u t h o r ( s )2 0 2 0DISCUSSION The obtained XRPD patterns indicate that all PFYWO samples adopt a cubic ( Fm/C223m) structure. Above x= 0.5, it was impossible to prepare single phase samples with the perovskite structure. Thefirst-principles calculations support the stabilization of the orderedphase by Yb substitution. However, the real structural mechanismof the ordering transformation in PFYWO remains unknown andimportant factors in triggering the Pm/C223m!Fm/C223mtransformation are still not fully clear. 18,38 In ordered PFYWO perovskites, the B(4b) position is occupied by a larger ferroelectrically “inactive ”cation (Yb3+) and the B(4a) position by a smaller ferroelectrically “active ”cation (W6+) and the intermediate oxygen anions will be displaced toward the B(4a) position. This anion displacement probably induces movement of the Pb cation along <111> toward the B(4b) position in order to accommodate the lone pair of electrons of Pb2+cations. This coop- erative displacement mechanism can then provide for extended fer-roelectric coupling. The length of ferroelectric correlations in B(4a) position could be tailored by controlling the concentration of theferroelectrically active W 6+cations. Because of the overall 2:1 B-site chemistry in PFWO-type perovskites, the degree of chemical disor- der plays a central role in frustrating long-range ferroelectric cou-pling. Following conclusions made in Refs. 39and 40, we can suggest that both normal ferroelectric and relaxor ferroelectric have the same polarization mechanism, i.e., inherent correlated cation and neighboring O anion displacements forming dipole clusters [orpolar nano regions (PNR)]. The role of the substituting ions is notto induce the PNR ’s but rather to suppress homogeneous strain distortion, transverse correlation of PNR ’s, and a transition into a long-range ordered ferroelectric state. A possible reason for the onset of ferroelectricity for x= 0.1 could hence be related to the remarkable increase in the concentration of ferroelectric-active W 6+ cations at the B(4a) position observed in Fig. 4 for that composi- tion in addition to the cationic ordering. The abnormally large atomic displacement parameters for Pb cations (thermal parameter BPb=3.7Å2) provide strong evidence for the presence of disordered atomic displacements. Such behavioris typical for Pb-based perovskites containing Pb 2+on the A-site due to the presence of a stereo active electron lone pair on FIG. 8. (a) Magnetic field dependence of magnetization M(H) represented in μB/f.u. recorded at constant temperature T = 5 K for different Yb doping concentrations 0≤x≤0.5. The inset shows a zoomed view of the main panel; see Fig. SM3 in the supplementary material for the magnetization data in emu/g. (b) The compositional dependence of M( μB/f.u.) at T = 5 K and H = 5 T (blue color), and excess moment extracted from the Fe occupancies [(Fe( 4a)−Fe(4b)) × 5μB] × 1/2 is shown using the red color circular symbols (see the main text and Figs. SM4 and SM5 in the supplementary material for details). FIG. 9. T emperature dependence of (a) dielectric permittivity εr(T) at two selected frequencies f= 2 kHz (filled symbols) and 0.2 MHz (open symbols)and (b) dielectric loss tan δ(T) for a selected frequency ( f= 2 kHz) recorded under zero magnetic field for0≤x≤0.5.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 134102 (2020); doi: 10.1063/5.0011576 128, 134102-7 ©A u t h o r ( s )2 0 2 0Pb2+cations.41Although a Pb deficiency appears unlikely because of the preparation method, the Pb stoichiometry was tested by refining its occupancy factor. The refined value indicated full siteoccupancy with an unchanged value of B Pb. Therefore, the meaning of such a large Pb displacement parameter has beenexamined. It is known 41,42that improvements can be obtained by considering Pb located in a multi-minimum potential around its Wyckoff position. We have attempted to model these displacementsby moving the Pb 2+cation away from the high-symmetry site in a disordered fashion. We have tested the three kinds of Pb disorder,i.e., along <100>, <110>, and <111> directions. The difference between these models is not very pronounced (the reliable factors, R Bare 2.73%, 2.54%, and 2.23%, respectively). The magnitude of the displacement of the Pb cation from its high-symmetry positionis around 0.25(2) Å independent of Yb concentration and theobserved short-range Pb 2+displacements appear randomly distrib- uted, mostly displaced along <111>. This displacement can be explained as a consequence of the repulsion between the non-bonded 6 s lone electron pair of Pb and the Pb –O bonds of its own coordination polyhedron, supposing that the lone pair is directed toward one of the vertices of the cube. The Pb shift along the <111> direction increases slightly the angles between the lone pairand all the Pb –O bonds, minimizing the repulsion effects. It was established that for Pb cation coordinated to 12 oxygen anions,there exist only two crystallographic directions, [l00] and [111], along which the non-bonded pair can be directed. 43 In an earlier study of Sc doped PFWO (PFSWO) investigated by both XRPD and NPD,18whose data were jointly refined to determine occupancies with great accuracy. We have observed agood match between the values of the magnetic moments deter- mined from NPD and those obtained by the simple multiplication of the determined Fe occupancies by 5 μ B(the expected moment value for Fe3+), which we denote “μ(Fe-occupancies × 5 μB)”(see Fig. SM4 in the supplementary material , data obtained from Ref. 18). We hence use the occupancies from XRPD data to predict the respective magnetic moments (occupancies × 5 μB)i n PFYWO. Figure SM5 in the supplementary material shows the compositional dependence of B-site magnetic moment evaluated using the Fe-occupancies determined from the XRPD data (Fe-occupancies × 5 μB). The blue color and red color solid symbols show the 4band 4a-site moment, respectively. The blackcolor dashed line represents the average moment determined by the formula [2/3 × (1 −x)×5μB], which is consistent with the average of 4aand 4b-site moment (solid symbols). The difference in4aand 4b-site moment is shown using hollow symbols. The compositional dependence of M( μB/f.u.) at T = 5 K and H = 5 T is shown in Fig. 8(b) (blue color). The value of M( μB/f.u.) increases initially with increasing x; after a certain composition x>0.2, the M(μB/f.u.) decreases continuously. As shown in Fig. 8(b) ,t h i s behavior is consistent with the moment extracted from theFe-occupancies shown by the red color circular symbols. As compared to PFSWO where the cationic order is strongly dependent on heat treatments, 18the magnetic properties and phase stability of PFYWO remain independent of such effects. The com-parison of magnetic properties between quenched and slowlycooled samples is shown in Fig. SM6 in the supplementary material . The magnetic properties of these differently synthesized samples are almost identical indicating ordered PFYWO samples without any special heat treatment. Although several relaxation mechanisms have been proposed in the literature (see details in Ref. 44), it is still rather difficult to identify the origin of the relaxor behavior in these systems, which stems from the local nanoscale crystal structure ordering and asso-ciated structural relaxation as well as inherent displacive cation dis-order. The dielectric data from Fig. 9(a) are plotted in Fig. 11 as the temperature dependence of normalized dielectric permittivity ε r/εr(Tm) for a selected frequency f= 0.02 MHz under zero mag- netic field. Between the two extreme compositions, x= 0.1 appears as a boundary composition below and above which the dielectricbehavior appears differently. The transition from a single dielectricpeak to a twin peak was noticed from x=0 t o x= 0.1. For x= 0.1, the first peak at 192 K ( f= 1 kHz) shows a frequency dispersion and the second peak observed at 280 K is frequency independent,suggesting a re-entrant ferroelectric behavior. 21This temperature is in the vicinity of T mag.21Below x≤0.1, all samples exhibit mag- netic field dependence where the magnitude of εrdecreases with the application of the magnetic field. A similar H dependence is observed for x= 0.2, whereas for samples with x> 0.2, εris inde- pendent of the superimposed magnetic field. The magnetic fielddependence ε rand tan δof two compositions x= 0.2 and 0.3 are shown in Fig. SM7 in the supplementary material . Data for the x= 0.4 and 0.5 compositions are also added in Fig. SM8 in the FIG. 10. T emperature dependence of (a) εr(T) and (b) tan δ(T) recorded at various frequencies(200 Hz ≤f≤2 MHz) for x= 0.2. The circular symbols in the inset of (a) show the logarithmic variation of τas a function of 1/(T m−TVF) and the solid continuous line is the best-fit to the Vogel –Fulcher equation.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 134102 (2020); doi: 10.1063/5.0011576 128, 134102-8 ©A u t h o r ( s )2 0 2 0supplementary material for completeness. The evolution of the magnetic and dielectric properties of the system upon doping is summarized in the electronic phase diagram depicted inFig. 12 . The magnetic ordering temperature T magmonotonously decreases as xincreases, while T m, characteristic of the ferroelec- tric relaxor behavior, slightly increases. The parameters extracted from the Volger –Fulcher analysis of the dielectric data a r ea l s oi n c l u d e di nt h em a i nf r a m ea n dt h ei n s e to f Fig. 12 . While T VFshows a relatively mild variation with the Yb content, the activation energy E awas found to increase by nearly an order of magnitude as xincreases, τ0decreasing by several orders of magnitude (see Table SM5 in the supplementary material for numerical values). CONCLUSIONS Structural, magnetic, and dielectric properties of stoichiomet- ric and single phased Pb(Fe 1−xYbx)2/3W1/3O3(0≤x≤0.5) ceram- ics have been investigated. Cationic order was observed over thefull concentration range of Yb susbstitution (0.1 ≤x≤0.5). In this “random site ”structure, one position is occupied solely by Fe(Yb), but the other contains a random mixture of W and the remaining Fe(Yb) cations. The ordering of the structure is supported byresults from first-principles electronic structure calculations. It isinteresting to note that the effect of Yb and Sc substituents withinconcentration range up to 50% on the ordering in PFWO is very similar. As a result of the ordering and unbalanced occupancies of the magnetic Fe 3+cations on the B-sites of the ordered structure, the antiferromagnetic state of the undoped compound is switchedto a ferrimagnetic state with substantial magnitude of the excessmoment. Our results also show that Yb doping induces significant changes of the dielectric properties of PFWO, particularly in the low doping ( x∼0.1) region where re-entrant ferroelectric-like behavior is observed. SUPPLEMENTARY MATERIAL See the supplementary material for experimental details con- cerning the synthesis and analysis of the materials, and additionaltables and figures. The tables list the results of the analysis of thecomposition (EDS), structural, magnetic, and dielectric properties of the materials. The figures illustrate the microstructure (SEM images) and give more details on the magnetic and dielectric prop-erties of the compounds. ACKNOWLEDGMENTS We thank the Stiftelsen Olle Engkvist Byggmästare, the Swedish Research Council (VR), and Russian Foundation for BasicResearch (No. 18-03-00245) for financially supporting this work.We thank Pedro Berastegui, Uppsala University, for his expert help.D.W. acknowledges the China Scholarship Council (Grant No. 201706210084) for financial support. D.W. and B.S. acknowledge SNIC-UPPMAX, SNIC-HPC2N, and SNIC-NSC centers under theSwedish National Infrastructure for Computing (SNIC) resourcesfor the allocation of time in high-performance supercomputers. Moreover, supercomputing resources from PRACE DECI-15 project DYNAMAT are gratefully acknowledged. FIG. 11. T emperature dependence of the normalized dielectric permittivity εr/ εr(Tm) for a selected frequency f= 0.02 MHz under zero magnetic field. FIG. 12. Electronic phase diagram for the Pb(Fe 1−xYbx)2/3W1/3O3series: compositional dependence of the temperature onset of (ferri)magnetic order (Tmag), temperature of maximum of dielectric anomaly T matf= 0.2 MHz, and dielectric freezing temperature (T VF) obtained from the Volger –Fulcher analy- sis. The inset shows the corresponding activation energy (E a) and pre- exponential factor ( τ0).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 134102 (2020); doi: 10.1063/5.0011576 128, 134102-9 ©A u t h o r ( s )2 0 2 0DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1N. A. Spaldin and R. Ramesh, Nat. Mater. 18, 203 (2019). 2J. Varignon, N. C. Bristowe, E. Bousquet, and P. Ghosez, Phys. Sci. Rev. 5, 20190069 (2020). 3S. Krohns and P. Lunkenheimer, Phys. Sci. Rev. 4, 20190015 (2019). 4H. Palneedi, V. Annapureddy, S. Priya, and J. Ryu, Actuators 5, 9 (2016). 5B. 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Zhu, H. Wu, J. Zhuang, and Z. G. Ye, Sci. Rep. 6, 22327 (2016).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 134102 (2020); doi: 10.1063/5.0011576 128, 134102-10 ©A u t h o r ( s )2 0 2 0

5.0016108.pdf

AIP Advances 10, 065232 (2020); https://doi.org/10.1063/5.0016108 10, 065232 © 2020 Author(s).Experimental analysis of the spin–orbit coupling dependence on the drift velocity of a spin packet Cite as: AIP Advances 10, 065232 (2020); https://doi.org/10.1063/5.0016108 Submitted: 01 June 2020 . Accepted: 10 June 2020 . Published Online: 25 June 2020 N. M. Kawahala , F. C. D. Moraes , G. M. Gusev , A. K. Bakarov , and F. G. G. Hernandez COLLECTIONS Paper published as part of the special topic on Chemical Physics , Energy , Fluids and Plasmas , Materials Science and Mathematical Physics AIP Advances ARTICLE scitation.org/journal/adv Experimental analysis of the spin–orbit coupling dependence on the drift velocity of a spin packet Cite as: AIP Advances 10, 065232 (2020); doi: 10.1063/5.0016108 Submitted: 1 June 2020 •Accepted: 10 June 2020 • Published Online: 25 June 2020 N. M. Kawahala,1 F. C. D. Moraes,1 G. M. Gusev,1 A. K. Bakarov,2 and F. G. G. Hernandez1,a) AFFILIATIONS 1Instituto de Física, Universidade de São Paulo, São Paulo, SP 05508-090, Brazil 2Institute of Semiconductor Physics and Novosibirsk State University, Novosibirsk 630090, Russia a)Author to whom correspondence should be addressed: felixggh@if.usp.br ABSTRACT Spin transport was studied in a two-dimensional electron gas hosted in a wide GaAs quantum well occupying two subbands. Using space and time Kerr rotation microscopy to image drifting spin packets under an in-plane accelerating electric field, optical injection and detection of spin polarization were achieved in a pump–probe configuration. The experimental data exhibited high spin mobility and long spin lifetimes allowing us to obtain the spin–orbit fields as a function of the spin velocities. Surprisingly, above moderate electric fields of 0.4 V/cm with velocities higher than 2 μm/ns, we observed a dependence of both bulk and structure-related spin–orbit interactions on the velocity mag- nitude. A remarkable feature is the increase in the cubic Dresselhaus term to approximately half of the linear coupling when the velocity is raised to 10 μm/ns. In contrast, the Rashba coupling for both subbands decreases to about half of its value in the same range. These results yield new information on the application of drift models in spin–orbit fields and about limitations for the operation of spin transistors. ©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/10.1063/5.0016108 .,s I. INTRODUCTION Over the past few decades, the quest to build spintronic analogs to conventional charge-based electronic devices has moti- vated intense research.1–5Paramount to this search is the spin tran- sistor, proposed by Datta and Das,6that uses a gate-tunable Rashba spin–orbit interaction (SOI)7for the electric manipulation of the spin state inside a ballistic channel. Later studies included the Dres- selhaus SOI8so that a non-ballistic transistor robust against spin- independent scattering could be realized.9–12For instance, when both Rashba and Dresselhaus spin-orbit couplings (SOCs) have equal magnitudes ( α=β), a uniaxial spin–orbit field is formed and the spin polarization could be preserved during transport.13–16 In order to bring further robustness to more realistic spin transistors, we have to account for unwanted effects caused when applying in-plane electric fields inside the transistor channel as, for example, heating by the current. A recent study in a single subband system showed that the heating of the electron system leads to a drift-induced enhancement of the cubic Dresselhaus SOI.17It wasalso shown that carrier heating strongly increases the diffusion coef- ficient.18Moreover, cubic fields introduced the temporal oscillations of the spin polarization during transport by drift and cause spin dephasing.19Alternately, new device architectures using external magnetic fields have been considered to overcome such dephasing in systems set to the persistent spin helix regime.20 In this work, we are interested in the investigation of drift- induced SOC modifications by exploring a two-dimensional elec- tron gas (2DEG) hosted in a wide GaAs quantum well (QW) with two-occupied subbands. Previous studies in such multilayer sys- tems show high charge mobility and long spin lifetimes21as well as the possibility to generate current-induced spin polarization.22–24 Employing optical techniques for the injection and detection of spin polarization, it was possible to image drifting spin packets and to obtain the spin mobility and spin–orbit field anisotropies. Further increasing the drift velocities, we have observed the enhancement of the cubic Dresselhaus SOI in agreement with the single subband case.17However, we have also observed an unexpected decrease in the Rashba SOI. These findings establish limitations to the AIP Advances 10, 065232 (2020); doi: 10.1063/5.0016108 10, 065232-1 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv assumption of constant spin–orbit couplings independent of the velocity range of the spin transistor operation. II. EXPERIMENTAL MEASUREMENTS The sample used in the investigations was a 45 nm wide QW, symmetrically doped with Si, and grown in the [001](z)direction. The top left image in Fig. 1(a) shows the calculated band profile of the QW. As the charge distribution experiences a soft barrier inside the well, caused by the Coulomb repulsion of the electrons, the elec- tronic system is configured with symmetric and antisymmetric wave functions for the two lowest subbands with a separation of ΔSAS = 2 meV. The Shubnikov-de Hass oscillations provided the values ofn1= 3.7×1011cm−2andn2= 3.3×1011cm−2for each subband density and the low-temperature charge mobility was given as μc = 2.2×106cm2/V s.21To induce the drift transport required for the spin–orbit field measurements, a cross-shaped device was fabricated with a width of w= 270 μm and channels along the [1¯10](x)and [110](y)directions, where lateral Ohmic contacts were deposited l = 500 μm apart so that in-plane voltages could be applied. A simple scheme of this device is also shown in Fig. 1(a). To perform spin polarization measurements using time- resolved Kerr rotation as a function of space and time, a mode- locked Ti:sapphire laser with a repetition rate of 76 MHz tuned to 819 nm was split into pump and probe beams. A photoelastic modu- lator was used to control the circular polarization of the pump beam and the optically generated spin polarization was then measured byanalyzing the rotation of the reflected linearly polarized probe beam. The intensity of the probe beam was also modulated by an opti- cal chopper for cascaded lock-in detection. The time delay of the probe pulse relative to the pump was controllable by mechanically adjusting the length of the pump path. The incidence of the probe beam was fixed to the center of the cross-shaped device, while the pump beam could be moved with a scanning mirror, allowing to spatially map the drifting spin packets. All presented measurements were performed at 10 K. The spin–orbit fields are obtained by measuring the Kerr rota- tion signal when varying the strength of the external magnetic field (Bext), applied in the plane of the QW, for a fixed space and time sep- aration between pump and probe pulses. We model the dependence of the Kerr rotation angle ΘKas follows: ΘK(Bext)=∑ nAncos[gμB ̷h(Δt+ntrep)√ (Bext+BSO,∥)2+B2 SO,⊥], (1) where gis the electron g-factor, μBis the Bohr magneton,̵his the reduced Planck constant, Δtis the time separation between pump and probe pulses, trepis the time interval between the subsequent laser pulses, and BSO,∥(BSO,/⊙◇⊞) is the component of the spin–orbit field parallel (perpendicular) to Bext. Considering the device geome- try, the resulting field is BSO,/⊙◇⊞orBSO,∥when the drifting electric-field is applied in the channel oriented parallel or perpendicular to Bext, respectively. Next, we will refer to the magnitude of the spin–orbit FIG. 1 . (a) Device geometry, contacts configuration, and a simple scheme of the pump-probe technique used for opti- cal measurements. In addition, the top left image shows the potential profile and the subbands charge density of the two- subband QW. (b) External magnetic field scan of the Kerr rotation signal measured at the spatial overlap of pump and probe beams for three different values of the in-plane electric field applied to the y- oriented channel. Solid lines are fittings using Eq. (1). (c) Spatial distribution of the spin polarization amplitudes, show- ing fittings with the expected Gaussian profiles (solid lines), and the displace- ment caused when applying the same electric fields displayed in (b). AIP Advances 10, 065232 (2020); doi: 10.1063/5.0016108 10, 065232-2 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv field simply as BSOindependently of its direction. For spin life- times that are longer than trep, the Kerr rotation measured at a given instant depends not only on the spin polarization injected by the last pump pulse but also on the remaining polarization from the previ- ous ones. Although the contribution of each nth pulse is accounted for by the sum in Eq. (1), only some of those terms are non-negligible since Anrapidly decreases as ngrows. All measurements presented in this work were made with a fixed long time delay of Δt= 12.9 ns between injection and detection. Figure 1(b) shows the Kerr rotation signals measured when scanning in a short range of Bextfor three different values of the in- plane electric field ( E) applied in the y-oriented channel (set along Bext). For each curve, the data were taken at the overlap position between pump and probe beams [peak amplitude in Fig. 1(c)]. Solid lines correspond to the fittings of Eq. (1) to the experimental data. Note that while the components of BSOparallel and perpendicu- lar to Bextshift laterally the data or decrease the magnitude of the center peak [as in Fig. 1(b)], the g-factor changes the frequency of the peaks. Thus, the field scan experiment determines the g fac- tor and the spin–orbit field components separately as opposite to time-resolved measurements. Repeating field scans for several positions of the pump–probe separation ( d), the fitted amplitudes allow us to map the spin distribution and to track the drifting spin package. For instance, Fig. 1(c) shows the extracted spin polarization amplitudes for the same electric field values in Fig. 1(b). The curves exhibit Gaussian profiles, expected as both pump and probe beams are also Gaus- sian, displaced from each other. Subtracting the center position for these distributions from the value at E= 0 to calculate the spin package displacement Δd, the drift velocities can be computed as v =Δd/Δt. Furthermore, the spin mobility ( μs) can also be obtained from the relation v( E) =μsE. Figure 2(a) shows the drift velocities obtained from the measurements performed for positive and negative val- ues of Eapplied in the x- (red circles) and y-oriented (blue squares) channels. The solid lines are linear fits with slope μx s= (5.94 ± 0.21)×105cm2/V s and μy s= (4.40 ±0.15)×105cm2/V s. A simi- lar mobility anisotropy was previously reported in this system and related to the existence of anisotropic spin–orbit fields. The spinmobility was found to be controlled by the SOCs setting the field along the direction perpendicular to the drift velocity.21 Figure 2(b) shows the variation of the electron g-factor modu- lus with the in-plane electric field. Note that, for both channel direc- tions, this modification is symmetric on the field polarity (velocity direction). A similar behavior was measured in InGaAs epilayers.25 However, the high mobility in our system lets us produce a stronger variation of 0.02 when increasing the drift velocity from zero to 10 μm/ns in considerably smaller electric fields. The mechanism behind this g-factor dependence still requires investigation. Next, we will focus on the dependence of the spin–orbit fields with an increasing drift velocity. As commented above, the Bextscans give an independent determination of the spin–orbit field compo- nents by fitting Eq. (1). Using the measured relation between v and Ein Fig. 2(a), Fig. 3(a) shows the BSOas a function of the drift veloc- ities vx(red circles) and vy(blue squares) when Eis applied in the x- and y-oriented channels, respectively. These strong anisotropic fields can be expressed using a model21in which the components of the spin–orbit field changes linearly with the transverse v as ⟨Bx SO⟩=[m ̷hgμB2 ∑ ν=1(+αν+β∗ ν)]vy, ⟨By SO⟩=[m ̷hgμB2 ∑ ν=1(−αν+β∗ ν)]vx,(2) where νis the subband index = 1, 2, m= 0.067 m 0is the effective electron mass for GaAs, ανis the Rashba SOC for each subband and β∗ ν=β1,ν−2β3,νdepends on the linear ( β1,ν) and cubic Dresselhaus (β3,ν) SOCs. The SOCs sum over the subband index indicates that the dynamics is governed by the average spin–orbit fields because the studied sample has an electron system with a strong inter-subband scattering.26 Figure 3(a) displays a non-monotonic function for BSO, which is opposite to the prediction of Eq. (2). The solid line is a lin- ear fit to the data points in the low-velocity range. The departure from the model means that the SOCs inside the brackets change with an increasing drift velocity. Those parameters are graphically represented by the local curve slope and can be inferred by the ratios FIG. 2 . (a) Drift velocity v of the spin packet for in-plane electric fields applied inx- and y-oriented channels. The lines are linear fits from which the spin mobility values were extracted. (b) Dependence of the g-factor with the drift velocities in each sample channel. AIP Advances 10, 065232 (2020); doi: 10.1063/5.0016108 10, 065232-3 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 3 . (a) Spin–orbit field Bx(y) SOas a function of the drift velocity vy(x). (b) Ratio BSO/v, computed for each data point in (a), as a function of the drift velocities in each of the device channels. (c) Projection of the dependence of the SOCs on v computed by plugging the linear fits obtained in (b) into Eq. (3). by=Bx SO/vyandbx=By SO/vx, plotted in Fig. 3(b) as a function of v. Clearly, bxand by(and the SOCs) remain constant only up to approximately 2 μm/ns followed by a decrease with an increase in v, that is stronger for bythan for bx. Finally, to translate the spin–orbit field dependence into the variation of the SOCs, we assumed a linear relation between band v in the high-speed regime (v >2μm/ns). Using this relation, illus- trated by the fit lines in Fig. 3(b), we can isolate the Rashba and Dresselhaus terms from busing the following equations: 2 ∑ ν=1β∗ ν=̷hgμB 2m(by+bx),2 ∑ ν=1αν=̷hgμB 2m(by−bx), (3) constructed from the definition of the ratios bxandbyabove. Projections from Eq. (3) are shown in Fig. 3(c). The sum of the Rashba coefficients decreases to half of its value at low velocities (0.2 meV Å) when increasing v up to 10 μm/ns. As αdepends on the QW symmetry, the observed dependence with an in-plane electric field is unexpected and may be related to the same mechanism affect- ing the g-factor, which is still not understood. On the other hand, the variation of β∗=β1−2β3is even stronger. Considering that the lin- ear Dresselhaus coefficient depends on the QW width, it is expected that the measured modification of β∗reflects exclusively an increase ofβ3as v grows. Similar enhancement of the cubic Dresselhaus SOCwas previously discussed in single subband systems and associated with heating due to the high currents used to induce high drift veloc- ities, as β3depends on the average kinetic energy.17,18Remarkably, we measured a strong enhancement of the cubic Dresselhaus term up to approximately half of the linear coupling when β∗∼0 at 10 μm/ns. III. CONCLUSIONS Cubic spin–orbit fields impose relevant constrains in spin tran- sistor proposals that target extended coherence, thus demanding particular attention. Here, we addressed this issue in the investiga- tion of a 2DEG confined in a GaAs QW with two occupied subbands. Applying in-plane electric fields that enabled the drift transport of spin packets along the device channels, we measured velocities as high as 10 μm/ns and a spin mobility in the range of 105cm2/V s. We observed the dependence of the spin–orbit couplings on the spin packet drift velocity. We found two regimes: (i) For low veloc- ities (v<2μm/ns), the SOCs are independent of the drift velocity and the spin–orbit fields increase with increasing velocity, (ii) For high velocities (v >2μm/ns), the Rashba SOC decreases, while the cubic Dresselhaus SOC increases and the spin–orbit fields become weaker with an increase in the velocity. Our findings indicate that limitations on the transport velocities should be considered when implementing spin transistors in multilayer systems. AIP Advances 10, 065232 (2020); doi: 10.1063/5.0016108 10, 065232-4 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv ACKNOWLEDGMENTS We acknowledge financial support from the São Paulo Research Foundation (FAPESP), Grant Nos. 2009/15007-5, 2013/03450-7, 2014/25981-7, 2015/16191-5, 2016/50018-1, and 2018/06142-5, and the National Council for Scientific and Technological Development (CNPq), Grant Nos. 301258/2017-1 and 131114/2017-4. REFERENCES 1S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnár, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, “Spintronics: A spin-based electronics vision for the future,” Science 294, 1488–1495 (2001). 2I. Žuti ´c, J. Fabian, and S. Das Sarma, “Spintronics: Fundamentals and applica- tions,” Rev. Mod. Phys. 76, 323–410 (2004). 3D. D. Awschalom and M. E. Flatté, “Challenges for semiconductor spintronics,” Nat. Phys. 3, 153–159 (2007). 4J. Wunderlich, B.-G. Park, A. C. Irvine, L. P. Zârbo, E. Rozkotová, P. Nemec, V. Novák, J. Sinova, and T. Jungwirth, “Spin hall effect transistor,” Science 330, 1801–1804 (2010). 5J. C. Egues, G. Burkard, and D. Loss, “Datta-Das transistor with enhanced spin control,” Appl. Phys. Lett. 82, 2658–2660 (2003). 6S. Datta and B. Das, “Electronic analog of the electro-optic modulator,” Appl. Phys. Lett. 56, 665–667 (1990). 7Y. A. Bychkov and E. I. Rashba, “Oscillatory effects and the magnetic suscepti- bility of carriers in inversion layers,” J. Phys. C: Solid State Phys. 17, 6039–6045 (1984). 8G. Dresselhaus, “Spin-orbit coupling effects in zinc blende structures,” Phys. Rev. 100, 580–586 (1955). 9J. Schliemann, J. C. Egues, and D. Loss, “Nonballistic spin-field-effect transistor,” Phys. Rev. Lett. 90, 146801 (2003). 10M. Ohno and K. Yoh, “Datta-Das-type spin-field-effect transistor in the nonbal- listic regime,” Phys. Rev. B 77, 045323 (2008). 11Y. Kunihashi, M. Kohda, H. Sanada, H. Gotoh, T. Sogawa, and J. Nitta, “Pro- posal of spin complementary field effect transistor,” Appl. Phys. Lett. 100, 113502 (2012). 12M. Kohda and G. Salis, “Physics and application of persistent spin helix state in semiconductor heterostructures,” Semicond. Sci. Technol. 32, 073002 (2017). 13B. A. Bernevig, J. Orenstein, and S.-C. Zhang, “Exact SU(2) symmetry and per- sistent spin helix in a spin-orbit coupled system,” Phys. Rev. Lett. 97, 236601 (2006).14J. D. Koralek, C. P. Weber, J. Orenstein, B. A. Bernevig, S.-C. Zhang, S. Mack, and D. D. Awschalom, “Emergence of the persistent spin helix in semiconductor quantum wells,” Nature 458, 610–613 (2009). 15M. P. Walser, C. Reichl, W. Wegscheider, and G. Salis, “Direct mapping of the formation of a persistent spin helix,” Nat. Phys. 8, 757–762 (2012). 16J. Schliemann, “Colloquium: Persistent spin textures in semiconductor nanos- tructures,” Rev. Mod. Phys. 89, 011001 (2017). 17Y. Kunihashi, H. Sanada, Y. Tanaka, H. Gotoh, K. Onomitsu, K. Nakagawara, M. Kohda, J. Nitta, and T. Sogawa, “Drift-induced enhancement of cubic dres- selhaus spin-orbit interaction in a two-dimensional electron gas,” Phys. Rev. Lett. 119, 187703 (2017). 18F. Passmann, A. D. Bristow, J. N. Moore, G. Yusa, T. Mano, T. Noda, M. Betz, and S. Anghel, “Transport of a persistent spin helix drifting transverse to the spin texture,” Phys. Rev. B 99, 125404 (2019). 19P. Altmann, F. G. G. Hernandez, G. J. Ferreira, M. Kohda, C. Reichl, W. Wegscheider, and G. Salis, “Current-controlled spin precession of quasis- tationary electrons in a cubic spin-orbit field,” Phys. Rev. Lett. 116, 196802 (2016). 20S. Anghel, F. Passmann, K. J. Schiller, J. N. Moore, G. Yusa, T. Mano, T. Noda, M. Betz, and A. D. Bristow, “Spin-locked transport in a two-dimensional electron gas,” Phys. Rev. B 101, 155414 (2020). 21M. Luengo-Kovac, F. C. D. Moraes, G. J. Ferreira, A. S. L. Ribeiro, G. M. Gusev, A. K. Bakarov, V. Sih, and F. G. G. Hernandez, “Gate control of the spin mobil- ity through the modification of the spin-orbit interaction in two-dimensional systems,” Phys. Rev. B 95, 245315 (2017). 22F. G. G. Hernandez, L. M. Nunes, G. M. Gusev, and A. K. Bakarov, “Observation of the intrinsic spin hall effect in a two-dimensional electron gas,” Phys. Rev. B 88, 161305 (2013). 23F. G. G. Hernandez, G. M. Gusev, and A. K. Bakarov, “Resonant optical control of the electrically induced spin polarization by periodic excitation,” Phys. Rev. B 90, 041302 (2014). 24F. G. G. Hernandez, S. Ullah, G. J. Ferreira, N. M. Kawahala, G. M. Gusev, and A. K. Bakarov, “Macroscopic transverse drift of long current-induced spin coherence in two-dimensional electron gases,” Phys. Rev. B 94, 045305 (2016). 25M. Luengo-Kovac, M. Macmahon, S. Huang, R. S. Goldman, and V. Sih, “ g- factor modification in a bulk ingaas epilayer by an in-plane electric field,” Phys. Rev. B 91, 201110 (2015). 26G. J. Ferreira, F. G. G. Hernandez, P. Altmann, and G. Salis, “Spin drift and diffusion in one- and two-subband helical systems,” Phys. Rev. B 95, 125119 (2017). AIP Advances 10, 065232 (2020); doi: 10.1063/5.0016108 10, 065232-5 © Author(s) 2020

5.0003985.pdf

J. Chem. Phys. 153, 024110 (2020); https://doi.org/10.1063/5.0003985 153, 024110 © 2020 Author(s).Hole–hole Tamm–Dancoff-approximated density functional theory: A highly efficient electronic structure method incorporating dynamic and static correlation Cite as: J. Chem. Phys. 153, 024110 (2020); https://doi.org/10.1063/5.0003985 Submitted: 06 February 2020 . Accepted: 16 June 2020 . Published Online: 09 July 2020 Christoph Bannwarth , Jimmy K. Yu , Edward G. Hohenstein , and Todd J. Martínez ARTICLES YOU MAY BE INTERESTED IN TeraChem: Accelerating electronic structure and ab initio molecular dynamics with graphical processing units The Journal of Chemical Physics 152, 224110 (2020); https://doi.org/10.1063/5.0007615 Recent developments in the PySCF program package The Journal of Chemical Physics 153, 024109 (2020); https://doi.org/10.1063/5.0006074 DFTB+, a software package for efficient approximate density functional theory based atomistic simulations The Journal of Chemical Physics 152, 124101 (2020); https://doi.org/10.1063/1.5143190The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Hole–hole Tamm–Dancoff-approximated density functional theory: A highly efficient electronic structure method incorporating dynamic and static correlation Cite as: J. Chem. Phys. 153, 024110 (2020); doi: 10.1063/5.0003985 Submitted: 6 February 2020 •Accepted: 16 June 2020 • Published Online: 9 July 2020 Christoph Bannwarth,1,2 Jimmy K. Yu,1,2,3 Edward G. Hohenstein,1,2 and Todd J. Martínez1,2,a) AFFILIATIONS 1Department of Chemistry and The PULSE Institute, Stanford University, Stanford, California 94305, USA 2SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, USA 3Biophysics Program, Stanford University, Stanford, California 94305, USA a)Author to whom correspondence should be addressed: toddjmartinez@gmail.com ABSTRACT The study of photochemical reaction dynamics requires accurate as well as computationally efficient electronic structure methods for the ground and excited states. While time-dependent density functional theory (TDDFT) is not able to capture static correlation, complete active space self-consistent field methods neglect much of the dynamic correlation. Hence, inexpensive methods that encompass both static and dynamic electron correlation effects are of high interest. Here, we revisit hole–hole Tamm–Dancoff approximated ( hh-TDA) density func- tional theory for this purpose. The hh-TDA method is the hole–hole counterpart to the more established particle–particle TDA ( pp-TDA) method, both of which are derived from the particle–particle random phase approximation ( pp-RPA). In hh-TDA, the N-electron electronic states are obtained through double annihilations starting from a doubly anionic ( N+2 electron) reference state. In this way, hh-TDA treats ground and excited states on equal footing, thus allowing for conical intersections to be correctly described. The treatment of dynamic cor- relation is introduced through the use of commonly employed density functional approximations to the exchange-correlation potential. We show that hh-TDA is a promising candidate to efficiently treat the photochemistry of organic and biochemical systems that involve several low-lying excited states—particularly those with both low-lying ππ∗andnπ∗states where inclusion of dynamic correlation is essential to describe the relative energetics. In contrast to the existing literature on pp-TDA and pp-RPA, we employ a functional-dependent choice for the response kernel in pp- and hh-TDA, which closely resembles the response kernels occurring in linear response and collinear spin-flip TDDFT. Published under license by AIP Publishing. https://doi.org/10.1063/5.0003985 .,s I. INTRODUCTION Linear response time-dependent density functional theory (TDDFT) is the most commonly used electronic structure method for excited states due to its low computational cost and rel- ative accuracy for absorption spectra involving valence excited states.1Unfortunately, due to the difficulties that presently available approximate exchange-correlation (XC) functionals encounter with near-degeneracies and static correlation, it is unsuitable for pho- tochemical problems involving a conical intersection between theground and first excited states.2While complete active space self- consistent field (CASSCF) methods can treat static electron cor- relation, they struggle to describe dynamic correlation.3Adding a correction to recover dynamic correlation (as in multirefer- ence perturbation theory4or multireference configuration inter- action5) significantly increases the computational cost of the method and renders nonadiabatic dynamics simulations com- putationally intractable for many interesting medium to large sized molecules (although such dynamics is feasible for small molecules6–12). J. Chem. Phys. 153, 024110 (2020); doi: 10.1063/5.0003985 153, 024110-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp There has long been interest in the development of inexpen- sive excited-state methods that can simultaneously treat dynamic and static correlation. One approach to this problem is to aug- ment traditional multiconfigurational methods with semiempirical parameters. These parameters can be as simple as a constant scaling parameter applied to the energy13or as complex as a full semiempir- ical treatment of the Hamiltonian matrix elements.14,15While often successful, these methods may require cumbersome system-specific parameterization.15–17Alternatively, conventional multiconfigura- tional wavefunctions can be combined with a Kohn–Sham (KS) den- sity functional theory (DFT)-based treatment of dynamic electron correlation.18,19Many of these methods use conventional multicon- figurational wavefunctions to capture static correlation and include a DFT-based treatment of dynamic correlation. Methods of this type include range-separated wavefunction/DFT methods,20,21mul- ticonfigurational pair-density functional theory22,23(MC-PDFT), and combinations of DFT and configuration interaction,24,25such as in DFT/MRCI.26,27Alternative approaches attempt to incorpo- rate multireference character directly in the DFT formalism by using an unconventional reference28or appealing to ensemble densities. Spin-flip TDDFT (SF-TDDFT)29is an example of a scheme based on an unconventional reference. Spin contamination problems have plagued DFT-based spin–flip methods;30,31however, this has recently been addressed by combining SF-TDDFT with ad hoc cor- rections from DFT/MRCI.32Ensemble formulations of DFT include the spin-restricted ensemble-referenced Kohn–Sham (REKS) meth- ods.33To date, REKS methods need to be formulated specifically for the chosen ensemble (defined by an active space, as in CASSCF), and a general formulation applicable to arbitrary active spaces is lack- ing. Current implementations including gradients and nonadiabatic couplings are only applicable to an active space of two electrons in two orbitals.34,35 The recent development of particle–particle random phase approximation ( pp-RPA) methods has opened new possibilities for inexpensive excited-state methods. The original aim of pp-RPA was to provide ground state correlation energies via an adiabatic con- nection fluctuation dissipation theorem (ACFDT) approach.36–41 While the particle–hole RPA ( ph-RPA) ACFDT approach recov- ers the ring channel of the correlation energy from the cou- pled cluster doubles (CCD) equations,42the ladder channel of the CCD correlation energy is obtained from the pp-RPA ACFDT approach.36,41Yang and co-workers also highlighted pp-RPA and its Tamm–Dancoff approximated pp-TDA variant as effective meth- ods to compute electronic ground and excited state energies.43–52 Starting from a doubly cationic ( N−2)-electron reference, the N- electron ground state and excited states generated by excitations from the highest occupied molecular orbital (HOMO) are recov- ered by performing two-electron attachments.45,49,50This allows the treatment of the N-electron ground and excited states on equal foot- ing (derived as simultaneous eigenvalues of a common Hamilto- nian) at a computational cost comparable to the simplest excited state methods, e.g., TDDFT/ ph-RPA and configuration interaction singles (CIS). Because the ground and excited states are treated on equal footing, pp-TDA based on an ( N−2)-electron reference is able to predict the correct topography around conical intersections, as has been shown explicitly for H 3and NH 3.53This is a major advance over conventional ph-TDDFT and CIS methods, which cannot reproduce conical intersections involving the ground state.2In 2014, Peng et al. derived the pp-RPA equations from linear response theory by choosing a pairing field perturbation (termed TDDFT-P, see below in Sec. II A).54This formally allowed for the combination of pp-RPA and DFT references. Hence, the effect of dynamic correlation on the orbitals is incorporated through the exchange-correlation (XC) potential, while the pp-RPA and pp-TDA schemes ensure that ground and excited states are treated on equal footing and therefore can treat exact degeneracies correctly (see also Sec. II D). In a procedure complementary to pp-TDA and pp-RPA based on an ( N–2)-electron reference, the N-electron ground state and excited states can also be generated through double annihilations from a double anionic ( N+2)-electron reference in which the low- est unoccupied molecular orbital (LUMO) is populated with two additional electrons. This ( N+2)-electron pp-RPA scheme and its corresponding hole–hole ( hh) Tamm–Dancoff approximation ( hh- TDA) were first presented by Yang and co-workers.49Although thepp-RPA and pp-TDA methods based on an ( N–2)-electron ref- erence have now become quite established,43–53less attention has been paid to the hh-TDA method based on an ( N+2)-electron refer- ence. Yang and co-workers applied the hh-TDA method to oxygen and sulfur atoms49(noting “relatively large errors” in the results), but we have found no published reports of further developments or applications of hh-TDA to molecules. In this paper, we suggest that the hh-TDA method is worthy of renewed attention. Similar to thepp-TDA method, hh-TDA can effectively capture dynamic and static correlation in ground and low-lying excited states, including near- and exact degeneracies. Furthermore, the active orbital space in the ( N+2)-electron-based hh-TDA appears to be suitable for the description of excited states in many organic molecules that are inac- cessible to ( N−2)-electron pp-TDA. Examples include molecules with low-lying n π∗andππ∗excited states that cannot be simul- taneously described within pp-TDA (which, by construction, can only describe excited states where an electron is excited from the HOMO). In contrast to previous work on pp-RPA, for both pp- and hh-TDA, we formulate the response kernel in a functional-specific way that resembles the kernels occurring in linear density matrix response theory. This new formulation of the response kernel is compared to the previously used functional-independent variant. We put particular emphasis on the utility of the hh-TDA method in the treatment of organic and biologically relevant systems involving bothππ∗andnπ∗transitions. II. THEORY A. Particle–particle random-phase approximation from pairing field perturbations The particle–particle random phase approximation ( pp-RPA) equations were derived by Peng and co-workers54by means of coupled time-dependent perturbation theory in analogy to well- established linear response theory by choosing a pairing field pertur- bation within the framework of Hartree–Fock (HF)/Kohn–Sham– Bogoliubov theory (termed TDDFT-P).55The ground state for this non-interacting particle system is defined by the zero-temperature grand potential as54–57 J. Chem. Phys. 153, 024110 (2020); doi: 10.1063/5.0003985 153, 024110-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Ω[γ,κ]=Ts[γ,κ]+Vext[γ]+Dext[κ]+EJXC[γ,κ]−μN. (1) Here, Ts[γ,κ] is the independent particle kinetic energy and Vext[γ] is the external potential energy, which contains the nuclear-electron attraction. Dext[κ] denotes the external pairing potential, and the last term preserves the total electron number. EJXC[γ,κ] is the mean-field potential energy due to the electrons and includes the particle–hole and particle–particle channels via the one-particle density matrix γ and the pairing matrix κ, respectively. The indices J, X, and C denote the Coulomb, exchange, and correlation components of this func- tional, respectively. The one-particle density matrix in the canonical molecular spin orbital basis is given by γpq=⟨Ψ∣a† paq∣Ψ⟩=δpq,∀p,q∈{↑,↓}, (2) while the pairing matrix (or anomalous density matrix) is defined as κpq=⟨Ψ∣apaq∣Ψ⟩,κ∗ pq=⟨Ψ∣a† pa† q∣Ψ⟩,∀p∈{↑}andq∈{↓}. (3) Ψrefers to the single-determinant ground state wavefunction of the non-interacting Kohn–Sham (KS) system. Here and in the follow- ing,i,j,k,ldenote occupied, a,b,c,ddenote unoccupied, and p,q, r,sdenote general molecular orbitals. For the hypothetical true functional EJXC[γ,κ], the real-space density and anomalous density of the non-interacting KS system are identical to the respective quantities of the true interacting system. In the work of Peng et al. ,54a non-superconducting system with no external pairing field in the ground state Hamiltonian is used. In that case, Dext[κ] = 0. As a result, the mean-field potential energy is free from indirect electron–electron interactions (e.g., phonon- mediated interactions)55and contains only direct electron–electron interaction contributions via the (anti-symmetrized) Coulomb oper- ator and XC potential. Based on the ground state defined by Eq. (1), Peng et al. derived the pp-RPA method by perturbing the ground state with an external pairing field. The TDDFT-P equations, which describe the coupled response to the pairing field perturbation, are given as54 −[ω−(εi+εk−2μ)]δκik(ω)=δDext ik(ω)+δDJXC ik(ω), (4) [ω−(εa+εc−2μ)]δκac(ω)=δDext ac(ω)+δDJXC ac(ω). (5) Here,δκpq(ω) are the first order changes in the pairing matrix induced by the external frequency-dependent pairing field perturba- tionδDext pq(ω),ωis the frequency of the perturbation, ϵpis the orbital energy, and μis the chemical potential or Fermi energy. The pair- ing matrix response Lpq,rsof the mean field potential is contained in δDJXC pq(ω), δDJXC pq(ω)=−Nocc ∑ j>lLpq,jlδκjl(ω)−Nvirt ∑ b>dLpq,bdδκbd(ω). (6) In our formulation of pp-RPA, the choice of Lpq,rsdiffers from the one presented by Yang and co-workers,44,45,54as discussed in detail below.B. The adiabatic particle–particle linear-response kernel For a non-superconducting system in the absence of Dext[κ], the pairing matrix and its associated XC potential are zero in the elec- tronic ground state. Consequently, only the γ-dependent terms in Eq. (1) survive, and minimizing the zero-temperature grand poten- tial becomes equivalent to minimizing the standard KS or Hartree– Fock (HF) energy expression. It is only after taking the second derivatives with respect to κthat the particle–particle channels of the mean-field potential as in Eqs. (4) and (5) become non-zero. In the framework of ab initio wavefunction theory, the pp-RPA equa- tions are based on a HF reference, and the mean-field potential takes the form of an anti-symmetrized Coulomb integral.41In that case, theppand particle–hole ( ph) channels of the mean-field potential are formally equivalent. We note, however, that only the exchange- type integral survives the spin integration in the ppchannel (see the supplementary material). In the context of approximate KS DFT, there may exist some liberty in choosing the explicit form of this response kernel. Given that contemporary semi-local density functional approximations (DFA) are defined for κ= 0, the pairing matrix response of the density functional drops out, EDFA XC[γ,κ=0]=EDFA XC[γ]→∂2EXC[γ] ∂κ∗pq∂κrs=0. (7) Yang and co-workers have already employed this approximation for the XC density functional response. Furthermore, they chose to use the HF-type mean-field response, i.e., a bare anti-symmetrized Coulomb integral, as the response kernel.54It is therefore indepen- dent of the underlying density functional approximation (DFA). With this kernel, the chosen DFA only affects the orbitals and orbital energies that enter the pp-RPA calculation, but not the ppresponse expression itself. In this work, we employ a different choice for the ppresponse kernel. Given the formally equivalent phandppresponse kernels when using a HF reference,41which may be regarded as a special choice of the KS system, we draw analogy from the well-known ph linear response TDDFT. In spin-preserving1,58and spin–flip (SF)29 formulations of TDDFT, the response kernel reflects the underlying ground state DFA in the way the non-local Fock exchange enters the response. Hence, we choose to use the same modification of the non- local Fock exchange, i.e., the global scaling and/or range-separation employed by the corresponding DFA, Lpq,rs=[ps∣qr]−aFR X[pr∣qs]−aSRA X[pr∣qs]SRA. (8) The first term on the right-hand side corresponds to the Coulomb- type two-electron integral (Mulliken notation for spin MOs), whereas the remaining terms denote the modified exchange-type two-electron integrals. aFR Xis the functional-specific scaling factor for the global [full-range (FR)] exchange integral, [pr∣qs]=∬ϕ∗ p(x1)ϕr(x1)1 r12ϕ∗ q(x2)ϕs(x2)dx1dx2. (9) Here,ϕrefers to a molecular spin orbital and x1/x2denote all spatial and spin coordinates of electrons 1/2. The short range-attenuated J. Chem. Phys. 153, 024110 (2020); doi: 10.1063/5.0003985 153, 024110-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp (SRA) part of the exchange integral employs the modified Coulomb operator and is given as [pr∣qs]SRA=∬ϕ∗ p(x1)ϕr(x1)erf(ωSRAr12) r12ϕ∗ q(x2)ϕs(x2)dx1dx2. (10) Like Yang and co-workers, we also assume the pairing matrix response of the XC functional to be zero [see Eq. (7)], allowing the response kernel to become frequency independent. Due to spin inte- gration (note the definition of the pairing matrix), the Coulomb part in the ppresponse kernel is zero; hence, Lp¯q,r¯s=−aFR X(pr∣qs)−aSRA X(pr∣qs)SRA. (11) Here, we have used the Mulliken notation for spatial MOs and over- barred orbital indices to highlight orbitals corresponding to βspin. Our choice for the response kernel [Eq. (11)] is identical to the one in collinear SF-TDDFT but with inverted sign [note that Lpq,rsis subtracted in Eq. (14) below]. This can be understood from a sec- ond quantization picture in which both the spin–flip and pp-RPA approaches apply creation/annihilation operators acting on differ- ent spin, i.e., a† ↑a↓in spin–flip and a† ↑a† ↓/a↑a↓in the pp/hhchannels. Therefore, only the exchange integral does not vanish upon spin integration, as in pp-RPA, and the flipped sign results from the com- mutation relation of the annihilation and creation operators (see the supplementary material for a comparison). SF-TDDFT can be derived from linear-response phTD-DFT if changes in the density matrix that do not preserve the ˆSzexpectation value are allowed.29 Due to its origin from an analogy to linear response SF-TDDFT, we refer to our choice of the response kernel, Eq. (11), as the lin- ear response-type kernel in the following to distinguish it from the DFA-independent, HF-like kernel used by Yang and co-workers.49 C. The hole–hole Tamm–Dancoff approximated pp-RPA method Using the response kernel in Eq. (11), we can rewrite Eqs. (4) and (5) in matrix notation as [(AppBph BhpAhh)−(ω 0 0−ω)](κpp(ω) κhh(ω))=(δDpp(ω) δDhh(ω)). (12) Here, the superscripts ppandhhdefine the particle–particle (dou- ble creation in the virtual space) and hole–hole (double annihila- tion in the occupied space) blocks, respectively. Appis of dimension n2 virt×n2 virt, and Ahhis of dimension n2 occ×n2 occ. Their elements are given as [App]ac,bd=Aac,bd=δabδcd(ϵa+ϵc−2μ)−Lac,bd (13) and [Ahh]ik,jl=Aik,jl=−δijδkl(ϵi+ϵk−2μ)−Lik,jl. (14) Bhp= [Bph]Tis of dimension n2 occ×n2 virtand is given asBik,bd=−Lik,bd. (15) If we now let the perturbation [right-hand side in Eq. (12)] go to zero, we find that there are non-trivial solutions to this sys- tem of equations, which correspond to the poles of the particle– particle/hole–hole response function. This leads to the non- Hermitian eigenvalue problem (AppBph BhpAhh)(X Y)=(X Y)(ω0 0−ω). (16) Employing the commonly used notation,54the matrices XandY contain the eigenvectors, which correspond to the ppandhhchanges in the pairing matrix κ.ωis a diagonal matrix, which contains the eigenvalues of the non-Hermitian eigenvalue problem. If we neglect the coupling matrices BhpandBphbetween the ppand hhblocks, we obtain the so-called Tamm–Dancoff approximated (TDA) eigenvalue problems, AppX=Xωpp(17) for the pp-TDA case and AhhY=Yωhh(18) for the hh-TDA problem. Note that within TDA, the constant diagonal shift of ∓2μin Eqs. (13) and (14) can be removed from the definition of the corre- sponding matrix elements. Both of these eigenvalue problems are guaranteed to have only real eigenvalues. This makes them more robust than the full pp-RPA case, which can have complex solu- tions that spoil the potential energy surfaces (PES) around coni- cal intersections (as noted in the context of coupled cluster the- ory59,60). While the pp-TDA method based on an ( N–2)-electron reference has been extensively investigated as a method to describe low-lying excited states45,49and S 0/S1conical intersections,53the hh-TDA method has been less thoroughly explored for the molec- ular electronic structure.49In this work, we suggest that hh-TDA should be revisited as an efficient DFT-based method capable of computing low-lying excited states even in the presence of conical intersections. Before benchmarking the hh-TDA method along with our choice for the response kernel [Eq. (11)], we will briefly discuss some practical aspects of the pp-TDA and hh-TDA methods in Sec. II D. D. Practical considerations of the pp-TDA andhh-TDA methods In both the pp-TDA and hh-TDA methods, the N-electron tar- get system is described by first solving the ground state electronic structure for a system that differs by two electrons from the N- electron target system. This corresponds to a double cation ( N–2 electrons) in the case of pp-TDA or a double anion ( N+2 electrons) in the case of hh-TDA. The ground and excited states of the N- electron target system are then obtained by creation ( pp-TDA) or annihilation ( hh-TDA) of two electrons (see Fig. 1). The inability to describe either doubly excited states or conical intersections involving the ground state are well-known shortcom- ings of standard single-reference methods such as linear-response J. Chem. Phys. 153, 024110 (2020); doi: 10.1063/5.0003985 153, 024110-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . Graphical representation of the pp-TDA (left) and hh-TDA (right) meth- ods to obtain the N-electron ground state and excited states. In the pp-TDA case (left), the orbitals are obtained from a SCF calculation on the ( N−2)-electron system. Addition (green) of two electrons into the virtual space recovers the N- electron ground and excited states. The closed-shell N-electron ground state obtained from pp-TDA is dominated by the configuration that corresponds to a double creation in the LUMO [with respect to the ( N−2)-electron system]. In the hh-TDA case (right), the orbitals are obtained from a SCF calculation on the ( N+2)-electron system. Annihilation (blue) of the two electrons in the occupied orbital space then recovers the N-electron ground and excited states. Here, the N-electron ground state is dominated by the configuration that cor- responds to a double annihilation in the HOMO [with respect to the ( N+2)- electron reference]. (ph) TD-DFT and CIS. These are both rectified by pp- and hh-TDA. Since the ground and the excited states are solutions of the same eigenvalue problem, near- and exact degeneracies (such as conical intersections) can be properly described. Furthermore, some dou- bly excited states can be computed (see Fig. 2). At the same time, the preceding ground state calculation of the double ionic state is of single-reference complexity and incorporates dynamic correla- tion by virtue of the XC potential. Therefore, both pp- and hh-TDA schemes are, in principle, able to capture static and dynamic cor- relation, making them promising methods for use in nonadiabatic dynamics simulations. Furthermore, in contrast to the SF-TDDFT scheme, pure eigenfunctions of ˆS2are obtained trivially in both pp- andhh-TDA. However, both methods also have some obvious shortcom- ings in common. First, the excited state expansion space is highly restricted. Taking the N-electron target system as reference, pp- TDA only includes excitations from the HOMO, while hh-TDA is restricted to excitations tothe LUMO. Second, both methods takea detour by computing the orbitals for a reference that differs in its total charge from the target state, as SF-TDDFT does with a differing spin state. As a result, two-state degeneracies of the N-electron state can be treated at the cost of a single reference method. However, the orbitals are not optimized for the N-electron system. This may have a significant implication for dynamics simulations: a degeneracy on the (N+2) or ( N−2) surface itself can occur and cause instabilities in the SCF procedure. Zhang et al. noted that the choice of a ( N+2) reference may be hampered by the existence of unbound orbitals.50Presumably, this is one of the reasons that the hh-TDA method has not received serious attention in molecular electronic structure so far. There are two dis- tinct consequences of unbound orbitals that can be envisioned. The first consequence is quite practical, namely, that it may be difficult to converge the SCF procedure for the ( N+2) reference. In the usual hh-TDA method, non-convergence of the SCF procedure for the (N+2) reference would be fatal. The second consequence is some- what more formal. Even if SCF convergence can be achieved, orbitals with positive orbital energies do not correspond to bound electrons. They can thus be very sensitive to the employed basis set. Indeed, with a sufficiently flexible basis set, they would be expected to be highly delocalized continuum functions and poor representatives of a low-lying valence excited state. Regarding the first consideration, we note that poor conver- gence of the SCF procedure will be exacerbated if the long-range potential is incorrect. DFAs with 100% long-range Fock exchange are known to capture the asymptotic potential correctly. Accord- ingly, using DFAs with 100% long-range exchange, we have not observed any instability of the ( N+2)-electron SCF procedure for the molecules we tested. Even the green fluorescent protein chro- mophore (HBI) anion with a net charge of −3 for the reference state converges without difficulty when an asymptotically correct, range- separated DFA is used. However, the hh-TDA method is likely to encounter serious SCF convergence difficulties for DFAs that do not incorporate long-range exact exchange. It has been previously observed that finite-basis set DFT is capa- ble of providing reasonable electron affinities computed as energy differences EA = EN−EN+1, even when the anion has unbound occu- pied orbitals.61This suggests that the second consideration is largely formal. However, there are also some practical concerns. The success ofhh-TDA in describing excited states is predicated on the HOMO orbital of the ( N+2)-electron determinant being a good approxi- mation to the bound orbital that predominantly contributes to the low-lying N-electron excited states. This will certainly not hold if the HOMO is a good approximation to a continuum orbital, which is the expected outcome in a sufficiently large basis set. We take a practical approach here and recommend that diffuse basis sets should be avoided in the context of hh-TDA. For the finite orbital basis sets of double- or triple-zeta quality used in this study, there are very few cases where the hh-TDA scheme based on an ( N+2)- electron reference fails, in spite of the presence of occupied orbitals with positive energies. This implies that the HOMO of the ( N+2)- electron system is a good approximation to the LUMO of the N- electron system, albeit with shifted orbital energies. This shift is effectively removed during the hh-TDA step, leading to a reasonable description of the relative energies between the N-electron ground and low-lying excited states. We will further elaborate on this issue in Sec. IV E, but we emphasize again that diffuse basis sets should J. Chem. Phys. 153, 024110 (2020); doi: 10.1063/5.0003985 153, 024110-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Schematic representation of the excitation types generated in hh-TDA with respect to the N-electron ground state. The “LUMO” and “HOMO” orbitals are defined with respect to the N-electron ground state determinant. usually be avoided in combination with the ( N+2)-electron refer- ence calculation. Forpp-TDA, there may also exist a practical implication with respect to the underlying DFA. It was shown that pp-RPA based on a HF reference leads to less accurate singlet-triplet splittings,48 slower basis set convergence,45and overall less accurate excitation energies for small molecules45than pp-RPA based on DFAs with lit- tle or no exact exchange. Similar results for excitation energies were reported for pp-TDA.45A convenient test case is the ππ∗state of ethylene, where both hh-TDA and pp-TDA provide a suitable active orbital space. Accurate methods such as coupled cluster and mul- tireference perturbation theory agree that this should be the lowest valence excited state. For ethylene, we observe a stronger depen- dence on the amount of Fock exchange in the DFA for pp-TDA (Table S13 in the supplementary material). The ππ∗state is the fourth excited state for pp-TDA-HF, while it is the second excited state for hh-TDA-HF. It is well-known that adding Fock exchange to a DFA increases the differential treatment of virtual and occu- pied orbitals. In molecular systems,62the virtual orbitals in HF are best suited to describe electron attachment, while virtual orbitals in (semi-)local DFT are more appropriate to describe electronic excitations.63This implies that DFAs with large amounts of exact exchange will raise the energy of the HOMO of the N-electron target system in pp-TDA. Therefore, when using practical DFAs, the optimal amount of exact exchange could be different for pp- TDA and hh-TDA, and this point deserves further study. In this work, we focus on DFAs for the linear-response-type kernel hh-TDA framework. We will note (details in Table S13) that the ordering of the ππ∗state for ethylene in pp-TDA is improved by using our lin- ear response-type kernel compared to the previously introduced functional-independent HF-type response kernel.45,49,54For clarity, it should be noted that purely (semi-)local DFAs have a vanishing response kernel in our linear-response-type formulation, i.e., App andAhhare diagonal in that case. In the HF-type response kernel,the bare anti-symmetrized Coulomb integral is used as the response kernel, regardless of the underlying DFA. Inhh-TDA, within the limitations of the given basis set (see above), the LUMO (with respect to the N-electron target system) is generated on equal footing with the occupied orbitals for HF or any DFA. Another difference between hh- and pp-TDA is the dimen- sion of the Amatrix to be diagonalized. In pp-TDA, this dimension increases strongly with the size of the atomic orbital (AO) basis. For hh-TDA, the dimensions are independent of the basis set size (unless effective core potentials are used). We mention for the sake of completeness that pp-TDA-HF is equivalent to full configuration interaction (FCI) for any two- electron system, while hh-TDA-HF is equivalent to FCI for an N- electron system in a basis that provides exactly two virtual molecu- lar spin orbitals. We have confirmed this via calculations on H 2in an STO-3G basis set (see the supplementary material). Thus, both schemes have similarities to CASCI methods with a fixed active space (i.e., Nelectrons in N/2 + 1 orbitals for hh-TDA and two electrons in nvirt+ 1 orbitals for pp-TDA). One can speculate that the ability of the pp-TDA and hh-TDA to describe static correla- tion arises from their similarity to CASCI methods. For the same reason, however, neither pp-TDA nor hh-TDA is size consistent. As long as no degeneracy of the LUMO/LUMO+1 ( hh-TDA) or the HOMO/HOMO–1 ( pp-TDA) is present, this is not expected to have practical implications for the description of relative energies between electronic states. III. COMPUTATIONAL DETAILS Thehh-TDA and pp-TDA methods have been implemented in the electronic structure code TeraChem.64,65Allhh-TDA and pp- TDA calculations in this work are performed with this development version of TeraChem. To compare with the pp-TDA data in the literature, we have implemented our choice for the response ker- nel [Eq. (11)] as well as the HF-type response kernel of Yang and J. Chem. Phys. 153, 024110 (2020); doi: 10.1063/5.0003985 153, 024110-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp co-workers.54We confirmed that our implementation is correct by comparison of pp-TDA excitation energies with values reported in the literature45as well as comparisons between CASCI and hh/pp- TDA using Hartree–Fock reference states. In the functional assessment for vertical excitation energies with hh-TDA in Sec. IV A, we considered the global hybrid function- als B3LYP,66–69PBE0,70,71and BHLYP66,67,72as well as Hartree–Fock. Thus, we cover different amounts of non-local Fock exchange in the mean field Hamiltonian (20%, 25%, 50%, and 100%, respectively). Additionally, we test different range-separated hybrid functionals, namely, CAM-B3LYP,66–69,73ωPBEh,74and the B9775type function- als:ωB97,76ωB97X,76andωB97X-D3.77For clarity, it is empha- sized that the latter two are different in their functional parameters (see the supplementary material) and hence should give rise to dif- ferent electronic structures. The DFA assessment is restricted to the linear response-type kernel variant of hh-TDA [Eq. (11)] For the assessment of different functionals, we employed the spherical split-valence atomic orbital def2-SV(P)78,79basis set by Ahlrichs and co-workers. All SCF calculations used the converged Hartree–Fock orbitals as guess orbitals. The systems considered in this work are given in the supplementary material. When comparing pp-TDA and hh-TDA in Sec. IV B, we use the same spherical TZVP basis set80that has been used in the orig- inal benchmark papers by Thiel and co-workers.81Both schemes are employed in a setting, in which the occupied and active orbitals have been generated on equal footing, i.e., where these orbitals experience a mean-field potential with the same number of elec- trons. The pp-TDA is hence combined with the generalized gra- dient approximation (GGA) functional PBE70(see also Sec. II D). hh-TDA is combined with the range-separated functional ωB97X,76 which is shown to be a well-performing combination in Sec. IV A (and in the supplementary material). We consider both the linear response-type (this work) and the HF-type (Yang and co-workers) response kernels for pp-TDA and hh-TDA. Additionally, we have performed phTamm–Dancoff approximated TDDFT calculations with theωB97X functional. For reference, we use the best estimates for excitation energies of Thiel and co-workers81and complement these with equation-of-motion singles and doubles coupled cluster (EOM-CCSD) calculations (the same basis set) as implemented in Q-Chem v5.0.0.82 In Sec. IV C 1, we calculate the ethylene potential energy sur- face (PES) with hh-TDA-ωB97X (linear response kernel) using the Cartesian def2-SVP78,79basis set. The scans along the pyramidaliza- tion and torsion angles were performed without relaxation of the remaining degrees of freedom. The excited state energies of thymine at three critical struc- tures are considered in Sec. IV C 2. We use hh-TDA in combi- nation with the ωPBE functional and the Cartesian 6-31G∗∗basis set.83–85A range separation parameter ω= 0.2 a.u. was selected after a coarse scan and comparison against high level reference values for this molecule. We calculated these reference values with EOM- CCSD/aug-cc-pVDZ with Q-Chem. While the Franck–Condon (FC), the S 1minimum, and the S 1/S2minimum energy conical inter- section (MECI) were optimized with hh-TDA, we used the respec- tive coupled cluster (CC) geometries of Ref. 86 for single-points at the EOM-CCSD/aug-cc-pVDZ level of theory. The S 2(ππ∗) sad- dle point geometry therein is claimed to be “in close proximity” to the conical intersection seam.86Due to the lack of a properlyCC-optimized MECI structure, we use that structure as a substitute and estimate the MECI energy by averaging the S 1and S 2energies computed with EOM-CCSD/aug-cc-pVDZ. In Sec. IV C 3, we apply hh-TDA with the ωB9776functional in combination with the Cartesian 6-31G∗∗basis set83–85and compare to results obtained with multireference configuration interaction methods including single and double excitations (MRSDCI).87 IV. RESULTS A. Functional benchmarking for hh-TDA The performance of density functionals for the calculation of vertical excitation energies in the linear-response phtime-dependent density functional theory framework is well known88and the role of Fock exchange is fairly well understood.1In contrast, for the hh- TDA method proposed in this work, the impact of Fock exchange and the use of an anionic reference on the excitation energies is not known. Therefore, we start by benchmarking several common DFAs along with Hartree–Fock (HF) in the calculation of vertical excitation energies with hh-TDA. To gain more insight, we consider the lowest vertical excita- tion energies using molecules and reference data from previously published datasets where highly accurate excitation energies are available.81,89,90We classify the excitation type into different cate- gories: intermolecular charge-transfer (CT) and predominantly local excitations. We also consider molecules that have push–pull type excitations, i.e., excitations with partial intramolecular CT charac- ter. We find that the functionals behave similarly for this set as for the local excitations. Thus, we have added them to the latter set and distinguish only intermolecular CT and intramolecular excita- tions (see the supplementary material for separate results of the local and the push–pull set). In the latter set, we have selected molecules for which hh-TDA, across different DFAs, produces the same char- acter for the lowest vertical excitation as the reference method (SCS-CC291). We have not considered any purely semi-local GGA-type DFAs since these show severe convergence problems in the self-consistent field (SCF) procedure of the double anionic reference (see also Sec. II D). Although we also expect that global hybrids will not be optimal in that case, for completeness, we have included some prototypical global hybrid functionals that differ in the amount of Fock exchange. All functionals considered are listed in Sec. III, but we restrict the discussion in this paper to the globally constant Fock-exchange DFAs B3LYP, BHLYP, and HF (these functionals contain 20%, 50%, and 100% of Fock exchange, respectively). Fur- thermore, we find that all tested range-separated DFAs with 100% long-range exchange perform similarly; therefore, we restrict the current discussion to the ωB97X-D3 functional. The results for the other functionals that were tested can be found in the supplementary material. Figure 3 shows the Gaussian error distributions based on the mean deviation (MD) and standard deviation (SD) obtained for the aforementioned reference datasets of excitation energies. While hh- TDA-HF yields CT excitation energies that are on average under- estimated by almost 1 eV, the excitation energies for locally excited states show a systematic overestimation of about the same magni- tude. Furthermore, the error is significantly more systematic for the J. Chem. Phys. 153, 024110 (2020); doi: 10.1063/5.0003985 153, 024110-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Gaussian error distribution functions for hh-TDA with different density functional approximations in the calculation of vertical excitation energies (VEEs). The spherical def2-SV(P)78,79basis set and the LR kernel in the hh-TDA calculation have been used throughout. The centers of the Gaussians correspond to the mean deviation (MD), whereas the width of the Gaussian corresponds to the standard deviation (SD), both in eV. (a) Lowest vertical excitations of single molecules with no or little intramolecular CT character. The individual MDs and SDs in eV are B3LYP ( −0.27, 0.47), BHLYP (0.33, 0.40), HF (1.03, 0.75), and ωB97X-D3 (0.24, 0.37), with N = 27. (b) Intermolecular CT excitations of organic bimolecular complexes. The individual MDs and SDs in eV are B3LYP (1.15, 0.46), BHLYP (0.54, 0.13), HF ( −0.87, 0.12), and ωB97X-D3 (0.52, 0.11), with N = 6. See the supplementary material for details on the benchmark sets and results with other functionals. intermolecular CT excitations. For DFAs with low amounts of Fock exchange, i.e., B3LYP, these trends are reversed, however, with a smaller magnitude for the underestimation of local excitation ener- gies. It is noteworthy that the DFAs behave quite different than in a phlinear response TD-DFT framework. In the latter, DFAs with low amounts of Fock exchange significantly underestimate CT excitations.89The BHLYP functional, which is in between in terms of the percentage of Fock exchange, shows more consistent errors for both sets (MD ≈0.4 eV–0.5 eV), as well as more narrow error distributions. The range-separated DFA ωB97X-D3 behaves even slightly better than BHLYP. This, along with the fact that the 100% asymptotic Fock exchange makes these DFAs less prone to SCF convergence issues with the ( N+2)-electron reference, suggests the use of asymptotically correct range-separated hybrid functionals in combination with the hh-TDA scheme. In addition to benchmarking the DFAs for CT and local states, we investigate state splittings between ππ∗and locally excited states with different character (mostly nπ∗). For this purpose, we have selected a set of systems mostly comprised of molecules from Thiel’s 2008 benchmark set.81,90Problematic systems, which are impossi- ble to treat with hh-TDA due to the restricted orbital space, were discarded from the benchmark set. These are systems in which either the population of the frontier orbitals in the ( N+2)-electron system is not clear (degenerate LUMOs in the N-electron deter- minant) or the excited states cannot predominantly be described by a single orbital–orbital transition (benzene, naphthalene, pyr- role, and s-triazine). Furthermore, butadiene, hexatriene, and octate- traene are not considered here but in Sec. IV B. The results are visualized in Fig. 4, while the respective symmetry labels and results for other density functionals can be found in the supplementary material. First, we stress the relevance of high amounts of Fock exchange in the DFA, which we mentioned earlier in Sec. II D.We find that B3LYP and BHLYP, but also a few range-separated functionals (CAM-B3LYP or ωPBEh, see supplementary material), show incorrect orbital occupations for some carbonyl systems in the ( N+2)-electron SCF calculation, i.e., the HOMO of the ( N+2)- electron system does not correspond to the LUMO obtained from FIG. 4 . Energetic splitting between a low-lying ππ∗and another (usually nπ∗) excited state. The excitation energies were computed at the hh-TDA level of theory (LR kernel) employing the spherical def2-SV(P) basis set. A π-system is classi- fied as “aromatic,” if a near-degeneracy of the N-electron system LUMO can be expected, due to the symmetry of the molecule (cf. Frost circle representation). An asterisk marks missing data due to an incorrect LUMO (with respect to the N-electron system) occupation in the ( N+2)-electron SCF calculation. The refer- ence values are taken from Ref. 81 (aspirin from Ref. 90). The respective state symmetries are given in the supplementary material. J. Chem. Phys. 153, 024110 (2020); doi: 10.1063/5.0003985 153, 024110-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp an SCF calculation on the N-electron system. Hence, the excited states for these systems cannot be described properly with these functionals. We also find that the global hybrid functionals perform poorly for the state splittings compared to Hartree–Fock or range-separated functionals (e.g., ωB97X-D3). Except in two cases (n π∗/ππ∗in aspirin and σπ∗/ππ∗in cyclopropane), the latter two perform similarly. Both describe the nπ∗splittings reasonably well, while systematically overestimated ππ∗excitations are found for aromatic systems, regardless of which DFA is used. This seems to be caused by the absence of accessible higher lying anti-bonding πorbitals in the expansion space, which would lower these excitations and improve the energetic splittings with respect to the nπ∗excitations. This is also observed for the splittings between the different ππ∗states in cyclopentadiene and norbornadiene, presumably for the same reasons. Interestingly, reduced short-range Fock exchange leads to increasedσπ∗excitation energies in cyclopropene, which then lead to qualitatively incorrect energy splittings with respect to the ππ∗ state. This is observed for all DFAs, and only HF ranks these states qualitatively correctly. Overall, the results indicate that the hh-TDA scheme in com- bination with a range-separated hybrid functional appears to be a promising method for describing the ground and low-lying excited states of organic molecules on equal footing. B. Comparison of pp-TDA and hh-TDA with linear response-type and full exchange response kernels In this section, we compare the hh-TDA and pp-TDA schemes as well as the two different response kernel choices. In Table I, we have tabulated the statistical data for excitation energies computed for a subset of excitations from the Thiel benchmark set81with the linear response (LR) and HF-type kernels for hh-TDA-ωB97X and pp-TDA-PBE. We note that our pp-TDA-PBE (HF) results are in agreement with the ones presented earlier by Yang and co-workers (note that we use a slightly different basis set here).45The statisti- cal data are listed in Table I, while detailed values of the individ- ual systems and states are given in the supplementary material. A significant difference between the hh-TDA and pp-TDA schemesbecomes apparent when looking at systems that involve both ππ∗ and other types of excitations (i.e., nπ∗orσπ∗). Here, we find that hh-TDA typically provides the better option since the configura- tion space in hh-TDA is more suited to describe both excitation types simultaneously. This manifests itself in a larger number of states on this set, which can actually be described by hh-TDA. The nucleobases and carbonyl systems are particularly good cases for hh-TDA (see the supplementary material). For pp-TDA-PBE, SCF convergence problems for these systems eliminate the possibility to include them in the analysis. This seems to be due to (nearly) degenerate frontier molecular orbitals in the ( N−2) reference and was confirmed using alternative electronic structure software (Turbomole version 6.4).74,92The HF-type variant performs better forpp-TDA-PBE than the LR variant, the latter being equivalent to plain PBE orbital energy differences. For hh-TDA-ωB97X, this is reversed and the LR-type kernel performs better, as reflected by smaller mean (absolute) and standard deviations. Both pp-TDA- PBE schemes on average show underestimated excitation energies, whereas the opposite is true for hh-TDA-ωB97X. From all of the considered pp/hhschemes, hh-TDA-ωB97X (LR) performs the best on this set. Compared to regular ph-TDA-ωB97X and EOM-CCSD, hh- TDA-ωB97X (LR) provides almost comparable accuracy, with a slightly less systematic error distribution as reflected by a SD, which is twice as large as for ph-TDA-ωB97X. In summary, we find that hh-TDA provides a reasonable method typically with a broader applicability to many organic systems compared to pp-TDA. In Sec. IV C, we will only consider the LR-type kernel variant of hh- TDA and investigate potential energy surface properties beyond the Franck–Condon point. C. Potential energy surfaces with hh-TDA 1. Ethylene As addressed in Sec. II D, one of the main advantages of the hh- TDA method is the ability to accurately treat degeneracies involv- ing the ground state and low-lying excited states. The topologically correct description of the potential energy surfaces in regions of TABLE I . Statistical data for excitation energies (in eV) computed with hh-TDA and pp-TDA using either the linear response (LR) type response kernel (this work) or the Hartree–Fock (HF) response kernel employed by Yang and co-workers. Particle–hole TD-DFT [Tamm–Dancoff approximated (TDA)] calculations and EOM-CCSD calculations are provided for comparison. All calculations use the spherical TZVP basis set. We chose DFAs which we expect to perform best for both schemes, i.e., a range-separated one forhh-TDA and a GGA for pp-TDA. The structures and reference values are taken from Ref. 81. If states could not be described by a method, they were discarded from the statistical analysis. The preceding PBE SCF calculations for the ( N−2)-electron reference failed to converge for eight systems, precluding a total of 23 states to be considered forpp-TDA-PBE (see the supplementary material for details). EOM-CCSD TDA- ωB97X hh-TDA- ωB97X (LR) hh-TDA- ωB97X (HF) pp-TDA-PBE (LR) pp-TDA-PBE (HF) MDa0.43 0.38 0.36 0.43 −0.25 −0.30 SDb0.32 0.30 0.59 0.69 1.14 0.69 MADc0.44 0.40 0.54 0.67 0.98 0.59 Statesd51 48 48 48 18 18 aMean deviation. bStandard deviation. cMean absolute deviation. dNumber of states included in the statistical data. J. Chem. Phys. 153, 024110 (2020); doi: 10.1063/5.0003985 153, 024110-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp degeneracy is especially important in nonadiabatic photochemical dynamics where population transfer between adiabatic potential surfaces is often mediated by these seams of conical intersection. A prototypical example of such an intersection can be accessed through torsion and pyramidalization degrees of freedom in ethy- lene; this particular case has previously been used as a model sys- tem for assessing the ability of electronic structure methods to describe statically correlated character and ground-excited state degeneracies.2 Typical low-cost single reference excited state methods, such as CIS and TDDFT, suffer from instability related to HOMO– LUMO orbital degeneracies as an MECI involving the ground state and an excited state is approached. As a result, these meth- ods cannot properly describe the twisted-pyramidalized MECI in ethylene. By contrast, hh-TDA is well-suited to describing conical intersection topologies, as demonstrated in combination with the ωB97X functional in Fig. 5(a), in which the double cone topol- ogy characterizes the vicinity of the S 0/S1MECI. Here, the poten- tial energy surfaces are plotted along torsion and pyramidalization coordinates, which are the branching plane coordinates correspond- ing to the twisted-pyramidalized minimum energy conical inter- section (MECI) between the S 0ground state and the ππ∗excited state. Moreover, CIS and TDDFT cannot describe double excita- tions, which then leads to a description of the S 1minimum at a purely twisted geometry (90○torsion, no pyramidalization).2Meth- ods that incorporate doubly excited states describe the S 1mini- mum as simultaneously twisted (90○torsion) and pyramidalized. hh-TDA captures this feature of the S 1potential energy surface (PES), as demonstrated by the appearance of the minimum at atwisted (90○) and pyramidalized (60○) geometry that lies 4.76 eV above the ground state minimum. This is detailed in Fig. 5(b). The S 1 minimum computed by the the-state-averaged, extended multi-state complete active space second-order perturbation method [SA3- XMS-CAS(2,2)PT2]93on the same set of ethylene coordinates is found at 90○torsion and 72○pyramidalization and lies 5.46 eV above the ground state minimum [additional information regarding the SA3-XMS-CAS(2,2)PT2 comparison can be found in the supple- mentary material; these potential energy surfaces can be compared to previous results obtained with quasidegenerate multi-reference perturbation theory2]. These results indicate that hh-TDA describes the potential energy surfaces reasonably well and thus is a promis- ing candidate for applications in efficient nonadiabatic dynam- ics simulations. This finding complements earlier studies on H 3 and NH 3, which showed that pp-TDA is able to describe conical interactions.53 2. Thymine The thymine molecule is a biologically relevant prototype for internal conversion dynamics between nπ∗andππ∗states. At the Franck–Condon point, highly accurate coupled cluster (CC) com- putations predict that thymine has nearly degenerate S 1and S 2states ofnπ∗andππ∗character, respectively.86Furthermore, accurate CC-based methods predict a reaction coordinate from the Franck– Condon point to the MECI between the S 1and S 2states without barriers or the presence of minima on the S 2state.86At lower lev- els of theory—particularly those that do not include dynamic elec- tron correlation, such as CASSCF—it is common to find a larger (greater than 1 eV) splitting between the S 1and S 2states at the Franck–Condon point as well as a minimum on the S 2state.94,95It FIG. 5 . (a) Global features of the S 0ground (red) and S 1excited (blue) state PESs of ethylene computed at the hh-TDA-ωB97X/def2-SVP level in a rigid scan over the branching plane coordinates of pyramidalization and torsion (visualized in the inset). The linear response-type kernel has been employed. hh-TDA describes the S 0/S1 degeneracy in ethylene, as demonstrated here with the appearance of the double cone feature. Ethylene reaches the conical intersection geometry once distorted to 90○torsion and at 60○pyramidalization. This point coincides with the predicted S 1minimum, lying 4.76 eV above the ground state minimum. (b) The S 1PES from (a) represented as a contour plot. The shape of the S 1PES is relatively well-described by hh-TDA-ωB97X compared to previously reported quasidegenerate multi-reference perturbation theory results.2The S 1minimum appears at a simultaneously twisted and pyramidalized geometry. For comparison, the location of the S 1minimum obtained from SA3-XMS-CAS(2,2)PT2 (see the supplementary material for details) that lies 5.46 eV above the S 0ground state is marked by a light-yellow X. J. Chem. Phys. 153, 024110 (2020); doi: 10.1063/5.0003985 153, 024110-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp remains unclear how these properties of the thymine PES affect its photodynamics; highly efficient electronic structure methods that agree more closely with the accurate CC-based methods would be required to answer these questions. We find hh-TDA to be a promis- ing candidate for use in nonadiabatic dynamics simulations of thymine. The relative excited state energies computed with hh-TDA with theωPBE(ω= 0.2 au) functional and EOM-CCSD (see Sec. III) at three different thymine structures are shown in Fig. 6. The verti- cal excitation energies to the S 1and S 2states are underestimated by about 0.5 eV–0.8 eV by hh-TDA-ωPBE(ω= 0.2 a.u.) compared to EOM-CCSD. The latter themselves are, however, higher by about 0.2 eV–0.3 eV compared to the higher-level CC3 data.86The S 1/S2 splittings are reproduced to within 0.17 eV–0.22 eV by hh-TDA. As a result, hh-TDA does a particularly good job describing the relative energetics of the S 1and S 2states at stationary points rel- evant to the photodynamics of thymine. The stabilization of the S1(nπ∗) minimum is correct to 0.04 eV compared to both the Franck–Condon point and S 1/S2MECI. Furthermore, no minimum is present on the S 2PES of thymine between the Franck–Condon point and S 1/S2MECI. The relative energy of that intersection with FIG. 6 . Relative energies (in eV) of the lowest three singlet electronic states of thymine computed at the hh-TDA-ωPBE(ω= 0.2 a.u.)/6-31G∗∗(blue) and EOM- CCSD/aug-cc-pVDZ (red) levels of theory. Energies are computed at three critical points relevant to the excited state dynamics of thymine: the S 0minimum (FC), the minimum energy conical intersection between the S 1and S 2states (S 1/S2MECI), and the S 1minimum (S 1min). The arrows indicate vertical excitations to S 2(solid) and S 1(dashed). Geometries are optimized for hh-TDA and taken from Ref. 86 for EOM-CCSD (see Sec. III for technical details). All energies are given relative to the respective S 0energy at the Franck −Condon point. Higher level vertical excitation energies at the CC3 level of theory are 0.20 eV–0.26 eV lower compared to our EOM-CCSD/aug-cc-pVDZ (cf. Ref. 86) and thus in better agreement with the hh- TDA-ωPBE(ω= 0.2 a.u.)/6-31G∗∗results.respect to the S 2energy at the Franck–Condon point is also in good agreement with EOM-CCSD. Remembering that the computational cost of hh-TDA scales formally as O(N4) but, in practice, as O(N2) with system size (see Sec. IV D), while EOM-CCSD formally scales asO(N6), the agreement between these methods is quite remark- able. What remains to be done in the future studies is to apply hh- TDA directly in a nonadiabatic dynamics simulation of thymine and assess the performance against experimental ultrafast spectroscopic observables. 3. Malonaldehyde Malonaldehyde is a molecule that is a prototype for excited- state proton transfer, nπ∗/ππ∗internal conversion dynamics, and photoisomerization. Here, we apply hh-TDA with the ωB9776func- tional and compare to the results obtained with multireference con- figuration interaction methods including single and double excita- tions (MRSDCI, see Fig. 7).87Again, we find good agreement in the absolute excitation energies and in the splitting between nπ∗and ππ∗states at the Franck–Condon point. However, the relative ener- getics of the S 1and S 2states at important stationary points show some potential problems. The minimum on the S 1(nπ∗) state is a bit too stable relative to both the Franck–Condon point and the S 0/S1 MECI. This indicates that the population may become spuriously trapped on the S 1state leading to longer excited-state lifetimes and perhaps increased involvement of triplet states (in view of the El- Sayed rules and the fact that the electronic wavefunction has nπ∗ character in the region). In addition, the S 1/S2MECI reached by proton transfer is energetically unfavorable. This may indicate that the proton transfer reaction will play a different role in the inter- nal conversion dynamics than MRSDCI would predict. The issues encountered by hh-TDA for malonaldehyde seem to be related to the restriction on the orbital space to include only one π∗orbital. For this molecule, a second π∗orbital may be essential to an accu- rate description of the photodynamics. Although we expect that malonaldehyde is not an ideal application of the hh-TDA method, nonadiabatic dynamics simulations, in which the molecular sym- metry is lifted, would provide a more definitive assessment of the performance of hh-TDA for this molecule. D. Computational cost of hh-TDA In Fig. 8, we report the computational time required to solve for the ground and first excited state of the Nile blue chromophore in aqueous solution using the hh-TDA and TDA-TDDFT methods with the Cartesian def2-SVP basis set and the ωPBE functional with ω= 0.8 a.u. The principal conclusions that should be drawn from these timings are that the hh-TDA and TDA-TDDFT methods are of similar computational cost and exhibit similar scaling behavior with the system size, i.e., O( N2) in our AO direct implementation (see below). In TDDFT, the solution of the KS equations defines the ground state, whereas in hh-TDA, both the ground and excited states are obtained as eigenvectors of the TDA response matrix. We find that multiplication of trial vectors against the response matrix is less expensive in the hh-TDA method (because there are no con- tributions from the Coulomb-type integrals or derivatives of the exchange-correlation potential). In cases where only one or two excited states are required, it should be expected that the cost of J. Chem. Phys. 153, 024110 (2020); doi: 10.1063/5.0003985 153, 024110-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 7 . Relative energies (in eV) of the lowest three singlet electronic states of malonaldehyde for critical geometries computed at the hh-TDA-ωB97/6-31G∗∗ (blue normal print) and MRSDCI/6-31G∗//SA3-CAS(4,4)SCF/6-31G∗(black ital- ics, taken from Ref. 87) levels of theory. In the latter, all configurations generated in a (4,4) active space served as references in the MRSDCI calculations. The Franck–Condon, S 1minimum, and S 0/S1minimum energy conical intersection geometries are optimized without constraints. The S 2C2vminimum and S 1/S2 MECI are minima in the C 2vand C spoint groups, respectively. Only the energy levels computed at the hh-TDA-ωB97/6-31G∗∗level of theory are plotted, and the vertical excitations at the Franck–Condon point to S 2and S 1are illustrated as solid and dashed arrows, respectively. TDA-TDDFT and hh-TDA will be quite similar, with the particulars of a given molecule determining which method will be faster. For example, in the present case, TDDFT required extra guess vectors in the Davidson diagonalization in order to avoid solving for higher- lying charge transfer excited states rather than the locally excited state of Nile blue. Formally, the hh-TDA method appears to scale as O( o6), where orepresents the number of occupied orbitals. However, since only a few electronic states are actually of interest, the full diagonaliza- tion of the response matrix can be avoided, and the scaling of the method would be O( o4)—the cost of the multiplication of a trial vec- tor by the response matrix. However, this ignores the cost of forming the response matrix in the molecular orbital basis. In practice, our implementation of hh-TDA is fully AO integral direct and scales, formally, as O( N4), where Nis the number of AO basis functions. The advantage of the AO formulation of hh-TDA is that sparsity can be exploited and the contractions between trial vectors and the response matrix scale more like O( N2). This is clearly demonstrated in Fig. 8, where the apparent quadratic scaling of the method allows us to apply it to systems with more than 2100 atoms in a polar- ized double- ζbasis set (roughly 18 000 basis functions). It should be noted that the underlying SCF procedure costs approximately the FIG. 8 . Timings of hh-TDA/def2-SVP and TDA-TDDFT/def2-SVP computations of the ground and first excited state of Nile blue including microsolvation by 50–700 water molecules. The hh-TDA/def2-SVP computation is decomposed into the time spent in the KS SCF for the ( N+2)-electron system and the “post-SCF” portion of the computation that comprises the Davidson diagonalization to obtain the lowest two eigenpairs as defined in Eq. (18). The largest system considered contained more than 2100 atoms and nearly 18 000 basis functions. The timings were per- formed using eight NVIDIA V100 GPUs and eight Intel Xeon E5-2698 CPU cores clocked at 2.2 GHz. Best fit power series are given for each of the methods. For both methods, the ωPBE functional with a range-separation parameter of 0.8 a.u. was used. same as the solution of the eigenvalue problem (for two states) and the SCF procedure scales slightly worse with the system size. E. The unbound orbital problem and basis set requirements Following the remarks above on the basis set requirements for hh-TDA due to the ( N+2)-electron reference, all calculations in this paper used double- and triple-zeta basis sets without diffuse func- tions. In order to explore the basis set dependence more thoroughly, we collect the lowest excitation energies computed with hh-TDA- ωB97X and different basis sets STO-3G, def2-SVP, def2-SVPD, def2- TZVP, and def2-QZVP (the latter two without f and g functions) for a few molecular systems (acetamide, p-benzoquinone, butadi- ene, ethylene, formaldehyde, norbornadiene, uracil, and water) in Table S11 of the supplementary material. As could be expected, these results show that excitation energies obtained with the STO-3G basis set are widely disparate from larger basis sets. However, with the exception of acetamide, the excitation energies vary mostly by less than 0.2 eV over the other choices of basis sets without diffuse func- tions. This indicates that the presence of unbound orbitals need not be a significant problem (as long as diffuse basis sets are avoided, see results for def2-SVPD). Acetamide was among the problematic systems mentioned in Sec. IV A, where DFAs with insufficient exact exchange pop- ulated an incorrect orbital in the ( N+2)-electron reference. Sim- ilar occupation problems are observed for the def2 basis sets considered here, leading to strong basis set dependence of the exci- tation energy. However, this seems to be distinct from the unbound J. Chem. Phys. 153, 024110 (2020); doi: 10.1063/5.0003985 153, 024110-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp orbital problem and is instead related to the limited active space in hh-TDA. Based on this assessment and the promising results obtained in this work, we therefore recommend the use of hh-TDA in com- bination with asymptotically correct range-separated hybrid DFAs and double- and triple-zeta basis sets without diffuse basis functions. The latter can be added if the excited states to be probed by hh-TDA are actually of Rydberg character (cf. results for water). In the future, the unbound orbital issue could be addressed by using an N-electron reference in the orbital generation. Such an approach was already applied in one of the early works on pp-RPA.49We plan to inves- tigate this issue in order to facilitate the routine use of hh-TDA in geometry optimizations and nonadiabatic dynamics simulations. V. CONCLUSIONS We have shown that the hole–hole Tamm–Dancoff approxima- tion ( hh-TDA) to the particle–particle random-phase approxima- tion ( pp-RPA) represents an efficient DFT-based electronic structure scheme for the electronic ground and low-lying excited states. The method shares similarities with the previously presented particle– particle ( pp-TDA) approach. We find some advantages of the hh- TDA scheme, particularly in its ability to describe different low- lying excitation types ( nπ∗,σπ∗, andππ∗). Starting from an ( N+2)- electron reference determinant, the ground and the low-lying excited states of the N-electron target system are obtained by a double annihilation of two electrons. Since the ground and excited states are obtained from the same eigenvalue problem, static corre- lation cases can be handled. Dynamic correlation is included in the orbitals by virtue of the density functional. We have also introduced an alternative choice for the response kernel that differs from the functional-independent Hartree–Fock-type kernel employed in the previous works on pp-RPA and pp-TDA.54Our choice is functional- dependent and in line with the response kernels appearing in particle–hole linear response theory. To avoid SCF convergence problems of the ( N+2)-electron reference, the method should be combined with range-separated density functionals that have 100% asymptotic Fock exchange. We recommend avoidance of diffuse atom-centered basis sets since these can lead to continuum-like orbitals in the ( N+2)-electron reference. We then do not observe any SCF convergence prob- lems and find that the hh-TDA can describe low-lying excited states reasonably well at a computational complexity comparable to a configuration interaction singles (CIS) calculation. As with any density functional approximation-based method, case-specific functional assessment is recommended. The range-separation parameter and amount of short-range Fock exchange can be viewed as parameters to optimize the method performance for a specific molecule (as is often done by variation of the active space in CAS methods). On the other hand, the main limitation of the hh-TDA is the restricted “virtual” orbital space, which formally consists exclusively of the lowest unoccupied orbital (with respect to the N-electron system). This precludes its application to systems with degenerate LUMOs, such as benzene. In spite of this shortcoming, the hh-TDA method is well-suited for the calculation of ground and low-lying excited states for many organic molecules. It offers an inexpensivealternative to the existing and widely used CASSCF methods, with the advantage that dynamic correlation effects are included at the orbital level by means of the used density functional. The method is implemented in the GPU-accelerated electronic structure code TeraChem. Future directions will include the com- putation of nonadiabatic couplings and testing the method in nona- diabatic dynamics simulations. In this context, it will be important to address the problem of potentially occupying unbound orbitals in the ( N+2)-electron SCF calculation. Generating orbitals for an N- electron reference with fractional orbital occupation and using them in an ( N+2)-electron-type hh-TDA procedure could remove this issue entirely, while guaranteeing stability in nonadiabatic dynamics simulations. This will be explored in the future work. SUPPLEMENTAL MATERIAL See the supplementary material for a zip-file containing hh- TDA optimized geometries of thymine (Fig. 6) and malonaldehyde (Fig. 7) and a pdf file containing detailed results on the DFA assess- ment (Sec. IV A) and the pp/hh-TDA benchmarks (Sec. IV B) and a potential curve of ethylene (at 90○torsion). ACKNOWLEDGMENTS This work was supported by the AMOS program of the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Chemical Sciences, and Biosciences Division and by the Department of Energy, Laboratory Directed Research and Development program at the SLAC National Accelerator Laboratory under Contract No. DE-AC02-76SF00515. T.J.M. is a co-founder of PetaChem, LLC. C.B. thanks the German National Academy of Sciences Leopoldina for support through the Leopoldina Fellowship Program (Project No. LPDS 2018-09). The authors thank Deniz Tuna for helpful discussions and for early testing of the method. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1A. Dreuw and M. Head-Gordon, “Single-reference ab initio methods for the calculation of excited states of large molecules,” Chem. Rev. 105, 4009–4037 (2005). 2B. G. Levine, C. Ko, J. Quenneville, and T. J. 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5.0007440.pdf

J. Appl. Phys. 128, 035108 (2020); https://doi.org/10.1063/5.0007440 128, 035108 © 2020 Author(s).An anomalously high Seebeck coefficient and power factor in ultrathin Bi2Te3 film: Spin–orbit interaction Cite as: J. Appl. Phys. 128, 035108 (2020); https://doi.org/10.1063/5.0007440 Submitted: 12 March 2020 . Accepted: 03 July 2020 . Published Online: 20 July 2020 Mujeeb Ahmad , Khushboo Agarwal , and B. R. Mehta ARTICLES YOU MAY BE INTERESTED IN Intermediate-band-assisted near-field thermophotovoltaic devices with InAs, GaSb, and Si based absorbers Journal of Applied Physics 128, 035105 (2020); https://doi.org/10.1063/5.0010965 Anomalous Nernst effect in ferromagnetic Mn 5Ge3Cx thin films on insulating sapphire Journal of Applied Physics 128, 033905 (2020); https://doi.org/10.1063/5.0014815 Spin–orbit torque based physical unclonable function Journal of Applied Physics 128, 033904 (2020); https://doi.org/10.1063/5.0013408An anomalously high Seebeck coefficient and power factor in ultrathin Bi 2Te3film: Spin –orbit interaction Cite as: J. Appl. Phys. 128, 035108 (2020); doi: 10.1063/5.0007440 View Online Export Citation CrossMar k Submitted: 12 March 2020 · Accepted: 3 July 2020 · Published Online: 20 July 2020 Mujeeb Ahmad, Khushboo Agarwal, and B. R. Mehtaa) AFFILIATIONS Thin Film Laboratory, Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India a)Author to whom correspondence should be addressed: brmehta@physics.iitd.ac.in ABSTRACT The present study reports a strong thickness-dependence and anomalously large enhancement in the values of the Seebeck coefficient and electrical conductivity in Bi 2Te3films at ultralow thickness. An opposite sign of the Hall coefficient (negative) and Seebeck coefficient (posi- tive) is observed in an ultrathin Bi 2Te3film (65 nm) as compared to the normally observed identical sign in the case of Bi 2Te3thin films (520 nm). A simultaneous enhancement in the values of electrical conductivity and the Seebeck coefficient results in a giant enhancement in the value of power factor from 1.86 mW/m K2to 18.0 mW/m K2at 416 K, with a reduction in thickness. X-ray photoelectron spectro- scopy investigation reveals the absence of any significant change in stoichiometry and chemical bonding upon reduction of thickness.Magnetoresistance vs magnetic field data show a sharp dip at the lower magnetic field values, indicating a weak antilocalization effect in thecase of the ultrathin film sample suggesting the role of strong spin –orbit coupling toward the carrier filtering effect resulting in enhance- ment of thermoelectric properties. Observation of the large Seebeck coefficient and the power factor at lower thickness values and its rela- tionship with spin –orbit coupling is an important result, both for practical applications and for better understanding of the thermoelectric properties. Published under license by AIP Publishing. https://doi.org/10.1063/5.0007440 INTRODUCTION Thermoelectric materials have drawn much interest over the last century as an effective technique to convert waste heat energy into electricity and vice versa. 1The thermoelectric efficiency of a material is defined by a figure of merit zT = ( σS2/κe+κl)⋅T, where S is the Seebeck coefficient, σis the electrical conductivity, κeis the electronic thermal conductivity, κlis the lattice thermal conductiv- ity, and T is the temperature.2,3To achieve better zT values, high electrical conductivity, large Seebeck coefficient, and lower thermalconductivity values are required. However, it is challenging todecouple these irreconcilable parameters due to the conflicting cor-relation of the transport parameters ( σ, S, and κ), resulting in diffi- culty in the enhancement of zT values. Various methods and concepts have been applied to enhance thermoelectric perfor-mance, such as doping, nanocomposite, band engineering, alloying,and energy carrier filtering. 4–9Typically, materials with high values of electrical conductivity demonstrate high values of thermalconductivity. A material with a large Seebeck coefficient is desirable for enhancing the value of the figure of merit (zT). The topological insulators offer not only high Seebeck coefficient values but alsotune the electron and phonon transport in the materials. Becausethe electron transport is immune to the backscattering from thenonmagnetic impurity and defects, the phonons transport is ham- pered from these impurities and defects. 10Topological insulators show the insulating bulk gap and gapless surface states, simultane-ously. The surface states create band inversion, spin –orbit coupling, and bulk bands of opposite parities and are topologically protectedby time-reversal symmetry of the materials. 11,12Bulk and surface states contribute to the charge transport in parallel and have a unique effect on electronic properties quite distinct from that ofconventional materials. 13,14The material properties of the topologi- cal insulators and thermoelectric materials are quite similar interms of the narrow bandgap and heavy elements. Therefore, a large number of topological materials are excellent thermoelectric candidates, such as Sb 2Te3,B i2Te3, SnTe, and Bi 2Se3, etc.15–19Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 035108 (2020); doi: 10.1063/5.0007440 128, 035108-1 Published under license by AIP Publishing.In the present study, the thermoelectric properties of Bi 2Te3 samples in the temperature range of 300 –484 K were studied. Anomalous and large Seebeck coefficient values (positive) wereobserved in the case of ultrathin films, indicating that the hole con-tribution to the Seebeck voltage is dominant in comparison to elec-tron contribution. The magnetoresistance (MR) measurements were performed to understand this anomaly in the electronic prop- erties of ultrathin Bi 2Te3samples. Furthermore, the presence of strong spin –orbit coupling in the case of ultra-thin Bi 2Te3samples was analyzed by the weak antilocalization (WAL) effect describedusing the Hikamie –Larkine –Nagaoka (HLN) model. 20The enhancement of thermoelectric properties was understood in terms of spin –orbit coupling. EXPERIMENTAL SECTION Bi2Te3thin films have been RF sputter deposited onto a silicon substrate with a 300 nm thick SiO 2surface layer. The working pressure and base pressure of the growth chamber fordeposition of all the samples were 7.5 × 10 −3mbar and 7.0 × 10−6mbar, respectively. To avoid contamination, the sub- strates were cleaned using the RCA method. The deposition of the thin film was carried out at an RF power of 20 W and a sub- strate temperature of 150 °C. Before deposition, the target waspre-sputtered for 20 min for reducing any effect of surface con-tamination. The deposition rate of Bi 2Te3was maintained at 13 nm/min, which was confirmed from optical profiler and x-ray reflectivity (XRR) measurements. For uniform deposition, a rotating substrate assembly was employed in the sputteringsystem. In this investigation, two thin film samples with thick-nesses of 65 nm and 520 nm prepared by the magneto RF sput-tering method have been studied. The 65 nm thin sample will be termed as ultrathin and topological (UTT), while the 520 nm thick sample will be termed as thin film and normal (TFN).X-ray diffractograms (XRDs) were carried out to study the orien-tation and phase of the deposited thin film, and XRR was used to measure the thickness and roughness by using Philips X ’Pert Pro-PW 3040. The structural and phonon modes of Bi 2Te3thin films were investigated by Raman measurements using aRenishaw inVia confocal Raman microscope at 532 nm laserwavelength with 1800 lines per mm grating. The surface topogra- phy of the deposited Bi 2Te3thin film samples was studied using a dual-beam field emission scanning electron microscope (FESEM)(Model No. Quanta 3D FEG, FEI). The magnetoresistance (MR)measurements were carried out using a physical property mea-surement system (Quantum Design), and electrical contacts were prepared using silver epoxy. The perpendicular magnetic field was applied to the sample in the range of +7 T to −7 T. Linseis LSR-3 simultaneously measured the Seebeck coefficient and elec-trical resistivity in the temperature range of 315 –480 K with ±1 ° C temperature accuracy. The electrical conductivity value of both the Bi 2Te3thin film samples was measured using the van der Pauw method. The morphology and the grain size were investi-gated using atomic force microscopy (Bruker, Dimensionsystem). A silicon cantilever was used for AFM measurements, which has a radius of curvature of 30 nm and a resonance fre- quency of 320 kHz, and the scan speed was kept at 0.6 Hz duringmeasurements. X-ray photoelectron spectroscopy (XPS) measure- ments were carried out using the PHI Quantera SXM system. The monochromatic x-ray excitation Al K αline (1486.6 eV) was used as an incident source. The detector collects photoelectronsfrom the small portion of the sample. The acquired spectra werecarbon corrected using amorphous carbon (C 1s peak at 284.6 eV), which is existing in the samples, and data analysis was done using XPSPEAK software. Survey spectra were obtained inthe binding energies ranging from 1 to 1200 eV. The backgroundof the peaks was subtracted before fitting the peaks usingTougaard ’sm e t h o d . 21Gaussian –Lorentzian functions software combination was carried out for fitting the peaks.22 RESULT AND DISCUSSION The thickness measurements of the samples were carried out using the x-ray reflectivity (XRR) technique. The XRR fringes confirmed the thickness of UTT samples to be ∼65 nm (with a roughness value of 2 nm), indicating relatively smoothmorphology, especially in the case of an ultrathin film sample asshown in Fig. 1(a) . The TFN sample thickness was found to be ∼520 nm, as determined by the optical profiler. The XRD spectra of both the samples show polycrystalline nature, and the observed features correspond to (006), (015), (1010), (1110),(0015), (205), (0210), and (0021) planes that match well with therhombohedral phase of Bi 2Te3(JCPDS CAS No. 15-0863),23,24 as shown in Fig. 1(b) . The intensity corresponding to the (015) plane is strong, as compared with the other peaks in the TFN sample. However, the intensity of the (006) plane is strong ascompared with the other peaks in the UTT sample. The XRDpeak intensities for the TFN sample were much higher as com-pared to the UTT sample. The lattice parameters for both Bi 2Te3 thin film samples were calculate d using the well-known Millar equation. The calculated lattice parameters (a = b = 4.385 Å andc = 30.42 Å) for the TFN sample and (a = b = 4.358 Å andc = 30.42 Å) for the UTT sample are comparable with the JCPDS data and earlier reports. 25–27Goto et al.28have observed that the XRD peak positions were shifted toward lower angles. They havealso reported the appearance of some additional peaks as a typeof carrier changes from n to p-type in Bi 2Te3samples. However, in the present study, no such shifting in peak positions was observed in the XRD spectra of both thin film samples. The ran- domly oriented grain size of deposited Bi 2Te3samples (TFN and UTT) was estimated using the Scherrer formula.29,30The grain size of the TFN and UTT samples was found to be 21.39 nm and9.33 nm, respectively. Raman measurements can provide information on the struc- tural changes that result due to thickness, including the appearanceof the impurity phase at the molecular level and surface phononmodes. The Bi 2Te3structure has a total of 15 phonon modes (12 optical and 3 acoustic modes) near to the center of the Brillion zone. The 12 optical branches are described as 2A2u+2 E u+2 A 1g+2 E gsymmetry. E gand A 1gmodes are Raman active modes, while E uand A umodes are IR active modes.31,32The Raman spectrum of UTT and TFN samples are shown in Fig. 1(c) . In the TFN samples, three Raman peaks were observed, which cor- respond to the A 1g,E2 g, and A2 1gmodes and are positioned at 68,Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 035108 (2020); doi: 10.1063/5.0007440 128, 035108-2 Published under license by AIP Publishing.100, and 131 cm−1, respectively. These Raman peak positions match well with the previously reported peak positions for bulkcrystalline Bi 2Te3.33,34However, an additional peak appeared at ∼89 cm−1in the Raman spectrum, indicating Te –Te interaction. In the E gmodes, the atoms have in-plane vibration, while in the A g mode, the atoms vibrate perpendicular to the basal plane. TheRaman peak intensities of the E gmodes are higher as compared to the A gmodes, which also matches with previous reports for bulk Bi2Te3.35Raman spectra of the UTT sample also possess the same Raman active modes as the TFN sample, except an additional peak at 119 cm−1. This additional peak corresponds to the A 1umode and is observed due to the low thickness. In the bulk Bi 2Te3, the A1umode is only IR active, but in ultrathin Bi 2Te3films, this mode is Raman active due to symmetry breaking.36,37Cheng and Ren34 reported that due to spin –orbit coupling, a strong inharmonic potential exists around Bi atoms in Bi 2Te3ultrathin films, which may be responsible for the symmetry breaking. Previous reportsobserved that the intensity of the A 1umode (out of plane) and the intensity ratio of the A2 1gmode and E2 gmode increases with the decrease in the thickness of the Bi 2Te3thin films.31,36,38The Raman spectra confirm that the UTT sample is crystalline, and theresults are consistent with the reported results for the ultrathinsample. The surface morphology of both the samples was analyzedusing AFM and FESEM measurements. The AFM measurements were performed in the tapping mode having a nm resolution, and the AFM images were analyzed using WSxM software. The surfacetopography images of the UTT and TFN samples show that theUTT sample has lower values of roughness as compared to theTFN sample, as shown in Figs. 2(a) and 2(b). The root mean square roughness (R rms) value of the UTT and TFN samples is cal- culated to be 3.5 nm and 32.8 nm, respectively. The grain size ofthe prepared samples has been calculated by drawing a line on the2D AFM micrograph. The average grain size of UTT and TFNsamples was found to be 25 –35 nm and 80 –120 nm, respectively. FESEM measurements were carried out for further investigating the microstructure and surface morphology of the prepared samples.Figures 2(c) and2(d) show the uniform surface morphology of the UTT and TFN samples. Similarly, the grain size of the prepared samples was also measured from the FESEM images by drawing a line on the microstructure, and it was found that the grain size ofthe UTT sample is about 15 –30 nm and that of the TFN sample is about 100 –130 nm. It is evident from the AFM and FESEM mea- surements that the grain size of the Bi 2Te3increases with an increase in the thickness of the film resulting in a higher R rmsvalue for the TFN sample as compared to the UTT sample.39The value of the grain size calculated from AFM and FESEM measurements are similar. However, the values of the grain size calculated fromXRD are different from those observed from AFM and FESEMmeasurements. This variation may be attributed to the fact that inthe case of XRD measurement, the information is gathered from the entire sample, whereas FESEM and AFM being morphological characterizations provide information from the crystallites present FIG. 1. (a) X-ray reflectivity spectra of UTT sample, (b) x-ray diffractogram, and (c) Raman spectra of UTT and TFN sample. FIG. 2. AFM topography images of (a) UTT and (b) TFN samples; FESEM images of (c) UTT and (d) TFN samples.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 035108 (2020); doi: 10.1063/5.0007440 128, 035108-3 Published under license by AIP Publishing.at the surface only. Raj et al.40also observed that the grain size measured from XRD is different from the AFM measurement due to differences in the measurement. Several studies have predictedthat the grain size of the sample plays a crucial role in the thermo-electric properties. 41,42As the grain size decreases, the number of grain boundaries subsequently increases in the sample. These grain boundaries act as a barrier that allows high energy charge carriers and restricts low energy charge carriers.43,44Thereby, the average energy of the charge carriers is increased, leading to enhancementin the value of the Seebeck coefficient. Due to restriction on themotion of the low energy charge carrier at the grain boundary, a decrease in the value of carrier concentration is accompanied by increased carrier mobility values. 45,46Moreover, the grain boundary resistance also decreases due to the reduction in the grain size,thereby resulting in increased values of electrical conductivity. Theearlier reported literature shows that with an increase in grain size, segregation of the impurity at the grain boundary increases, which increases the grain boundary resistance. 47Chiang et al.48reported that the grain boundary resistance is 103times lower at nanocrystal- line sizes. The effect of thickness of Bi 2Te3films on thermoelectric prop- erties were investigated in detail. Figure 3 shows the results of ther- moelectric measurements for both the samples in the temperaturerange 320 –484 K. Figure 3(a) shows that the electrical conductivity values for both the samples increase with an increase in the tem- perature up to 455 K; after 455 K, the electrical conductivity values were observed to saturate, indicating a heavily doped semiconduct-ing behavior. 49The value of electrical conductivity observed for the UTT sample ( ∼558.94 S/cm at 455 K) was higher as compared to the TFN sample ( ∼488.78 S/cm at 455 K). Figure 3(b) shows the temperature dependence Seebeck coefficient of the UTT and TFN samples. At higher temperatures, the Seebeck coefficient valuedecreases due to the bipolar condition process and intrinsic excita- tion.50The value of the Seebeck coefficient for the UTT and TFN samples at 416 K is +569.86 μV/K and 206.57 μV/K, respectively. It is important to note that the UTT sample shows a positive value ofthe Seebeck coefficient in contrast to the TFN sample, which showsa negative value of the Seebeck coefficient. The temperature- dependent power factor for both the samples is shown in Fig. 3(c) in the temperature range of 300 –484 K. The power factor value in the UTT sample (18.0 mW/m K 2) is one order higher than in the TFN sample (1.86 mW/m K2) at the 416 K. This remarkable enhancement in the power factor value of the UTT sample is due to high electrical conductivity and Seebeck coefficient, indicating a transformative change in the electronic nature on going to ultralowthickness as compared to the previously reported values. 51,52The electrical conductivity and Seebeck coefficient values are plotted asa function of the thickness of the Bi 2Te3film in Fig. 3(d) , which shows that the electrical conductivity value increases with a decrease in the thickness of the Bi 2Te3thin film. The highest elec- trical conductivity value was observed for the 65 nm Bi 2Te3thin film and the lowest at 520 nm Bi 2Te3thin film sample. Seebeck measurements showed that the Bi 2Te3thin film with a thickness greater than 130 nm has a negative Seebeck coefficient value. Upon further decrease in the thickness of Bi 2Te3thin film, positive Seebeck coefficient values were observed [as shown in Fig. 3(d) ]. The value of the Seebeck coefficient drastically increases with a decrease in the thickness of the Bi 2Te3thin film. There exist few experimental and theoretical reports on the increase in the Seebeckcoefficient with a reduction in the thickness. Yang et al. 53reported a high and positive Seebeck coefficient (+475 μV/K) with a reduc- tion in the thickness of Bi 2Te3thin film due to lowering in Fermi level position. Xu et al.54theoretically reported that the ultrathin Bi2Te3film might have a large and positive Seebeck coefficient FIG. 3. Thermoelectric properties of UTT and TFN sample in the tempera- ture range 320 –484 K: (a) Electrical conductivity ( σ), (b) Seebeck coeffi- cient (S), (c) Power factor, and (d) the variation of electrical conductivity and Seebeck coefficient with respect to thethickness of the Bi 2Te3thin film.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 035108 (2020); doi: 10.1063/5.0007440 128, 035108-4 Published under license by AIP Publishing.value due to the topological surface state effect. It was also theoreti- cally predicted by Hicks and Dresselhaus55that the reduction in the thickness of the film might result in an enhancement in theSeebeck coefficient value due to the quantum confinement effects. It is important to discuss further the origin of anomalous and large Seebeck coefficient values in the ultrathin thin film. There can be two possible reasons for the anomalous nature of the Seebeck coefficient in the UTT sample: one may be due to a significantchange in stoichiometry, 28,56and another may be attributed to the role of spin –orbit coupling.54,57XPS measurements were performed to estimate the stoichiometry of the UTT and TFN samples. The XPS spectra of both the thin film samples were fitted well with the mixing of Gaussian –Lorentzian components using XPSPEAK soft- ware.40The binding energy (BE) 284.6 eV was adopted for uninten- tionally absorbed carbon (C 1s) species and is used to calibrate thedata. 22,58The XPS survey spectrum of both the samples was observed for the binding energy range from 0 to 1100 eV, as shown inFigs. 4(a) and 4(d). The survey spectra of both samples show that Bi, Te, C, and O elements coexist, and no other impuritieswere present in the samples. Figures 4(b) and4(c) represented the scan of Bi 4f and Te 3d core level for the UTT sample, respectively. Figure 4(b) shows the high-resolution scan of Bi 4f doublet peakscentered at 157.2 eV and 162.5 eV, corresponding to Bi +34f7/2and Bi+34f5/2, respectively, further confirming the formation of the Bi2Te3phase .The peaks centered at 158.5 eV and 164.0 eV, corre- sponding to the oxide peaks, possibly present on the surface.Furthermore, the Te 3d peak contains three sub-peaks. Two ofthem represent the Bi 2Te3phase, centered at 572.0 eV and 582.3 eV, corresponding to Te−23d5/2and Te−23d5/2, respectively, while the third peak, centered at 574 eV, corresponds to Te+4 (TeO 2) due to the surface oxide. The Bi 4f and Te 3d XPS peaks were observed at the same binding energies as were observed in theearlier reports, 28,58–60further confirming the Bi 2Te3phase forma- tion in the UTT sample. Similarly, Figs. 4(e) and 4(f) show the high-resolution scan of Bi 4f and Te 3d for the TFN sample. In theTFN sample, Bi 4f orbital possesses six characteristic XPS peaks, asshown in Fig. 4(e) . The XPS peaks at 157.0 eV and 162.3 eV corre- spond to Bi +34f7/2and Bi+34f5/2, indicating the Bi 2Te3phase for- mation. The other four peaks at the higher binding energy represent the surface oxide in the Bi 2Te3thin film. The XPS peaks centered at 158.4 eV and 163.7 eV, corresponds to the Bi+5in the Bi2O3phase. Moreover, the position of the peak at 158.2 eV and 163.4 eV represent BiOx phase formation due to the surface oxide. Figure 4(f) shows the Te 3d scan, which consists of two major FIG. 4. The XPS analysis spectra of the deposited Bi 2Te3thin film: (a) represents the survey spectrum for the UTT samples. High resolution XPS spectra for (b) Bi 4f and (c) T e 3d from the UTT sample. Similarly, (d) represents the survey spectrum for the TFN sample and high-resolution XPS spectra for (e) Bi 4f and (f ) T e 3d from the TFN sample.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 035108 (2020); doi: 10.1063/5.0007440 128, 035108-5 Published under license by AIP Publishing.peaks and one shoulder peak. The two high-intensity XPS peaks positioned at 571.8 eV and 582.1 eV correspond to Te−23d7/2and Te−23d5/2, respectively. The lower intensity peak centered at 574.5 eV represents Te+4in the TeO 2phase formation.26,61The peak positions of bismuth and tellurium in the TFN sample areobserved in the same position as in the UTT sample and agree well with the previously reported values. A qualitative study of Bi and Te atomic concentrations was carried using the expression 62 %xa¼AaχaPn i¼1Ai/χi, where A i(i = 1, 2, 3, …, n) represents the area under the element andχiis the sensitivity factor of the elements. The sensitivity factor of the Bi and Te is 9.14 and 9.51, respectively.62The calculated atomic percentage ratios Bi:Te in the UTT and TFN samples are 1:1.38 and 1:1.33, respectively, which are very close to the earlier reported values. Goto et al.28have reported p-type conduction with an excess of Te in Bi 2Te3, whereas n-type conduction was observed to be Bi rich Bi 2Te3, and the Bi:Te ratio was observed to be 1:0.8 for n-type and 1:10 for p-type conduction. However, no such sig- nificant change was observed in the Bi:Te ratio for the UTT (1:1.38) and TFN (1:1.33) samples. Thus, in the present case, spin – orbit coupling is responsible for the anomalously high Seebeckcoefficient value in the UTT sample and will be discussed further. To understand the difference in the carrier type for UTT and TFN samples, transport properties of the two samples were mea- sured using the Hall effect measurements. It was observed that theUTT sample has very high mobility (330.95 cm 2V−1S−1) and lower carrier concentration (7.76 × 1018cm−3) as compared to the TFN sample (2.40 cm2V−1S−1and 9.35 × 1020cm−3, respectively) at 320 K as mentioned in Table I . The enhancement is the Hall mobility, and reduction in the carrier concentration may be due tothe small grain size as compared to the TFN sample leading to thecarrier filtering effect. 63The electrical conductivity and Seebeck coefficient of the UTT sample observed at 320 K is 410.91 S/cm and 400.19 μV/K, respectively. On the other hand, in the case of the TFN sample, the electrical conductivity and Seebeck coefficientvalues (at 320 K) are 360.52 S/cm and −175.51 μV/K, respectively. The higher mobility and lower carrier concentration in the case of the UTT sample are playing an essential role in enhancing the Seebeck coefficient values along with electrical conductivityvalues. 45,64,65A negative value of the Hall coefficient was observed for both the samples, indicating electron dominated conduction.Therefore, a positive value of the Seebeck coefficient indicates a sig- nificant change in the electron and phonon transport at ultralow thickness. For further understanding of the observed inherent changes in the electronic and thermoelectric nature, the magnetoresistance(MR) of the UTT and TFN samples were investigated at varying magnetic field and temperature values. The magnetic field (B) values were varied from −7 T to +7 T with field applied perpendic- ular to the samples. The MR is given by (R B−R0) × 100/R 0, where RBis resistance in the presence of the magnetic field and R 0is resistance at zero magnetic fields.66The resistance of the UTT sample sharply decreases when the magnetic field value goes to zero and forms a cusp of MR at the lower magnetic field values,which indicates the weak antilocalization (WAL) effect in the UTTsample as shown in Fig. 5(a) . 67,68The WAL effect is the quantum interface phenomena of the two-phase coherence of the electron waves due to high spin –orbit coupling. Therefore, the WAL effect is the signature of the strong spin –orbit coupling, which reduces the backscattering at the low magnetic field values due to the time-reversal symmetry. Thus, the value of the resistance sharplydecreases at low magnetic field values. 69,70On the other hand, the resistance of the TFN sample does not decrease sharply when the magnetic field tends to zero, indicating that the weak anti-localization effect is less dominant in TFN samples, as shown inFig. 5(b) . 71–73 For further confirming the WAL effect in the deposited samples, the value of magnetoconductance [ ΔσXX=σ(B)−σ(0)] can be fitted at the low magnetic field (B) using the Hikamie – Larkine –Nagaoka (HLN) model for WAL effect and is defined as20,70,74 ΔσXX¼αe2 2hπ2lnBw B/C0ψ1 2þBw B/C18/C19 /C20/C21 , (1) where B w¼/C22h (4eL2 w)with L was the phase coherence length and ψis the digamma function. α=−0.5 is an ideal value corresponding to a weak antilocalization effect, and α=−1.0 is related to the weak localization (WL) effect.19,67,75Additionally, the temperature dependence of L walso represents the dimensionality of the system, e.g., for a 2D system L w∼T−1/2and for a 3D system L w∼T−3/4.76 To understand the charge transport nature in Bi 2Te3samples, the HLN model has been used to fit experimental data of magnetocon-ductance for UTT and TFN samples and the corresponding αand L wvalues were calculated using Eq. (1). The magnetoconductance value was observed to vary with the magnetic field in the range of −0.5 T to 0.5 T at 2 K, as shown in Figs. 5(c) and 5(d). The observed data of magnetoconductance with the magnetic field (B),for the UTT sample, were fitted in Eq. (1)up to 0.5 T, where the weak antilocalization effect is observed to dominate. The observedvalues of α=−0.515 and L w= 120 nm at 2 K confirms the weak antilocalization effect in the UTT sample, as shown in Fig. 4(c) . The L wvalue decreases from 120 nm to 48.5 nm with an increase in the temperature from 2 K to 8 K, which is larger than the thicknessof the UTT sample, indicating 2D WAL characteristics. 77 TABLE I. The value of carrier concentration, electrical conductivity, Seebeck coefficient, Hall mobility, and power factor at 320 K for both the samples. Sample name Thickness (nm) Concentration (cm−3) Hall mobility (cm V−1S−1) σ(S/cm) S ( μV/K) σS2(mW/m K2) UTT 65 7.76 × 1018330.95 410.91 400.19 6.58 TFN 520 9.35 × 10202.40 360.52 −175.51 1.11Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 035108 (2020); doi: 10.1063/5.0007440 128, 035108-6 Published under license by AIP Publishing.Furthermore, the temperature-dependent power law (L w∼T−0.43) confirms the 2D WAL characteristic in the case of sample UTT32 (0.43), which is very close to the ideal value (0.5) in the tempera-ture range 2 K –8K . 78The value of αis independent of the temper- ature in the range of 2 K –8 K. In the case of the TFN sample, the weak antilocalization effect is present only at the 2 K temperaturewith α=−2.1 and L w= 75 nm, which indicates the weak localiza- tion (WL) effect, as shown in Fig. 5(d) . The WAL effect in the UTT sample was not observable with an increase in the tempera-ture. This is mainly due to the domination of electron –phonon scattering, and also the contribution of the impurity states increas- ing, and the MR curve exhibits a combined response from elec- tron–electron as well as electron –phonon interactions. 79MR measurements do not provide any evidence of topological surfacestates, but they suggest a spin –orbit interaction and electron – phonon scattering in ultrathin film samples, which may play an important role in governing the thermoelectric properties. 78 Angle-resolved photoemission spectroscopy (ARPES) is an impor- tant technique for confirming the presence of surface states withinthe bandgap. However, ARPES techniques are applicable to singlecrystals only and cannot be applied in the present case. 18,80 The weak antilocalization effect in the UTT sample is a signa- ture of strong spin –orbit coupling.17,78It is important to under- stand the role of the weak antilocalization effect in terms ofthermoelectric properties. In the WAL effect, the electrons are pro-tected from backscattering due to strong spin –orbit coupling result- ing in the enhancement of the electrical conductivity of the materials. 68,81Therefore, the higher electrical conductivity in the case of the UTT sample as compared to the TFN sample can beattributed to the dominance of the WAL effect in the case of the UTT sample. The Seebeck coefficient of the samples is the observed voltage induced by the applied temperature gradient. Thetemperature gradient creates electrons and phonons to flow, which leads to the two components contributing to the Seebeck coeffi- cient: one is termed as diffusion Seebeck coefficient S d,and the other is termed as the phonon drag Seebeck coefficient S g.78While Sdis associated only with the charge carrier diffusion, S gresults from the flowing phonons that drag electrons through electron – phonon interactions. The phonon drag effect is dominant at a low carrier concentration value and is mostly suppressed in highercarrier concentrations. According to Hall measurements, the UTTsample has a low carrier concentration and high mobility as com-pared to the TFN sample. Therefore, the phonon drag effect is strongly dominant in the UTT sample and enhances the Seebeck coefficient values. Wang et al. 82have demonstrated the correlation of phonon drag with the highest peak value in the Seebeck coeffi-cient. They have also claimed that the Seebeck coefficient values ofBi 2Te3can have large values for very thin films but decrease very fast upon increasing the film thickness due to electron –phonon coupling. Therefore, the significant contribution of S dand S gin the UTT sample results in remarkable enhancement in the totalSeebeck coefficient value. The spin –orbit coupling tunes the density of states (DOS) and directly manipulates the charge carrier availability near the Fermi levels. 14,83By applying the temperature gradient to the sample, the charge carriers present, below andabove the Fermi level, respond and contribute oppositely to theSeebeck coefficient value. If the charge carrier below the Fermi level (i.e., holes) contributed more as compared to the charge carrier above the Fermi level (i.e., electrons), then the sign of theSeebeck coefficient is found to be positive; otherwise, it is negative.Xuet al. 54observed that the scattering time of the charge carriers present below the Fermi level is large as compared to the carriers present above the Fermi level due to the strong spin –orbit coupling. This difference in the scattering time of the charge carriers led toan abrupt change in the density of states. 54,84Therefore, the polar- ity of the Seebeck coefficient is positive in the UTT sample due to achange in the density of states. 1,54Tayari et al.85have also reported the origin of the large Seebeck coefficient value in a p-type 2D SnSe due to the anisotropy behavior of the density of states and dif-ferent scattering times. The spin –orbit interaction and phonon – electron interaction are the key points to understand electronic andthermoelectric properties in ultrathin film Bi 2Te3samples and thus are found to be very useful in enhancing the thermoelectric perfor- mance. The origin of the anomalous Seebeck effect and an increasein the thermoelectric properties may be investigated in ultrathinfilms and epitaxial layers of other topological insulator materials. CONCLUSIONS In the present work, the role of thickness toward enhancing the Seebeck coefficient and electrical conductivity is observed inBi 2Te3films. Anomalous and large Seebeck coefficient with high electrical conductivity values is observed in the ultrathin film sample. On the other hand, the thicker sample shows a low Seebeck coefficient with negative signs and lower electrical conduc-tivity values. The XPS measurements confirm that the large andanomalous Seebeck coefficient values in the ultrathin film sample are not due to the change in stoichiometry. Moreover, they are related to the strong spin –orbit interaction and electron –phonon FIG. 5. Normalized MR with varying magnetic field (B) in the temperature range 2K–8 K: (a) UTT sample and (b) TFN sample. HLN fitting (solid black line) at 2 K data around low magnetic field region: (c) UTT sample and (d) TFN sample.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 035108 (2020); doi: 10.1063/5.0007440 128, 035108-7 Published under license by AIP Publishing.scattering, which are confirmed from MR measurements. The MR results show that weak antilocalization effects dominate in the UTT sample in comparison to the TFN sample, and the valuesα=−0.515 and L w= 120 nm clearly confirm the strong spin –orbit coupling and electron –phonon scattering in ultrathin film samples. The enormous increase in the value of the power factor of the ultrathin film sample is essential from the application point of view and a basic understanding of the thermoelectric properties of topo-logical insulators. ACKNOWLEDGMENTS B. R. Mehta acknowledges support from the Schlumberger Chair Professorship and the project funded by DST [Project Nos. DST/NM/NS/2018/234(G) and INT/NOR/RCN/ICT/P-04/2018]. Mujeeb Ahmad acknowledges NRF IIT Delhi and the Departmentof Physics IIT Delhi for providing necessary facilities. 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5.0023343.pdf

J. Chem. Phys. 153, 104501 (2020); https://doi.org/10.1063/5.0023343 153, 104501 © 2020 Author(s).A soft chemistry approach to the synthesis of single crystalline and highly pure (NH4)CoF3 for optical and magnetic investigations Cite as: J. Chem. Phys. 153, 104501 (2020); https://doi.org/10.1063/5.0023343 Submitted: 28 July 2020 . Accepted: 20 August 2020 . Published Online: 08 September 2020 Stefanie Siebeneichler , Katharina V. Dorn , Volodymyr Smetana , Martin Valldor , and Anja- Verena Mudring COLLECTIONS Paper published as part of the special topic on Special Collection in Honor of Women in Chemical Physics and Physical Chemistry The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp A soft chemistry approach to the synthesis of single crystalline and highly pure (NH 4)CoF 3 for optical and magnetic investigations Cite as: J. Chem. Phys. 153, 104501 (2020); doi: 10.1063/5.0023343 Submitted: 28 July 2020 •Accepted: 20 August 2020 • Published Online: 8 September 2020 Stefanie Siebeneichler,1 Katharina V. Dorn,1 Volodymyr Smetana,1 Martin Valldor,2 and Anja-Verena Mudring1,a) AFFILIATIONS 1Department of Materials and Environmental Chemistry (MMK), Stockholm University, Svante Arrhenius väg 16 C, 10691 Stockholm, Sweden 2Centre for Materials Science and Nanotechnology (SMN), Department of Chemistry, University of Oslo, Postbox 1033, Blindern 0315, Oslo, Norway 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: anja-verena.mudring@mmk.su.se ABSTRACT A new ionothermal synthesis utilizing 1-alkyl-pyridinium hexafluorophosphates [C xPy][PF 6] (x= 2, 4, 6) led to the formation of highly crystalline single-phase ammonium cobalt trifluoride, (NH 4)CoF 3. Although ammonium transition-metal fluorides have been extensively studied with respect to their structural and magnetic properties, multiple aspects remain unclear. For that reason, the obtained (NH 4)CoF 3has been investigated over a broad temperature range by means of single-crystal and powder x-ray diffraction as well as magnetization and specific heat measurements. In addition, energy-dispersive x-ray and vibrational spectroscopy as well as thermal analysis measurements were undertaken. (NH 4)CoF 3crystallizes in the cubic perovskite structure and undergoes a structural distortion to a tetragonal phase at 127.7 K, which also is observable in the magnetic susceptibility measurements, which has not been observed before. A second magnetic phase transition occurring at 116.9 K is of second-order character. The bifurcation of the suscep- tibility curves indicates a canted antiferromagnetic ordering. At 2.5 K, susceptibility measurements point to a third phase change for (NH 4)CoF 3. ©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/10.1063/5.0023343 .,s I. INTRODUCTION The family of ammonium metal trifluorides with the formula (NH 4)MF3has been intensively studied for several decades.1–16The most popular synthetic approach for the (NH 4)MF3series ( M= Mg, Mn, Fe, Co, Ni, Cu, Zn, and Cd) was precipitation in methano- lic solution,1–3,5,7–9,11,14while syntheses in agar–agar gel,4,8,9,14 aqueous solution,12HF solution,15hydrothermal syntheses,11,13,14 and high temperature solid-state reactions10also were successful for selected systems. Structural characterization has mostly been performed with the aid of powder and single-crystal x-ray diffractionmethods revealing the cubic CaTiO 3-type structure4,9,13,15,17,18at room temperature for the majority of the systems. Up to now, the low temperature behavior of the (NH 4)MF3series, which varies depending on M, was much less investigated.5,10,13,14 In contrast to conventional solid-state or solvothermal synthe- ses, ionothermal reactions19with task specific ionic liquids (ILs)20 combine several advantages: The IL can serve as the reaction and crystallization medium, as the element source (hence, be the reac- tion partner), and as the mineralizer. Furthermore, as compared to classical solid-state and molten salt reactions, much lower heat- ing temperatures and thermally less stable starting materials can be J. Chem. Phys. 153, 104501 (2020); doi: 10.1063/5.0023343 153, 104501-1 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp used. Moreover, ionothermal approaches bear the advantage over conventional organic solvents and water that the chemical and physicochemical characteristics can be readily tuned through the choice of cations and anions.21,22The application of ILs has particu- larly been found advantageous in the syntheses of open-framework materials and modification of their composition and structural architectures as templating agents.23–35 Recently, we successfully applied task-specific ILs based on tetrafluoroborate and hexafluorophosphate anions that can act as the fluorine source in the synthesis, particularly for the metal fluorides and oxide nano-materials for energy-related applica- tions.36–38Here, we report the ionothermal syntheses of (NH 4)CoF 3 using the 1-alkyl-pyridinium hexafluorophosphate ILs [C 2Py][PF 6], [C4Py][PF 6], and [C 6Py][PF 6], C 2= ethyl, C 4= butyl and C 6= hexyl, which act not only as solvents but also most importantly as fur- ther fluoride sources, hence mineralizer—allowing us to obtain high quality single-crystals under mild conditions. II. MATERIALS AND METHODS A. Synthesis Single-phase and highly crystalline samples of (NH 4)CoF 3 were prepared under ionothermal conditions using the ILs 1-ethyl-, 1-butyl-, and 1-hexyl-pyridinium hexafluorophosphate, [C xPy][PF 6] (x= 2, 4, and 6), respectively, as, both, solvent [ Tm(ILs) = 323 K–379 K) and reactant. A ground mixture of the par- ticular IL ([C 2Py][PF 6], 0.253 g (1 mmol); [C 4Py][PF 6], 0.282 g (1 mmol); [C 6Py][PF 6], 0.309 g (1 mmol); 99%, IOLITEC Ionic Liq- uid Technologies GmbH, Heilbronn, Germany), cobalt(II) acetate tetrahydrate ([Co(CH 3COO) 2⋅4 H 2O], 0.498 g (2 mmol), 98%, ThermoFisher GmbH, Kandel, Germany), and ammonium flu- oride [NH 4F, 0.556 g (15 mmol), >98% Sigma-Aldrich, Stein- heim, Germany] was filled into a 15 ml Teflon vessel, which was closed by its screwi cap and annealed at a maximum tempera- ture of 453 K for seven days. A heating rate of 1 K min−1and a cooling rate of 0.03 K min−1(2.08 K h−1) were applied (see Fig. S1 of the supplementary material). The product was washed in 30 ml of ethylene glycol (99.8%, Sigma-Aldrich, St. Louis, USA) for 5 min under stirring, filtered off, and dried in air on filter paper. B. Powder x-ray diffraction (PXRD) Intensity datasets for powder x-ray diffraction of (NH 4)CoF 3 were recorded with a Panalytical X’Pert PRO diffractometer in BRAGG –BRETANO geometry with Cu Kαradiation (λ1= 1.540 59 Å, λ2= 1.544 43 Å) and a PW3015/20 X’Celerator detector at room temperature (RT). Figure 1 shows the R IETVELD39–41refined and sim- ulated pattern as compared with the measured powder x-ray data. Due to the synthetic approach via derivatives of pyridinium hex- afluorophosphate ILs [C 2Py][PF 6], [C 4Py][PF 6], and [C 6Py][PF 6], the growth of platelet-like crystals (see the supplementary material, Fig. S2) is favored affecting the crystallographic texture. Conse- quently, a preferred orientation correction had to be applied during the R IETVELD refinements, despite thorough grinding of the sample. The ground samples were carefully spread on a silicon disc coated with grease for the measurement. Previous studies on the perovskite FIG. 1 . RIETVELD39–41refinement for (NH 4)CoF 3from 15○to 90○2θmeasured at RT with Cu Kαradiation (λ1= 1.540 59 Å, λ2= 1.544 43 Å). SrSnO 3have shown that the growth of various crystal morphologies can be promoted using different ionic liquids.42 C. Single-crystal x-ray diffraction (SXRD) The crystal structural measurements were carried out on a Bruker D8 Venture diffractometer equipped with a Photon 100 CMOS detector and a I μS microfocus source using Mo Kαradia- tion (λ= 0.710 73 Å) at room temperature and at 100 K. Inten- sity datasets of reflections and scaling were integrated using SAINT within the APEX3 software package.43SADABS44was applied for the absorption corrections. The crystal structure solution was per- formed by direct methods using SHELXT.45SHELXL-201346was used for the subsequent difference F OURIER analyses and least squares refinement. All non-hydrogen atoms were refined anisotropically, whereas hydrogen atoms were assigned geometrically. D. Infrared (IR) spectroscopy The infrared spectrum of (NH 4)CoF 3(see Fig. 4) was recorded at room temperature on a Bruker Alpha-P spectrometer (Bruker Nordic, Sweden) equipped with a single reflection diamond ATR (attenuated total reflectance) accessory (Platinum ATR) in a range of 400 cm–1–4000 cm–1. E. Single-crystal R AMAN spectroscopy Single-crystal R AMAN measurements (see Fig. 4) were performed at room temperature with a Horiba LabRAM HR system (Horiba Europe, Sweden) using a green LASER ( λ= 532 nm), a CCD detec- tor, and a Si standard (520.5 cm–1line) for calibration. Datasets were collected in a range of 400 cm–1–4000 cm–1. F. Scanning electron microscopy (SEM) and energy-dispersive spectroscopy (EDX) Scanning electron micrographs and energy-dispersive x-ray spectroscopy were recorded on a JEOL JSM-7000F (Jeol, Japan) equipped with a S CHOTTKY -type field emission gun. The sample was applied on a conductive carbon pad (Ted Pella, USA/Canada). EDX measurements were conducted at an acceleration voltage of 15 keV. Co and MgF 2were used as reference materials for quantification. A SEM image emphasizing the size and crystal habit, EDX maps for N, J. Chem. Phys. 153, 104501 (2020); doi: 10.1063/5.0023343 153, 104501-2 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Co, and F, an EDX spectrum as well as the result of quantification for validation of the formula (NH 4)CoF 3are given in Figure S2 (see the supplementary material). G. Thermal analysis Combined thermogravimetry (TG) and differential thermal analysis (DTA) measurements (see Fig. S3 of the supplementary material) were conducted on a TGA-50 thermogravimetric analyzer (Shimadzu Corp., Japan) in air with a heating rate of 10 K cm–1. Phase transition behavior was measured by utilizing differential scanning calorimetry (DSC 214 Polyma, Netzsch, Germany) in a sealed aluminum pan under nitrogen flow (40 ml min–1) and with a heating rate of 10 K min–1(see Fig. S4 of the supplementary material). H. Ultraviolet–Visible–Near Infrared (UV–Vis–NIR) absorption spectroscopy The UV–Vis–NIR absorption spectrum of powdered bulk (NH 4)CoF 3(see Fig. 5) was measured at room temperature in a range of 200 nm–1800 nm on an Agilent Cary 5000 spectrometer (Agilent, USA) via the internal diffuse reflection accessory (Harrick, USA). Barium sulfate, Ba[SO 4], was used as the standard white ref- erence. Measurement artifacts occur at 800 nm and 1200 nm due to the detector changes in the device. I. Magnetic measurements For the magnetic measurements (see Figs. 6 and 7; supplemen- tary material, Fig. S6), polycrystalline powder of washed and dried (NH 4)CoF 3was placed in a polypropylene capsule, which was subse- quently mounted into a brass sample holder. All temperature depen- dent magnetization were measured in a static ( DC) field of 0.1 T. As the sample signal was reasonable, no diamagnetic correction of the data was applied. The isothermal magnetization of one sample was investigated up to 7 T. All data were obtained by means of the Quantum Design Physical Property Measurement System (PPMS, Quantum Design, USA). J. Specific heat measurements To obtain specific heat data from the sample powder, a small amount of dry powder was cold pressed into a thin plate, which was cut into a shape that fits the sample holder of a Physical Property Measurement System (PPMS, Quantum Design, USA). A droplet of Apiezon N grease was applied on the holder (sap- phire plate) before measuring the addenda. Subsequently, the sam- ple plate was carefully placed on the holder to let the grease assure the thermal contact. The specific heat of the sample was investigated between 2 K and 300 K with focus (more data points) in the vicinity of the 127 K phase transition: in the range 100 K–145 K, the measurements were performed at zero and 5 T magnetic fields on both heating and cooling. The standard pro- gram for non-adiabatic thermal relaxation was employed; how- ever, double measurements at each temperature were performed, and the heat flash corresponded to 1% of the system absolute temperature.III. RESULTS AND DISCUSSION A. SXRD Based on single-crystal x-ray diffraction data, it had already been reported in the 1950s that (NH 4)CoF 3adopts at room temper- ature the classical cubic perovskite structure.1The low temperature (LT) behavior was investigated by means of magnetization1–16and heat capacity measurements,5,10,13,14indicating a first-order phase transition around 124 K. Following PXRD studies gave evidence of tetragonal distortion for (NH 4)CoF 3as well as (NH 4)MnF 3and (NH 4)ZnF 3.47It was proposed that (NH 4)CoF 3crystallizes in its low temperature phase in the highest symmetry tetragonal sub- group P4/mmm , and atomic coordinates for the hydrogen posi- tions were suggested based on geometric criteria.47This type of first-order phase transition was suggested also for compounds with other transition metals because of similar behavior but without crys- tallographic evidence in most cases. However, single-crystal neu- tron diffraction on (NH 4)MnF 3did indicate a structural transi- tion, but an orthorhombic distortion was put forward rather than a tetragonal.13 Using ionothermal syntheses allowed us to obtain single- crystals large enough to perform a more detailed structural analy- sis, especially at low temperatures (see Table I), which confirm that (NH 4)CoF 3undergoes a transition from Pm¯3mtoP4/mmm (see TABLE I . Crystallographic details and refinement parameters for RT- and LT- (NH 4)CoF 3. Compound RT-(NH 4)CoF 3LT-(NH 4)CoF 3 Chemical formula (NH 4)CoF 3 (NH 4)CoF 3 CCDC 1 960 551 1 960 552 Formula weight, g mol–1133.97 133.97 Space group, Z Pm ¯3m, 1 P4/mmm , 1 Space group No. 221 123 a, Å 4.127(2) 4.080(2) c, Å 4.147(3) V, Å370.30(8) 69.03(8) Temperature, K 293(2) 100(2) Density (calculated), g cm–33.165 3.223 Absorption coefficient, μ, mm–15.956 6.065 F(000) 65 65 θrange for data collection, deg 4.94 – 35.73 5.00 – 30.12 Index ranges −4≤h≤6 −5≤h≤4 −6≤k≤4 −5≤k≤5 −5≤l≤6 −5≤l≤5 Intensity data collected 544 406 Independent reflections 54 82 Rint 0.048 0.051 Completeness, % 100 98.8 Data/Restraints/Parameters 54/1/6 82/2/10 Goodness-of-fit ( F2) 1.46 1.31 R1,ωR2[I0>2σ(I)] 0.027; 0.036 0.027; 0.031 R1,ωR2(all data) 0.037; 0.037 0.064; 0.065 Largest diff. peak/hole (e Å–3) 0.891/ −0.806 0.921/ −0.768 J. Chem. Phys. 153, 104501 (2020); doi: 10.1063/5.0023343 153, 104501-3 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Structure of LT-(NH 4)CoF 3(T= 100 K; ellipsoid representation at 50% probability). Unit cell (a) and representation with outlined CoF 6octahedra (b), both with an idealized representation of the (NH 4)-group. Fig. 2) without any evidences of any lower symmetry. The struc- tural change is accompanied by a slight elongation of the CoF 6 octahedron (Fig. 3), a so-called J AHN–T ELLER distortion. The Co–F interatomic distance at room temperature is 2.0636(8) Å, and at 100 K, it is 2.040(1) Å along the aaxis and 2.074(2) Å along thecaxis, corresponding to an average Co–F distance of 2.057 Å. The cell volume decreases from 70.30(8) Å3to 69.03(8) Å3. The nitrogen atom of the ammonium cation is found at the cen- ter of the fluoride-cuboctahedron; however, the attached hydro- gen atoms could not be properly located in the difference F OURIER map. This is mainly due to their weak contribution to the total electron density but also due to their occupancy disorder in both modifications. A related (cubic-to-tetragonal) but ferroelectric phase transi- tion is reported in the archetype perovskite BaTiO 3,48–51though at much higher temperature. This behavior differs from the con- figurational, non-ferroelectric transitions in the (NH 4)MF3series: (NH 4)NiF 3exhibits the CaTiO 3-type as the LT form and the hexag- onal BaTiO 3-type as the high temperature (HT) form;15(NH 4)FeF 3 undergoes a structural transition at 143 K presumably to a tetragonal modification,52while (NH 4)CuF 3is tetragonal ( P4/mbm , NaNbO 3 structure type) at room temperature,12and no low-temperature phase transition has been reported. The crystal structures of sev- eral HT-K MF3(M= Mn, Fe, Co, Ni, Cu, Zn, and Cd) are also reported to belong to the CaTiO 3structure type with suggested FIG. 3 . Interatomic distances (in Å) observed in the CoF 6octahedra of the HT- [RT, (a)] and LT- [ T= 100 K, (b)] (NH 4)CoF 3(ellipsoid representation at 50% probability).low-temperature tetragonal distortions.53Both Mn and Cd repre- sentatives exhibit furthermore orthorhombic GdFeO 3-type as the LT modification with both NH 4+and also K+.10,13,54,55 B. TG/DTA/DSC The thermogram (see Fig. S3 of the supplementary material) of (NH 4)CoF 3shows that the compound is stable in air up to ∼523 K. Between 523 K and 1073 K, a one-step weight loss of 27.8 wt. % is observed, which corresponds to a decomposition according to (NH 4)CoF 3(s)→NH 3(g)+ HF (g)+ CoF 2(s)with a calculated weight loss to account for the release of NH 3+ HF according to 27.6 wt. %, which is well in agreement with the experimental data. In addition, powder x-ray diffraction was conducted to confirm the formation of the cobalt difluoride. These results are in good accordance with the literature.3The DSC thermogram indicates a structural phase transi- tion at 123.7 K upon cooling and at 129.8 K upon heating (see Fig. S4 of the supplementary material). C. IR/R AMAN Both the IR and the R AMAN spectrum (Fig. 4) comprise the expected internal modes of the NH 4units of the title compound in the regions 1200 cm–1–1750 cm–1and 2750–3500 cm–1. An assign- ment of the N–H modes took place on the basis of the literature describing vibrational spectroscopic measurements for (NH 4)MF3 (M= Mg,56,57Mn,3,7–9,14,56,58–60Fe,52Co,3,7Ni,3,56Zn,3,7–9,14,57–59,61 Cd3,57). In the literature, the splitting of the ν2mode is usually visualized using polarized R AMAN spectroscopy.8,9,14 D. UV–Vis–NIR The UV–Vis–NIR spectrum (Fig. 5) exhibits the major tran- sitions expected for d7-Co2+cations in an octahedral coordina- tion environment.62–68In the visible region of the absorption spec- trum of (NH 4)CoF 3, two transitions,4T1g(F)→4T1g(P) and4T1g(F) →4A2g(F), can be assigned to the wavelengths 537 nm (18 622 cm–1, FIG. 4 . IR ( top) and R AMAN (bottom ) spectrum of (NH 4)CoF 3(assignments of the N–H modes according to Refs. 3, 14, and 56). J. Chem. Phys. 153, 104501 (2020); doi: 10.1063/5.0023343 153, 104501-4 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . UV–Vis–NIR absorption spectrum of bulk (NH 4)CoF 3at room temperature. The measurement artifacts originate from detector changes. shoulder at 487 nm = 20 533 cm–1) and 750 nm (13 333 cm–1), respectively. The main visible absorption band at a wavelength of ∼537 nm (≈2.31 eV) is in agreement with the optical band gap of 2.2 eV–2.4 eV for the KCoF 3.67,69The broad single band with a maxi- mum of about 1538 nm (6502 cm−1) in the IR region can be assigned to the4T1g(F)→4T2g(F) transition. E. Magnetic properties The temperature dependence of the magnetic susceptibility χ for (NH 4)CoF 3is shown in Fig. 6. The susceptibility changes almost FIG. 6 . Temperature dependent susceptibility of bulk (NH 4)CoF 3measured at 0.1 T from 2 K–225 K. The anomalies at 127.7 K ( lower right inset ), 116.9 K, and 2.5 K (left inset ) are indicated. The top right inset shows the inverse susceptibility from 200 K–400 K and a C URIE–WEISSfit between 300 K and 400 K.linearly with the temperature above 130 K, rather than reciprocal. A CURIE–W EISSfit on the inverse susceptibility χ−1in the temperature range 300 K–400 K yielded a W EISSconstant of θCW=−192.37 K and a magnetic moment of 5.39 μBfor Co2+(seetop right inset , Fig. 6). The latter is much larger than the spin-only value (3.87 μB) for Co2+and suggests a large orbital contribution to the moment. The strongly negative W EISSconstant and the deviation of the inverse susceptibility from a linear behavior indicate considerable anti- ferromagnetic interactions. Measurements up to 400 K show that strong short-range antiferromagnetic interactions are retained even well above room temperature. As the sample never really enters the C URIE–W EISSregime, the magnetic moment and W EISSconstant should be taken with caution. Three events are visible in the magnetic data of (NH 4)CoF 3: at 127.7 K, 116.9 K, and 2.5 K (see Fig. 6). The signal at 127.7 K is relatively broad and weak and has not been reported previ- ously. It is worth noting its temperature proximity to the previ- ously reported cubic to tetragonal phase transition5,70and hints at both being related. Differential scanning calorimetry (DSC) and spe- cific heat CPdata both support this. Spikes in the DSC data at 123.7 K on cooling and 129.8 K on warming, respectively, indi- cate a structural phase transition in the exactly same temperature region as the event observed in the magnetic susceptibility curves (see Fig. S4 of the supplementary material). Specific heat data CP (see Fig. 7 and supplementary material, Fig. S5) shows a strong anomaly at 124.5 K, which clearly indicates a first-order phase transition. AtTN= 116.9 K, the susceptibility curves show the onset of long-range magnetic ordering (see Fig. 6), just a few degrees below the structural transition. At first glance, this contradicts previous studies. B ARTOLOMÉ et al. reported the structural transition at 124.5 K and saw no evidence of any decoupled magnetic phase changes in the recorded specific heat data. Therefore, it was concluded that the magnetic transition occurs as a result of the structural distor- tion.5,6,70However, close examination of the reported susceptibility FIG. 7 . Temperature dependent specific heat of bulk (NH 4)CoF 3measured at 0 T and 5 T from 100 K to 145 K. The arrow indicates a small shoulder a few degrees below the main peak. Its approximate position is highlighted by the first derivative. J. Chem. Phys. 153, 104501 (2020); doi: 10.1063/5.0023343 153, 104501-5 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp data reveals that the onset of the magnetic ordering also occurred at temperatures slightly below the structural transition.6 Figure S5 depicts CPdata collected from (NH 4)CoF 3between 2 K and 300 K and indicates the occurrence of two anomalies at 124.5 K and 2.5 K, respectively. The first has a unusually high amplitude, which arises from the ammonium cation’s hindered rota- tions.5A similar behavior has also been observed for other ammo- nium containing perovskites such as (NH 4)MF3(M= Mn, Zn).5,70 Interestingly, the peak’s maxima occurs at exactly the same tem- perature, as reported by B ARTOLOMÉ et al.5,6,70The second event at 2.5 K will be discussed later. For easier comparison of the vibra- tion background, its phonon contribution was calculated using the DEBYE function with 15 degrees of freedom.5In order to do this, the DEBYE temperature θDwas first determined by fitting the analyti- cal low temperature approximation of the D EBYE model ( T≪θD)71 to the data from 5 K to 50 K. The fit yielded θD= 392 K, which is in good agreement with the value reported by B ARTOLOMÉ et al.5 θDwas then further used to numerically calculate the D EBYE func- tion.71The resulting curve is plotted in Fig. S5 ( red line ) and fits well with the data reported for (NH 4)ZnF 3.5The remaining back- ground is associated with the librational motions of ammonium. For (NH 4)CoF 3, BARTOLOMÉ et al. combined the calculated phononic and the librational contributions. The latter was estimated from the Zn-analog.5 Now that it is clear that our specific heat data matches the one reported, we will take a closer look at the region between 100 K and 145 K (see Fig. 7). The measurements were performed both on cool- ing and warming, but no thermal hysteresis could be observed.70 BARTOLOMÉ et al. explained the slightly lower transition temperature in their susceptibility with respect to their specific heat curves by a strongly field dependence.6However, the data in Fig. 7 shows no evidence of such behavior and further supports the possibility of a decoupled magnetic phase transition. Instead, a small shoulder appears at approximately 115 K (as indicated by the arrow) and shows the first proof of an independent magnetic transition. The first derivative reveals that the maxima of this shoulder must lie somewhere above 112.5 K and below the structural transition. How- ever, a more accurate determination of the transition temperature is not possible without full knowledge of the non-trivial background. The complex background and asymmetric peak shape also prevent a proper double-peak fit. A decoupled magnetic transition would be of second order and therefore small compared to the large peak at 124.5 K. Furthermore, the combination of its close proximity to the dominating peak as well as the non-trivial background could easily bury a small anomaly and explain why it has not been observed pre- viously. Since there is no evidence of impurities in our samples and the temperature range of this shoulder coincides with the onset of magnetic long-range order in the susceptibility curves, we conclude that the magnetic transition indeed occurs independently from the structural transition. In context of other perovskites, (NH 4)CoF 3follows the phase transition behavior of (NH 4)MnF 35,47,70,72and (NH 4)FeF 3,52which undergo structural distortions at 182 K and 143 K, respectively, fol- lowed by separate magnetic orderings at 75 K and 98 K, respectively. In KCoF 3,73–75RbCoF 374,75and TlCoF 375simultaneous magnetic and structural transitions are caused by magnetostriction at 115 K, 101 K, and 94 K, respectively. Furthermore, the tetragonal distor- tions in (NH 4)CoF 3andACoF 3(A= K, Rb, Tl) are fundamentallydifferent. The tetragonal elongation ( c/a>1) in (NH 4)CoF 3lifts the degeneracy of the t 2glevels and quenches the orbital contribution of the magnetic moment. A tetragonal compressive distortion ( c/a<1) as observed in ACoF 3(A= K, Rb, Tl), however, leaves the orbital momentum unquenched. Thus, the electronic configuration and the resulting magnetic properties of (NH 4)CoF 3andACoF 3(A= K, Rb, Tl) should differ. Going back to the susceptibility data in Fig. 6, the clear differ- ence in zero field-cooled (ZFC) and field-cooled (FC) curves hints at a complex magnetic structure. As all the magnetic Co ions have d7configuration and are connected via fluoride ions by a 180○ bridge, according to the K ANAMORI –G OODENOUGH rules,76,77an anti- ferromagnetic ordering is expected. The ZFC measurement supports such a magnetic structure; the FC data, however, indicates the exis- tence of a small uncompensated magnetic moment. A small hys- teresis in the isothermal magnetization measured at 2 K and 50 K further supports this (see Fig. S6 of the supplementary material). In accordance with this description, the magnetization does not reach saturation in fields up to 7 T. The highest magnetization reached cor- responds to a magnetic moment of 0.11 μBper formula unit, which is much smaller than the full magnetization of 3 μB. The observed magnetization data matches that of a canted anti- ferromagnet. However, such a state arising from a Co atom at the 1aWYCKOFF position is incompatible with the magnetic subgroups of the tetragonal space group P4/mmm . The real magnetic struc- ture must have a larger unit cell, a lower symmetry, or both in order to describe both the antiferromagnetic and ferromagnetic compo- nents. In previous powder neutron diffraction measurements, the existence of an antiferromagnetic ordering was supported by the occurrence of additional magnetic reflections at low temperatures, hinting at a type G structure. Still the magnetic structure could not be fully solved. The small tetragonal splitting of peaks did not allow us to draw conclusions on the directions of the magnetic moments, and the minor ferromagnetic component could not be observed.6,47 The latter is indeed a problem for very small canting angles due to the low signals in neutron diffraction. Canted antiferromagnetism has been observed in other perovskites, such as KMnF 3, NaFeF 3, NaCoF 3and NaNiF 3, and appears due to their lower orthorhombic symmetry.78 The slightly negative susceptibility values observed in the low temperature ZFC measurement (see Fig. 6) originated from an experimental issue: there is a small, negative residual field that is trapped in the superconducting coil during the zero-field-cooling procedure, despite the use of a set-zero convergence mode. At 2.5 K, (NH 4)CoF 3exhibits a minor kink in both suscep- tibility curves, as well as the specific heat data (see Fig. S5 of the supplementary material). B ARTOLOMÉ et al. observed a peak at approximately 2 K in their specific heat data;5however, they attributed its origin to an unknown impurity, which was inferred from a paramagnetic tail in their magnetic susceptibility data.6As our data point toward a very high purity of the measured sample, and regardless of the preparation method and starting materials this magnetic effect is observed, we conclude that this transition is intrin- sic to (NH 4)CoF 3. It could originate from a second magnetic phase or be of S CHOTTKY -type, i.e., local redistribution of electron among thed-orbitals or arise from a further distortion. However, future investigations using a local probe, e.g., nuclear magnetic resonance (NMR) spectroscopy, have to settle that issue. J. Chem. Phys. 153, 104501 (2020); doi: 10.1063/5.0023343 153, 104501-6 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp IV. CONCLUSIONS Ammonium cobalt trifluoride (NH 4)CoF 3was synthesized via a new ionothermal approach in which 1-alkyl-pyridinium hexaflu- orophosphate [C xPy][PF 6] (x= 2, 4, 6) was applied as both the solvent and the reactant. This synthesis route led to a highly crys- talline and single-phase product at comparatively low tempera- ture. The compound undergoes the phase transition from Pm¯3m [a= 4.127(2) Å, V = 70.30(8) Å3, Z = 1] to P4/mmm [a= 4.080(2), c = 4.147(3) Å, 69.03(8) Å3,Z= 1]. Thermal decomposition and struc- tural phase transition were investigated via TG/DTA and DSC meth- ods. (NH 4)CoF 3decomposes above 523 K and experiences a struc- tural change at 123.7 K upon cooling and at 129.8 K upon heating. The magnetic properties were studied using temperature dependent susceptibility, isothermal magnetization measurements, and com- plemented by specific heat data. Three anomalies were observed in the susceptibility curves upon cooling: At 127.7 K, the structural dis- tortion from cubic to tetragonal is observable in the susceptibility curves, which was confirmed by both DSC and specific heat mea- surements. Contrary to what has been reported previously in the literature, this transition is not field dependent. Just a few degrees below at 116.9 K, the measured susceptibility data indicate the onset of a canted antiferromagnetic ordering. In the same temperature range, a small shoulder is observable in the specific heat data, which has not been reported previously. We argue that it originates from a second-order peak, which is half-hidden in the non-trivial vibration background and the dominating first-order peak. Consequently, the structural and magnetic transitions do not occur simultaneously but are decoupled from each other. Although previously published as an impurity effect, the magnetic anomaly at 2.5 K should be an intrinsic effect of (NH 4)CoF 3and could indicate a second magnetic phase, a SCHOTTKY anomaly, or a further structural distortion. SUPPLEMENTARY MATERIAL Additional information can be found in the supplementary material: Figure S1: Temperature program for the syntheses of (NH 4)CoF 3, Figure S2: SEM image of (NH 4)CoF 3including the EDX mapping analysis area ( top left ), elemental maps of N ( top right ), Co ( mid left ), and F ( mid right ) of a selected rectangular sec- tion as well as an EDX spectrum at an acceleration voltage of 15 kV with assignment of the relevant emission lines and the quantitative analysis ( bottom ), Figure S3: TG ( pink ) and DTA ( blue) curve for (NH 4)CoF 3, Figure S4: DSC thermogram of (NH 4)CoF 3. The arrows indicate the sign of the temperature gradient, Figure S5: Temper- ature dependent specific heat of bulk (NH 4)CoF 3measured at 0 T from 2 K to 300 K. Two anomalies at 124.5 K and 2.5 K ( inset ) occur. Thered line gives an estimate for the lattice phonon contribution to the signal, Figure S6: Field dependent magnetization at 2 K ( blue), 50 K ( red), and 300 K ( green ): The applied field was ramped utilizing the following sequence of turning points: 0, 7, −7, and 7 T. AUTHORS’ CONTRIBUTIONS S.S., K.V.D., V.S., M.V., and A.-V.M. helped in data collection and interpretation; S.S., K.V.D., M.V., and V.S. helped in investiga- tion; A.-V.M. provided the resources; S.S., K.V.D., V.S., M.V., and A.-V.M. helped in writing; and A.-V.M helped in supervision of this paper.ACKNOWLEDGMENTS The authors thank Olivier Renier for conducting the EDX measurements. This research was funded by the Swedish Foundation for Strate- gic Research (SSF) within the Swedish national graduate school in neutron scattering (SwedNess). Dedicated to Professor Patricia Ann Thiel. 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5.0019264.pdf

Appl. Phys. Lett. 117, 071901 (2020); https://doi.org/10.1063/5.0019264 117, 071901 © 2020 Author(s).Highly efficient nondoped bilayer organic light-emitting diodes based on triphenyl phosphine oxide protected iridium complexes Cite as: Appl. Phys. Lett. 117, 071901 (2020); https://doi.org/10.1063/5.0019264 Submitted: 23 June 2020 . Accepted: 06 August 2020 . Published Online: 19 August 2020 Ying Wei , Wenjing Kan , Jing Zhang , and Hui Xu COLLECTIONS This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN Room temperature infrared detectors made of PbTe/CdTe multilayer composite Applied Physics Letters 117, 072102 (2020); https://doi.org/10.1063/5.0018686 Revealing mechanism of obtaining the valence band maximum via photoelectron spectroscopy in organic halide perovskite single crystals Applied Physics Letters 117, 071602 (2020); https://doi.org/10.1063/5.0016223 Intra- and inter-conduction band optical absorption processes in β-Ga2O3 Applied Physics Letters 117, 072103 (2020); https://doi.org/10.1063/5.0016341Highly efficient nondoped bilayer organic light- emitting diodes based on triphenyl phosphine oxide protected iridium complexes Cite as: Appl. Phys. Lett. 117, 071901 (2020); doi: 10.1063/5.0019264 Submitted: 23 June 2020 .Accepted: 6 August 2020 . Published Online: 19 August 2020 Ying Wei,a)Wenjing Kan, Jing Zhang, and Hui Xua) AFFILIATIONS School of Mechanical and Electrical Engineering and Key Laboratory of Functional Inorganic Material Chemistry, Ministry of Education, Heilongjiang University, 74 Xuefu Road, Harbin 150080, People’s Republic of China a)Authors to whom correspondence should be addressed: ywei@hlju.edu.cn and hxu@hlju.edu.cn ABSTRACT Nondoped phosphorescent organic light-emitting diodes (OLEDs) with simplified stacks are desired for practical displaying and lighting applications. However, doping emissive layers are commonly adopted due to serious triplet quenching of phosphors. Here, we demonstrate high-efficiency nondoped bilayer OLEDs based on triphenylphosphine oxide protected iridium(III) complexes. The host feature of peripheral phosphine oxide groups effectively suppresses intermolecular interaction induced quenching. As consequence, the maximum external quan-tum efficiency beyond 20% and near-zero roll-offs at 1000 nits were achieved, comparable to the best nondoped phosphorescence devices. Published under license by AIP Publishing. https://doi.org/10.1063/5.0019264 Organic light-emitting diodes (OLEDs) attract much attention, owing to its merits of flexibility, energy conservation, environmental friendliness, and so on. 1However, for commercial applications, large- scale production of low-cost OLEDs is still a big challenge.2Since the statistical proportion of triplet exciton is 75% of electrogenerated exci-tons, triplet harvesting is the key issue determining device efficiencies. Phosphorescence (PH) originates from radiation of the first triplet state (T 1); therefore, phosphors can harvest 100% excitons, based on singlet-to-triplet intersystem crossing, rendering /C24100% internal quantum efficiency (IQE, gIQE)f o rt h e i rO L E D s .3However, the involvement of triplet exciton in emission induces collosion-based triplet quenching, e.g., triplet-triplet annihilation (TTA) and triplet polaron quenching (TPQ).4Most of the efficient phosphorescence OLEDs (PHOLED) adopted doping emissive layers (EML).5By using pure-organic host matrixes to disperse PH complexes and multi-stack structure, external quantum efficiencies (EQE, gEQE)o fP H O L E D sc a n be improved to more than 20%.6Nevertheless, it is still desired to con- struct host-free PHOLEDs with simplified structures and comparable electroluminescent (EL) performance.7 In order to confine charges and excitons on PH dopants, host matrixes should have the deeper highest occupied molecular orbitals (HOMO) and the shallower lowest unoccupied molecular orbitals (LUMO), as well as the higher first singlet (S 1) and T 1energy levels. Therefore, hole (HTL) and electron transporting layers (ETL) areessential to modify carrier injection into EMLs and carrier transport of whole devices. Actually, compared to their hosts, PH complexes can provide more suitable HOMO and LUMO energy levels matching with work functions of electrodes. So, the key issue is how to suppress triplet quenching effects in host-free EMLs. Wang and Wong et al. demonstrated an effective “self-host” design by incorporating carrier transporting and dendritic groups with host feature into iridium(III) complexes.8Our recent works showed that through linking peripheral carrier transporting groups and emitting iridium(III) complex cores with aliphatic chains, the charge-exciton interactions can be preventedto reducing TPQ, namely, so-called “charge-exciton separation” strategy. 9For this strategy, the carrier transporting channels on fron- tier molecular orbitals and triplet exciton locations of phosphors were separated to avoid direct interactions between charge and exciton, thereby mitigating TPQ at molecular level. Nonetheless, the involve- ment of aliphatic chains weakens electrical properties and evaporabil- ity of the complexes, limiting gEQEof their spin-coated nondoped bilayer devices within 15%. In this case, if the carrier injection and recombination sites of the complexes can be separated by direct modi- fication with carrier-transporting aromatic groups, the device perfor- mance can be dramatically improved. Figure 1 shows the molecular structures of Ir(PBI) 3,I r ( P B I ) 2PBIPO, and Ir(PBIPO) 3. PBIPO was reported as an electron-transporting material in our previous work, whose charged PBI moiety can be Appl. Phys. Lett. 117, 071901 (2020); doi: 10.1063/5.0019264 117, 071901-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplseparated from the EML jETL interface by its TPPO group, thereby reducing interfacial TPQ.10Moreover, hosts containing phosphine oxide (PO) groups also displayed the superiority in suppressing intermolecular interaction-induced quenching. These results sug- gested the similar steric effect of triphenylphosphine oxide (TPPO) moieties in Ir(PBI) 2PBIPO and Ir(PBIPO) 3. In this sense, the inter- molecular interactions of emitting core in Ir(PBIPO) 3should be the weakest, owing to its TPPO groups at three different directions. Meanwhile, besides steric hindrance of one TPPO group, the asym-metric configuration of Ir(PBI) 2PBIPO is also beneficial to prevent regular molecular packing. Density functional theory (DFT) simulation shows the LUMO of Ir(PBI) 3is thoroughly contributed by 2-phenylbenzimidazole of its ligand, without any contributions from N-phenyl ( Fig. 2 ). However, the strong electron-withdrawing effect of P @O induces the LUMOs of Ir(PBI) 2PBIPO and Ir(PBIPO) 3largely dispersed on P@O-substituted N-phenyl. The LUMO and the HOMO ofIr(PBI) 2PBIPO are, respectively, located on PBI and PBIPO, due to their different electron-deficient degrees. Meanwhile, two N-phenyls of Ir(PBIPO) 3are involved in the LUMO, relatively decreasing the contribution proportion of 2-phenylbenzimidazole. Triplet excitationsof these complexes are investigated with time-dependent DFT (TD-DFT) simulation by natural transition orbital (NTO) method. Holes and particles of S 0!T1transitions for the complexes are quite similar, which are mainly located on 2-phenylbenzimidazoles of their ligands. PBIPO of Ir(PBI) 2PBIPO is completely excluded from holes and particles. Therefore, the triplet transition character of Ir(PBI) 2PBIPO is almost identical to Ir(PBI) 3,w h i l e N-phenyls of Ir(PBIPO) 3make minor contributions to particles. Because the fron- tier molecule orbitals (FMOs) and NTOs of Ir(PBI) 3are consistent, its triplet excitation mainly involves in HOMO–LUMO transition with a weight ( rH-L) of 61%. On the contrary, the exclusion of PBIPO from NTOs sharply decreases rH-L in the triplet excitation of Ir(PBI) 2PBIPO to 7%. The symmetric configuration of Ir(PBIPO) 3 improves the consistence of FMOs and NTOs, but its rH-Lis still reduced to 17%. Nevertheless, the T 1energy levels of these complexes are similar. In consequence, the introduction of TPPO moieties sepa- rates the LUMOs of Ir(PBI) 2PBIPO and Ir(PBIPO) 3from their T 1 states, without changing excitation behaviors. In accord with TD-DFT results, the complexes show the similar electronic absorption spectra in CH 2Cl2(10/C06mol l/C01) with the identi- cal peaks and profiles, reflecting their consistent excitation characters [Fig. 3(a) ]. In solution, PL spectra of Ir(PBI) 2PBIPO and Ir(PBI) 3are almost overlapped with the peaks at 516 nm, corresponding to T 1 energy levels of 2.4 eV. In contrast, emission peak of Ir(PBIPO) 3shifts to 529 nm, corresponding to a reduced T 1energy level of 2.3 eV. gPL values of these complexes in solution are also comparable as 76%, 82%, and 80%, respectively. In neat films, emissions from Ir(PBI) 3, FIG. 1. Molecular structures of three iridium complexes: phosphine oxide–free Ir(PBI) 3, mono-phosphine-oxide Ir(PBI) 2PBIPO, and tri-phosphine-oxide Ir(PBIPO) 3. FIG. 2. TDDFT-simulated T 1energy levels and “hole” and “particle” contours of S 0!T1excitations for Ir(PBI) 3, Ir(PBI) 2PBIPO and Ir(PBIPO) 3based on nature transition orbital (NTO) method. DFT-simulated HOMO and LUMO contours are presented for comparison. ETandrH-Lrefer to T 1energy level and weight in transition.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 071901 (2020); doi: 10.1063/5.0019264 117, 071901-2 Published under license by AIP PublishingIr(PBI) 2PBIPO, and Ir(PBIPO) 3are peaked at 539 nm, corresponding to emission bathochromic shifts of 23 nm for Ir(PBI) 3and Ir(PBI) 2PBIPO and 10 nm for Ir(PBIPO) 3, respectively. It is noted that solid-state emission of Ir(PBI) 2PBIPO is 3 nm narrower than that of Ir(PBI) 3. Furthermore, gPLof Ir(PBI) 3film is sharply decreased to 18%, which is only two fifths of that of Ir(PBI) 2PBIPO film (46%). In contrast, gPLof Ir(PBIPO) 3film is completely unchanged as 81%. Therefore, it is convincing that peripheral TPPO groups can effectively suppress the intermolecular interaction induced quenching in neat films. According to cyclic voltammetry (CV) analysis, the HOMO energy levels of Ir(PBI) 3and Ir(PBI) 2PBIPO are equivalent as /C05.30 eV, which are 0.1 eV shallower than that of Ir(PBIPO) 3 (/C05.41 eV) [ Fig. 3(b) ]. Their LUMO energy levels are /C02.90,/C02.90, and/C02.98 eV, estimated with optical bandgaps. Therefore, FMO energy levels of these iridium(III) complexes can effectively matchwith work functions of the PEDOT:PSS anode ( /C05.2 eV) and the LUMO energy levels of conventional electron transporting materials (/C243.0 eV). The carrier transporting abilities of the complexes were investigated, based on the single-layer single-carrier-transporting devi-ces (Fig. S2). At the same voltages, the electron-only current densities(J)o fI r ( P B I ) 3,I r ( P B I ) 2PBIPO, and Ir(PBIPO) 3based devices were gradually increased, reflecting the enhanced electron transportation by peripheral P ¼O groups with electron-withdrawing effect. In contrast,the hole-only Jof Ir(PBI) 3based devices was remarkably higher than those of Ir(PBI) 2PBIPO and Ir(PBIPO) 3based analogs. For Ir(PBI) 2PBIPO, its asymmetrical structure hindered the regular molec- ular alignment and, therefore, weakened carrier hopping. The encap- sulation effect of P @Og r o u p si nI r ( P B I P O ) 3rendered the reduced hole transportation. According to space-charge limited current model,the electron mobility of Ir(PBI) 3and Ir(PBI) 2PBIPO was comparable as 4.5/C210/C05V/C01s/C01and 4.9 /C210/C05V/C01s/C01, respectively, which was lower than Ir(PBIPO) 3(5.6/C210/C05V/C01s/C01). On the contrary, hole mobility of Ir(PBI) 3reached to 6.5 /C210/C05V/C01s/C01,w h i c hw a s slightly higher than Ir(PBI) 2PBIPO (5.7 /C210/C05V/C01s/C01), but more t h a nt w i c eo ft h a to fI r ( P B I P O ) 3(2.8/C210/C05V/C01s/C01). Therefore, the peripheral P ¼O groups in Ir(PBI) 2PBIPO and Ir(PBIPO) 3remark- ably influenced their carrier-transporting abilities. Although Ir(PBI) 2PBIPO showed the most balanced intrinsic charge mobility, Ir(PBIPO) 3can improve the carrier balance in OLEDs featuring with heavy p-doping and hole as majority carrier. We further fabricated bilayer PHOLEDs using neat complex films as EML, with the structure of ITO jPEDOT:PSS (50 nm) jiridium(III) complex (20 nm) jDBFDPO (30 nm), in which PEDOT:PSS is poly(3,4-ethylenedioxythiophene)-poly(styrenesulfo-nate) as the anode, and DBFDPO is 4,6-bis(diphenylphosphoryl)di- benzofuran as ETL [inset in Fig. 4(a) ]. The improved energy level matching leads to barrier-free charge injection in EMLs. In accordwith optical results, EL spectra of Ir(PBI) 3,I r ( P B I ) 2PBIPO, and FIG. 3. (a) Electronic absorption and photoluminescence (PL) spectra of Ir(PBI) 3, Ir(PBI) 2PBIPO, and Ir(PBIPO) 3in CH 2Cl2(10/C06mol l/C01), and PL spectra of neat complex films prepared by vacuum evaporation (100 nm); (b) cyclic voltammograms of the complexes in CH 2Cl2measured at room temperature with a scan rate of 0.1 V s/C01. FIG. 4. EL performance of non-doped bilayer PHOLEDs based on Ir(PBI) 3, Ir(PBI) 2PBIPO, and Ir(PBIPO) 3. (a) Luminance-current density ( J)-voltage charac- teristics of the devices. Insets show the device structure and energy level diagram,and EL spectra at 1000 nits; (b) efficiency vs luminance correlations of the devices.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 071901 (2020); doi: 10.1063/5.0019264 117, 071901-3 Published under license by AIP PublishingIr(PBIPO) 3based green-emitting PHOLEDs were identical with peak wavelengths at 508 nm, revealing EL emissions of Ir(PBI) 2PBIPO and Ir(PBIPO) 3from their Ir(PBI) 3cores. At the same driving voltages, Jof Ir(PBI) 3,I r ( P B I ) 2PBIPO, and Ir(PBIPO) 3based OLEDs was gradually and remarkably decreased, in accord with the situation of hole-only Jfor these complexes [ Fig. 4(a) ]. In this sense, hole was the majority carrier in these OLEDs. However, the highest electron mobility of Ir(PBIPO) 3balanced charge carrier flux dramatically improved carrier recombination and utilizationefficiencies. Therefore, the maximum brightness of all the devices was comparable and more than 10 4cd m/C02. As the result, the maxi- mum efficiencies of Ir(PBIPO) 3-based devices were 69.3 cd A/C01, 47.1 lm W/C01and 20.8%, corresponding to an exciton utilization effi- ciency ( gEUE) value of 100% by taking the outcoupling ratio of 25% for glass substrate [ Fig. 4(b) ]. In contrast, the efficiencies of Ir(PBI) 2PBIPO-based devices were sharply decreased by more than two folds, accompanied by a smaller gEUEvalue of 75%. Ir(PBI) 3-based devices suffered the most serious quenching, rendering the efficiencies as one fifth of those of Ir(PBIPO) 3-based devices and a gEUEvalue as low as 33%. It indicates that compared to Ir(PBI) 3, intermolecular interaction suppression by TPPO effectively mitigated TTA in Ir(PBI) 2PBIPO and Ir(PBIPO) 3based devices, resulting in the doubled and tripled gEUE. Furthermore, it is known that efficiency roll-off is the combination result of TTA and TPQ. Compared to reported non- doped PHOLEDs, apparent EQE roll-offs of Ir(PBI) 3and Ir(PBI) 2PBIPO were limited since even at low luminance, the aggrega- tion induced quenching and TPQ in neat Ir(PBI) 3and Ir(PBI) 2PBIPO films were already serious, which also rendered their lower gPL. Nonetheless, EQE roll-off of Ir(PBIPO) 3based devices at 1000 cd m/C02 was as small as 0.5%, which is only one tenth of that of Ir(PBI) 2PBIPO based devices (6%). The negligible EQE roll-off demonstrated the effectively mitigated triplet-triplet and charge-exciton interactions in EML of Ir(PBIPO) 3, respectively, attributed to its TPPO-encapsulated phosphor core and separated FMO and excited state locations.11 In summary, we demonstrated the high-efficiency nondoped bilayer PHOLEDs with the maximum gEQEbeyond 20%, 100% gEUE, and near-zero EQE roll-offs. It shows that TPPO substitution can not only protect phosphorescent core from TTA, but also change the FMO locations for FMO-excited state separation. 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Lett. 117, 071901 (2020); doi: 10.1063/5.0019264 117, 071901-4 Published under license by AIP Publishing

5.0012636.pdf

APL Mater. 8, 081109 (2020); https://doi.org/10.1063/5.0012636 8, 081109 © 2020 Author(s).Eu3+-doped Bi7O5F11 microplates with simultaneous luminescence and improved photocatalysis Cite as: APL Mater. 8, 081109 (2020); https://doi.org/10.1063/5.0012636 Submitted: 03 May 2020 . Accepted: 28 July 2020 . Published Online: 19 August 2020 Donglei Wei , Yanlin Huang , and Hyo Jin Seo ARTICLES YOU MAY BE INTERESTED IN Suitability of binary oxides for molecular-beam epitaxy source materials: A comprehensive thermodynamic analysis APL Materials 8, 081110 (2020); https://doi.org/10.1063/5.0013159 Below bandgap photoluminescence of an AlN crystal: Co-existence of two different charging states of a defect center APL Materials 8, 081107 (2020); https://doi.org/10.1063/5.0012685 Role of the ferroelastic strain in the optical absorption of BiVO 4 APL Materials 8, 081108 (2020); https://doi.org/10.1063/5.0011507APL Materials ARTICLE scitation.org/journal/apm Eu3+-doped Bi 7O5F11microplates with simultaneous luminescence and improved photocatalysis Cite as: APL Mater. 8, 081109 (2020); doi: 10.1063/5.0012636 Submitted: 3 May 2020 •Accepted: 28 July 2020 • Published Online: 19 August 2020 Donglei Wei,1 Yanlin Huang,2and Hyo Jin Seo1,a) AFFILIATIONS 1Department of Physics, Pukyong National University, Busan 608-737, South Korea 2College of Chemistry, Chemical Engineering and Materials Science, Soochow University, Suzhou 215123, China a)Author to whom correspondence should be addressed: hjseo@pknu.ac.kr. Tel.:+82-51-629 5568. Fax: +82-51-62955494 ABSTRACT Doping of rare-earth ions in a host is one of the important strategies to modify the microstructure and electrical and optical properties. This work demonstrated the significant improvement of luminescence and photocatalytic performances of Bi 7O5F11via Eu3+doping. Bi 7O5F11has a typical Sillén–Aurivillius structure, which shows an intrinsic luminescence band peaked at 527 nm with a decay time of 0.041 μs. The intrinsic emission quenches in Bi 7O5F11:Eu3+, which shows characteristic transitions from5D0,1,2,3 levels to7FJ(J = 0–4) ground states. An experimental red-LED lamp was successfully fabricated by encapsulating Bi 7O5F11:Eu3+with a transparent resin. Bi 7O5F11has poor photocatalytic ability, which just can happen under UV light irradiation. The fast decay time (0.041 μs) of Bi 7O5F11causes an efficient recombination of the light- induced charges, resulting in a lower photocatalytic effect. Bi 7O5F11:Eu3+shows the improved photocatalytic abilities compared with pure Bi7O5F11. 4f levels of Eu3+provide a longer decay time (1 ms) for the excited states of Bi 7O5F11, which prevents the recombination of the light-induced charges. Importantly, Eu3+doping moves the required wavelength in photocatalytic reactions from UV light (pure Bi 7O5F11) to visible wavelength in Bi 7O5F11:Eu3+. Bi 7O5F11:Eu3+could be further investigated to develop a multifunctional bismuth material such as dielectric, photoelectric, and photochemical abilities. ©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/10.1063/5.0012636 .,s I. INTRODUCTION Bismuth compounds with a layered structure exhibit mul- tifunctional performances such as luminescence, dielectric, and photochemical applications. The special 6s2configuration of Bi3+ induces efficient excitation and polarizability in the lattices.1–3In particular, Sillen phases have gained significant interest in their interesting structures and photochemical properties. Its framework is constructed by [Bi 2O2]2+and halide atom (F, Cl, Br, and I) lay- ers. High polarization can be induced by the (Bi 2O2)2+layers, which are greatly beneficial to its piezoelectric and optical properties.4–6In recent years, a great deal of work has been reported for these mate- rials such as BiOX (X = Cl, Br, and I),4ABiO 2Cl (A2+= Cd, Ca, Pb, and Sr),7etc. This work reports Eu3+-activated Bi 7F11O5microplates with the red-luminescence and photocatalysis performances. Bi 7F11O5was first reported by Laval et al. ,8which crystallizes in an acentric space group of C2. Hu et al.9have reported that Bi 7F11O5has a SHG response, which is 2.7 times bigger than KDP. The excellent SHG is ascribed to the stereochemically active lone Bi3+pairs in Bi 7F11O5. The photocatalytic activities of single-phased Bi 7F11O510and het- erostructured Bi 7F11O5/BiOCl11have been reported for the degra- dation of methyl orange (MO). Wang et al.12reported the near-IR up-conversion fluorescent (800 nm) in Yb3+/Tm3+codoped Bi7F11O5. It was suggested that Bi 7F11O5can potentially act as a temperature sensor in biological tissues and cells. We paid attention to simultaneous luminescence and enhanced photocatalysis via Eu3+doping in Bi 7F11O5due to the following motivations. First, as a luminescence material, RE3+activated Bi- layered phosphor has its advantages such as the highly anisotropic lattices, structural variability, and ability.13RE3+activated BiOCl is the most often reported candidate for potential luminescence APL Mater. 8, 081109 (2020); doi: 10.1063/5.0012636 8, 081109-1 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm applications.12,14,15Compared with BiOCl, Bi 7F11O5shows more special polarization anisotropy with its layered lattices to accom- modate RE3+.9,12Fast s2↔sp electronic transitions in Bi 7F11O5are parity allowed involving the outer valence shell of the 6s2electron, which has a strong interaction with its surrounding.1Therefore, red- luminescence could be expected in Eu3+-doped Bi 7F11O5due to the strong splitting for Eu3+ions. Second, similar to BiOCl, Bi 7F11O5has a layered structure with great spontaneous polarization along [001], which can effi- ciently separate light-produced charges in the lattices. In Bi 7F11O5, it could be predicted that RE3+ions such as Eu3+can induce non- uniform charge distributions between the [Bi 2O2] and F−layers. The improved photocatalysis could be expected in Eu3+-doped Bi 7F11O5. Importantly, Eu3+doping could move the optical absorption from the UV to visible wavelength region.10 Third, Bi-photocatalysts have a shortcoming, that is, photo- created charges usually have fast recombination.16The lifetime of an excited state is vital to efficiently separate e–h pairs. The allowed transition by spectroscopic selection rules of Bi3+has a much shorter decay time than forbidden transitions (4f →4f tran- sitions in RE ions), e.g., 2 ms for Eu3+against 500 ns for Bi3+.1 Bi-compounds present a fast decay time with a nanosecond order. For example, Bi 4Ge3O1217and α-Bi 2O318have the lifetimes of 0.43 ±0.08 μs and 0.104 μs (300 K), respectively, while, BiOCl shows a fast lifetime of 5.89 ns at 300 K.19In order to prolong decay times of excited states, it is possible to dope Eu3+in Bi 7F11O5, which provides 4f levels with longer lifetime in μs or ms orders. In this way, the photocatalysis could be enhanced due to the pro- longed lifetimes of excited states. In the present references, sev- eral Bi-compounds have reported the enhanced photocatalysis by Eu3+doping such as BiOCl:Eu3+,20Bi2O3:Eu3+,21and BiVO 4:Eu3+,22 etc. However, the reported photocatalysts did not show efficient red-luminescence, and the mechanism on dynamic luminescence was not elucidated. In this work, it can be expected that the photocatalytic abilities of Bi 7F11O5could be improved via Eu3+ doping. Single-phased Bi 7−7xEu7xO5F11(x= 0, 0.01, 0.03, 0.05, 0.08) was synthesized by the hydrothermal method. The Rietveld refinements were conducted to investigate its structural details. The band energy and electronic structures were studied. The degradations for Rho- damine B (RhB) solutions were measured. The luminescence, decay time, and improved photocatalysis of Bi 7−7xEu7xO5F11were studied and discussed. II. EXPERIMENTAL Single-phased Bi 7−7xEu7xO5F11(x= 0, 0.01, 0.03, 0.05, 0.08) was synthesized via the hydrothermal reaction route. The raw materi- als and reagent dosages are listed in Table S1 of the supplementary material. Typically, the mixtures of Bi(NO 3)3⋅5H 2O, NaF, EuF 3, and deionized H 2O (50 ml) were sealed in an autoclave lined with a Teflon liner. Then, some ammonia (2 mol/l) was added into the solu- tions dropwise to reach pH = 1. Autoclaves were put in an oven and heated at 170○Cfor 24 h, and then, the system was naturally cooled to 300 K. The solutions were filtered to get the precipitates, which were slowly washed with alcohol and deionized water. The final pow- der products could be obtained after drying the precipitates at 50○C in air atmosphere.The XRD patterns were measured on a Rigaku D/Max diffrac- tometer. The incident x ray was generated by a Cu-K αtarget. The morphology nature was detected by using a scanning elec- tron microscope (SEM, Hitachi SU-1510, 15 keV). Meanwhile, the face-scanning of x-ray energy-dispersive spectrometry (EDS) mea- surements was applied to study the detailed elemental ratios. The diffused reflectance was performed via a UV–Vis spectrophotome- ter (Shimadzu, UV-2550). The photoluminescence was investigated on the Perkin-Elmer LS-50B. The dynamic curves were measured under 266 nm excitation from a fourth harmonic generation of a pulsed YAG:Nd laser. The pulse energy of the laser was 5 mJ, and its repetition rate is 10 Hz. The signals were dispersed on a monochro- mator connected with a photomultiplier tube (PMT) (Hamamatsu R928). All the measurements mentioned above were conducted at room temperature. In a typical photodegradation of RhB dye solutions (300 ml and 10 mg l−1), 0.05 g of Bi 7−7xEu7xO5F11was used. The reaction was conducted in a 500 ml reactor. The incident wavelength was big- ger than 420 nm generated by an optical cut filter from an Xe lamp (300 W). The dye solutions were extracted with a certain period of time for optical absorption. The photodegradation effects were determined by the equation [1 −(At/A0)]×100%. A0is the initial absorption intensity at time 0, and Atis the absorption intensity of RhB after time t. III. RESULTS AND DISCUSSIONS A. Phase formation and morphology Phase formations of the samples were confirmed by XRD measurements, as shown in Fig. 1(a). The XRD patterns of Bi7−7xEu7xO5F11(x= 0, 0.01, 0.03, 0.05, 0.08) fit well with the standard PDF card No: 50-0003. All samples were obtained with the single-phased structure of Bi 7O5F11. The synthesis experiments also confirmed that the tolerable maximum doping in Bi 7O5F11is 8 mol. % ( x= 0.08). Impurity phases were inevitably observed when the doping exceeds 8 mol. %. The Rietveld refinements (see Fig. S1 of the supplementary material) were conducted to establish the detailed crystal structure. In the refinements, Eu3+was suggested to substitute Bi3+due to the similar size and valence. The refined parameters and atomic posi- tions are listed in Tables S2 and S3 of the supplementary material, respectively. The unit cell shows a linear shrinkage with the increase in Eu3+doping ( x) induced by the different sizes between Eu3+(1.066 Å) and Bi3+(1.17 Å) (see Fig. S2 of the supplementary material). The result follows the so-called Vegard’s law. In summary, XRD analysis concluded that Bi 7−7xEu7xO5F11has a monoclinic structure with space group C2. The structure can be regarded to derive from the fluorite-type via the accommodation of the excess of anions within infinite cis-chains of BiF 6O2poly- hedra parallel to [010] [Fig. 1(b)].8The lattices show a distinct layered structure constructed by F−anion layers and the covalent cation of (Bi 2O2)2+.12There are four kinds of Bi3+ions in the lat- tices with a different coordination by F−and O2−(see Fig. S3 of the supplementary material). Most importantly, each Bi3+is greatly distorted resulting in great dipole moments,9which is beneficial to the energy splitting of RE3+and improvement of photocatalytic response. APL Mater. 8, 081109 (2020); doi: 10.1063/5.0012636 8, 081109-2 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm FIG. 1 . XRD patterns of Bi 7−7xEu7xO5F11(x= 0, 0.01, 0.03, 0.05, 0.08) compared with PDF#50-0003 (a) and the unit cell of Bi 7O5F11projected along the a axis (b). SEM images of Bi 7−7xEu7xO5F11(x= 0, 0.01, 0.03, 0.05, 0.08) were measured to detect morphological features (see Fig. S4 of the supplementary material). There are many aggregations of well- crystallized microplates, which show smooth and clear surfaces. The plate has a thickness of 300 nm and a side length of several μm to 10μm. With Eu3+doping, the morphological characteristics were not changed in the doped samples. To detect the elemental compo- sitions of Bi 7−7xEu7xO5F11, EDS measurements were conducted (see Fig. S5 of the supplementary material). The quantitative mole ratio between Bi and Eu measured via the elemental face-scanning agrees with the stoichiometric requirements in each chemical formula. This indicates that preparation experiments are very successful. XPS was used to study the elemental information of pure and Eu-doped samples. The survey spectrum of Bi 7O5F11shows the ele- ments of Bi, F, and O, while extra XPS signals of Eu were detected for Bi7−7xEu7xO5F11(x= 0.03, 0.08, 0.08) (see Fig. S6 of the supplemen- tary material). High-resolution XPS spectra of F-1s, Bi-4f, Eu-3d, and O-1s were also measured, as shown in Fig. S7 of the supple- mentary material. There are similar profiles and the same bindingenergy, confirming the existence of F−, Bi3+, Eu3+, and O2−ions in the lattices. Raman spectra of Bi 7−7xEu7xO5F11(x= 0, 0.03, 0.05, 0.08) microplates were measured (see Fig. S8 of the supplementary mate- rial) with three main peaks. The stretching and bending modes below 400 cm−1attribute to Bi–F–Bi and Bi–O–Bi, respectively. The weak peak at 444 cm−1can be assigned to ν(Bi–F) vibrations.9There is a weak phonon energy ( <500 cm−1) in Bi 7F11O5. The results indi- cate that much more phonons are required in Bi 7−7xEu7xO5F11to fill the gap of 2000 cm−1between5D1and5D0(as well as between5D2 and5D1). This is discussed in Sec. III B for its emission from5D1,0 states. B. Optical absorption and band structure The bandgap and electronic structures were investigated via density functional theory (DFT) calculations [Figs. 2(a) and 2(b)]. The maximum of the valence band (VB) and the minimum of the conduction band (CB) are at the Zpoint, indicating a direct tran- sition. This is in agreement with the experimental results obtained from the optical absorption in Figs. 2(a) and 2(b). The VB of Bi7O5F11has a maximum at 0 eV as labeled at the “A” point. The lowest-energy transition occurs at the “B” position with the changes in the k-vector. The smallest transition energy is at 2.71 eV. There is a heavy dispersion in the electron transition bands, such as CB com- positions above 2.71 eV and VB compositions from −2 eV to 0 eV. This characteristic is beneficial for photocatalytic abilities. It has been established for post-transition lone-pair oxides, such as Bi3+and Pb2+,23that (i) interactions between the “ p” orbit (oxygen) and the “ s” orbit (cation) create the antibonding states with a large degree of cationic “ s” character at the VB top and (ii), inter- actions between antibonding orbitals and empty cationic p states (Bi3+: 6s26p0) generate the lone pairs within distorted coordina- tion. Figure 2(b) shows the calculated partial and total density of states for Bi 7O5F11. The VBs are composed of O-2p, F-2p, and Bi- 6p states from −2.0 eV to the Fermi level. In addition, CBs (bottom to 6.0 eV) contain the dominant Bi-6p states. Usually, electronic transitions between the VB top and the CB bottom mainly deter- mine the optical properties. The calculation suggests that the heavy covalent interactions between anions (O2−/F−) and Bi3+greatly con- tribute to the band energy and its optical anisotropy. The hybridized 2p orbitals of O and F contribute to the optical properties to some degree.9The calculations get the conception that the band-to-band transitions of Bi 7O5F11are mainly formed by (F2p, O2p, Bi6s) →Bi6p. Bi7−7xEu7xO5F11(x= 0, 0.01, 0.03, 0.05, 0.08) shows an effi- cient absorption in UV and near-UV wavelength regions [Fig. 2(c)]. The absorption peaks from the 4 f→4ftransitions of Eu3+were not obviously observed. Compared with the abrupt profile of the pure sample [see the inset of Fig. 2(c)], the absorption edges of Eu3+- doped Bi 7O5F11show more and more deviation due to the extra absorption near the band tails. This region just coincides with the transition from excited states5D1,2to ground states (7F0–6) of Eu3+.24 It is reasonably suggested that5D1,2levels and possible defects could contribute to the deviation, as shown in the inset of Fig. 2(c). The band energy ( Eg) was calculated via the following formula: αhυ=A(hυ−Eg)n/2, (1) APL Mater. 8, 081109 (2020); doi: 10.1063/5.0012636 8, 081109-3 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm FIG. 2 . Band diagram representation between −2 eV and 6 eV (a), the calcu- lated partial density of states of Bi 7O5F11 (b), UV–Vis optical absorption (c), and the estimation for the band energy on Eq. (1) for n= 4 (d) of Bi 7−7xEu7xO5F11 (x= 0, 0.01, 0.03, 0.05, 0.08). where h,ν, and αare Planck’s constant, incident absorption fre- quency, and coefficient, respectively. Figure 2(d) displays the plots of (hυ) vs ( αhυ)2/nfitting into n = 1. The model shows that the samples have an indirect transition nature. The band energies for Bi7−7xEu7xO5F11are 3.85 eV ( x= 0), 3.81 eV ( x= 0.01), 3.64 eV (x= 0.03), and 3.53 eV ( x= 0.05). It is consistent with the reported values of 3.83 eV,113.81 eV,93.52 eV,12and 3.98 eV.10 C. Intrinsic luminescence transition The following equations are generally used to estimate the energy band potentials and positions of a semiconductor: ECB=X−Ee−0.5EgandEVB=X−Ee+ 0.5 Eg. Here, Ee(∼4.5 eV vs SHE) is the energy of free electrons on the hydrogen scale, and Xis the absolute electronegativity. The estimated values are presented in Fig. 3(a). The CB bottom and VB top locate at +0.12 eV and +3.97 eV in Bi 7O5F11, respectively. The CB is from Bi-6p mixed with O-2p orbitals. The O-2p, F-2p, and Bi-6s orbitals built VB components. In Bi 7−7xEu7xO5F11(x= 0.08), the CB goes down to 0.28 eV, and the top of VB lifts up to 3.81 eV. Compared with the pure sample, Eu3+-doped Bi 7O5F11shows the deviated absorption edge created by some donor defects. When undoped Bi 7O5F11is excited by UV light greater than its band energy, an electron (e−) in the VB rises up to the CB, leav- ing a hole (h+) behind in the valence band. The e−–h+pair can recombine radiatively resulting in intrinsic luminescence, nonradia- tively react with multi-phonons and defects creating heat release, or move to the surface taking part in the photocatalytic process. When Bi7O5F11:Eu3+is irradiated with UV light, a photogenerated electronin the CB relaxes to5D1and5D0states, where the electron transition occurs to the7FJ(J = 0–4) ground state of Eu3+corresponding to red- luminescence [Fig. 3(a)]. The photo-created hole in the VB could have a recombination with an electron after excitation or move to the particle surfaces taking part in a photocatalytic reaction when Bi7O5F11:Eu3+is applied as a photocatalyst. The ground state of the lowest configuration 6s2of Bi3+is 1S0, while the excited configuration 6s6p gives rise to a triplet state (3P0,1,2) and a singlet state1P1in the order of increasing energy [Fig. 3(b)].1S0→3P1(the so-called A-band) and1S0→3P2 (B-band) are spin forbidden. A-band is allowed by spin–orbit cou- pling between1P1and3P1and, in some cases, due to vibronic mixing of3P2with1P1.B-band is very low. The highest energy transi- tion1S0–1P1is an allowed electronic–dipole transition, which has a position in the VUV (high energy) region.1,25 In this work, we observed the intrinsic luminescence of Bi7O5F11at 300 K. It is known that the emission of Bi-compounds, unfortunately, usually quenches at room temperature.26According to the structure of Bi 7O5F11, the connections between the Bi3+ions are Bi–O–Bi, Bi–F–Bi, and Bi–O–Bi bridges. This characteristic is in favor of intrinsic luminescence because of the diluted activator concentration (Bi3+) in the lattices. Figure 3(c) exhibits the excitation and emission at 300 K. The corresponding Stokes shift is 18 400 cm−1, and the FWHM is 6700 cm−1. This big Stokes shift value is typical for Bi- containing oxides25,27such as BiPO 4(1.9 eV, 15 337 cm−1),28 K3Bi5(PO 4)6(2.2 eV, 17 741 cm−1),28and K 2Bi(PO 4) (WO 4) (2.2 eV, 17 741 cm−1),28etc. Usually, stokes shift can be related to the energy band structure of the system. The absorption depends on the excited APL Mater. 8, 081109 (2020); doi: 10.1063/5.0012636 8, 081109-4 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm FIG. 3 . Band structure and suggested energy positions of pure and Eu3+-doped Bi7O5F11(a), sketch of transitions for Bi3+and spectroscopic rules (b), pho- toluminescence spectra ( λem= 527 nm andλex= 254 nm) (c), and decay curves monitored at λem= 650 nm and 450 nm (λex= 266 nm) (d) of Bi 7O5F11. states, and the emission depends on the ground states. The bismuth compounds usually show broad emission and excitation bands with large Stokes shifts. The large Stokes shifts can be related to the asym- metrical coordination of the 6 s2ions in the lattices.25,27It could be tentatively proposed that there are asymmetrically surrounding and structural distortions of Bi3+ions in Bi 7O5F11lattices, which give rise to luminescence characterized by a broad emission band with a large Stokes shift. Decay lifetime is one of important physical parameters, which characterize the recombination between the induced electrons and the holes in a solid. Figure 3(d) shows the decay curve of 450 nm and 650 nm by monitoring two different positions on both sides of the spectra. The curves have a nonexponential characteristic following the biexponential decay, y(t)=A1exp(−t/τ1)+A2exp(−t/τ2), (2) where τ1and τ2are the fast and slow decay lifetimes, respectively. The lifetime, here, could correspond to the average time in which the molecule stays in the excited state before it returns back to the fundamental state. The average lifetime ( τave) was estimated by the following equation: τ=A1τ2 1+A2τ2 2 A1τ1+A1τ2. (3) The two emission wavelengths 450 nm and 650 nm have similar life- times of 0.039 μs and 0.041 μs, respectively. This is in the same orderas the reported Bi-compounds such as Bi 4Ge3O12(0.43 ±0.08 μs),17 α-Bi 2O3(0.104 μs),18and BiOCl (5.89 ns).19Figure 4(a) shows the chromaticity diagram (CIE 1931) representing the emission color of Bi7−7xEu7xO5F11(x= 0, 0.01, 0.03, 0.05, 0.08). The phosphor under the UV-lamp and the working LED-device shows the red color, as shown in Fig. 4(b). The luminescence of Bi 7O5F11was also detected under x-ray excitation. However, no clear emission signals were detected indicating a poor scintillating ability. D. Luminescence of Eu3+-activator The normalized excitation of Bi 7−7xEu7xO5F11shows two kinds of absorption [Fig. 5(a)]: one is the narrow excitation peaks (350 nm–480 nm) from 4 f–4ftransitions of Eu3+; another is the broadband (200 nm–350 nm) from the CT transition of Eu3+ to O2−. Compared with the CT band, 4 f–4ftransitions in the near region become stronger with Eu3+doping. This confirms that Bi7−7xEu7xO5F11can harvest both the UV and visible light for luminescence and photocatalysis. The sharp peaks were observed corresponding to 5D0,1→7FJradiative transitions with J= 0–4. The experimental integrated intensities were shown in the normalized emission of Bi7−7xEu7xO5F11[Fig. 5(b)]. The emission gradually increased by enhancing Eu3+doping [see the inset of Fig. 5(b)]. The strongest emission intensity was detected with x= 0.08 (8.0 mol. %). First, except for the5D0transition, the emission from the higher level, APL Mater. 8, 081109 (2020); doi: 10.1063/5.0012636 8, 081109-5 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm FIG. 4 . CIE chromaticity diagram of Bi 7−7xEu7xO5F11(x= 0, 0.01, 0.03, 0.05, 0.08) (a), and digital photos of the representative sample (x = 0.08) taken in natural appearance, under UV lamp, and as-packaged and working LED-lamp status (b). e.g.,5D1,2,3, was detected. The lifetimes easily distinguish the lumi- nescence from5D1,2,3 and5D0transitions.5D0states have a life- time in the order of millisecond; however,5D1,2,3states have a fast decay in an order of microsecond.29Figure 5(c) shows the typical decay monitored at 534 nm (5D1→7F1), which has a lifetime about tens of microsecond. This confirms that the emission is from5D1 transitions. Figure 5(d) shows the curves of5D0, which present a longer decay time (0.95 ms–1.52 ms). A small rise time in the ini- tial part of the curves was estimated to be 0.04 μs, which is com- parable to the decay time of intrinsic emission (0.039 μs–0.041 μs) [Fig. 3(d)]. This indicates the energy transfer from the host to the activators of Eu3+ions. Usually in Eu3+doped phosphors, the lumi- nescence from5D1,2,3can be easily quenched compared to5D0due to cross-relaxation processes such as (5D2→5D0)↔(7F0→7F5) or (5D1→5D0)↔(7F0→7F3). However, in Bi 7−7xEu7xO5F11, emis- sion from the upper states (5D1,2,3) appeared. This indicates thatthe concentration quenching has a little influence on the5D1,2,3 transition. The high energy phonons in a host can usually quench the luminescence from all emitting levels via fast multi-phonon relax- ations. For example, vibration energy of a P–O bond is high (∼1000 cm−1) in phosphate. This implies that only 2–3 phonons are required to bridge up the gap (2000 cm−1) between5D1and 5D0(or5D2and5D1). Therefore, Eu3+in phosphate presents only 5D0→7F0,1,2,3,4 transitions. As shown in Fig. S8 of the supple- mentary material, Bi 7O5F11has a strong Raman signal in the low energy region. By taking the mode of 130 cm−1, at least 15 phonons are required to fill the energy gap of 2000 cm−1. This explains the observation of5D1,2,3transitions in Bi 7−7xEu7xO5F11. The electrons in the CB of Eu3+-doped Bi 7O5F11will relax to 5D0,1,2,3 , where they stay temporarily for a longer decay time (tens of milliseconds to several microseconds) than that situation on the CB of the pure sample (0.041 μs). The holes in the VB of Eu3+- doped Bi 7O5F11also obtain the corresponding long lifetimes, which provide the longer time for light-produced charges to move to the particle surface. This greatly benefits for the photocatalytic behav- ior in Eu3+-doped Bi 7O5F11. The absolute quantum efficiency (QE) was measured by the integrated sphere method. The QE value of 26.8% was measured in Bi 7−7xEu7xO5F11(x= 0.08). Eu3+-doped Bi7O5F11shows a gray white color under daylight. A red color could be seen under an UV lamp, as shown in Fig. 4(b). An experimental LED-lamp was fabricated by encapsulating the phosphor with some transparent resin. The red-lighting LED could be obtained when the electricity is switched on. The result suggests that the phosphors are competent for the LED-lamp application. E. Photochemical abilities Figure 6(a) shows the photocatalytic effects of Bi 7−7xEu7xO5F11. There is a tiny effect ( <7%) in the blank test, indicating a negligible self-bleaching of RhB. Eu3+doping resulted in the improved pho- tocatalysis. Bi 7−7xEu7xO5F11(x= 0.08) shows the best effect on the photodegradation. The pseudo-first-order rate equation was used to determine the kinetic constant ( k), ln(C0/Ct)=kt, (4) where C0is the initial concentration and Ctrepresents the level at time ( t). Figure 6(b) shows the linear relationship between ln( C0/Ct) andt, suggesting the pseudo-first-order nature of the reactions. The kinetic constants were compared in Fig. 6(c). Under the same exper- imental conditions, Eu3+-doped samples showed the significantly improved abilities. Several reactive species have been reported in photocatalysis, such as holes (h+), hydroxyl radicals (•OH), and superoxide rad- ical anions (O 2•−), etc. Figure 6(d) shows the quenching experi- ments with the existence of trappings. The results show that the addition of AO (h+scavenger) and t-BuOH (•OH scavenger) obvi- ously depresses the degradation. This indicates that•OH and h+ play the important roles in the reactions. The luminescence of Bi7−7xEu7xO5F11(x= 0.08) was measured with three recycling tests to study its durability (see Fig. S9 of the supplementary material). There are similar5D0,1→7FJ(J= 0–4) transitions before and after the degradation. The emission intensity only shows some decrease APL Mater. 8, 081109 (2020); doi: 10.1063/5.0012636 8, 081109-6 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm FIG. 5 . The excitation (a), normalized emission spectra (b), decay curves of 5D1(534 nm) (c), and5D0states (615 nm) (d) of Bi 7−7xEu7xO5F11(x= 0.01, 0.03, 0.05, 0.08). The inset in (b) shows the experimental spectral intensity. with the recycling times. The results show that the samples keep the good crystal structure with the photodegradation application. The proposed mechanism is elucidated in Fig. 7. In pure Bi7O5F11[Fig. 7(a)], the band-to-band absorption after theincident excitation results in an exciton, that is, an e−in the CB and a h+in the VB. The dominant physical processes could be closely related to luminescence and photocatalysis. One is the radiative recombination between the e−–h+pair, which is responsible for FIG. 6 . Degradation efficiencies (a), the kinetic calculations (b), comparison of kinetic constant (c) of Bi 7−7xEu7xO5F11 (x= 0, 0.01, 0.03, 0.05, 0.08), and the quenching experiments with the exis- tence of the selected scavengers (d). APL Mater. 8, 081109 (2020); doi: 10.1063/5.0012636 8, 081109-7 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm FIG. 7 . Proposed mechanism for the photocatalysis of Bi 7O5F11(a) and the improved effects of Eu3+doping via prolonging the lifetime of light-created charges in Eu3+doped samples (b). the luminescence transition. Another is the migration of the light- produced e−and h+to the surfaces, where the photocatalytic reac- tions could take place. There is a competition between the two pro- cesses.30The longer decay time of the excited states could provide more opportunities for the immigration of the light-created charges. As shown in Fig. 3(a), it is difficult for an electron to take part in the photocatalytic reaction because its potential energy (0.12 eV vs NHE) is more positive to produce a superoxide radical anion. The decay measurements of pure Bi 7O5F11[Fig. 3(d)] verify a fast life- time of 0.041 μs. It indicates that a hole could have a recombination with an electron in 42 ns after excitation. This is a short time for a hole to move to the surfaces. The intrinsic emission is quenched in Bi 7O5F11:Eu3+due to the energy transfer from the host to Eu3+ions. This process relates to the relaxation of an electron in the CB to5D1and5D0states followedby a red-luminescence transition to the7FJ(J = 0–4) ground state of Eu3+ions. The competitive process is also the migration of the photogenerated charges to the surface for photocatalysis reactions when Bi 7O5F11:Eu3+is applied as a photocatalyst [Fig. 7(b)]. In Eu3+- doped Bi 7O5F11, there are many midgap states formed by the rich 4f energy levels such as5D0,1,2and (7F0–6), which have a longer lifetime of about 1 ms. An excited e−was manipulated: “jumping” from the VB to CB, relaxing to5D0,1,2,3 , and then, having a transition to7F0–6 ground states. In other words, the lifetime of a h+in the VB can be significantly prolonged from 0.041 μs in pure Bi 7O5F11to 1 ms in Bi 7O5F11:Eu3+. It can leisurely move to the surfaces of the par- ticles. Consequentially, the improved photocatalysis of Eu3+-doped Bi7O5F11was realized. IV. CONCLUSIONS In conclusion, we synthesized Bi 7O5F11:Eu3+by the hydrother- mal reaction method. The samples show well-crystallized microplates with a thickness of 300 nm and a side length of several μm. Under UV excitation, Bi 7O5F11shows an intrinsic luminescence band peaked at 527 nm with a decay lifetime of 0.041 μs at 300 K. Eu3+ doping quenches this intrinsic emission, while it introduces the tran- sitions from both the5D0and5D1,2,3 levels to7FJ(J = 0–4) in Bi7O5F11:Eu3+. An experimental red-LED lamp was fabricated by encapsulating Bi 7O5F11:Eu3+with a transparent resin. The photo- catalytic effects on photodegradation of RhB dye were improved in Bi 7O5F11:Eu3+. Importantly, Eu3+doping in Bi 7O5F11moves the wavelength required in photocatalysis from the UV to visible region by introducing 4f levels. The fast decay time (0.041 μs) of Bi 7O5F11 causes an efficient recombination of the light-induced charges. How- ever, in Eu3+-doped Bi 7O5F11, 4f levels inserted in the bandgap provide longer lifetimes of 1 ms for the excited states. This effect delays the recombination of e−and h+, resulting in higher photo- catalytic efficiencies. By considering its simultaneous luminescence and photocatalysis, Bi 7O5F11:Eu3+could be further investigated for a multifunctional material. SUPPLEMENTARY MATERIAL See the supplementary material for supporting figures associ- ated with this article. ACKNOWLEDGMENTS This research was supported by the National Research Foun- dation of Korea (NRF) grant funded by the Korean government (MSIT) (Grant No. 2020R1F1A1049740). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1G. Blasse and B. Grabmaier, Luminescent Materials (Springer, Berlin, 1994). 2A. B. Djuriši ´c, Y. He, and A. M. Ng, APL Mater. 8(3), 030903 (2020). 3X. Chen, D. Wang, Y. Huang, Y. Zhang, C. Li, S. Wang, Y. Liu, and X. Zhang, APL Mater. 8(3), 031112 (2020). APL Mater. 8, 081109 (2020); doi: 10.1063/5.0012636 8, 081109-8 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm 4X. Huang, C.-Y. Niu, J. Zhang, A. Wang, Y. Jia, and Y. Song, APL Mater. 7(8), 081110 (2019). 5M. Osada and T. Sasaki, APL Mater. 7(12), 120902 (2019). 6H. Wang, X. Yan, M. Zhao, J. 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5.0007599.pdf

J. Appl. Phys. 128, 054301 (2020); https://doi.org/10.1063/5.0007599 128, 054301 © 2020 Author(s).Reduction of surface spin-induced electron spin relaxations in nanodiamonds Cite as: J. Appl. Phys. 128, 054301 (2020); https://doi.org/10.1063/5.0007599 Submitted: 14 March 2020 . Accepted: 15 July 2020 . Published Online: 03 August 2020 Zaili Peng , Jax Dallas , and Susumu Takahashi ARTICLES YOU MAY BE INTERESTED IN Identification of the orientation of a single NV center in a nanodiamond using a three- dimensionally controlled magnetic field Applied Physics Letters 116, 264002 (2020); https://doi.org/10.1063/5.0009698 Developing silicon carbide for quantum spintronics Applied Physics Letters 116, 190501 (2020); https://doi.org/10.1063/5.0004454 Spin coherence and depths of single nitrogen-vacancy centers created by ion implantation into diamond via screening masks Journal of Applied Physics 127, 244502 (2020); https://doi.org/10.1063/5.0012187Reduction of surface spin-induced electron spin relaxations in nanodiamonds Cite as: J. Appl. Phys. 128, 054301 (2020); doi: 10.1063/5.0007599 View Online Export Citation CrossMar k Submitted: 14 March 2020 · Accepted: 15 July 2020 · Published Online: 3 August 2020 Zaili Peng,1 Jax Dallas,1and Susumu Takahashi1,2,a) AFFILIATIONS 1Department of Chemistry, University of Southern California, Los Angeles, California 90089, USA 2Department of Physics and Astronomy, University of Southern California, Los Angeles, California 90089, USA a)Author to whom correspondence should be addressed: susumu.takahashi@usc.edu ABSTRACT Nanodiamonds (NDs) hosting nitrogen-vacancy (NV) centers are promising for applications of quantum sensing. Long spin relaxation times ( T1and T2) are critical for high sensitivity in quantum applications. It has been shown that fluctuations of magnetic fields due to surface spins strongly influence T1and T2in NDs. However, their relaxation mechanisms have yet to be fully understood. In this paper, we investigate the relation between surface spins and T1and T2of single-substitutional nitrogen impurity (P1) centers in NDs. The P1 centers located typically in the vicinity of NV centers are a great model system to study the spin relaxation processes of the NV centers. Byemploying high-frequency electron paramagnetic resonance spectroscopy, we verify that air annealing removes surface spins efficiently and significantly reduces their contribution to T 1. Published under license by AIP Publishing. https://doi.org/10.1063/5.0007599 I. INTRODUCTION Diamond is a fascinating material in physics, chemistry, and biology. For example, a negatively charged nitrogen-vacancy (NV)center in diamond is a promising platform for fundamental sciences and applications of quantum sensing because of its unique magnetic and optical properties as well as a long coherence time atroom temperature. 1–11Magnetic sensing using a single NV center has been utilized to improve the sensitivity of electron paramagneticresonance (EPR) spectroscopy to the level of a single spin. 12–20 NV-detected EPR allows the detection of external spins existing around the NV center within several nanometers. NV-based sensingis also useful to detect electric field, temperature, strain, and pHvalue in a nanoscale volume. 21–24In NV-detected magnetic sensing, a magnetic field is detected through the measurement of the spin relaxation times of NVs such as T2and T1. For example, in NV-based AC magnetic sensing measurement using a spin echosequence, the detectable magnetic field is proportional to 1 =ffiffiffiffiffiT 2p.25 A small number of Gd3þspins has been detected through sensing of fluctuating magnetic fields from Gd3þspins.14In this case, the detec- tion is achieved by measuring changes of T1relaxation time and the detectable magnetic field is proportional to 1 =T1. Thus, long T1and T2times are desired for high detection sensitivity.In NV-based magnetic sensing applications, it is also critical to position the NV center near a target of the magnetic field sensing. NVs located near the diamond surface and NVs in nano- diamonds (NDs) will, therefore, be an ideal platform for the appli-cations. However, T 1and T2relaxation times of those NVs are often significantly reduced by surface defects and impurities includ- ing dangling bonds, graphite layers, and transition metals.14,17,26–36 For instance, it has been reported that shorter T1and T2were observed from shallow NVs.31,37It has also been reported that T1 of NVs in NDs is shorter than T1in bulk diamond. The recent study showed that T1of NV centers is shorter in a smaller size of NDs, and the result implies a decoherence process due to surfaceimpurities, although the surface impurities were not measured inthe study. 14Moreover, control of the diamond surface enables the determination of spin relaxation mechanisms, subsequently improv- ing the sensitivity of the NV-based magnetic sensing techniques.The recent experiment by Tsukahara et al. showed that air anneal- ing efficiently removes graphite layers compared with tri-acid clean- ing and increases the T 2time 1.4 times longer.38 In this paper, we investigate the relation between surface spins and T1and T2of single-substitutional nitrogen impurity (P1) centers in NDs using high-frequency (HF) EPR spectroscopy.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 054301 (2020); doi: 10.1063/5.0007599 128, 054301-1 Published under license by AIP Publishing.Our previous study on NDs suggested that the surface spins are dangling bonds located in the surface shell with a thickness of /difference9 nm.36Therefore, the present study aims to remove the surface spins by etching of NDs more than 9 nm and improves the spinrelaxation times. Although T 1and T2of NV centers are the primary interest for the quantum sensing applications, there are a few advantages to studying the spin relaxation on P1 centers over NV centers. First, NV centers are located near P1 centers, shownby the detection of their magnetic dipole coupling via double elec-tron–electron resonance spectroscopy. 12,20,39Therefore, their T1 and T2times are similar and the relaxation mechanisms are often common.6Second, as shown in the previous study,36EPR signals of both P1 and surface spins are observable in the same measure-ment. This allows us to determine the amount of surface spins andto study the spin relaxations using the same ND sample. In theexperiment, we employ air annealing to etch the diamond surface efficiently. The performance of the air annealing is confirmed by dynamic light scattering (DLS) and 230 GHz EPR experiments.The result of the DLS characterization shows a uniform etchingand a linear etching rate of ND samples. We also confirm thereduction of the surface spins after the annealing process with high resolution 230 GHz EPR spectral analysis. Then, we investi- gate T 1of P1 centers after the annealing using 115 GHz pulsed EPR spectroscopy. The 115 GHz EPR configuration is advanta-geous for pulsed EPR experiment because of its higher output power. The temperature- and size-dependence study elucidates the surface spin-induced T 1process. From the result, we find that air annealing significantly reduces the presence of surface spins,but a small fraction remains, even after the thickness of NDs isreduced more than 9 nm. We also find that the surface spin con- tribution on T 1is suppressed by a factor of 7 :5+5:4 after annealing at 550/C14C for 7 h. With the same annealing condition, T2is improved by a factor of 1 :2+0:2. II. MATERIALS AND METHODS A. Nanodiamond Five different sizes of diamond powders were investigated in the present study. The samples include micrometer-sized diamondpowders (10 +1μm) (Engis Corporation) and four different sizes of NDs (Engis Corporation and L.M. Van Moppes and Sons SA).The mean diameters of the ND samples specified by the manufac- turers are 550 +100 nm, 250 +80 nm, 100 +30 nm, and 50+20 nm. All diamond powders were manufactured by mechan- ical milling or grinding of type-Ib diamond crystals. The concen-tration of nitrogen related impurities in the ND powders is in theorder of 10 –100 parts per million (ppm) carbon atoms. B. Air annealing The air annealing process was performed using a tube furnace (MTI Corporation) where a ND sample was positioned in a quartz tube located in the cylindrical access of the furnace. For the prepa-ration of the annealing process, the ND sample was placed in a5 ml of acetone. The ND sample in acetone was then mixed by uti- lizing ultrasound sonication for 10 min at room temperature in order to achieve uniform dispersion. After the ultrasoundsonication, the sample solution was placed in a crucible and kept in a fume hood overnight (without application of heating) in order to evaporate acetone from the crucible. In the air annealing process, thetemperature of the furnace was first stabilized at the annealing tem-perature (550 /C14C in the present case), and then the ND sample in the crucible was inserted at the center of the quartz tube. In order to improve homogeneity of the application of the air annealing over the ND powders, the NDs were mixed by a lab spatula periodicallyduring the annealing (typically mixed for 30 s every 10 min). We alsolimited the initial amount of ND samples to be approximately 30 mgfor the homogeneous application of the air annealing. C. Dynamic light scattering The size of a diamond powder sample was characterized by dynamic light scattering (DLS) (Wyatt Technology). A diamondpowder sample of /difference1 mg was suspended in methanol and soni- cated for 2 h before the measurement of DLS. The DLS measure- ment was performed with a 632 nm incident laser and a 163 :5 /C14of detection angle. The second correlation data were analyzed usingthe constrained regularization method to obtain particle sizes. D. HF EPR spectroscopy HF (230 GHz and 115 GHz) EPR experiments were performed using a home-built system at University of Southern California.The HF EPR spectrometer consists of a high-frequency high-power solid-state source, quasioptics, a corrugated waveguide, a 12.1 T superconducting magnet, and a superheterodyne detection system.The output power of the source system is 100 mW at 230 GHz and480 mW at 115 GHz. A sample on a metallic end-plate at the endof the corrugated waveguide is placed at the center of the 12.1 T EPR superconducting magnet. Details of the system have been described elsewhere. 40In the present study, the diamond powder sample was placed in a Teflon sample holder (5 mm diameter), typ-ically containing the diamond powders of 5 mg. 41For cw EPR experiments, the microwave power and magnetic field modulation strength were adjusted to maximize the intensity of EPR signals without distortion of the signals.36A typical modulation amplitude was 0.02 mT at a modulation frequency of 20 kHz. III. RESULTS AND DISCUSSION We employed air annealing for the removal of the surface spins in the present study. In the air annealing, the surface removalis caused by etching by oxygen where oxygen molecules oxidizecarbon and form gaseous CO and CO 2. We first compared the weight of the ND sample before and after the annealing process. Figure 1(a) shows the ND normalized weight as a function of the annealing time. In the experiment, the annealing was done at anannealing temperature of 550 /C14C. The result shows linear reduction in ND weight with increased annealing time. The size of the ND samples was then characterized using DLS. As shown in Fig. 1(b) , the ND size decreased from dpeak¼53:4 nm to 22.4 nm after the annealing for 9 h. The observed reduction and narrow distributionof the size indicate a successful and uniform application of the annealing to NDs. Figure 1(c) shows the ND size as a function of the annealing duration. We observed a linear relationship betweenJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 054301 (2020); doi: 10.1063/5.0007599 128, 054301-2 Published under license by AIP Publishing.the size reduction and the annealing duration. A reduction rate of 3.5 nm/h was obtained from the linear fit. Next, we characterized paramagnetic spins existing in NDs using 230 GHz EPR spectroscopy. Figure 2(a) shows 230 GHz continuous-wave EPR spectra on 50-nm ND samples before andafter the air annealing. The measurements were performed at room temperature. As shown in Fig. 2(a) , all spectra contain a pro- nounced and broad EPR signal at 8.206 T and a narrow EPR signalat/difference8:207 T. From the EPR spectral analysis shown in the inset of Fig. 2(a) , we identified that the EPR signal at 8.207 T is from P1 centers while the signal at 8.206 T is from the surface spins (dangling bonds). The result is consistent with the previous HF EPR study. 36As shown in Fig. 2(a) , the intensity of the EPR signals from the surface spins decreases significantly after the annealing. Ingeneral, the EPR intensity is related to the spin population, and we,therefore, analyzed the EPR intensity ratio between the surface spins ( I S, where S represents surface spins) and P1 ( IP1) to deter- mine their spin population ratio. For example, we obtained IS=IP1 to be 61 and 5 with no annealing and 9 h annealing, respectively. The result from the EPR intensity and DLS analyses was summa-rized in Fig. 2(b) . Since our previous HF EPR study of the non-annealed NDs showed the core –shell structure with the shell thickness ( t) of 9 nm, 36we first consider the core –shell model to understand the size dependence of the EPR intensity. In thecore–shell model, the EPR intensity ratio IX IP1/C18/C19 coreshell¼ρXVX ρP1VP1¼ρX½4π=3fðd=2Þ3/C0ðd=2/C0tÞ3g/C138 ρP1½4π=3ðd=2/C0tÞ3/C138, where ρX(ρP1) is the density of the surface spins (P1 spins) and VX (VP1) is the volume of the surface spin (P1 spin) locations. The cal- culated ( IS=IP1)coreshell is shown in Fig. 2(b) .H o w e v e r ,w eo b s e r v e da poor agreement with the experimental data in the range of d,35. There may be two possible reasons to explain the result. First, asreported previously, 42–46the etching rate of the air annealing depends on a crystallographic axis. It has been shown that the etching rate of the (111) plane is a couple of times faster than the(100) plane. 47However, this can explain only the dependence of EPR but not the dependence of DLS. Another possible reason is the crea-tion of a small amount of surface spins during air annealing. For instance, it has been reported that dangling bonds were created by air annealing, especially when the surface termination was dominatedby C –H bonds. 48In the latter scenario, the surface spins in the non- annealed NDs (dangling bonds) are located in the shell, and then airannealing removes the dangling bonds in the shell as well as creates a small amount of dangling bonds on the surface [see Fig. 2(a) ]. To take into account the surface spins created by air annealing, weadded a contribution from the surface spin model with which(I s=IP1)surface ¼ρs=ρP1[4π(d=2)2]=[4π=3(d=2)3)]/ρs=d.ρsis the surface spin density, treating as a constant here. As shown in Fig. 2(b) , the sum of the core –shell and surface models agrees with the experimental result, supporting the latter case. Next, we measured the spin relaxation times ( T1and T2)o f the ND samples. The experimental results of the 50-nm ND sample is shown in Fig. 3(a) . The measurements of the T1and T2 relaxation times of P1 centers were carried out using the inversion FIG. 1. Overview of the air annealing experiment. (a) The normalized weight as a function of the annealing duration with annealing at 550/C14C. The red solid line shows a linear fit to obtain the rate of weight reduction. The weight reductionrate was 0 :12 h/C01. Each sample was weighed five times. The error bar repre- sents the standard deviation of the measurements. (b) DLS results for the size characterization of the ND samples before and after the annealing for 5, 7, and 9 h. The diameter at the maximum in the distribution ( dpeak) is indicated. The obtained polydispersity index (PDI) were 0.11, 0.07, 0.06, and 0.08 for theno-annealing sample and the annealing for 5, 7, and 9 h samples, respectively. (c)d peak as a function of the annealing duration. The red solid line represents the result of a linear fit. The error bar represents the standard deviation (calcu-lated by d peakffiffiffiffiffiffiffi PDIp ).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 054301 (2020); doi: 10.1063/5.0007599 128, 054301-3 Published under license by AIP Publishing.FIG. 3. T emperature dependence of T1and T2of P1 centers in NDs. (a) The T1measurement using the inversion recovery measurement. The measurement was performed at 200 K. The pulse sequence is Pπ/C0T/C0Pπ=2/C0τ/C0Pπ/C0τ/C0echo, where Pπ=2and Pπare π=2- and π-pulses, respectively, τis a fixed evolution time and an evolution time Tis varied in the measurement. In the measurement, the pulse lengths of Pπ=2and Pπwere 300 ns and 500 ns, τ¼1:2μs, and the repetition time was = 10 ms. The inset shows the spin echo measurement at 200 K to obtain T2. The pulse sequence is Pπ=2/C0τ/C0Pπ/C0τ/C0echo, where τis varied in the measurement. The pulse parameters for the T2measurement were Pπ=2¼300 ns, Pπ¼500 ns, and the repetition time = 10 ms. The errors associated with T1 and T2were obtained by computing the standard error. (b) T emperature depen- dence of 1 =T1on various sizes of NDs. The solid circles are experimental data and the solid lines are fits using Eq. (1). (c) T2on various sizes of NDs. The error bars are smaller than the dots representing the T2value. FIG. 2. cw EPR analysis of 50-nm NDs before and after the air annealing. (a) Signal intensity as a function of magnetic fields in T esla with no annealing, annealing for 5 h and 7 h. The solid green lines represent the experimental data. The inset on the top right shows contributions of P1 and surface spins (S)on the EPR spectrum, which were extracted from the EPR spectral analysis.Drawings representing NDs under the annealing process are also shown in the inset. The red arrows in the drawing represent the P1 centers, and the blue arrows represent surface spins. (b) The EPR intensity ratio I s=IP1as a function of the diameter ( dpeak). The blue solid circles with error bars represent Is=IP1 obtained from EPR spectral analysis. The details of the EPR spectral analysis is described in the supplementary material . The gray dashed line is the simulated (Is=IP1)coreshell . The green dashed line is the simulated ( Is=IP1)surface .Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 054301 (2020); doi: 10.1063/5.0007599 128, 054301-4 Published under license by AIP Publishing.recovery and the spin echo sequences, respectively. The T1and T2 measurements of P1 centers were performed at a microwave frequency of 115 GHz and 4.103 T, corresponding to the centerpeak of the P1 EPR spectrum. By fitting the change of the spinecho intensity with a single exponential function, we obtained T 1 to be 0 :382+0:080 ms, and T2is 0 :413+0:007μs as shown in the inset of Fig. 3(a) (see the supplementary material for the description of the T1and T2determination). Moreover, we mea- sured temperature dependence of T1and T2.Figure 3(b) and Table I summarize the result of the T1measurements as a function of temperature. We observed that T1times increase drastically by decreasing temperature. In addition, the temperature dependence is strongly correlated with the size of the diamond powder. To under-stand the temperature dependence, we first considered a contribu-tion of the spin-lattice relaxation observed in bulk diamond.According to the previous studies of T 1of bulk diamond, the tem- perature dependence of T1is well explained by a spin –orbit induced tunneling model,6,49in which a spin flip event occurs due to the tunneling between P1 ’s molecular axis orientations. Using the spin –orbit induced tunneling model, we write that 1 =T1is pro- portional to T5, namely, 1 =T1¼CT5, where the T-linear term in the spin –orbit induced tunneling model49is omitted because of its negligible contribution in the present temperature range. By fitting the experimental data of the 10- μm diamond to the T5model, we obtained C¼(2:96+0:52)/C210/C010s/C01K/C05, which is in a good agreement with previous finding.6,49The result was obtained from a weighted fit analysis in order to take into accountthe uncertainty in T 1values (see the supplementary material for the details). Furthermore, in cases of smaller diamond samples[from 550-nm to 50-nm NDs in Fig. 3(b) ], we observed a strong deviation from the T 5model and found that 1 =T1at low tempera- tures highly correlates with the size of NDs. Recent investigation ofshallow NV centers as well as NV centers in a single ND showedthat T 1in NDs is attributed to surface spins.14,17,30,31Since the surface spins were also detected from the same ND samples in our experiment, it is likely that the surface spins also influence T1of P1 centers in NDs. In order to take into account relaxation processesfrom both the surface spins and the spin –orbit induced tunneling, we consider the following for 1 =T 1: 1 T1¼CT5þΓs, (1) where Γsis the 1 =T1contribution from surface spins, originated by fluctuations of the magnetic dipole fields from the surface spins. Γs is assumed to be independent of temperature in a temperature range of the present experiment. In this 1 =T1model, when temper- ature increases, the first term (the spin –orbit induced tunneling contribution) increases. Therefore, when a sample has a significantcontribution from the surface spin relaxation, 1 =T 1has less pro- nounced temperature dependence. We performed a weighted fit analysis on 50-nm, 100-nm, 250-nm, and 550-nm ND samples todetermine their Γ s(see the supplementary material for the details). As shown in Fig. 3(b) , we found a good agreement between the temperature dependence of 1 =T1and the model. For example, we obtained that Γsof the 50-nm ND sample was 2430 +650 s/C01. FIG. 4. T emperature dependence of T1and T2of P1 centers in the annealed ND samples (initial diameter = 50 nm). (a) T1of the annealed ND samples as a function of temperature. Experimental data points are indicated by blue circles and red triangles for air annealing at 550/C14C for 5 h and 7 h, respectively. The blue and red solid lines are corresponding fits utilizing Eq. (1).T1data with no annealing (gray solid line) and the data of a bulk diamond (green solid line) are shown. The arrow represents the reduction of Γs. (b)Γsas a function of the ND diameter. The red solid line shows the fit result to the ρs=d4model for NDs without annealing. The green solid line shows the ρs=d4line simulated for the annealed NDs. The orange arrow represents the reduction of Γs. The error bar in the Γsis included and obtained by computing the 95 % confidence interval. (c)T2as a function of temperature for the non-annealed and annealed samples.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 054301 (2020); doi: 10.1063/5.0007599 128, 054301-5 Published under license by AIP Publishing.Furthermore, we investigated the temperature- and size- dependence of T2relaxation time of P1 centers. In contrast to the result of T1,T2of P1 centers in the studied NDs does not show noticeable temperature dependence [see Fig. 3(c) ].Table II shows the summary of the temperature- and size-dependence of T2as well as the mean T2(T2) which was obtained from a weighted fit analysis in order to take into account the errors in the T2values (see the supplementary material for the details of the T2analysis). T2for 50-nm ND and 550-nm samples were 0 :474+0:060μs and 2:03+0:10μs, respectively. Therefore, T2of the 50-nm NDs is approximately 4.3 times shorter than that of 550-nm NDs. The result indicates the effect of the surface spins on T2. On the other hand, T2of 550-nm and 10 /C0μm are similar ( /difference2μs). This is proba- bly because couplings to neighboring P1 centers dominates their T2 processes. Finally, we study the spin relaxation times ( T1and T2) of the annealed NDs. As shown in Fig. 4(a) ,T1times in the annealed diamond became longer after the annealing in the measured tem-perature range. In addition, as shown in Fig. 4(a) , the T 1times of the annealed NDs are still shorter than that of bulk diamond, implying the existence of remaining surface spins. To extract the contribution of the surface spins, we employed Eq. (1)to determine Γs. From the analysis, we indeed found that Γsin the annealed NDs are smaller than that of the non-annealed samples. The obtained Γsare 531+217 s/C01and 325+217 s/C01for the NDs annealed at 550/C14C for 5 h and 7 h, respectively, which are 4 :6+ 2:2 and 7 :5+5:4 times smaller than that of the non-annealed50-nm NDs as shown in Fig. 4(b) (see the supplementary material for the calculation of the Γsimprovement factor). We next discuss a model of the surface spin-induced T1(Γs). As reported previously,14,30,50by considering fluctuating magnetic fields ( Bdip) from surface spins, Γsis proportional to the variance ( hB2 dipi) and the spin density ( ρs). By assuming that surface spins cover the whole surface uniformly,B 2 dip(r! P1)/Ð Sρsb2 dip(r!/C0r! P1)dS, where the radius vectors ( r! and r! P1) define the locations of the surface and P1 spins relative to the center of the ND, respectively. bdip(r!) is the magnetic dipole field from the surface spins. By taking into account the quantiza- tion axis of P1 and the surface spins along the external magneticfield and considering a spherical shape of NDs and a spatiallyuniform ρ s,b2 dip(r!) is proportional to 1 =d6and the surface integral is proportional to d2, where dis a diameter of a ND, the magnetic field fluctuations [ B2 dip(r!)] is, therefore, proportional to ρs=d4and Γsis also proportional to ρs=d4. The Γsvalues obtained from the temperature dependence T1inFig. 3(b) were plotted as a function of the ND size in Fig. 4(b) . We found a good agreement between the obtained Γsvalue and the 1 =d4size dependence. Thus, the result supports the T1relaxation mechanism in NDs due to the surface spins. Furthermore, as shown in Fig. 4(b) ,Γsof the annealed NDs are very different from the 1 =d4line of the non- annealed NDs, indicating significant reduction of the surface spin density. Using the same model ( Γs/ρs=d4), we estimated that ρs for the annealed NDs is /difference100 times smaller than that of the non- annealed NDs [ Fig. 4(b) ].TABLE I. Summary of T1analyses. For the Γsanalysis, Eq. (1)and C= 2.96 × 10−10(s−1K−5) were used. T1andΓsare shown with three significant figures. The errors in T1represent the standard error of the mean. The errors in Γswere calculated as the 95% confidence interval. T1(ms) Sample 100 K 150 K 200 K 250 K 300 K Γs(s−1) 50 nm 0.581 ± 0.339 0.519 ± 0.341 0.382 ± 0.080 0.132 ± 0.034 0.320 ± 0.020 2430 ± 650 100 nm 1.74 ± 0.12 1.68 ± 0.18 1.19 ± 0.23 1.02 ± 0.15 0.668 ± 0.141 587 ± 63250 nm 25.6 ± 1.2 19.3 ± 0.7 7.80 ± 0.20 2.14 ± 0.07 1.26 ± 0.03 34.0 ± 16.9 550 nm 84.9 ± 7.4 42.5 ± 2.2 10.3 ± 0.4 2.58 ± 0.03 1.35 ± 0.03 8.12 ± 22.03 10μm 1200 ± 580 62.2 ± 5.0 12.0 ± 0.4 3.15 ± 0.06 1.36 ± 0.03 … Annealed (5 h) 2.46 ± 0.50 1.41 ± 0.26 1.31 ± 0.35 0.885 ± 0.188 0.679 ± 0.240 531 ± 217Annealed (7 h) 4.37 ± 1.37 1.86 ± 0.48 1.37 ± 0.57 1.06 ± 0.27 0.847 ± 0.595 325 ± 217 TABLE II. Summary of T2analyses. T2and T2are represented by three significant figures. The errors in T2represent the standard error of the mean. The errors in T2were calculated as the 95% confidence interval. T2(μs) Sample 100 K 150 K 200 K 250 K 300 K T2(μs) 50 nm 0.518 ± 0.002 0.440 ± 0.002 0.413 ± 0.007 0.398 ± 0.008 0.211 ± 0.036 0.474 ± 0.060 100 nm 0.756 ± 0.007 0.871 ± 0.011 0.835 ± 0.017 0.942 ± 0.020 1.15 ± 0.01 0.882 ± 0.214250 nm 1.48 ± 0.01 1.42 ± 0.01 1.36 ± 0.01 1.41 ± 0.01 1.02 ± 0.01 1.34 ± 0.23550 nm 1.89 ± 0.02 2.03 ± 0.01 2.00 ± 0.01 2.15 ± 0.01 1.98 ± 0.01 2.03 ± 0.1010μm 1.93 ± 0.02 1.84 ± 0.02 1.72 ± 0.01 1.59 ± 0.01 1.53 ± 0.01 1.65 ± 0.17 Annealed (5 h) 0.510 ± 0.091 0.957 ± 0.208 0.341 ± 0.252 0.551 ± 0.208 0.869 ± 0.095 0.675 ± 0.274 Annealed (7 h) 0.520 ± 0.041 0.509 ± 0.036 0.634 ± 0.051 0.604 ± 0.015 0.600 ± 0.023 0.589 ± 0.048Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 054301 (2020); doi: 10.1063/5.0007599 128, 054301-6 Published under license by AIP Publishing.In addition, the T2of the annealed diamond was studied. As shown in Fig. 4(c) , similarly to the non-annealed NDs, T2of the annealed diamond showed no temperature dependence. The meanT 2times were 0 :675+0:274μs and 0 :589+0:048μs after the 5 and 7 h annealing, respectively, showing that the extension ofT 2by a factor of 1 :4+0:6 and 1 :2+0:2, respectively (see the supplementary material for the calculation of the T2improvement factor). This improvement is due to the reduction of the surfacespins. The observed T 2improvement is comparable with the previ- ously reported result.38By considering the T2results of the non- annealed NDs, we speculate that the T2relaxation in the annealed NDs is caused by couplings to residual surface spins and P1 centers. IV. SUMMARY In summary, we investigated the relationship between the surface spins and the spin relaxation times ( T1and T2)o fP 1 centers in NDs. We reduced the amount of the surface spins usingair annealing. The amount of the surface spins was characterized by HF EPR analysis. The pulsed HF EPR experiment extracted the contribution of the surface spins on the T 1relaxation successfully. We found clear correlation between the amount of the surfacespins and T 1. In addition, the present study showed the improve- ment of T1and T2by removing the surface spins. The finding of the present investigation sets the basis to suppress the spin relaxa- tion process due to the surface spins in NDs which is critical forNV-based sensing applications. The present method is also poten-tially applicable to improve spin and optical properties of othernanomaterials. SUPPLEMENTARY MATERIAL See the supplementary material for the T 1and T2determina- tion method, the EPR spectral analysis, and the analyses of the temperature- and size-dependent T1andT2. ACKNOWLEDGMENTS We thank Benjamin Fortman for useful discussion of the EPR data analyses. This work was supported by the National ScienceFoundation (NSF) (Nos. DMR-1508661 and CHE-1611134), theUSC Anton B. Burg Foundation, and the Searle Scholars Program. This material is also based upon work supported by the Chemical Measurement and Imaging program in the National ScienceFoundation Division of Chemistry under Grant No. CHE-2004252(with partial co-funding from the Quantum Information Scienceprogram in the Division of Physics) (S.T.). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1A. Gruber, A. Dräbenstedt, C. Tietz, L. Fleury, J. Wrachtrup, and C. Von Borczyskowski, “Scanning confocal optical microscopy and magnetic resonance on single defect centers, ”Science 276, 2012 (1997).2J. Wrachtrup and F. Jelezko, “Processing quantum information in diamond, ” J. Phys. Condens. Matter 18, S807 (2006). 3R. J. Epstein, F. M. Mendoza, Y. K. Kato, and D. D. Awschalom, “Anisotropic interactions of a single spin and dark-spin spectroscopy in diamond, ”Nat. Phys. 1, 94 (2005). 4T. Gaebel, M. Domhan, I. 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5.0017892.pdf

J. Chem. Phys. 153, 054501 (2020); https://doi.org/10.1063/5.0017892 153, 054501 © 2020 Author(s).Raman studies of hydrogen trapped in As4O6·2H2 at high pressure and low temperature Cite as: J. Chem. Phys. 153, 054501 (2020); https://doi.org/10.1063/5.0017892 Submitted: 10 June 2020 . Accepted: 09 July 2020 . Published Online: 03 August 2020 Piotr A. Guńka , Li Zhu , Timothy A. 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Strobel,2 and Janusz Zachara1 AFFILIATIONS 1Faculty of Chemistry, Warsaw University of Technology, Noakowskiego 3, 00-664 Warszawa, Poland 2Earth and Planets Laboratory, Carnegie Institution for Science, 5251 Broad Branch Road NW, Washington, DC 20015, USA a)Author to whom correspondence should be addressed: piogun@ch.pw.edu.pl ABSTRACT Raman spectroscopic measurements of the arsenolite–hydrogen inclusion compound As 4O6⋅2H 2were performed in diamond anvil cells at high pressure and variable temperature down to 80 K. The experimental results were complemented by ab initio molecular dynamics simula- tions and phonon calculations. Observation of three hydrogen vibrons in As 4O6⋅2H 2is reported in the entire temperature and pressure range studied (up to 24 GPa). While the experiments performed with protium and deuterium at variable temperatures allowed for the assignment of two vibrons as Q1(1) and Q1(0) transitions of ortho andpara spin isomers of hydrogen trapped in the inclusion compound, the origin of the third vibron could not be unequivocally established. Low-temperature spectra revealed that the lowest-frequency vibron is actually composed of two overlapping bands of AgandT2gsymmetries dominated by H 2stretching modes as predicted by our previous density functional theory calculations. We observed low-frequency modes of As 4O6⋅2H 2vibrations dominated by H 2“librations,” which were missed in a previous study. A low-temperature fine structure was observed for the J= 0→2 and J= 1→3 manifolds of hydrogen trapped in As 4O6⋅2H 2, indicating the lifting of degeneracy due to an anisotropic environment. A non-spherical distribution was captured by molecular dynamics simulations, which revealed that the trajectory of H 2molecules is skewed along the crystallographic ⟨111⟩direction. Last but not least, low-temperature synchrotron powder x-ray diffraction measurements on As 4O6⋅2H 2revealed that the bulk structure of the compound is preserved down to 5 K at 1.6 GPa. Published under license by AIP Publishing. https://doi.org/10.1063/5.0017892 .,s I. INTRODUCTION Hydrogen is an emerging fuel of the future. It is ecological, as only water vapor is formed after its combustion, and very light offer- ing high density of energy if a proper method of storage is applied.1 Nonetheless, there are still many challenges related to cheap hydro- gen production and efficient storage. The basic research into inter- molecular interactions of H 2molecules with host materials to eluci- date factors responsible for their strength and directionality and, in turn, to help scientists and engineers design suitable hydrogen stor- age materials is necessary. Hydrogen inclusion compounds are good model compounds for such research. A few years ago, we discovered an arsenolite–hydrogen inclusion compound As 4O6⋅2H 2that forms above 1.5 GPa and characterized its structure and properties using x-ray diffraction, Raman spectroscopy, and density functional the- ory (DFT) computations.2The compound crystallizes in the cubic Fd¯3mspace group (no. 227, origin choice 2) with As 4O6and H 2molecules each centered on special Wyckoff positions. Arsenolite molecules sit at the 8 bsite with ¯43m(Td) point group symmetry, while hydrogen molecules are located at the 16 csite with ¯3m(D3d) point group symmetry (see Fig. 1). We observed that the H 2Q1(1) vibron is softened in the inclusion compound compared to fluid H2over the entire pressure range studied, and the vibron contin- ues to soften with increasing pressure. Interestingly, the vibron was observed to split into three components up to 4 GPa and into two components up to 10 GPa even though structural and computa- tional results suggest that H 2molecules occupy only one Wyckoff site. Multiple and/or split hydrogen vibrons have been observed for related inclusion compounds and may indicate a variety of physi- cal phenomena. For instance, clathrate hydrates of hydrogen exhibit multiple H 2vibrons due to the fact that various numbers of H 2 molecules occupy large and small cavities therein and, additionally, each of these bands are split to reflect the population hydrogen’s two J. Chem. Phys. 153, 054501 (2020); doi: 10.1063/5.0017892 153, 054501-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . Crystal structure of the As 4O6⋅2H2arsenolite–hydrogen inclusion com- pound.2As4O6molecules are presented using a wireframe model with arsenic and oxygen denoted by green and red, respectively. Hydrogen molecules are pre- sented using a ball-and-stick model and are enclosed within gray spheres centered at the 16 cWyckoff site. spin isomers.3,4Owing to the fact that1H nuclei are fermions, the total wavefunction of a hydrogen molecule must be antisymmetric with respect to nuclei permutation. This, combined with the sym- metry of the electronic part of the wavefunction, leads to the fact that hydrogen at room temperature (RT) is composed of a mix- ture of two spin isomers that cannot interconvert quickly into one another. ortho -1H2is the isomer with the symmetric nuclear spin wavefunction ( I= 1) and occurs only in rotational states described by odd rotational ( J) quantum numbers, whereas para -1H2exhibits an antisymmetric nuclear spin wavefunction ( I= 0) with even J. The situation is different for deuterium, in which the2H nuclei are bosons. The ortho -2H2isomer exhibits a symmetric nuclear spin wavefunction ( I= 0 or 2) and even J, whereas para -2H2possesses an antisymmetric nuclear wavefunction ( I= 1) and odd J. The equi- librium distribution of ortho topara H2is a result of an interplay between the degeneracy of nuclear spin and rotational wavefunc- tions as well as Boltzmann factor. In summary, the ortho –para ratio for2H2(deuterium) at RT is 2:1 and increases upon cooling, whereas the ratio at RT for1H2(protium) is 3:1 and decreases upon cool- ing.3,5It is noteworthy that the ortho –para conversion is forbidden by selection rules and requires the exchange of angular momentum, which is typically a very slow process. The presence of an exchange catalyst, containing paramagnetic oxides like commercially available Fe2O3⋅2%H 2O or CrO ⋅SiO 2, is often used to increase conversion rates.6 Other examples of systems with multiple or split H 2vibrons include compounds such as Ar(H 2)2,7Xe(H 2)x,8,9SiH 4(H2)2,10,11 GeH 4(H2)2,12,13(H2S)2H2,14(H2O)2H2,15Kr(H 2)4,16and H 2dis- solved in solid Ne.17The complex Raman spectra observed in com- pounds such as SiH 4(H2)2, which exhibits at least seven vibrons, were explained based on slight perturbations of freely rotating H 2 molecules leading to overall lowering of the crystallographic sym- metry,11similar to the statistical distribution of multiple local envi- ronments for H 2dissolved in rare gas solids.17In other compounds, such as high-pressure (HP) (H 2S)2(H2), multiple hydrogen vibrons reflect different crystallographic sites for hydrogen molecules.14Forthe case of Xe(H 2)x, a hydrogen 3 ×3×3 supercell was postu- lated.8,9,14 Herein, we present a detailed high-pressure (HP) Raman study of the arsenolite inclusion compound spanning from room temper- ature (RT) to low temperature (LT) both with hydrogen and deu- terium to elucidate the physical origins of the observed hydrogen vibrons in As 4O6⋅2H 2. This paper is organized as follows: Sec. II explains the experimental and computational approach that we have applied, Sec. III describes the results from our studies, and, finally, Sec. IV contains discussion and conclusions. II. EXPERIMENTAL AND COMPUTATIONAL DETAILS Symmetric diamond anvil cells (DACs) typically equipped with diamonds having 400- μm culets together with Re gaskets pre- indented to a thickness of 50 μm–60 μm and with laser-drilled 250- μm circular holes were used. The DACs were capable of in situ high-pressure and low-temperature Raman measurements. Thin pellets of arsenolite powder were placed in the gasket hole along- side with 2–3 ruby chips used for pressure determination from ruby fluorescence shift.18,19The pressure scale of Dewaele et al. along with the temperature corrections from Datchi et al. was applied, and fitting was carried out using the T-Rax program.20,21The DACs were subsequently gas-loaded with protium1H2or deuterium2H2, and Raman measurements were carried out. The samples were excited using a 532-nm diode laser. The laser was focused onto the sample using a 20 ×(10×in the case of LT measurements) objective lens. The backscattered light was detected using a Prince- ton Instrument spectrograph SP2750. An 1800-grooves/mm grat- ing was used to disperse the Raman light onto a liquid-nitrogen- cooled CCD, which enabled a spectral resolution of ∼1 cm−1. Initial experiments indicated that the As 4O6⋅2H 2inclusion com- pound is sensitive to intense laser light by the observation of elemen- tal arsenic formed upon irradiation via the reduction of arsenolite by hydrogen. Consequently, subsequent measurements were per- formed with a laser power near 1 mW to record Raman spectra with a sufficient signal-to-noise ratio and to avoid sample dam- age. Neon emission lines were used to calibrate the spectrometer. The spectra were fitted with Lorentzian peaks using the fityk 1.3.1 program.22In some LT runs, Fe 2O3powder, left in the air for a cou- ple of days to absorb moisture, was also loaded into DACs to help facilitate the ortho –para conversion of hydrogen. Computations of Raman modes were carried out according to the procedure outlined in Ref. 2. HP, LT powder x-ray diffraction measurements on the arseno- lite hydrogen inclusion compound were carried out at beam- line 16-BMD, HPCAT, at the Advanced Photon Source syn- chrotron. A helium-cooled cryostat was used with DACs equipped with 600- μm culets to help minimize the pressure variation with temperature. Diffraction data were collected with a MAR 345 image plate. The raw images were integrated with the Diop- tas 0.5.0 software, and Pawley fitting was carried out using GSAS-II.23,24 Ab initio molecular dynamics simulations for the As 4O6⋅2H 2 inclusion compound were performed with the NPT (N is the num- ber of particles, P is the pressure, and T is the temperature) ensemble implemented in Vienna ab initio Simulation Package (VASP) code with Langevin dynamics.25,26The all-electron projector-augmented J. Chem. Phys. 153, 054501 (2020); doi: 10.1063/5.0017892 153, 054501-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp wave (PAW) potential was adopted with the PAW potentials taken from the VASP library, where 4 s24p3, 2s22p4, and 1 s1are treated as valence electrons for As, O, and H atoms, respectively.27The plane wave was expanded to an energy cutoff of 400 eV, and Bril- louin zone sampling with a 2 ×2×2 Monkhorst–Pack k-mesh was employed.28Exchange and correlation effects were treated in the Perdew–Burke–Ernzerhof parameterization of the generalized gra- dient approximation.29Dispersion interactions were accounted for using Grimme’s D3 correction with standard damping function— D3(zero).30,31Trajectories of atomic movements were visualized using the VMD software.32III. RESULTS Variable-temperature and pressure Raman spectra of powdered As4O6⋅2H 2are shown in Figs. 2 and 3. We found that the forma- tion rate of the arsenolite–hydrogen inclusion compound is strongly dependent on the pressure at which the compound is formed. We were able to observe its formation for protium only above 1.5 GPa. The reaction takes tens of minutes to complete at this pressure, while at 1.8 GPa, it is completed within a few minutes (see Fig. S1 for the evolution of spectra as a function of time at ∼1.5 GPa). The formation of the inclusion compound is evidenced by a discon- tinuous change in the frequencies of As 4O6molecular vibrations FIG. 2 . Raman spectra of As 4O6⋅21H2and As 4O6⋅22H2in the 50 cm−1–900 cm−1spectral range. The evolution of As 4O6⋅21H2spectra as a function of pressure at room temperature (a) and of temperature at 3.6(3) GPa (b). The evolution of As 4O6⋅22H2spectra as a function of pressure at room temperature (c) and of temperature at 2.6(3) GPa (d). Labels next to spectra denote pressure or temperature. Thinner lines slightly below inclusion compound spectra correspond to the spectra of fluid/solid H 2recorded at the same conditions. J. Chem. Phys. 153, 054501 (2020); doi: 10.1063/5.0017892 153, 054501-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Raman spectra of As 4O6⋅21H2and As 4O6⋅22H2in the H 2vibron spec- tral range. The evolution of the As 4O6⋅21H2spectra as a function of pressure at room temperature (a) and of temperature at 3.6(3) GPa (b). The evolution of the As 4O6⋅22H2spectra as a function of pressure at room temperature (c) and of temperature at 2.6(3) GPa (d). Labels next to spectra denote pressure or temperature. and by the appearance of H 2vibrons in the Raman spectrum. The evolution of Raman spectra with pressure at room temperature, depicted in Figs. 2(a) and 3(a), confirms our earlier observations from the Raman spectra recorded for the single crystalline sample of the inclusion compound. However, careful acquisition of spectra from a thicker powder sample and subsequent fitting of the spectra revealed the presence of three1H2vibrons over the entire pressure range studied [up to 24.3(2) GPa] for the inclusion compound (see Figs. S2 and S3 for the frequency dependence of modes with pres- sure). Our interpretation of the previous Raman spectra pointed to the presence of three vibrons below ∼5 GPa and of two vibrons above this pressure.2For clarity of presentation and discussion, the vibrons will be hereafter referred to as ν1,ν2, and ν3, and their numbering corresponds to increasing wavenumber. It is noteworthy that (1) all three components soften with pressure, while the opposite trend is observed for fluid/solid H 2; (2) the frequency difference in between ν1and ν2decreases with pressure, and these bands merge above∼8 GPa (above this pressure, fits of the spectra with two peaks are sig- nificantly better than with one peak only); (3) ν3becomes very broad with diminished relative intensity at high pressure, and its frequency dependence with pressure deviates from the other two bands. The last two observations explain why we missed the presence of all three components at higher pressures in our initial study on As 4O6⋅2H 2. In order to further understand the vibrational spectra observed in the inclusion compound, we carried out LT Raman measurements with protium, as shown in Figs. 2(a), 2(b), 3(a), and 3(b). Starting in the lower-frequency region below 1000 cm−1, vibrational modes associated with the arsenolite host lattice are clearly identified by comparison with spectra from the empty structure and with DFT computations. Given that all Raman measurements were performed in excess fluid hydrogen, roton excitations [e.g., S0(0) and S0(1)] from the pure fluid (or solid) are also observed in each spectrum, but are easily distinguished by comparing the spectra of pure hydro- gen at the same conditions. With decreasing temperature, all Raman peak widths decrease as vibrations become more localized. Below ∼200 K, the rotational fine structure becomes apparent within the J= 0→2 and J= 1→3 manifolds near 350 cm−1and 600 cm−1, respectively, and these bands broaden with increasing pressure. By 78 K, the splitting of these bands is significant [denoted by aster- isks in Fig. 2(b)] and indicates at least partially hindered rotation of hydrogen molecules within the inclusion compound. These quan- tum rotor excitations are not captured by static phonon compu- tations. Notably, computations do predict two “librational” modes with symmetries T2gandEg, in which hydrogen atomic displace- ments dominate the overall character. These modes might be related to the two broad bands observed at ∼130 cm−1and 340 cm−1at 1.3 GPa and LT based on their appearance upon formation of the inclusion compound and their extreme pressure dependence, which agrees with the calculations. Turning to the high-frequency region, all vibron peaks become sharper as temperature decreases (see Fig. S4), and the ν1vibron clearly exhibits a shoulder on the high-frequency side indicating that it is, in fact, comprised of two very closely spaced ν1aand ν1b peaks. Given that DFT computations also predict two closely spaced Raman vibrons, one of which is of Agand the other of T2gsymme- try, it appears that the symmetry of the crystal structure explains the origin of these bands (see Tables S1 and S2). The relative intensi- ties of the three vibrons also reveal distinct trends with temperature. At a constant pressure of 3.6(3) GPa, the relative intensity ratio ofν1to all other vibrons (defined as the area of ν1aand ν1bover the total area of all vibrons) varies significantly, while the relative intensity ratio of the ν2vibron to all other vibrons remains con- stant at ∼0.25 upon cooling from RT to 80 K. The relative intensity of the ν3vibron diminishes from 0.36 at RT to 0.04 at ∼80 K (see Figs. 4 and S5 for an extended version including relative fluid/solid H2roton intensities). The relative intensities of the ν1:ν3vibrons recorded at RT do not exhibit any clear trends as a function of pressure. To further understand the observed vibrational spectra, we car- ried out analogous experiments with deuterium. In this case, the interaction potential should remain essentially constant, while the large isotopic mass shift and inverted nuclear spin isomers allow for unambiguous spectral assignments. A comparison of the spec- tra of the deuterium and protium inclusion compounds in the low-frequency range, shown in Figs. 2(a) and 2(c), allowed us to J. Chem. Phys. 153, 054501 (2020); doi: 10.1063/5.0017892 153, 054501-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . Relative intensities of inclusion compound vibrons for protium (a) and deu- terium (b) plotted as a function of temperature at 3.6(3) GPa and 2.6(3) GPa, respectively. confirm all the host modes that were predicted by us previously using CRYSTAL09 calculations and also suggest that the calcu- lated Egand T2glibration modes of H 2molecules are related to the observed modes that exhibit very strong pressure dependence (see Fig. S2). While the frequencies of modes dominated by As 4O6 molecules are predicted very well, the computed absolute frequen- cies of modes with dominant contributions from H 2molecules must be shifted by −160 cm−1and−90 cm−1for protium and deuterium, respectively, to compare with experimental values, but their evolu- tion with pressure is predicted properly. Interestingly, we were able to obtain the inclusion compound with deuterium at only 1.09(13) GPa (compared with ∼1.5 GPa for hydrogen), which was evidenced by a broad2H2vibron in the inclusion compound clearly separated from the fluid2H2vibron. Attempts to obtain the inclusion com- pound with protium at similar pressure were not successful, even after 17 h at 1.03(9) GPa. The2H2vibron in the inclusion compound is also composed of three components ( ν1–ν3), but at lower frequencies due to the higher reduced mass of deuterium [Fig. 3(c)]. While the absolute frequency difference between these bands is lower than that observed for pro- tium, fits of the2H2spectra recorded at RT with three Lorentzianpeaks are always significantly better than with fits using one or two peaks. Moreover, the presence of three vibrons becomes evident at LT when the peaks become sharper [see Figs. 3(d) and S4]. The observed frequencies of the2H2vibrons, both in the inclusion com- pound and in the fluid/solid, are in very good agreement with the frequencies of the respective1H2vibrons when scaled by a factor of 1.39, which is close to the theoretical reduced mass ratio of 1.41 (see Fig. S6). At ∼2.5 GPa, the temperature dependence of the rela- tive intensities of the2H2vibrons in the inclusion compound sim- ilar to1H although the trends for ν1and ν2are obscured above 200 K due to significant overlap of these peaks [see Figs. 3(d), 4, and S5]. The relative intensity of ν3falls from 0.55 at 260 K to 0.07 at 85 K, while the relative intensity of ν2grows from RT to 200 K when it reaches ∼0.60 and remains constant down to 85 K. Simi- lar to protium, ν1for deuterium grows in intensity as temperature is lowered, while ν3decreases in intensity over the same tempera- ture range. We were not able to detect a shoulder on ν1in the case of deuterium, but we suspect that this is caused by the very close spacing of peaks, even at 85 K. It is noteworthy that the relative intensities of ν1and ν2are inverted for deuterium as compared to protium. Our attempts to determine the LT stability of the inclusion compound at ambient pressure were complicated by irreversible sticking of the DACs that prevented full pressure release at LT. We were, however, able to completely release the pressure during one LT experiment with deuterium, which revealed that the inclusion com- pound remains (meta)stable at ambient pressure and 85 K for at least 21 h (see Fig. S7 for the recorded Raman spectra). Noting the observed vibron intensity trends between protium and deuterium and their frequency separations relative to the pure fluid/solid, we decided to carry out an additional HP–LT Raman experiment over a longer time to examine possible ortho –para con- version and make definitive peak assignments. While ortho –para conversion rates vary drastically under pressure,33,34conversion in other inclusion compounds is typically slow,3,4and we therefore utilized a catalyst to possibly help expedite conversion. Our goal was to study the inclusion compound at the lowest pressure possi- ble to minimize pressure-induced intermolecular coupling, which causes dramatic changes in the relative intensities of the J= 0 and J= 1 vibrons.35Due to the problems associated with releas- ing the DAC pressure at LT described above, we were only able to decrease the pressure down to 1.29(5) GPa, which was suffi- cient to clearly observe the Q1(1) and Q1(0) vibrons of solid1H2 [see Fig. 5(d)]. After 66 h at 84(1) K and 1.29(5) GPa, the relative intensi- ties of the solid rotons and vibrons appear to reach constant val- ues, indicating the approach to thermodynamic equilibrium [see Figs. 5(b) and 5(c)]. The relative intensities of the H 2vibrons for the inclusion compound also reached a plateau albeit within a shorter time span of 42 h. While the intensities of the solid rotons change significantly over time and reflect an ortho concentration moving toward thermal equilibrium, the intensity ratio of the Q1(1):Q1(0) fluid vibrons only decreases ∼3% reflecting the stronger vibron scat- tering for the J= 1 state. Given that the ν1andν2relative intensities from the compound change by this same amount and exhibit a sim- ilar spacing, we assign the ν1and ν2vibrons as Q1(1) and Q1(0) vibrons of H 2trapped in the inclusion compound, respectively. This assignment is also supported by the inverse intensity trends observed J. Chem. Phys. 153, 054501 (2020); doi: 10.1063/5.0017892 153, 054501-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . Relative intensities of inclusion compound1H2vibrons (a), of solid1H2vibrons (b) and of solid1H2rotons (c) plotted as a function of time at 85(2) K and 1.29(5) GPa. The Raman spectrum of the inclusion compound in the vibron spectral range at these conditions at the beginning of the experiment (d). Vertical and dotted lines correspond to the AgandT2gmodes, respectively, for ortho andpara H2, whereas the dashed lines illustrate that for ν3, there is no ortho –para pair. for deuterium reflecting the difference between ortho andpara for the different spin isomers. We note that while the ν1and ν2rela- tive intensities mirror the ortho andpara behavior of the fluid of time, the relative intensity of ν3remained constant near 0.15 over the entire period [see Fig. 5(a)]. In order to analyze the Raman spectra that we recorded at LT, it is important to know whether any temperature-induced phase transitions occur upon cooling. We therefore collected powder x-ray diffraction patterns of the As 4O6⋅21H2inclusion compound at 1.6 GPa at LT down to 5 K at the 16-BMD beamline of the Advanced Photon Source. The analysis of the obtained diffraction patterns revealed that there is no phase transition down to this tempera- ture. It is noteworthy that the x-rays used (wavelength of 0.41 Å)scatter mostly from arsenic atoms with negligible diffraction from H2molecules. The diffraction patterns rule out a lowering of the symmetry of the host structure formed by As 4O6molecules, but do not rule out possible ordering of H 2molecules in supercells or low- ering of the symmetry of the whole structure caused by H 2molecules (see Fig. S8). IV. DISCUSSION AND CONCLUSIONS Detailed analysis of H 2rotons from the inclusion compound is challenging due to the overlap with fluid/solid H 2present in the cell and overlap in the spectral range where strong As 4O6molec- ular vibrations are present [see Figs. 2(b) and S7(a)]. Nonetheless, J. Chem. Phys. 153, 054501 (2020); doi: 10.1063/5.0017892 153, 054501-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp it is clear that a low-temperature fine structure is observed for the J= 0→2 and J= 1→3 manifolds, indicating the lifting of degeneracy due to an anisotropic environment (which was observed for other inclusion compounds, e.g., for hydrogen clathrate hydrates).3,4The energy splitting for these bands is quite significant. Partially resolved components for the J= 0→2 transitions centered about 360 cm−1 appear to span 40 cm−1, while components appear to span more than 200 cm−1for the J= 1→3 transitions. At least partially hin- dered rotation, similar to the 2D rotor behavior for hydroquinone clathrate,36could potentially explain such splitting. While being incapable of capturing the quantum nature of the rotor, molecu- lar dynamics simulations performed on the As 4O6⋅2H 2compound do indeed indicate a non-spherical classical probability distribution of H 2molecules at the 16 cWyckoff site wherein the distribution is skewed along the crystallographic ⟨111⟩direction (Fig. S9). Previ- ously, we suggested that H 2molecules might be aligned along the crystallographic ⟨111⟩direction based on the calculated energies of an H 2molecule in different orientations enclosed in an (As 4O6)6 cluster simulating the environment of the molecule at the 16 cWyck- off site in As 4O6⋅2H 2.2Interestingly, the roton bands appeared much weaker in the spectra of the deuterium inclusion compound, but their presence could be detected in the quenched sample at 85 K [Fig. S7(a)]. Two additional modes were observed at low frequency and appear to be associated with hydrogen (deuterium). For1H2, these bands appear at ∼130 cm−1and 340 cm−1at 1.3 GPa and 85 K. As the pressure is increased, they exhibit very strong pressure depen- dence that agrees with the T2gandEglibrational modes predicted by DFT computations. It is conceivable that these modes are actu- ally also associated with the J= 0→2 manifold, but a shift to 130 cm−1would represent an unprecedented energy splitting for a molecular inclusion compound. Additional experiments at lower temperature, including inelastic neutron scattering, are needed to fully understand the coupled translational–rotational behavior of hydrogen within As 4O6⋅2H 2. The hydrogen vibrons in As 4O6⋅2H 2are significantly softened from the fluid/solid hydrogen vibrons. From previously character- ized H 2van der Waals compounds, we infer a trend that when- ever H 2forms an inclusion compound with an electron donat- ing substance, the H 2vibron is red-shifted due to the partial donation of electron density to the H 2σ∗antibonding orbital. This is the case for the herein studied As 4O6⋅2H 2as well as (H2S)2H2,14(H2Se) 2H2,37hydrogen clathrate hydrates,3,4(HI)(H 2)2 and (HI)(H 2)13,38in which all host molecules contain atoms with stereoactive lone electron pairs that donate some of its electron den- sity to the H 2molecule. The inclusion compounds with He, Ne, Ar, and Kr exhibit blue-shifted H 2vibron, indicating repulsive interac- tions with hydrogen molecules and no charge transfer from noble gas atoms to H 2molecules.7,16,17The effect is reversed in the inclu- sion compound of hydrogen with xenon where the H 2vibron is red-shifted, indicating the Xe valence electron density is partially transferred to the H 2antibonding orbital.8,9This is in agreement with the fact that Xe is known for being much more reactive than lighter noble gases and forming numerous chemical compounds. An interesting observation can be made in the series of H 2van der Waals compounds with CH 4, SiH 4, and GeH 4.10,12,39While the H 2 vibron is blue-shifted in the inclusion compound with methane, it is red-shifted for silane and germane. This is caused by theinversion of the X–H bond polarity. The C–H bonds are polarized toward carbon leaving partial positive charge on H atoms making donation of electron density to the H 2σ∗orbital impossible. From Si on, the X–H bonds are polarized toward H atoms leading to inter- actions similar to dihydrogen bonds that result in electron density being partially transferred to the H 2σ∗orbital and weakening the H–H bond.40In the hydrogen–nitrogen van der Waals compounds (N2)6(H2)7and (N 2)(H 2)2, the H 2vibron is blue-shifted, indicat- ing only repulsive interactions between N 2and H 2molecules.41,42 It is noteworthy that the H 2vibron is softened when protium or deuterium is dissolved in solid matrices of Ar, Kr, and Xe at ambi- ent pressure and LT (see Ref. 43 and references therein). Calcula- tions revealed that the vibron is softened whenever the H 2molecule is located in the region of space where the intermolecular inter- action potential is attractive.43Taking this into account, we may conclude that there are two general mechanisms responsible for H 2 vibron softening. One stems from the local intermolecular poten- tial and attraction between host molecules/atoms and H 2molecules, while the other comes from partial charge transfer between host and H 2molecules. The latter mechanism dominates repulsive inter- actions between the host and guest at HP as has been shown for As 4O6⋅2H 2.2 As for the multiple hydrogen vibrons observed for As 4O6⋅2H 2, our carefully collected spectra allowed us to conclude that there are three H 2vibrons from the inclusion compound up to the highest studied pressure. We assign ν1and ν2asQ1(1) and Q1(0) based on (1) the time dependence of their relative intensities at LT indi- cating slow ortho –para conversion for protium and (2) the change in their relative intensities when going from protium to deuterium reflecting the difference in degeneracy of ortho andpara spin iso- mers between1H2and2H2. As for the shoulder present on ν1in LT protium measurements, we attribute the ν1amain peak and theν1bshoulder to Agand T2gvibrations, respectively, both of which were predicted by DFT computations and may both exhibit ortho –para contributions in the observed spectra [see dotted lines in Fig. 5(d)]. The origin of ν3is more difficult. Our first hypothesis was that H2occupies an additional site in the As 4O6⋅2H 2crystal structure. This explanation seems unlikely to be the case since there should be a second pair of vibrons corresponding to ortho andpara isomers occupying the second site, which is not the case here [see dashed vertical lines in Fig. 5(d)]. The occupation of a second site is also inconsistent with our MD simulations. We simulated the As 4O6⋅2H 2 crystal structure with H 2molecules initially positioned on the 16 c site at RT and multiple pressure points (2 GPa, 4 GPa, and 6 GPa), and we did not observe H 2molecules migrate to any other sites (see Fig. S9). When H 2molecules were positioned on the 8 asite, they quickly migrate to the 16 csite, substantiating its energetic favorabil- ity. A possible temperature-dependent occupancy distribution of the 16csite (leading to multiple effective local environments) also seems unlikely based on the absence of ortho –para pairs, the behavior with pressure, and previous single-crystal XRD measurement that give no indication for non-stoichiometry. It is tempting to assign ν3as an additional rovibrational contri- bution [e.g., Q1(3)] based on the temperature dependence of the rel- ative intensities of ν1andν3both for protium and deuterium (Fig. 4). Here, the depopulation of a higher-energy rotational state is consis- tent with the observed temperature dependence, as is the apparent J. Chem. Phys. 153, 054501 (2020); doi: 10.1063/5.0017892 153, 054501-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp transfer of intensity to the next low-lying transition with the same nuclear spin state. However, such an assignment would require an unusual rearrangement of energy levels such that Q1(3) is observed at ahigher frequency than Q1(0) and Q1(1). This assignment is also unlikely in that Q1(2) is never observed and Q1(3) should not be populated at 80 K [the intensity of solid hydrogen S0(3) is zero, indicating that the J= 3 rotational state is not occupied in the pure phase]. It is noteworthy that the ν3vibron is significantly broader than theν1andν2vibrons, which may suggest a different physical origin. The observed temperature dependence of the ν3vibron is consistent with a Boltzmann factor and could indicate possible coupling with a phonon mode. For example, a librational mode or other transla- tional state could potentially couple with symmetric stretching to yield a separate vibron. The final understanding of this vibron will require additional measurements to reveal the detailed energy levels of H 2trapped within arsenolite. HP solid-state1H nuclear magnetic resonance (NMR), lower-temperature Raman, and inelastic neutron scattering measurements are all being pursued to reveal a greater understanding of the dynamics. SUPPLEMENTARY MATERIAL See the supplementary material for additional Raman spectra, trajectories from MD simulations, As 4O6⋅2H 2powder diffraction patterns, and tables of calculated Raman frequencies for As 4O6⋅21H2 and As 4O6⋅22H2. ACKNOWLEDGMENTS This work was supported by the Polish National Agency for Academic Exchange. Computations were performed using the Wro- claw Centre for Networking and Supercomputing (http://wcss.pl), Grant No. 260, and the Memex cluster of Carnegie Institution for Science. T.A.S. acknowledges the support from the U.S. Army Research Office under Grant No. W911NF-17-1-0604. Portions of this work were performed at HPCAT (Sector 16), Advanced Pho- ton Source (APS), Argonne National Laboratory. We thank Curtis Kenney-Benson and Dmitry Popov for assistance with cryostat work and x-ray diffraction data collection during beamtime. HPCAT operations are supported by DOE–NNSA’s Office of Experimen- tal 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 Contract No. DE-AC02-06CH11357. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1S. Sharma and S. K. Ghoshal, Renewable Sustainable Energy Rev. 43, 1151 (2015). 2P. A. Gu ´nka, M. Hapka, M. Hanfland, G. Chałasi ´nski, and J. Zachara, J. Phys. Chem. C 123, 16868 (2019). 3T. A. Strobel, E. D. 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A. Strobel, P. Ganesh, M. Somayazulu, P. R. C. Kent, and R. J. Hemley, Phys. Rev. Lett. 107, 255503 (2011). 15T. A. Strobel, M. Somayazulu, S. V. Sinogeikin, P. Dera, and R. J. Hemley, J. Am. Chem. Soc. 138, 13786 (2016). 16A. K. Kleppe, M. Amboage, and A. P. Jephcoat, Sci. Rep. 4, 4989 (2014). 17P. Loubeyre, R. LeToullec, and J. P. Pinceaux, Phys. Rev. B 45, 12844 (1992). 18H. K. Mao, J. Xu, and P. M. Bell, J. Geophys. Res.: Solid Earth 91, 4673, https://doi.org/10.1029/jb091ib05p04673 (1986). 19K. Syassen, High Pressure Res. 28, 75 (2008). 20A. Dewaele, M. Torrent, P. Loubeyre, and M. Mezouar, Phys. Rev. B 78, 104102 (2008). 21F. Datchi, A. Dewaele, P. Loubeyre, R. Letoullec, Y. Le Godec, and B. Canny, High Pressure Res. 27, 447 (2007). 22M. Wojdyr, J. Appl. Crystallogr. 43, 1126 (2010). 23C. Prescher and V. B. Prakapenka, High Pressure Res. 35, 223 (2015). 24B. H. Toby and R. B. Von Dreele, J. Appl. Crystallogr. 46, 544 (2013). 25E. Hernández, J. Chem. Phys. 115, 10282 (2001). 26G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996). 27G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999). 28H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1976). 29J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). 30S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, J. Chem. Phys. 132, 154104 (2010). 31S. Grimme, S. Ehrlich, and L. Goerigk, J. Comput. Chem. 32, 1456 (2011). 32W. Humphrey, A. Dalke, and K. Schulten, J. Mol. Graphics 14, 33 (1996). 33A. Driessen, E. van der Poll, and I. F. Silvera, Phys. Rev. B 30, 2517 (1984). 34J. H. Eggert, E. Karmon, R. J. Hemley, H.-K. Mao, and A. F. Goncharov, Proc. Natl. Acad. Sci. U. S. A. 96, 12269 (1999). 35J. H. Eggert, R. J. Hemley, H. K. Mao, and J. L. Feldman, AIP Conf. Proc. 309, 845 (1994). 36T. A. Strobel, A. J. Ramirez-Cuesta, L. L. Daemen, V. S. Bhadram, T. A. Jenkins, C. M. Brown, and Y. Cheng, Phys. Rev. Lett. 120, 120402 (2018). 37E. J. Pace, J. Binns, M. Peña Alvarez, P. Dalladay-Simpson, E. Gregoryanz, and R. T. Howie, J. Chem. Phys. 147, 184303 (2017). 38J. Binns, P. Dalladay-Simpson, M. Wang, G. J. Ackland, E. Gregoryanz, and R. T. Howie, Phys. Rev. B 97, 024111 (2018). 39M. S. Somayazulu, L. W. Finger, R. J. Hemley, and H. K. Mao, Science 271, 1400 (1996). 40R. Custelcean and J. E. Jackson, Chem. Rev. 101, 1963 (2001). 41D. K. Spaulding, G. Weck, P. Loubeyre, F. Datchi, P. Dumas, and M. Hanfland, Nat. Commun. 5, 5739 (2014). 42D. Laniel, V. Svitlyk, G. Weck, and P. Loubeyre, Phys. Chem. Chem. Phys. 20, 4050 (2018). 43J. Vitko and C. F. Coll, J. Chem. Phys. 69, 2590 (1978). J. Chem. Phys. 153, 054501 (2020); doi: 10.1063/5.0017892 153, 054501-8 Published under license by AIP Publishing

5.0013807.pdf

J. Chem. Phys. 153, 124305 (2020); https://doi.org/10.1063/5.0013807 153, 124305 © 2020 Author(s).A study of the translational temperature dependence of the reaction rate constant between CH3CN and Ne+ at low temperatures Cite as: J. Chem. Phys. 153, 124305 (2020); https://doi.org/10.1063/5.0013807 Submitted: 14 May 2020 . Accepted: 10 September 2020 . Published Online: 30 September 2020 Kunihiro Okada , Kazuhiro Sakimoto , Yusuke Takada , and Hans A. Schuessler ARTICLES YOU MAY BE INTERESTED IN Threshold photoelectron spectroscopy of the HO 2 radical The Journal of Chemical Physics 153, 124306 (2020); https://doi.org/10.1063/5.0022410 Reactions of translationally cold trapped CCl+ with acetylene (C 2H2) The Journal of Chemical Physics 152, 234310 (2020); https://doi.org/10.1063/5.0008656 Vibrational predissociation in the bending levels of the state of C 3Ar The Journal of Chemical Physics 153, 124303 (2020); https://doi.org/10.1063/5.0015592The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp A study of the translational temperature dependence of the reaction rate constant between CH 3CN and Ne+at low temperatures Cite as: J. Chem. Phys. 153, 124305 (2020); doi: 10.1063/5.0013807 Submitted: 14 May 2020 •Accepted: 10 September 2020 • Published Online: 30 September 2020 Kunihiro Okada,1,a) Kazuhiro Sakimoto,1Yusuke Takada,1and Hans A. Schuessler2 AFFILIATIONS 1Department of Physics, Sophia University, 7-1 Kioicho, Chiyoda, Tokyo 102-8554, Japan 2Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843-4242, USA a)Author to whom correspondence should be addressed: okada-k@sophia.ac.jp ABSTRACT We have measured the translational temperature dependence of the reaction rate constant for CH 3CN + Ne+→products at low temperatures. A cold Ne+ensemble was embedded in Ca+Coulomb crystals by a sympathetic laser cooling technique, while cold acetonitrile (CH 3CN) molecules were produced by two types of Stark velocity filters to widely change the translational temperatures. The measured reaction rate constant gradually increases with the decrease in the translational temperature of the velocity-selected CH 3CN molecules from 60 K down to 2 K, and thereby, a steep increase was observed at temperatures lower than 5 K. A comparison between experimental rate constants and the ion–dipole capture rate constants by the Perturbed Rotational State (PRS) theory was performed. The PRS capture rate constant reproduces well the reaction rate constant at a few kelvin but not for temperatures higher than 5 K. The result indicates that the reaction probability is small compared to typical ion–polar molecule reactions at temperatures above 5 K. Published under license by AIP Publishing. https://doi.org/10.1063/5.0013807 .,s I. INTRODUCTION Modern astronomical observatories, such as at the Atacama Large Millimeter/submillimeter Array (ALMA), enable us to observe with high spatial resolution the details of chemical compositions and their distribution in interstellar clouds and protoplanetary disks.1–3 Recently, many astronomical observations urge astrochemists to update the existing astrochemical models of these celestial bodies. For this purpose, further detailed information about gas-phase (also grain-surface) chemical processes is necessary to be acquired.4–7In the gas-phase processes, the information on the reaction tempera- ture dependence and branching ratios of ion–molecule reactions is in high demand since ion–molecule reactions play important roles in interstellar chemistry.8 In general, the ion–molecule reactions at low temperatures are well described by the Langevin model9in which the rate constants are independent of the reaction temperatures. The tem- perature dependence of the rate constants in ion–polar molecule reactions is considered based on the classical ion–dipole capturemodels.4,5These simple classical models give only capture-rate con- stants but not reaction branching ratios and are not necessarily sufficient for considering the chemical composition of relatively active interstellar matter such as falling gas forming a disk around a protostar.1 Many astrochemical relevant measurements of ion–molecule reactions have been performed so far using various experimental techniques,10–18and some important findings were obtained. For instance, deuteration fractionation and isomerization of molecules are important information to know physical and chemical condi- tions of interstellar clouds.19–21More accurate temperature depen- dences of the reaction rate constants are also necessary to refine the reaction networks in interstellar chemical models.4,5Interest- ingly, strong temperature dependences of the reaction rate con- stants in NH+ 3+ H 2→ NH+ 4+ H and C 3H+ n+ H 2(n= 0, 1) were found at low temperatures.14,22–28The former slow reaction is known to be key to the gas-phase ammonia synthesis in inter- stellar space and has been extensively studied experimentally and theoretically. J. Chem. Phys. 153, 124305 (2020); doi: 10.1063/5.0013807 153, 124305-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Despite the above vigorous past studies, the obtained knowl- edge is still insufficient to describe the chemical compositions of interstellar matter obtained by state-of-the-art observatories. Most of the previous measurements were performed for restricted ion– molecule reactions and in the limited temperature range near room temperature.4,5Particularly, reaction rate measurements between an ion and polar molecule have not been extensively performed using drift tube methods and cryogenic multipole ion traps at low temper- atures because a polar molecular beam is very hard to focus and con- trol. Therefore, there is a wide demand for detailed low-temperature measurements. Recent experimental developments combining a linear Paul trap (LPT) for the generation of Coulomb crystals with a Stark velocity filter29and a Stark decelerator30for producing cold polar molecules enabled us to measure cold ion–polar molecule reac- tions at low temperatures under ultra-high vacuum conditions.31–33 We have also demonstrated reaction-rate measurements between sympathetically cooled N 2H+and Ne+ions and velocity-selected CH 3CN molecules,34,35where CH 3CN is known to be one of the important interstellar molecules detected in a protoplanetary disk.2Since one of the important purposes of the present study is to specify the effect of the molecular rotation on the ion– polar molecule reaction, the rotationless atomic form is appro- priate as the ions. The existence of the Ne+ion has been con- firmed by x-ray absorption in the interstellar medium,36and the Ne+reaction could be important in a future model of interstellar chemistry. Although only the translational temperatures of polar molecules are cold in the above measurements, this general technique has a potential to enable a systematic study of cold ion–polar molecule reactions by changing both the translational and rotational tem- peratures at the same time by combining the above experimen- tal technique with a buffer-gas cooled molecular source.37–39In this context, we have developed a temperature variable wavy Stark velocity filter40by which we can generate a slow polar molecu- lar beam at translational temperatures from a few to a 100 K. Combining a cryogenic gas cell with the wavy Stark velocity filter enables us to generate slow polar molecules with cold rotational temperatures. In this paper, we report the measurement of the translational temperature dependence of the reaction rate constant in CH 3CN + Ne+→products at low temperatures by combining a sympathetic laser cooling technique with the Stark velocity filter. The results were compared to capture rate constants of the collisions between a polar molecule and an ion. The capture rate constants were eval- uated by using perturbed rotational state (PRS) theory41,42in which an adiabatic approximation is introduced for describing the molec- ular rotation during the collision. The PRS approach was found to be very useful for investigating the rotational excitation of polar molecules by low-energy ion impacts41–43and for understanding the significance of the dipole interaction in the ion–molecule chemi- cal reaction.42,44,45The idea based on the adiabatic approximation was afterward applied to various ion–molecule systems by a lot of workers.46–49 This paper is organized as follows: In Sec. II, we describe our setup and the measurement method of the reaction rate con- stants. Specifically, the structure and characteristics of the wavy Stark velocity filter combined with a cryogenic linear Paul trapapparatus are presented. We also present a summary of the charac- terization of the velocity-selected CH 3CN molecules generated from two types of Stark velocity filters. A short description of how the average kinetic energy of sympathetically cooled Ne+ions is eval- uated is also presented. In Sec. III, we outline the PRS theory and show some examples of calculated ion–dipole capture cross sections. The procedure to calculate the state-specific PRS capture rate con- stants is given. Then, we describe how to derive the rotational state distribution of CH 3CN from Monte–Carlo trajectory calculations of the velocity-selected CH 3CN and how the PRS ion–dipole capture rate constants are obtained from the state-specific rate constants. Next, in Sec. IV, we present the experimental results of the reac- tion rate measurements and discuss the comparison between the experimental rate constants with the PRS theory. In Sec. V, we fur- ther discuss the relation of our study to astrochemistry. Finally, in Sec. VI, we conclude our results and present an outlook for future experiments. II. EXPERIMENTAL A. Apparatus We use two types of Stark velocity filter for producing slow CH 3CN molecules in order to cover the wide translational temper- ature range. One is a standard type Stark velocity filter with a 90○ deflection section, which can provide a cold CH 3CN beam at a few kelvin. The details of the combined standard Stark velocity filter– ion trap apparatus are described in a previous paper.34The other filter is a wavy Stark velocity filter, which has a wavy deflection sec- tion and allows selection of slightly higher translational temperature molecules over a wide range of temperatures in the tens of kelvin range. The detailed description of the wavy Stark velocity filter has also been given previously.40 Here, we briefly describe a combined wavy Stark velocity filter– ion trap apparatus. The schematic of the setup is shown in Fig. 1. A gas reservoir of CH 3CN is connected to a gas cell via a Teflon tube with an inner diameter of 2 mm, while the gas cell has an exit aperture with the diameter of 1.5 mm. The exit aperture is posi- tioned closely at the input of the wavy Stark velocity filter. The absolute pressure of CH 3CN is typically set to 4.8 Pa, which is monitored at the reservoir by a capacitance manometer (PFEIFFER CMR 364). As the gas cell is thermally contacted to a cryocooler (ULVAC S030), we can cool the gas cell down to 30 K if neces- sary. The temperature of the cell is monitored by a silicon-diode sensor. The vacuum chamber containing the gas cell is always kept the lower pressure than 1 ×10−5Pa during the experiment to avoid the loss of slow molecules due to elastic collisions with residual gas molecules. The extracted polar molecules from the gas cell are guided by the wavy Stark velocity filter, which has a wavy deflection section with a large curvature radius of R= 1000 mm and a deflection angle of θ= 5○. A lower left photograph in Fig. 1 shows the wavy deflection section, which is enclosed in a stainless-steel chamber with a dimension of 355 ×667×300 mm3. The pressure of the chamber is of the order of 10−6Pa during the experiment. The molecular beam guide of the velocity filter consists of four stain- less steel rods with a diameter of 2 mm and the distance between the adjacent rods is designed to be 1 mm. The maximum guide J. Chem. Phys. 153, 124305 (2020); doi: 10.1063/5.0013807 153, 124305-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . Schematic arrangement and pho- tographs of the combined wavy Stark velocity filter–ion trap apparatus. The lin- ear ion trap (LPT) and the electron gun are enclosed in double radiation shields, which are thermally contacted to the 10 K cryocooler. The temperature of the inner radiation shield is monitored by a sili- con diode sensor. The temperature can be changed by using a ceramic heater if necessary. A typical operating temper- ature is about 13 K in this work. The detailed dimensions and setting param- eters are described in the text. voltages applied to the quadrupole electrodes is ±2.0 kV, which cor- responds to the nominal electric field of 40 kV/cm. Implementing three stages of differential pumping in the wavy Stark velocity filter, we achieve the ultra-high vacuum of 1.5 ×10−8Pa in the ion trap chamber. The lower right photograph in Fig. 1 shows a cryogenic Lin- ear Paul Trap (LPT) enclosed in double radiation shields, which are thermally contacted to a cryocooler (Iwatani D510, 10 K). As shown in the photograph, we equipped the setup with an electron gun and electron optics for producing Ne+ions by electron impact ioniza- tion. The LPT consists of beryllium copper rods with a diameter of 8 mm and the inner radius r0of the quadrupole is 3.5 mm. The distance between the end plate electrodes is 10 mm and the static voltages Vzof a few volts are applied to both end electrodes. The LPT is driven by a commercially available radio frequency generator (Stahl-Electronics HF-DR 4.5-900 FL). The driving frequency and the amplitude in balance mode are typically 3.47 MHz and 52 V, respectively. In this case, the radial pseudopotential for Ca+(Ne+) is 1.1(2.2) eV. Because the base copper plate of the LPT and the elec- tron gun is cooled down to be about 13 K during measurement, we can avoid undesirable reactions between cold ions and stray polar molecules, which are possibly leaked from the area upstream of the wavy Stark velocity filter. To study the translational reaction temperature dependence of the reaction rate constant between CH 3CN and Ne+, we have to characterize the velocity-selected CH 3CN at the posi- tion of the ion trap. Therefore, we have determined the number density of the velocity-selected CH 3CN by the correlation mea- surement using a quadrupole mass spectrometer and a ultrahigh vacuum pressure gauge. The detailed method is described in thesupplementary material and the previous papers.34,40A summary of the characterization of the velocity-selected CH 3CN molecules is given in Fig. 2. Additionally, the kinetic ion temperature of sympathetically cooled Ne+ions is needed to determine the translational reac- tion temperature for CH 3CN + Ne+→products. Although the energy distribution of sympathetically cooled ions is probably not thermal,50the average ion kinetic energy ( Tion) of the trapped ions was used as a guide for evaluating the translational reac- tion temperature for the present experimental study.33,34Tionwas determined by comparing the observed laser-induced fluorescence (LIF) images of dual-species Coulomb crystals with the simulated images obtained by classical molecular dynamics (MD) simula- tions, which consider the time-dependent rf fields, laser cooling, and collision heating. The detailed method has been discussed in the previous paper35and a summary is given in the supplementary material. Describing the results, the average kinetic energy of Ne+ions slightly decreases with the decrease in the number of Ne+ions and does not strongly depend on the number of Ca+ions and the shape of the crystal. The average kinetic energy difference between the initial and final number of Ne+ions is expected to be lower than 1.0 K. According to the present experimental conditions, the aver- age kinetic energy is confidently evaluated to be Tion= 1.8(0.5) K. The reaction translational temperature Ttris approximately determined by Ttr=Tionμ Mion+TPMμ MPM, (1) J. Chem. Phys. 153, 124305 (2020); doi: 10.1063/5.0013807 153, 124305-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Summary of the characteri- zation of the velocity-selected CH 3CN molecules. (a) A plot of the number den- sitynof the velocity-selected CH 3CN at the position of the ion trap as a function of the nominal guide voltages | VG| for both the standard type and wavy Stark velocity filters. (b) A plot of the longi- tudinal temperature TPMof the velocity- selected CH 3CN as a function of | VG|. The solid line between plot points is a guide for the eyes for both (a) and (b) (for details, see the supplementary material). where μ=MionMPM/(Mion+MPM) is the reduced mass of the ion (Mion) and the molecule ( MPM).32,34,35 B. Reaction rate measurement In the present experiment, first, we start with preparing a Ca+ Coulomb crystal by laser cooling trapped Ca+, and then, we intro- duce neon gas of typically 5 ×10−6Pa into the ion trap chamber and produce Ne+ions by electron impact ionization. The electron beam energy is 250 eV and the electron current is about 0.5 nA at the ion trap. After a few minutes, a mixed ion Coulomb crystal consisting of Ca+and Ne+is produced. Since the trapping potential for Ne+ions is deeper than that of Ca+, the Ne+ions distribute around the trap axis and a dark area appears in the LIF image of the Ca+Coulomb crystal. The detailed procedure to determine the relative number of sympathetically cooled ions has been discussed previously,34,35and the summary is given in the supplementary material. In brief, the absolute number of sympathetically crystallized Ne+ions is pro- portional to the volume of the dark area in the LIF image of a dual-species Coulomb crystal because of an approximately constant number density of cold ions in an LPT. Therefore, the relative num- ber of Ne+ions can be obtained by measuring the volume of the dark area in the LIF images under the assumption of the cylindri- cal symmetry of the spatial distribution of Ne+ions.35Finally, the relative number of Ne+as a function of the reaction time with the velocity-selected CH 3CN molecules provides us the reaction rate of the CH 3CN + Ne+→products. The decrease in the dark area after interacting with the slow CH 3CN suggests that a charge trans- fer reaction producing CH 3CN+and/or the proton transfer reaction producing CH 2CN+occur.35It should be noted that other reac- tions producing lighter molecular ions than Ca+are not excluded. However, most of the possible light products will be lost from the ion trap by them gaining high kinetic energy because the magni- tudes of the reaction enthalpies are large compared to their trap depths (see the supplementary material). Additionally, as shown in Fig. 6, the reaction rate was certainly characterized by a sin- gle exponential decay of the number of crystallized Ne+ions, and this suggests that a specific reaction path in which Ne+and most of the produced ions are lost dominates the reaction rate. Thus, it is concluded that we were able to measure the reaction rate ofCH 3CN + Ne+→products with reliable accuracy based on the comparison of the gain of the kinetic energy of the reaction prod- ucts with the depth of the trapping potential. The determination of the reaction branching ratio by time-of-flight mass spectrometry of ejected mixed ion Coulomb crystals17,51,52is the next step of our project. III. THE PRS ION–DIPOLE CAPTURE RATE CONSTANT We have calculated capture cross sections for ion collisions with CH 3CN by the perturbed rotational state (PRS) theory.41–45Here, we assume that the CH 3CN molecule is a symmetric-top rigid rotor and that the intermolecular interaction Vbetween the CH 3CN molecule and an ion is given by the asymptotic forms of the dipole and the spherically symmetric polarization potentials. Introducing the rota- tional constants Arot,Brot, and Crot=Arot, we can write the kinetic energy of the free rotation in the form ˜Hrot=Brot̷h−2˜J2+(Arot−Brot)̷h−2˜J2 0, (2) where ˜Jis the angular momentum vector of the molecule, ˜J0 is its projection on the symmetry axis of the molecule, and the tilde denotes the quantum mechanical operator. The free rotational energy is given by EJK=BrotJ(J+ 1)+(Arot−Brot)K2, (3) where ( J,K) are the rotational quantum numbers. In the PRS the- ory, the adiabatic picture is assumed for the rotational motion dur- ing the collision. We numerically diagonalize the Hamiltonian ˜HPRS =V+˜Hrotat each fixed intermolecular distance R.41,42LetεJKM(R) be the diagonalized eigenvalue, which may be called an adiabatic potential. The adiabatic potential εJKM(R) has the asymptotic form εJKM(R)/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne→ R→∞EJKand can be identified by the quantum numbers (J,K,M), with Mbeing the magnetic quantum number along the intermolecular axis. We assume that the intermolecular motion is effectively governed by the adiabatic potential. As R→∞,εJKM(R) −EJK(M≠0) becomes ∝R−2forK≠0 and∝R−4forK= 0.43,45 Therefore, we can expect a drastic effect on the intermolecular motion in low-energy collisions of the symmetric-top molecule having K≠0. J. Chem. Phys. 153, 124305 (2020); doi: 10.1063/5.0013807 153, 124305-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Examples of M-average ion– dipole capture cross sections for the CH3CN-ion system. The rotational level (J,K) corresponding to each curve is indicated in the figure. By using the PRS adiabatic potential, we can obtain the orbiting impact parameter bJKM(E) as a function of collision energy E. The capture cross section (called the “hitting cross section” in Ref. 44) is defined by σ(J,K,M)=π[bJKM(E)]2. Since the orientation of the molecular axis of a velocity-selected CH 3CN is random to the position of the sympathetically cooled Ne+ions under the present experimental conditions, we need the M-average capture cross sec- tions σ(J,K) = (2 J+ 1)−1∑Mσ(J,K,M). Figure 3 shows exam- ple plots of the M-average cross sections as a function of collision energy. We calculated the ion–dipole capture cross sections up to J= 60 for the present system: About 30 000 of the cross sections curves were considered. Since the cross sections can be calculatedonly for a finite number of energies, the cross section values needed for the calculation of rate constants were obtained through spline interpolation. The state-specific capture rate constants k(J,K) are calculated by convolution of the collision energy dependence of the cross sec- tions with the collision energy distribution between CH 3CN and Ne+. The latter can be evaluated by Ttrobtained from Eq. (1) by applying the Maxwell Boltzmann distribution ftr(E). We calculate the state-specific capture rate constant according to k(J,K)=∫Emax Eminσ(J,K)ftr(E)√ EdE, (4) FIG. 4 . Comparison between experiment (lower) and simulation (upper) for the longitudinal velocity distributions of the velocity-selected CH 3CN. Each number indicated in the upper graphs shows a translational temperature derived from the nonlinear fit of P1d[for (a)–(e)] or Pcold(v, 2) [for (f)] of the equations in the supplementary material to the simulation results. The experimentally measured TPMshown in the lower figures are the values in Fig. 2(b). J. Chem. Phys. 153, 124305 (2020); doi: 10.1063/5.0013807 153, 124305-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp where Emax (Emin) is the maximum (minimum) energy for the interpolated data. To obtain the average capture rate constant kPRScorresponding to the experimental condition from k(J,K), we need the informa- tion about rotational state distribution ρ(J,K) of the velocity-selected acetonitrile molecules. Since we could not directly measure ρ(J,K) of the velocity-selected CH 3CN molecules, it was evaluated by Monte– Carlo trajectory simulations. The details of the simulation method were described in our previous papers.34,40 First, we calculated the Stark shift energies of all rotational levels of CH 3CN up to J= 60 as a function of the applied electric field strength and record least-square fit coefficients of each Stark shift curve fitted by the fourth-order polynomial func- tion. Then, we randomly set the initial position and velocity of a polar molecule using uniform random numbers according to the Maxwell–Boltzmann distribution at the assumed temperature. We performed the numerical integration of the equation of motion for each polar molecule using the fourth order Runge–Kutta algorithm. The force acting on a polar molecule is obtained by the gradient of the fitted Stark shift curve, which depends on a molecular posi- tion under quadrupole fields in the guide electrodes. Finally, we recorded the position, velocity, intensity, and the rotational levelof the velocity-selected molecule at some certain distances from the beam guide exit. We considered the rotational state distribu- tion of the source gas and the degeneracy arising from nuclear spin statistics as the weight to the intensity of the velocity-selected polar molecules. Repeating the same calculations for all rotational levels considered, we obtained the renormalized ρ(J,K) and the velocity distribution. Figure 4 shows the comparison between simulation and exper- iment for the longitudinal velocity distributions of the velocity- selected CH 3CN molecules. The corresponding simulation results of rotational level distributions are shown in the upper graphs of Fig. 5. It is observed that the rotational levels with the smaller Stark shift lead to a smaller population. In the lower graphs in Fig. 5, we also show the thermal rotational distributions, which are obtained by assuming that the rotational temperatures are the same as the translational ones. Since a good agreement was obtained between the simulation and experiment for the longitudinal velocity distribu- tion, we assumed that the simulation result of ρ(J,K) reproduces the actual rotational level distribution of the velocity-selected CH 3CN molecules. Using the simulation results of ρ(J,K), we can calcu- late the velocity-selected PRS capture rate constant by the following equation: FIG. 5 . Simulation results of rotational level populations ρ(J,K) of the velocity-selected and thermal CH 3CN molecules. The upper graphs show the simulation results of the velocity-selected molecules with the translational temperatures Ttrof (a) 11(1) K and (b) 63(2) K. The lower graphs are the populations with the corresponding thermal conditions, i.e., the rotational temperature Trotis the same as the translational one. J. Chem. Phys. 153, 124305 (2020); doi: 10.1063/5.0013807 153, 124305-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp kPRS=∑ J∑ Kρ(J,K)k(J,K). (5) IV. RESULTS AND DISCUSSIONS To measure the translational temperature dependence of the reaction rate constant of the CH 3CN + Ne+→product, we used two types of the Stark velocity filters as mentioned above. A cold CH 3CN molecular beam of a few kelvin was generated by the standard type Stark velocity. On the other hand, higher translational temperatures can be obtained using the wavy Stark velocity filter. We changed the guide voltages | VG| of the wavy Stark velocity filter from 0.4 kV to 2.0 kV, which resulted in the change in the translational temperature of the velocity-selected CH 3CN from 13(2) K to 60(1) K under the present experimental conditions. In the reaction rate measurements, we first generated a two- component ion Coulomb crystal consisting of Ca+and Ne+ions and observed the decrease in the dark area occupied by Ne+ions with increasing reaction time with the velocity selected CH 3CN molecules. The experimental details are provided in the supplemen- tary material. Figure 6 shows plots of the relative number of Ne+ions as a function of the reaction time at some different translational tem- peratures of the velocity-selected CH 3CN molecules. As the number density of the velocity-selected molecules drastically increases with the increase in the guide voltages | VG| in the case of the wavy Stark velocity filter, the reaction rate becomes faster as | VG| increases in the measurements. It should be noted that our previous experiments suggested that a small number of residual ions, possibly NeH+, were present.35The fraction of the residuals in the dark area was less than 5% of the initial number of Ne+ions after a long reaction time of over 3000 s. Therefore, in the present reaction rate measurement, the sys- tematic error due to the residual ions is evaluated to be 5%, and we included this error into the uncertainty of each measured reaction rate. FIG. 6 . Reaction rate measurements between the velocity-selected CH 3CN molecules and cold Ne+ions. The relative number of sympathetically cooled Ne+ ions are plotted as a function of the reaction time. The measurements (a)–(c) were performed by the wavy Stark velocity filter while (d) by the standard type. The applied guide voltages | VG| are (a) 0.4 kV, (b) 0.8 kV, (c) 2.0 kV, and (d) 3.0 kV, respectively.The pseudo-first-order reaction rate γis transformed to the reaction rate constant kby the relation of k=γ/n, where nis the number density of the velocity-selected CH 3CN at the ion trap position, as shown in Fig. 2. Figure 7 shows the translational tem- perature dependence of the reaction rate constant obtained by the present experiment. We also plot the velocity-selected PRS capture rate constant kPRSdescribed in Sec. III. A strong translational temperature dependence on the exper- imental reaction rate constant is observed particularly at the lower temperature than 5 K. In contrast, the velocity-selected PRS rate constants show a weak temperature dependence over the whole range of temperatures. The PRS result is consistent with the exper- imental value at a few kelvin but deviates significantly at the higher translational temperatures. It can be noted that the experimental rate constants are even smaller than the Langevin rate constant kL at temperatures higher than 10 K. We regard the capture rate constant kcapas the upper limit of the reaction rate constant. Hence, we can set k=Preactkcap, with Preact≤1 being the reaction probability. Figure 7 shows Preact≃1 atTtr= 2.7 K and Preact≪1 atTtr>5 K. Unexpected small reaction rate constants (namely, Preact≪1) at room temperature have been reported in similar reactions such as Ne++ (H 2O and NH 3).53,54 In the present experiment, as the rotational state distribution ρ(J, K) of the velocity-selected molecules is not strongly dependent on the applied guide voltages (see Fig. 5), the rapid variation in kat Ttr>5 K is not due to the difference in the rotational state distri- butions. Further experiments at different source gas temperatures may be interesting for understanding the reaction mechanism. We have a plan to perform such experiments using a cold buffer gas FIG. 7 . Comparison of the experimental rate constants with various capture rate constants: (a) the velocity-selected PRS capture rate constant, (b) present experi- mental work, (c) the thermal PRS capture rate constant (see the text), (d) Langevin rate constant,9and (e) Su–Chesnavich expression of the ion–dipole capture rate constant obtained by classical trajectory calculations.55The lines between plot points are guides for the eyes. J. Chem. Phys. 153, 124305 (2020); doi: 10.1063/5.0013807 153, 124305-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp cell for systematically changing ρ(J,K). One possibility to explain Preact≪1 and its strong translational temperature dependence is the presence of a small potential barrier along the reaction path. In this type of reaction, a quantum chemistry calculation23,27has shown that the rate constant has a minimum value at a certain tempera- ture ( T0) associated with the potential barrier height: the increase in the rate constant with decreasing Ttrbelow T0occurs owing to the tunneling effect. Such temperature dependence was experimen- tally observed in NH+ 3+ H 2→NH+ 4+ H.24–26,28Also in the present system, the reaction rate measurements above 20 K might demon- strate the increase with increasing Ttr, and then, it would give us the information about the height of the potential barrier. It is very important to theoretically determine the potential energy surface relevant to intermediate and transitions states of the CH 3CN–Ne+ system. In the future, we plan to carry out the quantum chemistry calculation. V. FURTHER DISCUSSION In Fig. 7, we plot the thermal PRS capture rate constants calcu- lated by assuming the thermal distribution of the translational and rotational motion, as well as those obtained by the Su–Chesnavich expression, which was obtained by performing classical trajectory Monte–Carlo calculations.55Su and Chesnavich have found that the thermal ion–dipole capture rate constant kth(T) divided by the Langevin rate constant kLcan be empirically given for various kinds of the ion–molecule system by kth(T)/kL=0.4767 x+ 0.62, x≥2, (6) which is only a function of x=D/(2αkBT)1/2, with Dbeing the dipole moment, αbeing the polarizability, and Tbeing the temperature. This is a very simple relation and has been occa- sionally used in estimating unknown values of reaction rate con- stants in astrochemical studies.4,5Figure 7 shows that the ther- mal PRS capture rate constants are in good agreement with the values obtained by Eq. (6) for the present reaction system. Simi- lar results were obtained for other polar molecules in a previous study.56 However, Fig. 7 further shows that the thermal PRS capture rate constants are completely different from the velocity-selected PRS rate constants in which the experimental rotational state dis- tribution is taken into account. This reflects the fact that the rota- tional states of the molecules under the experimental condition do not have the thermal distribution and that the capture rate constant strongly depends on the rotational state. It should be noted furthermore that the reaction rate constant can be quite large for symmetric-top molecules having K≠0 (Fig. 3). Thus, the reaction rate constant of the CH 3CN–Ne+system is certainly very sensitive to the rotational state distribution of CH 3CN at the translational temperatures of a few kelvin. If the rotational state distribution of molecules were far from the thermal one in inter- stellar clouds, Eq. (6) could not supply valid data for the reaction rate constants of the interstellar polar molecules. Our study clearly shows the importance of getting reliable information on the rota- tional state distribution of polar molecules under interstellar cloud conditions.VI. CONCLUSION We have measured the translational temperature dependence of the reaction-rate constant of the CH 3CN + Ne+→products by applying two types of the Stark velocity filters, i.e., the standard type Stark velocity filter and the wavy Stark velocity filter. A steep increase in the reaction rate constant has been observed at temperatures lower than 5 K. To clarify the translational temperature dependence of the reaction probability, the PRS capture rate constants have been evaluated. The PRS theory reproduces well the experimental reac- tion rate constant at a few kelvin but not higher than 5 K. The difference between experiment and theory indicating the smaller reaction probability than unity may show the presence of a poten- tial barrier in the reaction path. Further measurements at different source gas temperatures and at higher translational temperatures are expected to reveal the reaction mechanisms. Such measurements can be performed by using a cold buffer gas cell37and the wavy Stark velocity filter with a larger curvature radius.40The experiment using an upgraded buffer gas cell connected to the wavy Stark velocity filter will be performed in the future work. The present research method combining the reaction rate mea- surements with the theoretical calculations of the PRS capture rate constants is very effective to elucidate both the translational and rotational temperature dependence of the reaction probability in cold ion–polar molecule reactions related to interstellar chemistry. 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5.0022060.pdf

J. Chem. Phys. 153, 104113 (2020); https://doi.org/10.1063/5.0022060 153, 104113 © 2020 Author(s).Transferable interactions of Li+ and Mg2+ ions in polarizable models Cite as: J. Chem. Phys. 153, 104113 (2020); https://doi.org/10.1063/5.0022060 Submitted: 16 July 2020 . Accepted: 21 August 2020 . Published Online: 11 September 2020 Vered Wineman-Fisher , Julián Meléndez Delgado , Péter R. Nagy , Eric Jakobsson , Sagar A. Pandit , and Sameer Varma ARTICLES YOU MAY BE INTERESTED IN Improved description of ligand polarization enhances transferability of ion–ligand interactions The Journal of Chemical Physics 153, 094115 (2020); https://doi.org/10.1063/5.0022058 Understanding how water models affect the anomalous pressure dependence of their diffusion coefficients The Journal of Chemical Physics 153, 104510 (2020); https://doi.org/10.1063/5.0021472 Ion transport in small-molecule and polymer electrolytes The Journal of Chemical Physics 153, 100903 (2020); https://doi.org/10.1063/5.0016163The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Transferable interactions of Li+and Mg2+ions in polarizable models Cite as: J. Chem. Phys. 153, 104113 (2020); doi: 10.1063/5.0022060 Submitted: 16 July 2020 •Accepted: 21 August 2020 • Published Online: 11 September 2020 Vered Wineman-Fisher,1Julián Meléndez Delgado,1 Péter R. Nagy,2 Eric Jakobsson,3Sagar A. Pandit,4 and Sameer Varma1,4,a) AFFILIATIONS 1Department of Cell Biology, Microbiology and Molecular Biology, University of South Florida, Tampa, Florida 33620, USA 2Department of Physical Chemistry and Materials Science, Budapest University of Technology and Economics, P.O. Box 91, H-1521 Budapest, Hungary 3National Center for Supercomputing Applications, Center for Biophysics and Computational Biology, Department of Molecular and Integrative Physiology, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA 4Department of Physics, University of South Florida, Tampa, Florida 33620, USA a)Author to whom correspondence should be addressed: svarma@usf.edu ABSTRACT Therapeutic implications of Li+, in many cases, stem from its ability to inhibit certain Mg2+-dependent enzymes, where it interacts with or substitutes for Mg2+. The underlying details of its action are, however, unknown. Molecular simulations can provide insights, but their reliability depends on how well they describe relative interactions of Li+and Mg2+with water and other biochemical groups. Here, we explore, benchmark, and recommend improvements to two simulation approaches: the one that employs an all-atom polarizable molec- ular mechanics (MM) model and the other that uses a hybrid quantum and MM implementation of the quasi-chemical theory (QCT). The strength of the former is that it describes thermal motions explicitly and that of the latter is that it derives local contributions from electron densities. Reference data are taken from the experiment, and also obtained systematically from CCSD(T) theory, followed by a benchmarked vdW-inclusive density functional theory. We find that the QCT model predicts relative hydration energies and structures in agreement with the experiment and without the need for additional parameterization. This implies that accurate descriptions of local interactions are essential. Consistent with this observation, recalibration of local interactions in the MM model, which reduces errors from 10.0 kcal/mol to 1.4 kcal/mol, also fixes aqueous phase properties. Finally, we show that ion–ligand transferability errors in the MM model can be reduced significantly from 10.3 kcal/mol to 1.2 kcal/mol by correcting the ligand’s polarization term and by introducing Lennard-Jones cross-terms. In general, this work sets up systematic approaches to evaluate and improve molecular models of ions binding to proteins. Published under license by AIP Publishing. https://doi.org/10.1063/5.0022060 .,s I. INTRODUCTION Li+is an essential nutrient, but at high concentrations, it is toxic.1At low-to-intermediate concentrations, Li+is considered a therapeutic agent. It is a generally accepted first line therapy for bipolar disorder,2and it is also useful as an augmentation to other antidepressants in the treatment of unipolar depression.3Several lines of evidence also suggest that Li+may be a component of ther- apy or prevention against neurodegenerative disorders4and can- cer.5Our analysis of the human “Li+interactome,” which is a network of genes interacting with Li+-sensitive genes, shows a strongmutual enrichment with KEGG (Kyoto Encyclopedia of Genes and Genomes) pathways associated with neurodegenerative diseases6 and cancer.7 The primary mode by which Li+affects the physiological func- tion is the one in which it inhibits the activities of certain Mg2+- dependent phosphoryl-transfer enzymes, including phosphatases, kinases, and adenylyl cyclases.1,8,9Based on the structural and bio- chemical studies on phosphatases,10–12Li+is expected to bind to these enzymes by competing against Mg2+and substituting for it in the enzyme’s catalytic core (competitive binding). Alternatively, NMR13and quantum chemical14studies show that in solution, free J. Chem. Phys. 153, 104113 (2020); doi: 10.1063/5.0022060 153, 104113-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp ATP loads Li+and Mg2+simultaneously, leading to the proposition that Li+could also bind to the catalytic core without altering the numbers of bound Mg2+(cooperative binding). Regardless of Li+’s binding mode, the molecular mechanism of how Li+affects their activities remains unknown. In fact, in a screening of over 70 human kinases at elevated Li+concentrations, Bain et al.15reported that many were affected to varying degrees, but there is no explanation for these variations. Molecular simulations can, in principle, provide the neces- sary energetic basis to understand these competitive and coop- erative binding effects of Li+. This, however, requires accurate and computationally efficient descriptions of relative interactions of Li+and Mg2+with water and other relevant organic groups. Here, we explore and validate two simulation models: the one that employs the all-atom polarizable AMOEBA molecular mechanics (MM) force field16and the other that is based on a hybrid quan- tum and implicit solvent implementation of the quasi-chemical theory (QCT).17,18The advantage of the former method over the latter is that it explicitly describes thermal motions and long ranged electrostatics. The advantage of the latter approach is that it derives all local contributions from a broad range of molecu- lar forces, including charge-redistribution, polarization, and dis- persion, directly from self-consistent electron densities, rather than through their classical interpretations in MM models. In addi- tion, the quasi-chemical formulation inherently provides an under- standing of solvation and binding in terms of contributions from ligand density and number, as well as local and long-ranged effects. Toward this end, we first benchmark a vdW-inclusive quantum density functional theory (DFT) against the experiment and high- level quantum mechanical calculations. We then use it to obtain reference data for gas phase interactions of ions with water and two other biochemical groups, methanol and N-methyl acetamide, which are representative small molecules of the hydroxyl and car- bonyl functional groups found in Mg-binding sites in proteins.1We use these reference data alongside experiments to recommend the necessary improvements to both the MM and QCT models in the gas and condensed phases. II. METHODS A. Reference energies Coupled cluster single double and perturbative triple excita- tion [CCSD(T)] theory is significantly more expensive than density functional theory (DFT), and so we restrict its application to obtain reference information to benchmark a vdW-inclusive DFT. Specif- ically, we employ CCSD(T) in combination with complete basis set (CBS) extrapolation19,20to compute Mg2+–water binding energies. We employ Dunning’s correlation-consistent basis sets augmented with diffuse functions (aug-cc-pV XZ, X = T, Q, 5) for the first row elements and the corresponding core-valence basis sets21for Mg2+. Sub-valence electrons of Mg2+are correlated in the CCSD(T) calcu- lations, while deep-core electrons of all atoms are kept frozen. The basis set incompleteness error (BSIE) of the CBS(Q,5) interaction energies are estimated as the difference of the CCSD(T) energies obtained with CBS(T,Q) and CBS(Q,5). The local natural orbital(LNO) scheme22,23is employed to accelerate the CCSD(T) calcula- tions as implemented in the MRCC package.24,25Approximation-free CCSD(T) energy and the corresponding local error estimates are evaluated using the Tight and very Tight LNO-CCSD(T) thresh- old sets,23,26according to the extrapolation scheme of Ref. 26. The cumulative BSIE and the local error estimates indicate that the LNO- CCSD(T)/CBS(Q,5) interaction energies are within ±0.4 kcal/mol of the approximation-free CCSD(T)/CBS ones for all studied com- plexes. The DFT that we benchmark is PBE0+vdW.27,28The PBE0 hybrid functional contains 25% exact exchange and is supplemented by Tkatchenko–Scheffler self-consistent corrections for dispersion (vdW). All PBE0+vdW calculations are performed using the FHI- AIMS package29with “really tight” basis sets. Total energies are con- verged to within 10−6eV and electron densities are converged to within 10−5electrons. Geometry optimizations are carried out with a force criterion of 10−3eV/Å and the PBE0+vdW functional. The ion–ligand cluster geometries used in CCSD(T) are those obtained from PBE0+vdW optimizations. B. Molecular dynamics simulations All MD simulations are performed using TINKER16version 7.1. Integration is carried out using the RESPA integrator with an outer time step of 1 fs.16The Bussi thermostat30and Monte Carlo barostat31,32with a coupling constant of 0.1 ps are employed to control temperature (T = 298 K) and pressure (P = 1 bar), respec- tively. Electrostatics is treated using the polarization modulation ellipsometry (PME) approach with a direct space cutoff of 9 Å. The convergence cutoff for induced dipoles is set to 0.01 D, and the van der Waals interactions are computed explicitly within a radius of 9 Å. III. FREE ENERGIES A. Condensed phase—Bennett’s acceptance ratio Hydration free energies of ions in explicit solvent are com- puted using Bennett’s acceptance ratio (BAR).30The conformational ensembles needed for BAR are obtained from molecular dynamics. Bennett shows that an almost optimal solution to estimate the free energy difference ( ΔG) between equally sampled states AandBis obtained by (a) minimizing the variance of the FEP average, which is done by choosing the Fermi–Dirac distribution f(x) = 1/(1 + ex) as weighting factor, and (b) offsetting the energy by a scalar cin such a way that the error is minimized. cis determined from the following equation in a self-consistent manner to ensure c≈ΔG: e−β(ΔG−c)=⟨f(β(UB−UA−c))⟩A ⟨f(β(UA−UB−c))⟩B. (1) In the equation above, β= 1/kbT.UaandUbare the potential ener- gies obtained for the same configuration but computed using func- tions describing states AandB, respectively. The triangular brackets represent averages over configurational space sampled in states, A or B, indicated by subscripts. To obtain an overlap between UAandUB, and thus reduce the variance in the estimation of ΔG,30we compute the solvation free J. Chem. Phys. 153, 104113 (2020); doi: 10.1063/5.0022060 153, 104113-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp energies in multiple steps.33,34We first scale down the charges and polarizabilities of ions. In systems containing Li+, this is done in 10 steps withλ= {1, 0.9, . . ., 0.1, 0}, and in systems containing Mg2+, this is done in 20 steps with λ= {1, 0.95, . . ., 0.05, 0}. Each step is simulated for 200 ps under NVT conditions, and the final 100 ps are used to calculate ΔG. Each system consists of 1500 water molecules, and either a single cation or a salt molecule (LiCl or MgCl 2). For a given sys- tem, the starting configurations of all its λ-simulations are identical. They are taken from a trajectory equilibrated under NPT conditions for 500 ps. The final 200 ps of this trajectory are used for calcu- lating average box lengths, and the snapshot that has the closest box length to this average is used as the starting configuration for λ-simulations. Following the computation of ΔGusing BAR, the correction term−RTlnCl/Cg= 1.9 kcal/mol is added to adjust for ion concen- tration differences between gas ( Cg= 0.041M) and condensed ( Cl = 1M) phases.18 B. Condensed phase—Implicit solvent model Hydration free energies of ion–ligand clusters in implicit sol- vent are obtained from two solutions to the static Poisson model, the one in a dielectric medium of ϵ= 1 and the other in a dielec- tric medium of ϵ= 78.5, that is, ΔGaq=G(78.5) −G(1). Poisson’s equations are set up numerically by describing atoms using the ParSE (Parameters for Solvation Energy) parameter set35and defin- ing dielectric boundaries using a solvent probe radius of 1.4 Å about the ParSE atomic radii. The region not occupied by the solvent is assigned a dielectric of ϵ= 2, consistent with ParSE parameteriza- tion. Poisson’s equation is solved using a multigrid approach imple- mented in the APBS v 1.3 package.36For all calculations, the finest grid spacing is set at 0.138 Å, and the spatial extent of the outer grid size is set at 100 Å. Reducing the outer grid size by half changes energies by less than 0.5 kcal/mol. C. Gas phase—Harmonic approximation The thermal component to the Gibbs free energy of a clus- ter in the gas phase is estimated using the ideal gas thermody- namic relationship, Gcorr=Fcorr+ 1/β, where Fcorris the correc- tion to the Helmholtz free energy. Assuming that the coupling between translational, vibrational, and rotational degrees of freedom can be neglected, Fcorris estimated as a sum of their independent contributions,37that is, Fcorr=Ftrans +Fvib+Frot, where Ftrans=−1/β⎡⎢⎢⎢⎢⎣ln(m 2π̵h2β)3/2 + ln1 βP+ 1⎤⎥⎥⎥⎥⎦, (2) Fvib=E+3N−6 ∑ i[̵hωi 2+ 1/βln(1−exp−β̵hωi)], (3) and Frot=−3 2βln[2 β̵h2(IAIBIC)1/3π1/3]. (4) In the expressions above, Nis the number of atoms in the molecule, IA,IB, and ICare the molecular moments of inertia, P= 1 atm, misthe molecular mass, and ωiare the harmonic vibrational frequen- cies obtained from a Hessian analysis of the PBE0 energy surface. Wave functions are described using the 6-311++G∗∗basis set, and we note that switching to a Dunning’s correlation consistent basis set (aug-cc-pVDZ) increases ion–ligand binding free energies by an average of 0.7 kcal/mol. These calculations are performed using Gaussian 09.38 IV. RESULTS A. Ion–water interactions in the gas phase 1. Reference energies We have demonstrated previously that the PBE0 density functional27augmented with self-consistent dispersion corrections (PBE0+vdW)28yields interaction energies of Na+and K+ions with water, methanol, and formamide molecules in excellent agree- ment with CCSD(T) and Quantum Monte Carlo (QMC).34,39We have also noted previously40that under a harmonic approxima- tion, PBE0+vdW predicts gas phase ion–water cluster enthalpies and free energies consistent with the experiment. Therefore, we con- tinue to use PBE0+vdW to obtain reference data for interactions of Li+ions with water, methanol, and N-methylacetamide (NMA) molecules. Table I compares predictions of Mg2+–water binding ener- gies from PBE0+vdW against LNO-CCSD(T). Ion–water binding energies are defined as ΔE=EAW n−n×EW−EA, (5) where EAW n,EW, and EAare, respectively, the electronic energies of ion–water clusters, isolated water molecule, and isolated ions fol- lowing independent energy optimizations. nis the number of water molecules in the ion–water cluster. With a mean absolute error (MAE)<0.5 kcal/mol, the correspondence is excellent. Based on our earlier work,39and the error measures corresponding to the LNO approximations being below 0.15 kcal/mol, we expect that the LNO scheme22,23employed to accelerate CCSD(T) retains its intrinsic accuracy. Table II compares experimental gas phase ion–water binding free energies against those obtained from PBE0+vdW. The gas phase ion–water binding free energies are computed by adding harmonic and analytical thermal corrections Gibb’s energy (see methods) to ΔEobtained from Eq. (5). Note that the 5- and 6-fold clusters of TABLE I . Comparison of Mg2+–water binding energies computed using LNO- CCSD(T) and PBE0+vdW. Binding energies are normalized by the number of water molecules in clusters and are in units of kcal/mol. n LNO-CCSD(T) PBE0+vdW 1 −82.4 −83.6 2 −77.8 −78.6 3 −71.6 −72.1 4 −65.8 −65.9 5 −59.2 −59.2 6 −54.4 −54.3 J. Chem. Phys. 153, 104113 (2020); doi: 10.1063/5.0022060 153, 104113-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II . Comparison of experimental gas phase free energies against those obtained from PBE0+vdW with harmonic and analytical thermal components to Gibb’s energy. All energies are in kcal/mol. n Expt. PBE0+vdW Li+1 −25.5 −28.0 2 −18.9 −21.8 3 −13.3 −13.0 4 −7.5 −7.5 5 −4.5 −2.1 6 −2.5 −2.6 Mg2+6 −16.0 −17.0 7 −12.8 −10.9 8 −10.9 −9.6 Li+as well as the 7- and 8-fold clusters of Mg2+used in comput- ing free energies do not contain all waters in their respective first coordination shells (see Fig. S1 of supplementary material). This is because their binding energies were less favorable compared to those in which a subset of waters were outside their inner shells, just as we noted previously for Na+and K+ions.18For ion–water clusters in which all waters are interacting directly, the computed values are on average overestimated by 1.2 kcal/mol. For the 7- and 8-fold Mg2+–water clusters in which one and two waters are, respectively, in the ion’s second shell, the binding free energies are off by a similar magnitude, but generally underestimated. Given that the computed ΔEin Table I agree well with LNO-CCSD(T), it appears that the discrepancies in free energies result from the harmonic approximation employed in computing the vibrational contributions to free energies. Nevertheless, the errors introduced by harmonic approximations are still small enough to keep thecomputed free energies close to the desired chemical accuracy of 1 kcal/mol. 2. Recalibration of ionic descriptors in MM model Table III shows that the ion–water binding energies pre- dicted from the original AMOEBA model16,41are substantially off with respect to our reference values. Additionally, the error is systematic—the predicted values are underestimated, and the extent of underestimation also grows with cluster size. For the 6-fold Li+ clusters, the error reaches 7 kcal/mol, and for the Mg2+cluster, the error gets larger than 20 kcal/mol. We, therefore, re-calibrate the descriptors of Li+and Mg2+ions against our new reference data and include as part of the target set all of the PBE0+vdW data listed in Table III. In the AMOEBA model, vdW interactions are described using a buffered 14-7 function, Uvdw=ϵij(1.07 ρij+ 0.07)7⎛ ⎝1.12 ρ7 ij+ 0.12−2⎞ ⎠, (6) whereϵijin kcal/mol is the potential well depth and ρij=rij/r0 ij, where rijin Å is the distance between sites iandjandr0 ijis the minimum energy distance. Re-optimizing the vdW descriptors of Li+against the new target data reduces the MAE in binding energies from 7.4 kcal/mol to 2.3 kcal/mol, but slightly decreases the Li+-oxygen distance. The ion–water cluster geometries energy optimized using the recalibrated model, however, remains similar to those obtained from PBE0+vdW (Fig. S1 of the supplementary material). The recal- ibrated descriptors of Li+(ϵ,r0), which we refer to as the Pol∗model, are (0.059, 1.906), while the original vdW descriptors were (0.08, 2.38). In the case of Mg2+, we also recalibrate a dimensionless parameter in its polarization term, as done in the development of the original and improved descriptors.41–43AMEOBA employs a Thole approach44to prevent polarization catastrophe, wherein TABLE III . Ion–water binding energies ( ΔEin kcal/mol) and optimum distances ( din Å) prior to (Orig16,41and Orig∗42,43) and after recalibration (Pol∗) of ion descriptors against PBE0+vdW. The ion–water binding energy is defined using Eq. (5), and d is the distance between the ion and the oxygen atom of water. MAE is an abbreviation for mean absolute error. PBE0+vdW Orig Orig∗Pol∗ Ion No. of waters ΔE d ΔE d ΔE d ΔE d Li+1 −36.0 1.82 −33.2 1.82 −36.9 1.72 2 −66.9 1.85 −61.1 1.88 −66.9 1.78 3 −90.6 1.89 −81.9 1.94 −88.3 1.86 4 −107.7 1.95 −97.7 2.01 −104.1 1.93 5 −116.6 2.06 −107.2 2.07 −112.5 1.99 6 −125.7 2.13 −117.8 2.15 −122.6 2.10 Mg2+1 −83.6 1.91 −78.0 1.88 −83.3 −83.6 1.84 2 −157.1 1.93 −146.4 1.91 −155.0 −157.1 1.87 3 −216.2 1.96 −201.2 1.96 −211.1 −215.7 1.91 4 −263.7 1.99 −245.5 2.00 −255.7 −263.0 1.95 5 −295.8 2.05 −276.1 2.04 −285.1 −294.8 1.99 6 −325.5 2.09 −304.4 2.11 −312.8 −324.4 2.05 MAE 10.0 0.02 6.5 1.4 0.05 J. Chem. Phys. 153, 104113 (2020); doi: 10.1063/5.0022060 153, 104113-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp electrostatic interactions are damped in the short range. Damping is applied to only one of the two sites of an interaction pair using ρ=3a 4πe−ar3 ij/√αiαj, where rijis the distance between two sites with atomic polarizabilities α, and “ a” is a dimensionless width param- eter of the damped charge distribution that controls the damping strength. The parameter ais typically assigned a value of 0.39 to reproduce molecular polarizabilities and cluster energies of water and other molecules.33,44–47The data in Table III show that recali- bration ofϵ,r0, and aresults in a significant improvement in Mg2+– water binding energies. The mean error in binding energy reduces from 15.0 kcal/mol to 0.6 kcal/mol, but just as in the case of Li+, the Mg2+-oxygen distances decrease slightly (Table III) with mini- mal effect on cluster geometry (Fig. S1 of the supplementary mate- rial). The recalibrated descriptors of Mg2+(ϵ,r0,a), which we also refer to as the Pol∗model, are (0.45, 2.05, 0.085), while the original descriptors were (0.3, 2.94, 0.0952). B. Ion–water interactions in aqueous phase 1. Explicit solvent polarizable MM model To evaluate the effect of MM model recalibration on aque- ous phase properties, we first use BAR30to compute ion hydration free energies. However, instead of comparing hydration free ener- gies of individual cations to the experiment, we focus on comparing their relative hydration free energies. This is because experimental estimates of the former quantity depend on extra-thermodynamic assumptions that yield a wide spread in their estimates. In con- trast, estimates of the relative hydration free energies are free from such assumptions.48As such, Li+/Mg2+competitive binding to pro- teins requires accurate estimates of the latter quantity. The extra- thermodynamic assumptions are essentially needed to separate out the energetics of salt dissolution into their constituent cationic and anionic contributions.49,50Two assumptions are commonly used, TATB and CPA. TATB assumes that the magnitudes of the solvation energies of tetraphenylarsonium (TA) and tetraphenylborate (TB)are equal.49,51,52CPA refers to the cluster-pair approximation53in which the energetics of cations and anions in their individual sol- vent clusters are expected to converge toward each other rapidly following a monotonous trend, although recent studies suggest a more complex convergence.54These two assumptions, for exam- ple, produce Li+hydration free energies that differ by 10 kcal/mol (Table IV). We carry out free energy calculations in two different ways. In one set, we simulate a single cation in a periodic box of 1500 waters and subject it to perturbation in BAR calculations. In the second set, we simulate a salt (LiCl or MgCl 2) in a periodic box of 1500 waters, and subject all ions in the box (two in the case of LiCl and three in the case of MgCl 2) to simultaneous perturbations in BAR calculations so that the net charge of the periodic box remains neutral and constant in allλ-simulations.55 The results of these free energy calculations are provided in Table IV. For each case, we report three different hydration free energy estimates, corresponding to different adjustments to air– water interface potentials ψ. Values denoted by ψ0are obtained directly from BAR calculations in bulk water, and so they do not include interface potential effects. This value may be compared directly to experimental values obtained using the TATB scheme, as this scheme is not expected to include air–water interface poten- tial effects.48,56We note, however, that there is no consensus on either the magnitude or the sign of air–water interface potential.57–61 Nevertheless, for the sake of comparison, we provide hydration free energy estimates in Table IV that are adjusted for two different inter- face potentials, ψ=−0.4 V60andψ= +0.1 V.58,61Adjustments are made as ΔGψ=eFψ+ΔGψ=0, where “ e” is the charge of the ion in electron units and “ F” is Faraday’s constant. When we simulate a single cation in a periodic box, we find that recalibration does improve Li+→Mg2+free energy differences; however, an error of 5–6 kcal/mol remains with respect to the exper- iment. This error, however, vanishes when we simulate and grow neutral salts, and compute 2Li+→Mg2+free energy differences. The error in the calculation of Li+→Mg2+free energies can, therefore, at least partly be attributed to integrating charges in a finite periodic TABLE IV . Hydration free energies (in kcal/mol) of ions from experiments, QCT QM/MM model, and all-atom MM models. Error estimates for single ion hydration energies in the MM model, which are obtained using Monte Carlo bootstrapping,16are all ≤0.1 kcal/mol. ψ0refers to hydration free energies lacking air–water interface potential effects. ψ−0.4 andψ+0.1refer to hydration free energies adjusted for two different surface potentials, ψ=−0.4 V60andψ=−0.1 V,58,61respectively. Note that experimental TATB estimates are not expected to include interface potential effects, and, therefore, can be compared to ψ0values.48,60Experimental CPA estimates contain interface potential effects and so should be compared to ψ−0.4andψ+0.1. Polarizable MM Orig Pol∗QCT QM/MM Experiment ψ0ψ−0.4ψ+0.1ψ0ψ−0.4ψ+0.1ψ0ψ−0.4ψ+0.1 TATBaTATBbCPAc Li+−105.5 −114.7 −103.2 −111.9 −121.1 −110.6 −116.9 −126.1 −114.6 −116.9 −113.5 −126.5 Mg2+−408.7 −427.1 −404.1 −428.8 −446.2 −424.2 −443.3 −461.7 −438.7 −439.1 −437.4 Li+→Mg2+−303.2 −312.4 −300.9 −316.9 −326.1 −314.6 −326.4 −335.6 −324.1 −322.2 −323.9 2LiCl→MgCl 2−200.9 −200.9 −200.9 −206.4 −206.4 −206.4 −209.5 −209.5 −209.5 −205.3 −210.4 aReference 52. bReference 51. cReference 53. J. Chem. Phys. 153, 104113 (2020); doi: 10.1063/5.0022060 153, 104113-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp boundary system, as demonstrated previously.55Note that the 2Li+ →Mg2+free energy difference is insensitive to the assumption of the interface potential. To examine the structure of water around ions, we compute the radial distributions of water oxygens around ions. We compute these from the final 4.5 ns of 5 ns long MD trajectories of single cations in water generated under isobaric and isothermal conditions. The results are shown in Fig. 1. We note first that recalibration makes the inner shell of both ions tighter, consistent with the enhanced stability of ion–water interactions in the recalibrated model. The positions of the first peak also change, but remain within the experimental ranger of 1.90 Å–1.95 Å for Li+and 2.00 Å–2.12 Å for Mg2+.62The coordi- nation number, that is, the number of waters within the first minima, remains unchanged at 6.0 for Mg2+and decreases from 4.2 to 4.0 for Li+. The coordination numbers from the recalibrated model match with those from the experiment and ab inito molecular dynamics simulations.62,63 Overall, we find that recalibrating ion–water interactions sub- stantially improves their relative ion hydration free energies, but have a little effect on the local structure, which as such was in agreement with the experiment. 2. QCT QM/MM model The potential distribution theorem17,64defines the excess chemical potential or the hydration Gibbs free energy of a solute asΔG=−1/βln⟨e−βΔU⟩, where ΔUis the interaction energy of FIG. 1 . Radial distribution functions g(r) and running integrations n(r) of the water oxygens around ions determined from all-atom MM models. The standard devi- ation represented as a gray shade on the Pol∗model is computed from block averaging.the solute with the surrounding waters. Introducing a conditional probability that the solute interacts with nwaters (W) within an arbitrarily defined sub-volume Γaround the ion leads to the rela- tionship17,18,48 ΔG=−1/βln[∑ n≥0K0 n[⟨e−βΔUAWn⟩Γ [⟨e−βΔUW⟩]n](CW)n]. (7) Here, K0 nis the “ideal” equilibrium constant of the association reac- tion A + nW⇌AW n, as it does not include any effects of the com- plementary region ΓCoutside the inner-shell domain. The specific effects of ΓCare incorporated through the ensemble averages. At the same time, K0 nis also not the gas phase equilibrium constant because it is obtained under the condition that the association equilibrium occurs within a predefined sub-volume, whereas the gas phase equi- librium constant is free from such an imposition. ⟨e−βΔUAWn⟩Γis the ensemble average of the distribution that is the product of the distributions for the water molecules in ΓCand for the complex enclosed within the region Γ.⟨e−βΔUW⟩is essentially the excess chem- ical potential of a water molecule. Finally, Caqis the concentration of water molecules in the aqueous phase. In accord with the quasi-chemical formulation in Eq. (7), we calculate the hydration free energy of an ion, ΔG= min{ ΔGn}, by summing up four terms,18 ΔGn=ΔGAW n+ΔGaq AW n−nΔGaq W+ΔGconc. (8) We compute ΔGAW n=−1/βlnK0 nfrom quantum DFT so that all local interactions of ions with water are treated at the electronic level. ΔGaq AW nand the hydration free energy of a water molecule, ΔGaq W, are computed from solutions to Poisson’s equation. The sub-volume Γ needed for these calculations is taken as a sphere around the ion with a radius equal to the first minimum in the RDF of water oxygens around the ion in the aqueous phase. We use the RDFs from the MD simulations reported above. Finally, ΔGconc=−1/βln(Caq/Cg), where the aqueous and gas phase concentrations of water are taken asCaq= 55.6M and Cg= 0.041M. The results of these calculations are provided in Fig. 2. For Li+, the preferred coordination number is n= 4 that yields the smallest ΔGnand a hydration free energy ΔG=−116.9 kcal/mol. Similarly, for Mg2+, the preferred coordination number is 6 and ΔG=−443.3 kcal/mol. The predicted preferred coordination num- bers as well as relative hydration free energies of these ions match experimental estimates.51,52,63Absolute and relative free energies are provided in Table IV, and also adjusted for surface potentials in the same manner as described in Subsection IV A 2. Note that the 7- and 8-fold Mg–water clusters reported in Table II are not used in these calculations because not all waters in these clusters are within the pre-defined sub-volumes (Fig. S1 of supplementary material). Note also that the 5- and 6-fold Li–water clusters used in Table II are not the same as the ones used here—the clusters used here have all waters in lithium’s pre-defined sub-volume (Fig. S1 of supplementary material). We also observed from Fig. 2 that for both ions, the overall shapes of their ΔGnprofiles closely resemble their respective local cluster energies ΔGAW nprofiles, suggesting that the hydration prop- erties of the these ions are dictated primarily by how they interact with waters in their inner coordination shells. Additionally, the local J. Chem. Phys. 153, 104113 (2020); doi: 10.1063/5.0022060 153, 104113-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Quasi-chemical components in the calculation of the hydration free energies of Li+and Mg2+ions. interactions ΔGAW ncontribute to more than half of the hydration free energy, and for both ions, this contribution is about 60%. These are precisely the reasons why getting local interactions, including many-body effects, is critical for predicting hydration free ener- gies. This is also perhaps why a recalibration of local interactions of ions with water in the MM model (Sec. IV B 1) leads to signifi- cant improvements and reproduction of experimental hydration free energies. C. Transferability of ionic interactions Improving interactions of ions with water in the MM model does not, by itself, guarantee meaningful predictions of interactions of ions with other biochemical groups. To evaluate such transferabil- ity, we consider the substitution reaction below, AW n+nX⇌AX n+nW, (9) and determine the associated substitution energy as ΔEsub=EAXn−nEX−EAW n+nEW. (10) Here “A” refers to either a Li+or Mg2+ion, “W” refers to water, and “X” refers to methanol or NMA, which are molecules representative of two different chemical groups, hydroxyls and carbonyls, that are found in cation binding sites in proteins. Figure 3 shows that the original MM model performs poorly in comparison to our reference data. The MAE for Li+is 3.9 kcal/mol and that of Mg2+is much higher at 13.2 kcal/mol. Improving interac- tions of these ions with water, as we did above, will not improve their interactions with methanol and NMA, as we demonstrated recently in the case of other monovalent cations.34 One approach to improve transferability in MM models is to define cross-terms or separate sets of non-bonded (NB) descriptors for every distinct pair of ion and its coordinating chemical group (ligand).43,65–71In most applications,65–71the error corrections in this NB-fix approach are assigned to the Lennard-Jones (LJ) term; however, there is no supporting information of this term being the source of error. FIG. 3 . Water→methanol and water →NMA substitution energies ( ΔEsub) obtained before and after recalibration. Pol∗refers to our recalibrated model of ligands39corrected for their dipolar field response, and Pol∗+NB-fix refers to the model in which the remaining error in the Pol∗model is corrected for by ignoring Lorentz–Berthelot type ion–ligand LJ combination rules and introducing specific ion–ligand cross-terms. An alternative approach is to determine the error source and fix the underlying physics. We demonstrated recently34that one such error source for transferability in the AMEOBA model is its polar- ization term. Specifically, we noted that the contribution of polariza- tion to a ligand’s binding energy was erroneous at the kind of high electric fields present near monovalent cations, although the model performs well in low dipolar electric fields where all MM models are calibrated and benchmarked. We have also shown recently39that when the polarization descriptors of ligands are themselves cal- ibrated to satisfy reference data at high fields, their interactions also improve with ions, and without compromising performance at low fields. Methanol and NMA were among the two ligands we recalibrated as part of this effort. Figure 3 reports the performance of these recalibrated models (Pol∗) in predicting transferability. We find that the use of these model does reduce the Li+MAE to 1.9 kcal/mol, but that of Mg2+, J. Chem. Phys. 153, 104113 (2020); doi: 10.1063/5.0022060 153, 104113-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp despite improvement, does stay quite large at 9.3 kcal/mol. Presum- ably, some physics essential to describing interactions of divalent cations is still misrepresented or missing in the MM model. This is supported by the observation that the residual errors in the recal- ibrated model are systematic, as in the binding energies of Mg2+ to both methanol and NMA are underestimated. Nevertheless, as we show in Fig. 3, these errors can be eliminated by the NB-fix approach, where we generate separate sets of LJ cross-terms for each ion–ligand pair (Table S1 of the supplementary material), instead of computing them from Lorentz–Berthelot type LJ combination rules. After applying the NB-fix approach, transferability MAE reduces to 0.6 kcal/mol and 1.1 kcal/mol, respectively, for Li+and Mg2+ions. V. CONCLUSIONS This work serves as a key step in the development of molec- ular simulations models needed to enable future investigations of relative binding effects of Li+and Mg2+to proteins. We report CCSD(T) reference energies for Mg2+–water clusters and find that the vdW-corrected PBE0 density functional reproduces them. We also show that the vdW-corrected PBE0 density functional also reproduces experimental gas phase ion–water binding free energies of Li+and Mg2+ions. These results are consistent with our pre- vious benchmarks on interactions of Na+and K+ions with vari- ous small molecules.34,39,40Using these reference data, we evaluate two molecular simulation models, the one that employs an all-atom polarizable molecular mechanics (MM) force field and the other that is based on a hybrid quantum and implicit solvent implemen- tation of the quasi-chemical theory. Recalibration of the polarizable MM model substantially improves interactions of Li+and Mg2+with water, with the mean absolute error reducing from 10.0 kcal/mol to 1.4 kcal/mol. Re-parameterization of local ion–water interactions also improves and yields relative hydration free energies of these ions in agreement with the experiment. The QCT QM/MM model, which describes all ion–water local interactions at the QM level, also reproduces the experimental relative hydration free energies of these ions and without any additional parameterization. The QCT analy- sis of energetic components also reveals that local interactions con- tribute substantially to hydration free energies, which provides ratio- nale for why improvements in local interactions in the MM model lead to subsequent improvements in its prediction of hydration free energies. As expected though,34improvements in ion–water interac- tions do not automatically improve interactions of ions with other small molecules. Nevertheless, we show that transferability errors in the MM model can be reduced substantially from 10.3 kcal/mol to 1.2 kcal/mol by correcting the field response of the MM model’s polarization term and explicitly defining Lennard-Jones cross-terms for each ion–ligand pair. The need for introducing Lennard-Jones cross terms, especially for Mg2+ions, suggests that there is still an important underlying physics to be explored for doubly charged cations. In general, this work sets up approaches needed to both eval- uate and improve molecular models of ions binding to proteins. The reference data generated here can also be used to evaluate and improve other MM models models, which can then be employed to study the binding of Li+and Mg2+ions to biomolecules with greaterreliability. Finally, we expect that the Li+study alone will also bene- fit lithium-battery technology and the study of other lithium-based materials.72 SUPPLEMENTARY MATERIAL The supplementary material contains one table and one figure. ACKNOWLEDGMENTS The authors acknowledge the use of computer time from Research Computing at USF and the DECI resource Saga with sup- port from the PRACE aisbl. The authors acknowledge funding from NIH Grant No. R01GM118697. P.R.N. is grateful for financial sup- port of NKFIH, Grant No. KKP126451, and ÚNKP-19-4-BME-418 New National Excellence Program of the Ministry for Innovation and Technology and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1E. Jakobsson, O. Argüello-Miranda, S.-W. Chiu, Z. Fazal, J. Kruczek, S. Nunez- Corrales, S. Pandit, and L. Pritchet, “Towards a unified understanding of lithium action in basic biology and its significance for applied biology,” J. Membr. Biol. 250, 587–604 (2017). 2H. Grunze, E. Vieta, G. M. Goodwin, C. Bowden, R. W. Licht, J.-M. Azorin, L. Yatham, S. Mosolov, H.-J. Möller, and S. 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5.0011786.pdf

Appl. Phys. Lett. 117, 072405 (2020); https://doi.org/10.1063/5.0011786 117, 072405 © 2020 Author(s).Effects of synthetic antiferromagnetic coupling on back-hopping of spin-transfer torque devices Cite as: Appl. Phys. Lett. 117, 072405 (2020); https://doi.org/10.1063/5.0011786 Submitted: 24 April 2020 . Accepted: 07 August 2020 . Published Online: 20 August 2020 Kuan-Ming Chen , Chih-Wei Cheng , Jeng-Hua Wei , Yu-Chen Hsin , and Yuan-Chieh Tseng ARTICLES YOU MAY BE INTERESTED IN A four-state magnetic tunnel junction switchable with spin–orbit torques Applied Physics Letters 117, 072404 (2020); https://doi.org/10.1063/5.0014771 Field-free switching of magnetic tunnel junctions driven by spin–orbit torques at sub-ns timescales Applied Physics Letters 116, 232406 (2020); https://doi.org/10.1063/5.0011433 Spin–orbit torque driven multi-level switching in He+ irradiated W–CoFeB–MgO Hall bars with perpendicular anisotropy Applied Physics Letters 116, 242401 (2020); https://doi.org/10.1063/5.0010679Effects of synthetic antiferromagnetic coupling on back-hopping of spin-transfer torque devices Cite as: Appl. Phys. Lett. 117, 072405 (2020); doi: 10.1063/5.0011786 Submitted: 24 April 2020 .Accepted: 7 August 2020 . Published Online: 20 August 2020 Kuan-Ming Chen,1Chih-Wei Cheng,1 Jeng-Hua Wei,2Yu-Chen Hsin,2and Yuan-Chieh Tseng1,a) AFFILIATIONS 1Department of Materials Science and Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan 2Electronic and Optoelectronic System Research Laboratories, Industrial Technology Research Institute (ITRI), Hsinchu 31040, Taiwan a)Author to whom correspondence should be addressed: yctseng21@mail.nctu.edu.tw ABSTRACT A synthetic antiferromagnetic (SAF) layer is a key component in spin-transfer torque magneto-resistive random-access memory devices. This study reveals that slight fluctuations in SAF coupling at the margin of the reference layer and hard layer (i.e., concurrent reversal) canlead to write errors in the form of back-hopping (BH). It appears that variable BH behavior can be attributed to competition betweenantiparallel ( AP)!parallel ( P) and P!APtransitions associated with SAF coupling. Our conclusions are supported by careful analysis of switching phase diagrams and measurements of self-heating and voltage-controlled magnetic anisotropy. We also observed that one form of coupling provided higher perpendicular magnetic anisotropic energy and thermal stability, which is likely due to the Dzyaloshinskii–Moriyainteraction (DMI) effect. Thus, minimizing variations in DMI by optimizing SAF coupling is crucial for minimizing write error rates. Published under license by AIP Publishing. https://doi.org/10.1063/5.0011786 Spin-transfer torque (STT) magneto-resistive random-access mem- o r yi sas t r o n gc a n d i d a t e ,a m o n ge m e r g i n gm e m o r yt e c h n o l o g i e s . 1,2 Despite extensive research in this field, issues pertaining to reliability stillimpede the commercialization of these devices. One notable issue is anescalation in the write error rate at elevated voltages, referred to as back-hopping (BH). 3,4BH is the result of competition between forward [anti- parallel ( AP)!parallel ( P)] and backward ( P!AP) transitions arising from a decrease in uniaxial anisotropy ( Hk) at elevated voltages.5,6 Hkreduction is generally related to electric-field/voltage and self- heating (SH)7,8effects. One example can be seen in the voltage- controlled magnetic anisotropy (VCMA) at the ferromagnet/oxideinterface, in situations where H kis affected by charge modifica- tion.9Researchers have reported that the high current density and low thermal conductivity of MgO10induce SH effects, which can undermine perpendicular magnetic anisotropy.11BH can also be caused by unexpected flipping of the reference layer (RL).12 Researchers have made great strides in developing the synthetic antiferromagnetic layer (SAF) to enhance RL-robustness (i.e., pre-venting RL-flipping); 13however, the combined effects of SAF cou- pling, VCMA, and SH on BH are yet to elucidated. The primaryissue is the fact that any modification to coupling increases thelikelihood of unexpected state switching. The issue of BH is furthercomplicated by the effects of VCMA and SH at elevated voltages.Our objective in this study was to gain a more comprehensive understanding of these phenomena. The stacking of a perpendicular magnetic tunnel junction (pMTJ) is as follows: [Co(4A ˚)/Pt(3A ˚)] 7/Co(6A ˚)/Ru(8A ˚)/[Co(4A ˚)/Pt(3A ˚)]3/ Co(4A ˚)/CoFeB(3A ˚)/Ta(2A ˚)/CoFeB(8A ˚)/MgO(10A ˚)/CoFeB(11.5A ˚)/ Ta(3A ˚)/CoFeB(7.5A ˚)/capping layer. The pMTJ was deposited via magnetron sputtering on a thermally oxidized Si substrate at room temperature and then annealed at 300/C14C for 2 h. E-beam lithogra- phy was used to pattern the pMTJ into pillar-like structures with adiameter of 180 nm. Finally, reactive ion etching was used to createa step-etched structure with the end point at the MgO barrier. For the sake of simplification, the stack can be divided into a hard layer (HL; [Co/Pt] 7/Co), a reference layer (RL, [Co/Pt] 3/Co/CoFeB/Ta/ CoFeB), and a free layer (FL; CoFeB/Ta/CoFeB), as shown inFig. 1(a) . The [Co(4A ˚)/Pt(3A ˚)] 7/Co(6A ˚)/Ru(8A ˚)/[Co(4A ˚)/Pt(3A ˚)]3/ Co(4A ˚) stack crossing the HL and RL layers is referred to as the SAF layer. Figure 1(a) illustrates the degree of magnetization ( M)v sHloop (64000 Oe) in the pMTJ following multiple reversals. The most distinctive feature is the concurrent reversal of the HL and RL layerscorresponding to a reversal from configurations 3–4. As shown inFig. 1(b) , plotting resistance ( R)v sHloops revealed five apparent tran- sitions between antiparallel ( R ap) and parallel ( Rp) resistance states. Appl. Phys. Lett. 117, 072405 (2020); doi: 10.1063/5.0011786 117, 072405-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplThese states correspond well to those observed in M–H in terms of the switching field ( Hsw). The multiple transitions originate from rotatable HL and RL14as a compromise between the Zeeman energy, exchange coupling, and the respective magnetic anisotropyof the two layers. 15The inset in Fig. 1(b) shows minor R–H loops (61000 Oe) with chiral exchange-bias ( Hex), which can be attrib- uted to SAF flipping. Henceforth, the two states are defined as cou-pling A (CPA) and coupling B (CPB), due to their intimateconnection to the SAF configuration. Figure 2(a) plots Rvs voltage ( V)c u r v e si nt h ec a s ew h e r eC P A / CPB transitions from the antiparallel ( AP) to the parallel ( P) state. Under CPA, the AP!Ptransition occurred at 0.47 V under H appl¼100 Oe (Note that this Happlis meant to compensate for the stray field). We then established the P(low R) state as the original state and performed a negative bias sweep in expectation that doing so would induce pMTJ to undergo a P!APtransition. However, P !APoccurred at /C00.7 V and then immediately bounced back to P under the effects of a continual increase in negative bias. Thisphenomenon, which is anomalous to pure STT switching, is referredto as BH. 3,4Note that under CPB, Happlhad to be adjusted to /C080 Oe in order to obtain BH voltages similar to those obtained under CPA. We sought to verify these results by establishing the AP(high R)s t a t e as the original state under zero bias and increasing the negative bias under Happl¼0. We expected that the APstate would be sustained under the increase in negative bias; however, undesired AP!Pshifts occurred at /C00.52 V and /C00.6 V, under CPA and CPB, respectively. Note that the phenomenon occurred during switching tests on morethan 200 MTJs. Figures 3(a) and3(b) present switching phase diagrams (SPDs) of CPA and CPB ( supplementary material I), respectively. The SPD of CPA and CPB overlapped almost perfectly when the sign of the applied field ( H appl) to one or the other was reversed ( supplementary material II). This means that the switching characteristics observed in the SPDs, such as the stray field of /C24250 Oe (red dashed line) and asymmetric distribution of zones, are intrinsic to the structure exam-ined in this study. In the absence of H appl(as indicated by the black dashed line), switching could be controlled by pure voltage. Specifically, AP!Pwas favored when Vapplwas increased, which sug- gests that this process is STT-driven; however, AP!Poccurred again FIG. 1. (a) Out-of-plane M–H loop obtained from the as-grown pMTJ sheet film where five magnetic configurations related to FL, RL, and HL rotations correspondto five step features of loop, and the inset presents the schematic of pMTJ stacking;(b) main panel: R–H major loops (red: forward; blue: backward) measured between 64000 Oe, containing five resistance states corresponding to those in (a), and the inset presents R–H minor loops with opposite SAF configurations defined as CPA and CPB. FIG. 2. (a)R–V curves measured at positive and negative Vapplfor CPA and CPB with original states set at AP(high R) and P(low R), respectively, and Happlvaried to obtain similar BH behavior for CPA/CPB; (b) R–V curves measured at positive and negative voltages without Happl, but with original states set at AP(high R). The phenomenon occurred in more than 200 MTJs’ switching test.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 072405 (2020); doi: 10.1063/5.0011786 117, 072405-2 Published under license by AIP Publishingwhen Vapplwas decreased. This is consistent with the anomalous switching behavior observed in Fig. 2(b) and appears not to be an STT-driven process. There are two mechanisms that could be respon- sible for this anomalous switching behavior: (i) P!AP(AP!P) boundary leaning back,11which is likely due to a reduction in Hkat elevated voltages; and (ii) the stray field effect ( Hstray), as evidenced by aþ250 Oe shift in the center of the SPD resulting from an uncompen- sated stray field emitted by the SAF. Our SPD findings are supported by the switching probability (Psw)s h o w ni n Figs. 3(c)–3(f) (supplementary material III). In the case of CPA, there was a clear tendency toward AP!Ptransitions follow- ing the increase in Vappland Happl,a ss h o w ni n Fig. 3(c) where the zone changed from purple (high Psw)t ob l a c k( l o w Psw). Nevertheless, under reverse bias, the P!APtransition was characteristic of a decrease in Pswunder increased voltage and Happl[Fig. 3(e) ], indicating the emergence of BH. Careful analysis of Figs. 3(b)–3(f) revealed the same tendency in the case of CPB. We found that in terms of Vappl, the Psw-declining point closely resembled the SPD triple point (the point where the AP,AP/P,a n d Pstates meet). This suggests that BHarose from a competition between forward and backward transi- tions. Interestingly, we found that under CPB, BH tended to occur under a far lower magnetic field [ /C24/C030 Oe, Pswinflection in Fig. 3(f) ] than under CPA [ /C24160 Oe, Pswinflection in Fig. 3(e) ], despite the fact that both the SPD triple points were located at /C24675 Oe. Supplementary material IV presents evidence that altering the range of the sweeping field modified the SAF spin configuration. This suggests that an additional anisotropic term must exist within the system.These results can be attributed to the long-range Dzyaloshinskii–Moriya interaction (DMI)16–21across the Ru layer within the SAF. Hsw(P!AP) histograms of 200 R–H loops [Fig. S4(c)] revealed that the distribution for CPA was broader than that forCPB, which provides evidence of weaker coupling (i.e., weaker DMI)associated with CPA than with CPB. Figures 4(a) and4(b) present the normalized- H swvsHxunder CPA and CPB, respectively. The decrease inHswfollowing an increase in Hxcan be attributed to a reduction in the energy barrier to magnetization reversal imposed by the externalfield. This suggests that CPA/CPB likely possessed dissimilar DMI-induced tilts in local magnetization, resulting in different H sw-Hx behaviors. This would explain why Happlh a dt ob ev a r i e di no r d e rf o r CPA/CPB to trigger BH. Subtracting Hsw(P!AP)from Hsw(AP!P)gives usDHsw,a ss h o w ni n Fig. 4(c) . Note that when Hx¼61000 Oe, DHsw FIG. 3. SPD of (a) CPA and (b) CPB with insets illustrating CPA /CPB spin configu- rations where the black dashed line tracks switching associated with Vapplunder zero magnetic field and red dashed line highlights the vertical shift in SPD undereffects of H stray; (c) PswofAP!Pfor CPA; (d) PswofAP!PCPB; (e) Pswof P!APfor CPA; (f) PswofP!APfor CPB where the colored bar indicates magnitude of Pswand purple ( AP!P)/orange ( P!AP) and black represent suc- cessful and failed switching, respectively. FIG. 4. Normalized- HswvsHxfor (a) CPA and (b) CPB; (c) DHsw[(Hsw(P!AP)- Hsw(AP!P))/2] vs Hxfor CPA (blue) and CPB (red) up to Happl¼61000 Oe.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 072405 (2020); doi: 10.1063/5.0011786 117, 072405-3 Published under license by AIP Publishingwas larger under CPB than under CPA. This suggests that preference for magnetization tilting was stronger under CPB, likely due to theinfluence of DMI. We also performed SH analysis to characterize the effects of cou- pling on BH. Figures 5(a) and5(b) present R–H loops at various tem- peratures and voltages, with the linear fitting of H exvsDT[inset of Fig. 5(a) ]a n d HexvsV2[inset of Fig. 5(b) ], respectively. Note that by precisely following the analytical methods outlined in Ref. 22,w e obtained the following power law relation: DT¼193V2.A ss h o w ni n Fig. 5(c) ,c o m b i n i n g DT–V andDT–H relations allowed us to obtain estimates of Hswinduced only by SH [henceforth referred to as Hsw (SH)] [ supplementary material V). Hsw(SH) is presented as (H–H stray)/Hcin order to exclude the Hstrayeffect, and coercive-field (Hc) normalization makes it possible to formulate a semi-quantitativecomparison. Under CPA and CPB, we clearly observed a reduction in Hsw(SH) corresponding to an increase in Vappl.T h e AP!Pcurves obtained under CPA were similar to those obtained under CPB; how-ever, the two sets of P!APcurves differed. This discrepancy can be attributed to the fact that the strong magnetic field required to alignthePstates increased the likelihood of concurrent reversal, compared to that required for APstate alignment. The difference in spin canting in CPA/CPB is presumably associated with the DMI chirality effect, which is magnified in the Pstate when the effects of temperature are taken into account. Note that when V applwas increased, Hsw(SH)o f P!APdropped more gradually under CPB than under CPA. This indicates that the thermal stability against an SH-induced reduction inH kis greater under CPB than under CPA, thanks to stronger DMI. The 130-nm MTJ presented similar results with less deviation inAP!P/P!AP(supplementary material VI), due to the reduced SH effects associated with a smaller MTJ. 22 Finally, we examined the effects of voltage on SAF coupling. R–H loops were collected at various Vapplvalues to output the cumulative distribution function (CDF)23–25ofHswin order to estimate the VCMA coefficient ( n). In Fig. 6(a) , the CDF is plotted as a function of positive/negative voltage and fitted using the following equation: FIG. 5. (a)R–H loops with the temperature dependence with the inset showing lin- ear fitting of HexvsDT; (b) R–H loops with the Vappldependence with the inset showing linear fitting of Hexvs V2(only CPA presented for clarity); (c) Hsw(SH) with Vappland coupling dependence. FIG. 6. (a) CDF with Vappland polarity dependence where circles on the left and right correspond to AP!PandP!APtransitions, respectively; (b) anisotropic energy ( Keff/C1tCoFeB ) as a function of electric field where blue and red data points correspond to CPA and CPB, respectively.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 072405 (2020); doi: 10.1063/5.0011786 117, 072405-4 Published under license by AIP PublishingCDF %ðÞ¼100/C2 1/C0exp( /C0Hkf0ffiffiffipp 2Rffiffiffiffi Dp /C2erfcffiffiffiffi Dp 1/C0H/C0Hstray Hk/C18/C19/C20/C21 )! ; (1) where f0is the attempt frequency ( /C251G H z ) , Ris the rate at which the field was swept (50 Oe/s), and Dindicates the thermal stabil- ity. Fitting was used to estimate Hkat various Vapplvalues, which was then converted to anisotropic energy ( Keff/C1tCoFeB ) vs electric field, as shown in Fig. 6(b) , where the slope of the linear function isn. We observed variations in the degree to which Vapplreduced Hk as a function of coupling, which can be expressed using variable n. This provides direct evidence of the VCMA dependence on the cou- pling effects in STT systems. Under zero bias, CPB yielded a larger Keff than did CPA, indicating that the intrinsic anisotropic energy of CPB was greater than that of CPA. We established that spin–orbit couplingdetermines the strength of perpendicular magnetic anisotropy as wellas DMI. 26This indicates that the DMI of CPB is stronger than that of CPA and stronger DMI is better able to preserve the perpendicularmagnetic moment. Within the STT-MRAM framework, BH is generally regarded as a reduction in H kat elevated voltages; however, its sensitivity to inter-layer coupling is poorly understood. In this study, we revealedthat this type of write error does indeed exist. We also discoveredthat perturbations in RL/HL coupling resulted in unexpected switch-ing behavior. Under the effects of VCMA and SH, this switchingbehavior led ultimately to BH. We also determined that in this sce-nario, DMI determines the spin configuration. Thus, precisely con-trolling DMI by optimizing the design of the SAF could go a long way toward improving write precision and is, therefore, worthy of further research. See the supplementary material for the details of R–V,R–H,S P D , switching probability, DT-V,a n d H sw-DTmeasurements. This work was financially supported by the “Center for the Semiconductor Technology Research” from The Featured AreasResearch Center Program within the framework of the HigherEducation Sprout Project by the Ministry of Education (MOE) inTaiwan. This work was also supported in part by the Ministry ofScience and Technology, Taiwan, under Grant No. MOST 109-2634-F-009-029/109-2218-E-009-017/109-2639-E-009-001. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request.REFERENCES 1S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno, Nat. Mater. 9, 721 (2010). 2K. L. Wang, J. G. Alzate, and P. K. Amiri, J. Phys. D: Appl. Phys. 46, 074003 (2013). 3C. Yoshida, T. Tanaka, T. Ataka, and A. Furuya, IEEE Trans. Magn. 55,1 (2019). 4J. 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H. Kim, D. S. Han, J. Jung, K. Park, H. J. M. Swagten, J. S. Kim, and C. Y. You, Appl. Phys. Express 10, 103003 (2017).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 072405 (2020); doi: 10.1063/5.0011786 117, 072405-5 Published under license by AIP Publishing

5.0005754.pdf

J. Chem. Phys. 153, 034107 (2020); https://doi.org/10.1063/5.0005754 153, 034107 © 2020 Author(s).NECI: N-Electron Configuration Interaction with an emphasis on state-of-the-art stochastic methods Cite as: J. Chem. Phys. 153, 034107 (2020); https://doi.org/10.1063/5.0005754 Submitted: 24 February 2020 . Accepted: 24 June 2020 . Published Online: 16 July 2020 Kai Guther , Robert J. Anderson , Nick S. Blunt , Nikolay A. Bogdanov , Deidre Cleland , Nike Dattani , Werner Dobrautz , Khaldoon Ghanem , Peter Jeszenszki , Niklas Liebermann , Giovanni Li Manni , Alexander Y. Lozovoi , Hongjun Luo , Dongxia Ma , Florian Merz , Catherine Overy , Markus Rampp , Pradipta Kumar Samanta , Lauretta R. Schwarz , James J. Shepherd , Simon D. Smart , Eugenio Vitale , Oskar Weser , George H. Booth , and Ali Alavi COLLECTIONS Paper published as part of the special topic on Electronic Structure Software Note: This article is part of the JCP Special Topic on Electronic Structure Software. ARTICLES YOU MAY BE INTERESTED IN Recent developments in the PySCF program package The Journal of Chemical Physics 153, 024109 (2020); https://doi.org/10.1063/5.0006074 Massively Parallel Quantum Chemistry: A high-performance research platform for electronic structure The Journal of Chemical Physics 153, 044120 (2020); https://doi.org/10.1063/5.0005889 The ORCA quantum chemistry program package The Journal of Chemical Physics 152, 224108 (2020); https://doi.org/10.1063/5.0004608The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp NECI: N-Electron Configuration Interaction with an emphasis on state-of-the-art stochastic methods Cite as: J. Chem. Phys. 153, 034107 (2020); doi: 10.1063/5.0005754 Submitted: 24 February 2020 •Accepted: 24 June 2020 • Published Online: 16 July 2020 Kai Guther,1,a)Robert J. Anderson,2Nick S. Blunt,3 Nikolay A. Bogdanov,1Deidre Cleland,4 Nike Dattani,5 Werner Dobrautz,1 Khaldoon Ghanem,1 Peter Jeszenszki,6,7 Niklas Liebermann,1 Giovanni Li Manni,1Alexander Y. Lozovoi,1Hongjun Luo,1Dongxia Ma,1Florian Merz,8Catherine Overy,3 Markus Rampp,9 Pradipta Kumar Samanta,1 Lauretta R. Schwarz,1,3James J. Shepherd,10 Simon D. Smart,3 Eugenio Vitale,1Oskar Weser,1 George H. Booth,2 and Ali Alavi1,3,b) AFFILIATIONS 1Max Planck Institute for Solid State Research, Heisenbergstr. 1, 70569 Stuttgart, Germany 2Department of Physics, King’s College London, Strand, London WC2R 2LS, United Kingdom 3Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom 4CSIRO Data61, Docklands, VIC 3008, Australia 5Department of Electrical and Computer Engineering, University of Waterloo, 200 University Avenue, Waterloo, Ontario N2L 3G1, Canada 6Centre for Theoretical Chemistry and Physics, NZ Institute for Advanced Study, Massey University, Auckland, New Zealand 7Dodd-Walls Centre for Photonic and Quantum Technologies, P.O. Box 56, Dunedin 9056, New Zealand 8Lenovo HPC and AI Innovation Center, Meitnerstr. 9, 70563 Stuttgart, Germany 9Max Planck Computing and Data Facility (MPCDF), Gießenbachstr. 2, 85748 Garching, Germany 10Department of Chemistry and Informatics Institute, University of Iowa, Iowa City, Iowa 52242, USA Note: This article is part of the JCP Special Topic on Electronic Structure Software. a)Electronic mail: k.guther@fkf.mpg.de b)Author to whom correspondence should be addressed: a.alavi@fkf.mpg.de ABSTRACT We present NECI , a state-of-the-art implementation of the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) algorithm, a method based on a stochastic application of the Hamiltonian matrix on a sparse sampling of the wave function. The program utilizes a very powerful parallelization and scales efficiently to more than 24 000 central processing unit cores. In this paper, we describe the core functionalities of NECI and its recent developments. This includes the capabilities to calculate ground and excited state energies, properties via the one- and two-body reduced density matrices, as well as spectral and Green’s functions for ab initio and model systems. A number of enhancements of the bare FCIQMC algorithm are available within NECI , allowing us to use a partially deterministic formulation of the algorithm, working in a spin-adapted basis or supporting transcorrelated Hamiltonians. NECI supports the FCIDUMP file format for integrals, supplying a convenient interface to numerous quantum chemistry programs, and it is licensed under GPL-3.0. Published under license by AIP Publishing. https://doi.org/10.1063/5.0005754 .,s I. INTRODUCTION NECI started off in the late 1990s as an exact diagonaliza- tion code for model quantum dots1,2and has evolved into a code to perform stochastic diagonalization of large fermionic systemsin finite but large quantum chemical basis sets using the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) algo- rithm.3This algorithm samples Slater determinant (i.e., antisym- metrized) Hilbert spaces using signed walkers by propagation of the walkers through stochastic application of the second-quantized J. Chem. Phys. 153, 034107 (2020); doi: 10.1063/5.0005754 153, 034107-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Hamiltonian onto the walker population. In philosophy, it is similar to the continuum real-space Diffusion Monte Carlo (DMC) algo- rithm. However, unlike DMC, no fixed node approximation needs to be applied. Instead, the nodal structure of the wave function, as encoded by the signed coefficients of the sampled Slater determi- nants (SDs), emerges from the dynamics of the simulation itself. However, being based on an FCI parameterization of the wave func- tion, the FCIQMC method exhibits a steep scaling with the number of electrons and is thus only suited for relatively small chemical sys- tems compared to those accessible to DMC. While the common energy measures in FCIQMC methods, namely, the projected, trial energies (cf. Sec. IV) and the energy “shift,” are not variational, a variational energy can be computed from two parallel FCIQMC calculations either directly (cf. Sec. VI) or via the reduced density matrix (RDM) based energy estimator (cf. Sec. VII). There are also similarities between the FCIQMC approach and the Auxiliary-Field Quantum Monte Carlo (AFQMC) method,4–6 both being stochastic projector techniques formulated in second quantized spaces. The latter however works in an over-complete space of non-orthogonal Slater determinants and relies on the phase- less approximation7to eliminate the phase problem associated with the Hubbard–Stratonovich transformation of the Coulomb inter- action kernel, the quality of this approximation being reliant on the trial wave function used to constrain the path. The objective of AFQMC is the measurement of observables such as the energy by sampling over the Hubbard–Stratonovich fields. On the other hand, FCIQMC works in a fixed Slater determinant space and relies on walker annihilation to overcome the fermion sign prob- lem. The phase-less approximation renders the AFQMC method polynomial scaling with an uncontrolled approximation, while i- FCIQMC, which is, in principle, an exact method, remains expo- nential scaling. Finally, FCIQMC provides a direct measure of the CI amplitudes of the many-body wave function expressed in the given orbital basis from which observables can be computed including elements of the reduced density matrices (which do not commute with the Hamiltonian) via pure estimators. Exact symmetry constraints, including total spin, can be incorporated into the formalism.8In this sense, the FCIQMC method is closer in spirit to multi-reference CI methods used in quantum chem- istry to study multi-reference problems rather than the AFQMC method. In its original formulation, the algorithm is guaranteed to con- verge onto the ground-state wave function in the long imaginary- time propagation limit, provided that a sufficient number of walkers are used. This number is generally found to scale with the Hilbert space size and is a manifestation of the sign-problem in this method, essentially implying an exponential memory cost in order to guar- antee stable convergence onto the exact solution. In the subsequent development of the initiator method (i-FCIQMC),9this condition was relaxed to allow for stable simulations at relatively low walker populations, much smaller than the full Hilbert space size, albeit at the cost of a systematically improvable bias. While the initiator adap- tation removes the strict need for a minimum walker number, it does not eliminate the exponential scaling of the method such that calcu- lations become more and more challenging with increasing system size. To give an idea of the capabilities of the NECI implementation, estimates for the accessible system sizes are given below. The rate of convergence of the initiator error with the walker number hasbeen found to be slow for large systems. This is a manifestation of size-inconsistency error, which generally plagues linear configura- tion interaction methods. A very recent development of the adaptive shift method10mitigates this error substantially, enabling near-FCI quality results to be obtained for systems as large as benzene. The development of the semi-stochastic method by Petruzielo et al.11and its further refinements12dramatically reduced the stochastic noise and hence improved the efficiency of the method. The FCIQMC algorithm, as well as its semi-stochastic and ini- tiator versions, is scalable on large parallel machines, thanks to the fact that walker distribution can be distributed over many proces- sors with a relatively small communication overhead. The methods, however, are not embarrassingly parallel owing to the annihilation step of the algorithm (see also Fig. 1). For this reason, parallelization over very large numbers of processors is a highly non-trivial task, but substantial progress has been made, and here, we show that efficient parallelization up to more than 24 000 central processing unit (CPU) cores can be achieved with the current NECI code. The FCIQMC method has been generalized to excited states13 of the same symmetry as the ground state and to the calculation of pure one- and two-particle reduced density matrices via the “replica- trick”14–17(and more recently, three- and four-particle RDMs18). The availability of RDMs enabled the development of the stochas- tic complete active space self-consistent field (CASSCF) method19,20 for treating extremely large active spaces. More recently, a fully spin-adapted formulation of FCIQMC has been implemented based on the Graphical Unitary Group Approach (GUGA),8which over- comes the previous limitations of spin-adaptation, which severely limited the number of open-shell orbitals that could be handled. Other advanced developments of FCIQMC in the NECI code include real-time propagation and application to spectroscopy,21Krylov- space FCIQMC,12and the similarity transformed FCIQMC,22–25 which allows the direct incorporation of Jastrow and similar fac- tors depending on explicit electron–electron variables into the wave function. A number of stochastic methods have been developed as an extension or variation of the FCIQMC approach. These include den- sity matrix quantum Monte Carlo (DMQMC), which allows the exact thermal density matrix to be sampled at any given temperature and also allows straightforward estimation of general observables, including those which do not commute with the Hamiltonian.16,26 Applications of DMQMC include providing accurate data for the warm dense electron gas.27Although not implemented in NECI , DMQMC is available in the HANDE-QMC code.28 The FCIQMC method has led to the development of a num- ber of highly efficient deterministic selected CI methods, including the adaptive sampling CI method of Head-Gordon and co-workers29 who also established the connection with the much older configura- tion interaction by perturbation with multiconfigurational zeroth- order wavefunction selected by iterative process (CIPSI) method of Huron et al.30but with a modified search procedure, while the heat- bath CI method of Holmes and co-workers31was developed from the heat-bath excitation generation for FCIQMC32together with an initiator-like criterion to select the connected determinants with extreme efficiency. Later, a sign-problem-free semi-stochastic eval- uation of the Epstein–Nesbet perturbation energy was developed by Sharma et al.33to compute the missing dynamical correlation energy at the second-order in a memory and CPU efficient manner. Other J. Chem. Phys. 153, 034107 (2020); doi: 10.1063/5.0005754 153, 034107-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . Flow chart showing the basic initiator-FCIQMC implementation in NECI . Marked in red are steps that require synchronization between the MPI tasks and thus are not trivially parallelizable. J. Chem. Phys. 153, 034107 (2020); doi: 10.1063/5.0005754 153, 034107-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp highly related developments of FCIQMC originate in the numerical analysis literature, including the fast randomized iteration,34further developments by Greene and co-workers,35and co-ordinate descent FCI of Wang and co-workers.36 Depending on the utilized features, the number of electrons and accessible basis sizes can vary. The i-FCIQMC implementa- tion including the semi-stochastic version is highly scalable and has been successfully applied to Hilbert space sizes of up to 10108 with 54 electrons.37Atomic basis sets up to aug-cc-pCV8Z for first- row atoms (1138 spin orbitals) are treatable.38The reduced density matrices can routinely be calculated for use in accurate stochastic- MCSCF19for active spaces containing up to 40 electrons and 38 spatial orbitals.39,40Real-time calculations are computationally more demanding but can still be performed for first-row dimers using cc- pVQZ basis sets.21For the similarity transformed FCIQMC method, the limiting factor is not the convergence of the FCIQMC, but the storage of the three-body interaction terms imposing a limit of ∼100 spatial orbitals on the currently available hardware.24The optimized implementations for the application to lattice model systems, such as the Hubbard41(in a real- and momentum space formulation), t−J, and Heisenberg models for a variety of lattice geometries, are implemented in NECI . The applicability of FCIQMC to the Hubbard model strongly depends on the interaction strength U/t. For the very weakly correlated regime U/t<1, FCIQMC is employable up to 70 lattice sites42using a momentum-space basis. In the interesting, yet most problematic, intermediate interaction strength regime in two dimensions, the transcorrelated (similarity-transformed) FCIQMC is necessary to obtain reliable energies in systems up to 50 sites (at and near half-filling).23 The FCIQMC algorithm as implemented in NECI is based on a sparse representation of the wave function and a stochastic application of the Hamiltonian. We start with the full wave function ∣ψFCI⟩=∑ iCi∣Di⟩ (1) with coefficients Ciin a many-body basis ∣Di⟩.NECI supports Slater determinants or CSFs as a many-body basis; for simplicity, for now, the usage of determinants is assumed, but the algorithm is analogous to CSFs (see also Sec. XII B 2). The FCIQMC wave function is not normalized. The ground state of a Hamiltonian ˆHis now obtained by iterative imaginary time-evolution, with the propagator expanded to the first order using a discrete time step Δτsuch that ∣ψ(τ+Δτ)⟩=(1−Δτ(ˆH−S(τ)))∣ψ(τ)⟩, (2) which converges to the ground state of ˆHforτ→∞for∣Δτ∣<2 W, where Wis the difference between the largest and smallest eigen- values of ˆH.43Here, S(τ) is a diagonal shift applied to ˆH, which is iteratively updated to match the ground-state energy. The full wave function is stored in a compressed manner, where only coefficients above a given threshold value Cminare stored. Coef- ficients smaller than Cminare stochastically rounded. That is, in every iteration, a wave function given by coefficients Ci(τ) is stored such that Ci(τ)=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩Ci if∣Ci∣>Cmin sign(Ci)Cmin else w. prob∣Ci∣ Cmin 0 otherwise,(3)where⟨Ci(τ)⟩=Ci. This compression is applied in every step of the algorithm that affects the coefficients. The value | Ci(τ)| is referred to as the walker number of the determinant ∣Di⟩, so∣Di⟩is said to have |Ci(τ)| walkers assigned. Applying the Hamiltonian to this compressed wave function is done by separating it into a diagonal and an off-diagonal part as ∣ψ(τ+Δτ)⟩=∑ i(1−Δτ(Hii−S(τ)))Ci(τ)∣Di⟩ ⌟⟨⟨⟪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⟫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⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟨⟨⟪rl⟫r⟩⟩⟩⟪ (b) Death step(c) Annihilation step ↓ ∑ i∑ j≠iΔτHjiCi(τ)∣Dj⟩ ⌟⟨⟨⟪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⟫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⟫r⟩⟩⟩⟪ (a) Spawn step(4) and then performed in the three labeled steps (a)–(c). First, in the spawning step , the off-diagonal part is evaluated by stochastically sampling the sum over jand storing the resulting spawned wave function as a separate entity, as described in the flow chart in Fig. 1. Then, in the death step , the diagonal contribution is evaluated deter- ministically, following a stochastic rounding of the resulting coeffi- cients. This step is performed in-place, since the coefficients of the previous iterations are not required anymore. Finally, the spawned wave function from the off-diagonal part is added in the annihilation step, summing up all contributions from the spawned wave func- tion to each determinant. NECI implements the initiator method too, which labels a class of determinants as initiators , typically those with an associated walker number above a given threshold, and effec- tively zeroes all matrix elements between non-initiator determinants and determinants with Ci(τ) = 0. The implementation thereof is also sketched in Fig. 1. In the context of FCIQMC calculations, the core functionality ofNECI consists of a highly parallelizable implementation of the ini- tiator FCIQMC method9for both real and complex Hamiltonians. There is both a generic interface for ab initio systems, specialized implementations for the Hubbard and Heisenberg models, and the uniform electron gas. The interface for passing input information on the system to NECI is discussed in Sec. XIV. To enable continua- tion of calculations at a later point, NECI can write the instantaneous wave function and current parameters—such as the shift value—to disk, saving the current state of the calculation. TheNECI program44itself is written in Fortran and requires extended Fortran 2003 support, which is the default for current For- tran compilers. Parallelization is achieved using the Message Passing Interface (MPI),45and support for MPI 3.0 or newer is required. NECI further requires the BLAS46and LAPACK47linear algebra libraries, which are available in numerous packages. The usage of the HDF5 library48for parallel I/O is supported, but not required. If used, the linked HDF5 library has to be built with Fortran support and for parallel applications. For installation, cmake is required, as well as the fypp Fortran preprocessor.49For pseudo-random num- ber generation, the double precision SIMD oriented fast Mersenne Twister (dSFMT)50,51implementation of the Mersenne Twister method52is used. The stable version of the program can be obtained from github at https://github.com/ghb24/NECI_STABLE, licensed under the GNU General Public License 3.0. Some advanced or experimental features are only contained in the development version (for access to the development version, contact the corresponding J. Chem. Phys. 153, 034107 (2020); doi: 10.1063/5.0005754 153, 034107-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp authors). All features presented here are eventually to be integrated to the stable version. The detailed instructions on the installation can be found in the documentation that is available together with the code. In the following, various important features of NECI are explained in detail. An overview of excitation generation, a fun- damental part of every FCIQMC calculation, is given in Sec. II. Then, the semi-stochastic approach (Sec. III), the estimation of energy and use of trial wave functions (Sec. IV), the recently pro- posed adaptive shift method to reduce the initiator error (Sec. V) and perturbative corrections to this error (Sec. VI), the sampling of the reduced density matrices, which is crucial for interfacing the FCIQMC method with other algorithms (Sec. VII), the calcu- lation of excited states (Sec. VIII), static response functions (Sec. IX) and the real-time FCIQMC method (Sec. X), the transcorrelated approach (Sec. XI), and the available symmetries, including the total spin conservation utilizing GUGA (Sec. XII), are discussed. Finally, the scalability of NECI is explored (Sec. XIII) and the interfaces for usage with other code are presented (Sec. XIV), in particular for the stochastic-MCSCF method (Sec. XV). II. EXCITATION GENERATION A key component of the FCIQMC algorithm is the sam- pling of the Hamiltonian matrix elements in the spawning step, where the Hamiltonian is applied stochastically. This requires an efficient algorithm to randomly generate the connected determi- nants with a known probability pgenfor any given determinant, referred to as excitation generation. This typically means making a symmetry constrained choice of (up to) two occupied orbitals in a determinant and (up to) two orbitals to replace them with, such that the corresponding Hamiltonian matrix element is non- zero. If the spin-adapted functions are used rather than determi- nants, the connectivity rules change, but the main principles are the same. The spawning probability for a spawn from a determinant ∣Di⟩ to a determinant ∣Dj⟩is, in practice, given by ps=Δτ∣Hij∣ pgen(j∣i). (5) This means that the purpose of pgen(j|i) of selecting ∣Dj⟩from∣Di⟩in the spawning probability psis to allow the flexibility in the selection of determinants ∣Dj⟩from∣Di⟩so that, irrespective of how we choose ∣Dj⟩from∣Di⟩, the rate at which transitions occur is not biased by the selection algorithm. In other words, if a particular determinant ∣Dj⟩ is only selected rarely from ∣Di⟩(i.e., with low generation probabil- ity), then the acceptance of the move (i.e., the spawning probability) will be with correspondingly high probability (i.e., proportional to the inverse of the generation probability). Conversely, if a deter- minant ∣Dj⟩is selected with relatively high generation probability from∣Di⟩, then its acceptance probability will be correspondingly low. In other words, from the point of view of the exactness of the FCIQMC algorithm, the precise manner in which excitations are made is immaterial: as long as the probability pgen(j|i)>0 when | Hij| >0, the algorithm will ensure that transitions from Di→Djoccur at a rate proportional to | Hij|, and hence, the walker dynamics converges onto the exact ground-state solution of the Hamiltonian matrix.However, from the point of view of efficiency , different algorithms to generate excitations are by no means equivalent. That is, events with a very large∣Hij∣ pgen(j∣i)can lead to very large spawns and thus endanger the stability of an i-FCIQMC calcula- tion. For time-step optimization, NECI offers a general histogram- ming method, which determines the time step from a histogram of∣Hij∣ pgen(j∣i),8and an optimized special case thereof, which only takes into account the maximal ratio.53If required, internal weights of the excitation generators such a bias toward double excitations are then optimized in the same fashion to maximize the time step. However, as a result, the time step and thus the overall effi- ciency of the simulation are driven by the worst-cases of the∣Hij∣ pgen(j∣i) ratio discovered within the explored Hilbert space. Thus, an opti- mal excitation generator should create excitations with a probability distribution to the Hamiltonian matrix elements such that ∣Hij∣ pgen(j∣i)≈const. (6) This is the optimal probability distribution since then, the accep- tance rate is solely determined by the time step.32 NECI supports a variety of algorithms to perform excitation generation, with the most notable being the pre-computed heat-bath (PCHB) sampling (a variant of the heat-bath sampling presented in Ref. 32, as described in Subsection 3 of the Appendix), the on-the- fly Cauchy–Schwartz method54(described in Subsection 2 of the Appendix), the pre-computed Power–Pitzer method,55and lattice- model excitation generators both for real-space and momentum- space lattices. Additionally, a three-body excitation generator and a uniform excitation generator are available, which are essential for treating systems with the transcorrelated ansatz when including three-body interactions. As heat-bath excitation generation can have high memory requirements, it might be impractical for some systems. There, the on-the-fly Cauchy–Schwartz method can maintain very good∣Hij∣ pgen(j∣i) ratios without significant memory cost, albeit at O(N)compu- tational cost, Nbeing the number of orbitals, and possibly with dynamic load imbalance. The details of the Cauchy–Schwartz exci- tation generation are discussed in the Appendix. III. SEMI-STOCHASTIC FCIQMC In many chemical systems, the wave function is dominated by a relatively small number of determinants. In a stochastic algo- rithm, the efficiency can be improved substantially by treating these determinants in a partially deterministic manner. Petruzielo et al. suggested a semi-stochastic algorithm,11where the FCIQMC projection operator ˆP=∑ijPij∣Di⟩⟨Dj∣is applied exactly within a small but important subspace, which we call the deterministic space, D. Specifically, we write ˆP=ˆPD+ˆPS, (7) where ˆPD=∑ i∈D,j∈DPij∣Di⟩⟨Dj∣. (8) J. Chem. Phys. 153, 034107 (2020); doi: 10.1063/5.0005754 153, 034107-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The ˆPDoperator therefore accounts for all spawnings that are both from and to determinants in D. The stochastic projection operator, ˆPS, contains all remaining terms. The matrix elements of ˆPDare cal- culated and stored in a fixed array, and applied exactly each iteration by a matrix–vector multiplication. The operator ˆPSis then applied stochastically by the usual FCIQMC spawning algorithm. The semi-stochastic adaptation requires storing the Hamilto- nian matrix within D, which we denote HD. InNECI ,HDis stored in a sparse format, distributed across all processes. To calculate HD, we have implemented the fast generation scheme of Li et al.56This approach has allowed us to use deterministic spaces containing up to∼107determinants. However, a more typical size of Dis between 104and 105. Ideally, a deterministic space of a given size ( ND) should be chosen to contain the determinants with the largest value of | Ci| in the exact FCI wave function. This optimal choice is not possi- ble in practice, but various approaches exist to make an approximate selection. Petruzielo and co-workers suggested using selected config- uration interaction (SCI) to make the selection.11Within NECI , the most common approach is to choose the NDdeterminants that have the largest weight in the FCIQMC wave function at a given itera- tion.12Therefore, a typical FCIQMC simulation in NECI will be per- formed until convergence (at some iteration number Nconv.) using the fully stochastic algorithm, at which point the semi-stochastic approach is turned on, selecting the NDmost populated determi- nants in the instantaneous wave function to form D. The appropriate parameters ( NDandNconv.) are specified in the NECI input file. NECI supports performing periodic re-evaluation of the NDmost popu- lated determinants, updating the deterministic space Dwith a given frequency. Using the semi-stochastic adaptation with a moderate deter- ministic space (on the order of ∼104) can improve the efficiency of FCIQMC by multiple orders of magnitudes. This is particularly true in weakly correlated systems. The semi-stochastic approach can also be used in NECI when sampling the reduced density matri- ces (RDMs), as described in Sec. VII. Here, contributions to RDMs are included exactly between all pairs of determinants within D. It has been shown that this can substantially reduce the error on RDM-based estimators.12Using the semi-stochastic adaptation in NECI disables the load-balancing unless a periodic update of Dis performed. IV. TRIAL WAVE FUNCTIONS The most common energy estimator used in FCIQMC is the reference-based projected estimator, ERef=⟨DRef∣ˆH∣Ψ⟩ ⟨DRef∣Ψ⟩, (9) where | DRef⟩is an appropriate reference determinant (usually the Hartree–Fock determinant). In case ∣Ψ⟩is an eigenstate, this yields the exact energy, but, in general, it is a non-variational estimator. This is the default estimator for the energy and can be obtained with a minimal overhead. NECI has the option to use the projected estimators based on more accurate trial wave functions, which can significantly reduce statistical error in energy estimates. For this reason, we define atrial subspace T, which is spanned by NTdeterminants. Similarly to the deterministic space, Tshould ideally be formed from the deter- minants with the largest contribution in the FCI wave function or some good approximation to these determinants. Projecting ˆHinto Tgives us a NT×NTmatrix, which we denote HT, whose eigen- states can be used as trial wave functions for more accurate energy estimators. Let us denote an eigenstate of HTby∣ΨT⟩=∑i∈TCT i∣Di⟩ with eigenvalue ET. Then, a trial function-based estimator can be defined as ETrial=⟨ΨT∣ˆH∣Ψ⟩ ⟨ΨT∣Ψ⟩(10) =ET+∑j∈CCjVj ∑i∈TCiCT i. (11) Here,Cis the space of all determinants connected to Tby a single application of ˆH(not including those in T).Cidenotes walker coef- ficients in the FCIQMC wave function, and Vjis defined within Cas Vj=∑ i∈T⟨Dj∣ˆH∣Di⟩CT i,∣Dj⟩∈C,∣Di⟩∈T. (12) To calculate the estimator ETrial, we therefore require several large arrays: first, HT, which is stored in a sparse format, in the same manner as the deterministic Hamiltonian in the semi-stochastic scheme; second, ∣ψT⟩, which must be calculated by the Lanczos or Davidson algorithm; third, V, which is a vector in the entire Cspace. The number of coefficients to store in Cis larger than in Tby a sig- nificant amount, typically by several orders of magnitude. Indeed, storing Vcan become the largest memory requirement. Because of this, using trial wave functions is typically more memory intensive inNECI than using the semi-stochastic approach for a given space size. We therefore suggest using a smaller trial space, T, compared to the deterministic space, D. Note that the initiator error on ETrial is not the same as the initiator error on ERef. For example, ETrial becomes exact as ∣ΨT⟩ approaches the FCI wave function. However, for practical trial wave functions, the two energy estimates typically give similar initiator errors for ground-state energies in our experience. An exception occurs for excited states (see Sec. VIII). In this case, the wave func- tion is usually not well approximated by a single reference determi- nant, and ETrial with an appropriate Tyields a great improvement, both for the statistical and initiator errors. V. ADAPTIVE SHIFT The initiator criterion9is important in making FCIQMC a practical method allowing us to achieve convergence at a dramat- ically lower number of walkers than the full FCIQMC.3However, this approximation introduces a bias in the energy when an insuffi- cient number of walkers are used. This bias can be attributed to the fact that non-initiators are systematically undersampled due to the lack of feedback from their local Hilbert space. To correct this, we can allow each non-initiator determinant ∣Di⟩to have its own local shift Si(τ) as an appropriate fraction of the full shift S(τ), Si(τ)=fi×S(τ). (13) J. Chem. Phys. 153, 034107 (2020); doi: 10.1063/5.0005754 153, 034107-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The fraction fiis computed by monitoring which spawns are accepted due to the initiator criterion and accumulating positive weights over the accepted and rejected ones, fi=∑j∈accepted wij ∑j∈allwij. (14) These weights wijare derived from perturbation theory57where the first-order contribution of determinant ∣Di⟩to the amplitude of determinant ∣Dj⟩is used as a weight for spawns from ∣Di⟩to∣Dj⟩, wij=∣Hij∣ Hjj−E0. (15) It is worth noting that, regardless of how the weights are chosen, expression (14) guarantees that initiators get the full shift. In addi- tion, as the number of walkers increases, the local Hilbert space of a non-initiator becomes more and more populated, restoring the full method in the large walker limit. We call the above approach for unbiasing the initiator approxi- mation, the adaptive-shift method.10In Fig. 2, the exemplary results (from Ref. 10) from using the adaptive shift method are displayed, comparing the total energies of the butadiene molecule in ANO-L- pVDZ basis (22 electrons in 82 spatial orbitals), obtained with the normal initiator method and the adaptive shift method using three different values of the initiator parameter na: 3, 10, and 20. The adap- tive shift results are in good agreement with other benchmark values from Density Matrix Renormalization Group (DMRG), coupled- cluster singles doubles triples perturbative quadruples [CCSDT(Q)], and extrapolated HCIPT2. In contrast, the normal initiator method has a bias of over 10 mH. Also note how by using the adaptive shift, the results become, to a large extent, independent of the initiator parameter na. FIG. 2 . Example of application of the adaptive shift method: total energies of buta- diene for the normal initiator and the adaptive shift method, as a function of the number of walkers, for three values of the initiator parameter na. The adaptive shift results converge to −155.5581(2) E h,−155.5583(2) E h, and−155.5578(2) E hfor naof 3, 10, and 20, respectively. The DMRG value of −155.5573 E h, obtained with a bond dimension of 6000,58the CCSDT(Q) value of −155.5576 E h, and the extrapolated HCIPT2 value of −155.5582(1) E h59are in good agreement with that. Reproduced with permission from Ghanem et al. , J. Chem. Phys. 151, 224108 (2019). Copyright 2019 AIP Publishing LLC.VI. PERTURBATIVE CORRECTIONS TO INITIATOR ERROR An alternative approach to removing initiator error in NECI is through a perturbative correction.60In the initiator approximation, spawning events from non-initiators to unoccupied determinants are typically discarded. These discarded events make up a signifi- cant fraction of all spawning attempts made, which, in turn, accounts for much of the total simulation time. While it is necessary to dis- card these spawned walkers to prevent disastrous noise from the sign problem,61this step is extremely wasteful. These discarded walkers actually contain significant informa- tion, which can be used to greatly increase the accuracy of the ini- tiator FCIQMC approach. Specifically, these walkers may sample up to double excitations from the currently occupied determinants (a similar argument can be used to justify the above adaptive shift approach). In analogy with a comparable approach taken in selected CI methods, these discarded walkers can be used to sample a second- order correction to the energy from Epstein–Nesbet perturbation theory. The correction is calculated by ΔE2=1 (Δτ)2∑ i∈rejectedS1 iS2 i E0−Hii. (16) Here,Δτis the time step, E0is the i-FCIQMC estimate of the energy, and Sr iis the total spawned weight onto the determinant | Di⟩in replica r(the replica approach will be discussed in more detail in Sec. VII). This correction requires that two replica FCIQMC simu- lations are being performed simultaneously to avoid biases in this estimator. The summation here is performed over all spawning attempts, which are discarded on both replicas simultaneously. This must only be applied to correct the variational energy esti- mator from i-FCIQMC. Such variational energies in NECI can be calculated either directly62,63or from the two-body reduced density matrices, which may be sampled in FCIQMC. This perturbative correction is essentially free to accumulate, since all spawned walkers contributing to Eq. (16) are created regardless. The only significant extra cost comes from the require- ment to perform two replica simulations. However, for large sys- tems, the noise on this correction can become significant, which necessitates further running time to reduce statistical errors. This correction has proven extremely successful in practice, particularly for weakly correlated systems, where it is typical to see 80%–90% of remaining initiator error removed.60,62,63 VII. DENSITY MATRIX SAMPLING AND PURE EXPECTATION VALUES While the total energy is an important quantity to extract from quantum systems, a more complete characterization of a system requires the ability to extract information about other expectation values. If these expectation values are derived from operators that do not commute with the Hamiltonian of the system, then a “pro- jected” estimate of the expectation value akin to Eq. (9) is not possi- ble, and alternatives within FCIQMC are required in order to com- pute them. This is the case for many key quantities such as nuclear derivatives (forces on atoms), dipole moments, and higher-order J. Chem. Phys. 153, 034107 (2020); doi: 10.1063/5.0005754 153, 034107-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp electrical moments, as well as other observables such as pair distri- bution functions.64They all can be obtained via the corresponding n-body reduced density matrix ( n-RDM), where nis the rank of the operator in question, which fully characterizes the correlated distribution and coherence of nelectrons relative to each other. This information can also be used to calculate quantum informa- tion measures, which are not observables but which characterize the entanglement within a system, such as correlation entropies.15 To characterize the strength of coupling between different states under certain operators, e.g., the oscillator strength of optical exci- tations, as well as obtaining other dynamical information requires computing transition density matrices (tRDMs) between stochastic samples of different states, which can be sampled within FCIQMC using the excited state feature discussed in Sec. VIII.17,65Further- more, the two states considered may not sample eigenstates of the system, but one of them can be a response state of the system, then the resulting tRDMs characterize the response of a system to a perturbation, corresponding to a higher derivative of the energy such as the polarizability of the system, which will be addressed in Sec. IX.66Finally, RDMs can also be used to characterize the expectation value of an effective Hamiltonian in a subspace of a system.67,68This effective Hamiltonian can include effects such as electronic correlations coupling the space to a wider external set of states. The plurality of electronic structure methods of this kind, such as explicitly correlated “F12” corrections for basis set incom- pleteness,69–71multi-configurational self-consistent field,19,20inter- nally contracted multireference perturbation theories,18embed- ding methods,72,73and the Multi-Configuration Pair-Density Func- tional Theory (MC-PDFT),74further attests the importance of faithful and efficient sampling of RDMs in electronic structure theory. All expectation values of interest can be derived from contractions with a general reduced density matrix object, defined as ΓA,B i1i2...in,j1j2...jn=⟨ΨA∣ˆa† i1ˆa† i2...ˆa† inˆajnˆajn−1...ˆaj1∣ΨB⟩, (17) where ndenotes the “rank” of the RDM and the choice of the states Aand Bdefines the type of RDM, as described above. In this section, we focus on the sampling of the 2-RDM. This is generally the most common RDM required, as most expectation values of interest are (up to) two-body operators, including the total energy of the system. Furthermore, within FCIQMC, the fact that the rank of the RDM required is then the same as the rank of the Hamiltonian, which is sampled within the stochastic dynamics, leads to a novel algorithm, which ensures that the overhead to compute the 2-RDM is relatively small and manageable.15 Expanding the expression for the 2-RDM in terms of the exact FCI wave function [Eq. (1)], we find ΓA,B kl,mn=∑ i,jCA∗ iCB j⟨Di∣ˆa† kˆa† lˆanˆam∣Dj⟩, (18) where i,jindex the many-electron Slater determinants and k,l,m, ndenote single-particle orbitals. We will focus on the case where we are sampling | ΨA⟩= |ΨB⟩= |Ψ0⟩, the ground state of the system, since the same basic principles are applied to sampling the tRDMs, where the other walker distribution may represent an excited stateor a response state, with more details for these cases considered in Refs. 17 and 66. The expectation values derived from these RDMs describe “pure” expectation values to distinguish them from the projective estimate of expectation values given in Eq. (9). There are some features of the form of Eq. (18) that should be noted. First, the 2-RDM requires the sampled amplitudes on all determinants in the space connected to each other via (up to) dou- ble electron substitutions. This means that this expectation value requires a global sampling of connections in the entire Hilbert space, in contrast to the projected energy estimate, which requires only a consideration of the determinant amplitudes that are con- nected directly through ˆHto the reference determinant (or small trial wave function, see Sec. IV). Second, it is seen that the pairs of determinants in Eq. (18) are exactly the same as the pairs of determinants connected, in general, through the Hamiltonian oper- ator used to sample the FCIQMC dynamics in Eq. (4), assuming that the matrix element is not zero due to (accidental) symme- try between the determinants. This allows an algorithm to sam- ple the 2-RDM concurrently with the sampling of the Hamilto- nian required for the spawning steps between occupied determinant pairs. A final point to note is that the n-RDM is a non-linear func- tional of the FCI amplitudes—specifically being a quadratic form. Within the FCIQMC sampling, the Ciamplitudes are stochastic variables represented as walkers [ Ci(τ)], which, at any one iter- ation, are, in general, very different from the true Cibut, when averaged over long times, have an expected mean amplitude, which is the same as (or a very good approximation to) Ci. However, due to this non-linearity in the form of the 2-RDM, the average of the sampled amplitude product is not equal to the product of the average amplitude, ⟨C∗ i(τ)Cj(τ)⟩τ≠⟨C∗ i(τ)⟩τ⟨Cj(τ)⟩τ, as it neglects the (co-)variance between the sampled determinant ampli- tudes. Initial applications of RDM sampling in FCIQMC neglected these correlations in the sampling of the RDMs, which signifi- cantly hampered the results, especially for the diagonal elements of the RDMs.69The result is that even if each determinant were correctly sampled on average, the stochastic error in the sam- pling would manifest as a systematic error in the RDMs, and thus, only give correct results in the large walker limit, but not the large sampling limit, even if the wave function were correctly resolved. The resolution to this problem came via the “replica trick,”15,16 which changes the quadratic RDM functional into a bilinear one.14 This formally removes the systematic error in the RDM sampling, at the expense of requiring a second walker distribution. The premise is to ensure that these two walker distributions are entirely inde- pendent and propagated in parallel, sampling the same (in this instance ground-state) distribution. This ensures an unbiased sam- pling of the desired RDM, by ensuring that each RDM contribution is derived from the product of an uncorrelated amplitude from each replica walker distribution. The sampling algorithm then proceeds by ensuring that during the spawning step, the current amplitudes are packaged and communicated along with any spawned walkers. During the annihilation stage, these amplitudes are then multiplied by the amplitude on the child determinant from the other replica dis- tribution, and this product then contributes to all n-RDMs, which are accumulated and equal to the rank of the excitation or higher. In this way, the efficient and parallel annihilation algorithm is used J. Chem. Phys. 153, 034107 (2020); doi: 10.1063/5.0005754 153, 034107-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp to avoid the latency of additional communication operations, with the necessary packaging of the amplitude and specification of the parent determinant along with each spawned walker being the only additional overhead. The NECI implementation allows for up to 20 replicas to be run, which exceeds any needs arising in the context of RDM calculation. Full details about the ground-state 2-RDM sampling algorithm can be found in Ref. 15; however, we mention a few salient addi- tional details here. The RDMs are stored in fully distributed and sparse data structures, allowing the accumulation of RDMs for very large numbers of orbitals. The sampling of the RDMs is also not inherently Hermitian. While the sampling within FCIQMC obeys detailed balance, the flux of walkers spawned from ∣Di⟩→∣Dj⟩ is only equal to the reverse flux on average, and therefore, the stochastic noise ensures that the swapping of the two states does not give identical accumulated RDM amplitudes for finite sam- pling (note that for transition RDMs, this is not expected, with more details in Ref. 17). Similarly, the states sampled in FCIQMC are not normalized, and therefore, neither are the sampled RDMs. Both of these aspects are addressed at the end of the calcula- tion, where the RDMs are explicitly made Hermitian via averaging appropriate entries, and the normalization is constrained by ensur- ing that the trace of the RDMs gives the appropriate number of electrons.15 The dominant cost of RDM sampling in large systems comes from the sampling of elements defined by pairs of creation and anni- hilation operators with the same orbital index. These correspond to tuples of occupied orbitals common to both | Di⟩and | Dj⟩states. We term these contributions promotions , as they contribute to a rank of a RDM greater than the excitation level between | Di⟩and |Dj⟩. For instance, single excitation spawning events need to con- tribute to all N−1 elements of the 2-RDM corresponding to com- mon occupied orbitals in the two determinants. The most extreme case comes from the “diagonal” contributions to the RDMs, where i=j, which requires N(N−1)/2 contributions to the 2-RDM to be included where each index defining the RDM element corre- sponds to the same occupied orbital in the two determinants. To mitigate this cost, these diagonal elements are stored locally on each MPI process and only infrequently accumulated at the end of an RDM “sampling block,” or when the determinant becomes unoccupied, with the amplitude averaged over the sampling block. This substantially reduces the frequency of the O(N2)operations required to sample these promoted contributions from the diagonal of Eq. (18). Other efficiency boosting modifications to the algorithm, such as the semi-stochastic adaptation12(detailed in Sec. III), are also seamlessly integrated with the RDM accumulation. Within the deterministic core space, the RDM contributions are exactly accu- mulated along with the exact propagation, with the connections from the deterministic to the stochastic spaces sampled in the stan- dard fashion. This combination of RDM sampling with the semi- stochastic algorithm can greatly reduce the stochastic errors in the RDMs by ensuring that contributions from large weighted deter- minant amplitudes are explicitly and deterministically included. Furthermore, the reference determinant and its direct excitations are also exactly accumulated. This is partly because these are likely important contributions, but principally, if the reference is a Hartree–Fock determinant, then the coupling to its singleexcitations via the Hamiltonian will be zero due to Brillouin’s the- orem. These single excitations will nevertheless contribute to the RDMs and therefore are included explicitly. The sampling of RDMs with a rank greater than two is also now possible within the FCIQMC algorithm and NECI code. The importance of these quantities is primarily in their use in inter- nally contracted multireference perturbation theories, although a number of other uses for these quantities also exist.18These meth- ods allow for the FCIQMC dynamics to only consider an active orbital subspace, hugely reducing both the full Hilbert space of the stochastic dynamics and the required time step, while the accumu- lation of up to 4-RDMs (or contracted lower-order intermediates for efficiency) allows for a rigorous coupling of the strong corre- lation in the low-energy active space to the dynamic correlation in the wider “external” space via post-processing of these higher-body RDMs with integrals of the external space. Sampling of higher-body RDMs cannot use the identical algorithm to the 2-RDMs, since it now requires the product of determinant amplitudes separated by up to 4-electron excitations, which are not explicitly sampled via the standard FCIQMC propagation algorithm. To allow for this sam- pling, we include an additional spawning step per walker of excita- tions with a rank between three and n, where nis the rank of the highest RDM accumulated. This additional spawning is controlled with a variable stochastic resolution, ensuring that the frequency of these samples is relatively rare to control the cost of sampling of these excitations [approximately only one higher-body spawn for every 10–20 traditional (up to two-body) spawning attempts]. There is no time step associated with these excitations, and every attempt is “successful,” transferring information about higher-body correlations in the system and contributing to these higher-body excitations, but not modifying the distribution of the sampled wave function. However, the dominant cost of sampling these higher- body RDMs is not the sampling events themselves, but rather the promotion of lower-rank excitations to these higher-body inter- mediates. Nevertheless, the faithful sampling of these higher-body properties has allowed for the stochastic estimate of fully internally contracted perturbation theories in large active spaces, with a sim- ilar number of walkers required to sample the 2-RDM in an active space.18 VIII. EXCITED STATE CALCULATIONS In many applications, besides ground-state energies, the prop- erties of excited states are of interest. If states in different symmetry sectors are targeted, this can be easily achieved by performing sepa- rate calculations in each sector, yielding the ground state with a given symmetry. If, however, several eigenstates with the same symme- try are required, then this approach is not sufficient. The FCIQMC method is not inherently limited to ground-state calculations and can employ a Gram–Schmidt orthogonalization technique to calcu- late a set of orthogonal eigenstates.13,17The obtained states will then be the lowest energy states with a given symmetry. Calculating eigenstates sequentially and orthogonalizing against all previously calculated states carry the problem of only orthogonal- izing against a single snapshot of the wave function, which will lead to a biased estimate of the excited states. Instead, calculating all states in parallel and orthogonalizing after each iteration give much better results. J. Chem. Phys. 153, 034107 (2020); doi: 10.1063/5.0005754 153, 034107-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The required modifications to the algorithm are minimal. To calculate a set of meigenstates, mFCIQMC calculations are run in parallel, with the additional step of performing the instantaneous orthogonalization between the mstates, performed at the end of each iteration. The orthogonalization requires O(m2)operations and uses one global communication per state. To run mparallel calculations, the replica feature presented in Sec. VII is used to effi- ciently sample a number of states in parallel. After each FCIQMC iteration, for each state, the contributions from all states of lower energies are projected out. The update step for the nth wave function ∣ψn⟩is then modified to ∣ψn(τ+Δτ)⟩=ˆOn(τ+Δτ)(1−Δτ(ˆH−Sn(τ)))∣ψn(τ)⟩, (19) with the orthogonalization operator for the nth state, ˆOn(τ)=1−∑ m<n∣ψm(τ)⟩⟨ψm(τ)∣ ⟨ψm(τ)∣ψm(τ)⟩. (20) With this definition of the orthogonalization operator, the ground- state FCIQMC wave function ( n= 0) is left unaffected. The first excited state ( n= 1) is then orthogonalized against the ground state (using the updated wave functions at τ+Δτ, after annihilation has been performed). The second excited state is orthogonalized against both the ground and first excited states, and so on. To enforce the FCIQMC wave function discretization, after performing the orthogonalization, all determinants with a coeffi- cient smaller than the minimal threshold (typically 1) are stochasti- cally rounded (either down to 0 or up to 1, in an unbiased manner). This is required to prevent proliferation of very small walkers, which adversely affects the wave function compression. IX. RESPONSE THEORY WITHIN FCIQMC TO CALCULATE STATIC MOLECULAR PROPERTIES Response theory is a well-established formalism to calculate molecular properties using quantum chemical methods.75–78It is, in general, formulated for a time-dependent field, which allows us to compute both static and dynamic molecular properties. However, it is currently only implemented for a static field within NECI .66 Calculation of molecular properties using response theory relies on the evaluation of the response vectors that are the first or higher order wave functions of the system in the presence of an external perturbation ˆV. According to Wigner’s “(2n + 1)” rule, response vectors up to order nare required to obtain response prop- erties up to order 2 n+ 1.77For calculating second-order proper- ties such as dipole polarizability, the first-order response vector, C(1), needs to be obtained along with the zero-order wave func- tion parameter C(0). While C(0)uses the original FCIQMC working equation (4), C(1)is updated according to ΔC(1) i=−Δτ∑ j(Hij−S(τ))C(1) 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⟫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⟫r⟩⟩⟩⟪ Hamiltonian dynamics−ΔταVijC(0) 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⟫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⟩⟩⟩⟪ Perturbation dynamics. (21) The response vector is discretized into signed walkers in the same way it is done for C(0). The dynamics of the response-statewalker is simulated according to Eq. (21) using an additional pair of replica, and it works in parallel with the dynamics of the zero- order state. Additional spawning and death steps are devised for the response-state walker dynamics due to the presence of the perturba- tion, alongside the original spawning and death steps in the dynam- ics. The dependence of the response state on the zero-order states comes from these two aforementioned additional steps. A Gram– Schmidt orthogonalization is applied to the response-state walker distribution with respect to the zero-order walker distribution at each iteration using the same functionality as described in Sec. VIII. This ensures the orthogonality of the response vectors with respect to all lower-order wave function parameters. The norm of the response walkers is fixed by the choice of the normalization of the zero-order walkers, and it can, in principle, grow at a much faster rate than the zero-order norm. Therefore, in Eq. (21), we introduce the parameter αto control the norm of the response walkers and to reduce the computational effort expended in simulating their dynamics. We aim at matching the number of response-state walkers [ N(1) w] with the number of zero-order walkers [N(0) w] by updating αperiodically as α=N(0) w N(1) w. (22) Once the walker number stabilizes, the value of αis kept fixed while accumulating statistics. As αscales the norm of the response vec- tor, it needs to be taken into account while evaluating response properties. Response properties are then obtained from the transition reduced density matrices (tRDMs) that are stochastically accu- mulated following Eq. (18). For example, dipole polarizability is obtained from the one-electron tRDMs between the zero- and first- order wave functions as αxy=−1 2∑ pq[ˆxpqγy p,q+ˆypqγx p,q], (23) with theγy p,qbeing calculated from the two-electron tRDM as γy pq=1 (N−1)∑ a[1 α1Γ(0)(1) pa,qa[1]+1 α2Γ(0)(1) pa,qa[2]]. (24) Due to the use of two replica per state while sampling both zero- and first-order states, a statistically independent and unbiased estima- tor of tRDMs can be constructed in two alternative ways, which are denoted here as “[1]” and “[2].” The perturbation used in the com- putation of the tRDMs in Eq. (24) is the dipole operator ˆy. The factor 1 αappears due to the re-scaling of the response vector following Eq. (21). X. REAL-TIME FCIQMC For the purpose of obtaining spectroscopic data or targeting highly excited states, the calculation of orthogonal sets of eigen- states quickly becomes unfeasible, as to obtain a certain eigenstate, all eigenstates of lower energy with the same symmetry have to be computed as well. Spectral functions and the resulting excitation energies can however be calculated using real-time evolution of the wave function, yielding time-resolved Green’s functions, which con- tain information on the full spectrum. In addition to the stochastic J. Chem. Phys. 153, 034107 (2020); doi: 10.1063/5.0005754 153, 034107-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . (a) Energy over iteration for an excited state calculation with NECI for the beryllium atom targeting two states in the B 1girreducible representation (irrep) of the D2h symmetry group (corresponding to P-states). The two states have triplet/singlet character, and the energy difference is 105.5 mH. (b) Spectral decomposition of a 2 s→2p excited state of the beryllium atom created using real-time evolution with NECI , containing the two lowest energy P-states, which correspond to the states targeted in (a). The gap between the two states is 106.6 mH, agreeing with the excited state calculation within the spectral resolution of 2.1 mH. The zero of the energy axis corresponds to the cation ground-state energy. The output files are available in the supplementary material.81In the experiment, a value of 93.8 mH is observed for this energy gap.82 imaginary time evolution of a wave function using in the calculation of individual states, NECI supports performing real-time and arbi- trary complex-time calculations, evolving the wave function along- side a complex time trajectory.21As Green’s functions are quadratic in the coefficients of the wave function and averaging over multi- ple iterations is not an option when evolving a wave function with a real-time component, running multiple calculations in parallel akin to excited state calculations discussed in Sec. VIII is mandatory, as is running with complex coefficients. The real-time propagation can be used to obtain energy gaps from spectral densities and thus target excited states. In contrast to the direct calculation of excited states, these have not to be calculated one by one and in order of ascend- ing energy, however. In Fig. 3, a simple example for applying both the excited-state search and the real-time evolution to the Beryllium atom in a cc-pVDZ basis set to obtain the singlet–triplet gap of the lowest P-state is given. An issue with running real-time calculations is the difficulty of population control, as the death step is essentially replaced by a rotation in the complex plane. This issue can be mit- igated by a rotation of the trajectory, evolving along a trajectory in the complex plane. NECI supports an automated trajectory selection that updates the angle αof the time trajectory in the complex plane to maintain a constant population. The Green’s function obtained in the complex plane can then be used to obtain real-frequency spectral functions using analytic continuation,79,80with the analytic continuation being significantly easier and more information being recoverable the closer to the real axis the trajectory is.21As, in con- trast to the projector FCIQMC, errors arising from the expansion of the propagator are a concern when running complex-time calcu- lations, NECI uses a second-order Runge–Kutta integrator here to sufficiently reduce the time-step error. XI. TRANSCORRELATED METHOD The computational cost of a full CI method usually scales expo- nentially with respect to the size of the basis set. On the other hand, the low regularity of wave functions (characterized by the electroniccusp83) causes a very slow convergence toward the basis set limit. For calculations aiming at highly accurate results, it is very helpful to speed up such slow convergence. A Jastrow84ansatz offers a way to factor out the cusp from the wave function, ∣Ψ⟩=eˆT∣Φ⟩, (25) where ˆT=∑i<ju(ri,rj)is a symmetric function [ u(ri,rj) =u(rj,ri)] over electron-pairs and ∣Φ⟩is an anti-symmetric many-body func- tion. By including the cusp term | ri−rj|/2 in u(ri,rj), the regularity of∣Φ⟩is improved at least by one order over ∣Ψ⟩.85We can also include other terms in u(ri,rj) to capture as much dynamic correla- tions as possible. By using variational quantum Monte Carlo (VMC) methods, the pair correlation function u(ri,rj) can be obtained for a single determinant ∣Φ⟩(e.g.,∣ΦHF⟩) or a linear combination of a small number of determinants (e.g., a small CAS wave function). The transcorrelated method of Boys and Handy86provides a simple and efficient way to treat the Jastrow ansatz, where the original Schrödinger equation is transformed into a non-Hermitian eigenvalue problem, ˜H∣Φ⟩=E∣Φ⟩,˜H=e−ˆTˆHeˆT. (26) The advantage of this form of ˆTis that the similarity transformation leads to an expansion, which terminates at the second order, ˜H=ˆH+[ˆH,ˆT]+1 2[[ˆH,ˆT],ˆT] (27) =ˆH−∑ i(1 2∇2 iˆT+(∇iˆT)∇i+1 2(∇iˆT)2) (28) =ˆH−∑ i<jˆK(ri,rj)−∑ i<j<kˆL(ri,rj,rk). (29) J. Chem. Phys. 153, 034107 (2020); doi: 10.1063/5.0005754 153, 034107-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The similarity transformation introduces a novel two-body operator ˆKand a three-body potential ˆL, ˆK(ri,rj)=1 2(∇2 iu(ri,rj)+∇2 ju(ri,rj)+(∇iu(ri,rj))2 +(∇ju(rj,ri))2)+(∇iu(ri,rj))⋅∇i +(∇ju(ri,rj))⋅∇j), (30) ˆL(ri,rj,rk)=∇iu(ri,rj)⋅∇iu(ri,rk)+∇ju(rj,ri)⋅∇ju(rj,rk) +∇ku(rk,ri)⋅∇ku(rk,rj). (31) The whole transcorrelated Hamiltonian can be written in the second quantized form as ˜H=∑ pqσhp qa† pσaqσ+1 2∑ pqrs(Vpq rs−Kpq rs)∑ στa† pσa† qτasτarσ −1 6∑ pqrstuLpqr stu∑ στλa† pσa† qτa† rλauλatτasσ, (32) where hp q=⟨ϕp∣h∣ϕq⟩and Vpq rs=⟨ϕpϕq∣r−1 12∣ϕrϕs⟩are the one- and two- body terms of the molecular Hamiltonian, while Kpq rs=⟨ϕpϕq∣ˆK∣ϕrϕs⟩ and Lpqr stu=⟨ϕpϕqϕr∣ˆL∣ϕsϕtϕu⟩originate from the ˆKand ˆLoperators. This transcorrelated method has been investigated by FCIQMC using NECI , as it can essentially speed up the convergence with respect to basis sets. On the other hand, the effective Hamilto- nian is non-Hermitian and contains up to three-body potentials. Luo and Alavi explored a transcorrelated approach where only up to two-body potentials are involved.22The performance on uni- form electron gases indicates that this approach could be devel- oped into an efficient FCIQMC method for plane wave basis sets in the future. For general molecular systems, the full transcorre- lated Hamiltonian (32) is implemented in NECI , where ˆTis fixed and treated as an input function, while ∣Φ⟩is sampled by the FCIQMC algorithm. The lack of a lower bound of the energy due to the non-Hermiticity of the similarity transformed Hamiltonian poses a severe problem for variational approaches. However, as a projec- tive technique, FCIQMC does not have an inherent problem sam- pling the ground-state right eigenvector by repetitive application of the projector (2) and obtaining the corresponding ground-state eigenvalue. The matrix elements Kpq rsand Lpqr stuare pre-calculated and have to be supplied as an input. The matrix elements of Kcan be passed combined with the Coulomb integrals, while the matrix elements ofLare passed in a separate input file. This treatment is efficient for small atomic and molecular systems, but for large systems, the storage of the Lmatrix becomes a bottleneck. Here, efficient low rank tensor product expansion of Lmight in the future make it practical to treat even larger systems. NECI supports storage of L in a dense and a sparse format as well as on-the-fly calculation ofLpqr stufrom a tensor decomposition. Additionally, major techni- cal changes to the FCIQMC implementation are required for sam- pling up to triple excitations, which generally leads to reduced time steps. The development of efficient excitation generation, which can alleviate the time-step bottleneck, is the subject of current work. FIG. 4 . Exemplary application of the transcorrelated method: errors in the total energies of the first-row atoms, in hartree, for the two correlation functions and the F12 methodology. Reproduced with permission from Cohen et al. , J. Chem. Phys. 151, 061101 (2019). Copyright 2019 AIP Publishing LLC. This method has been tested on the first row atoms,24which shall serve as an example here. Two different correlation fac- tors obtained by Schmidt and Moskowitz87based on variance- minimization VMC, which contain 7 and 17 terms of polynomial type basis functions, have been employed there. The 7-term factor (SM7) contains mainly electron–electron correlation terms together with some electron–nuclear terms, while the 17-term factor (SM17) uses more terms to describe also the electron–electron–nuclear cor- relation. For the full CI expansion of ∣Φ⟩, three different basis sets, cc-pVDZ, cc-pVTZ, and cc-pVQZ, respectively, have been used. In Fig. 4, the convergence of the total energies errors is displayed for the two different correlation factors, in comparison with the coupled- cluster singles doubles perturbative triples [CCSD(T)]-F12 method. This demonstrates that improving the correlation factor can lead to a significant speed up of the basis set convergence. Using the 17-term factor, the CBS limit results can already be reached (within errors <1 mH) using cc-pVQZ basis sets. XII. SYMMETRIES AND SPIN-ADAPTED FCIQMC Symmetry is a concept of paramount importance in the description and understanding of physical and chemical processes. According to Noether’s theorem, there is a direct connection between conserved quantities of a system and its inherent symme- tries. Thus, identifying them allows a deeper insight into the physical mechanisms of studied systems. Moreover, the usage of symmetries in electronic structure calculations enables a much more efficient formulation of the problem at hand. The Hamiltonian formulated in a basis respecting these symmetries has a block-diagonal struc- ture, with zero overlap between states belonging to different “good” quantum numbers. This greatly reduces the necessary computa- tional effort to solve these problems and allows much larger systems to be studied. J. Chem. Phys. 153, 034107 (2020); doi: 10.1063/5.0005754 153, 034107-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp A. Common symmetries utilized in electronic structure calculations and NECI There are several symmetries that are commonly used in elec- tronic structure calculations due to the above-mentioned benefits and their ease of implementation. Our FCIQMC code NECI is no exception in this regard. 1. Conservation of the ˆSzspin-projection As mentioned in Sec. I, FCIQMC is usually formulated in a complete basis of Slater determinants (SDs). SDs are eigenfunctions of the total ˆSzoperator, and consequently, if the studied Hamilto- nian, ˆH, is spin-independent (no applied magnetic field and spin– orbit interaction), it commutes with ˆSz, [ˆH,ˆSz] = 0. The conservation of the mseigenvalue in a FCIQMC calculation thus follows quite nat- urally: the initial chosen mssector, determined by the starting SD used, will never be left by the random excitation generation process sketched in Sec. II. No terms in the spin-conserving Hamiltonian will ever cause any state in the simulation to have a different msvalue than the initial one. As a consequence, the sampled wave function will always be an eigenfunction of ˆSzwith a chosen ms, determined at the start of a calculation. 2. Discrete and point group symmetries in FCIQMC NECI is also capable of utilizing Abelian point group sym- metries, with D 2hbeing the “largest” spatial group (similar to other quantum chemistry packages, e.g., Molcas88andMolpro89,90), momentum conservation (due to translational invariance) in the Hubbard model, and uniform electron gas calculations and preser- vation of the mleigenvalues of the orbital angular momentum oper- ator ˆLz(the underlying molecular orbitals have to be constructed as an eigenfunction of ˆLz). This is implemented via a symmetry- conserving excitation generation step and is explained in more detail in Subsection 1 a of the Appendix. B. Total spin conservation One important symmetry of spin-preserving, nonrelativistic Hamiltonians is the global SU(2) spin-rotation symmetry. However, despite the theoretical benefits, the total SU(2) spin symmetry is not as widely used as other symmetries, like translational or point group symmetries, due to their usually impractical and complicated implementation. There are two kinds of implementations of the total spin con- servation in our FCIQMC code NECI . One approximate is based on Half-Projected Hartree–Fock (HPHF) functions .44,91–94Their ratio- nale relies on the fact that for an even number of electrons, every spin state ∣S⟩contains degenerate eigenfunctions with ms= 0. Using time-reversal symmetry arguments, a HPHF function can be constructed as ∣Hi⟩=⎧⎪⎪⎪⎨⎪⎪⎪⎩∣Di⟩ for fully close-shell determinants 1√ 2(∣Di⟩±∣Di⟩) otherwise,(33) where∣Di⟩indicates the spin-flipped version of ∣Di⟩. Depending on the sign of the open-shell coupled determinants, ∣Hi⟩are eigenfunc- tions of ˆS2with odd (−) or even (+) eigenvalue S. The use of HPHF is restricted to systems with an even number of electrons and can onlytarget the lowest even- and odd- Sstate. Thus, it cannot differentiate between, e.g., a singlet S= 0 and a quintet S= 2 state. 1. The (graphical) unitary group approach (GUGA) Our full implementation of the total spin conservation is based on the graphical unitary group approach (GUGA). It relies on the observation that the spin-free excitation operators Êijin the spin-free formulation of the electronic Hamiltonian, ˆH=n ∑ ijtijˆEij+n ∑ ijklVijkl(ˆEijˆEkl−δjkˆEil), (34) have the same commutation relations, [ˆEij,ˆEkl]=δjkˆEil−δilˆEkj, (35) as the generators of the unitary group U(n). This connection allows the usage of the Gel’fand–Tsetlin (GT) basis,95–97which is irre- ducible and invariant under the action of the operators Êij, in elec- tronic structure calculations. The GT basis is a general basis for any irrep of U(n), but Paldus98–100realized that only a special subset is relevant for the electronic problem (34) due to the Pauli exclu- sion principle. Based on Paldus’s work, Shavitt101further developed an even more compact representation by introducing the graphical extension of the UGA. This leads to the most efficient encoding of a spin-adapted GT basis state (CSF) in the form of a step-vector ∣d⟩. This step-vector representation has the same storage cost of two bits per spatial orbital as Slater determinants. The entries of this step- vector encode the change of the total number of electrons ΔNiand the change of the total spin ΔSiof subsequent spatial orbitals i. This is summarized in Table I. All possible CSFs for a chosen number of spatial orbitals N, number of electrons n, and total spin Sare then given by all step-vectors ∣d⟩=∣d1,d2,...,dN⟩fulfilling the restrictions, N ∑ i=1Δni=n,N ∑ i=1ΔSi=S, and Sk=k ∑ i=1ΔSi≥0. (36) The last restriction in Eq. (36) corresponds to the fact that the (intermediate) total spin must never be less than 0. The most important finding of Paldus and Shavitt102,103was that the Hamiltonian matrix elements—more specifically the cou- pling coefficients between two CSFs, e.g., ⟨m′∣ˆEij∣m⟩—can be entirely formulated within the framework of the GUGA, without any refer- ence and thus necessity to transform to a Slater determinant-based formulation. Although CSFs can be expressed as a linear combina- tion of SDs, the complexity of this transformation scales exponential with the number of open-shell orbitals of a specific CSF.104Thus, it is TABLE I . Possible step-values diand the corresponding change in the number of electrons ΔNiand the total spin ΔSiof subsequent spatial orbitals i. di ΔNi ΔSi 0 0 0 1 1 1/2 2 1 −1/2 3 2 0 J. Chem. Phys. 153, 034107 (2020); doi: 10.1063/5.0005754 153, 034107-13 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp prohibitively hard to rely on such a transformation, and for already more than ≈15 electrons, a formulation without any reference to SDs is much more preferable. Furthermore, Shavitt and Paldus102,103were able to find a very efficient formulation of the coupling coefficients as a product of terms via the graphical extension of the UGA. Matrix elements between two given CSFs only depend on the shape of the loop enclosed by their graphical representation, as depicted in Fig. 5. The coupling coefficient of the one-body operator Êijis given by ⟨m′∣ˆEij∣m⟩=j ∏ k=iW(Qk;d′ k,dk,ΔSk,Sk), (37) where the product terms depend on the step-values of the two CSFs, d′ kand dk, the difference in the current spin ΔSk(with the restriction S′ k−Sk=±1/2), and the intermediate spin Skof∣m⟩at orbital k.Qk in Eq. (37) depends on the shape of the loop formed by ∣m⟩and∣m′⟩ at level kand was tabulated in, e.g., Ref. 102. Additionally, the two CSFs,∣m⟩and∣m′⟩, must coincide outside the range ( i,j) for Eq. (37) to be non-zero. 2. Spin-adapted excitation generation—GUGA-FCIQMC The compact representation of spin-adapted basis functions in the form of step-vectors and the product form of the coupling coef- ficients (37) allow for a very efficient implementation in our stochas- tic FCIQMC code NECI . As mentioned in Sec. II, the excitation generation step is at the heart of any FCIQMC code. FIG. 5 . Graphical representation of the coupling coefficient between two CSFs, ⟨m∣ˆEij∣m′⟩.The main difference to a SD-based implementation of FCIQMC, apart from the more involved matrix element calcula- tion (37), is the higher connectivity within a CSF basis. For a given excitation operator Êij, with spatial orbital indices ( i,j), there is usu- ally more than one possible excited CSF ∣m′⟩when applied to ∣m⟩, ˆEij∣m⟩=∑kck∣m′ k⟩. All valid spin-recouplings within the excitation range ( i,j) can have a non-zero coupling coefficient as well. This fact is usually the prohibiting factor in spin-adapted approaches. However, there is a quite virtuous combination of the concepts of FCIQMC and the GUGA formalism, as one only needs to pick one possible excitation from ∣m⟩to∣m′⟩in the excitation generation step of FCIQMC (see Sec. II). We resolved this issue, by randomly choosing one possible valid branch in the graphical representation, depicted in Fig. 5, for randomly chosen spatial orbital indices i,j(,k,l). Additionally, we weight the random moves according to the expected magnitude of the coupling coefficients8,105to ensure pgen(m′|m)∝|Hm′m|. This approach avoids the possible exponential scaling as a function of the open-shell orbitals of the connected states within a CSF-based approach. However, this comes with the price of reduced generation prob- abilities and consequently a lower imaginary time step, as men- tioned in Sec. II. Combined with an additional effort of calculating these random choices in the excitation generation and the on-the-fly matrix element computation, the GUGA-FCIQMC implementation has a worse scaling with the number of spatial orbitals Ncompared to a Slater determinant-based implementation.8 However, the benefits of using a spin-adapted basis are a reduced Hilbert space size ,elimination of spin-contamination in the sampled wave function and, most importantly, the spin-adapted FCIQMC implementation via the GUGA allows targeting specific spin states, which are otherwise not attainable with a SD-based implementation, as discussed in Ref. 8. The unique specification of a target spin allows resolving near degenerate spin states, and consequently, the numerical results can be interpreted more clearly. This enables more insight in the intri- cate interplay of nearly degenerate spin states and their effect on the chemical and physical properties of matter. 3. Example: Hydrogen chain in a minimal basis The GUGA-FCIQMC method has been benchmarked105by applying it to a linear chain of Lequidistant hydrogen atoms106 recently studied to test a variety of quantum chemical methods,107 which shall serve as an example here. Using a minimal STO-6G basis, there is only one orbital per H atom and the system resembles a one-dimensional Hubbard model41,108–110with long-range inter- action. Studying a system of hydrogen atoms removes complexities such as core electrons or relativistic effects and thus is a convenient benchmark system for quantum chemical methods. For large equidistant separation of the H atoms, a local- ized basis, obtained with the default Boys-localization in Molpro’s LOCALI routine, with singly occupied orbitals centered at each hydrogen is more appropriate than a HF basis. Thus, this is an opti- mally difficult benchmark system of the GUGA-FCIQMC method, since the complexity of a spin-adapted basis depends on the number of open-shell orbitals, which is maximal for this system. Particularly, targeting the low-spin eigenstates of such highly open-shell systems J. Chem. Phys. 153, 034107 (2020); doi: 10.1063/5.0005754 153, 034107-14 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 6 . Schematic representation of a one-dimensional hydrogen chain of Lhydrogen atoms with equal separation r. TABLE II . Example for application of GUGA-FCIQMC: difference of the energy per siteE/Lof a hydrogen chain for a different number Lof H atoms and total spin Sin a STO-6G basis set at the stretched bond distance of r= 3.6 a0compared with the DMRG107,111,112reference results.107The GUGA-FCIQMC results were obtained without the initiator approximation.9Reproduced with permission from W. Dobrautz, “Development of full configuration interaction quantum Monte Carlo meth- ods for strongly correlated electron systems,” Ph.D. thesis, University of Stuttgart, 2019. L S Eref(Eh) EFCIQMC (Eh) ΔE(mE h) 20 0 −0.481 979 −0.481 978 (1) −0.001(1) 20 1 −0.481 683 −0.481 681 (11) −0.002(11) 20 2 −0.480 766 −0.480 764 (18) −0.002(18) 30 0 −0.482 020 −0.481 972 (31) −0.047(31) poses a difficult challenge within a spin-adapted formulation. This situation is depicted schematically in Fig. 6. We studied this system to show that we are able to treat systems with up to 30 open-shell orbitals with our stochastic implementation of the GUGA approach.105We calculated the S= 0, 1, and 2 (only S= 0 for L= 30) energy per atom up to L= 30 H atoms in a minimal STO-6G basis at the stretched r= 3.6 a0geometry107and compared it with the DMRG107,111–114reference results. The results are shown in Table II, where we see excellent agreement within chemical accuracy with the reference results. An important fact is the order of the orbitals though. Similar to the DMRG method, it is most beneficial to order the orbitals accord- ing to their overlap, since the number of possible spin recouplings depends on the number of open shell orbitals in the excitation range. If we make a poor choice in the ordering of orbitals, excitations between physically adjacent and thus strongly overlapping orbitalsare accompanied by numerous possible spin-recouplings in the exci- tation range, if stored far apart in the list of orbitals. This behavior is thoroughly discussed in Ref. 115. XIII. PARALLEL SCALING When applying for access to large computing clusters, it is often necessary to demonstrate that the software being used (in this case NECI ) is capable of using the hardware efficiently. Ideally, the speed- up relative to using some base number of compute cores should grow perfectly linearly with the number of cores. In 2014, Booth et al.94 presented an example with 500 ×106walkers in which no devia- tion from a linear speed-up is notable when compared using 512 cores to using 32, and even at 2048 cores, a speed-up by a factor of 57.5 was reported, which is 90% of the ideal speed-up factor of 64. In that work, the largest number of cores explored was 2048. By comparing the performance for a calculation with 100 ×106 walkers and 500 ×106walkers, the same figure showed that the speed-up became closer to the ideal speed-up when the number of walkers was increased, suggesting that when using even more walk- ers, the efficiency comes even closer to 100% of the ideal speed-up factor. Since 90% of the ideal speed-up factor was achieved in 2014 with only 500 ×106walkers on 2048 cores, and large compute clus- ters nowadays tend to have tens of thousands of cores available, we report scaling data for a much larger number of walkers on up to 24 800 cores in Table III. The calculations were done using the integrals in FCIDUMP format for the (54e, 54o) active space first described in Ref. 116 for the FeMoco molecule, and the output files are provided in the supplementary material.117 The scaling analysis presented in Table III was done with 32×109walkers on each of the two replicas used for the RDM sam- pling. Calculations at 32 ×109walkers are expensive, so we only completed enough iterations to determine an accurate estimate of TABLE III . Efficiency of parallelization for a CAS(54,54) of the FeMoco molecule. In both 32 ×109walkers cases, the time per iteration is averaged over more than 250 iterations, and in both cases, the unbiased sample variance over the 250 + iterations is less than 0.5 s. For comparison, the time per iteration for 8 ×109walkers, which was used to obtain the energy reported in Table IV, is given. Calculations were run on 512, 620, and 400 nodes with Intel Xeon Gold 6148 Skylake processors with 20 cores at 2.4 GHz and 96 GB of DDR4 RAM, and all nodes were in a single island with a 100 Gb/s OmniPath interconnect between the nodes. Hyperthreading was not used. No. of No. of Average time Ratio no. of Ratio of average Efficiency of walkers cores per iteration (s) cores time/iteration parallelization (%) 32×10919 960 23.5 1.242 1.246 99.68 32×10924 800 18.8 8×10916 000 7.3 ... ... ... J. Chem. Phys. 153, 034107 (2020); doi: 10.1063/5.0005754 153, 034107-15 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the average runtime per iteration for the scaling analysis, and not enough iterations to accurately estimate the energy. One may ask whether or not the scaling observed in Table III was performed for a reasonable number of walkers for this active space. To answer this question, we compare in Table IV the best (non-extrapolated) DMRG and sHCI-PT2 energies in the litera- ture118to energies obtained with i-FCIQMC at only 8 ×109walk- ers/replica and find that the i-FCIQMC-RDM and i-FCIQMC-PT2 energies are closer together than the sHCI-VAR and sHCI-PT2 energies, indicating that the i-FCIQMC energies are closer to the true FCI limit where the difference between variational and PT2 energies should vanish. The DMRG result lies about half-way between the two i-FCIQMC results, but fairly well below the lower of the sHCI results (a forthcoming publication specifically about the FeMoco system is planned in which more details will be presented, but the purpose of this paper is to give an overview of the NECI code). Furthermore, comparing the time per iteration between 8 ×109 and 32×109walkers shows that a high parallel efficiency is also achieved at a lower walker number. The determinants in NECI are stored using a hash table, making i-FCIQMC linearly scaling in the walker number,94so the ideal time per iteration with 32 ×109walkers at 19 960 cores according to the result for 8 ×109 walkers at 16 000 cores would be 23.4 s, which is only marginally smaller than the reported 23.5 s. Note, however, that this is the relative efficiency between large scale calculations, which demon- strates performance gain from extending parallelization at large scales, not from parallelization over the entire range of scales, which is addressed to some extent by the chromium dimer example below. In the case of the chromium dimer (cc-pVDZ, 28 electrons cor- related in 76 spatial orbitals) considered in Fig. 7, the average time per iteration per walker ranges from 3.18 ×10−9s at 640 cores to 2.51×10−10s at 10 240 cores and 1.53 ×10−10s at 20 480 cores, corresponding to a parallel speed-up of 82.1% from 10 240 to 20 480 cores and an overall speed-up of 65.2% over the full range. The devi- ation from ideal scaling almost exclusively stems from the communi- cation of the spawns, at lower walker numbers, the communicative overhead is more significant, reducing the parallel efficiency com- pared to the FeMoco example. Nevertheless, a very high yield can be obtained from scaling up the number of cores, even for already large scales. TABLE IV . Best non-extrapolated energies obtained for the CAS(54,54) of the FeMoco molecule with three different methods. DMRG and sHCI energies were cal- culated in Ref. 118, and i-FCIQMC results were obtained in this work with 8 ×109 walkers on each of the two replicas for the RDM sampling. Method Total energy i-FCIQMC-RDM −13 482.174 95(4) i-FCIQMC-PT2 −13 482.178 45(40) sHCI-VAR −13 482.160 43 sHCI-PT2 −13 482.173 38 DMRG −13 482.176 81 FIG. 7 . Total time and time lost due to load imbalance for running 100 iterations with 1.6 ×109walkers for the Cr 2/cc-pVDZ (28e in 76o) on 640–20 480 cores (not counting initialization). The calculations were run on Intel Xeon Gold 6148 Skylake processors with a 100 Gb/s OmniPath node interconnect. The code was compiled using the Intel Fortran compiler, version 19.0.4. A semi-stochastic core- space of 50 000 determinants was used, and PCHB excitation generation. For the largest number of cores, the time step is 3.68 ×10−4with an average accep- tance ratio of 12.51%, which is representative for all numbers of cores. The load imbalance time is measured as the accumulated difference between the maximum and average time per iteration across MPI tasks. This figure was generated using Matplotlib .121 A. Load balancing The parallel efficiency of NECI is made possible by treating static load imbalance. NECI contains a load-balancing feature,28 which is enabled by default and periodically re-assigns some deter- minants to other processors in order to maintain a constant num- ber of walkers per processor. As shown in Fig. 7 for the given benchmarks, no significant load imbalance occurs up to (includ- ing) 20 480 cores.119,120The initialization of a simulation does not feature the same speed-up due to I/O operations and initial com- munication such as trial wave function setup and core space gen- eration. However, since it does not play a significant role for extended calculations, we consider only the time spent in the actual iterations. XIV. INTERFACING NECI The ongoing development of NECI is focused on an efficiently scaling solver for the CI-problem. It is not desirable to reimplement functionality that is already available in the existing quantum chem- istry codes. Since the CI-problem is defined by the electronic inte- grals and subsequent methods depend on the results of the CI-step, namely, the reduced density matrices, it is easily possible to replace a CI-solver for the existing quantum chemistry code with NECI . To use NECI , only an input file and a FCIDUMP file,122which is the widely understood file format for the electronic integrals, are required. After running NECI , the stochastically sampled reduced density matrices are available as an input for further calculations in other codes. It is possible to link NECI as a library and call it directly or to run it as an external process and do the communication with J. Chem. Phys. 153, 034107 (2020); doi: 10.1063/5.0005754 153, 034107-16 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp explicit copying of files. The first alternative will be referred to as embedded; the second is the decoupled form. Due to the stochastic nature of the Monte Carlo algorithm, it is not yet possible to use NECI as a black box CI-solver for larger sys- tems. In this case, it is recommended to use the decoupled form for a better manual control of the convergence. Another advantage of the decoupled form is the combination of NECI with different quantum chemical algorithms or implementations that do not benefit from massive parallelization as much as NECI . This way it is possible to switch from serial or single node execution to multiple nodes in the CI-step. So far, NECI has been coupled with Molpro ,89,123Molcas 8,88OpenMolcas ,124PySCF ,125andVASP .126 XV. STOCHASTIC-MCSCF The stochastic multi-configurational self-consistent field (MCSCF) procedure emerges from the combination of conven- tional MCSCF methodologies with FCIQMC as the CI-eigensolver. Stochastic-MCSCF approaches greatly enlarge the applicability of FCIQMC to strongly correlated molecular systems of practical inter- est in chemical science. To date, two implementations of stochastic-MCSCF have been made available based on the interface of NECI withOpenMolcas19,124 (and Molcas 888) and PySCF .20,125As they are both based on the complete active space (CAS) concept, they are also often referred to as stochastic-CASSCF methods. The stochastic-CASSCF implemented in PySCF is based on a second-order CASSCF algorithm,127which decouples the orbital optimization problem from the active space CI problem, allowing for easy interfacing with NECI . At each macro-iteration , a FCIQMC simulation is performed at the current point of orbital expansion, and density matrices are stochastically sampled (see Sec. VII). These are then passed back to PySCF, which updates the orbital coefficients accordingly, using either a 1-step127or 2-step approach.128 The stochastic-CASSCF implemented in OpenMolcas is based on the quasi-second-order super-CI orbital optimization. Optimal orbitals (in the variational sense) are found by solving the super- CI secular equations in the ∣Super−CI⟩basis, defined by the CAS wave function at the point of expansion, |0 ⟩, and all its possible single excitations, ∣Super−CI⟩=∣0⟩+∑ p>qχpq(ˆEpq−ˆEqp)∣0⟩. (38) The wave function is improved by mixing single excitations to the |0 ⟩ wave function. As the CASSCF optimization proceeds, the χpqcoef- ficients decrease until they vanish, and |0 ⟩will reveal the variational stationary point. Third-order density matrix elements of the exact super-CI approach are avoided by utilizing an effective one-electron Hamiltonian, as discussed in more detail in Ref. 19. A flow chart of stochastic-CASSCF describing the various steps of the CASSCF wave function optimization is given in Fig. 8. The stochastic-CASSCF approach has successfully been applied to a number of challenging chemical problems. The accuracy of the method has been demonstrated on simple test cases, such as benzene and naphthalene20and more complex molecular systems, namely, coronene,20free-base porphyrin, and Mg–porphyrin.19 More recently, the method has also been applied to understand FIG. 8 . Flow chart summarizing the stochastic-CASSCF steps. The blue boxes rep- resent parts of the algorithm performed at the OpenMolcas orPySCF interfacing software. The center yellow box shows the two crucial FCIQMC steps, stochastic optimization of the CAS-CI wave function, and sampling of the one- and two-body reduced density matrices. When embedded schemes are employed, additional external potentials are added within the interfacing software when generating the FCIDUMP file. Post-CASSCF procedures, such as the MC-PDFT methodology, follow the stochastic-CASSCF approach within the interfacing software. the mechanism stabilizing intermediate spin states in Fe(II)– porphyrin,39,40the study of a [Fe(III)2S2(SCH 3)2]2−iron–sulfur model system in its oxidized form,115and new superexchange paths in corner-sharing cuprates.129 To date, only state specific stochastic-CASSCF optimizations have been reported. However, state-average stochastic-CASSCF optimizations are a straightforward extension that can be reached by taking the advantage of the NECI capability to optimize excited state wave functions, as discussed in Sec. VIII. The stochastic-CASSCF method can also be coupled to the adaptive shift approach discussed in Sec. V with a great enhancement in performance. XVI. CONCLUSION With NECI , we present a state-of-the-art FCIQMC program capable of running a large variety of versions of the FCIQMC algo- rithm. This includes the semi-stochastic FCIQMC feature, energy estimation using trial wave functions, the stochastic sampling of J. Chem. Phys. 153, 034107 (2020); doi: 10.1063/5.0005754 153, 034107-17 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the reduced density matrices, and excited state calculations. Fur- ther features of NECI ’s FCIQMC implementation discussed are the real-time FCIQMC method and the adaptive shift method, as well as a spin-adapted formulation of the algorithm and support for the transcorrelated Hamiltonians. We demonstrated the scalability of the program to up to 24 800 cores, showing that the code can run efficiently on large-scale machines. Finally, we highlighted the interoperability of NECI with other quantum chemistry software, in particular OpenMolcas andPySCF , which can be used to run stochastic-CASSCF calculations. SUPPLEMENTARY MATERIAL Example FCIQMC output files for excited state calculations (output_file_excited_state_be2_b1g.txt andstats_file _excited_state_be2_b1g.txt ) and real-time calculations including the resulting spectrum ( output_file_real_time_be2 _b1g.txt and fft_spectrum_be2_b1g.txt ) for the exam- ples presented in Sec. III are available in the supplementary material. Furthermore, it contains the output files for scaling (output_file_scaling_with_ ∗_cores.txt andoutput_file _energy_with_8b_walkers.txt ) and load imbalance analy- sis ( output_file_load_imbalance_n ∗.txt ) and also exem- plary output and integral files for a similarity transformed FCIQMC calculation of the neon atom in a cc-pVDZ basis (tcdump_Ne_st_pVDZ.h5 andFCIDUMP_Ne_st_pVDZ integral files and output_file_Ne_st_pVDZ.txt and stats_file_Ne_st _pVDZ.txt files). All output files contain the corresponding FCIQMC input. ACKNOWLEDGMENTS The early development of NECI was supported by the EPSRC under Grant Nos. EP/J003867/1 and EP/I014624/1. We would like to thank Olle Gunnarsson, David Tew, Daniel Kats, Aron Cohen, and Vamshi Katakuri for insightful discussions. The high performance benchmarks discussed in Sec. XIII were run on the MPCDF (Max Planck Computing and Data Facility) sys- tem Cobra. PJ was supported by the Marsden Fund of New Zealand (Contract No. MAU1604), from government funding managed by the Royal Society Te Ap ¯arangi. APPENDIX: STOCHASTIC EXCITATION GENERATION AND pgen In this appendix, we will consider in some detail the pro- cess of (random) excitation generation in FCIQMC—a crucial yet rather flexible aspect of the algorithm. We will consider some gen- eral aspects, such as implementation of Abelian symmetries in the excitation process and non-uniform excitation generation, as is often desirable in quantum chemical Hamiltonians. There are other classes of systems (such as Hubbard models, transcorrelated Hamil- tonians, and spin models, such as Heisenberg systems) for which more specialized considerations are necessary for efficient excitation generation, but we will not consider them here. The first general point about excitation generation (by which we mean starting from a given determinant ∣Di⟩, we randomly pick either one or two electrons and a corresponding number of holes to substitute them with, to create a second determinant ∣Dj⟩) is that if|Hij|>0, then the probability [ pgen(j|i)] to select ∣Dj⟩and∣Di⟩must also be greater than 0. Furthermore, pgen(j|i) must be computable , and in general, the effort to do so will depend on the algorithm chosen to execute the excitation process. Let us discuss in more detail the process of stochastic excita- tion generation, and its impact on pgen. Suppose we are simulating a system of nelectrons in 2 Nspin orbitals { ϕ1,...,ϕ2N}. A given deter- minant∣Di⟩can be defined by its occupation number representation, I=∣n1,...,n2N⟩, which is a binary string such that ni= 1, if orbital i is occupied (“an electron in ∣Di⟩”), and ni= 0, if it is unoccupied (“a hole in∣Di⟩”). Each orbital carries a spin quantum number σ(ϕi) and may also carry a symmetry label, Γ(ϕi). These are both discrete sym- metries, with σ=±1/2, and Γ=Γ1,...,ΓG, where Gis the number of irreducible representations available in the point-group of the sys- tem under consideration. We will only consider Abelian groups so that the product of symmetry labels uniquely specifies another sym- metry label. This simplifies the task of selecting excitations, although it does not necessarily exploit the full symmetry of the problem. 1. Uniform excitation generation Now we wish to perform a stochastic excitation generation, which we will initially consider without the use of any symme- try/spin information. For example, we can select a pair of electrons, i,j(with i<j) in∣Di⟩, at random, and a pair of holes a,b(with a <b) and perform the transition ij→ab. The corresponding matrix element is Hab ij=⟨ij∣ab⟩−⟨ij∣ba⟩≡⟨ij∥ab⟩. (A1) We will denote the electron-pair simply as ijand the hole-pair as ab. For this simple procedure, it is clear that the probability to choose∣Dj⟩from∣Di⟩is simply pgen(j∣i)=(n 2)−1 (2N−n 2)−1 , (A2) from which it follows that pgen(j|i)∼(nN)−2. This procedure does not take symmetry or spin quantum numbers into account, and it is quite possible that the corresponding Hamiltonian matrix ele- ment is zero. To ensure that we do not generate such excitations, we need to select the hole pairs so that the following two conditions are met: σ(ϕi)+σ(ϕj)=σ(ϕa)+σ(ϕb), (A3) Γ(ϕi)×Γ(ϕj)=Γ(ϕa)×Γ(ϕb). (A4) These restriction greatly impact the way in which we will select i,j and a,b, and the resulting generation probability. a. Imposing symmetries via conditional probabilities One way to impose symmetries in excitation generation while keeping track of the generation probabilities is via the notion of con- ditional probabilities. For example, rather than drawing ( ij) and ( ab) independently, with probability p(ab,ij) =p(ab)p(ij), one can instead draw ( ab) given that one has already drawn ( ij); the probability for this process is given by p(ab,ij)=p(ab∣ij)p(ij), (A5) J. Chem. Phys. 153, 034107 (2020); doi: 10.1063/5.0005754 153, 034107-18 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp where p(ij) is the probability to select ( ij) in the first place. If ( ij) has a particular characteristic that confers a physical (e.g., symmetry- related) constraint on ( ab), this can be implemented at the stage in which we select ( ab):(ab) need only be selected from among those hole-pairs for which the constraint is satisfied. For example, if the electrons ( ij) have opposite spins, then the holes ( ab) must also have opposite spins. The smaller number of possibilities in choosing the abpair then leads to a larger p(ab|ij) compared to p(ab), which can be thought of as a renormalization of the latter probability to take into account the constraint. The concept of conditional probabilities can be further extended so that the pair ( ij) itself is made to satisfy a particular con- dition. Suppose we introduce a set of conditions {C1,C2,...}such that the union of all such conditions is exhaustive. It is possible to draw conditional probabilities with respect to such conditions. For example, C1=“electron pair have the same spin”, (A6) C2=“electron pair have opposite spins”, (A7) and then, one can write p(ab,ij)=p(ab,ij∣C1)p(C1)+p(ab,ij∣C2)p(C2), (A8) with p(C1)+p(C2)=1. (A9) Here, p(C1), the probability to select the same-spin excitations, can be chosen arbitrarily, which then fixes p(C2)according to the above. The advantage of this formulation is that we can skew the selec- tion of electron-pairs, for example, toward opposite spin excitations if that proves advantageous, and to be able to compute the resulting probabilities. Furthermore, we can write p(ab,ij∣C1)=p(ab∣ij)p(ij∣C1), (A10) p(ab,ij∣C2)=p(ab∣ij)p(ij∣C2), (A11) which allows us to select a pair of electrons satisfying condition C1 and, subsequently, draw a pair of holes, given that one has selected an electron-pair with the same spin (which implies that the hole-pair must be chosen to have the same spin as the electron-pair). 2. Cauchy–Schwartz excitation generation Let us now consider how to generate the hole-pairs in a non- uniform manner, to reflect the fact that, in ab initio Hamiltonians, the matrix elements vary strongly in magnitude. Since the spawn- ing probability is proportional to the ratio | Hij|/pgen(j|i), it is clearly desirable to generate excitations that make this ratio as uniform as possible, ideally with pgen(j|i)∝|Hij|. In this way, one would ensure a relatively uniform probability of successful spawning, which ide- ally would be close to one, implying a low rejection rate. Keeping the discussion focused on double excitations (the generalization to single excitations being straightforward), the question that arises is: how best can one select ijand absuch that pgen(j|i)∝|Hij| to a good approximation and pgenremains exactly computable without exces- sive cost. We will see that there is a compromise to be made. One can ensure precise proportionality between pgen(j|i) and | Hij|, butonly at prohibitive cost. Alternatively, one might be able to select ijand abto affect the transition ∣Di⟩→∣Dj⟩based on computation- ally inexpensive heuristics to provide approximate proportionality, which will nevertheless allow for a large overall improvement in efficiency. To ensure exact proportionality between pgen(j|i) and | Hij|, it is necessary to enumerate all electron-pairs and hole-pairs, which are possible from ∣Di⟩, and to construct the cumulative proba- bility function (CPF) from which the desired distribution can be straightforwardly sampled. The (unnormalized) CPF is Fab,ij[D]=ij ∑ ee′∈Dab ∑ hh′∈D∣⟨ee′∥hh′⟩∣. (A12) In this expression, the sum over ee′runs over all enumerated electron-pairs in Dup to ij, and similarly for the hole-pairs (up to ab). The CPF is a non-decreasing function of its discrete arguments, and its inverse transform enables one to select aband ijwith proba- bility proportional to | ⟨ij∥ab⟩|. From the point of generation proba- bilities, this is the ideal excitation generator, allowing for a uniform spawning probability (which can be made to equal unity, implying zero rejection rates). Unfortunately, the CPF costs O(n2N2)to set up (for each determinant ∣Di⟩), making it prohibitive, in practice. To make practical progress, we need an approximate distribu- tion function, which is much cheaper to calculate. Two observations can be made in this relation. First, if the two electrons have differ- ent spins, then the Hamiltonian matrix element consists of only one rather than two terms. This is because upon excitation ij→ab, the two holes must match the spins of the two electrons. For example, σ(a) =σ(i) andσ(b) =σ(j). In this case, the Hamiltonian matrix element reduces to Hij=⟨ij∣ab⟩ (A13) with the exchange term ⟨ij|ba⟩= 0. With this simpler matrix element, we now ask, given that we have chosen an electron-pair ij, how can we select the hole- pair abso that, with high probability, the resulting matrix element ⟨ij|ab⟩is large? At this point, we can appeal to the Cauchy–Schwarz inequality, which provides a strict upper bound, ⟨ij∣ab⟩≤√ ⟨ii∣aa⟩⟨jj∣bb⟩. (A14) This suggests that, as long as ⟨ij|ab⟩is non-zero by symmetry, it may be advantageous to select the hole aso that⟨ii|aa⟩is large, and the hole bso that⟨jj|bb⟩large. Because iand jhave different spins, the selection of aand bwill be independent of each other, with a, for example, being chosen from the α-spin holes available and bfrom theβ-spin holes. To do this, we set up two CPFs, Fa[i∈αD]=a ∑ h∈αD√ ⟨hh∣ii⟩, (A15) Fb[j∈βD]=b ∑ h∈βD√ ⟨hh∣jj⟩, (A16) where the sums over hruns over the αorβholes inD. (The notation i∈αDmeans anα-electron in D, and h∈αDmeans anα-hole in D.) Unlike Eq. (A12), these CPFs cost only O(N)to set up and allow (via J. Chem. Phys. 153, 034107 (2020); doi: 10.1063/5.0005754 153, 034107-19 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp their inverse transforms) the selection of aand bwith probabilities proportional to√ ⟨aa∣ii⟩and√ ⟨jj∣bb⟩, respectively. The Cauchy–Schwarz bound on an individual four-index inte- gral provides a very useful factorized approximation for the purposes of excitation generation, especially for opposite-spin excitations. The case for the same-spin excitations is less favorable because it involves the difference between two four-index integrals, and in this case, we must obtain an upper bound for this difference expressed in a factorized form. We use the following much less tight upper bound: ∣⟨ij∣ab⟩−⟨ij∣ba⟩∣≤[√ ⟨aa∣ii⟩+√ ⟨aa∣jj⟩] (A17) ×[√ ⟨bb∣ii⟩+√ ⟨bb∣jj⟩]. (A18) In practice, we must draw two holes aand bfrom the same set of holes, avoiding the possibility of drawing the same hole twice. Because we would like to avoid setting up a two-dimensional CPF (which would cost O(N2)), we create a one-dimensional CPF in order to draw hole aand then remove this hole in the CPF before drawing the second hole. In other words, we set up two related CPFs, Fa[ij∈D]=a ∑ h√ ⟨ii∣hh⟩+√ ⟨jj∣hh⟩, (A19) F′ b[ij∈D]=⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩Fb[ij∈D∣a] ifb<a Fb[ij∈D]−√ ⟨ii∣aa⟩ −√ ⟨jj∣aa⟩ifb≥a,(A20) drawing hole afrom Faand hole bfrom F′ b. Our exploration of excitation generation has led us to dis- cover many highly performing schemes. The Cauchy–Schwarz (CS) scheme presented above is a good starting point, but it has a number of weaknesses that can be further addressed. In particular, as noted above, the upper bound obtained is particularly poor for double exci- tations with the same spin, and in general, the specified bound can be too loose. Fortunately, the selection of the second hole, b, is made once the first hole, a, has already been chosen, and as such the exact double excitation Hamiltonian matrix elements can be used at this stage such that an updated CPF for selecting the second electron is given by Fb[ij∈D∣a]=b ∑ h∈D h≠a√ ∣⟨ij∣ab⟩−⟨ij∣ba⟩∣. (A21) This Part-Exact (PE) scheme no longer provides a strict bound, but by better representing the cancellation of terms present in these matrix elements, it provides a substantially better approximation. More crucially, it improves the prediction of the elements that were previously handled least effectively and thus relaxes the time-step constraints on the overall calculation. Due to the increase in computational cost involved in con- structing two lists, and the additional normalization of the probabili- ties required by causing the two selections not to be made in the same manner, this update to the scheme increases computational cost periteration. In almost all systems examined, this is far outweighed by the time-step changes, especially in systems with large basis sets or with translational symmetry. However, it is possible to find systems where the pure CS scheme is more optimal. a. Preparing for excitation generation For determinant D, to pick an excited determinant, first con- struct a table of hole occupancies for each spin and irreducible rep- resentation so that nσΓ[D] is the number of holes with spin σin irrep Γavailable in D. This is an O(n)process. We next decide whether we wish to make a single excitation or a double excitation from D. A single excitation is chosen with prob- ability psing, a parameter that can be optimized, as the simulation proceeds to maximize the acceptance ratio and time step of the sim- ulation. The probability to create a double excitation is chosen such that the maximal ratios∣Hij∣ pgen(j∣i)for single and double excitations are equal, which, for ab initio systems, typically means double excita- tions dominate. To a first approximation psing=nN/(nN+n2N2), which is, in general, a small number on the order of ( nN)−1. The probability of attempting a double excitation is then pdoub= 1−psing. b. Single excitations If a single excitation is being attempted, first select an electron (say i) at random, with probability n−1. The spinσ=σ(i) and irrep Γ=Γ(ϕi) of the electron determines the spin and irrep of the hole. To select the hole a, run over all nσΓholes available in Dwith spin and symmetry σΓ, and compute the (unnormalized) cumulative probability function, F(1) a[i∈D]=a ∑ h∈σΓD∣⟨Dh i∣ˆH∣D⟩∣, (A22) where∣Dh i⟩is a single-excitation i→hfrom∣D⟩and⟨Dh i∣ˆH∣D⟩is the Hamiltonian matrix element between them. The normalization of the CPF is given by the last element in the array, Σi=F(1) nσΓ[i∈D], (A23) where nσΓis the number of holes available with spin σin irrep Γin D. Using F(1) a, select hole a(with probability ∣⟨Da i∣H∣D⟩∣) by inverting the CPF. This is selected by generating a uniform random number ξ in the interval [0, Σi) and determining the index of asuch that the condition F(1) a−1<ξ≤F(1) a (A24) is met. The overall generation probability for this excitation is pgen(a,i)=p(a∣i)×p(i)×psing, (A25) where p(a∣i)=θa Σi, (A26) θa=F(1) a−F(1) a−1, (A27) p(i)=n−1. (A28) J. Chem. Phys. 153, 034107 (2020); doi: 10.1063/5.0005754 153, 034107-20 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp This completes the selection of a singly excited determinant. The computation of F(1) ais an order O(nN)operation [with O(N)holes being summed over and each Hamiltonian matrix element being O(n)to compute]. Although this is expensive, the generation of sin- gle excitations turns out overall to be a small fraction of the total cost largely because the relatively small number of times such excitations are attempted. c. Double excitations with opposite spin electron-pairs If a double excitation is being attempted, then first a pair of electrons needs to be selected. The first electron, i, should be selected uniformly at random. Following this, the CPF Fj[i∈D]=j ∑ k∈D k≠i⟨ik∣ik⟩×{popp ifi,kopp 1−popp otherwise(A29) should be constructed, where poppis a optimizable biasing factor toward excitations with electrons having opposite spins. The second electron is selected through the inversion of the CPF. If the two selected electrons have opposite spins, then the first hole to be chosen is, by convention, always a βelectron and the second hole always α. This choice is entirely arbitrary, and in some high-spin systems, it may make sense to reverse this selection. Considering all available orbitals of this spin, the CPF F(β) a[i∈D]=a ∑ h∈βD√ ⟨hh∣ii⟩ (A30) is constructed, where iis taken to be the electron from the selected pair withβspin and the hole selected by inverting the CPF. Once this first electron has been chosen, the symmetry of the target orbital is now fixed by the constraint that Γa⊗Γ′=Γi⊗Γj. This greatly restricts the number of holes that must be considered when constructing the final CPF, F(αΓ′a) b[ij∈D]=b ∑ h∈αΓ′D√ ∣⟨ij∣ab⟩∣. (A31) Note that with the conventional choice of orbital iabove,⟨ji|ah⟩= 0, and can thus be excluded. The second hole is then also obtained by inverting the CPF. The generation probability is then given by pgen(ab,ij)=p(ij)p(a∣i)p(b∣ija)pdoub, (A32) p(ij)=1 N⎛ ⎜ ⎝θ(i) j Σi+θ(j) i Σj⎞ ⎟ ⎠, (A33) θ(i) j=Fj[i∈D]−Fj−i[i∈D], (A34) p(a∣i)=θa Σ(β)(i), (A35) θa=F(β) a−F(β) a−1, (A36) p(b∣ija)=θb Σ(αΓ′)(ija), (A37) θb=F(αΓ′a) b−F(αΓ′a) b−1, (A38)where Σi,Σ(β)(i),Σ(αΓ′)(ija) are the normalizations of Fj,F(β) a,F(αΓ′a) b, respectively, and are given by the final entries of the corresponding arrays. The asymmetric selection of αandβholes is somewhat pecu- liar. It should be noted that it is possible to make this selection symmetrically, considering allavailable holes in the selection of the first hole and then renormalizing the probabilities to account for the possibility of selecting bfirst. The symmetric scheme increases computational cost substantially (twice as many holes need to be considered in the CPF, and a further CPF must be calculated for the renormalization). It also makes the overall time-step behavior worse as, although it improves the general smoothness, for the worst-case scenario with a very rarely selected excitation with very different a, band b,aprobabilities, the denominator is increased substantially by considering more orbitals while leaving the numerator essentially unchanged. d. Double excitations with same-spin electron-pairs If the pair of electrons, selected as described above, has the same spin, the process needs to account for the fact that the holes can be selected in either order and the probabilities need to be adjusted to compensate. Now, considering only holes with the same spin as the two electrons, construct the CPF F(σ) a[ij∈D]=a ∑ h∈σD√ ⟨hh∣ii⟩+√ ⟨hh∣jj⟩. (A39) Hole acan then be selected through inversion of this CPF, which fixes the symmetry of hole bsuch that Γa⊗Γb=Γi⊗Γj. The CPF for selecting the second hole can then be constructed from the (much smaller) set of holes with the appropriate symmetry such that F(σΓba) b[ij∈D]=b ∑ h∈αΓbD h≠a√ ∣⟨ij∣ab⟩−⟨ij∣ba⟩∣. (A40) The second hole, b, can then be selected through inversion of this CPF. It is important to note that as the selection of the first hole includes all holes of the hole with the given spin, the selection of the holes could have been made in the reverse order, and this needs to be taken into account in the generation probability, which is given by pgen(ab,ij)=[p(a∣ijb)p(b∣ij)+p(b∣ija)p(a∣ij)]p(ij)pdouble , (A41) where p(ij)=1 N⎛ ⎜ ⎝θ(i) j Σi+θ(j) i Σj⎞ ⎟ ⎠, (A42) θ(i) j=Fj[i∈D]−Fj−1[i∈D], (A43) p(a∣ij)=θa Σa, (A44) θa=F(σ) a−F(σ) a−1, (A45) J. Chem. Phys. 153, 034107 (2020); doi: 10.1063/5.0005754 153, 034107-21 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp p(b∣ija)=θ(a) b Σ(a) b, (A46) θ(a) b=F(σΓba) b−F(σΓba) b−1, (A47) andΣi,Σa,Σ(a) bare the normalizations of the three CPFs, given by their final elements. Note that in the implementation, the normaliza- tions of four CPFs must be calculated to be able to calculate p(a|ijb) and p(b|ija). 3. Pre-computed heat-bath sampling While the Cauchy–Schwartz excitation generator has negli- gible memory cost, picking an excitation requires O(N)steps, each involving Hamiltonian matrix elements, making the proce- dure expensive. The pre-computed heat-bath algorithm employed inNECI is a simple approximation derived from the heat-bath sam- pling32and offers a much faster excitation generation, at the cost of increased memory requirement. The heat-bath probability dis- tribution can also be used to determine a cutoff in a deterministic scheme, leading to the heat-bath CI (HCI) method.31The sampling can either use uniform single excitations or the weighted scheme outlined in Subsection 2 b of the Appendix and approximates the exact heat-bath sampling of double-excitations by uniformly picking the occupied orbitals and then picking two target orbitals simulta- neously weighted with the Hamiltonian matrix element. Since the double excitations play the largest role in excitation generation, and the singles’ matrix elements depend on the determinants in addition to the excitation, it is typically most efficient to generate only double excitations in a weighted fashion, resulting in an excel- lent trade-off between optimal weights and the cost of excitation generation. To create a double excitation using pre-computed heat-bath generation, first, two occupied orbitals i,jare chosen uniformly at random using a bias toward spin-opposite excitations, which is determined similar to the bias toward double excitations. This works analogously to the Cauchy–Schwartz excitation generation outlined in Subsection 2 of the Appendix. Then, a pair a,bof orbitals is chosen using pre-computed weights, p(ab∣ij)=∣Hab ij∣ ∑a′b′∣Ha′b′ ij∣, (A48) where Hab ijis the matrix element for a double excitation from orbitals i,jto orbitals a,b. These are independent of the determinant and thus can be pre-computed at memory cost O(M4). Then, pairs of orbitals can be picked using these weights via alias sampling130in O(1)time. If one of the picked orbitals a,bis occupied, or all matrix elements Hab ijare zero, the excitation is immediately rejected, otherwise, we continue with the FCIQMC scheme. As is desirable to use spatial orbital indices to save memory, but the matrix element depends on the relative spin of the orbitals in the case of a spin-opposite excitation since it determines if an exchange integral is used, for each pair of spatial orbitals i,j, three probabil- ity distributions are generated, one for the spin-parallel case, one for the spin-opposite case without exchange, and one for the spin- opposite case with exchange. Between the latter two, we then choosethe exchange case with probability, pexch(ij)=∑ab∣Haβbα iαjβ∣ ∑ab∣Haβbα iαjβ∣+∣Haαbβ iαjβ∣. (A49) The denominator is the same as the denominator in Eq. (A48) for spin orbitals, while the numerator is the denominator in Eq. (A48) for spatial orbitals in the exchange case. The bias pexch, hence, relates the spatial orbital distributions to the original distribution (A48). This approach is tailored for rapid excitation generation, as the process is, in principle, O(1)while yielding acceptance rates comparable to the on-the-fly Cauchy–Schwartz generation. Due to implementational details of NECI , the uniform selection of elec- trons scales linearly with the number of electrons, which, however, does not constitute a bottleneck in practical applications. The rapid excitation generation has important consequences for the scalabil- ity of the algorithm, since the stochastic nature of the algorithm can give rise to dynamic load imbalance if the time taken for exci- tation generation can vary significantly depending on determinant and electron/orbital selection. DATA AVAILABILITY The data that support the findings of this study are available within the article (and its supplementary material). 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Walker, “An efficient method for generating discrete random variables with general distributions,” ACM Trans. Math. Software 3, 253–256 (1977). J. Chem. Phys. 153, 034107 (2020); doi: 10.1063/5.0005754 153, 034107-25 Published under license by AIP Publishing

5.0016400.pdf

Appl. Phys. Lett. 117, 082404 (2020); https://doi.org/10.1063/5.0016400 117, 082404 © 2020 Author(s).Realization of mutual synchronization of spin torque nano-oscillators under room temperature by noise reduction technique Cite as: Appl. Phys. Lett. 117, 082404 (2020); https://doi.org/10.1063/5.0016400 Submitted: 03 June 2020 . Accepted: 12 August 2020 . Published Online: 25 August 2020 Lang Zeng , Xiaojun Xu , Hao-Hsuan Chen , Yan Zhou , Deming Zhang , Yijiao Wang , Youguang Zhang , and Weisheng Zhao ARTICLES YOU MAY BE INTERESTED IN Enhancement of the spin–orbit torque efficiency in W/Cu/CoFeB heterostructures via interface engineering Applied Physics Letters 117, 082409 (2020); https://doi.org/10.1063/5.0015557 Noise reduction of spin torque oscillator by phase-locked loop with combinational frequency tuning method Applied Physics Letters 117, 072407 (2020); https://doi.org/10.1063/5.0019390 Evolution of strong second-order magnetic anisotropy in Pt/Co/MgO trilayers by post- annealing Applied Physics Letters 117, 082403 (2020); https://doi.org/10.1063/5.0018924Realization of mutual synchronization of spin torque nano-oscillators under room temperature by noise reduction technique Cite as: Appl. Phys. Lett. 117, 082404 (2020); doi: 10.1063/5.0016400 Submitted: 3 June 2020 .Accepted: 12 August 2020 . Published Online: 25 August 2020 Lang Zeng,1,2,3,a) Xiaojun Xu,1,3Hao-Hsuan Chen,1,3Yan Zhou,4 Deming Zhang,3,5Yijiao Wang,1 Youguang Zhang,5and Weisheng Zhao1,2,3,b) AFFILIATIONS 1Fert Beijing Institute and School of Microelectronics, Beihang University, Beijing 100191, China 2Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing 100191, China 3Hefei Innovation Research Institute, Beihang University, Hefei 230013, China 4School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong 518172, China 5School of Electronic and Information Engineering, Beihang University, Beijing 100191, China a)Author to whom correspondence should be addressed: zenglang@buaa.edu.cn b)Electronic mail: weisheng.zhao@buaa.edu.cn ABSTRACT The pragmatic use of Spin Torque Nano-Oscillators (STNOs) in real electronic systems is severely hindered due to their low output power and poor noise figure. The most accepted and promising way to increase the output power and decrease the signal noise of STNOs is throughtheir mutual synchronization. However, it is confused that the mutual synchronization of STNOs is very difficult to achieve at room temper- ature although the non-linear nature of STNOs contributes to the large frequency range of injection locking. In this work, first, it is revealed that the difficulty of STNOs’ mutual synchronization stems from the high output signal noise of STNOs. Based on this observation, a noisereduction technique is invented, which introduces a Band Pass Filter in the coupling loop of STNOs. Using the noise reduction technique, itis demonstrated that even non-identical STNOs can be mutually synchronized at room temperature. Published under license by AIP Publishing. https://doi.org/10.1063/5.0016400 The modern telecommunication system requires stable genera- tion of high quality RF signals on chip. 1Now, it is almost solely depen- dent on CMOS Voltage Controlled Oscillators (VCOs) for such low noise RF signal generation.2,3However, although VCOs are already widely used in practice, the large areal consumption and small fre-quency tuning range maybe a severe issue in further development ofthe next generation telecommunication system. 4,5Among several promising nano-oscillators, Spin Torque Nano-Oscillators (STNOs)stand out due to their small size ( /C24100 nm), large frequency tuning range ( /C24GHz), CMOS compatibility, and mature fabrication process. 6–11Nonetheless, the STNO still suffers from low output power and poor noise figure, which can be mainly attributed to itssmall size. The output power of STNOs can be increased by variousapproaches, such as utilizing the vortex type structure, increase in theactive area, and placing an additional resonator. 10,12,13It is community consensus that mutual synchronization of STNO arrays is the mostefficient and promising approach to increase the output power and decrease the signal noise simultaneously.14–17 The mutual synchronization of STNOs was theoretically investi- gated under several theory frameworks, especially the Nonlinear Auto-Oscillator Theory (NAOT).18–22In this theory, the mutual syn- chronization of STNOs is treated well based on the theoretical deriva-tion of the injection locking phenomenon. The impact of the mutualcoupling strength and intrinsic phase difference on the feasibility oftwo STNOs’ synchronization is discussed, and several experimental results are explained well by this theory. The mutual synchronization of STNOs is also investigated by numerical simulations. Grollier et al. proves that several electrical connected STNOs in series can be syn-chronized with common stimulated microwave currents. 23The delay of the coupling between STNOs, which actually affects the phase dif-ference, plays an important role in the mutual synchronization. 24–26 However, mutual synchronization of STNOs of uniform magnetic Appl. Phys. Lett. 117, 082404 (2020); doi: 10.1063/5.0016400 117, 082404-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apldynamics with thermal noise at room temperature is rarely considered in neither theory nor numerical simulations. The experimental demonstration of mutual synchronization of STNO arrays has shown rapid progress in the past ten years.9In 2016, one-dimensional STNO arrays driven by spin–orbit torque generated by lateral charge currents in a heavy metal layer with a strong spin Hall effect achieve mutual synchronization with the distance as long as 4lm.27Later in 2019, the mutual synchronization extends to two- dimensional 8 /C28 STNO arrays.28However, the structure of the STNO used in the experiment is nano-constriction, which implies non-uniform magnetization dynamics and energy loss. Moreover, the output power increases linearly with the number of mutually synchro-nized nano-constrictions N, not theoretically predicted N 2.17,20,21 Thus, it is inferred that the mutual synchronization in the STNO array is weak and vulnerable. The mutual synchronization of STNOs with uniform magnetization rotation is still lacking. In this work, first, the reason why mutual synchronization of STNOs is so hard is analyzed and asserted. Then, a noise reductiontechnique is proposed and verified. Using such a noise reduction tech- nique, mutual synchronization of non-identical STNOs at room tem- perature is achieved. The STNO used in this work is shown in the inset of Fig. 1(a) .I t is CoFeB/MgO/CoFeB sandwich structure which is almost the same as Magnetic Tunneling Junction (MTJ). 6–11T h ef r e el a y e ro ft h eS T N Oi s perpendicularly polarized, while the reference layer is in-plane magne- tization. We developed a dedicated in-house STNO simulator, which numerically solves the Landau-Lifshitz-Gilbert (LLG) equation under macrospin approximation with careful consideration of thermal noise.29–31The device parameters are extracted from experiment, and reproduction of the experimental measurement is verified.32,33The details of our in-house simulator and the parameters used in the simu- lation can be found in the supplementary material . In our previous work,29,30it is shown that injection locking can be induced with external microwave current or a magnetic field in x- and y-directions. Detailed theoretical and numerical analyses indi- cate the injection locking range with the microwave magnetic field in x- and y-directions is much wider than that with microwave current. It is well known that the wider the injection locking range, the easier the mutual synchronization of STNOs. Therefore, the magnetic fieldin the x-direction is chosen as an exemplary coupling mechanism in this study. The injection locking by the microwave magnetic field in the x-direction is plotted in Fig. 1(a) . The oscillation frequency of STNOs follows the frequency of the external signal from 3.7 GHz to 5.5 GHz when the amplitude of the microwave magnetic field is 150 Oe. With such a small magnetic field, the locking range is as wide as 1.8 GHz. In Fig. 1(b) , the oscillation trajectory of STNO magnetiza- tion and the magnetic field are shown, while both the oscillation fre- quency is 4.6 GHz. It is observed that the intrinsic phase difference is p for phase locking by the magnetic field in the x-direction, and this agrees exactly with our previous theoretical derivation. 29,30 Based on the analysis of injection locking by the magnetic field in the x-direction, a mutual synchronization setup is proposed as shown inFig. 2(a) . Two STNOs are chosen for demonstration although the number of STNOs in our proposed setup has no limitation. The AC part of the STNO output power is extracted by Bias Tee and fed into a spiral coil with an amplifier to generate the microwave magnetic field.In order to achieve strong synchronization, phase coherence should beguaranteed. 14,15,20,24–26Since the intrinsic phase difference is p,a n additional pphase, which is introduced by applying the magnetic field in the negative x-direction, fulfills phase coherence. The amplitude of the magnetic field in the x-direction from the spiral coil is assumed to be 150 Oe. With such a mutual synchronization setup, two non-identical STNOs with 10% variation in the effective anisotropy fieldcan be synchronized when thermal noise at room temperature is nottaken into account. As shown in Fig. 2(b) , the power spectrum of the two free running STNOs is shown as m x3and m x4, while the power spectrum of the two STNOs after synchronization is shown as m x1 and m x2. For the clear presentation of mutual synchronization, the power spectrum in both linear and logarithmical plots is shown. Themutual synchronization is obvious in Fig. 2(b) . However, when ther- mal noise at room temperature is considered, the mutual synchroniza- tion always fails even when the magnetic field increases tounreasonably large magnitude. The reason why mutual synchronization cannot be accomplished at room temperature is not well studied in the literature although such a difficulty is commonly seen in both experiment and simulation. Part FIG. 1. (a) Injection locking caused by the external microwave magnetic field of 150 Oe in the x-direction. The locking range is about 1.8 GHz, which is quite large due to the strong non-linearity of STNOs. (b) The oscillation trajactory of magnetic moment in the x-direction and the external microwave magnetic field in the x-direction. It is obvious that these two signals have a phase difference of p.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 082404 (2020); doi: 10.1063/5.0016400 117, 082404-2 Published under license by AIP Publishingof the reason for this situation is the complexity of analyzing several mutually interacting STNOs. We need to make the scenario easier foranalysis. Since the injection locking is closely related to the mutualsynchronization, we can analyze the injection locking with noiseinstead. Distinct from the typical injection locking, we add different levels of white noise into the pure sine microwave magnetic field in the x-direction. The simulation results are shown in Fig. 3 . InFig. 3(a) , the power spectrum of a free running STNO at room temperature is shown. It is observed that the oscillation frequency iswidely spreading over a frequency range of 2 GHz. The impact of ther- mal noise on STNO oscillation is very strong since our device is small (/C2450 nm). However, when applying a pure sine microwave magnetic field without white noise to the STNO, the injection locking still canhappen even when the STNO suffers from thermal noise. The powerspectrum becomes very narrow in Fig. 3(b) compared to that in Fig. 3(a) , which indicates very good phase locking. As shown in Fig. 3(c) ,when the Signal to Noise Ratio (SNR) of the external microwave mag- netic field decreases to 8 dB with increasing white noise, the injectionlocking again can be observed. The effect of white noise embedded in the magnetic field can be seen in Fig. 3(c) with the decreasing peak at locking frequency and increasing noise at non-locking frequency. Thewhite noise increases further to SNR ¼0d Ba ss h o w ni n Fig. 3(d) .T h i s time the injection locking cannot be sustained, and the spectrum of the STNO is very noisy again and without apparent oscillating frequency, which is similar to Fig. 3(a) . From the above analysis, it is concluded that the mutual synchronization of STNOs is mainly hindered by the large noise of the STNO output signal. Such a conclusion can be understood from another point of view. In the injection locking, the STNO output signal will follow the injectedsignal since the phase difference of these two signals should be constant. When the injected signal has some level of noise, the STNO output sig- n a lw i l la l s oh a v es o m el e v e lo fn o i s eb yt h ep h a s ef o l l o w i n g .Furthermore, the injection locking is broken due to too large noise. It ispostulated that too large noise of the STNO output signal impedes the mutual synchronization, and mutual synchronization may be rebuilt if the noise of the output signal can be reduced. The simplest way toreduce noise is using a filter. Naturally, a mutual synchronization set-up is proposed in Fig. 4(a) , which contains a Band Pass Filter (BPF) in the FIG. 2. (a) The mutual synchronization setup for two STNOs by magnetic field cou- pling. Since the intrinsic phase difference of magnetic field locking is p, the mag- netic field coupling is exerted with a negative sign along the x-direction to introduce an additional pphase for coherence synchronization. (b) Two non-identical STNOs with 10% variation in the effective anisotropy field can be synchronized without con-sideration of thermal noise. The upper half plots the STNO power spectrum linearly, while the lower half plots it logarithmically.FIG. 3. (a) The power spectrum of the free running STNO at room temperature in linear and logarithmic plots, respectively. (b) The power spectrum of STNO injection locked with the external microwave magnetic field in the x-direction whose SNR¼1 at room temperature. (c) The power spectrum of STNO injection locked with the external microwave magnetic field in the x-direction whose SNR ¼8d B a t room temperature. (d) The power spectrum of the STNO is not injection locked with the external microwave magnetic field in the x-direction whose SNR ¼0d B a t room temperature.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 082404 (2020); doi: 10.1063/5.0016400 117, 082404-3 Published under license by AIP Publishingcoupling loop. The amplitude-frequency response of the digital BPF uti- lized in our macromagnetic simulation is plotted in Fig. 4(b) .I ti sw o r t h noting that the amplitude-frequency response of the digital BPF is verysimilar to that of Surface Acoustic Wave (SAW) filter, which is com-monly used in modern mobile devices. 34Integration of such a BPF with STNOs maybe difficult and needs further investigation. The simulation results for the mutual synchronization set-up with the noise reduction technique are shown in Fig. 5 .I nFigs. 5(a) and5(b), the power spectrum of two identical STNOs mutually cou- pled without and with the noise reduction technique is plotted, whilethe spectrum of two non-identical STNOs with 10% variation in theeffective anisotropy field mutually coupled without and with the noisereduction technique is plotted in Figs. 5(c) and5(d). Three important observations are (1) Without the noise reduction technique, twoidentical STNOs cannot be synchronized. Nevertheless, even two non-identical STNOs can be synchronized with the noise reduction tech- nique. (2) The power spectrum peak is more than 10 times higher when STNOs’ mutual synchronization is achieved with the noisereduction technique compared to that of the non-synchronizedsituation. The power increases even more than N 2times, where Nis the number of synchronized STNOs. This indicates that the mutualsynchronization is so well and strong beyond the theoretical predic-tion 19–21and is contrast to the weak synchronization observed in the STNO array.27,28(3) The total output power of the STNO can be cal- culated as Ptot¼ð1 0pðfÞdf; (1) where p(f)i st h ep o w e rs p e c t r u ma ss h o w ni n Fig. 5 . As we compute, the total power of mutually synchronized STNOs is even 10% largerthan that of the non-synchronized situation. The introduction of BPFdoes not make any energy loss. The effect of the performance of BPF on the STNO mutual syn- chronization is further investigated in detail and shown in Fig. 6(a) . For better judgment of how good the mutual synchronization isachieved, we proposed a Figure of Merit (FoM) defined as R occupied ¼ðf0þ5M f0/C05MpðfÞdf=Ptot; (2) FIG. 4. (a) The mutual synchronization setup for two STNOs by magnetic field cou- pling with a Band Pass Filter (BPF) in the coupling loop for noise reduction. (b) Theamplitude-frequency response of the digital BPF used in our macro-magnetic simulation.FIG. 5. (a) The power spectrum of two identical STNOs by magnetic field coupling without a BPF at room temperature. (b) The power spectrum of two identicalSTNOs by magnetic field coupling with a BPF at room temperature. (c) The powerspectrum of two non-identical STNOs by magnetic field coupling without a BPF at room temperature. (d) The power spectrum of two non-identical STNOs by mag- netic field coupling with a BPF at room temperature.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 082404 (2020); doi: 10.1063/5.0016400 117, 082404-4 Published under license by AIP Publishingwhere f0is the frequency of the centric highest peak in the power spec- trum and the output power in a very narrow frequency range of10 MHz is normalized by P totas the FoM Roccupied . The larger the Roccupied , the better the mutual synchronization is. As we observed, mutual synchronization can be achieved when Roccupied >10%. From Fig. 6(a) , it is shown that for bandwidths of 200 and 400 MHz, /C020 dB out-of-band suppression can already lead to mutual synchronization,while for bandwidths of 800 and 1000 MHz, /C080 dB out-of-band sup- pression is not enough for synchronization. The out-of-band suppres- sion should be smaller than /C060 dB for the bandwidth of 600 MHz. It is concluded that the effect of the bandwidth is more profound thanthat of out-of-band suppression. The effect of magnetic field amplitudeis also investigated. As shown in Fig. 6(b) , the mutual synchronization can be achieved when the amplitude is larger than 100 Oe with theBPF in Fig. 4(b) . In this work, the main obstacles for STNOs’ mutual synchroniza- tion are revealed by noisy injection locking. The noise reduction tech-nique is proposed for the facilitation of mutual synchronization ofSTNOs at room temperature. BPF is chosen as an example of the noisereduction technique, and mutual synchronization of two non-identical STNOs at room temperature is demonstrated. For the full integration of STNOs’ synchronized arrays on a chip, other noise reduction tech- niques are worth investigating, especially the spin wave filter made of magnonic crystals. See the supplementary material for a detailed description of our in-house STNO simulator and digital Band Pass Filter. The authors wish to acknowledge the support from the National Key R&D Program of China (No. 2018YFB0407602), the International mobility project under Grant No. B16001, the National Key Technology Program of China under Grant No. 2017ZX01032101, the National Postdoctoral Program for Innovation Talents under Grant No. BX20180028, and the China Postdoctoral Science Foundation funded project under Grant No. 2018M641153. Yan Zhouacknowledges support from the National Natural Science Foundation of China (Grant Nos. 11974298 and 61961136006), the Shenzhen Fundamental Research Fund (Grant No. JCYJ20170410171958839), and the Shenzhen Science and Technology Program (Grant No. KQTD20180413181702403). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1B. Razavi and R. Behzad, RF Microelectronics (Prentice Hall, New Jersey, 1998), Vol. 2. 2N. M. Nguyen and R. G. Meyer, IEEE J. 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5.0011411.pdf

AIP Advances 10, 095113 (2020); https://doi.org/10.1063/5.0011411 10, 095113 © 2020 Author(s).The in-plane magnetic anisotropy in manganite film: A novel magnetic anisotropy induced by structural anisotropy Cite as: AIP Advances 10, 095113 (2020); https://doi.org/10.1063/5.0011411 Submitted: 21 April 2020 . Accepted: 23 August 2020 . Published Online: 08 September 2020 Haiou Wang , and Weishi Tan 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 Manipulation of acoustic localizations based on defect mode coupling in a corrugated waveguide AIP Advances 10, 095109 (2020); https://doi.org/10.1063/5.0019744 Influence of copper damper on bifurcation vibration in superconductive magnetic levitation system AIP Advances 10, 095111 (2020); https://doi.org/10.1063/5.0020898 Development of x-ray phase tomographic microscope based on Talbot interferometer at BL37XU, SPring-8 AIP Advances 10, 095115 (2020); https://doi.org/10.1063/5.0016318AIP Advances ARTICLE scitation.org/journal/adv The in-plane magnetic anisotropy in manganite film: A novel magnetic anisotropy induced by structural anisotropy Cite as: AIP Advances 10, 095113 (2020); doi: 10.1063/5.0011411 Submitted: 21 April 2020 •Accepted: 23 August 2020 • Published Online: 8 September 2020 Haiou Wang1,a) and Weishi Tan2,3 AFFILIATIONS 1Institute of Material Physics, Hangzhou Dianzi University, Hangzhou 310018, China 2All-Solid-State Energy Storage Materials and Devices Key Laboratory of Hunan Province, College of Information and Electronic Engineering, Hunan City University, Yiyang 413002, China 3Key Laboratory of Soft Chemistry and Functional Materials, Ministry of Education, Department of Applied Physics, Nanjing University of Science and Technology, Nanjing 210094, China a)Author to whom correspondence should be addressed: wanghaiou@hdu.edu.cn ABSTRACT Magnetic anisotropy plays an important role in the development of manganite-based magnetic devices. Establishing a deeper understanding of the anisotropy in manganites is useful for controlling their magnetic properties. We have studied the structure, lattice strain, and magnetic properties of La 0.5Ba0.5MnO 3(LBMO-5) films with the thickness of 12 nm–96 nm, grown on the (001) SrTiO 3(STO) substrate. The LBMO-5 films are grown with high crystalline quality. The orientation relationship between the LBMO-5 film and the STO substrate [(001) f//(001) s, (010) f//(010) s, and (100) f//(100) s] exists at the film/substrate interface. With increasing the thickness of the film to 48 nm, the LBMO-5 film is fully strain relaxed. The LBMO-5 film shows the out-of-plane magnetic anisotropy (OMA) along the three mutually perpendicular crystalline axis directions. Usually, the manganite films with tetragonal distortion show both the OMA and the in-plane magnetic isotropy. However, the in-plane magnetic anisotropy (IMA) along the two mutually perpendicular in-plane directions ([010] fand [100] f) is clearly observed in LBMO-5 films. We attribute this new observation of the IMA to the in-plane structural anisotropy along the two distinct crystalline axes. Our studies provide helpful guidance for the understanding and the tuning of the IMA. ©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/10.1063/5.0011411 .,s I. INTRODUCTION Perovskite manganites have attracted great interest since their potential applications in spintronic devices such as magnetic memory and sensor devices.1–3Controlling magnetic properties including magnetic anisotropy and domain structure is very impor- tant for the realization of spintronic devices.4Usually, mag- netic anisotropy can affect the magnetization curve and coercivity of perovskite manganite films.5,6The types and origin of magnetic anisotropy are abundant and worth investigating. Besides magnetocrystalline anisotropy, lattice strain,7,8lattice defects at the interface,9and geometric anisotropy can also influence the magnetic properties of manganite films. Moreover, manganitefilms with different orientations possess the anisotropic magnetic behaviors.10 Perovskite manganite La 0.5Ba0.5MnO 3(LBMO-5) has a cubic (space group Pm-3m) crystal structure.11,12The LBMO-5 compound is a multiphase ferromagnetic (FM) insulator,13and it exhibits fer- romagnetism below its Curie temperature near 300 K.14Moreover, magnetic anisotropy and the magnetoresistance (MR) effect have been found in LBMO-5. The magnetic properties of LBMO-5 films can be effectively influenced by controlling the external field, strain, and lattice distortion.14–16In LBMO-5, the lattice, spin, charge, and orbital degrees of freedom are coupled to one another. In views of these physical properties mentioned above, LBMO-5 has been chosen to be an ideal material for studies. AIP Advances 10, 095113 (2020); doi: 10.1063/5.0011411 10, 095113-1 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv Many reports on magnetic anisotropy show that mag- netic anisotropy includes magnetocrystalline anisotropy,17–19strain anisotropy,7,8and exchange coupling anisotropy.20In these reports, magnetic anisotropy has been studied in detail. However, to the best of our knowledge, these studies only consider the imposition of the magnetic field, strain, and the coupling effect between FM and antiferromagnetic (AFM) phases. A systemic study on the in-plane structure (e.g., in-plane strain, in-plane orientation, and in-plane crystalline quality) dependence of magnetic anisotropy remains lacking. Specifically, few studies have focused on the link between the in-plane structure and magnetic properties. In this work, the out-of-plane and in-plane lattice parame- ters are measured, and then, lattice strain is investigated. Usually, the manganite films with tetragonal distortion show both the out- of-plane magnetic anisotropy (OMA) and the in-plane magnetic isotropy. However, the LBMO-5 film exhibits not only the OMA but also the in-plane magnetic anisotropy (IMA) along [010] fand [100] f directions. We attribute the observation of the IMA to the in-plane structural anisotropy. Because the origin of the IMA is not consistent with the previous reports,7,8,17–20it is a novel magnetic anisotropy. The effect of the in-plane structural anisotropy on magnetic prop- erties is interesting and deserves studies. It is useful for developing in-plane magnetic sensor devices. II. EXPERIMENTAL DETAILS LBMO-5 films were fabricated on (001) SrTiO 3(STO) sub- strates by the pulsed laser deposition (PLD) of the bulk LBMO-5 target with details described in the previous report.21The LBMO- 5 target material possessed a cubic unit cell (space group Pm-3m) with the lattice parameter of 0.3927 nm, which could be confirmed by the results of the Rietveld refinement of x-ray diffraction (XRD) (not shown here). The STO substrate possessed a cubic structure (space group Pm-3m) with the lattice parameter of 0.3905 nm.The thickness of LBMO-5 films ( dLBMO-5 ) changed from 12 nm to 96 nm, which was controlled by the deposition time. X-ray diffrac- tion (XRD), grazing incidence x-ray diffraction (GIXRD), and graz- ing incidence x-ray reflectivity (GIXRR) were applied to characterize the structure. In order to observe clear and distinct diffraction peaks, these experiments were carried out at the Synchrotron Radiation Facil- ity with high resolution. The high-resolution x-ray diffraction (HRXRD), GIXRD, and GIXRR data were collected on the X-ray Diffuse Scattering Station at the Beijing Synchrotron Radiation Facility (BSRF). The wavelength of x-rays was 0.153 684 nm, and the energy resolution was 4.4 ×10−4. The surface morphology was studied by atomic force microscopy. The crystalline structure (orien- tation) dependence of magnetic properties was measured in a Phys- ical Property Measurement System (PPMS) with a vibrating sample magnetometer (VSM). The IMA induced by structural differences was characterized by the temperature dependence of magnetization (M–T curves) and the field dependence of magnetization (M–H loops). III. RESULTS AND DISCUSSION Figures 1(a) and 1(b) depict the HRXRD of LBMO-5 films with the thickness of 12 nm–96 nm. Figure 1(b) shows an enlarged draw- ing of partial XRD shown in Fig. 1(a). Besides the (002) peaks of the film and the substrate, the interference peaks of films can be observed clearly. The LBMO-5 films are well grown along (001) ori- entation. The out-of-plane lattice parameter can be calculated by the position of reflection peaks of LBMO-5 films. Figure 1(c) shows the thickness dependence of out-of-plane lattice parameters of LBMO-5 films.21AtdLBMO-5 = 12 nm, the out- of-plane lattice parameter ( c) has changed from the lattice parameter of STO ( cSTO= 0.3905 nm) to 0.3907 nm, indicating that the out-of- plane lattice strain begins to be relaxed. Here, the thickness value FIG. 1 . (a) High resolution XRD θ-2θscans of the (002) plane for LBMO-5 films with various thicknesses of 12 nm– 96 nm on the STO substrate. (b) The enlarged drawing of partial XRD. (c) The thickness dependence of the out-of- plane lattice parameter of LBMO-5 films. The lattice param- eters of cubic LBMO-5 bulk and cubic STO substrate are also presented with dashed lines.21(d) High resolution XRD θ-2θscan of the (002) plane for the LBMO-5 film with the thickness of 48 nm on the STO substrate. In the inset of (d), the high resolution XRD φ-scan patterns for (103) planes of the LBMO-5 (48 nm) film and STO substrate starting from the same φazimuth are shown. AIP Advances 10, 095113 (2020); doi: 10.1063/5.0011411 10, 095113-2 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv (12 nm) where strain starts to relax is consistent with Matthews– Blakeslee critical thickness theory.22With increasing dLBMO-5 to 48 nm, the out-of-plane lattice parameter of the LBMO-5 film (cLBMO-5 film ) is near to that of the bulk target ( cLBMO-5 bulk ), sug- gesting that the out-of-plane lattice strain has been fully relaxed. Figure 1(d) shows the HRXRD of the LBMO-5 (48 nm) film on STO around the (002) diffraction plane. Besides the (002) peaks of the LBMO-5 film and STO substrate, one can observe clearly the interference peaks of LBMO-5 films. By the spacing of interference peaks marked by the arrows, the actual dLBMO-5 is determined to be about 48.6 nm, which is in accord with the nominal thickness of 48 nm. The actual thickness can be calculated by the formula21 T=λ/(2ΔαcosθB), where Δαis the interference fringe spacing in angle space and θBis the Bragg angle. The crystalline quality of the LBMO-5 film is investigated by the full width at half maximum (FWHM) of the on axis rocking curves. The FWHM of the (002) peak of the LBMO-5 (48 nm) film is about 0.06○, which is compa- rable to that of the MgO substrate (the FWHM of the substrate is 0.04○). This indicates the high crystalline quality of LBMO-5 films. The inset of Fig. 1(d) shows the XRD φ-scan patterns for (103) planes of the LBMO-5 film and STO substrate. The fourfold symmetry can be observed in the LBMO-5/STO. The diffraction peaks of (103) planes of LBMO-5 and STO appear at the same φangle, confirm- ing the in-plane orientation relationship of LBMO-5 [100]//STO [100]. All the above results of HRXRD in θ-2θand φ-scan modes [see Figs. 1(a)–1(d)] confirm the high crystalline quality of LBMO-5 films. GIXRD patterns of (200) and (020) planes for LBMO-5 (48 nm) films are shown in Figs. 2(a) and 2(b), respectively.21The rela- tionship between the penetration depth of x-rays and the graz- ing incidence angle ( φGI) is reported in a related report.23Withincreasing φGI, the penetration depth is increased, and therefore, the peak intensity of STO increases gradually; meanwhile, the relative intensity of peak I reduced gradually. With φGIlarger than 0.3○, peak I begins to disappear in Figs. 2(a) and 2(b). The peak I results from the structural changes in the depth of 7.1 nm distance near the sur- face of the film. In Fig. 2(b), two peaks appeared at the low angle side of the STO (020) peak. With increasing φGI, the left peak among the two peaks slightly shifts to a higher angle and the relative diffrac- tion intensity weakens. The change in the position and intensity of diffraction peaks demonstrates that the left peak develops from the LBMO-5 film. The right peak is named as peak II. With increasing φGIto 0.25○, the peak II begins to appear. This suggests that the peak II is related to the structural change at the LBMO-5/STO interface. According to some reports on the interface of oxide films,20,24–26 peak II develops from the LBMO-5 sublayer with a lower electron density at the LBMO-5/STO interface. Remarkably, the crystalline quality of LBMO-5 films in two mutually perpendicular in-plane directions is completely different, as shown in Figs. 2(a) and 2(b). In addition, according to Figs. 1, 2(a), and 2(b), we find the orien- tation relationship between the film and substrate: [(001) f//(001) s, (010) f//(010) s, and (100) f//(100) s]. The in-plane lattice parameter of the LBMO-5 film at different depths can be calculated based on the peak positions of GIXRD in Fig. 2(b). Figure 2(c) shows the in-plane lattice parameter of the LBMO-5 (48 nm) film on STO.21 The in-plane lattice parameter increases with the increase in dis- tance apart from the LBMO-5/STO interface. The distance [abscissa of Fig. 2(c)] actually equals to the film’s thickness during deposi- tion. With increasing the distance to 43 nm, the in-plane lattice parameter reaches a maximum saturation, indicating that the in- plane lattice strain is fully relaxed. Moreover, the out-of-plane lattice strain is fully relaxed with increasing dLBMO-5 to 48 nm [see Fig. 1(c)]. FIG. 2 . GIXRD patterns with different grazing incidence angles of the (a) (200) and (b) (020) plane for LBMO-5 (48 nm) films on the STO substrate.21(c) In- plane lattice parameter of the LBMO-5 (48 nm)/STO film vs the distance apart from the film/substrate interface. The dis- tance (abscissa) in fact equals to the film thickness during the deposition of LBMO-5 on the STO substrate.21(d) An atomic force microscopy image of the LBMO-5/STO film with dLBMO-5 to be 48 nm. The gray scale range is from 0μm to 2 μm. AIP Advances 10, 095113 (2020); doi: 10.1063/5.0011411 10, 095113-3 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv Therefore, the 48-nm-thick LBMO-5 film should be fully strain relaxed. Note that due to the full relaxation of strain, the 48-nm- thick film is an ideal choice for studying strain relaxation and the in-plane structure. Figure 2(d) shows the atomic force microscopy measurement of the LBMO-5 (48 nm) film on STO. The gray scale range is from 0 μm to 2 μm. One can find a smooth surface with a root mean square ( rms) roughness of 1.026 nm. The LBMO-5 film surface has steps, which are a clear imprint of the STO surface. The film shows this atomically flat stepped surface, with only unit cell high steps. The result of the GIXRR of the LBMO-5 (48 nm) film on STO shows that there is a surface layer (about 7 nm) in LBMO-5 (48 nm)/STO, which is consistent with the result of GIXRD shown in Fig. 2(b). Moreover, the result of the GIXRR shows a smooth sur- face ( rmssurface roughness is 1.03 nm), which is consistent with the result of atomic force microscopy shown in Fig. 2(d). The GIXRR is described in the supplementary material. The in-plane structural anisotropy of the LBMO-5 film [see Figs. 2(a) and 2(b)] needs to be discussed. The STO sub- strate is cubic; then, there needs to be something driving the difference between [010] and [100] structural directions. The atomic force microscopy in Fig. 2(d) shows a regular step- terrace structure on the LBMO-5 surface, which is a replica- tion of the surface of the substrate. The anisotropy in [010] and [100] structural directions could be related with substrate ter- race steps. Substrate step-induced anisotropy has been reported by Mathews et al.27 Figure 3 shows the magnetic hysteresis loops at 5 K for the LBMO-5 (48 nm) film on STO. The magnetization vs field ( M–H ) loop is measured by applying the field in three mutually perpendic- ular directions (H//[001] s, H//[010] s, and H//[100] s). The schematic of the film –substrate direction arrangement is shown in the inset of Fig. 3(a). It shows the diversity in magnetic measurements along the three different directions. The IMA and OMA coexist in LBMO-5 films. Moreover, the value of saturation magnetization ( Ms) in the three magnetization ([001] s, [100] s, and [010] s) directions is almost equal. At a low temperature of 5 K, the coercive field ( HC) in three mutually perpendicular directions is different, as shown in Fig. 3(b). The anisotropy is obvious. The film exhibits a small HC∼300 Oe in the H//[010] sdirection. HCincrease slightly to ∼480 Oe in the H//[001] sdirection. In the H//[100] sdirection, the LBMO-5 film shows a large HC∼600 Oe. The origin of the magnetic anisotropy needs to be investigated in detail. Comparing to the in-plane ([100] sand [010] s) direction, the magnetic moment along the [001] sdirection exhibits a different magnetization, which can be called OMA. This kind of anisotropy of magnetization is mainly attributed to the crystallographic align- ment of the LBMO-5 film. The OMA induced by the crystallographic alignment has been reported in some similar systems.17–19More- over, the MnO 6octahedron tilt in LBMO-5 films on STO can induce tetragonal distortion in LBMO-5, which has been confirmed in our previous report.21Usually, the IMA (in H//[010] sand H//[100] s directions) should not be observed in the tetragonal-distortion man- ganite film. However, we find the IMA in the tetragonal-distortion LBMO-5 film. It is very interesting. Here, this anisotropy has noth- ing to do with the crystalline alignment. Furthermore, because the lattice strain in the LBMO-5 (48 nm) film is fully relaxed [see Figs. 1(c) and 2(c)], the anisotropy cannot be ascribed to strain and strain anisotropy. In views of the observation of the in-plane FIG. 3 . (a) MvsHloops at a low temperature of 5 K for the LBMO-5 (48 nm) film. Measurements were carried out by applying the external field parallel to [001] S, [010] S, and [100] S. The schematic of the film –substrate direction arrangement is shown as an inset of (a). (b) Low-field magnetization behaviors. structural anisotropy [see Figs. 2(a) and 2(b)], we conclude that the in-plane structural anisotropy leads to the IMA along the two mutually perpendicular in-plane ([010] fand [100] f) directions. Figure 4 shows the M–H loops at 5 K for LBMO-5 films with the thickness of 60 nm. Magnetization is measured by applying a field in three mutually perpendicular directions (H//[001]s, H//[010]s, and H//[100]s). The low-field magnetization behavior is shown in the inset of Fig. 4. It shows the diversity in magnetic measurements FIG. 4 .MvsHloops at a low temperature of 5 K for the LBMO-5 (60 nm) film. Measurements were carried out by applying the external field parallel to [001] S, [010] S, and [100] S. The inset shows low-field magnetization behaviors. AIP Advances 10, 095113 (2020); doi: 10.1063/5.0011411 10, 095113-4 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv along the three different directions. The film with a thickness of 60 nm (higher than 48 nm) also owns both the IMA and OMA. This further demonstrates that the 48-nm-thick film is an ideal choice for studying magnetic anisotropy. Figures 5(a)–5(c) show the zero field-cooling (ZFC) and field- cooling (FC) in 500 Oe field magnetization vs temperature ( M–T ) curves for LBMO-5 (48 nm) films, measured in H//[001] s, H//[100] s, and H//[010] sdirections, respectively. Magnetic moment increases with decreasing the temperature from 400 K. The LBMO-5 film exhibits a paramagnetic (PM) to FM transition at Curie temperature TCnear 300 K in different directions. At low temperatures, the ZFC and FC M–T curves exhibit normal FM behaviors. Figures 5(d)–5(f) show the temperature derivative of magnetization (d M/dT) vs tem- perature curves, measured in H//[001] s, H//[100] s, and H//[010] s directions, respectively. TCcan be obtained in the plot of the tem- perature derivative of magnetization (d M/dT) vs temperature, as shown in Figs. 5(d)–5(f). At low temperatures, FC and ZFC M–T curves separate and then converge at around 250 K–260 K for differ- ent directions. The bifurcation of ZFC and FC M–T curves (freezing temperature T fmarked by the arrow) is usually considered as the typical signature of the spin glass state due to the magnetic frus- tration between the FM double exchange and AFM super-exchange interactions.28,29 The temperature dependence of saturation magnetization Ms and coercive field HCis shown in Figs. 6(a)–6(c). MsandHCshow an increasing trend with the decrease in temperature from 300 K to 5 K. The value of Msin the three magnetization directions is almost equal. Moreover, at 5 K, HCshows a value of 480 Oe in the out-of-plane (H//[001] s) direction [see Fig. 6(a)], a value of 300 Oe in the in-plane (H//[010] s) direction [see Fig. 6(b)], and a huge increase, reaching a value of 600 Oe in another in-plane (H//[010] s) direction [see Fig. 6(c)]. The sudden, huge increase in HCin the in-plane (H//[100] s) direction is related to the variation of the in- plane structure in the LBMO-5 film, as shown in Figs. 2(a) and 2(b). FIG. 5 .MvsTcurves after ZFC and FC in the 500 Oe field process for LBMO-5 (48 nm) films. Measurements were carried out by applying the applied field parallel to (a) H//[001] S, (b) H//[100] S, and (c) H//[010] Sdirections. During the magnetic measurements, the applied field is fixed as 500 Oe. The d M/dTvsTcurves are measured in (d) H//[001] S, (e) H//[100] S, and (f) H//[010] Sdirections. FIG. 6 . Temperature dependence of HCandMSof LBMO-5 (48 nm) films. Magnetic characterizations are measured in three mutually perpendicular orientations: (a) H//[001] S, (b) H//[010] S, and (c) H//[100] S. Generally, magnetic anisotropy in the film is decided by the balance between the magnetocrystalline energy and both magnetostriction and magneto-static energies.30Here, the observation of the IMA is very interesting and important for manganite-based film devices, and our studies suggest that the in-plane crystallization can control the IMA of the film. IV. CONCLUSIONS High-quality LBMO-5 films have been grown on (001) STO substrates by PLD. With increasing dLBMO-5 to 48 nm, the LBMO- 5 film is fully strain relaxed. Remarkably, magnetic anisotropy has been observed in the LBMO-5 film, especially the observation of the IMA. The origin of the IMA is ascribed to the in-plane struc- tural variation of LBMO-5 films. Our research findings offer a deeper understanding of magnetic anisotropy, as well as a potential method of tuning the anisotropy. AIP Advances 10, 095113 (2020); doi: 10.1063/5.0011411 10, 095113-5 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv SUPPLEMENTARY MATERIAL See the supplementary material for additional structures, GIXRD experimental details, and GIXRR patterns. ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Grant Nos. 11604067 and U1832143). The authors would like to thank colleagues from the Beijing Synchrotron Radiation Facility (BSRF) and Shanghai Synchrotron Radiation Facility (SSRF) for their help in XRD experiments. The author (H.W.) would like to thank his wife (Yashuang Gu) for her construc- tive suggestions in terms of graph presentation. 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 (1994). 2J.-H. Park, E. Vescovo, H.-J. Kim, C. Kwon, R. Ramesh, and T. Venkatesan, Nature 392, 794 (1998). 3E. Dagotto, Science 309, 257 (2005). 4A.-M. Haghiri-Gosnet and J.-P. Renard, J. Phys. D: Appl. Phys. 36, R127 (2003). 5L. You, B. Wang, X. Zou, Z. S. Lim, Y. Zhou, H. Ding, L. Chen, and J. Wang, Phys. Rev. B 88, 184426 (2013). 6L. You, C. Lu, P. Yang, G. Han, T. Wu, U. Luders, W. Prellier, K. Yao, L. Chen, and J. Wang, Adv. 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J. Vac. Sci. Technol. A 38, 061201 (2020); https://doi.org/10.1116/6.0000465 38, 061201 © 2020 Author(s).Introductory guide to the application of XPS to epitaxial films and heterostructures Cite as: J. Vac. Sci. Technol. A 38, 061201 (2020); https://doi.org/10.1116/6.0000465 Submitted: 17 July 2020 . Accepted: 08 September 2020 . Published Online: 01 October 2020 Scott A. Chambers , Le Wang , and Donald R. Baer COLLECTIONS Paper published as part of the special topic on Special Topic Collection: Reproducibility Challenges and Solutions This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN Experimental determination of electron attenuation lengths in complex materials by means of epitaxial film growth: Advantages and challenges Journal of Vacuum Science & Technology A 38, 043409 (2020); https:// doi.org/10.1116/6.0000291 Practical guides for x-ray photoelectron spectroscopy: Quantitative XPS Journal of Vacuum Science & Technology A 38, 041201 (2020); https://doi.org/10.1116/1.5141395 Atomic layer deposition of ruthenium using an ABC-type process: Role of oxygen exposure during nucleation Journal of Vacuum Science & Technology A 38, 062402 (2020); https:// doi.org/10.1116/6.0000434Introductory guide to the application of XPS to epitaxial films and heterostructures Cite as: J. Vac. Sci. Technol. A 38, 061201 (2020); doi: 10.1116/6.0000465 View Online Export Citation CrossMar k Submitted: 17 July 2020 · Accepted: 8 September 2020 · Published Online: 1 October 2020 Scott A. Chambers,1,a) Le Wang,1 and Donald R. Baer2 AFFILIATIONS 1Physical and Computational Sciences Directorate, Pacific Northwest National Laboratory, Richland, Washington 99352, USA 2Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, Richland, Washington 99352, USA Note: This paper is part of the Special Topic Collection on Reproducibility Challenges and Solutions. a)Electronic mail: Scott.chambers@pnnl.gov ABSTRACT XPS is an important characterization method for epitaxial films and heterostructures. Although standard approaches for XPS data collection and analysis provide useful information such as average composition and the presence of contaminants, more in-depth analyses provideinformation about the film structure, surface termination, built-in electric potentials, and band offsets. The high degree of structural order in these materials enables such information to be extracted from spectral data but also adds complications to the analysis. This guide high- lights three topics of importance in this field: (i) the impacts of crystallinity on XPS signals and quantification, (ii) the unexpected spectralline shapes that can occur in unusual or novel materials, and (iii) the ability of XPS to yield information about built-in potentials and bandoffsets. Concepts are demonstrated using complex oxide heterostructures. Although these topics are highly relevant to epitaxial films andheterostructures, they also apply to single crystals of complex materials. Published under license by AVS. https://doi.org/10.1116/6.0000465 I. INTRODUCTION Epitaxial films and heterostructures are important in many areas of technology and are the subject of widespread research activity. X-ray photoelectron spectroscopy (XPS) is an important tool for obtaining information about these films and is especiallypowerful when used as an in situ probe. In this context, in situ means that the XPS capability is in the same ultrahigh vacuum environment as the deposition tool but not necessarily in the samechamber. While the most definitive studies done to date involve in situ application of XPS, much useful information can also be gleaned by using XPS as an ex situ probe. Many important uses of XPS for analysis of epitaxial thin films are common to those of other materials systems as well. These include determining the chemical states, identifying the pres- ence of impurities on or near the surface, and measuring layerthicknesses. However, the epitaxial and very thin nature of many of these films, along with the presence of one or more solid/solid interface, impacts the nature of the XPS signals and limits the use-fulness of some of the standard analysis approaches, which usually assume a homogenous material within the analysis depth and donot consider diffraction effects. 1,2Nevertheless, the ways in which the detailed nature of these films impact the signals can be turnedaround to access the richness in XPS data that are generally ignoredduring routine measurements and analyses. These include direct measurement of the built-in electrostatic potentials and band offsets that result from the heterojunction formation during filmgrowth, as well as the determination of local structural environ-ments of specific elements by means of x-ray photoelectrondiffraction. This article is part of a collection of guides and tutorials intended to provide a basic understanding of important topicsinvolving the application of XPS. 3The intent of this guide is to highlight some of the ways in which the properties of heterostruc-tures influence XPS spectra along with the challenges they present to the analysis. Also highlighted are the opportunities these chal- lenges provide for gaining insights into these structures that areoften missed when using standard analysis methods. Theoreticalunderstanding and computer models of the complex physics ofphotoemission and photoelectron propagation in solids enable XPS to provide a level of information of which many practitioners are unaware. Although extracting the highest level of information mayTUTORIAL avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000465 38,061201-1 Published under license by A VS.require experimental care and detailed modeling, knowledge of the effects, how they impact the data, and what information might be obtained are useful for all XPS analysts. More details will be foundin the referenced literature and in a chapter on XPS in the bookMaterials Characterization Methods of Epitaxial Films andHeterostructures , which examines these topics in much greater detail than attempted in this short guide. 4 In this guide, we describe how the nature of epitaxial thin films and heterostructures impact XPS spectra and their interpreta-tion in three areas: (i) signal intensity and compositional analysis,(ii) spectral shape and measured binding energies, and (iii) the importance of built-in potentials at interfaces. Although the topics discussed are highly relevant to epitaxial films and heterostructures,aspects of the discussion are also important for XPS of single crys-tals, advanced complex materials, and other types of thin films. II. SIGNAL INTENSITY AND COMPOSITIONAL ANALYSIS The standard equations for quantification of material compo- sition using XPS effectively assume that the elemental distributionis uniform over the analysis volume. Thus, in effect, the atomnumber density for a given element is assumed to be constant over the probe depth. If a structure is known to consist of a single film on a substrate with an interface at a depth less than the XPS probedepth, other analysis equations may be used as summarized byShard. 2However, neither approach is accurate for multilayer epitax- ial heterostructures, which may have many layers within the XPS sampling depth as well as electron scattering effects related to the highly order structures.5Thus, there are two important ways in which the constant density and the thin overlayer assumptions failfor epitaxial films, both related to specific impacts of the structuralforms and elemental distributions therein. Section II A describes the effect of crystallographic order, which introduces orientation effects on signal intensities. Section II B summarizes an approach to determining the composition of complex materials, which avoidsthe use of sensitivity factors and deals with epitaxial samples for which multiple layers my exist within the XPS probe depth. The XPS probe depth is typically defined as some multiple of the electron attenuation length ( λ) for the peak in question. In the simplest model that takes into account only isotropic inelastic scat-tering, the intensity falls off as exp( −t/λcosθ), where tis the film thickness and θis the electron exit angle relative to the surface normal. Since 99% of the intensity originates over a depth of 4.6 λ, and most XPS systems with monochromatic x-ray sources are sen-sitive to the ∼1% level, 4.6 λis a useful working definition of the probe depth. However, for spectrometers without x-ray monochro- mators which thus exhibit inherently higher background levels, 3 λ, which encompasses 95% of the total intensity, may be moreappropriate. The electron attenuation length and, therefore, the probe depth vary with electron kinetic energy ( E k)a sEn kwhere nis mate- rial specific but typically ranges from ∼0.7 to ∼0.8 for most inor- ganic materials and Ekvalues in excess of a few hundred eV. Attenuation lengths can be estimated from fundamental materialproperties using databases generated and maintained by NIST. 6,7 More accurate values for a particular material can be measured by depositing epitaxial films of that material with different thicknesson a substrate not containing the elements in the film, as described in detail elsewhere.8 To provide some sense of the analysis depth and relation to layered structures, the ordered layered structure of SrTiO 3(STO) is shown in Fig. 1 . The spacing between the Ti and Sr layers is roughly 0.2 nm (or a unit cell of ≈0.4 nm), and the electron attenu- ation length λfor the Ti 2p electron generated using an Al kαx-ray source is roughly 1.5 nm.8Using an analysis depth of 4.6 λ, an XPS signal would come from roughly 1.5 nm * 4.6/0.2 nm ≈40 layers, half containing Sr and half containing Ti. As noted above, the signal from the deepest layers would be very small. Differences between the quantification of such materials using the “uniform ” composition and considering actual elemental distribution are dis-cussed in Secs. II A andII B. A. Impacts of highly ordered elemental distribution In addition to the unknowns associated with several aspects of standard quantitative analysis, 1,2the assumption that the distribu- tion of elements can be described by the same density distributionfor all elements is not appropriate or accurate for complex epitaxial materials. For instance, consider materials consisting of sublatticescontaining different elements that alternate with the film depth. Cubic perovskites oriented in the (001) direction, such as STO (001) as shown in Fig. 1 , are one example of such materials. For STO(001), the terminating layer can be either TiO 2or SrO. As pointed out by Chambers and Sushko9with results shown in Fig. 2(a) , equations appropriate for materials with a uniform distri- bution of elements may be replaced by sums over sublattices for the different elements in each layer of the film that enables differ-ences in signal intensities for TiO 2and SrO terminated films to be calculated. However, simple layer-specific sums that take into account isotropic inelastic scattering do not account for photoelectron dif- fraction effects that occur in epitaxial and single-crystal materials,as seen in Fig. 2(b) . As Chambers notes, 4there is a significant modulation of the photoelectron intensities with the exit angle due to elastic scattering and interference of outgoing photoelectron and Auger electron waves for epitaxial films and bulk single FIG. 1. Schematic diagram of SrTiO 3(001) showing the layered structure along with TiO 2and SrO terminated surfaces.TUTORIAL avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000465 38,061201-2 Published under license by A VS.crystals.5,10–12Experimental data for the various elements in an STO(001) crystal are shown as a function of the polar angle in Fig. 2(b) to illustrate this point. As can be seen, diffraction effects can be significant. A useful feature is the so-called “forward focus- ing”peak, which results from Coulombic attraction between outgo- ing photoelectrons and ion cores along the exit path. This effect leads to large scattering amplitudes and small scattering phase shifts for low scattering angles, resulting in zeroth-order intensitymaxima (see the inset in Fig. 2 ). Such features are particularly notable along close-packed, low-index directions such as [001] and [011] in the perovskite lattice and serve as a convenient means oforienting the crystal. Higher-order diffraction features are also gen-erated, with constructive interference occurring at emission anglesnot corresponding to low-index directions (see the inset). Most of these data were collected with a full angular acceptance angle of 14° in the analyzer. However, in normal operation, many modern ana-lyzers have angular acceptances ranging from 30° to 60° and thesepartially smooth out diffraction effects, as seen by comparing the O1s scans for acceptance angles of 14° and 30° in Fig. 2(b) . Diffraction effects can impact the quantitative analysis and orienta- tion effects and need to be considered in data acquisition and anal-ysis. Using larger acceptance angles and consistently using awell-defined emission direction can yield consistent measurements.See the study by Chambers and Du 8for a more detailed discussion of the interplay between elastic and inelastic scattering and its impact on the quantitative XPS analysis. Although diffraction mod-ulation can be a complication for some types of analyses, it canalso be used to extract important structural information. Fittingangular distributions of Auger and photoelectron signals with model scattering calculations enables surface and epitaxial film structures to be determined at the atomistic level. 5,10–12 B. Alternative approach to determining composition The molecular beam epitaxy (MBE) community has developed alternative approaches to deal with the quantification of the com-position of epitaxial films. For epitaxial films that consist of solid solutions of the form N xMp−xOq, Chambers et al .13used an approach to determine the mole fractions of N and M that doesnot require knowing cross sections or sensitivity factors. Therequirements for the use of this method are that at least one endmember (e.g., M pOq) be available for XPS and that N substitutes for M as x increases. This approach has also been applied to the epitaxial growth of La 1−xSrxFeO 3(LSFO) on SrTiO 3(001) by Wang et al.14and is reported in the supplementary material of their paper. For their measurement, LFO was the stable end memberthat was used as the standard for the ferrite perovskite with 100% of the A site occupied by La. The basis of the measurement is that as Sr is added, the concentration of La and thus the La signal willdecrease. As the O concentration is constant, the signal amplitudeof La referenced to that of O is determined as xis increased. The mole fraction of Sr ( x) is then determined by the relative peak areas inEq. (1) , x¼1/C0 ALa3d5/2 AO1 s/C16/C17 LSFO ALa3d5/2 AO1 s/C16/C17 LFO: (1) By using this method, it is essential that the spectra be mea- sured with the same analyzer and sample orientation conditions and that the peak areas be determined in a consistent mannerincluding background removal. Based on the experience with mul-tiple materials and growth conditions, it is estimated that this approach can be used to verify film stoichiometry to well within ∼10%. FIG. 2. Calculated (a) and measured (b) core-level polar intensity profiles for TiO 2-terminated SrTiO3(001). The experimental profiles have been vertically offset for clarity. The intensity modulations seen in experiment are due to elasticscattering and interference of outgoing photoelectron waves, which are notincluded in simple model calculations. Large scattering amplitudes and small phase shifts at low scattering angles lead to strong constructive interference in the forward direction for close-packed chains of atoms, such as those occurringalong [011] and [001] in the cubic perovskite lattice. However, higher-order inter-ferences also modulate photoelectron intensities in emission directions away from [011] and [001]. Diffraction effects can be substantial when the analyzer cone of acceptance is small; in this case, the full angle of acceptance was 14°and there is a ∼30% intensity increase across [001] for O 1s, Ti 2p, and Sr 3d. These effects are only partially averaged out when analyzers with larger cones of acceptance are used, as seen in the O 1s scan collected with a full angle of 30°. Reprinted with permission from Chambers, “X-ray photoelectron spectro- scopy, ”inMaterials Characterization Methods of epitaxial Films and Heterostructures , edited by S. A. Chambers and A. A. Demkov (World Scientific, Singapore, 2020). Copyright 2020, World Scientific.TUTORIAL avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000465 38,061201-3 Published under license by A VS.III. SPECTRAL SHAPE AND BINDING ENERGY Often the objective of growing epitaxial films is to synthesize new materials or material combinations. It is important to recog-nize that photoelectron peak shapes for such materials may deviate from the patterns that many XPS analysts have come to expect for more common materials, particularly when they involve transitionmetal cations with unpaired delectrons in the valence band. It is most useful for analysts to recognize that many-body and final-stateeffects can be important and have a significant impact on peak structure and shape. 4,15Too often in the literature, unexpected peak features have been incorrectly identified as new or additionalchemical states. 16The series of Ti 2p spectra shown in Fig. 3 illus- trates this effect. Figures 3(a) ,3(c), and 3(d) show a progression of measured binding energies that might be expected when increasing the oxidation state from 0 to 3+ to 4+.17,18However, the Ti formal charge for the material shown in Fig. 3(b) is also 3+, but this mate- rial has fundamentally different crystallographic and electronicstructures than that for the material in Fig. 3(c) . 17,19Such examples highlight the importance of reference materials and careful exami- nation of relevant spectra from the literature. IV. BAND BENDING AND BAND OFFSETS Interfaces play a dominant role in the properties of many epi- taxial heterostructure materials systems. In addition to the impor-tant uses of XPS to determine the composition and chemical statechanges in epitaxial films and heterostructures, it is possible to obtain information about electronic structure at interfaces, which is critical for many potential applications. Significantly, XPS has beenfruitfully used to monitor variations in the electrostatic potentialwith depth, making it possible to measure things such as bandbending, heterojunction band discontinuities, and Schottky barrier heights. From the 1980s, several groups used soft x-ray photoemission at synchrotron radiation facilities to investigate changes in bandbending that occur at compound semiconductor surfaces upon sub- monolayer deposition of metals. 20–22At the same time, Schottky barrier height formation and interface chemistry were beingexplored as metal films in the one-to-several monolayer range weredeposited and probed with both synchrotron and lab-based x-raysources. 23–52Although most of these studies involved disordered metal films deposited at room temperature, others utilized metals that were lattice matched to the semiconductor at some level anddeposited at elevated temperature, leading to heteroepitaxialgrowth. 44,53–58Other groups combined III –V compound semicon- ductor MBE capability with in situ XPS to directly measure valence and conduction band offsets for a wide range of semiconductor heterojunctions.59–74In what follows, we present these methods and illustrate with a recent example from the world of complexoxide heterojunctions. Electrostatic potentials present at interfaces include built-in potentials such as those that occur at p-njunctions and band bending due to charge accumulation near surfaces or interfaces.The series of energy diagrams in Fig. 4 show separated materials [panel (a)], materials in contact but without Fermi-level equilibra- tion [panel (b)], and materials in contact with Fermi-level equili- bration and the establishment of a space-charge region at theinterface [panel (c)]. The situation that occurs when one of the materials is a thin layer is illustrated in Fig. 4(d) . In this last case, there is charge accumulation and band bending at the interface andat the surface of the film. The variation in potentials caused bycharge accumulation at interfaces and surfaces can alter XPS peakshapes and shift peak energies. When such potential gradients exist, photoelectrons emitted in each layer originate from a slightly different potential and thus appear at shifted binding energies andwith asymmetrically broadened line shapes. To provide specific examples of these effects, we consider data from measurements for p-La 0.88Sr0.12FeO 3 on n-SrTi 0.99Nb0.01O3(001) [LSFO/Nb:STO] by Wang et al .14These FIG. 3. Ti 2p spectra from (a) Ti metal, (b) UHV-cleaved single-crystal Ti 2O3, (c) LaTiO 3(001), and (d) SrTiO 3(001). Spectra (b) and (c) are both measured on materials with 100% Ti+3but have very different line shapes and peak energies due to the different lattices and electronic structures. Reprinted with permission from Chambers, “X-ray photoelectron spectroscopy, ”inMaterials Characterization Methods of epitaxial Films and Heterostructures , edited by S. A. Chambers and A. A. Demkov (World Scientific, Singapore, 2020).Copyright 2020, World Scientific.TUTORIAL avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000465 38,061201-4 Published under license by A VS.authors investigated the built-in potential, band alignment, and electrocatalytic activity of this heterojunction. The built-in potentialthat forms at a p-njunction would be useful for diverting photo- generated electrons into Nb:STO and photogenerated holes into the LSFO thin film to drive water oxidation. These interface potentials are“built-in ”as a result of the material structure and are not exter- nally applied. Epitaxial films of 3, 5, 9, and 35 unit cell (UC) thick-nesses of LSFO were grown on Nb:STO by oxygen plasma assistedmolecular beam epitaxy. This work demonstrates that the core-level binding energy differences across the buried interface, obtained by fitting the spectra in the usual way as has traditionally been used todetermine valence band offsets (VBOs), lead to incorrect resultsbecause of the built-in potential in the film. This material system is particularly interesting in that Sr is present in both the film and substrate, and some Ti diffuses outinto the film from the substrate. Significantly, the differences in binding energy between the film and substrate species for both Sr3d and Ti 2p are due at least in part to the difference in electro-static potential between the two materials. Spectra for three ele- ments are shown in Fig. 5 for three different LSFO thicknesses. There are several interesting features that emerge as the film thick-ness increases: (i) the Ti 2p peak shape remains essentially cons-tant, but is different than that of pure Ti 4+in STO; (ii) a second set of Sr 3d peaks appears at a lower binding energy; and (iii) the width and position of the La 4d peak changes with film thickness. As discussed by Wang et al.14the Sr 3d spectra were fit with two pairs of spin –orbit doublet peaks. The most intense signals were assigned to Sr+2in the substrate and the smaller peaks to Sr+2in the LSFO. The binding energy differences are attributed to the band discontinuity at the interface and built-in potential FIG. 4. Schematic energy diagrams of two materials: (a) isolated, (b) in contact, but not at equilibrium, (c) as equilibrium is established at the interface, and (d) a generic band profile for a single semiconductor heterojunction formed by growing a thin film of material 2 on material 1, showing band bending and band offsets that can be measured by XPS.TUTORIAL avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000465 38,061201-5 Published under license by A VS.in the film. Although the Ti 2p spectrum seems little changed with LSFO thickness, some Ti di ffuses into the LSFO producing a second small peak at an energy consistent with the presence ofTi 3+in the film, which, in turn, has a different potential than Nb: STO. The absence of broadening in the Ti 2p4+spectrum sug- gests that no built-in potential of any significant magnitude is present in Nb:STO. The variation in La 4d peak width is attrib-uted to the presence of a large bui lt-in potential in the thin film as discussed in additional deta il below. This dataset from the LSFO/STO system demonstrates the impact of electrostatic poten- tial at interfaces to shift photoelectron spectra and provides the basis for a method to extract useful information about thebuilt-in potential. In order to determine band offsets at semiconductor inter- faces, the default approach is to use the method developed by Kraut et al. 75,76This method involves determining the energy dif- ference between an appropriate core level and the top of thevalence band in bulk or thick-film samples of each material. Thesenumbers are then combined with binding energies of the same core levels in the heterojunction when a film of one material is grown on a substrate of the other [case (d) in Fig. 4 ] to yield the VBO ( ΔE V), ΔEV¼(ECL1–EV1)pure1/C0(ECL2–EV2)pure2–(ΔECL)HJ:(2) Here, the terms ( ECL–EV) are the binding energy differences between the chosen core level and the valence band maximum ineach of the two materials as pure phases, and ΔECLis the core-level binding energy difference across the heterojunction (see Fig. 4 ). Using the bandgaps of the two materials, the conduction band offset can be determined by ΔEc¼ΔEVþ(Eg1/C0Eg2): (3) In developing this method, it was implicitly assumed by Kraut et al. that there is no measurable band bending on either side of the interface and, therefore, that a single core-level binding energyis representative of all layers in each material. With this assump-tion, and the assumptions of an atomically abrupt junction and no interface chemistry (leading to a unique phase not found in the bulk spectra), the band alignment can be determined using Eqs. (2) and(3). However, the first of these assumptions is not valid for the LSFO/Nb:STO system because of the presence of the sizeablebuilt-in potential within the LSFO film. This potential drop across the LSFO film was detected not only by core-level broadening but also by changes in the x-rayexcited valence band spectra, specifically the dependence of thevalence band maximum (VBM, E V) on the thickness. Chambers et al.77examined three approaches for accurately determining the VBM and found that extrapolation of the linear portion of the leading edge to the energy axis is valid for many systems. This FIG. 6. Valence band XPS for pure Nb:STO(001) and La 0.88Sr0.12FeO 3/Nb:STO (001) heterojunctions with films of thickness equal to 3, 5, 9, and 35 UC. Thesolid lines indicate the linear part of the leading edge that can be extrapolated to binding energy axis to determine the valence band maxima ( E V) shown in Table I and used in Eq. (2) .D a t af r o m Ref. 14 . The 35 UC film is a proxy for bulk LSFO because its thickness ( ∼14 nm) exceeds the XPS probe depth such that no XPS signals from the underlying STO are detected. The monotonic increase in VBM with film thickness reveals the presence of a built-in potential. Adapted with permission from Wang et al ., Appl. Phys. Lett. 112(26), 261601 (2018). Copyright 2018, AIP Publishing LLC. FIG. 5. Ti 2p, Sr 3d, and La 4d photoelectron spectra for La 0.88Sr0.12FeO 3on Nb:STO heterojunctions at different film thicknesses. The fits shown werechosen solely to reproduce the experimental spectra and identify single features (marked by arrows) for band offset determination. The fitting parameters (except for amplitude) were kept constant for all thicknesses. The stars mark peaks thatoriginate in the films. Adapted with permission from Wang et al ., Appl. Phys. Lett. 112(26), 261601 (2018). Copyright 2018, AIP Publishing LLC.TUTORIAL avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000465 38,061201-6 Published under license by A VS.method was applied to the VB spectra for the LSFO/Nb:STO heter- ojunctions in the Wang et al. study, as shown in Fig. 6 . The EV values from Nb:STO and the 35 UC film were used for materials 1 and 2 in Eq. (2) above. It is then possible to determine ΔEVvalues for several different pairs of core photoelectron lines, and the results are quite consistent, as shown in Table I . However, as noted inRef. 14 , these values are incorrect because they are based on core-level binding energies averaged over all layers when, in fact,they need to be based on binding energies for the layers directly at the interface because of the presence of the built-in potential. In order to find the binding energies for La in the interfacial layers, the composite La 4d spectra in Fig. 5 were modeled as sums over layers across which a built-in potential is present. Such modelsare shown in Fig. 7 for 3, 6, and 9 UC films. The nearly flatband spectrum measured for the 35 UC film was assigned as a “basis spectrum ”for each layer in the thinner films. The relative intensi- ties were attenuated based on the depths of the various layers, andthe energy shifts between layers were optimized to yield the best fit to the measured heterojunction spectrum. The potential drop per layer is relatively small for the 3 UC film, is significantly larger forthe 5 UC film, and then decreases in magnitude as the film getsthicker. The conduction band offsets are determined from Eq. (3) using values for the bandgaps obtained from other types of mea- surements. 14,78Using the binding energies of the interfacial layers for all three heterojunctions, the resulting energy diagram is quitedifferent than the flatband picture that emerges using the averageTABLE I. Core-level binding energies and valence band offsets (in eV) for the LSFO/Nb:STO heterojunctions. [Taken from Wang et al ., Appl. Phys. Lett. 112(26), 261601 (2018). Copyright 2018, AIP Publishing LLC.]a ETi2p3/2ESr3d 5/2 EFe2p3/2ELa4d 5/2 EV ΔEV(Fe&Ti) ΔEV(Fe&Sr) ΔEV(La&Ti) ΔEV(La&Sr) ΔEV Nb:STO 459.07 133.69 —— 3.20 LSFO —— 709.34 101.05 0.29 3 UC 459.29 133.91 710.62 102.40 1.48 1.8 1.8 1.8 1.8 1.85 UC 459.24 133.85 709.87 101.65 0.77 2.6 2.6 2.5 2.5 2.5 9 UC 459.23 133.84 709.50 101.25 0.37 2.9 2.9 2.9 2.9 2.9 a(1) All samples were conductive and did not exhibit any charging artifacts during XPS. Therefore, the binding energies are relative to the Fermi level of the grounded heterojunctions. The binding energy and dispersion scales for the spectrometer were calibrated using the Ag 3d 5/2core peak at 368.21(2) eV and the Fermi level at 0.00(2) eV from a polycrystalline Ag foil; (2) EVis the valence band maximum relative to the Fermi level for each specimen, and ΔEVis the valence band offset for the three thin-film heterojunctions; (3) For the three thin-film heterojunctions, the listed Ti 2p 3/2and Sr 3d 5/2binding energies are those for the peaks associated with photoemission from the substrate; (4) As a specific example, for the Ti 2p 3/2and Fe 2p 3/2core levels, the VBO was determined using the formula ΔEV(Fe&Ti)¼(EFe2p3/2–EV)LSFO/C0(ETi2p3/2–EV)Nb:STO–(EFe2p3/2–ETi2p3/2)HJ. Analogous formulas (not shown) were used for the other pairs of core levels; (5) The experimental uncertainties are 0.02 eV for Ti 2p 3/2and Sr 3d 5/2, 0.05 eV for Fe 2p 3/2and La 4d 5/2, 0.05 eV for Ev, and 0.1 eV for ΔEV. FIG. 7. La 4d spectra (open circles, blue on line) and simulations (solid curves, red on line) made by assigning a flatband reference spectrum to each layer wit ha n assumed potential profile across all layers. The spectrum for each sample was fit to a sum of individual spectra in the various layers, each attenuated to account for depth below the surface, and shifted in energy due to the potential drop. Note that the 3 UC film exhibits a narrower peak than the 5 UC film, and that the 9 UC film has a some- what narrower peak than the 5 UC film. The spectra for the individual layers are shown for each heterojunction. Reprinted with permission from Wang et al ., Appl. Phys. Lett. 112(26), 261601 (2018). Copyright 2018, AIP Publishing LLC.TUTORIAL avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000465 38,061201-7 Published under license by A VS.La 4d 5/2binding energies shown in Table I . The final energy diagram is shown in Fig. 8 . Instead of a constant VBO of 1.8 eV for heterojunctions consisting of 3, 5, and 9 UC thick films with flat-bands throughout, we see upward band bending with increasing filmthickness, culminating in the VBM being quite close to the Fermi level at t≥9 UC, consistent with the VB spectra shown in Fig. 6 . The above analysis was carried out by manually modeling linear potential drops with either one (3 and 5 UC) or two (9 UC)segments across the film. However, a sophisticated algorithm that performs a comprehensive search over all binding energies for the best fit set of layer-resolved spectra has been recently devel-oped. 79,80This approach is particularly useful for analyzing hard x-ray XPS data for which many more layers contribute to the totalspectrum. The use of XPS to obtain electronic structure information for heterojunctions is much less common than measurements of com-position and chemical bonding but is nevertheless quite important.Indeed, XPS, when properly interpreted, yields electronic structureinformation not readily available by any other technique. The dis- cussion here has focused on measurements of complex oxide heter- ostructures. A creative application of XPS has also been used toextract important information that is difficult to obtain informationsuch as dielectric, piezoelectric, and ferroelectric properties offilms, along with the electric potential of selected layers in inor- ganic films and potential profiles across organic films. 81–84 V. SUMMARY AND CONCLUSIONS XPS is an important analysis tool for epitaxial thin films. In addition to what might be identified as standard XPS analysis toyield information about contamination, average elemental composi- tion, and chemical states present, XPS can also be used to extract important information related to the crystallographic structure,layer stoichiometry, and band edge profiles. The structural order inthese films adds a complication to the analysis that requires appro-priate experimental planning and deeper levels of data analysis. The less commonly utilized advantages of XPS to obtain infor- mation about the electronic properties of films and interfaces havebeen highlighted. An example of obtaining information aboutbuilt-in potentials and band offsets demonstrates how such infor-mation can be obtained. Although each of the uses of XPS described in this guide is relevant to epitaxial films and heterostructures, aspects of themethods presented are relevant to other single-crystal materials,advanced complex materials and structures, and electronic charac-teristics of other types of thin overlayers. ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy, Office of Science, Division of Materials Sciences and Engineeringunder Award No. 10122 and was performed in the EnvironmentalMolecular Sciences Laboratory (Grid. No. 436923.9), a national sci-entific user facility sponsored by the Department of Energy ’sO f f i c e of Biological and Environmental Research and located at the PNNL. DATA AVAILABILITY Data sharing is not applicable to this article as no new data were created or analyzed in this study. REFERENCES 1C. R. Brundle and B. V. Crist, J. Vac. Sci. Technol. A 38, 041001 (2020). 2A. G. Shard, J. Vac. Sci. Technol. 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A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000465 38,061201-9 Published under license by A VS.

5.0002194.pdf

AIP Conference Proceedings 2220 , 110029 (2020); https://doi.org/10.1063/5.0002194 2220 , 110029 © 2020 Author(s).Growth and characterization of MnBi2Te4 magnetic topological insulator Cite as: AIP Conference Proceedings 2220 , 110029 (2020); https://doi.org/10.1063/5.0002194 Published Online: 05 May 2020 A. Saxena , P. Rani , V. Nagpal , S. Patnaik , and V. P. S. Awana ARTICLES YOU MAY BE INTERESTED IN Signatures of temperature driven antiferromagnetic transition in the electronic structure of topological insulator MnBi 2Te4 APL Materials 8, 021105 (2020); https://doi.org/10.1063/1.5142846 The mechanism exploration for zero-field ferromagnetism in intrinsic topological insulator MnBi 2Te4 by Bi 2Te3 intercalations Applied Physics Letters 116, 221902 (2020); https://doi.org/10.1063/5.0009085 Precise resistance measurement of quantum anomalous Hall effect in magnetic heterostructure film of topological insulator Applied Physics Letters 116, 143101 (2020); https://doi.org/10.1063/1.5145172Growth and C haracterization of MnBi 2Te4 Magnetic Topological Insulator A. Saxena1,2,a), P. Rani1, V. Nagpal3, S. Patnaik3 and V.P.S. Awana1,2 1CSIR -National Physical Laboratory, K.S. Krishnan Marg, New Delhi -110012, India. 2Academy council of scientific and industrial Research, Ghaziabad U.P. -201002 , India 3School of Physical Sciences, Jawaharlal Nehru University, New Delhi -110067 ,India a)Corresponding Author : kingofrmu@gmail.com Abstract . We report successful growth of magne tic topological insulator (MTI) MnBi 2Te4singlecrystalby solid state reaction route via self flux m ethod. The phase formation of MnBi 2Te4singlecrystal is strongly dependent on the heat treatment .MnBi 2Te4is grown in various phases i.e., MnBi 4Te7, MnBi 6Te10 and MnTe as seen in powder X -ray diffraction (PXRD) of crushed resultant crystal. The Rietveld analysis shows some impurity lines along with the main phase MnBi 2Te4. Low temperature (10K) magneto -resistance (MR) in applied magnetic field of up to 6 Tesla exhibited –ve MR below 0.5 Tesla and +ve for higher fields. The studied MnBi 2Te4, MTI crystal could be a possible candidate for Quantum Anomalous Hall (QAH) effect. Here w e are reporting a newly discovered magnetic topological insulator MnBi 2Te4having non-trivial symmetry as well as strong Spin -Orbit Coupling forQAH effect . INTRODUCTION Topological Insulators (TIs) are the materials which have conducting state on surface and its bulk behave like insulator [1, 2]. The existence of magnetism in topological in sulator has become promising field to facilitatet he exotic phenomena viz. quantized magneto electric coupling and the axion insulator state . MnBi 2Te4 is the recently proposed candidate as magnetic topological insulator (MTI) for Quantum Anomalous Hall Effect (QAHE) to occur due to its strong Spin -orbit coupling . MnBi 2Te4forms an interlayer Anti-ferromagnetism ( AFM ) state, in which ferromagnetic (FM) Mn layers of neighbouring blocks are coupled anti -parallel to each other, converti ng the material in a three -dimensional (3D) Anti -ferromagnetic TI. The reason of Magnetism is due to 3d metal which inserted into 3D topological insulator layers [3]. In case of MTIs, a magnetic layer or element is inserted among the running TI unit cells o f bulk 3D topological insulators such as 3d metal doped Bi 2Se3, Bi 2Te3 and Sb 2Te3 [4- 6]. The insertion of magnetic layer along running 3D bulk topological insulators shifts the Dirac position and thus alters the quantum transport properties of theparent system [7 -10]. One of the most fascinating properties of the MTIs is the appearance of Quantum Anomalous Hall (QAH) effect [7 -9]. QAH happens due to the finite Hall voltage created due to magnetic polarization and spin -orbit coupling, while the external magnetic field is absent. QAH is found to be in integer multiple of e2/h which is cal led Landau Level [9]. MnBi 2Te4 can be formulated as Bi2Te3 + MnTe in which the MnTe, the magnetically ordered layer being inserted in the periodic structure at van der Waals gaps in Bi 2Se33D bulk topological insulator and later [14]. MnBi 2Te4 is the first 3D antiferromagnetic topological insulator [1 1-13, 15 ]. Interestingly, there are only some reports on MnBi 2Te4, in p articular the single crystals [15 ]. EXPERIMENTAL High -quality MnBi 2Te4 single crystal has been grown by self -flux method through the conventional solid -state reaction route. High purity powders of Mn, Te and Bi were taken as starting material, weighed in stoichiometric ratio ground properly in glove box which is filled with argon (Ar) gas (or inert atmo sphere) to avoid oxidation. Ground powder of MnBi 2Te4was then palletized in rectangular form using hydraulic press then sealed into a quartz tube under the vacuum pressure 10-5mBar. The vacuum sealed sample then kept into electric furnace with the rate of 2oC/min. up to 1000oC, kept there for 12 h and then slowly cooled down to 600oC at a rate of 1oC/h, again sample has been hold there for 12 h. Further, the furnace was allowed to cool naturally (20oC/Hour) to room temperature. Finally silvercolored shiny c rystals were obtained (shown in Fig ure. 1(b)) which is mechanically 3rd International Conference on Condensed Matter and Applied Physics (ICC-2019) AIP Conf. Proc. 2220, 110029-1–110029-4; https://doi.org/10.1063/5.0002194 Published by AIP Publishing. 978-0-7354-1976-6/$30.00110029-1cleaved for further characterizations. The heat treatment diagram of as grown single crystal of MnBi 2Te4 is shown in Fig.1 (a).XRD pattern were performed on a Rigaku Made Mini Flex -II X-ray diffractometer while the magneto resistance measurements were carried out on a Physical Property Measurement System (PPMS -10Tesla) using a close cycle refrigerator . (a) (b) FIGURE 1.(a) Schematic heat treatment of MnBi 2Te4 single crystal growth (b) As grown single crystal of same . RESULTS AND DISCUSSI ON Figure 2(a) depicts the X-ray diffraction pattern on flake of as grown MnBi 2Te4 single crystal , which determines the crystallographic structure of the compound. The sharp XRD peaks of (0, 0, 4n) indicate the sample has been grown along the [00l] plane which confirms the highly crystalline nat ure of the studied sample . The single phase fullprof Rie tveld analysis is carried out on the observed pattern of crushed piece of MnBi 2Te4 crystal shown in Fig ure 2(b). The PXRD pattern is similar to that as observed in some recent reports for MnBi 2Te4 self flux grown crystal. The main phase MnBi 2Te4 is found to be crystallized in rhombohedral structure having Space group R -3m. The lattice parameters are a =b=4.3882(1)Å and c=42.7125(1)Å. Some extra peaks of other phases i.e., MnBi 6Te10, MnBi 4Te7, MnTe and Bi 2Te3 are also observed with t he main phase which we have reported elsewhere [16]. 10 20 30 40 50 60 70 800.05.0x1041.0x1051.5x1052.0x105 Intensity (a.u.) 2 (Degree )MnBi2Te40 0 8 0 0 20 0 0 24 0 0 28 (a) (b) FIGURE 2 . (a) X -ray diffraction pattern of flake of MnBi 2Te4 single crystal (b) R ietveld refined PXRD p attern of MnBi 2Te4 crystal and inset shows the unit cell of MnBi 2Te4. Inset of Figure 2(b) represents the unit cells for major ity phase MnBi 2Te4 drawn by VESTA software . The simplified general formula for these homologous series of compounds is given in the form of MnTe+nBi 2Te3, with n = 0, 1, 2, 3 as MnBi 2Te4, MnBi 4Te7 and MnBi 6Te10. So, principally it depends upon the fact that after how many unit cells of Bi 2Te3, a MnTe magnetic layer is inserted . Figure 3 shows the magneto -resistance percentage (MR %)versus applied magnetic field (H) at different temperature (10K, 20K, 50K and 200K) in applied magnetic field of up to 6Tesla for MnBi 2Te4 single crystal . The MR (%) is calculated using the formula MR = [ρ(H) - ρ(0)]/ρ(0)*100 , where ρ(H) is the resistivity in the presence of magnetic field and ρ(0) is resistivity in zer o magnetic field. We observed non -saturating nearly linear positive MR value reaching up to 1.4% at 10K, whereas it reduces to around 0.8% when the te mperature is 110029-2increased to 200K which is enormously less compared to as obtained in case of Bi2Te3 single crystal [17], shows the strong effect of magnetic ordering of Mn in MnTe layer of MnBi 2Te4 crystal. Inset of Figure 3 depicts the zoomed part of magneto -resistance measurement of MnBi 2Te4 crystal in magnetic field range of ±1Tesla at 10K temperature. From the figure, it can be concluded that MR% decreases with increase magnetic field up to 0.2 Tesla , shows the –ve MR% on further increase in magnetic field the MR% increases and shows +ve MR % just above 0.5 Tesla, which further followed by almost linear behavior up to 1 Tesla magnetic field. It is also clear from the figure 3, that the MR% versus magnetic field plot for MnBi 2Te4 crystal at low temperature shows almost linear behavior up to lo w field range of 0.2Tesla and shows the increment in MR% above the 0.2Tesla, which may be lead toward the quantum transport property like QAH. For confirmation about the observation of Quantum Anomalous Hall effect, detailed experiments are underway in dif ferent protocols in magnetic range up to 0.5 Tesla in very small step interval of field for MnBi 2Te4 crystal. -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-0.20.00.20.40.60.81.01.21.4 -0.9 -0.6 -0.3 0.0 0.3 0.6 0.90.000.020.040.060.08MR (%) H (Tesla)T = 10KMR (%) H (Tesla) 10K 20K 50K 200K MnBi2Te4 FIGURE 3. MR(%) as a function of magnetic field for the MnBi 2Te4 single crystal at different temperatures . CONCLUSION In summary, in this the short communication we discuss the structural and magneto -resistance properties of the magnetic topological insulator MnBi 2Te4. Detailed growth parameters have been discussed through XRD analysis.As l ong heat treatments are required to grow high quality single crystals so our crystal quality are defensible . Low temper ature (10K) MR under magnetic field of up to 6 Tesla exhibited –ve MR below 0.5 Tesla and +ve for higher fields. The studied MnBi 2Te4, MTI crystal could be a possible candidate for Quantum Anomalous Hall (QAH) effect. ACKNOWLEDGMENTS We thanks for CSIR -NPL & JNU for all experimental observation of data. Also we are thankful of DST for Fellowship facility. Currently we are working on it f or better result and discovering new MTI. REFERENCES 1. C. L. Kane and E. J. Mele, Phys . Rev. Lett. 95, 146802 -1-146802 -4 (2005) . 2. F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 -2018 (1988) . 3. L. Fu, C. L. Kane and E. J. Mele, Phys. Rev. Lett. 98, 106803 -1-106803 -4 (2007) . 4. Y. L. Chen, et al. Science , 325, 178 -181 (2009) . 5. R. Sultana, P. Neha, R. Goyal, S. Patnaik and V. P. S. Awana, J. Magn. Mag. Mater. 428, 213 -218 (2017) . 6. R. Sultana, G. Gurjar, S. Patnaik and V. P. S. Awana, Mat. Res. Exp. 5, 046107 (2018) . 7. C. X. Liu, S. C. Zhang and X. L. Qi, Annual Review of Condensed Matter 7, 301 -321 (2016) . 8. C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, L. Minghua, O. 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5.0013094.pdf

Appl. Phys. Lett. 117, 113105 (2020); https://doi.org/10.1063/5.0013094 117, 113105 © 2020 Author(s).Twin defect-triggered deformations and Bi segregation in GaAs/GaAsBi core–multishell nanowires Cite as: Appl. Phys. Lett. 117, 113105 (2020); https://doi.org/10.1063/5.0013094 Submitted: 08 May 2020 . Accepted: 04 September 2020 . Published Online: 18 September 2020 Teruyoshi Matsuda , Kyohei Takada , Kohsuke Yano , Satoshi Shimomura , Yumiko Shimizu , and Fumitaro Ishikawa ARTICLES YOU MAY BE INTERESTED IN Single quantum dot-in-a-rod embedded in a photonic nanowire waveguide for telecom band emission Applied Physics Letters 117, 113102 (2020); https://doi.org/10.1063/5.0020681 Strain-induced structural transition of rutile type ReO 2 epitaxial thin films Applied Physics Letters 117, 111903 (2020); https://doi.org/10.1063/5.0006373 Investigation of carrier compensation traps in n−-GaN drift layer by high-temperature deep- level transient spectroscopy Applied Physics Letters 117, 112103 (2020); https://doi.org/10.1063/5.0019576Twin defect-triggered deformations and Bi segregation in GaAs/GaAsBi core–multishell nanowires Cite as: Appl. Phys. Lett. 117, 113105 (2020); doi: 10.1063/5.0013094 Submitted: 8 May 2020 .Accepted: 4 September 2020 . Published Online: 18 September 2020 Teruyoshi Matsuda,1Kyohei Takada,1Kohsuke Yano,1Satoshi Shimomura,1Yumiko Shimizu,2 and Fumitaro Ishikawa1,a) AFFILIATIONS 1Graduate School of Science and Engineering, Ehime University, 3 Bunkyo-cho, Matsuyama, Ehime 790-8577, Japan 2Toray Research Center, 3-3-7 Sonoyama, Otsu, Shiga 520-8567, Japan a)Author to whom correspondence should be addressed: ishikawa.fumitaro.zc@ehime-u.ac.jp ABSTRACT We investigated microstructural deformations and Bi segregation in GaAs/GaAsBi/GaAs core–multishell heterostructures, which were triggered by the existence of twin defects. We observed Bi segregation at the interface of the twin defect interface in the GaAsBi shell. The phenomenon produced a horizontally spread Bi-accumulated nanostructure in the nanowire, which is probably induced by the large latticemismatch between GaAs and GaAsBi. Bi is expected to penetrate through the twin defect interface, which results in the existence of Bi alongtwin defects and also inside the GaAs core. The existence of twin defects induced structural deformations and resulted in the formation of corrugated complex sidewall surfaces on the nanowire. Published under license by AIP Publishing. https://doi.org/10.1063/5.0013094 In the recent decades, III–V semiconductor nanowires (NWs) have attracted considerable interest. 1–3The introduction of epitaxial heterostructures into NWs allows us to realize integrated systems on a Si platform, which are based on III–V compounds and exhibit superior electronic and optical functions.1–9Optical devices, which are based on III–V GaAs, such as lasers and optical amplifiers operating in thenear-infrared regime, suffer from intrinsic losses related to Auger recombination. To circumvent this, the use of a dilute bismide GaAsBi alloy has recently gained intensive attention. 11–15The introduction of Bi increases the splitting of the valence band energy levels between the heavy hole band and the spin–orbit band.12–15Recently, we obtained GaAs/GaAsBi heterostructure NWs on Si using molecular beam epitaxy.3,16–18GaAsBi NWs exhibit specific Bi-induced structural features and have a rough surface with corrugations, which induces the synthesis of nanostructures in NWs.16,18,19These structural modi- fications were considered to be induced by the large lattice mismatch and resulting strain accumulation between GaAs and the GaAsBi alloy.20–24In addition, for GaAs NWs, there have been several reports concerning twin defect formation and its impact on structural and electronic properties.25–29Specifically, the intentionally controlled twin defect formation, through growth conditions and doping, have resulted in superlattice structures with high optical and electronicfunctions.25–29Within those, for the self-catalyzed Ga-induced vapor–liquid–solid (VLS) growth of GaAs NWs by molecular beam epitaxy, the twin defect formation and its control by doping has been reported.29In this study, we report the structural analysis of GaAs/ GaAsBi/GaAs core–multishell heterostructure NWs showing specific structural deformation and Bi accumulation with respect to the exis-tence of twin defects. The samples were grown by molecular beam epitaxy using con- stituent self-catalyzed VLS growth on phosphorous-doped n-type Si(111) substrates. 16–18,30–32The samples were heated to 580/C14Cu n d e r an As 4beam equivalent pressure (BEP) of 1.0 /C210/C05mbar. The Ga supply was set to obtain a planar growth rate of 1.0 ML/s on GaAs (001) by providing group-V As-rich growth conditions. Bi BEP was adjusted to 5.4 /C210/C07mbar. First, GaAs core nanowires were grown for 15 min; then, the growth was interrupted for 10 min, and the sub- strate temperature was reduced to 550/C14C to crystallize the Ga catalyst. After the crystallization of the catalyst, the lateral growth became dominant and was expected to yield core–shell-type NWs. Then, wesupplied Ga flux for 15 min to form the GaAs shell and introduced growth interruption by further reducing the substrate temperature to 350 /C14C for the subsequent growth of GaAsBi. Next, we provided Ga and Bi fluxes for 15 min (to form the GaAsBi shell) and subsequently Appl. Phys. Lett. 117, 113105 (2020); doi: 10.1063/5.0013094 117, 113105-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplGa flux for 15 min without growth interruption (to form the outer- most GaAs shell). Thus, NWs consisted of the GaAs core with aGaAs/GaAsBi/GaAs core–multishell structure. 16–18The widths of the GaAs core, GaAsBi shell, and outermost GaAs were expected to be 100, 80, and 80 nm, respectively. The average Bi concentration in theGaAsBi shell was expected to be 2% indicated by the microscopic ele-mental mapping using energy dispersive X-ray spectrometry (EDS)and the optical emission wavelength using low temperature cathodolu-minescence, as already reported in our previous study. 16 The NWs were investigated by cross-sectional scanning trans- mission electron microscopy (STEM) on radially sliced single NW samples prepared by focused ion beam processing (Helios660, FEI, USA). STEM measurements were carried out on a transmissionelectron microscopy system (JEM-ARM200F Dual-X, JEOL,Japan) operating at 200 kV with an EDS attachment employing a100 mm 2silicon drift detector (JED-2300, JEOL). STEM images were obtained in both bright-field (BF) and high-angle annulardark-field (HAADF) modes. 18,32 Figure 1(a) shows the radial cross-sectional BF-STEM image of a single nanowire. We observed the formation of nanowires on the Sisubstrates. The NWs exhibited a preferential vertical alignment. 33The length of the wires was 7 lm, and the diameter was approximately 400 nm.16,18As seen in the figure, the nanowire exhibits a corrugated surface morphology and a hemispherical shape at the top, which isinduced by the effect of Bi introduction, as has been previouslyreported by our group. 16,18Figure 1(b) s h o w sh i g h e rm a g n i fi c a t i o n BF- and HAADF-STEM images of the area near the tip of the wire. As seen in the figure in the BF-STEM image, we observe many horizon- tally spread dark lines with various lengths inside the NW. The darklines clearly correspond to the bright contrast in the HAADF-STEMimage. The brighter contrast in the HAADF image suggests the exis- tence of heavy elements at the spot; thus, we can make a sensitive investigation of elemental distribution such as phase segregation. 34 Figure 1(c) shows higher magnification BF- and HAADF-STEM images of the area near the tip delimited by the red rectangle in (b).Dashed vertical white lines in (b) and (c) indicate interfaces between GaAs core/GaAsBi shell/GaAs outer shells. The red arrows in (c) indi- cate the observed horizontal dark lines in BF-STEM images. The more detailed series of STEM images for various magnifications around the investigated area are shown in Fig. S1 of the supplementary material . The indicated areas suggest the formation of the core–multishell struc- ture by the existence of comparatively darker regions in the BF-STEM image or vice versa in the HAADF-STEM image. Specifically, we observed horizontal dark lines with larger thicknesses at the area of the GaAsBi shell. In the GaAs core, we observed fewer lines with smaller thickness, which resulted in the brighter contrast of the inside core area and allowed us to recognize the existence of the GaAs core. Thissuggests that horizontal lines are penetrated into the GaAs core. At the outer GaAs shell, the number of dark lines is drastically reduced, which allows us to observe the area with small dark lines. The horizontal lines observed in Figs. 1(b) and1(c)are related to the twin defects in the nanowire. 35,36The density of the twin defects is obviously larger in the upper part of the nanowire. The lower part of the wire exhibits a smaller twin defect density than the upper part, which is in agreement with the previous reports.37–43However, the variation of the linewidth, as well as the clear correspondence of the flipped contrast between the BF and HAADF-STEM images, has never been reported.35–43 Figure 2(a) shows the radial cross-sectional HAADF-STEM image and EDS elemental mapping of Bi at the upper part of the nano- wire with horizontally spread contrast modulations shown in Fig. 1 . As seen in the HAADF image in Fig. 2(a) , the horizontal bright lines, at the position of the dark lines in BF images in Figs. 1(a)–1(c) ,a g r e e with the strong intensity observed in the Bi intensity in EDS elemental mapping. Thus, the lines definitely originate from the existence of Bi. The bright contrast in the HAADF image can be obtained by the large atomic number. The local positions clearly agree between the horizon- tal lines in the HAADF image and those of Bi intensity in EDS ele- mental mapping. Hence, the lines originate from the area with Bisegregation. Figure 2(b) shows a higher magnification HAADF-STEM image and the EDS elemental mapping of Ga, As, Bi, and their super- impositions at the area with a strong Bi intensity delimited by the red rectangle in the EDS image of (a). Because the image was observed from the [1 10] direction, we can identify the existence of twin defects and crystal polytypes from the atomic arrangement.41,44Of note, the analyzed NW primarily consists of zinc blende, which is considered tobe related to our growth conditions under high As pressure. 44The HAADF image in Fig. 2(b) shows the twin defect in this area. At the twin defect interface, the brighter contrast can be observed, which sug- gests the existence of heavy elements at the interface.34The EDS ele- mental maps of Ga, As, and Bi clearly show the existence Bi at thetwin defect interface. The segregation of Bi is significant because we cannot observe Bi intensity in other areas outside the twin defect inter- face. Thus, Bi primarily exists at the twin defect interface. The Bi con- centration at the interface was about 4%, while the rest of the area contained a concentration of about 1.3% (Fig. S2 in the supplementary material ). Owing to the large difference in the lattice mismatch FIG. 1. (a) Radial cross-sectional BF-STEM image of the NW and (b) a higher magnification BF- and HAADF-STEM images of the area near the tip [marked bythe red rectangle in (a)], (c) higher magnification BF- and HAADF-STEM images ofthe area near the tip [marked by the red rectangle in (b)]. The images were observed from the [1 10] direction. The dashed vertical white lines in (b) and (c) show the interfaces between GaAs core/GaAsBi shell/GaAs outer shells expectedfrom the contrast difference in the image. The red arrows in (c) indicate theobserved horizontal dark lines in the BF-STEM images.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 113105 (2020); doi: 10.1063/5.0013094 117, 113105-2 Published under license by AIP Publishingbetween GaAs and GaAsBi alloys, Bi preferentially segregates to reduce the strain energy.10–15,20–22The atomically layered arrangement of highly mismatched systems (e.g., InAs/GaAs45and InN/GaN46)h a s resulted in a high-quality quantum structure. In our case, Bi preferen-tially segregates at the twin defect interface, which results in the forma-tion of the twin defect interface with accumulated Bi. Because horizontal lines strongly penetrated inside the GaAs core from the GaAsBi shell [which was not clear for the outer GaAs shell, asobserved in Figs. 1(b) and1(c)], the structure was possibly forming until the end of the growth of the GaAsBi shell. Thus, Bi possibly pene-trated through the twin defects into the GaAs core during the GaAsBi shell growth. The horizontally arranged twin defects also affect the outer surface morphology of the NWs. Figure 3(a) shows the radial cross- sectional BF-STEM image of the upper part of the nanowire withhorizontally spread contrast modulations and surface deformations,which were observed in Figs. 1(a)–1(c) . A twisted or zig-zag feature at the nanowire sidewall is observed, whose characteristic corner is indi- cated by the red arrows in the figure. The twisted surface is not as reg- ular as the one, which has been reported for intentionally obtainedcontrolled twinning superlattice in GaAs nanowires. 25–28However, the angle related to the {111} plane is similar to the one in our study.26 Figures 3(b) and3(c)show the BF-STEM and HAADF-STEM images at higher magnification, respectively. The partial dislocation, whichinduces the zig-zag structured twin superlattice, was also observed inour structure close to the sidewall surface, as shown in Figs. 3(b) and3(c). At the defects, the strain-hardening effects are obtained by the blockage of partial dislocations emitted from the free surface by coherent twin boundaries. 26 In summary, we investigated specific microstructural features of the GaAs/GaAsBi/GaAs core–multishell heterostructure triggered bythe existence of twin defects. We observed Bi segregation at the FIG. 2. (a) Radial cross-sectional HAADF-STEM image and EDS elemental mapping of Bi at an upper part of the nanowire with horizontally spread contrast modula tions shown in Fig. 1 ; (b) higher magnification HAADF-STEM image and elemental mapping of Ga, As, Bi, and their superimpositions at the area delimited by the red rectangle i n (a), as observed from the [1 10] direction. The arrow in the HAADF image indicates the twin defect interface. FIG. 3. (a) Radial cross-sectional BF-STEM image at an upper part of the nanowire with horizontally spread contrast modulations and surface deformations sh own in Fig. 1 ; (b) higher magnification BF-STEM image of the area indicated by the red rectangle in (a); (c) higher magnification HAADF-STEM image of the area indicate d by the red rectan- gle in (b), as observed from the [1 10] direction. The red arrows in (a) indicate the position of surface structural modification. The red and green filled circles in (c) indicate Ga and As, respectively.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 113105 (2020); doi: 10.1063/5.0013094 117, 113105-3 Published under license by AIP Publishinginterface of the twin defect interface in the GaAsBi shell. The phenom- enon produced a horizontally spread Bi-accumulated structure in the nanowire, which was probably induced by the large lattice mismatch between GaAs and GaAsBi. Bi penetrated through the twin defectinterface, which resulted in the presence of Bi along the twin defectand inside the GaAs core. The existence of twin defects induced struc- tural deformations, which produced corrugated complex sidewall sur- faces on nanowires. See the supplementary material f o rm o r ed e t a i l e ds e r i e so ft h e STEM images for various magnifications around the investigated areaforFigs. 1(a)–1(c) and quantitative Bi concentration at the twin inter- face corresponding to Fig. 2(b) . This work was partly supported by KAKENHI (Grant Nos. 19H00855 and 16H05970) from the Japan Society of Promotion of Science. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1M. S. Gudiksen, L. J. Lauhon, J. Wang, D. C. Smith, and C. M. Lieber, Nature 415, 617 (2002). 2H. Zhang, C.-X. Liu, S. Gazibegovic, D. Xu, J. A. Logan, G. Wang, N. van Loo, J. D. S. Bommer, M. W. A. de Moor, D. Car, R. L. M. Op het Veld, P. J. van Veldhoven, S. Koelling, M. A. Verheijen, M. Pendharkar, D. J. Pennachio, B. Shojaei, J. S. Lee, C. J. Palmstrøm, E. P. A. M. Bakkers, S. Das Sarma, and L. P.Kouwenhoven, Nature 556, 74 (2018). 3Novel Compound Semiconductor Nanowires: Materials, Devices, and Applications , edited by F. Ishikawa and I. A. Buyanova (Pan Stanford Publishing, Singapore, 2017). 4T. Ma ˚rtensson, C. P. T. Svensson, B. A. Wascaser, M. W. Larsson, W. Seifert, K. Deppert, A. Gustafsson, and L. R. 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5.0017882.pdf

Appl. Phys. Lett. 117, 052408 (2020); https://doi.org/10.1063/5.0017882 117, 052408 © 2020 Author(s).Current-induced bulk magnetization of a chiral crystal CrNb3S6 Cite as: Appl. Phys. Lett. 117, 052408 (2020); https://doi.org/10.1063/5.0017882 Submitted: 10 June 2020 . Accepted: 24 July 2020 . Published Online: 07 August 2020 Yoji Nabei , Daichi Hirobe , Yusuke Shimamoto , Kohei Shiota , Akito Inui , Yusuke Kousaka , Yoshihiko Togawa , and Hiroshi M. Yamamoto ARTICLES YOU MAY BE INTERESTED IN Magnetic transition behavior and large topological Hall effect in hexagonal Mn 2−xFe1+xSn (x = 0.1) magnet Applied Physics Letters 117, 052407 (2020); https://doi.org/10.1063/5.0011570 Large topological Hall effect in an easy-cone ferromagnet (Cr 0.9B0.1)Te Applied Physics Letters 117, 052409 (2020); https://doi.org/10.1063/5.0018229 Room-temperature multiferroic behavior in layer-structured Aurivillius phase ceramics Applied Physics Letters 117, 052903 (2020); https://doi.org/10.1063/5.0017781Current-induced bulk magnetization of a chiral crystal CrNb 3S6 Cite as: Appl. Phys. Lett. 117, 052408 (2020); doi: 10.1063/5.0017882 Submitted: 10 June 2020 .Accepted: 24 July 2020 . Published Online: 7 August 2020 YojiNabei,1,2 Daichi Hirobe,1,2 Yusuke Shimamoto,3 Kohei Shiota,3Akito Inui,3Yusuke Kousaka,3 Yoshihiko Togawa,3and Hiroshi M. Yamamoto1,2,a) AFFILIATIONS 1Research Center of integrative Molecular Systems, Institute for Molecular Science, Okazaki 444-8585, Japan 2Department of Structural Molecular Science, SOKENDAI (Graduate University for Advanced Studies), Okazaki 444-8585, Japan 3Department of Physics and Electronics, Osaka Prefecture University, Sakai 599-8531, Japan a)Author to whom correspondence should be addressed: yhiroshi@ims.ac.jp ABSTRACT Current-induced magnetization has been investigated in a monoaxial chiral crystal CrNb 3S6by means of superconducting quantum interference device magnetometry. We found that bulk magnetization was generated by applying electric current along the principal axis of the monoaxial chiral crystal and that the magnetization changed linearly with the current. Directly detecting such magnetization enables one to estimate the number of spin-polarized electrons. Using this number, we evaluated the spin polarization rate within the framework ofBoltzmann’s equation. We also observed that the current-induced magnetization increased in the vicinity of the phase boundary betweenparamagnetic and forced ferromagnetic phases, which could be attributed to the enhancement of spin fluctuation. We discuss these observa-tions based on a chirality-induced spin selectivity effect enhanced by exchange interactions. Published under license by AIP Publishing. https://doi.org/10.1063/5.0017882 Chirality-induced spin selectivity (CISS) has been found for chi- ral molecular systems, 1,2as demonstrated by spin-polarized photoelec- tron emission3,4and magnetoresistance5–8experiments. These observations indicate that chiral systems emit a charge current, which is spin-polarized along the flow direction, and that the spin polariza- tion is opposite between right- and left-handed structures. The essence of this effect is likely to consist in spatial symmetry breaking by chiralstructure as well as time-reversal symmetry breaking by electric cur- rent via energy dissipation; 9–12the mechanism, however, remains to be clarified. From the perspective of applications, the CISS effect has been used as proof of concept of magnetoresistive memory and enatio- separation.13–16However, the CISS effect has been limited to thin- layer devices with tunneling conduction. This limitation prevents one from studying the spin polarization process inside chiral materials. Recently, the concept of CISS has been extended to a chiral inor- ganic bulk crystal CrNb 3S6,17which is useful to envisage its application to solid state devices. In those experiments, the spin polarization was detected as a voltage drop via the inverse spin Hall effect (ISHE).18–20 The discovery that this chiral crystal exhibits a CISS effect with metal-lic conduction motivates us to examine a process of CISS inside the material. However, the ISHE-based experiment detects spin accumula-tion at the interface between the chiral crystal and the detectionelectrode, unable to present detailed information on the spin polariza- tion inside the bulk. Locating the spin polarization will help not onlyto demonstrate the bulk CISS effect but also to elucidate the CISS mechanism. For instance, topological insulators exhibit spin polariza- tion only on the surface, which corroborates the mechanism of bulk- surface correspondence. 21,22In addition, in the previous transport experiments,17the conversion efficiency between spin and charge degrees of freedom could not be determined, which prevented one from obtaining the absolute value of current-induced spin polariza- tion. Determining this value will also be important in understanding the CISS effect. In this Letter, we report on a current-induced CISS magnetiza- tion in a monoaxial chiral dichalcogenide CrNb 3S6,w h i c hw a s detected by superconducting quantum interference device (SQUID) magnetometry. We found that an observable magnetization was gen- erated parallel to electric current along the principal axis and that thepolarity was reversed by reversing the current direction. These results can be attributed to a bulk magnetization of a spin-polarized state gen- erated by the CISS. Furthermore, magnetization measurements allowed us to obtain the exact number of spin-polarized electrons and to compare it with the number of conducting electrons calculated byBoltzmann’s transport equation. We also examined the temperature Appl. Phys. Lett. 117, 052408 (2020); doi: 10.1063/5.0017882 117, 052408-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apland magnetic field dependences of the CISS magnetization, thereby finding its enhancement in the vicinity of the phase boundary betweenparamagnetic and forced ferromagnetic phases. CrNb 3S6crystallizes in the hexagonal space group P6322. Coordinating atoms to realize the principal sixfold screw axis results inah e l i c a ls t r u c t u r ea ss h o w ni n Fig. 1(a) . Because of large spin–orbit interaction alongside the chiral crystal structure, CrNb 3S6exhibits a magnetic phase transition to chiral helimagnetic order at critical tem-perature T c/C24130 K under zero magnetic fields.23–25Below Tc,i t undergoes subsequent phase transitions into conical and forced ferro-magnetic phases when applying Halong the caxis (critical field H c ¼19.5 kOe at 10 K).26 Bulk crystals of CrNb 3S6were grown by a chemical-vapor trans- port method following previous reports.23,27The obtained crystals had thin plates with the caxis along the thickness direction. In this report, we present representative data of the crystal with dimensions of1.0 mm /C21.0 mm /C20.2 mm ( caxis). The magnetic field ( H)a n dt e m - perature ( T) dependences of the magnetization provided a H–Tphase diagram which is consistent with that reported elsewhere [also seeFig. 3(a) ]. 28We note that selectively growing pure enantiomers is gen- erally difficult. However, we expect that the present sample exhibitsreasonable enantio excess because of disproportion between the twoenantiomers. This is because a chirality-induced transport effect, calledthe electrical magnetochiral effect, has been found for bulk crystalsgrown by the same method as that used in the present study. 28 To detect the CISS, we investigated the magnetization of CrNb 3S6by SQUID (Quantum Design, Inc.) while applying electric current to CrNb 3S6. The SQUID measurement was made with the sample on a printed circuit board (PCB) [see Fig. 1(b) ]. A PCB (FR-4 No. 32, Sunhayato) was cut into a cuboid of 200 /C26/C21.6 mm3,t h e length of which was larger than a pickup-coil length of 30 mm. Sincethe detection by pickup coil is sensitive to a spatial variation of a mag-netic moment, the PCB magnetic components were hardly detectablein the SQUID measurement along the pickup-coil length. The surfaceof the patterned PCB was polished using alumina powder #1000 (Sankyo Fuji Star), followed by ultrasonication to remove magnetic residues. A CrNb 3S6sample was attached to the PCB so that the caxis was along the length of the PCB. Gold wires with a diameter of0.05 mm were attached to the top and bottom facets of the CrNb 3S6 plate to apply electric current using a source measure unit (Keithley 2400). The magnetic moment along the caxis was measured while applying electric current along the caxis in the presence of the mag- netic field in the same direction. We note that the Oersted field due tothe current generated a background signal in the magnetization mea-surement. However, such a signal should be negligibly small because the Oersted field is predominantly at right angles to the detection axis of SQUID equipment in the present configuration. It was estimated tobe as small as 3.0 /C210 /C07emu at 30 mA with a reference sample of an achiral indium. Figure 2(a) shows M–Hcurves at T¼300 K at currents of þ30 and/C030 mA (3.0 /C2104A/m2).Mincreases with Hbetween þ20 kOe and/C020 kOe for each current. Remarkably, the M–H curve is shifted vertically by current application and the shift changes sign upon reversing the current direction. We note that Joule heating by current application cannot explain this shift since the heating simply changesthe absolute value of Mand yields a H-odd change in M.To eliminate this component, we consider a H-even component of M,w h i c hi s defined as DM even¼MþHðÞ þM/C0HðÞ/C2/C3 2: (1) DMevenwill reflect a CISS-induced magnetization because the spin- polarization direction of CISS is independent of the direction of H.29 DMevenis plotted as a function of HinFig. 2(b) .W ef o u n dt h a t DMevenappears only when current is applied and that DMevenchanges sign when the current direction is reversed while being independent ofH. These results show that spin polarization is generated by charge current in the present chiral system and that its direction is FIG. 1. (a) Crystal structure of CrNb 3S6. (b) Schematic illustration of the experimental setup for the SQUID measurement (right) and the optical micrograph of a CrNb 3S6thin plate sample (left). Scale bar , 0.5 mm. M,I, and Hdenote the magnetization, electric current, and magnetic field, respectively.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 052408 (2020); doi: 10.1063/5.0017882 117, 052408-2 Published under license by AIP Publishingdetermined by the current direction. This is consistent with the CISS symmetry reported experimentally.17We note that a background signal due to the Oersted field was insignificant because its value(/C243.0/C210 /C07emu at 30 mA) was one order of magnitude smaller than the unnormalized value of DMeven(/C241.5/C210/C06emu). We investigated the current dependence of DMevenby applying currents of 615 and645 mA. Figure 2(c) shows DMevenvsHat each current, whose values depend only on the applied current, not on H. Figure 2(d) shows the current dependence of DMevenatH¼10 kOe. DMevenis proportional to the applied current within þ45 mA and /C045 mA, which is also consistent with a current-linear CISS in the same compound.17Our results for magnetization measurements pro- vide solid evidence of a bulk nature of the CISS. We also note that, thanks to its current linearity, the CISS-driven bulk magnetization will be able to be understood using linear response theory, thereby helpingto theoretically elucidate the CISS. Another benefit of directly measuring the magnetization is the capability to obtain the exact number of spin-polarized electrons.From measured DM even, we see that the number of polarized electrons was 3.0 /C210/C04per unit cell when 30 mA was applied. It is notable that in the ISHE experiment,17a linear relationship between input and out- put continued up to a current density of 8.6 /C2108A/m2,w h i c hi s about 104times larger than those used in the present experiment. If this linear response continues to such a current density in SQUIDexperiment as well, the number of spin-polarized electrons increasesby 10 4times. This is equivalent to a spin-polarized electron of order unity per unit cell. Although there has been no saturation observed inthe transport measurement yet, it seems possible to realize a half- metal-like state by applying a sufficient current to chiral systems. We will discuss a spin polarization rate ( SP) within the frame- work of Boltzmann’s transport equation with relaxation time approxi- mation. Let us tentatively define SPas SP%ðÞ¼Np Nc/C2100 : (2) Here, Npis the number of spin-polarized electrons and Ncis the num- ber of conducting electrons that yield a drift current. To calculate Nc, we simply assume that the carrier concentration is the same for all thecrystal axes. Accordingly, the mobility along the caxis, l c, can be expressed as lc¼RH;ab=qc.RH;abis the Hall coefficient within the ab plane, and qcis the resistivity along the caxis. RH,ab/C241/C210/C02 cm3C/C01(Ref. 30)a n d qc/C243/C2104lXcm (Ref. 31)a t3 0 0K . Ncwas then estimated to be 2 /C210/C09per unit cell with an electric current of 30 mA by calculating an electric field-induced shift of the Fermi energy level, dEF,a sdEF¼/C22hkFlcE(/C22h,kF,a n dEare the reduced Planck con- stant, the Fermi wavenumber, and an electric field, respectively).32 Simply substituting this value in Eq. (2)yielded SPof 107%, which is far above 100%. Such a large value indicates that spin polarization originates not only from the conducting electrons carrying a drift cur-rent but also from the other electrons irrelevant to the conduction. This additional polarization could be explained by exchange interac- tion of the conducting electrons with the others including both itinerant and localized electrons: The conducting electrons become spin-polarized by the CISS, followed by spin polarization of the otherelectrons via exchange interaction. This mechanism seems to be related to spin-transfer torque, which is usually detected in multilayer devices with magnetic fixed and free layers being spaced by a non-magnetic layer. In the present study, however, electrons that generate and receive spin torque coexist in the same material, which is different from typical spin transfer torque. We note that a previous CISS experi-ment 13demonstrated replacement of a magnetic fixed layer with a layer of chiral molecules in typical spin transfer torque experiments. In addi- tion, our conjecture about the additional polarization may be associated with the mechanism proposed in a theoretical paper,33which relates the high spin polarization of CISS to electron correlation. Indeed, CrNb 3S6 appears to be a strongly correlated system with a large exchange interac- tion, which was pointed out by a photoemission experiment.34We admit, however, that further studies are needed to support the above-mentioned mechanism of additional spin polarization with exchange interaction. One way of examining the effect of exchange interaction is by applying pressure to crystals, thanks to the tunability of electron correlation. We also investigated the temperature and magnetic field depend- ences of DM even, which are shown in Fig. 3(a) as aH–T color contour map together with the phase diagram of CrNb 3S6. The boundary between chiral conical and FM (forced ferromagnetic) phases was determined both from inflection points in M–H curves and from kinks inM–T curves. The FM-to-PM (paramagnetic) phase boundary is known to be almost independent of Hand is approximated by a straight line at Tc¼132 K.28In this plot, DMevenw a sd e fi n e da sh a l fo f a difference between those values at I¼þ30 mA and /C030 mA to con- sider a current-odd component. In the PM phase, DMevenwas nearly independent of Tfrom 300 K down to 150 K. In contrast, DMeven increased in the vicinity of the PM-to-FM phase boundary and thenFIG. 2. (a)M–H curves under I¼þ 30 and /C030 mA within a range of 64 kOe. The inset shows the M–Hcurves within a range of 630 kOe. (b) DMeven-Hcurves under I¼630 and 0 mA. The definition of DMevenis given in Eq. (1). (c)DMeven-H curves under I¼645 and615 mA. The error bars in (b) and (c) represent the 68% confidence level ( 6s.d.). The magnitude of the standard deviations was smaller than the marker size except for H¼2 kOe. (d) Current dependence of DMevenatH¼10 kOe. The line shows linear fitting as a function of I.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 052408 (2020); doi: 10.1063/5.0017882 117, 052408-3 Published under license by AIP Publishingdecreased upon entering the FM or conical phase. This behavior is exemplified by a T-scan at H¼14 kOe at I¼þ30 mA as shown in Fig. 3(b) . The magnitude of DMevenexhibited its maximum around TC and decreased upon cooling into the FM or conical phase. Magnetic phase transition commonly accompanies the enhancement of relevant spin correlation. Such enhanced correlation can increase a contribu- tion from localized spins to DMevenand give a peak structure observed at the PM-to-FM phase boundary. Importantly, the signals show almost the same value in the high- and low-temperature regions far away from the phase transition. Thisresult suggests that the obtained value of DM even/C243:0/C210/C04lB/C1 u:c/C01is insensitive to the magnetic order. Thus, this value can be attributed to itinerant electrons affected by CISS and exchange interac- tion with the spin-polarized conducting electrons that yield a drift cur- rent. We evaluated this interaction as mean field Hmby using the Pauli paramagnetic model: v¼M=Hm¼2l2 BDðEFÞ,w h e r e v,lB, andDðEFÞare the susceptibility, the Bohr magneton, and the density of states at the Fermi level, respectively. For the magnitude of DðEFÞ, we referred to the one obtained with the first-principles calculation.35 Finally, Hmturned out to be /C241 kOe (0.1 T in SI units) at I¼30 mA. If the current-linear relation continues up to the order of 108A/m2 similar to the previous experiment, Hmincreases up to 104kOe (103T). From an application perspective, exchange interaction alongsidespin correlation may be used for highly efficient CISS devices and sensors. We admit that, however, further investigations are needed to interpret the above results in depth. For instance, we have notconsidered current-induced orbital magnetization 36,37so far, which originates from the microscopic helical motion of electrons in chiral structures. Separating out this contribution, if any, will be an impor-tant future task. In conclusion, we detected current-induced magnetization in a chiral crystal CrNb 3S6by SQUID magnetometry, which was attributed to CISS. Our result has shown that the CISS in CrNb 3S6stems from spin polarization within the bulk. Owing to its current linearity, the CISS-driven magnetization will be able to be understood using well- established linear response theory, thereby helping to elucidate thebulk CISS. Additionally, the present magnetization measurements allow us to estimate the number of spin-polarized electrons. The esti- mated number exceeded the number of conducting electrons thatcarry a drift current. It was also discussed based on the field and tem- perature dependences. These results required us to consider exchange interaction as a possible cause of additional spin polarization of elec-trons other than the conducting electrons carrying a drift current. Furthermore, the enhancement of the CISS magnetization takes place in the vicinity of the phase boundary between the paramagnetic andmagnetically ordered phases. We anticipate that exchange interaction and spin correlation could serve as a guiding principle for enhancing the CISS effect in bulk materials. We acknowledge support from Grants-in-Aid for Scientific Research (Nos. 17H02767, 19K03751, 19H00891, 19K21039, and 20K20903) and the Research Grant of Specially Promoted Research Program by Toyota RIKEN. The authors are grateful to theEquipment Development Center, the Institute for Molecular Science, for technical assistance, especially T. Toyoda, S. Kimura, and T. Kondo. This work was partly conducted in the InstrumentCenter (Institute for Molecular Science), supported by the Nanotechnology Platform Program (Molecule and Material Synthesis) of the Ministry of Education, Culture, Sport, Science andTechnology (MEXT), Japan. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1R. Naaman and D. H. Waldeck, J. Phys. Chem. Lett. 3, 2178 (2012). 2R. Naaman, Y. Paltiel, and D. H. Waldeck, Nat. Rev. Chem. 3, 250 (2019). 3B. Gohler, V. Hamelbeck, T. Z. Markus, M. Kettner, G. F. Hanne, Z. Vager, R. Naaman, and H. Zacharias, Science 331, 894 (2011). 4M. Kettner, B. Gohler, H. Zacharias, D. Mishra, V. Kiran, R. Naaman, C. Fontanesi, D. H. Waldeck, S. Sek, J. Pawlowski, and J. Juhaniewicz, J. Phys. Chem. C 119, 14542 (2015). 5Z. Xie, T. Z. Markus, S. R. Cohen, Z. Vager, R. Gutierrez, and R. Naaman, Nano Lett. 11, 4652 (2011). 6A. C. Aragones, E. Medina, M. 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5.0012103.pdf

J. Appl. Phys. 128, 034304 (2020); https://doi.org/10.1063/5.0012103 128, 034304 © 2020 Author(s).Two-dimensional gallium and indium oxides from global structure searching: Ferromagnetism and half metallicity via hole doping Cite as: J. Appl. Phys. 128, 034304 (2020); https://doi.org/10.1063/5.0012103 Submitted: 28 April 2020 . Accepted: 22 June 2020 . Published Online: 17 July 2020 Ruishen Meng , Michel Houssa , Konstantina Iordanidou , Geoffrey Pourtois , Valeri Afanasiev , and André Stesmans COLLECTIONS Paper published as part of the special topic on 2D Quantum Materials: Magnetism and Superconductivity Note: This paper is part of the Special Topic on 2D Quantum Materials: Magnetism and Superconductivity This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN Searching for stable 2D gallium and indium oxides Scilight 2020 , 291113 (2020); https://doi.org/10.1063/10.0001655 Oxygen vacancies: The (in)visible friend of oxide electronics Applied Physics Letters 116, 120505 (2020); https://doi.org/10.1063/1.5143309 Synthesis, properties, and applications of 2D amorphous inorganic materials Journal of Applied Physics 127, 220901 (2020); https://doi.org/10.1063/1.5144626Two-dimensional gallium and indium oxides from global structure searching: Ferromagnetism and half metallicity via hole doping Cite as: J. Appl. Phys. 128, 034304 (2020); doi: 10.1063/5.0012103 View Online Export Citation CrossMar k Submitted: 28 April 2020 · Accepted: 22 June 2020 · Published Online: 17 July 2020 Ruishen Meng,1,a) Michel Houssa,1,a) Konstantina Iordanidou,2 Geoffrey Pourtois,3Valeri Afanasiev,1 and André Stesmans1 AFFILIATIONS 1Department of Physics and Astronomy, KU Leuven, Celestijnenlaan 200D, Leuven B-3001, Belgium 2Department of Physics, University of Oslo, NO-0316 Oslo, Norway 3imec, Kapeldreef 75, B-3001 Leuven, Belgium Note: This paper is part of the Special Topic on 2D Quantum Materials: Magnetism and Superconductivity a)Authors to whom correspondence should be addressed: ruishen.meng@kuleuven.be andmichel.houssa@kuleuven.be ABSTRACT There has been tremendous research effort in hunting for novel two-dimensional (2D) materials with exotic properties, showing great promise for various potential applications. Here, we report the findings about a new hexagonal phase of 2D Ga 2O3and In 2O3,w i t hh i g he n e r g e t i c stability, using a global searching method based on an evolutionary algorithm, combined with density functional theory calculations. Theirstructural and thermal stabilities are investigated by the calculations of their phonon spectra and by ab initio molecular dynamics simulations. They are predicted to be intrinsically non-magnetic stable semiconductors, with a flatband edge around the valence band top, leading to itiner- ant ferromagnetism and half-metallicity upon hole doping. Bilayer Ga 2O3is also studied and found to exhibit ferromagnetism without extra hole doping. The Curie temperature of these materials, estimated using Monte Carlo simulations based on the Heisenberg model, is around40–60 K upon a moderate hole doping density. Published under license by AIP Publishing. https://doi.org/10.1063/5.0012103 I. INTRODUCTION Computational screening of novel materials with intriguing properties has been attracting enormous research attention recently. 1–4Among this research, the prediction of new two- dimensional (2D) materials, especially those with magnetic proper-ties, has also been widely explored. 5–8These investigations have facilitated and accelerated the experimental discovery of materials fora variety of applications. Nevertheless, some of the predicted materi- als, found from property-oriented materials searching, may turn out to be unstable and very challenging to grow or synthesize. 9–13 Therefore, stricter criteria are necessary for materials prediction, to make sure the candidate materials have global minimum of freeenergy, as well as high structural and thermal stabilities. Various 2D ferromagnetic materials, for example, CrX 3(X = Cl, Br, I),14–16Cr2Ge2Te6,16,17Fe3GeTe 218,19monolayers, etc., have been theoretically predicted and experimentally synthesized. Furtherstrategies, such as electrostatic doping, electric field or strain effects, and van der Waals stacking, have been intensively exploited to tune the magnetic properties of those monolayers.15,17,18,20–25Apart from 2D materials with intrinsic magnetic properties, efforts have alsobeen made to induce ferromagnetism in some nonmagnetic 2Dsemiconductors by, e.g., hole doping. 26–29The spontaneous ferro- magnetic order in those intrinsically nonmagnetic 2D materials orig-inates from the exchange splitting of the electronic states at the topof the valence band (VB), where a sharp Van Hove singularity existsin the density of states (DOS), leading to the Stoner instability.Therefore, these nonmagnetic 2D materials could also be promisingfor application in novel spintronic devices. In this work, we explore all possible 2D phases of gallium oxides and indium oxides by means of the Universal Structure Predictor: Evolutionary Xtallography (USPEX) method, on the basisof first-principles calculations, to find out the most energeticallyJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 034304 (2020); doi: 10.1063/5.0012103 128, 034304-1 Published under license by AIP Publishing.stable atomic configurations, i.e., M 2O3(M = Ga, In). Their stability is demonstrated by the formation energies, phonon dispersions, and by molecular dynamics simulations. We find that M 2O3monolayers are nonmagnetic semiconductors, while hole doping can turn theminto ferromagnetic materials over a wide range of hole concentra-tions. Their Curie temperatures ( T c) under different hole doping concentrations are also estimated. II. COMPUTATIONAL METHODS The search for the possible structures of 2D gallium oxides and indium oxides, through an ab initio evolutionary algorithm, was performed by USPEX,30,31interfaced with the Vienna ab initio simulation package (VASP).32,33The variable-composition search- ing was considered, with the total number of atoms in the 2D crys-tals fixed between 2 and 8, and their vacuum spaces restricted to be within 20 Å. 40 groups of symmetries were used to produce a random symmetric structure generator for the initial population.Then, the full structure relaxations were performed, and the moststable and metastable atomic structures were screened and inheritedinto the next generation, by comparing their formation enthalpy. The population kept evolving until the most stable configuration stayed unchanged for a further ten generations, in order to findthe globally lowest energy state. All density functional theory(DFT) calculations were performed using VASP, with electron –ion interaction described by projector augmented wave (PAW) pseudo- potentials. 32The generalized gradient approximation (GGA), parameterized by the Perdew –Burke –Ernzerhof (PBE) approach,34 was used as the exchange correlation functional. The energy cutoff of 550 eV and k-point meshes of 8 × 8 × 1 and 32 × 32 × 1 wereused for structural optimizations and self-consistent calculations, respectively. A total energy convergence criterion of 10 −5eV and a force convergence criterion of 0.005 eV/Å were chosen for completerelaxations of the lattice constants and the atomic positions. TheHeyd –Scuseria –Ernzerhof functional (HSE06), 35which mixes 25% nonlocal exchange with the PBE functional, was used for more accurate electronic band structure and density of states (DOS)calculations. Additionally, the optimized exchange van derWaals functional, optB86-vdW, 36was used to take into account the van der Waals interactions in the calculations of the Ga 2O3 bilayers (BLs). Phonon dispersion curves were calculated using the PHONOPY package37on the basis of the Density Functional Perturbation Theory (DFPT). The ab initio molecular dynamics (AIMD) simulations were carried out using 5 × 5 × 1 supercells (with 125 atoms) and were equilibrated at 300 K for 10 ps with a time step of 2 fs. The constant-temperature and volume canonicalensemble (NVT) was selected, and the temperature was controlledby means of a Nosé thermostat. 38,39 The Curie temperatures were estimated using Monte Carlo simulations, as implemented in the VAMPIRE package.40 Rectangular supercells of 100 × 60 √3 × 1 were used for the simula- tions, since only orthogonal lattice vectors are supported. The spinswere thermalized for 10 000 equilibrium steps, followed by 20 000 averaging steps for the calculation of the thermal equilibrium mag- netization at every temperature.III. RESULTS AND DISCUSSION The convex hulls, defined as the enthalpy of formation vs the composition of gallium oxides and indium oxides with all possiblestoichiometries, are plotted in the left panel of Figs. 1(a) and1(b), respectively. The formation enthalpy per atom, △H, which can be used to determine the relative stability between different composi-tions and structures, was calculated by ΔH¼(E MxOy–xEM–yEO2/2)/( xþy), (1) where EMxOyis the total energy of M xOy(M = Ga, In), EMis the energy of the metal atom in its stable bulk structure, EO2corre- sponds to the energy of an isolated oxygen molecule in its para- magnetic ground state. Any structure with its formation enthalpyon the convex hull is considered to be thermodynamically stableand experimentally synthesizable. The most stable structures of Ga–O and In –O, highlighted by the red dots in the convex hulls, are Ga 2O3and In 2O3, respectively. Specifically, their △Hcan, respectively, reach −1.95 eV/atom and −1.52 eV/atom. In addition, the GaS-like GaO monolayer is also found to be stable, since itsformation enthalpy ( −1.65 eV/atom) is on the convex hull. In this paper, we only focus on the most stable M 2O3monolayers. The relaxed crystal structures of single-layer M 2O3are given in the right panel of Fig. 1 , which consist of five atomic sublayers stacked in the sequence of O –M–O–M–O in the out-of-plane direction. The calculated in-plane lattice constants of Ga 2O3and In2O3monolayers are 3.10 Å and 3.04 Å, respectively. The Ga 2O3 monolayer is non-centrosymmetric (polar), similarly to In 2X3 (X = S, Se) monolayers.41,42On the other hand, the In 2O3mono- layer is found to be centrosymmetric (non-polar). The oxygenatoms on the top and bottom layer are bonded to three neighboring metal atoms, while the central oxygen atoms bond to four and six metal atoms for Ga 2O3and In 2O3monolayers, respectively. The indium atoms are sixfold coordinated in In 2O3, while in Ga 2O3, the gallium atoms in the upper and lower layers are sixfold and four- fold coordinated, respectively. Phonon dispersion calculations, together with AIMD simula- tions, were carried out to evaluate the dynamic stability as well asthe room-temperature stability of the two M 2O3monolayers; the results are presented in Fig. 2 . In the phonon dispersion spectra, merely no imaginary frequency is observed, which suggests that both M 2O3monolayers are dynamically stable. During the AIMD simulations, with the temperature fixed at 300 K, the total energiesof M 2O3monolayers oscillate within narrow ranges, i.e., ∼3 eV/cell. In the addition, the atoms only slightly oscillate around their equi- librium positions, and the structural integrity is well retained after 10 ps. Given their low formation enthalpy, dynamical stability,together with their thermal stability, the growth of M 2O3monolay- ers appears highly possible. From the electronic band structures shown in Fig. 3 , one can see that both M 2O3monolayers are semiconductors with indirect bandgaps. Specifically, the bandgap of the Ga 2O3monolayer is ∼1.43 eV at the PBE level ( ∼2.72 eV at the HSE06 level, see Fig. S1 in the supplementary material ); the bottom of the conduction band (CB) is contributed by the sand pzorbital of oxygen atoms and s orbitals of gallium atoms (denoted as O- s,O - pz, and Ga- s,Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 034304 (2020); doi: 10.1063/5.0012103 128, 034304-2 Published under license by AIP Publishing.respectively), whereas the top of the valence band (VB) is mainly contributed by the O- pyand O- pzorbitals. In particular, the valence band maximum (VBM) essentially originates from the pz orbital of the top oxygen atoms, as clearly shown in the partial charge density plot. Concerning the In 2O3monolayer, the bandgap calculated at the PBE level is ∼1.64 eV ( ∼2.93 eV at the HSE06 level, see Fig. S1 in the supplementary material ). The bottom CB originates mainly from the hybridization of O- pzand In- sorbitals, while the top VB is merely derived from the O- pzorbital. It is evident that the partial charge density of the VBM is only located on the oxygen atoms of the In 2O3monolayer. In the DOS of the M 2O3monolayers, there is a sharp peak in the edge of valence band. This sharp peak is related to so-called Van Hove singularities. When the Fermi level approaches singular- ity, a transition from a paramagnetic phase to a ferromagneticphase could occur, as reported in several previous works.26–29 Therefore, in the following, we will study hole doping of M 2O3 monolayers, using spin-polarized DFT simulations, to investigate their possible magnetic properties. Since dipole corrections are only implemented for charged cubic systems in VASP, they could not be included in our simulations. The effect of the dipole corrections onthe band structure of a (undoped) Ga 2O3monolayer was tested, and no noticeable difference was observed, compared to the onewithout dipole corrections. By performing the relaxation of the charged cells, the effect of hole doping on the atomic structures is first investigated. The struc-tural variation of the M 2O3monolayers upon hole doping is sum- marized in Fig. S2 in the supplementary material . The thickness of the Ga 2O3monolayer shrinks when it is hole doped, while for In2O3monolayer, the thickness first decreases when the doping FIG. 1. Convex hulls of different stoichiometries searched by USPEX, and the corresponding atomic structures of the most energetically favorable candidat es for 2D binary (a) Ga –O and (b) In –O materials. The red dots on the Convex hulls indicate the predicted stable phases. The unit cell is highlighted by yellow solid lines.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 034304 (2020); doi: 10.1063/5.0012103 128, 034304-3 Published under license by AIP Publishing.density is lower than ∼4×1 014cm−2and then increases and becomes larger than the original thickness before doping for a doping densityhigher than ∼7×1 0 14cm−2.I ng e n e r a l ,t h eM 2O3monolayers do not suffer pronounced deformation upon hole doping, as the variation of the bond lengths and thicknesses is typically less than 5%, even at a high hole doping concentration of 1015cm−2, except for the In –In distance ( ∼8% elongation) and a specific O –Ga bond length ( ∼12% elongation) (see Fig. S2 in the supplementary material ). Therefore, we can conclude that hole doping has a limited effect on the atomic structures of the M 2O3monolayers.The computed magnetic moments and spin polarization ener- gies of M 2O3monolayers are shown in the left panel of Fig. 4 ,a sa function of the hole doping density. The spin polarization energy isdefined as the total energy of the non-magnetic state minus the one of the ferromagnetic state. Thus, a positive spin polarization energy indicates that the ferromagnetic state is more stable. The paramag-netic to ferromagnetic transition occurs at a hole concentration aslow as 5 × 10 12cm−2. In this case, the magnetic moment is ∼1.0μB/ hole (where μBis the Bohr magneton), i.e., the injected holes are fully spin polarized. The spin polarization energy increases FIG. 2. Phonon dispersion spectra, and total energy evolution during molecular dynamics simulations, together with the atomic structures obtained after 1 0 ps, for (a) Ga2O3and (b) In 2O3monolayers at 300 K.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 034304 (2020); doi: 10.1063/5.0012103 128, 034304-4 Published under license by AIP Publishing.monotonously with the hole density, up to a hole concentration of ∼1015cm−2for both M 2O3monolayers. The spin-polarized DOS of M 2O3monolayers under different doping levels are presented in the right panel of Fig. 4 , where the energy splitting between spin up and spin down states can beobserved. The Fermi level crosses the spin down states, while the up-spins are fully occupied at the Fermi level, making these two materials half-metallic upon hole doping. Obviously, the energysplitting increases with the increase of the hole density, i.e., from∼0.3 eV ( ∼0.2 eV) at 2 × 10 14cm−2to∼0.8 eV (0.6 eV) at 6×1 014cm−2for the Ga 2O3(In2O3) monolayer, respectively. In addition, the valence band top still mainly consists of the porbitals of the oxygen atoms when the hole doping concentration increases.Therefore, the oxygen atoms mostly contribute to the ferromag-netic order in the M 2O3monolayers.Similar to the situation in In 2X3(X = S, Se) monolayers, the centrosymmetry of the Ga 2O3monolayer is broken, due to the dif- ferent coordination environment of the Ga atoms, which resultsfrom the asymmetric positions of O atoms in the middle layer.This leads to an intrinsic dipole, pointing from the bottom to thetop surface along the out of plane direction. It has been reported that the dipole moment will be almost doubled in In 2X3(X = S, Se) bilayers.42Consequently, we also considered Ga 2O3bilayers to investigate the influence of this enhanced polarization on the mag-netic properties. We constructed six different stacking patterns, asshown in Fig. S3 in the supplementary material , and the most stable one was studied further. Using the optB86-vdW functional and spin-polarized calculations for the atomic relaxation, the inter-layer distance is calculated to be 2.29 Å, and the lattice constant is3.10 Å. Surprisingly, we found that the Ga 2O3bilayer exhibits FIG. 3. Flatband structures, density of states, and partial charge densities of the conduction band minimum (CBM) (blue) and VBM (red) for (a) Ga 2O3and (b) In 2O3 monolayers. The isosurface value is set to 0.2 e/Å3.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 034304 (2020); doi: 10.1063/5.0012103 128, 034304-5 Published under license by AIP Publishing.ferromagnetism without hole doping, with a magnetic moment of ∼0.03μBper unit cell, which is comparable to a hole doping density of ∼4×1 013cm−2. To find out the underlying mechanism responsible for the spontaneous ferromagnetism, we computed thepartial band structures, partial charge densities, as well as thecharge density difference of the Ga 2O3bilayer, presented in Fig. 5 . Clearly, the bottom CB of the double layer is solely contributed by the bottom layer, while the top VB is originating from the top layer, which gives rise to the typical type-II band alignment. Inaddition, the CB crosses the Fermi level, making the bilayer ametal. This is also confirmed by the HSE band structure calcula-tions, as shown in Fig. S1(c) in the supplementary material . On the other hand, the porbitals of the topmost oxygen atoms of the toplayer (O t) mostly contribute to the top VB, as shown in the band structure; this is also confirmed by the partial charge density plot shown in Fig. 5(b) . Note that these topmost oxygen atoms mainly contribute to the magnetic moment of the Ga 2O3bilayers. The intrinsic dipole present in the bilayer is induced by electrons trans-ferred from the top layer to the bottom layer, resulting in the “hole doping ”of the top layer, and the occurrence of a ferromagnetic order in the material; this is also reflected by the charge density dif- ference plot in Fig. 5(c) . The magnetic properties of the Ga 2O3bilayer, induced by increasing further the hole doping density, was also investigated.The computed magnetic moment, shown in Fig. 5(d) , is larger than 1.0μB/hole when the hole doping density is less than or equal to FIG. 4. Magnetic moment and spin polarization energy as a function of hole doping concentration, and spin-polarized partial density of states under differe nt hole doping densities of (a) Ga 2O3and (b) In 2O3monolayers.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 034304 (2020); doi: 10.1063/5.0012103 128, 034304-6 Published under license by AIP Publishing.1×1 014cm−2and then it decreases gradually to ∼1.0μB/hole when the doping density is further increased. Nevertheless, the magneticmoment gradually declines again when the hole doping islarger than ∼5×1 0 14cm−2, and eventually reaches zero at ∼7×1 014cm−2. Thereafter, the magnetic moment oscillates between zero and one when the hole density further increases. Bycomparing the DOS at hole concentrations of 4 × 10 14cm−2, 7×1 014cm−2, and 8 × 1014cm−2in the right panel of Fig. 5(d) ,i ti s evident that the upmost oxygen atoms of the bottom layer (O b) play more and more important roles when the hole concentrationis larger than ∼4×1 014cm−2, as its contribution to the DOS becomes closer to the Fermi level. Thereafter, at ∼7×1 014cm−2, the DOS contributed by the O tand O bis merely the same, but with opposite spin direction, which is especially noticeable around the Fermi level. This leads to an antiferromagnetic state in the system. Finally, at ∼8×1 014cm−2, the DOS contributed respec- tively by the O tand O balmost overlap. In this case, the ferromag- netic state is more energetically favorable. The Curie temperatures ( Tc) of the M 2O3monolayers as well as the Ga 2O3bilayer were next computed, using Monte Carlo FIG. 5. (a) Flatband structures, (b) partial charge densities of the CBM (blue) and VBM (red), (c) charge density difference (yellow and cyan represent charg e accumula- tion and depletion, respectively), and (d) magnetic moment and spin polarization energy as a function of hole doping concentration and spin-polariz ed partial density of states under different hole doping densities of a Ga 2O3bilayer. The isovalues of partial charge densities and charge density differences are fixed to 0.005 e/Å3and 0.2 e/ Å3, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 034304 (2020); doi: 10.1063/5.0012103 128, 034304-7 Published under license by AIP Publishing.simulations. Based on the 2D Ising model, their spin Hamiltonian is given by H¼/C0X ijhiJ1Si!/C1~Sj/C0X ijhihiJ2Si!/C1~Sj, (2) where ⟨ij⟩and ⟨⟨ij⟩⟩are the summation over the pairs of nearest and second nearest oxygen atoms, respectively. J1and J2are the nearest-neighbor and second nearest-neighbor exchange interac-tions, respectively. S i!(or~Sj) is a unit vector. For Ga 2O3monolayer and bilayer, we only considered the nearest-neighbor exchange parameter, since only the topmost oxygen atoms contribute to the magnetic moment, while for the In 2O3monolayer, we considered both exchange parameters, which are derived from the two Oatoms in the same or different planes. These exchange parameterscan be evaluated using the energy difference between the ferromag- netic and antiferromagnetic phases, obtained from first-principles calculations. Details about the construction of the ferromagneticand antiferromagnetic configurations as well as the calculation ofthe exchange parameters are given in Figs. S4 and S5 in thesupplementary material . The values of the exchange parameters under different hole doping concentrations are summarized in Table S3 in the supplementary material . It is clear that a higher hole density gives rise to larger exchange parameters. The temperature-dependent normalized magnetization curves, computed from Monte Carlo simulations under different hole doping densities, are displayed in Fig. 6 . The magnetization remains high in the low temperature range, and then drops tonearly zero at a certain critical (Curie) temperature. Thetemperature-dependent magnetization is fitted using the Curie –Bloch equation in the classical limit, m(T)¼1/C0 T Tc/C18/C19β , (3) where Tis the temperature and βis a critical exponent. Obviously, Tcis proportional to the exchange interactions and the hole doping concentrations in M 2O3monolayers and Ga 2O3bilayer. At the same doping level, the In 2O3monolayer has the highest Tc, fol- lowed by the Ga 2O3bilayer, and then by the Ga 2O3monolayer. At a hole concentration of ∼2×1 014cm−2, the fitted Tcis∼20 K/24 K/ 27 K for Ga 2O3monolayer/Ga 2O3bilayer/In 2O3monolayer, respec- tively, while it increases to 36 K/39 K/43 K when the hole doping density reaches ∼4×1 014cm−2.A t∼6×1 014cm−2,Tcis 45 K and 62 K for Ga 2O3and In 2O3monolayers, respectively, being compa- rable to the ones of monolayer CrI 3(45 K)16and bilayer Cr 2Ge2Te6 (30 K).43Although the second nearest-neighbor exchange interac- tion ( J2) is several times larger in In 2O3as compared to Ga 2O3, their computed Curie temperatures are comparable. It may be due to the “competition ”between J1and J2, arising from the weaker antiferromagnetic interaction and stronger ferromagnetic interac-tion in the In 2O3monolayer, leading to a lower Tcthan expected. The effect of spin –orbit coupling (SOC) is also tested. The band structures and temperature-dependent normalized magnetizationcurve of the Ga 2O3monolayer with and without SOC at the PBE level are provided in Fig. S6 in the supplementary material . No sig- nificant difference is observed for the valence band edge, since it is mainly contributed by the O- porbitals. Likewise, the SOC also has limited effect on the Tc. IV. CONCLUSIONS In summary, by combining first-principles calculations with the evolutionary algorithm USPEX, we discovered energeticallystable phases of 2D Ga 2O3and In 2O3. Calculations of their phonon dispersion curves and ab initio molecular dynamics simulations confirmed their structural and thermal stability. Both Ga 2O3and In2O3monolayers are predicted to be intrinsically non-magnetic semiconductors, with flatband edges at the top of their valencebands, resulting in sharp peaks in their electronic density of states. A spontaneous ferromagnetic phase transition occurs upon hole doping of these 2D materials, which turns them into half-metalsover a wide range of hole densities. The bilayer structure of Ga 2O3was also investigated and it showed ferromagnetism without external hole doping, resulting from the non-centrosymmetric structure of 2D Ga 2O3. In this case, the spontaneous ferromagnetic order is triggered by the chargetransfer taking place between the two layers of the Ga 2O3bilayer. Using Monte Carlo simulations, based on the Heisenberg model,the estimated Curie temperatures of the Ga 2O3and In 2O3mono- layers and the Ga 2O3bilayer are proportional to the hole doping density and are comparable to the ones of CrI 3monolayer and 2D Cr2Ge2Te6bilayer at moderate hole doping densities. We hope that our work can inspire the synthesis of these new 2D materials and the experimental study of their predicted magnetic properties upon hole doping. FIG. 6. Monte Carlo simulations of the normalized magnetization of M 2O3 monolayers and Ga 2O3bilayer (BL) as a function of temperature, for various hole doping densities, as indicated in the caption.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 034304 (2020); doi: 10.1063/5.0012103 128, 034304-8 Published under license by AIP Publishing.SUPPLEMENTARY MATERIAL See the supplementary material for detailed information about the computed hybrid band structures of the M 2O3monolayers and the Ga 2O3bilayer, variation rates of the bond lengths and bond angles of the charged M 2O3monolayers after atomic relaxations, schematics of the various magnetic structures, and details about thecalculations of the exchange interactions, as well as their computedvalues. ACKNOWLEDGMENTS Part of this work was financially supported by the KU Leuven Research Fund, Project No. C14/17/080; part of the computationalresources and services used in this work were provided by the VSC(Flemish Supercomputer Center), funded by the Research Foundation Flanders (FWO) and the Flemish Government — Department EWI. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1F. Tang, H. C. Po, A. Vishwanath, and X. Wan, Nature 566, 486 (2019). 2M. G. Vergniory, L. Elcoro, C. Felser, N. Regnault, B. A. Bernevig, and Z. Wang, Nature 566, 480 (2019). 3T. Zhang, Y. Jiang, Z. Song, H. Huang, Y. He, Z. Fang, H. Weng, and C. Fang, Nature 566, 475 (2019). 4A. Jain, Y. Shin, and K. A. Persson, Nat. Rev. Mater. 1, 15004 (2016). 5N. Mounet, M. Gibertini, P. Schwaller, D. Campi, A. Merkys, A. Marrazzo, T. Sohier, I. E. Castelli, A. Cepellotti, G. Pizzi, and N. Marzari, Nat. 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5.0010478.pdf

J. Appl. Phys. 128, 043901 (2020); https://doi.org/10.1063/5.0010478 128, 043901 © 2020 Author(s).Spin wave generation via localized spin–orbit torque in an antiferromagnet-topological insulator heterostructure Cite as: J. Appl. Phys. 128, 043901 (2020); https://doi.org/10.1063/5.0010478 Submitted: 10 April 2020 . Accepted: 08 July 2020 . Published Online: 23 July 2020 Xinyi Xu , Yuriy G. Semenov , and Ki Wook Kim ARTICLES YOU MAY BE INTERESTED IN Anisotropy in antiferromagnets Journal of Applied Physics 128, 040901 (2020); https://doi.org/10.1063/5.0006077 Skyrmion-based spin-torque nano-oscillator in synthetic antiferromagnetic nanodisks Journal of Applied Physics 128, 033907 (2020); https://doi.org/10.1063/5.0013402 Electrical control of antiferromagnets for the next generation of computing technology Applied Physics Letters 117, 010501 (2020); https://doi.org/10.1063/5.0013917Spin wave generation via localized spin –orbit torque in an antiferromagnet-topological insulator heterostructure Cite as: J. Appl. Phys. 128, 043901 (2020); doi: 10.1063/5.0010478 View Online Export Citation CrossMar k Submitted: 10 April 2020 · Accepted: 8 July 2020 · Published Online: 23 July 2020 Xinyi Xu,1Yuriy G. Semenov,1,2and Ki Wook Kim1,3,a) AFFILIATIONS 1Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, North Carolina 27695, USA 2V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, Kyiv 03680, Ukraine 3Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA a)Author to whom correspondence should be addressed: kwk@ncsu.edu ABSTRACT The spin –orbit torque induced by a topological insulator (TI) is theoretically examined for spin wave generation in a neighboring antiferromagnetic thin film. The investigation is based on the micromagnetic simulation of Néel vector dynamics and the analysis of trans- port properties in the TI. The results clearly illustrate that propagating spin waves can be achieved in the antiferromagnetic thin-film strip through localized excitation, traveling over a long distance. The oscillation amplitude gradually decays due to the non-zero damping as theNéel vector precesses around the magnetic easy axis with a fixed frequency. The frequency is also found to be tunable via the strength of thedriving electrical current density. While both the bulk and the surface states of the TI contribute to induce the effective torque, the calcula- tion indicates that the surface current plays a dominant role over the bulk counterpart except in the heavily degenerate cases. Compared to the more commonly applied heavy metals, the use of a TI can substantially reduce the threshold current density to overcome the magneticanisotropy, making it an efficient choice for spin wave generation. The Néel vector dynamics in the nano-oscillator geometry are examinedas well. Published under license by AIP Publishing. https://doi.org/10.1063/5.0010478 I. INTRODUCTION Spin waves have recently attracted much attention as a poten- tial carrier of information with low energy dissipation. They cantransport a pure spin current without involving the charge flowalong with the possibility to manipulate the amplitude and phase. 1,2The excitation of spin waves has been achieved via the optical, thermal, or electrical methods.3–6The mechanisms that take advantage of the electrically induced effective torque offer oneof the most efficient approaches without introducing bulky externalantenna. 7,8In a heterostructure consisting of a magnetic layer and a strongly spin –orbit coupled (SOC) material, the spin –orbit torque (SOT) resulting from the spin dependent electrontrajectories can manipulate the magnetic state of the magnet. 7 For instance, auto-oscillations can be driven via the SOT, resultingin the propagating spin waves as has been demonstrated experi- mentally in a ferromagnetic structure. 9The spin wave frequency, although limited by the characteristic properties of the waveguide,is proportional to the strength of the SOT. Evidently, a SOC layer that can realize a stronger torque for a given driving current density is highly desirable for a range of applications with an obvious ramification on the effectiveness of electrical control. Heavy metals such as Pt and W are the commonly used SOC materials with the spin-Hall angle θSHtypically in the range of 0.012 –0.12,10requiring a relatively high driving current density (/difference107A/cm2) to overcome the magnetic anisotropy. In compari- son, recent investigations have indicated that a topological insulator (TI) can induce the SOT with a higher efficiency than the heavymetals —an order of magnitude higher at room temperature. 11–13 The experimentally determined values of θSH.1 have been reported in literature studies,14,15making the TI a promising alter- native to the more conventional heavy metals.16It is well known that the TIs are characterized by the insulating bulk and two-dimensional (2D) semi-metallic surface states. In particular, the topologically protected surface electronic states with a linearJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 043901 (2020); doi: 10.1063/5.0010478 128, 043901-1 Published under license by AIP Publishing.dispersion exhibit a strong magnetoelectric effect via the inherent spin-momentum interlock.17While the bulk states are frequently overlooked, the bias-driven in-plane current tends to flow both inthe bulk and on the surface of the TI since the energy separationbetween them is relatively small. In fact, the electrical current inthe bulk can induce the spin torque in a manner similar to that in a heavy metal (i.e., the spin-Hall effect). 18,19Evidently, the bulk contribution depends on the details of the band structure includ-ing the band separation and the position of the Fermi level.As the Fermi level can be modulated externally, their impacton the induced SOT can show a range of responses, necessitating careful consideration. In this work, we theoretically explore the generation of traveling spin waves in a magnetic strip by exploiting the localizedexcitation through a TI layer in place of a heavy metal. Both thebulk and surface states are examined as the source of the SOT for a range of parameters such as the damping constant and the Fermi level position. The thin-film waveguide structure based on anA-type antiferromanget (AFM) with uniaxial easy-axis anisotropyis considered as the primary example, while the analysis can bereadily extended to the ferromangets (FMs). Compared to the FM counterparts, the AFMs are associated with higher resonance frequencies up to THz, faster switching speeds, and higher energyefficiencies. 20Furthermore, they are not subject to the demagnet- ization field, offering a promising medium for spin wave propaga- tion. The investigation takes advantage of the micromagnetic simulations along with the analysis on the TI properties. Theresults clearly illustrate the possibility of efficient spin wave genera-tion that can travel over a long distance as well as its efficientcontrol via the TI induced SOT. The role of the bulk vs surface cur- rents is also elucidated which appears to be consistent with a recent experimental report. 13 II. THEORETICAL MODEL The dynamics of magnetization can be described by using the Landau –Lifshitz –Gilbert (LLG) equation as @m @t¼/C0γm/C2Heffþαm/C2@m @tþTSOC: (1) Here, mdenotes the reduced or normalized local magnetization vector, γis the gyromagnetic ratio, αrefers to the Gilbert damping constant, and the macroscopic effective field Heff/@H @mis obtained from the Hamiltonian Hof the considered system that accounts for the exchange interaction and the anisotropy energy. In the case of an AFM, sublattices Aand Bare antiferromagnetically coupled to each other via the exchange interaction, giving the normalized Néelvector nasm A/C0mB. The last term TSOCin Eq. (1)corresponds to the SOT induced by an SOC material (in this case, a TI) that canbe further separated into the anti-damping torque T adand the field-like torque Tfldepending on the actual physical processes (i.e.,TSOC¼TadþTfl). A schematic of the AFM/TI heterostructure under consider- ation is shown in Fig. 1 . As depicted, an electrical current can flow in both the TI bulk and surface along the xaxis when an appropri- ate bias voltage is applied. The driving current is assumed to bedirect (i.e., dc) in comparison to the approach with a radio fre- quency source explored in the recent studies.21The induced SOT can excite the localized oscillations of the Néel vector in the AFMregion directly in contact with the TI, which may subsequently beguided along the AFM strip with the easy axis aligned in the samedirection (i.e., y). To ensure the uniformity in the excited waves, the thickness of the magnetic film in the zdirection needs to be sufficiently smaller than the typical AFM exchange length of tensof nanometers. 22An insulating or dielectric material is preferred for the AFM strip to avoid the current shunting (thus, loss) in theexcitation region. Two key mechanisms that can lead to the desired Néel vector oscillations are the spin-Hall effect in the TI bulk and the proximity effect driven by the coupling with the magnet onthe surface. 18,23 The spin-Hall effect taking place in the bulk generates a spin current in the zdirection. Subsequent injection into the adjacent magnetic layer leads to the Néel vector dynamics by exerting ananti-damping torque in the following form: 24 TB ad¼γ/C22h 2qμ0MsdθSHJB[m/C2(^σ/C2m)], (2) where q(.0) is the electron charge, μ0the vacuum permeability, Msthe sublattice magnetization, dthe thickness of the AFM layer, JBthe bulk current density, and ^σthe unit vector of the spin current polarization ( k+^ywhen the electrons flow along the +x direction). The product of spin-Hall angle θSHand JBgives the spin current density. The field-like torque that can also arise from the TI bulk current is neglected for its relatively minor contributionto the desired magnetization rotation or oscillation. The spin-Hallangle θ SHof 0 :15 is chosen in Eq. (2)at room temperature, which is comparable to an upper-end value for the heavy metals.13,14 Note that this choice for θSH(i.e., 0 :15) is to account for only the FIG. 1. Schematic illustration of the AFM/TI heterostructure under consideration (not to scale). The driving current flows along the xaxis in the TI (via the surface JSand bulk JB), inducing the spin current and the corresponding SOT in the AFM thin film with easy-axis ( y) anisotropy. The Néel vectors in the region of the AFM driven by the SOT undergo rotations and form spin wavesthat propagate along the strip. The use of an A-type AFM provides ferromag- netic intra-plane coupling and, thus, net non-zero magnetization at the interface with the TI.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 043901 (2020); doi: 10.1063/5.0010478 128, 043901-2 Published under license by AIP Publishing.bulk effect. An effective value extracted for the entire TI can be much larger as discussed briefly in the introduction. The bulk con- tribution is included in the calculations unless mentioned explicitlyotherwise. Concerning the effect of the TI surface, it is likely that multi- ple mechanisms of differing microscopic origins 18,25,26manifest simultaneously through the self-consistent interaction with the magnet. As their comprehensive account is yet to be achieved, weadopt an empirical treatment based on an experimentally observedphenomenon —the proximity induced anomalous Hall effect. 25,27 The resulting y-directional Hall current ( Jy) introduces an additional spin polarization component that can lead to the anti-damping behavior. A simple expression is used to phenomenologicallydescribe J yin terms of the x-directional driving surface current JS, i.e.,Jy¼/C0βzmzJS.25Here, mzsignifies the zcomponent of AFM sublattice magnetization at the interface with the TI (which is non-zero in the A-type) and βzis the ratio between the two current components that can be determined experimentally ( /C250:06).28–30 Given the empirical nature of the treatment, it may be possible to adjust this parameter βzto reflect the influence of other surface driven anti-damping mechanisms. While the anomalous Hall effect is deemed absent in a collinear AFM due to the symmetry, the A-type AFMs with ferromagnetic intra-plane coupling can actuallyhave the magnetization on a surface essentially analogous to theFMs (i.e., non-zero). Accordingly, the proximity interaction at the interface in the present case can be described as in the FM/TI bilayer structure studied earlier. 18,25,26Combined with the field-like contri- bution, the total SOT induced by the surface states can then bewritten as 25 TS¼TS adþTS fl¼/C0γG qμ0MsdvFδtJSm/C2(βzmz^xþ^y), (3) where Gis the TI/magnet exchange coupling energy and vFthe Fermi velocity of the TI surface states. The angular dependence inthe anti-damping term is apparent from the expression, where the effective torque (i.e., T S ad) becomes zero when mz¼0. A similar dependence was reported in the literature.31Higher order terms such as those discussed in Ref. 32are not considered. Note that JSin Eq.(3)is modified to take a 3D equivalent form for direct compari- son with JB. More specifically, the actual surface current density (given by per unit length) is divided by the thickness δtof the TI layer to convert it to a per-unit-area quantity (i.e., a 3D currentdensity). 13 The desired Néel vector dynamics are analyzed by numeri- cally solving the LLG equation based on Object Oriented MicroMagnetic Framework (OOMMF).33The AFM thin-film nanostrip is assumed to have the dimensions of 3 /C2600/C21n m3 with a perfectly absorbing boundary at both ends of the strip (i.e., no reflection to avoid the unnecessary complication), of whichthe first 90 nm at the left end is excited by the SOT through the interaction with a TI layer in contact. The easy y-axis anisotropy ofK y¼20 kJ/m3is adopted along with the exchange stiffness Aex¼5 pJ/m, Ms¼350 kA/m, and α¼0:001. These values are within the range for a dielectric AFM such as NiO. As for the TI, a relatively thick film of 10 nm ( ¼δt) is consid- ered to ensure decoupling between top and bottom surfaces.The Fermi velocity vFis set at 4 :5/C2107cm/s for the surface states and the separation between the bulk conduction band minimum εCand the Dirac point εDat 0 :2 eV, both of which are typical for well-known TIs such as Bi 2Se3. A parabolic energy dispersion is used for the bulk conduction band with an effective massm *¼0:15me, where medenotes the electron rest mass.34The exchange coupling strength Gwith the neighboring AFM is taken to be 40 meV. Transport properties in the TI layer are evaluated byadopting a simple ohmic relation J¼qμ nnEwith the mobilities μS n/C25μB n/C25103cm2/V s at room temperature.35While the electric field Ecan clearly be modulated by the external bias as well, a cons- tant value of 12 kV/cm is assumed to avoid proliferation of adjust- able variables. Accordingly, JBandJSare determined by the carrier densities, which can in turn be specified as a function of a singleparameter, i.e., the Fermi level position ε F. III. RESULTS AND DISCUSSION Details of the excited spin wave dynamics and properties are illustrated in Fig. 2 . With the application of a sufficiently large SOT att¼0, the spins are driven away from the easy y-axis ( nset ini- tially in the þydirection) to oscillate around it in the excitation region, which then propagates along the channel. The precessionangle does not reach 90 /C14(i.e.,jnyj.0) with non-zero damping (α=0) as the growth in the axial tilt via the anti-damping spin torque is compensated by the interaction with the magneticmoments in the unexcited part of the AFM strip. 8The precession can be around either the þy[Figs. 2(a) and 2(d)]o r/C0yaxis FIG. 2. (a) and (b) Snap shots of the steady-state Néel vector obtained along the AFM waveguide for two different directions of the spin current polarization ^σ (+y). With εF/C0εC¼30 meV , the excitation current density (magnitude) cor- responds to JB¼5:3/C2106A/cm2and JS¼4:1/C2107A/cm2. (c) Calculated spin wave dispersion relation. (d) and (e) Trajectories of the Néel vector in the excitation region with ^σ¼+^y. The initial state is aligned along the þyeasy axis. In (e) [as well as in (b)], the Néel vector reversal to /C0yis observed before the precession.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 043901 (2020); doi: 10.1063/5.0010478 128, 043901-3 Published under license by AIP Publishing.[which is preceded by the Néel vector flip; Figs. 2(b) and 2(e)] depending on the polarity of the excitation current (thus, ^σ). Figures 2(a) and 2(b) show the Néel vector state obtained along the AFM strip at t¼0:1 ns, by which time a steady spin wave is established. The Fermi level εFis assumed to be at 30 meV above εC, which corresponds to the excitation current density (magnitude) of JB¼5:3/C2106A/cm2and JS¼4:1/C2107A/cm2. The observed decay in the magnitude of nzillustrates the continued reduction in the angle of rotation around the yaxis due to non-zero damping. Nevertheless, the traveling wave maintainsconstant wavelength and frequency. The 3D illustrations of the Néel vector trajectories in the excitation region are plotted in Figs. 2(d) and 2(e), which correspond to the above mentioned cases of ^σ¼þ^yand/C0^y, respectively. The calculated dispersion relation between the angular frequency ω(=2πf) and the wavevec- tork(=2π=λ) is essentially linear as indicated in Fig. 2(c) , whose slope amounts to the magnon velocity v m(¼1:4/C2106cm/s). The velocity vmis given by the characteristics of the AFM material asγffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi HexAex=Msp , where the exchange field Hexcan be expressed further in terms of Aexand Ms.36A larger current density (thus, the SOT) leads to a higher oscillation frequency of the Néel vector or the spin wave as well as a shorter wavelength.37 It is interesting to systematically examine the role of the often neglected bulk contribution to the SOT. Figure 3(b) shows JS (line 1) and JB(line 2) calculated as a function of the Fermi level position with respect to the bulk conduction band minimum (i.e., Δε¼εF/C0εC). When Δεis negative (thus, εFin the bulk bandgap), JBbecomes very small and is thus not considered. As εF moves above εC, the current flow in the bulk goes up faster than the surface counterpart resulting in the gradual increase of the ratio JB=JS(line 3). Since the Fermi level in a TI is typically chosen to be not far from the Dirac point εD, the surface current density is expected to be generally more pronounced than the bulk contribu-tion in a sufficiently thin structure. Note that this observation isbased on the assumption of ε C/C0εD¼0:2 eV. A smaller separation between the bulk and surface bands can enhance the significance ofJB. Nevertheless, the addition of even a relatively small bulk current may considerably alter the effectiveness of a TI as a SOCmaterial. For a more definitive understanding, a comparative analy- sis of the SOTs generated by the two mechanisms (i.e., the spin-Hall and anomalous Hall effects) is desired. However, a directterm-by-term comparison is challenging due to the differences inthe functional dependence. When such details are ignored, Eqs. (2) and(3)suggest that the ratio between the bulk and surface induced anti-damping torques scales roughly as TB ad TSad/C12/C12/C12/C12/C12/C12/C12/C12/C25/C22hvF 2GθSH βzδtJB JS/C12/C12/C12/C12/C12/C12/C12/C12: (4) The prefactor in front of J B=JSis estimated to be around 1 with the numerical values discussed earlier. Accordingly, the Fermi levelposition [via Fig. 3(b) ] appears to provide at least an approximate indicator for the contribution of the bulk states although the actual impact on the Néel vector dynamics is determined by the physical details of the SOT processes.For a quantitative evaluation of the macroscopic response, the frequencies of the excited spin waves are compared as a function of the Fermi level position Δεby using the micromagnetic simula- tions. Figure 3(c) plots the results with and without the consider- ation of J B. As expected, the frequency of the eigen mode increases withΔε(thus the current density). An interesting point to note is that the addition of JBappears to induce a comparatively larger jump in the frequency with a steeper slope than the corresponding change in the total current density. For instance, the oscillation fre-quency at Δε¼30 meV goes up by about 40% (from 0.58 THz to 0.8 THz), while J Bonly adds 13% to the total driving current. The results clearly indicate that the bulk current, although only at a fraction in the magnitude of the surface term, can make a consider- able contribution to the spin wave generation, highlighting theefficient nature of the SOT induced by the spin-Hall effect in theAFM/TI heterostructure. With both the surface and bulk contribu-tions, the oscillation frequency can reach 1 THz at the current density of /difference6/C210 7A/cm2, which is nearly an order of magnitude smaller than the value estimated with a heavy metal as the SOCmaterial. 8Once Δεbecomes sufficiently negative, the spin wave excitation is no longer possible. The corresponding threshold current density ( /difference107A/cm2) is associated with the surface induced SOT via the anomalous Hall effect. FIG. 3. (a) Typical electronic band structure of a TI. The bulk states are approx- imated by parabolic energy bands, while the surface states are described by a linear dispersion with the Fermi velocity vF.Δεindicates the difference between the Fermi level εFand the bulk conduction band minimum εC. (b) Current density in the TI as a function of the Fermi level position. Line 1: surface currentdensity J Sin a 3D equivalent form, line 2: bulk current density JB, and line 3: ratio of JB=JS. (c) Excited spin wave frequency in the AFM as functions of the Fermi level position Δεwith and without the SOT contributed by the bulk current (lines 1 and 2, respectively). For convenience, the current density valuecorresponding to each Δεis also provided for both J S(blue) and JB(red). The upper and lower panels correspond to the cases of ^σ¼þ^yand/C0^y, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 043901 (2020); doi: 10.1063/5.0010478 128, 043901-4 Published under license by AIP Publishing.Along with electrical generation of traveling spin waves, spin torque nano-oscillators are another spintronic application that can take advantage of the AFM/TI system. In this case, the AFM doesnot take a long strip form, rather the entire magnetic material isinterfaced with and thus subject to excitation by the TI. Unlike thestructure described in Fig. 1 , a hard-axis anisotropy is assumed in theyaxis ( K y¼/C0160 kJ/m3) normal to the directions of the driving current ( x) and the spin current ( z). This hard-axis config- uration is known to enable a low-threshold condition for oscilla-tions as the Néel vector does not encounter the anisotropy energybarrier to rotate on the easy x–zplane. In addition, αis considered a variable while the rest of the parameters such as A exand Ms remain unchanged. The Néel vector is assumed to be initially aligned in the þxdirection. Figure 4 summarizes the multiplicity of the Néel vector dynamics in the α/C0Δε(thus, J) parameter space. Two phase maps [Figs. 4(a) and 4(b)] correspond to the cases of TI electrons flowing in the +xdirections (thus, ^σof+^y), respectively. Region 1 shows the conditions where the induced torque is insufficient toovercome the damping, resulting in no appreciable change in themagnetic state. Once the induced SOT becomes sufficiently large to overcome the damping and other energy barriers, the Néel vector isdriven from its initial orientation and rotates in the easy x–zplane (i.e., auto-oscillation; region 2). A further increase in the driving current can lead to a stable 90 /C14rotation, aligning it along the injected spin polarization despite the hard-axis anisotropy in thesame direction (region 3). In the case of Fig. 4(b) with ^σ¼/C0^y,a n additional dynamical behavior is observed along with the auto-oscillations and 90 /C14rotations. More specifically, 180/C14reversal of the Néel vector ( þx!/C0 x) can be realized in region 4. The cor- responding schematics are given in Figs. 4(c) –4(f), respectively. At first glance, it is not intuitively obvious for the system to have a directional preference in the x–zplane enabling the stable rotation to /C0xsince no magnetic anisotropy is specified other than the hard yaxis. However, the axial symmetry is broken by the explicit dependence of the surface-state anti-damping torque onthezcomponent of the magnetization unlike the SOT induced by a heavy metal (and thus the TI bulk states). Since this term T S ad reduces to zero as the magnetization at the interface orients normal to the zaxis [i.e., mz/C250; see Eq. (3)], the+xdirections serve in effect as the easy axis, offering bistable configurations. In compari-son, this reversal from þxto/C0xis not observed in Fig. 4(a) since the torque is induced toward the þxaxis (i.e., the initial orientation), which is the opposite direction to that experience in Fig. 4(b) . As such, the orientation remains unchange until strong excitationsufficiently disrupts the state leading to the auto-oscillation. Whenthe Néel vector is initially in the /C0xdirection, the dependence on the polarity of the driving current or electron flow is also reversed. In such a setup, the equivalent of Fig. 4(a) shows region 4 while that ofFig. 4(b) does not. These results clearly suggest the possibility to deterministically encode the Néel vector orientation: in the α/C0Δε parameter space corresponding to region 4, the final state always aligns with the direction of the driving current irrespective of the initial orientation ( +x). 23Note that uncontrollable flip-flop ’s occur near the boundary with region 2 (i.e., just before theauto-oscillations) in both Figs. 4(a) and 4(b). As it involves a rather narrow range, this feature is not shown in the phase maps in order not to complicate the picture. Similarly, no rapid change takes place across the boundary between regions 2 and 3. Thesteady oscillations do not disappear suddenly. Instead, the preces-sion angle around the +yaxis gradually collapses from 90 /C14(i.e., thex–zplane) to nearly 0/C14. The oscillation frequency is clearly a strong function of αandΔε(i.e., the driving current density) with a wide tunable range. The desired frequencies near the THz appearto require αsmaller than those plotted ( ,0:005). Along with the dynamic modulation of Δε, the numerical value for αcan also be tai- lored through doping or by introducing auxiliary layers. 38,39 IV. SUMMARY The feasibility of spin wave generation via the SOT induced in an adjacent TI is theoretically demonstrated in an easy-axis AFM.The results from the numerical simulations clearly show that the spin auto-oscillations can be achieved in the thin-film AFM strip through localized excitation, traveling over a long distance asthe angle of precession gradually collapses due to the non-zerodamping. The calculations also elucidate the dependence of the Néel vector dynamics on the relevant physical parameters including the TI electronic properties, highlighting the potential significance FIG. 4. Néel vector dynamics in the AFM/TI spin torque nano-oscillator. A hard-axis anisotropy is assumed for the AFM in the yaxis ( Ky¼/C0 160 kJ/m3) normal to the directions of the driving current ( x) and the spin current ( z). The entire magnet is subject to excitation by the SOT . (a) and (b) Phase maps for various regions of operation in the α/C0Δεparameter space with the spin current polarization ^σ¼þ^yand/C0^y, respectively. Regions 1, 2, and 3 corre- spond to the conditions of no appreciable change in the magnetic state,auto-oscillation, and 90 /C14switching toward ^σ. In the case of (b) with ^σ¼/C0^y, 180/C14rotation of the Néel vector ( þx!/C0 x) is observed before the auto-oscillations (region 4). (c) –(f) Schematic illustration of the Néel vector motions characteristic to regions 2 –4.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 043901 (2020); doi: 10.1063/5.0010478 128, 043901-5 Published under license by AIP Publishing.of the bulk states as the source of the SOT in the heavily degenerate conditions. Along with the propagating spin waves, the application of the AFM/TI bilayer structure as a spin torque nano-oscillator isalso illustrated. With the contributions from both the stronglyspin–orbit coupled surface and bulk states, the TIs offer a highly efficient alternative to the conventional heavy metals for the SOT. ACKNOWLEDGMENTS This work was supported, in part, by the U.S. Army Research Office (No. W911NF-16-1-0472). DATA AVAILABILITY The data that support the findings of this study are available within the article. REFERENCES 1A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nat. Phys. 11, 453 (2015). 2V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. D Appl. Phys. 43, 264001 (2010). 3B. Lenk, H. Ulrichs, F. Garbs, and M. Münzenberg, Phys. Rep. 507, 107 (2011). 4S. Seki, T. Ideue, M. Kubota, Y. Kozuka, R. Takagi, M. Nakamura, Y. Kaneko, M. Kawasaki, and Y. Tokura, Phys. Rev. Lett. 115, 266601 (2015). 5S. Cherepov, P. Khalili Amiri, J. G. Alzate, K. 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Appl. Phys. Lett. 117, 010501 (2020); https://doi.org/10.1063/5.0013917 117, 010501 © 2020 Author(s).Electrical control of antiferromagnets for the next generation of computing technology Cite as: Appl. Phys. Lett. 117, 010501 (2020); https://doi.org/10.1063/5.0013917 Submitted: 18 May 2020 . Accepted: 25 June 2020 . Published Online: 08 July 2020 O. J. Amin , K. W. Edmonds , and P. Wadley ARTICLES YOU MAY BE INTERESTED IN Spintronics with compensated ferrimagnets Applied Physics Letters 116, 110501 (2020); https://doi.org/10.1063/1.5144076 Spintronics on chiral objects Applied Physics Letters 116, 120502 (2020); https://doi.org/10.1063/1.5144921 Spin–orbit torque driven multi-level switching in He+ irradiated W–CoFeB–MgO Hall bars with perpendicular anisotropy Applied Physics Letters 116, 242401 (2020); https://doi.org/10.1063/5.0010679Electrical control of antiferromagnets for the next generation of computing technology Cite as: Appl. Phys. Lett. 117, 010501 (2020); doi: 10.1063/5.0013917 Submitted: 18 May 2020 .Accepted: 25 June 2020 . Published Online: 8 July 2020 O. J. Amin, K. W. Edmonds, and P. Wadleya) AFFILIATIONS School of Physics and Astronomy, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom a)Author to whom correspondence should be addressed: Peter.Wadley@nottingham.ac.uk ABSTRACT Antiferromagnets are a class of magnetically ordered material with zero net magnetization. A swathe of recent experimental studies have shown that electrical control of antiferromagnetic order is possible by two distinct mechanisms: field-like and damping-like torques. Thiscould be revolutionary for the next generation of computing technologies, as the properties of antiferromagnets are advantageous for highspeed, high density memory applications. Here, we review the electrical control of antiferromagnets via field-like and damping-like torques as well as distinguishing from pervasive non-magnetic effects that have raised questions about the origins of electrically measured switching signals. Published under license by AIP Publishing. https://doi.org/10.1063/5.0013917 Antiferromagnetic (AF) spintronics has gained a lot of interest since the prediction 1and experimental demonstration2of electrical switching by current-induced torques in AF materials. The surge in interest is justifiable as switchable AF devices offer properties that areattractive for high speed, high density memory applications. 3,4These include a non-volatile order, terahertz frequency dynamics, no device crosstalk, and robustness against external magnetic fields. Furthermore, AF order is found in a wide range of materials, from insulators andsemiconductors to conductors and superconductors. They may also be coexistent with symmetries and topologies not found in their ferromag- netic counterparts. 5 Key to fast and efficient switching of AF moments are current- induced N /C19eel-order spin–orbit torques (NSOTs), where the torques induce the same sense rotation of each AF sublattice. It is well estab- lished that in conductors lacking spatial inversion symmetry, a charge current can induce a non-equilibrium spin polarization,6and the resulting effective magnetic field can manipulate the magnetization in a ferromagnetic crystal.7,8Analogous effects can arise in AF crystals with particular symmetries. CuMnAs9and Mn 2Au10are collinear antiferromagnets in which the Mn spin sublattices form space-inversion partners, such that a charge current generates an alternating spin polarization that coincides with the sublattices. 1This is illustrated inFig. 1(a) . The resulting local effective fields are directed perpendicu- lar to the current direction.2,11 Current-induced torques have also been realized in heterostruc- tures of insulating AF layers and Pt, a heavy metal with a large spinHall conductivity. Charge current in the Pt layer injects a spin current into the AF layer due to the spin Hall effect,12as depicted in Fig. 1(b) . Once injected into the AF, the spin current triggers excitations of the magnetic order by inducing damping-like torques, which are an even function of the sublattice magnetic moment.1,13,14Making use of an external spin current allows for a broader choice of AF material as itdoes not require specific crystal symmetry. However, damping-like torque switching is more complex than the N /C19eel order field-like torque due to the dependence of the effective magnetic field on the sublattice magnetic moment orientation. 15 Experimental demonstrations of current-induced switching in AF films typically utilized 8-arm or 4-arm device structures, as illus- trated in Fig. 2 . These geometries allow current pulses to be applied along the biaxial easy magnetic axes of the AF film, with the aim of inducing 90/C14reorientation of the magnetic moments. For the 8-arm device, current pulses are applied between orthogonal pairs of con- tacts, whereas for the 4-arm device, to achieve a net current at 45/C14to the arms in the center of the device, pulses are sent through all four contacts. Reorientation of the AF moments is detected in the trans- verse voltage, V xy, with a weak probing current applied at 45/C14to the current pulse directions. This measures the anisotropic magnetoresis- tance (AMR) for conducting AFs or spin Hall magnetoresistance (SHMR) for Pt-capped insulating AFs. Both effects have the same symmetry, which is related to the difference in conductivity for prob- ing currents parallel and perpendicular to the magnetic momentorientation. Appl. Phys. Lett. 117, 010501 (2020); doi: 10.1063/5.0013917 117, 010501-1 Published under license by AIP PublishingApplied Physics Letters PERSPECTIVE scitation.org/journal/aplThe first experimental demonstration of reversible current- induced AF switching and electrical readout was performed in biaxial CuMnAs using the geometry illustrated in Fig. 2(a) .2Later studies showed reversible and reproducible electrical switching with currentpulse lengths down to /C24100 ps 16as well as with terahertz electromag- netic pulses.17The same approach was used to demonstrate reproduc- ible electrical switching in Mn 2Au11,18,19and in a wide range of insulating AF/Pt bilayers.15,20–23Table I summarizes the different physical systems studied using the current amplitudes, pulse lengths,and measurement techniques included. Multi-level switching has been observed in AF systems, where multiple current pulses or increasing pulse lengths result in an increas- ing electrical readout signal. This may be attributed to the multi-domain nature of the switching, and the memristive response suggests potential applications in neuromorphic computing. 24However, quali- tatively similar “sawtooth-like” switching signals have also been observed in non-magnetic thin films,25,26and in these cases, the mea- sured signal has been attributed to electromigration,20,24,27annealing,23 or anisotropic thermal gradients.25Electromigration and joule heating may result in an anisotropic conductivity with the same symmetry as AMR and SHMR. In contrast, thermal gradients may be important on short timescales but are unlikely to produce signals that persist for lon-ger than a few seconds. It is important to distinguish between AF switching and spurious non-magnetic effects that might contribute to the measured readout signal. Magnetic field-dependent and temperature-dependent studies provide methods to do so; for example, in a-Fe 2O3, a large applied field fixes the orientation of the AF magnetic moments and suppresses the switching signal.20,22Similarly, the AF metal dichalcogenide, Fe1=3NbS 2, shows a switching signal only below its N /C19eel temperature (42 K) and the signal is suppressed by applying a large magnetic field.28It has also been shown that electric field-induced strain in Mn 2Au can alter the ability to electrically switch between magnetic easy axes due to the changing magnetic anisotropy energy barrier. This is supported by studies of switching in layers grown by different crystallography.18,29 A more direct way to assess the effect of current-induced tor- ques in AFs is imaging the sub-micron AF domain structure using x-ray photoemission electron microscopy (XPEEM). Utilizing the polarization-dependent absorption of soft x-rays is an established technique for measuring AF domain structures with a spatial reso- lution of a few tens of nm.30Current-induced modification of individual AF domains has been demonstrated using XPEEM in several studies,2,19,21,31–33but it is essential to establish the revers- ibility and reproducibility of the microscopic domain switching in order to compare with electrical measurements. This was done in CuMnAs, and it was shown that for 8 successive pairs of orthogo- nal pulses, reproducible switching occurs of the AF domain orientation that is averaged over a 10 lm area, which is clearly correlated with the electrical readout signal.31However, at a sub-micron level, the switching behavior was inhomogeneous and stochastic, highlighting the complex behavior of AF multi-domain switching. In CuMnAs films with larger domains, reproducible current-induced motion of 90/C14domain walls between two pinning points was observed, with the direction of the domain wall motion determined by the polarity of the current pulse.32Several examples of combined XPEEM and electrical switching studies in CuMnAs are shown in Fig. 3 , which includes reversible domain switching, 90/C14domain wall manipulation, an example of electromigration, and the time evolution of the transverse electrical signal after orthogonal current pulses. XPEEM is an extremely powerful technique for imaging the AF domain structure, but it is not easily accessible. This has led to the development of several benchtop tools for imaging the AF order. A promising approach generates thermoelectric voltage with local ther- mal gradients using a focused laser beam.34,35Spin Seebeck imaging of AF domains in Pt/NiO bilayers revealed both domain rotation and domain wall motion induced by current pulses.35Another method uses scanning nitrogen-vacancy center (NV) magnetometry to mea- sure the very weak magnetic stray fields from the layered AF structure, FIG. 2. Schematics of pulsing device geometries. Orthogonal current pulses (red and black arrows) rotate the magnetic moments 90/C14between the magnetic easy axes. After each pulse, reorientation of the magnetic moments is probed by apply-ing a relatively small current (blue arrow) at 45/C14to the pulse directions and measur- ing the transverse voltage, V xy. (a) 8-arm geometry, where current pulses are applied between orthogonal pairs of contacts. (b) 4-arm geometry, where orthogo-nal current pulses, 45/C14to the main arms, in the center of the device are achieved by pulsing between all four contacts. FIG. 1. Illustrations of current-induced torques in antiferromagnetic materials. (a) Magnetic Mn sites in the tetragonal CuMnAs unit cell (shown as purple sphereswith red arrows indicating the spin direction) couple antiferromagnetically betweensublattices (blue and yellow planes), which are space-inversion partners. An applied current pulse generates a staggered effective field coinciding with the magnetic sublattices (blue and yellow arrows). This creates a field-like torque on the Mnspins, rotating them with the same sense perpendicular to the pulse direction. (b) Ina bilayer of Pt and NiO, a charge current flowing through the Pt layer (silver spheres) generates a spin current, perpendicular to the charge current direction, due to the spin Hall effect. The spin current is injected into the antiferromagneticNiO layer that rotates the Ni spins (green spheres with red arrows) via anti-damping torques.Applied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 117, 010501 (2020); doi: 10.1063/5.0013917 117, 010501-2 Published under license by AIP Publishingand using this method, current-induced switching in CuMnAs was imaged.36A suppression of the magnetic stray field contrast was observed in the first few seconds following a current pulse, with a recovery on a timescale of hours or days. This behavior, attributed to a fragmentation of relatively large AF domains into nanoscale textures,was found to be correlated with transient changes in the electrical readout signal. 36,37These transient electrical signals can be “frozen in” at lower temperatures, with a characteristic activation energy ofaround 0.7–0.8 eV. 37Moreover, the electrical signal can be much larger than the typical /C240:2% AMR found in CuMnAs38and can also be induced by 100 fs laser pulses.37Ab initio calculations indicate that in tetragonal CuMnAs, the resistivity can depend strongly on the local magnetic order.39A connection between the electrical readout, with a large and temperature-dependent relaxation behavior, and the nano- scale AF order can be inferred. However, its micromagnetic origins and relationship to magnetoelastic stresses and defects remain unknown. A benchtop technique for imaging AF domains, which has had recent success, involves the magneto-optical birefringence effect. This method was used to image NiO and CoO thin films,40,41as well as a NiO/Pt, 8-arm device in combination with the measurement of the SHMR signal after orthogonal pulses.42A clear distinction was made between electrical signals of magnetic and non-magnetic origin, and the magnetic signal was shown to be linearly proportional to the area of domain switching in the center of the device. This result confirmsthe origin of the electrical signal as corresponding to AF domainswitching, allowing reversible electrical writing, and reading to be unambiguously demonstrated in this AF insulator system. Potential applications of AF spintronic devices are substantial. Their ultra-fast dynamics under field-like and damping-like torqueslend themselves to the development of nano-oscillators and detectorsin the “terahertz gap” between electrical and optical sources. 43,44The ability to control AF domains electrically extends their usefulness to magnetoelectric memories45and magnonic logic devices,46and it is anticipated that purely AF magnetic random-access memory (MRAM) will exceed the performance of ferromagnetic MRAM,47 which is one of the leading candidates for “beyond Moore’s law” information technologies.17The demonstrated multi-level behavior of electrical switching in AFs allows them to be used as artificial neurons in spiking neural networks (SNNs), something that was previouslyunattainable using silicon-based hardware, and is of interest due to their ability for writing and speech recognition. 24 While much of the experimental work so far has demonstrated the rotation of the AF domains due to spin–orbit torques, non- magnetic contributions to the electrical signals can occur. This hasincreased the importance of magnetic field and temperature- dependent studies, as well as direct magnetic imaging, in elucidating the role of current-induced spin–orbit torques. However, characteriza-tion of the AF order with high spatial and temporal resolution remains a considerable challenge, especially for buried layers and interfaces. 48 Overall, the micromagnetics and timescales of switching need to be investigated further.TABLE I. A summary of experimental electrical switching studies in antiferromagnetic materials, showing material systems, ranges of pulse amplitudes and p ulse widths investi- gated, and detection methods. In the detection method column, the abbreviations are as follows: anisotropic magnetoresistance (AMR), spin Hall mag netoresistance (SHMR), x- ray magnetic linear dichroism (XMLD), photoemission electron microscopy (PEEM), and nitrogen-vacancy center magnetometry (NV magnetometry). Material Jpulse(MA cm/C02) Pulse width Detection method References CuMnAs/GaAs 4.5 50–275 ms Transverse AMR 2 CuMnAs/GaAs 6.1 50 ms XMLD PEEM 31 CuMnAs/GaP 160 100 ps–10 ms Transverse AMR 16 CuMnAs/GaP 9 50 ms Transverse AMR, XMLD PEEM 32 CuMnAs/GaP 12 10 ls Resistivity, Optical reflectivity, NV magnetometry, and XMLD PEEM 37 CuMnAs/GaP 24 10 ls Transverse AMR and NV magnetometry 36 Mn 2Au(001)/Al 2O3 18 1 ms Transverse AMR 11 Mn 2Au(110)/MgO 20 100 lm Transverse AMR 19 Mn 2Au(103,101,204)/MgO 21 1 ms Transverse AMR 18 Mn 2Au(001)/Al 2O3 13 1 ms XMLD PEEM 33 Mn 2Au(103)/PMN-PT 12 500 ms Transverse AMR and XMLD 29 Pt/Mn 2Au(103)/MgO 71 1 ms Transverse AMR and SHMR 49 Pt/NiO(001)/SrTiO 3 40 1 ms SHMR 15 Pt/NiO(111)/Pt/MgO 54 3 s SHMR 20 Pt/NiO(001)/MgO 150 1 ms SHMR and XMLD PEEM 22 Pt/NiO(001)/MgO 115–150 100 ls SHMR and Magneto-optical birefringence effect 42 Pt/a-Fe2O3/a-Al2O3 150 10 ms SHMR 21 Pt/a-Fe2O3/a-Al2O3 90 1 ms SHMR 23 Fe1=3NbS 2 0.054 10 ms Transverse AMR 28 Ta/MnN/Pt 80–100 4 ls SHMRs 50 Pt/NiO, Pt 40 10 ms SHMR 26 Nb/MgO 110 1–10 ms Transverse resistivity 25Applied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. 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Black, “Electromigration—A brief survey and some recent results,” IEEE Trans. Electron Devices 16, 338–347 (1969). 28N. L. Nair, E. Maniv, C. John, S. Doyle, J. Orenstein, and J. G. Analytis, “Electrical switching in a magnetically intercalated transition metal dichalcogenide,” Nat. Mater. 19, 153–157 (2020). FIG. 3. Examples of the effects of electrical pulsing in biaxial CuMnAs. In (a), XPEEM images are taken after applying 6.1 MA cm/C02current pulses along two orthogonal directions to a film with a granular domain microstructure. The accompa- nying XMLD signal shows the average domain changes after 16 successive pulses.In (b), XPEEM is used to demonstrate switching of a micrometer-sized domain for4.5 MA cm /C02current pulses with opposite polarity. (c) demonstrates a nonreprodu- cible electromigration effect in a CuMnAs device after a 27 MA cm/C02pulse. (d) shows the time evolution of the transverse electrical signal in a cross-shapedCuMnAs device. The dashed red and black lines on the plot correspond to theapplication of 12 MA cm /C02current pulses, applied in the direction of the red and black arrows shown in the pulsing geometries to the right of the plot. The measure- ment geometry is also shown, with the blue arrow indicating the direction of theprobe current.Applied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 117, 010501 (2020); doi: 10.1063/5.0013917 117, 010501-4 Published under license by AIP Publishing29X. Chen, X. Zhou, R. Cheng, C. Song, J. Zhang, Y. Wu, Y. Ba, H. Li, Y. Sun, Y. You et al. , “Electric field control of N /C19eel spin–orbit torque in an anti- ferromagnet,” Nat. Mater. 18, 931–935 (2019). 30A. Scholl, J. St €ohr, J. L €uning, J. W. Seo, J. Fompeyrine, H. Siegwart, J.-P. Locquet, F. Nolting, S. Anders, E. Fullerton et al. , “Observation of antiferro- magnetic domains in epitaxial thin films,” Science 287, 1014–1016 (2000). 31M. Grzybowski, P. Wadley, K. Edmonds, R. 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M. Stiehl, J. T. Heron, R. Need, B. J. Kirby, D. H. Low, K. C. Nowack, D. G. Schlom et al. , “Spin Seebeck imaging of spin-torque switching in antiferromagnetic Pt/NiO heterostructures,” Phys. Rev. X 9, 041016 (2019). 36M. W €ornle, P. Welter, Z. Ka /C20spar, K. Olejn /C19ık, V. Nov /C19ak, R. Campion, P. Wadley, T. Jungwirth, C. Degen, and P. Gambardella, “Current-induced frag-mentation of antiferromagnetic domains,” preprint arXiv:1912.05287 (2019). 37Z. Ka /C20spar, M. Sur /C19ynek, J. Zub /C19acˇ, F. Krizek, V. Nov /C19ak, R. P. Campion, M. S. W€ornle, P. Gambardella, X. Marti, P. N /C20emec, K. W. Edmonds, S. Reimers, O. J. Amin, F. Maccherozzi, S. S. Dhesi, P. Wadley, J. Wunderlich, K. Olejn /C19ık, and T. Jungwirth, “High electrical and optical resistive switching in a thin-film anti- ferromagnet,” arXiv:1909.09071 (2019).38M. Wang, C. Andrews, S. Reimers, O. Amin, P. Wadley, R. Campion, S. Poole, J. Felton, K. Edmonds, B. 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Saleem, Y. You, F. Pan, and C. Song, “From fieldlike torque to antidamping torque in antiferromagneticMn 2Au,” Phys. Rev. Appl. 11, 054030 (2019). 50M .D u n z ,T .M a t a l l a - W a g n e r ,a n dM .M e i n e r t ,“ S p i n - o r b i tt o r q u ei n d u c e de l e c t r i - cal switching of antiferromagnetic MnN,” Phys. Rev. Res. 2, 013347 (2020).Applied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 117, 010501 (2020); doi: 10.1063/5.0013917 117, 010501-5 Published under license by AIP Publishing

5.0012914.pdf

Appl. Phys. Lett. 117, 032903 (2020); https://doi.org/10.1063/5.0012914 117, 032903 © 2020 Author(s).Experimental observation of ferroelectricity in ferrimagnet MnCr2S4 Cite as: Appl. Phys. Lett. 117, 032903 (2020); https://doi.org/10.1063/5.0012914 Submitted: 19 May 2020 . Accepted: 03 July 2020 . Published Online: 22 July 2020 J. X. Wang , L. Lin , C. Zhang , H. F. Guo , and J.-M. Liu Experimental observation of ferroelectricity in ferrimagnet MnCr 2S4 Cite as: Appl. Phys. Lett. 117, 032903 (2020); doi: 10.1063/5.0012914 Submitted: 19 May 2020 .Accepted: 3 July 2020 . Published Online: 22 July 2020 J. X.Wang,1,a) L.Lin,2C.Zhang,1H. F. Guo,1and J.-M. Liu2,a) AFFILIATIONS 1School of Physics and Engineering, Henan University of Science and Technology, Henan Key Laboratory of Photoelectric Energy Storage Materials and Applications, Luoyang 471003, China 2Laboratory of Solid State Microstructures and Innovation Centre of Advanced Microstructures, Nanjing University, Nanjing 210093, China a)Authors to whom correspondence should be addressed: enyl@163.com andliujm@nju.edu.cn ABSTRACT Ferrimagnetic spinel compounds AB 2X4(A and B are the magnetic transition elements) are considered to be promising candidates for multiferroics with large magnetization and polarization. In this work, we synthesize polycrystalline spinel MnCr 2S4and characterize the magnetic and ferroelectric properties. Two well-defined ferroelectric phase transitions are demonstrated. The first one occurs at the Cr3þ ferromagnetic phase transition temperature of TC¼65 K, and the other takes place at the Yafet–Kittel (YK) magnetic phase transition tem- perature of TYK/C255 K. It is suggested that ferroelectricity in the YK phase is driven by the noncollinear triangular YK spin orders and can be greatly tuned by an external magnetic field. Between TYKand TCranges, another opposite electric polarization sublattice appears, which is enhanced by the external magnetic field just near TC, revealing that this opposite electric polarization is likely related to magnetostriction and the magnetic field can enhance the lattice distortions near TC. Thus, this work paves the way for exploiting ferrimagnetic multiferroicity although more studies are needed to clarify the ferroelectricity mechanism in the Cr3þferromagnetic phase. Published under license by AIP Publishing. https://doi.org/10.1063/5.0012914 Multiferroicity, as one of the favorable topics in condensed mat- ter and materials physics, has been receiving intensive attention sincethe two landmark peer-reviewed works on TbMnO 3and BiFeO 3in 2003, thanks to their unusual physics and potential applications.1,2 Two classes of multiferroics have been identified to date.3–8For Type I multiferroics, the coexistence of magnetism and ferroelectricity comesfrom separate functional units and may not necessarily offer strong magnetoelectric (ME) coupling, while the ferroelectricity of Type II multiferroics is induced by particular magnetic orders and shows anunprecedented sensitivity to applied magnetic fields. For theseso-called spin driven ferroelectrics (FEs), representative examples areboracites (Ni 3B7O13I, Co 3B7O13I, etc.), magnetite Fe 3O4,r a r ee a r t h manganites ReMnO 3and ReMn 2O5(R¼Gd—Ho), etc.However, it is a combination of the ferroelectric order and anti-ferromagnetic ordercoexisting in discovered Type II multiferroics so far, with small electricpolarization and nearly zero magnetization. 9,10Therefore, coexistence of ferroelectricity (or ferrielectricity) and ferromagnetism (or ferrimag- netism) becomes very precious, and many scientists are trying to finda new kind of multiferroics with high multiferroic phase transitiontemperature and large electric and magnetic polarization. 11–13Cr-based chalcogenide spinels with general formula ACr 2X4 (A¼C d ,H g ,F e ,C o ,M n ,a n dC u ;X ¼O, S, and Se) host rich physical properties due to the coexistence of frustration and strong coupling among spin, charge, orbital, and lattice degrees of freedom,14such as complex orbital states and colossal magnetoresistance in FeCr 2S4,15 negative thermal expansion in ZnCr 2S4,13colossal magnetocapacitance in CdCr 2S4and HgCr 2S4,16,17and multiferroicity in CdCr 2S4, HgCr 2S4,a n dF e C r 2S4.16–19For multiferroicity, the ferromagnetic CdCr 2S4and HgCr 2S4show a relaxor ferroelectric behavior with a colossal magnetocapacitive effect. The relaxor ferroelectricity wasclaimed to originate from the off-center position of the Cr 3þion.17 Improper ferroelectricity in the multiferroic FeCr 2S4stems from both the noncollinear conical spin order associated with the spin-orbital coupling and Jahn–Teller distortion below the Fe orbital ordering tem-perature of 8.5 K, and the electric polarization ( P)r e a c h e st h es a t u r a t e d value of /C2470lC/m 2below 4 K.19However, the multiferrioic phase transition temperature is very low and the value of Pis not large enough. Therefore, along this group of materials, we pay our attentionto MnCr 2S4, a spinel with two magnetic sublattices, for designing mul- tiferroic with net large magnetization and polarization. Appl. Phys. Lett. 117, 032903 (2020); doi: 10.1063/5.0012914 117, 032903-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplMnCr 2S4is a representative material for Yafet–Kittel (YK for short) ferrimagnetic spinel. The competition of magnetic exchanges within and between Mn and Cr ( JMn–Mn ,JMn–Cr ,a n d JCr–Cr)i st h ek e y essence of its physical properties. Since the ferromagnetic Cr–Cr inter- action JCr–Cr is stronger than the magnetic exchanges JMn–Mn ,JMn–Cr , and JMn–Mn –JMn–Cr , two consecutive magnetic phase transitions at TC /C2565 K (ferromagnetic order of Cr3þspins) and at TYK/C255K ( i n t h e coplanar YK phase, the net magnetization moment of the two canted Mn-sublattice spins is antiparallel to that of the Cr spins) have been evidenced by previous magnetization and neutron diffraction investi- gations.20–25AtTC, the Cr sublattice orders ferromagnetically, whereas the Mn sublattice still remains quasiparamagnetic. In the quasipara- magnetic state, the longitudinal component of the magnetic moment of the Mn sublattice is ordered antiparallel to the Cr sublattice, whereas the transverse component of the moment of the Mn sublattice is fully disordered. On decreasing temperature, the Mn ions gradually align in the exchange field of the Cr sublattice with an effective antifer- romagnetic orientation with respect to the Cr magnetization. At TYK, the coplanar YK magnetic structure sets in, in which the two Mn- sublattice spins are canted and their net magnetization moment is antiparallel to the Cr spins. Recently, theoretical and experimental studies have demonstrated that a pair of canted spins can give rise to an electric dipole moment P; specifically, spin-driven ferroelectricity has been confirmed in the YK phase in MnCr 2S4.26,27Moreover, the magnetostrictive effect has been observed below the Curie temperature TCin the ferrimagnetic MnCr 2S4single crystals, and spin-lattice coupling contributes to stabi- lizing the multiferroic phase.28,29In addition, structural transition was observed at ferrimagnetic transition temperature in spinel compound NiCr 2O4and ferroelectric (FE) polarization appears at TCin the ferri- magnetic FeCr 2O4and CoCr 2O4.30–32Relaxor ferroelectricity has been found in the ferromagnetic spin state of CdCr 2S4and HgCr 2S4.16,17 Therefore, it is also possible for MnCr 2S4to be ferroelectric both in its ferromagnetic phase and in the YK phase. In this work, we pay our attention to the low- Tpyroelectric prop- erties of MnCr 2S4in addition to other structural and magnetic charac- terization studies. Our results provide clear evidence that MnCr 2S4 shows a coexistence of relatively high FE polarization and magnetiza- tion in the whole temperature range below TC. More importantly, MnCr 2S4shows strong magnetoelectric coupling in the YK phase not only below TYK(5 K) but also near the ferromagnetic phase transition temperature TC, and the FE mechanisms are subsequently proposed based on the relevant data. Polycrystalline MnCr 2S4was prepared by the solid-state reaction method from the required molar ratio mixture of Mn, Cr, and S powders. The structure and phase purity were examined by x-ray dif- fraction (XRD) with Cu K aradiation at room temperature. The T-dependent specific heat ( Cp) and magnetization ( M), as well as the magnetic hysteresis loops, were measured using a physical property measurement system (PPMS) and a superconducting quan- tum interference device (SQUID), both from the Quantum Design, Inc. PPMS was also used for temperature and magnetic field control during the overall dielectric and pyroelectric measurement process. Thee–Tdata were collected using a HP4294 impedance analyzer. The pyroelectric current (magnetoelectric current) was measured upon heating in zero electric field employing a Keithley 6517A electrometer after cooling the sample in the poling electric field of 10 kV/cm andshortening at the lowest temperature of 2 K for about 60 min. The P–T data were calculated by integrating the pyroelectric current Iwith time. P–Eloops at various temperatures were obtained by the Positive- Up-Negative-Down (PUND) method. As we know, MnCr 2S4crystallizes in a normal cubic spinel structure (space group Fd3m ) at room temperature, with Mn2þ(3d5, S¼5/2) and Cr3þ(3d3,S¼3/2) ions occupying the tetrahedral Aand octahedral Bsites, respectively.20The measured high resolution h–2h XRD pattern of polycrystalline MnCr 2S4we fabricated is presented in Fig. 1(a) . The sample is of single phase and identified as cubic Fd3m symmetry with unit cell parameter a¼b¼c¼10.11989 A ˚obtained from Rietveld refinement. Next, let us consider the magnetic properties of the single phase MnCr 2S4.T h e Tdependence of the measured magnetization Mand specific heat normalized by temperature Cp/Tare shown in Fig. 1(b) for MnCr 2S4polycrystalline samples in the Trange of 2K<T<100 K. Upon cooling, a kink appears at 65 K and a small plateau develops below 5 K in the field-cooling (FC) and zero-field-cooling (ZFC) M–Tcurves, while the M–Tcurves are shown as a platform with rounded corners between 5 K and 65 K. Two k-type anomalies at T C¼65 K and TYK¼5K a r e o b s e r v e d i n t h e Cp/T–T FIG. 1. (a) The XRD spectra, whole pattern fitting, and Rietveld refinement of pow- der MnCr 2S4at room temperature. (b) M–Tcurves in the ZFC and FC modes and specific heat Cp/TvsTplot. (c) Tdependences of polarization Pand dielectric con- stant e0at a frequency of f¼10 kHz without the external magnetic field.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 032903 (2020); doi: 10.1063/5.0012914 117, 032903-2 Published under license by AIP Publishingcurve. The upper anomaly at /C2465 K marks the onset of ferromagnetic ordering of Cr3þions and quasiparamagnetic Mn2þions, while the lower one at TYK¼5 K indicates the YK type two-dimensional non- collinear structure with a canted spin structure of the Mn2þions. The transition between different magnetic order occurs via successive steps,suggested by the rounded corners in the M–Tcurves and the gradual variation near the specific heat peaks. These results are in good agree- ment with previous studies. 15–20The upper magnetization kink char- acterizes a massive scale, almost simultaneous polarization ofchromium moments. The lower magnetization plateau highlights the frozen triangular magnetic structure of the YK type below 5 K. Besides, the YK spin configuration, a kind of spin-superfluid phase, isstable at a low external magnetic field ( H<11 T). 27,29So, taking into account both the multiferroic spin-superfluid phase and magnetostric- tion below TCin MnCr 2S4, we need further exploration of multifer- roicity in a wide Trange. The dielectric and FE properties are the main focus of this effort. First, the Tdependence of Punder zero external magnetic field [Fig. 1(c) ] shows that the remnant polarization exhibits a maximum value Pmax¼192lC/m2atT¼2 K, and we also observe that Pvan- ishes at TC(TFE¼TC). In other words, polycrystalline MnCr 2S4is shown to be magnetoelectric material below TC. The reliability of the measured P(T) is evidenced by the pyroelectric current Ias a function ofTwith different warming rates: 2, 4, 6 K/min without the applied magnetic field, as shown in Fig. 2(a) . One can see that a discharging electric current appears below TFEand a charging electric current below TYKin the three I(T) curves. The peak positions of the three I(T) curves are almost identical, and the pyroelectric current intensity increases about linearly with the heating rate. This indicates that thepyroelectric current mainly comes from ferroelectric polarization. We also measured the P(T) curves under positive and negative poling elec- tric fields E¼61 0k V / c mw i t hah e a t i n gr a t eo f4K / m i n .T h ep o l a r i t y ofPis reversed by the sign of E,a ss h o w ni n Fig. 2(b) . All those illus- trate that ferroelectricity really exists in MnCr 2S4below TC. Then, a small but clear anomaly was identified at 4.5 K in the T dependence of the dielectric constant e(T) curve at a frequency of f¼10 kHz under zero external magnetic field [shown in Fig. 1(c) ], indicating a FE transition, coinciding with TYK. However, there is no dielectric anomaly at TC, and the dielectric constant grows continu- ously up to the highest test temperature limit of 100 K, 35 K above TC [shown in Fig. 1(c) ]. We think that the dielectric anomaly at TCmay be buried deep into the high dielectric background brought by the thermal activation of defects in the polycrystalline sample. Although we have shown enough evidence to prove the ferroelec- tricity in the ferrimagnetic MnCr 2S4,t h e M–Hloops and P–Eloops are the most important symbols of ferromagnetism and ferroelectric-ity, respectively. M–Hloops and P–Eloops of MnCr 2S4atT¼5K , 30 K, 50 K are plotted in Figs. 3(a) and3(b). The perfect shape of the M–Hloops reconfirms the ferrimagnetism of MnCr 2S4below TCand the reduced magnetism below TY-K. Because of the not large enough resistivity of MnCr 2S4at the test temperatures of T¼50 K and 30 K, theP–Ecurve cannot show us perfect loops as that of typical ferroelec- trics, but eye-like shaped loops until T¼5K .19AtT¼5 K, MnCr 2S4 shows a near perfect P–Eloop, and the measured ferroelectric polari- zation 186 lC/m2is almost the same as that obtained by the pyroelec- tric method (192 lC/m2), implying that P–Eloops obtained by the PUND method are quite reliable. The PUND data confirm the FIG. 2. Reliability verification of the pyroelectric data. (a) The pyroelectric currents with different warming rates of 2 K/min, 4 K/min, and 6 K/min without the external magnetic field. (b) The symmetric FE polarization under positive and negative pol-ing electric fields E¼610 kV/cm, respectively, at a warming rate of 4 K/min with- out the external magnetic field. FIG. 3. Ferroelectricity in ferrimagnetic MnCr 2S4.M–Hloops (a) and P–Eloops obtained by the PUND method (b) at 5 K, 30 K, and 50 K.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 032903 (2020); doi: 10.1063/5.0012914 117, 032903-3 Published under license by AIP Publishingferroelectricity of ferrimagnetic MnCr 2S4again as revealed by the pyroelectric current method. Here, it should be noted that this ferroelectric phase transition temperature TEF¼65 K is about 35 K higher than that of o-ReMnO 3 and o-ReMn 2O5,9and the polarization value is close to that of o-DyMnO 3. In addition, the nonzero discharging electric current below TFEand charging electric current peak below TYKin the I(T) curve suggest that the direction of Pin the ferroelectric sublattice highlighted below TYKis in contrast to that of the ferroelectric sublat- tice above TYK. Thus, we expect to have much richer underlying phys- ics on the ferroelectricity in MnCr 2S4, especially the ferroelectricity related to the interesting canted Mn spins in the YK phase. Finally, we demonstrate the magnetoelectric effect to help us determine the ferroelectricity mechanism in MnCr 2S4. For clarity, the IH(T) data under different magnetic fields Hare plotted in Fig. 4(a) in two separate Tranges. The first view comes into our sights is theremarkable responses of the lower charging electric current peak to H. With the increasing applied magnetic field, the lower negative peak position shifts to higher temperatures, which is good agreement with the temperature dependences of the magnetization in different mag- netic fields for a MnCr 2S4single crystal reported by Tsurkan et al. in 2003.20These data imply that the ferroelectricity in the YK phase is spin driven, and the ferroelectricity or the YK type magnetic structure is stabilized by the applied magnetic field. As revealed by the theoreti- cal literature and magnetization experiments, a pair of magnetic atoms with canted spin can give rise to an electric dipole moment Pno mat- ter how high the symmetry of the atoms plus environment, and spin- lattice coupling is expected to play an essential role in ferroelectric properties of MnCr 2S4. Thus, we think that spin current or the inverse Dzyaloshinskii–Moriya (DM) interaction associated with pairs of Mn(1) and Mn(2) spins, which is connected by the exchange path via intermediate Cr and S atoms, can be used to explain the ferroelectricity origin in the YK phase of MnCr 2S4.A ss h o w ni n Fig. 4(b) , suppose that the chromium spins are aligned in the external magnetic field along [111], all pairs of Mn(1) and Mn(2) spins will exhibit the same vector chirality S1/C2S2, and according to the electric polarization Pij produced by DM interaction, ~Pi;iþ1¼~ei;iþ1/C2~si/C2~siþ1 ðÞ ;the ferro- electric polarization of exchange paths of 1-2-3 and 4-2-5 is presented asP123and P425and the nonzero net polarization is along the [110] direction.26For the lower negative current peak, associated with the YK magnetic phase, shifts to higher temperature with the magnetic field, we contribute it to the fact that the canted spin angle of manganese ions in the YK phase continuously increases up to the applied magnetic field about 11 T revealed by Tsurkan, which is shown inFig. 4(c) .25,27 Unlike the lower current peak, the upper pyroelectric current peak displays a different magnetoelectric coupling behavior in the r i g h tp a r to f Fig. 4(a) .T h e I(T) curves remain the same, regardless of the magnitude of the magnetic field below the temperature of 55 K, about 10 K lower than the TCof MnCr 2S4. However, once the temper- ature Texceeds 55 K, with the increasing applied magnetic field, the I(T) curve turns more gently and the nonzero Ivalues extend to the higher T. In view of the directly proportional relationship between the electric polarization Pand the area under the I(T)c u r v e ,w ec o m e to the conclusion that the ferroelectricity is enhanced and the ferro- electric phase transition temperature is raised by the external magnetic field. Along with the magnetoelectric response of the upper currentpeak and well-proven magnetostriction and spin-lattice coupling effect by the optical fiber Bragg grating method and high-field magnetization measurements, respectively, we speculated that the upper I(T)p e a k probably derives from magnetostriction; furthermore, the ferroelec- tricity enhancement near T Cby the applied magnetic field can be attributed to the field-induced further lattice distortions.28,29 So far, ferroelectricity in MnCr 2S4has been revealed by dielectric anomaly, negative pyroelectric current at TYK, and the prominent pos- itive pyroelectric current at TC. Next, the ferroelectricity was con- firmed by the linear increase in pyroelectric currents of warming ratesof 2 K/min, 4 K/min, and 6 K/min and the symmetric switching behav- ior of the polarization at equal voltage of plus and minus. Then, ferro- electricity in the ferromagnetic MnCr 2S4is further confirmed by M–H loops and P–Eloops at T¼5 K, 30 K, and 50 K. Finally, the ferroelec- tricity generation mechanisms in MnCr 2S4are proposed based on the magnetoelectric coupling effect. There are two ferroelectric sublattices FIG. 4. (a) Magnetoelectric coupling in MnCr 2S4. Pyroelectric current at different magnetic fields in two separate Tranges of 2 K /C20T/C2012 K and 15 K /C20T/C20100 K. (b) Lattice and magnetic structure of MnCr 2S4in the YK phase (suppose that Chromium spins are aligned along [111]), as well as the ferroelectric polarization arising from the exchange paths 1-2-3 and 4-2-5 between the two manganese sub- lattices Mn(1) and Mn(2) based upon DM interaction. (c) Schematic diagram of theangle between Mn(1) and Mn(2) ain the YK phase as a function of external mag- netic field H(H/C209T).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 032903 (2020); doi: 10.1063/5.0012914 117, 032903-4 Published under license by AIP Publishingin MnCr 2S4. One sublattice has its polarization generated by mag- netostriction below TC. When Tis decreased to TYK, the other fer- roelectric sublattice with polarization in an opposite direction appears, and its polarization is attributed to the spin canted Mn2þ ion in the YK phase. In summary, extensive multiferroic measurements have been car- ried out on MnCr 2S4, and the origin of ferroelectricity has been dis- cussed. We argued that magnetostriction-induced ferroelectricityappears in MnCr 2S4once Tfalls below TC,w h i l e Tis lower than TYK, another ferroelectric sublattice with reverse polarization rises above the former one. This new emerging ferroelectric sublattice is driven bythe YK type magnetic structure. Furthermore, ferroelectricity inMnCr 2S4can be enhanced by the applied magnetic field for the increased canted spin angle of Mn2þions in the YK phase under the magnetic field and the field-induced lattice distortions. This work was supported by the Natural Science Foundation of China (Grant Nos. 11804080, 11874031, and 11834002) and the NSFC-Henan Joint Fund (Grant No 162300410089). DATA AVAILABILITY The data that support the findings of this study are available within this article. REFERENCES 1T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, and Y. Tokura, Nature 426, 55 (2003). 2J. Wang, J. B. Neaton, H. Zheng, V. Nagarajan, S. B. Ogale, B. Liu, D. Viehland, V. Vaithyanathan, D. G. Schlom, U. V. Waghmare, N. A. Spaldin, K.M. Rabe, M. Wuttig, and R. Ramesh, Science 299, 1719 (2003). 3K. F. Wang, J.-M. Liu, and Z. F. Ren, Adv. Phys. 58(4), 321 (2009). 4S. Dong, J.-M. Liu, S. W. Cheong, and Z. F. Ren, Adv. Phys. 64, 519 (2015). 5K. Xu, X.-Z. Lu, and H. J. Xiang, npj Quantum Mater. 2, 1 (2017). 6S.-W. Cheong, D. Talbayev, V. Kiryukhin, and A. Saxena, npj Quantum Mater. 3, 19 (2018). 7K. Yoo, B. Koteswararao, J. Kang, A. Shahee, W. Nam, F. Balakirev, V. S. Zapf, N. Harrison, A. Guda, N. Ter-Oganessian, and K. Hoon Kim, npj Quantum Mater. 3, 45 (2018). 8C. L. Lu, M. H. Wu, L. Lin, and J.-M. Liu, Natl. Sci. Rev. 6, 653 (2019).9T. Goto, T. Kimura, G. Lawes, A. P. Ramirez, and Y. Tokura, Phys. Rev. Lett. 92, 257201 (2004). 10N. Hur, S. Park, P. A. Sharma, S. Guha, and S.-W. Cheong, Phys. Rev. Lett. 93, 107207 (2004). 11H. Schmid and E. Asche, J. Phys. C 7, 2697 (1974). 12G. Q. Zhang, S. Dong, Z. B. Yan, Y. Y. Guo, Q. F. Zhang, S. Yunoki, E. Dagotto, and J.-M. Liu, Phys. Rev. B 84, 174413 (2011). 13F .W a n g ,T .Z o u ,L .Q .Y a n ,Y .L i u ,a n dY .S u n , Appl. Phys. Lett. 100, 122901 (2012). 14C.-C. Gu, X.-L. Chen, and Z.-R. Yang, Electric, Transports and Magnetic Properties of Cr-Based Chalcogenide Spinels (INTECH Publisher, 2017). 15J. Hemberger, P. Lunkenheimer, and R. Fichtl, Phase Transitions 79(12), 1065 (2006). 16J. Hemberger, H.-A. Krug von Nidda, V. Tsurkan, and A. Loidl, Phys. Rev. Lett. 98, 147203 (2007). 17J. Hemberger, P. Lunkenheimer, R. Fichtl, H.-A. Krug von Nidda, V. Tsurkan, and A. Loidl, Nature 434, 364 (2005). 18S. Weber, P. Lunkenheimer, R. Fichtl, J. Hemberger, V. Tsurkan, and A. Loidl, Phys. Rev. Lett. 96, 157202 (2006). 19L. Lin, H. X. Zhu, X. M. Jiang, K. F. Wang, S. Dong, Z. B. Yan, J. G. Wan, and J.-M. Liu, Sci. Rep. 4, 6530 (2015). 20V. Tsurkan, M. M €ucksch, V. Fritsch, J. Hemberger, M. Klemm, S. Klimm, S. K}orner, H.-A. Krug von Nidda, D. Samusi, E.-W. Scheidt, A. Loidl, S. Horn, and R. Tidecks, Phys. Rev. B 68, 134434 (2003). 21N. Menyuk, K. Dwight, and A. Wold, J. Appl. Phys. 36, 1088 (1965). 22J. Denis, Y. Allain, and R. Plumier, J. Appl. Phys. 41, 1091 (1970). 23K. Ohgush, Y. Okimoto, T. Ogasawara, S. Miyasaka, and Y. Tokura, J. Phys. Soc. Jpn. 77(3), 034713 (2008). 24R. Plumier and M. Sougi, Mater. Sci. Forum 133-136 , 523 (1993). 25V. Tsurkan, S. Zherlitsyn, L. Prodan, V. Felea, P. T. Cong, Y. Skourski, Z. Wang, J. Deisenhofer, H. K. von Nidda, J. Wosnitza, and A. Loidl, Sci. Adv. 3, e1601982 (2017). 26T. A. Kaplan and S. D. Mahanti, Phys. Rev. B 83, 174432 (2011). 27A. Ruff, Z. S. Wang, S. Zherlitsyn, J. Wosnitza, S. Krohns, H.-A. K. von Nidda, P. Lunkenheimer, V. Tsurkan, and A. Loidl, Phys. Rev. B 100, 014404 (2019). 28M. Nauciei-Bloch, A. Castets, and R. Plumier, Phys. Lett. A 39(4), 311 (1972). 29A. Miyata, H. Suwa, T. Nomura, L. Prodan, V. Felea, Y. Skourski, J. Deisenhofer, H.-A. Krug von Nidda, O. Portugall, S. Zherlitsyn, V. Tsurkan, J. Wosnitza, and A. Loidl, Phys. Rev. B 101, 054432 (2020). 30M. R. Suchomel, D. P. Shoemaker, L. Ribaud, M. C. Kemei, and R. Seshadri, Phys. Rev. B 86, 054406 (2012). 31K. Singh, A. Maignan, C. Simon, and C. Martin, Appl. Phys. Lett. 99, 172903 (2011). 32Y. Yamasaki, S. Miyasaka, Y. Kaneko, J.-P. He, T. Arima, and Y. Tokura, Phys. Rev. Lett. 96, 207204 (2006).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 032903 (2020); doi: 10.1063/5.0012914 117, 032903-5 Published under license by AIP Publishing

5.0012120.pdf

J. Chem. Phys. 153, 074112 (2020); https://doi.org/10.1063/5.0012120 153, 074112 © 2020 Author(s).Complex excited state polarizabilities in the ADC/ISR framework Cite as: J. Chem. Phys. 153, 074112 (2020); https://doi.org/10.1063/5.0012120 Submitted: 28 April 2020 . Accepted: 31 July 2020 . Published Online: 21 August 2020 Maximilian Scheurer , Thomas Fransson , Patrick Norman , Andreas Dreuw , and Dirk R. Rehn ARTICLES YOU MAY BE INTERESTED IN Electronic structure software The Journal of Chemical Physics 153, 070401 (2020); https://doi.org/10.1063/5.0023185 Dyson-orbital concepts for description of electrons in molecules The Journal of Chemical Physics 153, 070902 (2020); https://doi.org/10.1063/5.0016472 Simple hydrogenic estimates for the exchange and correlation energies of atoms and atomic ions, with implications for density functional theory The Journal of Chemical Physics 153, 074114 (2020); https://doi.org/10.1063/5.0017805The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Complex excited state polarizabilities in the ADC/ISR framework Cite as: J. Chem. Phys. 153, 074112 (2020); doi: 10.1063/5.0012120 Submitted: 28 April 2020 •Accepted: 31 July 2020 • Published Online: 21 August 2020 Maximilian Scheurer,1,a) Thomas Fransson,1,2 Patrick Norman,3 Andreas Dreuw,1 and Dirk R. Rehn1,b) AFFILIATIONS 1Interdisciplinary Center for Scientific Computing, Ruprecht-Karls University, Im Neuenheimer Feld 205, D-69120 Heidelberg, Germany 2Department of Physics, AlbaNova University Center, Stockholm University, Stockholm SE-106 91, Sweden 3Department of Theoretical Chemistry and Biology, School of Engineering Sciences in Chemistry, Biotechnology and Health, KTH Royal Institute of Technology, SE-106 91 Stockholm, Sweden a)Electronic mail: maximilian.scheurer@iwr.uni-heidelberg.de b)Author to whom correspondence should be addressed: rehn@uni-heidelberg.de ABSTRACT We present the derivation and implementation of complex, frequency-dependent polarizabilities for excited states using the algebraic– diagrammatic construction for the polarization propagator (ADC) and its intermediate state representation. Based on the complex polar- izability, we evaluate C6dispersion coefficients for excited states. The methodology is implemented up to third order in perturbation theory in the Python-driven adcc toolkit for the development and application of ADC methods. We exemplify the approach using illustrative model systems and compare it to results from other ab initio methods and from experiments. Published under license by AIP Publishing. https://doi.org/10.1063/5.0012120 .,s I. INTRODUCTION Response theory offers a framework to derive and compute a multitude of molecular properties.1,2Through its general formu- lation, it can be applied to both density functional theory (DFT) and ab initio methods. Within the algebraic–diagrammatic con- struction scheme for the polarization propagator (ADC),3such response properties can elegantly and easily be derived owing to its Hermitian formulation using the formalism of the interme- diate state representation (ISR).4,5The latter offers direct access to excited state wave functions and operators, which makes it straightforward to implement method-independent spectral rep- resentations of molecular response functions. This strategy has been successfully employed to evaluate several response proper- ties with ADC, e.g., static and frequency-dependent polarizabil- ities of the electronic ground state,5–7two-photon absorption,8 and resonant inelastic x-ray scattering cross sections.9As a matter of fact, only minor programming effort is needed once the required building blocks of the ISR are in place. Here, we expandupon previous work by addressing frequency-dependent electric dipole polarizabilities for electronically excited states within the ADC/ISR framework. These can be evaluated from either the excited state linear response function or the double residue of the ground state cubic response function.10–12Following the former route via damped response theory2offers access to one-photon absorption cross sections as well as C6dispersion coefficients. We have implemented a protocol for the calculation of these properties through third order of perturbation theory in our recently published Python-driven toolkit for ADC method devel- opment, adcc ,13which made the implementation particularly straightforward. Over the past few decades, numerous quantum chemical meth- ods have been employed for the calculation of frequency-dependent excited state polarizabilities, including wave function methods such as Hartree–Fock,10,14–16coupled-cluster,11,12,17–21and multi- reference approaches,15,16,18,22as well as DFT methods.20,23–25Calcu- lations of excited state C6dispersion coefficients are more scarce,24 and this property is also difficult to determine experimentally. J. Chem. Phys. 153, 074112 (2020); doi: 10.1063/5.0012120 153, 074112-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Generally, two possible approaches to calculate molecular proper- ties exist: either by derivatives of the energy or through expecta- tion values. A comparative analysis of these approaches has recently been conducted for ADC methods.26For excited state polarizabil- ities, both approaches have been reported for equation-of-motion coupled-cluster with singles and doubles (EOM-CCSD).12,21,27,28 The ISR-based ansatz described in this work is comparable to the expectation-value coupled-cluster approach to molecular properties, and both methods will be analyzed and compared to experimental data where available. The remainder of this paper is structured as follows: First, we outline the theoretical derivation and implementation. Second, the employed computational methodology is presented, followed by the results of test calculations on s-tetrazine, pyrimidine, formaldehyde, naphthalene, uracil, and p-nitroaniline for static excited state polar- izabilities and computations of excited state C6dispersion coeffi- cients for pyridine, pyrazine, and s-tetrazine. Finally, a brief out- look for future applications using the presented methodology is given. II. THEORETICAL DERIVATION AND IMPLEMENTATION The theoretical background of ADC and the ISR formalism has already been presented in great detail.3,4,29,30Here, we will out- line the necessary building blocks for our approach to excited state polarizabilities. Note that the same notation conventions as in Ref. 4 are used. Most important is the Hermitian eigenvalue problem MY =YΩ, involving the ADC matrix M, the matrix of eigenvec- torsY= {yn}, and the matrix Ωwith excitation energies ωnon the diagonal. The ADC matrix corresponds to the matrix representation of the shifted Hamiltonian, MIJ=⟨˜ΨI∣ˆH−E0∣˜ΨJ⟩, in the basis of intermediate states {∣˜ΨJ⟩}. The IS basis is constructed by applying excitation operators to the correlated ground state and subsequently orthogonalizing the obtained correlated excited states.4A key fea- ture, which we can exploit for the derivation of molecular response properties, is the IS expansion of an excited state wave function, given by ∣Ψn⟩=∑ JYJn∣˜ΨJ⟩. (1) This IS formalism also allows one to express other operators in the same basis. For example, a general one-particle operator, ˆd=∑pqdpqc† pcq, can be represented in the IS basis as BIJ=⟨˜ΨI∣ˆd∣˜ΨJ⟩−δIJ⟨Ψ0∣ˆd∣Ψ0⟩, (2) where the matrix is again shifted by the ground state expectation value on the diagonal. With the same “recipe,” the modified tran- sition moments from ground to excited states can be formulated asFJ=⟨˜ΨJ∣ˆd∣Ψ0⟩. Further details and discussion of ISR equations can be found in Ref. 4. The quantities introduced above suffice to conveniently express various ground-state response functions in the ADC/ISR formalism. To extend this approach to the excited state, we introduce the modified quantity Mf=M−1ωf−ωfyfy† f, (3) which shifts the diagonal by ωfand projects the fth eigenstate out of the matrix Mand the analogously modified ISR operator matrix,Bf=B−ωfyfy† f. (4) These quantities replace the original ADC matrix Mand operator matrix Bin the corresponding excited state response function. From time-dependent perturbation theory, the electric dipole polarizability of an electronic state | ΨN⟩can be obtained as a frequency-dependent response function,5 αN AB(ω)=−⟨ΨN∣ˆμA(̵hω−ˆH+EN)−1ˆμB∣ΨN⟩ +⟨ΨN∣ˆμB(̵hω+ˆH−EN)−1ˆμA∣ΨN⟩, (5) with the dipole operator ˆμand the respective Cartesian component A orB. This expression is valid both for the electronic ground state |0 ⟩ and for electronically excited states | f⟩. Equation (5) can be recast to the sum-over-states (SOS) expression or spectral representation to yield the polarizability of an excited state fas αf AB(ω)=1 ̵h∑ n≠f[⟨f∣ˆμA∣n⟩⟨n∣ˆμB∣f⟩ ωn−ωf−ω−iγn+⟨f∣ˆμB∣n⟩⟨n∣ˆμA∣f⟩ ωn−ωf+ω+iγn]. (6) The response function from Eq. (5) was further made resonant- convergent by introducing γn, which is related to the inverse, finite life time of excited state | n⟩. We have used a common damping parameterγn=γfor every state.9Note that the prefactor1̵his omit- ted in the following. The transition moments arising in Eq. (6) can be expressed using the ISR according to ⟨f∣ˆμA∣n⟩=∑ IY† If⟨˜ΨI∣ˆμA∑ JYJn∣˜ΨJ⟩ =∑ IJY† IfBA IJYJn=y† fBAyn (7) for excited states fand n. The electric dipole operator is used for the matrix elements of B, and for notational brevity, the super- script indicates the Cartesian component the matrix refers to. If n corresponds to the electronic ground state, one finds ⟨f∣ˆμA∣0⟩=∑ IY† If⟨˜ΨI∣ˆμA∣0⟩ =∑ IY† IfFA I=y† fFA. (8) Inserting the previous two expressions into Eq. (6), we arrive at αf AB(ω)=∑ n≠f, n≠0⎡⎢⎢⎢⎢⎣y† fBA fyny† nBB fyf ωn−ωf−ω−iγ+y† fBB fyfy† fBA fyf ωn−ωf+ω+iγ⎤⎥⎥⎥⎥⎦ +y† fFAFB†yf −ωf−ω−iγ+y† fFBFA†yf −ωf+ω+iγ. (9) The final step to arrive at a programmable expression is to substitute the excitation vectors ynand the denominator with an expression involving the inverse ADC matrix M−1, i.e., αf AB(ω)=y† fBA f(Mf−ω−iγ)−1BB fyf+y† fBB f(Mf+ω+iγ)−1BA fyf +y† fFAFB†yf −ωf−ω−iγ+y† fFBFA†yf −ωf+ω+iγ. (10) J. Chem. Phys. 153, 074112 (2020); doi: 10.1063/5.0012120 153, 074112-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Instead of full matrix inversion, a system of linear equations is solved, which for the first term corresponds to (Mf−ω−iγ)xB f=BB fyf, (11) yielding the response vector xf. Subsequently, the polarizability can be evaluated as αf AB(ω)=y† fBA fxB f+y† fBB fxA f+y† fFAFB†yf −ωf−ω−iγ+y† fFBFA†yf −ωf+ω+iγ. (12) From the complex, frequency-dependent polarizability, C6disper- sion coefficients can be obtained, as described in Ref. 6. A. Implementation Complex algebra is avoided by recasting the equation to a double-dimensional form (Mf−1ω 1γ 1γ −Mf+1ω)(xR xI)=(Bfyf 0), (13) where the solution vector contains a real and an imaginary block xRandxI, respectively.9,31,32The right-hand side is treated in the same manner, where the imaginary part is zero for real-valued operators. Only the matrix-vector products of the unshifted ADC/ISR matrices with a trial vector rare available in our code and the mod- ified matrices MfandBfare implemented by projecting out all components along the eigenvector xfafter the matrix multiplica- tion, i.e., Mfr=Mr−ωfr−xfx† fMr x† fxf. (14) We employ a conjugate gradient (CG) algorithm33with a Jacobi pre- conditioner to solve Eq. (13) for a given right-hand side. For the static polarizability, the problem in Eq. (13) reduces to MfxR=Bfyf, (15) which we solve using a standard Jacobi algorithm including Ander- son mixing (sometimes called DIIS mixing).34The implementation was achieved using our recently published adcc toolkit.13All addi- tional working equations (e.g., the Bmatrix) were implemented on the C++ layer and are conveniently exposed to the Python layer. Since only the full matrix representation for Bis given in Ref. 4, the necessary matrix-vector product was derived and the programmable expressions are shown in the Appendix. Iterative solvers and evaluation of the final polarizability expressions are written entirely in Python. Complex frequency-dependent polariz- abilities were implemented as well.6To test our implementation, Eq. (6) was evaluated for small test systems (H 2O/6-31G, LiH/STO- 3G), where a full ADC(2) matrix diagonalization is easily achiev- able. The results from evaluating the SOS expression were then compared to the result from the linear solvers and were found to agree, which confirmed the validity of our implementation (data not shown).III. COMPUTATIONAL DETAILS Geometries for s-tetrazine, pyrimidine, uracil, and p-nitroaniline (PNA) were obtained from Ref. 12. For s-tetrazine, static polariz- abilities of the ground state and the 11B1uexcited state were com- puted using the Sadlej-pVTZ basis set35and the geometry of the corresponding electronic state, as described in Ref. 12. Results were obtained using ADC(2), ADC(2)-x, and ADC(3/2), as now imple- mented in adcc . SCF results were obtained using pyscf .36,37In all ADC calculations, the second-order ISR was employed.4,6,26In combination with third-order ADC, this results in the ADC(3/2) approximation. For consistency, the calculations using the EOM- CCSD derivative and expectation-value approaches were repeated from Ref. 12, employing the Q-Chem 5.2 program package.38The same methods were employed to compute static polarizabilities of the pyrimidine ground state and the 11B2excited state. Formalde- hyde and naphthalene were optimized at the MP(2)/cc-pVTZ level of theory using Q-Chem 5.2,38where the former was placed in thexzplane and the latter in the xyplane. For formaldehyde, the polarizabilities of the ground state and the 11B1excited state were computed, whereas the ground state and 11B3ustate polarizabili- ties were obtained for naphthalene using all three ADC methods and CCSD with the aug-cc-pVDZ basis set. For uracil and PNA, the aug-cc-pVDZ basis set was employed to compute the polariz- abilities of the ground state and 11A′′and 21A′, as well as 21A1 excited states, respectively. For the computation of C6dispersion coefficients, pyridine and pyrazine geometries were optimized at the MP(2)/cc-pVTZ level of theory using Q-Chem 5.2,38whereas the previously mentioned ground state geometry was taken for s- tetrazine. Ground and excited state C 6dispersion coefficients for these three molecules were computed according to the procedure outlined in Ref. 6. For pyridine, the 11B2excited state was consid- ered and 11B3ufor pyrazine and s-tetrazine. Results were analyzed using cclib39andpandas40,41and plotted using matplotlib42and seaborn .43 IV. NUMERICAL CASE STUDIES To illustrate our implementation of excited state polarizabil- ities and C6coefficients, we have computed these properties for s-tetrazine, pyrimidine, uracil, and PNA. The static polarizabili- ties of these molecules were previously investigated using EOM- CCSD.12We compare our ADC results with this study, analyzing the ADC/ISR results at different levels of perturbation theory, in contrast to EOM-CCSD results using a derivative or expectation- value scheme. In addition, static polarizabilities of formaldehyde and naphthalene are compared to experimental data and among the employed methodologies. Anticipated trends for excited state polarizabilities have been thoroughly discussed in Ref. 12: In brief, states with a large exciton size tend to have larger polarizabilities than the electronic ground state (e.g., Rydberg states), whereas the opposite should be the case for excited states with a large perma- nent dipole moment, e.g., charge-transfer (CT) states. From the SOS expression of the polarizability [Eq. (6)], it also becomes clear that low-lying dipole-allowed excited states should possess larger polarizabilities than the electronic ground state due to coupling to the ground state and higher-lying excited states.12To discuss the different computational methods, we use absolute differences J. Chem. Phys. 153, 074112 (2020); doi: 10.1063/5.0012120 153, 074112-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp of isotropic polarizabilities, αiso=1 3(αxx+αyy+αzz). The largest differences can be expected in comparison with derivative-based CCSD (CCSD Der.), since this approach follows a different strat- egy than expectation-value-based methods. Thus, we use the devi- ation defined as δDer.= |αiso(CCSD Der.) −αiso(expectation-value method)| to discuss the results. We stress, however, that CCSD Der. results are not considered superior, i.e., they are not considered as the theoretical best estimate for excited state polarizabilities in this work. A.s-Tetrazine and pyrimidine Table I shows the Cartesian components of the static polariz- abilities for the ground and 11B1ustates of s-tetrazine, as obtained using ADC and CCSD methods.12As previously stated, most CCSD results discussed herein have already been reported12and have only been amended by CCSD expectation-value (E.V.) results for completeness. For the ground state static polarizabilities, all methods yield results of comparable magnitude, with differences δDer.of 3.78 a.u., 6.86 a.u., 2.53 a.u., and 1.61 a.u. for ADC(2), ADC(2)-x, ADC(3/2), and CCSD E.V., respectively. For excited state polarizabilities, the differences are larger, with a decreasing discrepancy trend for the ADC hierarchy of 14.91 a.u. for ADC(2), 8.58 a.u. for ADC(2)-x, and 6.64 a.u. for ADC(3/2). By comparison, the deviation for CCSD E.V. is 11.32 a.u. As such, we note that ADC(3/2) is in closest agree- ment with the derivative-based EOM-CCSD result. An experimental result for the anisotropy of the polarizability [ Δα=1 2(αyy+αxx)−αzz] is reported as 5.4 a.u. and 45.2 a.u. for the ground state and the lowest singlet state, respectively.44All computational methods overshoot the anisotropy for the electronic ground state [30.2 a.u., 31.5 a.u., 28.4 a.u., 25.6 a.u., and 26.9 a.u. for ADC(2), ADC(2)-x, ADC(3/2), CCSD Der., and CCSD E.V., respectively]. However, the anisotropy of the lowest singlet state static polarizability matches the experi- mental result reasonably well [43.1 a.u., 43.8 a.u., 40.2 a.u., 41.1 a.u., and 37.5 a.u. for ADC(2), ADC(2)-x, ADC(3/2), CCSD Der., and CCSD E.V., respectively]. For the pyrimidine molecule, a similar trend is observed for the agreement between computational results, which are shown in Table II. Here, we find discrepancies δDer.for ground state polariz- abilities of 4.31 a.u., 7.23 a.u., 2.55 a.u., and 1.78 a.u. for for ADC(2), ADC(2)-x, ADC(3/2), and CCSD E.V., respectively. Deviations for the excited state (here 11B2) to CCSD Der. are decreasing fromADC(2) (13.65 a.u.) and ADC(3/2) (9.90 a.u.) to ADC(2)-x (9.56 a.u.), and the difference to CCSD E.V. lies between ADC(3/2) and ADC(2) (12.68 a.u.). Both excited states are of n→π∗character, and the largest increase in polarizability is found for in-plane com- ponents. This is consistent for all ADC methods in comparison to EOM-CCSD.12While all methods based on ISR/expectation values are capable of predicting these trends for s-tetrazine and pyrimi- dine correctly, one notices that trends for out-of-plane components (αzzandαyyfors-tetrazine and pyrimidine, respectively) are not in agreement with the EOM-CCSD derivative approach. However, this observation is made for ADC and CCSD E.V., and thus, the effect is solely related to the ansatz to compute the polarizability and not to the method itself. Note that coupling to the electronic ground state is negligible for the reported polarizabilities, since the n→π∗states are dipole-forbidden. Another observation that requires discussion is the reduced discrepancy to CCSD Der. results with increasing order of perturbation theory from ADC(2) toward ADC(3/2). In a recent study, it has been demonstrated that ADC(3/2) yields orbital relaxation effects for p−hexcited states through higher order of perturbation theory by including 2 p−2hstates.26This explains the trends observed for the employed ADC schemes. Our numerical results thus yield the anticipated behavior of the respective ADC schemes and provide values comparable to the related CCSD E.V. approach. B. Formaldehyde and naphthalene Two more molecules for which experimental data for excited state polarizabilities are available are presented in the follow- ing. First, we examine formaldehyde for which the computational results are displayed in Table III. For all methods, a rather small ground state polarizability is found, which largely increases when the molecules is in the 11B1excited state. An approximately 20- fold increase in isotropic polarizability is present for all compu- tational methods. This is consistent with the experiment, which reports an isotropic polarizability for the ground state as 18.9 a.u. and that of 11B1as∼410±180 a.u.15,45As such, all computational result are well within the range of the experimentally obtained val- ues. The large increase in polarizability can be rationalized from the Rydberg-type excitation of the state at hand, which possesses a large exciton size. This also explains why all components of the polarizability tensor are larger compared to the electronic ground state. TABLE I . Static polarizabilities of the s-tetrazine ground and excited 11B1ustates.a Ground state 11B1u State ( αxx,αyy,αzz) αiso (αxx,αyy,αzz) αiso Eexc ADC(2) (66.08, 61.26, 33.50) 53.61 (39.06, 78.55, 15.71) 44.44 2.20 ADC(2)-x (69.38, 64.96, 35.72) 56.69 (50.64, 80.08, 21.60) 50.77 1.31 ADC(3/2) (64.10, 59.57, 33.41) 52.36 (57.28, 74.94, 25.90) 52.70 2.18 CCSD Der. (60.73, 56.02, 32.73) 49.83 (66.03, 80.09, 31.93) 59.35 2.39 CCSD E.V. (62.82, 58.01, 33.48) 51.44 (49.28, 71.75, 23.05) 48.03 2.39 aPolarizability components ( αAA,αiso) in a.u. and excitation energies ( Eexc) in eV. J. Chem. Phys. 153, 074112 (2020); doi: 10.1063/5.0012120 153, 074112-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II . Static polarizabilities of the pyrimidine ground state and 11B2.a Ground state 11B2 State ( αxx,αyy,αzz) αiso (αxx,αyy,αzz) αiso Eexc ADC(2) (73.48, 38.69, 76.22) 62.80 (118.89, 26.65, 38.79) 61.45 4.32 ADC(2)-x (77.21, 39.99, 79.97) 65.72 (114.28, 31.22, 51.10) 65.53 3.44 ADC(3/2) (71.24, 38.27, 73.61) 61.04 (104.45, 33.64, 57.48) 65.19 4.50 CCSD Der. (67.79, 37.50, 70.18) 58.49 (111.76, 42.13, 71.38) 75.09 4.59 CCSD E.V. (70.05, 38.25, 72.52) 60.27 (102.93, 33.16, 51.14) 62.41 4.59 aPolarizability components ( αAA,αiso) in a.u. and excitation energies ( Eexc) in eV. TABLE III . Static polarizabilities of the formaldehyde ground state and 11B1.a Ground state 11B1 State ( αxx,αyy,αzz)αiso (αxx,αyy,αzz) αiso Eexc ADC(2) (17.94, 12.88, 24.90) 18.57 (712.91, 243.79, 310.03) 422.25 6.27 ADC(2)-x (18.40, 13.20, 25.65) 19.09 (641.26, 250.44, 314.84) 402.18 5.98 ADC(3/2) (17.31, 12.68, 23.47) 17.82 (678.81, 281.39, 432.19) 464.13 7.57 CCSD Der. (17.23, 12.50, 22.58) 17.44 (680.32, 272.78, 384.16) 445.75 7.05 CCSD E.V. (17.39, 12.66, 22.98) 17.67 (688.15, 272.27, 388.23) 449.55 7.05 aPolarizability components ( αAA,αiso) in a.u., excitation energies ( Eexc) in eV. Next, we consider the ground state and 11B3upolarizabil- ity of naphthalene. The corresponding results from computations and experiment are shown in Table IV. The percentaged devia- tions from experimental values are depicted in Fig. 1. The per- formance of the computational methods compared to experiment is rather heterogeneous for the ground state polarizability. The largest overestimation for in-plane components αxxandαyyis found for ADC(2)-x with ∼20%, whereas CCSD Der. agrees best with the experimental results for these components. Deviations from the experimental αzzresult are below 5%, except for CCSD Der., which underestimates the component by ∼6%. As such, all employed methods except for ADC(2)-x yield reliable static polarizabilities for the electronic ground state of naphthalene. In the experiment,a small increase was observed for the static polarizability compo- nents of the 11B3ustate compared to the ground state, the largest of which is found for αxx. For the polarizabilities based on expecta- tion values, this trend could not be observed in the computational results. Especially, the αyyandαzzcomponents are largely under- estimated by expectation-value-based methods, the most extreme being ADC(2) with more than −40% deviation for αzz. On the contrary, derivative-based EOM-CCSD is capable of describing the trend of small increases in the components correctly. Here, the deviations are below 5% for αxxandαyyand∼−9% forαzz. Thus, one can conclude that in this case, amplitude relaxation effects seem to be especially important to model the polarizabilities correctly. TABLE IV . Static polarizabilities of the naphthalene ground and 11B3ustates.a Ground state 11B3u State ( αxx,αyy,αzz) αiso (αxx,αyy,αzz) αiso Eexc ADC(2) (182.04, 133.68, 69.29) 128.34 (178.48, 73.79, 44.13) 98.80 4.45 ADC(2)-x (194.22, 140.69, 71.34) 135.42 (164.97, 88.51, 50.91) 101.47 3.46 ADC(3/2) (177.81, 129.95, 68.62) 125.46 (170.71, 98.76, 56.14) 108.53 4.16 CCSD Der. (166.78, 123.14, 66.67) 118.86 (195.12, 121.47, 70.32) 128.97 4.41 CCSD E.V. (173.89, 128.11, 68.22) 123.41 (164.30, 88.58, 54.89) 102.59 4.41 Experimentb(162.0, 119.5, 70.9) 117.4 (186.9, 120.1, 76.9) 128.0 4.02 aPolarizability components ( αAA,αiso) in a.u. and excitation energies ( Eexc) in eV. bReferences 46 and 14. J. Chem. Phys. 153, 074112 (2020); doi: 10.1063/5.0012120 153, 074112-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . Deviations of computed polarizability components from the experimental value in percent for the ground and 11B3ustates of naphthalene. C. Uracil and p-nitroaniline As another example, the ground state and lowest singlet n→π∗(11A′′) andπ→π∗(11A′) states of uracil are considered, with results presented in Table V. For the electronic ground state, all five methods again yield comparable results. Similar to s-tetrazine and pyrimidine, the n→π∗states have slightly increased polarizabilities for in-plane components αxxandαyy, when derivative-based EOM- CCSD is used. This is not the case for the expectation-value methods. In fact, all ADC and the EOM-CCSD E.V. results show a notice- able drop in polarizability for this state of uracil. Surprisingly, this discrepancy is not reduced when employing ADC(3/2) but instead becomes even larger. EOM-CCSD E.V. here yields values similar to ADC(3/2). For the π→π∗transition, both dipole moments and polarizabilities show a large increase.12For this state, the αxxcom- ponent of the polarizability increases the most due to the large coupling matrix element to the ground state. Again, ADC(3/2) andEOM-CCSD E.V. show a much smaller increase for this component than EOM-CCSD Der. Results for PNA are shown in Table VI, including polarizabil- ity components of the electronic ground state and the lowest singlet excitedπ→π∗state (21A1). Ground state polarizabilities are again similar. The probed singlet state corresponds to a strong intramolec- ular charge-transfer (CT) excitation.12As such, the corresponding dipole moment increases upon excitation, yielding a species with more ionic character than in the ground state. The excitation still shows a large oscillator strength, i.e., transition dipole moment along thez-axis. Therefore, the polarizability of the 11A1largely increases in theαzzcomponent for all presented methods—particularly using CCSD Der. ADC(3/2) and EOM-CCSD E.V. behave similarly for the excited state, with a deviation δDer.to EOM-CCSD Der. of 27.37 a.u. and 29.04 a.u., respectively. Discrepancies of ADC(2) and ADC(2)-x are larger by ∼10 a.u., amounting to 39.17 a.u. and 40.91 a.u., TABLE V . Static polarizabilities of the uracil ground state, 11A′′, and 21A′.a Ground state 11A′′21A′ State ( αxx,αyy,αzz)αiso (αxx,αyy,αzz)αiso Eexc (αxx,αyy,αzz) αiso Eexc ADC(2) (105.23, 80.00, 43.37) 76.20 (86.44, 67.30, 27.08) 60.28 4.73 (138.88, 81.77, 35.46) 85.37 5.32 ADC(2)-x (110.75, 83.17, 44.48) 79.47 (85.17, 73.16, 33.17) 63.83 3.97 (147.08, 87.72, 38.37) 91.06 4.61 ADC(3/2) (98.47, 76.27, 42.30) 72.35 (74.72, 70.74, 36.03) 60.50 5.36 (105.12, 86.93, 42.27) 78.11 5.38 CCSD Der. (95.82, 74.62, 41.59) 70.67 (98.08, 88.29, 45.19) 77.19 5.22 (133.94, 102.78, 51.69) 96.14 5.58 CCSD E.V. (98.75, 76.41, 42.42) 72.53 (71.49, 69.43, 34.55) 58.49 5.22 (112.82, 85.86, 42.97) 80.55 5.58 aPolarizability components ( αAA,αiso) in a.u. and excitation energies ( Eexc) in eV. J. Chem. Phys. 153, 074112 (2020); doi: 10.1063/5.0012120 153, 074112-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE VI . Static polarizabilities of the PNA ground state and 21A1.a Ground state 21A1 State ( αxx,αyy,αzz) αiso (αxx,αyy,αzz) αiso Eexc ADC(2) (118.84, 58.94, 168.68 ) 115.49 (68.26, 49.52, 196.97 ) 104.92 4.30 ADC(2)-x (125.29, 60.62, 183.45 ) 123.12 (82.44, 50.60, 176.49 ) 103.18 3.56 ADC(3/2) (112.76, 58.02, 162.94 ) 111.24 (86.53, 59.25, 204.37 ) 116.72 4.23 CCSD Der. (106.38, 56.95, 152.90 ) 105.41 (109.60, 83.60, 239.08 ) 144.09 4.62 CCSD E.V. (110.76, 58.20, 157.68 ) 108.88 (74.95, 69.30, 200.89 ) 115.05 4.62 aPolarizability components ( αAA,αiso) in a.u., excitation energies ( Eexc) in eV. respectively. Hence, the π→π∗intramolecular CT state shows the largest differences between derivative- and expectation-value-based methods studied here. The effects of full amplitude response in the case of a CT excitation seem to have a large impact on the excited state polarizability of the respective state. As such, care should be taken in these cases. Nevertheless, ADC methods are capable of cap- turing the trend of an increasing αzzcomponent for CT excitations correctly, whereas methods such as time-dependent DFT tend to fail in this case.47,48 To summarize this brief study of static excited state polariz- abilities, our presented findings match both the expected trends and previously published results, suggesting that our implementation is comparable to related methodologies using an expectation-value- based ansatz. We have shown that the agreement between meth- ods solely depends on the approach to evaluate the polarizability, and not whether ADC or CC is chosen as the underlying quantum chemical method. Hence, it would also be interesting to see how derivative-based ADC excited state polarizabilities would compare to derivative-based EOM-CCSD. In this case, amplitude-relaxedsecond derivatives of the ADC excited state energy would need to be derived and implemented, which is beyond the scope of this work. In addition, note that the ISR-based ansatz requires much less compu- tational effort, yielding excited state polarizabilities for the price of ground state polarizabilities, once the excited states are determined. In a derivative-based approach, however, more response equations need to be solved.12 D.C6dispersion coefficients for excited states Until now, we have only considered static polarizabilities of excited states, which do not require solutions of the complex response function. For C6dispersion coefficients, however, the isotropic average of the molecular dipole polarizability as a function of purely imaginary frequencies is needed to compute the inter- action between two systems through the Casimir–Polder poten- tial.6,49With the Python function to solve Eq. (13) in place, the required Gauss–Legendre integration can be easily carried out using built-in NumPy functions,50as shown in the code snippet in Fig. 2. FIG. 2 . Python function to compute the C6dispersion coefficients with adcc . J. Chem. Phys. 153, 074112 (2020); doi: 10.1063/5.0012120 153, 074112-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE VII .C6dispersion coefficients of electronic ground and excited states employing ADC and CAS. C6dispersion coefficient (a.u.) System State ADC(2) ADC(2)-x ADC(3/2) CASa Pyridine Ground state 1717.23 1770.02 1682.04 1374 11B2 923.17 1100.90 1248.39 1278 Pyrazine Ground state 1512.76 1559.15 1478.81 1245 11B3u 743.57 849.04 1007.38 1147 s-tetrazine Ground state 1161.05 1197.54 1129.03 919.6 11B3u 499.53 575.77 687.24 835.0 aObtained from Ref. 24. This code example shows again how well adcc integrates with the Python ecosystem, making it possible to quickly implement new features with only minor effort. In addition, the rich feature set of NumPy makes it possible to write code that strongly resembles the text book equations. Using the code shown in Fig. 2, we computed C6dispersion coefficients for excited states of pyridine, pyrazine, and s-tetrazine, which were previously studied with multi-configurational complete active space (CAS) calculations using a derivative-based approach.24 The results for ADC(2), ADC(2)-x, ADC(3/2), and CAS are sum- marized in Table VII. The ground state C6dispersion coefficients for these molecules computed with ADC methods are all larger than the respective CAS results by ∼18% to 30%. This deviation, how- ever, becomes smaller from ADC(2) to ADC(3/2). In the case of excited state C6coefficients, one observes the opposite: all values obtained with ADC methods are smaller than the corresponding CAS value. In the case of pyrimidine, this amounts to dispersion coefficients that are 27%, 13%, and 2% smaller for ADC(2), ADC(2)- x, and ADC(3/2), respectively. This trend is even more pronounced for pyrazine, where we find decreases with respect to CAS results by 35%, 26%, and 12% for the ADC method hierarchy, and the results are quite similar also for s-tetrazine with discrepancies of 40%, 31%, and 18%. These deviations to CAS results can be explained through the different methodologies used in the approaches. The C6coeffi- cients are just derived properties of the excited state polarizabilities, which have been extensively discussed and compared in previous sections. Hence, for the expectation-value-based ADC results, which only describe relaxation effects through the ADC matrix itself, it is expected to see differences compared to results from fully relaxed multi-configurational CAS. This behavior of the ADC hierarchy of methods is again corroborated by the fact that ADC(3/2) results, which include the most relaxation, deviate least from CAS val- ues. Thus, the same behavior that was already observed for static excited state polarizabilities in previous sections is observed also for response properties derived from complex frequency-dependent polarizabilities. V. CONCLUSIONS We have presented the first derivation and implementation of complex, frequency-dependent excited state polarizabilities for ADC using an ISR-based ansatz. The derivation elegantly illustrateshow ADC/ISR response properties for excited state can be treated in general. The presented derivation further demonstrates how the ADC/ISR framework, in combination with the adcc toolkit, can be used to perform rapid prototyping of new response property functionalities. Since only response functions for ground state and ground-to-excited-state properties were studied in the ADC/ISR for- mulation so far, our presented theoretical methodology serves as a blueprint paving the way for arbitrary response functions. To ver- ify that the implemented methodology is consistent with similar methodologies, we have presented calculations of static excited state polarizabilities in comparison with EOM-CCSD. Furthermore, we have computed C6dispersion coefficients of excited states derived from complex, frequency-dependent polarizabilities and compared them to CAS results. The general formulation and implementation inadcc also allow us to evaluate the respective response prop- erty for open-shell molecules on top of an unrestricted Hartree– Fock reference or for systems with few-reference character using the spin–flip ADC ansatz.51From a practical point of view, our method could serve as a ab initio benchmark method for excited state polarizabilities and dispersion coefficients obtained from DFT- based methodologies. These are needed to parametrize force fields for classical molecular dynamics simulations of excited states.52Per- forming atomistic decompositions of our derived properties is, how- ever, beyond the scope of this article. We hope that our presented methodology will serve as a useful template for similar response properties in future work. ACKNOWLEDGMENTS M.S. was supported by the DFG by means of the research training group “CLiC” (GRK 1986, Complex Light Control) and by the Heidelberg Graduate School of Mathematical and Computa- tional Methods for the Sciences (Grant No. GSC220). T.F. and P.N. acknowledge financial support from the Swedish Research Coun- cil (Grant Nos. 2017-00356 and 2018-04343). The authors thank M. Hodecker for helpful discussions. APPENDIX: MATRIX-VECTOR PRODUCT OF A SECOND-ORDER ISR ONE-PARTICLE OPERATOR Based on the original equations from the work of Schirmer and Trofimov,4we derived the matrix-vector product of the ISR J. Chem. Phys. 153, 074112 (2020); doi: 10.1063/5.0012120 153, 074112-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp one-particle operator with a vector vthrough second order in per- turbation theory, i.e., Bv=r. The Bmatrix possesses a block struc- ture similar to that of the ADC matrix4such that the vectors v andrcontain a singles and a doubles block. In the following, the indices i,j,k,l,m,nrefer to occupied molecular orbitals, a,b,c,d, e,frefer to virtual ones, and p,q,r,sare general molecular orbital indices. Within the equations, anti-symmetrized two-electron inte- grals⟨pq||rs⟩occur, together with the T2-amplitudes defined as tab ij=⟨ij∥ab⟩ εa+εb−εi−εj,whereεpdenotes the energy of HF orbital p. Furthermore, the MP(2) density matrix contribution53is defined as ρ(2) ia=−1 2(εa−εi)⎡⎢⎢⎢⎢⎣∑ jbctbc ij⟨ja∥bc⟩+∑ jkbtab jk⟨jk∥ib⟩⎤⎥⎥⎥⎥⎦. The permutation operator ˆPpr,qspermutes the index pairs ( p,q) and (r,s). Collecting all terms, one finds the result for the singles block of the matrix-vector product as rai=∑ cdacvci−∑ kdikvak−∑ cvci⎛ ⎝∑ jρ(2) jadcj+ρ(2) jcdaj⎞ ⎠−∑ kvak(∑ bρ(2) ibdbk+ρ(2) kbdbi) −1 4∑ cvci⎛ ⎝∑ efmntef mn(taf mndec+tcf mndea)⎞ ⎠+∑ cvci⎛ ⎝−1 2∑ efmntce mntaf mndef+∑ fmnjtcf mntaf jndjm⎞ ⎠ +1 4∑ kvak⎛ ⎝∑ efmntef mn(tef indkm+tef kndim)⎞ ⎠+∑ kvak⎛ ⎝−∑ edfntef kntdf inded+1 2∑ efmntef kntef imdmn⎞ ⎠ +1 2∑ ckvck⎛ ⎝(1 +ˆPai,ck)⎛ ⎝∑ efntef kntaf indec−∑ fmntcf mntaf indkm⎞ ⎠⎞ ⎠+∑ ckvck⎛ ⎝−∑ fmntcf kntaf imdmn+∑ efntce kntaf indef⎞ ⎠ −2∑ dlvadil⎛ ⎝dld−∑ fntdf lndfn⎞ ⎠+ 2∑ clvcail⎛ ⎝dlc−∑ fntcf lndfn⎞ ⎠−2∑ dklvadkl∑ eted kldei−2∑ cdlvcdil∑ ntcd nldan. The doubles part of the vector is given by rabij=−vai⎛ ⎝djb−∑ fntbf jndfn⎞ ⎠+vaj⎛ ⎝dib−∑ fntbf indfn⎞ ⎠ +vbi⎛ ⎝dja−∑ fntaf jndfn⎞ ⎠−vbj⎛ ⎝dia−∑ fntaf indfn⎞ ⎠ −∑ kvak∑ eteb ijdek+∑ kvbk∑ etea ijdek −∑ cvci∑ ntab njdcn+∑ cvcj∑ ntab nidcn + 2∑ cdacvcbij−dbcvcaij−2∑ kdkivabkj−dkjvabki. 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5.0012415.pdf

J. Appl. Phys. 128, 045705 (2020); https://doi.org/10.1063/5.0012415 128, 045705 © 2020 Author(s).Electronic structure and spontaneous magnetization in Mn-doped SnO2 Cite as: J. Appl. Phys. 128, 045705 (2020); https://doi.org/10.1063/5.0012415 Submitted: 30 April 2020 . Accepted: 16 July 2020 . Published Online: 29 July 2020 Rezq Naji Aljawfi , Mahmoud Abu-Samak , Mohammed A. Swillam , Keun Hwa Chae , Shalendra Kumar , and John A. McLeod Electronic structure and spontaneous magnetization in Mn-doped SnO 2 Cite as: J. Appl. Phys. 128, 045705 (2020); doi: 10.1063/5.0012415 View Online Export Citation CrossMar k Submitted: 30 April 2020 · Accepted: 16 July 2020 · Published Online: 29 July 2020 Rezq Naji Aljawfi,1,a) Mahmoud Abu-Samak,2Mohammed A. Swillam,3 Keun Hwa Chae,4 Shalendra Kumar,5 and John A. McLeod6,b) AFFILIATIONS 1Department of Physics, Ibb University, Ibb 70270, Yemen 2Department of Physics, Al-Hussein Bin Talal University, P.O. Box 20, Ma ’an 71111, Jordan 3Department of Physics, American University in Cairo, P.O. Box 74, New Cairo 11835, Egypt 4Advanced Analysis Center, Korea Institute of Science and Technology, Seoul 136-791, Republic of Korea 5Department of Physics, College of Science, King Faisal University, P.O Box 400, Hoff, Al-Asha 31982, Saudi Arabia 6Jiangsu Key Laboratory for Carbon-Based Functional Materials & Devices, Institute of Functional Nano and Soft Materials (FUNSOM), Joint International Research Laboratory of Carbon-Based Functional Materials and Devices, Soochow University, Suzhou, Jiangsu 215123, China a)Author to whom correspondence should be addressed: rizqnaji@yahoo.com b)Present Address: Department of Electrical and Computer Engineering, Western University, 1151 Richmond St., London, ON, Canada N6A 5B9. Electronic mail: jmcleod@suda.edu.cn andjmcleod7@uwo.ca ABSTRACT Mn-doped SnO 2is a promising dilute magnetic semiconductor; however, there are many inconsistent reports on the magnetic ordering in the literature. We investigate the magnetic ordering and the local electronic structure in stoichiometric and Mn-doped (with Mn concentra- tions of 1 at.%, 3 at.%, and 6 at.%) SnO 2using magnetization measurements, Mn L 2,3-edge and O K-edge x-ray absorption fine structure measurements, and density functional theory and model Hamiltonian calculations. We find that paramagnetic and ferromagnetic behavioris present as a function of Mn concentration and, in particular, that paramagnetic, ferromagnetic, and antiferromagnetic order coexist inde-pendently in Mn(6%):SnO 2. Simultaneously, we find that Mn2þ,M n3þ, and Mn4þalso coexist in Mn(6%):SnO 2. These findings demon- strate the care needed to study Mn:SnO 2and point to the wealth of magnetic behaviors that might be realized with careful control of synthesis conditions. Published under license by AIP Publishing. https://doi.org/10.1063/5.0012415 I. INTRODUCTION Dilute magnetic semiconductors (DMSs) have attracted con- tinual attention due to their promise as active materials for next- generation computing and optoelectronics.1,2Tin dioxide is of particular interest, as it has high thermal and chemical stability atroom temperature, 3is optically transparent,4can be electrically conductive,4,5and has weak natural ferromagnetism.5,6The natural ferromagnetism of SnO 2can be enhanced by doping with magnetic transition metals.7Ferromagnetic (FM) order, with a Curie temper- ature ( TC) above room temperature, is a fundamental requirement of a DMS;8,9so, detailed understanding of the magnetic ordering in the host material is essential.Of all the transition metal elements, Mn is often the first choice of dopant for tuning magnetic properties.10,11Mn is of par- ticular interest for SnO 2since it has high solubility in the host lattice.9Multiple reports have demonstrated the FM order in Mn-doped SnO 2(Mn:SnO 2),2,12–18and evidence points to the coexistence of Mn2þand Mn3þ.7,13,19,20This single-species, hetero- valent doping has been suggested as the origin of the FM order,induced by spin-splitting at the Fermi level in the electronic densityof states local to Mn, due to the Stoner effect. 8,9 However, despite years of study, the origin of ferromagnetism in Mn:SnO 2is still not completely understood. The situation is further complicated as numerous carefully performed experimentsJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 045705 (2020); doi: 10.1063/5.0012415 128, 045705-1 Published under license by AIP Publishing.have nevertheless produced inconsistent results. Although many reports find FM behavior at room temperature, as noted above, some researchers have found paramagnetic (PM) behavior at roomtemperature. 20–22It has even been argued that Mn-doping does not create any new magnetic ordering; rather it only modifies intrinsicmagnetic order native to SnO 2.16Average magnetic moments of around 0.18 μB/Mn,213.8μB/Mn and 5.2 μB/Mn for Mn2þand Mn3þ, respectively,20from 19.5 μB/Mn to 0.2 μB/Mn dependent on Mn concentration,7and 0.27 μB/Mn12are all reported from various characterization measurements. Evidence has also been presentedthat the magnetic moment in Mn:SnO 2is time-dependent, as the magnetic moment has been observed to increase after a few months of aging in ambient conditions.18This last finding is attributed to an increase in oxygen vacancies (V O),18at certain points to the difficulty in achieving consistent results if the magnetic order in Mn:SnO 2is subject to time- and environment-dependent evolution. In addition to conventional magnetization measurements, Mn: SnO 2has been characterized by an Mn K-edge x-ray absorption fine structure (XAFS)12,13,20and M L 2,3-edge x-ray photoelectron spectroscopy.14,19The former is bulk sensitive, whereas the latter is surface sensitive. Both techniques are useful probes of the local electronic structure and can reveal the valence state of Mn present; however, Mn K-edge XAFS is a somewhat indirect probe sincetransitions to unoccupied Mn 3d orbitals are dipole forbidden,while Mn L 2,3-edge XPS suffers from a low signal-to-noise ratio due to the relatively low concentration of Mn typically present. Electron paramagnetic resonance (EPR) has also been employed tocharacterize Mn:SnO 2. These measurements suggest the coexistence of Mn4þand Mn2þ;23however, EPR cannot verify the presence of Mn3þsince that species is EPR-silent. Mn:SnO 2has also been investigated with first principles calculations.13,17,20These calculations predict that Mn is most likely to substitute for Sn (as opposed to being interstitial) int h ep r e s e n c eV O,13and that robust ferromagnetism requires both V Oand Mn-doping.17The predicted valence state of Mn is not reported, as it is very difficult to rigorously calculate the valence state of an impurity in a crystal. These calculations arealso only accurate to the extent that the chosen Mn:SnO 2struc- tures are realistic. Certainly, the choices made by the authorsseem very reasonable, but there is always the possibility that they are not exhaustive. In particular, the reports of coexisting Mn 2þand Mn3þmay suggest that at least two different environ- ments are present. Owing to these difficulties and inconsistencies, a soft x-ray Mn L 2,3-edge XAFS study of Mn:SnO 2is of great interest. Transition metal L 2,3-edge XAFS is now a well-established tech- nique for characterizing the local electronic structure,24and as tran- sitions from the 2p core level to the 3d unoccupied states aredipole-allowed, it is a more direct probe of the valence state. 25 Furthermore, it is possible to very accurately simulate transitionmetal L 2,3-edge XAFS with model Hamiltonian methods involving explicit charge transfer between ligands and the metal, as well ascrystal field-driven multiplets. 26–28This allows very detailed analy- sis of Mn L 2,3XAFS spectra, as the entire spectral shape is consid- ered rather than simply an energy shift. We, therefore, investigate the magnetic and electronic struc- ture of Mn:SnO 2using magnetization measurements and MnL2,3-edge XAFS, among other complementary characterization techniques. We find evidence for multiple magnetic phases and multiple Mn valences coexisting in Mn:SnO 2. Our findings help make sense of previous contradictory reports and demonstrate thebreadth of magnetic behavior that may be expected from Mn:SnO 2. In particular, our findings include those that have been previously reported, but also extend to new phenomena. Namely, we find PM and FM order, and that Mn2þand Mn3þcoexist, as was previously found. However, our detailed analysis of magnetization and MnL 2,3-edge XAFS points to an additional antiferromagnetic (AFM) phase and the presence of Mn4þas well. II. MATERIALS AND METHODS Nanostructured pristine and Mn:SnO 2(doped at 1 at.%, 3 at.%, and 6 at.%) were synthesized by co- precipitation using theconventional route, in which manganese and tin in nitrate form were taken as the precursors. Salts with the appropriate stoichiome- try were mixed in 50 ml of double distilled water using an adjust-able hotplate magnetic stirrer at 70 /C14C. After an hour of stirring, a potassium hydroxide (KOH) solution was dissolved separately andadded drop-wise under constant stirring until the pH reached 10. The solution was then centrifuged at 7000 rpm. The (Mn,Sn) 2O6 (OH) 4precipitate was collected and washed several times with de-ionized water and methanol and then dried at 80/C14C. Finally, the product was ground and annealed 500/C14C in a digital program- mable furnace. The Mn:SnO 2products were characterized using various techniques. X-ray diffraction (XRD) patterns were acquired usinga Philips X-pert x-ray diffracto meter with a radiation of wave- length 1.5418 Å (Cu K α), and these patterns were analyzed by the Reitveld refinement implemented using the FULLPROF suite. 29Near-edge x-ray absorption fine structure (NEXAFS) spectra at the Mn L 3,2- and O K-edges were measured at the soft x-ray beamline 10D XAS KIST (Korea Institute of Science andTechnology) at the Pohang Accelerator Laboratory (PAL). Magnetization measurements were carried out at 300 K using a quantum design physical properties measurement setup (PPMS).Raman spectroscopy was performed using an NSR-3300micro-Raman spectrometer (Jasco), and the samples were excitedwith a laser of wavelength 532.24 nm. Density functional theory (DFT) calculations were performed for bulk SnO 2, MnO, Mn 2O3, and MnO 2(using experimentally determined crystal structures30–33); SnO 2with a (110) surface (and around 10 Å of vacuum); and Mn-substituted SnO 2at concentra- tion levels of 3.125%, 6.25%, and 12.5% [including 6.25% Mn near a (110) surface] using the all-electron, full-potential, linearized augmented-plane wave method implemented by WIEN2k.34The DFT calculations for reference oxides were performed on a 1000special k-point grid; for Mn-substituted models, the number ofk-points was reduced proportionally to the increase in the volume of the supercell structure employed. The Perdew –Burke –Ernzerhoff generalized gradients approximation (PBE-GGA) was used. 35An onsite Hubbard potential Uof 3 eV,36or the modified Becke – Johnson functional (mBJ),37was employed to localize the Mn 3d orbitals. All calculations were self-consistent to energy, charge, and force tolerances of 0.0001 Ryd, 0.001 e, and 0.1 mRyd/bohr,Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 045705 (2020); doi: 10.1063/5.0012415 128, 045705-2 Published under license by AIP Publishing.respectively. The atomic positions for all structures were optimized to a tolerance of 5.0 mRyd/bohr using the MSR1a method.38 Mn L 3,2XAFS spectra were simulated using ligand –metal charge transfer crystal field (LMCT-CF) multiplet calculationsimplemented by Quanty. 27,28The Slater integrals (F dd,Fpd, and F pd) were reduced to 80% of the Hartree –Fock value. A variety of D 4h crystal field parameters ( Dq,Dt, and Ds) were investigated to best fit the experimental data (see Discussion for more details). Theligand –metal charge transfer (LMCT) parameters were chosen as follows: Δ¼2:0 eV; V(a 1g)¼2:0 eV, 2.5 eV, 3.0 eV for Mn2þ, Mn3þ, and Mn4þ, respectively; Va2g¼Va1g;Vb1g¼Veg¼0:5Va1g; and the ligand crystal field parameters DqL,DtL, and DsLwere1 2 which were the respective crystal field parameters for the metal site. These values were selected based on “physical reasonableness ”and on our experience with XAFS. These values can furthermore bevaried significantly (often by 50% or more) with little influence on the simulated spectra. O K XAFS spectra were simulated using the electronic density of states (DOS) calculated from the DFT calcula-tions mentioned above and implemented by the XSPEC packagefor WIEN2k. 39Calculations with, and without, an explicit core hole in the O 1s level were performed. For calculations with a core hole, a sufficiently large structure such that all core holes were sep- arated by at least 6 Å was employed. III. RESULTS AND DISCUSSION Our nanostructured SnO 2and Mn:SnO 2crystallizes in the rutile phase with negligible impurity phases. Figure 1 shows the measured XRD patterns and corresponding Rietveld refinementsfor SnO 2and Mn:SnO 2at 3% and 6% concentrations. All measure- ments fit the rutile space group (P4 2/mnm, JCPDS Card No. 88-0287) with the lattice constants, as shown in Fig. 1 . Scherrer ’s formula40suggests that the average crystallite size is D¼40 nm for SnO 2andD¼7 nm for Mn(6%)Mn:SnO 2. Mn-doping is expected to promote an increased oxygen deficiency ( δ), which should scale asδ≃D/C02. Our XRD data suggest that the oxygen deficiency is around 0.05% in our pristine SnO 2and 2% in Mn(6%):SnO 2. Lattice contraction from Mn substitution is unlikely to directlycause the observed decrease in crystallite size concurrent with theincrease in Mn content, as the ionic radii of Mn 2þand Mn3þare very close to that of Sn4þ(0.67 Å, 0.65 Å, and 0.69 Å for Mn3þ, Mn2þ, and Sn4þ, respectively).41 We see no hint of any secondary phase within the accuracy limit of our XRD measurements. It is, however, difficult to rule outthe presence of small amounts of segregated amorphous Mn oxides or hydroxides, a topic that we address in more detail below. To further investigate the crystallinity and influence of Mn-doping, we perform Raman spectroscopy on Mn(6%):SnO 2,a s shown in Fig. 2 . Rutile SnO 2has 18 normal lattice vibrational modes, with the irreducible representation Γ¼Γþ 1(A1g)þΓþ 2(A2g)þΓþ 3(B1g)þΓþ 4(B2g) þΓ/C0 5(Eg)þ2Γ/C0 1(A2u)þ2Γ/C0 4(B1u)þ4Γ/C0 5(Eu), (1) where, as usual, all symbols indexed “g”represent Raman active modes and all Erepresentations are doubly degenerate.42,43The Raman spectrum of Mn(6%):SnO 2has multiple overlapping features, owing to the disorder in the sample. To better resolvethese features, we fit the spectrum with eight Gaussian compo-nents. This method has previously demonstrated good results for nanoscale SnO 2.42Some of these features agree with those observed in the Raman spectra of rutile SnO 2and correspond to the Eg,A1g, andB2gvibrational modes.44The most intense part of the Raman spectrum, however, is related to disordered SnO 2.42This is FIG. 1. XRD patterns, Reitveld refinements, and corresponding residuals for SnO 2, Mn(3%):SnO 2, and Mn(6%):SnO 2. FIG. 2. Raman spectrum of Mn(6%):SnO 2. The spectrum has been fit with eight Gaussian components, those representing key vibrational modes innanoscale and rutile SnO 2are highlighted in green and orange, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 045705 (2020); doi: 10.1063/5.0012415 128, 045705-3 Published under license by AIP Publishing.commonly observed in nanoscale SnO 2, and the intensity of these features increases with decreasing crystallite size.42,45 From previous reports, the approximate ratio of the intensity of the S1 and S2 disorder features with that of A1gis consistent with the small crystallite size of D¼7 nm predicted by our XRD analysis for Mn(6%):SnO 2. The remaining features with smaller Raman shifts are not directly attributable to any distinct source (although the SnO 2B1gmode may be hiding within)44and are also commonly observed.42,45Our measurements are also consistent with those previously reported for Mn:SnO 2.46 TheA1gandEgfeatures in the Raman spectra are of particular interest for Mn(6%):SnO 2, since the former is due to in-plane oscil- lation of oxygen atoms relative to the c-axis of the Jahn –Teller dis- torted SnO 6octahedron, while the latter is due to the out-of-plane oscillation of the same.44These modes are consequently very sensi- tive to the oxygen local environment. The relatively low intensity of these features compared to what is expected for pristine rutile SnO 2 suggests that Mn(6%):SnO 2is rich in V O. As octahedral Mn is fre- quently found with lower valency than 4+, it is expected that Mn:SnO 2is conducive to forming additional V O. Finally, it is worth noting that the Raman spectrum of Mn (OH) 2exhibits an intense feature related to the A1gmode of vibra- tion near 400 cm/C01,47precisely at the local minimum of the Raman spectrum from Mn(6%):SnO 2. This helps to rule out the presence of Mn hydroxides as a contaminant phase. This is an important consideration, as using KOH in sample preparation does make it possible for an unwanted Mn hydroxide to form. The magnetization of Mn:SnO 2is significantly influenced by the Mn-doping concentration, as shown in Fig. 3 . The magnetic susceptibility of pure SnO 2decreases with increasing magnetic field [seeFig. 3(b) ]. Pure SnO 2is weakly ferromagnetic when significant VOis present,6but close to stoichiometric SnO 2is diamagnetic.14 The diamagnetic behavior observed for SnO 2inFig. 3(b) is further justification for the very minor oxygen deficiency predicted abovefrom our XRD data. Mn(1%):SnO 2exhibits paramagnetic (PM) behavior, as the magnetic susceptibility increases linearly with increasing magnetic field. No sign of saturation is observed in ourmeasurements; similar behavior is reported for paramagnetic Mn:SnO 2, which saturates at M.100 emu/mol for H.40 kOe, well outside the range of our measurements.20This strong paramagnetic behavior suddenly changes higher Mn-doping, as Mn(3%):SnO 2 exhibits easily saturated PM or weak FM behavior [with a very small coercivity, see Fig. 3(b) ] at low fields and with a small satura- tion magnetization. This suggests the coexistence of FM and PM phases or possibly a super-PM phase. Interestingly, the PM compo- nent in Mn(1%):SnO 2clearly induces a larger magnetization than in Mn(3%):SnO 2, as it does not saturate. The magnetic susceptibil- ity of Mn(3%):SnO 2is higher, it simply saturates at a much lower field. From the arguments about the concentration of V Opresented above (in the context of the XRD measurements), possibly the increased V Ocaused by a larger concentration of Mn leads to a shift from the PM to the FM order being dominant.13,17Finally, Mn(6%):SnO 2clearly demonstrates an FM hysteresis loop, but does not clearly exhibit saturation. FM behavior in Mn(6%):SnO 2with significant hysteresis and large saturation magnetization ( Ms) has been reported,18but in that case, the hysteresis loop clearly extends to the saturation point,whereas in Fig. 3(a) , the hysteresis appears “pinched off ”well before saturation. Based on the magnetization of Mn(1%):SnO 2 and Mn(3%):SnO 2, it seems reasonable to model the magnetization in Mn(6%):SnO 2as coexisting PM and FM phases, which we model as the superposition of a line and a shifted Brillouin func- tion12,48 M(H)¼χPMHþM0,PM ðÞ þ MsBJμFM kBT[H/C0Hc]/C18/C19 , (2) where χPMis the susceptibility of the PM phase, M0,PMis any resid- ual magnetization in the PM phase (expected to be close to zero),M sis the saturation magnetization of the FM phase, BJ(x) is the Brillouin function for angular momentum J,Hcis the magnetic coercivity of the FM phase, and μFMis the effective magnetic moment of the ferromagnetic domain. We fit the rising and falling portions of the magnetization hysteresis loop separately. Equation (2)is used to fit the magnetization of Mn(6%):SnO 2 for a continuous range of angular momenta J. From the r-factors of the fit, shown in Fig. 3(c) , it is clear that the best fit is obtained when J!1[the dotted line in Fig. 3(c) ]. In this limit, the Brillouin function becomes the Langevin function, BJ!1(x)¼L(x). In fact, this result is quite reasonable, as it corresponds to randomlyoriented ferromagnetic nanodomains 49,50and has been observed in semiconducting oxides where native magnetism is defect-driven.51 In particular, the coexistence of paramagnetic and Langevin-type FIG. 3. Magnetization Mof SnO 2and Mn:SnO 2at room temperature. (a) Magnetization of Mn(6%)SnO 2as a function of magnetic field (H) with fit and fit components. (b) Magnetization of SnO 2, Mn(1%):SnO 2, and Mn(3%):SnO 2as a function of magnetic field (H). (c) The r-factor (lower is better) of the fitted M(H) as a function of J, and the dotted line shows the limit as J!1.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 045705 (2020); doi: 10.1063/5.0012415 128, 045705-4 Published under license by AIP Publishing.ferromagnetic phases was previously found in undoped, oxygen- deficient SnO 2nanoparticles.18 The magnetization of Mn(6%):SnO 2is, therefore, fit using Eq.(2)in the limit J!1, and the fit is shown in Fig. 3(a) .W e find (within error) essentially the same PM and FM behaviorfor each portion of the loop, as expected. We obtain χ PM¼0:47+0:01 ðÞ /C210/C03emu/mol Oe, Ms¼1:11+0:03 emu/ mol, and M0,PM¼3:7+0:5 ðÞ /C2 10/C02emu/mol. M0,PMis reassur- ingly very close to zero: it should be identically zero for noise-freedata from a perfect PM phase, but noise in the data, minor offset inthe measuring equipment, and possibly even trace residual magne- tism from incomplete saturation may be responsible for the small deviation from zero obtained in our fit. The effective magneticmoment is μ FM¼12 800+400μBandμFM¼11 900+800μBfor rising and falling M(H) curves, respectively. Rather, large effective magnetic moments are frequently observed in materials with nano- domains of ferromagnetism.18,49,50The magnetic coercivity is Hc¼272+7 Oe and Hc¼/C0160+10 Oe for rising and falling M(H) curves, respectively. These magnitudes are not identical, pos- sibly because we did not achieve complete saturation in both PMand FM phases with our measurements. From the fitted FM phase, we calculate the residual magnetism to be M r¼/C00:9+0:1 emu/ mol for the rising curve and Mr¼0:5+0:1 emu/mol for the falling curve. The main difference in these two residual magnetismvalues comes from the different magnetic coercivities. To more clearly identify the magnetic phase in Mn(6%):SnO 2, we measure the magnetization as a function of temperature, asshown in Fig. 4 . Field cooling (FC) and zero field cooling (ZFC) were performed over the range of 5 –350 K under a field of 100 Oe(also used during cooling for FC). For both FC and ZFC, as shown in the inset of Fig. 4 , the magnetization rapidly increases with decreasing temperature, and no obvious sign of saturation isobserved. This is indicative of a PM phase. However, as remnantmagnetization clearly exists at room temperature, and as there is adifference between ZFC and FC below 340 K, an FM phase exists as well. Similar FC/ZFC phenomena have been observed in other DMS materials. 52 The inverse susceptibility χ/C01(T) linearly trends to zero as T!0, as shown in Fig. 4 . There is also a linear trend at higher temperatures, in which the extrapolated line has a negative inter- cept with the temperature axis. The former linear behavior is indicative of a PM phase, and the latter is indicative of an AFMphase. 52This is not necessarily inconsistent with the coexisting PM and FM phases found in the magnetization data shown inFig. 3 . The hysteresis observed shows that the Curie temperature T Cof the FM phase is larger than room temperature. Above TC, we expect χ/C01(T) of an FM phase to also have linear dependence with a positive intercept, but below TC, there is no simple rela- tionship between susceptibility and temperature. We argue thats i n c et h eF Mp h a s ei sm i n o rc o m p a r e dt ot h eP Mp h a s e( Fig. 3 shows that the FM phase is easily saturated at low fields, while the PM phase shows no sign of saturation), the magnetization of theFM phase may be close to saturation and, consequently, a veryweak function of temperature. Mn(6%):SnO 2may, in fact, consist of separate PM, FM, and AFM phases, with a magnetic suscepti- bility given by χ¼MPM(T)þMAFM(T)þMFM(T) H ¼CPM TþCAFM TþθþχFM, (3) where we used the Curie law for the PM phase (with Curie cons- tant CPM), the Curie –Weiss law for the AFM phase (with Curie constant CAFM and Curie –Weiss temperature θ), and approxi- mated the susceptibility of the FM phase below TCas a constant. As shown in Fig. 4 ,E q . (3)provides an excellent fit to the measured magnetic susceptibility of Mn(6%):SnO 2.B yf i t t i n g the inverse of Eq. (3)to our measured χ/C01(T), we find CPM¼0:298 +0:006 emu K/mol Oe, C¼0:73+0:03 emu K/mol Oe, θ¼111 +9K ,a n d χFM¼1:99+0:05 ðÞ /C210/C03emu/mol Oe. The Curie –Weiss law for the AFM order is only valid above the Néel temperature TN, and we have applied Eq. (3)over the entire temperature range. No apparent phase transition is visible inM(T); however, the Curie –Weiss temperature θis generally larger than T N, sometimes significantly so. For example, θis larger than TNby more than a factor of five for bulk MnO,53so it is likely TN is rather small for Mn(6%):SnO 2—especially given the difficulties in realizing the AFM order in nanoscale domains.54,55It is also not unexpected that we do not see clear evidence of the AFM order in M(H) as shown in Fig. 3 , as the AFM order does not provide a dis- tinct signature from the PM order as a function of the externalfield H, especially for T.T N. Our magnetization measurements provide strong evidence for multiple magnetic phases in Mn(6%):SnO 2, and our fitting suggests that PM, FM, and AFM phases all exist simultaneously. This makes FIG. 4. FC inverse susceptibility χ/C01at 100 Oe as a function of temperature for Mn(6%):SnO 2, including linear trends and the fit using Eq. (3). Note that the linear trends are only for visualization; they do not correspond to the actuallinear components from the fit (naturally, these components taken individuallyare substantially offset from the data and have rather different slopes than the trends shown here, since a good fit is only obtained by taking them all together). FC and ZFC magnetization at 100 Oe as a function of temperature is shown inthe inset.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 045705 (2020); doi: 10.1063/5.0012415 128, 045705-5 Published under license by AIP Publishing.investigating the valency of the Mn dopants particularly important, as it is perhaps somewhat unlikely that homovalent doping can provide such a diverse range of magnetic behavior. To this end, wemeasure Mn L 2,3-edge XAFS for Mn(1%):SnO 2, Mn(3%):SnO 2, and Mn(6%):SnO 2, as shown in Fig. 5 . These spectra are all very similar, exhibiting the Mn L 3transition near 638 eV and the L 2 transition near 649 eV, as expected. What is noteworthy is the rather small ratio of the intensity of the L 3transition to the total spectral intensity I(L3), compared to the spectra from other oxides containing Mn.26I(L3) is statistically expected to be 0.66, as 4 electrons are present in the 2p 3=2core shell out of 6 electrons in total; however, in real materials, I(L3) often deviates significantly from this value. The shape of the Mn L2,3-edge spectra is strongly dependent on the local symmetry; however, I(L3) is primarily influenced only by the spin state of the absorbing atom, with a lower branching ratio occurring for low-spin ground states.56The low I(L3) from our Mn L 2,3-edge spectra strongly suggests that high-spin Mn3þis not the sole species of Mn ions in our samples. However, we stress that apartfrom a small I(L 3)=I(L2), the shape of the spectrum is reminiscent of Mn in an oxygen-coordinated distorted octahedral environment (as expected if Mn substitutes for Sn). In particular, the spectral shape does not exhibit the fine structure seen in bulk Mn-oxideslike MnO, Mn 2O3, and MnO 2.57,58This suggests that a significant amount of Mn-clustering does not occur in our samples.To quantitatively analyze the XAFS spectra shown in Fig. 5 , we perform LMCT-CF calculations for Mn in D 4hsymmetry. In perfect octahedral (O h) symmetry, the Mn 3d orbitals are split into two levels: t2g(3d xy,3 d xz, and 3d yz) and eg(3d z2and 3d x2/C0y2), the separation between these orbitals is the CF parameter 10 Dq.I nD 4h symmetry, as is found in SnO 2, one bond axis in the MnO 6cluster is stretched compared to the other two. This causes the Mn 3d orbitals to split into four levels: a1g(3d z2),b1g(3d x2/C0y2),b2g(3d xy), andeg(3d xzand 3d yz). This further splitting introduces two addi- tional CF parameters ( Dsand Dt), and the relationship between these parameters and the energy levels is Ea1g¼E0/C02Dsþ6Dq/C06Dt, Eb1g¼E0þ2Dsþ6Dq/C0Dt, Eb2g¼E0þ2Ds/C04Dq/C0Dt, Eeg¼E0/C0Ds/C04Dqþ4Dt, where E0is the average energy of the 3d orbitals. We search the space of CF parameters by choosing a splitting ΔEbetween the average of the a1g,b1glevels and the average of the b2g,eglevels (i.e., equivalent to 10 Dq in O hsymmetry) in the range 0:5e V/C20ΔE/C204:0 eV, searched in increments of 0.1 eV. In this manner, we investigate both I(L3) and the shape of the spectra simultaneously. For a hydrogenic atom in an ideal crystal field, the CF parameters are defined in terms of the effective charge of the ligand ions, the bond lengths of the ligands, and Slater inte- grals.59Although we do not necessarily expect such a simple rela- tionship for a real system, we use ΔE, the bond length obtained from our fully relaxed DFT calculations, and tabulate Slater inte-grals for various valencies of Mn, 60to estimate Dq,Dt,Ds,a n d perform a fine-grained parameter search in this vicinity. The resulting spectra are calculated with LMCT-CF theory for Mn2þ, Mn3þ,a n dM n4þ. For each combination of CF parameters, a linear superposition of spectra for Mn2þ,M n3þ,a n dM n4þis fit to the measured data.61The best fit from this procedure is shown inFig. 5 . The CF parameters that provide the best fit are (Dq,Ds,Dt)¼(0:272, 0 :110, 0 :138) eV, (0 :327, 0 :176, 0 :146) eV, and (0 :327, 0 :219, 0 :146) eV for Mn2þ,M n3þ,a n dM n4þ,r e s p e c - tively. The estimated concentrations of Mn2þ,M n3þ,a n dM n4þ found from this fitting are presented in Table I . The Mn L 2,3-edge XAFS spectra, therefore, suggest that a sig- nificant composition of Mn2þ,M n3þ, and Mn4þis present in all of our samples. Compared to the previous EPR study, we find a signif-icantly higher concentration of Mn 4þ;23however, this discrepancy FIG. 5. Mn L 2,3-edge XAFS measured for Mn(1%):SnO 2, Mn(3%):SnO 2, and Mn(6%):SnO 2and simulated for Mn2þ,M n3þ, and Mn4þin D 4hsymmetry with LMCT-CF theory. The fitted spectra are the superposition of Mn2þ,M n3þ, and Mn4þcurves as shown, with an additional background composed of arctangent step functions for the L 3and L 2absorption edges.TABLE I. Estimated Mn2+,M n3+, and Mn4+composition obtained from fitting measured Mn L 2,3-edge XAFS using spectra simulated with LMCT-CF theory. Sample Mn2+(%) Mn3+(%) Mn4+(%) Mn(1%):SnO 2 37 ± 15 24 ± 17 39 ± 22 Mn(3%):SnO 2 24 ± 5 42 ± 9 34 ± 8 Mn(6%):SnO 2 27 ± 5 34 ± 8 39 ± 9Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 045705 (2020); doi: 10.1063/5.0012415 128, 045705-6 Published under license by AIP Publishing.may arise from the different synthesis methods employed. Combined with the multiple magnetic phases predicted from our analysis of the magnetization in our samples, it seems likely thatthe different valencies present may contribute to different magneticphases. Our LMCT-CF calculations further provide estimates of theground state of the Mn ions as 4F7 2for both Mn2þand Mn4þand 5D3for Mn3þ. It is noteworthy that these calculations predict that Mn2þis in the low-spin state (expected ground state from Hund ’s rules is6S5 2for Mn2þ), and both Mn3þand Mn4þhave higher J than expected. The predicted ionic magnetic moments are, there-fore, μ¼gS¼2:214μ Bfor Mn2þand Mn4þand μ¼4μBfor Mn3þ. From the PM Curie constant found from fitting χ/C01(T), we estimate the effective magnetic moment of the PM phase asμ eff¼ffiffiffiffiffiffiffiffiffiffiffi8CPMpμB¼1:54+0:02 ðÞ μB. This is slightly smaller than the predicted moment of Mn2þand Mn4þ. present, there is little we can meaningfully say about which valency of Mn contribute to which magnetic phase. To the extent magnetic ordering in Mn oxides is relevant, note that homovalentMnO, Mn 2O3, and MnO 2all exhibit AFM order.62–64Heterovalent Mn 3O4, however, is a ferrimagnet,65raising the possibility that the FM hysteresis observed in Fig. 3 is rather more of a heterovalent ferrimagnetic phase (it is very difficult to distinguish between FM and ferrimagnetic phases in our data, since there is also PM andAFM phases present). Of course, there is no reason to expect diluteMn ions in an SnO 2host to mimic the magnetic properties of a bulk Mn oxide. Based on previous reports, it is most reasonable to attribute mixed Mn2þ,M n3þto the FM order.8,9 It is worth noting that fitting M(H) for Mn(6%):SnO 2pro- duces a similar Msto that from Mn(3%):SnO 2(refer back to Fig. 3 ), suggesting the same mechanism for FM in both samples. We should be cautious connected magnetic behavior to Mn valency, given the rather large margin of error in the Mn composi-tion (refer Table I ), but we may tentatively suggest that unpaired Mn 2þis connected to the PM phases seen in Mn(1%):SnO 2and Mn(6%):SnO 2, as the samples with the largest Mn2þcomposition relative to Mn3þdemonstrate the strongest PM order, while Mn4þ may perhaps be connected with the AFM phase [which is largely invisible in room-temperature M(H) measurements]. Our DFT calculations of Mn substitution in SnO 2in bulk and at the surface, with and without V O, all suggest Mn with a local magnetic moment of around μ/C252:8μBfor both FM and AFM order. Furthermore, these calculations predict that FMorder has slightly lower energy than AFM (by 6 meV/f.u. or less)at 12.5% or 6.25% doping, but at 3.125% doping AFM is favored by 0.2 meV/f.u. The rather low energy barrier between FM and AFM phases, and its sensitivity to Mn concentration perhaps helpexplain why we observe multiple magnetic phases in Mn(6%):SnO 2, and Mn-concentration dependent magnetization curves [refer Fig. 3(b) ]. However, the independence of local magnetic moments from magnetic ordering or local geometry (surface vs bulk) does not agree with the different valencies we predict fromour Mn L 2,3-edge XAFS spectra. This is not entirely unexpected: although DFT at the GGA+ Ulevel of theory has demonstrated success predicting magnetic properties of some DMS,66this is far from universal, and more sophisticated methods are often neces- sary for accurate characterization of the local electronic structureto the dopant ion. 67DFT has shown success simulating the O K-edge XAFS for various metal oxides.68–71Since the local electronic structure of the ligand is more accurately treated with DFT than the strongly corre-lated local electronic structure to metal ions, we compare measuredO K-edge XAFS spectra of SnO 2and Mn(6%):SnO 2with simulated spectra in Fig. 6 . As expected, the spectra from Mn(6%):SnO 2are very similar to that of stoichiometric SnO 2. Since the shape of the O K-edge XAFS spectrum is dominated by the final state, it is oftennecessary to calculate the electronic structure with an explicit corehole in an O 1s core level. 68,69This necessitates a supercell calcula- tion, since the absorbing site with the core hole must be treated as a dilute impurity to avoid physically improbable correlations between core holes. However, for some oxides, a calculationwithout a core hole provides a sufficiently accurate simulation ofthe spectra, as the increased effective nuclear charge at the absorp-tion site is effectively screened. 72 With that in mind, the simulated O K-edge XAFS spectra of bulk SnO 2are shown in Fig. 6 calculated with (ch, for “core hole ”) and without (gs, for “ground state ”) an explicit core hole. It is clear that an explicit core hole shifts and sharpens features found in theground state electronic structure near the absorption edge (i.e., below 532 eV). This improves the energy alignment and shape of the spectral features relative to experiment; however, the spectrasimulated from the ground state still reasonably accurately repro-duce these features. We, therefore, only perform ground state (gs) calculations for Mn:SnO 2.73As shown in Fig. 6 , Mn-doping and FIG. 6. O K-edge XAFS measured for SnO 2and Mn(6%):SnO 2and simulated for SnO 2, Mn:SnO 2for FM or AFM order, with and without V O(MnSn 31O62and MnSn 31O64, respectively), and MnO, Mn 2O3, and MnO 2. Labels (ch) and (gs) indicate whether the simulation involved an explicit core hole or just the groundstate, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 045705 (2020); doi: 10.1063/5.0012415 128, 045705-7 Published under license by AIP Publishing.VOare predicted to have only a minor effect on the shape of the spectra, creating some broad and low-intensity features on the pre-edge below 530 eV, while the choice of magnetic ordering has anegligible effect on the shape of the spectra. Given that the simulated O K-edge XAFS spectrum for SnO 2 generally exhibits the features observed in the measured spectrum, but with some differences in relative intensity, width, and energy alignment, it is not surprising that the spectral signature ofMn-doping predicted by the simulated spectra of MnSn 31O64/C0xis not clearly visible in the measured spectrum of Mn(6%):SnO 2.I ti s worth noting, however, that the simulated spectra of Mn:SnO 2are also not a good match to the appropriately weighted superposition of the spectra for bulk SnO 2and the spectrum from a reference Mn oxide (the same goes for the measured spectrum from Mn(6%):SnO 2, for that matter), as the spectra of Mn oxides all contain numerous sharp spectra features that should be partially visible, even at only 6% concentration (see Fig. 6 ). This point is raised because we argue that using partial energy alignment of the Mn K-edge XAFS spectra of Mn:SnO 2with various reference Mn oxides, as has been previously employed,13,20 may not be the best way of determining the valency of Mn present in Mn:SnO 2. Mn K-edge XAFS is primarily a direct probe of unoc- cupied conduction band states with 4p-character; the unoccupied3d levels are only probed by a weaker, dipole-forbidden transition.In contrast, Mn L 2,3-edge XAFS is a direct probe of unoccupied 3d levels. Furthermore, the K transition is less sensitive than the L2, 3 transition to the multiplet effects that make the occupancy of the3d level have such a significant influence on the spectral shape.Finally, as shown by our O K XAFS and other DFT calculationsavailable in the literature, 13,17,20the local electronic structure to dilute Mn substitution in SnO 2does not have much in common with the electronic structure of stoichiometric Mn oxides. Giventhe complex magnetic ordering observed in our magnetizationmeasurements and suggested by our Mn L 2,3-edge XAFS measure- ments, Mn K-edge XAFS and DFT calculations may not be suffi- cient to characterize the magnetic order, valency, and electronic structure in Mn:SnO 2. IV. CONCLUSIONS In summary, we have investigated SnO 2, Mn(1%):SnO 2, Mn(3%):SnO 2, and Mn(6%):SnO 2with XRD, magnetization mea- surements, and Mn L 2,3-edge and O K-edge XAFS. We clearly observe PM and FM behavior, which is dependent on the Mn con-centration. We also present evidence that PM, FM, and AFM orderall coexist in Mn(6%):SnO 2, pointing to an even more complicated magnetic phase diagram than might be expected from previous studies. Our analysis of Mn L 2,3-edge XAFS further predicts the coexistence of Mn2þ,M n3þ, and Mn4þ, all in significant amounts, while our O K-edge XAFS and XRD measurements suggest that allMn species are in the SnO 2host lattice: no segregated Mn-oxide phase is present. Our Mn L 2,3-edge XAFS analysis also suggests that the strong crystal field in the host lattice leads to Mn2þin a low-spin state, rather than the high-spin state predicted by Hund ’s rules for a free atom. Taken together, our findings demonstrate that detailed analy- sis of the magnetic ordering and chemical state of Mn:SnO 2isnecessary, likely on a case-by-case basis since these factors are likely dependent on synthesis conditions and probably evolve with time (on the order of months).18Claims of monatomic doping or single magnetic phase behavior in Mn:SnO 2should be made with caution unless supported by multiple characterizationmeasurements. ACKNOWLEDGMENTS This work was funded in part by the National Natural Science Foundation of China (NSFC) (Project No. 21650410656). We also acknowledge support from the Collaborative Innovation Center of Suzhou Nano Science and Technology (NANO-CIC), SoochowUniversity. 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5.0005597.pdf

APL Photonics 5, 070802 (2020); https://doi.org/10.1063/5.0005597 5, 070802 © 2020 Author(s).Enhancing the modal purity of orbital angular momentum photons Cite as: APL Photonics 5, 070802 (2020); https://doi.org/10.1063/5.0005597 Submitted: 23 February 2020 . Accepted: 17 June 2020 . Published Online: 13 July 2020 Isaac Nape , Bereneice Sephton , Yao-Wei Huang , Adam Vallés , Cheng-Wei Qiu , Antonio Ambrosio , Federico Capasso , and Andrew Forbes ARTICLES YOU MAY BE INTERESTED IN Harmonic generation at the nanoscale Journal of Applied Physics 127, 230901 (2020); https://doi.org/10.1063/5.0006093 Super-resolution localization microscopy: Toward high throughput, high quality, and low cost APL Photonics 5, 060902 (2020); https://doi.org/10.1063/5.0011731 Reconstruction of angle-resolved backscattering through a multimode fiber for cell nuclei and particle size determination APL Photonics 5, 076105 (2020); https://doi.org/10.1063/5.0011500APL Photonics ARTICLE scitation.org/journal/app Enhancing the modal purity of orbital angular momentum photons Cite as: APL Photon. 5, 070802 (2020); doi: 10.1063/5.0005597 Submitted: 23 February 2020 •Accepted: 17 June 2020 • Published Online: 13 July 2020 Isaac Nape,1 Bereneice Sephton,1 Yao-Wei Huang,2,3Adam Vallés,1,4 Cheng-Wei Qiu,3 Antonio Ambrosio,5Federico Capasso,2 and Andrew Forbes1,a) AFFILIATIONS 1School of Physics, University of the Witwatersrand, Private Bag 3, Wits, Johannesburg 2050, South Africa 2Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA 3Department of Electrical and Computer Engineering, National University of Singapore, 117583, Singapore 4Molecular Chirality Research Center, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan 5CNST - Fondazione Istituto Italiano di Tecnologia, Via Giovanni Pascoli, 70, 20133 Milano MI, Italy a)Author to whom correspondence should be addressed: andrew.forbes@wits.ac.za ABSTRACT Orbital angular momentum (OAM) beams with topological charge ℓare commonly generated and detected by modulating an incoming field with an azimuthal phase profile of the form exp( iℓϕ) by a variety of approaches. This results in unwanted radial modes and reduced power in the desired OAM mode. Here, we show how to enhance the modal purity in the creation and detection of classical OAM beams and in the quantum detection of OAM photons. Classically, we combine holographic and metasurface control to produce high purity OAM modes and show how to detect them with high efficiency, extending the demonstration to the quantum realm with spatial light modulators. We demonstrate ultra-high purity OAM modes in orders as high as ℓ= 100 and a doubling of dimensionality in the quantum OAM spectrum from a spontaneous parametric downconversion source. Our work offers a simple route to increase the channel capacity in classical and quantum communication using OAM modes as a basis. ©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/10.1063/5.0005597 .,s I. INTRODUCTION It has been known for at least a century that photons could carry both a spin angular momentum and an orbital angular momentum (OAM), but the “creation” of the latter required rare quadrupole transitions in atoms to occur and so remained largely unstudied. Just over 25 years ago, Allen and co-workers1realized that OAM car- rying beams could be created in the laboratory using conventional optics in a deterministic manner. They noticed that a phase vortex of the form exp( iℓϕ) would give each photon a “twist” in wavefront, resulting in OAM of ℓ̵hper photon. Here, ϕis the azimuthal angle, and ℓis the helicity or topological charge (an integer), suggesting an optical element with a transmission function in the form of an azimuthally varying phase. Indeed, this was exactly how such beams were first created and detected in both the classical2,3and quantum4 regimes. Since then, the use of azimuthal phase elements for thecreation of such vortex beams has become ubiquitous, implemented by dynamic phase approaches on spatial light modulators (SLMs),5,6 as well as by geometric phase approaches using liquid crystals7–9 or metasurfaces (MSs),10and has been pioneered to harness the radial degree of freedom by merging these approaches in a single element.11,12These phase-only vortex beams have found a myriad of applications, making them highly topical forms of structured light.13–15 Although OAM modes can be detected by a suitable confor- mal mapping in so-called mode sorters,16the standard approach is to exploit the reciprocity of light and run the creation step in reverse, e.g., by intensity or phase flattening approaches.17Both work on the idea of unraveling the twisted wavefront of the OAM mode and then coupling the Gaussian-like beam into a single mode fiber (SMF) or measuring with a single pixel. Now, although the early successes with these approaches set the stage for further study and APL Photon. 5, 070802 (2020); doi: 10.1063/5.0005597 5, 070802-1 © Author(s) 2020APL Photonics ARTICLE scitation.org/journal/app application of OAM beams, the azimuthal phase approach to their creation and detection has some inherent inefficiencies: transverse solutions to the wave equation are 2D structures (in the paraxial limit) so that two indices are needed to describe them, both requir- ing control in the creation and detection steps. A consequence of using the azimuthal phase alone is that the radial mode content has been ignored, resulting in hypergeometric modes18with the desired p= 0 OAM mode having its power distributed across many radial modes.19This has deleterious effects for both radial mode purity as well as creation and detection efficiency. While the cre- ation can be dealt with by intra-cavity OAM approaches, which result in pure modes,20,21this does not address the detection issue nor the application to quantum states. It may be noted that ref- erence to purity throughout this paper refers to the radial mode purity, i.e., how much power is in, or detected in, the p= 0 OAM mode. While the effects of radial purity are low for small OAM charges, it becomes increasingly prominent for higher values with several approaches readily achieving high charges of ℓ= 100 and beyond using liquid crystal q-plates,22spiral phase plate mirrors,23 anisotropic diffractive waveplates,24and others in the phase-only regime. Moreover, due to applications such as working toward test- ing quantum foundations23,25and creating sensitive photonic polar- ization gears22now require higher OAM modes, it follows that the consequential purity is no longer negligible. Here, we pay atten- tion to complex amplitude detection of these modes and show that adjusting the scale in the detection step can be used for enhanced detection efficiencies. Advantages are shown both classically and quantum mechanically. Reciprocally, we also demonstrate the cre- ation of high purity OAM modes with complete azimuthal phase and radial mode control, thus creating a comprehensive approach to working with phase-only generation devices. To make clear that the solution is not device specific, we employ two fundamentally different phase-only devices with a metasur- face enhanced with holographically programmed dynamic phase to produce ultra-high purity classical OAM modes and a SLM-only solution in the quantum realm. The metasurface comprises nano- structured rods patterned for complete control of the OAM state of light, vector and scalar, while the hologram written to a spatial light modulator uses a dynamic phase to introduce complex amplitude modulation, thus engineering all degrees of freedom of the field. Moreover, the scheme is extendable to any OAM encoding device, not limited to SLMs or metasurfaces. Our results will be of particu- lar benefit to both the classical and quantum communication where a large alphabet with a good signal-to-noise ratio is essential. II. BACKGROUD Here, we illustrate the concept of enhancing the creation and detection of high purity OAM modes external to a laser, having already shown that it is possible directly from the source.21Impor- tantly, we provide the full details for the classical case and intro- duce a new quantum case for the enhanced purity of single photons and quantum states. The objective is to reduce the contribution of unwanted radial modes and to concentrate all the energy into the zero order radial mode, both at the creation and detection steps. But how to achieve this in a generic manner? To demonstrate this,we create high purity OAM modes in the Laguerre–Gaussian basis (natural OAM basis). Such transverse modes, ψp,ℓ(r,ϕ;w0), of radial order pand azimuthal index ℓare solutions to the wave equation in quadratic index media, including free-space and graded index fiber. The OAM content of the beam is associated with the azimuthal phase, exp( iℓϕ), while w0is a scale parameter, sometimes referred to as the embedded Gaussian beam size. A pure OAM mode in this basis, the mode we desire, would have no radial modes ( p= 0) and is described by ψ0,ℓ(r,ϕ;w0)∝(r√ 2 w0)∣ℓ∣ exp(−r2 w2 0)exp(−iℓϕ). (1) Note that ℓappears in both the amplitude and phase terms so that both the amplitude and phase modulation are required to produce a true OAM mode in this basis. In fact, it is the amplitude term that results in the intensity null so often associated with such modes. Yet, the conventional approach to producing OAM modes is to approxi- mate this by phase-only azimuthal modulation of a Gaussian beam, as shown in Fig. 1(a), resulting in ˜ψℓ(r,ϕ;w0)≈exp(−r2 w2 0)exp(−iℓϕ). (2) This vortex beam results in the generation of many radial modes, shown in Fig. 1(b), with low power content in the desired p= 0 mode. In fact, the modal powers for all pand a given ℓcan be expressed as ∣cpℓ∣2=(p+∣ℓ∣)! p!⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪Γ(p+∣ℓ∣ 2)Γ(∣ℓ∣ 2+ 1) Γ(∣ℓ∣ 2)Γ(p+∣ℓ∣+ 1)⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪2 . (3) Here, we propose how to overcome this in two approaches: one for creation and the other for detection. For detection, we observe that the OAM mode scales with the embedded Gaussian beam size ( w0). It has been traditional to use this scale in the detection since, by reciprocity, this is the scale at which the vortex beam was initially created. However, ideal ( p= 0) OAM modes would have a second moment radius of wℓ=w0√ ∣ℓ∣+ 1 that corresponds to an intensity ring of radius w0√ ℓ/2. This highlights that while the scale of the final OAM mode will certainly depend onw0, it will not be equal to it. We select an optimal scale for the detection in order to maximize modal power in the desired p= 0 mode when the initial mode is notamplitude modulated [shown in Eq. (2) and depicted in Fig. 1(a)]. In the case where a mode gener- ated without amplitude modulation is detected with a system that assumes a scale of w0, the measured power in the desired p= 0 mode approaches zero as the azimuthal index increases (see the sup- plementary material). This case would represent the usual method of detection, particularly in quantum experiments with OAM. For example, an input vortex mode of ℓ= 1 at this scale has only about 80% of its energy in the ψ0,1mode, with the rest spread across higher radial modes. Furthermore, an input vortex mode of ℓ= 10 has less than 1% of its energy in the ψ0,10mode. This places severe limits on the available alphabet using OAM as a basis. We ask what should the ℓdependent scale parameter be such that the expansion has the min- imum power in the unwanted radial modes, p≠0. The salient point is that higher modal power is physically available, but the detection APL Photon. 5, 070802 (2020); doi: 10.1063/5.0005597 5, 070802-2 © Author(s) 2020APL Photonics ARTICLE scitation.org/journal/app FIG. 1 . (a) A Gaussian beam modulated by an azimuthal phase, shown here as a metasurface, results in an OAM mode with many radial orders, shown in (b) as the sum of beams with several intensity rings. By combining phase control using holograms and metasurfaces, high purity OAM modes can be produced, as seen in (c). For detection of the modal composition of a beam, conjugate modulation of the mode being evaluated is applied, and the Fourier plane on-axis intensity is measured, as shown in (d). system needs to be optimized to recover it. Accordingly, we exploit the fact that changing the scale in size-dependent bases can alter the detected modal content.26Here, the correct scale to achieve this can be expressed analytically as wopt=w0√ ∣ℓ∣+ 1, (4) leading to an optimized modal power content in the p= 0 mode given by ∣cℓ(αopt)∣2=⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪2∣ℓ∣ 2∣ℓ∣αoptΓ(∣ℓ∣ 2) (1 +α2 opt)(∣ℓ∣ 2+1)√ Γ(∣ℓ∣+ 1)⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪2 (5) when αopt=1√ ∣ℓ∣+ 1. (6) The implication of this is best visualized in a quantum exper- iment where the experimenter has full control over the detection steps but often little control over the creation steps. Rather than detecting for an incoming mode size of w0, the detection system should be adjusted to a scale of wopt. The prediction is a signal enhancement for a larger available alphabet of OAM modes. While detection is enhanced by optimizing the available sig- nal with wopt, one cannot recover it to 100% as the vortex beam in Eq. (2) is not the desired mode. Amplitude modulation is needed. To illustrate this, we use metasurfaces to create OAM modes of very high order and then circumvent the spread of power into higher order radial modes by employing holographically encoded complex amplitude modulation to correct for the missing amplitude term. By combining a metasurface for azimuthal phase control with high fidelity (since the vortex can be very well defined at the nano-scale) with holographic complex amplitude modulation, we are able to demonstrate ultra-high purity OAM states, even at very high OAM, as given by Eq. (1), with all the power in the p= 0 mode, illustrated in Fig. 1(c). Here, the creation step is summarized that, when reversed,allows a detection route for the enhanced signal over conventional approaches, as illustrated in Fig. 1(d). Finally, in the quantum case, the situation is a little different. There is no “initial” beam but only what one detects, which is related to a pseudo initial beam in the form of the pump mode at the crystal. Assuming that the pump is Gaussian and the detection includes sin- gle mode fibers (the ubiquitous case), we derive the modal powers to be given by ∣cℓ(αopt)∣2=⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪2∣ℓ∣+3 2√ 1 ∣ℓ∣!Γ(∣ℓ∣ 2+ 1)(1 α2 opt+1 η2+ 2)−(∣ℓ∣ 2+1) π3/2w2 0ηα∣ℓ∣+1 opt⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪2 , (7) with αopt=η√ (2η2+ 1)(∣ℓ∣+ 1), (8) whereη=wp w0is the pump field and fiber mode size ratio. This equation can easily be generalized to other pump geometries and detection modalities. III. EXPERIMENTAL A. The dielectric metasurface The metasurface device used for the creation of the clas- sical OAM modes was a dielectric metasurface made of amor- phous TiO 2nano-posts with a rectangular section [the inset of Fig. 2(a)] on a quartz substrate. Each post has a height of 600 nm, while the width and length ( wx,wy) change in order to impart a different phase delay to the propagating visible light at 532 nm. More specifically, each nano-post imparts an over- all phase delay δto the propagating light and a phase delay difference Δχbetween the field xand ycomponents. This last term only depends upon the shape of the post and is called the form birefringence : symmetric sections such as squares or circles do not show the form birefringence. Note that amorphous TiO 2 used for these nano-structures is not optically birefringent at this APL Photon. 5, 070802 (2020); doi: 10.1063/5.0005597 5, 070802-3 © Author(s) 2020APL Photonics ARTICLE scitation.org/journal/app FIG. 2 . (a) Schematic of a J-plate design. The J-plate imprints two kinds of helical phase profiles for x- and y-incident polarization, resulting in output beams with the orbital angular momentum m̵hand n̵h, where mand ncan be any independent integers. The metasurface elements are rectangular nano-posts made of amor- phous TiO 2with a fixed height of h= 600 nm. By changing the width along the x- and y-direction ( wxand wy), the nano-posts impart phase delays given by δxand δy. Optical micrographs (b) and scanning electron micrographs (SEMs) [(c) and (d)] of a representative J-plate with the OAM number ( m,n) = (10, 100). The SEMs show a top view (c) and angled view (d) of the device center. wavelength. Our metasurface combines both the propagation phase and PB-phase to convert any two orthogonal polarization states of the incident light into the conjugate helical modes with any arbitrary value of orbital angular momentum mandn, not just opposite val- ues. This is possible by controlling the PB-phase, the overall phase, and the form birefringence of each element. Figure 2(a) shows the schematic image of the central part of a representative J-plate that converts two orthogonal linear polar- ization states (with the field oriented along xand y, respectively) into helical modes with the same polarization state and orbital angu- lar momentum ( m,n) = (10, 100). Figure 2(b) shows an optical microscope image of a J-plate producing helical modes with the orbital angular momentum 10 and 100 for the x- and y-linearly polarized incident light, respectively. Figures 2(c) and 2(d) show the scanning electron micrographs (SEMs) of the realized device. Note that the nano-posts have different shapes and orientations to implement the necessary phase gradient. The J-plate is fabricated using electron-beam lithography, followed by atomic layer deposi- tion and etching. Note that the implemented azimuthal phase gra- dient is visible in the optical image of the device as color varia- tion. In fact, nano-posts with different shapes have different scat- tering resonance frequencies, resulting in a colorful optical image. For instance, in Fig. 2(b), it is easy to distinguish 10 sectors, each made of 10 inner sectors. This means that for the helical mode resulting from the incident y-polarized light, in order togenerate a beam with the orbital angular momentum 100, a 2 π azimuthal phase variation must be accumulated in an angle of just 2π/100 (3.6○). It is worth noting that the high azimuthal gradient necessary to impart a topological charge (100) to a propagating beam can be realized by means of SLMs. However, the typical pitch of such device is about 8 μm. In our metasurfaces, the unit cell dimen- sion is 420 nm that represents a 20 times increase in resolution with respect to a SLM, crucial for high quality vortices. Moreover, a SLM does not affect the light polarization and cannot be used alone to encode spin-to-orbital angular momentum devices. Thus, we elect to use this metasurface approach because the azimuthal phase is well defined down to nano-meter scales for high quality azimuthal OAM modes. This allows us to emphasis the point that while the creation element is as ideal as can be, the purity of the resulting modes is nevertheless very low. It follows that while this work could be implemented with any phase-only device such as spiral phase plates or only a SLM (which is the case in the quan- tum experiment), we elect to employ the metasurface in the classical FIG. 3 . Illustrations of the (a) classical and (b) quantum experimental setups. SLM: spatial light modulator functioning in the reflective mode but drawn here as trans- missive; F, L: lens; CCD: charge-coupled device; NC: nonlinear crystal; BS: beam splitter; M: mirror; SMF: single mode fiber; APD: avalanche photodiode; BPF: bandpass filter; s: signal; i: idler; C.C.: coincidence counter, and MS: metasur- face. Overhead insets show representative phase maps for the respective beam manipulations carried out at each point. APL Photon. 5, 070802 (2020); doi: 10.1063/5.0005597 5, 070802-4 © Author(s) 2020APL Photonics ARTICLE scitation.org/journal/app case. The issue here is in the physics of the situation and not its implementation. B. Enhanced generation scheme To test the concept of enhanced purity in the creation and the enhanced signal in the detection, we used the experimental setups shown in Figs. 3(a) and 3(b) for classical and quantum light, respectively. In the classical experiment, a 532 nm wave- length horizontally polarized laser beam was generated through the phase and amplitude modulation on a HOLOEYE PLUTO spa- tial light modulator (SLM), allowing one to obtain any desired beam size and phase. The resulting mode traversed the metasur- face (MS), which was further relayed with a 4f system ( F1andF2 with focal lengths of 200 mm) onto another SLM for analysis. The measurement was performed by detecting the on-axis intensity in the Fourier plane of the second SLM ( F3with a focal length of 300 mm), yielding the modal spectrum (see modal decomposition in the supplementary material). Phase plots above the optical elements in Fig. 3(a) exemplify the phase transformations imparted to the beam for amplitude correction of the MS and subsequent analysis thereof. C. Enhanced two photon detection scheme The setup used for measurements on the entangled photons is illustrated in Fig. 3(b). Our type I PPKTP crystals produced collinearly propagating photon pairs entangled in the OAM degree of freedom at 810 nm (see the supplementary material). The photon pairs were separated, in path, using a 50:50 beam splitter (BS). From the crystal plane, each pair was imaged onto a SLM with a 4f sys- tem ( L1andL2having focal lengths of 100 mm and 500 mm each)and subsequently imaged ( L3andL4having focal lengths of 750 mm and 2 mm each) into single mode fibers (SMF) for detection with APDs. The photons from each arm were correlated in time using a coincidence counter (C.C.) with a 25 ns coincidence window. All measurements were obtained by recording the photons with an inte- gration time of 5 s. A fixed ratio between the mode fiber radius ( w0) of the SMF and the pump photon of η= 1.5 was used during the experiment. IV. RESULTS A. Enhanced generation of OAM modes In Figs. 4(a) and 4(b), we compute the predicted radial modal power weighing (| cp|2) for ℓ= 10 and ℓ= 1 beams, respectively, as a function of ratio, α=wℓ/w0. Accordingly, in each case, the beam waist size is varied on the projection hologram for the detection of each radial mode ranging from p= 0 to p= 10. Our aim is to engineer the mode by the metasurface and holographic control to produce | c0|2→1, i.e., 100% of the power in the desired OAM mode in order to emphasize the effect of amplitude control in the gener- ation process. We observe that if no initial amplitude control of the mode is performed and the traditional scale is used ( wℓ=w0) in the detection, then | c0|2≈0.8 ( ℓ= 1) and | c0|2≈10−3(ℓ= 10). This is confirmed by the experiment: the panels shown in (i) for both ℓ= 1 and ℓ= 10 modes show that the measured power in the radial modes is as predicted by theory. The blue bars represent the experimen- tal data, and the black dots represent the theory. For example, the cross section highlighted by the gray panel at wℓ=w0in the theory plot of Fig. 4(a) is shown again in associated panel (i) as black dots (theory) with experimental data as blue bars. The measured power is spread across many unwanted radial modes, as predicted. There is FIG. 4 . Normalized detection probabilities with variation in the p-mode index and α=wℓ/w0for (a) ℓ= 10 and (b) ℓ= 1. Measured p-mode spectra (blue bars) are given for detection with (i) the unadjusted mode size ( wℓ=w0) and (ii) the optimal encoded mode size ( wℓ=wopt) shown with the theoretical predictions (black dots). The normalization was performed with respect to the p-mode distribution. False color map images of an ℓ= 100 generated beam with (c) phase-only modulation and (d) amplitude corrected phase modulation are shown. APL Photon. 5, 070802 (2020); doi: 10.1063/5.0005597 5, 070802-5 © Author(s) 2020APL Photonics ARTICLE scitation.org/journal/app a second cross section highlighted by the gray panel at wopt, corre- sponding to the peak in the theory plot of Fig. 4(a). The associated panel is in part (ii), again shown as black dots (theory) with exper- imental data as blue bars. Now, the power in the p= 0 mode has substantially increased from <1% to about 50%. This confirms that optimizing the scale at the detection results in significant signal-to- noise enhancements, crucial for optical communication with OAM modes. The same trend is seen in (b) with its associated panels (i) and (ii), again with excellent agreement between theory and exper- iment. A decrease in the ratio αalso results in an oscillation in the modal power as a function of radial index, with no power in the odd pmodes. This truncated radial mode expansion becomes more evi- dent as the scale outdistances w0due to the pparameter restriction (p/2∈N) when describing the resulting mode in the LG basis, as theoretically predicted.27 So far, only the detection step has been optimized. Next, we wish to apply full amplitude and phase control with our holograph- ically enhanced metasurfaces. To do so, we structure the ampli- tude only and pass the modified profile through the metasurface to impart the phase profile, akin to the method used by Ref. 28 and in a similar concept in the considerations of Refs. 29 and 30. The results for the two beams are discussed here and shown in Fig. 5(a) FIG. 5 . The p-mode spectrum when full amplitude and phase control is applied for (a)ℓ= 10 and (b) ℓ= 1. The bars represent the theory, while the points represent the experiment. Insets show the processed transverse spatial distributions. The small intensity oscillations are due to the metasurface fins.forℓ= 10 and Fig. 5(b) for ℓ= 1. The modal purity is as close to 100% as we can determine within experimental uncertainty: all the power is now in the desired ℓwith the p= 0 mode, quantita- tively characterized by | c0|2. Using ℓ= 10 as an example, the initial modal power following traditional schemes was | c0|2≈4×10−3, becoming | c0|2= 0.5 after detection optimization, and | c0|2≈1 after full mode control. This represents about a 125 ×and 250×signal enhancement. The impact of our approach becomes more enhanced with an increase in ℓ. To illustrate this, we perform these experi- ments with a metasurface to produce ℓ= 100 modes (the highest reported to date by metasurface and metamaterials). Figure. 4(c) shows such a beam created by only modulating the phase, while (d) shows the same beam but produced by our metasurface together with complex amplitude modulation. Even visually, one can see the numerous rings in (c), which “collapse” into a single ring in (d): a high purity OAM mode with an astonishing power enhancement in the desired mode. It is pertinent to point out that allphase-only approaches that only modulate a Gaussian beam by an azimuthal phase will produce the rings seen in (c) and is not a factor of the device efficiency. In most reports, the rings are removed by opti- cal filtering prior to recording and hence resulting in unwanted losses. B. Enhanced quantum detection of OAM modes The aforementioned tests modulated and detected classical light fields. To complement the classical results, we also perform a quantum experiment, as shown in Fig. 3(b). It may be noted that while the modes ( ℓ,p) are discrete, surface plots were used as a guide so that the trend is easily seen [similar to the lines in Fig. 6(f)], and thus, the discrete values are interpolated to yield the graphs. Here, entangled photons were created using a PPKTP crystal via sponta- neous parametric down-conversion (SPDC). Each of the entangled bi-photons were relay imaged to projective optics, with one set pro- jecting onto a vortex state by an azimuthal phase profile, and the other performing full ℓcontrol of both the amplitude and the phase. This is the quantum equivalent of the classical prepare and measure experiment already reported in Sec. IV A. However, as is custom- ary in such quantum experiments, the detection system includes a single mode fiber so that only the p= 0 mode can be filtered out. Figure 6(a) shows the experimental results for measuring the OAM spiral bandwidth as a function of the scale in the detection (α=wℓ/w0), with the corresponding theoretical plot shown in (b). Clearly, they are in excellent agreement. Cross sections of the power in a specific ℓmode, quantified by | cℓ|2, as a function of changing the scaleα, are shown in (c)–(e) for ℓ= 1, 2, and 5, respectively. The theory (black dots) is in good agreement with the experimental data (blue bars). This confirms that the concept we have outlined is equally impactful with quantum light. We predict and observe that the coincidence rate can be dramatically increased by our approach, resulting in a visible spiral bandwidth beyond that possible with- out scale adjustment: the Schmidt number more than doubles from K= 3 to K= 7, as shown in Fig. 6(f), demonstrating more than a doubling of the quantum dimensionality. This represents a signifi- cant increase in the available dimensions in our OAM Hilbert space and is a consequence of transferring power into the desired mode, thus lifting it from the noise. This experiment also serves to illus- trate that it is not the manner of implementing the azimuthal phase APL Photon. 5, 070802 (2020); doi: 10.1063/5.0005597 5, 070802-6 © Author(s) 2020APL Photonics ARTICLE scitation.org/journal/app FIG. 6 . (a) Experimental and (b) simulated coincidence measurements based on the detection probabilities of OAM photons measured with the encoded holograms. (c) The normalized coincidences as a function of α=wℓ/w0for (c) ℓ= 1, (d) ℓ= 2, and (e) ℓ= 5. The bars represent the experimental data, while the points represent the simulation. (f) Spiral-bandwidth plots obtained from the diagonal of the inset mode projections for (left) w0and (right) wopt. Insets are density plots representing measured coincidences for OAM projections on the SLMs when the encoded LG beam size is w0(black squares) and the optimal mode size is ( wopt) (blue triangles). Each data point was measured over a 5 s time interval. pattern that matters—here, only SLMs were used due to the unsuit- ability of the metasurface for the quantum photon wavelengths. For completeness, we point out that we have not tested the quantum correlations in the new subspaces and so make no statements on the entanglement. However, since the SPDC process produces pure states,31–35the full quantum toolkit can be applied to the new sub- spaces revealed from the noise, including testing for entanglement in high-dimensions.36 V. DISCUSSION AND CONCLUSION In many applications, the detection of OAM modes implicitly assumes a projection onto a Gaussian-like mode ( p≈0), by coupling into single mode fibers, by OAM mode sorting, or by the match filters used in MUX/DEMUX optical communication systems. In such cases, the detection approach we outlined here will yield an improved signal-to-noise ratio and thus a larger available alphabet, as we have demonstrated with both the classical and quantum states. For completeness, we point out that if the application and/or detec- tion simply integrates out the radial modes, e.g., accumulates all power from a particular azimuthal mode, then there will be neither a gain nor a loss from our approach.We employed the metasurface in the classical experiment as an example of a phase-only device, which we note could have been replaced with any equivalent element to illustrate our concept. For example, a second SLM could have achieved the same outcome, and this, in fact, was demonstrated with the quantum experiment. Here, a contrast in features between these two devices explicitly emphasizes the generality of the effect and thus universality of how our optimization may be of benefit. The choice of the metasurface allowed us to simply demonstrate a high OAM charge ( ℓ= 100) with a good purity, which was easily achievable with the greater resolu- tion. Additionally, due to the asymmetrical nature of the coupling in the device paired with the amplitude and phase control, which were achieved in separate steps, it would be possible to extend this to easily produce arbitrary high purity vector modes in line with Ref. 28. It may also be interesting to consider the combination of both techniques into a single integrated device in future studies, simi- lar to that achieved by Refs. 11 and 12 or potentially employing phase correction approaches37,38for enhanced purity in all degrees of freedom. In conclusion, we have outlined a simple approach for increas- ing the detection efficiency of classical and quantum experiments with OAM modes by a simple adjustment of the scale at which APL Photon. 5, 070802 (2020); doi: 10.1063/5.0005597 5, 070802-7 © Author(s) 2020APL Photonics ARTICLE scitation.org/journal/app the measurements are made. In the quantum case, this is all that is needed since there is no “initial” mode: the state is determined by the measurement and has no reality prior to that. In the classical case, to truly maximize the signal requires very high purity initial modes. We used metasurfaces that are holographically enhanced to control both the azimuthal and radial profiles of OAM modes to create high purity OAM beams up to ℓ= 100. This applied research paper will be important to applications that seek to exploit OAM as a basis in the classical and quantum experiments, particularly those pertaining to communication. SUPPLEMENTARY MATERIAL See the supplementary material for supporting information. ACKNOWLEDGMENTS I.N. and B.S. would like to acknowledge the Department of Sci- ence and Technology (South Africa) for funding and A.V. from the Claude Leon Foundation, F.C. is supported by funding from the Air Force Office of Scientific Research (Grant Nos. MURI: FA9550- 14-1-0389, FA9550-16-1-0156), and the King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) (Award No. OSR-2016-CRG5-2995). Y.-W.H. and C.-W.Q. are supported by the National Research Foundation, Prime Min- ister’s Office, Singapore under its Competitive Research Program (CRP Award No. NRF-CRP15-2015-03). This work was performed, in part, at the Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Coordinated Infrastructure (NNCI), which is supported by the NSF under Award No. 1541959. CNS is a part of Harvard University. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185 (1992). 2N. Heckenberg, R. McDuff, C. Smith, and A. 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5.0009353.pdf

J. Appl. Phys. 128, 023901 (2020); https://doi.org/10.1063/5.0009353 128, 023901 © 2020 Author(s).Magnetic anisotropy of doped Cr2O3 antiferromagnetic films evaluated by utilizing parasitic magnetization Cite as: J. Appl. Phys. 128, 023901 (2020); https://doi.org/10.1063/5.0009353 Submitted: 30 March 2020 . Accepted: 18 June 2020 . Published Online: 08 July 2020 Tomohiro Nozaki , Muftah Al-Mahdawi , Yohei Shiokawa , Satya Prakash Pati , Hiroshi Imamura , and Masashi Sahashi COLLECTIONS Paper published as part of the special topic on Antiferromagnetic Spintronics Note: This paper is part of the special topic on Antiferromagnetic Spintronics. 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Phys. 128, 023901 (2020); doi: 10.1063/5.0009353 View Online Export Citation CrossMar k Submitted: 30 March 2020 · Accepted: 18 June 2020 · Published Online: 8 July 2020 Tomohiro Nozaki,1,a) Muftah Al-Mahdawi,2,3 Yohei Shiokawa,4Satya Prakash Pati,4Hiroshi Imamura,1 and Masashi Sahashi4 AFFILIATIONS 1Spintronics Research Center, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki 305-8568, Japan 2Center for Science and Innovation in Spintronics (Core Research Cluster), Tohoku University, Sendai, Miyagi 980-8577, Japan 3Center for Spintronics Research Network, Tohoku University, Sendai, Miyagi 980-8577, Japan 4Department of Electronic Engineering, Tohoku University, Sendai, Miyagi 980-8579, Japan Note: This paper is part of the special topic on Antiferromagnetic Spintronics. a)Author to whom correspondence should be addressed: nozaki.tomohiro@aist.go.jp ABSTRACT In Cr 2O3thin films doped with Al or Ir, we have discovered a parasitic magnetization, accompanied by the antiferromagnetic order, with tunable direction and magnitude. In this study, by utilizing the parasitic magnetization, the antiferromagnetic anisotropy KAFof the doped Cr2O3thin films was evaluated. A much greater improvement of KAFwas obtained for Al-doped Cr 2O3films than that of bulk. The maximum KAFin this study was ∼9×1 04J/m3, obtained for the Al 3.7%-doped Cr 2O3film sample. The enhancement of the magnetic dipole anisotropy KMDdue to the site-selective substitution is speculated for the dominant origin of the enhancement. Furthermore, based on the obtained KAF, the influence of the parasitic magnetization on the exchange bias blocking temperature TBof the doped-Cr 2O3/Co exchange coupled system was discussed. TBgreatly increases when the parasitic magnetization is coupled antiparallel to ferromagnetic moment, such as Al-doped Cr 2O3/Co systems. Published under license by AIP Publishing. https://doi.org/10.1063/5.0009353 INTRODUCTION Along with the development of manipulation and detection techniques of antiferromagnet, antiferromagnetic spintronics havebeen attracting much attention. 1,2Antiferromagnetic insulators have also been attracted as a medium for a pure spin current having a long propagation distance.3–5Cr2O3is an antiferromag- netic insulator having uniaxial magnetic anisotropy, unlike cubicantiferromagnets such as NiO and CoO. Cr 2O3has an antiferro- magnetic structure of ↑↓↑↓ along the caxis and exhibits a linear magnetoelectric (ME) effect because the time reversal symmetryand the space reversal symmetry are broken. An electric 180°manipulation of the antiferromagnetic domain by utilizing the MEeffect and its detection via exchange bias has been demonstrated. 6–9 To realize voltage-controlled antiferromagnetic devices, the MEeffect of Cr 2O3thin films has been extensively studied10–12and various devices have been proposed.13–17For the practical device applications, evaluation of antiferromagnetic properties, such asantiferromagnetic anisotropy K AF, is indispensable. However, due to the vanishing net magnetization, to evaluate antiferromagneticproperties is not easy. K AFof Cr 2O3has been evaluated by antiferro- magnetic resonance18,19and spin-Hall magnetoresistance,20but the number of reports was limited. Recently, we discovered that Cr 2O3thin films doped with either Al or Ir exhibit a ferrimagnetic-like moment, which we called a para- sitic magnetization ( MCr2O3).21Figure 1 depicts the schematic spin structure of Al- and Ir-doped Cr 2O3films with averaged sublattice Cr moments. The magnitude and direction of MCr2O3can be con- trolled by the dopant contents and elements; the magnitude ofJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 023901 (2020); doi: 10.1063/5.0009353 128, 023901-1 Published under license by AIP Publishing.MCr2O3increases with increasing dopant contents. Al-doping gener- atesMCr2O3parallel to the Cr 2O3antiferromagnetic Néel vector (or antiferromagnetic staggered magnetization vector) L,w h e r e a s Ir-doping generates MCr2O3antiparallel to L.H e r e , Lis a unit vector representing the direction of the antiferromagnetic domain of Cr 2O3 (↑↓↑↓ or↓↑↓↑ ).14,21The most-likely origin of MCr2O3is a site- selective substitution of the dopant at one of the Cr sublattices, causing a ferrimagnetic-like moment due to an average imbalance ofthe sublattice moments. A non-equivalent site occupation energyduring the layer by layer growth of doped Cr 2O3films was specu- lated to be the origin of the metastable site-selective substitution. The unusual parasitic magnetization in doped Cr 2O3films enhances the controllability and detectability of its antiferromagnetic domain.For example, an antiferromagnetic single domain state can be easilyobtained by applying a magnetic field isothermally, whereas itcannot be obtained for antiferromagnets without parasitic magneti- zation unless utilizing the ME effect or exchange bias phenomena. In this study, we demonstrated the control and detection of L of doped Cr 2O3films by utilizing the parasitic magnetization. By applying magnetic fields along the hard anisotropy axis, we evalu-ated K AFof doped Cr 2O3films. We also verified that Lis parallel to MCr2O3under the magnetic field applied along the hard axis, by detecting the ME susceptibility α. Furthermore, based on the obtained KAF, we discussed the effect of the parasitic magnetization on the exchange bias blocking temperature TBin doped-Cr 2O3/Co exchange coupled systems. EXPERIMENTAL PROCEDURES Samples were fabricated using a radio frequency (RF) magne- tron sputtering system. We prepared doped Cr 2O3films by the reactive sputtering method with a power of 80 W in an Ar + O 2atmosphere at a substrate temperature of 773 K using a Cr target with Al chips or Al –Cr and Ir –Cr alloy target. For details, please also refer to Refs. 21and22. We confirmed the film composition by x-ray fluorescence (XRF) measurements. The magnetic properties ofsamples were measured by using a superconducting quantum inter-ference device (SQUID) magnetometer. The magnetization curves were measured either in a field applied out-of-plane ( H//c)o r in-plane ( H⊥c). For the evaluation of the magnetic properties of Cr 2O3films, such as KAF,A l 2O3sub/Pt (25 nm)/doped-Cr 2O3 (250 or 500 nm) structures were used. For the evaluation of the magnetic properties of Cr 2O3/Co exchange coupled systems, such as exchange bias Hexand TB,A l 2O3sub/Pt (25 nm)/doped-Cr 2O3 (250 nm)/Co (1 nm)/Pt (5 nm) structures were used. The ME suscept- ibility αij=dMi/dEjof Al 2O3sub/Pt (25 nm)/Al 3.7% doped-Cr 2O3 (500 nm)/Pt (5 nm) structure was obtained by measuring the ac magnetization induced by an ac electric field (2 V rmsor 40 kV/cm, 21 Hz) due to the inverse ME effect. The bottom and top Pt elec- trodes were patterned and the electric field was applied at thecross-junction of 8 mm 2area. For the details, see Ref. 23. The elec- tric field was fixed along the caxis, while the magnetic field was applied in the in-plane direction (hard axis). The ME susceptibility was measured along the applied magnetic field direction, using the SQUID magnetometer. RESULTS AND DISCUSSIONS Figures 2(b) and 2(c) show the out-of-plane and in-plane magnetization curves of Al 3.7%-doped and Ir 3.7%-doped Cr 2O3 film samples, respectively, measured at 50 K. The presumable mag-netic structure in the presence of an out-of-plane ( H op) and a large in-plane ( Hip) magnetic fields, in the case of Al-doped Cr 2O3,a r e depicted in Fig. 2(a) . Because of the ferrimagnetic-like magnetiza- tion of the doped Cr 2O3film samples, a clear hysteresis curve was obtained despite the absence of ferromagnet. The out-of-plane(in-plane) direction corresponds to the easy (hard) axis of doped Cr 2O3film samples. High squareness perpendicular magnetization curves with a squareness ratio of Mr/Ms≈1, originating from the uniaxial hexagonal structure, were obtained for doped Cr 2O3film samples. It is in contrast to that of an oxide ferrimagnet CoFe 2O4 exhibiting a high perpendicular magnetic anisotropy but a low squareness ratio.24,25The high squareness is a feature of this system. Assuming a sufficiently large Cr –Cr inter-sublattice exchange cou- pling energy compared to a Zeeman energy of MCr2O3,Lrotates simultaneously with MCr2O3, as depicted in Fig. 2(a) . In such a case, KAFof doped Cr 2O3films can be estimated in the same way as a fer- romagnet or ferrimagnet with a uniaxial anisotropy. A detailedexplanation is given in S1 in the supplementary material . Based on these assumptions, we estimated K AFof the doped Cr 2O3film samples from the magnetization curves. Figure 2(d) shows the temperature dependence of the effective antiferromagnetic anisot- ropy energy Keff AF¼μ0MCr2O3Hk/2 (open markers) and KAF ¼Keff AFþμ0M2 Cr2O3/2 (closed markers) of the doped Cr 2O3film samples, where MCr2O3and Hkare the saturation parasitic magneti- zation and the anisotropy field of the samples, respectively. MCr2O3 (in A/m) was calculated by dividing the obtained areal saturation magnetization by the Cr 2O3thickness, assuming the parasitic FIG. 1. Schematics of the parasitic magnetization MCr2O3of Al- and Ir-doped Cr2O3films. Averaged up (down) spin sublattice moments become larger than that of down (up) spin sublattice moments for Al- (Ir-) doped Cr 2O3films. As the results, the direction of MCr2O3(green arrows) against antiferromagnetic Néel vector L(blue arrows) differs by the dopant elements.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 023901 (2020); doi: 10.1063/5.0009353 128, 023901-2 Published under license by AIP Publishing.magnetization to be volume magnetization.21Due to the small mag- nitude of MCr2O3, the demagnetization energy μ0M2 Cr2O3/2 is negligi- ble, and KAF≈KAFeff.KAFtended to increase as the dopant contents increased. Furthermore, the Al-doped samples showed a larger KAF than the Ir-doped sample. The largest value in this study was about 9×1 04J/m3, obtained for the Al 3.7%-doped sample at 10 K. Figure 2(e) shows temperature dependence of MCr2O3of the samples. MCr2O3exhibited a similar trend with that of KAF;s a m p l ew i t ha large KAFtended to exhibit a large MCr2O3 .KAFof undoped and Al-doped Cr 2O3bulk samples has long been examined by antiferro- magnetic magnetic resonance experiments.18,19Figure 2(f) compares theKAFvalue between Al- and Ir-doped Cr 2O3thin film samples measured at 50 K (this work) and Al-doped Cr 2O3bulk samples measured at low temperatures.19While KAFof undoped Cr 2O3films without parasitic magnetization has not been evaluated, from theAl-content dependency of KAFof Al-doped Cr 2O3films, it is esti- mated to be the similar value as the bulk sample ( KAF∼2×1 04J/m3). An important finding is that KAFchanges more abruptly with the Al-contents in the thin film samples compared with the bulk samples. Changes in the magnetocrystalline anisotropy (the fine structure anisotropy) KFSof Cr 2O3by Al-doping has been investi- gated by first-principles calculations.22,26It was indicated that a large change in KFSwill not occur due to the electronic state change by Al impurity state or lattice compression by Al-doping.22By con- sidering the Al impurity induce local structural deformation, increase in KFShas been predicted.26The predicted increase in KFS possibly explains the increase in KAFin the Al-doped Cr 2O3bulk samples. However, it cannot explain the difference between theAl-doped Cr 2O3bulk and film samples. The magnetic dipole anisot- ropy KMDalso largely contribute to KAFof Cr 2O3. In undoped FIG. 2. (a) Schematics of the presumed magnetic structure of Al-doped Cr 2O3film in the presence of an out-of-plane ( Hop) and a large in-plane ( Hip) magnetic fields. The direction of MCr2O3,L, and His depicted by green, blue, and orange arrows, respectively. (b) and (c) Out of plane ( Hop, black line) and in-plane ( Hip, red line) magnetization curves of Al 3.7%- and Ir 3.7%-doped Cr 2O3film samples, respectively, measured at 50 K. (d) T emperature dependence of KAF(closed markers) and KAFeff(open markers) of Al 3.7%-, 2.8%-, 1.2%-, and Ir 3.7%-doped Cr 2O3film samples. (e) T emperature dependence of the parasitic magnetization MCr2O3of Al 3.7%-, 2.8%-, 1.2%-, and Ir 3.7%-doped Cr 2O3film samples. (f) Comparison of the dopant contents dependent KAFof Al-doped and Ir-doped Cr 2O3film samples (this work) and Al-doped Cr 2O3bulk samples (Ref. 19) at low temperatures.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 023901 (2020); doi: 10.1063/5.0009353 128, 023901-3 Published under license by AIP Publishing.Cr2O3bulk, KMDand KFSequally contribute to KAF.18Since KMDof Cr2O3is quite sensitive to the spin structure18and the Cr ion spatial position parameter w,18,27a large difference in KMDmay occur between the bulk and film samples. In the case of Cr 2O3bulk, KMD was estimated to decrease slightly by Al-doping, considering changes in the lattice parameter a,c,and spatial position was well the mag- netic dilution effect.19On the other hand, further changes in the KMD may occur if the site-selective substitution of Al occurs in Al-doped Cr2O3films; an increase in wdue to a cooperative lattice distortion can lead to an increase in KMD. Loss of magnetic moments of only one sublattice can lead to a further increase in KMD,a saq u i t el a r g e KMDchange is expected due to spin structure changes.18Thus, the site-selective substitution has the potential to explain the large KAFas well the appearance of parasitic magnetization, both of them areunique characteristics for the doped Cr 2O3film samples. Based on these assumptions, we speculated that the enhancement of KMDdue to the site-selective substitution largely contributes to the enhance- ment of KAFobserved in the Al-doped Cr 2O3films. In order to make sure that Lrotates together with MCr2O3,w e also measured the in-plane magnetic field dependence of the MEsusceptibility α, where the appearance in doped Cr 2O3has beenconfirmed before.21With applying an ac electric field Ein the out-of-plane ( caxis) direction and a magnetic field in the in-plane direction, the electric-field-induced ac magnetization in thein-plane direction ( α 31E) was measured. The schematics of applied field directions, α31Emeasurement direction, and the spin configu- ration of doped Cr 2O3in the in-plane magnetic field are depicted inFig. 3(a) . Note that the doped Cr 2O3in this study is caxis ori- ented textured films, similar to the sputtered undoped-Cr 2O3 films28and we measured the electric-field induced magnetization along the applied magnetic field direction. In this study, because L is tilted along the applied field direction, α31andα32are not clearly distinguished. This is in contrast to the measurement of α13(=α31) orα23(=α32) by utilizing a spin-flop phenomenon in the presence of high magnetic fields along the easy axis,29,30where Lis tilted along the baxis.31Based on our symmetry considerations, α31Ein theL⊥cconfiguration arise the same mechanism as α33Ein the L// cconfiguration, and these are equivalent. As shown in Fig. S2 (b) in the supplementary material , the temperature dependence of α31 of Al 3.7%-doped Cr 2O3films in a large in-plane magnetic field (5 T) is almost identical to that of α33of undoped Cr 2O3films.23 The result is consistent with the assumption. Figures 3(c) and3(d) FIG. 3. (a) Schematics of applied field directions, αEmeasurement direction, and the spin configuration of Al-doped Cr 2O3films in the in-plane magnetic field. (b) Magnified image of the CrO 6octahedra in the in-plane magnetic field. (c) and (d) T emperature dependence of α31-μ0Hcurves of the Al 3.7%-doped Cr 2O3film sample. (e) and (f) T emperature dependence of in-plane magnetization curve of the Al 3.7%-doped Cr 2O3film sample.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 023901 (2020); doi: 10.1063/5.0009353 128, 023901-4 Published under license by AIP Publishing.show the in-plane magnetic field dependence of α31of the Al 3.7%-doped Cr 2O3film sample. Figures 3(e) and3(f) show the cor- responding in-plane magnetization curves of the Al 3.7%-dopedCr 2O3film sample, for comparison. If a same μ0Hkas the magnetiza- tion curve measurements can be obtained from α31Emeasurements, it become clearer that Lrotates together with the MCr2O3 . In the range of 250 K to 175 K [ Figs. 3(c) and3(e)], the α31-μ0Hcurves are similar to the magnetization curves. We can clearly see the satura-tion of α 31andμ0Hkis close to that of the magnetization curves. In this temperature range, the origin of α33is the electric-field modula- tion of the non-relativistic exchange interaction in the presence of thermally driven spin fluctuations.32–36Therefore, the normalized α31is the projection of Lon the in-plane direction, and the satura- tion corresponds to the L⊥cconfiguration. The agreement between μ0Hkobtained from the magnetization curve and that obtained from the α31-μ0Hcurve in Figs. 3(c) and3(e)validate the KAFmea- surement method of this paper. On the other hand, a further decrease in temperature causes the appearance of a large peak of∼−1.6 ps/m at around 1.5 T, slightly smaller than μ 0Hk,i nt h e α31-μ0Hcurves [ Fig. 3(d) ] which disturb estimating μ0Hk. The appearance of such an unusual contribution when Ldeviates from thecaxis was also reported before.37The appearance of such a complex curve only at low temperatures is likely related to an unex-plored part of the dominant α 33mechanism at that temperature range. It has been suggested that α33in the low temperature region is related to orbital effects,33,34,38such as an electric field induced g-factor shift and a Van Vleck-type paramagnetism. Figure 3(b) depicts the magnified images of CrO 6octahedra in the presence of in-plane magnetic fields, corresponding to Fig. 3(a) . The vertices of CrO 6octahedra are oriented at angles from the caxis. As Lis rotated, the g-factor shift would be affected due to the strong crystal field of O vertices through LScoupling; when L//candL⊥c,C r spins are surrounded in the similar crystal field. In both case, Crspins point to the center of the oxygen triangles. On the otherhand, at the intermediate Ldirection where the peak of the α 31was observed, Cr spins point to the vertices of the CrO 6, surrounded in the different crystal field. Therefore, we expect that at intermediateLangles, the orbital-driven αmechanism is enhanced. Interestingly, below 225 K, it seems a minor anomaly is found in the magnetiza- tion curve [ Figs. 3(e) and3(f)]. The corners shape gets round and the saturation of the magnetization become unclear, compared withthe magnetization curve at 250 K. The unclear saturation can beunderstood as the presence of an anisotropy field of a higher-order magnetic anisotropy. 39The minor anomaly in the magnetization curves possibly be related to the anomaly in the α, since the appear- ance of a higher-order anisotropy is related to spin –orbit interac- tions. Note that the unusual αpeak value, ∼1.6 ps/m, is very large compared to the ME effect of Cr 2O3obtained by orbital mecha- nisms, α33∼0.1 ps/m and α11∼0.4 ps/m. This result suggests a dis- covery of a new, large ME effect raised by the crystal field withunusual symmetries. As discussed so far, below 150 K, the estima-tion of μ 0Hkfrom α-μ0Hmeasurements is difficult. These results demonstrated the effectiveness of utilizing the parasitic magnetiza- tion for a tool to easily estimate the KAFof doped Cr 2O3. Next, we will discuss the influence of the parasitic magnetization on TBof the doped Cr 2O3/Co exchange coupled systems. Figures 4(a) –4(c) show the temperature dependence of μ0Hexand μ0Hcof Cr 2O3/Co exchange coupled systems with undoped, Al 1.2%-doped22and Ir 3.7%-doped Cr 2O3.W ed e f i n e TBas a tempera- ture that both abrupt disappearance of μ0Hexand increase of μ0Hc occurs. The obtained TBare 100 K, 250 K, and 50 K for the undoped, Al-doped, and Ir-doped systems, respectively; compared to the undoped system, the Al-doped system showed a significant improve- ment of TB, whereas the Ir-doped system showed a decrease in TB. TBof Cr 2O3/Co exchange coupled systems has been qualitatively understood based on the Meiklejohn –Bean free-energy model (MB model).40,41Recently, a generalized MB model which taking the hori- zontal antiferromagnetic domain into account was proposed.42The effect of the horizontal antiferromagnetic domain should be consid-ered, when the interfacial exchange coupling energy is sufficientlylarge. 43Based on the model, in a usual Cr 2O3/Co exchange coupled system (without parasitic magnetization), when the Co magnetization is reversed by the magnetic field, the simplified free energy in Cr 2O3, F,i sr e p r e s e n t e db yE q . (1), considering the antiferromagnetic FIG. 4 (a)–(c) T emperature dependence of exchange bias μ0Hex(closed markers) and coercivity μ0Hc(open makers) of the undoped, Al 1.2%-doped,22and Ir 3.7%-doped Cr 2O3/Co exchange coupled systems. The exchange bias blocking temperature TBis indicated by arrows.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 023901 (2020); doi: 10.1063/5.0009353 128, 023901-5 Published under license by AIP Publishing.anisotropy term and the exchange coupling term, F¼KAFtc AFsin2θþJintSFMSAFcosθ(undoped) : (1) Here, Jintis the interface exchange coupling constant, which is closely related to the experimentally obtained unidirectional anisot-ropy energy by exchange bias J K=μ0MCoHextCo(see also S3 in the supplementary material ).SFM,SAF,tCo, and MCoare the interface magnetization of ferromagnet and antiferromagnet, the thickness of Co, and the saturation magnetization of Co, respectively. tAFcis the effective Cr 2O3thickness which consider the temperature dependence of the horizontal Cr 2O3domain, not the geometrical film thickness tAF. The derivation of Eq. (1)is given in S3 in the supplementary material . The initial antiferromagnetic domain was destabilized by the exchange coupling from reversed Co [thesecond term in Eq. (1)]. When the first term in Eq. (1)is suffi- ciently large, the antiferromagnetic domain is rigid, producingfinite μ 0Hex(T<TB). With increasing temperature, the relative magnitude of the first and second terms in Eq. (1)changes. When the relative magnitude of the second term becomes sufficientlylarge, the antiferromagnetic domain of Cr 2O3will rotate with the Co magnetization, leading to the disappearance of μ0Hexand enhancement of μ0Hc(T>TB). A detailed explanation is given in S3 in the supplementary material . Thus, in a system with a negligi- ble parasitic magnetization, TBis related to the relative magnitude ofJKand KAFtAFc. In a system with doped Cr 2O3having a large par- asitic magnetization, the Zeeman energy of the parasitic magnetiza-tion have to be added to Eq. (1). Considering the direction of the parasitic magnetization in Fig. 1 , the free energy in Cr 2O3is repre- sented by the following equations: F¼KAFtc AFsin2θþJintSFMSAFcosθ /C0μ0MCr2O3HtAFcosθ(Al-doped), (2) F¼KAFtc AFsin2θþJintSFMSAFcosθ þμ0MCr2O3HtAFcosθ(Ir-doped) : (3) For the sign of the MCr2O3 , see also S1 in the supplementary material .T h es i g n so f Jintand MCr2O3in Eqs. (2)and(3)are consis- tent with those observed experimentally.21As represented by the equations, the initial domain is stabilized for the Al-doped systems,and further destabilized for the Ir-doped systems, by the Zeemanenergy of the parasitic magnetization [third term in Eqs. (2)and(3)]. As the results, much higher T Bwas obtained for the Al-doped system and lower TBwas obtained for the Ir-doped system com- pared with the undoped system. The substantial variation of TB among the three system despite the similar KAF[Fig. 2(d) ]s u g - gests the large contribution of the Zeeman effect to TBin these systems. Specifically, μ0MCr2O3HtAFand JKare calculated to be ∼0.95 and ∼0.43 mJ/m2for the Al 1.2%-doped system, and ∼0.33 and∼0.25 mJ/m2for the Ir 3.7%-doped system, at 50 K, when μ0H=μ0Hex. For the Al 1.2%-doped system, since μ0MCr2O3 HtAF exceeds JK, the initial domain is sufficiently stabilized by the Zeeman effect only, and an exchange bias can be observed up to ∼TNregardless of the magnitude of the KAF. This study revealed asignificant contribution of the Zeeman energy of the parasitic magnetization for the TBin doped Cr 2O3/Co exchange coupled systems; we have to take into account the direction and magnitudeof the parasitic magnetization to design the exchange coupledsystems. We found that by using an antiferromagnetic material(ferrimagnetic material) having a parasitic magnetization coupled antiparallel to a ferromagnetic material, such as Al-doped Cr 2O3, high TBexchange coupled systems can be easily obtained. CONCLUSION In this study, we investigated the antiferromagnetic anisotropy KAFof doped Cr 2O3films, by utilizing the parasitic magnetization. We demonstrated that KAFcan be easily estimated by using the parasitic magnetization, while typically the estimation of KAFof thin films is difficult. The in-plane magnetic field dependence of the magnetoelec-tric (ME) effect, α 31measurement results proved the usefulness of the parasitic magnetization to evaluate KAF.F u r t h e r m o r e ,w es h o w e dt h e importance of considering the Zeem an energy of the parasitic magne- tization, for designing doped-Cr 2O3/Co exchange coupled systems. When Al was doped for Cr 2O3thin film, larger enhancements ofKAFwere obtained compared with that of Cr 2O3bulk. The Al 3.7%-doped Cr 2O3sample exhibits KAF∼9×1 04J/m3. It is antici- pated that the site-selective substitution, specific for the dopedCr 2O3thin films, contribute to the drastic enhancement of KAF.I n addition, the parasitic magnetization allows us to access a new sym-metry for the spin configuration in Cr 2O3that realizes a new ME susceptibility. Furthermore, it is found that the parasitic magnetiza- tion obtained by Al-doping can be utilized to enhance the exchangebias blocking temperature T Bof Cr 2O3/Co exchange coupled systems. The parasitic magnetization in doped Cr 2O3thin films can be an important tool to elucidate/enhance the antiferromagnetic properties and contribute to the further development of the antifer-romagnetic spintronics. SUPPLEMENTARY MATERIAL See the supplementary material for S1 for the model of K AF estimation by utilizing parasitic magnetization, S2 for the magneto- electric effect of doped Cr 2O3in an in-plane magnetic field, and S3 for the stability of antiferromagnetic domains in the doped Cr 2O3/ Co exchange coupled system. ACKNOWLEDGMENTS This work was partly supported by the Center for Science and Innovation in Spintronics (CSIS), Center for SpintronicsResearch Network (CSRN), Tohoku University, JSPS KAKENHI (Grant No. 16H05975), and the ImPACT Program of the Council for Science, Technology and Innovation (Cabinet Office,Government of Japan). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 023901 (2020); doi: 10.1063/5.0009353 128, 023901-6 Published under license by AIP Publishing.REFERENCES 1V. Baltz, A. Manchon, M. Tsoi, T. 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5.0020431.pdf

J. Chem. Phys. 153, 054308 (2020); https://doi.org/10.1063/5.0020431 153, 054308 © 2020 Author(s).Spectroscopy and electronic structure of the hypermetallic oxide, MgOMg Cite as: J. Chem. Phys. 153, 054308 (2020); https://doi.org/10.1063/5.0020431 Submitted: 01 July 2020 . Accepted: 17 July 2020 . Published Online: 05 August 2020 Thomas D. Persinger , Daniel J. Frohman , Wafaa M. Fawzy , and Michael C. Heaven ARTICLES YOU MAY BE INTERESTED IN Photoelectron–photofragment coincidence spectroscopy of the mixed trihalides The Journal of Chemical Physics 153, 054304 (2020); https://doi.org/10.1063/5.0014253 Accurate quantum mechanical calculations on deuterated vinylidene isomerization The Journal of Chemical Physics 153, 054309 (2020); https://doi.org/10.1063/5.0015470 Energy transfer between vibrationally excited carbon monoxide based on a highly accurate six-dimensional potential energy surface The Journal of Chemical Physics 153, 054310 (2020); https://doi.org/10.1063/5.0015101The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Spectroscopy and electronic structure of the hypermetallic oxide, MgOMg Cite as: J. Chem. Phys. 153, 054308 (2020); doi: 10.1063/5.0020431 Submitted: 1 July 2020 •Accepted: 17 July 2020 • Published Online: 5 August 2020 Thomas D. Persinger, Daniel J. Frohman, Wafaa M. Fawzy,a) and Michael C. Heavenb) AFFILIATIONS Department of Chemistry, Emory University, Atlanta, Georgia 30322, USA a)Permanent address: Department of Chemistry, Murray State University, Murray, KY 42071, USA. b)Author to whom correspondence should be addressed: mheaven@emory.edu. Telephone: 404 727 6617 ABSTRACT Electronic spectra for the hypermetallic oxide MgOMg have been observed in the 21 100 cm−1–24 000 cm−1spectral range using laser induced fluorescence and two-photon resonantly enhanced ionization techniques. Rotationally resolved data confirmed the prediction of a ˜X1Σ+ g ground state. The spectrum was highly congested due to the optical activity of a low-frequency bending mode and the presence of three isotopologues with significant natural abundances. Ab initio calculations predict a bent equilibrium structure for the ˜A1B2upper state, con- sistent with the observation of a long progression of the bending vibration mode. However, the vibrational intervals were not reproduced by the theoretical calculations. In part, this discrepancy is attributed to strong vibronic coupling between multiple electronically excited states. Two-photon ionization measurements were used to determine an ionization energy of 6.5800(25) eV. Published under license by AIP Publishing. https://doi.org/10.1063/5.0020431 .,s INTRODUCTION The group IIA hypermetallic oxides (MOM, with M = Be, Mg, Ca, Sr, and Ba) have attracted theoretical interest concerning the nature of the bonding.1–9Initially, it appeared that the diatomic metal oxides would be stable, closed-shell molecules with the for- mal oxidation state M2+O2−. However, theoretical calculations show that the oxides are multi-reference in character1,10with ground states that are closer to M+O−. This facilitates the binding of a sec- ond metal atom to yield linear symmetric molecules of the form M+O2−M+. These hypermetallic oxides are also predicted to have multi-reference ground states.1,3–9The reason for this characteris- tic can be illustrated by considering MgOMg. The occupied frontier orbitals are primarily linear combinations of the Mg 3 sorbitals of the form σg= N g(3s1+ 3s2) and σu= N u(3s1−3s2), where N gand Nuare normalization constants.1,4As the Mg atoms are far apart, theσgand σuorbitals are close in energy. Plausible ground state configurations are σgσu,3Σ+ uor (aσ2 g+bσ2 u),1Σ+ g(where aand b are expansion coefficients). Due to the multi-reference nature of the 1Σ+ gstate, single-reference calculations find that the triplet is lower in energy. This ordering is reversed for high-level, multi-reference methods.1,4,7These characteristics hold for all of the group IIA hypermetallic oxides, with the ground states of the series M = Be toBa all predicted to be singlets.7The calculated singlet–triplet energy intervals are small, and Ostojic et al.7reported values that range from 280 cm−1to 656 cm−1. The spacing does not increase smoothly with increasing atomic number, and the largest gap is predicted for MgOMg.7 While the hypermetallic oxides have been detected by means of mass spectrometry, the spectroscopic data for these species are very limited. To date, BeOBe is the most extensively studied mem- ber of the family.8,11Cryogenic matrix and gas phase spectra have been reported for BeOBe.11Rotationally resolved near UV bands of the ˜A1Σ+ u–˜X1Σ+ gtransition, recorded in the gas phase,8provided definitive evidence that the ground state is ˜X1Σ+ g. Photoionization of BeOBe has also been examined.12It was found that the removal of an electron yields the ˜X2Σ+ gion, which arises from the σ1 gconfiguration [BeOBe ( aσ2 g+bσ2 u)→BeOBe+(σ1 g)]. The non-bonding character of theσgand σuorbitals was indicated by the minimal changes in the vibrational frequencies and bond length on ionization. Spectroscopic data have not been reported for MgOMg, though Ostojic et al.4speculate that the unassigned emission bands from a magnesium arc in the range 360 nm–400 nm13may be from MgOMg. Spectra for CaOCa and SrOSr are similarly lacking. The only published data attributed to CaOCa, an IR band recorded from a cryogenic matrix,14have been called into question by Ostojic et al.5 J. Chem. Phys. 153, 054308 (2020); doi: 10.1063/5.0020431 153, 054308-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Low-resolution electronic emission spectra have been reported for BaOBa.15,16The bands were detected in chemiluminescence from the reactions of Ba with CO, CO 2, or N 2O. The present paper reports rotationally resolved gas phase elec- tronic spectra, ionization energy (IE) measurements, and ab initio calculations for MgOMg. Hence, we provide a more detailed review of the previous work on this molecule. In their mass spectromet- ric study, Deng et al.17employed the reaction of Mg vapor (gener- ated by heating Mg/Sb mixtures in a crucible) with N 2O to produce Mg nOmclusters. A pulsed UV laser was used to ionize the prod- ucts, and strong signals from MgOMg+were observed. Deng et al.17 speculated that the prominent signal from MgOMg+was a direct reflection of a high concentration of neutral MgOMg in the reac- tion products prior to photoionization (there was no experimental validation). This was consistent with the prediction by Boldyrev et al.1that MgOMg is a deeply bound system. In fact, theoretical calculations indicated that the Mg–OMg bond was of compara- ble strength to that of diatomic MgO.1Following detection of the ion, Boldyrev et al.2carried out electronic structure calculations for MgOMg+. Several methods of calculation were investigated, with the highest level being coupled cluster with singles, doubles, and perturbative triples [CCSD(T)]. The 6-311+G(3df) basis sets were employed. They found a linear ˜X2Σ+ gground state, bonding com- parable to that of MgOMg, and a neutral molecule IE of 6.6 eV. Calculation of the MgOMg+vibrational frequencies was compli- cated as symmetry breaking occurred for the antisymmetric stretch mode. Boldyrev et al.2attributed this problem to an artificial local- ization of the unpaired electron at one end of the molecule, with the expectation that this defect could be removed by the application of higher–level correlation methods. More recently, Ostojic et al.4have computed three-dimensional potential energy surfaces (PESs) for the ˜X1Σ+ gand ˜a3Σ+ ustates of MgOMg. These calculations were carried out using the full valence— state averaged—complete active space self-consistent field (FV-SA- CASSCF) method followed by multi-reference configuration inter- action with singles and doubles (MRCISD) calculations. The MOL- PRO 2008.1 suite of programs was employed with basis sets of quadruple zeta quality (aug-cc-pCVQZ). The resulting potential energy surfaces were used in conjunction with the MORBID com- puter code to predict the low-energy vibrational manifolds for both states and to calculate the fundamental vibrational transition fre- quencies for various isotopologues.4For the ground state of the most abundant isotopologue (24Mg16O24Mg), they obtained fundamental vibrational frequencies of ν(σg) = 484.7 cm−1,ν(πu) = 77.1 cm−1, and ν(σu) = 915.0 cm−1. Ostojic et al.4also calculated vertical excitation energies for transitions from the ground state equilibrium structure (D∞hwith Re= 1.814 Å) to the lower energy singlet and triplet excited states. The FV-CASSCF/MRCISD method was used for this task. The first group of electronically excited singlet states was found in a relatively small energy range. Ostojic et al.4reported a vertical excitation energy of 23 789 cm−1for the first allowed singlet transi- tion, ˜A1Σ+ u–˜X1Σ+ g. Access to many of the other excited states (starting from ˜X1Σ+ g) was symmetry forbidden for the linear geometry, but some of these transitions become allowed when the symmetry of the molecule is lowered by bending. Ostojic et al.4briefly consid- ered these effects. Trial calculations indicated that significant state mixing occurred on bending by just 10○. They concluded that “acomplicated and irregular structure of the electronic spectrum of Mg 2O is expected.” The present study supports the conjecture of Ostojic et al.4 regarding the complicated electronic structure of MgOMg through experimental examination of rotationally resolved gas phase spectra. This was accomplished via laser induced fluorescence (LIF) and res- onantly enhanced multi-photo ionization (REMPI) measurements, in combination with jet expansion cooling of MgOMg. Furthermore, we support our experimental findings through ab initio computa- tions using MRCI calculations, which predict a bent equilibrium structure for the excited ˜A1B2state that correlates with ˜A1Σ+ uat the linear geometry. EXPERIMENTAL Two different experimental systems were used to record elec- tronic spectra for MgOMg. The initial set of measurements were carried out using the LIF apparatus described in Ref. 18. Magnesium oxides were generated using a Smalley-type laser ablation nozzle source.19Ablation of a rotating and translating Mg rod was per- formed using the 1064 nm pulses from a Nd/YAG laser (Quanta Ray DCR1A). The Nd/YAG laser enters adjacent and antiparallel to the nozzle stream, and with a right-angle prism, it is guided to the rod. The pulse energy delivered to the surface of the Mg rod was a crit- ical parameter. The Nd/YAG laser, running under conditions that produced a stable output power, was too intense for the production of MgOMg. A Glan–Thompson polarizer, held in a precision rota- tion mount, was used to provide a variable attenuation of the linearly polarized 1064 nm light. Frequent optimization of the laser abla- tion conditions was required. The ablated Mg vapor was entrained in a carrier flow of He that contained either 0.1% N 2O or 0.2% O 2, with a source pressure in the range of 8 atm–10 atm. This gas mix- ture was delivered by a synchronized pulsed valve (Parker–Hannifin Series 9). The ablation and reaction products were then superson- ically expanded through a 2 mm diameter orifice into a vacuum chamber that was configured for standard LIF measurements. Care- ful optimization of the gas pressure, composition, and laser ablation conditions was essential for the observation of MgOMg. The beam from a pulsed tunable dye laser (Lambda Physik EMG201/FL3002) crossed the jet expansion at a distance of approx- imately 2.5 cm from the nozzle orifice and was propagated along an axis that was perpendicular to the central axis of the expansion. Using just a diffraction grating for wavelength selection, the laser operated with a linewidth (FWHM) of 0.3 cm−1. To obtain rota- tionally resolved spectra, the linewidth was reduced to 0.06 cm−1 by the addition of an intracavity etalon. A two-lens telescope was situated in the vacuum chamber beneath the expansion, with a view- ing axis that was perpendicular to both the expansion and the probe laser axes. This was used to collect fluorescence and focus it onto a photomultiplier tube (PMT). The signal from the PMT was pro- cessed by using two boxcar integrators (Stanford Research Systems, SR250), time-gated to integrate over the fluorescence signal and the background signal, respectively. The outputs from the boxcar inte- grators were sent to an analog processor (Stanford Research Sys- tems, SR235) where the background signal was subtracted from the fluorescence signal. The subtracted signal was then sent to a com- puter, where it was recorded using a data acquisition card (National Instruments, USB-6210). A custom LabVIEW program was used J. Chem. Phys. 153, 054308 (2020); doi: 10.1063/5.0020431 153, 054308-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp to control the data acquisition and display. Timing for all of the instruments was maintained using a delay generator (Stanford Research Systems, DG535) that was referenced to the 10 Hz inter- nal frequency standard of the Nd:YAG laser. Wavelength calibration of the dye laser was accomplished by the simultaneous recording of the Ne optogalvanic spectrum using a Ne-filled Fe hollow cathode lamp. The Ne line positions were taken from the PGOPHER spectral simulation package.20 The second apparatus used for these measurements was equipped for both LIF and mass-resolved resonantly enhanced multi-photon ionization (REMPI) spectroscopy. The details of this system were presented in Refs. 21 and 22. It consisted of two differ- entially pumped vacuum chambers that were connected by using a skimmer. The first chamber housed a pulsed nozzle ablation source that was essentially the same as that used in the set-up described above. This source also used the Parker–Hannifin Series 9 solenoid valve, operated with a typical backing pressure of 3 atm–4 atm. This resulted in total gas pressures of ∼10−5Torr in the first cham- ber and 5 ×10−7Torr in the second chamber. LIF spectra were recorded in the first chamber, with the beam for laser excitation set to cross the gas expansion ∼7.5 cm away from the nozzle orifice. The fluorescence was detected by using a photomultiplier tube (Photo- nis XP2020) and recorded by using a digital oscilloscope that was operated in a signal averaging mode. For the recording of spectra, the time-gated area under the fluorescence decay curve was used to determine the relative intensity of the signal. The entire time- resolved signal was transferred to the computer that controlled the experiment for the subsequent determination of fluorescence decay lifetimes. The second vacuum chamber was equipped with a time-of- flight mass spectrometer for the detection of cations produced by two-photon ionization processes. Counter-propagating dye laser beams were overlapped in the ion source region of the mass spec- trometer. Nd/YAG-pumped dye lasers with pulse durations of ∼10 ns were used to drive sequential excitation and ionization processes. Electronic transitions of the neutral molecules were excited by the light from a Lambda Physik ScanMate Pro dye laser. Tunable near uv radiation was used for the ionization step. This was obtained by frequency doubling the output from a Continuum ND6000 dye laser. A short-pass optical filter was used to block the dye laser fundamental. Pulse timings were controlled by using pulsed delay generators (Stanford Research Systems DG353). Wavelength cal- ibrations of both dye lasers were established using a wavemeter (Bristol Instruments model 821) and optogalvanic detection of Ne transitions. RESULTS AND ANALYSIS LIF survey scans were carried out in the first apparatus over the wavelength range from 520 nm to 353 nm. Transitions of MgO were observed at longer wavelengths, while the spectrum below 380 nm was dominated by the well-known A1Σ+ u–X1Σ+ gbands of the Mg 2 dimer.23,24The spectrum in the range from 472 nm–413 nm was very congested with many blue-shaded bands that could not be ascribed to any known band system of magnesium oxides or Mg nclusters. We attribute these bands to MgOMg, based on both the observed rota- tional structure and subsequent REMPI measurements with mass- resolved ion detection. The experimental conditions for observingMgOMg were very sensitive to the intensity of the ablation laser. We typically used pulse energies in the range of 1 mJ–2 mJ, with the laser focused by a 30 cm focal length biconvex lens. Even within this parameter space, the optimal production of MgOMg was a sharply peaked function of the intensity. This characteristic resulted in large shot-to-shot variations in the MgOMg production, requiring signal averaging to obtain reasonable signal-to-noise ratios (typically 64 laser pulses per wavelength point). Figure 1 shows a section of the low-resolution LIF survey spec- trum. The features in this trace arise from MgOMg, and they all exhibit blue-shading. The level of congestion is high, but a promi- nent progression of bands with a spacing of about 110 cm−1is imme- diately apparent. The associated bands arising from two other iso- topologues were also easily recognized. Note that the most common isotopes of Mg are 24, 25, and 26, with relative abundances of 0.79, 0.10, and 0.11, respectively. Hence, the relative abundances for the isotopologues24MgO24Mg,25MgO24Mg, and26MgO24Mg are 0.62, 0.16, and 0.17. For the spectral region of Fig. 1, the24MgO24Mg– 25MgO24Mg and24MgO24Mg–26MgO24Mg isotope shifts were close to 14 cm−1and 27 cm−1. Figure 2 shows the isotopologue group- ing for the band near 21 985 cm−1. A rotationally resolved spectrum for the24MgO24Mg band, shown in Fig. 3, had a simple P and R branch structure that was straightforward to assign using combina- tion differences. The odd rotational levels of the ground state were absent, as expected for the linear symmetric molecule as24Mg has a nuclear spin of zero.25In this context, it is of interest to note that the unsymmetrical isotopologues do not have missing levels, resulting in rotational contours for the25MgO24Mg and26MgO24Mg bands in Fig. 2 that are more congested than that of the symmetrical variant. Longer wavelength LIF survey spectra were recorded in an attempt to locate an origin for the vibronic band systems. The lowest energy band that could be reliably attributed to MgOMg was found at 21 147.6 cm−1. However, we cannot assert that this was an ori- gin band as the pattern of transition intensities indicated a fall-off at lower energies due to diminishing Franck–Condon factors. Rota- tionally resolved spectra were recorded for many of the stronger FIG. 1 . Representative section of the low-resolution LIF spectrum of MgOMg. The numbered peaks are members of a vibrational progression for the24MgO24Mg isotopologue. J. Chem. Phys. 153, 054308 (2020); doi: 10.1063/5.0020431 153, 054308-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Expanded region of the LIF spectrum for the bands near 21 980 cm−1. Note that the rotational structure is partially resolved for the24MgO24Mg isotopologue as the ground state zero-point level does not support odd rotational levels. The simulation corresponds to a rotational temperature of 7 K. bands of24MgO24Mg. Simple blue-shaded P- and R-branch struc- tures were observed with the odd- Jrotational lines missing (Fig. 3 is typical). This was consistent with a ˜A1Σ+ u–˜X1Σ+ g(D∞h) or ˜A1B2– ˜X1Σ+ g(C2vexcited state with Ka= 0) transition. Rotational constants and band origins were obtained from these data using the spectral fitting capabilities of the PGOPHER program.20Simulations indi- cated low rotational temperatures that were in the range of 5 K– 15 K, with some day-to-day variability. Given the limited spectral resolution and the low rotational temperature, the centrifugal dis- tortion constants could not be determined with any statistical valid- ity. Hence, the fitting was carried out with just the band origins and upper and lower state B constants treated as variable param- eters. The results are collected in Table I. It was apparent that all FIG. 3 . Rotationally resolved spectrum for the 21 985.5 cm−1band of24MgO24Mg.TABLE I . Band origins and rotational constants for24MgO24Mg. Constants are given in cm−1. Errors for the band origins are ±1.0 cm−1. Errors for the least significant figures of the excited state rotational constants are given in parentheses. The average ground state rotational constant was B 0= 0.1095(10) cm−1. maν0 B′ 1 21 147.6 0.1361 (8) 2 21 275.1 0.1365 (13) 3 21 397.1 0.1330 (8) 4 21 521.2 0.1306 (11) 5 21 643.6 0.1329 (17) 6 21 765.9 0.1336 (14) 7 21 876.5 0.1323 (10) 8 21 985.5 0.1281 (10) 9 22 096.3 0.1275 (8) 10 22 207.5 0.1298 (15) 22 537.5 0.1296 (12) 22 605.4 0.1308 (10) 22 608.9 0.1308 (10) 22 632.0 0.1310 (10) aThe index mlabels successive vibrational levels that can be assigned to a bending pro- gression. The entries that do not have mvalues could not be assigned to the bending progression with confidence. of the bands analyzed were from the ground state zero-point level, so we report the average value for the ground state rotational con- stant. Due to the congestion and complexity of the spectrum, we have identified only one vibrational progression with any confi- dence. The members of this progression are identified in Table I and Fig. 1 using an arbitrary sequential index mthat begins with m= 1 for the first member of the progression. The band spac- ings start at about 127 cm−1and slowly decrease to 109 cm−1for the highest energy assignable features, as shown in Fig. 4. Assign- ments were not followed beyond 22 210 cm−1as the spectral con- gestion was too great. The spectrum was extremely congested over FIG. 4 . Energy spacings of successive members of the low-frequency vibrational progression. These are the differences between the m+ 1 and mbands, labeled bym. J. Chem. Phys. 153, 054308 (2020); doi: 10.1063/5.0020431 153, 054308-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the higher energy range examined (up to 24 000 cm−1). For exam- ple, at least six band heads for the main isotopologue were present in a rotationally resolved scan of the region from 22 690 cm−1to 22 730 cm−1. Fluorescence decay lifetimes were recorded using excitation of bands in the 22 700 cm−1–22 800 cm−1range. Each measurement was carried out with the laser tuned to an R-branch band head. Sin- gle exponential decay curves were observed, with lifetime all inside the range 38 ±2 ns. Provided that the decay rate was not influenced by the presence of non-radiative decay channels, the short lifetime was indicative of a fully allowed transition. Two-photon ionization techniques were used to verify that the spectral features originated from MgOMg and to determine the ionization energy (IE). REMPI spectra, with mass-resolved ion detection, were recorded for the more intense bands in the 21 800 cm−1–22 200 cm−1(2.70 eV–2.75 eV) range. The initial step for the experiments in the second apparatus was usually a mea- surement of the LIF signal in the source chamber, which was used to optimize the ablation source conditions. The optics were then moved to send two counter-propagating laser beams through the ion source region of the mass spectrometer. The ionizing laser pulse was timed to arrive at the center of the chamber about 5 ns after the excitation pulse. REMPI spectra were first observed using the out- put from a KrF excimer laser (5 eV photons) for ionization, as this was below the expected IE of MgOMg (predicted to be 6.6 eV), but would provide enough energy to ionize molecules that had been pro- moted to the excited states of interest. The stronger bands of the LIF spectrum were faithfully reproduced in the REMPI spectrum. How- ever, the power of the excitation laser had to be increased in order to obtain workable signals, and this caused some power broadening of the lines. Rotationally resolved REMPI data were not obtained, but the band contours were the same as those obtained at low-resolution using LIF detection. Photo-ionization efficiency (PIE) curves were recorded with the excitation laser tuned to the heads of the 21 876.5 cm−1, 21 985.5 cm−1, and 22 096.3 cm−1bands. Three different interme- diate levels were used to check that the IE obtained was independent of the excitation pathway. For these measurements, the KrF laser was replaced by a frequency-doubled dye laser, as described in the sec- tion titled “Experimental”. In each case, the wavelength of the second laser was scanned to locate the threshold energy for the appearance of MgOMg+ions. The result obtained using initial excitation of the 21 876.5 cm−1band is shown in Fig. 5. This particular curve included its own wavelength calibration. The sharp atomic line in this trace was caused by one-color, two-photon ionization of Ni atoms via the 3G4–3F4transition at 30 979.75 cm−1. Although the experiment used mass resolved detection of MgOMg+, the on-resonance signal from Ni+(mass 59) was so intense that the detector had not fully recov- ered by the time that the MgOMg+(mass 64) ions arrived. The Ni originated from unintentional ablation of the stainless-steel mount that held the Mg rod. By inspection, we estimate that the ionization threshold defined by Fig. 5 is at a second photon energy of 31 080 ± 20 cm−1, and the total photon energy was 52 956 ±20 cm−1. These data were taken with voltages applied to the grid plates of the mass spectrometer such that the local electric field was F= 364 V/cm. This field depresses the IE by 6√ F= 115 cm−1, so the field cor- rected value for the IE of MgOMg derived from Fig. 5 was 53 072 ±20 cm−1. Equivalent analyses of the PIE curves recorded using the FIG. 5 . Photoionization efficiency curve for24MgO24Mg, recorded with the first laser tuned to the band head for the 21 876.5 cm−1band. The x axis gives the energy of the second photon. This trace has been subjected to five-point smoothing to reduce the high frequency noise from the laser ablation source. 21 985.5 cm−1and 22 097.5 cm−1bands yielded IE’s of 53 077 cm−1 and 53 063 cm−1, respectively. Hence, our best estimate for the IE is 53 071(20) cm−1[6.5800(25) eV]. THEORETICAL CALCULATIONS The computational techniques used to explore the ground and excited state potential energy surfaces of MgOMg were closely sim- ilar to those used by Ostojic et al.4The majority of the calcu- lations were carried out using the state-averaged complete active space self-consistent field (SA-CASSCF) method followed by multi- reference configuration interaction calculations. The latter included the Davidson correction (MRCI + Q). The correlation-consistent polarized core-valence quadruple zeta (cc-pCVQZ) basis set of Prascher et al.26was used for Mg, and the aug-cc-pVQZ basis27was used for O. The Mg 1 s, 2s, and 2 p, and O 1 sorbitals were con- strained to be doubly occupied, but the electrons in these orbitals were included in the correlation treatment. The active space con- sisted of the Mg 3 sand 3 p, and O 2 sand 2 porbitals (10 electrons in 12 orbitals). Calculations were carried out using the C 2vand C s point groups. For C 2v, the active space consisted of 5a 1, 2b 1, 4b 2, and 1a 2orbitals, and this set become 9a′and 3a′′when the sym- metry was reduced to C s. Single point energy calculations were performed for 11 singlet states, which included the ground state. Geometry optimization calculations were attempted using numer- ical gradients, but this was only successful for the electronic ground state. Consequently, stationary points for the excited states were determined from surfaces defined by pointwise grid sampling. Typ- ically, the bond lengths were sampled using a step size of 0.02 Å, and the bond angle was varied with 5○intervals. All of the results reported here were obtained using the Molpro 2015 suite of programs.28 Comparison with the ground state properties reported by Osto- jicet al.4and Boldyrev et al.1was made to check that we were J. Chem. Phys. 153, 054308 (2020); doi: 10.1063/5.0020431 153, 054308-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp obtaining comparable results. In the following, the harmonic vibra- tional constants ω1,ω2, andω3correspond to the symmetric stretch, bend, and antisymmetric stretch modes. In the present work, we obtainedω1= 488 cm−1,ω2= 75 cm−1, andω3= 936 cm−1, in reasonable agreement with the values of ω1= 446,ω2= 86, andω3= 881 cm−11andω1= 485 cm−1,ω2= 77 cm−1, andω3 = 915 cm−1.4We obtained an equilibrium bond length of Re= 1.804 Å, in agreement with the previous predictions of 1.821 Å1 and 1.801 Å.4 For the linear symmetric geometry, Ostojic et al.4found that the first allowed vertical electronic transition was to the ˜A1Σ+ ustate, with an excitation energy of 23 789 cm−1. Consequently, we were particularly interested in the potential energy surface for this state, which becomes1B2and1A′, as the symmetry is lowered to C 2vand Cs. We obtained a vertical excitation energy for the linear molecule ˜A–˜Xtransition of 23 838 cm−1, in close agreement with the results of Ostojic et al.4However, our calculations showed that the linear excited state was unstable with respect to both bending and struc- tures where the bond lengths were inequivalent. The latter problem seems to be a variant of the symmetry-breaking reported by Boldyrev et al.2in their study of MgOMg+. As noted in the introduction, they found an artificial localization of the unpaired electron on one of the Mg atoms. Indicative of this same problem, our calculated dipole moment for the ˜A1Σ+ ustate with the bond lengths fixed at 1.80 Å was 16 D. The distance of 1.80 Å was used as this was the equilibrium dis- tance for the linear symmetric molecule, obtained from a scan with the two bond lengths constrained to be equal. A series of single-point energy calculations were carried out for the antisymmetric stretch of the˜Astate by setting the bond lengths to R1= 1.81 +δandR2= 1.81 −δ, withδsampled from 0.0 Å to 0.16 Å. This produced a minimum atδ= 0.09 that was 1290 cm−1below the symmetric molecule sad- dle point. Bending potential energy curves (PECs) were constructed with the two bond lengths held equal. These sweeps showed that the symmetric equilibrium bond length was a weak function of the bond angle, smoothly increasing to 1.88 Å for a bond angle of θ= 95○. Figure 6 shows the bending PEC with R= 1.88 Å. The minimum was at θ= 95○, lying 4410 cm−1below the energy of the linear struc- ture. We conclude that ˜A1B2state is bent, but it probably retains C 2v symmetry. Vibrational frequencies for the ˜Astate were estimated using PEC’s that were defined by one-dimensional cuts through the poten- tial energy surface. A cut along the symmetric stretch coordinate, made with the bond angle fixed at 180○, was fitted to a Morse potential. This defined the function V(R)=0.2132(1−Exp(−0.9718(R−3.42)))2 in atomic units. A cut through the bending potential made with R1=R2= 1.88 Å is shown in Fig. 6. This was adequately represented by the polynomial U(χ)=−0.0204 χ2+ 0.0104 χ4−0.004 84 χ6+ 0.001 10 χ8, where the energy is in atomic units and χis the angle π−θcon- verted to radians. The vibrational frequencies for MgOMg ˜Awere estimated using the second derivatives of cuts through the PES. This yielded harmonic vibrational constants of ω1= 572 cm−1, ω2= 281 cm−1, andω3= 670 cm−1. FIG. 6 . One-dimensional cut through the potential energy surface of the ˜Astate of MgOMg. The bond lengths were both set to 1.88 Å for this sweep of the angle π−θ. The smooth curve is a polynomial fit to the single point energies. See text for details. Clearly, the low-frequency progression observed in the elec- tronic spectrum was associated with excitation of the bending mode. To further explore the correspondence with the experimental data, we obtained the one-dimensional eigenvalues of the bending PEC using a matrix implementation of the Numerov method.29As the potential energy function is symmetric with respect to the angle χ, the excited bending states are alternatively symmetric and anti- symmetric with respect to this coordinate. The vibronic transitions observed for MgOMg originated from the ground state zero-point level, which imposes the selection rule that only the symmetric bend- ing levels of the ˜Astate will be active in the spectrum. Figure 7 shows a plot of the energy interval between successive symmetric levels with the increasing index ( n) for the vibrational state (starting from n= 0). This plot has the characteristic behavior noted by Dixon,30 with a minimum at an energy that is close to that of the barrier to FIG. 7 . Energy spacings of successive members of the low-frequency vibrational progression. These are for successive even symmetry vibrational levels (K a= 0) predicted for the bending potential energy curve shown in Fig. 6. J. Chem. Phys. 153, 054308 (2020); doi: 10.1063/5.0020431 153, 054308-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 8 . Potential energy surface cuts along the symmetric stretch coordinate with the bond angle fixed at θ= 180○. linearity. Below the barrier, the interval decreases with increasing excitation, and the rate of this decrease accelerates as the vibrational energy approaches the barrier. The minimum occurs for the levels that are closest to the barrier (Refs. 22 and 23 in Fig. 7), and then, the interval increases with increasing excitation. The bending intervals predicted for the ˜Astate were not consis- tent with the bending progression observed in the spectrum. Ostojic et al.4noted that mixing of ˜A1B2with other states of the same sym- metry could significantly complicate the spectrum. To illustrate the range of states that can be mixed by bending and vibronic coupling, Fig. 8 shows a cut through the potential energy surfaces along the symmetric stretch coordinate for the linear configuration. Many of the excited state curve intersections become avoided crossings when the symmetry is lowered to C 2v. The calculations of Boldyrev et al.2for the IE of MgOMg were in excellent agreement with the present experimental result, but we briefly reexamined the properties of MgOMg+˜Xdue to the diffi- culties reported in the earlier study. Boldyrev et al.2noted that the ab initio methods that they had applied produced broken symme- try structures and/or unphysical vibrational constants. The meth- ods used included coupled cluster calculations [CCSD(T)] with the Pople-style basis set 6-311+G(2df). In the present study, we observed symmetry-breaking for CASSCF/cc-pCVQZ calculations, but this problem was not encountered for the CCSD(T) method. The lat- ter predicted a symmetric linear equilibrium geometry with a bond length of 1.791 Å and harmonic vibrational constants ω1= 486 cm−1, ω2= 108 cm−1, andω3= 1038 cm−1. The IE for this level of theory was 53 330 cm−1(6.61 eV). DISCUSSION The spectra recorded in this study provide limited infor- mation for the electronic ground state of MgOMg as all of the rotationally resolved bands originated from the zero-point level. However, the simple P- and R-branch rotational structure offerssome useful insights. Only even rotational levels were observed, con- sistent with the computationally predicted1Σ+ gsymmetry for the ground state. The other possible candidate for the ground state, 3Σ+ u, was definitively eliminated, as was the case for BeOBe.8The ground state rotational constant was found to be 0.1095(10) cm−1, in agreement with the theoretical prediction (0.1088 cm−1).4 The vibronic bands of the ˜A−˜Xsystem were blue-shaded, with upper state rotational constants in the range of 0.128 cm−1– 0.137 cm−1. The theoretical calculations indicate that the Mg–O bond length is almost unchanged on electronic excitation, so the increase in the rotation constant is most probably associated with the change in the equilibrium bond angle. For the equilibrium geom- etry of the ˜Astate, the rotational constants are A = 0.8569 cm−1, B = 0.1800 cm−1, and C = 0.1487 cm−1. The asymmetry parame- ter isκ=−0.91, corresponding to a near-prolate top. The absence of Q-branch lines indicates that K a= 0 for the optically active lev- els. Hence, the upper state rotational energies can be approximated using the expression E ROT(J′) =¯BJ′(J′+ 1), where ¯B= (B + C)/2. The equilibrium value for ¯Bis 0.164 cm−1, while the measured values of near 0.13 cm−1correspond to vibrationally averaged bond angles of ∼115○. The progression of bending levels observed for the ˜Astate is consistent with a bent excited state, and the spectrum is most intense around 22 000 cm−1, which is about 1790 cm−1lower than the pre- dicted vertical excitation energy. The lowest energy band recorded was about 1000 cm−1below the most intense part of the spec- trum, suggesting that the depth of the bending potential exceeds this value. The measured intervals between successive bending lev- els are plotted against a vibrational index ( m) in Fig. 4. This plot exhibits a diminishing interval indicating that the observed levels are located below the barrier to linearity. This would put the low- est energy observed level at least 1470 cm−1below the barrier. The one-dimensional bending potential from the ab initio calculations is substantially deeper (3026 cm−1), but there is no reason to assume that the m= 1 level is the zero-point. However, the energy level pat- tern derived from the PEC shown in Fig. 6 differs markedly from the observed properties. The lower energy calculated vibrational inter- vals are much greater than the observed values, and the changes in the intervals with increasing vibrational excitation are more dra- matic. These discrepancies can be somewhat reduced by empirically adjusting the barrier height and shape of the potential, but all of the physically reasonable double-minimum potentials examined yielded interval variations near the barrier that were inconsistent with the data. Obvious limitations of this analysis are the use of a one- dimensional model and the neglect of interactions between closely nested electronically excited states. Programs for solving the three- dimensional Hamiltonian are readily available (e.g., DVR3D31), but this requires the generation of a reliable potential energy surface. Given the symmetry-breaking and state mixings encountered in the present ab initio calculations, we do not consider the results to be sufficiently reliable for this task. Finally, we consider the possibility that some of the unassigned emission bands reported by Pesic and Gaydon13originated from Mg 2O. The comparison between the present spectra with those of Pesic and Gaydon13is complicated by the very different conditions of the experiments. Pesic and Gaydon13used an electric arc dis- charge source consisting of Mg electrodes in the presence of about J. Chem. Phys. 153, 054308 (2020); doi: 10.1063/5.0020431 153, 054308-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 10 Torr of O 2. Hence, their spectra would have originated from a large number of excited state levels, including many that would not have been accessible in our low-temperature laser excitation spectra. For the energy range where the two datasets overlap, we could not find convincing evidence of correlations. ACKNOWLEDGMENTS This work was supported by the National Science Foundation under Grant No. CHE-1900555 and the Army Research Office under Grant No. W911NF-15-1-0121. We thank Dr. Michael N. Sullivan for his assistance in recording the initial laser induced fluorescence spectra. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1A. I. Boldyrev, I. L. Shamovskii, and P. v. R. Schleyer, J. Am. Chem. Soc. 114, 6469 (1992). 2A. I. Boldyrev, J. Simons, and P. v. R. Schleyer, Chem. Phys. Lett. 233, 266 (1995). 3A. I. Boldyrev and J. Simons, J. Phys. Chem. 99, 15041 (1995). 4B. Ostojic, P. R. Bunker, P. Schwerdtfeger, B. Assadollahzadeh, and P. Jensen, Phys. Chem. Chem. Phys. 13, 7546 (2011). 5B. Ostojic, P. R. Bunker, P. Schwerdtfeger, A. Gertych, and P. Jensen, J. Mol. 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T. Deng, Y. Okada, M. Foltin, and A. W. Castleman, J. Phys. Chem. 98, 9350 (1994). 18S. M. Bresler, J. R. Schmitz, M. C. Heaven, and R. W. Field, J. Mol. Spectrosc. 370, 111293 (2020). 19M. A. Duncan, Rev. Sci. Instrum. 83, 041101 (2012). 20C. M. Western, J. Quant. Spectrosc. Radiat. Transfer 186, 221 (2017). 21M. C. Heaven, Phys. Chem. Chem. Phys. 8, 4497 (2006). 22J. M. Merritt, V. E. Bondybey, and M. C. Heaven, Phys. Chem. Chem. Phys. 10, 5403 (2008). 23W. J. Balfour and A. E. Douglas, Can. J. Phys. 48, 901 (1970). 24H. Knoeckel, S. Ruehmann, and E. Tiemann, Eur. Phys. J. D 68, 293 (2014). 25G. Herzberg, Infrared and Raman Spectra of Polyatomic Molecules (Van Nos- trand Reinhold, 1945). 26B. P. Prascher, D. E. Woon, K. A. Peterson, T. H. Dunning, Jr., and A. K. Wilson, Theor. Chem. Acc. 128, 69 (2011). 27K. E. Yousaf and K. A. Peterson, Chem. Phys. Lett. 476, 303 (2009). 28H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, and M. Schütz, Wiley Interdiscip. Rev.: Comput. Mol. 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5.0013393.pdf

J. Chem. Phys. 152, 244704 (2020); https://doi.org/10.1063/5.0013393 152, 244704 © 2020 Author(s).First-principles investigation of a new 2D magnetic crystal: Ferromagnetic ordering and intrinsic half-metallicity Cite as: J. Chem. Phys. 152, 244704 (2020); https://doi.org/10.1063/5.0013393 Submitted: 13 May 2020 . Accepted: 04 June 2020 . Published Online: 23 June 2020 B. G. Li , Y. F. Zheng , H. Cui , P. Wang , T. W. Zhou , D. D. Wang , H. Chen , and H. K. Yuan The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp First-principles investigation of a new 2D magnetic crystal: Ferromagnetic ordering and intrinsic half-metallicity Cite as: J. Chem. Phys. 152, 244704 (2020); doi: 10.1063/5.0013393 Submitted: 13 May 2020 •Accepted: 4 June 2020 • Published Online: 23 June 2020 B. G. Li,1Y. F. Zheng,1H. Cui,2,3,a)P. Wang,1T. W. Zhou,1D. D. Wang,1H. Chen,1and H. K. Yuan1,a) AFFILIATIONS 1School of Physical Science and Technology, Southwest University, Chongqing 400715, China 2Shaanxi Key Laboratory of Industrial Automation, Shaanxi University of Technology, Hanzhong 723001, China 3School of Mechanical Engineering, Shaanxi University of Technology, Hanzhong 723001, China a)Authors to whom correspondence should be addressed: hongcui@snut.edu.cn and yhk10@swu.edu.cn ABSTRACT The development of two-dimensional (2D) magnetic materials with half-metallic characteristics is of great interest because of their promising applications in spintronic devices with high circuit integration density and low energy consumption. Here, by using density functional theory calculations, ab initio molecular dynamics, and Monte Carlo simulation, we study the stability, electronic structure, and magnetic properties of a OsI 3monolayer, of which crystalline bulk is predicted to be a van der Waals layered ferromagnetic (FM) semiconductor. Our results reveal that the OsI 3monolayer can be easily exfoliated from the bulk phase with small cleavage energy and is energetically and thermodynamically stable at room temperature. Intrinsic half-metallicity with a wide bandgap and FM ordering with an estimated TC= 35 K are found for the OsI 3 monolayer. Specifically, the FM ordering can be maintained under external biaxial strain from −2% to 5%. The in-plane magnetocrystalline anisotropy energy of the 2D OsI 3monolayer reaches up to 3.89 meV/OsI 3, which is an order larger than that of most magnetic 2D materials such as the representative monolayer CrI 3. The excellent magnetic features of the OsI 3monolayer therefore render it a promising 2D candidate for spintronic applications. Published under license by AIP Publishing. https://doi.org/10.1063/5.0013393 .,s I. INTRODUCTION Two-dimensional (2D) electronic materials have attracted much interest from researchers since the discovery of graphene in 20041because of their advantages that are essential in next- generation information processing owing to their 2D structures and novel electronic properties.2–4To date, a large class of 2D materi- als has been theoretically proposed and/or successively synthesized. These materials include graphene-like single elemental phospho- rene,5,6antimonene,6borophene,7,8silicene,9,10germanene,11and stanene,12as well as metal-free compounds such as h-BN,13g- C3N4,14h2D-C 2N,15–17and BCN.18,19Other 2D materials include transition metal-dichalcogenides (TMDs),20–22metal-nitrides, car- bides (MXenes),23and transition metal-halides (TMHs).24,25It is fas- cinating and intriguing that these 2D materials exhibit a diversity of physical properties for potential applications ranging from sensingand detection to electronics. Nanoscale materials that simultane- ously possess half-metallicity, large magnetic moments, and strong magnetic anisotropy are promising electrode materials for mag- netic tunnel junctions (MTJs) in spin transfer torque magnetic ran- dom access memory (STT-MRAM) and spin valves in spintronic devices.25–31Unfortunately, most of these pristine 2D materials are intrinsically nonmagnetic or only weakly magnetic, and thus unable to satisfy all the prerequisite requirements in one system. The appli- cations of 2D materials in spintronic devices are hence largely lim- ited. This has driven the scientific community to move forward in seeking new 2D magnetic materials with desirable features via the bottom-up approach.32–42 The Mermin–Wagner theorem43implies that thermal fluctua- tions prohibit long-range magnetic order in an isotropic 2D mag- netic system at finite temperatures unless the system is magneti- cally anisotropic so that an energy barrier exists against thermal J. Chem. Phys. 152, 244704 (2020); doi: 10.1063/5.0013393 152, 244704-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp fluctuations. Encouragingly, strong magnetic anisotropy can persist in magnetic van der Waals (vdW) materials down to a single mono- layer.44,45Several prototypical 2D-vdW layers have been experi- mentally validated in just the past three years, including RuCl 3,34 CrI 3,35Cr2Ge2Te6,36Fe3GeTe 2,37,38TMPX 3(TM = Fe, Ni; X = S, Se),46–48and TMSe 2(TM = V, Mn).49,50The successful experimen- tal realization of ferromagnetic (FM) 2D-vdW magnets has not only brought forth new ideas for the development of new 2D-vdW mag- netic materials but also offered an exciting new platform for 2D- vdW spintronics.25,32As the magnetic anisotropy originates from the spin–orbit coupling (SOC) interaction51or strong sp-dorbital hybridization between the bonded magnetic atoms,52a reasonable and essential way to develop new magnetic 2D-vdW materials is to reexamine the materials in which large magnetic anisotropy may be present. For example, it has been suggested that sizable mag- netic anisotropy and thus resultant magnetic moment can arise from the exchange anisotropy in CrI 3monolayers due to the SOC inter- action of I atoms that mediate the super-exchange between the Cr atoms.45,53 TM-trihalide OsI 3, an isostructural compound of TMHs 3,24,54,55 may offer a critical advantage for achieving sizable magnetic anisotropy and thus stable magnetic ordering in the crystalline bulk and in monolayers because both the heavy I and Os atoms are intrin- sically related to the remarkable SOC effect that may reduce the spin fluctuations. Furthermore, the dorbitals of the Os atom are partially filled with two more valence electrons compared to the Cr atom in the well-known CrI 3monolayer.35This may alter the band structure around the Fermi level and give rise to distinctive electronic struc- tures. In this work, we demonstrate the existence of a stable OsI 32D crystal on the basis of first-principle calculations as confirmed by energetic, dynamical, and thermal analyses. Our calculations sug- gest that monolayer OsI 3possesses ferromagnetic properties with strong magnetic anisotropy and modest Curie temperatures. More- over, the monolayer exhibits a half-metallic type of conductivity with 100% spin polarization around the Fermi level and has a wide bandgap. II. COMPUTATIONAL METHODS All calculations including the SOC effect were performed by using the Vienna ab initio Simulation Package (VASP) based on spin-polarized density functional theory (DFT).56–59The valence electrons of Os (5 d66s2) and I (5 s25p5) atoms were explicitly treated by projected augmented plane wave (PAW) pseudopotentials,60,61 for which a plane-wave basis set with a cutoff energy of 500 eV was selected. The Perdew–Burke–Ernzerhof (PBE) parameterization of the generalized gradient approximation (GGA)62,63was used to describe the exchange and correlation functions. In addition, we have included the vdW interaction contributed by dispersive forces (DFT-D2) proposed by Grimme.64The Monkhorst–Pack k-point sampling was performed with 12 ×12×1 and 6 ×6×3 grids for the monolayer and crystalline bulk, respectively. Bader charge anal- ysis was performed to obtain the charge transfer in the monolayer. The lattice constants and atomic coordinates were fully relaxed with a convergence criterion of 10−6eV for the self-consistent field (SCF) procedure. The maximum residual force on each atom was less than 0.001 eV Å−1.The electron correlation effect is important for TM atoms because of certain localizations of their delectrons in magnetic materials.65To better describe the strongly correlated 5 delectrons of Os atoms, the GGA plus Hubbard correction U (GGA+D2+U) method was employed.66,67The on-site Coulomb parameter U and the exchange parameter J can be combined into one parame- ter U eff= U−J. Here, U eff= 3.2–0.7 = 2.5 eV was adopted in our calculations.53,68Test calculations with different U = 2.3 eV –4.7 eV but the fixed J = 0.7 eV have been performed to vali- date our main results. As we expected, it has been found that the lattice constant, magnetic moment, and band structures change very little with U values. The OsI 3sheet was designed as a slab model within a 15 Å thick vacuum layer added in the direction normal to the hexagonal plane of the OsI 3layer to avoid inter- layer interaction due to adjacent imaging. An ab initio molecu- lar dynamics (AIMD) simulation under a statistical ensemble with a fixed particle number, volume, and temperature (NVT, 300 K) and phonon dispersion calculations with the density functional perturbation theory (DFPT) method were performed to evalu- ate the thermodynamic and dynamical stability, respectively. The Gamma point for the k-space mesh and periodic boundary condi- tions were employed for the AIMD simulations. The lattice dynam- ics were calculated by the PHONOPY code, as implemented in VASP.69 III. RESULTS AND DISCUSSION A. Crystalline bulk and cleavage energy Stable 2D monolayers can be bonded to one another through the weak vdW interaction to form a layered crystallographic phase. Members of the TM trihalide family TMHs 3(TM = Sc, Ti, V, Cr, Fe, Y, Mo, Ru, Rh, Ir; H = Cl, Br, I) have been found to adopt either the monoclinic AlCl 3phase ( C2/m) or the rhombohedral BiI 3 phase ( R¯3).24Although bulk crystalline OsI 3has not yet been exper- imentally synthesized, it is very likely to have similar structures to the other members of the family and possess C2/mandR¯3 phases. To make a systemic comparison of more structural types, another phase with the trigonal space group P3112 structure corresponding to the high-temperature phase of CrCl 3was also calculated. These 3D phase structures are formed by stacking the hexagonal honey- comb OsI 3sheets in different patterns (shifted along one of the in- plane directions). The OsI 3monolayer is a hexagonal honeycomb structure with one Os atom layer sandwiched between two I atom layers. After optimizing the lattice parameters and atomic positions under the constraint of symmetry for these considered structural phases, we found that R¯3 and P3112 phases have the lattice con- stants a=b= 6.81 Å, c= 18.99 Å and a=b= 6.75 Å, c= 19.09 Å, respectively. The C2/mphase is energetically more stable than R¯3 andP3112 phases by 0.03 eV and 0.09 eV/OsI 3, respectively. The top and side views of the C2/mstructure are shown in Fig. 1. The optimized lattice parameters are a= 6.80 Å, b= 6.84 Å, c= 11.69 Å, and β= 109○. These parameters are similar to a= 6.87 Å, b = 6.98 Å, c= 11.89 Å, and β= 109○of bulk monoclinic CrI 3.54Intra- layer ferromagnetic ordering and inter-layer ferromagnetic coupling were found. The structure possesses a magnetic moment of 0.87 μB (μs Os= 0.84 μBand μs I= 0.03 μB) and thus has a 3D ferromagnetic J. Chem. Phys. 152, 244704 (2020); doi: 10.1063/5.0013393 152, 244704-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . Top and side views of the structures of (a) OsI 3 monolayer and (b) crystalline bulk (C2/ mspace group), where a sheet of Os atoms is sandwiched between I atom sheets, and green and gray balls represent Os and I atoms, respectively. Unit cells are represented by the rectangles. (c) Exfoliation process of the monolayer from the bulk structure. ordering with a bandgap of ∼0.5 eV. The corresponding vdW gap was found to be 3.42 Å, which is close to the values in CrI 3(3.49 Å)54and RuI 3(3.35 Å),70and indicates the weak vdW interaction between the layers. Although there are two in-plane Os–Os distances of 3.93/3.89 Å, the honeycomb nets are nearly undistorted because of the small differences between these distances. This is unlike the heavier TM compounds MoCl 3and TcCl 3in which one of the two TM–TM distances is short and comparable to the bond length in the TM bulk. In these materials, the honeycomb net is broken into TM– TM dimers and the TM–TM distance difference is approximately one Angstrom.71 The most common approaches for obtaining 2D monolayers from the vdW crystalline phase are mechanical cleavage and liquid exfoliation. Regardless of the method used, the cleavage energy is a critical quantity for estimating how easily 2D monolayers can be obtained. Note that different stacking sequences (structural phases) of the same compound may result in small differences in the cleav- age energy due to small differences in the inter-layer distances and weak van der Waals interactions. Thus, we calculated the cleav- age energy of the C2/mstructure by using the approach named the “rigorous method” [depicted in Fig. 1(c)],72Ec(m) = [ Elayer(m) −Ebulk×m/n]/S, where Elayer(m) is the total energy of an iso- lated slab with mlayers in vacuum, Ebulk is the total energy of the unit cell of the bulk material composed of nlayers, and Sis the in-plane area. This method is different from the traditional “slab method”73,74because it can deal with complicated situations in a simple manner and does not necessarily consider the surface relaxation, in-plane lattice change, and structural reconstruction but only requires the energies of the crystalline bulk and isolated layer unit cells. Our calculated cleavage energy of 0.36 J/m2for the OsI 3 layer is equal to 0.36 J/m2in CrI 3,54,55RuI 3,70and graphene75and quite comparable to 0.33 J/m2–0.35 J/m2in TiI 3and VI 3.76There- fore, OsI 3monolayers can be easily exfoliated from the bulk phase. This indicates the possibility of experimentally producing 2D crystal layers.B. Monolayer structure and stability In the OsI 3monolayer, there are six I and two Os atoms in each unit cell, and the corresponding optimal lattice constants are a= 6.870 Å and b= 6.928 Å. This configuration belongs to the hexag- onal honeycomb structure with space group C2/ m(Fig. 1). The two Os–I bond lengths are 2.689 Å and 2.725 Å. The Os–Os distances are 3.969 Å and 3.996 Å. Note that TMI 3(TM = Ru, Cr, Ir, V, and Ti) are in the D3dconfiguration54,55,70,76but RuCl 3is in the C2/ mconfigu- ration.77Although the D3dconfiguration with the lattice parameters a=b= 6.973 Å, Os–I bond length of 2.701 Å and Os–Os dis- tance of 4.026 Å, was also found to be stable, it is 0.29 eV higher in energy. We have calculated the phonon dispersion spectra of the D3dand C2/ mconfigurations to confirm the lowest energy struc- ture of the OsI 3monolayer. The results show that a small imaginary frequency appears at the gamma point for the D3dconfiguration but not for the C2/ mconfiguration. To evaluate the sensitivity of the magnetic state to the lattice constant, we applied an equibiaxial compression and expansion along the aandbaxes and considered both ferromagnetic (FM) and antiferromagnetic (AFM) orderings. The strain is defined as ε= (a−a0)/a0×100% = ( b−b0)/b0 ×100%, where a(b) and a0(b0) are the unit cell lattice constants with and without tension, respectively. Here, positive and negative εrepresent tensile and compressive strain, respectively. In Fig. 2, the more stable FM state compared to the AFM state around the equilibrium position validates our conclusion that the lattice con- stant is optimal and the FM ordering is more preferable (inset chart). For all values of compressive and tensile strains, the OsI 3unit cell rigidly maintains a magnetic moment of 2 μB. The magnetic phase transition from a FM to an AFM phase under a large compressive strain is consistent with the behavior of monolayer CrI 3observed by Zheng et al .78 To investigate the thermodynamic stability of the OsI 3sheet, we calculated the cohesive energy Ec= [2EOs+ 6EI−E(unit)]/8 defined as the energy difference between the total energy of the OsI 3unit cell J. Chem. Phys. 152, 244704 (2020); doi: 10.1063/5.0013393 152, 244704-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Uniform contraction and expansion along the aand baxes of unit cell in FM and AFM ordering. The strain is expressed by ε= (a−a0)/a0×100% = ( b −b0)/b0×100%. The strain energy is relative to the lowest value of the equivalent unstrained structure. The inset chart shows the energy of FM ordering relative to AFM ordering at each value of specific strain. and the summed energy of the isolated Os and I atoms. The obtained Ec= 3.18 eV/atom of the 2D OsI 3sheet is larger than 2.45 eV/atom in CrI 355and 2.52 eV/atom in MoS 2,79indicating that OsI 3is ener- getically more stable than these realistic 2D materials. Thus, 2D OsI 3 sheets can be synthesized in the future experiments. We studied the lattice dynamics of the OsI 3sheet by calculat- ing its phonon dispersion. The phonon band structure and partial density of states (PDOS) are presented in Fig. 3. Similar to most 2D materials,80three distinct acoustic modes with low frequencies (less than 50 cm−1) can be clearly discerned in the phonon spec- trum. However, the in-plane longitudinal and transverse modes as FIG. 3 . Phonon spectrum curve of OsI 3sheet along a path from G (0, 0) to K ( −1/3, 2/3) to M (0, 1/2) to G in the two-dimensional Brillouin zone (left panel) and phonon partial density of states (PDOS) (right panel).well as the out-of-plane (ZA) mode present linear dispersions near the G point, which differ from the quadratic dispersion of the ZA mode presented by CrI 354,55and RuI 3monolayers.70The absence of imaginary modes in the entire Brillouin zone confirms that the OsI 3 sheet is dynamically stable and can exist as a freestanding 2D crystal. We note from the total and atom-resolved phonon PDOS that con- tributions from both the Os and I atoms are significant in the whole dispersion region. To examine the thermal stability of the OsI 3sheet, AIMD sim- ulations on an OsI 33×3 supercell (72 atoms in hexagonal config- uration) and a (3 ×2√ 3)R30○supercell (96 atoms in rectangular configuration) were performed under the NVT ensemble. A large supercell was used to reduce the constraints of the periodic boundary conditions and to provide more possibilities for structural recon- struction. After heating at room temperature (300 K) for 5 ps with a time step of 1 fs, no structural reconstruction occurred in both of our constructed supercells. It was found that each Os atom held a magnetic moment of around 1 μBduring the simulation. The atomic configuration snapshot of the 3 ×2√ 3R30osupercell at the end of the AIMD simulation in the inset of Fig. 4 shows that the super- cell maintained its initial rectangular lattice structure. This suggests that the monolayer is magnetically and thermally stable at room temperature. C. Magnetic properties Having confirmed the stability of the OsI 3sheet, we now inves- tigate its magnetic properties. To confirm the magnetic ground state, FM ordering and various AFM orderings (AFM-Neel, AFM-Zigzag and AFM-Stripy types shown in Fig. 5) were systematically exam- ined using the (1 ×√ 3)R30○rectangular configuration. The results demonstrate that the FM ordering is more stable than the AFM- Neel, AFM-Zigzag, and AFM-Stripy orderings by energy differences of 21.51 meV, 11.20 meV, and 13.96 meV, respectively. The rect- angular cell containing four Os and twelve I atoms presents a spin magnetic moment of 4 μB(μs Os= 0.938 μB, and μs I=−0.004/0.028 μB). FIG. 4 . AIMD simulation results for the total energy and temperature of OsI 3sheet with a ( 3×2√ 3)R30○structural cell (96 atoms in rectangular configuration). The inset charts are the side- and top-view snapshots of the atomic configuration at the end of the simulation. J. Chem. Phys. 152, 244704 (2020); doi: 10.1063/5.0013393 152, 244704-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . Magnetic configurations of OsI 3 sheet: (i) FM, (ii) AFM-Neel, (iii) AFM- zigzag, and (iv) AFM-stripy orderings. (v) Magnetic coupling interactions. Brown arrows represent the spin directions of the Os atoms. The rectangular cell used in the calculations is shown with dashed lines. The weak magnetic moment on the I atoms is associated with the spin polarization of 6 pelectrons induced by the Os atoms via strong Os–I bonding hybridization. When SOC was included in the non-SCF calculations, the spin magnetic moments were slightly changed. However, large orbital magnetic moments of 0.924 μB(μl Os= 0.211 μBand μl I= 0.004/0.008 μB) for [001] magnetization (normal to Os hexagonal plane), 0.818 μB(μl Os= 0.187 μBand μl I= 0.004/0.009 μB) for [010] magnetiza- tion, and 0.353 μB(μl Os= 0.076 μBand μl I= 0.003/0.01 μB) for [100] magnetization were discerned for this rectangular cell. This means that the orbital moment can contribute to almost 20% of the total magnetic moment. Comparing the total energies of the three magne- tizations, we found that the magnetic moment along the [010] direc- tion is more stable than the other directions. This indicates that the easy axis is along the [010] direction. By calculating the energy dif- ference between these magnetizations, i.e., the magnetic anisotropy energy, we obtained the in-plane (out-of-plane) magnetic anisotropy energy of −3.89 meV/OsI 3. Note that monolayer CrI 3was experi- mentally found to be ferromagnetic with an out-of-plane easy axis,35 and theoretical calculations have given a magnetic anisotropy energy of 0.69 meV/CrI 3.54,55,78 Since Os atoms are indirectly bonded in the OsI 3monolayer and the Os1–Os2 distance is relatively large, the direct-exchange interaction between them is expected to be rather weak even if it is not disregarded. Noting that an Os atom has five residual valence electrons and a spin magnetic moment of nearly 1 μB, the large energy separation between the closed Os- t↑ 2gsubshell and the empty Os- e↑ gsubshell suggests that the Os- t↓ 2gsubshell is occupied partially by two spin-down electrons (Fig. 6). If direct-exchange interaction between the Os1 and Os2 atoms does occur, the FIG. 6 . Partial density of states (PDOS) of I atoms (top-panel) and Os atoms (bottom-panel) in FM OsI 3sheet. The dashed black line indicates the Fermi level. Positive and negative PDOS represent the spin-up and spin-down channels, respectively. x′,y′,z′correspond to the crystallographic axis directions of OsI 3, while x,y,zcorrespond to the octahedral ligand field directions. J. Chem. Phys. 152, 244704 (2020); doi: 10.1063/5.0013393 152, 244704-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp spin-up electrons of the Os1- t3↑ 2gsubshell will hop to the Os2- t2↓ 2gspin- down subshell instead of the empty Os2- e0↑↓ gsubshell because of the orbital symmetry constraint (and also not to the Os2- t3↑ 2gspin-up subshell that is fully occupied). This hopping would lead to a neg- ative sign of the direct-exchange interaction and result in AFM cou- pling between the Os1 and Os2 atoms. Clearly, this is inconsistent with our DFT results of FM coupling in the OsI 3monolayer, which hence demonstrates that direct-exchange does not play a decisive role in the magnetic coupling. Therefore, the magnetic interaction may be through long-range coupling interactions such as the super- exchange, double-exchange, or Ruderman–Kittel–Kasuya–Yosida (RKKY) mechanisms. Previously, Wang et al .81have proposed that the FM ordering in a CrI 3monolayer is due to the super-exchange interaction via the orthogonal channels (near 90○Cr–I–Cr bonds) of the ( x2−y2)−(px,py)−(x2−y2) orbitals and ( x2−y2)−px −xyorbitals. Similar to the CrI 3monolayer, the Os atoms in the OsI 3monolayer can be FM-coupled through the shared coordinat- ing I anions (the Os–I–Os angle is 93○/94○) via the super-exchange interaction. In addition to the super-exchange interaction, the double- exchange interaction also seems appreciable. Under the Os–I–Os2 bonding interaction, the minority spin orbital I- p1↓ zoverlaps signif- icantly with the minority spin orbitals Os- t2↓ 2g(Fig. 6), forming a double-exchange interaction “green channel.” When an I- p1↓ zelec- tron is transferred to the Os2- t2↓ 2gorbitals to fully fill these three-fold degenerate orbitals and yield an empty I- p0↓ zorbital, one of the two Os1- t2↓ 2gelectrons simultaneously jumps to the empty I- p0↓ zorbital. This virtual process completes the FM double-exchange transition from t2↓ 2g(Os1)- p1↓ z(I)-t2↓ 2g(Os2) to t1↓ 2g(Os1)- p1↓ z(Os1)- t3↓ 2g(Os2) with- out the cost of any energy. To verify the RKKY mechanism, we calculated the exchange coupling constant in terms of the approximate relations Jex =−1 2nEex/S2andEex=EFM−EAFM, where n= 3 is the number of the nearest neighbor exchange pairs, the factor of 1/2 avoids dou- ble counting the exchange pairs, and S= 1/2 is the spin moment. We plot Eexin the inset of Fig. 2 as a function of the lattice constant (the Os–Os distance ris equivalent to a different lattice constant). The displayed variation suggests that it is unlikely to be due to a RKKY-type interaction because Jexshould exhibit a damped oscillatory behavior with the distance runder the RKKY interaction. To confirm the coexistence of AFM direct interaction, FM super-exchange, and FM double-exchange, we analyzed the vari- ation of Eexas a function of the lattice constant. When the lat- tice constant is expanded, we found that the Os–Os bond length is slightly increased but the Os–I bond length is barely changed. This will decrease the AFM direct interaction between the Os atoms but has a negligible effect on the FM indirect interaction through Os–I– Os bonding. It explains why FM coupling becomes more favorable with the increase in lattice constant, as shown in Fig. 2. In con- trast, when the lattice constant (Os–Os distance) is heavily com- pressed, the Os–Os direct interaction will play a dominant role in the AFM ordering. It can be seen from Fig. 7 that there is a close relation between the magnetic moments of the I and Os atoms, i.e., the spin polarization of the I atoms is dependent on that of the Os atoms. FIG. 7 . Magnetic moments contributed by Os-5 dand I-5 porbitals as functions of the lattice constant (Os–Os distance) in the unit cell. Strain (biaxis) values repre- sent the uniform contraction and expansion of the unit cell around the equivalent structure. D. Curie temperature We now study an important magnetic parameter, the Curie temperature ( TC), because it determines the thermal threshold for the magnetic ordering of the system to be disturbed from ferro- magnetic to paramagnetic and is an important criterion for judg- ing the potential application of magnetic materials. To precisely evaluate TC, one should consider the magnetic exchange coupling interactions between the non-equivalent magnetic atoms as much as possible. Because the calculated magnetic anisotropy energy is comparable to those reported for many 2D Ising-like ferromagnets, for example, monolayer CrI 3,82–85theTCof the OsI 3monolayer can be estimated by statistical Monte Carlo (MC) simulations based on the Ising model. In our calculations, we considered the mag- netic exchange interactions between the nearest, second-nearest, and third-nearest neighboring Os atoms, which are denoted by J1,J2, and J3, respectively. To obtain these exchange-coupling parame- ters, four types of magnetic configurations (FM, AFM-Neel, AFM- Zigzag, and AFM-Stripy) were examined in Fig. 5. Based on the Ising model, the spin Hamiltonian for the 2D honeycomb lattice can be written as Hspin=−J1∑ ijSiSj−J2∑ klSkSl−J3∑ mnSmSn. Here, Jis the magnetic exchange coupling parameter, Sis the spin quantum number of the magnetic Os site, and ( i,j), (k,l), and (m,n) stand for the nearest, second-nearest, and third-nearest Os sites, respectively. By mapping the DFT energies of various mag- netic states to the above Ising spin Hamiltonian, J1,J2, and J3can be calculated by the following equations:86 EFM/Neel =E0−(±3J1+ 6J2±3J3)∣⃗S∣2, EZigzag/Stripy =E0−(±J1−2J2∓3J3)∣⃗S∣2. J. Chem. Phys. 152, 244704 (2020); doi: 10.1063/5.0013393 152, 244704-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Taking S= 1/2 for the Os atoms from the DFT calculations, we calculated J1,J2, and J3to be 3.03 meV, 0.23 meV, and 0.55 meV, respectively. These positive values indicate that the magnetic cou- pling between the nearest, second-nearest, and third-nearest neigh- boring Os–Os pairs are indeed stable FM orderings. Previously, Wang et al .81found that the J1andJ2of a CrI 3monolayer are pos- itive, while J3is negative and much smaller than the former two in magnitude. With the obtained magnetic exchange parameters and mag- netic moments of each magnetic site, we performed MC simulations using the Metropolis algorithm to estimate TC. The MC simulations were run for 1 ×105loops on a 90 ×90×1 2D honeycomb supercell, which is large enough to minimize the constraints from the periodic boundary conditions. In Fig. 8(a), we plot the simulated magneti- zation per Os atom as a function of temperature. On the whole, the spin magnetic moments of Os atoms tend to be parallel (FM state) at low temperatures, producing a net magnetic moment of 1 μB/Os. The spin magnetic moments are aligned randomly, result- ing in net zero magnetization (AFM state) at high temperatures. The FIG. 8 . (a) Simulated magnetic moment per Os atom and (b) heat capacity Cvas functions of the temperature in the OsI 3monolayer. The nearest J1, second J2, and third J3exchange interaction are included for the Monte Carlo (MC) simulations and Curie temperature TCestimation.TABLE I . The relative energy ( ΔE, meV) of different AFM states with respect to the FM state for the (1 ×√ 3)R30○rectangular cell. TCwas obtained by considering the J1,J2, and J3exchange coupling interactions in the MC simulations. Configurations AFM-Neel AFM-Zigzag AFM-Stripy ΔE(meV) 21.51 11.20 13.96 Exchange coupling J1 J1+J2 J1+J2+J3 TC 20 K 22 K 35 K magnetic transition to zero net moment occurs at 20 K when only magnetic coupling between the nearest neighboring Os atoms was considered and at 35 K when all three coupling interactions were included (it can be seen from Table I). This suggests that the cou- pling interactions between the nearest neighboring and between the third-nearest neighboring Os atoms have equally important contri- butions to TC. In addition, the TCobtained from the heat capacity curve Cv=∂E/∂T[Fig. 8(b)] is consistent with the TCobtained from the magnetization curve. Although the TCof the OsI 3sheet is far below room temperature, the prospect for its applications in spin- tronics is still fascinating because TCcan be tuned by methods such as strain, electron/hole doping, and electric fields. Indeed, the TC of a CrI 3monolayer was predicted to steadily increase from 75 K to 150/300 K upon half-electron/half-hole doping.81TheTCof the hexagonal MnN monolayer has been tuned by equibiaxial strains.87 E. Electronic structure In Fig. 9, we display the spin-polarized band structure (left panel) and the spin-polarized total density of states (TDOS) of the OsI 3sheet (right panel). The black and red lines represent the spin- up (majority) and spin-down (minority) channels, respectively, and the dashed green line denotes the Fermi level ( Ef). It is clear that the minority states cross Efand exhibit metallic characteristics, while the majority states show a remarkable bandgap (1.83 eV) around Ef and exhibit insulating characteristics. Therefore, the OsI 3sheet has intrinsic half-metallicity and displays 100% spin polarization near Ef without any artificial modification. There is a bandgap of about 1 eV between the valence band maximum (VBM) at the M-point of the minority states and the conductor band minimum (CBM) between the G- and K-points of the majority states. To analyze the bonding interaction and the orbital contribu- tions to the band structure, the partial densities of states (PDOSs) are projected onto the individual orbitals of the Os and I atoms and are shown in Fig. 6. In a wide energy window ranging from −3 eV to 3 eV, the PDOSs of the Os and I atoms overlap sig- nificantly. The coincidence of the peak positions and gaps at the same energy levels suggests a strong Os–I bonding interaction. Bader charge analysis shows that each Os atom has lost 0.58 electrons and each I atom has attracted 0.18/0.22 electrons. Thus, there is a strong ionic bonding interaction between the Os and I atoms. Because the Os atoms are located in an octahedral crystal field coor- dinated by six I atoms, the five suborbitals of the 5 dshell should split into two groups, namely, a lower energy t2g-subshell ( dxy,dyz, dxz) atEfand a higher energy egsubshell ( dz2,dx2−y2) atE= 2 eV. However, the honeycomb lattice of the Os atoms has two in-plane Os–Os distances (3.969 Å and 3.996 Å), i.e., the Os atoms are in J. Chem. Phys. 152, 244704 (2020); doi: 10.1063/5.0013393 152, 244704-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 9 . Spin-polarized band structure (left panel) and spin- polarized total density of states (TDOS) of the OsI 3mono- layer (right panel). The black and red lines represent the spin-up (majority) and spin-down (minority) states, respec- tively, and the green dashed lines indicate the Fermi level. a pseudo-octahedral environment with C2hsymmetry, which splits the three-fold degenerate t2gorbitals into two peaks at −0.8 eV and Efin the spin-down channel. Note that the x′,y′,z′directions cor- responding to the OsI 3crystallographic axes do not exactly coin- cide with the octahedral ligand field directions in I–Os–I ( x,y,z). This leads to the seemingly “abnormal” results shown in Fig. 6: the t2g-subshell primarily consists of dx′y′(E=−0.8 eV), dx′2−y′2, and dz′2 orbitals ( E= 0 eV), while the eg-subshell primarily consists of dy′z′, dx′z′, and dx′2−y′2orbitals ( E= 2 eV). If the crystallographic axes are rotated to exactly match the octahedral ligand field directions with- out any changes to the relative atomic positions, the PDOS shapes and peak positions will remain the same, but the orbitals compos- ingt2gandegwill change to the “formally expressed orbitals.” In other words, the rotation does not affect the components of the t2g andegorbitals owing to the different expressions of the intrinsic orbitals. There are four points that should be emphasized in Fig. 6 to underline the electronic and magnetic properties of the OsI 3mono- layer: (i) The occupied I- p′ x(pz) orbital and the Os- dz′2,dx′z′,dx′2−y′2 (dxz,dyz) orbitals are strongly hybridized in the spin-down chan- nel at Ef, which contributes to the metallic features and the VBM edge states. The unoccupied I- pand Os- dorbitals are significantly hybridized in both spin channels and thus contribute equally to the CBM edge states at 2.0 eV. (ii) The spin-up/down orbitals of the Os- t3↑ 2g/t2↓ 2gsubshell are fully/partially occupied, resulting in one unpaired delectron contributing 1 μBof spin magnetic moment in the Os atom. (iii) An asymmetrical distribution of two spin-down electrons on the t2↓ 2gsubshell near Efgives rise to a large orbital mag- netic moment of the Os atoms. Although the spin-down I- p′ x(pz) orbital is partially occupied near Ef, its magnetic quantum num- ber ( ml= 0) results in almost zero orbital magnetic moment of the I atoms. (iv) According to second-order perturbation theory,88the dominant contribution of Os atoms to the large in-plane magnetic anisotropy energy stems from electron hopping between the occu- pied and unoccupied spin-down t2↓ 2gorbitals near Efdue to the SOCinteraction via the ⟨dxy↓∣lx∣d↓ xz;d↓ yz⟩(⟨m=−2↓|lx|m=±1↓⟩) coupling matrix elements as well as the electron hopping between the occu- pied t↓ 2gorbitals and unoccupied e↓ gorbitals via the ⟨d↓ xz;d↓ yz∣lx∣d↓ x2−y2⟩ (⟨m=±1↓|lx|m= 2↓⟩) coupling matrix elements; for the nonmetal- lic I atoms, the out-of-plane magnetic anisotropy energy contributed from the ⟨pz↓∣lx∣p↑ x;p↑ y⟩matrix and the in-plane magnetic anisotropy energy contributed from the ⟨pz↓∣lx∣p↓ x;p↓ y⟩matrix approach to each other because of their close energy differences between the occu- pied state ⟨pz↓∣and the unoccupied states ∣p↑ x;p↑ y⟩(∣p↓ x;p↓ y⟩) as well as the comparable PDOS magnitudes, resulting in small net magnetic anisotropy energy of I atoms. IV. CONCLUSION In conclusion, we found a stable structural configuration and unique electronic and magnetic properties of 2D OsI 3monolay- ers using first-principles calculations with the GGA+D2+U method. Based on the analysis of the cohesive energy, phonon dispersions, and AIMD simulation results, we confirmed that a OsI 3monolayer is energetically, dynamically, and thermally stable at room temper- ature. The OsI 3monolayer can be easily exfoliated from the crys- talline bulk owing to the low cleavage energy of 0.36 J/m2, which is equal to that of graphene. The calculations demonstrate that the OsI 3 monolayer is intrinsically half-metallic with a net magnetic moment of 1 μBper formula unit, a typical spin polarization of 100%, and a large pseudodirect bandgap of 1.83 eV near the Fermi energy. Statistical Monte Carlo simulations based on the Ising model eval- uated the magnetic transition temperature TCto be 35 K. Under a large range of biaxial strain, the OsI 3monolayer remains rigidly in the FM ground state and maintains a constant magnetic moment. Importantly, we revealed that the OsI 3monolayer has an in-plane magnetization easy axis and a magnetic anisotropy energy of up to 3.89 meV/OsI 3, which is one order larger than the value in CrI 3monolayers. Finally, the origins of the magnetic properties and J. Chem. Phys. 152, 244704 (2020); doi: 10.1063/5.0013393 152, 244704-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp electronic structure have been comprehensively analyzed to under- line their essential natures. The 2D OsI 3TCmonolayer should be a promising candidate for two-dimensional spintronic applications. ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Grant No. 11574253), the Natural Science Foundation of Chongqing (Grant No. CSTC-2017jcyjBX0035), and the Postgraduates’ Research and Innovation Project of Chongqing (Grant No. CYS18088). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666–669 (2004). 2A. K. Geim and I. V. Grigorieva, “Van der Waals heterostructures,” Nature 499, 419–425 (2013). 3K. S. Novoselov, A. Mishchenko, A. Carvalho, and A. H. Castro Neto, “2D materials and van der Waals heterostructures,” Science 353, aac9439 (2016). 4D. L. Duong, S. J. Yun, and Y. H. 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5.0008715.pdf

AIP Conference Proceedings 2249 , 030016 (2020); https://doi.org/10.1063/5.0008715 2249 , 030016 © 2020 Author(s).Four-body Faddeev-type calculation of the NNN system Cite as: AIP Conference Proceedings 2249 , 030016 (2020); https://doi.org/10.1063/5.0008715 Published Online: 27 July 2020 N. V. Shevchenko ARTICLES YOU MAY BE INTERESTED IN Generalized parton distributions and transverse momentum distribution of pion AIP Conference Proceedings 2249 , 030018 (2020); https://doi.org/10.1063/5.0008601 From N interactions to -nuclear quasi-bound states AIP Conference Proceedings 2249 , 030014 (2020); https://doi.org/10.1063/5.0008968 Determination of the pion-nucleon σ term from pionic atoms AIP Conference Proceedings 2249 , 030015 (2020); https://doi.org/10.1063/5.0008577Four-Body Faddeev-Type Calculation of the ¯KNNN System N.V . Shevchenkoa) Nuclear Physics Institute, ˇRež 250 68, Czech Republic a)Corresponding author:shevchenko@ujf.cas.cz Abstract. The paper is devoted to four-body Faddeev-type AGS equations, written down for the ¯KNNN system, which is a system consisting of an antikaon and three nucleons. The aim is to find possible quasi-bound state in the system and calculate its properties. INTRODUCTION It is known that ¯KNintegration is attractive, and this fact lead to suggestions, that a quasi-bound state can exist in systems consisting of antikaons and nucleons. In particular, deep and relatively narrow quasi-bound states were predicted in few-body antikaon-nucleon systems [1]. Special interest attracted the lightest three-body ¯KNN system [2]. Many theoretical calculations of the system were performed using different methods and inputs, see e.g. [3]. They predict quite diverse binding energies and widths of the state, but all of them agree, that the quasi-bound state really exists in spin-zero state of ¯KNN , usually denoted as Kpp. We performed a series of calculations of different properties and states of the three-body ¯KNN and ¯K¯KNsystems. In particular, we predicted Kppquasi-bound state binding energy and width using three different models of ¯KN interaction. The same was done for the ¯K¯KNsystem. We also demonstrated, that there is no quasi-bound states, caused by pure strong interactions, in another spin state of ¯KNN system, which is Kd. In addition, we calculated the near-threshold scattering amplitudes of Kelastic scattering on deuteron, including Kdscattering length. All these calculations were performed using dynamically exact Faddeev-type equations in AGS form with coupled ¯KNN and pSNchannels. Finally, we evaluated 1 slevel shift in kaonic deuterium, which is an atomic state, caused by presence of the strong ¯KNinteraction in comparison to the pure Coulomb state. Keeping in mind all our three-body antikaon-nuclear experience, we decided to move further and study four-body ¯KNNN system. Some calculations were already performed for it, but more accurate calculations are needed. We use four-body Faddeev-type equations in AGS form [4]. Only these dynamically exact equations in momentum representation can treat energy-dependent ¯KNpotentials, necessary for the this system, exactly. We solve the four- body equations using our two-body ¯KNpotentials constructed for the three-body AGS calculations of the ¯KNN system, described in [3]. FOUR-BODY FADDEEV-TYPE AGS EQUATIONS The four-body AGS equations contain not only two-body T-matrices, but also three-body transition operators. If separable potentials Va=lajgaihgajare used, the three-body transitions operators can be found from the three-body Faddeev-type equations in AGS form [5]: Xab(z) =Zab(z) +3 å g=1Zag(z)tg(z)Xgb(z) (1) with transition Xaband kernel Zaboperators are defined as Xab(z) =hgajG0(z)Uab(z)G0(z)jgbi; (2) Zab(z) = ( 1dab)hgajG0(z)jgbi: (3) Operator Uab(z)in Eq.(2) is a three-body transition operator of the general form, which describes process b+(ag)! a+ (bg), while G0(z)in Eqs.(2,3) is the three-body free Green function. Faddeev partition indices a;b=1;2;3 simultaneously define a particle ( a) and the remained pair ( bg). The operator ta(z)in Eq.(1) is an energy-dependent part of a separable two-body T-matrix Ta(z) =jgaita(z)hgaj;corresponding to the separable potential describing interaction in the ( bg) pair;jgaiis a form-factor. Proceedings of the 15th International Conference on Meson-Nucleon Physics and the Structure of the Nucleon AIP Conf. Proc. 2249, 030016-1–030016-5; https://doi.org/10.1063/5.0008715 Published by AIP Publishing. 978-0-7354-2008-3/$30.00030016-1The four-body Faddeev-type AGS equations [4], written for separable potentials, have a form ¯Usr ab(z) = ( 1dsr)(¯G01)ab(z) +å t;g;d(1dst)¯Tt ag(z)(¯G0)gd(z)¯Utr db(z); (4) ¯Usr ab(z) =hgajG0(z)Usr ab(z)G0(z)jgbi; (5) ¯Tt ab(z) =hgajG0(z)Ut ab(z)G0(z)jgbi; (6) (¯G0)ab(z) =dabta(z): (7) Here operators ¯Usr aband ¯Tt abcontain four-body Usr ab(z)and three-body Ut ab(z)transition operators of the general form, correspondingly. The lower indices a;bin Eqs.(4,5,6) similarly to the case of the three-body equations (1,2,3) define two-body subsystems of the full system. The upper indices t;s;rdefine a partition of the four-body system which can be of 3 +1 or 2 +2 type. The free Green function G0(z)now acts in four-body space. The four-body system of AGS equations Eq.(4) look similar to the three-body AGS system with arbitrary potentials. Due to this it was suggested in [6] to represent the ”effective three-body potentials” ¯Tt ab(z)in Eq. (4) in a separable form: ¯Tt ab(z) =j¯gt ai¯tt ab(z)h¯gt aj:After this the four-body equations can be written as [6] ¯Xsr ab(z) =¯Zsr ab(z) +å t;g;d¯Zst ag(z)¯tt gd(z)¯Xtr db(z) (8) with new four-body transition ¯Xsrand kernel ¯Zsroperators defined by ¯Xsr ab(z) =h¯gs aj¯G0(z)aa¯Usr ab(z)¯G0(z)bbj¯gr bi; (9) ¯Zsr ab(z) = ( 1dsr)h¯gs aj¯G0(z)abj¯gr bi: (10) SEPARABLE VERSIONS OF THE THREE-BODY AMPLITUDES Necessary for the ¯KNNN calculations ¯KNandNNpotentials, which we use, are separable ones by construction. Therefore, we need to contruct separable versions of three-body and 2+2 amplitudes only, entering the equations (8). Three-body Faddeev-type AGS equations written in momentum basis for s-wave interactions have a form: Xab(p;p0;z) =Zab(p;p0;z) +3 å g=14pZ¥ 0Zag(p;p00;z)tg(p00;z)Xgb(p00;p0;z)p002d p00; (11) were p;p0andzare three-body momenta and energy. Eigenvalues lnand eigenfunctions gna(p;z)of the system Eq.(11) can be evaluated from the system of equations gna(p;z) =1 ln3 å g=14pZ¥ 0Zag(p;p0;z)tg(p0;z)gng(p0;z)p02d p0(12) with normalization condition 3 å g=14pZ¥ 0gng(p0;z)tg(p0;z)gn0g(p0;z)p02d p0=dnn0: (13) Knowledge of the eigenvalues and eigenfunction allows us to write down Hilbert-Schmidt expansion of the kernel functions Z: Zab(p;p0;z) =¥ å n=1lngna(p)gnb(p0); (14) which leads to the separable three-body amplitude Xab(p;p0;z) =¥ å n=1ln 1lngna(p)gnb(p0): (15) 030016-2Since the kernel function Zand three-body amplitude Xare energy-dependent functions, it is necessary to solve Eq.(12) with Eq.(13) for every value of the three-body subsystem energy during the four-body calculations. Instead of this we will use Energy Dependent Pole Expansion/Approximation (EDPE/EDPA) method, suggested in [7] specially for the four-body AGS equations. It needs solution of the eigenequations Eq.(12) only once, for a fixed energy z, which usually is chosen to be the binding energy z=EB. After that energy dependent form-factors gna(p;z) =1 ln3 å g=14pZ¥ 0Zag(p;p0;z)tg(p0;EB)gng(p0;EB)p02d p0(16) and propagators (Q(z))1 mn=3 å g=14pZ¥ 0gmg(p0;z)tg(p0;EB)gng(p0;EB)p02d p03 å g=14pZ¥ 0gmg(p0;z)tg(p0;z)gng(p0;z)p02d p0(17) are calculated. Finally, the separable three-body amplitude has a form Xab(p;p0;z) =¥ å m;n=1gma(p;z)Qmn(z)gnb(p0;z): (18) If only one term is taken in the sums in Eq.(18), the Energy Dependent Pole Expansion turns into Energy Dependent Pole Approximation. It is seen, that EDPE method needs only one solution of the eigenvalue equations Eq.(12) and calculations of the integrals Eqs.(16,17) after that. According to the authors, the method is accurate already with one term (i.e. EDPA), and it converges faster than Hilbert-Schmidt expansion. In our four-body notations form-factors gna!¯gr a, and energy-dependent functions Qmn(z)!¯tr ab. FOUR-BODY EQUATIONS FOR THE ¯KNNN SYSTEM There are two partitions of 3 +1 type:j¯K+ (NNN )i,jN+ (¯KNN )i, - and one of the 2 +2 type:j(¯KN) + (NN)i, - for the¯KNNN system. At the begin we considered all three nucleons as different particles, so we started by writing down the four-body system of equations Eq.(8) for the following 18 channels sawith a=NNor¯KN: 1NN:j¯K+ (N1+N2N3)i;j¯K+ (N2+N3N1)i;j¯K+ (N3+N1N2)i; 2NN:jN1+ (¯K+N2N3)i;jN2+ (¯K+N3N1)i;jN3+ (¯K+N1N2)i; 2¯KN:jN1+ (N2+¯KN3)i;jN2+ (N3+¯KN1)i;jN3+ (N1+¯KN2)i; (19) jN1+ (N3+¯KN2)i;jN2+ (N1+¯KN3)i;jN3+ (N2+¯KN1)i; 3NN:j(N2N3) + ( ¯K+N1)i;j(N3N1) + ( ¯K+N2)i;j(N1N2) + ( ¯K+N3)i; 3¯KN:j(¯KN1) + (N2+N3)i;j(¯KN2) + (N3+N1)i;j(¯KN3) + (N1+N2)i In order to find the quasi-bound state energy the homogeneous system of equations Eq.(8) has to be solved. After antisymmetrization, necessary for a system with identical fermions, the system of equations can be written in a form: ˆX=ˆZˆtˆX: (20) If two-body interactions are spin- and isospin-independent, ˆZand ˆtare the 55 matrices containing the kernel operators ¯Zsr aand¯tr ab, correspondingly: ¯Zsr a=0 BBBB@0¯Z12 NN 0¯Z13 NN 0 ¯Z21 NN 0 0 ¯Z23 NN 0 0 0 ¯Z22 ¯KN0¯Z23 ¯KN¯Z31 NN¯Z32 NN 0 0 0 0 0 ¯Z32 ¯KN0 01 CCCCA; (21) 030016-3¯tr ab=0 BBBBB@¯t1 NN;NN 0 0 0 0 0 ¯t2 NN;NN¯t2 NN;¯KN0 0 0 ¯t2 ¯KN;NN¯t2 ¯KN;¯KN0 0 0 0 0 ¯t3 NN;NN¯t3 NN;¯KN 0 0 0 ¯t3 ¯KN;NN¯t3 ¯KN;¯KN1 CCCCCA; (22) while matrix ¯Xr ais a vector of unknown operators ¯Xr a=0 BBBB@¯X1 NN¯X2 NN¯X2 ¯KN¯X3 NN¯X3 ¯KN1 CCCCA: (23) However, necessary for the ¯KNNN system ¯KNandNNpotentials are isospin- and spin-dependent ones. In addition, ourNNinteraction model is a two-term potential. ˆZandˆt, entering the antisymmetrized equations Eq.(20) for such potentials are matrices 18 x18. TWO-BODY POTENTIALS, THREE-BODY SUBSYSTEMS AND 2+2PARTITION Two-body potentials Both ¯KNandNNpotentials are separable isospin- and spin-dependent ones in s-wave. We use three our separable antikaon-nucleon potentials constructed for our three-body calculations of the ¯KNN and ¯K¯KNsystems. They are: two phenomenological potentials with coupled ¯KNpSchannels, having one- or two-pole structure of the L(1405 ) resonance and a chirally motivated model with coupled ¯KNpSpLchannels and two-pole structure. All three potentials describe low-energy Kpscattering, namely: elastic Kp!Kpand inelastic Kp!MBcross-sections and threshold branching ratios g;Rc;Rn. They also reproduce 1 slevel shift of kaonic hydrogen caused by the strong ¯KNinteraction in comparison to the pure Coulomb level, measured by SIDDHARTA experiment [8]: DSIDD 1s=283 366 eV , GSIDD 1s=5418922 eV . All the experimental data are described by three our potentials with equally high accuracy. In addition, elastic pScross-sections with isospin IpSprovided by all three potentials have a bump in a region of the L(1405 )resonance (according to PDG [9]: MPDG L(1405 )=1405 :1+1:3 1:0MeV , GPDG L(1405 )=50:52:0 MeV). The three antikaon-nucleon potentials with coupled ¯KNpSchannels were used in three-body AGS equations with coupled ¯KNNpSNthree-body channels. By this the channel coupling was taken into account in a direct way. The four-body AGS equations, which we solve, are too complicated to do the same. Due to this we use the exact optical versions of our ¯KNpotentials. They have exactly the same elastic part of the potential as the potential with coupled channels, while all in-elasticity is taken into account in an energy-dependent imaginary part of the potential. It was demonstrated in our three-body calculations [3], that such potentials give very accurate results in comparison with the results obtained with the coupled-channel potentials. We constructed a new version of the two-term separable NNpotential. It reproduces Argonne v18 NNphase shifts with change of sign, which means it is repulsive at short distances. It provides the following singlet and triplet NN scattering lengths: aS=0=16:32 fm, aS=1=5:40 fm, and give the deuteron binding energy Edeu=2:225 MeV . Three-body subsystems and 2+2partition We are studying the ¯KNNN system with the lowest value of the four-body isospin I(4)=0, which can be denoted as Kppn. Its total spin S(4)is equal to one half, while the orbital momentum is zero, since all two-body interactions are chosen to be s-wave ones. For the ¯KNNN system with these quantum numbers the following three-body subsystems contribute: ¯KNN with isospin I(3)=1=2 and spin S(3)=0 (Kpp) or spin S(3)=1 (Kd). 030016-4NNN with isospin I(3)=1=2 and spin S(3)=1=2 (3H or3He). The three-body ¯KNN system with different quantum numbers was studied in our previous works, see Ref. [3]. In particular, quasi-bound state pole positions in the Kppsystem ( ¯KNN with isospin I(3)=1=2 and spin S(3)=0) were calculated, while no quasi-bound states caused by pure strong interactions were found in the Kdsystem ( ¯KNN with isospin I(3)=1=2 and spin S(3)=1). The codes for numerical solution of the three-body AGS equations for the ¯KNN systems were modified to construct separable versions of the three-body amplitudes, as described before. The three-body AGS equations Eq.(1) were written and numerically solved for the three-nucleon system NNN with our new two-term NNpotential as an input. The calculated binding energy was found to be 9 :95 MeV for both3H and3He nuclei since Coulomb interaction was not taken into account. The numerical code was afterwards changed for calculation of the separable version of the NNN amplitudes. Finally, the partition of the 2 +2 type ¯KN+NNis a system with two non-interacting pairs of particles. It was described by special three-body system of ASG equations , and its separable version was constructed in a similar way as for the ¯KNN andNNN three-body subsystems. CONCLUSION The four-body Faddeev-type AGS equations for search of the quasi-bound state in the ¯KNNN system were written down for distinguishable nucleons and antisymmetrized. All necessary additional three-body calculations of the NNN and ¯KN+NNsystems were performed, afterwards separable versions of all three-body and 2 +2 amplitudes were constructed. The code for numerical four-body calculations was written, its tuning and checks are in progress. ACKNOWLEDGMENTS The author is thankful to the Academy of Sciences of the Czech Republic for provided subsidy for DSc. researchers. The work was supported by GACR grant 19-19640S. REFERENCES 1. Y . Akaishi, T. Yamazaki, Phys. Rev. C 65, 044005 (2002). 2. T. Yamazaki, Y . Akaishi, Phys. Lett. B 535, 70 (2002). 3. N.V . Shevchenko, Few Body Syst. 58, 6 (2017). 4. P. Grassberger, W. Sandhas, Nucl. Phys. B 2, 181-206 (1967). 5. E.O. Alt, P. Grassberger, W. Sandhas, Nucl. Phys. B 2, 167-180 (1967). 6. A. Casel, H. Haberzettl, W. Sandhas, Phys. Rev. C 25, 1738 (1982). 7. S. Sofianos, N.J. McGurk, H. Fiedeldeldey, Nucl. Phys. A 318, 295 (1979). 8. M. Bazzi et al. (SIDDHARTA Collaboration), Phys. Lett. B 704, 113 (2011). 9. M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018). 030016-5

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J. Vac. Sci. Technol. B 38, 040602 (2020); https://doi.org/10.1116/6.0000222 38, 040602 © 2020 Author(s).Engineering anisotropic magnetoresistance of Hall bars with interfacial organic layers Cite as: J. Vac. Sci. Technol. B 38, 040602 (2020); https://doi.org/10.1116/6.0000222 Submitted: 09 June 2020 . Accepted: 10 June 2020 . Published Online: 23 June 2020 Jun Hong Park , Mario Ribeiro , Thi Kim Hang Pham , Nyun Jong Lee , Tai-woon Eom , Junhyeon Jo , Seung-Young Park , Sonny H. Rhim , Kohji Nakamura , Jung-Woo Yoo , and Tae Hee Kim ARTICLES YOU MAY BE INTERESTED IN Spin Hall magnetoresistance at Pt/CoFe 2O4 interfaces and texture effects Applied Physics Letters 105, 142402 (2014); https://doi.org/10.1063/1.4897544 Molecular tilting and columnar stacking of Fe phthalocyanine thin films on Au(111) Journal of Applied Physics 117, 17A735 (2015); https://doi.org/10.1063/1.4916302 Inverse spin-Hall effect induced by spin pumping in metallic system Journal of Applied Physics 109, 103913 (2011); https://doi.org/10.1063/1.3587173Engineering anisotropic magnetoresistance of Hall bars with interfacial organic layers Cite as: J. Vac. Sci. Technol. B 38, 040602 (2020); doi: 10.1116/6.0000222 View Online Export Citation CrossMar k Submitted: 9 June 2020 · Accepted: 10 June 2020 · Published Online: 23 June 2020 Jun Hong Park,1,2,3 Mario Ribeiro,1,2Thi Kim Hang Pham,2Nyun Jong Lee,4Tai-woon Eom,5Junhyeon Jo,6 Seung-Young Park,7Sonny H. Rhim,4Kohji Nakamura,8Jung-Woo Yoo,6and Tae Hee Kim1,2,a) AFFILIATIONS 1Center for Quantum Nanoscience, Institute for Basic Science (IBS), Seoul 03760, South Korea 2Department of Physics, Ewha Womans University, Seoul 03760, South Korea 3School of Materials Science and Engineering, Gyeongsang National University, Jinju 52828, South Korea 4Department of Physics and Energy Harvest Storage Research Center, University of Ulsan, Ulsan 44610, South Korea 5RNDWARE Co., Ltd., Daejeon 34133, South Korea 6School of Materials Science and Engineering-Low Dimensional Carbon Materials Center, Ulsan National Institute of Science and Technology, Ulsan 44919, South Korea 7Center for Scientific Instrumentation, Korea Basic Science Institute, Daejeon 34133, South Korea 8Department of Physics Engineering, Mie University, Tsu, Mie 514-8507, Japan a)Electronic mail: taehee@ewha.ac.kr ABSTRACT Tuning the magnetoresistance behavior of heterostructures composed of nonmagnetic and ferromagnetic (FM) materials is crucial for improving their applicability in electronic and spintronic devices. In this study, we investigate whether the integration of organic layers toNiFe/Pt junctions can result in the modification of the magnetic moment of the FM layer using iron phthalocyanines (FePc) and copperphthalocyanines (CuPc) as the interfacial layers for controlling the spin-charge conversion. Relaxation of the out-of-plane magnetic hard axisof the NiFe/Pt junctions is observed, as a result of the modification of the interfacial magnetic structure. The transport measurements of the fabricated hybrid Hall bar junctions with NiFe/FePc/Pt and NiFe/CuPc/Pt reveal that although the intrinsic anisotropic magnetoresistance of the present Hall bar is maintained with the integration of interfacial metal phthalocyanine (MPc) layers, a change in the magnetic responsealong the axis perpendicular to the in-plane of Hall bars is observed, owing to the insertion of the interfacial MPc layers. The present methodof interface engineering via integration of organic interfacial layers can act as a model system for controlling the spin-charge conversionbehavior of magnetic heterojunction toward the development of multifunctional molecular-engineered spintronic devices. Published under license by AVS. https://doi.org/10.1116/6.0000222 I. INTRODUCTION Magnetoresistance (MR) has long been playing a vital role in the field of electronics and spintronics, 1which has broad applica- tions in sensory systems,2–4memories,5–7and data storages.2,8 Specifically, anisotropic magnetoresistance (AMR), the direction of the external magnetic field, affects the trajectory of the conducingelectrons. 9–11In heterostructures comprised of ferromagnet (FM) and nonmagnetic (NM) layers, the tunability of AMR is well explored;11–14this is possible by the introduction of an interface and non-negligible spin-orbit coupling.Manipulating magnetism through the introduction of semi- conducting π-conjugated organic complexes has been investigated in studies in the field of solid state electronics and spintronics.15–19 Specifically, C60 or metal phthalocyanine (MPc) layers have been studied at the NM/FM interfacial layers,20–22while V[TCNE] X layers have been demonstrated as organic ferromagnets.15The deposition of organic molecules onto an inorganic layer does notinclude covalent bonding but van der Waals interactions withrather weak spin-orbit coupling. 18,23,24Despite the absence of covalent bonding, weak hybridization and the formation of theLETTER avs.scitation.org/journal/jvb J. Vac. Sci. T echnol. B 38(4) Jul/Aug 2020; doi: 10.1116/6.0000222 38,040602-1 Published under license by A VS.organic-inorganic interface introduce variations in the magnetism of FM. This demonstrates the tunability of magnetism through the use of organic molecules: non-negligible orbital overlap betweenorganic molecules and magnetic layers can shift the magneto-anisotropy to be in the perpendicular direction. 20,21,25,26The orbital overlap introduces a state near the Fermi level with nonbonding charge transfer, thereby altering the total magnetism of the system with localized modification of magnetism of the organic/FMinterfaces. 27–30For example, as presented in the previous reports, insertion of thick (over 30 nm) C60 layers between Ta and Feresults in a 27% reduction in the magnetic moment of the Fe layer, 21while the formation of a few layers of coverage of C60 on Co/Au(111) leads to a change in the total effective anisotropy coef-ficient of the Co film ( ΔK eff≈0.3 mJ m3).30This feature also reveals possible ways of manipulating magnetism by the introduction ofmolecules or functional groups. In the present report, layers of metal MPc are sandwiched in the organic-inorganic Hall bar junctions, and copper phthalocya-nine (CuPc) and iron phthalocyanine (FePc) molecules are depos-ited on the ultrathin NiFe layer, whereas Pt layers are employed asthe electric charge transfer layers on top of the deposited MPc layers. Although both CuPc and FePc share identical molecular structures, different metal atoms would inherently exhibit varia-tions in hybridization with the FM; 31–33therefore, when orbital hybridization is dominant to tune the magnetic moment of the fer- romagnetic hybrid systems, a difference in the introduced perpen- dicular magnetism is observed. Electronic transport measurementswith external magnetic fields reveal the increase of the perpendicu-lar resistive contribution of spin polarized electrons in both theCuPc and FePc inserted devices, relative to NiFe/Pt without MPc; contribution of perpendicular resistivity to AMR behavior along the z-direction increases by over 16% at a high magnetic field(5000 Oe) with the insertion of ultrathin MPc layers. Consequently,the present study demonstrates a potential pathway for interfacialengineering for spin polarized transport systems, with the use of the ultrathin organic layer. II. EXPERIMENT Stacking of Hall bar junctions is performed by in situ sequen- tial deposition in home-built UHV chambers (base pressure: 4×1 0 −10Torr) without breaking the vacuum. During the deposi- tion of samples using an electron beam source, all the junctions arepatterned in the chamber using the standard shadow mask tech-nique to minimize the introduction of unintentional contaminantduring the fabrication process; Si substrates are cleaned using acetone, ethanol, and 20% H 2SO4(diluted in H 2O) with sonication, and native oxide is subsequently removed by dipping the substratesin 10% HF diluted in H 2O. After this, the samples are transferred into the UHV chamber through a load lock chamber using a turbomolecular pump (base pressure: 2 × 10 −10Torr). Deposition of the NiFe alloy is performed using an electron beam source at 300 K. Subsequently, the sample is transferred into an organic depositionchamber (base pressure: 1 × 10 −8Torr) without breaking the vacuum to allow the deposition of the MPc layers. The effusion cell is used in the deposition of the MPc layers, where the samples are held at a temperature of 353 K. MPc was purchased from SigmaAldrich and purified through multiple sublimations before being deposited onto the samples. Next, the MPc/NiFe samples are placed back in the UHV chamber for the deposition of the Ptcontact layer. Notably, the deposition of these layers is limited to athickness of 1 nm. If the thickness of the deposited organic layer isgreater than 2 nm, the magnetic response of the organic/FM inter- face would get screened to the Pt electrodes. During deposition, quartz crystal microbalance is used for monitoring the film thick-ness and the rate of deposition. The transport measurements of the fabricated Hall bar junc- tions are performed at a temperature of 77 K; the anisotropic mag- netoresistance is measured using a physical property measurement system (Quantum Design), while magnetoresistance with the direc-tion of the magnetic field fixed with respect to that of the current isapplied using a home-built magnetic probe station (Keithley 6220precision current source and LakeShore Model 642 electromagnet power supply). For measuring magnetoresistance, the fabricated samples are transferred to a home-built cell, and transport mea-surements are performed. After mounting the samples, the cell iscooled to a temperature of 77 K, and the precise effect of the inser-tion ultrathin MPc layer on the magnetic response is observed, while suppressing the thermally activated phonon or noise. III. RESULTS AND DISCUSSION A. Insertion of ultrathin MPc layer into NM/FM To estimate the magnetic contribution of organic molecules in a hybrid junction, spin states and orbital hybridizations at central metal atoms of FePc and CuPc are compared, based on previous reports. 31–36As shown in Fig. 1(a) , FePc and CuPc have identical molecular structures; a transition metal atom is at the center, sur-rounded by four aromatic benzene rings. Regarding CuPc, the spinstate is 1/2 and the expansion of d-orbital is limited in the plane of the molecule. 31,34,36Conversely, in the case of FePc, the spin state FIG. 1. Schematic of metal phthalocyanines and spin configurations. (a) Molecular structure of MPc. (b) Hybridized orbitals of the central Fe atom inFePc. (c) Hybridized orbitals of the central Cu atom in CuPc.LETTER avs.scitation.org/journal/jvb J. Vac. Sci. T echnol. B 38(4) Jul/Aug 2020; doi: 10.1116/6.0000222 38,040602-2 Published under license by A VS.is 1, and the d-orbital not only expands in the plane, but also out of the plane, thereby exhibiting strong interactions with FM,31,32,35 when compared to the interaction of CuPc with FM, as shown in Fig. 1(b) ; as per the results of scanning tunneling microscopy of a previous study, the local density of state for Fe atoms of MPc isimaged as bright spots, and the height of the Fe atoms is approxi- mately 0.5 –1 Å more than that of the benzene rings. 37Conversely, the local density of state for Cu atoms of MPc is imaged as darkholes and the height of the Cu atoms is less than that of thebenzene rings. 31Under the present experimental settings, the mag- netic response of the MPc-integrated Hall bar can be observed via a Pt contact because the spin-polarized charge is transferred from NiFe to Pt via the 1-nm MPc layers. Therefore, the observation ofmagnetic response of FePc and CuPc layers would reveal whetherthe transport of spin related carrier is governed by the d-orbitalinteraction of center metal atoms in MPc or the molecular packing configurations, as discussed below. Vertical organic/inorganic hybrid junctions are fabricated on Si (001) substrates using molecular beam epitaxy in an ultrahighvacuum chamber, without exposing the sample to ambient condi-tions during the growth process. Therefore, it can be hypothesized that the possibility of the sample being contaminated by ambient air is ruled out. Although some carbon-based contaminants mayget introduced into the samples, the overall impact on the magne-toresistance behavior of the fabricated samples is negligible, because these carbon-based contaminants possess extremely weak orbital coupling or have insignificant magnetic contribution. It isalso noted that no preferential oxidation at room temperature iswell known for Py thin films; 38oxidation process is less efficient than by pure Ni or Fe, because of a weaker penetration of the mate- rial by oxygen. As shown in Figs. 2(a) and2(b), 1.5 nm NiFe and 3 nm Pt layers are employed as the FM and NM layers, respectively.It is noted that in the current study, ultrathin metal layers aredeposited to enhance the effect of the interface on AMR, and there-fore, the magnetic response of fabricated Hall bars that is mostly dominated by the bulk FM effect is suppressed. Three different ver- tical junctions are fabricated, consistent with the bilayer of NiFe/Pt,NiFe/FePc/Pt, and NiFe/CuPc/Pt. For organic/inorganic hybridjunctions, 1-nm CuPc and FePc organic layers are inserted betweenNiFe and Pt. Magnetism curves of the fabricated bilayer and hybrid junc- tions are measured in the in-plane and out of plane directionsusing a superconducting quantum interference device (SQUID) ata temperature of 77 K. Note that all the junctions in the present study are fabricated using ultrathin films; NiFe is 1.5 nm, while Pt is 3 nm. Therefore, the overall effect on the magnetoresistancebehavior is minimized. As shown in Figs. 2(c) and2(d), the easy axis of NiFe/Pt bilayer junction is observed along the plane of junc-tion; in-plane magnetization of the junction is fully saturated at approximately 200 Oe, while the saturation of magnetization in the direction of out of plane is observed at 4000 Oe, which is consistentwith the easy axis (hard axis) of the magnetization in-plane (out ofplane) of the FM layers. Notably, the demonstration of the observedeasy axis in the plane of FM layers is mostly caused by the shape anisotropy of the two-dimensional configuration of the deposited FM layers; the magnetic orientation of the FM layers along thein-plane directions could be reconstructed by the applied magneticfield with low exchange energy, while the magnetic orientation along the out of plane (perpendicular direction) could be pinned by the interface with Pt placed on top of FM or Al 2O3placed underneath FM.39,40Thus, a complete sweeping of the magnetic FIG. 2. Magnetism properties of hybrid junctions. (a) Schematic of the hybrid junctions. (b) Schematic of the interfacial MPc layers. (c) Magnetism curves ofthe junctions when the magnetic field is applied parallel to the plane of thelayers. (d) Magnetism curves of the junctions when the magnetic field is applied perpendicular to the plane of the layers.LETTER avs.scitation.org/journal/jvb J. Vac. Sci. T echnol. B 38(4) Jul/Aug 2020; doi: 10.1116/6.0000222 38,040602-3 Published under license by A VS.ordering of FM requires an additional magnetic force. After the 1-nm MPc layers are inserted, the in-plane magnetization remains nearly constant, thereby indicating that the MPc layers do notbother the in-plane magnetic ordering at the surface of FM.However, perpendicular magnetization is saturated at approximately1600 Oe, which is lower than the field of NiFe/Pt. This reduction in the saturation field indicates that the hard axis of the Hall bar junc- tions is relaxed by the insertion of the MPc layers, which induces anadditional magnetic contribution, including the reorientation of spinmoments by organic interlayers between the FM and NM, whileweakening the magnetic contribution induced by the shape anisot- ropy of the FM. It is noted that although both FePc and CuPc organic layers are deposited directly on the FM, H c(coercivity field) remains nearly constant, thereby indicating that the depositedorganic molecules do not induce the pinning of the domain-wallmotion or increasing of the magnetic exchange energy. The relaxation of the hard axis magnetization of FM via the insertion of interfacial organic layers can be understood with thehelp of previous reports on the hybridization of magnetic momen-tum at the organic/inorganic interface; 20,21,25,26,30deposition of C60 and thiolates induces the switching of the direction of the easy axis in the magnetization curves, thereby resulting in the reorienta- tion of magnetic momentums. Similarly, as the deposition organicmolecules on the FM surface leads to the formation of the organic/FM interface, the orbital of the organic molecules is coupled and hybridized with that of NiFe, thereby resulting in the exchange of charge carriers between the organic molecules and FM atoms. As aresult of the charge transfer at the hybrid interface, the verticalmagnetic structure of the devices is perpendicularly reconstructedto induce and reduce the magnetic moment in the organic layers and FM layers, respectively. Consequently, although the magnetic contribution of FM layers to the magnetism curves is weakened bya certain extent, additional magnetic momentum can be providedby ultrathin organic layers coupled with FM layers underneath. B. Magnetoresistance of hybrid Hall bar To track the change in magnetoresistance with the insertion of interfacial MPc layers to the bilayer metal junctions, theangular dependent transport resistance is measured while applyingconstant magnetic fields of 2000 and 7000 Oe. As depicted in Figs. 3(a) –3(c), a longitudinal electric current is applied along the Pt layer at a temperature of 77 K, while magnetic fields are appliedto the samples along the longitudinal (H//X), transverse (H//Y),and perpendicular (H//Z) directions, as shown in color coordina-tion of Fig. 3(d) .I nFigs. 3(e) –3(g), the behavior of AMR is demon- strated in all the fabricated Hall bar junctions, 9–11,41with reference to NiFe/Pt, NiFe/FePc/Pt, and NiFe/CuPc/Pt; the longitudinal resis-tance is maximized when the rotated magnetic field is parallel tothe direction of the electric current, while it is minimized when therotated magnetic field is transverse to the direction of the electric current in-plane. In addition, the rotation of the magnetic field along the direction of Z to X and Z to Y results in the ellipticalchange of magnetoresistance for all the samples. Therefore, it isconfirmed that the measured magnetoresistance depicted in Fig. 3 is governed by the angle of the applied magnetic field to the direc- tion of the electric current consistent with AMR. Although the Pt FIG. 3. Angular dependence of magnetic response with Hall bar junctions. Schematic of the hybrid Hall bar junctions when the applied magnetic field is rotated from (a) X to Y , (b) Z to X, and (c) Z to Y directions. (d) Color label cor-responded to the scan direction for (e), (f), and (g). (d) Angular dependence ofmagnetoresistance of NiFe/Pt at 2000 and 7000 Oe. (e) Angular dependence of magnetoresistance of NiFe/FePc/Pt at 2000 and 7000 Oe. (f) Angular depen- dence of magnetoresistance of NiFe/CuPc/Pt at 2000 and 7000 Oe.LETTER avs.scitation.org/journal/jvb J. Vac. Sci. T echnol. B 38(4) Jul/Aug 2020; doi: 10.1116/6.0000222 38,040602-4 Published under license by A VS.electrode and NiFe FM are decoupled by the insertion of organic interfacial layers for modifying local magnetic momentum, the intrinsic AMR behavior of the Hall bar junctions is nearly constant.It is notable that as the applied field is increased to 7000 Oe, whichexceeds the saturation field (>4000 Oe) as depicted in Fig 2 , the shape of the magnetoresistance curves changes from orthogonal to ellipsoidal, consistent with the completely aligned magnetization of the Hall bars with an external magnetic field. Magnetoresistance of the fabricated Hall bars is estimated by sweeping the intensity of the magnetic field at the directions of X,Y, and Z to elucidate the effect of the inserted layers on AMR. Figure 4(a) reveals the evolution of spin-related carrier transfer upon applying magnetic fields. In Fig. 4(b) , a typical AMR is dem- onstrated with a bilayer Hall bar of NiFe/Pt; a longitudinal magne-toresistance (R X) with H//X is nearly saturated at a high field, while the decrease of the magnetic field intensity induces a negative peak at approximately 0 Oe. The resistance of the bilayer Hall bar is maximized by the alignment of the magnetic field along the direc-tion of current with an increase in the rate of scattering of elec-trons. However, as the intensity of the magnetic field approaches0 Oe, the rate of electron scattering is suppressed, thereby resulting in the decrease of resistance which can be seen as negativelypointed peak. Conversely, although when applying a transverse mag- netic field to the direction (H//Y), a nearly constant magnetoresistance (R Y) is observed at the high field, the positive peak of magnetoresis- tance is observed around 0 Oe (similar to the behavior R X), which is consistent with the fact that magnetic field intensity decreases withthe increase in resistance. When the magnetoresistance (R Z) of the NiFe/Pt Hall bar, on application of the perpendicular magnetic field (H//Z) to the plane of the Hall bar, is placed between R X and R Y, the observed characteristics of this magnetoresistance can be expressed as R X>R Z>RY, which is consistent with the AMR observed in previous results.10,42Although the positively pointed peaks in R Zare observed at a low field, the position of the peak is shifted to ∼−500 and ∼500 Oe, as H is swept with forward and backward directions, respectively; it is hypothesized that during thesweeping of H, the magnetic structure of the NiFe layer is pinnedat the interface of NiFe/Pt or Al 2O3/NiFe, thereby requiring an additional magnetic force to orient the applied magnetic field along the Z direction,39,43consistent with the existence of the hard axis, as observed in Fig. 2 . However, as the intensity of the perpendicular magnetic field (H//Z) increases, the resistance of the NiFe/Pt Hallbar gradually increases and is nearly saturated when H is above −5000 and 5000 Oe. The fabricated Hall bars inserted with MPc interfacial layers show a similar behavior of magnetoresistance with reference to theNi/Pt Hall bar when H is applied in a similar manner. It is noted that the absolute magnetoresistance of the MPc-integrated Hall bars increases by a factor of three as shown in Figs. 4(c) and4(d);M P c molecules are semiconducting; therefore, the insertion of MPc layersinduces decoupling between the metal Pt and metal NiFe, as thisacts as electric resistors in the Hall bars for the spin related charge carriers. As shown in Figs. 4(c) and4(d), the AMR behavior is dem- onstrated to be maintaining the equation, R X>RZ>RY, for both the FePc and CuPc inserted junctions, respectively. Compared with thereference NiFe/Pt junctions, although the MR ratio of the hybridHall bar for the peaks at a low field decreases to less than half of that of NiFe/Pt, sharp peaks can be observed at a low field (around 0 Oe) for all the directions of H//X, H//Y, and H//Z. This indicates that thespin polarized electrons can be transported through the MPc layers,from the NiFe FM layer to the Pt electrode. However, the insertionof MPc interfacial layers induces a different behavior of R Zat a high field intensity from the R Zof the reference NiFe/Pt junction; R Zof NiFe/FePc/Pt in Fig. 4(c) is nearly constant in the range of 1500 (−1500) to 7000 ( −7000) Oe. Conversely, R Zof NiFe/CuPc/Pt in Fig. 4(d) gradually increases and is maximized at a magnetic field around 5000 ( −5000) Oe, which subsequently decreased, consistent with formation of broad secondary peaks at both negative and posi-tive high fields. In addition, the insertion of CuPc layers results in acloser position of R Zto R x, indicating the existence of additional contribution to perpendicular magnetoresistance. It is notable that the shape of the peak of FePc integrated Hall bar is shaper than that of the reference NiFe/Pt and CuPc integrated Hall bar. Moreover,the FePc integrated Hall bar shows a drastic drop in the magnetore-sistance value, indicated by orange arrows, near 0 Oe. Although itcan be hypothesized that FePc molecules possibly have additional magnetic contribution to spin-polarized carrier transport, finding its exact mechanism would require additional experiments andtheoretical approaches. FIG. 4. Magnetic field dependence of resistance of Hall bars. (a) Schematic of the hybrid Hall bar junctions with interfacial MPc layers. (b) Magnetoresistance of the reference Hall bar (NiFe/Pt). (c) Magnetoresistance of the Hall bar upon insertion of 1-nm FePc layers. (d) Magnetoresistance of the Hall bar upon inser-tion of 1-nm CuPc layers.LETTER avs.scitation.org/journal/jvb J. Vac. Sci. T echnol. B 38(4) Jul/Aug 2020; doi: 10.1116/6.0000222 38,040602-5 Published under license by A VS.C. Magnetic contribution of MPc layers to AMR To elucidate the additional contribution of the insertion of MPc layers to perpendicular magnetoresistance, the magnitude ofcontribution to the magnetoresistance of R X,RY, and R Zat 5000 and 7000 Oe is redefined; since all the fabricated Hall bars show AMR behavior of magnetoresistance with R X>R Z>RY, as shown inFig. 5(a) ,10,11the contributions at 5000 and 7000 Oe are defined as follows: α¼RX–RY,β¼RX–RZ,γ¼RZ–RY: (1) The ratios of β/αandγ/αare obtained to track the change in contribution made by the MPc layers to the perpendicular magne- toresistance, and this change is depicted in Fig. 5(b) . The results reveal that when compared to the reference NiFe/Pt, although theresistive contribution of both CuPc and FePc to AMR behavioralong the Z direction is larger than NiFe/Pt, the insertion of CuPccauses a slightly larger perpendicular magnetoresistance than the insertion of FePc; at 5000 Oe, the perpendicularly magneto-resistive contribution of the Hall bar, γ/α, increase from 33.9% to 50.4% with the insertion of 1-nm FePc layers between NiFe and Pt, whileγ/αincreases to 53.3% with the insertion of 1-nm CuPc layers. Therefore, it can be hypothesized that during charge transport between the NiFe FM layer and the Pt electrode, both the CuPc and FePc interfacial layers provide a perpendicular resistive channelfor the spin polarized electrons; as the electric current is applied inthe junctions along the X direction, the spin polarized electrons areaccumulated at the surface of the NiFe FM layer when a perpendic- ular H (H//Z) is applied. For the hybrid Hall bar junctions, the accumulated electrons are transported through the MPc layers tothe Pt electrode. Since FePc and CuPc have identical molecularstructures without the central metal species, 44it can be estimated that the both these molecules show similar molecular packing on the solid state NiFe FM. Previous studies have revealed that the firstmonolayer of MPc molecules is deposited on the metal surface withthe formation of a flat-lying monolayer to maximize the contactarea between the metal substrates and MPc molecules owing to thestrong substrate-molecules interactions. 31,33Thus, the orbital of the central metal in the MPc molecules can interact with the metal substrates underneath it. However, as additional MPc molecules aredeposited onto the MPc monolayer, molecules in the multilayers aretilted to the vacuum space surface. 45Therefore, it can be hypothe- sized that the molecular orientation transitions from flat to tilted in the vacuum space in 1 nm MPc layers. Consequently, it can be con- cluded that when the charge carrier is transported in the hybrid Hallbars, the scattering or trapping rate of spin polarized electrons inMPc layers is mostly altered by the molecular packing configuration,rather than by the configuration of orbital hybridization. IV. SUMMARY AND CONCLUSIONS The magnetic contribution of interfacial MPc layers to AMR is investigated using the fabrication of organic/inorganic hybridHall bar junctions. When compared with NiFe/Pt, the insertion ofboth CuPc and FePc layers between the Pt electrode and NiFe FM induced the relaxation of the hard axis along the perpendicular direction, which is consistent with the observations in the case ofmagnetization curves. This relaxation indicates that the magneticstructure of the FM layers is reconstructed by the MPc molecules.The transport measurements of the fabricated hybrid Hall bars when the magnetic field is applied reveal that although the integra- tion of MPc interfacial layers to the NiFe/Pt junctions maintainsthe AMR behaviors, both CuPc and FePc interfacial layers provideadditional resistive contribution to the AMR along perpendiculardirections. Consequently, the results obtained in this study regard- ing the different contributions of organic molecules to the spin transport behavior can provide a model system for understandingthe charge transport behaviors in organic layers and can be appliedfor the development of multifunctional molecular-engineered spin-tronic devices. ACKNOWLEDGMENTS This work was supported by the Institute for Basic Science, Republic of Korea (No. IBS-R027-D1). This work was also supportedby the National Research Council of Science & Technology (NST)(Grant No. CAP-16-01-KIST) by the Korean government (MSIP). J.H.P. would like to acknowledge support from the National Research Foundation of Korea (NRF) grant funded by the Koreangovernment (MSIT) (No. NRF-2018R1C1B5085644). REFERENCES 1Y. Hishiyama, Y. Kaburagi, and M. Inagaki, in Materials Science and Engineering of Carbon , edited by M. Inagaki and F. Kang (Butterworth-Heinemann, Oxford, 2016), p. 173. 2M. Djamal, Darsikin, T. Saragi, and M. 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5.0012475.pdf

J. Chem. Phys. 153, 034703 (2020); https://doi.org/10.1063/5.0012475 153, 034703 © 2020 Author(s).Optical properties of charged excitons in two-dimensional semiconductors Cite as: J. Chem. Phys. 153, 034703 (2020); https://doi.org/10.1063/5.0012475 Submitted: 04 May 2020 . Accepted: 24 June 2020 . Published Online: 15 July 2020 M. M. Glazov COLLECTIONS Paper published as part of the special topic on 2D Materials Note: This paper is part of the JCP Special Topic on 2D Materials. ARTICLES YOU MAY BE INTERESTED IN The shape of the electric dipole function determines the sub-picosecond dynamics of anharmonic vibrational polaritons The Journal of Chemical Physics 152, 234111 (2020); https://doi.org/10.1063/5.0009869The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Optical properties of charged excitons in two-dimensional semiconductors Cite as: J. Chem. Phys. 153, 034703 (2020); doi: 10.1063/5.0012475 Submitted: 4 May 2020 •Accepted: 24 June 2020 • Published Online: 15 July 2020 M. M. Glazova) AFFILIATIONS Ioffe Institute, 194021 St. Petersburg, Russia Note: This paper is part of the JCP Special Topic on 2D Materials. a)Author to whom correspondence should be addressed: glazov@coherent.ioffe.ru ABSTRACT Strong Coulomb interaction in atomically thin transition metal dichalcogenides makes these systems particularly promising for studies of excitonic physics. Of special interest are the manifestations of the charged excitons, also known as trions, in the optical properties of two- dimensional semiconductors. In order to describe the optical response of such a system, the exciton interaction with resident electrons should be explicitly taken into account. In this paper, we demonstrate that this can be done in both the trion (essentially, few-particle) and Fermi- polaron (many-body) approaches, which produce equivalent results, provided that the electron density is sufficiently low and the trion binding energy is much smaller than the exciton one. Here, we consider the oscillator strengths of the optical transitions related to the charged excitons, fine structure of trions, and Zeeman effect, as well as photoluminescence of trions illustrating the applicability of both few-particle and many-body models. Published under license by AIP Publishing. https://doi.org/10.1063/5.0012475 .,s I. INTRODUCTION Coulomb interaction is highly important in semiconductors. The concept of the small-radius excitons, i.e., electrons and holes tightly bound to neighboring lattice sites suggested by Frenkel1has been extended by Wannier2and Mott3who demonstrated that in a number of semiconductors, the hydrogen-like large radius excitons can be formed. The large radius excitons were discovered in cuprous oxide by Gross and Karryew4in 1952 and are actively studied since then. Excitons govern optical properties of bulk semiconductors and semiconductor nanostructures.5–7 Shortly after the discovery of large radius excitons, the atomic physics analogy has been extended, and the excitonic molecules, also termed biexcitons, and charged excitons, known as trions, have been predicted.8The latter three-particle complexes, negative and positive trions, are formed of two identical charge carriers and an unpaired one with an opposite sign: two electrons and a hole (X−) and two holes and an electron (X+). They are analogs of the hydrogenic ions H−(a proton and two electrons) and H+ 2(two protons and an elec- tron). The binding energies of these excitonic complexes are quite small in bulk semiconductors. The reduction in dimensionality and transition from the bulk form of materials to their two-dimensional(2D) counterparts—quantum wells—results in a substantial increase in the trion binding energies,9–11which led to the observation of tri- ons in CdTe12and GaAs13quantum wells and initiated extensive experimental and theoretical studies of these complexes in various semiconductor nanosystems.6,14 Recently emerged atomically thin semiconductors based on transition metal dichalcogenide monolayers (TMDC MLs) demon- strate spectacular optical properties and enhanced Coulomb effects.15,16The trions have been observed in these materials as well,17and their fine structure and dynamics are actively studied nowadays.18–21Multivalley band structure of the TMDC MLs makes it possible to observe charged biexcitons as well.22,23 There are, however, fundamental questions related to trion formation and their manifestations in optical properties of two- dimensional semiconductors. Indeed, the trions can be formed only in the presence of resident electrons, which makes it necessary to take into account the interaction of the exciton with the Fermi-sea of charge carriers rather than with a single electron. This many-body problem turns out to be extremely involved even in the limit of high carrier density.24,25Several approaches have been applied to study the interactions between excitons and free electrons in 2D struc- tures, including the direct calculation of optical susceptibility of the J. Chem. Phys. 153, 034703 (2020); doi: 10.1063/5.0012475 153, 034703-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp structure via the equations of motion or diagrammatic treatment of the electron–exciton correlations.26–31On the other hand, the prob- lem of an exciton interacting with the Fermi-sea of electrons resem- bles the famous polaron problem of an electron interacting with an ionic crystalline lattice32,33or an impurity atom immersed in a Fermi gas.34,35Thus, the concept of Fermi-polarons and dressed electron– exciton excitations has been put forward and applied to study the optical response of TMDC MLs.36–38 It is not, however, fully obvious that different approaches should provide the same results. One important issue is related to the trion or Fermi-polaron oscillator strength. Another problem is related with the manifestations of the trion or Fermi-polaron fine structure caused by the complex spin-valley band structure of TMDC MLs and magnetic field. In addition, the comparative anal- ysis of some of the basic kinetic properties of trions and Fermi- polarons, e.g., photoluminescence, is absent. Thus, it is instructive to provide a side-by-side derivation of these quantities in the two approaches: trion and Fermi-polaron, and demonstrate the conver- gence of these approaches, at least for a specific parameter range, i.e., very low density of electrons and linear response regime. This paper is aimed to fill this gap. II. MODEL We consider the excitonic effects in TMDC MLs within the effective mass approach, which provides a simplified but physicallytransparent picture of the Coulomb effects in semiconductors. The three-particle bound states of electrons and holes in the TMDC ML can be described within the effective mass approach by the wavefunction19,39 Ψi,j;k=eiKRφ(ρi,ρj)U(2) ij(ri,rj)U(1) k(rk). (1) Here, the subscripts iandjdenote the identical charge carriers, for example, two electrons e1ande2in the X−trion, and kdenotes the unpaired charge carrier, e.g., a hole in the X−trion, ri,j,kare the coor- dinates of these particles, ρiand ρjare the relative coordinates in the ML plane of identical particles with respect to the unpaired one, andRis the center of the mass position; hereafter, the normalization area is set to unity. In Eq. (1), U(2) ij(ri,rj)is the Bloch function of the electron pair (in the case of the X−trion), and U(1) k(rk)is the Bloch function of the unpaired hole, Kis the wavevector of the trion trans- lational motion, and φ(ρi,ρj) is the smooth envelope of the relative motion in the trion. In what follows, we consider only the ground state of the trion, focusing on the X−case. Thus, the envelope func- tionφ(ρi,ρj) is symmetric under permutations of electrons, ρ1↔ρ2, while the two-electron Bloch function U(2) ij(ri,rj)is odd and ensures the antisymmetry of the total wavefunction.19 Figure 1 illustrates the band structure of the TMDC monolayers with two valleys K±and the spin–orbit splitting in the conduction FIG. 1 . [(a) and (b)] Schematic band structure of a lightly doped TMDC ML. The vertical arrow shows the optical transition in σ+polarization, and the thick wavy line denotes the interaction of the photoexcited electron–hole complex with the Fermi-sea: (a) intravalley interaction and (b) intervalley interaction. Panels (c) and (d) show the corresponding intra- and intervalley trions. The spin–orbit split valence subband is not shown (it is much lower in energy). The bottom conduction subbands are shown by dashed lines and assumed to be the spin-unlike with the top valence band (we consider W-based MLs). The optical transitions in σ−polarization involve the excitation of the opposite valley K−. J. Chem. Phys. 153, 034703 (2020); doi: 10.1063/5.0012475 153, 034703-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp band; the spin–orbit splitting in the valence band is large and is not shown. We consider the W-based 2D TMDC where the spins of the bottom conduction band and top valence band are opposite,19,39–41 so the optical transition takes place to the excited spin subband of the conduction band, as shown in Figs. 1(a) and 1(b). In the pres- ence of doping with electron Fermi energy EFbeing much smaller than the conduction band spin–orbit splitting Δc, the photocreated exciton can interact with the electron gas in the same [ K+forσ+ excitation, Fig. 1(a)] or in the opposite [ K−, Fig. 1(b)] valley. In the trion picture, the exciton picks up the electron from the correspond- ing Fermi-sea and forms the intra- and intervalley trions shown in Figs. 1(c) and 1(d), respectively.19,42,43Most of the results are also relevant for the Mo-based TMDC MLs, where the optical transitions at the normal incidence of radiation involve the bottom conduc- tion subbands. In this situation, under moderate doping, only the intervalley interaction similar to that shown in Figs. 1(b) and 1(d) is important, which makes the trion fine structure quite simple. How- ever, a complication arises due to the fact that the photoelectron is excited to the already partially occupied band, and the state filling effects related to the Pauli-blocking could be of importance. The main conclusions of this work do not largely depend on the band structure model. In this section and in Sec. III, we disregard, for transparency of presentation, the spin/valley structure of the Bloch functions. We address the trion fine structure in Sec. IV. The smooth envelope function, φ(ρ1,ρ2) in Eq. (1), can be determined from the solution of the corresponding Schrödinger equation either variationally19,44or using exact analytical45,46or numerical47methods. In the Fermi-polaron picture, however, it is instructive to further simplify the model and consider the exciton as a rigid particle, which attracts the electron by short-range forces,29,37 see Ref. 47 for the detailed analysis and extensions of the model. To that end, we present the exciton–electron scattering amplitude in the form T(ε)=V0 1 +DV0[ln∣˜E−ε ε∣+ iπθ(ε)]=1 D1 ln[ε−˜E εexp(1 DV0)]. (2) Here, V0is the bare matrix element of the exciton–electron scat- tering being short-range in the model of the rigid exciton, ε is the kinetic energy of the relative electron–exciton motion, D =m/(2π̵h2)is the reduced electron–exciton density of states with m=memx/mtrbeing the reduced mass ( meis the electron effective mass, mhis the hole mass, mx=me+mhis the exciton mass, and mtr= 2me+mhis the trion mass), θ(x) is the Heaviside θ-function, θ(x) = 1 for x>0 and 0 otherwise, and ˜Eis the cutoff energy, ε≪˜E, which naturally arises in the 2D short-range scattering problem. The cutoff energy introduced in Eq. (2) is on the order of the exciton binding energy Ex: Forε≪Ex, the rigid exciton model is valid, but atε≳Ex, the internal structure of the exciton should be taken into account. At ε>0, the scattering amplitude contains both real and imaginary parts with the latter responsible for the real scattering processes, while at ε<0, the amplitude T(ε) is real. In derivation of Eq. (2), the phase-space filling effects are disregarded; this is just a solution of a two-body “electron+exciton” problem. We are interested in the situation where the electron–exciton interaction is attractive, V0<0. Thus, T(ε) has a pole at a certain neg- ativeεcorresponding to the bound trion state.29,48,49We introducethe trion binding energy from the condition T−1(−Etr) = 0, Etr=˜Eexp(1 DV0)≪˜E∼Ex (3) and recast Eq. (2) in the alternative form T(ε)=1 D1 ln(−Etr ε)≈D−1Etr ε+Etr. (4) The approximate equality holds at ε≈−Etr, i.e., in the vicinity of the trion pole. Note that the model formulated above is valid, provided thatEtr≪Exor∣DV0∣≪1. In this approach to the Fermi-polaron problem, the trion binding energy Etris the free parameter of the model, which should be taken from experiments or microscopic calculations. The relative motion bound state wavefunction reads Φ(ρ)∝K0(ρ/atr),atr=√̵h2 2mE tr, (5) where ρis the electron–exciton relative motion coordinate, K 0 is the modified Bessel function (Hankel function of an imagi- nary argument), and atris the effective trion radius. We also introduce the effective exciton radius by analogy with Eq. (5), ax=̵h/√ (2mE x)≪atr. It is instructive to compare the relative motion wavefunctions in the full model [Eq. (1)] and in the simplified model. For com- parison, we take the envelope function in Eq. (1) in the exponential form φ(ρ1,ρ2)∝e−ρ1/ax−ρ2/atr+e−ρ2/ax−ρ1/atr(6) (see Refs. 19 and 44 for discussion of applicability of such trial function,) and extract the probability density for the bound state as ∣Ψ(ρ)∣2=∫dρ′∣φ(ρ,ρ′)∣2. (7) The functions | Φ(ρ)|2and |Ψ(ρ)|2are plotted in Fig. 2 and quali- tatively agree with each other. Note that using more sophisticated FIG. 2 . Probability density for the exciton–electron relative motion. The blue solid line shows | Φ(ρ)|2calculated after Eq. (5) (a short-range interaction model used in our approach to the Fermi-polaron). The dark red dashed line shows | Ψ(ρ)|2 calculated after Eqs. (6) and (7) (a variational approach to the trion wavefunction). Exciton radius ax=atr/4. The shaded area shows the range of small ρ⩽ax, where Eq. (5) is inapplicable. J. Chem. Phys. 153, 034703 (2020); doi: 10.1063/5.0012475 153, 034703-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp form of the electron–hole scattering amplitude, one can reproduce the results of the trion variational calculations within the scattering amplitude approach.47 In summary, let us highlight the relations between the system parameters where the developed approach is valid. We consider here the 2D semiconductor with free electrons. Importantly, the following hierarchy of energies should take place: Ex≫Etr≫EF, (8a) i.e., the exciton binding energy (typically, hundreds of meV) should exceed by far the trion binding energy (typically, tens of meV), which, in its turn, should be much larger than the electron Fermi energy EF. This inequality can be translated to the equivalent relation between the length scales, ax≪atr≪1/kF, (8b) where kF=√ 2meEF/̵h2with mebeing the electron effective mass is the Fermi wavevector of the electron. Equation (8b) has a trans- parent physical meaning: The exciton can be considered as a small rigid particle, and the trion is formed by attaching the electron to this particle. Furthermore, the Coulomb binding of excitons and tri- ons should occur on the small length scales as compared with the characteristic wavelengths of the resident electrons, in order to be able to treat the exciton interaction with Fermi-sea perturbatively. It is noteworthy, however, that due to numerical factors, Eq. (8b) provides more stringent conditions than the relation between the energies [Eq. (8a)]. Furthermore, for simplicity, we assume that the temperature, T, expressed in the units of energy is low, kBT≪EF, (8c) i.e., the electrons are degenerate, and thermal excitations can be neglected. This condition is, in general, not mandatory and can be easily relaxed. Only quantitative changes are expected for kBT≪Etr. We note that the condition Ex≫EF, Eq. (8a), also implies that the electron–electron interactions in the Fermi-sea are parametrically strong. We disregard the effects of Wigner crystallization of elec- trons because typically there is a parameter range where such collec- tive effects are unimportant even at EF≪Ex.50Therefore, we assume that the electrons can be still treated as weakly interacting quasi- particles with all Coulomb effects included in renormalized values of their parameters (Fermi energy and effective mass). For instance, Refs. 26 and 31 consider the crossover between different regimes of the exciton–electron interactions with variation of electron density. We also note that due to the conditions Etr≫EF, one can, at least in the first approximation, disregard the state-filling effects in the case of Mo-based 2D TMDCs where the optical transition involves the trion formation in the already occupied valley. III. OSCILLATOR STRENGTH The key parameter controlling the optical response of the exci- tonic complexes in semiconductors is the oscillator strength, which describes the efficiency of the light–matter interaction. In this sec- tion, we calculate the oscillator strength in both the trion and Fermi-polaron approaches and compare the results. The resonant trion excitation can be considered as a process where (i) an exciton is created in the virtual intermediate state, and (ii) the exciton picks up an electron from the Fermi-sea to form atrion. Thus, a finite density of resident electrons is needed to make this process possible. A. Fermi-polaron approach In the Fermi-polaron approach, the optical response function can be readily expressed via the exciton Green’s function,51 Gx(ε;k)=1 ε−Ek−Σ(ε;k)+ iΓ. (9) Here, Ekis the exciton dispersion in the TMDC ML plane, kis the exciton in-plane wavevector, Γis the exciton damping rate caused, e.g., by the exciton–phonon interaction, inhomogeneous broaden- ing, etc.,52andΣ(ε;k) is the exciton-self energy resulting from the interaction with resident electrons. We take it in the simplest form using Eq. (2) (see Appendix A for more detailed discussion), Σ(ε;k)=T(ε)Ne, (10) where Neis the electron density. The corresponding optical suscep- tibility in a given circular polarization at the normal incidence of radiation can be written as Π(ω)=fxGx(̵hω−Eg+Ex; 0),fx=∣Mr∣2∣φx(0)∣2. (11) Here, fxis the effective exciton oscillator strength, Mris the matrix element of the interband transition (per photon), φx(ρ) is the exci- ton relative motion envelope function, and Egis the bandgap, see Refs. 53 and 54 for details. In this section, we disregard the spin and valley fine structures of the trion, and the role of these intrinsic degrees of freedom is discussed in detail in Sec. IV. In the vicinity of the exciton resonance where̵hω≈Eg−Ex, the self-energy is small, and excitonic states are almost unaffected by the electron gas with respect to the exciton binding energy and wavefunction,55see, however, more details below and Eq. (17) for the analysis of the oscillator strength, and Π(ω)≈fx ̵hω−Eg+Ex+ iΓ+Σ(̵hω−Eg+Ex). (12) The main important effect here is the exciton damping induced by the electron–exciton scattering: Qualitatively, it follows from Eq. (2) where T(ε) has an imaginary part at ε>0 responsible for the scatter- ing. Quantitative discussion of this and related issues is beyond the scope of the paper.37,56,57In addition, the exciton oscillator strength decreases as it is transferred to the attractive Fermi-polaron (a trion) (see below). The resonance in Π(ω) at̵hω≈Eg−Ex, and Eq. (12) is termed the repulsive Fermi-polaron (see below). Importantly, because Σ≠0, particularly due to the pole in Σ(ε) atε=−Etr, another resonance—termed the attractive Fermi- polaron—appears in the susceptibility at̵hω≈Eg−Ex−Etr. Indeed, making use of approximate Eq. (4), we arrive at (cf. Refs. 29, 36, and 37) Π(ω)≈ftr ̵hω−Eg+Ex+Etr+Ne/D+ iΓNe/D, (13) where the effective oscillator strength ftr=Ne DEtr∣Mr∣2∣φx(0)∣2=4πNea2 trfx (14) J. Chem. Phys. 153, 034703 (2020); doi: 10.1063/5.0012475 153, 034703-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp and fxis introduced in Eq. (11) and corresponds to the absence of doping. Thus, part of the exciton oscillator strength is shuf- fled toward the Fermi-polaron peak. The peak position is at ̵hω=Eg−Ex−Etr−Ne/D. The shift of the peak with respect to the trion energy ( Eg−Ex−Etr) is proportional to the electron Fermi energy. Namely, the quantity δ=Ne/Dcan be recast as δ=EFme m=EFmtr mx. (15) We recall that here mtr= 2me+mhis the trion translational mass, andmx=me+mhis the exciton translational mass. This shift is assumed to be small, δ≪Etr, cf. Eq. (8a); otherwise, the form of the self-energy used here is insufficient (see Appendix A for details). It is instructive to introduce, based on the considerations above, even more simplified model of the Fermi-polaron. To that end, we use an approximate form of the scattering amplitude (4) across the whole relevant energy range and present the exciton Green’s function in the form Gx(ε)=1 ε+ iΓ−NeD−1Etr ε+Etr+ iγ. (16) To shorten the notations and for simplicity, we put k= 0, but for generality, we introduced the trion damping γ. The Green’s function (16) describes two coupled oscillators: One describes the exciton, and another one describes the trion. The parameter g =√ NeD−1Etr∼√EFEtrplays a role of the coupling constant. We consider the regime where g≪Etr(analog of the weak coupling; oth- erwise, the simplifications behind the model make it inapplicable) and recast Eq. (16) in the form Gx(ε)≈1−D−1Ne/Etr ε−NeD−1+ iΓ+D−1Ne/Etr ε+Etr+NeD−1+ iγ. (17) Equation (17) makes it possible to introduce the notions of the attractive andrepulsive Fermi polarons as the poles of Gxatε≈−Etr (in the vicinity of the trion resonance) and ε≈0 (in the vicinity of the exciton resonance). The attractive polaron state stems from the bound trion and corresponds to the exciton strongly correlated with the resident electrons. The repulsive polaron states describe the continuum-like exciton–electron states, i.e., the exciton state perturbed by the Fermi-sea of electrons. B. Trion approach Now, let us calculate the oscillator strength in the trion approach. For rigorous calculation, one has to take into account the presence of the Fermi-sea explicitly. It can be conveniently done in the secondary quantization approach. Light–matter coupling Hamiltonian describing optical transi- tions at the normal incidence of radiation reads Hrad=Mr∑ ke,kha† keb† khδke+kh,0+ h.c., (18) where the operators ak(a† k) correspond to an electron, bk(b† k) cor- respond to the hole, and keandkhare the in-plane wavevectors of the electron and hole. As discussed above, we disregard the spin and valley structure of the electronic states (it is considered in detail in Sec. IV).It is convenient to calculate the matrix element of the exciton optical generation. Within the secondary quantization approach, the exciton wavefunction can be written as6 ∣X⟩=∑ ke,khFx(ke,kh)a† keb† kh∣vac⟩, (19) where | vac⟩is the state of the 2D crystal with empty conduction and filled valence bands and Fx(ke,kh) is the exciton envelope function in the kspace. The matrix element of the optical transition to the exciton state reads ⟨X∣Hrad∣vac⟩=Mr∑ kF∗(k,−k)=Mrφ∗ x(0). (20) In the last equation, we took into account the relation φx(ρ) =∑kF(k,−k) exp(−ikρ). Thus, the effective oscillator strength of the exciton is given by fx=∣Mr∣2∣φx(0)∣2, (21) in full agreement with Eq. (11). Let us now consider the X−trion. Its wavefunction in the k space can be written via the Fourier transform of Eq. (1), ∣T⟩=∑ k1,k2,khFtr(k1,k2,kh)a† k1˜a† k2b† kh∣vac⟩. (22) We used ˜a† k2to denote the creation operator of one of the electrons to highlight that it is in the different spin or valley state as compared with another electron. For the trion to be formed, an electron with the wavevector keshould be present in the system; thus, the initial state is ∣e⟩=˜a† ke∣vac⟩. (23) At small electron densities where EF≪Etrand, correspondingly, kF≪a−1 tr, the effects of Pauli blocking in the final (trion) state can be disregarded. Neglecting, as discussed before, the photon momentum, we cal- culate the matrix element of the Hamiltonian (18) and arrive at, in agreement with Refs. 58 and 59, ⟨T∣Hrad∣e⟩=δke,KMr∑ k2,khF∗(−kh,k2,kh) =δke,KMr∫φ∗(0,ρ)eiKρdρ. (24) We stress that due to the momentum conservation law, the in-plane wavevector of the electron is equal to the wavevector of the trion translational motion, ke=K. The TMDC ML susceptibility in the vicinity of the trion reso- nance can be evaluated taking into account all possible initial states for electrons and, correspondingly, all possible wavevectors of the trions in the final state. Making use of the Fermi’s golden rule, we recast the imaginary part of the susceptibility in the form −Im{Π(ω)}=∑ K∣Mr∣2nK∣∫dρφ(0,ρ)exp(−iKρ)∣2 ×γ (̵hω−Eg+Ex+Etr+δK)2+γ2, (25) with nKbeing the electron distribution function, Ne=∑KnK, and δK=̵h2K2 2memx mtr. (26) J. Chem. Phys. 153, 034703 (2020); doi: 10.1063/5.0012475 153, 034703-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp This quantity takes into account the energy and momentum conser- vation in the process of picking the electron from the Fermi-sea and forming the trion. Equation (25) is in agreement with Ref. 27. Neglecting the trion dispersion and assuming, similarly to Sec. III A [Eq. (8a)], that the electron Fermi energy is much smaller than the trion binding energy, we arrive at60 ft=∣Mr∣2∑ KnK∣∫dρφ(0,ρ)exp(−iKρ)∣2 ≈Ne∣Mr∣2∣∫φ(0,ρ)dρ∣2 , (27) where in the latter approximate equation, we have made a replace- ment exp(−iKρ)→1 valid at very low electron densities [ K∼kF ≪a−1 tr, Eq. (8b)]. Strictly speaking, the possibility to neglect δKin the denominator is possible if, in addition to Eq. (8), we assume that EF≲γ, i.e., if the broadening of the trion line is sufficiently large. To provide a link with the Fermi-polaron approach, we use the trial function (6) and evaluate the trion oscillator strength from Eq. (27) with the following result: ftr=Ne∣Mr∣2(a2 x+a2 tr)2 a2 xa2 tr 8+2a4 xa4 tr (ax+atr)4≈4πNea2 trfx. (28) The last approximate equality holds at atr≫ax[Eq. (8b)] and to derive it, we used the hydrogenic form of the exciton envelope func- tionφx(ρ)=√ 2/πa2xexp(−ρ/ax)with the same exciton radius. Noteworthily, Eq. (14) derived in the Fermi-polaron approach and Eq. (28) derived in the trion approach agree at atr≫ax. At a fixed electron density, the trion oscillator strength scales as (atr/ax)2(see Fig. 3). We stress that the agreement of Eqs. (14) and (28) is not a coincidence. In both cases, the process of virtual exciton creation by a photon and subsequent binding with the electron is described, which corresponds to the resonant excitation in the vicinity of the trion resonance. Qualitatively, this explains the ratio ftr/fx∼Nea2 tr since the trion formation is only possible if there is an electron in FIG. 3 . Effective trion oscillator strength as a function of the ratio of trion and exciton radii. The dark red dashed line shows the results of the calculation after Eq. (28), and the blue solid line shows a2 tr/a2 xasymptotics.the area∼a2 trin the vicinity of the exciton. This proportionality rela- tion is used in conventional semiconductor quantum wells to deter- mine the electron density optically.61,62Thus, oscillator strengths of the trion (an attractive polaron) optical transitions can be calcu- lated in any approach with the same result at small resident electron densities. It follows from Eqs. (25) and (26) that the presence of elec- trons broadens and shifts the trion resonance in the susceptibil- ity as compared to its initial position at̵hω−Eg+Ex+Etr. The origin of the shift is somewhat similar to the “polaron” shift in Eq. (15), and the magnitude of the effect is, however, dif- ferent. The difference is related with simplifications used here. An accurate comparison requires the calculation of the difference between the trion and exciton (attractive and repulsive polarons) positions in the spectra. This requires going beyond the approx- imate form for the scattering amplitude [Eq. (4)] used above in the Fermi-polaron approach and a self-consistent determination of the exciton self-energy (see Appendix A). In the trion approach, the electron–exciton and electron–trion interaction-induced renor- malizations of the exciton and trion energies were neglected and should be taken into account. This is beyond the scope of this work. IV. FINE STRUCTURE AND ZEEMAN EFFECT In this section, we address the trion and Fermi-polaron energy spectrum fine structures and the Zeeman effect in TMDC MLs. We start with the situation where the external magnetic field is absent. We recall that the short-range contributions to the electron–electron interaction and electron–hole interaction split the intra- and intervalley trion states.19,63,64We denote the intravalley state as X− 1, the intervalley state as X− 2, and their binding ener- gies (including the short-range contributions) as Etr,1and Etr,2, respectively. Accordingly, the trion radii are different as well and denoted as atr,1and atr,2, respectively. Thus, the trions/Fermi- polarons in tungsten-based MLs should appear as a doublet split by |Etr,1−Etr,2| with slightly different oscillator strengths of individual peaks. Figure 4 demonstrates the optical absorption spectrum as a function of energy and electron density. It is calculated by extending Eqs. (11) and (16) to account for two trion states, G+ x(ε)=1 ε+ iΓ−NeD−1Etr,1 ε+Etr,1+ iγ−NeD−1Etr,2 ε+Etr,2+ iγ, (29) with Nebeing the electron density per valley. The spectrum in Fig. 4 shows the strong excitonic feature (a repulsive polaron) and two low-energy trion features (attractive polarons). The appearance of the trion oscillator strength with an increase in the electron density is clearly seen, and it is described by the general model outlined in Sec. II. Note that with an increase in the Fermi energy, the indirect coupling between X− 1andX− 2appears via their interaction with exci- tons, making redistribution of the oscillator strengths non-trivial. Here, we abstain from detailed discussion of the oscillator strengths of the trion (an attractive polaron) features, see Ref. 64 for more detailed analysis at low densities. In addition, the polaron-like repul- sion of the peaks in the optical spectrum controlled by the parameter J. Chem. Phys. 153, 034703 (2020); doi: 10.1063/5.0012475 153, 034703-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . False color plot (log-scale of intensity) of the optical absorption spectrum given by−Im{Π(ω)} calculated after Eqs. (11) and (29) in the absence of the mag- netic field for varied electron density (per valley). Electron and hole masses are me=mh=m0/2 with m0being free electron mass, Etr,1= 25 meV, Etr,2= 35 meV (exaggerated for illustrative purposes), and γ=Γ= 1 meV. The inset shows the plot in the vicinity of the trion (Fermi-polaron) resonances at low doping in the linear scale. Energy is reckoned from the exciton resonance energy at negligible doping. δin Eq. (15) is clearly seen. We note that the presence of elec- trons can also affects the bandgap and exciton and trion binding energies and provides further modifications of the absolute posi- tions of the lines in the spectrum and also of the relative distance between the neutral and charged exciton (repulsive and attractive) polaron lines. Let us now discuss the Zeeman effect in the presence of an external magnetic field Bapplied along the ML normal. We assume that the field is sufficiently small to disregard the orbital effects of the field both on the excitons and trions and on the electrons. It is justi- fied at | eB/mec|τe≪1, whereτeis the electron scattering time or, at finite temperature T, at |eB/mec|≪kBT/̵h. Thus, the magnetic field produces the Zeeman splitting of the electron and hole states, lifting the Kramers degeneracy between the states in the opposite valleys, and, due to the splitting, the valley polarization of the resident elec- trons. For the valence band states, the Zeeman effect (in the electron representation) is described by the Landé factor gv, and the splitting is given by ΔZ,v=gvμBB. (30a) It is responsible for the energy shift of the valence band in the K+ valley with respect to the valence band in the K−valley. Note that ΔZ,v>0 corresponds to the K+valence band top being above the K− valence band. For the conduction band, there are two spin subbands. Thus, we introduce two Landé factors, gcand g′ c, responsible for the splitting of the Kramers-degenerate pairs of the top and bottom subbands in K±valleys, respectively, ΔZ,c=gcμBB, (30b) Δ′ Z,c=g′ cμBB. (30c)The sign convention is the same, and ΔZ,c>0 (Δ′ Z,c>0) corresponds to the K+state higher in energy as compared with the K−state in the corresponding subband. Since in our model the topmost subbands have the same spin as the valence band top, the splitting of the optical transitions is given by the combination of ΔZ,candΔZ,v, giving rise to the bright exciton Zeeman splitting,65 ΔZ,x=ΔZ,c−ΔZ,v=gxμBB, (31) with gx=gc−gv. As we also assume that EF≪Δc, only the bottom conduction sub- bands are occupied with the electrons. Correspondingly, the Zee- man effect in the bottom subbands gives rise to the electron val- ley polarization ( Ne,K±is the electron density in the corresponding valley), Pv=Ne,K+−Ne,K− Ne,K++Ne,K−=−1 2Δ′ Z,c EF. (32) In derivation of Eq. (32), we assumed that kBT≪EF[Eq. (8c)] and that | Δ′ Z,c|⩽2EF(EFcorresponds to the magnetic-field-less case). If the latter inequality is not satisfied, Pv=−signΔ′ Z,c. Here, we also neglect the exchange renormalization of the electron g-factor.66 A. Trion approach It follows from Sec. III B that in the course of the trion for- mation, an electron is picked up from the Fermi-sea. Similarly, the trion recombination returns an electron back. Thus, the splitting of the trion transition lines is given by the difference of the Zeeman splitting of the three-particle complex, X− 1orX− 2, ΔZ,tr,1=ΔZ,c+Δ′ Z,c−ΔZ,v, ΔZ,tr,2=ΔZ,c−Δ′ Z,c−ΔZ,v,(33) and that of the charge carrier, which remains in the system after the recombination. The latter is Δ′ Z,cif the electron remains in the K+- valley or−Δ′ Z,cif the electron remains, in the K−–valley Fig. 1. Thus, in both cases, the splitting of the trion line in the optical spectrum is the same as for the neutral exciton,65 Δtr,1=Δtr,2=ΔZ,c−ΔZ,v=gxμBB. (34) The effects of the Coulomb interaction and band nonparabolic- ity that could result in the renormalization of the trion g-factor as compared to that of the exciton are disregarded here. Addi- tional renormalization of the g-factor related to the fact that the electron is taken and returned from the Fermi-sea and having the same origin as the trion energy shift [Eq. (26)] is discussed below in Sec. IV B. Importantly, the Zeeman splitting of the resident electrons and corresponding valley polarization [Eq. (32)] results in the differ- ence of the oscillator strengths of the transitions. Particularly, in J. Chem. Phys. 153, 034703 (2020); doi: 10.1063/5.0012475 153, 034703-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp accordance with Eq. (28) [cf. Eq. (14)], for transitions active in the σ+ polarization, the oscillator strengths of X− 1andX− 2are proportional toNe,K+andNe,K−, respectively. Conversely, for transitions active in theσ−polarization, the oscillator strengths of X− 1and X− 2are proportional to Ne,K−andNe,K+, respectively. Thus, at a given circu- lar polarization, the oscillator strengths of the intra- and intervalley trions will demonstrate an opposite dependence on the magnetic field: One of the trions gains the oscillator strength due to the elec- tron valley polarization, while another one loses it. In the opposite polarization, the behavior is opposite.B. Fermi-polaron approach The trion picture outlined above is corroborated by the calcu- lation in the Fermi-polaron approach. Extending Eqs. (11) and (16) to allow for the valley degrees of freedom, polarization, and Zeeman effect, we arrive at the following expressions for the susceptibilites in σ±circular polarizations: Π±(ω)=fxG± x(̵hω−Eg+Ex; 0), (35) where the exciton Green’s functions read G+ x(ε)=1 ε−1 2ΔZ,x+ iΓ−Ne,K+D−1Etr,1 ε−1 2(ΔZ,tr,1−Δ′c)+Etr,1+ iγ−Ne,K−D−1Etr,2 ε−1 2(ΔZ,tr,2+Δ′c)+Etr,2+ iγ, (36a) G− x(ε)=1 ε+1 2ΔZ,x+ iΓ−Ne,K−D−1Etr,1 ε+1 2(ΔZ,tr,1−Δ′c)+Etr,1+ iγ−Ne,K+D−1Etr,2 ε+1 2(ΔZ,tr,2+Δ′c)+Etr,2+ iγ, (36b) and Ne,K±=Ne(1±Pv). Figure 5 demonstrates the circular dichroism of absorption, Pc(B,̵hω)=Im{Π+(ω)}−Im{Π−(ω)} Im{Π+(ω)}+ Im{Π−(ω)}, (37) calculated within the Fermi-polaron model. In this calculation, we took the set of g-factors: gv= 4,gc= 0, and g′ c= 2, which gives the exciton Landé factor gx=−4. We stress that the values of g-factors we use are selected here for illustrative purposes, see detailed discus- sions and microscopic approaches to calculate the Zeeman effect in Refs. 67–71. In Fig. 5, the features in the circular dichroism related to the exciton (a repulsive polaron) and trion (an attractive polaron) states are clearly seen. Let us analyze the cases of low and moderate elec- tron densities in more detail. At relatively low electron densities [Fig. 5(a)], the trion (an attractive polaron) features X− 1and X− 2 provide significant circular dichroism with opposite signs at the res- onances. The Zeeman splitting of the trions is not very prominent here. This is because for the considered set of parameters, the com- plete valley polarization of the resident electrons is achieved at a relatively low magnetic field of about 1.65 T, where the Zeeman split- ting of the resonances ( ≈0.38 meV) is smaller than the linewidth (1 meV). Thus, only one Zeeman component of each trion (an attractive polaron) is optically active, namely, the one that requires the electrons remaining in the occupied conduction subband. In contrast, at moderate electron densities [Fig. 5(b)], the electron val- ley polarization is far from complete even at the highest magnetic fields. In this case, both the Zeeman states of each X− 1,2trions are opti- cally active and are visible in the spectra, providing a sign-alternating behavior of the circular polarization at each resonance. The exci- ton oscillator strength just weakly depends on the electron subband occupations [cf. Eq. (17)], and both the Zeeman components of theexciton are present in the circular dichroism spectrum at both the low and moderate electron densities, Figs. 5(a) and 5(b). The inter- play of the resident electron valley polarization and Zeeman splitting of the excitonic species provides a complex dependence of Pcon the energy and field shown in Fig. 5. The situation could be even more involved in the case of photoluminescence experiments.72 It is noteworthy that the valley polarization of the elec- tron gas results in the renormalization of the trion g-factor. Indeed, the density-dependent shifts of the attractive polaron energy ∝Ne,K±D−1[see Eq. (17) and discussion in Sec. III A] in the pres- ence of the magnetic field differ for different Zeeman components, and the resulting corrections to the X− 1,2state Zeeman splittings are given by±PvNeD−1. These corrections could be sizable for moder- ate electron densities (at small densities, these corrections quickly saturate) and could explain the observed65,73,74differences between the bright exciton and trion g-factors. While these corrections are straightforwardly derived in the Fermi-polaron approach, they can also be estimated in the trion approach if one takes into account the fact that for the trion formation, the electron is picked up from the Fermi-sea, which results in the shift of the trion resonance [cf. Eq. (26)]. We stress that at low electron densities where both the approaches merge, this contribution to the g-factor could be impor- tant only at small magnetic fields. Again, the key features of the trion (an attractive polaron) fine structure can be evaluated in both the trion and Fermi-polaron models with the same result, provided that the resident electron density is low enough and the conditions (8) are satisfied. V. PHOTOLUMINESCENCE AT NON-RESONANT EXCITATION Above, we discussed resonant optical properties of TMDC MLs in the spectral range of neural and charged excitons. Particularly, −ImΠgiven by Eq. (11) provides the absorption spectrum via the J. Chem. Phys. 153, 034703 (2020); doi: 10.1063/5.0012475 153, 034703-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . False color plot of the circular dichroism of absorption, Pc(B,̵hω), calcu- lated after Eq. (37) for relatively low doping, Ne,tot=Ne,K++Ne,K−=2×1010 cm−2, (a) and for moderate doping, Ne,tot= 2×1011cm−2. The dotted lines show the positions of the Zeeman-split states calculated after Eqs. (31) and (34). The zero-field positions of the resonances are adjusted taking into account the ∝Ne,K±D−1shifts of the states, Eq. (17). The Landé factors used in the calcu- lation are as follows: gv= 4,gc= 0, and g′ c= 2. The remaining parameters of the calculation are the same as in Fig. 4. exciton Green’s function. An alternative experimental approach to study the Coulomb-bound electron–hole complexes is to observe photoluminescence (or resonant light scattering) under non- resonant excitation where the electron–hole pairs or excitons are formed with high excess energy and eventually relax to the low- energy radiative states. Below, we briefly discuss the trion formation process and photoluminescence effect from the trion and Fermi- polaron viewpoints. In this section, we disregard the complex band structure of the TMDC MLs.A. Trion approach Here, we analyze the formation of the trions from excitons in 2D TMDC where the energy difference between the exciton and trion states is close to the energy of the optical phonon,̵hΩ. We assume that the main process governing the trion photolumines- cence is related to the trion formation and its subsequent radia- tive recombination, leaving out the discussion of the thermaliza- tion issues.64We develop the model of the capture of the electron by an exciton to form a trion following the general approach in Ref. 75. The exciton–electron interaction is modeled as a zero-radius potential, Sec. II. A free electron wavefunction with the in-plane wavevector kreads Φk(ρ)=eikρ+fk√ πk 2iH(1) 0(kρ)≈eikρ+fkeikρ √ −iρ. (38) Here, for convenience, we used the scattering amplitude fkin the coordinate normalization,48 fk=−√ 2π kDT(ε),ε=̵h2k2 2m, (39) andT(ε) is given by Eq. (2). Note that in our model, the interac- tion takes place only in the channel with the angular momentum component lz= 0. It is instructive to check the orthogonality rela- tion between the bound [electron-in-trion, Eq. (5)] and free-electron [Eq. (38)] states, which is necessary to properly calculate the capture rate, ∫dρΦ0(ρ)Φk(ρ)={2√πæ k2+ æ2+ 2fk√ 2kæ k2+ æ2ln(ik æ)}=0. (40) Here, æ =a−1 tr. Making use of the explicit form of fk= −√ π/2kln−1(iæ/k)[Eqs. (4) and (39)], one can see that the expres- sion in curly brackets of Eq. (40) is identically zero. Under non-resonant excitation, the trion is formed when exci- ton captures the resident electrons and emits optical phonon to ensure the energy conservation. This process can be considered as a trapping of the electron by the effective potential well created by the exciton accompanied by the phonon emission. The matrix element of the optical phonon emission that couples free and bound states can be written in the simplest approximation as Mq k=C0(q)∫dρΦ0(ρ)e−iqρΦk(ρ), (41) with qbeing the phonon wavevector and C0(q) being a parameter (see Refs. 76 and 77 for the explicit form of the Fröhlich interaction in 2D systems). Note that at q= 0, the matrix element (41) vanishes due to the orthogonality of the wavefunctions. Now, we are able to calculate the trion formation rate (per exciton with the given energy Ek) making use of the Fermi’s golden rule,75 νx(Ek)=2π ̵hNe∑ q∣Mq k∣2δ(Ek−̵hΩ +Etr), (42) where the δ-function describes the energy conservation and we neglected the trion dispersion. Assuming that C0(q) weakly depends J. Chem. Phys. 153, 034703 (2020); doi: 10.1063/5.0012475 153, 034703-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp onqand replacing it by its q= 0 value, we can perform the summation in Eq. (42) over the phonon wavevector with the result νtr(Ek)=2π ̵h∣C0∣2IkNeδ(Ek−̵hΩ +Etr), (43) where Ik=∫dρ∣Φ0(ρ)Φk(ρ)∣2. (44) The trion generation rate is given by Wtr=∑ kνx(Ek)nx(Ek), (45) where nx(E) is the exciton distribution function formed as a result of the non-resonant excitation. The decay rate of the trion is 2 γ/̵h. Thus, the steady-state trion population is̵hW tr/(2γ). Correspond- ingly, the trion photoluminescence spectrum can be presented as [cf. Eq. (25)] I(̵hω)∝1 πγ (̵hω−Eg+Ex+Etr)2+γ2̵hW tr 2γ. (46) This treatment agrees with results of the approach developed in Ref. 78, where the processes of exciton recombination via capture to the localized electron centers have been considered. Equation (46) is valid, provided that the phonon-induced trion dissociation rate, Wdiss∝Wtrexp(−̵hΩ/kBT), is slow as compared with its decay rate 2γ/̵h. In Eq. (46), we neglected the energy shifts and recoil effects [cf. Eqs. (25) and (26)], see Refs. 59 and 79 for detail. As discussed before, the latter approximation is strictly justified at EF≲γ. B. Fermi-polaron approach In the Fermi-polaron approach, the trion generation and pho- toluminescence can be readily calculated using the Keldysh dia- gram technique following Refs. 80 and 81. We introduce the Green’s function G−+ x(ε,k)=n(ε)[G∗ x(ε,k)−Gx(ε,k)], (47) which accounts for the non-equilibrium distribution of the quasi- particles n(ε). The remaining Green’s functions in the Keldysh tech- nique in the lowest order in n(ε) readG−− x=GxandG++ x=−G∗ x. The photoluminescence spectrum is given as follows: I(̵hω)∝fxIm{G−+ x(̵hω−Eg−Ex, 0)}. (48) If the excitonic subsystem was in thermal quasi-equilibrium, n(ε)∝exp[(μc−ε)/kBT]. Below, like in Sec. V A, we focus on the non-equilibrium situation where the photoluminescence of Fermi- polarons (trions) is controlled by the optical phonon-induced tran- sitions. Following the rules of the Keldysh technique, we evaluate G−+ x(ε, 0)accounting from the phonon-assisted transitions from the higher-energy excitonic states in the first order, δG−+ x(ε, 0)=−G−− x(ε, 0)Σ−+G++ x(ε, 0) =1 ̵h∑ k,q∣Mq,eff k∣2Im{G−+ x(ε+̵hΩ,k)}∣Gx(ε, 0)∣2. (49) Here, we expressed the self-energy Σ−+via the Green’s function G−+ x and the effective matrix element (vortex) of the exciton–phononinteraction Mq,eff k, which should be calculated with an allowance for the exciton–electron interaction (see Appendix B). This approxima- tion corresponds to the neglect of the phonon-induced transitions to the higher energies. At ε≈−Etr, the Green’s function G−+ x(ε+̵hΩ,k) can be replaced by 2 πiδ(Ek−̵hΩ +Etr)n(Ek), while in the evaluation of∣Gx(ε, 0)∣2, one has to keep the contribution linear in Neresult- ing from the interference of the first and second terms in Eq. (17). Neglecting the term NeD−1 ein the denominator, we arrive at Eq. (46), where Wtr=Ne DEtr2π ̵h∑ k,q∣Mq,eff k∣2n(Ek)δ(Ek−̵hΩ +Etr). (50) To establish the agreement of the approaches, we need to calcu- lateMq,eff kand compare Eq. (50) with the result of the Fermi’s golden rule [Eq. (45)]. The calculations presented in Appendix B show that ∣Mq,eff k∣2=DEtr∣Mq k∣2. (51) Therefore, Eqs. (45) and (50) are consistent, and both the approaches provide the same result. Thus, it is a matter of convenience to select the approach to calculate the photoluminescence, provided that Eqs. (8) are fulfilled, and more complex processes of trion–electron scattering can be neglected. VI. CONCLUSION To conclude, we have demonstrated by several examples that optical properties of charged excitons in transition metal dichalco- genide monolayers can be described both in the trion and in the Fermi-polaron approaches, provided that the following hierarchy of energy is fulfilled: The exciton binding energy exceeds by far the trion binding energy, which, in its turn, exceeds the elec- tron Fermi energy. The direct analysis of (i) the optical transi- tion oscillator strengths, (ii) the spectrum fine structure and Zee- man effect, and (iii) the photoluminescence demonstrates that these effects can be adequately described both in the trion and in the Fermi-polaron pictures taking into account simplifications behind each approach. There are several interesting and important problems to be addressed in the future. On the one hand, it is desirable to explore the high electron density regime where the Fermi energy of the charge carriers is comparable (or exceeds) the trion binding energy. In addition, the description of the trion/Fermi-polaron transport properties, e.g., the effect of the photoconductivity [cf. Ref. 82] in the spectral range of charged excitons resonance, is an interesting and important problem to be addressed in the future. It is like- wise important to search for the experimentally accessible situations where the exciton–electron correlations in two-dimensional semi- conductors are so strong that the simplified approaches outlined above become inapplicable. ACKNOWLEDGMENTS The author is grateful to A. Chernikov, A. Imamoglu, D. Reich- man, M. A. Semina, T. Smolenski, and R. Schmidt for valuable dis- cussions. This work was partially supported by the Russian Science Foundation (Project No. 19-12-00051). J. Chem. Phys. 153, 034703 (2020); doi: 10.1063/5.0012475 153, 034703-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp APPENDIX A: EXCITON SELF-ENERGY Let us discuss in more detail the approximations behind Eq. (10), which states Σ(ε,k) =T(ε)Ne. Such an expression cor- responds to the Hartree type of the self-energy where the corre- lations and self-consistent effects are disregarded. If the exciton– electron interaction was weak and described by the matrix element V0and the perturbation theory in the electron–exciton interaction was applicable, this expression would correspond to the first-order perturbation theory, Σ(1)=V0Ne. (A1) Following general arguments,83one could replace in Eq. (A1) the perturbation matrix element V0by the full scattering amplitude T(ε) in order to account for the main contributions due to the higher orders in V0and arrive at Eq. (10). Although being the simplest possible approximation, Eq. (10) captures the key effects and allows for the fully analytical solu- tion of the problem. More sophisticated treatment of the prob- lem29,38,49,82,84–86demonstrates that qualitative differences could appear. The state-of-the-art approach82,86is to self-consistently solve the following set of equations: Σ(ε,k)=S−1∑ pnpT(ε+̵h2p2 2me,k+p), (A2a) T−1(ε,k)=−Dln(¯E Etr)−S−1∑ p(1−np)G(ε−̵h2p2 2me;k−p), (A2b) with Sbeing the normalization area and npbeing the electron distribution function. Note that neglecting npin Eq. (A2b), which could be justified forEF≪Ex,Etrwhere the main part of the integration involves empty states, substituting the bare exciton Green’s function, and taking k= 0, we arrive at Eq. (4). Furthermore, if in integrat- ing Eq. (A2a) we disregard the electron dispersion, we arrive at Eq. (10) (for the full analytical solution for T, see Refs. 37 and 86). This approximation, however, overestimates the exciton–electron interaction and, particularly, the polaron repulsion parameter δin Eq. (15). This is because at ε≈−Etr, the scattering amplitude has a pole and strongly depends on its arguments. One possible extension is to introduce an additional parameter of the theory 0 <ξ<1, which phenomenologically takes into account the self-consistent effects, and reduce the exciton self-energy as Σ(ε;k)=ξT(ε)Ne. (A3) In fact, this is equivalent to an artificial decrease in the electron density in the expressions presented in the main text. APPENDIX B: CALCULATION OF THE EFFECTIVE MATRIX ELEMENT OF FERMI-POLARON INTERACTION WITH PHONONS In order to determine Mq,eff k, we calculate the self-energies Σ−−(ε,k), describing the damping of excitons with ε≈0 due to the phonon emission, and Σ−+(ε,k), describing the generation of tri- ons withε≈−Etrdue to the phonon absorption. We need the fullexciton Green’s function Gx(ε;k,k′), which depends on the “initial” and “final” wavevectors of the excitons. For the case of an exciton interacting with a single electron, it reads29 G(1) x(ε;k,k′)=δk,k′Gε(k)+T(ε) SGε(k)Gε(k′), (B1) with T(ε) being the scattering amplitude, Sec. II, Sbeing the normal- ization area, and Gε(k)=1 ε−Ek+ iΓ. The self-energy of the exciton related to the phonon emission can be recast as Σ−−=∑ q∣C0(q)∣2Im⎧⎪⎪⎨⎪⎪⎩∑ k′,p′,p′ 1,p1δp1,p′ 1+qδp′,k−q ×[δk,k′+T(ε) SGε(k′)][δp,p1+T(ε) SGε(p1)] ×G(1) x(ε−̵hΩ,p′,p′ 1)⎫⎪⎪⎬⎪⎪⎭. (B2) Atε−̵hω≈−Etr, it is sufficient to account for the term ∝T(ε− ̵hω) inG(1) x(ε−̵hΩ,p′,p′ 1). Furthermore, allowing for trion damp- ing, we present T(ε−̵hΩ)=−iπD−1Etrδ(ε−̵hΩ +Etr)and perform summation over k′,p′,p′ 1,p1transforming the Green’s functions to the real space. Taking into account the finite density of electrons (replacement S−1→Ne) and using Eqs. (38), (40), and (41), we arrive at Σ−−(ε,k)=−πNe∑ q∣Mq k∣2δ(ε−̵hΩ +Etr). (B3a) On the other hand, Σ−−=∑ k,q∣Mq,eff k∣2Im{G−− x(ε−̵hΩ,k)}. (B3b) Equations (B3) are consistent, provided that ∣Mq,eff k∣2=DEtr∣Mq k∣2. (B4) Similar transformations allow us to present ( ε≈−Etr) Σ−+(ε, 0)=2π∑ k,qn(Ek)∣Mq k∣2δ(Ek−̵hΩ−ε), (B5a) which with an allowance for Eq. (B4) is consistent with the definition [Eq. (49)], Σ−+(ε, 0)=∑ k,q∣Mq,eff k∣2Im{G−+ x(ε+̵hΩ,k)}. (B5b) DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. J. Chem. Phys. 153, 034703 (2020); doi: 10.1063/5.0012475 153, 034703-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp REFERENCES 1J. Frenkel, “On the transformation of light into heat in solids. I,” Phys. Rev. 37, 17–44 (1931). 2G. H. Wannier, “The structure of electronic excitation levels in insulating crystals,” Phys. Rev. 52, 191–197 (1937). 3N. F. Mott, “On the absorption of light by crystals,” Proc. R. Soc. London, Ser. A 167, 384–391 (1938). 4E. F. Gross and N. A. Karrjew, “Light absorption by cuprous oxide crystal in infrared and visible part of the spectrum,” Dokl. Akad. Nauk SSSR 84, 471 (1952). 5Excitons , edited by E. I. Rashba and M. D. 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10.0001550.pdf

Low Temp. Phys. 46, 830 (2020); https://doi.org/10.1063/10.0001550 46, 830 © 2020 Author(s).Wannier-Stark ladder spectrum of Bloch oscillations of magneto-dipole spin waves in graded 1D magnonic crystals Cite as: Low Temp. Phys. 46, 830 (2020); https://doi.org/10.1063/10.0001550 Submitted: 19 June 2020 . Published Online: 27 August 2020 E. V. Tartakovskaya , A. S. Laurenson , and V. V. Kruglyak ARTICLES YOU MAY BE INTERESTED IN Non-uniform along thickness spin excitations in magnetic vortex-state nanodots Low Temperature Physics 46, 863 (2020); https://doi.org/10.1063/10.0001555 Aharonov–Casher effect and electric field control of magnetization dynamics Low Temperature Physics 46, 820 (2020); https://doi.org/10.1063/10.0001548 Dynamics of pair of coupled nonlinear systems. I. Magnetic systems Low Temperature Physics 46, 856 (2020); https://doi.org/10.1063/10.0001554Wannier-Stark ladder spectrum of Bloch oscillations of magneto-dipole spin waves in graded 1D magnonic crystals Cite as: Fiz. Nizk. Temp. 46,9 8 4 –990 (August 2020); doi: 10.1063/10.0001550 View Online Export Citation CrossMar k Submitted: 19 June 2020 E. V. Tartakovskaya,1,a)A. S. Laurenson,2and V. V. Kruglyak2 AFFILIATIONS 1Institute of Magnetism NAS of Ukraine and MES of Ukraine, Kyiv 03142, Ukraine 2University of Exeter, Stocker Road, Exeter, EX4 4QL, United Kingdom a)Author to whom correspondence should be addressed: olena.tartakivska@gmail.com ABSTRACT We have used the method of Wannier functions to calculate the frequencies and profiles of spin waves localized in one-dimensional mag- nonic crystals due to a gradient in the bias magnetic field. This localization of spin waves is analogous to the phenomenon of Bloch oscilla-tions of quantum-mechanical electrons in crystals in a uniform electric field. As a convenient yet realistic model, we consider backward volume magnetostatic spin waves in a film of yttrium-iron garnet in a bias magnetic field comprising spatially uniform, cosine and gradient contributions. The spin-wave spectrum is shown to have the characteristic form of a Wannier –Stark ladder. The analytical results are verified using those obtained using numerical micromagnetic simulations. The physics of spin-wave Bloch oscillations combines the topicsof magnonic crystals and graded magnonic index —the two cornerstones of modern magnonics. Published under license by AIP Publishing. https://doi.org/10.1063/10.0001550 The concept of elementary excitations is one of the corner- stones of modern physics, including such an important and quickly developing part of it as physics of superlattices and nanostructures.Just as the dynamics of crystal structures is determined by the spec-trum of phonons (quanta of normal modes of collective elastic vibrations of atoms), the dynamics of magnetically ordered systems is described using the concept of elementary magnetization excita-tions —spin waves (SWs) —and their quanta —magnons. SWs determine the high-frequency dynamics and relaxation of the mag-netization in magnetic materials, as well as their thermal and kinetic properties. 1–5 The behavior of plane waves in artificial periodic media, e.g., superlattices, is analogous to well-investigated case of electron wavesin crystals. For instance, the waves ’band structures are similar to the valence and the conduction bands in semiconductors. Hence, after application of the well-developed methods of quantum mechanics and solid-state physics to such new artificially nanostructured mate-rials, properties of elementary excitations in photonic, 6,7acoustic,8,9 and magnonic10–13crystals were successfully investigated. Among other interesting effects, such a well-known phenome- non as Bloch oscillations (localization)14is also not unique to elec- trons in crystals but can occur for any waves in periodic mediawith graded properties. Bloch oscillations were observed, for instance, in optical (photonic)15,16and acoustic (phononic)17,18 structures. A similar phenomenon was investigated in arrays of cold atoms19–21and in the systems with a strong spin-orbit cou- pling in gradient magnetic field.22,23However, neither experimental24–28nor theoretical29–31investigations did give the evidence of SW localization in realistic magnetic nanostructureswith graded properties. Actually, the task of studying Bloch oscilla-tions was not posed in these experimental works, so the tempera-ture 28and bias magnetic field24–27gradients were chosen too small for these oscillations to be detected. As to the theoretical studies, the possibility of existence of Bloch localization in magneticsystems was confirmed in principle, but only for models far fromrealistic, experimentally realisable magnonic crystals. In Refs. 29 and32, only nonlinear excitations were considered, whose behavior obeys laws different to those for linear elementary excitations of the SW type. In Refs. 30and31, concrete calculations of the excitations spectra in the form of a Wannier –Stark ladder were performed. However, only discrete models with exchange interactions betweenspins were considered. Yet, for the sizes of realistic nanostructures, the most suitable is the phenomenological model of a continuous medium dominated by the magneto-dipolar interaction. 1,2,33Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 46,000000 (2020); doi: 10.1063/10.0001550 46,000000-830 Published under license by AIP Publishing.In this article, we present results of analytical and numerical calculations that show Bloch oscillations and their spectra in the formof Wannier –Stark ladder for the backward volume magnetostatic spin waves (BVMSWs) in magnonic crystals with realistic sizes and geom-etry. We consider a thin film of yttrium-iron garnet (YIG) in a bias magnetic field parallel to the film ’s surface (along the xaxis). This external bias magnetic field is a sum of three terms: (1)as p a t i a l l y uniform term, (2)cosine term (which forms the analogue of superlat- tice), and (3)a slowly varying linear term. The geometry of the problem is presented in Fig. 1 . We assume that the sample is in the saturated state and that the static average magnetization of the film isco-directional with it. We treat the magnetization of the film as thesum of the saturation magnetization and a weakly excited term (i.e.,an SW), which has two spatial components [ μ y(x),μz(x)] exp ( iΩt). To find the frequencies of spin waves Ω, we use the Landau – Lifshitz equation, iΩμy(x)¼(ωHþωh/C1cos(Kx)þγμ0Gx)μz(x), /C0iΩμz(x)¼(ωHþωh/C1cos(Kx)þγμ0Gx)μz(x)/C0ωM^h(μy(x)),( (1) where the term γμ0Gxdescribes the field gradient, ωΗ=γμ0H,w h e r e His the spatially uniform component of the magnetic field, Mis the saturation magnetization, ωΜ=γΜ,μ0is the permeability, γis the gyromagnetic ratio, ais the period of the cosine static magnetic field, and values K=2π/a,a n d ωh=γμ0hcorrespond to the scale and the amplitude of the field modulation. The dynamical dipolar field is ^h(μy)¼/C0@ @yð dr0(μy/C1∇0)1 jr/C0r0j: (2) IfG= 0, the solution of Eq. (1)is a standard linear eigenfre- quency and eigenfunction problem. In this case, we denote the SWsolutions as my(k,x) and mz(k,x), and corresponding frequencies as ω(k). In accordance with the Bloch theorem for a periodic poten- tial, we can employ the usual for magnonics, photonics and pho-nonics presentation for elementary excitations, representing thetwo SW components as m y(k,x)¼e/C0ikxX nTn(k) exp i2πn ax/C18/C19 , mz(k,x)¼e/C0ikxX nDn(k) exp i2πn ax/C18/C19 :(3) Setting G= 0 in Eq. (1), we obtain an infinite system of linear algebraic equations for coefficients Dn(k) and Tn(k) iω(k)Tn(k)¼ωHDn(k)þωh 2(Dnþ1(k)þDn/C01(k)), /C0iω(k)Dn(k)¼Ξn(k)Tn(k)þωh 2(Tnþ1(k)þTn/C01(k)),8 >< >:(4) where Ξn(k)¼ωHþωM1/C0exp/C0k/C02πn a/C12/C12/C12/C12/C12/C12/C12/C12d/C18/C19 k/C0 2πn a/C12/C12/C12/C12/C12/C12/C12/C12d0 BB@1 CCA: (5) We consider the case when ωh/ω Η<< 1 . This allows us to approximate the full solution of the problem Eqs. (4)and(5)by a finite-sized subset of the basis states and leads to the standard diag- onalization of the characteristic matrix of finite size for Eq. (4).A s a result, we obtain the expected picture of the band dispersion ω (k), which is usual for crystals. The magnonic bands are ordered in frequency from top to down. As our numerical calculations show ( Fig. 2 , black dash lines), with parameters chosen here the periodic field modulationinduces a large first band gap, while the other (higher order) bandgaps are significantly smaller and the allowed bands are increas-ingly flat. So, we can try to limit allowed bands are increasingly flat. So, we can try to limit our model to the first and second bands only. In this approximation, the characteristic equation takes thefollowing simple biquadratic form ω(k) 2/C0ωHΞ0þω2 h 2/C20/C21 /C18/C19 ω(k)2/C0ωHΞ1þω2 h 2/C20/C21 /C18/C19 /C0[Ξ1þωH][Ξ0þωH]ω2 h 4¼0: (6) The magnonic dispersion relations for the first, ω+(k), and second, ω−(k), bands can be found analytically as ω+(k)¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2n [ωH(Ξ0þΞ1)þω2 h]+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ω2 H(Ξ0/C0Ξ1)2þ(ωH(Ξ0þΞ1)þΞ1Ξ0þω2 H)ω2 hq or : (7) FIG. 1. The geometry of the problem is shown. A thin magnetic film of thick- ness dis magnetised along the xaxis by a bias magnetic field comprising spa- tially uniform H, cosine hcos(Kx), and gradient Gxcontributions. SWs propagate (with a wave vector k) also along the xaxis (BVMSW geometry). The period of the cosine field contribution is a=3μm, and so, K=2π/a.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 46,000000 (2020); doi: 10.1063/10.0001550 46,000000-831 Published under license by AIP Publishing.Using Eq. (7), we can find the analytical expression for the gapΔgapbetween the first and the second bands as Δgap¼ωþπ a/C16/C17 /C0ω/C0π a/C16/C17 /C25ffiffiffiffiffiffiωH Ξr (ΞþωH)ωh 2ωH, (8) where Ξ=Ξ0(π/a)=Ξ1(π/a). In the first approximation by the small parameter ωh/ωΗ, the band gap is linear in ωh/ωΗ Figure 2 shows the magnonic dispersion relations in the first Brillouin zone, calculated for a uniform bias magnetic field of185 mT spatially modulated by an additional cosine static magneticfield with a period of 3 μm. The calculations are shown for different amplitudes of the field modulations: panels (a) and (b) correspond to the field amplitudes of μ 0h= 5 mT and μ0h= 10 mT, respectively.The SW branches are calculated for two different finite-sized subsets of the basic states: numerically with the extended scheme by Eqs. (4)and(5), shown by black dashed lines, and analytically for the first two bands by Eq. (7), shown by red solid lines. As expected, the analytical result gives a good approximation in thecase of μ 0h= 5 mT, while the discrepancy between analytical and numerical calculations increases for μ0h=1 0m T . At the next step, we use the eigenvalues ω(k) given by Eq. (7) and the corresponding eigenfunctions my(k,x),mz(k,x) given by Eq.(3)calculated for G≠0 to construct solutions of the problem with a nonzero field gradient. So, we return to Eq. (1)with G≠0. Our task is to find new eigenvalues Ωand new eigenfunctions, μy(x) and μz(x). Now the magnetic excitations in the sample cannot be presented in the form of the expansion (3). Indeed, firstly, such a representation is a consequence of the Bloch theorem, i.e., of theperiodicity of the potential, while this periodicity is broken when the gradient is nonzero. Secondly, the matrix elements of the new graded potential proportional to Gxdiverge if the eigenfunctions are not localized. So, we must use basis functions that are localizedin the real space. In this case, Wannier functions 34,35are a good choice. The scheme common in the problem of electron localization in crystals in a uniform electric field is applied to one band with anassumption of non-interacting bands. Due to the large first bandgap in our magnonic crystal, an interband tunnelling between the first and second band is negligible, which allows us to find the Wannier –Stark ladder spectrum in the isolated first band. We determine two sets of Wannier functions for both compo- nents of the SW in the usual way a(x/C0R)¼ a 2πðπ/a /C0π/adkeikRmy(k,x), b(x/C0R)¼a 2πðπ/a /C0π/adkeikRmz(k,x),(9) where my(k, x) and mz(k, z) are the solutions (3)of the eigenpro- blem with G=0, described above. Rn=n⋅a, where nare integers, are coordinates of the external field maxima, which are analoguesof the atomic positions in a crystal. The Wannier functions havesharp extrema near the corresponding R[Fig. 3(a) ]. This is the source of orthogonality of Wannier functions ð dxa(x/C0R 0)a*(x/C0R)¼fΔ(R/C0R0), ð dxb(x/C0R0)b*(x/C0R)¼gΔ(R/C0R0), ð dxa(x/C0R0)b*(x/C0R)¼sΔ(R/C0R0),(10) which we employ below. It follows from Eqs. (1),(4), and (9)that constants fandgare real, while constant sis imaginary. FIG. 2. The SW frequency is shown as a function of the wave number in the first Brillouin zone. The cosine static m agnetic field with a period of 3 μmh a s the modulation amplitude of μ0h= 5 mT (a) and μ0h= 10 mT (b). The black dash lines show the calculations by Eqs. (4)and (5)with the indices n varying from 0 –10 (only the first 7 bands are shown). The red solid lines show the analytical calculations by Eq. (7), i.e., n= 0,1. The grey stripe in panel (a) shows the first band gap, which is almost the same in the two approximations. The grey stripe and the red hatched stripe in (b) show the first band gap for numerical and analytical calculations, respectively. Both inpanels (a) and (b), the uniform bias magnetic field is μ 0Η= 185 mT , the satu- ration magnetization is M = 200 kA/m, γ/2π= 28 GHz/T , and the film thickness isd=1μm.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 46,000000 (2020); doi: 10.1063/10.0001550 46,000000-832 Published under license by AIP Publishing.Further, we represent unknown profiles μx(z) and μy(z)a s superpositions of Wannier functions (9): μy(x)¼X RA(R)a(x/C0R), μz(x)¼X RB(R)b(x/C0R):(11) To obtain the coefficients A(R) and B(R), we substitute Eq.(11) into Eq. (1). At the length scale of variation of Wannier functions, Gx≈GR, and so, integrating both parts of the equations byxand using Eq. (10), we rewrite Eq. (1)asiΩfA(R)¼ia 2πðπ/a -π/adke/C0ikRB(k)ωþ(k)þγμ0GRB(R)/C1s*, iΩgB(R)¼/C0ia 2πðπ/a -π/adke/C0ikRA(k)ωþ(k)þγμ0GRA (R)/C1s,8 >>>>>>>< >>>>>>>:(12) where ω +(k) is determined by Eq. (7). Ink-representation, Eq. (12) takes the following form ΩA(k)¼B(k)ωþ(k)-γμ0Gs* f@ @kB(k), ΩB(k)¼A(k)ωþ(k)-γμ0Gs* g@ @kA(k):8 >>< >>:(13) From Eq. (13), we obtain an equation for one of the unknown functions, for instance, for A(k) Ω2A(k)¼A(k)(ω(k))2þs g/C0s* f/C26/C27 γμ0GΩ@ @kA(k): (14) This is a differential equation of the first order with constant coeffi- cients, and so, it has a standard solution: A(k)¼~Aexpi γμ0Gα0ðk 0Ω2/C0(ωþ(k0))2 Ωdk00 @1 A, (15) where α¼/C0Ims g/C0s* fno The dispersion relation can be found from the periodic boun- dary condition A(k)¼Akþ2π an/C18/C19 , (16) where nare integer. From Eqs. (15) and(16), we obtain a quadratic equation Ω2/C0Ω/C1n/C1Gaγα-ω2 þ/C10/C11 ¼0, (17) where ω2 þ/C10/C11 ¼a πÐπ/a 0(ωþ(k0))2dk0: The solution of Eq. (17) is Ωn¼1 2n/C1γμ0Gaαþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (n/C1γμ0Gaα)2þ4ω2 þ/C10/C11q /C20/C21 , which can be written in the form of the Wannier-Stark ladder, Ωn/C25ffiffiffiffiffiffiffiffiffiffiffi ω2 þ/C10/C11q þ1 2n/C1γμ0Gaα, (18) if the gradient of external magnetic field is small enough, i.e., (γμ0Gaα)2/C284ω2 þ/C10/C11 : FIG. 3. (a) Examples of Wannier functions b(x−Ri) centred around the Ri(i=0 , ±1, ±2) points, calculated using Eq. (9), are shown for μ0h=1 0m T , μ0H= 185 mT , a=3μm,d=1μm,M= 200 kA/m. (b) The profiles of localized SWs, calculated using Eqs. (11) and (14) for the central level Ω0/C25ffiffiffiffiffiffiffiffiffiffiffi ω2 þ/C10/C11q of the corresponding Wannier-Stark ladder μ0h=5m T –red (Re( μz)) and dash black [ −Im(μy)] lines, μ0h= 10 mT - dash-dot magenta (Re( μz)) and dash green [−Im(μy)] lines), are shown for a field gradient of μ0G= 40 mT/mm.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 46,000000 (2020); doi: 10.1063/10.0001550 46,000000-833 Published under license by AIP Publishing.Equation (18) shows that an energy band of a magnonic crystal, with initial dispersion relation ω+(k), in a weakly graded field gives rise to the Wannier –Stark ladder with central level Ω0/C25ffiffiffiffiffiffiffiffiffiffiffiffiffi ω2 þ/C10/C11q and the distance between levels1 2γμ0Gaα.F o r instance, the Wannier –Stark ladder in the first band, as calculated using Eq. (18) forμ0G= 40 mT/mm and μ0h= 5mT, consists of a set of levels with the central level at Ω0/C257:498 GHz and with the distance between neighbouring levels δof about 3.6 MHz. For the amplitude of the field modulation of μ0h= 10mT and the same field gradient, these values become Ω0/C257:498 GHz and δ≈3.45 MHz. Figure 3(b) shows profiles of localized SWs that cor- responding to the central level of the Wannier –Stark ladder and the two values of the amplitude of the field modulation. The results of the analytical theory presented above are in agreement with those obtained from micromagnetic simulationsperformed using MuMax software. 36The simulations are run in the time domain and their results are converted into the frequency domain using standard Fourier techniques.37Fig. 4 shows the spatial maps of the SW amplitude distributions for different valueso the magnetic field gradient and excitation by a uniform micro-wave magnetic field with a spectrum centred at 18 GHz andspectral bandwidth of 10 MHz. This microwave field couples to the magnetization precession where the frequency matches either uniform ferromagnetic resonance (FMR) frequency (at about17.2 GHz) or that corresponding to the band edges in the Brillouinzone centre, i.e. k¼2πn/a,where nis an integer number. At zero gradient, only the FMR mode is excited and then very weakly. Atfinite values of the field gradient, the Wannier –Stark ladder spec- trum is formed. However, the individual levels are not very well resolved, owing to the very small frequency splitting between theneighbouring levels. In summary, we have used analytical theory based on the method of Wannier functions and numerical simulations to study the spectrum of BVMSW in magnonic crystals subjected to a graded magnetic field. Our results demonstrate that this field gradi-ent can lead to Bloch oscillations of localized SWs, with their spec-trum having the characteristic form of the Wannier –Stark ladder. Here, we have presented results for magnonic crystals formed byapplying using a cosine-modulated bias magnetic field to a thin film of YIG. Strictly speaking, such a bias magnetic field does not satisfy one of the Maxwell equations, div B= 0. The account of a corresponding out-of-plane non-uniform bias magnetic field, FIG. 4. Spatial maps of the SW amplitude distribution are shown forμ 0h=1 0m T , μ0H=0 . 5T , a=3μm, d=1μm, and M= 200 kA/m and the indicated values of the magnetic fieldgradient. The greyscale shows theresults of the numerical simulations (darker color corresponds to greater Fourier amplitude of spin waves).The dashed and dotted lines showthe top and bottom boundaries of the whole BVMSW band in a uniform film, while the dash-dotted line corre-sponds to the bottom edge of the firstmagnonic band estimated from the empty-lattice approximation.Low Temperature PhysicsARTICLE scitation.org/journal/ltp Low T emp. Phys. 46,000000 (2020); doi: 10.1063/10.0001550 46,000000-834 Published under license by AIP Publishing.which would ensure that the equation is satisfied, does not change substantially our theory. Moreover, the field should be treated as a general effective magnetic field, representing, e.g., modulatedanisotropy or exchange bias. 38We have also performed similar cal- culations and obtained similar results for other 1D magnonic crys-tals, e.g., those formed by arrays of long rectangular strips. The calculations can be generalised to other SW geometries, to the case of dipole-exchange SWs, to graded magnonic crystals formed viaspatial modulation (periodic and linear) of the magnonic indexthrough other mechanisms, 39and to the case of a spatial variation of the lattice constant a. ACKNOWLEDGMENTS The research leading to these results has received funding from the Engineering and Physical Sciences Research Council of the United Kingdom, via the EPSRC Centre for Doctoral Trainingin Metamaterials (Grant No. EP/L015331/1), and from theEuropean Union ’s Horizon 2020 research and innovation program under Marie Sklodowska-Curie Grant Agreement No. 644348 (MagIC). 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